diff --git a/.github/workflows/latest_import.yml b/.github/workflows/latest_import.yml
index aa7a52cd7f9811..6e9b5b0cf2cfd9 100644
--- a/.github/workflows/latest_import.yml
+++ b/.github/workflows/latest_import.yml
@@ -48,8 +48,10 @@ jobs:
 
     - name: add minImports linter option
       run: |
-        # set `linter.minImports option` to true in `lakefile`
-        sed -i -- '/^  -- '\`'latest_import.yml'\`' uses this comment/{s=^=  ⟨`linter.minImports, true⟩,\n=}' lakefile.lean
+        # we disable checking for backticks inside single quotes with the next line
+        # shellcheck disable=SC2016
+        # set `linter.minImports option` to true and `Elab.async` to false in `lakefile`
+        sed -i -- '/^  -- '\`'latest_import.yml'\`' uses this comment/{s=^=  ⟨`linter.minImports, true⟩,\n  ⟨`Elab.async, false⟩,\n=}' lakefile.lean
 
         # import the `minImports` linter in `Mathlib.Init`
         sed -i -z 's=^=import Mathlib.Tactic.Linter.MinImports\n=' Mathlib/Init.lean
diff --git a/.github/workflows/maintainer_bors.yml b/.github/workflows/maintainer_bors.yml
index 10a12e5f3a439f..1f6e215d7fa1ec 100644
--- a/.github/workflows/maintainer_bors.yml
+++ b/.github/workflows/maintainer_bors.yml
@@ -77,13 +77,15 @@ jobs:
           ( steps.user_permission.outputs.require-result == 'true' ||
             steps.merge_or_delegate.outputs.bot == 'true' ) }}
         id: remove_labels
-        name: Remove "awaiting-author"
+        name: Remove "awaiting-author" and "maintainer-merge"
         # we use curl rather than octokit/request-action so that the job won't fail
         # (and send an annoying email) if the labels don't exist
         run: |
-          curl --request DELETE \
-            --url "https://api.github.com/repos/${{ github.repository }}/issues/${{ github.event.issue.number }}${{ github.event.pull_request.number }}/labels/awaiting-author" \
-            --header 'authorization: Bearer ${{ secrets.TRIAGE_TOKEN }}'
+          for label in awaiting-author maintainer-merge; do
+            curl --request DELETE \
+              --url "https://api.github.com/repos/${{ github.repository }}/issues/${{ github.event.issue.number }}${{ github.event.pull_request.number }}/labels/${label}" \
+              --header 'authorization: Bearer ${{ secrets.TRIAGE_TOKEN }}'
+          done
 
       - name: Set up Python
         if: ${{ ! steps.merge_or_delegate.outputs.mOrD == '' &&
diff --git a/Mathlib.lean b/Mathlib.lean
index e5041a5fd5c1bb..f62f59976932f7 100644
--- a/Mathlib.lean
+++ b/Mathlib.lean
@@ -1,3 +1,4 @@
+import Std
 import Batteries
 import Mathlib.Algebra.AddConstMap.Basic
 import Mathlib.Algebra.AddConstMap.Equiv
@@ -355,6 +356,7 @@ import Mathlib.Algebra.Group.Submonoid.Defs
 import Mathlib.Algebra.Group.Submonoid.DistribMulAction
 import Mathlib.Algebra.Group.Submonoid.Finsupp
 import Mathlib.Algebra.Group.Submonoid.Membership
+import Mathlib.Algebra.Group.Submonoid.MulAction
 import Mathlib.Algebra.Group.Submonoid.MulOpposite
 import Mathlib.Algebra.Group.Submonoid.Operations
 import Mathlib.Algebra.Group.Submonoid.Pointwise
@@ -449,6 +451,7 @@ import Mathlib.Algebra.Homology.Embedding.StupidTrunc
 import Mathlib.Algebra.Homology.Embedding.TruncGE
 import Mathlib.Algebra.Homology.Embedding.TruncGEHomology
 import Mathlib.Algebra.Homology.Embedding.TruncLE
+import Mathlib.Algebra.Homology.Embedding.TruncLEHomology
 import Mathlib.Algebra.Homology.ExactSequence
 import Mathlib.Algebra.Homology.Factorizations.Basic
 import Mathlib.Algebra.Homology.Functor
@@ -1166,6 +1169,7 @@ import Mathlib.Analysis.BoxIntegral.Partition.Tagged
 import Mathlib.Analysis.BoxIntegral.UnitPartition
 import Mathlib.Analysis.CStarAlgebra.ApproximateUnit
 import Mathlib.Analysis.CStarAlgebra.Basic
+import Mathlib.Analysis.CStarAlgebra.CStarMatrix
 import Mathlib.Analysis.CStarAlgebra.Classes
 import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic
 import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances
@@ -1732,6 +1736,7 @@ import Mathlib.CategoryTheory.Category.ULift
 import Mathlib.CategoryTheory.ChosenFiniteProducts
 import Mathlib.CategoryTheory.ChosenFiniteProducts.Cat
 import Mathlib.CategoryTheory.ChosenFiniteProducts.FunctorCategory
+import Mathlib.CategoryTheory.ChosenFiniteProducts.InfSemilattice
 import Mathlib.CategoryTheory.Closed.Cartesian
 import Mathlib.CategoryTheory.Closed.Enrichment
 import Mathlib.CategoryTheory.Closed.Functor
@@ -1845,7 +1850,7 @@ import Mathlib.CategoryTheory.Galois.Prorepresentability
 import Mathlib.CategoryTheory.Galois.Topology
 import Mathlib.CategoryTheory.Generator.Abelian
 import Mathlib.CategoryTheory.Generator.Basic
-import Mathlib.CategoryTheory.Generator.Coproduct
+import Mathlib.CategoryTheory.Generator.Indization
 import Mathlib.CategoryTheory.Generator.Preadditive
 import Mathlib.CategoryTheory.Generator.Presheaf
 import Mathlib.CategoryTheory.Generator.Sheaf
@@ -1930,6 +1935,7 @@ import Mathlib.CategoryTheory.Limits.Indization.Equalizers
 import Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
 import Mathlib.CategoryTheory.Limits.Indization.IndObject
 import Mathlib.CategoryTheory.Limits.Indization.LocallySmall
+import Mathlib.CategoryTheory.Limits.Indization.ParallelPair
 import Mathlib.CategoryTheory.Limits.Indization.Products
 import Mathlib.CategoryTheory.Limits.IsConnected
 import Mathlib.CategoryTheory.Limits.IsLimit
@@ -1939,6 +1945,7 @@ import Mathlib.CategoryTheory.Limits.MorphismProperty
 import Mathlib.CategoryTheory.Limits.Opposites
 import Mathlib.CategoryTheory.Limits.Over
 import Mathlib.CategoryTheory.Limits.Pi
+import Mathlib.CategoryTheory.Limits.Preorder
 import Mathlib.CategoryTheory.Limits.Preserves.Basic
 import Mathlib.CategoryTheory.Limits.Preserves.Filtered
 import Mathlib.CategoryTheory.Limits.Preserves.Finite
@@ -2143,6 +2150,7 @@ import Mathlib.CategoryTheory.Preadditive.Projective
 import Mathlib.CategoryTheory.Preadditive.ProjectiveResolution
 import Mathlib.CategoryTheory.Preadditive.Schur
 import Mathlib.CategoryTheory.Preadditive.SingleObj
+import Mathlib.CategoryTheory.Preadditive.Transfer
 import Mathlib.CategoryTheory.Preadditive.Yoneda.Basic
 import Mathlib.CategoryTheory.Preadditive.Yoneda.Injective
 import Mathlib.CategoryTheory.Preadditive.Yoneda.Limits
@@ -2263,6 +2271,7 @@ import Mathlib.CategoryTheory.Subobject.Types
 import Mathlib.CategoryTheory.Subobject.WellPowered
 import Mathlib.CategoryTheory.Subpresheaf.Basic
 import Mathlib.CategoryTheory.Subpresheaf.Image
+import Mathlib.CategoryTheory.Subpresheaf.OfSection
 import Mathlib.CategoryTheory.Subpresheaf.Sieves
 import Mathlib.CategoryTheory.Subterminal
 import Mathlib.CategoryTheory.Sums.Associator
@@ -2500,6 +2509,7 @@ import Mathlib.Data.Complex.FiniteDimensional
 import Mathlib.Data.Complex.Module
 import Mathlib.Data.Complex.Order
 import Mathlib.Data.Complex.Orientation
+import Mathlib.Data.Complex.Trigonometric
 import Mathlib.Data.Countable.Basic
 import Mathlib.Data.Countable.Defs
 import Mathlib.Data.Countable.Small
@@ -3082,7 +3092,6 @@ import Mathlib.Deprecated.Cardinal.Finite
 import Mathlib.Deprecated.Cardinal.PartENat
 import Mathlib.Deprecated.Combinator
 import Mathlib.Deprecated.Equiv
-import Mathlib.Deprecated.Group
 import Mathlib.Deprecated.HashMap
 import Mathlib.Deprecated.LazyList
 import Mathlib.Deprecated.Logic
@@ -3090,11 +3099,6 @@ import Mathlib.Deprecated.MinMax
 import Mathlib.Deprecated.NatLemmas
 import Mathlib.Deprecated.Order
 import Mathlib.Deprecated.RelClasses
-import Mathlib.Deprecated.Ring
-import Mathlib.Deprecated.Subfield
-import Mathlib.Deprecated.Subgroup
-import Mathlib.Deprecated.Submonoid
-import Mathlib.Deprecated.Subring
 import Mathlib.Dynamics.BirkhoffSum.Average
 import Mathlib.Dynamics.BirkhoffSum.Basic
 import Mathlib.Dynamics.BirkhoffSum.NormedSpace
@@ -3231,8 +3235,8 @@ import Mathlib.Geometry.Manifold.IntegralCurve.Basic
 import Mathlib.Geometry.Manifold.IntegralCurve.ExistUnique
 import Mathlib.Geometry.Manifold.IntegralCurve.Transform
 import Mathlib.Geometry.Manifold.IntegralCurve.UniformTime
-import Mathlib.Geometry.Manifold.InteriorBoundary
-import Mathlib.Geometry.Manifold.IsManifold
+import Mathlib.Geometry.Manifold.IsManifold.Basic
+import Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary
 import Mathlib.Geometry.Manifold.LocalDiffeomorph
 import Mathlib.Geometry.Manifold.LocalInvariantProperties
 import Mathlib.Geometry.Manifold.MFDeriv.Atlas
@@ -3742,6 +3746,7 @@ import Mathlib.MeasureTheory.Covering.OneDim
 import Mathlib.MeasureTheory.Covering.Vitali
 import Mathlib.MeasureTheory.Covering.VitaliFamily
 import Mathlib.MeasureTheory.Decomposition.Exhaustion
+import Mathlib.MeasureTheory.Decomposition.IntegralRNDeriv
 import Mathlib.MeasureTheory.Decomposition.Jordan
 import Mathlib.MeasureTheory.Decomposition.Lebesgue
 import Mathlib.MeasureTheory.Decomposition.RadonNikodym
@@ -3768,7 +3773,9 @@ import Mathlib.MeasureTheory.Function.EssSup
 import Mathlib.MeasureTheory.Function.Floor
 import Mathlib.MeasureTheory.Function.Intersectivity
 import Mathlib.MeasureTheory.Function.Jacobian
-import Mathlib.MeasureTheory.Function.L1Space
+import Mathlib.MeasureTheory.Function.L1Space.AEEqFun
+import Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral
+import Mathlib.MeasureTheory.Function.L1Space.Integrable
 import Mathlib.MeasureTheory.Function.L2Space
 import Mathlib.MeasureTheory.Function.LocallyIntegrable
 import Mathlib.MeasureTheory.Function.LpOrder
@@ -4019,6 +4026,7 @@ import Mathlib.NumberTheory.LSeries.Nonvanishing
 import Mathlib.NumberTheory.LSeries.Positivity
 import Mathlib.NumberTheory.LSeries.PrimesInAP
 import Mathlib.NumberTheory.LSeries.RiemannZeta
+import Mathlib.NumberTheory.LSeries.SumCoeff
 import Mathlib.NumberTheory.LSeries.ZMod
 import Mathlib.NumberTheory.LegendreSymbol.AddCharacter
 import Mathlib.NumberTheory.LegendreSymbol.Basic
@@ -4155,6 +4163,7 @@ import Mathlib.Order.Compare
 import Mathlib.Order.CompleteBooleanAlgebra
 import Mathlib.Order.CompleteLattice
 import Mathlib.Order.CompleteLattice.Finset
+import Mathlib.Order.CompleteLattice.SetLike
 import Mathlib.Order.CompleteLatticeIntervals
 import Mathlib.Order.CompletePartialOrder
 import Mathlib.Order.CompleteSublattice
@@ -4355,7 +4364,10 @@ import Mathlib.Probability.Independence.ZeroOne
 import Mathlib.Probability.Integration
 import Mathlib.Probability.Kernel.Basic
 import Mathlib.Probability.Kernel.Composition.Basic
+import Mathlib.Probability.Kernel.Composition.CompNotation
 import Mathlib.Probability.Kernel.Composition.IntegralCompProd
+import Mathlib.Probability.Kernel.Composition.Lemmas
+import Mathlib.Probability.Kernel.Composition.MeasureComp
 import Mathlib.Probability.Kernel.Composition.MeasureCompProd
 import Mathlib.Probability.Kernel.Composition.ParallelComp
 import Mathlib.Probability.Kernel.CondDistrib
@@ -4384,7 +4396,9 @@ import Mathlib.Probability.Martingale.OptionalSampling
 import Mathlib.Probability.Martingale.OptionalStopping
 import Mathlib.Probability.Martingale.Upcrossing
 import Mathlib.Probability.Moments.Basic
+import Mathlib.Probability.Moments.ComplexMGF
 import Mathlib.Probability.Moments.IntegrableExpMul
+import Mathlib.Probability.Moments.MGFAnalytic
 import Mathlib.Probability.Notation
 import Mathlib.Probability.ProbabilityMassFunction.Basic
 import Mathlib.Probability.ProbabilityMassFunction.Binomial
@@ -4632,6 +4646,7 @@ import Mathlib.RingTheory.Localization.LocalizationLocalization
 import Mathlib.RingTheory.Localization.Module
 import Mathlib.RingTheory.Localization.NormTrace
 import Mathlib.RingTheory.Localization.NumDen
+import Mathlib.RingTheory.Localization.Pi
 import Mathlib.RingTheory.Localization.Submodule
 import Mathlib.RingTheory.MatrixAlgebra
 import Mathlib.RingTheory.Multiplicity
@@ -4860,6 +4875,7 @@ import Mathlib.SetTheory.Cardinal.SchroederBernstein
 import Mathlib.SetTheory.Cardinal.Subfield
 import Mathlib.SetTheory.Cardinal.ToNat
 import Mathlib.SetTheory.Cardinal.UnivLE
+import Mathlib.SetTheory.Descriptive.Tree
 import Mathlib.SetTheory.Game.Basic
 import Mathlib.SetTheory.Game.Birthday
 import Mathlib.SetTheory.Game.Domineering
diff --git a/Mathlib/Algebra/AddTorsor.lean b/Mathlib/Algebra/AddTorsor.lean
index c853185785a742..7977a04a58c145 100644
--- a/Mathlib/Algebra/AddTorsor.lean
+++ b/Mathlib/Algebra/AddTorsor.lean
@@ -38,6 +38,8 @@ multiplicative group actions).
 
 -/
 
+open scoped Pointwise
+
 /-- An `AddTorsor G P` gives a structure to the nonempty type `P`,
 acted on by an `AddGroup G` with a transitive and free action given
 by the `+ᵥ` operation and a corresponding subtraction given by the
@@ -162,8 +164,6 @@ theorem vadd_eq_vadd_iff_neg_add_eq_vsub {v₁ v₂ : G} {p₁ p₂ : P} :
 
 namespace Set
 
-open Pointwise
-
 theorem singleton_vsub_self (p : P) : ({p} : Set P) -ᵥ {p} = {(0 : G)} := by
   rw [Set.singleton_vsub_singleton, vsub_self]
 
@@ -231,6 +231,13 @@ theorem vadd_eq_vadd_iff_sub_eq_vsub {v₁ v₂ : G} {p₁ p₂ : P} :
 theorem vsub_sub_vsub_comm (p₁ p₂ p₃ p₄ : P) : p₁ -ᵥ p₂ - (p₃ -ᵥ p₄) = p₁ -ᵥ p₃ - (p₂ -ᵥ p₄) := by
   rw [← vsub_vadd_eq_vsub_sub, vsub_vadd_comm, vsub_vadd_eq_vsub_sub]
 
+namespace Set
+
+@[simp] lemma vadd_set_vsub_vadd_set (v : G) (s t : Set P) : (v +ᵥ s) -ᵥ (v +ᵥ t) = s -ᵥ t := by
+  ext; simp [mem_vsub, mem_vadd_set]
+
+end Set
+
 end comm
 
 namespace Prod
diff --git a/Mathlib/Algebra/Algebra/Pi.lean b/Mathlib/Algebra/Algebra/Pi.lean
index 188a138794019c..d04b77f4ffb485 100644
--- a/Mathlib/Algebra/Algebra/Pi.lean
+++ b/Mathlib/Algebra/Algebra/Pi.lean
@@ -48,6 +48,15 @@ theorem algebraMap_apply {_ : CommSemiring R} [_s : ∀ i, Semiring (f i)] [∀
     (a : R) (i : I) : algebraMap R (∀ i, f i) a i = algebraMap R (f i) a :=
   rfl
 
+variable {I} in
+instance (g : I → Type*) [∀ i, CommSemiring (f i)] [∀ i, Semiring (g i)]
+    [∀ i, Algebra (f i) (g i)] : Algebra (∀ i, f i) (∀ i, g i) where
+  algebraMap := Pi.ringHom fun _ ↦ (algebraMap _ _).comp (Pi.evalRingHom f _)
+  commutes' _ _ := funext fun _ ↦ Algebra.commutes _ _
+  smul_def' _ _ := funext fun _ ↦ Algebra.smul_def _ _
+
+example [∀ i, CommSemiring (f i)] : Pi.instAlgebraForall f f = Algebra.id _ := rfl
+
 -- One could also build a `∀ i, R i`-algebra structure on `∀ i, A i`,
 -- when each `A i` is an `R i`-algebra, although I'm not sure that it's useful.
 variable {I} (R)
@@ -70,6 +79,16 @@ def evalAlgHom {_ : CommSemiring R} [∀ i, Semiring (f i)] [∀ i, Algebra R (f
     toFun := fun f => f i
     commutes' := fun _ => rfl }
 
+@[simp]
+theorem algHom_evalAlgHom [CommSemiring R] [s : ∀ i, Semiring (f i)] [∀ i, Algebra R (f i)] :
+    algHom R f (evalAlgHom R f) = AlgHom.id R (Π i, f i) := rfl
+
+/-- `Pi.algHom` commutes with composition. -/
+theorem algHom_comp [CommSemiring R] [∀ i, Semiring (f i)] [∀ i, Algebra R (f i)]
+    {A B : Type*} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
+    (g : ∀ i, B →ₐ[R] f i) (h : A →ₐ[R] B) :
+    (algHom R f g).comp h = algHom R f (fun i ↦ (g i).comp h) := rfl
+
 variable (A B : Type*) [CommSemiring R] [Semiring B] [Algebra R B]
 
 /-- `Function.const` as an `AlgHom`. The name matches `Pi.constRingHom`, `Pi.constMonoidHom`,
diff --git a/Mathlib/Algebra/BigOperators/Fin.lean b/Mathlib/Algebra/BigOperators/Fin.lean
index 2a445b2244dcc9..4d62dc8fa144aa 100644
--- a/Mathlib/Algebra/BigOperators/Fin.lean
+++ b/Mathlib/Algebra/BigOperators/Fin.lean
@@ -55,7 +55,7 @@ theorem prod_univ_zero [CommMonoid β] (f : Fin 0 → β) : ∏ i, f i = 1 :=
 /-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`
 is the product of `f x`, for some `x : Fin (n + 1)` times the remaining product -/
 @[to_additive "A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of
-`f x`, for some `x : Fin (n + 1)` plus the remaining product"]
+`f x`, for some `x : Fin (n + 1)` plus the remaining sum"]
 theorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :
     ∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by
   rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb]
@@ -64,7 +64,7 @@ theorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (
 /-- A product of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)`
 is the product of `f 0` plus the remaining product -/
 @[to_additive "A sum of a function `f : Fin (n + 1) → β` over all `Fin (n + 1)` is the sum of
-`f 0` plus the remaining product"]
+`f 0` plus the remaining sum"]
 theorem prod_univ_succ [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :
     ∏ i, f i = f 0 * ∏ i : Fin n, f i.succ :=
   prod_univ_succAbove f 0
@@ -215,17 +215,7 @@ theorem partialProd_left_inv {G : Type*} [Group G] (f : Fin (n + 1) → G) :
 @[to_additive]
 theorem partialProd_right_inv {G : Type*} [Group G] (f : Fin n → G) (i : Fin n) :
     (partialProd f (Fin.castSucc i))⁻¹ * partialProd f i.succ = f i := by
-  obtain ⟨i, hn⟩ := i
-  induction i with
-  | zero => simp [-Fin.succ_mk, partialProd_succ]
-  | succ i hi =>
-    specialize hi (lt_trans (Nat.lt_succ_self i) hn)
-    simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi ⊢
-    rw [← Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn), ← Fin.succ_mk _ _ hn]
-    simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]
-    -- Porting note: was
-    -- assoc_rw [hi, inv_mul_cancel_left]
-    rw [← mul_assoc, mul_left_eq_self, mul_assoc, hi, inv_mul_cancel]
+  rw [partialProd_succ, inv_mul_cancel_left]
 
 /-- Let `(g₀, g₁, ..., gₙ)` be a tuple of elements in `Gⁿ⁺¹`.
 Then if `k < j`, this says `(g₀g₁...gₖ₋₁)⁻¹ * g₀g₁...gₖ = gₖ`.
diff --git a/Mathlib/Algebra/BigOperators/Pi.lean b/Mathlib/Algebra/BigOperators/Pi.lean
index 88511371f4a9ef..0db9fbd3746737 100644
--- a/Mathlib/Algebra/BigOperators/Pi.lean
+++ b/Mathlib/Algebra/BigOperators/Pi.lean
@@ -181,3 +181,22 @@ def Pi.monoidHomMulEquiv {ι : Type*} [Fintype ι] [DecidableEq ι] (M : ι →
       MonoidHom.mul_apply, mul_apply]
 
 end MulEquiv
+
+variable [Finite ι] [DecidableEq ι] {M : Type*}
+
+-- manually additivized to fix variable names
+-- See https://github.com/leanprover-community/mathlib4/issues/11462
+lemma Pi.single_induction [AddCommMonoid M] (p : (ι → M) → Prop) (f : ι → M)
+    (zero : p 0) (add : ∀ f g, p f → p g → p (f + g))
+    (single : ∀ i m, p (Pi.single i m)) : p f := by
+  cases nonempty_fintype ι
+  rw [← Finset.univ_sum_single f]
+  exact Finset.sum_induction _ _ add zero (by simp [single])
+
+@[to_additive (attr := elab_as_elim) existing]
+lemma Pi.mulSingle_induction [CommMonoid M] (p : (ι → M) → Prop) (f : ι → M)
+    (one : p 1) (mul : ∀ f g, p f → p g → p (f * g))
+    (mulSingle : ∀ i m, p (Pi.mulSingle i m)) : p f := by
+  cases nonempty_fintype ι
+  rw [← Finset.univ_prod_mulSingle f]
+  exact Finset.prod_induction _ _ mul one (by simp [mulSingle])
diff --git a/Mathlib/Algebra/Category/AlgebraCat/Basic.lean b/Mathlib/Algebra/Category/AlgebraCat/Basic.lean
index 87dd2422804195..808f200d06da42 100644
--- a/Mathlib/Algebra/Category/AlgebraCat/Basic.lean
+++ b/Mathlib/Algebra/Category/AlgebraCat/Basic.lean
@@ -61,15 +61,38 @@ variable {R} in
 structure Hom (A B : AlgebraCat.{v} R) where
   private mk ::
   /-- The underlying algebra map. -/
-  hom : A →ₐ[R] B
+  hom' : A →ₐ[R] B
 
 instance : Category (AlgebraCat.{v} R) where
   Hom A B := Hom A B
   id A := ⟨AlgHom.id R A⟩
-  comp f g := ⟨g.hom.comp f.hom⟩
+  comp f g := ⟨g.hom'.comp f.hom'⟩
 
-instance {M N : AlgebraCat.{v} R} : CoeFun (M ⟶ N) (fun _ ↦ M → N) where
-  coe f := f.hom
+instance : ConcreteCategory (AlgebraCat.{v} R) (· →ₐ[R] ·) where
+  hom := Hom.hom'
+  ofHom := Hom.mk
+
+variable {R} in
+/-- Turn a morphism in `AlgebraCat` back into an `AlgHom`. -/
+abbrev Hom.hom {A B : AlgebraCat.{v} R} (f : Hom A B) :=
+  ConcreteCategory.hom (C := AlgebraCat R) f
+
+variable {R} in
+/-- Typecheck an `AlgHom` as a morphism in `AlgebraCat`. -/
+abbrev ofHom {A B : Type v} [Ring A] [Ring B] [Algebra R A] [Algebra R B] (f : A →ₐ[R] B) :
+    of R A ⟶ of R B :=
+  ConcreteCategory.ofHom (C := AlgebraCat R) f
+
+variable {R} in
+/-- Use the `ConcreteCategory.hom` projection for `@[simps]` lemmas. -/
+def Hom.Simps.hom (A B : AlgebraCat.{v} R) (f : Hom A B) :=
+  f.hom
+
+initialize_simps_projections Hom (hom' → hom)
+
+/-!
+The results below duplicate the `ConcreteCategory` simp lemmas, but we can keep them for `dsimp`.
+-/
 
 @[simp]
 lemma hom_id {A : AlgebraCat.{v} R} : (𝟙 A : A ⟶ A).hom = AlgHom.id R A := rfl
@@ -90,11 +113,7 @@ lemma comp_apply {A B C : AlgebraCat.{v} R} (f : A ⟶ B) (g : B ⟶ C) (a : A)
 lemma hom_ext {A B : AlgebraCat.{v} R} {f g : A ⟶ B} (hf : f.hom = g.hom) : f = g :=
   Hom.ext hf
 
-/-- Typecheck an `AlgHom` as a morphism in `AlgebraCat R`. -/
-abbrev ofHom {R : Type u} [CommRing R] {X Y : Type v} [Ring X] [Algebra R X] [Ring Y] [Algebra R Y]
-    (f : X →ₐ[R] Y) : of R X ⟶ of R Y :=
-  ⟨f⟩
-
+@[simp]
 lemma hom_ofHom {R : Type u} [CommRing R] {X Y : Type v} [Ring X] [Algebra R X] [Ring Y]
     [Algebra R Y] (f : X →ₐ[R] Y) : (ofHom f).hom = f := rfl
 
@@ -127,12 +146,6 @@ lemma hom_inv_apply {A B : AlgebraCat.{v} R} (e : A ≅ B) (x : B) : e.hom (e.in
 instance : Inhabited (AlgebraCat R) :=
   ⟨of R R⟩
 
-instance : HasForget.{v} (AlgebraCat.{v} R) where
-  forget :=
-    { obj := fun R => R
-      map := fun f => f.hom }
-  forget_faithful := ⟨fun h => by ext x; simpa using congrFun h x⟩
-
 lemma forget_obj {A : AlgebraCat.{v} R} : (forget (AlgebraCat.{v} R)).obj A = A := rfl
 
 lemma forget_map {A B : AlgebraCat.{v} R} (f : A ⟶ B) :
diff --git a/Mathlib/Algebra/Category/Grp/AB.lean b/Mathlib/Algebra/Category/Grp/AB.lean
index dde16f6cf9ea29..3038d2cedbcace 100644
--- a/Mathlib/Algebra/Category/Grp/AB.lean
+++ b/Mathlib/Algebra/Category/Grp/AB.lean
@@ -32,14 +32,15 @@ noncomputable instance :
     intro x (hx : _ = _)
     dsimp at hx
     rcases Concrete.colimit_exists_rep S.X₂ x with ⟨j, y, rfl⟩
-    rw [← comp_apply, colimMap_eq, colimit.ι_map, comp_apply,
+    rw [← CategoryTheory.comp_apply, colimMap_eq, colimit.ι_map, CategoryTheory.comp_apply,
       ← map_zero (by exact colimit.ι S.X₃ j : (S.X₃).obj j →+ ↑(colimit S.X₃))] at hx
     rcases Concrete.colimit_exists_of_rep_eq.{u, u, u} S.X₃ _ _ hx
       with ⟨k, e₁, e₂, hk : _ = S.X₃.map e₂ 0⟩
-    rw [map_zero, ← comp_apply, ← NatTrans.naturality, comp_apply] at hk
+    rw [map_zero, ← CategoryTheory.comp_apply, ← NatTrans.naturality, CategoryTheory.comp_apply]
+      at hk
     rcases hS k hk with ⟨t, ht⟩
     use colimit.ι S.X₁ k t
-    erw [← comp_apply, colimit.ι_map, comp_apply, ht]
+    erw [← CategoryTheory.comp_apply, colimit.ι_map, CategoryTheory.comp_apply, ht]
     exact colimit.w_apply S.X₂ e₁ y)
 
 noncomputable instance :
diff --git a/Mathlib/Algebra/Category/Grp/LargeColimits.lean b/Mathlib/Algebra/Category/Grp/LargeColimits.lean
index 9f9628925dad5b..9d0a40892ca36e 100644
--- a/Mathlib/Algebra/Category/Grp/LargeColimits.lean
+++ b/Mathlib/Algebra/Category/Grp/LargeColimits.lean
@@ -40,7 +40,7 @@ lemma isColimit_iff_bijective_desc [DecidableEq J] :
     apply ofHom_injective
     refine hc.hom_ext (fun j ↦ ?_)
     ext x
-    rw [comp_apply, comp_apply, ← Quot.ι_desc _ c j x]
+    rw [CategoryTheory.comp_apply, CategoryTheory.comp_apply, ← Quot.ι_desc _ c j x]
     exact ULift.down_injective (DFunLike.congr_fun eq (Quot.ι F j x))
   · set c' : Cocone F :=
       { pt := AddCommGrp.of (ULift (AddCircle (1 : ℚ)))
diff --git a/Mathlib/Algebra/Category/ModuleCat/Basic.lean b/Mathlib/Algebra/Category/ModuleCat/Basic.lean
index 4c63c2f7e9aeae..1692cf88738586 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Basic.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Basic.lean
@@ -89,20 +89,40 @@ variable {R} in
 structure Hom (M N : ModuleCat.{v} R) where
   private mk ::
   /-- The underlying linear map. -/
-  hom : M →ₗ[R] N
+  hom' : M →ₗ[R] N
 
 instance moduleCategory : Category.{v, max (v+1) u} (ModuleCat.{v} R) where
   Hom M N := Hom M N
   id _ := ⟨LinearMap.id⟩
-  comp f g := ⟨g.hom.comp f.hom⟩
+  comp f g := ⟨g.hom'.comp f.hom'⟩
 
-instance {M N : ModuleCat.{v} R} : CoeFun (M ⟶ N) (fun _ ↦ M → N) where
-  coe f := f.hom
+instance : ConcreteCategory (ModuleCat.{v} R) (· →ₗ[R] ·) where
+  hom := Hom.hom'
+  ofHom := Hom.mk
 
 section
 
 variable {R}
 
+/-- Turn a morphism in `ModuleCat` back into a `LinearMap`. -/
+abbrev Hom.hom {A B : ModuleCat.{v} R} (f : Hom A B) :=
+  ConcreteCategory.hom (C := ModuleCat R) f
+
+/-- Typecheck a `LinearMap` as a morphism in `ModuleCat`. -/
+abbrev ofHom {X Y : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y]
+    (f : X →ₗ[R] Y) : of R X ⟶ of R Y :=
+  ConcreteCategory.ofHom (C := ModuleCat R) f
+
+/-- Use the `ConcreteCategory.hom` projection for `@[simps]` lemmas. -/
+def Hom.Simps.hom (A B : ModuleCat.{v} R) (f : Hom A B) :=
+  f.hom
+
+initialize_simps_projections Hom (hom' → hom)
+
+/-!
+The results below duplicate the `ConcreteCategory` simp lemmas, but we can keep them for `dsimp`.
+-/
+
 @[simp]
 lemma hom_id {M : ModuleCat.{v} R} : (𝟙 M : M ⟶ M).hom = LinearMap.id := rfl
 
@@ -137,15 +157,9 @@ lemma hom_surjective {M N : ModuleCat.{v} R} :
     Function.Surjective (Hom.hom : (M ⟶ N) → (M →ₗ[R] N)) :=
   hom_bijective.surjective
 
-/-- Typecheck a `LinearMap` as a morphism in `ModuleCat R`. -/
-abbrev ofHom {X Y : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y] [Module R Y]
-    (f : X →ₗ[R] Y) : of R X ⟶ of R Y :=
-  ⟨f⟩
-
 @[deprecated (since := "2024-10-06")] alias asHom := ModuleCat.ofHom
 
-/- Doesn't need to be `@[simp]` since the `simp` tactic applies this rewrite automatically:
-`ofHom` and `hom` are reducibly equal to the constructor and projection respectively. -/
+@[simp]
 lemma hom_ofHom {X Y : Type v} [AddCommGroup X] [Module R X] [AddCommGroup Y]
     [Module R Y] (f : X →ₗ[R] Y) : (ofHom f).hom = f := rfl
 
@@ -444,14 +458,17 @@ end
 
 variable (M N : ModuleCat.{v} R)
 
-/-- `ModuleCat.Hom.hom` as an isomorphism of monoids. -/
-@[simps]
-def endMulEquiv : End M ≃* (M →ₗ[R] M) where
+/-- `ModuleCat.Hom.hom` as an isomorphism of rings. -/
+@[simps!] def endRingEquiv : End M ≃+* (M →ₗ[R] M) where
   toFun := ModuleCat.Hom.hom
   invFun := ModuleCat.ofHom
   map_mul' _ _ := rfl
   left_inv _ := rfl
   right_inv _ := rfl
+  map_add' _ _ := rfl
+
+/-- `ModuleCat.Hom.hom` as an isomorphism of monoids. -/
+@[deprecated (since := "2025-01-23")] alias endMulEquiv := endRingEquiv
 
 /-- The scalar multiplication on an object of `ModuleCat R` considered as
 a morphism of rings from `R` to the endomorphisms of the underlying abelian group. -/
@@ -537,9 +554,9 @@ a morphism between the underlying objects in `AddCommGrp` and the compatibility
 with the scalar multiplication. -/
 @[simps]
 def homMk : M ⟶ N where
-  hom.toFun := φ
-  hom.map_add' _ _ := φ.map_add _ _
-  hom.map_smul' r x := (congr_hom (hφ r) x).symm
+  hom'.toFun := φ
+  hom'.map_add' _ _ := φ.map_add _ _
+  hom'.map_smul' r x := (congr_hom (hφ r) x).symm
 
 lemma forget₂_map_homMk :
     (forget₂ (ModuleCat R) AddCommGrp).map (homMk φ hφ) = φ := rfl
@@ -568,7 +585,7 @@ variable {R : Type*} [CommRing R]
 namespace ModuleCat
 
 /-- Turn a bilinear map into a homomorphism. -/
-@[simps]
+@[simps!]
 def ofHom₂ {M N P : ModuleCat.{u} R} (f : M →ₗ[R] N →ₗ[R] P) :
     M ⟶ of R (N ⟶ P) :=
   ofHom <| homLinearEquiv.symm.toLinearMap ∘ₗ f
diff --git a/Mathlib/Algebra/Category/ModuleCat/Biproducts.lean b/Mathlib/Algebra/Category/ModuleCat/Biproducts.lean
index 4387064834c7ea..cb8a2f9ed669a5 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Biproducts.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Biproducts.lean
@@ -86,7 +86,7 @@ variable {J : Type w} (f : J → ModuleCat.{max w v} R)
 /-- The map from an arbitrary cone over an indexed family of abelian groups
 to the cartesian product of those groups.
 -/
-@[simps]
+@[simps!]
 def lift (s : Fan f) : s.pt ⟶ ModuleCat.of R (∀ j, f j) :=
   ofHom
   { toFun := fun x j => s.π.app ⟨j⟩ x
diff --git a/Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean b/Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean
index b922ed2155138c..fa3ae76a99db89 100644
--- a/Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/ChangeOfRings.lean
@@ -298,7 +298,7 @@ def map' {M1 M2 : ModuleCat.{v} R} (l : M1 ⟶ M2) : obj' f M1 ⟶ obj' f M2 :=
 
 theorem map'_id {M : ModuleCat.{v} R} : map' f (𝟙 M) = 𝟙 _ := by
   ext x
-  simp [map']
+  simp [map', obj']
 
 theorem map'_comp {M₁ M₂ M₃ : ModuleCat.{v} R} (l₁₂ : M₁ ⟶ M₂) (l₂₃ : M₂ ⟶ M₃) :
     map' f (l₁₂ ≫ l₂₃) = map' f l₁₂ ≫ map' f l₂₃ := by
@@ -415,7 +415,7 @@ instance : CoeFun (obj' f M) fun _ => S → M :=
 
 /-- If `M, M'` are `R`-modules, then any `R`-linear map `g : M ⟶ M'` induces an `S`-linear map
 `(S →ₗ[R] M) ⟶ (S →ₗ[R] M')` defined by `h ↦ g ∘ h`-/
-@[simps]
+@[simps!]
 def map' {M M' : ModuleCat R} (g : M ⟶ M') : obj' f M ⟶ obj' f M' :=
   ofHom
   { toFun := fun h => g.hom.comp h
@@ -610,7 +610,7 @@ Given `R`-module X and `S`-module Y and a map `g : (extendScalars f).obj X ⟶ Y
 map `S ⨂ X → Y`, there is a `X ⟶ (restrictScalars f).obj Y`, i.e. `R`-linear map `X ⟶ Y` by
 `x ↦ g (1 ⊗ x)`.
 -/
-@[simps hom_apply]
+@[simps! hom_apply]
 def HomEquiv.toRestrictScalars {X Y} (g : (extendScalars f).obj X ⟶ Y) :
     X ⟶ (restrictScalars f).obj Y :=
   -- TODO: after https://github.com/leanprover-community/mathlib4/pull/19511 we need to hint `(Y := ...)`.
@@ -649,7 +649,7 @@ Given `R`-module X and `S`-module Y and a map `X ⟶ (restrictScalars f).obj Y`,
 `X ⟶ Y`, there is a map `(extend_scalars f).obj X ⟶ Y`, i.e `S`-linear map `S ⨂ X → Y` by
 `s ⊗ x ↦ s • g x`.
 -/
-@[simps hom_apply]
+@[simps! hom_apply]
 def HomEquiv.fromExtendScalars {X Y} (g : X ⟶ (restrictScalars f).obj Y) :
     (extendScalars f).obj X ⟶ Y := by
   letI m1 : Module R S := Module.compHom S f; letI m2 : Module R Y := Module.compHom Y f
@@ -734,7 +734,7 @@ def unit : 𝟭 (ModuleCat R) ⟶ extendScalars f ⋙ restrictScalars.{max v u
 
 /-- For any `S`-module Y, there is a natural `R`-linear map from `S ⨂ Y` to `Y` by
 `s ⊗ y ↦ s • y` -/
-@[simps hom_apply]
+@[simps! hom_apply]
 def Counit.map {Y} : (restrictScalars f ⋙ extendScalars f).obj Y ⟶ Y :=
   ofHom
   { toFun :=
@@ -873,8 +873,7 @@ lemma homEquiv_extendScalarsId (M : ModuleCat R) :
     (extendRestrictScalarsAdj (RingHom.id R)).homEquiv _ _ ((extendScalarsId R).hom.app M) =
       (restrictScalarsId R).inv.app M := by
   ext m
-  rw [extendRestrictScalarsAdj_homEquiv_apply, ← extendScalarsId_inv_app_apply]
-  erw [← comp_apply]
+  rw [extendRestrictScalarsAdj_homEquiv_apply, ← extendScalarsId_inv_app_apply, ← comp_apply]
   simp
 
 lemma extendScalarsId_hom_app_one_tmul (M : ModuleCat R) (m : M) :
diff --git a/Mathlib/Algebra/Category/ModuleCat/Differentials/Presheaf.lean b/Mathlib/Algebra/Category/ModuleCat/Differentials/Presheaf.lean
index da141704256b25..33b8dc3cb3281e 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Differentials/Presheaf.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Differentials/Presheaf.lean
@@ -181,7 +181,8 @@ commutative rings. -/
 noncomputable def relativeDifferentials' :
     PresheafOfModules.{u} (R ⋙ forget₂ _ _) where
   obj X := CommRingCat.KaehlerDifferential (φ'.app X)
-  map f := CommRingCat.KaehlerDifferential.map (φ'.naturality f)
+  -- Have to hint `g' := R.map f` below, or it gets unfolded weirdly.
+  map f := CommRingCat.KaehlerDifferential.map (g' := R.map f) (φ'.naturality f)
   -- Without `dsimp`, `ext` doesn't pick up the right lemmas.
   map_id _ := by dsimp; ext; simp
   map_comp _ _ := by dsimp; ext; simp
@@ -192,7 +193,8 @@ attribute [simp] relativeDifferentials'_obj
 lemma relativeDifferentials'_map_d {X Y : Dᵒᵖ} (f : X ⟶ Y) (x : R.obj X) :
     DFunLike.coe (α := CommRingCat.KaehlerDifferential (φ'.app X))
       (β := fun _ ↦ CommRingCat.KaehlerDifferential (φ'.app Y))
-      ((relativeDifferentials' φ').map f).hom (CommRingCat.KaehlerDifferential.d x) =
+      (ModuleCat.Hom.hom (R := ↑(R.obj X)) ((relativeDifferentials' φ').map f))
+        (CommRingCat.KaehlerDifferential.d x) =
         CommRingCat.KaehlerDifferential.d (R.map f x) :=
   CommRingCat.KaehlerDifferential.map_d (φ'.naturality f) _
 
diff --git a/Mathlib/Algebra/Category/ModuleCat/Limits.lean b/Mathlib/Algebra/Category/ModuleCat/Limits.lean
index 431ac7e1734a3d..81944fb1cb6489 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Limits.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Limits.lean
@@ -249,8 +249,7 @@ def directLimitIsColimit : IsColimit (directLimitCocone G f) where
       rfl
   fac s i := by
     ext
-    dsimp only [directLimitCocone, CategoryStruct.comp]
-    rw [LinearMap.comp_apply]
+    dsimp only [hom_comp, directLimitCocone, hom_ofHom, LinearMap.comp_apply]
     apply DirectLimit.lift_of
   uniq s m h := by
     have :
diff --git a/Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean b/Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean
index 89bdba339ff9f5..1f38171bfc7b0d 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean
@@ -244,7 +244,7 @@ lemma tensorLift_tmul (m : M₁) (n : M₂) :
 
 end
 
-lemma tensor_ext {f g : M₁ ⊗ M₂ ⟶ M₃} (h : ∀ m n, f (m ⊗ₜ n) = g (m ⊗ₜ n)) :
+lemma tensor_ext {f g : M₁ ⊗ M₂ ⟶ M₃} (h : ∀ m n, f.hom (m ⊗ₜ n) = g.hom (m ⊗ₜ n)) :
     f = g :=
   hom_ext <| TensorProduct.ext (by ext; apply h)
 
diff --git a/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean b/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean
index a619b7e79f8806..3fd18ca3d8336b 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean
@@ -42,12 +42,12 @@ structure PresheafOfModules where
   /-- the restriction maps of a presheaf of modules -/
   map {X Y : Cᵒᵖ} (f : X ⟶ Y) : obj X ⟶ (ModuleCat.restrictScalars (R.map f).hom).obj (obj Y)
   map_id (X : Cᵒᵖ) :
-    map (𝟙 X) =
-      (ModuleCat.restrictScalarsId' _ (congrArg RingCat.Hom.hom (R.map_id X))).inv.app _ := by
+    map (𝟙 X) = (ModuleCat.restrictScalarsId' (R.map (𝟙 X)).hom
+      (congrArg RingCat.Hom.hom (R.map_id X))).inv.app _ := by
         aesop_cat
   map_comp {X Y Z : Cᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) :
     map (f ≫ g) = map f ≫ (ModuleCat.restrictScalars _).map (map g) ≫
-      (ModuleCat.restrictScalarsComp' _ _ _
+      (ModuleCat.restrictScalarsComp' (R.map f).hom (R.map g).hom (R.map (f ≫ g)).hom
         (congrArg RingCat.Hom.hom <| R.map_comp f g)).inv.app _ := by aesop_cat
 
 namespace PresheafOfModules
diff --git a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Monoidal.lean b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Monoidal.lean
index cd94fb824ae020..869be578d46d6b 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Monoidal.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Monoidal.lean
@@ -53,7 +53,8 @@ noncomputable def tensorObj : PresheafOfModules (R ⋙ forget₂ _ _) where
   map_id X := ModuleCat.MonoidalCategory.tensor_ext (by
     intro m₁ m₂
     dsimp [tensorObjMap]
-    simp)
+    simp
+    rfl) -- `ModuleCat.restrictScalarsId'App_inv_apply` doesn't get picked up due to type mismatch
   map_comp f g := ModuleCat.MonoidalCategory.tensor_ext (by
     intro m₁ m₂
     dsimp [tensorObjMap]
@@ -65,7 +66,8 @@ variable {M₁ M₂ M₃ M₄}
 lemma tensorObj_map_tmul {X Y : Cᵒᵖ} (f : X ⟶ Y) (m₁ : M₁.obj X) (m₂ : M₂.obj X) :
     DFunLike.coe (α := (M₁.obj X ⊗ M₂.obj X :))
       (β := fun _ ↦ (ModuleCat.restrictScalars (R.map f).hom).obj (M₁.obj Y ⊗ M₂.obj Y))
-      ((tensorObj M₁ M₂).map f).hom (m₁ ⊗ₜ[R.obj X] m₂) = M₁.map f m₁ ⊗ₜ[R.obj Y] M₂.map f m₂ := rfl
+      (ModuleCat.Hom.hom (R := ↑(R.obj X)) ((tensorObj M₁ M₂).map f)) (m₁ ⊗ₜ[R.obj X] m₂) =
+    M₁.map f m₁ ⊗ₜ[R.obj Y] M₂.map f m₂ := rfl
 
 /-- The tensor product of two morphisms of presheaves of modules. -/
 @[simps]
@@ -73,8 +75,11 @@ noncomputable def tensorHom (f : M₁ ⟶ M₂) (g : M₃ ⟶ M₄) : tensorObj
   app X := f.app X ⊗ g.app X
   naturality {X Y} φ := ModuleCat.MonoidalCategory.tensor_ext (fun m₁ m₃ ↦ by
     dsimp
-    rw [tensorObj_map_tmul, ModuleCat.MonoidalCategory.tensorHom_tmul, tensorObj_map_tmul,
-      naturality_apply, naturality_apply])
+    rw [tensorObj_map_tmul]
+    -- Need `erw` because of the type mismatch in `map` and the tensor product.
+    erw [ModuleCat.MonoidalCategory.tensorHom_tmul, tensorObj_map_tmul]
+    rw [naturality_apply, naturality_apply]
+    simp)
 
 end Monoidal
 
diff --git a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Pushforward.lean b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Pushforward.lean
index 62c75c356bf580..80d229fe13f6cd 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Pushforward.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Pushforward.lean
@@ -28,15 +28,16 @@ namespace PresheafOfModules
 
 variable (F : C ⥤ D)
 
+set_option maxHeartbeats 400000 in -- Seems to be a timeout in type checking the definition
 /-- The pushforward functor on presheaves of modules for a functor `F : C ⥤ D` and
 `R : Dᵒᵖ ⥤ RingCat`. On the underlying presheaves of abelian groups, it is induced
 by the precomposition with `F.op`. -/
 def pushforward₀ (R : Dᵒᵖ ⥤ RingCat.{u}) :
     PresheafOfModules.{v} R ⥤ PresheafOfModules.{v} (F.op ⋙ R) where
   obj M :=
-    { obj := fun X ↦ ModuleCat.of _ (M.obj (F.op.obj X))
-      map := fun {X Y} f ↦ M.map (F.op.map f)
-      map_id := fun X ↦ by
+    { obj X := ModuleCat.of _ (M.obj (F.op.obj X))
+      map {X Y} f := M.map (F.op.map f)
+      map_id X := by
         refine ModuleCat.hom_ext
           -- Work around an instance diamond for `restrictScalarsId'`
           (@LinearMap.ext _ _ _ _ _ _ _ _ (_) (_) _ _ _ (fun x => ?_))
@@ -46,7 +47,7 @@ def pushforward₀ (R : Dᵒᵖ ⥤ RingCat.{u}) :
           -- Work around an instance diamond for `restrictScalarsId'`
           (@LinearMap.ext _ _ _ _ _ _ _ _ (_) (_) _ _ _ (fun x => ?_))
         exact (M.congr_map_apply (F.op.map_comp f g) x).trans (by simp) }
-  map {M₁ M₂} φ := { app := fun X ↦ φ.app _ }
+  map {M₁ M₂} φ := { app X := φ.app _ }
 
 /-- The pushforward of presheaves of modules commutes with the forgetful functor
 to presheaves of abelian groups. -/
diff --git a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean
index 4a8d0a5b700861..a1d35a63c69edb 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Presheaf/Sheafify.lean
@@ -106,7 +106,7 @@ lemma isCompatible_map_smul : ((r₀.smul m₀).map (whiskerRight φ (forget _))
       RingCat.comp_apply]
   have hb₀ : (φ.app (Opposite.op Z)) b₀ = (A.map (f₁.op ≫ g₁.op)) m := by
     dsimp [b₀]
-    erw [NatTrans.naturality_apply, hb₁, Functor.map_comp, comp_apply]
+    erw [NatTrans.naturality_apply, hb₁, Functor.map_comp, CategoryTheory.comp_apply]
   have ha₀' : (α.app (Opposite.op Z)) a₀ = (R.map (f₂.op ≫ g₂.op)) r := by
     rw [ha₀, ← op_comp, fac, op_comp]
   have hb₀' : (φ.app (Opposite.op Z)) b₀ = (A.map (f₂.op ≫ g₂.op)) m := by
@@ -161,7 +161,7 @@ def SMulCandidate.mk' (S : Sieve X.unop) (hS : S ∈ J X.unop)
     apply A.isSeparated _ _ (J.pullback_stable f.unop hS)
     rintro Z g hg
     dsimp at hg
-    erw [← comp_apply, ← A.val.map_comp, ← NatTrans.naturality_apply, M₀.map_smul]
+    erw [← CategoryTheory.comp_apply, ← A.val.map_comp, ← NatTrans.naturality_apply, M₀.map_smul]
     refine (ha _ hg).trans (app_eq_of_isLocallyInjective α φ A.isSeparated _ _ _ _ ?_ ?_)
     · rw [← RingCat.comp_apply, NatTrans.naturality, RingCat.comp_apply, ha₀]
       apply (hr₀ _ hg).symm.trans
@@ -298,9 +298,9 @@ lemma map_smul :
   apply A.isSeparated _ _ hS
   rintro Y f ⟨⟨r₀,
     (hr₀ : (α.app (Opposite.op Y)).hom r₀ = (R.val.map f.op).hom ((R.val.map π).hom r))⟩, ⟨m₀, hm₀⟩⟩
-  rw [← comp_apply, ← Functor.map_comp,
+  rw [← CategoryTheory.comp_apply, ← Functor.map_comp,
     map_smul_eq α φ r m (π ≫ f.op) r₀ (by rw [hr₀, Functor.map_comp, RingCat.comp_apply]) m₀
-      (by rw [hm₀, Functor.map_comp, comp_apply]),
+      (by rw [hm₀, Functor.map_comp, CategoryTheory.comp_apply]),
     map_smul_eq α φ (R.val.map π r) (A.val.map π m) f.op r₀ hr₀ m₀ hm₀]
 
 end Sheafify
diff --git a/Mathlib/Algebra/Category/ModuleCat/Products.lean b/Mathlib/Algebra/Category/ModuleCat/Products.lean
index 937317e6ea0a50..d4a7f6a1b8fcba 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Products.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Products.lean
@@ -86,7 +86,7 @@ def coproductCoconeIsColimit : IsColimit (coproductCocone Z) where
     ext : 1
     refine DirectSum.linearMap_ext _ fun i ↦ ?_
     ext x
-    simpa only [LinearMap.coe_comp, Function.comp_apply, toModule_lof] using
+    simpa only [LinearMap.coe_comp, Function.comp_apply, hom_ofHom, toModule_lof] using
       congr($(h ⟨i⟩) x)
 
 variable [HasCoproduct Z]
diff --git a/Mathlib/Algebra/Category/ModuleCat/Subobject.lean b/Mathlib/Algebra/Category/ModuleCat/Subobject.lean
index 5fb8394a9c465d..29c919d23199df 100644
--- a/Mathlib/Algebra/Category/ModuleCat/Subobject.lean
+++ b/Mathlib/Algebra/Category/ModuleCat/Subobject.lean
@@ -59,7 +59,7 @@ noncomputable def subobjectModule : Subobject M ≃o Submodule R M :=
               ofHom (underlyingIso (ofHom N.subtype)).symm.toLinearEquiv.toLinearMap := by
               ext x
               rfl
-          rw [this, hom_comp, LinearEquiv.range_comp]
+          rw [this, hom_comp, hom_ofHom, LinearEquiv.range_comp]
         · exact (Submodule.range_subtype _).symm
       map_rel_iff' := fun {S T} => by
         refine ⟨fun h => ?_, fun h => mk_le_mk_of_comm (↟(Submodule.inclusion h)) rfl⟩
diff --git a/Mathlib/Algebra/Category/Ring/Basic.lean b/Mathlib/Algebra/Category/Ring/Basic.lean
index eb6977db756324..8dd6aac24e41de 100644
--- a/Mathlib/Algebra/Category/Ring/Basic.lean
+++ b/Mathlib/Algebra/Category/Ring/Basic.lean
@@ -53,18 +53,37 @@ lemma of_carrier (R : SemiRingCat.{u}) : of R = R := rfl
 variable {R} in
 /-- The type of morphisms in `SemiRingCat`. -/
 @[ext]
-structure Hom (R S : SemiRingCat) where
+structure Hom (R S : SemiRingCat.{u}) where
   private mk ::
   /-- The underlying ring hom. -/
-  hom : R →+* S
+  hom' : R →+* S
 
 instance : Category SemiRingCat where
   Hom R S := Hom R S
   id R := ⟨RingHom.id R⟩
-  comp f g := ⟨g.hom.comp f.hom⟩
+  comp f g := ⟨g.hom'.comp f.hom'⟩
 
-instance {R S : SemiRingCat.{u}} : CoeFun (R ⟶ S) (fun _ ↦ R → S) where
-  coe f := f.hom
+instance : ConcreteCategory.{u} SemiRingCat (fun R S => R →+* S) where
+  hom := Hom.hom'
+  ofHom f := ⟨f⟩
+
+/-- Turn a morphism in `SemiRingCat` back into a `RingHom`. -/
+abbrev Hom.hom {R S : SemiRingCat.{u}} (f : Hom R S) :=
+  ConcreteCategory.hom (C := SemiRingCat) f
+
+/-- Typecheck a `RingHom` as a morphism in `SemiRingCat`. -/
+abbrev ofHom {R S : Type u} [Semiring R] [Semiring S] (f : R →+* S) : of R ⟶ of S :=
+  ConcreteCategory.ofHom (C := SemiRingCat) f
+
+/-- Use the `ConcreteCategory.hom` projection for `@[simps]` lemmas. -/
+def Hom.Simps.hom (R S : SemiRingCat) (f : Hom R S) :=
+  f.hom
+
+initialize_simps_projections Hom (hom' → hom)
+
+/-!
+The results below duplicate the `ConcreteCategory` simp lemmas, but we can keep them for `dsimp`.
+-/
 
 @[simp]
 lemma hom_id {R : SemiRingCat} : (𝟙 R : R ⟶ R).hom = RingHom.id R := rfl
@@ -85,10 +104,7 @@ lemma comp_apply {R S T : SemiRingCat} (f : R ⟶ S) (g : S ⟶ T) (r : R) :
 lemma hom_ext {R S : SemiRingCat} {f g : R ⟶ S} (hf : f.hom = g.hom) : f = g :=
   Hom.ext hf
 
-/-- Typecheck a `RingHom` as a morphism in `SemiRingCat`. -/
-abbrev ofHom {R S : Type u} [Semiring R] [Semiring S] (f : R →+* S) : of R ⟶ of S :=
-  ⟨f⟩
-
+@[simp]
 lemma hom_ofHom {R S : Type u} [Semiring R] [Semiring S] (f : R →+* S) : (ofHom f).hom = f := rfl
 
 @[simp]
@@ -120,12 +136,6 @@ lemma hom_inv_apply {R S : SemiRingCat} (e : R ≅ S) (s : S) : e.hom (e.inv s)
 instance : Inhabited SemiRingCat :=
   ⟨of PUnit⟩
 
-instance : HasForget.{u} SemiRingCat where
-  forget :=
-    { obj := fun R => R
-      map := fun f => f.hom }
-  forget_faithful := ⟨fun h => by ext x; simpa using congrFun h x⟩
-
 /-- This unification hint helps with problems of the form `(forget ?C).obj R =?= carrier R'`. -/
 unif_hint forget_obj_eq_coe (R R' : SemiRingCat) where
   R ≟ R' ⊢
@@ -154,8 +164,8 @@ instance hasForgetToAddCommMonCat : HasForget₂ SemiRingCat AddCommMonCat where
 @[simps]
 def _root_.RingEquiv.toSemiRingCatIso {R S : Type u} [Semiring R] [Semiring S] (e : R ≃+* S) :
     of R ≅ of S where
-  hom := ⟨e⟩
-  inv := ⟨e.symm⟩
+  hom := ofHom e
+  inv := ofHom e.symm
 
 instance forgetReflectIsos : (forget SemiRingCat).ReflectsIsomorphisms where
   reflects {X Y} f _ := by
@@ -197,18 +207,37 @@ lemma of_carrier (R : RingCat.{u}) : of R = R := rfl
 variable {R} in
 /-- The type of morphisms in `RingCat`. -/
 @[ext]
-structure Hom (R S : RingCat) where
+structure Hom (R S : RingCat.{u}) where
   private mk ::
   /-- The underlying ring hom. -/
-  hom : R →+* S
+  hom' : R →+* S
 
 instance : Category RingCat where
   Hom R S := Hom R S
   id R := ⟨RingHom.id R⟩
-  comp f g := ⟨g.hom.comp f.hom⟩
+  comp f g := ⟨g.hom'.comp f.hom'⟩
 
-instance {R S : RingCat.{u}} : CoeFun (R ⟶ S) (fun _ ↦ R → S) where
-  coe f := f.hom
+instance : ConcreteCategory.{u} RingCat (fun R S => R →+* S) where
+  hom := Hom.hom'
+  ofHom f := ⟨f⟩
+
+/-- Turn a morphism in `RingCat` back into a `RingHom`. -/
+abbrev Hom.hom {R S : RingCat.{u}} (f : Hom R S) :=
+  ConcreteCategory.hom (C := RingCat) f
+
+/-- Typecheck a `RingHom` as a morphism in `RingCat`. -/
+abbrev ofHom {R S : Type u} [Ring R] [Ring S] (f : R →+* S) : of R ⟶ of S :=
+  ConcreteCategory.ofHom (C := RingCat) f
+
+/-- Use the `ConcreteCategory.hom` projection for `@[simps]` lemmas. -/
+def Hom.Simps.hom (R S : RingCat) (f : Hom R S) :=
+  f.hom
+
+initialize_simps_projections Hom (hom' → hom)
+
+/-!
+The results below duplicate the `ConcreteCategory` simp lemmas, but we can keep them for `dsimp`.
+-/
 
 @[simp]
 lemma hom_id {R : RingCat} : (𝟙 R : R ⟶ R).hom = RingHom.id R := rfl
@@ -229,10 +258,7 @@ lemma comp_apply {R S T : RingCat} (f : R ⟶ S) (g : S ⟶ T) (r : R) :
 lemma hom_ext {R S : RingCat} {f g : R ⟶ S} (hf : f.hom = g.hom) : f = g :=
   Hom.ext hf
 
-/-- Typecheck a `RingHom` as a morphism in `RingCat`. -/
-abbrev ofHom {R S : Type u} [Ring R] [Ring S] (f : R →+* S) : of R ⟶ of S :=
-  ⟨f⟩
-
+@[simp]
 lemma hom_ofHom {R S : Type u} [Ring R] [Ring S] (f : R →+* S) : (ofHom f).hom = f := rfl
 
 @[simp]
@@ -264,12 +290,6 @@ lemma hom_inv_apply {R S : RingCat} (e : R ≅ S) (s : S) : e.hom (e.inv s) = s
 instance : Inhabited RingCat :=
   ⟨of PUnit⟩
 
-instance : HasForget.{u} RingCat where
-  forget :=
-    { obj := fun R => R
-      map := fun f => f.hom }
-  forget_faithful := ⟨fun h => by ext x; simpa using congrFun h x⟩
-
 /-- This unification hint helps with problems of the form `(forget ?C).obj R =?= carrier R'`.
 
 An example where this is needed is in applying
@@ -302,8 +322,8 @@ instance hasForgetToAddCommGrp : HasForget₂ RingCat AddCommGrp where
 @[simps]
 def _root_.RingEquiv.toRingCatIso {R S : Type u} [Ring R] [Ring S] (e : R ≃+* S) :
     of R ≅ of S where
-  hom := ⟨e⟩
-  inv := ⟨e.symm⟩
+  hom := ofHom e
+  inv := ofHom e.symm
 
 instance forgetReflectIsos : (forget RingCat).ReflectsIsomorphisms where
   reflects {X Y} f _ := by
@@ -345,18 +365,37 @@ lemma of_carrier (R : CommSemiRingCat.{u}) : of R = R := rfl
 variable {R} in
 /-- The type of morphisms in `CommSemiRingCat`. -/
 @[ext]
-structure Hom (R S : CommSemiRingCat) where
+structure Hom (R S : CommSemiRingCat.{u}) where
   private mk ::
   /-- The underlying ring hom. -/
-  hom : R →+* S
+  hom' : R →+* S
 
 instance : Category CommSemiRingCat where
   Hom R S := Hom R S
   id R := ⟨RingHom.id R⟩
-  comp f g := ⟨g.hom.comp f.hom⟩
+  comp f g := ⟨g.hom'.comp f.hom'⟩
 
-instance {R S : CommSemiRingCat.{u}} : CoeFun (R ⟶ S) (fun _ ↦ R → S) where
-  coe f := f.hom
+instance : ConcreteCategory.{u} CommSemiRingCat (fun R S => R →+* S) where
+  hom := Hom.hom'
+  ofHom f := ⟨f⟩
+
+/-- Turn a morphism in `CommSemiRingCat` back into a `RingHom`. -/
+abbrev Hom.hom {R S : CommSemiRingCat.{u}} (f : Hom R S) :=
+  ConcreteCategory.hom (C := CommSemiRingCat) f
+
+/-- Typecheck a `RingHom` as a morphism in `CommSemiRingCat`. -/
+abbrev ofHom {R S : Type u} [CommSemiring R] [CommSemiring S] (f : R →+* S) : of R ⟶ of S :=
+  ConcreteCategory.ofHom (C := CommSemiRingCat) f
+
+/-- Use the `ConcreteCategory.hom` projection for `@[simps]` lemmas. -/
+def Hom.Simps.hom (R S : CommSemiRingCat) (f : Hom R S) :=
+  f.hom
+
+initialize_simps_projections Hom (hom' → hom)
+
+/-!
+The results below duplicate the `ConcreteCategory` simp lemmas, but we can keep them for `dsimp`.
+-/
 
 @[simp]
 lemma hom_id {R : CommSemiRingCat} : (𝟙 R : R ⟶ R).hom = RingHom.id R := rfl
@@ -377,10 +416,7 @@ lemma comp_apply {R S T : CommSemiRingCat} (f : R ⟶ S) (g : S ⟶ T) (r : R) :
 lemma hom_ext {R S : CommSemiRingCat} {f g : R ⟶ S} (hf : f.hom = g.hom) : f = g :=
   Hom.ext hf
 
-/-- Typecheck a `RingHom` as a morphism in `CommSemiRingCat`. -/
-abbrev ofHom {R S : Type u} [CommSemiring R] [CommSemiring S] (f : R →+* S) : of R ⟶ of S :=
-  ⟨f⟩
-
+@[simp]
 lemma hom_ofHom {R S : Type u} [CommSemiring R] [CommSemiring S] (f : R →+* S) :
   (ofHom f).hom = f := rfl
 
@@ -413,12 +449,6 @@ lemma hom_inv_apply {R S : CommSemiRingCat} (e : R ≅ S) (s : S) : e.hom (e.inv
 instance : Inhabited CommSemiRingCat :=
   ⟨of PUnit⟩
 
-instance : HasForget.{u} CommSemiRingCat where
-  forget :=
-    { obj := fun R => R
-      map := fun f => f.hom }
-  forget_faithful := ⟨fun h => by ext x; simpa using congrFun h x⟩
-
 /-- This unification hint helps with problems of the form `(forget ?C).obj R =?= carrier R'`. -/
 unif_hint forget_obj_eq_coe (R R' : CommSemiRingCat) where
   R ≟ R' ⊢
@@ -449,8 +479,8 @@ instance hasForgetToCommMonCat : HasForget₂ CommSemiRingCat CommMonCat where
 def _root_.RingEquiv.toCommSemiRingCatIso
     {R S : Type u} [CommSemiring R] [CommSemiring S] (e : R ≃+* S) :
     of R ≅ of S where
-  hom := ⟨e⟩
-  inv := ⟨e.symm⟩
+  hom := ofHom e
+  inv := ofHom e.symm
 
 instance forgetReflectIsos : (forget CommSemiRingCat).ReflectsIsomorphisms where
   reflects {X Y} f _ := by
@@ -492,18 +522,37 @@ lemma of_carrier (R : CommRingCat.{u}) : of R = R := rfl
 variable {R} in
 /-- The type of morphisms in `CommRingCat`. -/
 @[ext]
-structure Hom (R S : CommRingCat) where
+structure Hom (R S : CommRingCat.{u}) where
   private mk ::
   /-- The underlying ring hom. -/
-  hom : R →+* S
+  hom' : R →+* S
 
 instance : Category CommRingCat where
   Hom R S := Hom R S
   id R := ⟨RingHom.id R⟩
-  comp f g := ⟨g.hom.comp f.hom⟩
+  comp f g := ⟨g.hom'.comp f.hom'⟩
 
-instance {R S : CommRingCat.{u}} : CoeFun (R ⟶ S) (fun _ ↦ R → S) where
-  coe f := f.hom
+instance : ConcreteCategory.{u} CommRingCat (fun R S => R →+* S) where
+  hom := Hom.hom'
+  ofHom f := ⟨f⟩
+
+/-- The underlying ring hom. -/
+abbrev Hom.hom {R S : CommRingCat.{u}} (f : Hom R S) :=
+  ConcreteCategory.hom (C := CommRingCat) f
+
+/-- Typecheck a `RingHom` as a morphism in `CommRingCat`. -/
+abbrev ofHom {R S : Type u} [CommRing R] [CommRing S] (f : R →+* S) : of R ⟶ of S :=
+  ConcreteCategory.ofHom (C := CommRingCat) f
+
+/-- Use the `ConcreteCategory.hom` projection for `@[simps]` lemmas. -/
+def Hom.Simps.hom (R S : CommRingCat) (f : Hom R S) :=
+  f.hom
+
+initialize_simps_projections Hom (hom' → hom)
+
+/-!
+The results below duplicate the `ConcreteCategory` simp lemmas, but we can keep them for `dsimp`.
+-/
 
 @[simp]
 lemma hom_id {R : CommRingCat} : (𝟙 R : R ⟶ R).hom = RingHom.id R := rfl
@@ -524,10 +573,7 @@ lemma comp_apply {R S T : CommRingCat} (f : R ⟶ S) (g : S ⟶ T) (r : R) :
 lemma hom_ext {R S : CommRingCat} {f g : R ⟶ S} (hf : f.hom = g.hom) : f = g :=
   Hom.ext hf
 
-/-- Typecheck a `RingHom` as a morphism in `CommRingCat`. -/
-abbrev ofHom {R S : Type u} [CommRing R] [CommRing S] (f : R →+* S) : of R ⟶ of S :=
-  ⟨f⟩
-
+@[simp]
 lemma hom_ofHom {R S : Type u} [CommRing R] [CommRing S] (f : R →+* S) :
   (ofHom f).hom = f := rfl
 
@@ -560,12 +606,6 @@ lemma hom_inv_apply {R S : CommRingCat} (e : R ≅ S) (s : S) : e.hom (e.inv s)
 instance : Inhabited CommRingCat :=
   ⟨of PUnit⟩
 
-instance : HasForget.{u} CommRingCat where
-  forget :=
-    { obj := fun R => R
-      map := fun f => f.hom }
-  forget_faithful := ⟨fun h => by ext x; simpa using congrFun h x⟩
-
 lemma forget_obj {R : CommRingCat} : (forget CommRingCat).obj R = R := rfl
 
 /-- This unification hint helps with problems of the form `(forget ?C).obj R =?= carrier R'`.
@@ -606,8 +646,8 @@ instance hasForgetToAddCommMonCat : HasForget₂ CommRingCat CommSemiRingCat whe
 def _root_.RingEquiv.toCommRingCatIso
     {R S : Type u} [CommRing R] [CommRing S] (e : R ≃+* S) :
     of R ≅ of S where
-  hom := ⟨e⟩
-  inv := ⟨e.symm⟩
+  hom := ofHom e
+  inv := ofHom e.symm
 
 instance forgetReflectIsos : (forget CommRingCat).ReflectsIsomorphisms where
   reflects {X Y} f _ := by
@@ -669,10 +709,10 @@ abbrev CommRingCatMax.{u1, u2} := CommRingCat.{max u1 u2}
 
 lemma RingCat.forget_map_apply {R S : RingCat} (f : R ⟶ S)
     (x : (CategoryTheory.forget RingCat).obj R) :
-    @DFunLike.coe _ _ _ HasForget.instFunLike f x = f x :=
+    (forget _).map f x = f x :=
   rfl
 
 lemma CommRingCat.forget_map_apply {R S : CommRingCat} (f : R ⟶ S)
     (x : (CategoryTheory.forget CommRingCat).obj R) :
-    @DFunLike.coe _ _ _ HasForget.instFunLike f x = f x :=
+    (forget _).map f x = f x :=
   rfl
diff --git a/Mathlib/Algebra/Category/Ring/Epi.lean b/Mathlib/Algebra/Category/Ring/Epi.lean
index ae3808ac124703..4ba86a9d7d7c35 100644
--- a/Mathlib/Algebra/Category/Ring/Epi.lean
+++ b/Mathlib/Algebra/Category/Ring/Epi.lean
@@ -23,8 +23,7 @@ lemma CommRingCat.epi_iff_tmul_eq_tmul {R S : Type u} [CommRing R] [CommRing S]
       ∀ s : S, s ⊗ₜ[R] 1 = 1 ⊗ₜ s := by
   constructor
   · intro H
-    #adaptation_note
-    /-- After https://github.com/leanprover/lean4/pull/6024
+    #adaptation_note /-- https://github.com/leanprover/lean4/pull/6024
     we need to add `(R := R) (A := S)` in the next line to deal with unification issues. -/
     have := H.1 (CommRingCat.ofHom <| Algebra.TensorProduct.includeLeftRingHom (R := R))
       (CommRingCat.ofHom <| (Algebra.TensorProduct.includeRight (R := R) (A := S)).toRingHom)
diff --git a/Mathlib/Algebra/DualNumber.lean b/Mathlib/Algebra/DualNumber.lean
index 6e2ef4a79c866f..2eca2386d15669 100644
--- a/Mathlib/Algebra/DualNumber.lean
+++ b/Mathlib/Algebra/DualNumber.lean
@@ -149,36 +149,24 @@ theorem lift_apply_apply (fe : {_fe : (A →ₐ[R] B) × B // _}) (a : A[ε]) :
 @[simp] theorem coe_lift_symm_apply (F : A[ε] →ₐ[R] B) :
     (lift.symm F).val = (F.comp (inlAlgHom _ _ _), F ε) := rfl
 
-#adaptation_note
-/--
-The new unused variable linter in
-https://github.com/leanprover/lean4/pull/5338
-flags `{fe : (A →ₐ[R] B) × B // _}`.
--/
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/5338
+The new unused variable linter flags `{fe : (A →ₐ[R] B) × B // _}`. -/
 set_option linter.unusedVariables false in
 /-- When applied to `inl`, `DualNumber.lift` applies the map `f : A →ₐ[R] B`. -/
 @[simp] theorem lift_apply_inl (fe : {fe : (A →ₐ[R] B) × B // _}) (a : A) :
     lift fe (inl a : A[ε]) = fe.val.1 a := by
   rw [lift_apply_apply, fst_inl, snd_inl, map_zero, zero_mul, add_zero]
 
-#adaptation_note
-/--
-The new unused variable linter in
-https://github.com/leanprover/lean4/pull/5338
-flags `{fe : (A →ₐ[R] B) × B // _}`.
--/
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/5338
+The new unused variable linter flags `{fe : (A →ₐ[R] B) × B // _}`. -/
 set_option linter.unusedVariables false in
 /-- Scaling on the left is sent by `DualNumber.lift` to multiplication on the left -/
 @[simp] theorem lift_smul (fe : {fe : (A →ₐ[R] B) × B // _}) (a : A) (ad : A[ε]) :
     lift fe (a • ad) = fe.val.1 a * lift fe ad := by
   rw [← inl_mul_eq_smul, map_mul, lift_apply_inl]
 
-#adaptation_note
-/--
-The new unused variable linter in
-https://github.com/leanprover/lean4/pull/5338
-flags `{fe : (A →ₐ[R] B) × B // _}`.
--/
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/5338
+The new unused variable linter flags `{fe : (A →ₐ[R] B) × B // _}`. -/
 set_option linter.unusedVariables false in
 /-- Scaling on the right is sent by `DualNumber.lift` to multiplication on the right -/
 @[simp] theorem lift_op_smul (fe : {fe : (A →ₐ[R] B) × B // _}) (a : A) (ad : A[ε]) :
diff --git a/Mathlib/Algebra/Free.lean b/Mathlib/Algebra/Free.lean
index c0663264e177d9..0fbfaa9678903a 100644
--- a/Mathlib/Algebra/Free.lean
+++ b/Mathlib/Algebra/Free.lean
@@ -80,8 +80,8 @@ theorem hom_ext {β : Type v} [Mul β] {f g : FreeMagma α →ₙ* β} (h : f 
 
 end FreeMagma
 
-#adaptation_note /-- around nightly-2024-02-25, we need to write `mul x y` in the second pattern,
-instead of `x * y`. --/
+#adaptation_note /-- nightly-2024-02-25
+we need to write `mul x y` in the second pattern, instead of `x * y`. --/
 /-- Lifts a function `α → β` to a magma homomorphism `FreeMagma α → β` given a magma `β`. -/
 def FreeMagma.liftAux {α : Type u} {β : Type v} [Mul β] (f : α → β) : FreeMagma α → β
   | FreeMagma.of x => f x
@@ -185,8 +185,8 @@ end Category
 
 end FreeMagma
 
-#adaptation_note /-- around nightly-2024-02-25, we need to write `mul x y` in the second pattern,
-  instead of `x * y`. -/
+#adaptation_note /-- nightly-2024-02-25
+we need to write `mul x y` in the second pattern, instead of `x * y`. -/
 /-- `FreeMagma` is traversable. -/
 protected def FreeMagma.traverse {m : Type u → Type u} [Applicative m] {α β : Type u}
     (F : α → m β) : FreeMagma α → m (FreeMagma β)
@@ -262,8 +262,8 @@ end Category
 
 end FreeMagma
 
-#adaptation_note /-- around nightly-2024-02-25, we need to write `mul x y` in the second pattern,
-instead of `x * y`. -/
+#adaptation_note /-- nightly-2024-02-25
+we need to write `mul x y` in the second pattern, instead of `x * y`. -/
 /-- Representation of an element of a free magma. -/
 protected def FreeMagma.repr {α : Type u} [Repr α] : FreeMagma α → Lean.Format
   | FreeMagma.of x => repr x
@@ -279,8 +279,8 @@ attribute [to_additive existing] FreeMagma.repr
 @[to_additive]
 instance {α : Type u} [Repr α] : Repr (FreeMagma α) := ⟨fun o _ => FreeMagma.repr o⟩
 
-#adaptation_note /-- around nightly-2024-02-25, we need to write `mul x y` in the second pattern,
-instead of `x * y`. -/
+#adaptation_note /-- nightly-2024-02-25
+we need to write `mul x y` in the second pattern, instead of `x * y`. -/
 /-- Length of an element of a free magma. -/
 def FreeMagma.length {α : Type u} : FreeMagma α → ℕ
   | FreeMagma.of _x => 1
diff --git a/Mathlib/Algebra/FreeMonoid/Basic.lean b/Mathlib/Algebra/FreeMonoid/Basic.lean
index 7340041857cfc8..596248f9831176 100644
--- a/Mathlib/Algebra/FreeMonoid/Basic.lean
+++ b/Mathlib/Algebra/FreeMonoid/Basic.lean
@@ -372,7 +372,7 @@ theorem map_apply_map_symm_eq {x : FreeMonoid β} (e : α ≃ β) :
 instance uniqueUnits : Unique (FreeMonoid α)ˣ where
   uniq u := Units.ext <| toList.injective <|
     have : toList u.val ++ toList u.inv = [] := DFunLike.congr_arg toList u.val_inv
-    (List.append_eq_nil.mp this).1
+    (List.append_eq_nil_iff.mp this).1
 
 @[to_additive (attr := simp)]
 theorem map_surjective {f : α → β} : Function.Surjective (map f) ↔ Function.Surjective f := by
diff --git a/Mathlib/Algebra/GCDMonoid/Finset.lean b/Mathlib/Algebra/GCDMonoid/Finset.lean
index 89d7b6ca4f3efc..245b1c3f9ec1b0 100644
--- a/Mathlib/Algebra/GCDMonoid/Finset.lean
+++ b/Mathlib/Algebra/GCDMonoid/Finset.lean
@@ -76,8 +76,7 @@ theorem lcm_insert [DecidableEq β] {b : β} :
 theorem lcm_singleton {b : β} : ({b} : Finset β).lcm f = normalize (f b) :=
   Multiset.lcm_singleton
 
--- Porting note: Priority changed for `simpNF`
-@[simp 1100]
+@[local simp] -- This will later be provable by other `simp` lemmas.
 theorem normalize_lcm : normalize (s.lcm f) = s.lcm f := by simp [lcm_def]
 
 theorem lcm_union [DecidableEq β] : (s₁ ∪ s₂).lcm f = GCDMonoid.lcm (s₁.lcm f) (s₂.lcm f) :=
@@ -149,8 +148,7 @@ theorem gcd_insert [DecidableEq β] {b : β} :
 theorem gcd_singleton {b : β} : ({b} : Finset β).gcd f = normalize (f b) :=
   Multiset.gcd_singleton
 
--- Porting note: Priority changed for `simpNF`
-@[simp 1100]
+@[local simp] -- This will later be provable by other `simp` lemmas.
 theorem normalize_gcd : normalize (s.gcd f) = s.gcd f := by simp [gcd_def]
 
 theorem gcd_union [DecidableEq β] : (s₁ ∪ s₂).gcd f = GCDMonoid.gcd (s₁.gcd f) (s₂.gcd f) :=
diff --git a/Mathlib/Algebra/GCDMonoid/Multiset.lean b/Mathlib/Algebra/GCDMonoid/Multiset.lean
index a50a1633317558..cbe9338671d3a8 100644
--- a/Mathlib/Algebra/GCDMonoid/Multiset.lean
+++ b/Mathlib/Algebra/GCDMonoid/Multiset.lean
@@ -63,11 +63,7 @@ theorem dvd_lcm {s : Multiset α} {a : α} (h : a ∈ s) : a ∣ s.lcm :=
 theorem lcm_mono {s₁ s₂ : Multiset α} (h : s₁ ⊆ s₂) : s₁.lcm ∣ s₂.lcm :=
   lcm_dvd.2 fun _ hb ↦ dvd_lcm (h hb)
 
-/- Porting note: Following `Algebra.GCDMonoid.Basic`'s version of `normalize_gcd`, I'm giving
-this lower priority to avoid linter complaints about simp-normal form -/
-/- Porting note: Mathport seems to be replacing `Multiset.induction_on s $` with
-`(Multiset.induction_on s)`, when it should be `Multiset.induction_on s <|`. -/
-@[simp 1100]
+@[simp]
 theorem normalize_lcm (s : Multiset α) : normalize s.lcm = s.lcm :=
   Multiset.induction_on s (by simp) fun a s _ ↦ by simp
 
@@ -139,9 +135,7 @@ theorem gcd_dvd {s : Multiset α} {a : α} (h : a ∈ s) : s.gcd ∣ a :=
 theorem gcd_mono {s₁ s₂ : Multiset α} (h : s₁ ⊆ s₂) : s₂.gcd ∣ s₁.gcd :=
   dvd_gcd.2 fun _ hb ↦ gcd_dvd (h hb)
 
-/- Porting note: Following `Algebra.GCDMonoid.Basic`'s version of `normalize_gcd`, I'm giving
-this lower priority to avoid linter complaints about simp-normal form -/
-@[simp 1100]
+@[simp]
 theorem normalize_gcd (s : Multiset α) : normalize s.gcd = s.gcd :=
   Multiset.induction_on s (by simp) fun a s _ ↦ by simp
 
diff --git a/Mathlib/Algebra/Group/Hom/Defs.lean b/Mathlib/Algebra/Group/Hom/Defs.lean
index 2ffc6457f0b71d..08c5149911c7e0 100644
--- a/Mathlib/Algebra/Group/Hom/Defs.lean
+++ b/Mathlib/Algebra/Group/Hom/Defs.lean
@@ -706,6 +706,15 @@ def MonoidHom.id (M : Type*) [MulOneClass M] : M →* M where
   map_one' := rfl
   map_mul' _ _ := rfl
 
+@[to_additive (attr := simp)]
+lemma OneHom.coe_id {M : Type*} [One M] : (OneHom.id M : M → M) = _root_.id := rfl
+
+@[to_additive (attr := simp)]
+lemma MulHom.coe_id {M : Type*} [Mul M] : (MulHom.id M : M → M) = _root_.id := rfl
+
+@[to_additive (attr := simp)]
+lemma MonoidHom.coe_id {M : Type*} [MulOneClass M] : (MonoidHom.id M : M → M) = _root_.id := rfl
+
 /-- Composition of `OneHom`s as a `OneHom`. -/
 @[to_additive "Composition of `ZeroHom`s as a `ZeroHom`."]
 def OneHom.comp [One M] [One N] [One P] (hnp : OneHom N P) (hmn : OneHom M N) : OneHom M P where
diff --git a/Mathlib/Algebra/Group/Pi/Lemmas.lean b/Mathlib/Algebra/Group/Pi/Lemmas.lean
index 697d42405d54c7..b73a9e5ea8db5b 100644
--- a/Mathlib/Algebra/Group/Pi/Lemmas.lean
+++ b/Mathlib/Algebra/Group/Pi/Lemmas.lean
@@ -338,7 +338,7 @@ theorem Pi.semiconjBy_iff {x y z : ∀ i, f i} :
     SemiconjBy x y z ↔ ∀ i, SemiconjBy (x i) (y i) (z i) := funext_iff
 
 @[to_additive]
-theorem Commute.pi {x y : ∀ i, f i} (h : ∀ i, Commute (x i) (y i)) : Commute x y := .pi h
+theorem Commute.pi {x y : ∀ i, f i} (h : ∀ i, Commute (x i) (y i)) : Commute x y := SemiconjBy.pi h
 
 @[to_additive]
 theorem Pi.commute_iff {x y : ∀ i, f i} : Commute x y ↔ ∀ i, Commute (x i) (y i) := semiconjBy_iff
diff --git a/Mathlib/Algebra/Group/Prod.lean b/Mathlib/Algebra/Group/Prod.lean
index aa462854b70861..f3a94d3d34aa6e 100644
--- a/Mathlib/Algebra/Group/Prod.lean
+++ b/Mathlib/Algebra/Group/Prod.lean
@@ -260,7 +260,7 @@ theorem Prod.semiconjBy_iff {x y z : M × N} :
 
 @[to_additive AddCommute.prod]
 theorem Commute.prod {x y : M × N} (hm : Commute x.1 y.1) (hn : Commute x.2 y.2) : Commute x y :=
-  .prod hm hn
+  SemiconjBy.prod hm hn
 
 @[to_additive]
 theorem Prod.commute_iff {x y : M × N} :
diff --git a/Mathlib/Algebra/Group/Submonoid/MulAction.lean b/Mathlib/Algebra/Group/Submonoid/MulAction.lean
new file mode 100644
index 00000000000000..7c37bc72c9f08b
--- /dev/null
+++ b/Mathlib/Algebra/Group/Submonoid/MulAction.lean
@@ -0,0 +1,69 @@
+/-
+Copyright (c) 2021 Eric Wieser. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eric Wieser
+-/
+import Mathlib.Algebra.Group.Submonoid.Defs
+import Mathlib.Algebra.Group.Action.Defs
+
+/-!
+# Actions by `Submonoid`s
+
+These instances transfer the action by an element `m : M` of a monoid `M` written as `m • a` onto
+the action by an element `s : S` of a submonoid `S : Submonoid M` such that `s • a = (s : M) • a`.
+
+These instances work particularly well in conjunction with `Monoid.toMulAction`, enabling
+`s • m` as an alias for `↑s * m`.
+-/
+
+namespace Submonoid
+
+variable {M' : Type*} {α β : Type*}
+
+section MulOneClass
+
+variable [MulOneClass M']
+
+@[to_additive]
+instance smul [SMul M' α] (S : Submonoid M') : SMul S α :=
+  SMul.comp _ S.subtype
+
+@[to_additive]
+instance smulCommClass_left [SMul M' β] [SMul α β] [SMulCommClass M' α β]
+    (S : Submonoid M') : SMulCommClass S α β :=
+  ⟨fun a _ _ => smul_comm (a : M') _ _⟩
+
+@[to_additive]
+instance smulCommClass_right [SMul α β] [SMul M' β] [SMulCommClass α M' β]
+    (S : Submonoid M') : SMulCommClass α S β :=
+  ⟨fun a s => smul_comm a (s : M')⟩
+
+/-- Note that this provides `IsScalarTower S M' M'` which is needed by `SMulMulAssoc`. -/
+instance isScalarTower [SMul α β] [SMul M' α] [SMul M' β] [IsScalarTower M' α β]
+      (S : Submonoid M') :
+    IsScalarTower S α β :=
+  ⟨fun a => smul_assoc (a : M')⟩
+
+section SMul
+variable [SMul M' α] {S : Submonoid M'}
+
+@[to_additive] lemma smul_def (g : S) (a : α) : g • a = (g : M') • a := rfl
+
+@[to_additive (attr := simp)]
+lemma mk_smul (g : M') (hg : g ∈ S) (a : α) : (⟨g, hg⟩ : S) • a = g • a := rfl
+
+end SMul
+end MulOneClass
+
+variable [Monoid M']
+
+/-- The action by a submonoid is the action by the underlying monoid. -/
+@[to_additive
+      "The additive action by an `AddSubmonoid` is the action by the underlying `AddMonoid`. "]
+instance mulAction [MulAction M' α] (S : Submonoid M') : MulAction S α where
+  one_smul := one_smul M'
+  mul_smul m₁ m₂ := mul_smul (m₁ : M') m₂
+
+example {S : Submonoid M'} : IsScalarTower S M' M' := by infer_instance
+
+end Submonoid
diff --git a/Mathlib/Algebra/Group/Submonoid/Operations.lean b/Mathlib/Algebra/Group/Submonoid/Operations.lean
index 9037ade39062b4..a997c5650f854b 100644
--- a/Mathlib/Algebra/Group/Submonoid/Operations.lean
+++ b/Mathlib/Algebra/Group/Submonoid/Operations.lean
@@ -8,6 +8,7 @@ import Mathlib.Algebra.Group.Action.Faithful
 import Mathlib.Algebra.Group.Nat.Defs
 import Mathlib.Algebra.Group.Prod
 import Mathlib.Algebra.Group.Submonoid.Basic
+import Mathlib.Algebra.Group.Submonoid.MulAction
 import Mathlib.Algebra.Group.TypeTags.Basic
 
 /-!
@@ -973,75 +974,10 @@ theorem Submonoid.equivMapOfInjective_coe_mulEquiv (e : M ≃* N) :
   ext
   rfl
 
-section Actions
-
-/-! ### Actions by `Submonoid`s
-
-These instances transfer the action by an element `m : M` of a monoid `M` written as `m • a` onto
-the action by an element `s : S` of a submonoid `S : Submonoid M` such that `s • a = (s : M) • a`.
-
-These instances work particularly well in conjunction with `Monoid.toMulAction`, enabling
-`s • m` as an alias for `↑s * m`.
--/
-
-
-namespace Submonoid
-
-variable {M' : Type*} {α β : Type*}
-
-section MulOneClass
-
-variable [MulOneClass M']
-
-@[to_additive]
-instance smul [SMul M' α] (S : Submonoid M') : SMul S α :=
-  SMul.comp _ S.subtype
-
-@[to_additive]
-instance smulCommClass_left [SMul M' β] [SMul α β] [SMulCommClass M' α β]
-    (S : Submonoid M') : SMulCommClass S α β :=
-  ⟨fun a _ _ => smul_comm (a : M') _ _⟩
-
-@[to_additive]
-instance smulCommClass_right [SMul α β] [SMul M' β] [SMulCommClass α M' β]
-    (S : Submonoid M') : SMulCommClass α S β :=
-  ⟨fun a s => smul_comm a (s : M')⟩
-
-/-- Note that this provides `IsScalarTower S M' M'` which is needed by `SMulMulAssoc`. -/
-instance isScalarTower [SMul α β] [SMul M' α] [SMul M' β] [IsScalarTower M' α β]
-      (S : Submonoid M') :
-    IsScalarTower S α β :=
-  ⟨fun a => smul_assoc (a : M')⟩
-
-section SMul
-variable [SMul M' α] {S : Submonoid M'}
-
-@[to_additive] lemma smul_def (g : S) (a : α) : g • a = (g : M') • a := rfl
-
-@[to_additive (attr := simp)]
-lemma mk_smul (g : M') (hg : g ∈ S) (a : α) : (⟨g, hg⟩ : S) • a = g • a := rfl
-
-instance faithfulSMul [FaithfulSMul M' α] : FaithfulSMul S α :=
+instance Submonoid.faithfulSMul {M' α : Type*} [MulOneClass M'] [SMul M' α] {S : Submonoid M'}
+    [FaithfulSMul M' α] : FaithfulSMul S α :=
   ⟨fun h => Subtype.ext <| eq_of_smul_eq_smul h⟩
 
-end SMul
-end MulOneClass
-
-variable [Monoid M']
-
-/-- The action by a submonoid is the action by the underlying monoid. -/
-@[to_additive
-      "The additive action by an `AddSubmonoid` is the action by the underlying `AddMonoid`. "]
-instance mulAction [MulAction M' α] (S : Submonoid M') : MulAction S α :=
-  MulAction.compHom _ S.subtype
-
-example {S : Submonoid M'} : IsScalarTower S M' M' := by infer_instance
-
-
-end Submonoid
-
-end Actions
-
 section Units
 
 namespace Submonoid
diff --git a/Mathlib/Algebra/GroupWithZero/Hom.lean b/Mathlib/Algebra/GroupWithZero/Hom.lean
index f30ace0a86cbce..5f4976f69371a0 100644
--- a/Mathlib/Algebra/GroupWithZero/Hom.lean
+++ b/Mathlib/Algebra/GroupWithZero/Hom.lean
@@ -40,13 +40,6 @@ open Function
 namespace NeZero
 variable {F α β : Type*} [Zero α] [Zero β] [FunLike F α β] [ZeroHomClass F α β] {a : α}
 
-#adaptation_note
-/--
-We name `neZero` so it can be used as a named argument,
-but since https://github.com/leanprover/lean4/pull/5338, this is considered unused,
-so we need to disable the linter.
--/
-set_option linter.unusedVariables false in
 lemma of_map (f : F) [neZero : NeZero (f a)] : NeZero a :=
   ⟨fun h ↦ ne (f a) <| by rw [h]; exact ZeroHomClass.map_zero f⟩
 
diff --git a/Mathlib/Algebra/GroupWithZero/WithZero.lean b/Mathlib/Algebra/GroupWithZero/WithZero.lean
index 43ea0028206c0d..9ad955134228e7 100644
--- a/Mathlib/Algebra/GroupWithZero/WithZero.lean
+++ b/Mathlib/Algebra/GroupWithZero/WithZero.lean
@@ -266,6 +266,10 @@ def unitsWithZeroEquiv : (WithZero α)ˣ ≃* α where
   right_inv _ := rfl
   map_mul' _ _ := coe_inj.mp <| by simp only [Units.val_mul, coe_unzero, coe_mul]
 
+theorem coe_unitsWithZeroEquiv_eq_units_val (γ : (WithZero α)ˣ) :
+    ↑(unitsWithZeroEquiv γ) = γ.val := by
+  simp only [WithZero.unitsWithZeroEquiv, MulEquiv.coe_mk, Equiv.coe_fn_mk, WithZero.coe_unzero]
+
 /-- Any group with zero is isomorphic to adjoining `0` to the units of itself. -/
 def withZeroUnitsEquiv {G : Type*} [GroupWithZero G]
     [DecidablePred (fun a : G ↦ a = 0)] :
diff --git a/Mathlib/Algebra/Homology/Additive.lean b/Mathlib/Algebra/Homology/Additive.lean
index 86d5e417b4051e..a5c9c7078277ab 100644
--- a/Mathlib/Algebra/Homology/Additive.lean
+++ b/Mathlib/Algebra/Homology/Additive.lean
@@ -170,7 +170,7 @@ theorem NatTrans.mapHomologicalComplex_comp (c : ComplexShape ι) {F G H : W₁
       NatTrans.mapHomologicalComplex α c ≫ NatTrans.mapHomologicalComplex β c := by
   aesop_cat
 
-@[reassoc (attr := simp 1100)]
+@[reassoc]
 theorem NatTrans.mapHomologicalComplex_naturality {c : ComplexShape ι} {F G : W₁ ⥤ W₂}
     [F.PreservesZeroMorphisms] [G.PreservesZeroMorphisms]
     (α : F ⟶ G) {C D : HomologicalComplex W₁ c} (f : C ⟶ D) :
diff --git a/Mathlib/Algebra/Homology/ConcreteCategory.lean b/Mathlib/Algebra/Homology/ConcreteCategory.lean
index f510e2876a6f3b..9c7f9f7bae473c 100644
--- a/Mathlib/Algebra/Homology/ConcreteCategory.lean
+++ b/Mathlib/Algebra/Homology/ConcreteCategory.lean
@@ -80,7 +80,7 @@ lemma δ_apply (x₃ : (forget₂ C Ab).obj (S.X₃.X i))
             AddMonoidHom.zero_apply])) := by
   refine hS.δ_apply' i j hij _ ((forget₂ C Ab).map (S.X₂.pOpcycles i) x₂) _ ?_ ?_
   · rw [← forget₂_comp_apply, ← forget₂_comp_apply,
-      HomologicalComplex.p_opcyclesMap, Functor.map_comp, comp_apply,
+      HomologicalComplex.p_opcyclesMap, Functor.map_comp, CategoryTheory.comp_apply,
       HomologicalComplex.homology_π_ι, forget₂_comp_apply, hx₂, HomologicalComplex.i_cyclesMk]
   · apply (Preadditive.mono_iff_injective (S.X₂.iCycles j)).1 inferInstance
     conv_lhs =>
diff --git a/Mathlib/Algebra/Homology/DerivedCategory/Ext/Basic.lean b/Mathlib/Algebra/Homology/DerivedCategory/Ext/Basic.lean
index 908ef5c957030b..4aae573cb738de 100644
--- a/Mathlib/Algebra/Homology/DerivedCategory/Ext/Basic.lean
+++ b/Mathlib/Algebra/Homology/DerivedCategory/Ext/Basic.lean
@@ -160,12 +160,12 @@ lemma mk₀_hom [HasDerivedCategory.{w'} C] (f : X ⟶ Y) :
     (mk₀ f).hom = ShiftedHom.mk₀ _ (by simp) ((singleFunctor C 0).map f) := by
   apply SmallShiftedHom.equiv_mk₀
 
-@[simp 1100]
+@[simp]
 lemma mk₀_comp_mk₀ (f : X ⟶ Y) (g : Y ⟶ Z) :
     (mk₀ f).comp (mk₀ g) (zero_add 0) = mk₀ (f ≫ g) := by
   letI := HasDerivedCategory.standard C; ext; simp
 
-@[simp 1100]
+@[simp]
 lemma mk₀_comp_mk₀_assoc (f : X ⟶ Y) (g : Y ⟶ Z) {n : ℕ} (α : Ext Z T n) :
     (mk₀ f).comp ((mk₀ g).comp α (zero_add n)) (zero_add n) =
       (mk₀ (f ≫ g)).comp α (zero_add n) := by
diff --git a/Mathlib/Algebra/Homology/Embedding/TruncGEHomology.lean b/Mathlib/Algebra/Homology/Embedding/TruncGEHomology.lean
index d39f0d57c60523..33aa50866daca8 100644
--- a/Mathlib/Algebra/Homology/Embedding/TruncGEHomology.lean
+++ b/Mathlib/Algebra/Homology/Embedding/TruncGEHomology.lean
@@ -10,8 +10,8 @@ import Mathlib.Algebra.Homology.QuasiIso
 
 /-! # The homology of a canonical truncation
 
-Given an embedding of complex shapes `e : Embedding c c'`, we shall
-relate the homology of `K : HomologicalComplex C c'` and of
+Given an embedding of complex shapes `e : Embedding c c'`,
+we relate the homology of `K : HomologicalComplex C c'` and of
 `K.truncGE e : HomologicalComplex C c'`.
 
 The main result is that `K.πTruncGE e : K ⟶ K.truncGE e` induces a
@@ -50,7 +50,8 @@ lemma hasHomology_of_not_mem_boundary (hj : ¬ e.BoundaryGE j) :
     (K.truncGE' e).HasHomology j :=
   hasHomology_sc'_of_not_mem_boundary K e _ j _ rfl rfl hj
 
-lemma quasiIsoAt_restrictionToTruncGE'_f (hj : ¬ e.BoundaryGE j)
+/-- `K.restrictionToTruncGE' e` is a quasi-isomorphism in degrees that are not at the boundary. -/
+lemma quasiIsoAt_restrictionToTruncGE' (hj : ¬ e.BoundaryGE j)
     [(K.restriction e).HasHomology j] [(K.truncGE' e).HasHomology j] :
     QuasiIsoAt (K.restrictionToTruncGE' e) j := by
   rw [quasiIsoAt_iff]
diff --git a/Mathlib/Algebra/Homology/Embedding/TruncLEHomology.lean b/Mathlib/Algebra/Homology/Embedding/TruncLEHomology.lean
new file mode 100644
index 00000000000000..4e0871f9545d5d
--- /dev/null
+++ b/Mathlib/Algebra/Homology/Embedding/TruncLEHomology.lean
@@ -0,0 +1,79 @@
+/-
+Copyright (c) 2024 Joël Riou. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joël Riou
+-/
+import Mathlib.Algebra.Homology.Embedding.TruncGEHomology
+import Mathlib.Algebra.Homology.Embedding.TruncLE
+
+/-! # The homology of a canonical truncation
+
+Given an embedding of complex shapes `e : Embedding c c'`,
+we relate the homology of `K : HomologicalComplex C c'` and of
+`K.truncLE e : HomologicalComplex C c'`.
+
+The main result is that `K.ιTruncLE e : K.truncLE e ⟶ K` induces a
+quasi-isomorphism in degree `e.f i` for all `i`. (Note that the complex
+`K.truncLE e` is exact in degrees that are not in the image of `e.f`.)
+
+All the results are obtained by dualising the results in the file `Embedding.TruncGEHomology`.
+
+-/
+
+open CategoryTheory Category Limits
+
+namespace HomologicalComplex
+
+variable {ι ι' : Type*} {c : ComplexShape ι} {c' : ComplexShape ι'}
+  {C : Type*} [Category C] [HasZeroMorphisms C]
+  (K L : HomologicalComplex C c') (φ : K ⟶ L) (e : c.Embedding c') [e.IsTruncLE]
+  [∀ i', K.HasHomology i'] [∀ i', L.HasHomology i']
+
+namespace truncLE'
+
+/-- `K.truncLE'ToRestriction e` is a quasi-isomorphism in degrees that are not at the boundary. -/
+lemma quasiIsoAt_truncLE'ToRestriction (j : ι) (hj : ¬ e.BoundaryLE j)
+    [(K.restriction e).HasHomology j] [(K.truncLE' e).HasHomology j] :
+    QuasiIsoAt (K.truncLE'ToRestriction e) j := by
+  dsimp only [truncLE'ToRestriction]
+  have : (K.op.restriction e.op).HasHomology j :=
+    inferInstanceAs ((K.restriction e).op.HasHomology j)
+  rw [quasiIsoAt_unopFunctor_map_iff]
+  exact truncGE'.quasiIsoAt_restrictionToTruncGE' K.op e.op j (by simpa)
+
+instance truncLE'_hasHomology (i : ι) : (K.truncLE' e).HasHomology i :=
+  inferInstanceAs ((K.op.truncGE' e.op).unop.HasHomology i)
+
+end truncLE'
+
+variable [HasZeroObject C]
+
+instance (i' : ι') : (K.truncLE e).HasHomology i' :=
+  inferInstanceAs ((K.op.truncGE e.op).unop.HasHomology i')
+
+lemma quasiIsoAt_ιTruncLE {j : ι} {j' : ι'} (hj' : e.f j = j') :
+    QuasiIsoAt (K.ιTruncLE e) j' := by
+  have := K.op.quasiIsoAt_πTruncGE e.op hj'
+  exact inferInstanceAs (QuasiIsoAt ((unopFunctor _ _ ).map (K.op.πTruncGE e.op).op) j')
+
+instance (i : ι) : QuasiIsoAt (K.ιTruncLE e) (e.f i) := K.quasiIsoAt_ιTruncLE e rfl
+
+lemma quasiIso_ιTruncLE_iff_isSupported :
+    QuasiIso (K.ιTruncLE e) ↔ K.IsSupported e := by
+  rw [← quasiIso_opFunctor_map_iff, ← isSupported_op_iff]
+  exact K.op.quasiIso_πTruncGE_iff_isSupported e.op
+
+lemma acyclic_truncLE_iff_isSupportedOutside :
+    (K.truncLE e).Acyclic ↔ K.IsSupportedOutside e := by
+  rw [← acyclic_op_iff, ← isSupportedOutside_op_iff]
+  exact K.op.acyclic_truncGE_iff_isSupportedOutside e.op
+
+variable {K L}
+
+lemma quasiIso_truncLEMap_iff :
+    QuasiIso (truncLEMap φ e) ↔ ∀ (i : ι) (i' : ι') (_ : e.f i = i'), QuasiIsoAt φ i' := by
+  rw [← quasiIso_opFunctor_map_iff]
+  simp only [← quasiIsoAt_opFunctor_map_iff φ]
+  apply quasiIso_truncGEMap_iff
+
+end HomologicalComplex
diff --git a/Mathlib/Algebra/Homology/HomologicalComplex.lean b/Mathlib/Algebra/Homology/HomologicalComplex.lean
index 9fc057de9025d3..f09f0593e1d0f5 100644
--- a/Mathlib/Algebra/Homology/HomologicalComplex.lean
+++ b/Mathlib/Algebra/Homology/HomologicalComplex.lean
@@ -469,7 +469,7 @@ theorem dFrom_comp_xNextIso {i j : ι} (r : c.Rel i j) :
 theorem dFrom_comp_xNextIsoSelf {i : ι} (h : ¬c.Rel i (c.next i)) :
     C.dFrom i ≫ (C.xNextIsoSelf h).hom = 0 := by simp [h]
 
-@[simp 1100]
+-- This is not a simp lemma; the LHS already simplifies.
 theorem dTo_comp_dFrom (j : ι) : C.dTo j ≫ C.dFrom j = 0 :=
   C.d_comp_d _ _ _
 
@@ -552,14 +552,12 @@ theorem next_eq (f : Hom C₁ C₂) {i j : ι} (w : c.Rel i j) :
 theorem comm_from (f : Hom C₁ C₂) (i : ι) : f.f i ≫ C₂.dFrom i = C₁.dFrom i ≫ f.next i :=
   f.comm _ _
 
-attribute [simp 1100] comm_from_assoc
 attribute [simp] comm_from_apply
 
 @[reassoc, elementwise]
 theorem comm_to (f : Hom C₁ C₂) (j : ι) : f.prev j ≫ C₂.dTo j = C₁.dTo j ≫ f.f j :=
   f.comm _ _
 
-attribute [simp 1100] comm_to_assoc
 attribute [simp] comm_to_apply
 
 /-- A morphism of chain complexes
diff --git a/Mathlib/Algebra/Homology/Homotopy.lean b/Mathlib/Algebra/Homology/Homotopy.lean
index bb13f21e29877f..8e8923988a0cf6 100644
--- a/Mathlib/Algebra/Homology/Homotopy.lean
+++ b/Mathlib/Algebra/Homology/Homotopy.lean
@@ -51,12 +51,12 @@ lemma dNext_eq_zero (f : ∀ i j, C.X i ⟶ D.X j) (i : ι) (hi : ¬ c.Rel i (c.
   dsimp [dNext]
   rw [shape _ _ _ hi, zero_comp]
 
-@[simp 1100]
+-- This is not a simp lemma; the LHS already simplifies.
 theorem dNext_comp_left (f : C ⟶ D) (g : ∀ i j, D.X i ⟶ E.X j) (i : ι) :
     (dNext i fun i j => f.f i ≫ g i j) = f.f i ≫ dNext i g :=
   (f.comm_assoc _ _ _).symm
 
-@[simp 1100]
+-- This is not a simp lemma; the LHS already simplifies.
 theorem dNext_comp_right (f : ∀ i j, C.X i ⟶ D.X j) (g : D ⟶ E) (i : ι) :
     (dNext i fun i j => f i j ≫ g.f j) = dNext i f ≫ g.f i :=
   (assoc _ _ _).symm
@@ -85,12 +85,12 @@ theorem prevD_eq (f : ∀ i j, C.X i ⟶ D.X j) {j j' : ι} (w : c.Rel j' j) :
   obtain rfl := c.prev_eq' w
   rfl
 
-@[simp 1100]
+-- This is not a simp lemma; the LHS already simplifies.
 theorem prevD_comp_left (f : C ⟶ D) (g : ∀ i j, D.X i ⟶ E.X j) (j : ι) :
     (prevD j fun i j => f.f i ≫ g i j) = f.f j ≫ prevD j g :=
   assoc _ _ _
 
-@[simp 1100]
+-- This is not a simp lemma; the LHS already simplifies.
 theorem prevD_comp_right (f : ∀ i j, C.X i ⟶ D.X j) (g : D ⟶ E) (j : ι) :
     (prevD j fun i j => f i j ≫ g.f j) = prevD j f ≫ g.f j := by
   dsimp [prevD]
@@ -426,21 +426,21 @@ section MkInductive
 
 variable {P Q : ChainComplex V ℕ}
 
-@[simp 1100]
+-- This is not a simp lemma; the LHS already simplifies.
 theorem prevD_chainComplex (f : ∀ i j, P.X i ⟶ Q.X j) (j : ℕ) :
     prevD j f = f j (j + 1) ≫ Q.d _ _ := by
   dsimp [prevD]
   have : (ComplexShape.down ℕ).prev j = j + 1 := ChainComplex.prev ℕ j
   congr 2
 
-@[simp 1100]
+-- This is not a simp lemma; the LHS already simplifies.
 theorem dNext_succ_chainComplex (f : ∀ i j, P.X i ⟶ Q.X j) (i : ℕ) :
     dNext (i + 1) f = P.d _ _ ≫ f i (i + 1) := by
   dsimp [dNext]
   have : (ComplexShape.down ℕ).next (i + 1) = i := ChainComplex.next_nat_succ _
   congr 2
 
-@[simp 1100]
+-- This is not a simp lemma; the LHS already simplifies.
 theorem dNext_zero_chainComplex (f : ∀ i j, P.X i ⟶ Q.X j) : dNext 0 f = 0 := by
   dsimp [dNext]
   rw [P.shape, zero_comp]
@@ -556,21 +556,21 @@ section MkCoinductive
 
 variable {P Q : CochainComplex V ℕ}
 
-@[simp 1100]
+-- This is not a simp lemma; the LHS already simplifies.
 theorem dNext_cochainComplex (f : ∀ i j, P.X i ⟶ Q.X j) (j : ℕ) :
     dNext j f = P.d _ _ ≫ f (j + 1) j := by
   dsimp [dNext]
   have : (ComplexShape.up ℕ).next j = j + 1 := CochainComplex.next ℕ j
   congr 2
 
-@[simp 1100]
+-- This is not a simp lemma; the LHS already simplifies.
 theorem prevD_succ_cochainComplex (f : ∀ i j, P.X i ⟶ Q.X j) (i : ℕ) :
     prevD (i + 1) f = f (i + 1) _ ≫ Q.d i (i + 1) := by
   dsimp [prevD]
   have : (ComplexShape.up ℕ).prev (i + 1) = i := CochainComplex.prev_nat_succ i
   congr 2
 
-@[simp 1100]
+-- This is not a simp lemma; the LHS already simplifies.
 theorem prevD_zero_cochainComplex (f : ∀ i j, P.X i ⟶ Q.X j) : prevD 0 f = 0 := by
   dsimp [prevD]
   rw [Q.shape, comp_zero]
diff --git a/Mathlib/Algebra/Homology/HomotopyCategory.lean b/Mathlib/Algebra/Homology/HomotopyCategory.lean
index 93400ddd35d940..b4ecba4a3831a2 100644
--- a/Mathlib/Algebra/Homology/HomotopyCategory.lean
+++ b/Mathlib/Algebra/Homology/HomotopyCategory.lean
@@ -121,7 +121,6 @@ def homotopyOutMap {C D : HomologicalComplex V c} (f : C ⟶ D) :
   apply homotopyOfEq
   simp
 
-@[simp 1100]
 theorem quotient_map_out_comp_out {C D E : HomotopyCategory V c} (f : C ⟶ D) (g : D ⟶ E) :
     (quotient V c).map (Quot.out f ≫ Quot.out g) = f ≫ g := by simp
 
diff --git a/Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean b/Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean
index 57a474997c34a4..aebf5de588e5a8 100644
--- a/Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean
+++ b/Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean
@@ -241,10 +241,10 @@ lemma inl_v_d (i j k : ℤ) (hij : i + (-1) = j) (hik : k + (-1) = i) :
   rw [homotopyCofiber.inlX_d φ j i k (by dsimp; omega) (by dsimp; omega)]
   abel
 
-@[reassoc (attr := simp 1100)]
+@[reassoc]
 lemma inr_f_d (n₁ n₂ : ℤ) :
     (inr φ).f n₁ ≫ (mappingCone φ).d n₁ n₂ = G.d n₁ n₂ ≫ (inr φ).f n₂ := by
-  apply Hom.comm
+  simp
 
 @[reassoc]
 lemma d_fst_v (i j k : ℤ) (hij : i + 1 = j) (hjk : j + 1 = k) :
diff --git a/Mathlib/Algebra/Homology/Opposite.lean b/Mathlib/Algebra/Homology/Opposite.lean
index 5bbd8fee09f160..cb42b8f38d5d44 100644
--- a/Mathlib/Algebra/Homology/Opposite.lean
+++ b/Mathlib/Algebra/Homology/Opposite.lean
@@ -7,6 +7,7 @@ import Mathlib.CategoryTheory.Abelian.Opposite
 import Mathlib.Algebra.Homology.Additive
 import Mathlib.Algebra.Homology.ImageToKernel
 import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
+import Mathlib.Algebra.Homology.QuasiIso
 
 /-!
 # Opposite categories of complexes
@@ -221,6 +222,86 @@ instance (K : HomologicalComplex Vᵒᵖ c) (i : ι) [K.HasHomology i] :
 
 variable {V c}
 
+@[simp]
+lemma quasiIsoAt_opFunctor_map_iff
+    {K L : HomologicalComplex V c} (φ : K ⟶ L) (i : ι)
+    [K.HasHomology i] [L.HasHomology i] :
+    QuasiIsoAt ((opFunctor _ _).map φ.op) i ↔ QuasiIsoAt φ i := by
+  simp only [quasiIsoAt_iff]
+  exact ShortComplex.quasiIso_opMap_iff ((shortComplexFunctor V c i).map φ)
+
+@[simp]
+lemma quasiIsoAt_unopFunctor_map_iff
+    {K L : HomologicalComplex Vᵒᵖ c} (φ : K ⟶ L) (i : ι)
+    [K.HasHomology i] [L.HasHomology i] :
+    QuasiIsoAt ((unopFunctor _ _).map φ.op) i ↔ QuasiIsoAt φ i := by
+  rw [← quasiIsoAt_opFunctor_map_iff]
+  rfl
+
+instance {K L : HomologicalComplex V c} (φ : K ⟶ L) (i : ι)
+    [K.HasHomology i] [L.HasHomology i] [QuasiIsoAt φ i] :
+    QuasiIsoAt ((opFunctor _ _).map φ.op) i := by
+  rw [quasiIsoAt_opFunctor_map_iff]
+  infer_instance
+
+instance {K L : HomologicalComplex Vᵒᵖ c} (φ : K ⟶ L) (i : ι)
+    [K.HasHomology i] [L.HasHomology i] [QuasiIsoAt φ i] :
+    QuasiIsoAt ((unopFunctor _ _).map φ.op) i := by
+  rw [quasiIsoAt_unopFunctor_map_iff]
+  infer_instance
+
+@[simp]
+lemma quasiIso_opFunctor_map_iff
+    {K L : HomologicalComplex V c} (φ : K ⟶ L)
+    [∀ i, K.HasHomology i] [∀ i, L.HasHomology i] :
+    QuasiIso ((opFunctor _ _).map φ.op) ↔ QuasiIso φ := by
+  simp only [quasiIso_iff, quasiIsoAt_opFunctor_map_iff]
+
+@[simp]
+lemma quasiIso_unopFunctor_map_iff
+    {K L : HomologicalComplex Vᵒᵖ c} (φ : K ⟶ L)
+    [∀ i, K.HasHomology i] [∀ i, L.HasHomology i] :
+    QuasiIso ((unopFunctor _ _).map φ.op) ↔ QuasiIso φ := by
+  simp only [quasiIso_iff, quasiIsoAt_unopFunctor_map_iff]
+
+instance {K L : HomologicalComplex V c} (φ : K ⟶ L)
+    [∀ i, K.HasHomology i] [∀ i, L.HasHomology i] [QuasiIso φ] :
+    QuasiIso ((opFunctor _ _).map φ.op) := by
+  rw [quasiIso_opFunctor_map_iff]
+  infer_instance
+
+instance {K L : HomologicalComplex Vᵒᵖ c} (φ : K ⟶ L)
+    [∀ i, K.HasHomology i] [∀ i, L.HasHomology i] [QuasiIso φ] :
+    QuasiIso ((unopFunctor _ _).map φ.op) := by
+  rw [quasiIso_unopFunctor_map_iff]
+  infer_instance
+
+lemma ExactAt.op {K : HomologicalComplex V c} {i : ι} (h : K.ExactAt i) :
+    K.op.ExactAt i :=
+  ShortComplex.Exact.op h
+
+lemma ExactAt.unop {K : HomologicalComplex Vᵒᵖ c} {i : ι} (h : K.ExactAt i) :
+    K.unop.ExactAt i :=
+  ShortComplex.Exact.unop h
+
+@[simp]
+lemma exactAt_op_iff (K : HomologicalComplex V c) {i : ι} :
+    K.op.ExactAt i ↔ K.ExactAt i :=
+  ⟨fun h ↦ h.unop, fun h ↦ h.op⟩
+
+lemma Acyclic.op {K : HomologicalComplex V c} (h : K.Acyclic) :
+    K.op.Acyclic :=
+  fun i ↦ (h i).op
+
+lemma Acyclic.unop {K : HomologicalComplex Vᵒᵖ c} (h : K.Acyclic) :
+    K.unop.Acyclic :=
+  fun i ↦ (h i).unop
+
+@[simp]
+lemma acyclic_op_iff (K : HomologicalComplex V c) :
+    K.op.Acyclic ↔ K.Acyclic :=
+  ⟨fun h ↦ h.unop, fun h ↦ h.op⟩
+
 /-- If `K` is a homological complex, then the homology of `K.op` identifies to
 the opposite of the homology of `K`. -/
 def homologyOp (K : HomologicalComplex V c) (i : ι) [K.HasHomology i] :
diff --git a/Mathlib/Algebra/Homology/ShortComplex/Ab.lean b/Mathlib/Algebra/Homology/ShortComplex/Ab.lean
index 393b22f24e512c..f1b0f96f3b7a08 100644
--- a/Mathlib/Algebra/Homology/ShortComplex/Ab.lean
+++ b/Mathlib/Algebra/Homology/ShortComplex/Ab.lean
@@ -37,7 +37,7 @@ variable (S : ShortComplex Ab.{u})
 
 @[simp]
 lemma ab_zero_apply (x : S.X₁) : S.g (S.f x) = 0 := by
-  rw [← comp_apply, S.zero]
+  rw [← CategoryTheory.comp_apply, S.zero]
   rfl
 
 /-- The canonical additive morphism `S.X₁ →+ AddMonoidHom.ker S.g` induced by `S.f`. -/
@@ -81,7 +81,7 @@ noncomputable def abCyclesIso : S.cycles ≅ AddCommGrp.of (AddMonoidHom.ker S.g
 lemma abCyclesIso_inv_apply_iCycles (x : AddMonoidHom.ker S.g) :
     S.iCycles (S.abCyclesIso.inv x) = x := by
   dsimp only [abCyclesIso]
-  rw [← comp_apply, S.abLeftHomologyData.cyclesIso_inv_comp_iCycles]
+  rw [← CategoryTheory.comp_apply, S.abLeftHomologyData.cyclesIso_inv_comp_iCycles]
   rfl
 
 /-- Given a short complex `S` of abelian groups, this is the isomorphism between
@@ -129,7 +129,7 @@ lemma ab_exact_iff_range_eq_ker : S.Exact ↔ S.f.range = S.g.ker := by
   · intro h
     refine le_antisymm ?_ h
     rintro _ ⟨x₁, rfl⟩
-    rw [AddMonoidHom.mem_ker, ← comp_apply, S.zero]
+    rw [AddMonoidHom.mem_ker, ← CategoryTheory.comp_apply, S.zero]
     rfl
   · intro h
     rw [h]
diff --git a/Mathlib/Algebra/Homology/ShortComplex/Abelian.lean b/Mathlib/Algebra/Homology/ShortComplex/Abelian.lean
index ad19544a15033f..806aabbe913404 100644
--- a/Mathlib/Algebra/Homology/ShortComplex/Abelian.lean
+++ b/Mathlib/Algebra/Homology/ShortComplex/Abelian.lean
@@ -50,10 +50,10 @@ lemma abelianImageToKernel_comp_kernel_ι :
 instance : Mono S.abelianImageToKernel :=
   mono_of_mono_fac S.abelianImageToKernel_comp_kernel_ι
 
-@[reassoc (attr := simp 1100)]
+@[reassoc]
 lemma abelianImageToKernel_comp_kernel_ι_comp_cokernel_π :
     S.abelianImageToKernel ≫ kernel.ι S.g ≫ cokernel.π S.f = 0 := by
-  simp only [abelianImageToKernel_comp_kernel_ι_assoc, kernel.condition]
+  simp
 
 /-- `Abelian.image S.f` is the kernel of `kernel.ι S.g ≫ cokernel.π S.f` -/
 noncomputable def abelianImageToKernelIsKernel :
diff --git a/Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean b/Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
index 92d14ea2564f50..610dbd89fe4145 100644
--- a/Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
+++ b/Mathlib/Algebra/Homology/ShortComplex/ConcreteCategory.lean
@@ -32,7 +32,7 @@ lemma ShortComplex.zero_apply
     [Limits.HasZeroMorphisms C] [(forget₂ C Ab).PreservesZeroMorphisms]
     (S : ShortComplex C) (x : (forget₂ C Ab).obj S.X₁) :
     ((forget₂ C Ab).map S.g) (((forget₂ C Ab).map S.f) x) = 0 := by
-  rw [← comp_apply, ← Functor.map_comp, S.zero, Functor.map_zero]
+  rw [← CategoryTheory.comp_apply, ← Functor.map_comp, S.zero, Functor.map_zero]
   rfl
 
 section preadditive
@@ -112,8 +112,8 @@ lemma i_cyclesMk [S.HasHomology] (x₂ : (forget₂ C Ab).obj S.X₂)
     (hx₂ : ((forget₂ C Ab).map S.g) x₂ = 0) :
     (forget₂ C Ab).map S.iCycles (S.cyclesMk x₂ hx₂) = x₂ := by
   dsimp [cyclesMk]
-  erw [← comp_apply, S.mapCyclesIso_hom_iCycles (forget₂ C Ab),
-    ← comp_apply, abCyclesIso_inv_apply_iCycles ]
+  erw [← CategoryTheory.comp_apply, S.mapCyclesIso_hom_iCycles (forget₂ C Ab),
+    ← CategoryTheory.comp_apply, abCyclesIso_inv_apply_iCycles ]
 
 end ShortComplex
 
@@ -172,13 +172,13 @@ lemma δ_apply' (x₃ : (forget₂ C Ab).obj D.L₀.X₃)
   · refine ((congr_hom (e.hom.naturality D.L₁.g) x₂).symm.trans ?_).trans
       (congr_hom (e.hom.naturality D.v₀₁.τ₃) x₃)
     dsimp
-    rw [comp_apply, comp_apply]
+    rw [CategoryTheory.comp_apply, CategoryTheory.comp_apply]
     erw [h₂]
     rfl
   · refine ((congr_hom (e.hom.naturality D.L₂.f) x₁).symm.trans ?_).trans
       (congr_hom (e.hom.naturality D.v₁₂.τ₂) x₂)
     dsimp
-    rw [comp_apply, comp_apply]
+    rw [CategoryTheory.comp_apply, CategoryTheory.comp_apply]
     erw [h₁]
     rfl
 
diff --git a/Mathlib/Algebra/Homology/Single.lean b/Mathlib/Algebra/Homology/Single.lean
index b6dc763c87a954..16d9d5943f7329 100644
--- a/Mathlib/Algebra/Homology/Single.lean
+++ b/Mathlib/Algebra/Homology/Single.lean
@@ -44,8 +44,8 @@ noncomputable def single (j : ι) : V ⥤ HomologicalComplex V c where
     split_ifs with h
     · subst h
       simp
-    · #adaptation_note /-- after nightly-2024-03-07, the previous sensible proof
-      `rw [if_neg h]; simp` fails with "motive not type correct".
+    · #adaptation_note /-- nightly-2024-03-07
+      the previous sensible proof `rw [if_neg h]; simp` fails with "motive not type correct".
       The following is horrible. -/
       convert (id_zero (C := V)).symm
       all_goals simp [if_neg h]
diff --git a/Mathlib/Algebra/Lie/CartanExists.lean b/Mathlib/Algebra/Lie/CartanExists.lean
index 4e3acbb133d815..4574da2c841879 100644
--- a/Mathlib/Algebra/Lie/CartanExists.lean
+++ b/Mathlib/Algebra/Lie/CartanExists.lean
@@ -119,8 +119,6 @@ variable {K L : Type*} [Field K] [LieRing L] [LieAlgebra K L] [Module.Finite K L
 
 open Module LieSubalgebra LieSubmodule Polynomial Cardinal LieModule engel_isBot_of_isMin
 
-#adaptation_note /-- otherwise there is a spurious warning on `contrapose!` below. -/
-set_option linter.unusedVariables false in
 /-- Let `L` be a Lie algebra of dimension `n` over a field `K` with at least `n` elements.
 Given a Lie subalgebra `U` of `L`, and an element `x ∈ U` such that `U ≤ engel K x`.
 Suppose that `engel K x` is minimal amongst the Engel subalgebras `engel K y` for `y ∈ U`.
diff --git a/Mathlib/Algebra/Lie/Semisimple/Basic.lean b/Mathlib/Algebra/Lie/Semisimple/Basic.lean
index 0697232c790f3f..28001afd616102 100644
--- a/Mathlib/Algebra/Lie/Semisimple/Basic.lean
+++ b/Mathlib/Algebra/Lie/Semisimple/Basic.lean
@@ -212,8 +212,7 @@ lemma finitelyAtomistic : ∀ s : Finset (LieIdeal R L), ↑s ⊆ {I : LieIdeal
   set K := s'.sup id
   suffices I ≤ K by
     obtain ⟨t, hts', htI⟩ := finitelyAtomistic s' hs'S I this
-    #adaptation_note
-    /-- Prior to https://github.com/leanprover/lean4/pull/6024
+    #adaptation_note /-- https://github.com/leanprover/lean4/pull/6024
     we could write `hts'.trans hs'.subset` instead of
     `Finset.Subset.trans hts' hs'.subset` in the next line. -/
     exact ⟨t, Finset.Subset.trans hts' hs'.subset, htI⟩
diff --git a/Mathlib/Algebra/LinearRecurrence.lean b/Mathlib/Algebra/LinearRecurrence.lean
index a07f5edb8399f6..b9fe57a324105f 100644
--- a/Mathlib/Algebra/LinearRecurrence.lean
+++ b/Mathlib/Algebra/LinearRecurrence.lean
@@ -92,11 +92,11 @@ theorem eq_mk_of_is_sol_of_eq_init {u : ℕ → α} {init : Fin E.order → α}
   rw [mkSol]
   split_ifs with h'
   · exact mod_cast heq ⟨n, h'⟩
-  rw [← tsub_add_cancel_of_le (le_of_not_lt h'), h (n - E.order)]
-  congr with k
-  have : n - E.order + k < n := by omega
-  rw [eq_mk_of_is_sol_of_eq_init h heq (n - E.order + k)]
-  simp
+  · rw [← tsub_add_cancel_of_le (le_of_not_lt h'), h (n - E.order)]
+    congr with k
+    have : n - E.order + k < n := by omega
+    rw [eq_mk_of_is_sol_of_eq_init h heq (n - E.order + k)]
+    simp
 
 /-- If `u` is a solution to `E` and `init` designates its first `E.order` values,
   then `u = E.mkSol init`. This proves that `E.mkSol init` is the only solution
diff --git a/Mathlib/Algebra/Module/Equiv/Basic.lean b/Mathlib/Algebra/Module/Equiv/Basic.lean
index b0228d7304c795..ddbbd94475a751 100644
--- a/Mathlib/Algebra/Module/Equiv/Basic.lean
+++ b/Mathlib/Algebra/Module/Equiv/Basic.lean
@@ -482,6 +482,41 @@ theorem symm_neg : (neg R : M ≃ₗ[R] M).symm = neg R :=
 
 end Neg
 
+section Semiring
+
+open LinearMap
+
+variable {M₂₁ M₂₂ : Type*} [Semiring R] [AddCommMonoid M₁] [AddCommMonoid M₂]
+  [AddCommMonoid M₂₁] [AddCommMonoid M₂₂] [Module R M₁] [Module R M₂] [Module R M₂₁] [Module R M₂₂]
+
+/-- A linear isomorphism between the domains and codomains of two spaces of linear maps gives a
+additive isomorphism between the two function spaces.
+
+See also `LinearEquiv.arrowCongr` for the linear version of this isomorphism. -/
+def arrowCongrAddEquiv (e₁ : M₁ ≃ₗ[R] M₂) (e₂ : M₂₁ ≃ₗ[R] M₂₂) :
+    (M₁ →ₗ[R] M₂₁) ≃+ (M₂ →ₗ[R] M₂₂) where
+  toFun f := e₂.comp (f.comp e₁.symm.toLinearMap)
+  invFun f := e₂.symm.comp (f.comp e₁.toLinearMap)
+  left_inv f := by
+    ext x
+    simp only [symm_apply_apply, Function.comp_apply, coe_comp, coe_coe]
+  right_inv f := by
+    ext x
+    simp only [Function.comp_apply, apply_symm_apply, coe_comp, coe_coe]
+  map_add' f g := by
+    ext x
+    simp only [map_add, add_apply, Function.comp_apply, coe_comp, coe_coe]
+
+/-- If `M` and `M₂` are linearly isomorphic then the endomorphism rings of `M` and `M₂`
+are isomorphic.
+
+See `LinearEquiv.conj` for the linear version of this isomorphism. -/
+def conjRingEquiv (e : M₁ ≃ₗ[R] M₂) : Module.End R M₁ ≃+* Module.End R M₂ where
+  __ := arrowCongrAddEquiv e e
+  map_mul' _ _ := by ext; simp [arrowCongrAddEquiv]
+
+end Semiring
+
 section CommSemiring
 
 variable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃]
@@ -494,24 +529,15 @@ def smulOfUnit (a : Rˣ) : M ≃ₗ[R] M :=
   DistribMulAction.toLinearEquiv R M a
 
 /-- A linear isomorphism between the domains and codomains of two spaces of linear maps gives a
-linear isomorphism between the two function spaces. -/
+linear isomorphism between the two function spaces.
+
+See `LinearEquiv.arrowCongrAddEquiv` for the additive version of this isomorphism that works
+over a not necessarily commutative semiring. -/
 def arrowCongr {R M₁ M₂ M₂₁ M₂₂ : Sort _} [CommSemiring R] [AddCommMonoid M₁] [AddCommMonoid M₂]
     [AddCommMonoid M₂₁] [AddCommMonoid M₂₂] [Module R M₁] [Module R M₂] [Module R M₂₁]
     [Module R M₂₂] (e₁ : M₁ ≃ₗ[R] M₂) (e₂ : M₂₁ ≃ₗ[R] M₂₂) : (M₁ →ₗ[R] M₂₁) ≃ₗ[R] M₂ →ₗ[R] M₂₂ where
-  toFun := fun f : M₁ →ₗ[R] M₂₁ ↦ (e₂ : M₂₁ →ₗ[R] M₂₂).comp <| f.comp (e₁.symm : M₂ →ₗ[R] M₁)
-  invFun f := (e₂.symm : M₂₂ →ₗ[R] M₂₁).comp <| f.comp (e₁ : M₁ →ₗ[R] M₂)
-  left_inv f := by
-    ext x
-    simp only [symm_apply_apply, Function.comp_apply, coe_comp, coe_coe]
-  right_inv f := by
-    ext x
-    simp only [Function.comp_apply, apply_symm_apply, coe_comp, coe_coe]
-  map_add' f g := by
-    ext x
-    simp only [map_add, add_apply, Function.comp_apply, coe_comp, coe_coe]
-  map_smul' c f := by
-    ext x
-    simp only [smul_apply, Function.comp_apply, coe_comp, map_smulₛₗ e₂, coe_coe]
+  __ := arrowCongrAddEquiv e₁ e₂
+  map_smul' c f := by ext; simp [arrowCongrAddEquiv]
 
 @[simp]
 theorem arrowCongr_apply {R M₁ M₂ M₂₁ M₂₂ : Sort _} [CommSemiring R] [AddCommMonoid M₁]
@@ -547,7 +573,11 @@ def congrRight (f : M₂ ≃ₗ[R] M₃) : (M →ₗ[R] M₂) ≃ₗ[R] M →ₗ
   arrowCongr (LinearEquiv.refl R M) f
 
 /-- If `M` and `M₂` are linearly isomorphic then the two spaces of linear maps from `M` and `M₂` to
-themselves are linearly isomorphic. -/
+themselves are linearly isomorphic.
+
+See `LinearEquiv.conjRingEquiv` for the isomorphism between endomorphism rings,
+which works over a not necessarily commutative semiring. -/
+-- TODO: upgrade to AlgEquiv (but this file currently cannot import AlgEquiv)
 def conj (e : M ≃ₗ[R] M₂) : Module.End R M ≃ₗ[R] Module.End R M₂ :=
   arrowCongr e e
 
@@ -582,12 +612,8 @@ isomorphism between `M₂ →ₗ[R] M` and `M₃ →ₗ[R] M`, if `M` is both an
 `S`-module and their actions commute. -/
 def congrLeft {R} (S) [Semiring R] [Semiring S] [Module R M₂] [Module R M₃] [Module R M]
     [Module S M] [SMulCommClass R S M] (e : M₂ ≃ₗ[R] M₃) : (M₂ →ₗ[R] M) ≃ₗ[S] (M₃ →ₗ[R] M) where
-  toFun f := f.comp e.symm.toLinearMap
-  invFun f := f.comp e.toLinearMap
-  map_add' _ _ := rfl
+  __ := e.arrowCongrAddEquiv (.refl ..)
   map_smul' _ _ := rfl
-  left_inv f := by dsimp only; apply DFunLike.ext; exact (congr_arg f <| e.left_inv ·)
-  right_inv f := by dsimp only; apply DFunLike.ext; exact (congr_arg f <| e.right_inv ·)
 
 end CommSemiring
 
diff --git a/Mathlib/Algebra/Module/LinearMap/Polynomial.lean b/Mathlib/Algebra/Module/LinearMap/Polynomial.lean
index 5637e7a41bdf33..a5f9f7bb8180f3 100644
--- a/Mathlib/Algebra/Module/LinearMap/Polynomial.lean
+++ b/Mathlib/Algebra/Module/LinearMap/Polynomial.lean
@@ -253,11 +253,8 @@ lemma polyCharpolyAux_baseChange (A : Type*) [CommRing A] [Algebra R A] :
     simp only [RingHom.coe_comp, RingHom.coe_coe, Function.comp_apply, map_X, bind₁_X_right]
     classical
     rw [toMvPolynomial_comp _ (basis A (Basis.end bₘ)), ← toMvPolynomial_baseChange]
-    #adaptation_note
-    /--
-    After https://github.com/leanprover/lean4/pull/4119 we either need to specify the `M₂` argument,
-    or use `set_option maxSynthPendingDepth 2 in`.
-    -/
+    #adaptation_note /-- https://github.com/leanprover/lean4/pull/4119
+    we either need to specify the `M₂` argument, or use `set_option maxSynthPendingDepth 2 in`. -/
     suffices toMvPolynomial (M₂ := (Module.End A (TensorProduct R A M)))
         (basis A bₘ.end) (basis A bₘ).end (tensorProduct R A M M) ij = X ij by
       rw [this, bind₁_X_right]
diff --git a/Mathlib/Algebra/Module/Submodule/Defs.lean b/Mathlib/Algebra/Module/Submodule/Defs.lean
index 55faba74532642..50ffade02ba270 100644
--- a/Mathlib/Algebra/Module/Submodule/Defs.lean
+++ b/Mathlib/Algebra/Module/Submodule/Defs.lean
@@ -5,6 +5,7 @@ Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
 -/
 import Mathlib.Algebra.Group.Subgroup.Defs
 import Mathlib.GroupTheory.GroupAction.SubMulAction
+import Mathlib.Algebra.Group.Submonoid.Basic
 
 /-!
 
diff --git a/Mathlib/Algebra/Order/CauSeq/Basic.lean b/Mathlib/Algebra/Order/CauSeq/Basic.lean
index fb76b07c2edb27..b7366261beeeae 100644
--- a/Mathlib/Algebra/Order/CauSeq/Basic.lean
+++ b/Mathlib/Algebra/Order/CauSeq/Basic.lean
@@ -404,7 +404,7 @@ theorem zero_limZero : LimZero (0 : CauSeq β abv)
 theorem const_limZero {x : β} : LimZero (const x) ↔ x = 0 :=
   ⟨fun H =>
     (abv_eq_zero abv).1 <|
-      (eq_of_le_of_forall_le_of_dense (abv_nonneg abv _)) fun _ ε0 =>
+      (eq_of_le_of_forall_lt_imp_le_of_dense (abv_nonneg abv _)) fun _ ε0 =>
         let ⟨_, hi⟩ := H _ ε0
         le_of_lt <| hi _ le_rfl,
     fun e => e.symm ▸ zero_limZero⟩
diff --git a/Mathlib/Algebra/Order/Field/Basic.lean b/Mathlib/Algebra/Order/Field/Basic.lean
index d2b9446b43e012..5d80dad8dfcfd6 100644
--- a/Mathlib/Algebra/Order/Field/Basic.lean
+++ b/Mathlib/Algebra/Order/Field/Basic.lean
@@ -414,7 +414,7 @@ theorem le_iff_forall_one_lt_le_mul₀ {α : Type*} [LinearOrderedSemifield α]
   obtain rfl|hb := hb.eq_or_lt
   · simp_rw [zero_mul] at h
     exact h 2 one_lt_two
-  refine le_of_forall_le_of_dense fun x hbx => ?_
+  refine le_of_forall_gt_imp_ge_of_dense fun x hbx => ?_
   convert h (x / b) ((one_lt_div hb).mpr hbx)
   rw [mul_div_cancel₀ _ hb.ne']
 
@@ -755,14 +755,14 @@ private lemma exists_mul_right_lt₀ {a b c : α} (hc : a * b < c) : ∃ b' > b,
   simp_rw [mul_comm a] at hc ⊢; exact exists_mul_left_lt₀ hc
 
 lemma le_mul_of_forall_lt₀ {a b c : α} (h : ∀ a' > a, ∀ b' > b, c ≤ a' * b') : c ≤ a * b := by
-  refine le_of_forall_le_of_dense fun d hd ↦ ?_
+  refine le_of_forall_gt_imp_ge_of_dense fun d hd ↦ ?_
   obtain ⟨a', ha', hd⟩ := exists_mul_left_lt₀ hd
   obtain ⟨b', hb', hd⟩ := exists_mul_right_lt₀ hd
   exact (h a' ha' b' hb').trans hd.le
 
 lemma mul_le_of_forall_lt_of_nonneg {a b c : α} (ha : 0 ≤ a) (hc : 0 ≤ c)
     (h : ∀ a' ≥ 0, a' < a → ∀ b' ≥ 0, b' < b → a' * b' ≤ c) : a * b ≤ c := by
-  refine le_of_forall_ge_of_dense fun d d_ab ↦ ?_
+  refine le_of_forall_lt_imp_le_of_dense fun d d_ab ↦ ?_
   rcases lt_or_le d 0 with hd | hd
   · exact hd.le.trans hc
   obtain ⟨a', ha', d_ab⟩ := exists_lt_mul_left_of_nonneg ha hd d_ab
diff --git a/Mathlib/Algebra/Order/Group/DenselyOrdered.lean b/Mathlib/Algebra/Order/Group/DenselyOrdered.lean
index a019c1739800a6..a1bd1983375e5f 100644
--- a/Mathlib/Algebra/Order/Group/DenselyOrdered.lean
+++ b/Mathlib/Algebra/Order/Group/DenselyOrdered.lean
@@ -68,7 +68,7 @@ private lemma exists_mul_right_lt [CommGroup α] [LT α] [DenselyOrdered α]
 lemma le_mul_of_forall_lt [CommGroup α] [LinearOrder α] [CovariantClass α α (· * ·) (· ≤ ·)]
     [DenselyOrdered α] {a b c : α} (h : ∀ a' > a, ∀ b' > b, c ≤ a' * b') :
     c ≤ a * b := by
-  refine le_of_forall_le_of_dense fun d hd ↦ ?_
+  refine le_of_forall_gt_imp_ge_of_dense fun d hd ↦ ?_
   obtain ⟨a', ha', hd⟩ := exists_mul_left_lt hd
   obtain ⟨b', hb', hd⟩ := exists_mul_right_lt hd
   exact (h a' ha' b' hb').trans hd.le
@@ -77,7 +77,7 @@ lemma le_mul_of_forall_lt [CommGroup α] [LinearOrder α] [CovariantClass α α
 lemma mul_le_of_forall_lt [CommGroup α] [LinearOrder α] [CovariantClass α α (· * ·) (· ≤ ·)]
     [DenselyOrdered α] {a b c : α} (h : ∀ a' < a, ∀ b' < b, a' * b' ≤ c) :
     a * b ≤ c := by
-  refine le_of_forall_ge_of_dense fun d hd ↦ ?_
+  refine le_of_forall_lt_imp_le_of_dense fun d hd ↦ ?_
   obtain ⟨a', ha', hd⟩ := exists_lt_mul_left hd
   obtain ⟨b', hb', hd⟩ := exists_lt_mul_right hd
   exact hd.le.trans (h a' ha' b' hb')
diff --git a/Mathlib/Algebra/Order/Monoid/Unbundled/ExistsOfLE.lean b/Mathlib/Algebra/Order/Monoid/Unbundled/ExistsOfLE.lean
index 1bd716addf3274..c2e25592e58cd5 100644
--- a/Mathlib/Algebra/Order/Monoid/Unbundled/ExistsOfLE.lean
+++ b/Mathlib/Algebra/Order/Monoid/Unbundled/ExistsOfLE.lean
@@ -69,7 +69,7 @@ variable [LinearOrder α] [DenselyOrdered α] [Monoid α] [ExistsMulOfLE α]
 
 @[to_additive]
 theorem le_of_forall_one_lt_le_mul (h : ∀ ε : α, 1 < ε → a ≤ b * ε) : a ≤ b :=
-  le_of_forall_le_of_dense fun x hxb => by
+  le_of_forall_gt_imp_ge_of_dense fun x hxb => by
     obtain ⟨ε, rfl⟩ := exists_mul_of_le hxb.le
     exact h _ (one_lt_of_lt_mul_right hxb)
 
diff --git a/Mathlib/Algebra/Polynomial/AlgebraMap.lean b/Mathlib/Algebra/Polynomial/AlgebraMap.lean
index 40e975a43fbfe1..2129497e1006c0 100644
--- a/Mathlib/Algebra/Polynomial/AlgebraMap.lean
+++ b/Mathlib/Algebra/Polynomial/AlgebraMap.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
 -/
 import Mathlib.Algebra.Algebra.Pi
+import Mathlib.Algebra.Algebra.Prod
 import Mathlib.Algebra.Algebra.Subalgebra.Basic
 import Mathlib.Algebra.Algebra.Tower
 import Mathlib.Algebra.MonoidAlgebra.Basic
@@ -368,6 +369,38 @@ theorem aeval_algebraMap_apply_eq_algebraMap_eval (x : R) (p : R[X]) :
     aeval (algebraMap R A x) p = algebraMap R A (p.eval x) :=
   aeval_algHom_apply (Algebra.ofId R A) x p
 
+/-- Polynomial evaluation on a pair is a product of the evaluations on the components. -/
+theorem aeval_prod (x : A × B) : aeval (R := R) x = (aeval x.1).prod (aeval x.2) :=
+  aeval_algHom (.fst R A B) x ▸ aeval_algHom (.snd R A B) x ▸
+    (aeval x).prod_comp (.fst R A B) (.snd R A B)
+
+/-- Polynomial evaluation on a pair is a pair of evaluations. -/
+theorem aeval_prod_apply (x : A × B) (p : Polynomial R) :
+    p.aeval x = (p.aeval x.1, p.aeval x.2) := by simp [aeval_prod]
+
+section Pi
+
+variable {I : Type*} {A : I → Type*} [∀ i, Semiring (A i)] [∀ i, Algebra R (A i)]
+variable (x : Π i, A i) (p : R[X])
+
+/-- Polynomial evaluation on an indexed tuple is the indexed product of the evaluations
+on the components.
+Generalizes `Polynomial.aeval_prod` to indexed products. -/
+theorem aeval_pi (x : Π i, A i) : aeval (R := R) x = Pi.algHom R A (fun i ↦ aeval (x i)) :=
+  (funext fun i ↦ aeval_algHom (Pi.evalAlgHom R A i) x) ▸
+    (Pi.algHom_comp R A (Pi.evalAlgHom R A) (aeval x))
+
+theorem aeval_pi_apply₂ (j : I) : p.aeval x j = p.aeval (x j) :=
+  aeval_pi (R := R) x ▸ Pi.algHom_apply R A (fun i ↦ aeval (x i)) p j
+
+/-- Polynomial evaluation on an indexed tuple is the indexed tuple of the evaluations
+on the components.
+Generalizes `Polynomial.aeval_prod_apply` to indexed products. -/
+theorem aeval_pi_apply : p.aeval x = fun j ↦ p.aeval (x j) :=
+  funext fun j ↦ aeval_pi_apply₂ x p j
+
+end Pi
+
 @[simp]
 theorem coe_aeval_eq_eval (r : R) : (aeval r : R[X] → R) = eval r :=
   rfl
diff --git a/Mathlib/Algebra/Polynomial/Basic.lean b/Mathlib/Algebra/Polynomial/Basic.lean
index 4a0d9b37f9a1d3..880597eb13c1f6 100644
--- a/Mathlib/Algebra/Polynomial/Basic.lean
+++ b/Mathlib/Algebra/Polynomial/Basic.lean
@@ -8,6 +8,7 @@ import Mathlib.Algebra.MonoidAlgebra.Defs
 import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
 import Mathlib.Data.Finset.Sort
 import Mathlib.Tactic.FastInstance
+import Mathlib.Algebra.Group.Submonoid.Operations
 
 /-!
 # Theory of univariate polynomials
diff --git a/Mathlib/Algebra/Polynomial/RingDivision.lean b/Mathlib/Algebra/Polynomial/RingDivision.lean
index eb2061427a125d..9be07ffaeff4d2 100644
--- a/Mathlib/Algebra/Polynomial/RingDivision.lean
+++ b/Mathlib/Algebra/Polynomial/RingDivision.lean
@@ -234,7 +234,6 @@ theorem rootMultiplicity_mul {p q : R[X]} {x : R} (hpq : p * q ≠ 0) :
     multiplicity_mul (prime_X_sub_C x) (finiteMultiplicity_X_sub_C _ hpq)]
 
 open Multiset in
-set_option linter.unusedVariables false in
 theorem exists_multiset_roots [DecidableEq R] :
     ∀ {p : R[X]} (_ : p ≠ 0), ∃ s : Multiset R,
       (Multiset.card s : WithBot ℕ) ≤ degree p ∧ ∀ a, s.count a = rootMultiplicity a p
@@ -245,11 +244,6 @@ theorem exists_multiset_roots [DecidableEq R] :
       have hpd : 0 < degree p := degree_pos_of_root hp hx
       have hd0 : p /ₘ (X - C x) ≠ 0 := fun h => by
         rw [← mul_divByMonic_eq_iff_isRoot.2 hx, h, mul_zero] at hp; exact hp rfl
-      #adaptation_note
-      /--
-      Since https://github.com/leanprover/lean4/pull/5338, this is considered unused,
-      because it is only used in the decreasing_by clause.
-      -/
       have wf : degree (p /ₘ (X - C x)) < degree p :=
         degree_divByMonic_lt _ (monic_X_sub_C x) hp ((degree_X_sub_C x).symm ▸ by decide)
       let ⟨t, htd, htr⟩ := @exists_multiset_roots _ (p /ₘ (X - C x)) hd0
diff --git a/Mathlib/Algebra/Ring/Subring/Basic.lean b/Mathlib/Algebra/Ring/Subring/Basic.lean
index 761bf9815c9242..8c113ce263a893 100644
--- a/Mathlib/Algebra/Ring/Subring/Basic.lean
+++ b/Mathlib/Algebra/Ring/Subring/Basic.lean
@@ -384,6 +384,15 @@ theorem center_eq_top (R) [CommRing R] : center R = ⊤ :=
 instance : CommRing (center R) :=
   { inferInstanceAs (CommSemiring (Subsemiring.center R)), (center R).toRing with }
 
+/-- The center of isomorphic (not necessarily associative) rings are isomorphic. -/
+@[simps!] def centerCongr (e : R ≃+* S) : center R ≃+* center S :=
+  NonUnitalSubsemiring.centerCongr e
+
+/-- The center of a (not necessarily associative) ring
+is isomorphic to the center of its opposite. -/
+@[simps!] def centerToMulOpposite : center R ≃+* center Rᵐᵒᵖ :=
+  NonUnitalSubsemiring.centerToMulOpposite
+
 end
 
 section DivisionRing
diff --git a/Mathlib/Algebra/Ring/Subsemiring/Basic.lean b/Mathlib/Algebra/Ring/Subsemiring/Basic.lean
index e38a8720a241af..65b72af50f1f40 100644
--- a/Mathlib/Algebra/Ring/Subsemiring/Basic.lean
+++ b/Mathlib/Algebra/Ring/Subsemiring/Basic.lean
@@ -262,6 +262,17 @@ This is not an instance as it forms a non-defeq diamond with
 abbrev center.commSemiring' : CommSemiring (center R) :=
   { Submonoid.center.commMonoid', (center R).toNonAssocSemiring with }
 
+variable {R}
+
+/-- The center of isomorphic (not necessarily associative) semirings are isomorphic. -/
+@[simps!] def centerCongr [NonAssocSemiring S] (e : R ≃+* S) : center R ≃+* center S :=
+  NonUnitalSubsemiring.centerCongr e
+
+/-- The center of a (not necessarily associative) semiring
+is isomorphic to the center of its opposite. -/
+@[simps!] def centerToMulOpposite : center R ≃+* center Rᵐᵒᵖ :=
+  NonUnitalSubsemiring.centerToMulOpposite
+
 end NonAssocSemiring
 
 section Semiring
@@ -285,7 +296,6 @@ instance decidableMemCenter {R} [Semiring R] [DecidableEq R] [Fintype R] :
 theorem center_eq_top (R) [CommSemiring R] : center R = ⊤ :=
   SetLike.coe_injective (Set.center_eq_univ R)
 
-
 end Semiring
 
 section Centralizer
diff --git a/Mathlib/AlgebraicGeometry/AffineScheme.lean b/Mathlib/AlgebraicGeometry/AffineScheme.lean
index a7a71a986a8cab..4772bc08cc2a91 100644
--- a/Mathlib/AlgebraicGeometry/AffineScheme.lean
+++ b/Mathlib/AlgebraicGeometry/AffineScheme.lean
@@ -630,7 +630,7 @@ lemma appLE_eq_away_map {X Y : Scheme.{u}} (f : X ⟶ Y) {U : Y.Opens} (hU : IsA
   letI := hV.isLocalization_basicOpen (f.appLE U V e r)
   ext : 1
   apply IsLocalization.ringHom_ext (.powers r)
-  rw [IsLocalization.Away.map, IsLocalization.map_comp,
+  rw [IsLocalization.Away.map, CommRingCat.hom_ofHom, IsLocalization.map_comp,
     RingHom.algebraMap_toAlgebra, RingHom.algebraMap_toAlgebra, ← CommRingCat.hom_comp,
     ← CommRingCat.hom_comp, Scheme.Hom.appLE_map, Scheme.Hom.map_appLE]
 
@@ -646,7 +646,7 @@ lemma app_basicOpen_eq_away_map {X Y : Scheme.{u}} (f : X ⟶ Y) {U : Y.Opens}
   haveI := h.isLocalization_basicOpen (f.app U r)
   ext : 1
   apply IsLocalization.ringHom_ext (.powers r)
-  rw [IsLocalization.Away.map, CommRingCat.hom_comp, RingHom.comp_assoc,
+  rw [IsLocalization.Away.map, CommRingCat.hom_comp, RingHom.comp_assoc, CommRingCat.hom_ofHom,
     IsLocalization.map_comp, RingHom.algebraMap_toAlgebra,
     RingHom.algebraMap_toAlgebra, ← RingHom.comp_assoc, ← CommRingCat.hom_comp,
     ← CommRingCat.hom_comp, ← X.presheaf.map_comp]
@@ -817,31 +817,35 @@ def SpecMapRestrictBasicOpenIso {R S : CommRingCat} (f : R ⟶ S) (r : R) :
       Arrow.mk (Spec.map <| CommRingCat.ofHom (Localization.awayMap f.hom r)) := by
   letI e₁ : Localization.Away r ≃ₐ[R] Γ(Spec R, basicOpen r) :=
     IsLocalization.algEquiv (Submonoid.powers r) _ _
-  letI e₂ : Localization.Away (f r) ≃ₐ[S] Γ(Spec S, basicOpen (f r)) :=
-    IsLocalization.algEquiv (Submonoid.powers (f r)) _ _
+  letI e₂ : Localization.Away (f.hom r) ≃ₐ[S] Γ(Spec S, basicOpen (f.hom r)) :=
+    IsLocalization.algEquiv (Submonoid.powers (f.hom r)) _ _
   refine Arrow.isoMk ?_ ?_ ?_
   · exact (Spec (.of S)).isoOfEq (comap_basicOpen _ _) ≪≫
-      (IsAffineOpen.Spec_basicOpen (f r)).isoSpec ≪≫ Scheme.Spec.mapIso e₂.toCommRingCatIso.op
+      (IsAffineOpen.Spec_basicOpen (f.hom r)).isoSpec ≪≫ Scheme.Spec.mapIso e₂.toCommRingCatIso.op
   · exact (IsAffineOpen.Spec_basicOpen r).isoSpec ≪≫ Scheme.Spec.mapIso e₁.toCommRingCatIso.op
   · have := AlgebraicGeometry.IsOpenImmersion.of_isLocalization
       (S := (Localization.Away r)) r
     rw [← cancel_mono (Spec.map (CommRingCat.ofHom (algebraMap R (Localization.Away r))))]
     simp only [Arrow.mk_left, Arrow.mk_right, Functor.id_obj, Scheme.isoOfEq_rfl, Iso.refl_trans,
-      Iso.trans_hom, Functor.mapIso_hom, Iso.op_hom, Iso.symm_hom,
-      Scheme.Spec_map, Quiver.Hom.unop_op, Arrow.mk_hom, Category.assoc,
-      ← Spec.map_comp]
-    show _ ≫ Spec.map (CommRingCat.ofHom
-        ((e₂.toRingHom.comp (Localization.awayMap f.hom r)).comp (algebraMap R _)))
-      = _ ≫ _ ≫ Spec.map (CommRingCat.ofHom (e₁.toRingHom.comp (algebraMap R _)))
-    rw [RingHom.comp_assoc]
-    conv => enter [1,2,1,1,2]; tactic => exact IsLocalization.map_comp _
-    rw [← RingHom.comp_assoc]
-    conv => enter [1,2,1,1,1]; tactic => exact e₂.toAlgHom.comp_algebraMap
-    conv => enter [2,2,2,1,1]; tactic => exact e₁.toAlgHom.comp_algebraMap
-    show _ ≫ Spec.map (f ≫ (Scheme.ΓSpecIso S).inv ≫
-      (Spec S).presheaf.map (homOfLE le_top).op) =
-      Spec.map f ∣_ basicOpen r ≫ _ ≫ Spec.map ((Scheme.ΓSpecIso R).inv ≫
-      (Spec R).presheaf.map (homOfLE le_top).op)
+      Iso.trans_hom, Functor.mapIso_hom, Iso.op_hom, Scheme.Spec_map, Quiver.Hom.unop_op,
+      Arrow.mk_hom, Category.assoc, ← Spec.map_comp]
+    conv =>
+      congr
+      · enter [2, 1]; tactic =>
+        show _ =
+          (f ≫ (Scheme.ΓSpecIso S).inv ≫ (Spec S).presheaf.map (homOfLE le_top).op)
+        ext
+        simp only [Localization.awayMap, IsLocalization.Away.map, AlgEquiv.toRingEquiv_eq_coe,
+          RingEquiv.toCommRingCatIso_hom, AlgEquiv.toRingEquiv_toRingHom, CommRingCat.hom_comp,
+          CommRingCat.hom_ofHom, RingHom.comp_apply, IsLocalization.map_eq, RingHom.coe_coe,
+          AlgEquiv.commutes, IsAffineOpen.algebraMap_Spec_obj]
+      · enter [2, 2, 1]; tactic =>
+        show _ = (Scheme.ΓSpecIso R).inv ≫ (Spec R).presheaf.map (homOfLE le_top).op
+        ext
+        simp only [AlgEquiv.toRingEquiv_eq_coe, RingEquiv.toCommRingCatIso_hom,
+          AlgEquiv.toRingEquiv_toRingHom, CommRingCat.hom_comp, CommRingCat.hom_ofHom,
+          RingHom.coe_comp, RingHom.coe_coe, Function.comp_apply, AlgEquiv.commutes,
+          IsAffineOpen.algebraMap_Spec_obj, homOfLE_leOfHom]
     simp only [IsAffineOpen.isoSpec_hom, homOfLE_leOfHom, Spec.map_comp, Category.assoc,
       Scheme.Opens.toSpecΓ_SpecMap_map_assoc, Scheme.Opens.toSpecΓ_top, Scheme.homOfLE_ι_assoc,
       morphismRestrict_ι_assoc]
diff --git a/Mathlib/AlgebraicGeometry/AffineSpace.lean b/Mathlib/AlgebraicGeometry/AffineSpace.lean
index e845edc4961402..b8779620b20642 100644
--- a/Mathlib/AlgebraicGeometry/AffineSpace.lean
+++ b/Mathlib/AlgebraicGeometry/AffineSpace.lean
@@ -89,7 +89,7 @@ def toSpecMvPolyIntEquiv : (X ⟶ Spec ℤ[n]) ≃ (n → Γ(X, ⊤)) where
       Iso.cancel_iso_inv_left, ← Iso.eq_inv_comp]
     apply of_mvPolynomial_int_ext
     intro i
-    rw [coe_eval₂Hom, eval₂_X]
+    rw [ConcreteCategory.hom_ofHom, coe_eval₂Hom, eval₂_X]
     rfl
   right_inv v := by
     ext i
@@ -203,7 +203,7 @@ def isoOfIsAffine [IsAffine S] :
         ext : 1
         apply ringHom_ext'
         · show _ = (CommRingCat.ofHom C ≫ _).hom
-          rw [CommRingCat.hom_comp, RingHom.comp_assoc, eval₂Hom_comp_C,
+          rw [CommRingCat.hom_comp, RingHom.comp_assoc, CommRingCat.hom_ofHom, eval₂Hom_comp_C,
             ← CommRingCat.hom_comp, ← CommRingCat.hom_ext_iff,
             ← cancel_mono (Scheme.ΓSpecIso _).hom]
           rw [← Scheme.comp_appTop, homOfVector_over, Scheme.comp_appTop]
@@ -212,7 +212,7 @@ def isoOfIsAffine [IsAffine S] :
           rw [← Scheme.comp_appTop_assoc, Scheme.isoSpec, asIso_inv, IsIso.hom_inv_id]
           simp
         · intro i
-          rw [CommRingCat.comp_apply, coe_eval₂Hom]
+          rw [CommRingCat.comp_apply, ConcreteCategory.hom_ofHom, coe_eval₂Hom]
           simp only [eval₂_X]
           exact homOfVector_appTop_coord _ _ _
 
@@ -328,7 +328,7 @@ lemma map_Spec_map {R S : CommRingCat.{max u v}} (φ : R ⟶ S) :
     conv_lhs => enter[2]; tactic => exact map_appTop_coord _ _
     conv_rhs => enter[2]; tactic => exact SpecIso_inv_appTop_coord _ _
     rw [SpecIso_inv_appTop_coord, ← CommRingCat.comp_apply, ← Scheme.ΓSpecIso_inv_naturality,
-        CommRingCat.comp_apply, map_X]
+        CommRingCat.comp_apply, ConcreteCategory.hom_ofHom, map_X]
 
 /-- The map between affine spaces over affine bases is
 isomorphic to the natural map between polynomial rings. -/
diff --git a/Mathlib/AlgebraicGeometry/Cover/MorphismProperty.lean b/Mathlib/AlgebraicGeometry/Cover/MorphismProperty.lean
index 87849e46136cac..042468ac880d9b 100644
--- a/Mathlib/AlgebraicGeometry/Cover/MorphismProperty.lean
+++ b/Mathlib/AlgebraicGeometry/Cover/MorphismProperty.lean
@@ -113,7 +113,7 @@ def Cover.bind [P.IsStableUnderComposition] (f : ∀ x : 𝒰.J, (𝒰.obj x).Co
     rcases (f (𝒰.f x)).covers y with ⟨z, hz⟩
     change x ∈ Set.range ((f (𝒰.f x)).map ((f (𝒰.f x)).f y) ≫ 𝒰.map (𝒰.f x)).base
     use z
-    erw [comp_apply]
+    erw [CategoryTheory.comp_apply]
     rw [hz, hy]
   map_prop _ := P.comp_mem _ _ ((f _).map_prop _) (𝒰.map_prop _)
 
diff --git a/Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean b/Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
index f7d458c2b79a60..2506ebe79df0c0 100644
--- a/Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
+++ b/Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean
@@ -139,7 +139,7 @@ theorem toΓSpecCApp_iff
   -- Porting Note: Type class problem got stuck in `IsLocalization.Away.AwayMap.lift_comp`
   -- created instance manually. This replaces the `pick_goal` tactics
   have loc_inst := IsLocalization.to_basicOpen (Γ.obj (op X)) r
-  refine CommRingCat.hom_ext_iff.trans ?_
+  refine ConcreteCategory.ext_iff.trans ?_
   rw [← @IsLocalization.Away.lift_comp _ _ _ _ _ _ _ r loc_inst _
       (X.isUnit_res_toΓSpecMapBasicOpen r)]
   --pick_goal 5; exact is_localization.to_basic_open _ r
diff --git a/Mathlib/AlgebraicGeometry/Gluing.lean b/Mathlib/AlgebraicGeometry/Gluing.lean
index bcdfd0adfd7723..2d70705beae896 100644
--- a/Mathlib/AlgebraicGeometry/Gluing.lean
+++ b/Mathlib/AlgebraicGeometry/Gluing.lean
@@ -227,7 +227,7 @@ theorem ι_eq_iff (i j : D.J) (x : (D.U i).carrier) (y : (D.U j).carrier) :
     (TopCat.GlueData.ι_eq_iff_rel
       D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData
       i j x y)
-  rw [← ((TopCat.mono_iff_injective D.isoCarrier.inv).mp _).eq_iff, ← comp_apply]
+  rw [← ((TopCat.mono_iff_injective D.isoCarrier.inv).mp _).eq_iff, ← CategoryTheory.comp_apply]
   · simp_rw [← D.ι_isoCarrier_inv]
     rfl -- `rfl` was not needed before https://github.com/leanprover-community/mathlib4/pull/13170
   · infer_instance
@@ -339,7 +339,7 @@ theorem fromGlued_injective : Function.Injective 𝒰.fromGlued.base := by
   intro x y h
   obtain ⟨i, x, rfl⟩ := 𝒰.gluedCover.ι_jointly_surjective x
   obtain ⟨j, y, rfl⟩ := 𝒰.gluedCover.ι_jointly_surjective y
-  rw [← comp_apply, ← comp_apply] at h
+  rw [← CategoryTheory.comp_apply, ← CategoryTheory.comp_apply] at h
   simp_rw [← Scheme.comp_base] at h
   rw [ι_fromGlued, ι_fromGlued] at h
   let e :=
@@ -389,7 +389,7 @@ instance : Epi 𝒰.fromGlued.base := by
   intro x
   obtain ⟨y, h⟩ := 𝒰.covers x
   use (𝒰.gluedCover.ι (𝒰.f x)).base y
-  rw [← comp_apply]
+  rw [← CategoryTheory.comp_apply]
   rw [← 𝒰.ι_fromGlued (𝒰.f x)] at h
   exact h
 
diff --git a/Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean b/Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean
index 4fa80786dbe0c9..a941013a085e5d 100644
--- a/Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean
+++ b/Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean
@@ -248,9 +248,8 @@ theorem exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isAffineOpen (X : Sch
     (H : x |_ (X.basicOpen f) = 0) :
     ∃ n : ℕ, f ^ n * x = 0 := by
   rw [← map_zero (X.presheaf.map (homOfLE <| X.basicOpen_le f : X.basicOpen f ⟶ U).op).hom] at H
-  #adaptation_note
-  /--
-  Prior to nightly-2024-09-29, we could use dot notation here:
+  #adaptation_note /-- nightly-2024-09-29
+  we could use dot notation here:
   `(hU.isLocalization_basicOpen f).exists_of_eq H`
   This is no longer possible;
   likely changing the signature of `IsLocalization.Away.exists_of_eq` is in order.
diff --git a/Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean b/Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean
index 9f07373d33dce9..625901a76887ed 100644
--- a/Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean
+++ b/Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean
@@ -158,7 +158,7 @@ theorem sourceAffineLocally_isLocal (h₁ : RingHom.RespectsIso P)
       simpa using hU
     rw [← f.appLE_congr _ rfl this (fun f => P f.hom),
       IsAffineOpen.appLE_eq_away_map f (isAffineOpen_top Y) U.2 _ r]
-    simp only
+    simp only [CommRingCat.hom_ofHom]
     apply (config := { allowSynthFailures := true }) h₂
     exact H U
   · introv hs hs' U
@@ -167,7 +167,7 @@ theorem sourceAffineLocally_isLocal (h₁ : RingHom.RespectsIso P)
     simp_rw [sourceAffineLocally_morphismRestrict] at hs'
     have := hs' r ⟨X.basicOpen (f.appLE ⊤ U le_top r.1), U.2.basicOpen (f.appLE ⊤ U le_top r.1)⟩
       (by simp [Scheme.Hom.appLE])
-    rwa [IsAffineOpen.appLE_eq_away_map f (isAffineOpen_top Y) U.2,
+    rwa [IsAffineOpen.appLE_eq_away_map f (isAffineOpen_top Y) U.2, CommRingCat.hom_ofHom,
       ← h₁.is_localization_away_iff] at this
 
 variable {P}
diff --git a/Mathlib/AlgebraicGeometry/Noetherian.lean b/Mathlib/AlgebraicGeometry/Noetherian.lean
index 7d1c4fff13747f..73a384e72fb398 100644
--- a/Mathlib/AlgebraicGeometry/Noetherian.lean
+++ b/Mathlib/AlgebraicGeometry/Noetherian.lean
@@ -191,9 +191,8 @@ lemma noetherianSpace_of_isAffineOpen (U : X.Opens) (hU : IsAffineOpen U)
     (Scheme.restrictFunctorΓ.app (op U)).symm.commRingCatIsoToRingEquiv
   exact @noetherianSpace_of_isAffine _ hU _
 
-/-- Any open immersion `Z ⟶ X` with `X` locally Noetherian is quasi-compact.
-
-[Stacks: Lemma 01OX](https://stacks.math.columbia.edu/tag/01OX) -/
+/-- Any open immersion `Z ⟶ X` with `X` locally Noetherian is quasi-compact. -/
+@[stacks 01OX]
 instance (priority := 100) {Z : Scheme} [IsLocallyNoetherian X]
     {f : Z ⟶ X} [IsOpenImmersion f] : QuasiCompact f := by
   apply (quasiCompact_iff_forall_affine f).mpr
@@ -206,9 +205,8 @@ instance (priority := 100) {Z : Scheme} [IsLocallyNoetherian X]
     · exact Set.inter_subset_left
   · exact Set.inter_subset_right
 
-/-- A locally Noetherian scheme is quasi-separated.
-
-[Stacks: Lemma 01OY](https://stacks.math.columbia.edu/tag/01OY) -/
+/-- A locally Noetherian scheme is quasi-separated. -/
+@[stacks 01OY]
 instance (priority := 100) IsLocallyNoetherian.quasiSeparatedSpace [IsLocallyNoetherian X] :
     QuasiSeparatedSpace X := by
   apply (quasiSeparatedSpace_iff_affine X).mpr
@@ -266,9 +264,8 @@ theorem isNoetherian_iff_of_finite_affine_openCover {𝒰 : Scheme.OpenCover.{v,
     · exact Scheme.OpenCover.compactSpace 𝒰
 
 open CategoryTheory in
-/-- A Noetherian scheme has a Noetherian underlying topological space.
-
-[Stacks, Lemma 01OZ](https://stacks.math.columbia.edu/tag/01OZ) -/
+/-- A Noetherian scheme has a Noetherian underlying topological space. -/
+@[stacks 01OZ]
 instance (priority := 100) IsNoetherian.noetherianSpace [IsNoetherian X] :
     NoetherianSpace X := by
   apply TopologicalSpace.noetherian_univ_iff.mp
@@ -284,9 +281,8 @@ instance (priority := 100) IsNoetherian.noetherianSpace [IsNoetherian X] :
   convert noetherianSpace_of_isAffineOpen U.1 U.2
   apply IsLocallyNoetherian.component_noetherian
 
-/-- Any morphism of schemes `f : X ⟶ Y` with `X` Noetherian is quasi-compact.
-
-[Stacks, Lemma 01P0](https://stacks.math.columbia.edu/tag/01P0) -/
+/-- Any morphism of schemes `f : X ⟶ Y` with `X` Noetherian is quasi-compact. -/
+@[stacks 01P0]
 instance (priority := 100) quasiCompact_of_noetherianSpace_source {X Y : Scheme}
     [NoetherianSpace X] (f : X ⟶ Y) : QuasiCompact f :=
   ⟨fun _ _ _ => NoetherianSpace.isCompact _⟩
@@ -325,9 +321,8 @@ theorem isNoetherian_Spec {R : CommRingCat} :
   ⟨fun _ => inferInstance,
    fun _ => inferInstance⟩
 
-/-- A Noetherian scheme has a finite number of irreducible components.
-
-[Stacks, Lemma 0BA8](https://stacks.math.columbia.edu/tag/0BA8) -/
+/-- A Noetherian scheme has a finite number of irreducible components. -/
+@[stacks 0BA8]
 theorem finite_irreducibleComponents_of_isNoetherian [IsNoetherian X] :
     (irreducibleComponents X).Finite := NoetherianSpace.finite_irreducibleComponents
 
diff --git a/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Basic.lean b/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Basic.lean
index 16327672ba49a1..8a215ab44a0221 100644
--- a/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Basic.lean
+++ b/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Basic.lean
@@ -179,7 +179,7 @@ lemma awayMap_awayToSection  :
   obtain ⟨⟨n, a, ⟨b, hb'⟩, i, rfl : _ = b⟩, rfl⟩ := mk_surjective a
   simp only [homOfLE_leOfHom, CommRingCat.hom_comp, RingHom.coe_comp, Function.comp_apply]
   erw [ProjectiveSpectrum.Proj.awayToSection_apply]
-  rw [val_awayMap_mk, Localization.mk_eq_mk', IsLocalization.map_mk',
+  rw [CommRingCat.hom_ofHom, val_awayMap_mk, Localization.mk_eq_mk', IsLocalization.map_mk',
     ← Localization.mk_eq_mk']
   refine Localization.mk_eq_mk_iff.mpr ?_
   rw [Localization.r_iff_exists]
diff --git a/Mathlib/AlgebraicGeometry/Spec.lean b/Mathlib/AlgebraicGeometry/Spec.lean
index fe628dda5c4cb5..26b9f817d1bf58 100644
--- a/Mathlib/AlgebraicGeometry/Spec.lean
+++ b/Mathlib/AlgebraicGeometry/Spec.lean
@@ -420,7 +420,7 @@ instance isLocalizedModule_toPushforwardStalkAlgHom :
     change PrimeSpectrum.basicOpen r ≤ U at hrU
     apply_fun (Spec.topMap (CommRingCat.ofHom (algebraMap R S)) _* (structureSheaf S).1).map
         (homOfLE hrU).op at e
-    simp only [Functor.op_map, map_zero, ← comp_apply, toOpen_res] at e
+    simp only [Functor.op_map, map_zero, ← CategoryTheory.comp_apply, toOpen_res] at e
     have : toOpen S (PrimeSpectrum.basicOpen <| algebraMap R S r) x = 0 := by
       refine Eq.trans ?_ e; rfl
     have :=
diff --git a/Mathlib/AlgebraicGeometry/StructureSheaf.lean b/Mathlib/AlgebraicGeometry/StructureSheaf.lean
index 55c3b14a35165f..9e79bc6c7f8b98 100644
--- a/Mathlib/AlgebraicGeometry/StructureSheaf.lean
+++ b/Mathlib/AlgebraicGeometry/StructureSheaf.lean
@@ -947,9 +947,8 @@ theorem localizationToStalk_stalkSpecializes {R : Type*} [CommRing R] {x y : Pri
         StructureSheaf.localizationToStalk R x := by
   ext : 1
   apply IsLocalization.ringHom_ext (S := Localization.AtPrime y.asIdeal) y.asIdeal.primeCompl
-  erw [RingHom.comp_assoc]
-  conv_rhs => erw [RingHom.comp_assoc]
-  dsimp [CommRingCat.ofHom, localizationToStalk, PrimeSpectrum.localizationMapOfSpecializes]
+  rw [CommRingCat.hom_comp, RingHom.comp_assoc, CommRingCat.hom_comp, RingHom.comp_assoc]
+  dsimp [localizationToStalk, PrimeSpectrum.localizationMapOfSpecializes]
   rw [IsLocalization.lift_comp, IsLocalization.lift_comp, IsLocalization.lift_comp]
   exact CommRingCat.hom_ext_iff.mp (toStalk_stalkSpecializes h)
 
diff --git a/Mathlib/AlgebraicTopology/SimplicialObject/Split.lean b/Mathlib/AlgebraicTopology/SimplicialObject/Split.lean
index 9061d57109b29b..8edfd646e7deed 100644
--- a/Mathlib/AlgebraicTopology/SimplicialObject/Split.lean
+++ b/Mathlib/AlgebraicTopology/SimplicialObject/Split.lean
@@ -376,7 +376,8 @@ theorem comp_f {S₁ S₂ S₃ : Split C} (Φ₁₂ : S₁ ⟶ S₂) (Φ₂₃ :
     (Φ₁₂ ≫ Φ₂₃).f n = Φ₁₂.f n ≫ Φ₂₃.f n :=
   rfl
 
-@[reassoc (attr := simp 1100)]
+-- This is not a `@[simp]` lemma as it can later be proved by `simp`.
+@[reassoc]
 theorem cofan_inj_naturality_symm {S₁ S₂ : Split C} (Φ : S₁ ⟶ S₂) {Δ : SimplexCategoryᵒᵖ}
     (A : Splitting.IndexSet Δ) :
     (S₁.s.cofan Δ).inj A ≫ Φ.F.app Δ = Φ.f A.1.unop.len ≫ (S₂.s.cofan Δ).inj A := by
diff --git a/Mathlib/AlgebraicTopology/SimplicialSet/Boundary.lean b/Mathlib/AlgebraicTopology/SimplicialSet/Boundary.lean
index 58805409d1cebd..e9f81272b8ff26 100644
--- a/Mathlib/AlgebraicTopology/SimplicialSet/Boundary.lean
+++ b/Mathlib/AlgebraicTopology/SimplicialSet/Boundary.lean
@@ -41,13 +41,6 @@ def boundary (n : ℕ) : SSet.{u} where
 /-- The boundary `∂Δ[n]` of the `n`-th standard simplex -/
 scoped[Simplicial] notation3 "∂Δ[" n "]" => SSet.boundary n
 
-#adaptation_note
-/--
-The new unused variable linter in
-https://github.com/leanprover/lean4/pull/5338
-flags `{ α : Δ[n].obj m // _ }`.
--/
-set_option linter.unusedVariables false in
 /-- The inclusion of the boundary of the `n`-th standard simplex into that standard simplex. -/
 def boundaryInclusion (n : ℕ) : ∂Δ[n] ⟶ Δ[n] where app m (α : { α : Δ[n].obj m // _ }) := α
 
diff --git a/Mathlib/AlgebraicTopology/SimplicialSet/Horn.lean b/Mathlib/AlgebraicTopology/SimplicialSet/Horn.lean
index ef8c001fd1d797..6eb839414c7599 100644
--- a/Mathlib/AlgebraicTopology/SimplicialSet/Horn.lean
+++ b/Mathlib/AlgebraicTopology/SimplicialSet/Horn.lean
@@ -35,13 +35,6 @@ def horn (n : ℕ) (i : Fin (n + 1)) : SSet where
 /-- The `i`-th horn `Λ[n, i]` of the standard `n`-simplex -/
 scoped[Simplicial] notation3 "Λ[" n ", " i "]" => SSet.horn (n : ℕ) i
 
-#adaptation_note
-/--
-The new unused variable linter in
-https://github.com/leanprover/lean4/pull/5338
-flags `{ α : Δ[n].obj m // _ }`.
--/
-set_option linter.unusedVariables false in
 /-- The inclusion of the `i`-th horn of the `n`-th standard simplex into that standard simplex. -/
 def hornInclusion (n : ℕ) (i : Fin (n + 1)) : Λ[n, i] ⟶ Δ[n] where
   app m (α : { α : Δ[n].obj m // _ }) := α
diff --git a/Mathlib/Analysis/Analytic/Basic.lean b/Mathlib/Analysis/Analytic/Basic.lean
index f607e676809c63..095e60deae80ab 100644
--- a/Mathlib/Analysis/Analytic/Basic.lean
+++ b/Mathlib/Analysis/Analytic/Basic.lean
@@ -503,13 +503,13 @@ theorem HasFPowerSeriesOnBall.comp_sub (hf : HasFPowerSeriesOnBall f p x r) (y :
 theorem HasFPowerSeriesWithinOnBall.hasSum_sub (hf : HasFPowerSeriesWithinOnBall f p s x r) {y : E}
     (hy : y ∈ (insert x s) ∩ EMetric.ball x r) :
     HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
-  have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy.2
+  have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_enorm_sub] using hy.2
   have := hf.hasSum (by simpa only [add_sub_cancel] using hy.1) this
   simpa only [add_sub_cancel]
 
 theorem HasFPowerSeriesOnBall.hasSum_sub (hf : HasFPowerSeriesOnBall f p x r) {y : E}
     (hy : y ∈ EMetric.ball x r) : HasSum (fun n : ℕ => p n fun _ => y - x) (f y) := by
-  have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy
+  have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_enorm_sub] using hy
   simpa only [add_sub_cancel] using hf.hasSum this
 
 theorem HasFPowerSeriesOnBall.radius_pos (hf : HasFPowerSeriesOnBall f p x r) : 0 < p.radius :=
@@ -544,7 +544,7 @@ lemma HasFPowerSeriesWithinOnBall.congr {f g : E → F} {p : FormalMultilinearSe
   · simpa using h''
   · apply h'
     refine ⟨hy, ?_⟩
-    simpa [edist_eq_coe_nnnorm_sub] using h'y
+    simpa [edist_eq_enorm_sub] using h'y
 
 /-- Variant of `HasFPowerSeriesWithinOnBall.congr` in which one requests equality on `insert x s`
 instead of separating `x` and `s`. -/
@@ -554,7 +554,7 @@ lemma HasFPowerSeriesWithinOnBall.congr' {f g : E → F} {p : FormalMultilinearS
     HasFPowerSeriesWithinOnBall g p s x r := by
   refine ⟨h.r_le, h.r_pos, fun {y} hy h'y ↦ ?_⟩
   convert h.hasSum hy h'y using 1
-  exact h' ⟨hy, by simpa [edist_eq_coe_nnnorm_sub] using h'y⟩
+  exact h' ⟨hy, by simpa [edist_eq_enorm_sub] using h'y⟩
 
 lemma HasFPowerSeriesWithinAt.congr {f g : E → F} {p : FormalMultilinearSeries 𝕜 E F} {s : Set E}
     {x : E} (h : HasFPowerSeriesWithinAt f p s x) (h' : g =ᶠ[𝓝[s] x] f) (h'' : g x = f x) :
@@ -576,7 +576,7 @@ theorem HasFPowerSeriesOnBall.congr (hf : HasFPowerSeriesOnBall f p x r)
     hasSum := fun {y} hy => by
       convert hf.hasSum hy using 1
       apply hg.symm
-      simpa [edist_eq_coe_nnnorm_sub] using hy }
+      simpa [edist_eq_enorm_sub] using hy }
 
 theorem HasFPowerSeriesAt.congr (hf : HasFPowerSeriesAt f p x) (hg : f =ᶠ[𝓝 x] g) :
     HasFPowerSeriesAt g p x := by
@@ -679,7 +679,7 @@ theorem HasFPowerSeriesWithinAt.mono_of_mem_nhdsWithin
   rcases hy with rfl | hy
   · simp
   apply mem_insert_of_mem _ (hr' ?_)
-  simp only [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, sub_zero, lt_min_iff, mem_inter_iff,
+  simp only [EMetric.mem_ball, edist_eq_enorm_sub, sub_zero, lt_min_iff, mem_inter_iff,
     add_sub_cancel_left, hy, and_true] at h'y ⊢
   exact h'y.2
 
@@ -912,7 +912,7 @@ theorem HasFPowerSeriesWithinOnBall.tendsto_partialSum_prod {y : E}
     (atTop ×ˢ 𝓝 y) (𝓝 0) by simpa using A.add this
   apply Metric.tendsto_nhds.2 (fun ε εpos ↦ ?_)
   obtain ⟨r', yr', r'r⟩ : ∃ (r' : ℝ≥0), ‖y‖₊ < r' ∧ r' < r := by
-    simp [edist_eq_coe_nnnorm] at hy
+    simp [edist_zero_eq_enorm] at hy
     simpa using ENNReal.lt_iff_exists_nnreal_btwn.1 hy
   have yr'_2 : ‖y‖ < r' := by simpa [← coe_nnnorm] using yr'
   have S : Summable fun n ↦ ‖p n‖ * ↑r' ^ n := p.summable_norm_mul_pow (r'r.trans_le hf.r_le)
@@ -991,7 +991,7 @@ theorem HasFPowerSeriesWithinOnBall.uniform_geometric_approx' {r' : ℝ≥0}
   have hr'0 : 0 < (r' : ℝ) := (norm_nonneg _).trans_lt yr'
   have : y ∈ EMetric.ball (0 : E) r := by
     refine mem_emetric_ball_zero_iff.2 (lt_trans ?_ h)
-    exact mod_cast yr'
+    simpa [enorm] using yr'
   rw [norm_sub_rev, ← mul_div_right_comm]
   have ya : a * (‖y‖ / ↑r') ≤ a :=
     mul_le_of_le_one_right ha.1.le (div_le_one_of_le₀ yr'.le r'.coe_nonneg)
@@ -1102,7 +1102,7 @@ theorem HasFPowerSeriesWithinOnBall.isBigO_image_sub_image_sub_deriv_principal
         zero_add, ← Subsingleton.pi_single_eq (0 : Fin 1) (y.1 - x), Pi.single,
         ← Subsingleton.pi_single_eq (0 : Fin 1) (y.2 - x), Pi.single, ← (p 1).map_update_sub,
         ← Pi.single, Subsingleton.pi_single_eq, sub_sub_sub_cancel_right]
-    rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ENNReal.coe_lt_coe] at hy'
+    rw [EMetric.mem_ball, edist_eq_enorm_sub, enorm_lt_coe] at hy'
     set B : ℕ → ℝ := fun n => C * (a / r') ^ 2 * (‖y - (x, x)‖ * ‖y.1 - y.2‖) * ((n + 2) * a ^ n)
     have hAB : ∀ n, ‖A (n + 2)‖ ≤ B n := fun n =>
       calc
@@ -1282,7 +1282,7 @@ theorem HasFPowerSeriesWithinOnBall.tendstoLocallyUniformlyOn'
   · ext z
     simp
   · intro z
-    simp [edist_eq_coe_nnnorm, edist_eq_coe_nnnorm_sub]
+    simp [edist_zero_eq_enorm, edist_eq_enorm_sub]
 
 /-- If a function admits a power series expansion at `x`, then it is the locally uniform limit of
 the partial sums of this power series on the disk of convergence, i.e., `f y`
diff --git a/Mathlib/Analysis/Analytic/CPolynomial.lean b/Mathlib/Analysis/Analytic/CPolynomial.lean
index 8c2c58d2957d56..838c7af23b5b86 100644
--- a/Mathlib/Analysis/Analytic/CPolynomial.lean
+++ b/Mathlib/Analysis/Analytic/CPolynomial.lean
@@ -278,7 +278,7 @@ theorem HasFiniteFPowerSeriesOnBall.eq_partialSum'
     ∀ y ∈ EMetric.ball x r, ∀ m, n ≤ m →
     f y = p.partialSum m (y - x) := by
   intro y hy m hm
-  rw [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, ← mem_emetric_ball_zero_iff] at hy
+  rw [EMetric.mem_ball, edist_eq_enorm_sub, ← mem_emetric_ball_zero_iff] at hy
   rw [← (HasFiniteFPowerSeriesOnBall.eq_partialSum hf _ hy m hm), add_sub_cancel]
 
 /-! The particular cases where `f` has a finite power series bounded by `0` or `1`. -/
@@ -300,8 +300,8 @@ theorem HasFiniteFPowerSeriesOnBall.bound_zero_of_eq_zero (hf : ∀ y ∈ EMetri
     rw [hf (x + y)]
     · convert hasSum_zero
       rw [hp, ContinuousMultilinearMap.zero_apply]
-    · rwa [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, add_comm, add_sub_cancel_right,
-        ← edist_eq_coe_nnnorm, ← EMetric.mem_ball]
+    · rwa [EMetric.mem_ball, edist_eq_enorm_sub, add_comm, add_sub_cancel_right,
+        ← edist_zero_eq_enorm, ← EMetric.mem_ball]
 
 /-- If `f` has a formal power series at `x` bounded by `0`, then `f` is equal to `0` in a
 neighborhood of `x`. -/
@@ -396,8 +396,8 @@ lemma changeOriginSeriesTerm_bound (p : FormalMultilinearSeries 𝕜 E F) {n : 
     (hn : ∀ (m : ℕ), n ≤ m → p m = 0) (k l : ℕ) {s : Finset (Fin (k + l))}
     (hs : s.card = l) (hkl : n ≤ k + l) :
     p.changeOriginSeriesTerm k l s hs = 0 := by
-  #adaptation_note
-  /-- `set_option maxSynthPendingDepth 2` required after https://github.com/leanprover/lean4/pull/4119 -/
+  #adaptation_note /-- https://github.com/leanprover/lean4/pull/4119
+  `set_option maxSynthPendingDepth 2` required after https://github.com/leanprover/lean4/pull/4119 -/
   set_option maxSynthPendingDepth 2 in
   rw [changeOriginSeriesTerm, hn _ hkl, map_zero]
 
@@ -504,8 +504,7 @@ theorem HasFiniteFPowerSeriesOnBall.changeOrigin (hf : HasFiniteFPowerSeriesOnBa
         FormalMultilinearSeries.sum (FormalMultilinearSeries.changeOrigin p y) z := by
       rw [mem_emetric_ball_zero_iff, lt_tsub_iff_right, add_comm] at hz
       rw [p.changeOrigin_eval_of_finite hf.finite, add_assoc, hf.sum]
-      refine mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt ?_ hz)
-      exact mod_cast nnnorm_add_le y z
+      exact mem_emetric_ball_zero_iff.2 ((enorm_add_le _ _).trans_lt hz)
     rw [this]
     apply (p.changeOrigin y).hasSum_of_finite fun _ => p.changeOrigin_finite_of_finite hf.finite
 
@@ -514,7 +513,7 @@ it is continuously polynomial at every point of this ball. -/
 theorem HasFiniteFPowerSeriesOnBall.cPolynomialAt_of_mem
     (hf : HasFiniteFPowerSeriesOnBall f p x n r) (h : y ∈ EMetric.ball x r) :
     CPolynomialAt 𝕜 f y := by
-  have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h
+  have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_enorm_sub] using h
   have := hf.changeOrigin this
   rw [add_sub_cancel] at this
   exact this.cPolynomialAt
diff --git a/Mathlib/Analysis/Analytic/ChangeOrigin.lean b/Mathlib/Analysis/Analytic/ChangeOrigin.lean
index 0e2e48a529be8c..a9dd1bdf959940 100644
--- a/Mathlib/Analysis/Analytic/ChangeOrigin.lean
+++ b/Mathlib/Analysis/Analytic/ChangeOrigin.lean
@@ -306,8 +306,8 @@ variable [CompleteSpace F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F}
 it also admits a power series on any subball of this ball (even with a different center provided
 it belongs to `s`), given by `p.changeOrigin`. -/
 theorem HasFPowerSeriesWithinOnBall.changeOrigin (hf : HasFPowerSeriesWithinOnBall f p s x r)
-    (h : (‖y‖₊ : ℝ≥0∞) < r) (hy : x + y ∈ insert x s) :
-    HasFPowerSeriesWithinOnBall f (p.changeOrigin y) s (x + y) (r - ‖y‖₊) where
+    (h : ‖y‖ₑ < r) (hy : x + y ∈ insert x s) :
+    HasFPowerSeriesWithinOnBall f (p.changeOrigin y) s (x + y) (r - ‖y‖ₑ) where
   r_le := by
     apply le_trans _ p.changeOrigin_radius
     exact tsub_le_tsub hf.r_le le_rfl
@@ -322,8 +322,7 @@ theorem HasFPowerSeriesWithinOnBall.changeOrigin (hf : HasFPowerSeriesWithinOnBa
         rw [insert_eq_of_mem hy] at this
         apply this
         simpa [add_assoc] using h'z
-      refine mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt ?_ hz)
-      exact mod_cast nnnorm_add_le y z
+      exact mem_emetric_ball_zero_iff.2 (lt_of_le_of_lt (enorm_add_le _ _) hz)
     rw [this]
     apply (p.changeOrigin y).hasSum
     refine EMetric.ball_subset_ball (le_trans ?_ p.changeOrigin_radius) hz
@@ -342,7 +341,7 @@ it is analytic at every point of this ball. -/
 theorem HasFPowerSeriesWithinOnBall.analyticWithinAt_of_mem
     (hf : HasFPowerSeriesWithinOnBall f p s x r)
     (h : y ∈ insert x s ∩ EMetric.ball x r) : AnalyticWithinAt 𝕜 f s y := by
-  have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_coe_nnnorm_sub] using h.2
+  have : (‖y - x‖₊ : ℝ≥0∞) < r := by simpa [edist_eq_enorm_sub] using h.2
   have := hf.changeOrigin this (by simpa using h.1)
   rw [add_sub_cancel] at this
   exact this.analyticWithinAt
diff --git a/Mathlib/Analysis/Analytic/Composition.lean b/Mathlib/Analysis/Analytic/Composition.lean
index ae4386b4831c7f..6844e170f22e95 100644
--- a/Mathlib/Analysis/Analytic/Composition.lean
+++ b/Mathlib/Analysis/Analytic/Composition.lean
@@ -742,7 +742,7 @@ theorem HasFPowerSeriesWithinAt.comp {g : F → G} {f : E → F} {q : FormalMult
     apply hδ
     have : y ∈ EMetric.ball (0 : E) δ :=
       (EMetric.ball_subset_ball (le_trans (min_le_left _ _) (min_le_right _ _))) hy
-    simpa [-Set.mem_insert_iff, edist_eq_coe_nnnorm_sub, h'y]
+    simpa [-Set.mem_insert_iff, edist_eq_enorm_sub, h'y]
   /- Now the proof starts. To show that the sum of `q.comp p` at `y` is `g (f (x + y))`,
     we will write `q.comp p` applied to `y` as a big sum over all compositions.
     Since the sum is summable, to get its convergence it suffices to get
@@ -775,7 +775,7 @@ theorem HasFPowerSeriesWithinAt.comp {g : F → G} {f : E → F} {q : FormalMult
     have : Tendsto (fun (z : ℕ × F) ↦ q.partialSum z.1 z.2)
         (atTop ×ˢ 𝓝 (f (x + y) - f x)) (𝓝 (g (f x + (f (x + y) - f x)))) := by
       apply Hg.tendsto_partialSum_prod (y := f (x + y) - f x)
-      · simpa [edist_eq_coe_nnnorm_sub] using fy_mem.2
+      · simpa [edist_eq_enorm_sub] using fy_mem.2
       · simpa using fy_mem.1
     simpa using this.comp A
   -- Third step: the sum over all compositions in `compPartialSumTarget 0 n n` converges to
@@ -806,9 +806,9 @@ theorem HasFPowerSeriesWithinAt.comp {g : F → G} {f : E → F} {q : FormalMult
           apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
           rw [Finset.prod_const, Finset.card_fin]
           gcongr
-          rw [EMetric.mem_ball, edist_eq_coe_nnnorm] at hy
+          rw [EMetric.mem_ball, edist_zero_eq_enorm] at hy
           have := le_trans (le_of_lt hy) (min_le_right _ _)
-          rwa [ENNReal.coe_le_coe, ← NNReal.coe_le_coe, coe_nnnorm] at this
+          rwa [enorm_le_coe, ← NNReal.coe_le_coe, coe_nnnorm] at this
     tendsto_nhds_of_cauchySeq_of_subseq cau compPartialSumTarget_tendsto_atTop C
   -- Fifth step: the sum over `n` of `q.comp p n` can be expressed as a particular resummation of
   -- the sum over all compositions, by grouping together the compositions of the same
diff --git a/Mathlib/Analysis/Analytic/Inverse.lean b/Mathlib/Analysis/Analytic/Inverse.lean
index 6ef7fff93cce3a..bb9bba9211dc66 100644
--- a/Mathlib/Analysis/Analytic/Inverse.lean
+++ b/Mathlib/Analysis/Analytic/Inverse.lean
@@ -592,9 +592,9 @@ lemma HasFPowerSeriesAt.tendsto_partialSum_prod_of_comp
           apply mul_le_mul_of_nonneg_left _ (norm_nonneg _)
           rw [Finset.prod_const, Finset.card_fin]
           gcongr
-          rw [EMetric.mem_ball, edist_eq_coe_nnnorm] at hy
+          rw [EMetric.mem_ball, edist_zero_eq_enorm] at hy
           have := le_trans (le_of_lt hy) (min_le_right _ _)
-          rwa [ENNReal.coe_le_coe, ← NNReal.coe_le_coe, coe_nnnorm] at this
+          rwa [enorm_le_coe, ← NNReal.coe_le_coe, coe_nnnorm] at this
     apply HasSum.of_sigma (fun b ↦ hasSum_fintype _) ?_ cau
     simpa [FormalMultilinearSeries.comp] using h0.hasSum hy0
   have B : Tendsto (fun (n : ℕ × ℕ) => ∑ i ∈ compPartialSumTarget 0 n.1 n.2,
@@ -686,7 +686,7 @@ theorem PartialHomeomorph.hasFPowerSeriesAt_symm (f : PartialHomeomorph E F) {a
   refine ⟨min r (p.leftInv i a).radius, min_le_right _ _,
     lt_min r_pos (radius_leftInv_pos_of_radius_pos h.radius_pos hp), fun {y} hy ↦ ?_⟩
   have : y + f a ∈ EMetric.ball (f a) r := by
-    simp only [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, sub_zero, lt_min_iff,
+    simp only [EMetric.mem_ball, edist_eq_enorm_sub, sub_zero, lt_min_iff,
       add_sub_cancel_right] at hy ⊢
     exact hy.1
   simpa [add_comm] using hr this
diff --git a/Mathlib/Analysis/Analytic/Meromorphic.lean b/Mathlib/Analysis/Analytic/Meromorphic.lean
index 86a7ea98ddcc01..ff69888be1dc23 100644
--- a/Mathlib/Analysis/Analytic/Meromorphic.lean
+++ b/Mathlib/Analysis/Analytic/Meromorphic.lean
@@ -278,6 +278,30 @@ lemma _root_.AnalyticAt.meromorphicAt_order {f : 𝕜 → E} {x : 𝕜} (hf : An
     rcases (hf.order_eq_nat_iff _).mp hn.symm with ⟨g, h1, h2, h3⟩
     exact ⟨g, h1, h2, h3.filter_mono nhdsWithin_le_nhds⟩
 
+/-- The order is additive when multiplying scalar-valued and vector-valued meromorphic functions. -/
+theorem order_smul {f : 𝕜 → 𝕜} {g : 𝕜 → E} {x : 𝕜}
+    (hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
+    (hf.smul hg).order = hf.order + hg.order := by
+  -- Trivial cases: one of the functions vanishes around z₀
+  cases' h₂f : hf.order with m h₂f
+  · simp only [top_add, order_eq_top_iff] at h₂f ⊢
+    filter_upwards [h₂f] with z hz using by simp [hz]
+  cases' h₂g : hg.order with n h₂f
+  · simp only [add_top, order_eq_top_iff] at h₂g ⊢
+    filter_upwards [h₂g] with z hz using by simp [hz]
+  -- Non-trivial case: both functions do not vanish around z₀
+  rw [← WithTop.coe_add, order_eq_int_iff]
+  obtain ⟨F, h₁F, h₂F, h₃F⟩ := (hf.order_eq_int_iff _).1 h₂f
+  obtain ⟨G, h₁G, h₂G, h₃G⟩ := (hg.order_eq_int_iff _).1 h₂g
+  use F • G, h₁F.smul h₁G, by simp [h₂F, h₂G]
+  filter_upwards [self_mem_nhdsWithin, h₃F, h₃G] with a ha hfa hga
+  simp [hfa, hga, smul_comm (F a), zpow_add₀ (sub_ne_zero.mpr ha), mul_smul]
+
+/-- The order is additive when multiplying meromorphic functions. -/
+theorem order_mul {f g : 𝕜 → 𝕜} {x : 𝕜} (hf : MeromorphicAt f x) (hg : MeromorphicAt g x) :
+    (hf.mul hg).order = hf.order + hg.order :=
+  hf.order_smul hg
+
 lemma iff_eventuallyEq_zpow_smul_analyticAt {f : 𝕜 → E} {x : 𝕜} : MeromorphicAt f x ↔
     ∃ (n : ℤ) (g : 𝕜 → E), AnalyticAt 𝕜 g x ∧ ∀ᶠ z in 𝓝[≠] x, f z = (z - x) ^ n • g z := by
   refine ⟨fun ⟨n, hn⟩ ↦ ⟨-n, _, ⟨hn, eventually_nhdsWithin_iff.mpr ?_⟩⟩, ?_⟩
diff --git a/Mathlib/Analysis/Analytic/OfScalars.lean b/Mathlib/Analysis/Analytic/OfScalars.lean
index 5674b07141092c..555c1c81c911a8 100644
--- a/Mathlib/Analysis/Analytic/OfScalars.lean
+++ b/Mathlib/Analysis/Analytic/OfScalars.lean
@@ -82,6 +82,11 @@ theorem ofScalars_apply_zero (n : ℕ) :
   rw [ofScalars]
   cases n <;> simp
 
+@[simp]
+lemma coeff_ofScalars {𝕜 : Type*} [NontriviallyNormedField 𝕜] {p : ℕ → 𝕜} {n : ℕ} :
+    (FormalMultilinearSeries.ofScalars 𝕜 p).coeff n = p n := by
+  simp [FormalMultilinearSeries.coeff, FormalMultilinearSeries.ofScalars, List.prod_ofFn]
+
 theorem ofScalars_add (c' : ℕ → 𝕜) : ofScalars E (c + c') = ofScalars E c + ofScalars E c' := by
   unfold ofScalars
   simp_rw [Pi.add_apply, Pi.add_def _ _]
diff --git a/Mathlib/Analysis/Analytic/Uniqueness.lean b/Mathlib/Analysis/Analytic/Uniqueness.lean
index 6c836c8afcab6f..7bd05ac5c6aea5 100644
--- a/Mathlib/Analysis/Analytic/Uniqueness.lean
+++ b/Mathlib/Analysis/Analytic/Uniqueness.lean
@@ -183,7 +183,7 @@ theorem eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux [CompleteSpace F] {f
     EMetric.mem_closure_iff.1 xu (r / 2) (ENNReal.half_pos hp.r_pos.ne')
   let q := p.changeOrigin (y - x)
   have has_series : HasFPowerSeriesOnBall f q y (r / 2) := by
-    have A : (‖y - x‖₊ : ℝ≥0∞) < r / 2 := by rwa [edist_comm, edist_eq_coe_nnnorm_sub] at hxy
+    have A : (‖y - x‖₊ : ℝ≥0∞) < r / 2 := by rwa [edist_comm, edist_eq_enorm_sub] at hxy
     have := hp.changeOrigin (A.trans_le ENNReal.half_le_self)
     simp only [add_sub_cancel] at this
     apply this.mono (ENNReal.half_pos hp.r_pos.ne')
diff --git a/Mathlib/Analysis/Analytic/Within.lean b/Mathlib/Analysis/Analytic/Within.lean
index fd9bd854f032d5..9ffa3528476c86 100644
--- a/Mathlib/Analysis/Analytic/Within.lean
+++ b/Mathlib/Analysis/Analytic/Within.lean
@@ -124,14 +124,14 @@ lemma hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall [CompleteSpac
   · intro h
     refine ⟨fun y ↦ p.sum (y - x), ?_, ?_⟩
     · intro y ⟨ys,yb⟩
-      simp only [EMetric.mem_ball, edist_eq_coe_nnnorm_sub] at yb
+      simp only [EMetric.mem_ball, edist_eq_enorm_sub] at yb
       have e0 := p.hasSum (x := y - x) ?_
       have e1 := (h.hasSum (y := y - x) ?_ ?_)
       · simp only [add_sub_cancel] at e1
         exact e1.unique e0
       · simpa only [add_sub_cancel]
-      · simpa only [EMetric.mem_ball, edist_eq_coe_nnnorm]
-      · simp only [EMetric.mem_ball, edist_eq_coe_nnnorm]
+      · simpa only [EMetric.mem_ball, edist_zero_eq_enorm]
+      · simp only [EMetric.mem_ball, edist_zero_eq_enorm]
         exact lt_of_lt_of_le yb h.r_le
     · refine ⟨h.r_le, h.r_pos, ?_⟩
       intro y lt
@@ -145,7 +145,7 @@ lemma hasFPowerSeriesWithinOnBall_iff_exists_hasFPowerSeriesOnBall [CompleteSpac
     rw [hfg]
     · exact hg.hasSum lt
     · refine ⟨ys, ?_⟩
-      simpa only [EMetric.mem_ball, edist_eq_coe_nnnorm_sub, add_sub_cancel_left, sub_zero] using lt
+      simpa only [EMetric.mem_ball, edist_eq_enorm_sub, add_sub_cancel_left, sub_zero] using lt
 
 /-- `f` has power series `p` at `x` iff some local extension of `f` has that series -/
 lemma hasFPowerSeriesWithinAt_iff_exists_hasFPowerSeriesAt [CompleteSpace F] {f : E → F}
diff --git a/Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean b/Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean
index e1c0e57e890c53..00668c051caf31 100644
--- a/Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean
+++ b/Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean
@@ -3,9 +3,8 @@ Copyright (c) 2021 Anatole Dedecker. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anatole Dedecker
 -/
-import Mathlib.Analysis.Normed.Order.Basic
-import Mathlib.Analysis.Asymptotics.Lemmas
-import Mathlib.Analysis.Normed.Module.Basic
+import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
+import Mathlib.Analysis.SpecificLimits.Basic
 
 /-!
 # A collection of specific asymptotic results
@@ -143,3 +142,19 @@ theorem Filter.Tendsto.cesaro {u : ℕ → ℝ} {l : ℝ} (h : Tendsto u atTop (
   h.cesaro_smul
 
 end Real
+
+section NormedLinearOrderedField
+
+variable {R : Type*} [NormedLinearOrderedField R] [OrderTopology R] [FloorRing R]
+
+theorem Asymptotics.isEquivalent_nat_floor :
+    (fun (x : R) ↦ ↑⌊x⌋₊) ~[atTop] (fun x ↦ x) := by
+  refine isEquivalent_of_tendsto_one ?_ tendsto_nat_floor_div_atTop
+  filter_upwards with x hx using by rw [hx, Nat.floor_zero, Nat.cast_eq_zero]
+
+theorem Asymptotics.isEquivalent_nat_ceil :
+    (fun (x : R) ↦ ↑⌈x⌉₊) ~[atTop] (fun x ↦ x) := by
+  refine isEquivalent_of_tendsto_one ?_ tendsto_nat_ceil_div_atTop
+  filter_upwards with x hx using by rw [hx, Nat.ceil_zero, Nat.cast_eq_zero]
+
+end NormedLinearOrderedField
diff --git a/Mathlib/Analysis/Asymptotics/TVS.lean b/Mathlib/Analysis/Asymptotics/TVS.lean
index 5d0db2d9253a32..59720fc3eebddc 100644
--- a/Mathlib/Analysis/Asymptotics/TVS.lean
+++ b/Mathlib/Analysis/Asymptotics/TVS.lean
@@ -193,6 +193,17 @@ lemma IsLittleOTVS.insert [TopologicalSpace α] {x : α} {s : Set α}
 lemma IsLittleOTVS.bot : f =o[𝕜;⊥] g :=
   fun u hU => ⟨univ, by simp⟩
 
+theorem IsLittleOTVS.add [TopologicalAddGroup E] [ContinuousSMul 𝕜 E]
+    {f₁ f₂ : α → E} {g : α → F} {l : Filter α}
+    (h₁ : f₁ =o[𝕜; l] g) (h₂ : f₂ =o[𝕜; l] g) : (f₁ + f₂) =o[𝕜; l] g := by
+  rw [(nhds_basis_balanced 𝕜 E).add_self.isLittleOTVS_iff (basis_sets _)]
+  rintro U ⟨hU, hUb⟩
+  rcases ((h₁.eventually_smallSets U hU).and (h₂.eventually_smallSets U hU)).exists_mem_of_smallSets
+    with ⟨V, hV, hVf₁, hVf₂⟩
+  refine ⟨V, hV, fun ε hε ↦ ?_⟩
+  filter_upwards [hVf₁ ε hε, hVf₂ ε hε] with x hx₁ hx₂
+  exact (egauge_add_add_le hUb hUb _ _).trans (max_le hx₁ hx₂)
+
 protected lemma IsLittleOTVS.smul_left (h : f =o[𝕜;l] g) (c : α → 𝕜) :
     (fun x ↦ c x • f x) =o[𝕜;l] (fun x ↦ c x • g x) := by
   unfold IsLittleOTVS at *
diff --git a/Mathlib/Analysis/BoxIntegral/Integrability.lean b/Mathlib/Analysis/BoxIntegral/Integrability.lean
index a8a89c77948867..695710a0bc2224 100644
--- a/Mathlib/Analysis/BoxIntegral/Integrability.lean
+++ b/Mathlib/Analysis/BoxIntegral/Integrability.lean
@@ -223,7 +223,7 @@ theorem IntegrableOn.hasBoxIntegral [CompleteSpace E] {f : (ι → ℝ) → E} {
   -- Choose `N` such that the integral of `‖f N x - g x‖` is less than or equal to `ε`.
   obtain ⟨N₀, hN₀⟩ : ∃ N : ℕ, ∫ x in I, ‖f N x - g x‖ ∂μ ≤ ε := by
     have : Tendsto (fun n => ∫⁻ x in I, ‖f n x - g x‖₊ ∂μ) atTop (𝓝 0) :=
-      SimpleFunc.tendsto_approxOn_range_L1_nnnorm hg.measurable hgi
+      SimpleFunc.tendsto_approxOn_range_L1_enorm hg.measurable hgi
     refine (this.eventually (ge_mem_nhds ε0')).exists.imp fun N hN => ?_
     exact integral_coe_le_of_lintegral_coe_le hN
   -- For each `x`, we choose `Nx x ≥ N₀` such that `dist (f Nx x) (g x) ≤ ε`.
diff --git a/Mathlib/Analysis/CStarAlgebra/CStarMatrix.lean b/Mathlib/Analysis/CStarAlgebra/CStarMatrix.lean
new file mode 100644
index 00000000000000..27bbb72bc3611b
--- /dev/null
+++ b/Mathlib/Analysis/CStarAlgebra/CStarMatrix.lean
@@ -0,0 +1,684 @@
+/-
+Copyright (c) 2025 Frédéric Dupuis. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Frédéric Dupuis
+-/
+
+import Mathlib.Analysis.CStarAlgebra.Module.Constructions
+import Mathlib.Analysis.Matrix
+import Mathlib.Topology.UniformSpace.Matrix
+
+/-!
+# Matrices with entries in a C⋆-algebra
+
+This file creates a type copy of `Matrix m n A` called `CStarMatrix m n A` meant for matrices with
+entries in a C⋆-algebra `A`. Its action on `C⋆ᵐᵒᵈ (n → A)` (via `Matrix.mulVec`) gives
+it the operator norm, and this norm makes `CStarMatrix n n A` a C⋆-algebra.
+
+## Main declarations
+
++ `CStarMatrix m n A`: the type copy
++ `CStarMatrix.instNonUnitalCStarAlgebra`: square matrices with entries in a non-unital C⋆-algebra
+    form a non-unital C⋆-algebra
++ `CStarMatrix.instCStarAlgebra`: square matrices with entries in a unital C⋆-algebra form a
+    unital C⋆-algebra
+
+## Implementation notes
+
+The norm on this type induces the product uniformity and bornology, but these are not defeq to
+`Pi.uniformSpace` and `Pi.instBornology`. Hence, we prove the equality to the Pi instances and
+replace the uniformity and bornology by the Pi ones when registering the
+`NormedAddCommGroup (CStarMatrix m n A)` instance. See the docstring of the `TopologyAux` section
+below for more details.
+-/
+
+open scoped ComplexOrder Topology Uniformity Bornology Matrix NNReal InnerProductSpace
+  WithCStarModule
+
+/-- Type copy `Matrix m n A` meant for matrices with entries in a C⋆-algebra. This is
+a C⋆-algebra when `m = n`. -/
+def CStarMatrix (m : Type*) (n : Type*) (A : Type*) := Matrix m n A
+
+namespace CStarMatrix
+
+variable {m n A B R S : Type*}
+
+section basic
+
+variable (m n A) in
+/-- The equivalence between `Matrix m n A` and `CStarMatrix m n A`. -/
+def ofMatrix {m n A : Type*} : Matrix m n A ≃ CStarMatrix m n A := Equiv.refl _
+
+@[simp]
+lemma ofMatrix_apply {M : Matrix m n A} {i : m} : (ofMatrix M) i = M i := rfl
+
+@[simp]
+lemma ofMatrix_symm_apply {M : CStarMatrix m n A} {i : m} : (ofMatrix.symm M) i = M i := rfl
+
+theorem ext_iff {M N : CStarMatrix m n A} : (∀ i j, M i j = N i j) ↔ M = N :=
+  ⟨fun h => funext fun i => funext <| h i, fun h => by simp [h]⟩
+
+@[ext]
+lemma ext {M₁ M₂ : CStarMatrix m n A} (h : ∀ i j, M₁ i j = M₂ i j) : M₁ = M₂ := ext_iff.mp h
+
+/-- `M.map f` is the matrix obtained by applying `f` to each entry of the matrix `M`. -/
+def map (M : CStarMatrix m n A) (f : A → B) : CStarMatrix m n B :=
+  ofMatrix fun i j => f (M i j)
+
+@[simp]
+theorem map_apply {M : CStarMatrix m n A} {f : A → B} {i : m} {j : n} : M.map f i j = f (M i j) :=
+  rfl
+
+@[simp]
+theorem map_id (M : CStarMatrix m n A) : M.map id = M := by
+  ext
+  rfl
+
+@[simp]
+theorem map_id' (M : CStarMatrix m n A) : M.map (·) = M := map_id M
+
+theorem map_map {C : Type*} {M : Matrix m n A} {f : A → B} {g : B → C} :
+    (M.map f).map g = M.map (g ∘ f) := by ext; rfl
+
+theorem map_injective {f : A → B} (hf : Function.Injective f) :
+    Function.Injective fun M : CStarMatrix m n A => M.map f := fun _ _ h =>
+  ext fun i j => hf <| ext_iff.mpr h i j
+
+/-- The transpose of a matrix. -/
+def transpose (M : CStarMatrix m n A) : CStarMatrix n m A :=
+  ofMatrix fun x y => M y x
+
+@[simp]
+theorem transpose_apply (M : CStarMatrix m n A) (i j) : transpose M i j = M j i :=
+  rfl
+
+/-- The conjugate transpose of a matrix defined in term of `star`. -/
+def conjTranspose [Star A] (M : CStarMatrix m n A) : CStarMatrix n m A :=
+  M.transpose.map star
+
+instance instStar [Star A] : Star (CStarMatrix n n A) where
+  star M := M.conjTranspose
+
+instance instInvolutiveStar [InvolutiveStar A] : InvolutiveStar (CStarMatrix n n A) where
+  star_involutive := star_involutive (R := Matrix n n A)
+
+instance instInhabited [Inhabited A] : Inhabited (CStarMatrix m n A) :=
+  inferInstanceAs <| Inhabited <| m → n → A
+
+instance instDecidableEq [DecidableEq A] [Fintype m] [Fintype n] :
+    DecidableEq (CStarMatrix m n A) :=
+  Fintype.decidablePiFintype
+
+instance {n m} [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] (α) [Fintype α] :
+    Fintype (CStarMatrix m n α) := inferInstanceAs (Fintype (m → n → α))
+
+instance {n m} [Finite m] [Finite n] (α) [Finite α] :
+    Finite (CStarMatrix m n α) := inferInstanceAs (Finite (m → n → α))
+
+instance instAdd [Add A] : Add (CStarMatrix m n A) :=
+  Pi.instAdd
+
+instance instAddSemigroup [AddSemigroup A] : AddSemigroup (CStarMatrix m n A) :=
+  Pi.addSemigroup
+
+instance instAddCommSemigroup [AddCommSemigroup A] : AddCommSemigroup (CStarMatrix m n A) :=
+  Pi.addCommSemigroup
+
+instance instZero [Zero A] : Zero (CStarMatrix m n A) :=
+  Pi.instZero
+
+instance instAddZeroClass [AddZeroClass A] : AddZeroClass (CStarMatrix m n A) :=
+  Pi.addZeroClass
+
+instance instAddMonoid [AddMonoid A] : AddMonoid (CStarMatrix m n A) :=
+  Pi.addMonoid
+
+instance instAddCommMonoid [AddCommMonoid A] : AddCommMonoid (CStarMatrix m n A) :=
+  Pi.addCommMonoid
+
+instance instNeg [Neg A] : Neg (CStarMatrix m n A) :=
+  Pi.instNeg
+
+instance instSub [Sub A] : Sub (CStarMatrix m n A) :=
+  Pi.instSub
+
+instance instAddGroup [AddGroup A] : AddGroup (CStarMatrix m n A) :=
+  Pi.addGroup
+
+instance instAddCommGroup [AddCommGroup A] : AddCommGroup (CStarMatrix m n A) :=
+  Pi.addCommGroup
+
+instance instUnique [Unique A] : Unique (CStarMatrix m n A) :=
+  Pi.unique
+
+instance instSubsingleton [Subsingleton A] : Subsingleton (CStarMatrix m n A) :=
+  inferInstanceAs <| Subsingleton <| m → n → A
+
+instance instNontrivial [Nonempty m] [Nonempty n] [Nontrivial A] : Nontrivial (CStarMatrix m n A) :=
+  Function.nontrivial
+
+instance instSMul [SMul R A] : SMul R (CStarMatrix m n A) :=
+  Pi.instSMul
+
+instance instSMulCommClass [SMul R A] [SMul S A] [SMulCommClass R S A] :
+    SMulCommClass R S (CStarMatrix m n A) :=
+  Pi.smulCommClass
+
+instance instIsScalarTower [SMul R S] [SMul R A] [SMul S A] [IsScalarTower R S A] :
+    IsScalarTower R S (CStarMatrix m n A) :=
+  Pi.isScalarTower
+
+instance instIsCentralScalar [SMul R A] [SMul Rᵐᵒᵖ A] [IsCentralScalar R A] :
+    IsCentralScalar R (CStarMatrix m n A) :=
+  Pi.isCentralScalar
+
+instance instMulAction [Monoid R] [MulAction R A] : MulAction R (CStarMatrix m n A) :=
+  Pi.mulAction _
+
+instance instDistribMulAction [Monoid R] [AddMonoid A] [DistribMulAction R A] :
+    DistribMulAction R (CStarMatrix m n A) :=
+  Pi.distribMulAction _
+
+instance instModule [Semiring R] [AddCommMonoid A] [Module R A] : Module R (CStarMatrix m n A) :=
+  Pi.module _ _ _
+
+@[simp, nolint simpNF]
+theorem zero_apply [Zero A] (i : m) (j : n) : (0 : CStarMatrix m n A) i j = 0 := rfl
+
+@[simp] theorem add_apply [Add A] (M N : CStarMatrix m n A) (i : m) (j : n) :
+    (M + N) i j = (M i j) + (N i j) := rfl
+
+@[simp] theorem smul_apply [SMul B A] (r : B) (M : Matrix m n A) (i : m) (j : n) :
+    (r • M) i j = r • (M i j) := rfl
+
+@[simp] theorem sub_apply [Sub A] (M N : Matrix m n A) (i : m) (j : n) :
+    (M - N) i j = (M i j) - (N i j) := rfl
+
+@[simp] theorem neg_apply [Neg A] (M : Matrix m n A) (i : m) (j : n) :
+    (-M) i j = -(M i j) := rfl
+
+/-! simp-normal form pulls `of` to the outside, to match the `Matrix` API. -/
+
+@[simp] theorem of_zero [Zero A] : ofMatrix (0 : Matrix m n A) = 0 := rfl
+
+@[simp] theorem of_add_of [Add A] (f g : Matrix m n A) :
+    ofMatrix f + ofMatrix g = ofMatrix (f + g) := rfl
+
+@[simp]
+theorem of_sub_of [Sub A] (f g : Matrix m n A) : ofMatrix f - ofMatrix g = ofMatrix (f - g) :=
+  rfl
+
+@[simp] theorem neg_of [Neg A] (f : Matrix m n A) : -ofMatrix f = ofMatrix (-f) := rfl
+
+@[simp] theorem smul_of [SMul R A] (r : R) (f : Matrix m n A) :
+    r • ofMatrix f = ofMatrix (r • f) := rfl
+
+instance instStarAddMonoid [AddMonoid A] [StarAddMonoid A] : StarAddMonoid (CStarMatrix n n A) where
+  star_add := star_add (R := Matrix n n A)
+
+instance instStarModule [Star R] [Star A] [SMul R A] [StarModule R A] :
+    StarModule R (CStarMatrix n n A) where
+  star_smul := star_smul (A := Matrix n n A)
+
+/-- The equivalence to matrices, bundled as a linear equivalence. -/
+def ofMatrixₗ [AddCommMonoid A] [Semiring R] [Module R A] :
+    (Matrix m n A) ≃ₗ[R] CStarMatrix m n A := LinearEquiv.refl _ _
+
+section zero_one
+
+variable [Zero A] [One A] [DecidableEq n]
+
+instance instOne : One (CStarMatrix n n A) := inferInstanceAs <| One (Matrix n n A)
+
+theorem one_apply {i j} : (1 : CStarMatrix n n A) i j = if i = j then 1 else 0 := rfl
+
+@[simp]
+theorem one_apply_eq (i) : (1 : CStarMatrix n n A) i i = 1 := Matrix.one_apply_eq _
+
+@[simp] theorem one_apply_ne {i j} : i ≠ j → (1 : CStarMatrix n n A) i j = 0 := Matrix.one_apply_ne
+
+theorem one_apply_ne' {i j} : j ≠ i → (1 : CStarMatrix n n A) i j = 0 := Matrix.one_apply_ne'
+
+instance instAddMonoidWithOne [AddMonoidWithOne A] : AddMonoidWithOne (CStarMatrix n n A) where
+
+instance instAddGroupWithOne [AddGroupWithOne A] : AddGroupWithOne (CStarMatrix n n A) where
+  __ := instAddGroup
+  __ := instAddMonoidWithOne
+
+instance instAddCommMonoidWithOne [AddCommMonoidWithOne A] :
+    AddCommMonoidWithOne (CStarMatrix n n A) where
+  __ := instAddCommMonoid
+  __ := instAddMonoidWithOne
+
+instance instAddCommGroupWithOne [AddCommGroupWithOne A] :
+    AddCommGroupWithOne (CStarMatrix n n A) where
+  __ := instAddCommGroup
+  __ := instAddGroupWithOne
+
+-- We want to be lower priority than `instHMul`, but without this we can't have operands with
+-- implicit dimensions.
+@[default_instance 100]
+instance {l : Type*} [Fintype m] [Mul A] [AddCommMonoid A] :
+    HMul (CStarMatrix l m A) (CStarMatrix m n A) (CStarMatrix l n A) where
+  hMul M N := ofMatrix (ofMatrix.symm M * ofMatrix.symm N)
+
+instance [Fintype n] [Mul A] [AddCommMonoid A] : Mul (CStarMatrix n n A) where mul M N := M * N
+
+end zero_one
+
+theorem mul_apply {l : Type*} [Fintype m] [Mul A] [AddCommMonoid A] {M : CStarMatrix l m A}
+    {N : CStarMatrix m n A} {i k} : (M * N) i k = ∑ j, M i j * N j k := rfl
+
+theorem mul_apply' {l : Type*} [Fintype m] [Mul A] [AddCommMonoid A] {M : CStarMatrix l m A}
+    {N : CStarMatrix m n A} {i k} : (M * N) i k = (fun j => M i j) ⬝ᵥ fun j => N j k := rfl
+
+@[simp]
+theorem smul_mul {l : Type*} [Fintype n] [Monoid R] [AddCommMonoid A] [Mul A] [DistribMulAction R A]
+    [IsScalarTower R A A] (a : R) (M : CStarMatrix m n A) (N : CStarMatrix n l A) :
+    (a • M) * N = a • (M * N) := Matrix.smul_mul a M N
+
+theorem mul_smul {l : Type*} [Fintype n] [Monoid R] [AddCommMonoid A] [Mul A] [DistribMulAction R A]
+    [SMulCommClass R A A] (M : Matrix m n A) (a : R) (N : Matrix n l A) :
+    M * (a • N) = a • (M * N) := Matrix.mul_smul M a N
+
+instance instNonUnitalNonAssocSemiring [Fintype n] [NonUnitalNonAssocSemiring A] :
+    NonUnitalNonAssocSemiring (CStarMatrix n n A) :=
+  inferInstanceAs <| NonUnitalNonAssocSemiring (Matrix n n A)
+
+instance instNonUnitalNonAssocRing [Fintype n] [NonUnitalNonAssocRing A] :
+    NonUnitalNonAssocRing (CStarMatrix n n A) :=
+  inferInstanceAs <| NonUnitalNonAssocRing (Matrix n n A)
+
+instance instNonUnitalSemiring [Fintype n] [NonUnitalSemiring A] :
+    NonUnitalSemiring (CStarMatrix n n A) :=
+  inferInstanceAs <| NonUnitalSemiring (Matrix n n A)
+
+instance instNonAssocSemiring [Fintype n] [DecidableEq n] [NonAssocSemiring A] :
+    NonAssocSemiring (CStarMatrix n n A) :=
+  inferInstanceAs <| NonAssocSemiring (Matrix n n A)
+
+instance instNonUnitalRing [Fintype n] [NonUnitalRing A] :
+    NonUnitalRing (CStarMatrix n n A) :=
+  inferInstanceAs <| NonUnitalRing (Matrix n n A)
+
+instance instNonAssocRing [Fintype n] [DecidableEq n] [NonAssocRing A] :
+    NonAssocRing (CStarMatrix n n A) :=
+  inferInstanceAs <| NonAssocRing (Matrix n n A)
+
+instance instSemiring [Fintype n] [DecidableEq n] [Semiring A] :
+    Semiring (CStarMatrix n n A) :=
+  inferInstanceAs <| Semiring (Matrix n n A)
+
+instance instRing [Fintype n] [DecidableEq n] [Ring A] : Ring (CStarMatrix n n A) :=
+  inferInstanceAs <| Ring (Matrix n n A)
+
+/-- `ofMatrix` bundled as a ring equivalence. -/
+def ofMatrixRingEquiv [Fintype n] [Semiring A] :
+    Matrix n n A ≃+* CStarMatrix n n A :=
+  { ofMatrix with
+    map_mul' := fun _ _ => rfl
+    map_add' := fun _ _ => rfl }
+
+instance instStarRing [Fintype n] [NonUnitalSemiring A] [StarRing A] :
+    StarRing (CStarMatrix n n A) := inferInstanceAs <| StarRing (Matrix n n A)
+
+instance instAlgebra [Fintype n] [DecidableEq n] [CommSemiring R] [Semiring A] [Algebra R A] :
+    Algebra R (CStarMatrix n n A) := inferInstanceAs <| Algebra R (Matrix n n A)
+
+/-- `ofMatrix` bundled as a star algebra equivalence. -/
+def ofMatrixStarAlgEquiv [Fintype n] [SMul ℂ A] [Semiring A] [StarRing A] :
+    Matrix n n A ≃⋆ₐ[ℂ] CStarMatrix n n A :=
+  { ofMatrixRingEquiv with
+    map_star' := fun _ => rfl
+    map_smul' := fun _ _ => rfl }
+
+lemma ofMatrix_eq_ofMatrixStarAlgEquiv [Fintype n] [SMul ℂ A] [Semiring A] [StarRing A] :
+    (ofMatrix : Matrix n n A → CStarMatrix n n A)
+      = (ofMatrixStarAlgEquiv : Matrix n n A → CStarMatrix n n A) := rfl
+
+end basic
+
+variable [Fintype m] [Fintype n] [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
+
+
+/-- Interpret a `CStarMatrix m n A` as a continuous linear map acting on `C⋆ᵐᵒᵈ (n → A)`. -/
+def toCLM : CStarMatrix m n A →ₗ[ℂ] (C⋆ᵐᵒᵈ (n → A)) →L[ℂ] (C⋆ᵐᵒᵈ (m → A)) where
+  toFun M := { toFun := (WithCStarModule.equivL ℂ).symm ∘ M.mulVec ∘ WithCStarModule.equivL ℂ
+               map_add' := M.mulVec_add
+               map_smul' := M.mulVec_smul
+               cont := by
+                 simp only [LinearMap.coe_mk, AddHom.coe_mk]
+                 exact Continuous.comp (by fun_prop) (by fun_prop) }
+  map_add' M₁ M₂ := by
+    ext
+    simp only [ContinuousLinearMap.coe_mk', LinearMap.coe_mk, AddHom.coe_mk, Function.comp_apply,
+      WithCStarModule.equivL_apply, WithCStarModule.equivL_symm_apply,
+      WithCStarModule.equiv_symm_pi_apply, ContinuousLinearMap.add_apply, WithCStarModule.add_apply]
+    rw [Matrix.add_mulVec, Pi.add_apply]
+  map_smul' c M := by
+    ext x i
+    simp only [ContinuousLinearMap.coe_mk', LinearMap.coe_mk, AddHom.coe_mk, Function.comp_apply,
+      WithCStarModule.equivL_apply, WithCStarModule.equivL_symm_apply,
+      WithCStarModule.equiv_symm_pi_apply, Matrix.mulVec, dotProduct,
+      WithCStarModule.equiv_pi_apply, RingHom.id_apply, ContinuousLinearMap.coe_smul',
+      Pi.smul_apply, WithCStarModule.smul_apply, Finset.smul_sum]
+    congr
+    ext j
+    rw [CStarMatrix.smul_apply, smul_mul_assoc]
+
+lemma toCLM_apply {M : CStarMatrix m n A} {v : C⋆ᵐᵒᵈ (n → A)} :
+    toCLM M v = (WithCStarModule.equiv _).symm (M.mulVec v) := rfl
+
+lemma toCLM_apply_eq_sum {M : CStarMatrix m n A} {v : C⋆ᵐᵒᵈ (n → A)} :
+    toCLM M v = (WithCStarModule.equiv _).symm (fun i => ∑ j, M i j * v j) := by
+  ext i
+  simp [toCLM_apply, Matrix.mulVec, dotProduct]
+
+/-- Interpret a `CStarMatrix m n A` as a continuous linear map acting on `C⋆ᵐᵒᵈ (n → A)`. This
+version is specialized to the case `m = n` and is bundled as a non-unital algebra homomorphism. -/
+def toCLMNonUnitalAlgHom : CStarMatrix n n A →ₙₐ[ℂ] (C⋆ᵐᵒᵈ (n → A)) →L[ℂ] (C⋆ᵐᵒᵈ (n → A)) :=
+  { toCLM (n := n) (m := n) with
+    map_zero' := by ext1; simp
+    map_mul' := by intros; ext; simp [toCLM] }
+
+lemma toCLMNonUnitalAlgHom_eq_toCLM {M : CStarMatrix n n A} :
+    toCLMNonUnitalAlgHom (A := A) M = toCLM M := rfl
+
+open WithCStarModule in
+@[simp high]
+lemma toCLM_apply_single [DecidableEq n] {M : CStarMatrix m n A} {j : n} (a : A) :
+    (toCLM M) (equiv _ |>.symm <| Pi.single j a) = (equiv _).symm (fun i => M i j * a) := by
+  ext
+  simp [toCLM_apply, EmbeddingLike.apply_eq_iff_eq, equiv, Equiv.refl]
+
+open WithCStarModule in
+lemma toCLM_apply_single_apply [DecidableEq n] {M : CStarMatrix m n A} {i : m} {j : n} (a : A) :
+    (toCLM M) (equiv _ |>.symm <| Pi.single j a) i = M i j * a := by simp
+
+open WithCStarModule in
+lemma mul_entry_mul_eq_inner_toCLM [DecidableEq m] [DecidableEq n] {M : CStarMatrix m n A}
+    {i : m} {j : n} (a b : A) :
+    star a * M i j * b
+      = ⟪equiv _ |>.symm (Pi.single i a), toCLM M (equiv _ |>.symm <| Pi.single j b)⟫_A := by
+  simp [mul_assoc, inner_def]
+
+lemma toCLM_injective [DecidableEq n] : Function.Injective (toCLM (A := A) (m := m) (n := n)) := by
+  rw [injective_iff_map_eq_zero]
+  intro M h
+  ext i j
+  rw [Matrix.zero_apply, ← norm_eq_zero, ← sq_eq_zero_iff, sq, ← CStarRing.norm_self_mul_star,
+    ← toCLM_apply_single_apply]
+  simp [h]
+
+open WithCStarModule in
+lemma inner_toCLM_conjTranspose_left {M : CStarMatrix m n A} {v : C⋆ᵐᵒᵈ (m → A)}
+    {w : C⋆ᵐᵒᵈ (n → A)} : ⟪toCLM Mᴴ v, w⟫_A = ⟪v, toCLM M w⟫_A := by
+  simp only [toCLM_apply_eq_sum, pi_inner, equiv_symm_pi_apply, inner_def, Finset.mul_sum,
+    Matrix.conjTranspose_apply, star_sum, star_mul, star_star, Finset.sum_mul]
+  rw [Finset.sum_comm]
+  simp_rw [mul_assoc]
+
+lemma inner_toCLM_conjTranspose_right {M : CStarMatrix m n A} {v : C⋆ᵐᵒᵈ (n → A)}
+    {w : C⋆ᵐᵒᵈ (m → A)} : ⟪v, toCLM Mᴴ w⟫_A = ⟪toCLM M v, w⟫_A := by
+  apply Eq.symm
+  simpa using inner_toCLM_conjTranspose_left (M := Mᴴ) (v := v) (w := w)
+
+/-- The operator norm on `CStarMatrix m n A`. -/
+noncomputable instance instNorm : Norm (CStarMatrix m n A) where
+  norm M := ‖toCLM M‖
+
+lemma norm_def {M : CStarMatrix m n A} : ‖M‖ = ‖toCLM M‖ := rfl
+
+lemma norm_def' {M : CStarMatrix n n A} : ‖M‖ = ‖toCLMNonUnitalAlgHom (A := A) M‖ := rfl
+
+set_option maxSynthPendingDepth 2 in
+lemma normedSpaceCore [DecidableEq n]: NormedSpace.Core ℂ (CStarMatrix m n A) where
+  norm_nonneg M := (toCLM M).opNorm_nonneg
+  norm_smul c M := by rw [norm_def, norm_def, map_smul, norm_smul _ (toCLM M)]
+  norm_triangle M₁ M₂ := by simpa [← map_add] using norm_add_le (toCLM M₁) (toCLM M₂)
+  norm_eq_zero_iff := by
+    simpa only [norm_def, norm_eq_zero, ← injective_iff_map_eq_zero'] using toCLM_injective
+
+open WithCStarModule in
+lemma norm_entry_le_norm [DecidableEq n] {M : CStarMatrix m n A} {i : m} {j : n} :
+    ‖M i j‖ ≤ ‖M‖ := by
+  suffices ‖M i j‖ * ‖M i j‖ ≤ ‖M‖ * ‖M i j‖ by
+    obtain (h | h) := eq_zero_or_norm_pos (M i j)
+    · set_option maxSynthPendingDepth 2 in
+      simp [h, norm_def]
+    · exact le_of_mul_le_mul_right this h
+  rw [← CStarRing.norm_self_mul_star, ← toCLM_apply_single_apply]
+  apply norm_apply_le_norm _ _ |>.trans
+  apply (toCLM M).le_opNorm _ |>.trans
+  simp [norm_def]
+
+open CStarModule in
+lemma norm_le_of_forall_inner_le {M : CStarMatrix m n A} {C : ℝ≥0}
+    (h : ∀ v w, ‖⟪w, toCLM M v⟫_A‖ ≤ C * ‖v‖ * ‖w‖) : ‖M‖ ≤ C := by
+  refine (toCLM M).opNorm_le_bound (by simp) fun v ↦ ?_
+  obtain (h₀ | h₀) := (norm_nonneg (toCLM M v)).eq_or_lt
+  · rw [← h₀]
+    positivity
+  · refine le_of_mul_le_mul_right ?_ h₀
+    simpa [← sq, norm_sq_eq] using h ..
+
+end CStarMatrix
+
+section TopologyAux
+/-
+## Replacing the uniformity and bornology
+
+Note that while the norm that we have defined on `CStarMatrix m n A` induces the product uniformity,
+it is not defeq to `Pi.uniformSpace`. In this section, we show that the norm indeed does induce
+the product topology and use this fact to properly set up the
+`NormedAddCommGroup (CStarMatrix m n A)` instance such that the uniformity is still
+`Pi.uniformSpace` and the bornology is `Pi.instBornology`.
+
+To do this, we locally register a `NormedAddCommGroup` instance on `CStarMatrix` which registers
+the "bad" topology, and we also locally use the matrix norm `Matrix.normedAddCommGroup`
+(which takes the norm of the biggest entry as the norm of the matrix)
+in order to show that the map `ofMatrix` is bilipschitz. We then finally register the
+`NormedAddCommGroup (C⋆ᵐᵒᵈ (n → A))` instance via `NormedAddCommGroup.ofCoreReplaceAll`.
+-/
+
+namespace CStarMatrix
+
+variable {m n A : Type*} [Fintype m] [Fintype n] [DecidableEq n]
+  [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
+
+private noncomputable def normedAddCommGroupAux : NormedAddCommGroup (CStarMatrix m n A) :=
+  .ofCore CStarMatrix.normedSpaceCore
+
+attribute [local instance] normedAddCommGroupAux
+
+private def normedSpaceAux : NormedSpace ℂ (CStarMatrix m n A) :=
+  .ofCore CStarMatrix.normedSpaceCore
+
+/- In this `Aux` section, we locally activate the following instances: a norm on `CStarMatrix`
+ which induces a topology that is not defeq with the matrix one, and the elementwise norm on
+ matrices, in order to show that the two topologies are in fact equal -/
+attribute [local instance] normedSpaceAux Matrix.normedAddCommGroup Matrix.normedSpace
+
+private lemma nnnorm_le_of_forall_inner_le {M : CStarMatrix m n A} {C : ℝ≥0}
+    (h : ∀ v w, ‖⟪w, CStarMatrix.toCLM M v⟫_A‖₊ ≤ C * ‖v‖₊ * ‖w‖₊) : ‖M‖₊ ≤ C :=
+  CStarMatrix.norm_le_of_forall_inner_le fun v w => h v w
+
+open Finset in
+private lemma lipschitzWith_toMatrixAux [DecidableEq m] :
+    LipschitzWith 1 (ofMatrixₗ.symm (R := ℂ) : CStarMatrix m n A → Matrix m n A) := by
+  refine AddMonoidHomClass.lipschitz_of_bound_nnnorm _ _ fun M => ?_
+  rw [one_mul, ← NNReal.coe_le_coe, coe_nnnorm, coe_nnnorm, Matrix.norm_le_iff (norm_nonneg _)]
+  exact fun _ _ ↦ CStarMatrix.norm_entry_le_norm
+
+open CStarMatrix WithCStarModule in
+private lemma antilipschitzWith_toMatrixAux :
+    AntilipschitzWith (Fintype.card m * Fintype.card n)
+      (ofMatrixₗ.symm (R := ℂ) : CStarMatrix m n A → Matrix m n A) := by
+  refine AddMonoidHomClass.antilipschitz_of_bound _ fun M => ?_
+  calc
+    ‖M‖ ≤ ∑ i, ∑ j, ‖M i j‖ := by
+      rw [norm_def]
+      refine (toCLM M).opNorm_le_bound (by positivity) fun v => ?_
+      simp only [toCLM_apply_eq_sum, equiv_symm_pi_apply, Finset.sum_mul]
+      apply pi_norm_le_sum_norm _ |>.trans
+      gcongr with i _
+      apply norm_sum_le _ _ |>.trans
+      gcongr with j _
+      apply norm_mul_le _ _ |>.trans
+      gcongr
+      exact norm_apply_le_norm v j
+    _ ≤ ∑ _ : m, ∑ _ : n, ‖ofMatrixₗ.symm (R := ℂ) M‖ := by
+      gcongr with i _ j _
+      exact ofMatrixₗ.symm (R := ℂ) M |>.norm_entry_le_entrywise_sup_norm
+    _ = _ := by simp [mul_assoc]
+
+private lemma uniformInducing_toMatrixAux [DecidableEq m] :
+    IsUniformInducing (ofMatrix.symm : CStarMatrix m n A → Matrix m n A) :=
+  AntilipschitzWith.isUniformInducing antilipschitzWith_toMatrixAux
+    lipschitzWith_toMatrixAux.uniformContinuous
+
+private lemma uniformity_eq_aux [DecidableEq m] :
+    𝓤 (CStarMatrix m n A) = (𝓤[Pi.uniformSpace _] :
+      Filter (CStarMatrix m n A × CStarMatrix m n A)) := by
+  have :
+    (fun x : CStarMatrix m n A × CStarMatrix m n A => ⟨ofMatrix.symm x.1, ofMatrix.symm x.2⟩)
+      = id := by
+    ext i <;> rfl
+  rw [← uniformInducing_toMatrixAux.comap_uniformity, this, Filter.comap_id]
+  rfl
+
+open Bornology in
+private lemma cobounded_eq_aux [DecidableEq m] :
+    cobounded (CStarMatrix m n A) = @cobounded _ Pi.instBornology := by
+  have : cobounded (CStarMatrix m n A) = Filter.comap ofMatrix.symm (cobounded _) := by
+    refine le_antisymm ?_ ?_
+    · exact antilipschitzWith_toMatrixAux.tendsto_cobounded.le_comap
+    · exact lipschitzWith_toMatrixAux.comap_cobounded_le
+  exact this.trans Filter.comap_id
+
+end CStarMatrix
+
+end TopologyAux
+
+namespace CStarMatrix
+
+section non_unital
+
+variable {A : Type*} [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
+
+variable {m n : Type*} [Fintype m] [Fintype n] [DecidableEq m]
+
+instance instTopologicalSpace : TopologicalSpace (CStarMatrix m n A) := Pi.topologicalSpace
+instance instUniformSpace : UniformSpace (CStarMatrix m n A) := Pi.uniformSpace _
+instance instBornology : Bornology (CStarMatrix m n A) := Pi.instBornology
+instance instCompleteSpace : CompleteSpace (CStarMatrix m n A) := Pi.complete _
+instance instT2Space : T2Space (CStarMatrix m n A) := Pi.t2Space
+instance instT3Space : T3Space (CStarMatrix m n A) := _root_.instT3Space
+
+instance instTopologicalAddGroup : TopologicalAddGroup (CStarMatrix m n A) :=
+  Pi.topologicalAddGroup
+
+instance instUniformAddGroup : UniformAddGroup (CStarMatrix m n A) :=
+  Pi.instUniformAddGroup
+
+instance instContinuousSMul {R : Type*} [SMul R A] [TopologicalSpace R] [ContinuousSMul R A] :
+    ContinuousSMul R (CStarMatrix m n A) := instContinuousSMulForall
+
+noncomputable instance instNormedAddCommGroup [DecidableEq n] :
+    NormedAddCommGroup (CStarMatrix m n A) :=
+  .ofCoreReplaceAll CStarMatrix.normedSpaceCore
+    CStarMatrix.uniformity_eq_aux.symm
+      fun _ => Filter.ext_iff.1 CStarMatrix.cobounded_eq_aux.symm _
+
+instance instNormedSpace [DecidableEq n] : NormedSpace ℂ (CStarMatrix m n A) :=
+  .ofCore CStarMatrix.normedSpaceCore
+
+noncomputable instance instNonUnitalNormedRing [DecidableEq n] :
+    NonUnitalNormedRing (CStarMatrix n n A) where
+  dist_eq _ _ := rfl
+  norm_mul _ _ := by
+    set_option maxSynthPendingDepth 2 in
+    simpa only [norm_def', map_mul] using norm_mul_le _ _
+
+open ContinuousLinearMap CStarModule in
+/-- Matrices with entries in a C⋆-algebra form a C⋆-algebra. -/
+instance instCStarRing [DecidableEq n] : CStarRing (CStarMatrix n n A) where
+  norm_mul_self_le M := by
+    have hmain : ‖M‖ ≤ √‖star M * M‖ := by
+      change ‖toCLM M‖ ≤ √‖star M * M‖
+      rw [opNorm_le_iff (by positivity)]
+      intro v
+      rw [norm_eq_sqrt_norm_inner_self, ← inner_toCLM_conjTranspose_right]
+      have h₁ : ‖⟪v, (toCLM Mᴴ) ((toCLM M) v)⟫_A‖ ≤ ‖star M * M‖ * ‖v‖ ^ 2 := calc
+          _ ≤ ‖v‖ * ‖(toCLM Mᴴ) (toCLM M v)‖ := norm_inner_le (C⋆ᵐᵒᵈ (n → A))
+          _ ≤ ‖v‖ * ‖(toCLM Mᴴ).comp (toCLM M)‖ * ‖v‖ := by
+                    rw [mul_assoc]
+                    gcongr
+                    rw [← ContinuousLinearMap.comp_apply]
+                    exact le_opNorm ((toCLM Mᴴ).comp (toCLM M)) v
+          _ = ‖(toCLM Mᴴ).comp (toCLM M)‖ * ‖v‖ ^ 2 := by ring
+          _ = ‖star M * M‖ * ‖v‖ ^ 2 := by
+                    congr
+                    simp only [← toCLMNonUnitalAlgHom_eq_toCLM, Matrix.star_eq_conjTranspose,
+                      map_mul]
+                    rfl
+      have h₂ : ‖v‖ = √(‖v‖ ^ 2) := by simp
+      rw [h₂, ← Real.sqrt_mul]
+      gcongr
+      positivity
+    rw [← pow_two, ← Real.sqrt_le_sqrt_iff (by positivity)]
+    simp [hmain]
+
+/-- Matrices with entries in a non-unital C⋆-algebra form a non-unital C⋆-algebra. -/
+noncomputable instance instNonUnitalCStarAlgebra [DecidableEq n] :
+    NonUnitalCStarAlgebra (CStarMatrix n n A) where
+  smul_assoc x y z := by simp
+  smul_comm m a b := (Matrix.mul_smul _ _ _).symm
+
+instance instPartialOrder [DecidableEq n] :
+    PartialOrder (CStarMatrix n n A) := CStarAlgebra.spectralOrder _
+instance instStarOrderedRing [DecidableEq n] :
+    StarOrderedRing (CStarMatrix n n A) := CStarAlgebra.spectralOrderedRing _
+
+end non_unital
+
+section unital
+
+variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
+
+variable {n : Type*} [Fintype n] [DecidableEq n]
+
+noncomputable instance instNormedRing : NormedRing (CStarMatrix n n A) where
+  dist_eq _ _ := rfl
+  norm_mul := norm_mul_le
+
+noncomputable instance instNormedAlgebra : NormedAlgebra ℂ (CStarMatrix n n A) where
+  norm_smul_le r M := by simpa only [norm_def, map_smul] using (toCLM M).opNorm_smul_le r
+
+/-- Matrices with entries in a unital C⋆-algebra form a unital C⋆-algebra. -/
+noncomputable instance instCStarAlgebra [DecidableEq n] : CStarAlgebra (CStarMatrix n n A) where
+
+end unital
+
+section
+
+variable {m n A : Type*} [NonUnitalCStarAlgebra A]
+
+lemma uniformEmbedding_ofMatrix :
+    IsUniformEmbedding (ofMatrix : Matrix m n A → CStarMatrix m n A) where
+  comap_uniformity := Filter.comap_id'
+  injective := fun ⦃_ _⦄ a ↦ a
+
+/-- `ofMatrix` bundled as a continuous linear equivalence. -/
+def ofMatrixL : Matrix m n A ≃L[ℂ] CStarMatrix m n A :=
+  { ofMatrixₗ with
+    continuous_toFun := continuous_id
+    continuous_invFun := continuous_id }
+
+lemma ofMatrix_eq_ofMatrixL :
+    (ofMatrix : Matrix m n A → CStarMatrix m n A)
+      = (ofMatrixL : Matrix m n A → CStarMatrix m n A) := rfl
+
+end
+
+end CStarMatrix
diff --git a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean
index c093bde67133cd..62ae49c6840de3 100644
--- a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean
+++ b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Order.lean
@@ -291,7 +291,8 @@ lemma le_iff_norm_sqrt_mul_rpow {a b : A} (hbu : IsUnit b) (ha : 0 ≤ a) (hb :
   lift b to Aˣ using hbu
   have hbab : 0 ≤ (b : A) ^ (-(1 / 2) : ℝ) * a * (b : A) ^ (-(1 / 2) : ℝ) :=
     conjugate_nonneg_of_nonneg ha rpow_nonneg
-  #adaptation_note /-- 2024-11-10 added `(R := A)` -/
+  #adaptation_note /-- 2024-11-10
+  added `(R := A)` -/
   conv_rhs =>
     rw [← sq_le_one_iff₀ (norm_nonneg _), sq, ← CStarRing.norm_star_mul_self, star_mul,
       IsSelfAdjoint.of_nonneg (R := A) sqrt_nonneg, IsSelfAdjoint.of_nonneg rpow_nonneg,
diff --git a/Mathlib/Analysis/CStarAlgebra/Module/Constructions.lean b/Mathlib/Analysis/CStarAlgebra/Module/Constructions.lean
index 64d2ba88bc9c29..872bf20de1001d 100644
--- a/Mathlib/Analysis/CStarAlgebra/Module/Constructions.lean
+++ b/Mathlib/Analysis/CStarAlgebra/Module/Constructions.lean
@@ -296,6 +296,13 @@ private lemma isBounded_pi_iff_aux (s : Set (C⋆ᵐᵒᵈ (Π i, E i))) :
 
 end Aux
 
+noncomputable instance : PseudoMetricSpace (C⋆ᵐᵒᵈ (Π i, E i)) :=
+  .ofSeminormedAddCommGroupCoreReplaceAll
+    normedSpaceCore.toCore uniformity_pi_eq_aux isBounded_pi_iff_aux
+
+noncomputable instance : SeminormedAddCommGroup (C⋆ᵐᵒᵈ (Π i, E i)) :=
+  .ofCoreReplaceAll normedSpaceCore.toCore uniformity_pi_eq_aux isBounded_pi_iff_aux
+
 noncomputable instance : NormedAddCommGroup (C⋆ᵐᵒᵈ (Π i, E i)) :=
   .ofCoreReplaceAll normedSpaceCore uniformity_pi_eq_aux isBounded_pi_iff_aux
 
diff --git a/Mathlib/Analysis/CStarAlgebra/Module/Synonym.lean b/Mathlib/Analysis/CStarAlgebra/Module/Synonym.lean
index 95c7421f62a168..02b57dc8603b2a 100644
--- a/Mathlib/Analysis/CStarAlgebra/Module/Synonym.lean
+++ b/Mathlib/Analysis/CStarAlgebra/Module/Synonym.lean
@@ -6,6 +6,8 @@ Authors: Jireh Loreaux
 import Mathlib.RingTheory.Finiteness.Defs
 import Mathlib.Topology.Bornology.Constructions
 import Mathlib.Topology.UniformSpace.Equiv
+import Mathlib.Topology.Algebra.Module.Equiv
+import Mathlib.Topology.Algebra.UniformGroup.Basic
 
 /-! # Type synonym for types with a `CStarModule` structure
 
@@ -75,11 +77,21 @@ instance instUnique [Unique E] : Unique (WithCStarModule E) := ‹Unique E›
 
 /-! ## `WithCStarModule E` inherits various module-adjacent structures from `E`. -/
 
+instance instZero [Zero E] : Zero (WithCStarModule E) := ‹Zero E›
+instance instAdd [Add E] : Add (WithCStarModule E) := ‹Add E›
+instance instSub [Sub E] : Sub (WithCStarModule E) := ‹Sub E›
+instance instNeg [Neg E] : Neg (WithCStarModule E) := ‹Neg E›
+instance instAddMonoid [AddMonoid E] : AddMonoid (WithCStarModule E) := ‹AddMonoid E›
+instance instSubNegMonoid [SubNegMonoid E] : SubNegMonoid (WithCStarModule E) := ‹SubNegMonoid E›
+instance instSubNegZeroMonoid [SubNegZeroMonoid E] : SubNegZeroMonoid (WithCStarModule E) :=
+  ‹SubNegZeroMonoid E›
+
 instance instAddCommGroup [AddCommGroup E] : AddCommGroup (WithCStarModule E) := ‹AddCommGroup E›
+
 instance instSMul {R : Type*} [SMul R E] : SMul R (WithCStarModule E) := ‹SMul R E›
+
 instance instModule {R : Type*} [Semiring R] [AddCommGroup E] [Module R E] :
-    Module R (WithCStarModule E) :=
-  ‹Module R E›
+    Module R (WithCStarModule E) := ‹Module R E›
 
 instance instIsScalarTower [SMul R R'] [SMul R E] [SMul R' E]
     [IsScalarTower R R' E] : IsScalarTower R R' (WithCStarModule E) :=
@@ -89,11 +101,6 @@ instance instSMulCommClass [SMul R E] [SMul R' E] [SMulCommClass R R' E] :
     SMulCommClass R R' (WithCStarModule E) :=
   ‹SMulCommClass R R' E›
 
-instance instModuleFinite [Semiring R] [AddCommGroup E] [Module R E] [Module.Finite R E] :
-    Module.Finite R (WithCStarModule E) :=
-  ‹Module.Finite R E›
-
-
 section Equiv
 
 variable {R E}
@@ -151,6 +158,12 @@ theorem equiv_symm_smul : (equiv E).symm (c • x') = c • (equiv E).symm x' :=
 
 end Equiv
 
+/-- `WithCStarModule.equiv` as an additive equivalence. -/
+def addEquiv [AddCommGroup E] : C⋆ᵐᵒᵈ E ≃+ E :=
+  { AddEquiv.refl _ with
+    toFun := equiv _
+    invFun := (equiv _).symm }
+
 /-- `WithCStarModule.equiv` as a linear equivalence. -/
 @[simps (config := .asFn)]
 def linearEquiv [Semiring R] [AddCommGroup E] [Module R E] : C⋆ᵐᵒᵈ E ≃ₗ[R] E :=
@@ -158,6 +171,16 @@ def linearEquiv [Semiring R] [AddCommGroup E] [Module R E] : C⋆ᵐᵒᵈ E ≃
     toFun := equiv _
     invFun := (equiv _).symm }
 
+lemma map_top_submodule {R : Type*} [Semiring R] [AddCommGroup E] [Module R E] :
+    (⊤ : Submodule R E).map (linearEquiv R E).symm = ⊤ := by
+  ext x
+  refine ⟨fun _  => trivial, fun _ => ?_⟩
+  rw [Submodule.mem_map]
+  exact ⟨linearEquiv R E x, by simp⟩
+
+instance instModuleFinite [Semiring R] [AddCommGroup E] [Module R E] [Module.Finite R E] :
+    Module.Finite R (WithCStarModule E) := inferInstanceAs (Module.Finite R E)
+
 /-! ## `WithCStarModule E` inherits the uniformity and bornology from `E`. -/
 
 variable {E}
@@ -170,9 +193,26 @@ instance [Bornology E] : Bornology (C⋆ᵐᵒᵈ E) := Bornology.induced <| equ
 def uniformEquiv [UniformSpace E] : C⋆ᵐᵒᵈ E ≃ᵤ E :=
   equiv E |>.toUniformEquivOfIsUniformInducing ⟨rfl⟩
 
+/-- `WithCStarModule.equiv` as a continuous linear equivalence between `C⋆ᵐᵒᵈ E` and `E`. -/
+@[simps! apply symm_apply]
+def equivL [Semiring R] [AddCommGroup E] [UniformSpace E] [Module R E] : C⋆ᵐᵒᵈ E ≃L[R] E :=
+  { linearEquiv R E with
+    continuous_toFun := UniformEquiv.continuous uniformEquiv
+    continuous_invFun := UniformEquiv.continuous uniformEquiv.symm }
+
 instance [UniformSpace E] [CompleteSpace E] : CompleteSpace (C⋆ᵐᵒᵈ E) :=
   uniformEquiv.completeSpace_iff.mpr inferInstance
 
+instance [AddCommGroup E] [UniformSpace E] [ContinuousAdd E] : ContinuousAdd (C⋆ᵐᵒᵈ E) :=
+  ContinuousAdd.induced (addEquiv E)
+
+instance [AddCommGroup E] [UniformSpace E] [UniformAddGroup E] : UniformAddGroup (C⋆ᵐᵒᵈ E) :=
+  UniformAddGroup.comap (addEquiv E)
+
+instance [Semiring R] [TopologicalSpace R] [AddCommGroup E] [UniformSpace E] [Module R E]
+    [ContinuousSMul R E] : ContinuousSMul R (C⋆ᵐᵒᵈ E) :=
+  ContinuousSMul.induced (linearEquiv R E)
+
 end Basic
 
 /-! ## Prod
diff --git a/Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean b/Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean
index 7cc8a905e12eca..9ed44eadb36478 100644
--- a/Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean
+++ b/Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean
@@ -279,11 +279,8 @@ theorem hasFTaylorSeriesUpToOn_succ_iff_left {n : ℕ} :
         rw [this]
         exact h.2.2
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/4119,
-without `set_option maxSynthPendingDepth 2` this proof needs substantial repair.
--/
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/4119
+without `set_option maxSynthPendingDepth 2` this proof needs substantial repair. -/
 set_option maxSynthPendingDepth 2 in
 -- Porting note: this was split out from `hasFTaylorSeriesUpToOn_succ_iff_right` to avoid a timeout.
 theorem HasFTaylorSeriesUpToOn.shift_of_succ
@@ -489,9 +486,8 @@ theorem iteratedFDerivWithin_succ_apply_right {n : ℕ} (hs : UniqueDiffOn 𝕜
         rw [fderivWithin_congr A (A x hx)]
       _ = (I ∘ fderivWithin 𝕜 (iteratedFDerivWithin 𝕜 n (fderivWithin 𝕜 f s) s) s x :
               E → E[×n + 1]→L[𝕜] F) (m 0) (tail m) := by
-        #adaptation_note
-        /--
-        After https://github.com/leanprover/lean4/pull/4119 we need to either use
+        #adaptation_note /-- https://github.com/leanprover/lean4/pull/4119
+        we need to either use
         `set_option maxSynthPendingDepth 2 in`
         or fill in an explicit argument as
         ```
diff --git a/Mathlib/Analysis/Calculus/Deriv/Basic.lean b/Mathlib/Analysis/Calculus/Deriv/Basic.lean
index 64f66f524e715c..9f7a1638661298 100644
--- a/Mathlib/Analysis/Calculus/Deriv/Basic.lean
+++ b/Mathlib/Analysis/Calculus/Deriv/Basic.lean
@@ -658,6 +658,9 @@ theorem deriv_const' : (deriv fun _ : 𝕜 => c) = fun _ => 0 :=
 theorem derivWithin_const : derivWithin (fun _ => c) s = 0 := by
   ext; simp [derivWithin]
 
+@[simp]
+theorem derivWithin_zero : derivWithin (0 : 𝕜 → F) s = 0 := derivWithin_const _ _
+
 end Const
 
 section Continuous
diff --git a/Mathlib/Analysis/Calculus/Deriv/Mul.lean b/Mathlib/Analysis/Calculus/Deriv/Mul.lean
index b97172a75dda63..3da9a8ed349913 100644
--- a/Mathlib/Analysis/Calculus/Deriv/Mul.lean
+++ b/Mathlib/Analysis/Calculus/Deriv/Mul.lean
@@ -246,6 +246,17 @@ theorem derivWithin_mul_const (hc : DifferentiableWithinAt 𝕜 c s x) (d : 𝔸
   · exact (hc.hasDerivWithinAt.mul_const d).derivWithin hxs
   · simp [derivWithin_zero_of_isolated hxs]
 
+lemma derivWithin_mul_const_field (u : 𝕜') :
+    derivWithin (fun y => v y * u) s x = derivWithin v s x * u := by
+  by_cases hv : DifferentiableWithinAt 𝕜 v s x
+  · rw [derivWithin_mul_const hv u]
+  by_cases hu : u = 0
+  · simp [hu]
+  rw [derivWithin_zero_of_not_differentiableWithinAt hv, zero_mul,
+      derivWithin_zero_of_not_differentiableWithinAt]
+  have : v = fun x ↦ (v x * u) * u⁻¹ := by ext; simp [hu]
+  exact fun h_diff ↦ hv <| this ▸ h_diff.mul_const _
+
 theorem deriv_mul_const (hc : DifferentiableAt 𝕜 c x) (d : 𝔸) :
     deriv (fun y => c y * d) x = deriv c x * d :=
   (hc.hasDerivAt.mul_const d).deriv
@@ -284,6 +295,11 @@ theorem derivWithin_const_mul (c : 𝔸) (hd : DifferentiableWithinAt 𝕜 d s x
   · exact (hd.hasDerivWithinAt.const_mul c).derivWithin hxs
   · simp [derivWithin_zero_of_isolated hxs]
 
+lemma derivWithin_const_mul_field (u : 𝕜') :
+    derivWithin (fun y => u * v y) s x = u * derivWithin v s x := by
+  simp_rw [mul_comm u]
+  exact derivWithin_mul_const_field u
+
 theorem deriv_const_mul (c : 𝔸) (hd : DifferentiableAt 𝕜 d x) :
     deriv (fun y => c * d y) x = c * deriv d x :=
   (hd.hasDerivAt.const_mul c).deriv
diff --git a/Mathlib/Analysis/Calculus/Deriv/Prod.lean b/Mathlib/Analysis/Calculus/Deriv/Prod.lean
index caa6c116835431..61e10a2f7aeefa 100644
--- a/Mathlib/Analysis/Calculus/Deriv/Prod.lean
+++ b/Mathlib/Analysis/Calculus/Deriv/Prod.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2019 Gabriel Ebner. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Gabriel Ebner, Yury Kudryashov
+Authors: Gabriel Ebner, Yury Kudryashov, Eric Wieser
 -/
 import Mathlib.Analysis.Calculus.Deriv.Basic
 import Mathlib.Analysis.Calculus.FDeriv.Prod
@@ -13,7 +13,7 @@ In this file we prove lemmas about derivatives of functions `f : 𝕜 → E × F
 `f : 𝕜 → (Π i, E i)`.
 
 For a more detailed overview of one-dimensional derivatives in mathlib, see the module docstring of
-`analysis/calculus/deriv/basic`.
+`Mathlib/Analysis/Calculus/Deriv/Basic.lean`.
 
 ## Keywords
 
@@ -90,3 +90,88 @@ theorem deriv_pi (h : ∀ i, DifferentiableAt 𝕜 (fun x => φ x i) x) :
   (hasDerivAt_pi.2 fun i => (h i).hasDerivAt).deriv
 
 end Pi
+
+
+/-!
+### Derivatives of tuples `f : 𝕜 → Π i : Fin n.succ, F' i`
+
+These can be used to prove results about functions of the form `fun x ↦ ![f x, g x, h x]`,
+as `Matrix.vecCons` is defeq to `Fin.cons`.
+-/
+section PiFin
+
+variable {n : Nat} {F' : Fin n.succ → Type*}
+variable [∀ i, NormedAddCommGroup (F' i)] [∀ i, NormedSpace 𝕜 (F' i)]
+variable {φ : 𝕜 → F' 0} {φs : 𝕜 → ∀ i, F' (Fin.succ i)}
+
+theorem hasStrictDerivAt_finCons {φ' : Π i, F' i} :
+    HasStrictDerivAt (fun x => Fin.cons (φ x) (φs x)) φ' x ↔
+      HasStrictDerivAt φ (φ' 0) x ∧ HasStrictDerivAt φs (fun i => φ' i.succ) x :=
+  hasStrictFDerivAt_finCons
+
+/-- A variant of `hasStrictDerivAt_finCons` where the derivative variables are free on the RHS
+instead. -/
+theorem hasStrictDerivAt_finCons' {φ' : F' 0} {φs' : Π i, F' (Fin.succ i)} :
+    HasStrictDerivAt (fun x => Fin.cons (φ x) (φs x)) (Fin.cons φ' φs') x ↔
+      HasStrictDerivAt φ φ' x ∧ HasStrictDerivAt φs φs' x :=
+  hasStrictDerivAt_finCons
+
+theorem HasStrictDerivAt.finCons {φ' : F' 0} {φs' : Π i, F' (Fin.succ i)}
+    (h : HasStrictDerivAt φ φ' x) (hs : HasStrictDerivAt φs φs' x) :
+    HasStrictDerivAt (fun x => Fin.cons (φ x) (φs x)) (Fin.cons φ' φs') x :=
+  hasStrictDerivAt_finCons'.mpr ⟨h, hs⟩
+
+theorem hasDerivAtFilter_finCons {φ' : Π i, F' i} {l : Filter 𝕜} :
+    HasDerivAtFilter (fun x => Fin.cons (φ x) (φs x)) φ' x l ↔
+      HasDerivAtFilter φ (φ' 0) x l ∧ HasDerivAtFilter φs (fun i => φ' i.succ) x l :=
+  hasFDerivAtFilter_finCons
+
+/-- A variant of `hasDerivAtFilter_finCons` where the derivative variables are free on the RHS
+instead. -/
+theorem hasDerivAtFilter_finCons' {φ' : F' 0} {φs' : Π i, F' (Fin.succ i)} {l : Filter 𝕜} :
+    HasDerivAtFilter (fun x => Fin.cons (φ x) (φs x)) (Fin.cons φ' φs') x l ↔
+      HasDerivAtFilter φ φ' x l ∧ HasDerivAtFilter φs φs' x l :=
+  hasDerivAtFilter_finCons
+
+theorem HasDerivAtFilter.finCons {φ' : F' 0} {φs' : Π i, F' (Fin.succ i)} {l : Filter 𝕜}
+    (h : HasDerivAtFilter φ φ' x l) (hs : HasDerivAtFilter φs φs' x l) :
+    HasDerivAtFilter (fun x => Fin.cons (φ x) (φs x)) (Fin.cons φ' φs') x l :=
+  hasDerivAtFilter_finCons'.mpr ⟨h, hs⟩
+
+theorem hasDerivAt_finCons {φ' : Π i, F' i} :
+    HasDerivAt (fun x => Fin.cons (φ x) (φs x)) φ' x ↔
+      HasDerivAt φ (φ' 0) x ∧ HasDerivAt φs (fun i => φ' i.succ) x :=
+  hasDerivAtFilter_finCons
+
+/-- A variant of `hasDerivAt_finCons` where the derivative variables are free on the RHS
+instead. -/
+theorem hasDerivAt_finCons' {φ' : F' 0} {φs' : Π i, F' (Fin.succ i)} :
+    HasDerivAt (fun x => Fin.cons (φ x) (φs x)) (Fin.cons φ' φs') x ↔
+      HasDerivAt φ φ' x ∧ HasDerivAt φs φs' x :=
+  hasDerivAt_finCons
+
+theorem HasDerivAt.finCons {φ' : F' 0} {φs' : Π i, F' (Fin.succ i)}
+    (h : HasDerivAt φ φ' x) (hs : HasDerivAt φs φs' x) :
+    HasDerivAt (fun x => Fin.cons (φ x) (φs x)) (Fin.cons φ' φs') x :=
+  hasDerivAt_finCons'.mpr ⟨h, hs⟩
+
+theorem hasDerivWithinAt_finCons {φ' : Π i, F' i} :
+    HasDerivWithinAt (fun x => Fin.cons (φ x) (φs x)) φ' s x ↔
+      HasDerivWithinAt φ (φ' 0) s x ∧ HasDerivWithinAt φs (fun i => φ' i.succ) s x :=
+  hasDerivAtFilter_finCons
+
+/-- A variant of `hasDerivWithinAt_finCons` where the derivative variables are free on the RHS
+instead. -/
+theorem hasDerivWithinAt_finCons' {φ' : F' 0} {φs' : Π i, F' (Fin.succ i)} :
+    HasDerivWithinAt (fun x => Fin.cons (φ x) (φs x)) (Fin.cons φ' φs') s x ↔
+      HasDerivWithinAt φ φ' s x ∧ HasDerivWithinAt φs φs' s x :=
+  hasDerivAtFilter_finCons
+
+theorem HasDerivWithinAt.finCons {φ' : F' 0} {φs' : Π i, F' (Fin.succ i)}
+    (h : HasDerivWithinAt φ φ' s x) (hs : HasDerivWithinAt φs φs' s x) :
+    HasDerivWithinAt (fun x => Fin.cons (φ x) (φs x)) (Fin.cons φ' φs') s x :=
+  hasDerivWithinAt_finCons'.mpr ⟨h, hs⟩
+
+-- TODO: write the `Fin.cons` versions of `derivWithin_pi` and `deriv_pi`
+
+end PiFin
diff --git a/Mathlib/Analysis/Calculus/FDeriv/Analytic.lean b/Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
index 09a1501984d890..00cea385a3f4a6 100644
--- a/Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
+++ b/Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
@@ -205,7 +205,7 @@ protected theorem HasFPowerSeriesOnBall.fderiv [CompleteSpace F]
       (h.r_pos.trans_le h.r_le)).mono h.r_pos h.r_le).comp_sub x
   dsimp only
   rw [← h.fderiv_eq, add_sub_cancel]
-  simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
+  simpa only [edist_eq_enorm_sub, EMetric.mem_ball] using hz
 
 /-- If a function has a power series within a set on a ball, then so does its derivative. -/
 protected theorem HasFPowerSeriesWithinOnBall.fderivWithin [CompleteSpace F]
@@ -220,7 +220,7 @@ protected theorem HasFPowerSeriesWithinOnBall.fderivWithin [CompleteSpace F]
       (h.r_pos.trans_le h.r_le)).mono h.r_pos h.r_le).comp_sub x
   · dsimp only
     rw [← h.fderivWithin_eq _ _ hu, add_sub_cancel]
-    · simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz.2
+    · simpa only [edist_eq_enorm_sub, EMetric.mem_ball] using hz.2
     · simpa using hz.1
 
 /-- If a function has a power series within a set on a ball, then so does its derivative. For a
@@ -315,7 +315,7 @@ theorem HasFPowerSeriesWithinOnBall.hasSum_derivSeries_of_hasFDerivWithinAt
   rw [← b.isEmbedding.hasSum_iff]
   have : HasFPowerSeriesWithinOnBall (a ∘ f) (a.compFormalMultilinearSeries p) s x r :=
     a.comp_hasFPowerSeriesWithinOnBall h
-  have Z := (this.fderivWithin hu).hasSum h'y (by simpa [edist_eq_coe_nnnorm] using hy)
+  have Z := (this.fderivWithin hu).hasSum h'y (by simpa [edist_zero_eq_enorm] using hy)
   have : fderivWithin 𝕜 (a ∘ f) (insert x s) (x + y) = a ∘L f' := by
     apply HasFDerivWithinAt.fderivWithin _ (hu _ h'y)
     exact a.hasFDerivAt.comp_hasFDerivWithinAt (x + y) hf'
@@ -339,7 +339,7 @@ protected theorem HasFPowerSeriesWithinOnBall.fderivWithin_of_mem_of_analyticOn
     (h : AnalyticOn 𝕜 f s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
     HasFPowerSeriesWithinOnBall (fderivWithin 𝕜 f s) p.derivSeries s x r := by
   refine ⟨hr.r_le.trans p.radius_le_radius_derivSeries, hr.r_pos, fun {y} hy h'y ↦ ?_⟩
-  apply hr.hasSum_derivSeries_of_hasFDerivWithinAt (by simpa [edist_eq_coe_nnnorm] using h'y) hy
+  apply hr.hasSum_derivSeries_of_hasFDerivWithinAt (by simpa [edist_zero_eq_enorm] using h'y) hy
   · rw [insert_eq_of_mem hx] at hy ⊢
     apply DifferentiableWithinAt.hasFDerivWithinAt
     exact h.differentiableOn _ hy
@@ -489,7 +489,7 @@ protected theorem HasFiniteFPowerSeriesOnBall.fderiv
       ((p.hasFiniteFPowerSeriesOnBall_changeOrigin 1 h.finite).mono h.r_pos le_top).comp_sub x
   dsimp only
   rw [← h.fderiv_eq, add_sub_cancel]
-  simpa only [edist_eq_coe_nnnorm_sub, EMetric.mem_ball] using hz
+  simpa only [edist_eq_enorm_sub, EMetric.mem_ball] using hz
 
 /-- If a function has a finite power series on a ball, then so does its derivative.
 This is a variant of `HasFiniteFPowerSeriesOnBall.fderiv` where the degree of `f` is `< n`
diff --git a/Mathlib/Analysis/Calculus/FDeriv/Prod.lean b/Mathlib/Analysis/Calculus/FDeriv/Prod.lean
index c5e33c862226a1..892ed8c6dfa95e 100644
--- a/Mathlib/Analysis/Calculus/FDeriv/Prod.lean
+++ b/Mathlib/Analysis/Calculus/FDeriv/Prod.lean
@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2019 Jeremy Avigad. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
+Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov, Eric Wieser
 -/
 import Mathlib.Analysis.Calculus.FDeriv.Linear
 import Mathlib.Analysis.Calculus.FDeriv.Comp
@@ -487,6 +487,187 @@ theorem fderiv_pi (h : ∀ i, DifferentiableAt 𝕜 (φ i) x) :
 
 end Pi
 
+/-!
+### Derivatives of tuples `f : E → Π i : Fin n.succ, F' i`
+
+These can be used to prove results about functions of the form `fun x ↦ ![f x, g x, h x]`,
+as `Matrix.vecCons` is defeq to `Fin.cons`.
+-/
+section PiFin
+
+variable {n : Nat} {F' : Fin n.succ → Type*}
+variable [∀ i, NormedAddCommGroup (F' i)] [∀ i, NormedSpace 𝕜 (F' i)]
+variable {φ : E → F' 0} {φs : E → ∀ i, F' (Fin.succ i)}
+
+theorem hasStrictFDerivAt_finCons {φ' : E →L[𝕜] Π i, F' i} :
+    HasStrictFDerivAt (fun x => Fin.cons (φ x) (φs x)) φ' x ↔
+      HasStrictFDerivAt φ (.proj 0 ∘L φ') x ∧
+      HasStrictFDerivAt φs (Pi.compRightL 𝕜 F' Fin.succ ∘L φ') x := by
+  rw [hasStrictFDerivAt_pi', Fin.forall_fin_succ, hasStrictFDerivAt_pi']
+  dsimp [ContinuousLinearMap.comp, LinearMap.comp, Function.comp_def]
+  simp only [Fin.cons_zero, Fin.cons_succ]
+
+/-- A variant of `hasStrictFDerivAt_finCons` where the derivative variables are free on the RHS
+instead. -/
+theorem hasStrictFDerivAt_finCons'
+    {φ' : E →L[𝕜] F' 0} {φs' : E →L[𝕜] Π i, F' (Fin.succ i)} :
+    HasStrictFDerivAt (fun x => Fin.cons (φ x) (φs x)) (φ'.finCons φs') x ↔
+      HasStrictFDerivAt φ φ' x ∧ HasStrictFDerivAt φs φs' x :=
+  hasStrictFDerivAt_finCons
+
+@[fun_prop]
+theorem HasStrictFDerivAt.finCons
+    {φ' : E →L[𝕜] F' 0} {φs' : E →L[𝕜] Π i, F' (Fin.succ i)}
+    (h : HasStrictFDerivAt φ φ' x) (hs : HasStrictFDerivAt φs φs' x) :
+    HasStrictFDerivAt (fun x => Fin.cons (φ x) (φs x)) (φ'.finCons φs') x :=
+  hasStrictFDerivAt_finCons'.mpr ⟨h, hs⟩
+
+theorem hasFDerivAtFilter_finCons
+    {φ' : E →L[𝕜] Π i, F' i} {l : Filter E} :
+    HasFDerivAtFilter (fun x => Fin.cons (φ x) (φs x)) φ' x l ↔
+      HasFDerivAtFilter φ (.proj 0 ∘L φ') x l ∧
+      HasFDerivAtFilter φs (Pi.compRightL 𝕜 F' Fin.succ ∘L φ') x l := by
+  rw [hasFDerivAtFilter_pi', Fin.forall_fin_succ, hasFDerivAtFilter_pi']
+  dsimp [ContinuousLinearMap.comp, LinearMap.comp, Function.comp_def]
+  simp only [Fin.cons_zero, Fin.cons_succ]
+
+/-- A variant of `hasFDerivAtFilter_finCons` where the derivative variables are free on the RHS
+instead. -/
+theorem hasFDerivAtFilter_finCons'
+    {φ' : E →L[𝕜] F' 0} {φs' : E →L[𝕜] Π i, F' (Fin.succ i)} {l : Filter E} :
+    HasFDerivAtFilter (fun x => Fin.cons (φ x) (φs x)) (φ'.finCons φs') x l ↔
+      HasFDerivAtFilter φ φ' x l ∧ HasFDerivAtFilter φs φs' x l :=
+  hasFDerivAtFilter_finCons
+
+theorem HasFDerivAtFilter.finCons
+    {φ' : E →L[𝕜] F' 0} {φs' : E →L[𝕜] Π i, F' (Fin.succ i)} {l : Filter E}
+    (h : HasFDerivAtFilter φ φ' x l) (hs : HasFDerivAtFilter φs φs' x l) :
+    HasFDerivAtFilter (fun x => Fin.cons (φ x) (φs x)) (φ'.finCons φs') x l :=
+  hasFDerivAtFilter_finCons'.mpr ⟨h, hs⟩
+
+theorem hasFDerivAt_finCons
+    {φ' : E →L[𝕜] Π i, F' i} :
+    HasFDerivAt (fun x => Fin.cons (φ x) (φs x)) φ' x ↔
+      HasFDerivAt φ (.proj 0 ∘L φ') x ∧ HasFDerivAt φs (Pi.compRightL 𝕜 F' Fin.succ ∘L φ') x :=
+  hasFDerivAtFilter_finCons
+
+/-- A variant of `hasFDerivAt_finCons` where the derivative variables are free on the RHS
+instead. -/
+theorem hasFDerivAt_finCons'
+    {φ' : E →L[𝕜] F' 0} {φs' : E →L[𝕜] Π i, F' (Fin.succ i)} :
+    HasFDerivAt (fun x => Fin.cons (φ x) (φs x)) (φ'.finCons φs') x ↔
+      HasFDerivAt φ φ' x ∧ HasFDerivAt φs φs' x :=
+  hasFDerivAt_finCons
+
+@[fun_prop]
+theorem HasFDerivAt.finCons
+    {φ' : E →L[𝕜] F' 0} {φs' : E →L[𝕜] Π i, F' (Fin.succ i)}
+    (h : HasFDerivAt φ φ' x) (hs : HasFDerivAt φs φs' x) :
+    HasFDerivAt (fun x => Fin.cons (φ x) (φs x)) (φ'.finCons φs') x :=
+  hasFDerivAt_finCons'.mpr ⟨h, hs⟩
+
+theorem hasFDerivWithinAt_finCons
+    {φ' : E →L[𝕜] Π i, F' i} :
+    HasFDerivWithinAt (fun x => Fin.cons (φ x) (φs x)) φ' s x ↔
+      HasFDerivWithinAt φ (.proj 0 ∘L φ') s x ∧
+      HasFDerivWithinAt φs (Pi.compRightL 𝕜 F' Fin.succ ∘L φ') s x :=
+  hasFDerivAtFilter_finCons
+
+/-- A variant of `hasFDerivWithinAt_finCons` where the derivative variables are free on the RHS
+instead. -/
+theorem hasFDerivWithinAt_finCons'
+    {φ' : E →L[𝕜] F' 0} {φs' : E →L[𝕜] Π i, F' (Fin.succ i)} :
+    HasFDerivWithinAt (fun x => Fin.cons (φ x) (φs x)) (φ'.finCons φs') s x ↔
+      HasFDerivWithinAt φ φ' s x ∧ HasFDerivWithinAt φs φs' s x :=
+  hasFDerivAtFilter_finCons
+
+@[fun_prop]
+theorem HasFDerivWithinAt.finCons
+    {φ' : E →L[𝕜] F' 0} {φs' : E →L[𝕜] Π i, F' (Fin.succ i)}
+    (h : HasFDerivWithinAt φ φ' s x) (hs : HasFDerivWithinAt φs φs' s x) :
+    HasFDerivWithinAt (fun x => Fin.cons (φ x) (φs x)) (φ'.finCons φs') s x :=
+  hasFDerivWithinAt_finCons'.mpr ⟨h, hs⟩
+
+theorem differentiableWithinAt_finCons :
+    DifferentiableWithinAt 𝕜 (fun x => Fin.cons (φ x) (φs x)) s x ↔
+      DifferentiableWithinAt 𝕜 φ s x ∧ DifferentiableWithinAt 𝕜 φs s x := by
+  rw [differentiableWithinAt_pi, Fin.forall_fin_succ, differentiableWithinAt_pi]
+  simp only [Fin.cons_zero, Fin.cons_succ]
+
+/-- A variant of `differentiableWithinAt_finCons` where the derivative variables are free on the RHS
+instead. -/
+theorem differentiableWithinAt_finCons' :
+    DifferentiableWithinAt 𝕜 (fun x => Fin.cons (φ x) (φs x)) s x ↔
+      DifferentiableWithinAt 𝕜 φ s x ∧ DifferentiableWithinAt 𝕜 φs s x :=
+  differentiableWithinAt_finCons
+
+@[fun_prop]
+theorem DifferentiableWithinAt.finCons
+    (h : DifferentiableWithinAt 𝕜 φ s x) (hs : DifferentiableWithinAt 𝕜 φs s x) :
+    DifferentiableWithinAt 𝕜 (fun x => Fin.cons (φ x) (φs x)) s x :=
+  differentiableWithinAt_finCons'.mpr ⟨h, hs⟩
+
+theorem differentiableAt_finCons :
+    DifferentiableAt 𝕜 (fun x => Fin.cons (φ x) (φs x)) x ↔
+      DifferentiableAt 𝕜 φ x ∧ DifferentiableAt 𝕜 φs x := by
+  rw [differentiableAt_pi, Fin.forall_fin_succ, differentiableAt_pi]
+  simp only [Fin.cons_zero, Fin.cons_succ]
+
+/-- A variant of `differentiableAt_finCons` where the derivative variables are free on the RHS
+instead. -/
+theorem differentiableAt_finCons' :
+    DifferentiableAt 𝕜 (fun x => Fin.cons (φ x) (φs x)) x ↔
+      DifferentiableAt 𝕜 φ x ∧ DifferentiableAt 𝕜 φs x :=
+  differentiableAt_finCons
+
+@[fun_prop]
+theorem DifferentiableAt.finCons
+    (h : DifferentiableAt 𝕜 φ x) (hs : DifferentiableAt 𝕜 φs x) :
+    DifferentiableAt 𝕜 (fun x => Fin.cons (φ x) (φs x)) x :=
+  differentiableAt_finCons'.mpr ⟨h, hs⟩
+
+theorem differentiableOn_finCons :
+    DifferentiableOn 𝕜 (fun x => Fin.cons (φ x) (φs x)) s ↔
+      DifferentiableOn 𝕜 φ s ∧ DifferentiableOn 𝕜 φs s := by
+  rw [differentiableOn_pi, Fin.forall_fin_succ, differentiableOn_pi]
+  simp only [Fin.cons_zero, Fin.cons_succ]
+
+/-- A variant of `differentiableOn_finCons` where the derivative variables are free on the RHS
+instead. -/
+theorem differentiableOn_finCons' :
+    DifferentiableOn 𝕜 (fun x => Fin.cons (φ x) (φs x)) s ↔
+      DifferentiableOn 𝕜 φ s ∧ DifferentiableOn 𝕜 φs s :=
+  differentiableOn_finCons
+
+@[fun_prop]
+theorem DifferentiableOn.finCons
+    (h : DifferentiableOn 𝕜 φ s) (hs : DifferentiableOn 𝕜 φs s) :
+    DifferentiableOn 𝕜 (fun x => Fin.cons (φ x) (φs x)) s :=
+  differentiableOn_finCons'.mpr ⟨h, hs⟩
+
+theorem differentiable_finCons :
+    Differentiable 𝕜 (fun x => Fin.cons (φ x) (φs x)) ↔
+      Differentiable 𝕜 φ ∧ Differentiable 𝕜 φs := by
+  rw [differentiable_pi, Fin.forall_fin_succ, differentiable_pi]
+  simp only [Fin.cons_zero, Fin.cons_succ]
+
+/-- A variant of `differentiable_finCons` where the derivative variables are free on the RHS
+instead. -/
+theorem differentiable_finCons' :
+    Differentiable 𝕜 (fun x => Fin.cons (φ x) (φs x)) ↔
+      Differentiable 𝕜 φ ∧ Differentiable 𝕜 φs :=
+  differentiable_finCons
+
+@[fun_prop]
+theorem Differentiable.finCons
+    (h : Differentiable 𝕜 φ) (hs : Differentiable 𝕜 φs) :
+    Differentiable 𝕜 (fun x => Fin.cons (φ x) (φs x)) :=
+  differentiable_finCons'.mpr ⟨h, hs⟩
+
+-- TODO: write the `Fin.cons` versions of `fderivWithin_pi` and `fderiv_pi`
+
+end PiFin
+
 end CartesianProduct
 
 end
diff --git a/Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean b/Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean
index 4254d8378f1d65..78250c59bb934c 100644
--- a/Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean
+++ b/Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean
@@ -64,11 +64,8 @@ end Module
 
 namespace FormalMultilinearSeries
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/4481
-the `simpNF` linter incorrectly claims this lemma can't be applied by `simp`.
--/
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/4481
+the `simpNF` linter incorrectly claims this lemma can't be applied by `simp`. -/
 @[simp, nolint simpNF]
 theorem zero_apply (n : ℕ) : (0 : FormalMultilinearSeries 𝕜 E F) n = 0 := rfl
 
diff --git a/Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean b/Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean
index bd390d2a8ebcb5..d51fff9a736c09 100644
--- a/Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean
+++ b/Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean
@@ -290,3 +290,21 @@ theorem iteratedDeriv_eq_iterate : iteratedDeriv n f = deriv^[n] f := by
 derivative. -/
 theorem iteratedDeriv_succ' : iteratedDeriv (n + 1) f = iteratedDeriv n (deriv f) := by
   rw [iteratedDeriv_eq_iterate, iteratedDeriv_eq_iterate]; rfl
+
+lemma AnalyticAt.hasFPowerSeriesAt {𝕜 : Type*} [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
+    [CharZero 𝕜] {f : 𝕜 → 𝕜} {x : 𝕜} (h : AnalyticAt 𝕜 f x) :
+    HasFPowerSeriesAt f
+      (FormalMultilinearSeries.ofScalars 𝕜 (fun n ↦ iteratedDeriv n f x / n.factorial)) x := by
+  obtain ⟨p, hp⟩ := h
+  convert hp
+  obtain ⟨r, hpr⟩ := hp
+  ext n u
+  have h_fact_smul := hpr.factorial_smul 1
+  simp only [FormalMultilinearSeries.apply_eq_prod_smul_coeff, Finset.prod_const, Finset.card_univ,
+    Fintype.card_fin, smul_eq_mul, nsmul_eq_mul, one_pow, one_mul] at h_fact_smul
+  simp only [FormalMultilinearSeries.apply_eq_prod_smul_coeff,
+    FormalMultilinearSeries.coeff_ofScalars, smul_eq_mul, mul_eq_mul_left_iff]
+  left
+  rw [div_eq_iff, mul_comm, h_fact_smul, ← iteratedDeriv_eq_iteratedFDeriv]
+  norm_cast
+  exact Nat.factorial_ne_zero _
diff --git a/Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean b/Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean
index fb639723208c38..e797df9f8f6a72 100644
--- a/Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean
+++ b/Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean
@@ -152,6 +152,18 @@ lemma iteratedDeriv_sub (hf : ContDiff 𝕜 n f) (hg : ContDiff 𝕜 n g) :
     iteratedDerivWithin_sub (Set.mem_univ _) uniqueDiffOn_univ
       (contDiffOn_univ.mpr hf) (contDiffOn_univ.mpr hg)
 
+theorem iteratedDeriv_const_smul {n : ℕ} {f : 𝕜 → F} (h : ContDiffAt 𝕜 n f x) (c : 𝕜) :
+    iteratedDeriv n (c • f) x = c • iteratedDeriv n f x := by
+  simpa only [iteratedDerivWithin_univ] using
+    iteratedDerivWithin_const_smul (Set.mem_univ x) uniqueDiffOn_univ
+      c (contDiffWithinAt_univ.mpr h)
+
+theorem iteratedDeriv_const_mul {n : ℕ} {f : 𝕜 → 𝕜} (h : ContDiffAt 𝕜 n f x) (c : 𝕜) :
+    iteratedDeriv n (fun z => c * f z) x = c * iteratedDeriv n f x := by
+  simpa only [iteratedDerivWithin_univ] using
+    iteratedDerivWithin_const_mul (Set.mem_univ x) uniqueDiffOn_univ
+      c (contDiffWithinAt_univ.mpr h)
+
 theorem iteratedDeriv_comp_const_smul {n : ℕ} {f : 𝕜 → F} (h : ContDiff 𝕜 n f) (c : 𝕜) :
     iteratedDeriv n (fun x => f (c * x)) = fun x => c ^ n • iteratedDeriv n f (c * x) := by
   funext x
@@ -159,16 +171,10 @@ theorem iteratedDeriv_comp_const_smul {n : ℕ} {f : 𝕜 → F} (h : ContDiff 
     iteratedDerivWithin_comp_const_smul (Set.mem_univ x) uniqueDiffOn_univ (contDiffOn_univ.mpr h)
       c (Set.mapsTo_univ _ _)
 
-@[deprecated (since := "2024-12-20")]
-alias iteratedDeriv_const_smul := iteratedDeriv_comp_const_smul
-
 theorem iteratedDeriv_comp_const_mul {n : ℕ} {f : 𝕜 → 𝕜} (h : ContDiff 𝕜 n f) (c : 𝕜) :
     iteratedDeriv n (fun x => f (c * x)) = fun x => c ^ n * iteratedDeriv n f (c * x) := by
   simpa only [smul_eq_mul] using iteratedDeriv_comp_const_smul h c
 
-@[deprecated (since := "2024-12-20")]
-alias iteratedDeriv_const_mul := iteratedDeriv_comp_const_mul
-
 lemma iteratedDeriv_comp_neg (n : ℕ) (f : 𝕜 → F) (a : 𝕜) :
     iteratedDeriv n (fun x ↦ f (-x)) a = (-1 : 𝕜) ^ n • iteratedDeriv n f (-a) := by
   induction' n with n ih generalizing a
diff --git a/Mathlib/Analysis/Calculus/TangentCone.lean b/Mathlib/Analysis/Calculus/TangentCone.lean
index 68f5491fa0644a..e3061bb13fff2c 100644
--- a/Mathlib/Analysis/Calculus/TangentCone.lean
+++ b/Mathlib/Analysis/Calculus/TangentCone.lean
@@ -70,15 +70,17 @@ def UniqueDiffOn (s : Set E) : Prop :=
 
 end TangentCone
 
-variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
-variable {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
-variable {G : Type*} [NormedAddCommGroup G] [NormedSpace ℝ G]
-variable {𝕜} {x y : E} {s t : Set E}
+variable {𝕜}
+variable {E F G : Type*}
 
 section TangentCone
 
 -- This section is devoted to the properties of the tangent cone.
+
 open NormedField
+section TVS
+variable [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E]
+variable {x y : E} {s t : Set E}
 
 theorem mem_tangentConeAt_of_pow_smul {r : 𝕜} (hr₀ : r ≠ 0) (hr : ‖r‖ < 1)
     (hs : ∀ᶠ n : ℕ in atTop, x + r ^ n • y ∈ s) : y ∈ tangentConeAt 𝕜 s x := by
@@ -96,6 +98,14 @@ theorem tangentCone_mono (h : s ⊆ t) : tangentConeAt 𝕜 s x ⊆ tangentConeA
   rintro y ⟨c, d, ds, ctop, clim⟩
   exact ⟨c, d, mem_of_superset ds fun n hn => h hn, ctop, clim⟩
 
+end TVS
+
+section Normed
+variable [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+variable [NormedAddCommGroup F] [NormedSpace 𝕜 F]
+variable [NormedAddCommGroup G] [NormedSpace ℝ G]
+variable {x y : E} {s t : Set E}
+
 /-- Auxiliary lemma ensuring that, under the assumptions defining the tangent cone,
 the sequence `d` tends to 0 at infinity. -/
 theorem tangentConeAt.lim_zero {α : Type*} (l : Filter α) {c : α → 𝕜} {d : α → E}
@@ -337,6 +347,8 @@ theorem tangentCone_eq_univ {s : Set 𝕜} {x : 𝕜} (hx : (𝓝[s \ {x}] x).Ne
   · convert tendsto_const_nhds (α := ℕ) (x := y) with n
     simp [mul_assoc, inv_mul_cancel₀ (d_ne n)]
 
+end Normed
+
 end TangentCone
 
 section UniqueDiff
@@ -346,6 +358,9 @@ section UniqueDiff
 
 This section is devoted to properties of the predicates `UniqueDiffWithinAt` and `UniqueDiffOn`. -/
 
+section TVS
+variable [AddCommGroup E] [Module 𝕜 E] [TopologicalSpace E]
+variable {x y : E} {s t : Set E}
 
 theorem UniqueDiffOn.uniqueDiffWithinAt {s : Set E} {x} (hs : UniqueDiffOn 𝕜 s) (h : x ∈ s) :
     UniqueDiffWithinAt 𝕜 s x :=
@@ -364,6 +379,13 @@ theorem uniqueDiffOn_empty : UniqueDiffOn 𝕜 (∅ : Set E) :=
 theorem UniqueDiffWithinAt.congr_pt (h : UniqueDiffWithinAt 𝕜 s x) (hy : x = y) :
     UniqueDiffWithinAt 𝕜 s y := hy ▸ h
 
+end TVS
+
+section Normed
+variable [NormedAddCommGroup E] [NormedSpace 𝕜 E]
+variable [NormedAddCommGroup F] [NormedSpace 𝕜 F]
+variable {x y : E} {s t : Set E}
+
 theorem UniqueDiffWithinAt.mono_nhds (h : UniqueDiffWithinAt 𝕜 s x) (st : 𝓝[s] x ≤ 𝓝[t] x) :
     UniqueDiffWithinAt 𝕜 t x := by
   simp only [uniqueDiffWithinAt_iff] at *
@@ -457,6 +479,11 @@ theorem UniqueDiffOn.univ_pi (ι : Type*) [Finite ι] (E : ι → Type*)
     (h : ∀ i, UniqueDiffOn 𝕜 (s i)) : UniqueDiffOn 𝕜 (Set.pi univ s) :=
   UniqueDiffOn.pi _ _ _ _ fun i _ => h i
 
+end Normed
+
+section RealNormed
+variable [NormedAddCommGroup G] [NormedSpace ℝ G]
+
 /-- In a real vector space, a convex set with nonempty interior is a set of unique
 differentiability at every point of its closure. -/
 theorem uniqueDiffWithinAt_convex {s : Set G} (conv : Convex ℝ s) (hs : (interior s).Nonempty)
@@ -479,6 +506,10 @@ theorem uniqueDiffOn_convex {s : Set G} (conv : Convex ℝ s) (hs : (interior s)
     UniqueDiffOn ℝ s :=
   fun _ xs => uniqueDiffWithinAt_convex conv hs (subset_closure xs)
 
+end RealNormed
+
+section Real
+
 theorem uniqueDiffOn_Ici (a : ℝ) : UniqueDiffOn ℝ (Ici a) :=
   uniqueDiffOn_convex (convex_Ici a) <| by simp only [interior_Ici, nonempty_Ioi]
 
@@ -532,4 +563,6 @@ theorem uniqueDiffWithinAt_or_nhdsWithin_eq_bot (s : Set 𝕜) (x : 𝕜) :
     apply neBot_of_le (hf := h)
     exact nhdsWithin_mono _ diff_subset
 
+end Real
+
 end UniqueDiff
diff --git a/Mathlib/Analysis/Complex/Basic.lean b/Mathlib/Analysis/Complex/Basic.lean
index 73c4f7ae5cdfa4..3d4217f58338eb 100644
--- a/Mathlib/Analysis/Complex/Basic.lean
+++ b/Mathlib/Analysis/Complex/Basic.lean
@@ -5,7 +5,7 @@ Authors: Sébastien Gouëzel
 -/
 import Mathlib.Data.Complex.Module
 import Mathlib.Data.Complex.Order
-import Mathlib.Data.Complex.Exponential
+import Mathlib.Data.Complex.Trigonometric
 import Mathlib.Analysis.RCLike.Basic
 import Mathlib.Topology.Algebra.InfiniteSum.Field
 import Mathlib.Topology.Algebra.InfiniteSum.Module
@@ -177,11 +177,11 @@ theorem norm_int_of_nonneg {n : ℤ} (hn : 0 ≤ n) : ‖(n : ℂ)‖ = n := by
 lemma normSq_eq_norm_sq (z : ℂ) : normSq z = ‖z‖ ^ 2 := by
   rw [normSq_eq_abs, norm_eq_abs]
 
-@[continuity]
+@[continuity, fun_prop]
 theorem continuous_abs : Continuous abs :=
   continuous_norm
 
-@[continuity]
+@[continuity, fun_prop]
 theorem continuous_normSq : Continuous normSq := by
   simpa [← normSq_eq_abs] using continuous_abs.pow 2
 
diff --git a/Mathlib/Analysis/Complex/CauchyIntegral.lean b/Mathlib/Analysis/Complex/CauchyIntegral.lean
index 99b3e0d3c22dc6..41557430b9bf36 100644
--- a/Mathlib/Analysis/Complex/CauchyIntegral.lean
+++ b/Mathlib/Analysis/Complex/CauchyIntegral.lean
@@ -343,7 +343,7 @@ theorem circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable_of
     (hd : ∀ z ∈ (ball c R \ {c}) \ s, DifferentiableAt ℂ f z) (hy : Tendsto f (𝓝[{c}ᶜ] c) (𝓝 y)) :
     (∮ z in C(c, R), (z - c)⁻¹ • f z) = (2 * π * I : ℂ) • y := by
   rw [← sub_eq_zero, ← norm_le_zero_iff]
-  refine le_of_forall_le_of_dense fun ε ε0 => ?_
+  refine le_of_forall_gt_imp_ge_of_dense fun ε ε0 => ?_
   obtain ⟨δ, δ0, hδ⟩ : ∃ δ > (0 : ℝ), ∀ z ∈ closedBall c δ \ {c}, dist (f z) y < ε / (2 * π) :=
     ((nhdsWithin_hasBasis nhds_basis_closedBall _).tendsto_iff nhds_basis_ball).1 hy _
       (div_pos ε0 Real.two_pi_pos)
@@ -536,7 +536,7 @@ theorem hasFPowerSeriesOnBall_of_differentiable_off_countable {R : ℝ≥0} {c :
   hasSum := fun {w} hw => by
     have hw' : c + w ∈ ball c R := by
       simpa only [add_mem_ball_iff_norm, ← coe_nnnorm, mem_emetric_ball_zero_iff,
-        NNReal.coe_lt_coe, ENNReal.coe_lt_coe] using hw
+        NNReal.coe_lt_coe, enorm_lt_coe] using hw
     rw [← two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable
       hs hw' hc hd]
     exact (hasFPowerSeriesOn_cauchy_integral
diff --git a/Mathlib/Analysis/Complex/Liouville.lean b/Mathlib/Analysis/Complex/Liouville.lean
index aee36657dab9e8..75d46597fd3d1e 100644
--- a/Mathlib/Analysis/Complex/Liouville.lean
+++ b/Mathlib/Analysis/Complex/Liouville.lean
@@ -89,7 +89,7 @@ theorem liouville_theorem_aux {f : ℂ → F} (hf : Differentiable ℂ f) (hb :
     exact
       ⟨max C 1, lt_max_iff.2 (Or.inr zero_lt_one), fun z =>
         (hC (f z) (mem_range_self _)).trans (le_max_left _ _)⟩
-  refine norm_le_zero_iff.1 (le_of_forall_le_of_dense fun ε ε₀ => ?_)
+  refine norm_le_zero_iff.1 (le_of_forall_gt_imp_ge_of_dense fun ε ε₀ => ?_)
   calc
     ‖deriv f c‖ ≤ C / (C / ε) :=
       norm_deriv_le_of_forall_mem_sphere_norm_le (div_pos C₀ ε₀) hf.diffContOnCl fun z _ => hC z
diff --git a/Mathlib/Analysis/Complex/PhragmenLindelof.lean b/Mathlib/Analysis/Complex/PhragmenLindelof.lean
index ee37b9bc62b367..1d6cb0068727fe 100644
--- a/Mathlib/Analysis/Complex/PhragmenLindelof.lean
+++ b/Mathlib/Analysis/Complex/PhragmenLindelof.lean
@@ -116,7 +116,7 @@ theorem horizontal_strip (hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo a b))
   cases' hza with hza hza; · exact hle_a _ hza.symm
   cases' hzb with hzb hzb; · exact hle_b _ hzb
   wlog hC₀ : 0 < C generalizing C
-  · refine le_of_forall_le_of_dense fun C' hC' => this (fun w hw => ?_) (fun w hw => ?_) ?_
+  · refine le_of_forall_gt_imp_ge_of_dense fun C' hC' => this (fun w hw => ?_) (fun w hw => ?_) ?_
     · exact (hle_a _ hw).trans hC'.le
     · exact (hle_b _ hw).trans hC'.le
     · refine ((norm_nonneg (f (a * I))).trans (hle_a _ ?_)).trans_lt hC'
diff --git a/Mathlib/Analysis/Complex/TaylorSeries.lean b/Mathlib/Analysis/Complex/TaylorSeries.lean
index dbd3037467b3ac..4c7a2f85e0047c 100644
--- a/Mathlib/Analysis/Complex/TaylorSeries.lean
+++ b/Mathlib/Analysis/Complex/TaylorSeries.lean
@@ -24,9 +24,8 @@ see `Complex.hasSum_taylorSeries_of_entire`, `Complex.taylorSeries_eq_of_entire`
 `Complex.taylorSeries_eq_of_entire'`.
 -/
 
-#adaptation_note
-/--
-Due to https://github.com/leanprover/lean4/issues/5126, we've had to change all
+#adaptation_note /-- https://github.com/leanprover/lean4/issues/5126
+we've had to change all
 the `⦃⦄` stricit implicit variable statements in this file to normal `{}` implicit variables.
 
 Once this issue is fixed, we should change them back. For now it doesn't break anything downstream.
diff --git a/Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean b/Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean
index c92c0c0ce665bc..eba4e36e6505b5 100644
--- a/Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean
+++ b/Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean
@@ -157,11 +157,8 @@ theorem cmp_dist_eq_cmp_dist_coe_center (z w : ℍ) (r : ℝ) :
       ((mul_neg_of_pos_of_neg w.im_pos (sinh_neg_iff.2 hr₀)).trans_le dist_nonneg).cmp_eq_gt.symm]
   have hr₀' : 0 ≤ w.im * Real.sinh r := by positivity
   have hzw₀ : 0 < 2 * z.im * w.im := by positivity
-  #adaptation_note
-  /--
-  After the bug fix in https://github.com/leanprover/lean4/pull/6024,
-  we need to give Lean the hint `(y := w.im * Real.sinh r)`.
-  -/
+  #adaptation_note /-- https://github.com/leanprover/lean4/pull/6024
+  we need to give Lean the hint `(y := w.im * Real.sinh r)`. -/
   simp only [← cosh_strictMonoOn.cmp_map_eq dist_nonneg hr₀,
     ← (pow_left_strictMonoOn₀ two_ne_zero).cmp_map_eq dist_nonneg (y := w.im * Real.sinh r) hr₀',
     dist_coe_center_sq]
diff --git a/Mathlib/Analysis/Convex/Basic.lean b/Mathlib/Analysis/Convex/Basic.lean
index cbf1474ae125fc..1894c691b8c59e 100644
--- a/Mathlib/Analysis/Convex/Basic.lean
+++ b/Mathlib/Analysis/Convex/Basic.lean
@@ -624,8 +624,8 @@ theorem single_mem_stdSimplex (i : ι) : Pi.single i 1 ∈ stdSimplex 𝕜 ι :=
 theorem ite_eq_mem_stdSimplex (i : ι) : (if i = · then (1 : 𝕜) else 0) ∈ stdSimplex 𝕜 ι := by
   simpa only [@eq_comm _ i, ← Pi.single_apply] using single_mem_stdSimplex 𝕜 i
 
-#adaptation_note /-- as of `nightly-2024-03-11`, we need a type annotation on the segment in the
-following two lemmas. -/
+#adaptation_note /-- nightly-2024-03-11
+we need a type annotation on the segment in the following two lemmas. -/
 
 /-- The edges are contained in the simplex. -/
 lemma segment_single_subset_stdSimplex (i j : ι) :
diff --git a/Mathlib/Analysis/Convex/Birkhoff.lean b/Mathlib/Analysis/Convex/Birkhoff.lean
index d69061ff3199b0..3b8c6303e749c5 100644
--- a/Mathlib/Analysis/Convex/Birkhoff.lean
+++ b/Mathlib/Analysis/Convex/Birkhoff.lean
@@ -5,6 +5,7 @@ Authors: Bhavik Mehta
 -/
 
 import Mathlib.Analysis.Convex.Combination
+import Mathlib.Analysis.Convex.Extreme
 import Mathlib.Combinatorics.Hall.Basic
 import Mathlib.Data.Matrix.DoublyStochastic
 import Mathlib.Tactic.Linarith
@@ -18,10 +19,11 @@ import Mathlib.Tactic.Linarith
   combination of permutation matrices.
 * `doublyStochastic_eq_convexHull_perm`: The set of doubly stochastic matrices is the convex hull
   of the permutation matrices.
+* `extremePoints_doublyStochastic`: The set of extreme points of the doubly stochastic matrices is
+  the set of permutation matrices.
 
 ## TODO
 
-* Show that the extreme points of doubly stochastic matrices are the permutation matrices.
 * Show that for `x y : n → R`, `x` is majorized by `y` if and only if there is a doubly stochastic
   matrix `M` such that `M *ᵥ y = x`.
 
@@ -169,4 +171,28 @@ theorem doublyStochastic_eq_convexHull_permMatrix :
     obtain ⟨w, hw1, hw2, hw3⟩ := exists_eq_sum_perm_of_mem_doublyStochastic hM
     exact mem_convexHull_of_exists_fintype w (·.permMatrix R) hw1 hw2 (by simp) hw3
 
+/--
+The set of extreme points of the doubly stochastic matrices is the set of permutation matrices.
+-/
+theorem extremePoints_doublyStochastic :
+    Set.extremePoints R (doublyStochastic R n) = {σ.permMatrix R | σ : Equiv.Perm n} := by
+  refine subset_antisymm ?_ ?_
+  · rw [doublyStochastic_eq_convexHull_permMatrix]
+    exact extremePoints_convexHull_subset
+  rintro _ ⟨σ, rfl⟩
+  refine ⟨permMatrix_mem_doublyStochastic, fun x₁ hx₁ x₂ hx₂ hσ ↦ ?_⟩
+  suffices ∀ i j : n, x₁ i j = x₂ i j by
+    obtain rfl : x₁ = x₂ := by simpa [← Matrix.ext_iff]
+    simp_all
+  intro i j
+  have h₁ : σ.permMatrix R i j ∈ openSegment R (x₁ i j) (x₂ i j) :=
+    image_openSegment _ (entryLinearMap R R i j).toAffineMap x₁ x₂ ▸ ⟨_, hσ, rfl⟩
+  by_contra! h
+  have h₂ : openSegment R (x₁ i j) (x₂ i j) ⊆ Set.Ioo 0 1 := by
+    rw [openSegment_eq_Ioo' h]
+    apply Set.Ioo_subset_Ioo <;>
+    simp_all [nonneg_of_mem_doublyStochastic, le_one_of_mem_doublyStochastic]
+  specialize h₂ h₁
+  aesop
+
 end LinearOrderedField
diff --git a/Mathlib/Analysis/Convex/EGauge.lean b/Mathlib/Analysis/Convex/EGauge.lean
index 5083bf5351560e..6e85fb91802748 100644
--- a/Mathlib/Analysis/Convex/EGauge.lean
+++ b/Mathlib/Analysis/Convex/EGauge.lean
@@ -32,15 +32,15 @@ section SMul
 /-- The Minkowski functional for vector spaces over normed fields.
 Given a set `s` in a vector space over a normed field `𝕜`,
 `egauge s` is the functional which sends `x : E`
-to the infimum of `‖c‖₊` over `c` such that `x` belongs to `s` scaled by `c`.
+to the infimum of `‖c‖ₑ` over `c` such that `x` belongs to `s` scaled by `c`.
 
-The definition only requires `𝕜` to have a `NNNorm` instance
+The definition only requires `𝕜` to have a `ENorm` instance
 and `(· • ·) : 𝕜 → E → E` to be defined.
 This way the definition applies, e.g., to `𝕜 = ℝ≥0`.
 For `𝕜 = ℝ≥0`, the function is equal (up to conversion to `ℝ`)
 to the usual Minkowski functional defined in `gauge`. -/
-noncomputable def egauge (𝕜 : Type*) [NNNorm 𝕜] {E : Type*} [SMul 𝕜 E] (s : Set E) (x : E) : ℝ≥0∞ :=
-  ⨅ (c : 𝕜) (_ : x ∈ c • s), ‖c‖₊
+noncomputable def egauge (𝕜 : Type*) [ENorm 𝕜] {E : Type*} [SMul 𝕜 E] (s : Set E) (x : E) : ℝ≥0∞ :=
+  ⨅ (c : 𝕜) (_ : x ∈ c • s), ‖c‖ₑ
 
 variable (𝕜 : Type*) [NNNorm 𝕜] {E : Type*} [SMul 𝕜 E] {c : 𝕜} {s t : Set E} {x : E} {r : ℝ≥0∞}
 
@@ -52,13 +52,13 @@ lemma egauge_anti (h : s ⊆ t) (x : E) : egauge 𝕜 t x ≤ egauge 𝕜 s x :=
 
 variable {𝕜}
 
-lemma egauge_le_of_mem_smul (h : x ∈ c • s) : egauge 𝕜 s x ≤ ‖c‖₊ := iInf₂_le c h
+lemma egauge_le_of_mem_smul (h : x ∈ c • s) : egauge 𝕜 s x ≤ ‖c‖ₑ := iInf₂_le c h
 
-lemma le_egauge_iff : r ≤ egauge 𝕜 s x ↔ ∀ c : 𝕜, x ∈ c • s → r ≤ ‖c‖₊ := le_iInf₂_iff
+lemma le_egauge_iff : r ≤ egauge 𝕜 s x ↔ ∀ c : 𝕜, x ∈ c • s → r ≤ ‖c‖ₑ := le_iInf₂_iff
 
 lemma egauge_eq_top : egauge 𝕜 s x = ∞ ↔ ∀ c : 𝕜, x ∉ c • s := by simp [egauge]
 
-lemma egauge_lt_iff : egauge 𝕜 s x < r ↔ ∃ c : 𝕜, x ∈ c • s ∧ ‖c‖₊ < r := by
+lemma egauge_lt_iff : egauge 𝕜 s x < r ↔ ∃ c : 𝕜, x ∈ c • s ∧ ‖c‖ₑ < r := by
   simp [egauge, iInf_lt_iff]
 
 lemma egauge_union (s t : Set E) (x : E) : egauge 𝕜 (s ∪ t) x = egauge 𝕜 s x ⊓ egauge 𝕜 t x := by
@@ -67,7 +67,7 @@ lemma egauge_union (s t : Set E) (x : E) : egauge 𝕜 (s ∪ t) x = egauge 𝕜
 
 lemma le_egauge_inter (s t : Set E) (x : E) :
     egauge 𝕜 s x ⊔ egauge 𝕜 t x ≤ egauge 𝕜 (s ∩ t) x :=
-  max_le_iff.2 ⟨egauge_anti _ inter_subset_left _, egauge_anti _ inter_subset_right _⟩
+  max_le (egauge_anti _ inter_subset_left _) (egauge_anti _ inter_subset_right _)
 
 end SMul
 
@@ -87,25 +87,24 @@ section Module
 variable {𝕜 : Type*} [NormedDivisionRing 𝕜] {E : Type*} [AddCommGroup E] [Module 𝕜 E]
     {c : 𝕜} {s : Set E} {x : E}
 
-/-- If `c • x ∈ s` and `c ≠ 0`, then `egauge 𝕜 s x` is at most `((‖c‖₊⁻¹ : ℝ≥0) : ℝ≥0∞).
+/-- If `c • x ∈ s` and `c ≠ 0`, then `egauge 𝕜 s x` is at most `(‖c‖₊⁻¹ : ℝ≥0)`.
 
 See also `egauge_le_of_smul_mem`. -/
-lemma egauge_le_of_smul_mem_of_ne (h : c • x ∈ s) (hc : c ≠ 0) :
-    egauge 𝕜 s x ≤ ↑(‖c‖₊⁻¹ : ℝ≥0) := by
+lemma egauge_le_of_smul_mem_of_ne (h : c • x ∈ s) (hc : c ≠ 0) : egauge 𝕜 s x ≤ (‖c‖₊⁻¹ : ℝ≥0) := by
   rw [← nnnorm_inv]
   exact egauge_le_of_mem_smul <| (mem_inv_smul_set_iff₀ hc _ _).2 h
 
-/-- If `c • x ∈ s`, then `egauge 𝕜 s x` is at most `(‖c‖₊ : ℝ≥0∞)⁻¹.
+/-- If `c • x ∈ s`, then `egauge 𝕜 s x` is at most `‖c‖ₑ⁻¹`.
 
 See also `egauge_le_of_smul_mem_of_ne`. -/
-lemma egauge_le_of_smul_mem (h : c • x ∈ s) : egauge 𝕜 s x ≤ (‖c‖₊ : ℝ≥0∞)⁻¹ := by
+lemma egauge_le_of_smul_mem (h : c • x ∈ s) : egauge 𝕜 s x ≤ ‖c‖ₑ⁻¹ := by
   rcases eq_or_ne c 0 with rfl | hc
   · simp
   · exact (egauge_le_of_smul_mem_of_ne h hc).trans ENNReal.coe_inv_le
 
 lemma mem_of_egauge_lt_one (hs : Balanced 𝕜 s) (hx : egauge 𝕜 s x < 1) : x ∈ s :=
   let ⟨c, hxc, hc⟩ := egauge_lt_iff.1 hx
-  hs c (mod_cast hc.le) hxc
+  hs c (by simpa [enorm] using hc.le) hxc
 
 lemma egauge_eq_zero_iff : egauge 𝕜 s x = 0 ↔ ∃ᶠ c : 𝕜 in 𝓝 0, x ∈ c • s := by
   refine (iInf₂_eq_bot _).trans ?_
@@ -128,25 +127,25 @@ lemma egauge_le_one (h : x ∈ s) : egauge 𝕜 s x ≤ 1 := by
 variable {𝕜}
 
 lemma le_egauge_smul_left (c : 𝕜) (s : Set E) (x : E) :
-    egauge 𝕜 s x / ‖c‖₊ ≤ egauge 𝕜 (c • s) x := by
+    egauge 𝕜 s x / ‖c‖ₑ ≤ egauge 𝕜 (c • s) x := by
   simp_rw [le_egauge_iff, smul_smul]
   rintro a ⟨x, hx, rfl⟩
   apply ENNReal.div_le_of_le_mul
-  rw [← ENNReal.coe_mul, ← nnnorm_mul]
+  rw [← enorm_mul]
   exact egauge_le_of_mem_smul <| smul_mem_smul_set hx
 
 lemma egauge_smul_left (hc : c ≠ 0) (s : Set E) (x : E) :
-    egauge 𝕜 (c • s) x = egauge 𝕜 s x / ‖c‖₊ := by
+    egauge 𝕜 (c • s) x = egauge 𝕜 s x / ‖c‖ₑ := by
   refine le_antisymm ?_ (le_egauge_smul_left _ _ _)
   rw [ENNReal.le_div_iff_mul_le (by simp [*]) (by simp)]
   calc
-    egauge 𝕜 (c • s) x * ‖c‖₊ = egauge 𝕜 (c • s) x / ‖c⁻¹‖₊ := by
-      rw [nnnorm_inv, ENNReal.coe_inv (by simpa), div_eq_mul_inv, inv_inv]
+    egauge 𝕜 (c • s) x * ‖c‖ₑ = egauge 𝕜 (c • s) x / ‖c⁻¹‖ₑ := by
+      rw [enorm_inv (by simpa), div_eq_mul_inv, inv_inv]
     _ ≤ egauge 𝕜 (c⁻¹ • c • s) x := le_egauge_smul_left _ _ _
     _ = egauge 𝕜 s x := by rw [inv_smul_smul₀ hc]
 
 lemma le_egauge_smul_right (c : 𝕜) (s : Set E) (x : E) :
-    ‖c‖₊ * egauge 𝕜 s x ≤ egauge 𝕜 s (c • x) := by
+    ‖c‖ₑ * egauge 𝕜 s x ≤ egauge 𝕜 s (c • x) := by
   rw [le_egauge_iff]
   rintro a ⟨y, hy, hxy⟩
   rcases eq_or_ne c 0 with rfl | hc
@@ -157,51 +156,69 @@ lemma le_egauge_smul_right (c : 𝕜) (s : Set E) (x : E) :
     simp only [mul_smul, hxy, inv_smul_smul₀ hc]
 
 lemma egauge_smul_right (h : c = 0 → s.Nonempty) (x : E) :
-    egauge 𝕜 s (c • x) = ‖c‖₊ * egauge 𝕜 s x := by
+    egauge 𝕜 s (c • x) = ‖c‖ₑ * egauge 𝕜 s x := by
   refine le_antisymm ?_ (le_egauge_smul_right c s x)
   rcases eq_or_ne c 0 with rfl | hc
   · simp [egauge_zero_right _ (h rfl)]
-  · rw [mul_comm, ← ENNReal.div_le_iff_le_mul (.inl <| by simpa) (.inl ENNReal.coe_ne_top),
-      ENNReal.div_eq_inv_mul, ← ENNReal.coe_inv (by simpa), ← nnnorm_inv]
+  · rw [mul_comm, ← ENNReal.div_le_iff_le_mul (.inl <| by simpa) (.inl enorm_ne_top),
+      ENNReal.div_eq_inv_mul, ← enorm_inv (by simpa)]
     refine (le_egauge_smul_right _ _ _).trans_eq ?_
     rw [inv_smul_smul₀ hc]
 
 end Module
 
+section VectorSpace
+
+variable {𝕜 : Type*} [NormedField 𝕜] {E : Type*} [AddCommGroup E] [Module 𝕜 E]
+
+theorem egauge_add_add_le {U V : Set E} (hU : Balanced 𝕜 U) (hV : Balanced 𝕜 V) (a b : E) :
+    egauge 𝕜 (U + V) (a + b) ≤ max (egauge 𝕜 U a) (egauge 𝕜 V b) := by
+  refine le_of_forall_lt' fun c hc ↦ ?_
+  simp only [max_lt_iff, egauge_lt_iff] at hc ⊢
+  rcases hc with ⟨⟨x, hx, hxc⟩, ⟨y, hy, hyc⟩⟩
+  wlog hxy : ‖x‖ ≤ ‖y‖ generalizing a b x y U V
+  · simpa only [add_comm] using this hV hU b a y hy hyc x hx hxc (le_of_not_le hxy)
+  refine ⟨y, ?_, hyc⟩
+  rw [smul_add]
+  exact add_mem_add (hU.smul_mono hxy hx) hy
+
+end VectorSpace
+
 section SeminormedAddCommGroup
 
 variable (𝕜 : Type*) [NormedField 𝕜] {E : Type*} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E]
 
-lemma div_le_egauge_closedBall (r : ℝ≥0) (x : E) : ‖x‖₊ / r ≤ egauge 𝕜 (closedBall 0 r) x := by
+lemma div_le_egauge_closedBall (r : ℝ≥0) (x : E) : ‖x‖ₑ / r ≤ egauge 𝕜 (closedBall 0 r) x := by
   rw [le_egauge_iff]
   rintro c ⟨y, hy, rfl⟩
   rw [mem_closedBall_zero_iff, ← coe_nnnorm, NNReal.coe_le_coe] at hy
-  simp only [nnnorm_smul, ENNReal.coe_mul]
+  rw [enorm_smul]
   apply ENNReal.div_le_of_le_mul
   gcongr
+  rwa [enorm_le_coe]
 
-lemma le_egauge_closedBall_one (x : E) : ‖x‖₊ ≤ egauge 𝕜 (closedBall 0 1) x := by
+lemma le_egauge_closedBall_one (x : E) : ‖x‖ₑ ≤ egauge 𝕜 (closedBall 0 1) x := by
   simpa using div_le_egauge_closedBall 𝕜 1 x
 
-lemma div_le_egauge_ball (r : ℝ≥0) (x : E) : ‖x‖₊ / r ≤ egauge 𝕜 (ball 0 r) x :=
+lemma div_le_egauge_ball (r : ℝ≥0) (x : E) : ‖x‖ₑ / r ≤ egauge 𝕜 (ball 0 r) x :=
   (div_le_egauge_closedBall 𝕜 r x).trans <| egauge_anti _ ball_subset_closedBall _
 
-lemma le_egauge_ball_one (x : E) : ‖x‖₊ ≤ egauge 𝕜 (ball 0 1) x := by
+lemma le_egauge_ball_one (x : E) : ‖x‖ₑ ≤ egauge 𝕜 (ball 0 1) x := by
   simpa using div_le_egauge_ball 𝕜 1 x
 
 variable {𝕜}
 variable {c : 𝕜} {x : E} {r : ℝ≥0}
 
 lemma egauge_ball_le_of_one_lt_norm (hc : 1 < ‖c‖) (h₀ : r ≠ 0 ∨ ‖x‖ ≠ 0) :
-    egauge 𝕜 (ball 0 r) x ≤ ‖c‖₊ * ‖x‖₊ / r := by
+    egauge 𝕜 (ball 0 r) x ≤ ‖c‖ₑ * ‖x‖ₑ / r := by
   letI : NontriviallyNormedField 𝕜 := ⟨c, hc⟩
   rcases (zero_le r).eq_or_lt with rfl | hr
   · rw [ENNReal.coe_zero, ENNReal.div_zero (mul_ne_zero _ _)]
     · apply le_top
     · simpa using one_pos.trans hc
-    · simpa [← NNReal.coe_eq_zero] using h₀
+    · simpa [enorm, ← NNReal.coe_eq_zero] using h₀
   · rcases eq_or_ne ‖x‖ 0 with hx | hx
-    · have hx' : ‖x‖₊ = 0 := by rwa [← coe_nnnorm, NNReal.coe_eq_zero] at hx
+    · have hx' : ‖x‖ₑ = 0 := by simpa [enorm, ← coe_nnnorm, NNReal.coe_eq_zero] using hx
       simp [egauge_eq_zero_iff, hx']
       refine (frequently_iff_neBot.2 (inferInstance : NeBot (𝓝[≠] (0 : 𝕜)))).mono fun c hc ↦ ?_
       simp [mem_smul_set_iff_inv_smul_mem₀ hc, norm_smul, hx, hr]
@@ -210,10 +227,10 @@ lemma egauge_ball_le_of_one_lt_norm (hc : 1 < ‖c‖) (h₀ : r ≠ 0 ∨ ‖x
         egauge 𝕜 (ball 0 r) x ≤ ↑(‖a‖₊⁻¹) :=
           egauge_le_of_smul_mem_of_ne (mem_ball_zero_iff.2 har) ha₀
         _ ≤ ↑(‖c‖₊ * ‖x‖₊ / r) := by rwa [ENNReal.coe_le_coe, div_eq_inv_mul, ← mul_assoc]
-        _ ≤ ‖c‖₊ * ‖x‖₊ / r := ENNReal.coe_div_le.trans <| by rw [ENNReal.coe_mul]
+        _ ≤ ‖c‖ₑ * ‖x‖ₑ / r := ENNReal.coe_div_le.trans <| by simp [ENNReal.coe_mul, enorm]
 
 lemma egauge_ball_one_le_of_one_lt_norm (hc : 1 < ‖c‖) (x : E) :
-    egauge 𝕜 (ball 0 1) x ≤ ‖c‖₊ * ‖x‖₊ := by
+    egauge 𝕜 (ball 0 1) x ≤ ‖c‖ₑ * ‖x‖ₑ := by
   simpa using egauge_ball_le_of_one_lt_norm hc (.inl one_ne_zero)
 
 end SeminormedAddCommGroup
diff --git a/Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean b/Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
index 616027a97b9a72..f6b41f8197a662 100644
--- a/Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
+++ b/Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
@@ -300,7 +300,7 @@ theorem lintegral_pow_le_pow_lintegral_fderiv_aux [Fintype ι]
     {p : ℝ} (hp : Real.IsConjExponent #ι p)
     {u : (ι → ℝ) → F} (hu : ContDiff ℝ 1 u)
     (h2u : HasCompactSupport u) :
-    ∫⁻ x, (‖u x‖₊ : ℝ≥0∞) ^ p ≤ (∫⁻ x, ‖fderiv ℝ u x‖₊) ^ p := by
+    ∫⁻ x, ‖u x‖ₑ ^ p ≤ (∫⁻ x, ‖fderiv ℝ u x‖ₑ) ^ p := by
   classical
   /- For a function `f` in one variable and `t ∈ ℝ` we have
   `|f(t)| = `|∫_{-∞}^t Df(s)∂s| ≤ ∫_ℝ |Df(s)| ∂s` where we use the fundamental theorem of calculus.
@@ -311,43 +311,41 @@ theorem lintegral_pow_le_pow_lintegral_fderiv_aux [Fintype ι]
   have : (1 : ℝ) ≤ ↑#ι - 1 := by
     have hι : (2 : ℝ) ≤ #ι := by exact_mod_cast hp.one_lt
     linarith
-  calc ∫⁻ x, (‖u x‖₊ : ℝ≥0∞) ^ p
-      = ∫⁻ x, ((‖u x‖₊ : ℝ≥0∞) ^ (1 / (#ι - 1 : ℝ))) ^ (#ι : ℝ) := by
+  calc ∫⁻ x, ‖u x‖ₑ ^ p
+      = ∫⁻ x, (‖u x‖ₑ ^ (1 / (#ι - 1 : ℝ))) ^ (#ι : ℝ) := by
         -- a little algebraic manipulation of the exponent
         congr! 2 with x
         rw [← ENNReal.rpow_mul, hp.conj_eq]
         field_simp
-    _ = ∫⁻ x, ∏ _i : ι, (‖u x‖₊ : ℝ≥0∞) ^ (1 / (#ι - 1 : ℝ)) := by
+    _ = ∫⁻ x, ∏ _i : ι, ‖u x‖ₑ ^ (1 / (#ι - 1 : ℝ)) := by
         -- express the left-hand integrand as a product of identical factors
         congr! 2 with x
         simp_rw [prod_const, card_univ]
         norm_cast
-    _ ≤ ∫⁻ x, ∏ i, (∫⁻ xᵢ, ‖fderiv ℝ u (update x i xᵢ)‖₊) ^ ((1 : ℝ) / (#ι - 1 : ℝ)) := ?_
-    _ ≤ (∫⁻ x, ‖fderiv ℝ u x‖₊) ^ p := by
+    _ ≤ ∫⁻ x, ∏ i, (∫⁻ xᵢ, ‖fderiv ℝ u (update x i xᵢ)‖ₑ) ^ ((1 : ℝ) / (#ι - 1 : ℝ)) := ?_
+    _ ≤ (∫⁻ x, ‖fderiv ℝ u x‖ₑ) ^ p := by
         -- apply the grid-lines lemma
         apply lintegral_prod_lintegral_pow_le _ hp
         have : Continuous (fderiv ℝ u) := hu.continuous_fderiv le_rfl
         fun_prop
   -- we estimate |u x| using the fundamental theorem of calculus.
   gcongr with x i
-  calc (‖u x‖₊ : ℝ≥0∞)
-    _ ≤ ∫⁻ xᵢ in Iic (x i), ‖deriv (u ∘ update x i) xᵢ‖₊ := by
-        apply le_trans (by simp) (HasCompactSupport.ennnorm_le_lintegral_Ici_deriv _ _ _)
+  calc ‖u x‖ₑ
+    _ ≤ ∫⁻ xᵢ in Iic (x i), ‖deriv (u ∘ update x i) xᵢ‖ₑ := by
+        apply le_trans (by simp) (HasCompactSupport.enorm_le_lintegral_Ici_deriv _ _ _)
         · exact hu.comp (by convert contDiff_update 1 x i)
         · exact h2u.comp_isClosedEmbedding (isClosedEmbedding_update x i)
-    _ ≤ ∫⁻ xᵢ, (‖fderiv ℝ u (update x i xᵢ)‖₊ : ℝ≥0∞) := ?_
+    _ ≤ ∫⁻ xᵢ, ‖fderiv ℝ u (update x i xᵢ)‖ₑ := ?_
   gcongr
   · exact Measure.restrict_le_self
   intro y
   dsimp
-  gcongr
   -- bound the derivative which appears
-  calc ‖deriv (u ∘ update x i) y‖₊ = ‖fderiv ℝ u (update x i y) (deriv (update x i) y)‖₊ := by
+  calc ‖deriv (u ∘ update x i) y‖ₑ = ‖fderiv ℝ u (update x i y) (deriv (update x i) y)‖ₑ := by
         rw [fderiv_comp_deriv _ (hu.differentiable le_rfl).differentiableAt
           (hasDerivAt_update x i y).differentiableAt]
-    _ ≤ ‖fderiv ℝ u (update x i y)‖₊ * ‖deriv (update x i) y‖₊ :=
-        ContinuousLinearMap.le_opNNNorm ..
-    _ ≤ ‖fderiv ℝ u (update x i y)‖₊ := by simp [deriv_update, Pi.nnnorm_single]
+    _ ≤ ‖fderiv ℝ u (update x i y)‖ₑ * ‖deriv (update x i) y‖ₑ := ContinuousLinearMap.le_opENorm _ _
+    _ ≤ ‖fderiv ℝ u (update x i y)‖ₑ := by simp [deriv_update, Pi.enorm_single]
 
 variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E]
   [FiniteDimensional ℝ E] (μ : Measure E) [IsAddHaarMeasure μ]
@@ -374,8 +372,8 @@ Lebesgue integral of the Fréchet derivative of `u`. -/
 theorem lintegral_pow_le_pow_lintegral_fderiv {u : E → F}
     (hu : ContDiff ℝ 1 u) (h2u : HasCompactSupport u)
     {p : ℝ} (hp : Real.IsConjExponent (finrank ℝ E) p) :
-    ∫⁻ x, (‖u x‖₊ : ℝ≥0∞) ^ p ∂μ ≤
-    lintegralPowLePowLIntegralFDerivConst μ p * (∫⁻ x, ‖fderiv ℝ u x‖₊ ∂μ) ^ p := by
+    ∫⁻ x, ‖u x‖ₑ ^ p ∂μ ≤
+      lintegralPowLePowLIntegralFDerivConst μ p * (∫⁻ x, ‖fderiv ℝ u x‖ₑ ∂μ) ^ p := by
   /- We reduce to the case where `E` is `ℝⁿ`, for which we have already proved the result using
   an explicit basis in `MeasureTheory.lintegral_pow_le_pow_lintegral_fderiv_aux`.
   This proof is not too hard, but takes quite some steps, reasoning about the equivalence
@@ -403,30 +401,28 @@ theorem lintegral_pow_le_pow_lintegral_fderiv {u : E → F}
   have hv : ContDiff ℝ 1 v := hu.comp e.symm.contDiff
   have h2v : HasCompactSupport v := h2u.comp_homeomorph e.symm.toHomeomorph
   have :=
-  calc ∫⁻ x, (‖u x‖₊ : ℝ≥0∞) ^ p ∂(volume : Measure (ι → ℝ)).map e.symm
-      = ∫⁻ y, (‖v y‖₊ : ℝ≥0∞) ^ p := by
+  calc ∫⁻ x, ‖u x‖ₑ ^ p ∂(volume : Measure (ι → ℝ)).map e.symm
+      = ∫⁻ y, ‖v y‖ₑ ^ p := by
         refine lintegral_map ?_ e.symm.continuous.measurable
         borelize F
         exact hu.continuous.measurable.nnnorm.coe_nnreal_ennreal.pow_const _
-    _ ≤ (∫⁻ y, ‖fderiv ℝ v y‖₊) ^ p :=
-        lintegral_pow_le_pow_lintegral_fderiv_aux hp hv h2v
-    _ = (∫⁻ y, ‖(fderiv ℝ u (e.symm y)).comp (fderiv ℝ e.symm y)‖₊) ^ p := by
+    _ ≤ (∫⁻ y, ‖fderiv ℝ v y‖ₑ) ^ p := lintegral_pow_le_pow_lintegral_fderiv_aux hp hv h2v
+    _ = (∫⁻ y, ‖(fderiv ℝ u (e.symm y)).comp (fderiv ℝ e.symm y)‖ₑ) ^ p := by
         congr! with y
         apply fderiv_comp _ (hu.differentiable le_rfl _)
         exact e.symm.differentiableAt
-    _ ≤ (∫⁻ y, ‖fderiv ℝ u (e.symm y)‖₊ * ‖(e.symm : (ι → ℝ) →L[ℝ] E)‖₊) ^ p := by
+    _ ≤ (∫⁻ y, ‖fderiv ℝ u (e.symm y)‖ₑ * ‖(e.symm : (ι → ℝ) →L[ℝ] E)‖ₑ) ^ p := by
         gcongr with y
-        norm_cast
         rw [e.symm.fderiv]
-        apply ContinuousLinearMap.opNNNorm_comp_le
-    _ = (‖(e.symm : (ι → ℝ) →L[ℝ] E)‖₊ * ∫⁻ y, ‖fderiv ℝ u (e.symm y)‖₊) ^ p := by
+        apply ContinuousLinearMap.opENorm_comp_le
+    _ = (‖(e.symm : (ι → ℝ) →L[ℝ] E)‖ₑ * ∫⁻ y, ‖fderiv ℝ u (e.symm y)‖ₑ) ^ p := by
         rw [lintegral_mul_const, mul_comm]
         refine (Continuous.nnnorm ?_).measurable.coe_nnreal_ennreal
         exact (hu.continuous_fderiv le_rfl).comp e.symm.continuous
-    _ = (‖(e.symm : (ι → ℝ) →L[ℝ] E)‖₊ ^ p : ℝ≥0) * (∫⁻ y, ‖fderiv ℝ u (e.symm y)‖₊) ^ p := by
-        rw [ENNReal.mul_rpow_of_nonneg _ _ h0p, ← ENNReal.coe_rpow_of_nonneg _ h0p]
+    _ = (‖(e.symm : (ι → ℝ) →L[ℝ] E)‖₊ ^ p : ℝ≥0) * (∫⁻ y, ‖fderiv ℝ u (e.symm y)‖ₑ) ^ p := by
+        rw [ENNReal.mul_rpow_of_nonneg _ _ h0p, enorm_eq_nnnorm, ← ENNReal.coe_rpow_of_nonneg _ h0p]
     _ = (‖(e.symm : (ι → ℝ) →L[ℝ] E)‖₊ ^ p : ℝ≥0)
-        * (∫⁻ x, ‖fderiv ℝ u x‖₊ ∂(volume : Measure (ι → ℝ)).map e.symm) ^ p := by
+        * (∫⁻ x, ‖fderiv ℝ u x‖ₑ ∂(volume : Measure (ι → ℝ)).map e.symm) ^ p := by
         congr
         rw [lintegral_map _ e.symm.continuous.measurable]
         have : Continuous (fderiv ℝ u) := hu.continuous_fderiv le_rfl
@@ -449,7 +445,7 @@ theorem eLpNorm_le_eLpNorm_fderiv_one  {u : E → F} (hu : ContDiff ℝ 1 u) (h2
     {p : ℝ≥0} (hp : NNReal.IsConjExponent (finrank ℝ E) p) :
     eLpNorm u p μ ≤ eLpNormLESNormFDerivOneConst μ p * eLpNorm (fderiv ℝ u) 1 μ := by
   have h0p : 0 < (p : ℝ) := hp.coe.symm.pos
-  rw [eLpNorm_one_eq_lintegral_nnnorm,
+  rw [eLpNorm_one_eq_lintegral_enorm,
     ← ENNReal.rpow_le_rpow_iff h0p, ENNReal.mul_rpow_of_nonneg _ _ h0p.le,
     ← ENNReal.coe_rpow_of_nonneg _ h0p.le, eLpNormLESNormFDerivOneConst, ← NNReal.rpow_mul,
     eLpNorm_nnreal_pow_eq_lintegral hp.symm.pos.ne',
@@ -535,16 +531,16 @@ theorem eLpNorm_le_eLpNorm_fderiv_of_eq_inner  {u : E → F'}
     have : (p : ℝ) * (n - 1) - (n - p) = n * (p - 1) := by ring
     field_simp [this]; ring
   have h4γ : (γ : ℝ) ≠ 0 := (zero_lt_one.trans h1γ).ne'
-  by_cases h3u : ∫⁻ x, ‖u x‖₊ ^ (p' : ℝ) ∂μ = 0
+  by_cases h3u : ∫⁻ x, ‖u x‖ₑ ^ (p' : ℝ) ∂μ = 0
   · rw [eLpNorm_nnreal_eq_lintegral h0p', h3u, ENNReal.zero_rpow_of_pos] <;> positivity
-  have h4u : ∫⁻ x, ‖u x‖₊ ^ (p' : ℝ) ∂μ ≠ ∞ := by
-    refine lintegral_rpow_nnnorm_lt_top_of_eLpNorm'_lt_top
+  have h4u : ∫⁻ x, ‖u x‖ₑ ^ (p' : ℝ) ∂μ ≠ ∞ := by
+    refine lintegral_rpow_enorm_lt_top_of_eLpNorm'_lt_top
       ((NNReal.coe_pos.trans pos_iff_ne_zero).mpr h0p') ?_ |>.ne
     rw [← eLpNorm_nnreal_eq_eLpNorm' h0p']
     exact hu.continuous.memℒp_of_hasCompactSupport (μ := μ) h2u |>.eLpNorm_lt_top
-  have h5u : (∫⁻ x, ‖u x‖₊ ^ (p' : ℝ) ∂μ) ^ (1 / q) ≠ 0 :=
+  have h5u : (∫⁻ x, ‖u x‖ₑ ^ (p' : ℝ) ∂μ) ^ (1 / q) ≠ 0 :=
     ENNReal.rpow_pos (pos_iff_ne_zero.mpr h3u) h4u |>.ne'
-  have h6u : (∫⁻ x, ‖u x‖₊ ^ (p' : ℝ) ∂μ) ^ (1 / q) ≠ ∞ :=
+  have h6u : (∫⁻ x, ‖u x‖ₑ ^ (p' : ℝ) ∂μ) ^ (1 / q) ≠ ∞ :=
     ENNReal.rpow_ne_top_of_nonneg (div_nonneg zero_le_one hq.symm.nonneg) h4u
   have h7u := hu.continuous -- for fun_prop
   have h8u := (hu.fderiv_right (m := 0) le_rfl).continuous -- for fun_prop
@@ -553,20 +549,21 @@ theorem eLpNorm_le_eLpNorm_fderiv_of_eq_inner  {u : E → F'}
   have h2v : HasCompactSupport v := h2u.norm.rpow_const h4γ
   set C := eLpNormLESNormFDerivOneConst μ n'
   have :=
-  calc (∫⁻ x, ‖u x‖₊ ^ (p' : ℝ) ∂μ) ^ (1 / (n' : ℝ)) = eLpNorm v n' μ := by
+  calc (∫⁻ x, ‖u x‖ₑ ^ (p' : ℝ) ∂μ) ^ (1 / (n' : ℝ)) = eLpNorm v n' μ := by
         rw [← h2γ, eLpNorm_nnreal_eq_lintegral hn.symm.pos.ne']
-        simp (discharger := positivity) [v, Real.nnnorm_rpow_of_nonneg, ENNReal.rpow_mul,
+        simp (discharger := positivity) [v, Real.enorm_rpow_of_nonneg, ENNReal.rpow_mul,
           ← ENNReal.coe_rpow_of_nonneg]
     _ ≤ C * eLpNorm (fderiv ℝ v) 1 μ := eLpNorm_le_eLpNorm_fderiv_one μ hv h2v hn
-    _ = C * ∫⁻ x, ‖fderiv ℝ v x‖₊ ∂μ := by rw [eLpNorm_one_eq_lintegral_nnnorm]
-    _ ≤ C * γ * ∫⁻ x, ‖u x‖₊ ^ ((γ : ℝ) - 1) * ‖fderiv ℝ u x‖₊ ∂μ := by
+    _ = C * ∫⁻ x, ‖fderiv ℝ v x‖ₑ ∂μ := by rw [eLpNorm_one_eq_lintegral_enorm]
+    _ ≤ C * γ * ∫⁻ x, ‖u x‖ₑ ^ ((γ : ℝ) - 1) * ‖fderiv ℝ u x‖ₑ ∂μ := by
       rw [mul_assoc, ← lintegral_const_mul γ]
       gcongr
-      simp_rw [← mul_assoc, ← ENNReal.coe_rpow_of_nonneg _ (sub_nonneg.mpr h1γ.le)]
-      exact ENNReal.coe_le_coe.mpr <| nnnorm_fderiv_norm_rpow_le (hu.differentiable le_rfl) h1γ
+      simp_rw [← mul_assoc]
+      exact enorm_fderiv_norm_rpow_le (hu.differentiable le_rfl) h1γ
+      dsimp [enorm]
       fun_prop
-    _ ≤ C * γ * ((∫⁻ x, ‖u x‖₊ ^ (p' : ℝ) ∂μ) ^ (1 / q) *
-        (∫⁻ x, ‖fderiv ℝ u x‖₊ ^ (p : ℝ) ∂μ) ^ (1 / (p : ℝ))) := by
+    _ ≤ C * γ * ((∫⁻ x, ‖u x‖ₑ ^ (p' : ℝ) ∂μ) ^ (1 / q) *
+        (∫⁻ x, ‖fderiv ℝ u x‖ₑ ^ (p : ℝ) ∂μ) ^ (1 / (p : ℝ))) := by
         gcongr
         convert ENNReal.lintegral_mul_le_Lp_mul_Lq μ
           (.symm <| .conjExponent <| show 1 < (p : ℝ) from hp) ?_ ?_ using 5
@@ -574,12 +571,12 @@ theorem eLpNorm_le_eLpNorm_fderiv_of_eq_inner  {u : E → F'}
         · borelize F'
           fun_prop
         · fun_prop
-    _ = C * γ * (∫⁻ x, ‖fderiv ℝ u x‖₊ ^ (p : ℝ) ∂μ) ^ (1 / (p : ℝ)) *
-      (∫⁻ x, ‖u x‖₊ ^ (p' : ℝ) ∂μ) ^ (1 / q) := by ring
+    _ = C * γ * (∫⁻ x, ‖fderiv ℝ u x‖ₑ ^ (p : ℝ) ∂μ) ^ (1 / (p : ℝ)) *
+      (∫⁻ x, ‖u x‖ₑ ^ (p' : ℝ) ∂μ) ^ (1 / q) := by ring
   calc
     eLpNorm u p' μ
-      = (∫⁻ x, ‖u x‖₊ ^ (p' : ℝ) ∂μ) ^ (1 / (p' : ℝ)) := eLpNorm_nnreal_eq_lintegral h0p'
-    _ ≤ C * γ * (∫⁻ x, ‖fderiv ℝ u x‖₊ ^ (p : ℝ) ∂μ) ^ (1 / (p : ℝ)) := by
+      = (∫⁻ x, ‖u x‖ₑ ^ (p' : ℝ) ∂μ) ^ (1 / (p' : ℝ)) := eLpNorm_nnreal_eq_lintegral h0p'
+    _ ≤ C * γ * (∫⁻ x, ‖fderiv ℝ u x‖ₑ ^ (p : ℝ) ∂μ) ^ (1 / (p : ℝ)) := by
       rwa [← h2q, ENNReal.rpow_sub _ _ h3u h4u, ENNReal.div_le_iff h5u h6u]
     _ = eLpNormLESNormFDerivOfEqInnerConst μ p *  eLpNorm (fderiv ℝ u) (↑p) μ := by
       suffices (C : ℝ) * γ = eLpNormLESNormFDerivOfEqInnerConst μ p by
@@ -683,8 +680,8 @@ theorem eLpNorm_le_eLpNorm_fderiv_of_le [FiniteDimensional ℝ F]
   have hp' : p'⁻¹ = p⁻¹ - (finrank ℝ E : ℝ)⁻¹ := by
     rw [inv_inv, NNReal.coe_sub]
     · simp
-    · #adaptation_note
-      /-- This should just be `gcongr`, but this is not working as of nightly-2024-11-20.
+    · #adaptation_note /-- nightly-2024-11-20
+      This should just be `gcongr`, but this is not working as of nightly-2024-11-20.
       Possibly related to #19262 (since this proof fails at `with_reducible_and_instances`). -/
       exact inv_anti₀ (by positivity) h2p.le
   have : (q : ℝ≥0∞) ≤ p' := by
@@ -694,8 +691,8 @@ theorem eLpNorm_le_eLpNorm_fderiv_of_le [FiniteDimensional ℝ F]
     · dsimp
       have : 0 < p⁻¹ - (finrank ℝ E : ℝ≥0)⁻¹ := by
         simp only [tsub_pos_iff_lt]
-        #adaptation_note
-        /-- This should just be `gcongr`, but this is not working as of nightly-2024-11-20.
+        #adaptation_note /-- nightly-2024-11-20
+        This should just be `gcongr`, but this is not working as of nightly-2024-11-20.
         Possibly related to #19262 (since this proof fails at `with_reducible_and_instances`). -/
         exact inv_strictAnti₀ (by positivity) h2p
       positivity
diff --git a/Mathlib/Analysis/LocallyConvex/Polar.lean b/Mathlib/Analysis/LocallyConvex/Polar.lean
index af5dc28df843c0..d49aedb9e44496 100644
--- a/Mathlib/Analysis/LocallyConvex/Polar.lean
+++ b/Mathlib/Analysis/LocallyConvex/Polar.lean
@@ -139,7 +139,7 @@ variable (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜)
 theorem polar_univ (h : SeparatingRight B) : B.polar Set.univ = {(0 : F)} := by
   rw [Set.eq_singleton_iff_unique_mem]
   refine ⟨by simp only [zero_mem_polar], fun y hy => h _ fun x => ?_⟩
-  refine norm_le_zero_iff.mp (le_of_forall_le_of_dense fun ε hε => ?_)
+  refine norm_le_zero_iff.mp (le_of_forall_gt_imp_ge_of_dense fun ε hε => ?_)
   rcases NormedField.exists_norm_lt 𝕜 hε with ⟨c, hc, hcε⟩
   calc
     ‖B x y‖ = ‖c‖ * ‖B (c⁻¹ • x) y‖ := by
diff --git a/Mathlib/Analysis/Normed/Algebra/Spectrum.lean b/Mathlib/Analysis/Normed/Algebra/Spectrum.lean
index 4f081227992a49..649fa425cabf54 100644
--- a/Mathlib/Analysis/Normed/Algebra/Spectrum.lean
+++ b/Mathlib/Analysis/Normed/Algebra/Spectrum.lean
@@ -184,10 +184,8 @@ theorem exists_nnnorm_eq_spectralRadius_of_nonempty [ProperSpace 𝕜] {a : A} (
 
 theorem spectralRadius_lt_of_forall_lt_of_nonempty [ProperSpace 𝕜] {a : A} (ha : (σ a).Nonempty)
     {r : ℝ≥0} (hr : ∀ k ∈ σ a, ‖k‖₊ < r) : spectralRadius 𝕜 a < r :=
-  sSup_image.symm.trans_lt <|
-    ((spectrum.isCompact a).sSup_lt_iff_of_continuous ha
-          (ENNReal.continuous_coe.comp continuous_nnnorm).continuousOn (r : ℝ≥0∞)).mpr
-      (by dsimp only [(· ∘ ·)]; exact mod_cast hr)
+  sSup_image.symm.trans_lt <| ((spectrum.isCompact a).sSup_lt_iff_of_continuous ha
+    continuous_enorm.continuousOn (r : ℝ≥0∞)).mpr (by simpa using hr)
 
 open ENNReal Polynomial
 
diff --git a/Mathlib/Analysis/Normed/Field/Ultra.lean b/Mathlib/Analysis/Normed/Field/Ultra.lean
index 8ef392b9b1afd4..79e9e92787437d 100644
--- a/Mathlib/Analysis/Normed/Field/Ultra.lean
+++ b/Mathlib/Analysis/Normed/Field/Ultra.lean
@@ -76,7 +76,7 @@ lemma isUltrametricDist_of_forall_pow_norm_le_nsmul_pow_max_one_norm
   -- that avoids explicitly mentioning `m`-th roots.
   -- First note it suffices to show that `‖x + 1‖ ≤ a` for all `a : ℝ` with `max ‖x‖ 1 < a`.
   rw [max_comm]
-  refine le_of_forall_le_of_dense fun a ha ↦ ?_
+  refine le_of_forall_gt_imp_ge_of_dense fun a ha ↦ ?_
   have ha' : 1 < a := (max_lt_iff.mp ha).left
   -- `max 1 ‖x‖ < a`, so there must be some `m : ℕ` such that `m + 1 < (a / max 1 ‖x‖) ^ m`
   -- by the virtue of exponential growth being faster than linear growth
diff --git a/Mathlib/Analysis/Normed/Group/Basic.lean b/Mathlib/Analysis/Normed/Group/Basic.lean
index 0b033f8d126185..d773e1f72c4f5c 100644
--- a/Mathlib/Analysis/Normed/Group/Basic.lean
+++ b/Mathlib/Analysis/Normed/Group/Basic.lean
@@ -686,8 +686,7 @@ alias nndist_eq_nnnorm := nndist_eq_nnnorm_sub
 theorem nndist_one_right (a : E) : nndist a 1 = ‖a‖₊ := by simp [nndist_eq_nnnorm_div]
 
 @[to_additive (attr := simp)]
-theorem edist_one_right (a : E) : edist a 1 = ‖a‖₊ := by
-  rw [edist_nndist, nndist_one_right]
+lemma edist_one_right (a : E) : edist a 1 = ‖a‖ₑ := by simp [edist_nndist, nndist_one_right, enorm]
 
 @[to_additive (attr := simp) nnnorm_zero]
 theorem nnnorm_one' : ‖(1 : E)‖₊ = 0 := NNReal.eq norm_one'
@@ -800,44 +799,51 @@ lemma nnnorm_div_eq_nnnorm_right {x : E} (y : E) (h : ‖x‖₊ = 0) : ‖x / y
 lemma nnnorm_div_eq_nnnorm_left (x : E) {y : E} (h : ‖y‖₊ = 0) : ‖x / y‖₊ = ‖x‖₊ :=
   NNReal.eq <| norm_div_eq_norm_left _ <| congr_arg NNReal.toReal h
 
-@[to_additive ofReal_norm_eq_coe_nnnorm]
-theorem ofReal_norm_eq_coe_nnnorm' (a : E) : ENNReal.ofReal ‖a‖ = ‖a‖₊ :=
-  ENNReal.ofReal_eq_coe_nnreal _
-
 /-- The non negative norm seen as an `ENNReal` and then as a `Real` is equal to the norm. -/
 @[to_additive toReal_coe_nnnorm "The non negative norm seen as an `ENNReal` and
 then as a `Real` is equal to the norm."]
 theorem toReal_coe_nnnorm' (a : E) : (‖a‖₊ : ℝ≥0∞).toReal = ‖a‖ := rfl
 
-@[to_additive]
-theorem edist_eq_coe_nnnorm_div (a b : E) : edist a b = ‖a / b‖₊ := by
-  rw [edist_dist, dist_eq_norm_div, ofReal_norm_eq_coe_nnnorm']
-
-@[to_additive edist_eq_coe_nnnorm]
-theorem edist_eq_coe_nnnorm' (x : E) : edist x 1 = (‖x‖₊ : ℝ≥0∞) := by
-  rw [edist_eq_coe_nnnorm_div, div_one]
-
 open scoped symmDiff in
 @[to_additive]
 theorem edist_mulIndicator (s t : Set α) (f : α → E) (x : α) :
     edist (s.mulIndicator f x) (t.mulIndicator f x) = ‖(s ∆ t).mulIndicator f x‖₊ := by
   rw [edist_nndist, nndist_mulIndicator]
 
-@[to_additive]
-theorem mem_emetric_ball_one_iff {r : ℝ≥0∞} : a ∈ EMetric.ball (1 : E) r ↔ ↑‖a‖₊ < r := by
-  rw [EMetric.mem_ball, edist_eq_coe_nnnorm']
-
 end NNNorm
 
 section ENorm
 
 @[to_additive (attr := simp) enorm_zero] lemma enorm_one' : ‖(1 : E)‖ₑ = 0 := by simp [enorm]
 
+@[to_additive ofReal_norm_eq_enorm]
+lemma ofReal_norm_eq_enorm' (a : E) : .ofReal ‖a‖ = ‖a‖ₑ := ENNReal.ofReal_eq_coe_nnreal _
+
+@[deprecated (since := "2025-01-17")] alias ofReal_norm_eq_coe_nnnorm := ofReal_norm_eq_enorm
+@[deprecated (since := "2025-01-17")] alias ofReal_norm_eq_coe_nnnorm' := ofReal_norm_eq_enorm'
+
 instance : ENorm ℝ≥0∞ where
   enorm x := x
 
 @[simp] lemma enorm_eq_self (x : ℝ≥0∞) : ‖x‖ₑ = x := rfl
 
+@[to_additive]
+theorem edist_eq_enorm_div (a b : E) : edist a b = ‖a / b‖ₑ := by
+  rw [edist_dist, dist_eq_norm_div, ofReal_norm_eq_enorm']
+
+@[deprecated (since := "2025-01-17")] alias edist_eq_coe_nnnorm_sub := edist_eq_enorm_sub
+@[deprecated (since := "2025-01-17")] alias edist_eq_coe_nnnorm_div := edist_eq_enorm_div
+
+@[to_additive]
+theorem edist_one_eq_enorm (x : E) : edist x 1 = ‖x‖ₑ := by rw [edist_eq_enorm_div, div_one]
+
+@[deprecated (since := "2025-01-17")] alias edist_eq_coe_nnnorm := edist_zero_eq_enorm
+@[deprecated (since := "2025-01-17")] alias edist_eq_coe_nnnorm' := edist_one_eq_enorm
+
+@[to_additive]
+theorem mem_emetric_ball_one_iff {r : ℝ≥0∞} : a ∈ EMetric.ball 1 r ↔ ‖a‖ₑ < r := by
+  rw [EMetric.mem_ball, edist_one_eq_enorm]
+
 end ENorm
 
 @[to_additive]
@@ -1319,11 +1325,15 @@ lemma enorm_of_nonneg (hr : 0 ≤ r) : ‖r‖ₑ = .ofReal r := by
 @[simp] lemma nnnorm_abs (r : ℝ) : ‖|r|‖₊ = ‖r‖₊ := by simp [nnnorm]
 @[simp] lemma enorm_abs (r : ℝ) : ‖|r|‖ₑ = ‖r‖ₑ := by simp [enorm]
 
-theorem ennnorm_eq_ofReal (hr : 0 ≤ r) : (‖r‖₊ : ℝ≥0∞) = ENNReal.ofReal r := by
-  rw [← ofReal_norm_eq_coe_nnnorm, norm_of_nonneg hr]
+theorem enorm_eq_ofReal (hr : 0 ≤ r) : ‖r‖ₑ = .ofReal r := by
+  rw [← ofReal_norm_eq_enorm, norm_of_nonneg hr]
+
+@[deprecated (since := "2025-01-17")] alias ennnorm_eq_ofReal := enorm_eq_ofReal
 
-theorem ennnorm_eq_ofReal_abs (r : ℝ) : (‖r‖₊ : ℝ≥0∞) = ENNReal.ofReal |r| := by
-  rw [← Real.nnnorm_abs r, Real.ennnorm_eq_ofReal (abs_nonneg _)]
+theorem enorm_eq_ofReal_abs (r : ℝ) : ‖r‖ₑ = ENNReal.ofReal |r| := by
+  rw [← enorm_eq_ofReal (abs_nonneg _), enorm_abs]
+
+@[deprecated (since := "2025-01-17")] alias ennnorm_eq_ofReal_abs := enorm_eq_ofReal_abs
 
 theorem toNNReal_eq_nnnorm_of_nonneg (hr : 0 ≤ r) : r.toNNReal = ‖r‖₊ := by
   rw [Real.toNNReal_of_nonneg hr]
@@ -1331,12 +1341,10 @@ theorem toNNReal_eq_nnnorm_of_nonneg (hr : 0 ≤ r) : r.toNNReal = ‖r‖₊ :=
   rw [coe_mk, coe_nnnorm r, Real.norm_eq_abs r, abs_of_nonneg hr]
   -- Porting note: this is due to the change from `Subtype.val` to `NNReal.toReal` for the coercion
 
-theorem ofReal_le_ennnorm (r : ℝ) : ENNReal.ofReal r ≤ ‖r‖₊ := by
-  obtain hr | hr := le_total 0 r
-  · exact (Real.ennnorm_eq_ofReal hr).ge
-  · rw [ENNReal.ofReal_eq_zero.2 hr]
-    exact bot_le
--- Porting note: should this be renamed to `Real.ofReal_le_nnnorm`?
+theorem ofReal_le_enorm (r : ℝ) : ENNReal.ofReal r ≤ ‖r‖ₑ := by
+  rw [enorm_eq_ofReal_abs]; gcongr; exact le_abs_self _
+
+@[deprecated (since := "2025-01-17")] alias ofReal_le_ennnorm := ofReal_le_enorm
 
 end Real
 
diff --git a/Mathlib/Analysis/Normed/Lp/PiLp.lean b/Mathlib/Analysis/Normed/Lp/PiLp.lean
index 0c311a281bb35c..dfd02e7678382b 100644
--- a/Mathlib/Analysis/Normed/Lp/PiLp.lean
+++ b/Mathlib/Analysis/Normed/Lp/PiLp.lean
@@ -95,13 +95,10 @@ variable [Semiring 𝕜] [∀ i, SeminormedAddCommGroup (β i)]
 variable [∀ i, Module 𝕜 (β i)] (c : 𝕜)
 variable (x y : PiLp p β) (i : ι)
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/4481
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/4481
 the `simpNF` linter incorrectly claims this lemma can't be applied by `simp`.
 
-(It appears to also be unused in Mathlib.)
--/
+(It appears to also be unused in Mathlib.) -/
 @[simp, nolint simpNF]
 theorem zero_apply : (0 : PiLp p β) i = 0 :=
   rfl
diff --git a/Mathlib/Analysis/Normed/Module/Dual.lean b/Mathlib/Analysis/Normed/Module/Dual.lean
index 8a947e649a0b2e..279e098d12e2c2 100644
--- a/Mathlib/Analysis/Normed/Module/Dual.lean
+++ b/Mathlib/Analysis/Normed/Module/Dual.lean
@@ -240,7 +240,7 @@ theorem polar_ball {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [Normed
   apply le_antisymm
   · intro x hx
     rw [mem_closedBall_zero_iff]
-    apply le_of_forall_le_of_dense
+    apply le_of_forall_gt_imp_ge_of_dense
     intro a ha
     rw [← mem_closedBall_zero_iff, ← (mul_div_cancel_left₀ a (Ne.symm (ne_of_lt hr)))]
     rw [← RCLike.norm_of_nonneg (K := 𝕜) (le_trans zero_le_one
diff --git a/Mathlib/Analysis/Normed/Operator/BoundedLinearMaps.lean b/Mathlib/Analysis/Normed/Operator/BoundedLinearMaps.lean
index 9ad44b68f8091d..72c464afef4f11 100644
--- a/Mathlib/Analysis/Normed/Operator/BoundedLinearMaps.lean
+++ b/Mathlib/Analysis/Normed/Operator/BoundedLinearMaps.lean
@@ -193,12 +193,9 @@ theorem isBoundedLinearMap_prod_multilinear {E : ι → Type*} [∀ i, Seminorme
   (ContinuousMultilinearMap.prodL 𝕜 E F G).toContinuousLinearEquiv
     |>.toContinuousLinearMap.isBoundedLinearMap
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/6024
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/6024
 we needed to add the named arguments `(ι := ι) (G := F)`
-to `ContinuousMultilinearMap.compContinuousLinearMapL`.
--/
+to `ContinuousMultilinearMap.compContinuousLinearMapL`. -/
 /-- Given a fixed continuous linear map `g`, associating to a continuous multilinear map `f` the
 continuous multilinear map `f (g m₁, ..., g mₙ)` is a bounded linear operation. -/
 theorem isBoundedLinearMap_continuousMultilinearMap_comp_linear (g : G →L[𝕜] E) :
diff --git a/Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean b/Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean
index 59589fe9246181..4ce0cf8cdda550 100644
--- a/Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean
+++ b/Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean
@@ -145,8 +145,7 @@ variable {E : Type u} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
 open ContinuousLinearEquiv Submodule
 
 theorem coord_norm' {x : E} (h : x ≠ 0) : ‖(‖x‖ : 𝕜) • coord 𝕜 x h‖ = 1 := by
-  #adaptation_note
-  /--
+  #adaptation_note /-- https://github.com/leanprover/lean4/pull/4119
   `set_option maxSynthPendingDepth 2` required after https://github.com/leanprover/lean4/pull/4119
   Alternatively, we can add:
   ```
diff --git a/Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean b/Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean
index 8193941231cfa9..471a7a72a27a8a 100644
--- a/Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean
+++ b/Mathlib/Analysis/NormedSpace/OperatorNorm/Bilinear.lean
@@ -146,11 +146,8 @@ def flip (f : E →SL[σ₁₃] F →SL[σ₂₃] G) : F →SL[σ₂₃] E →SL
     ‖f‖ fun y x => (f.le_opNorm₂ x y).trans_eq <| by simp only [mul_right_comm]
 
 private theorem le_norm_flip (f : E →SL[σ₁₃] F →SL[σ₂₃] G) : ‖f‖ ≤ ‖flip f‖ :=
-  #adaptation_note
-  /--
-  After https://github.com/leanprover/lean4/pull/4119 we either need
-  to specify the `f.flip` argument, or use `set_option maxSynthPendingDepth 2 in`.
-  -/
+  #adaptation_note /-- https://github.com/leanprover/lean4/pull/4119
+  we either need to specify the `f.flip` argument, or use `set_option maxSynthPendingDepth 2 in`. -/
   f.opNorm_le_bound₂ (norm_nonneg f.flip) fun x y => by
     rw [mul_right_comm]
     exact (flip f).le_opNorm₂ y x
diff --git a/Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean b/Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean
index 45424cd36fd7de..aec0cfaa5eab13 100644
--- a/Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean
@@ -73,3 +73,47 @@ theorem AnalyticOn.cpow (fs : AnalyticOn ℂ f s) (gs : AnalyticOn ℂ g s)
 theorem AnalyticOnNhd.cpow (fs : AnalyticOnNhd ℂ f s) (gs : AnalyticOnNhd ℂ g s)
     (m : ∀ z ∈ s, f z ∈ slitPlane) : AnalyticOnNhd ℂ (fun z ↦ f z ^ g z) s :=
   fun z n ↦ (fs z n).cpow (gs z n) (m z n)
+
+section ReOfReal
+
+variable {f : ℂ → ℂ} {s : Set ℝ} {x : ℝ}
+
+lemma AnalyticAt.re_ofReal (hf : AnalyticAt ℂ f x) :
+    AnalyticAt ℝ (fun x : ℝ ↦ (f x).re) x :=
+  (Complex.reCLM.analyticAt _).comp (hf.restrictScalars.comp (Complex.ofRealCLM.analyticAt _))
+
+lemma AnalyticAt.im_ofReal (hf : AnalyticAt ℂ f x) :
+    AnalyticAt ℝ (fun x : ℝ ↦ (f x).im) x :=
+  (Complex.imCLM.analyticAt _).comp (hf.restrictScalars.comp (Complex.ofRealCLM.analyticAt _))
+
+lemma AnalyticWithinAt.re_ofReal (hf : AnalyticWithinAt ℂ f (ofReal '' s) x) :
+    AnalyticWithinAt ℝ (fun x : ℝ ↦ (f x).re) s x :=
+  ((Complex.reCLM.analyticWithinAt _ _).comp hf.restrictScalars (mapsTo_image f _)).comp
+    (Complex.ofRealCLM.analyticWithinAt _ _) (mapsTo_image ofReal s)
+
+lemma AnalyticWithinAt.im_ofReal (hf : AnalyticWithinAt ℂ f (ofReal '' s) x) :
+    AnalyticWithinAt ℝ (fun x : ℝ ↦ (f x).im) s x :=
+  ((Complex.imCLM.analyticWithinAt _ _).comp hf.restrictScalars (mapsTo_image f _)).comp
+    (Complex.ofRealCLM.analyticWithinAt _ _) (mapsTo_image ofReal s)
+
+lemma AnalyticOn.re_ofReal (hf : AnalyticOn ℂ f (ofReal '' s)) :
+    AnalyticOn ℝ (fun x : ℝ ↦ (f x).re) s :=
+  ((Complex.reCLM.analyticOn _).comp hf.restrictScalars (mapsTo_image f _)).comp
+    (Complex.ofRealCLM.analyticOn _) (mapsTo_image ofReal s)
+
+lemma AnalyticOn.im_ofReal (hf : AnalyticOn ℂ f (ofReal '' s)) :
+    AnalyticOn ℝ (fun x : ℝ ↦ (f x).im) s :=
+  ((Complex.imCLM.analyticOn _).comp hf.restrictScalars (mapsTo_image f _)).comp
+    (Complex.ofRealCLM.analyticOn _) (mapsTo_image ofReal s)
+
+lemma AnalyticOnNhd.re_ofReal (hf : AnalyticOnNhd ℂ f (ofReal '' s)) :
+    AnalyticOnNhd ℝ (fun x : ℝ ↦ (f x).re) s :=
+  ((Complex.reCLM.analyticOnNhd _).comp hf.restrictScalars (mapsTo_image f _)).comp
+    (Complex.ofRealCLM.analyticOnNhd _) (mapsTo_image ofReal s)
+
+lemma AnalyticOnNhd.im_ofReal (hf : AnalyticOnNhd ℂ f (ofReal '' s)) :
+    AnalyticOnNhd ℝ (fun x : ℝ ↦ (f x).im) s :=
+  ((Complex.imCLM.analyticOnNhd _).comp hf.restrictScalars (mapsTo_image f _)).comp
+    (Complex.ofRealCLM.analyticOnNhd _) (mapsTo_image ofReal s)
+
+end ReOfReal
diff --git a/Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean b/Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean
index 3ef729e02e9f10..4e8ea696d800af 100644
--- a/Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean
+++ b/Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean
@@ -81,6 +81,14 @@ theorem integrableOn_Ioi_rpow_iff {s t : ℝ} (ht : 0 < t) :
     exact Real.rpow_le_rpow_of_exponent_le x_one h
   exact not_IntegrableOn_Ioi_inv this
 
+theorem integrableAtFilter_rpow_atTop_iff {s : ℝ} :
+    IntegrableAtFilter (fun x : ℝ ↦ x ^ s) atTop ↔ s < -1 := by
+  refine ⟨fun ⟨t, ht, hint⟩ ↦ ?_, fun h ↦
+    ⟨Set.Ioi 1, Ioi_mem_atTop 1, (integrableOn_Ioi_rpow_iff zero_lt_one).mpr h⟩⟩
+  obtain ⟨a, ha⟩ := mem_atTop_sets.mp ht
+  refine (integrableOn_Ioi_rpow_iff (zero_lt_one.trans_le (le_max_right a 1))).mp ?_
+  exact hint.mono_set <| fun x hx ↦ ha _ <| (le_max_left a 1).trans hx.le
+
 /-- The real power function with any exponent is not integrable on `(0, +∞)`. -/
 theorem not_integrableOn_Ioi_rpow (s : ℝ) : ¬ IntegrableOn (fun x ↦ x ^ s) (Ioi (0 : ℝ)) := by
   intro h
@@ -127,6 +135,24 @@ theorem integrableOn_Ioi_cpow_iff {s : ℂ} {t : ℝ} (ht : 0 < t) :
     simp [Complex.abs_cpow_eq_rpow_re_of_pos this]
   rwa [integrableOn_Ioi_rpow_iff ht] at B
 
+theorem integrableOn_Ioi_deriv_ofReal_cpow {s : ℂ} {t : ℝ} (ht : 0 < t) (hs : s.re < 0) :
+    IntegrableOn (deriv fun x : ℝ ↦ (x : ℂ) ^ s) (Set.Ioi t) := by
+  have h : IntegrableOn (fun x : ℝ ↦ s * x ^ (s - 1)) (Set.Ioi t) := by
+    refine (integrableOn_Ioi_cpow_of_lt ?_ ht).const_mul _
+    rwa [Complex.sub_re, Complex.one_re, sub_lt_iff_lt_add, neg_add_cancel]
+  refine h.congr_fun (fun x hx ↦ ?_) measurableSet_Ioi
+  rw [Complex.deriv_ofReal_cpow_const (ht.trans hx).ne' (fun h ↦ (Complex.zero_re ▸ h ▸ hs).false)]
+
+theorem integrableOn_Ioi_deriv_norm_ofReal_cpow {s : ℂ} {t : ℝ} (ht : 0 < t) (hs : s.re ≤ 0):
+    IntegrableOn (deriv fun x : ℝ ↦ ‖(x : ℂ) ^ s‖) (Set.Ioi t) := by
+  rw [integrableOn_congr_fun (fun x hx ↦ by
+    rw [deriv_norm_ofReal_cpow _ (ht.trans hx)]) measurableSet_Ioi]
+  obtain hs | hs := eq_or_lt_of_le hs
+  · simp_rw [hs, zero_mul]
+    exact integrableOn_zero
+  · replace hs : s.re - 1 < - 1 := by rwa [sub_lt_iff_lt_add, neg_add_cancel]
+    exact (integrableOn_Ioi_rpow_of_lt hs ht).const_mul s.re
+
 /-- The complex power function with any exponent is not integrable on `(0, +∞)`. -/
 theorem not_integrableOn_Ioi_cpow (s : ℂ) :
     ¬ IntegrableOn (fun x : ℝ ↦ (x : ℂ) ^ s) (Ioi (0 : ℝ)) := by
diff --git a/Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean b/Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean
index 595b8aa9b7f348..a773a828ece6ab 100644
--- a/Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean
+++ b/Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean
@@ -132,9 +132,9 @@ theorem integrable_one_add_norm {r : ℝ} (hnr : (finrank ℝ E : ℝ) < r) :
   constructor
   · apply Measurable.aestronglyMeasurable (by fun_prop)
   -- Lower Lebesgue integral
-  have : (∫⁻ a : E, ‖(1 + ‖a‖) ^ (-r)‖₊ ∂μ) = ∫⁻ a : E, ENNReal.ofReal ((1 + ‖a‖) ^ (-r)) ∂μ :=
-    lintegral_nnnorm_eq_of_nonneg fun _ => rpow_nonneg (by positivity) _
-  rw [hasFiniteIntegral_iff_nnnorm, this]
+  have : (∫⁻ a : E, ‖(1 + ‖a‖) ^ (-r)‖ₑ ∂μ) = ∫⁻ a : E, ENNReal.ofReal ((1 + ‖a‖) ^ (-r)) ∂μ :=
+    lintegral_enorm_of_nonneg fun _ => rpow_nonneg (by positivity) _
+  rw [hasFiniteIntegral_iff_enorm, this]
   exact finite_integral_one_add_norm hnr
 
 theorem integrable_rpow_neg_one_add_norm_sq {r : ℝ} (hnr : (finrank ℝ E : ℝ) < r) :
diff --git a/Mathlib/Analysis/SpecialFunctions/Log/Summable.lean b/Mathlib/Analysis/SpecialFunctions/Log/Summable.lean
index 3c1969101ea969..0bb757f1e28ba1 100644
--- a/Mathlib/Analysis/SpecialFunctions/Log/Summable.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Log/Summable.lean
@@ -22,7 +22,7 @@ products.
 
 open Filter Function Complex Real
 
-open scoped Interval Topology BigOperators Nat Classical Complex
+open scoped Interval Topology BigOperators Nat Complex
 
 variable {ι : Type*}
 
diff --git a/Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean b/Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
index 0407d95d6e2ccc..30cb5251b049e8 100644
--- a/Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
@@ -42,6 +42,13 @@ theorem cpow_eq_zero_iff (x y : ℂ) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by
   simp only [cpow_def]
   split_ifs <;> simp [*, exp_ne_zero]
 
+theorem cpow_ne_zero_iff {x y : ℂ} :
+    x ^ y ≠ 0 ↔ x ≠ 0 ∨ y = 0 := by
+  rw [ne_eq, cpow_eq_zero_iff, not_and_or, ne_eq, not_not]
+
+theorem cpow_ne_zero_iff_of_exponent_ne_zero {x y : ℂ} (hy : y ≠ 0) :
+    x ^ y ≠ 0 ↔ x ≠ 0 := by simp [hy]
+
 @[simp]
 theorem zero_cpow {x : ℂ} (h : x ≠ 0) : (0 : ℂ) ^ x = 0 := by simp [cpow_def, *]
 
diff --git a/Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean b/Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
index 901b5b2038c028..88657855528d15 100644
--- a/Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
@@ -623,6 +623,13 @@ theorem deriv_rpow_const (hf : DifferentiableAt ℝ f x) (hx : f x ≠ 0 ∨ 1 
     deriv (fun x => f x ^ p) x = deriv f x * p * f x ^ (p - 1) :=
   (hf.hasDerivAt.rpow_const hx).deriv
 
+theorem deriv_norm_ofReal_cpow (c : ℂ) {t : ℝ} (ht : 0 < t) :
+    (deriv fun x : ℝ ↦ ‖(x : ℂ) ^ c‖) t = c.re * t ^ (c.re - 1) := by
+  rw [EventuallyEq.deriv_eq (f := fun x ↦ x ^ c.re)]
+  · rw [Real.deriv_rpow_const (Or.inl ht.ne')]
+  · filter_upwards [eventually_gt_nhds ht] with x hx
+    rw [Complex.norm_eq_abs, Complex.abs_cpow_eq_rpow_re_of_pos hx]
+
 lemma isTheta_deriv_rpow_const_atTop {p : ℝ} (hp : p ≠ 0) :
     deriv (fun (x : ℝ) => x ^ p) =Θ[atTop] fun x => x ^ (p-1) := by
   calc deriv (fun (x : ℝ) => x ^ p) =ᶠ[atTop] fun x => p * x ^ (p - 1) := by
diff --git a/Mathlib/Analysis/SpecialFunctions/Pow/Real.lean b/Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
index a49168a3cabb83..0a8b78fdc039e3 100644
--- a/Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
@@ -318,6 +318,12 @@ theorem abs_cpow_eq_rpow_re_of_nonneg {x : ℝ} (hx : 0 ≤ x) {y : ℂ} (hy : r
     abs (x ^ y) = x ^ re y := by
   rw [abs_cpow_of_imp] <;> simp [*, arg_ofReal_of_nonneg, _root_.abs_of_nonneg]
 
+open Filter in
+lemma norm_ofReal_cpow_eventually_eq_atTop (c : ℂ) :
+    (fun t : ℝ ↦ ‖(t : ℂ) ^ c‖) =ᶠ[atTop] fun t ↦ t ^ c.re := by
+  filter_upwards [eventually_gt_atTop 0] with t ht
+  rw [Complex.norm_eq_abs, abs_cpow_eq_rpow_re_of_pos ht]
+
 lemma norm_natCast_cpow_of_re_ne_zero (n : ℕ) {s : ℂ} (hs : s.re ≠ 0) :
     ‖(n : ℂ) ^ s‖ = (n : ℝ) ^ (s.re) := by
   rw [norm_eq_abs, ← ofReal_natCast, abs_cpow_eq_rpow_re_of_nonneg n.cast_nonneg hs]
@@ -962,6 +968,35 @@ lemma norm_log_natCast_le_rpow_div (n : ℕ) {ε : ℝ} (hε : 0 < ε) : ‖log
 
 end Complex
 
+namespace  Asymptotics
+
+open Filter
+
+variable {E : Type*} [SeminormedRing E] (a b c : ℝ)
+
+theorem IsBigO.mul_atTop_rpow_of_isBigO_rpow {f g : ℝ → E}
+    (hf : f =O[atTop] fun t ↦ (t : ℝ) ^ a) (hg : g =O[atTop] fun t ↦ (t : ℝ) ^ b)
+    (h : a + b ≤ c) :
+    (f * g) =O[atTop] fun t ↦ (t : ℝ) ^ c := by
+  refine (hf.mul hg).trans (Eventually.isBigO ?_)
+  filter_upwards [eventually_ge_atTop 1] with t ht
+  rw [← Real.rpow_add (zero_lt_one.trans_le ht), Real.norm_of_nonneg (Real.rpow_nonneg
+    (zero_le_one.trans ht) (a + b))]
+  exact Real.rpow_le_rpow_of_exponent_le ht h
+
+theorem IsBigO.mul_atTop_rpow_natCast_of_isBigO_rpow {f g : ℕ → E}
+    (hf : f =O[atTop] fun n ↦ (n : ℝ) ^ a) (hg : g =O[atTop] fun n ↦ (n : ℝ) ^ b)
+    (h : a + b ≤ c) :
+    (f * g) =O[atTop] fun n ↦ (n : ℝ) ^ c := by
+  refine (hf.mul hg).trans (Eventually.isBigO ?_)
+  filter_upwards [eventually_ge_atTop 1] with t ht
+  replace ht : 1 ≤ (t : ℝ) := Nat.one_le_cast.mpr ht
+  rw [← Real.rpow_add (zero_lt_one.trans_le ht), Real.norm_of_nonneg (Real.rpow_nonneg
+    (zero_le_one.trans ht) (a + b))]
+  exact Real.rpow_le_rpow_of_exponent_le ht h
+
+end Asymptotics
+
 /-!
 ## Square roots of reals
 -/
@@ -1068,14 +1103,6 @@ theorem isRat_rpow_neg {a b : ℝ} {nb : ℕ}
     IsRat (a ^ b) num den := by
   rwa [pb.out, Real.rpow_intCast]
 
-#adaptation_note
-/--
-Since https://github.com/leanprover/lean4/pull/5338,
-the unused variable linter can not see usages of variables in
-`haveI' : ⋯ =Q ⋯ := ⟨⟩` clauses, so generates many false positives.
--/
-set_option linter.unusedVariables false
-
 /-- Evaluates expressions of the form `a ^ b` when `a` and `b` are both reals. -/
 @[norm_num (_ : ℝ) ^ (_ : ℝ)]
 def evalRPow : NormNumExt where eval {u α} e := do
diff --git a/Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean b/Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean
index 049879b8369028..e56f20010f8cf1 100644
--- a/Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean
+++ b/Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean
@@ -3,7 +3,7 @@ Copyright (c) 2020 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 -/
-import Mathlib.Data.Complex.Exponential
+import Mathlib.Data.Complex.Trigonometric
 import Mathlib.Data.Complex.Module
 import Mathlib.RingTheory.Polynomial.Chebyshev
 
diff --git a/Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean b/Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean
index 5812a21f568584..b81f5bddd383a9 100644
--- a/Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean
+++ b/Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean
@@ -128,12 +128,6 @@ section Five
 variable {R₁ R₂ : ComposableArrows C 4} (hR₁ : R₁.Exact) (hR₂ : R₂.Exact) (φ : R₁ ⟶ R₂)
 include hR₁ hR₂
 
-#adaptation_note /-- nightly-2024-03-11
-We turn off simprocs here.
-Ideally someone will investigate whether `simp` lemmas can be rearranged
-so that this works without the `set_option`,
-*or* come up with a proposal regarding finer control of disabling simprocs. -/
-set_option simprocs false in
 /-- The five lemma. -/
 theorem isIso_of_epi_of_isIso_of_isIso_of_mono (h₀ : Epi (app' φ 0)) (h₁ : IsIso (app' φ 1))
     (h₂ : IsIso (app' φ 3)) (h₃ : Mono (app' φ 4)) : IsIso (app' φ 2) := by
diff --git a/Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean b/Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean
index 9110373e9c67a4..dac1bafa0aea28 100644
--- a/Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean
+++ b/Mathlib/CategoryTheory/Abelian/InjectiveResolution.lean
@@ -298,7 +298,6 @@ lemma ofCocomplex_d_0_1 :
     (ofCocomplex Z).d 0 1 = d (Injective.ι Z) := by
   simp [ofCocomplex]
 
-#adaptation_note /-- Since nightly-2024-03-11, this takes forever now -/
 lemma ofCocomplex_exactAt_succ (n : ℕ) :
     (ofCocomplex Z).ExactAt (n + 1) := by
   rw [HomologicalComplex.exactAt_iff' _ n (n + 1) (n + 1 + 1) (by simp) (by simp)]
diff --git a/Mathlib/CategoryTheory/Abelian/Transfer.lean b/Mathlib/CategoryTheory/Abelian/Transfer.lean
index 20ca075bae7579..114d4d2a53902d 100644
--- a/Mathlib/CategoryTheory/Abelian/Transfer.lean
+++ b/Mathlib/CategoryTheory/Abelian/Transfer.lean
@@ -8,6 +8,7 @@ import Mathlib.CategoryTheory.Abelian.Basic
 import Mathlib.CategoryTheory.Adjunction.Limits
 import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
 import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
+import Mathlib.CategoryTheory.Preadditive.Transfer
 
 /-!
 # Transferring "abelian-ness" across a functor
@@ -202,21 +203,14 @@ lemma inverse_map_add {P Q : ShrinkHoms C} (f g : P ⟶ Q) :
 
 variable (C)
 
-noncomputable instance preadditive :
-    Preadditive.{w} (ShrinkHoms C) where
-  homGroup := homGroup
-  add_comp _ _ _ _ _ _ := by
-    apply (inverse C).map_injective
-    simp only [inverse_map_add, Functor.map_comp, Preadditive.add_comp]
-  comp_add _ _ _ _ _ _ := by
-    apply (inverse C).map_injective
-    simp only [inverse_map_add, Functor.map_comp, Preadditive.comp_add]
-
-instance : (inverse C).Additive where
-  map_add := by apply inverse_map_add
-
-instance : (functor C).Additive where
-  map_add := by apply functor_map_add
+instance preadditive : Preadditive.{w} (ShrinkHoms C) :=
+  .ofFullyFaithful (equivalence C).fullyFaithfulInverse
+
+instance : (inverse C).Additive :=
+  (equivalence C).symm.fullyFaithfulFunctor.additive_ofFullyFaithful
+
+instance : (functor C).Additive :=
+  (equivalence C).symm.additive_inverse_of_FullyFaithful
 
 instance hasLimitsOfShape (J : Type*) [Category J]
     [HasLimitsOfShape J C] : HasLimitsOfShape.{_, _, w} J (ShrinkHoms C) :=
diff --git a/Mathlib/CategoryTheory/Action/Basic.lean b/Mathlib/CategoryTheory/Action/Basic.lean
index 1c61d3fcf4c62b..f68dccaad46f25 100644
--- a/Mathlib/CategoryTheory/Action/Basic.lean
+++ b/Mathlib/CategoryTheory/Action/Basic.lean
@@ -42,7 +42,7 @@ namespace Action
 
 variable {V}
 
-@[simp 1100]
+@[simp]
 theorem ρ_one {G : MonCat.{u}} (A : Action V G) : A.ρ 1 = 𝟙 A.V := by rw [MonoidHom.map_one]; rfl
 
 /-- When a group acts, we can lift the action to the group of automorphisms. -/
diff --git a/Mathlib/CategoryTheory/Adjunction/Lifting/Left.lean b/Mathlib/CategoryTheory/Adjunction/Lifting/Left.lean
index 432970c912ce8e..508de191a1af9d 100644
--- a/Mathlib/CategoryTheory/Adjunction/Lifting/Left.lean
+++ b/Mathlib/CategoryTheory/Adjunction/Lifting/Left.lean
@@ -122,13 +122,6 @@ variable [HasReflexiveCoequalizers A]
 noncomputable def constructLeftAdjointObj (Y : B) : A :=
   coequalizer (F'.map (U.map (adj₁.counit.app Y))) (otherMap _ _ adj₁ adj₂ Y)
 
-#adaptation_note
-/--
-The new unused variable linter in
-https://github.com/leanprover/lean4/pull/5338
-flags `{ z : F.obj (U.obj X) ⟶ R.obj Y // _ }`.
--/
-set_option linter.unusedVariables false in
 /-- The homset equivalence which helps show that `R` is a right adjoint. -/
 @[simps!]
 noncomputable def constructLeftAdjointEquiv [∀ X : B, RegularEpi (adj₁.counit.app X)] (Y : A)
diff --git a/Mathlib/CategoryTheory/Category/Basic.lean b/Mathlib/CategoryTheory/Category/Basic.lean
index e47e8bcebd744b..247f3da889691e 100644
--- a/Mathlib/CategoryTheory/Category/Basic.lean
+++ b/Mathlib/CategoryTheory/Category/Basic.lean
@@ -23,7 +23,7 @@ Introduces notations in the `CategoryTheory` scope
 
 Users may like to add `g ⊚ f` for composition in the standard convention, using
 ```lean
-local notation g ` ⊚ `:80 f:80 := category.comp f g    -- type as \oo
+local notation:80 g " ⊚ " f:80 => CategoryTheory.CategoryStruct.comp f g    -- type as \oo
 ```
 
 ## Porting note
diff --git a/Mathlib/CategoryTheory/ChosenFiniteProducts/InfSemilattice.lean b/Mathlib/CategoryTheory/ChosenFiniteProducts/InfSemilattice.lean
new file mode 100644
index 00000000000000..ae74a6c833fadf
--- /dev/null
+++ b/Mathlib/CategoryTheory/ChosenFiniteProducts/InfSemilattice.lean
@@ -0,0 +1,42 @@
+/-
+Copyright (c) 2025 Sina Hazratpour. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Sina Hazratpour
+-/
+
+import Mathlib.CategoryTheory.ChosenFiniteProducts
+import Mathlib.CategoryTheory.Limits.Preorder
+
+/-!
+# Chosen finite products for the preorder category of a meet-semilattice with a greatest element
+
+The preorder category of a meet-semilattice `C` with a greatest element has chosen finite products.
+
+A symmetric monoidal structure on the preorder category is automatically provided by the
+instances `monoidalOfChosenFiniteProducts` and `ChosenFiniteProducts.instSymmetricCategory`.
+
+-/
+
+namespace CategoryTheory
+
+open Category MonoidalCategory
+
+universe u
+
+variable (C : Type u) [SemilatticeInf C] [OrderTop C]
+
+namespace SemilatticeInf
+
+/-- Chosen finite products for the preorder category of a meet-semilattice with a greatest element-/
+noncomputable scoped instance chosenFiniteProducts : ChosenFiniteProducts C where
+  terminal := ⟨_, Preorder.isTerminalTop C⟩
+  product X Y := ⟨_,  Preorder.isLimitBinaryFan X Y⟩
+
+lemma tensorObj {C : Type u} [SemilatticeInf C] [OrderTop C] {X Y : C} : X ⊗ Y = X ⊓ Y := rfl
+
+lemma tensorUnit {C : Type u} [SemilatticeInf C] [OrderTop C] :
+    𝟙_ C = ⊤ := rfl
+
+end SemilatticeInf
+
+end CategoryTheory
diff --git a/Mathlib/CategoryTheory/CofilteredSystem.lean b/Mathlib/CategoryTheory/CofilteredSystem.lean
index a3088857c8cf8e..103f619ddd6525 100644
--- a/Mathlib/CategoryTheory/CofilteredSystem.lean
+++ b/Mathlib/CategoryTheory/CofilteredSystem.lean
@@ -170,15 +170,11 @@ def toPreimages : J ⥤ Type v where
     rw [← mem_preimage, preimage_preimage, mem_preimage]
     convert h (g ≫ f); rw [F.map_comp]; rfl
   map_id j := by
-    #adaptation_note /-- nightly-2024-03-16: simp was
-    simp (config := { unfoldPartialApp := true }) only [MapsTo.restrict, Subtype.map, F.map_id] -/
-    simp only [MapsTo.restrict, Subtype.map_def, F.map_id]
+    simp +unfoldPartialApp only [MapsTo.restrict, Subtype.map, F.map_id]
     ext
     rfl
   map_comp f g := by
-    #adaptation_note /-- nightly-2024-03-16: simp was
-    simp (config := { unfoldPartialApp := true }) only [MapsTo.restrict, Subtype.map, F.map_comp] -/
-    simp only [MapsTo.restrict, Subtype.map_def, F.map_comp]
+    simp +unfoldPartialApp only [MapsTo.restrict, Subtype.map, F.map_comp]
     rfl
 
 instance toPreimages_finite [∀ j, Finite (F.obj j)] : ∀ j, Finite ((F.toPreimages s).obj j) :=
@@ -253,15 +249,11 @@ def toEventualRanges : J ⥤ Type v where
   obj j := F.eventualRange j
   map f := (F.eventualRange_mapsTo f).restrict _ _ _
   map_id i := by
-    #adaptation_note /--- nightly-2024-03-16: simp was
-    simp (config := { unfoldPartialApp := true }) only [MapsTo.restrict, Subtype.map, F.map_id] -/
-    simp only [MapsTo.restrict, Subtype.map_def, F.map_id]
+    simp +unfoldPartialApp only [MapsTo.restrict, Subtype.map, F.map_id]
     ext
     rfl
   map_comp _ _ := by
-    #adaptation_note /-- nightly-2024-03-16: simp was
-    simp (config := { unfoldPartialApp := true }) only [MapsTo.restrict, Subtype.map, F.map_comp] -/
-    simp only [MapsTo.restrict, Subtype.map_def, F.map_comp]
+    simp +unfoldPartialApp only [MapsTo.restrict, Subtype.map, F.map_comp]
     rfl
 
 instance toEventualRanges_finite [∀ j, Finite (F.obj j)] : ∀ j, Finite (F.toEventualRanges.obj j) :=
diff --git a/Mathlib/CategoryTheory/Comma/Final.lean b/Mathlib/CategoryTheory/Comma/Final.lean
index 6c77ce36c3afbc..23e5a766393055 100644
--- a/Mathlib/CategoryTheory/Comma/Final.lean
+++ b/Mathlib/CategoryTheory/Comma/Final.lean
@@ -5,7 +5,9 @@ Authors: Jakob von Raumer
 -/
 import Mathlib.CategoryTheory.Functor.KanExtension.Adjunction
 import Mathlib.CategoryTheory.Limits.IsConnected
+import Mathlib.CategoryTheory.Limits.Sifted
 import Mathlib.CategoryTheory.Filtered.Final
+import Mathlib.CategoryTheory.Filtered.Flat
 import Mathlib.CategoryTheory.Grothendieck
 import Mathlib.CategoryTheory.Comma.StructuredArrow.CommaMap
 
@@ -19,9 +21,12 @@ final and `A` is connected.
 We then use this in a proof that derives finality of `map` between two comma categories
 on a quasi-commutative diagram of functors, some of which need to be final.
 
+Finally we prove filteredness of a `Comma L R` and finality of `snd L R`, given that `R` is final
+and `A` and `B` are filtered.
+
 ## References
 
-* [M. Kashiwara, P. Schapira, *Categories and Sheaves*][Kashiwara2006], Lemma 3.4.3 & 3.4.4
+* [M. Kashiwara, P. Schapira, *Categories and Sheaves*][Kashiwara2006], Lemma 3.4.3 -- 3.4.5
 -/
 
 universe v₁ v₂ v₃ v₄ v₅ v₆ u₁ u₂ u₃ u₄ u₅ u₆
@@ -80,9 +85,9 @@ instance final_fst [R.Final] : (fst L R).Final := by
   exact (Functor.associator _ _ _).symm.trans (Iso.compInverseIso (mapFst _ _))
 
 instance initial_snd [L.Initial] : (snd L R).Initial := by
-  haveI : ((opFunctor L R).leftOp ⋙ fst R.op L.op).Final :=
+  have : ((opFunctor L R).leftOp ⋙ fst R.op L.op).Final :=
     final_equivalence_comp (opEquiv L R).functor.leftOp (fst R.op L.op)
-  haveI : (snd L R).op.Final := final_of_natIso (opFunctorCompFst _ _)
+  have : (snd L R).op.Final := final_of_natIso (opFunctorCompFst _ _)
   apply initial_of_final_op
 
 /-- `Comma L R` with `L : A ⥤ T` and `R : B ⥤ T` is connected if `R` is final and `A` is
@@ -109,9 +114,9 @@ end NonSmall
 
 Let `F`, `G`, `R` and `R'` be final and `B` be filtered. Then, the induced functor between the comma
 categories of the first and second row of the diagram is final. -/
-lemma map_final {A : Type u₁} [Category.{v₁} A] {B : Type u₂} [Category.{u₂} B] {T : Type u₃}
+lemma map_final {A : Type u₁} [Category.{v₁} A] {B : Type u₂} [Category.{v₂} B] {T : Type u₃}
     [Category.{v₃} T] {L : A ⥤ T} {R : B ⥤ T} {A' : Type u₄} [Category.{v₄} A'] {B' : Type u₅}
-    [Category.{v₅} B'] {T' : Type u₂} [Category.{u₂} T'] {L' : A' ⥤ T'} {R' : B' ⥤ T'} {F : A ⥤ A'}
+    [Category.{v₅} B'] {T' : Type u₆} [Category.{v₆} T'] {L' : A' ⥤ T'} {R' : B' ⥤ T'} {F : A ⥤ A'}
     {G : B ⥤ B'} {H : T ⥤ T'} (iL : F ⋙ L' ≅ L ⋙ H) (iR : G ⋙ R' ≅ R ⋙ H) [IsFiltered B]
     [R.Final] [R'.Final] [F.Final] [G.Final] :
     (Comma.map iL.hom iR.inv).Final := ⟨fun ⟨i₂, j₂, u₂⟩ => by
@@ -132,6 +137,57 @@ lemma map_final {A : Type u₁} [Category.{v₁} A] {B : Type u₂} [Category.{u
   rw [IsIso.Iso.inv_inv]
   infer_instance⟩
 
+section Filtered
+
+variable {A : Type u₁} [Category.{v₁} A]
+variable {B : Type u₂} [Category.{v₂} B]
+variable {T : Type u₃} [Category.{v₃} T]
+variable (L : A ⥤ T) (R : B ⥤ T)
+
+attribute [local instance] map_final in
+/-- Let `A` and `B` be filtered categories, `R : B ⥤ T` be final and `L : A ⥤ T`. Then, the
+comma category `Comma L R` is filtered. -/
+instance isFiltered_of_final [IsFiltered A] [IsFiltered B] [R.Final] : IsFiltered (Comma L R) := by
+  haveI (a : A) : IsFiltered (Comma (fromPUnit (L.obj a)) R) :=
+    R.final_iff_isFiltered_structuredArrow.mp inferInstance (L.obj a)
+  have (a : A) : (fromPUnit (Over.mk (𝟙 a))).Final := final_const_of_isTerminal Over.mkIdTerminal
+  let η (a : A) : fromPUnit (Over.mk (𝟙 a)) ⋙ Over.forget a ⋙ L ≅ fromPUnit (L.obj a) :=
+    NatIso.ofComponents (fun _ => Iso.refl _)
+  have (a : A) := IsFiltered.of_final (map (L := fromPUnit (L.obj a)) (F := 𝟭 T) (η a).hom
+    ((Iso.refl (𝟭 B ⋙ R)).inv))
+  have : RepresentablyCoflat (fst L R) :=
+    ⟨fun a => IsFiltered.of_equivalence (CostructuredArrow.ofCommaFstEquivalence L R a).symm⟩
+  apply isFiltered_of_representablyCoflat (fst L R)
+
+attribute [local instance] isFiltered_of_final in
+/-- Let `A` and `B` be cofiltered categories, `L : A ⥤ T` be initial and `R : B ⥤ T`. Then, the
+comma category `Comma L R` is cofiltered. -/
+lemma isCofiltered_of_initial [IsCofiltered A] [IsCofiltered B] [L.Initial] :
+    IsCofiltered (Comma L R) :=
+ IsCofiltered.of_equivalence (Comma.opEquiv _ _).symm
+
+attribute [local instance] final_of_isFiltered_of_pUnit in
+/-- Let `A` and `B` be filtered categories, `R : B ⥤ T` be final and `R : A ⥤ T`. Then, the
+projection `snd L R : Comma L R ⥤ B` is final. -/
+instance final_snd [IsFiltered A] [IsFiltered B] [R.Final] : (snd L R).Final := by
+  let iL : star.{1} A ⋙ 𝟭 _ ≅ L ⋙ star _ := Iso.refl _
+  let iR : 𝟭 B ⋙ star.{1} B ≅ R ⋙ star _ := Iso.refl _
+  have := map_final iL iR
+  let s := (equivProd (𝟭 _) (star B)).trans <| prod.leftUnitorEquivalence B
+  let iS : map iL.hom iR.inv ⋙ s.functor ≅ snd L R :=
+    NatIso.ofComponents (fun _ => Iso.refl _) (fun f => by simp [iL, iR, s])
+  apply final_of_natIso iS
+
+/-- Let `A` and `B` be cofiltered categories, `L : A ⥤ T` be initial and `R : B ⥤ T`. Then, the
+projection `fst L R : Comma L R ⥤ A` is initial. -/
+instance initial_fst [IsCofiltered A] [IsCofiltered B] [L.Initial] : (fst L R).Initial := by
+  have : ((opFunctor L R).leftOp ⋙ snd R.op L.op).Final :=
+    final_equivalence_comp (opEquiv L R).functor.leftOp _
+  have : (fst L R).op.Final := final_of_natIso <| opFunctorCompSnd _ _
+  apply initial_of_final_op
+
+end Filtered
+
 end Comma
 
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Comma/StructuredArrow/CommaMap.lean b/Mathlib/CategoryTheory/Comma/StructuredArrow/CommaMap.lean
index 5a2d0d2c797989..9a0d5b34cc35b6 100644
--- a/Mathlib/CategoryTheory/Comma/StructuredArrow/CommaMap.lean
+++ b/Mathlib/CategoryTheory/Comma/StructuredArrow/CommaMap.lean
@@ -23,7 +23,7 @@ noncomputable section
 variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D]
   {T : Type u₃} [Category.{v₃} T] {L : C ⥤ T} {R : D ⥤ T}
   {C' : Type u₄} [Category.{v₄} C'] {D' : Type u₅} [Category.{v₅} D'] {T' : Type u₆}
-  [Category.{u₆} T'] {L' : C' ⥤ T'} {R' : D' ⥤ T'} {F₁ : C ⥤ C'} {F₂ : D ⥤ D'} {F : T ⥤ T'}
+  [Category.{v₆} T'] {L' : C' ⥤ T'} {R' : D' ⥤ T'} {F₁ : C ⥤ C'} {F₂ : D ⥤ D'} {F : T ⥤ T'}
   (α : F₁ ⋙ L' ⟶ L ⋙ F) (β : R ⋙ F ⟶ F₂ ⋙ R')
 
 /-- The functor establishing the equivalence `StructuredArrow.commaMapEquivalence`. -/
diff --git a/Mathlib/CategoryTheory/Endofunctor/Algebra.lean b/Mathlib/CategoryTheory/Endofunctor/Algebra.lean
index fe740cc36ff77b..aaf7e3bd093855 100644
--- a/Mathlib/CategoryTheory/Endofunctor/Algebra.lean
+++ b/Mathlib/CategoryTheory/Endofunctor/Algebra.lean
@@ -17,7 +17,6 @@ coalgebras over `G`.
 
 ## TODO
 
-* Prove the dual result about the structure map of the terminal coalgebra of an endofunctor.
 * Prove that if the countable infinite product over the powers of the endofunctor exists, then
   algebras over the endofunctor coincide with algebras over the free monad on the endofunctor.
 -/
@@ -382,6 +381,35 @@ def equivOfNatIso {F G : C ⥤ C} (α : F ≅ G) : Coalgebra F ≌ Coalgebra G w
   counitIso :=
     (functorOfNatTransComp _ _).symm ≪≫ functorOfNatTransEq (by simp) ≪≫ functorOfNatTransId
 
+namespace Terminal
+
+variable {A} (h : @Limits.IsTerminal (Coalgebra F) _ A)
+
+/-- The inverse of the structure map of an terminal coalgebra -/
+@[simp]
+def strInv : F.obj A.1 ⟶ A.1 :=
+  (h.from ⟨F.obj A.V, F.map A.str⟩).f
+
+theorem right_inv' :
+    ⟨A.str ≫ strInv h, by rw [Category.assoc, F.map_comp, strInv, ← Hom.h] ⟩ = 𝟙 A :=
+  Limits.IsTerminal.hom_ext h _ (𝟙 A)
+
+theorem right_inv : A.str ≫ strInv h = 𝟙 _ :=
+  congr_arg Hom.f (right_inv' h)
+
+theorem left_inv : strInv h ≫ A.str = 𝟙 _ := by
+  rw [strInv, ← (h.from ⟨F.obj A.V, F.map A.str⟩).h, ← F.map_id, ← F.map_comp]
+  congr
+  exact right_inv h
+
+/-- The structure map of the terminal coalgebra is an isomorphism,
+hence endofunctors preserve their terminal coalgebras
+-/
+theorem str_isIso (h : Limits.IsTerminal A) : IsIso A.str :=
+  { out := ⟨strInv h, right_inv _, left_inv _⟩  }
+
+end Terminal
+
 end Coalgebra
 
 namespace Adjunction
diff --git a/Mathlib/CategoryTheory/Galois/Topology.lean b/Mathlib/CategoryTheory/Galois/Topology.lean
index 17d6329fb58cdc..7020a117b13ee3 100644
--- a/Mathlib/CategoryTheory/Galois/Topology.lean
+++ b/Mathlib/CategoryTheory/Galois/Topology.lean
@@ -17,7 +17,7 @@ embedding of `Aut F` into `∀ X, Aut (F.obj X)` where
 
 ## References
 
-- Stacks Project: Tag 0BMQ
+- [Stacks 0BMQ](https://stacks.math.columbia.edu/tag/0BMQ)
 
 -/
 
diff --git a/Mathlib/CategoryTheory/Generator/Basic.lean b/Mathlib/CategoryTheory/Generator/Basic.lean
index 049514c52cef97..70168c6d7626de 100644
--- a/Mathlib/CategoryTheory/Generator/Basic.lean
+++ b/Mathlib/CategoryTheory/Generator/Basic.lean
@@ -527,15 +527,21 @@ theorem isSeparator_coprod_of_isSeparator_right (G H : C) [HasBinaryCoproduct G
     (hH : IsSeparator H) : IsSeparator (G ⨿ H) :=
   (isSeparator_coprod _ _).2 <| IsSeparating.mono hH <| by simp
 
+lemma isSeparator_of_isColimit_cofan {β : Type w} {f : β → C}
+    (hf : IsSeparating (Set.range f)) {c : Cofan f} (hc : IsColimit c) : IsSeparator c.pt := by
+  refine (isSeparator_def _).2 fun X Y u v huv => hf _ _ fun Z hZ g => ?_
+  obtain ⟨b, rfl⟩ := Set.mem_range.1 hZ
+  classical simpa using c.ι.app ⟨b⟩ ≫= huv (hc.desc (Cofan.mk _ (Pi.single b g)))
+
 theorem isSeparator_sigma {β : Type w} (f : β → C) [HasCoproduct f] :
     IsSeparator (∐ f) ↔ IsSeparating (Set.range f) := by
-  refine
-    ⟨fun h X Y u v huv => ?_, fun h =>
-      (isSeparator_def _).2 fun X Y u v huv => h _ _ fun Z hZ g => ?_⟩
-  · refine h.def _ _ fun g => colimit.hom_ext fun b => ?_
-    simpa using huv (f b.as) (by simp) (colimit.ι (Discrete.functor f) _ ≫ g)
-  · obtain ⟨b, rfl⟩ := Set.mem_range.1 hZ
-    classical simpa using Sigma.ι f b ≫= huv (Sigma.desc (Pi.single b g))
+  refine ⟨fun h X Y u v huv => ?_, fun h => isSeparator_of_isColimit_cofan h (colimit.isColimit _)⟩
+  refine h.def _ _ fun g => colimit.hom_ext fun b => ?_
+  simpa using huv (f b.as) (by simp) (colimit.ι (Discrete.functor f) _ ≫ g)
+
+theorem IsSeparating.isSeparator_coproduct {β : Type w} {f : β → C} [HasCoproduct f]
+    (hS : IsSeparating (Set.range f)) : IsSeparator (∐ f) :=
+  (isSeparator_sigma _).2 hS
 
 theorem isSeparator_sigma_of_isSeparator {β : Type w} (f : β → C) [HasCoproduct f] (b : β)
     (hb : IsSeparator (f b)) : IsSeparator (∐ f) :=
diff --git a/Mathlib/CategoryTheory/Generator/Coproduct.lean b/Mathlib/CategoryTheory/Generator/Coproduct.lean
deleted file mode 100644
index 0e948301d48193..00000000000000
--- a/Mathlib/CategoryTheory/Generator/Coproduct.lean
+++ /dev/null
@@ -1,43 +0,0 @@
-/-
-Copyright (c) 2024 Joël Riou. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Joël Riou
--/
-
-import Mathlib.CategoryTheory.Generator.Basic
-import Mathlib.CategoryTheory.Limits.Shapes.Products
-
-/-!
-# The coproduct of a separating family of objects is separating
-
-If a family of objects `S : ι → C` in a category with zero morphisms
-is separating, then the coproduct of `S` is a separator in `C`.
-
--/
-
-universe w v u
-
-namespace CategoryTheory.IsSeparating
-
-open Limits
-
-variable {C : Type u} [Category.{v} C] [HasZeroMorphisms C]
-  {ι : Type w} {S : ι → C}
-
-open Classical in
-lemma isSeparator_of_isColimit_cofan
-    (hS : IsSeparating (Set.range S)) {c : Cofan S} (hc : IsColimit c) :
-    IsSeparator c.pt := by
-  intro X Y f g h
-  apply hS
-  rintro _ ⟨i, rfl⟩ α
-  let β : c.pt ⟶ X := Cofan.IsColimit.desc hc
-      (fun j ↦ if hij : i = j then eqToHom (by rw [hij]) ≫ α else 0)
-  have hβ : c.inj i ≫ β = α := by simp [β]
-  simp only [← hβ, Category.assoc, h c.pt (by simp) β]
-
-lemma isSeparator_coproduct (hS : IsSeparating (Set.range S)) [HasCoproduct S] :
-    IsSeparator (∐ S) :=
-  isSeparator_of_isColimit_cofan hS (colimit.isColimit _)
-
-end CategoryTheory.IsSeparating
diff --git a/Mathlib/CategoryTheory/Generator/Indization.lean b/Mathlib/CategoryTheory/Generator/Indization.lean
new file mode 100644
index 00000000000000..4c79e7d781802e
--- /dev/null
+++ b/Mathlib/CategoryTheory/Generator/Indization.lean
@@ -0,0 +1,37 @@
+/-
+Copyright (c) 2025 Markus Himmel. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Markus Himmel
+-/
+import Mathlib.CategoryTheory.Generator.Basic
+import Mathlib.CategoryTheory.Limits.Indization.Category
+
+/-!
+# Separating set in the category of ind-objects
+
+We construct a separating set in the category of ind-objects.
+
+## Future work
+
+Once we have constructed zero morphisms in the category of ind-objects, we will be able to show
+that under sufficient conditions, the category of ind-objects has a separating object.
+
+-/
+
+universe v u
+
+namespace CategoryTheory
+
+open Limits
+
+section
+
+variable {C : Type u} [Category.{v} C]
+
+theorem Ind.isSeparating_range_yoneda : IsSeparating (Set.range (Ind.yoneda : C ⥤ _).obj) := by
+  refine fun X Y f g h => (cancel_epi (Ind.colimitPresentationCompYoneda X).hom).1 ?_
+  exact colimit.hom_ext (fun i => by simp [← Category.assoc, h])
+
+end
+
+end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Generator/Presheaf.lean b/Mathlib/CategoryTheory/Generator/Presheaf.lean
index d12ab50d4882a9..61042589d44a2e 100644
--- a/Mathlib/CategoryTheory/Generator/Presheaf.lean
+++ b/Mathlib/CategoryTheory/Generator/Presheaf.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joël Riou
 -/
 
-import Mathlib.CategoryTheory.Generator.Coproduct
+import Mathlib.CategoryTheory.Generator.Basic
 import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
 
 /-!
diff --git a/Mathlib/CategoryTheory/GradedObject.lean b/Mathlib/CategoryTheory/GradedObject.lean
index 60536139bb7171..0b144c74283155 100644
--- a/Mathlib/CategoryTheory/GradedObject.lean
+++ b/Mathlib/CategoryTheory/GradedObject.lean
@@ -216,11 +216,8 @@ theorem shiftFunctor_map_apply {β : Type*} [AddCommGroup β] (s : β)
 instance [HasZeroMorphisms C] (β : Type w) (X Y : GradedObject β C) :
   Zero (X ⟶ Y) := ⟨fun _ => 0⟩
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/4481
-the `simpNF` linter incorrectly claims this lemma can't be applied by `simp`.
--/
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/4481
+the `simpNF` linter incorrectly claims this lemma can't be applied by `simp`. -/
 @[simp, nolint simpNF]
 theorem zero_apply [HasZeroMorphisms C] (β : Type w) (X Y : GradedObject β C) (b : β) :
     (0 : X ⟶ Y) b = 0 :=
diff --git a/Mathlib/CategoryTheory/GradedObject/Monoidal.lean b/Mathlib/CategoryTheory/GradedObject/Monoidal.lean
index b10fab11e47d85..3467d00d9731fd 100644
--- a/Mathlib/CategoryTheory/GradedObject/Monoidal.lean
+++ b/Mathlib/CategoryTheory/GradedObject/Monoidal.lean
@@ -312,9 +312,8 @@ lemma associator_naturality (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (f₃
     [HasGoodTensor₁₂Tensor Y₁ Y₂ Y₃] [HasGoodTensorTensor₂₃ Y₁ Y₂ Y₃] :
     tensorHom (tensorHom f₁ f₂) f₃ ≫ (associator Y₁ Y₂ Y₃).hom =
       (associator X₁ X₂ X₃).hom ≫ tensorHom f₁ (tensorHom f₂ f₃) := by
-        #adaptation_note
-        /-- this used to be aesop_cat, but that broke with
-        https://github.com/leanprover/lean4/pull/4154 -/
+        #adaptation_note /-- https://github.com/leanprover/lean4/pull/4154
+        this used to be aesop_cat -/
         ext x i₁ i₂ i₃ h : 2
         simp only [categoryOfGradedObjects_comp, ιTensorObj₃'_tensorHom_assoc,
           associator_conjugation, ιTensorObj₃'_associator_hom, assoc, Iso.inv_hom_id_assoc,
diff --git a/Mathlib/CategoryTheory/Limits/ConcreteCategory/WithAlgebraicStructures.lean b/Mathlib/CategoryTheory/Limits/ConcreteCategory/WithAlgebraicStructures.lean
index d5e68a50703d7c..1b1231d4bd3c37 100644
--- a/Mathlib/CategoryTheory/Limits/ConcreteCategory/WithAlgebraicStructures.lean
+++ b/Mathlib/CategoryTheory/Limits/ConcreteCategory/WithAlgebraicStructures.lean
@@ -74,7 +74,7 @@ lemma colimit_no_zero_smul_divisor
   rw [← this, ← map_smul (colimit.ι F j).hom] at hx
   obtain ⟨j', i, h⟩ := Concrete.colimit_rep_eq_zero (hx := hx)
   obtain ⟨j'', H⟩ := H
-  simpa [elementwise_of% (colimit.w F), this, map_zero] using congr(colimit.ι F _
+  simpa [elementwise_of% (colimit.w F), ← this, map_zero] using congr(colimit.ι F _
     $(H (IsFiltered.sup {j, j', j''} { ⟨j, j', by simp, by simp, i⟩ })
       (IsFiltered.toSup _ _ <| by simp)
       (F.map (IsFiltered.toSup _ _ <| by simp) x)
diff --git a/Mathlib/CategoryTheory/Limits/Cones.lean b/Mathlib/CategoryTheory/Limits/Cones.lean
index 57bf33aa6b3ab0..c5a23c859f29df 100644
--- a/Mathlib/CategoryTheory/Limits/Cones.lean
+++ b/Mathlib/CategoryTheory/Limits/Cones.lean
@@ -169,8 +169,6 @@ theorem Cocone.w {F : J ⥤ C} (c : Cocone F) {j j' : J} (f : j ⟶ j') :
   rw [c.ι.naturality f]
   apply comp_id
 
-attribute [simp 1001] Cocone.w_assoc
-
 end
 
 variable {F : J ⥤ C}
diff --git a/Mathlib/CategoryTheory/Limits/Final.lean b/Mathlib/CategoryTheory/Limits/Final.lean
index 93331eeffeb40a..3431f905c7954d 100644
--- a/Mathlib/CategoryTheory/Limits/Final.lean
+++ b/Mathlib/CategoryTheory/Limits/Final.lean
@@ -335,11 +335,8 @@ section
 variable (G)
 
 /-- When `F : C ⥤ D` is final, and `G : D ⥤ E` has a colimit, then `F ⋙ G` has a colimit also and
-`colimit (F ⋙ G) ≅ colimit G`
-
-https://stacks.math.columbia.edu/tag/04E7
--/
-@[simps! (config := .lemmasOnly)]
+`colimit (F ⋙ G) ≅ colimit G`. -/
+@[simps! (config := .lemmasOnly), stacks 04E7]
 def colimitIso [HasColimit G] : colimit (F ⋙ G) ≅ colimit G :=
   asIso (colimit.pre G F)
 
@@ -684,11 +681,8 @@ section
 variable (G)
 
 /-- When `F : C ⥤ D` is initial, and `G : D ⥤ E` has a limit, then `F ⋙ G` has a limit also and
-`limit (F ⋙ G) ≅ limit G`
-
-https://stacks.math.columbia.edu/tag/04E7
--/
-@[simps! (config := .lemmasOnly)]
+`limit (F ⋙ G) ≅ limit G`. -/
+@[simps! (config := .lemmasOnly), stacks 04E7]
 def limitIso [HasLimit G] : limit (F ⋙ G) ≅ limit G :=
   (asIso (limit.pre G F)).symm
 
diff --git a/Mathlib/CategoryTheory/Limits/Indization/Category.lean b/Mathlib/CategoryTheory/Limits/Indization/Category.lean
index 7b2b555d9cb66e..c1cf76d2a04fe0 100644
--- a/Mathlib/CategoryTheory/Limits/Indization/Category.lean
+++ b/Mathlib/CategoryTheory/Limits/Indization/Category.lean
@@ -7,6 +7,7 @@ import Mathlib.CategoryTheory.Functor.Flat
 import Mathlib.CategoryTheory.Limits.Constructions.Filtered
 import Mathlib.CategoryTheory.Limits.FullSubcategory
 import Mathlib.CategoryTheory.Limits.Indization.LocallySmall
+import Mathlib.CategoryTheory.Limits.Indization.ParallelPair
 import Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
 import Mathlib.CategoryTheory.Limits.Indization.Products
 
@@ -30,7 +31,8 @@ Adopting the theorem numbering of [Kashiwara2006], we show that
 * if `C` has finite coproducts, then `Ind C` has small coproducts (6.1.18(ii)),
 * if `C` has products indexed by `α`, then `Ind C` has products indexed by `α`, and the functor
   `Ind C ⥤ Cᵒᵖ ⥤ Type v` creates such products (6.1.17)
-* the functor `C ⥤ Ind C` preserves small limits.
+* the functor `C ⥤ Ind C` preserves small limits,
+* if `C` has coequalizers, then `Ind C` has coequalizers (6.1.18(i)).
 
 More limit-colimit properties will follow.
 
@@ -43,7 +45,7 @@ Note that:
 * [M. Kashiwara, P. Schapira, *Categories and Sheaves*][Kashiwara2006], Chapter 6
 -/
 
-universe v u
+universe w v u
 
 namespace CategoryTheory
 
@@ -180,14 +182,14 @@ noncomputable def Ind.limCompInclusion {I : Type v} [SmallCategory I] [IsFiltere
   _ ≅ (whiskeringRight _ _ _).obj yoneda ⋙ colim :=
     isoWhiskerRight ((whiskeringRight _ _ _).mapIso (Ind.yonedaCompInclusion)) colim
 
-instance {α : Type v} [SmallCategory α] [FinCategory α] [HasLimitsOfShape α C] {I : Type v}
+instance {α : Type w} [SmallCategory α] [FinCategory α] [HasLimitsOfShape α C] {I : Type v}
     [SmallCategory I] [IsFiltered I] :
     PreservesLimitsOfShape α (Ind.lim I : (I ⥤ C) ⥤ _) :=
   haveI : PreservesLimitsOfShape α (Ind.lim I ⋙ Ind.inclusion C) :=
     preservesLimitsOfShape_of_natIso Ind.limCompInclusion.symm
   preservesLimitsOfShape_of_reflects_of_preserves _ (Ind.inclusion C)
 
-instance {α : Type v} [SmallCategory α] [FinCategory α] [HasColimitsOfShape α C] {I : Type v}
+instance {α : Type w} [SmallCategory α] [FinCategory α] [HasColimitsOfShape α C] {I : Type v}
     [SmallCategory I] [IsFiltered I] :
     PreservesColimitsOfShape α (Ind.lim I : (I ⥤ C) ⥤ _) :=
   inferInstanceAs (PreservesColimitsOfShape α (_ ⋙ colim))
@@ -209,11 +211,43 @@ instance {α : Type v} [Finite α] [HasColimitsOfShape (Discrete α) C] :
   -- ```
   -- from the fact that finite limits commute with filtered colimits and from the fact that
   -- `Ind.yoneda` preserves finite colimits.
-  apply hasColimitOfIso iso.symm
+  exact hasColimitOfIso iso.symm
 
 instance [HasFiniteCoproducts C] : HasCoproducts.{v} (Ind C) :=
   have : HasFiniteCoproducts (Ind C) :=
     ⟨fun _ => hasColimitsOfShape_of_equivalence (Discrete.equivalence Equiv.ulift)⟩
   hasCoproducts_of_finite_and_filtered
 
+/-- Given an `IndParallelPairPresentation f g`, we can understand the parallel pair `(f, g)` as
+the colimit of `(P.φ, P.ψ)` in `Ind C`. -/
+noncomputable def IndParallelPairPresentation.parallelPairIsoParallelPairCompIndYoneda
+    {A B : Ind C} {f g : A ⟶ B}
+    (P : IndParallelPairPresentation ((Ind.inclusion _).map f) ((Ind.inclusion _).map g)) :
+    parallelPair f g ≅ parallelPair P.φ P.ψ ⋙ Ind.lim P.I :=
+  ((whiskeringRight WalkingParallelPair _ _).obj (Ind.inclusion C)).preimageIso <|
+    diagramIsoParallelPair _ ≪≫
+      P.parallelPairIsoParallelPairCompYoneda ≪≫
+      isoWhiskerLeft (parallelPair _ _) Ind.limCompInclusion.symm
+
+instance [HasColimitsOfShape WalkingParallelPair C] :
+    HasColimitsOfShape WalkingParallelPair (Ind C) := by
+  refine ⟨fun F => ?_⟩
+  obtain ⟨P⟩ := nonempty_indParallelPairPresentation (F.obj WalkingParallelPair.zero).2
+    (F.obj WalkingParallelPair.one).2 (Ind.inclusion _ |>.map <| F.map WalkingParallelPairHom.left)
+    (Ind.inclusion _ |>.map <| F.map WalkingParallelPairHom.right)
+  exact hasColimitOfIso (diagramIsoParallelPair _ ≪≫ P.parallelPairIsoParallelPairCompIndYoneda)
+
+/-- A way to understand morphisms in `Ind C`: every morphism is induced by a natural transformation
+of diagrams. -/
+theorem Ind.exists_nonempty_arrow_mk_iso_ind_lim {A B : Ind C} {f : A ⟶ B} :
+    ∃ (I : Type v) (_ : SmallCategory I) (_ : IsFiltered I) (F G : I ⥤ C) (φ : F ⟶ G),
+      Nonempty (Arrow.mk f ≅ Arrow.mk ((Ind.lim _).map φ)) := by
+  obtain ⟨P⟩ := nonempty_indParallelPairPresentation A.2 B.2
+    (Ind.inclusion _ |>.map f) (Ind.inclusion _ |>.map f)
+  refine ⟨P.I, inferInstance, inferInstance, P.F₁, P.F₂, P.φ, ⟨Arrow.isoMk ?_ ?_ ?_⟩⟩
+  · exact P.parallelPairIsoParallelPairCompIndYoneda.app WalkingParallelPair.zero
+  · exact P.parallelPairIsoParallelPairCompIndYoneda.app WalkingParallelPair.one
+  · simpa using
+      (P.parallelPairIsoParallelPairCompIndYoneda.hom.naturality WalkingParallelPairHom.left).symm
+
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Limits/Indization/Equalizers.lean b/Mathlib/CategoryTheory/Limits/Indization/Equalizers.lean
index ef9fd1afa688dc..880fad04850627 100644
--- a/Mathlib/CategoryTheory/Limits/Indization/Equalizers.lean
+++ b/Mathlib/CategoryTheory/Limits/Indization/Equalizers.lean
@@ -4,16 +4,15 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Markus Himmel
 -/
 import Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
+import Mathlib.CategoryTheory.Limits.FullSubcategory
+import Mathlib.CategoryTheory.Limits.Indization.ParallelPair
 
 /-!
 # Equalizers of ind-objects
 
-The eventual goal of this file is to show that if a category `C` has equalizers, then ind-objects
-are closed under equalizers. For now, it contains one of the main prerequisites, namely we show
-that under sufficient conditions the limit of a diagram in `I ⥤ C` taken in `Cᵒᵖ ⥤ Type v` is an
-ind-object.
+We show that if a category `C` has equalizers, then ind-objects are closed under equalizers.
 
-## References
+# References
 * [M. Kashiwara, P. Schapira, *Categories and Sheaves*][Kashiwara2006], Section 6.1
 -/
 
@@ -60,4 +59,21 @@ theorem isIndObject_limit_comp_yoneda_comp_colim
 
 end
 
+/-- If `C` has equalizers. then ind-objects are closed under equalizers.
+
+This is Proposition 6.1.17(i) in [Kashiwara2006].
+-/
+theorem closedUnderLimitsOfShape_walkingParallelPair_isIndObject [HasEqualizers C] :
+    ClosedUnderLimitsOfShape WalkingParallelPair (IsIndObject (C := C)) := by
+  apply closedUnderLimitsOfShape_of_limit
+  intro F hF h
+  obtain ⟨P⟩ := nonempty_indParallelPairPresentation (h WalkingParallelPair.zero)
+    (h WalkingParallelPair.one) (F.map WalkingParallelPairHom.left)
+    (F.map WalkingParallelPairHom.right)
+  exact IsIndObject.map
+    (HasLimit.isoOfNatIso (P.parallelPairIsoParallelPairCompYoneda.symm ≪≫
+      (diagramIsoParallelPair _).symm)).hom
+    (isIndObject_limit_comp_yoneda_comp_colim (parallelPair P.φ P.ψ)
+      (fun i => isIndObject_limit_comp_yoneda _))
+
 end CategoryTheory.Limits
diff --git a/Mathlib/CategoryTheory/Limits/Indization/ParallelPair.lean b/Mathlib/CategoryTheory/Limits/Indization/ParallelPair.lean
new file mode 100644
index 00000000000000..68c898bc326347
--- /dev/null
+++ b/Mathlib/CategoryTheory/Limits/Indization/ParallelPair.lean
@@ -0,0 +1,171 @@
+/-
+Copyright (c) 2025 Markus Himmel. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Markus Himmel
+-/
+import Mathlib.CategoryTheory.Comma.Final
+import Mathlib.CategoryTheory.Limits.Indization.IndObject
+
+/-!
+# Parallel pairs of natural transformations between ind-objects
+
+We show that if `A` and `B` are ind-objects and `f` and `g` are natural transformations between
+`A` and `B`, then there is a small filtered category `I` such that `A`, `B`, `f` and `g` are
+commonly presented by diagrams and natural transformations in `I ⥤ C`.
+
+
+## References
+* [M. Kashiwara, P. Schapira, *Categories and Sheaves*][Kashiwara2006], Proposition 6.1.15 (though
+  our proof is more direct).
+-/
+
+universe v₁ v₂ v₃ u₁ u₂ u₃
+
+namespace CategoryTheory
+
+open Limits
+
+variable {C : Type u₁} [Category.{v₁} C]
+
+/-- Structure containing data exhibiting two parallel natural transformations `f` and `g` between
+presheaves `A` and `B` as induced by a natural transformation in a functor category exhibiting
+`A` and `B` as ind-objects. -/
+structure IndParallelPairPresentation {A B : Cᵒᵖ ⥤ Type v₁} (f g : A ⟶ B) where
+  /-- The indexing category. -/
+  I : Type v₁
+  /-- Category instance on the indexing category. -/
+  [ℐ : SmallCategory I]
+  [hI : IsFiltered I]
+  /-- The diagram presenting `A`. -/
+  F₁ : I ⥤ C
+  /-- The diagram presenting `B`. -/
+  F₂ : I ⥤ C
+  /-- The cocone on `F₁` with apex `A`. -/
+  ι₁ : F₁ ⋙ yoneda ⟶ (Functor.const I).obj A
+  /-- The cocone on `F₁` with apex `A` is a colimit cocone. -/
+  isColimit₁ : IsColimit (Cocone.mk A ι₁)
+  /-- The cocone on `F₂` with apex `B`. -/
+  ι₂ : F₂ ⋙ yoneda ⟶ (Functor.const I).obj B
+  /-- The cocone on `F₂` with apex `B` is a colimit cocone. -/
+  isColimit₂ : IsColimit (Cocone.mk B ι₂)
+  /-- The natural transformation presenting `f`. -/
+  φ : F₁ ⟶ F₂
+  /-- The natural transformation presenting `g`. -/
+  ψ : F₁ ⟶ F₂
+  /-- `f` is in fact presented by `φ`. -/
+  hf : f = IsColimit.map isColimit₁ (Cocone.mk B ι₂) (whiskerRight φ yoneda)
+  /-- `g` is in fact presented by `ψ`. -/
+  hg : g = IsColimit.map isColimit₁ (Cocone.mk B ι₂) (whiskerRight ψ yoneda)
+
+instance {A B : Cᵒᵖ ⥤ Type v₁} {f g : A ⟶ B} (P : IndParallelPairPresentation f g) :
+    SmallCategory P.I := P.ℐ
+instance {A B : Cᵒᵖ ⥤ Type v₁} {f g : A ⟶ B} (P : IndParallelPairPresentation f g) :
+    IsFiltered P.I := P.hI
+
+namespace NonemptyParallelPairPresentationAux
+
+variable {A B : Cᵒᵖ ⥤ Type v₁} (f g : A ⟶ B) (P₁ : IndObjectPresentation A)
+  (P₂ : IndObjectPresentation B)
+
+/-- Implementation; see `nonempty_indParallelPairPresentation`. -/
+abbrev K : Type v₁ :=
+  Comma ((P₁.toCostructuredArrow ⋙ CostructuredArrow.map f).prod'
+    (P₁.toCostructuredArrow ⋙ CostructuredArrow.map g))
+    (P₂.toCostructuredArrow.prod' P₂.toCostructuredArrow)
+
+/-- Implementation; see `nonempty_indParallelPairPresentation`. -/
+abbrev F₁ : K f g P₁ P₂ ⥤ C := Comma.fst _ _ ⋙ P₁.F
+/-- Implementation; see `nonempty_indParallelPairPresentation`. -/
+abbrev F₂ : K f g P₁ P₂ ⥤ C := Comma.snd _ _ ⋙ P₂.F
+
+/-- Implementation; see `nonempty_indParallelPairPresentation`. -/
+abbrev ι₁ : F₁ f g P₁ P₂ ⋙ yoneda ⟶ (Functor.const (K f g P₁ P₂)).obj A :=
+  whiskerLeft (Comma.fst _ _) P₁.ι
+
+/-- Implementation; see `nonempty_indParallelPairPresentation`. -/
+noncomputable abbrev isColimit₁ : IsColimit (Cocone.mk A (ι₁ f g P₁ P₂)) :=
+  (Functor.Final.isColimitWhiskerEquiv _ _).symm P₁.isColimit
+
+/-- Implementation; see `nonempty_indParallelPairPresentation`. -/
+abbrev ι₂ : F₂ f g P₁ P₂ ⋙ yoneda ⟶ (Functor.const (K f g P₁ P₂)).obj B :=
+  whiskerLeft (Comma.snd _ _) P₂.ι
+
+/-- Implementation; see `nonempty_indParallelPairPresentation`. -/
+noncomputable abbrev isColimit₂ : IsColimit (Cocone.mk B (ι₂ f g P₁ P₂)) :=
+  (Functor.Final.isColimitWhiskerEquiv _ _).symm P₂.isColimit
+
+/-- Implementation; see `nonempty_indParallelPairPresentation`. -/
+def ϕ : F₁ f g P₁ P₂ ⟶ F₂ f g P₁ P₂ where
+  app h := h.hom.1.left
+  naturality _ _ h := by
+    have := h.w
+    simp only [Functor.prod'_obj, Functor.comp_obj, prod_Hom, Functor.prod'_map, Functor.comp_map,
+      prod_comp, Prod.mk.injEq, CostructuredArrow.hom_eq_iff, CostructuredArrow.map_obj_left,
+      IndObjectPresentation.toCostructuredArrow_obj_left, CostructuredArrow.comp_left,
+      CostructuredArrow.map_map_left, IndObjectPresentation.toCostructuredArrow_map_left] at this
+    exact this.1
+
+theorem hf : f = IsColimit.map (isColimit₁ f g P₁ P₂)
+    (Cocone.mk B (ι₂ f g P₁ P₂)) (whiskerRight (ϕ f g P₁ P₂) yoneda) := by
+  refine (isColimit₁ f g P₁ P₂).hom_ext (fun i => ?_)
+  rw [IsColimit.ι_map]
+  simpa using i.hom.1.w.symm
+
+/-- Implementation; see `nonempty_indParallelPairPresentation`. -/
+def ψ : F₁ f g P₁ P₂ ⟶ F₂ f g P₁ P₂ where
+  app h := h.hom.2.left
+  naturality _ _ h := by
+    have := h.w
+    simp only [Functor.prod'_obj, Functor.comp_obj, prod_Hom, Functor.prod'_map, Functor.comp_map,
+      prod_comp, Prod.mk.injEq, CostructuredArrow.hom_eq_iff, CostructuredArrow.map_obj_left,
+      IndObjectPresentation.toCostructuredArrow_obj_left, CostructuredArrow.comp_left,
+      CostructuredArrow.map_map_left, IndObjectPresentation.toCostructuredArrow_map_left] at this
+    exact this.2
+
+theorem hg : g = IsColimit.map (isColimit₁ f g P₁ P₂)
+    (Cocone.mk B (ι₂ f g P₁ P₂)) (whiskerRight (ψ f g P₁ P₂) yoneda) := by
+  refine (isColimit₁ f g P₁ P₂).hom_ext (fun i => ?_)
+  rw [IsColimit.ι_map]
+  simpa using i.hom.2.w.symm
+
+attribute [local instance] Comma.isFiltered_of_final in
+/-- Implementation; see `nonempty_indParallelPairPresentation`. -/
+noncomputable def presentation : IndParallelPairPresentation f g where
+  I := K f g P₁ P₂
+  F₁ := F₁ f g P₁ P₂
+  F₂ := F₂ f g P₁ P₂
+  ι₁ := ι₁ f g P₁ P₂
+  isColimit₁ := isColimit₁ f g P₁ P₂
+  ι₂ := ι₂ f g P₁ P₂
+  isColimit₂ := isColimit₂ f g P₁ P₂
+  φ := ϕ f g P₁ P₂
+  ψ := ψ f g P₁ P₂
+  hf := hf f g P₁ P₂
+  hg := hg f g P₁ P₂
+
+end NonemptyParallelPairPresentationAux
+
+theorem nonempty_indParallelPairPresentation {A B : Cᵒᵖ ⥤ Type v₁} (hA : IsIndObject A)
+    (hB : IsIndObject B) (f g : A ⟶ B) : Nonempty (IndParallelPairPresentation f g) :=
+  ⟨NonemptyParallelPairPresentationAux.presentation f g hA.presentation hB.presentation⟩
+
+namespace IndParallelPairPresentation
+
+/-- Given an `IndParallelPairPresentation f g`, we can understand the parallel pair `(f, g)`
+as the colimit of `(P.φ, P.ψ)` in `Cᵒᵖ ⥤ Type v`. -/
+noncomputable def parallelPairIsoParallelPairCompYoneda {A B : Cᵒᵖ ⥤ Type v₁} {f g : A ⟶ B}
+    (P : IndParallelPairPresentation f g) :
+    parallelPair f g ≅ parallelPair P.φ P.ψ ⋙ (whiskeringRight _ _ _).obj yoneda ⋙ colim :=
+  parallelPair.ext
+    (P.isColimit₁.coconePointUniqueUpToIso (colimit.isColimit _))
+    (P.isColimit₂.coconePointUniqueUpToIso (colimit.isColimit _))
+    (P.isColimit₁.hom_ext (fun j => by
+      simp [P.hf, P.isColimit₁.ι_map_assoc, P.isColimit₁.comp_coconePointUniqueUpToIso_hom_assoc,
+        P.isColimit₂.comp_coconePointUniqueUpToIso_hom]))
+    (P.isColimit₁.hom_ext (fun j => by
+      simp [P.hg, P.isColimit₁.ι_map_assoc, P.isColimit₁.comp_coconePointUniqueUpToIso_hom_assoc,
+        P.isColimit₂.comp_coconePointUniqueUpToIso_hom]))
+
+end IndParallelPairPresentation
+
+end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Limits/Preorder.lean b/Mathlib/CategoryTheory/Limits/Preorder.lean
new file mode 100644
index 00000000000000..605ebbea32e56b
--- /dev/null
+++ b/Mathlib/CategoryTheory/Limits/Preorder.lean
@@ -0,0 +1,75 @@
+/-
+Copyright (c) 2025 Sina Hazratpour. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Sina Hazratpour, Joël Riou
+-/
+
+import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
+
+/-!
+# (Co)limits in a preorder category
+
+We provide basic results about the nullary and binary (co)products in the associated category of
+a preordered type.
+
+-/
+
+universe u
+
+open CategoryTheory Limits
+
+namespace Preorder
+
+variable (C : Type u)
+
+section OrderBot
+
+variable [Preorder C] [OrderBot C]
+
+/-- The least element in a preordered type is initial in the category
+associated to this preorder. -/
+def isInitialBot : IsInitial (⊥ : C) := IsInitial.ofUnique _
+
+instance : HasInitial C := hasInitial_of_unique ⊥
+
+end OrderBot
+
+section OrderTop
+
+variable [Preorder C] [OrderTop C]
+
+/-- The greatest element of a preordered type is terminal in the category
+associated to this preorder. -/
+def isTerminalTop : IsTerminal (⊤ : C) := IsTerminal.ofUnique _
+
+instance : HasTerminal C := hasTerminal_of_unique ⊤
+
+end OrderTop
+
+section SemilatticeInf
+
+variable {C} [SemilatticeInf C]
+
+/-- The infimum of two elements in a preordered type is a binary product in
+the category associated to this preorder. -/
+def isLimitBinaryFan (X Y : C) :
+    IsLimit (BinaryFan.mk (P := X ⊓ Y) (homOfLE inf_le_left) (homOfLE inf_le_right)) :=
+  BinaryFan.isLimitMk (fun s ↦ homOfLE (le_inf (leOfHom s.fst) (leOfHom s.snd)))
+    (by intros; rfl) (by intros; rfl) (by intros; rfl)
+
+end SemilatticeInf
+
+section SemilatticeSup
+
+variable {C} [SemilatticeSup C]
+
+/-- The supremum of two elements in a preordered type is a binary coproduct
+in the category associated to this preorder. -/
+def isColimitBinaryCofan (X Y : C) :
+    IsColimit (BinaryCofan.mk (P := X ⊔ Y) (homOfLE le_sup_left) (homOfLE le_sup_right)) :=
+  BinaryCofan.isColimitMk (fun s ↦ homOfLE (sup_le (leOfHom s.inl) (leOfHom s.inr)))
+    (by intros; rfl) (by intros; rfl) (by intros; rfl)
+
+end SemilatticeSup
+
+end Preorder
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Countable.lean b/Mathlib/CategoryTheory/Limits/Shapes/Countable.lean
index 6f0db0a360ddbb..9064ae11e1cb93 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Countable.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Countable.lean
@@ -219,9 +219,7 @@ instance sequentialFunctor_initial : (sequentialFunctor J).Initial where
     · right
       exact ⟨CostructuredArrow.homMk (homOfLE h).op rfl⟩
 
-/--
-This is proved in https://stacks.math.columbia.edu/tag/0032
--/
+@[stacks 0032]
 proof_wanted preorder_of_cofiltered (J : Type*) [Category J] [IsCofiltered J] :
     ∃ (I : Type*) (_ : Preorder I) (_ : IsCofiltered I) (F : I ⥤ J), F.Initial
 
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean b/Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
index 6e11689c3fc8bb..6dc28a06e73c91 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
@@ -420,7 +420,8 @@ def kernelCompMono {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [HasKernel f] [Mono g
         simp)
   inv := kernel.lift _ (kernel.ι _) (by simp)
 
-#adaptation_note /-- nightly-2024-04-01 The `symm` wasn't previously necessary. -/
+#adaptation_note /-- nightly-2024-04-01
+The `symm` wasn't previously necessary. -/
 instance hasKernel_iso_comp {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso f] [HasKernel g] :
     HasKernel (f ≫ g) where
   exists_limit :=
@@ -874,7 +875,8 @@ theorem cokernel_not_mono_of_nonzero (w : f ≠ 0) : ¬Mono (cokernel.π f) := f
 theorem cokernel_not_iso_of_nonzero (w : f ≠ 0) : IsIso (cokernel.π f) → False := fun _ =>
   cokernel_not_mono_of_nonzero w inferInstance
 
-#adaptation_note /-- nightly-2024-04-01 The `symm` wasn't previously necessary. -/
+#adaptation_note /-- nightly-2024-04-01
+The `symm` wasn't previously necessary. -/
 -- TODO the remainder of this section has obvious generalizations to `HasCoequalizer f g`.
 instance hasCokernel_comp_iso {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [HasCokernel f] [IsIso g] :
     HasCokernel (f ≫ g) where
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Preorder/Basic.lean b/Mathlib/CategoryTheory/Limits/Shapes/Preorder/Basic.lean
index fd2ca7692b49c4..cf21ab0f2fbac8 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Preorder/Basic.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Preorder/Basic.lean
@@ -3,7 +3,7 @@ Copyright (c) 2024 Joël Riou. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Joël Riou
 -/
-import Mathlib.CategoryTheory.Limits.Shapes.Terminal
+import Mathlib.CategoryTheory.Limits.Preorder
 
 /-!
 # Limits and colimits indexed by preorders
@@ -27,12 +27,6 @@ section OrderBot
 
 variable [OrderBot J]
 
-/-- The least element in a preordered type is initial in the category
-associated to this preorder. -/
-def isInitialBot : IsInitial (⊥ : J) := IsInitial.ofUnique _
-
-instance : HasInitial J := hasInitial_of_unique ⊥
-
 instance : HasLimitsOfShape J C := ⟨fun _ ↦ by infer_instance⟩
 
 end OrderBot
@@ -41,12 +35,6 @@ section OrderTop
 
 variable [OrderTop J]
 
-/-- The greatest element of a preordered type is terminal in the category
-associated to this preorder. -/
-def isTerminalBot : IsTerminal (⊤ : J) := IsTerminal.ofUnique _
-
-instance : HasTerminal J := hasTerminal_of_unique ⊤
-
 instance : HasColimitsOfShape J C := ⟨fun _ ↦ by infer_instance⟩
 
 end OrderTop
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Mono.lean b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Mono.lean
index 315be3a49b8d54..9574c460133413 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Mono.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Mono.lean
@@ -160,11 +160,7 @@ instance hasPullback_of_right_factors_mono : HasPullback i (f ≫ i) := by
 
 instance pullback_snd_iso_of_right_factors_mono :
     IsIso (pullback.snd i (f ≫ i)) := by
-  #adaptation_note /-- nightly-testing 2024-04-01
-  this could not be placed directly in the `show from` without `dsimp` -/
   have := limit.isoLimitCone_hom_π ⟨_, pullbackIsPullbackOfCompMono (𝟙 _) f i⟩ WalkingCospan.right
-  dsimp only [cospan_right, id_eq, eq_mpr_eq_cast, PullbackCone.mk_pt, PullbackCone.mk_π_app,
-    Functor.const_obj_obj, cospan_one] at this
   convert (congrArg IsIso (show _ ≫ pullback.snd (𝟙 Z) f = _ from this)).mp inferInstance
   · exact (Category.id_comp _).symm
   · exact (Category.id_comp _).symm
@@ -176,11 +172,7 @@ instance hasPullback_of_left_factors_mono : HasPullback (f ≫ i) i := by
 
 instance pullback_snd_iso_of_left_factors_mono :
     IsIso (pullback.fst (f ≫ i) i) := by
-  #adaptation_note /-- nightly-testing 2024-04-01
-  this could not be placed directly in the `show from` without `dsimp` -/
   have := limit.isoLimitCone_hom_π ⟨_, pullbackIsPullbackOfCompMono f (𝟙 _) i⟩ WalkingCospan.left
-  dsimp only [cospan_left, id_eq, eq_mpr_eq_cast, PullbackCone.mk_pt, PullbackCone.mk_π_app,
-    Functor.const_obj_obj, cospan_one] at this
   convert (congrArg IsIso (show _ ≫ pullback.fst f (𝟙 Z) = _ from this)).mp inferInstance
   · exact (Category.id_comp _).symm
   · exact (Category.id_comp _).symm
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Types.lean b/Mathlib/CategoryTheory/Limits/Shapes/Types.lean
index 5d87af2751e5f3..99b3d3e5f71a0b 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/Types.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/Types.lean
@@ -55,6 +55,8 @@ example [UnivLE.{v, u}] : HasProducts.{v} (Type u) := inferInstance
 instance : HasProducts.{v} (Type v) := inferInstance
 
 /-- A restatement of `Types.Limit.lift_π_apply` that uses `Pi.π` and `Pi.lift`. -/
+-- The increased `@[simp]` priority here results in a minor speed up in
+-- `Mathlib.CategoryTheory.Sites.EqualizerSheafCondition`.
 @[simp 1001]
 theorem pi_lift_π_apply {β : Type v} [Small.{u} β] (f : β → Type u) {P : Type u}
     (s : ∀ b, P ⟶ f b) (b : β) (x : P) :
@@ -69,7 +71,7 @@ theorem pi_lift_π_apply' {β : Type v} (f : β → Type v) {P : Type v}
   simp
 
 /-- A restatement of `Types.Limit.map_π_apply` that uses `Pi.π` and `Pi.map`. -/
-@[simp 1001]
+@[simp]
 theorem pi_map_π_apply {β : Type v} [Small.{u} β] {f g : β → Type u}
     (α : ∀ j, f j ⟶ g j) (b : β) (x) :
     (Pi.π g b : ∏ᶜ g → g b) (Pi.map α x) = α b ((Pi.π f b : ∏ᶜ f → f b) x) :=
diff --git a/Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean b/Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean
index 349e66c0e30f53..04dfbf1805cdec 100644
--- a/Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean
+++ b/Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean
@@ -517,11 +517,9 @@ abbrev wideEqualizer.ι : wideEqualizer f ⟶ X :=
 abbrev wideEqualizer.trident : Trident f :=
   limit.cone (parallelFamily f)
 
-@[simp]
 theorem wideEqualizer.trident_ι : (wideEqualizer.trident f).ι = wideEqualizer.ι f :=
   rfl
 
-@[simp 1100]
 theorem wideEqualizer.trident_π_app_zero :
     (wideEqualizer.trident f).π.app zero = wideEqualizer.ι f :=
   rfl
@@ -543,11 +541,11 @@ abbrev wideEqualizer.lift [Nonempty J] {W : C} (k : W ⟶ X) (h : ∀ j₁ j₂,
     W ⟶ wideEqualizer f :=
   limit.lift (parallelFamily f) (Trident.ofι k h)
 
-@[reassoc (attr := simp 1100)]
+@[reassoc]
 theorem wideEqualizer.lift_ι [Nonempty J] {W : C} (k : W ⟶ X)
     (h : ∀ j₁ j₂, k ≫ f j₁ = k ≫ f j₂) :
-    wideEqualizer.lift k h ≫ wideEqualizer.ι f = k :=
-  limit.lift_π _ _
+    wideEqualizer.lift k h ≫ wideEqualizer.ι f = k := by
+  simp
 
 /-- A morphism `k : W ⟶ X` satisfying `∀ j₁ j₂, k ≫ f j₁ = k ≫ f j₂` induces a morphism
     `l : W ⟶ wideEqualizer f` satisfying `l ≫ wideEqualizer.ι f = k`. -/
@@ -604,11 +602,9 @@ abbrev wideCoequalizer.π : Y ⟶ wideCoequalizer f :=
 abbrev wideCoequalizer.cotrident : Cotrident f :=
   colimit.cocone (parallelFamily f)
 
-@[simp]
 theorem wideCoequalizer.cotrident_π : (wideCoequalizer.cotrident f).π = wideCoequalizer.π f :=
   rfl
 
-@[simp 1100]
 theorem wideCoequalizer.cotrident_ι_app_one :
     (wideCoequalizer.cotrident f).ι.app one = wideCoequalizer.π f :=
   rfl
@@ -631,11 +627,11 @@ abbrev wideCoequalizer.desc [Nonempty J] {W : C} (k : Y ⟶ W) (h : ∀ j₁ j
     wideCoequalizer f ⟶ W :=
   colimit.desc (parallelFamily f) (Cotrident.ofπ k h)
 
-@[reassoc (attr := simp 1100)]
+@[reassoc]
 theorem wideCoequalizer.π_desc [Nonempty J] {W : C} (k : Y ⟶ W)
     (h : ∀ j₁ j₂, f j₁ ≫ k = f j₂ ≫ k) :
-    wideCoequalizer.π f ≫ wideCoequalizer.desc k h = k :=
-  colimit.ι_desc _ _
+    wideCoequalizer.π f ≫ wideCoequalizer.desc k h = k := by
+  simp
 
 /-- Any morphism `k : Y ⟶ W` satisfying `∀ j₁ j₂, f j₁ ≫ k = f j₂ ≫ k` induces a morphism
     `l : wideCoequalizer f ⟶ W` satisfying `wideCoequalizer.π ≫ g = l`. -/
diff --git a/Mathlib/CategoryTheory/Monad/Monadicity.lean b/Mathlib/CategoryTheory/Monad/Monadicity.lean
index 591201fa317258..b18ae5ed47930c 100644
--- a/Mathlib/CategoryTheory/Monad/Monadicity.lean
+++ b/Mathlib/CategoryTheory/Monad/Monadicity.lean
@@ -79,13 +79,6 @@ def comparisonLeftAdjointObj (A : adj.toMonad.Algebra)
     [HasCoequalizer (F.map A.a) (adj.counit.app _)] : D :=
   coequalizer (F.map A.a) (adj.counit.app _)
 
-#adaptation_note
-/--
-The new unused variable linter in
-https://github.com/leanprover/lean4/pull/5338
-flags `{ f : F.obj A.A ⟶ B // _ }`.
--/
-set_option linter.unusedVariables false in
 /--
 We have a bijection of homsets which will be used to construct the left adjoint to the comparison
 functor.
@@ -337,8 +330,8 @@ instance [CreatesColimitOfIsSplitPair G] : ∀ (A : Algebra adj.toMonad),
   fun _ => CreatesColimitOfIsSplitPair.out _ _
 
 /--
-Beck's monadicity theorem. If `G` has a left adjoint and creates coequalizers of `G`-split pairs,
-then it is monadic.
+**Beck's monadicity theorem**: if `G` has a left adjoint and creates coequalizers of `G`-split
+pairs, then it is monadic.
 This is the converse of `createsGSplitCoequalizersOfMonadic`.
 -/
 def monadicOfCreatesGSplitCoequalizers [CreatesColimitOfIsSplitPair G] :
@@ -351,7 +344,7 @@ def monadicOfCreatesGSplitCoequalizers [CreatesColimitOfIsSplitPair G] :
   have : ReflectsColimitOfIsSplitPair G := ⟨by intros; infer_instance⟩
   exact monadicOfHasPreservesReflectsGSplitCoequalizers adj
 
-/-- An alternate version of Beck's monadicity theorem. If `G` reflects isomorphisms, preserves
+/-- An alternate version of **Beck's monadicity theorem**: if `G` reflects isomorphisms, preserves
 coequalizers of `G`-split pairs and `C` has coequalizers of `G`-split pairs, then it is monadic.
 -/
 def monadicOfHasPreservesGSplitCoequalizersOfReflectsIsomorphisms [G.ReflectsIsomorphisms]
diff --git a/Mathlib/CategoryTheory/Monoidal/Mod_.lean b/Mathlib/CategoryTheory/Monoidal/Mod_.lean
index cfb9163cb552cb..570447a4586ad9 100644
--- a/Mathlib/CategoryTheory/Monoidal/Mod_.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Mod_.lean
@@ -88,14 +88,11 @@ def forget : Mod_ A ⥤ C where
 
 open CategoryTheory.MonoidalCategory
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/6053
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/6053
 we needed to increase the `maxHeartbeats` limit if we didn't write an explicit proof for
 `map_id` and `map_comp`.
 
-This may indicate a configuration problem in Aesop.
--/
+This may indicate a configuration problem in Aesop. -/
 /-- A morphism of monoid objects induces a "restriction" or "comap" functor
 between the categories of module objects.
 -/
diff --git a/Mathlib/CategoryTheory/Monoidal/Mon_.lean b/Mathlib/CategoryTheory/Monoidal/Mon_.lean
index f87726f0ec8203..b03e0f9b350c37 100644
--- a/Mathlib/CategoryTheory/Monoidal/Mon_.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Mon_.lean
@@ -246,14 +246,11 @@ namespace CategoryTheory.Functor
 
 variable {C} {D : Type u₂} [Category.{v₂} D] [MonoidalCategory.{v₂} D]
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/6053
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/6053
 we needed to increase the `maxHeartbeats` limit if we didn't write an explicit proof for
 `map_id` and `map_comp`.
 
-This may indicate a configuration problem in Aesop.
--/
+This may indicate a configuration problem in Aesop. -/
 -- TODO: mapMod F A : Mod A ⥤ Mod (F.mapMon A)
 /-- A lax monoidal functor takes monoid objects to monoid objects.
 
diff --git a/Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean b/Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
index 09afedcc3b9aa8..86dba8755a920d 100644
--- a/Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
+++ b/Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean
@@ -148,12 +148,10 @@ attribute [instance] HasLeftDual.exact
 
 open ExactPairing HasRightDual HasLeftDual MonoidalCategory
 
-#adaptation_note
-/--
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/4596
 The overlapping notation for `leftDual` and `leftAdjointMate` become more problematic in
 after https://github.com/leanprover/lean4/pull/4596, and we sometimes have to disambiguate with
-e.g. `(ᘁX : C)` where previously just `ᘁX` was enough.
--/
+e.g. `(ᘁX : C)` where previously just `ᘁX` was enough. -/
 
 @[inherit_doc] prefix:1024 "ᘁ" => leftDual
 @[inherit_doc] postfix:1024 "ᘁ" => rightDual
diff --git a/Mathlib/CategoryTheory/Preadditive/Transfer.lean b/Mathlib/CategoryTheory/Preadditive/Transfer.lean
new file mode 100644
index 00000000000000..273c431bbe7b02
--- /dev/null
+++ b/Mathlib/CategoryTheory/Preadditive/Transfer.lean
@@ -0,0 +1,63 @@
+/-
+Copyright (c) 2025 Jakob von Raumer. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jakob von Raumer
+-/
+import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
+import Mathlib.Algebra.Equiv.TransferInstance
+
+/-!
+# Pulling back a preadditive structure along a fully faithful functor
+
+A preadditive structure on a category `D` transfers to a preadditive structure on `C` for a given
+fully faithful functor `F : C ⥤ D`.
+-/
+namespace CategoryTheory
+
+open Limits
+
+universe v₁ v₂ u₁ u₂
+
+variable {C : Type u₁} [Category.{v₁} C]
+variable {D : Type u₂} [Category.{v₂} D] [Preadditive D]
+variable {F : C ⥤ D} (hF : F.FullyFaithful)
+
+namespace Preadditive
+
+
+/-- If `D` is a preadditive category, any fully faithful functor `F : C ⥤ D` induces a preadditive
+structure on `C`. -/
+def ofFullyFaithful : Preadditive C where
+  homGroup P Q := hF.homEquiv.addCommGroup
+  add_comp P Q R f f' g := hF.map_injective (by simp [Equiv.add_def])
+  comp_add P Q R f g g' := hF.map_injective (by simp [Equiv.add_def])
+
+end Preadditive
+
+open Preadditive
+namespace Functor.FullyFaithful
+
+/-- The preadditive structure on `C` induced by a fully faithful functor `F : C ⥤ D` makes `F` an
+additive functor. -/
+lemma additive_ofFullyFaithful :
+    letI : Preadditive C := Preadditive.ofFullyFaithful hF
+    F.Additive :=
+  letI : Preadditive C := Preadditive.ofFullyFaithful hF
+  { map_add := by simp [Equiv.add_def] }
+
+end Functor.FullyFaithful
+
+namespace Equivalence
+
+/-- The preadditive structure on `C` induced by an equivalence `e : C ≌ D` makes `e.inverse` an
+additive functor. -/
+lemma additive_inverse_of_FullyFaithful (e : C ≌ D) :
+    letI : Preadditive C := ofFullyFaithful e.fullyFaithfulFunctor
+    e.inverse.Additive :=
+  letI : Preadditive C := ofFullyFaithful e.fullyFaithfulFunctor
+  letI : e.functor.Additive := e.fullyFaithfulFunctor.additive_ofFullyFaithful
+  e.inverse_additive
+
+end Equivalence
+
+end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Shift/ShiftSequence.lean b/Mathlib/CategoryTheory/Shift/ShiftSequence.lean
index 78359306417b7b..fce6ef65bd5b59 100644
--- a/Mathlib/CategoryTheory/Shift/ShiftSequence.lean
+++ b/Mathlib/CategoryTheory/Shift/ShiftSequence.lean
@@ -85,17 +85,17 @@ def shiftIso (n a a' : M) (ha' : n + a = a') :
     shiftFunctor C n ⋙ F.shift a ≅ F.shift a' :=
   ShiftSequence.shiftIso n a a' ha'
 
-@[reassoc (attr := simp 1100)]
+@[reassoc (attr := simp)]
 lemma shiftIso_hom_naturality {X Y : C} (n a a' : M) (ha' : n + a = a') (f : X ⟶ Y) :
     (shift F a).map (f⟦n⟧') ≫ (shiftIso F n a a' ha').hom.app Y =
       (shiftIso F n a a' ha').hom.app X ≫ (shift F a').map f :=
   (F.shiftIso n a a' ha').hom.naturality f
 
-@[reassoc (attr := simp 1100)]
+@[reassoc]
 lemma shiftIso_inv_naturality {X Y : C} (n a a' : M) (ha' : n + a = a') (f : X ⟶ Y) :
     (shift F a').map f ≫ (shiftIso F n a a' ha').inv.app Y =
-      (shiftIso F n a a' ha').inv.app X ≫ (shift F a).map (f⟦n⟧') :=
-  (F.shiftIso n a a' ha').inv.naturality f
+      (shiftIso F n a a' ha').inv.app X ≫ (shift F a).map (f⟦n⟧') := by
+  simp
 
 variable (M)
 
diff --git a/Mathlib/CategoryTheory/Shift/ShiftedHom.lean b/Mathlib/CategoryTheory/Shift/ShiftedHom.lean
index 09bea544345c60..0f182e8d965ba7 100644
--- a/Mathlib/CategoryTheory/Shift/ShiftedHom.lean
+++ b/Mathlib/CategoryTheory/Shift/ShiftedHom.lean
@@ -92,7 +92,7 @@ lemma comp_mk₀_id {a : M} (f : ShiftedHom X Y a) (m₀ : M) (hm₀ : m₀ = 0)
     f.comp (mk₀ m₀ hm₀ (𝟙 Y)) (by rw [hm₀, zero_add]) = f := by
   simp [comp_mk₀]
 
-@[simp 1100]
+@[simp]
 lemma mk₀_comp_mk₀ (f : X ⟶ Y) (g : Y ⟶ Z) {a b c : M} (h : b + a = c)
     (ha : a = 0) (hb : b = 0) :
     (mk₀ a ha f).comp (mk₀ b hb g) h = mk₀ c (by rw [← h, ha, hb, add_zero]) (f ≫ g) := by
diff --git a/Mathlib/CategoryTheory/Sites/Canonical.lean b/Mathlib/CategoryTheory/Sites/Canonical.lean
index 8dd61b381382af..be99c646d4406d 100644
--- a/Mathlib/CategoryTheory/Sites/Canonical.lean
+++ b/Mathlib/CategoryTheory/Sites/Canonical.lean
@@ -139,11 +139,9 @@ theorem isSheafFor_trans (P : Cᵒᵖ ⥤ Type v) (R S : Sieve X)
     rw [this]
     apply hR' hf
 
-/-- Construct the finest (largest) Grothendieck topology for which the given presheaf is a sheaf.
-
-This is a special case of https://stacks.math.columbia.edu/tag/00Z9, but following a different
-proof (see the comments there).
--/
+/-- Construct the finest (largest) Grothendieck topology for which the given presheaf is a sheaf. -/
+@[stacks 00Z9 "This is a special case of the Stacks entry, but following a different
+proof (see the Stacks comments)."]
 def finestTopologySingle (P : Cᵒᵖ ⥤ Type v) : GrothendieckTopology C where
   sieves X S := ∀ (Y) (f : Y ⟶ X), Presieve.IsSheafFor P (S.pullback f : Presieve Y)
   top_mem' X Y f := by
@@ -164,11 +162,9 @@ def finestTopologySingle (P : Cᵒᵖ ⥤ Type v) : GrothendieckTopology C where
       rw [pullback_id, pullback_comp] at this
       apply this
 
-/--
-Construct the finest (largest) Grothendieck topology for which all the given presheaves are sheaves.
-
-This is equal to the construction of <https://stacks.math.columbia.edu/tag/00Z9>.
--/
+/-- Construct the finest (largest) Grothendieck topology for which all the given presheaves are
+sheaves. -/
+@[stacks 00Z9 "Equal to that Stacks construction"]
 def finestTopology (Ps : Set (Cᵒᵖ ⥤ Type v)) : GrothendieckTopology C :=
   sInf (finestTopologySingle '' Ps)
 
diff --git a/Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean b/Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean
index 5137ac167f17ea..820d2cdb384bee 100644
--- a/Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean
+++ b/Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean
@@ -87,13 +87,7 @@ instance : haveI := F.reflects_precoherent;
     F.IsDenseSubsite (coherentTopology C) (coherentTopology D) where
   functorPushforward_mem_iff := by
     rw [eq_induced F]
-    #adaptation_note
-    /--
-    This proof used to be `rfl`,
-    but has been temporarily broken by https://github.com/leanprover/lean4/pull/5329.
-    It can hopefully be restored after https://github.com/leanprover/lean4/pull/5359
-    -/
-    exact Iff.rfl
+    rfl
 
 lemma coverPreserving : haveI := F.reflects_precoherent
     CoverPreserving (coherentTopology _) (coherentTopology _) F :=
@@ -191,13 +185,7 @@ instance : haveI := F.reflects_preregular;
     F.IsDenseSubsite (regularTopology C) (regularTopology D) where
   functorPushforward_mem_iff := by
     rw [eq_induced F]
-    #adaptation_note
-    /--
-    This proof used to be `rfl`,
-    but has been temporarily broken by https://github.com/leanprover/lean4/pull/5329.
-    It can hopefully be restored after https://github.com/leanprover/lean4/pull/5359
-    -/
-    exact Iff.rfl
+    rfl
 
 lemma coverPreserving : haveI := F.reflects_preregular
     CoverPreserving (regularTopology _) (regularTopology _) F :=
diff --git a/Mathlib/CategoryTheory/Sites/CoverLifting.lean b/Mathlib/CategoryTheory/Sites/CoverLifting.lean
index b6098d47f159a5..db34551ea03238 100644
--- a/Mathlib/CategoryTheory/Sites/CoverLifting.lean
+++ b/Mathlib/CategoryTheory/Sites/CoverLifting.lean
@@ -211,11 +211,9 @@ variable [∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F]
 
 /-- If `G` is cocontinuous, then `G.op.ran` pushes sheaves to sheaves.
 
-This is SGA 4 III 2.2. An alternative reference is
-https://stacks.math.columbia.edu/tag/00XK (where results
-are obtained under the additional assumption that
-`C` and `D` have pullbacks).
--/
+This is SGA 4 III 2.2. -/
+@[stacks 00XK "Alternative reference. There, results are obtained under the additional assumption
+that `C` and `D` have pullbacks."]
 theorem ran_isSheaf_of_isCocontinuous (ℱ : Sheaf J A) :
     Presheaf.IsSheaf K (G.op.ran.obj ℱ.val) := by
   rw [Presheaf.isSheaf_iff_multifork]
diff --git a/Mathlib/CategoryTheory/Sites/CoverPreserving.lean b/Mathlib/CategoryTheory/Sites/CoverPreserving.lean
index fce2b79404c900..f7d3acf8d8072d 100644
--- a/Mathlib/CategoryTheory/Sites/CoverPreserving.lean
+++ b/Mathlib/CategoryTheory/Sites/CoverPreserving.lean
@@ -154,11 +154,8 @@ theorem compatiblePreservingOfDownwardsClosed (F : C ⥤ D) [F.Full] [F.Faithful
 
 variable {F J K}
 
-/-- If `F` is cover-preserving and compatible-preserving,
-then `F` is a continuous functor.
-
-This result is basically <https://stacks.math.columbia.edu/tag/00WW>.
--/
+/-- If `F` is cover-preserving and compatible-preserving, then `F` is a continuous functor. -/
+@[stacks 00WW "This is basically this Stacks entry."]
 lemma Functor.isContinuous_of_coverPreserving (hF₁ : CompatiblePreserving.{w} K F)
     (hF₂ : CoverPreserving J K F) : Functor.IsContinuous.{w} F J K where
   op_comp_isSheaf_of_types G X S hS x hx := by
diff --git a/Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean b/Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean
index 406251e3f3f1bf..c52235dba062f2 100644
--- a/Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean
+++ b/Mathlib/CategoryTheory/Sites/EqualizerSheafCondition.lean
@@ -45,8 +45,9 @@ noncomputable section
 
 /--
 The middle object of the fork diagram given in Equation (3) of [MM92], as well as the fork diagram
-of <https://stacks.math.columbia.edu/tag/00VM>.
+of the Stacks entry.
 -/
+@[stacks 00VM "This is the middle object of the fork diagram there."]
 def FirstObj : Type max v u :=
   ∏ᶜ fun f : ΣY, { f : Y ⟶ X // R f } => P.obj (op f.1)
 
@@ -78,8 +79,9 @@ instance : Inhabited (FirstObj P ((⊥ : Sieve X) : Presieve X)) :=
 
 /--
 The left morphism of the fork diagram given in Equation (3) of [MM92], as well as the fork diagram
-of <https://stacks.math.columbia.edu/tag/00VM>.
+of the Stacks entry.
 -/
+@[stacks 00VM "This is the left morphism of the fork diagram there."]
 def forkMap : P.obj (op X) ⟶ FirstObj P R :=
   Pi.lift fun f => P.map f.2.1.op
 
@@ -176,15 +178,17 @@ namespace Presieve
 variable [R.hasPullbacks]
 
 /--
-The rightmost object of the fork diagram of https://stacks.math.columbia.edu/tag/00VM, which
+The rightmost object of the fork diagram of the Stacks entry, which
 contains the data used to check a family of elements for a presieve is compatible.
 -/
-@[simp] def SecondObj : Type max v u :=
+@[simp, stacks 00VM "This is the rightmost object of the fork diagram there."]
+def SecondObj : Type max v u :=
   ∏ᶜ fun fg : (ΣY, { f : Y ⟶ X // R f }) × ΣZ, { g : Z ⟶ X // R g } =>
     haveI := Presieve.hasPullbacks.has_pullbacks fg.1.2.2 fg.2.2.2
     P.obj (op (pullback fg.1.2.1 fg.2.2.1))
 
-/-- The map `pr₀*` of <https://stacks.math.columbia.edu/tag/00VL>. -/
+/-- The map `pr₀*` of the Stacks entry. -/
+@[stacks 00VM "This is the map `pr₀*` there."]
 def firstMap : FirstObj P R ⟶ SecondObj P R :=
   Pi.lift fun fg =>
     haveI := Presieve.hasPullbacks.has_pullbacks fg.1.2.2 fg.2.2.2
@@ -193,7 +197,8 @@ def firstMap : FirstObj P R ⟶ SecondObj P R :=
 instance [HasPullbacks C] : Inhabited (SecondObj P (⊥ : Presieve X)) :=
   ⟨firstMap _ _ default⟩
 
-/-- The map `pr₁*` of <https://stacks.math.columbia.edu/tag/00VL>. -/
+/-- The map `pr₁*` of the Stacks entry. -/
+@[stacks 00VM "This is the map `pr₁*` there."]
 def secondMap : FirstObj P R ⟶ SecondObj P R :=
   Pi.lift fun fg =>
     haveI := Presieve.hasPullbacks.has_pullbacks fg.1.2.2 fg.2.2.2
@@ -254,10 +259,11 @@ variable {B : C} {I : Type} (X : I → C) (π : (i : I) → X i ⟶ B)
 -- TODO: allow `I : Type w`
 
 /--
-The middle object of the fork diagram of <https://stacks.math.columbia.edu/tag/00VM>.
+The middle object of the fork diagram of the Stacks entry.
 The difference between this and `Equalizer.FirstObj P (ofArrows X π)` arises if the family of
 arrows `π` contains duplicates. The `Presieve.ofArrows` doesn't see those.
 -/
+@[stacks 00VM "The middle object of the fork diagram there."]
 def FirstObj : Type w := ∏ᶜ (fun i ↦ P.obj (op (X i)))
 
 @[ext]
@@ -268,10 +274,11 @@ lemma FirstObj.ext (z₁ z₂ : FirstObj P X) (h : ∀ i, (Pi.π _ i : FirstObj
   exact h i
 
 /--
-The rightmost object of the fork diagram of https://stacks.math.columbia.edu/tag/00VM.
+The rightmost object of the fork diagram of the Stacks entry.
 The difference between this and `Equalizer.Presieve.SecondObj P (ofArrows X π)` arises if the
 family of arrows `π` contains duplicates. The `Presieve.ofArrows` doesn't see those.
 -/
+@[stacks 00VM "The rightmost object of the fork diagram there."]
 def SecondObj : Type w  :=
   ∏ᶜ (fun (ij : I × I) ↦ P.obj (op (pullback (π ij.1) (π ij.2))))
 
diff --git a/Mathlib/CategoryTheory/Sites/Grothendieck.lean b/Mathlib/CategoryTheory/Sites/Grothendieck.lean
index 816a9f82f51ce0..d581e6a006f6af 100644
--- a/Mathlib/CategoryTheory/Sites/Grothendieck.lean
+++ b/Mathlib/CategoryTheory/Sites/Grothendieck.lean
@@ -265,13 +265,7 @@ lemma mem_sInf (s : Set (GrothendieckTopology C)) {X : C} (S : Sieve X) :
 @[stacks 00Z7]
 theorem isGLB_sInf (s : Set (GrothendieckTopology C)) : IsGLB s (sInf s) := by
   refine @IsGLB.of_image _ _ _ _ sieves ?_ _ _ ?_
-  · #adaptation_note
-    /--
-    This proof used to be `rfl`,
-    but has been temporarily broken by https://github.com/leanprover/lean4/pull/5329.
-    It can hopefully be restored after https://github.com/leanprover/lean4/pull/5359
-    -/
-    exact Iff.rfl
+  · rfl
   · exact _root_.isGLB_sInf _
 
 /-- Construct a complete lattice from the `Inf`, but make the trivial and discrete topologies
diff --git a/Mathlib/CategoryTheory/Sites/IsSheafFor.lean b/Mathlib/CategoryTheory/Sites/IsSheafFor.lean
index 0d7effb4b905f3..e229cdcd97a5f1 100644
--- a/Mathlib/CategoryTheory/Sites/IsSheafFor.lean
+++ b/Mathlib/CategoryTheory/Sites/IsSheafFor.lean
@@ -83,9 +83,10 @@ A presheaf is a sheaf (resp, separated) if every *compatible* family of elements
 (resp, at most one) amalgamation.
 
 This data is referred to as a `family` in [MM92], Chapter III, Section 4. It is also a concrete
-version of the elements of the middle object in https://stacks.math.columbia.edu/tag/00VM which is
+version of the elements of the middle object in the Stacks entry which is
 more useful for direct calculations. It is also used implicitly in Definition C2.1.2 in [Elephant].
 -/
+@[stacks 00VM "This is a concrete version of the elements of the middle object there."]
 def FamilyOfElements (P : Cᵒᵖ ⥤ Type w) (R : Presieve X) :=
   ∀ ⦃Y : C⦄ (f : Y ⟶ X), R f → P.obj (op Y)
 
@@ -416,8 +417,8 @@ This version is also useful to establish that being a sheaf is preserved under i
 presheaves.
 
 See the discussion before Equation (3) of [MM92], Chapter III, Section 4. See also C2.1.4 of
-[Elephant]. This is also a direct reformulation of <https://stacks.math.columbia.edu/tag/00Z8>.
--/
+[Elephant]. -/
+@[stacks 00Z8 "Direct reformulation"]
 def YonedaSheafCondition (P : Cᵒᵖ ⥤ Type v₁) (S : Sieve X) : Prop :=
   ∀ f : S.functor ⟶ P, ∃! g, S.functorInclusion ≫ g = f
 
diff --git a/Mathlib/CategoryTheory/Sites/LocallyInjective.lean b/Mathlib/CategoryTheory/Sites/LocallyInjective.lean
index 64b1d4140a669e..9f60ac73c6f1a7 100644
--- a/Mathlib/CategoryTheory/Sites/LocallyInjective.lean
+++ b/Mathlib/CategoryTheory/Sites/LocallyInjective.lean
@@ -243,8 +243,8 @@ lemma mono_of_isLocallyInjective [IsLocallyInjective φ] : Mono φ := by
   infer_instance
 
 instance {F G : Sheaf J (Type w)} (f : F ⟶ G) :
-    IsLocallyInjective (imageSheafι f) := by
-  dsimp [imageSheafι]
+    IsLocallyInjective (Sheaf.imageι f) := by
+  dsimp [Sheaf.imageι]
   infer_instance
 
 end Sheaf
diff --git a/Mathlib/CategoryTheory/Sites/LocallySurjective.lean b/Mathlib/CategoryTheory/Sites/LocallySurjective.lean
index e3c5a1393050fd..390f9e08622c43 100644
--- a/Mathlib/CategoryTheory/Sites/LocallySurjective.lean
+++ b/Mathlib/CategoryTheory/Sites/LocallySurjective.lean
@@ -50,7 +50,7 @@ def imageSieve {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) {U : C} (s : G.obj (op U)) :
     rw [op_comp, G.map_comp, comp_apply, ← ht, elementwise_of% f.naturality]
 
 theorem imageSieve_eq_sieveOfSection {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) {U : C} (s : G.obj (op U)) :
-    imageSieve f s = (imagePresheaf (whiskerRight f (forget A))).sieveOfSection s :=
+    imageSieve f s = (Subpresheaf.range (whiskerRight f (forget A))).sieveOfSection s :=
   rfl
 
 theorem imageSieve_whisker_forget {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) {U : C} (s : G.obj (op U)) :
@@ -92,13 +92,13 @@ instance {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) [IsLocallySurjective J f] :
   imageSieve_mem s := imageSieve_mem J f s
 
 theorem isLocallySurjective_iff_imagePresheaf_sheafify_eq_top {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) :
-    IsLocallySurjective J f ↔ (imagePresheaf (whiskerRight f (forget A))).sheafify J = ⊤ := by
-  simp only [Subpresheaf.ext_iff, funext_iff, Set.ext_iff, top_subpresheaf_obj,
+    IsLocallySurjective J f ↔ (Subpresheaf.range (whiskerRight f (forget A))).sheafify J = ⊤ := by
+  simp only [Subpresheaf.ext_iff, funext_iff, Set.ext_iff, Subpresheaf.top_obj,
     Set.top_eq_univ, Set.mem_univ, iff_true]
   exact ⟨fun H _ => H.imageSieve_mem, fun H => ⟨H _⟩⟩
 
 theorem isLocallySurjective_iff_imagePresheaf_sheafify_eq_top' {F G : Cᵒᵖ ⥤ Type w} (f : F ⟶ G) :
-    IsLocallySurjective J f ↔ (imagePresheaf f).sheafify J = ⊤ := by
+    IsLocallySurjective J f ↔ (Subpresheaf.range f).sheafify J = ⊤ := by
   apply isLocallySurjective_iff_imagePresheaf_sheafify_eq_top
 
 theorem isLocallySurjective_iff_whisker_forget {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) :
@@ -248,7 +248,7 @@ lemma isLocallySurjective_comp_iff
     infer_instance
 
 instance {F₁ F₂ : Cᵒᵖ ⥤ Type w} (f : F₁ ⟶ F₂) :
-    IsLocallySurjective J (toImagePresheafSheafify J f) where
+    IsLocallySurjective J (Subpresheaf.toRangeSheafify J f) where
   imageSieve_mem {X} := by
     rintro ⟨s, hs⟩
     refine J.superset_covering ?_ hs
@@ -257,20 +257,20 @@ instance {F₁ F₂ : Cᵒᵖ ⥤ Type w} (f : F₁ ⟶ F₂) :
 
 /-- The image of `F` in `J.sheafify F` is isomorphic to the sheafification. -/
 noncomputable def sheafificationIsoImagePresheaf (F : Cᵒᵖ ⥤ Type max u v) :
-    J.sheafify F ≅ ((imagePresheaf (J.toSheafify F)).sheafify J).toPresheaf where
+    J.sheafify F ≅ ((Subpresheaf.range (J.toSheafify F)).sheafify J).toPresheaf where
   hom :=
-    J.sheafifyLift (toImagePresheafSheafify J _)
+    J.sheafifyLift (Subpresheaf.toRangeSheafify J _)
       ((isSheaf_iff_isSheaf_of_type J _).mpr <|
         Subpresheaf.sheafify_isSheaf _ <|
           (isSheaf_iff_isSheaf_of_type J _).mp <| GrothendieckTopology.sheafify_isSheaf J _)
   inv := Subpresheaf.ι _
   hom_inv_id :=
-    J.sheafify_hom_ext _ _ (J.sheafify_isSheaf _) (by simp [toImagePresheafSheafify])
+    J.sheafify_hom_ext _ _ (J.sheafify_isSheaf _) (by simp [Subpresheaf.toRangeSheafify])
   inv_hom_id := by
     rw [← cancel_mono (Subpresheaf.ι _), Category.id_comp, Category.assoc]
     refine Eq.trans ?_ (Category.comp_id _)
     congr 1
-    exact J.sheafify_hom_ext _ _ (J.sheafify_isSheaf _) (by simp [toImagePresheafSheafify])
+    exact J.sheafify_hom_ext _ _ (J.sheafify_isSheaf _) (by simp [Subpresheaf.toRangeSheafify])
 
 section
 
@@ -328,8 +328,8 @@ instance isLocallySurjective_of_iso [IsIso φ] : IsLocallySurjective φ := by
   infer_instance
 
 instance {F G : Sheaf J (Type w)} (f : F ⟶ G) :
-    IsLocallySurjective (toImageSheaf f) := by
-  dsimp [toImageSheaf]
+    IsLocallySurjective (Sheaf.toImage f) := by
+  dsimp [Sheaf.toImage]
   infer_instance
 
 variable [J.HasSheafCompose (forget A)]
@@ -339,11 +339,11 @@ instance [IsLocallySurjective φ] :
   (Presheaf.isLocallySurjective_iff_whisker_forget J φ.val).1 inferInstance
 
 theorem isLocallySurjective_iff_isIso {F G : Sheaf J (Type w)} (f : F ⟶ G) :
-    IsLocallySurjective f ↔ IsIso (imageSheafι f) := by
+    IsLocallySurjective f ↔ IsIso (Sheaf.imageι f) := by
   dsimp only [IsLocallySurjective]
-  rw [imageSheafι, Presheaf.isLocallySurjective_iff_imagePresheaf_sheafify_eq_top',
+  rw [Sheaf.imageι, Presheaf.isLocallySurjective_iff_imagePresheaf_sheafify_eq_top',
     Subpresheaf.eq_top_iff_isIso]
-  exact isIso_iff_of_reflects_iso (f := imageSheafι f) (F := sheafToPresheaf J (Type w))
+  exact isIso_iff_of_reflects_iso (f := Sheaf.imageι f) (F := sheafToPresheaf J (Type w))
 
 instance epi_of_isLocallySurjective' {F₁ F₂ : Sheaf J (Type w)} (φ : F₁ ⟶ F₂)
     [IsLocallySurjective φ] : Epi φ where
@@ -369,7 +369,7 @@ lemma isLocallySurjective_iff_epi {F G : Sheaf J (Type w)} (φ : F ⟶ G)
   · intro
     infer_instance
   · intro
-    have := epi_of_epi_fac (toImageSheaf_ι φ)
+    have := epi_of_epi_fac (Sheaf.toImage_ι φ)
     rw [isLocallySurjective_iff_isIso φ]
     apply isIso_of_mono_of_epi
 
diff --git a/Mathlib/CategoryTheory/Sites/NonabelianCohomology/H1.lean b/Mathlib/CategoryTheory/Sites/NonabelianCohomology/H1.lean
index b15c8b4f9117fa..4452691a4e8ba5 100644
--- a/Mathlib/CategoryTheory/Sites/NonabelianCohomology/H1.lean
+++ b/Mathlib/CategoryTheory/Sites/NonabelianCohomology/H1.lean
@@ -57,11 +57,8 @@ instance : Group (ZeroCochain G U) := Pi.group
 
 namespace Cochain₀
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/4481
-the `simpNF` linter incorrectly claims this lemma can't be applied by `simp`.
--/
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/4481
+the `simpNF` linter incorrectly claims this lemma can't be applied by `simp`. -/
 @[simp, nolint simpNF]
 lemma one_apply (i : I) : (1 : ZeroCochain G U) i = 1 := rfl
 
diff --git a/Mathlib/CategoryTheory/Sites/Pretopology.lean b/Mathlib/CategoryTheory/Sites/Pretopology.lean
index 6285e9b9a6bde9..1732a166ae6f3a 100644
--- a/Mathlib/CategoryTheory/Sites/Pretopology.lean
+++ b/Mathlib/CategoryTheory/Sites/Pretopology.lean
@@ -52,12 +52,10 @@ three axioms:
 In some sense, a pretopology can be seen as Grothendieck topology with weaker saturation conditions,
 in that each covering is not necessarily downward closed.
 
-See: https://ncatlab.org/nlab/show/Grothendieck+pretopology, or
-https://stacks.math.columbia.edu/tag/00VH, or [MM92] Chapter III, Section 2, Definition 2.
-Note that Stacks calls a category together with a pretopology a site, and [MM92] calls this
-a basis for a topology.
--/
-@[ext]
+See: https://ncatlab.org/nlab/show/Grothendieck+pretopology or [MM92] Chapter III,
+Section 2, Definition 2. -/
+@[ext, stacks 00VH "Note that Stacks calls a category together with a pretopology a site,
+and [MM92] calls this a basis for a topology."]
 structure Pretopology where
   coverings : ∀ X : C, Set (Presieve X)
   has_isos : ∀ ⦃X Y⦄ (f : Y ⟶ X) [IsIso f], Presieve.singleton f ∈ coverings X
diff --git a/Mathlib/CategoryTheory/Sites/Sheaf.lean b/Mathlib/CategoryTheory/Sites/Sheaf.lean
index 9b7c318a4906ac..b213f408e9c076 100644
--- a/Mathlib/CategoryTheory/Sites/Sheaf.lean
+++ b/Mathlib/CategoryTheory/Sites/Sheaf.lean
@@ -68,10 +68,8 @@ variable (J : GrothendieckTopology C)
 
 -- We follow https://stacks.math.columbia.edu/tag/00VL definition 00VR
 /-- A sheaf of A is a presheaf P : Cᵒᵖ => A such that for every E : A, the
-presheaf of types given by sending U : C to Hom_{A}(E, P U) is a sheaf of types.
-
-https://stacks.math.columbia.edu/tag/00VR
--/
+presheaf of types given by sending U : C to Hom_{A}(E, P U) is a sheaf of types. -/
+@[stacks 00VR]
 def IsSheaf (P : Cᵒᵖ ⥤ A) : Prop :=
   ∀ E : A, Presieve.IsSheaf J (P ⋙ coyoneda.obj (op E))
 
@@ -586,34 +584,35 @@ section
 variable [HasProducts.{max u₁ v₁} A]
 variable [HasProducts.{max u₁ v₁} A']
 
-/--
-The middle object of the fork diagram given in Equation (3) of [MM92], as well as the fork diagram
-of <https://stacks.math.columbia.edu/tag/00VM>.
--/
+/-- The middle object of the fork diagram given in Equation (3) of [MM92], as well as the fork
+diagram of the Stacks entry. -/
+@[stacks 00VM "The middle object of the fork diagram there."]
 def firstObj : A :=
   ∏ᶜ fun f : ΣV, { f : V ⟶ U // R f } => P.obj (op f.1)
 
-/--
-The left morphism of the fork diagram given in Equation (3) of [MM92], as well as the fork diagram
-of <https://stacks.math.columbia.edu/tag/00VM>.
--/
+/-- The left morphism of the fork diagram given in Equation (3) of [MM92], as well as the fork
+diagram of the Stacks entry. -/
+@[stacks 00VM "The left morphism the fork diagram there."]
 def forkMap : P.obj (op U) ⟶ firstObj R P :=
   Pi.lift fun f => P.map f.2.1.op
 
 variable [HasPullbacks C]
 
-/-- The rightmost object of the fork diagram of https://stacks.math.columbia.edu/tag/00VM, which
+/-- The rightmost object of the fork diagram of the Stacks entry, which
 contains the data used to check a family of elements for a presieve is compatible.
 -/
+@[stacks 00VM "The rightmost object of the fork diagram there."]
 def secondObj : A :=
   ∏ᶜ fun fg : (ΣV, { f : V ⟶ U // R f }) × ΣW, { g : W ⟶ U // R g } =>
     P.obj (op (pullback fg.1.2.1 fg.2.2.1))
 
-/-- The map `pr₀*` of <https://stacks.math.columbia.edu/tag/00VM>. -/
+/-- The map `pr₀*` of the Stacks entry. -/
+@[stacks 00VM "The map `pr₀*` there."]
 def firstMap : firstObj R P ⟶ secondObj R P :=
   Pi.lift fun _ => Pi.π _ _ ≫ P.map (pullback.fst _ _).op
 
-/-- The map `pr₁*` of <https://stacks.math.columbia.edu/tag/00VM>. -/
+/-- The map `pr₁*` of the Stacks entry. -/
+@[stacks 00VM "The map `pr₁*` there."]
 def secondMap : firstObj R P ⟶ secondObj R P :=
   Pi.lift fun _ => Pi.π _ _ ≫ P.map (pullback.snd _ _).op
 
diff --git a/Mathlib/CategoryTheory/Sites/Subsheaf.lean b/Mathlib/CategoryTheory/Sites/Subsheaf.lean
index 3be19de81341bc..03d5262b3fe05d 100644
--- a/Mathlib/CategoryTheory/Sites/Subsheaf.lean
+++ b/Mathlib/CategoryTheory/Sites/Subsheaf.lean
@@ -194,16 +194,17 @@ variable (J)
 
 /-- A morphism factors through the sheafification of the image presheaf. -/
 @[simps!]
-def toImagePresheafSheafify (f : F' ⟶ F) : F' ⟶ ((imagePresheaf f).sheafify J).toPresheaf :=
-  toImagePresheaf f ≫ Subpresheaf.homOfLe ((imagePresheaf f).le_sheafify J)
+def Subpresheaf.toRangeSheafify (f : F' ⟶ F) : F' ⟶ ((Subpresheaf.range f).sheafify J).toPresheaf :=
+  toRange f ≫ Subpresheaf.homOfLe ((range f).le_sheafify J)
+
 
 variable {J}
 
 /-- The image sheaf of a morphism between sheaves, defined to be the sheafification of
 `image_presheaf`. -/
 @[simps]
-def imageSheaf {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Sheaf J (Type w) :=
-  ⟨((imagePresheaf f.1).sheafify J).toPresheaf, by
+def Sheaf.image {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Sheaf J (Type w) :=
+  ⟨((Subpresheaf.range f.1).sheafify J).toPresheaf, by
     rw [isSheaf_iff_isSheaf_of_type]
     apply Subpresheaf.sheafify_isSheaf
     rw [← isSheaf_iff_isSheaf_of_type]
@@ -211,34 +212,35 @@ def imageSheaf {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Sheaf J (Type w) :=
 
 /-- A morphism factors through the image sheaf. -/
 @[simps]
-def toImageSheaf {F F' : Sheaf J (Type w)} (f : F ⟶ F') : F ⟶ imageSheaf f :=
-  ⟨toImagePresheafSheafify J f.1⟩
+def Sheaf.toImage {F F' : Sheaf J (Type w)} (f : F ⟶ F') : F ⟶ Sheaf.image f :=
+  ⟨Subpresheaf.toRangeSheafify J f.1⟩
 
 /-- The inclusion of the image sheaf to the target. -/
 @[simps]
-def imageSheafι {F F' : Sheaf J (Type w)} (f : F ⟶ F') : imageSheaf f ⟶ F' :=
+def Sheaf.imageι {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Sheaf.image f ⟶ F' :=
   ⟨Subpresheaf.ι _⟩
 
+
 @[reassoc (attr := simp)]
-theorem toImageSheaf_ι {F F' : Sheaf J (Type w)} (f : F ⟶ F') :
-    toImageSheaf f ≫ imageSheafι f = f := by
+theorem Sheaf.toImage_ι {F F' : Sheaf J (Type w)} (f : F ⟶ F') :
+    toImage f ≫ imageι f = f := by
   ext1
-  simp [toImagePresheafSheafify]
+  simp [Subpresheaf.toRangeSheafify]
 
-instance {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Mono (imageSheafι f) :=
+instance {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Mono (Sheaf.imageι f) :=
   (sheafToPresheaf J _).mono_of_mono_map
     (by
       dsimp
       infer_instance)
 
-instance {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Epi (toImageSheaf f) := by
+instance {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Epi (Sheaf.toImage f) := by
   refine ⟨@fun G' g₁ g₂ e => ?_⟩
   ext U ⟨s, hx⟩
   apply ((isSheaf_iff_isSheaf_of_type J _).mp G'.2 _ hx).isSeparatedFor.ext
   rintro V i ⟨y, e'⟩
   change (g₁.val.app _ ≫ G'.val.map _) _ = (g₂.val.app _ ≫ G'.val.map _) _
   rw [← NatTrans.naturality, ← NatTrans.naturality]
-  have E : (toImageSheaf f).val.app (op V) y = (imageSheaf f).val.map i.op ⟨s, hx⟩ :=
+  have E : (Sheaf.toImage f).val.app (op V) y = (Sheaf.image f).val.map i.op ⟨s, hx⟩ :=
     Subtype.ext e'
   have := congr_arg (fun f : F ⟶ G' => (Sheaf.Hom.val f).app _ y) e
   dsimp at this ⊢
@@ -246,9 +248,9 @@ instance {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Epi (toImageSheaf f) := by
 
 /-- The mono factorization given by `image_sheaf` for a morphism. -/
 def imageMonoFactorization {F F' : Sheaf J (Type w)} (f : F ⟶ F') : Limits.MonoFactorisation f where
-  I := imageSheaf f
-  m := imageSheafι f
-  e := toImageSheaf f
+  I := Sheaf.image f
+  m := Sheaf.imageι f
+  e := Sheaf.toImage f
 
 /-- The mono factorization given by `image_sheaf` for a morphism is an image. -/
 noncomputable def imageFactorization {F F' : Sheaf J (Type (max v u))} (f : F ⟶ F') :
@@ -257,14 +259,13 @@ noncomputable def imageFactorization {F F' : Sheaf J (Type (max v u))} (f : F 
   isImage :=
     { lift := fun I => by
         haveI M := (Sheaf.Hom.mono_iff_presheaf_mono J (Type (max v u)) _).mp I.m_mono
-        haveI := isIso_toImagePresheaf I.m.1
-        refine ⟨Subpresheaf.homOfLe ?_ ≫ inv (toImagePresheaf I.m.1)⟩
+        refine ⟨Subpresheaf.homOfLe ?_ ≫ inv (Subpresheaf.toRange I.m.1)⟩
         apply Subpresheaf.sheafify_le
         · conv_lhs => rw [← I.fac]
-          apply imagePresheaf_comp_le
+          apply Subpresheaf.range_comp_le
         · rw [← isSheaf_iff_isSheaf_of_type]
           exact F'.2
-        · apply Presieve.isSheaf_iso J (asIso <| toImagePresheaf I.m.1)
+        · apply Presieve.isSheaf_iso J (asIso <| Subpresheaf.toRange I.m.1)
           rw [← isSheaf_iff_isSheaf_of_type]
           exact I.I.2
       lift_fac := fun I => by
@@ -273,10 +274,17 @@ noncomputable def imageFactorization {F F' : Sheaf J (Type (max v u))} (f : F 
         generalize_proofs h
         rw [← Subpresheaf.homOfLe_ι h, Category.assoc]
         congr 1
-        rw [IsIso.inv_comp_eq, toImagePresheaf_ι] }
+        rw [IsIso.inv_comp_eq, Subpresheaf.toRange_ι] }
 
 instance : Limits.HasImages (Sheaf J (Type max v u)) :=
-  ⟨@fun _ _ f => ⟨⟨imageFactorization f⟩⟩⟩
+  ⟨fun f => ⟨⟨imageFactorization f⟩⟩⟩
+
+@[deprecated (since := "2025-01-25")] alias toImagePresheafSheafify :=
+  Subpresheaf.toRangeSheafify
+@[deprecated (since := "2025-01-25")] alias imageSheaf := Sheaf.image
+@[deprecated (since := "2025-01-25")] alias toImageSheaf := Sheaf.toImage
+@[deprecated (since := "2025-01-25")] alias imageSheafι := Sheaf.imageι
+@[deprecated (since := "2025-01-25")] alias toImageSheaf_ι := Sheaf.toImage_ι
 
 end Image
 
diff --git a/Mathlib/CategoryTheory/Subpresheaf/Basic.lean b/Mathlib/CategoryTheory/Subpresheaf/Basic.lean
index 080283e8b18806..890d33e462b17e 100644
--- a/Mathlib/CategoryTheory/Subpresheaf/Basic.lean
+++ b/Mathlib/CategoryTheory/Subpresheaf/Basic.lean
@@ -1,11 +1,11 @@
 /-
 Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Andrew Yang
+Authors: Andrew Yang, Joël Riou
 -/
 import Mathlib.CategoryTheory.Elementwise
-import Mathlib.Data.Set.Basic
-
+import Mathlib.Data.Set.Lattice
+import Mathlib.Order.CompleteLattice
 
 /-!
 
@@ -47,15 +47,99 @@ variable {F F' F'' : Cᵒᵖ ⥤ Type w} (G G' : Subpresheaf F)
 instance : PartialOrder (Subpresheaf F) :=
   PartialOrder.lift Subpresheaf.obj (fun _ _ => Subpresheaf.ext)
 
-instance : Top (Subpresheaf F) :=
-  ⟨⟨fun _ => ⊤, @fun U _ _ x _ => by simp⟩⟩
+instance : CompleteLattice (Subpresheaf F) where
+  sup F G :=
+    { obj U := F.obj U ⊔ G.obj U
+      map _ _ := by
+        rintro (h|h)
+        · exact Or.inl (F.map _ h)
+        · exact Or.inr (G.map _ h) }
+  le_sup_left _ _ _ := by simp
+  le_sup_right _ _ _ := by simp
+  sup_le F G H h₁ h₂ U := by
+    rintro x (h|h)
+    · exact h₁ _ h
+    · exact h₂ _ h
+  inf S T :=
+    { obj U := S.obj U ⊓ T.obj U
+      map _ _ h := ⟨S.map _ h.1, T.map _ h.2⟩}
+  inf_le_left _ _ _ _ h := h.1
+  inf_le_right _ _ _ _ h := h.2
+  le_inf _ _ _ h₁ h₂ _ _ h := ⟨h₁ _ h, h₂ _ h⟩
+  sSup S :=
+    { obj U := sSup (Set.image (fun T ↦ T.obj U) S)
+      map f x hx := by
+        obtain ⟨_, ⟨F, h, rfl⟩, h'⟩ := hx
+        simp only [Set.sSup_eq_sUnion, Set.sUnion_image, Set.preimage_iUnion,
+          Set.mem_iUnion, Set.mem_preimage, exists_prop]
+        exact ⟨_, h, F.map f h'⟩ }
+  le_sSup _ _ _ _ _ := by aesop
+  sSup_le _ _ _ _ _ := by aesop
+  sInf S :=
+    { obj U := sInf (Set.image (fun T ↦ T.obj U) S)
+      map f x hx := by
+        rintro _ ⟨F, h, rfl⟩
+        exact F.map f (hx _ ⟨_, h, rfl⟩) }
+  sInf_le _ _ _ _ _ := by aesop
+  le_sInf _ _ _ _ _ := by aesop
+  bot :=
+    { obj U := ⊥
+      map := by simp }
+  bot_le _ _ := bot_le
+  top :=
+    { obj U := ⊤
+      map := by simp }
+  le_top _ _ := le_top
+
+namespace Subpresheaf
+
+lemma le_def (S T : Subpresheaf F) : S ≤ T ↔ ∀ U, S.obj U ≤ T.obj U := Iff.rfl
+
+variable (F)
+
+@[simp] lemma top_obj (i : Cᵒᵖ) : (⊤ : Subpresheaf F).obj i = ⊤ := rfl
+@[simp] lemma bot_obj (i : Cᵒᵖ) : (⊥ : Subpresheaf F).obj i = ⊥ := rfl
+
+variable {F}
+
+lemma sSup_obj (S : Set (Subpresheaf F)) (U : Cᵒᵖ) :
+    (sSup S).obj U = sSup (Set.image (fun T ↦ T.obj U) S) := rfl
+
+lemma sInf_obj (S : Set (Subpresheaf F)) (U : Cᵒᵖ) :
+    (sInf S).obj U = sInf (Set.image (fun T ↦ T.obj U) S) := rfl
+
+@[simp]
+lemma iSup_obj {ι : Type*} (S : ι → Subpresheaf F) (U : Cᵒᵖ) :
+    (⨆ i, S i).obj U = ⋃ i, (S i).obj U := by
+  simp [iSup, sSup_obj]
+
+@[simp]
+lemma iInf_obj {ι : Type*} (S : ι → Subpresheaf F) (U : Cᵒᵖ) :
+    (⨅ i, S i).obj U = ⋂ i, (S i).obj U := by
+  simp [iInf, sInf_obj]
+
+@[simp]
+lemma max_obj (S T : Subpresheaf F) (i : Cᵒᵖ) :
+    (S ⊔ T).obj i = S.obj i ∪ T.obj i := rfl
+
+@[simp]
+lemma min_obj (S T : Subpresheaf F) (i : Cᵒᵖ) :
+    (S ⊓ T).obj i = S.obj i ∩ T.obj i := rfl
+
+lemma max_min (S₁ S₂ T : Subpresheaf F) :
+    (S₁ ⊔ S₂) ⊓ T = (S₁ ⊓ T) ⊔ (S₂ ⊓ T) := by
+  aesop
+
+lemma iSup_min {ι : Type*} (S : ι → Subpresheaf F) (T : Subpresheaf F) :
+    (⨆ i, S i) ⊓ T = ⨆ i, S i ⊓ T := by
+  aesop
 
 instance : Nonempty (Subpresheaf F) :=
   inferInstance
 
 /-- The subpresheaf as a presheaf. -/
 @[simps!]
-def Subpresheaf.toPresheaf : Cᵒᵖ ⥤ Type w where
+def toPresheaf : Cᵒᵖ ⥤ Type w where
   obj U := G.obj U
   map := @fun _ _ i x => ⟨F.map i x, G.map i x.prop⟩
   map_id X := by
@@ -72,7 +156,7 @@ instance {U} : CoeHead (G.toPresheaf.obj U) (F.obj U) where
 
 /-- The inclusion of a subpresheaf to the original presheaf. -/
 @[simps]
-def Subpresheaf.ι : G.toPresheaf ⟶ F where app _ x := x
+def ι : G.toPresheaf ⟶ F where app _ x := x
 
 instance : Mono G.ι :=
   ⟨@fun _ _ _ e =>
@@ -81,7 +165,7 @@ instance : Mono G.ι :=
 
 /-- The inclusion of a subpresheaf to a larger subpresheaf -/
 @[simps]
-def Subpresheaf.homOfLe {G G' : Subpresheaf F} (h : G ≤ G') : G.toPresheaf ⟶ G'.toPresheaf where
+def homOfLe {G G' : Subpresheaf F} (h : G ≤ G') : G.toPresheaf ⟶ G'.toPresheaf where
   app U x := ⟨x, h U x.prop⟩
 
 instance {G G' : Subpresheaf F} (h : G ≤ G') : Mono (Subpresheaf.homOfLe h) :=
@@ -92,7 +176,7 @@ instance {G G' : Subpresheaf F} (h : G ≤ G') : Mono (Subpresheaf.homOfLe h) :=
           Subtype.ext <| (congr_arg Subtype.val <| (congr_fun (congr_app e U) x :) :)⟩
 
 @[reassoc (attr := simp)]
-theorem Subpresheaf.homOfLe_ι {G G' : Subpresheaf F} (h : G ≤ G') :
+theorem homOfLe_ι {G G' : Subpresheaf F} (h : G ≤ G') :
     Subpresheaf.homOfLe h ≫ G'.ι = G.ι := by
   ext
   rfl
@@ -103,7 +187,7 @@ instance : IsIso (Subpresheaf.ι (⊤ : Subpresheaf F)) := by
   rw [isIso_iff_bijective]
   exact ⟨Subtype.coe_injective, fun x => ⟨⟨x, _root_.trivial⟩, rfl⟩⟩
 
-theorem Subpresheaf.eq_top_iff_isIso : G = ⊤ ↔ IsIso G.ι := by
+theorem eq_top_iff_isIso : G = ⊤ ↔ IsIso G.ι := by
   constructor
   · rintro rfl
     infer_instance
@@ -113,12 +197,12 @@ theorem Subpresheaf.eq_top_iff_isIso : G = ⊤ ↔ IsIso G.ι := by
     rw [← IsIso.inv_hom_id_apply (G.ι.app U) x]
     exact ((inv (G.ι.app U)) x).2
 
-theorem Subpresheaf.nat_trans_naturality (f : F' ⟶ G.toPresheaf) {U V : Cᵒᵖ} (i : U ⟶ V)
+theorem nat_trans_naturality (f : F' ⟶ G.toPresheaf) {U V : Cᵒᵖ} (i : U ⟶ V)
     (x : F'.obj U) : (f.app V (F'.map i x)).1 = F.map i (f.app U x).1 :=
   congr_arg Subtype.val (FunctorToTypes.naturality _ _ f i x)
 
-@[simp]
-theorem top_subpresheaf_obj (U) : (⊤ : Subpresheaf F).obj U = ⊤ :=
-  rfl
+end Subpresheaf
+
+@[deprecated (since := "2025-01-23")] alias top_subpresheaf_obj := Subpresheaf.top_obj
 
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Subpresheaf/Image.lean b/Mathlib/CategoryTheory/Subpresheaf/Image.lean
index 367029af657f05..4247b888fb3af6 100644
--- a/Mathlib/CategoryTheory/Subpresheaf/Image.lean
+++ b/Mathlib/CategoryTheory/Subpresheaf/Image.lean
@@ -1,20 +1,18 @@
 /-
 Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Andrew Yang
+Authors: Andrew Yang, Joël Riou
 -/
-
 import Mathlib.CategoryTheory.Subpresheaf.Basic
 import Mathlib.CategoryTheory.Limits.FunctorCategory.EpiMono
 
 /-!
 # The image of a subpresheaf
 
-Given a morphism of presheaves of types `f : F' ⟶ F`, we define `imagePresheaf f`
-as a subpresheaf of `F`.
-
-## TODO (@joelriou)
-* introduce `Subpresheaf.image` and `Subpresheaf.preimage`
+Given a morphism of presheaves of types `p : F' ⟶ F`, we define its range
+`Subpresheaf.range p`. More generally, if `G' : Subpresheaf F'`, we
+define `G'.image p : Subpresheaf F` as the image of `G'` by `f`, and
+if `G : Subpresheaf F`, we define its preimage `G.preimage f : Subpresheaf F'`.
 
 -/
 
@@ -22,63 +20,171 @@ universe w v u
 
 namespace CategoryTheory
 
-variable {C : Type u} [Category.{v} C] {F F' F'' : Cᵒᵖ ⥤ Type w} {G : Subpresheaf F}
+variable {C : Type u} [Category.{v} C] {F F' F'' : Cᵒᵖ ⥤ Type w}
 
-/-- If the image of a morphism falls in a subpresheaf, then the morphism factors through it. -/
-@[simps!]
-def Subpresheaf.lift (f : F' ⟶ F) (hf : ∀ U x, f.app U x ∈ G.obj U) : F' ⟶ G.toPresheaf where
-  app U x := ⟨f.app U x, hf U x⟩
-  naturality := by
-    have := elementwise_of% f.naturality
-    intros
-    refine funext fun x => Subtype.ext ?_
-    simp only [toPresheaf_obj, types_comp_apply]
-    exact this _ _
+namespace Subpresheaf
 
-@[reassoc (attr := simp)]
-theorem Subpresheaf.lift_ι (f : F' ⟶ F) (hf : ∀ U x, f.app U x ∈ G.obj U) :
-    G.lift f hf ≫ G.ι = f := by
-  ext
-  rfl
+section range
 
-/-- The image presheaf of a morphism, whose components are the set-theoretic images. -/
+/-- The range of a morphism of presheaves of types, as a subpresheaf of the target. -/
 @[simps]
-def imagePresheaf (f : F' ⟶ F) : Subpresheaf F where
-  obj U := Set.range (f.app U)
+def range (p : F' ⟶ F) : Subpresheaf F where
+  obj U := Set.range (p.app U)
   map := by
     rintro U V i _ ⟨x, rfl⟩
-    have := elementwise_of% f.naturality
-    exact ⟨_, this i x⟩
+    exact ⟨_, FunctorToTypes.naturality  _ _ p i x⟩
+
+variable (F) in
+lemma range_id : range (𝟙 F) = ⊤ := by aesop
 
 @[simp]
-theorem imagePresheaf_id : imagePresheaf (𝟙 F) = ⊤ := by
-  ext
-  simp
+lemma range_ι (G : Subpresheaf F) : range G.ι = G := by aesop
+
+end range
+
+section lift
 
-/-- A morphism factors through the image presheaf. -/
+variable (f : F' ⟶ F) {G : Subpresheaf F} (hf : range f ≤ G)
+
+/-- If the image of a morphism falls in a subpresheaf, then the morphism factors through it. -/
 @[simps!]
-def toImagePresheaf (f : F' ⟶ F) : F' ⟶ (imagePresheaf f).toPresheaf :=
-  (imagePresheaf f).lift f fun _ _ => Set.mem_range_self _
+def lift : F' ⟶ G.toPresheaf where
+  app U x := ⟨f.app U x, hf U (by simp)⟩
+  naturality _ _ g := by
+    ext x
+    simpa [Subtype.ext_iff] using FunctorToTypes.naturality _ _ f g x
 
 @[reassoc (attr := simp)]
-theorem toImagePresheaf_ι (f : F' ⟶ F) : toImagePresheaf f ≫ (imagePresheaf f).ι = f :=
-  (imagePresheaf f).lift_ι _ _
-
-theorem imagePresheaf_comp_le (f₁ : F ⟶ F') (f₂ : F' ⟶ F'') :
-    imagePresheaf (f₁ ≫ f₂) ≤ imagePresheaf f₂ := fun U _ hx => ⟨f₁.app U hx.choose, hx.choose_spec⟩
-
-instance isIso_toImagePresheaf {F F' : Cᵒᵖ ⥤ Type w} (f : F ⟶ F') [hf : Mono f] :
-  IsIso (toImagePresheaf f) := by
-  have : ∀ (X : Cᵒᵖ), IsIso ((toImagePresheaf f).app X) := by
-    intro X
-    rw [isIso_iff_bijective]
-    constructor
-    · intro x y e
-      have := (NatTrans.mono_iff_mono_app f).mp hf X
-      rw [mono_iff_injective] at this
-      exact this (congr_arg Subtype.val e :)
-    · rintro ⟨_, ⟨x, rfl⟩⟩
-      exact ⟨x, rfl⟩
-  apply NatIso.isIso_of_isIso_app
+theorem lift_ι : lift f hf ≫ G.ι = f := rfl
+
+end lift
+
+section range
+
+variable (p : F' ⟶ F)
+
+/-- Given a morphism `p : F' ⟶ F` of presheaves of types, this is the morphism
+from `F'` to its range. -/
+def toRange :
+    F' ⟶ (range p).toPresheaf :=
+  lift p (by rfl)
+
+@[reassoc (attr := simp)]
+lemma toRange_ι : toRange p ≫ (range p).ι = p := rfl
+
+lemma toRange_app_val {i : Cᵒᵖ} (x : F'.obj i) :
+    ((toRange p).app i x).val = p.app i x := by
+  simp [toRange]
+
+@[simp]
+lemma range_toRange : range (toRange p) = ⊤ := by
+  ext i ⟨x, hx⟩
+  dsimp at hx ⊢
+  simp only [Set.mem_range, Set.mem_univ, iff_true]
+  simp only [Set.mem_range] at hx
+  obtain ⟨y, rfl⟩ := hx
+  exact ⟨y, rfl⟩
+
+lemma epi_iff_range_eq_top :
+    Epi p ↔ range p = ⊤ := by
+  simp [NatTrans.epi_iff_epi_app, epi_iff_surjective, Subpresheaf.ext_iff, funext_iff,
+    Set.range_eq_univ]
+
+instance : Epi (toRange p) := by simp [epi_iff_range_eq_top]
+
+instance [Mono p] : IsIso (toRange p) := by
+  have := mono_of_mono_fac (toRange_ι p)
+  rw [NatTrans.isIso_iff_isIso_app]
+  intro i
+  rw [isIso_iff_bijective]
+  constructor
+  · rw [← mono_iff_injective]
+    infer_instance
+  · rw [← epi_iff_surjective]
+    infer_instance
+
+lemma range_comp_le (f : F ⟶ F') (g : F' ⟶ F'') :
+    range (f ≫ g) ≤ range g := fun _ _ _ ↦ by aesop
+
+end range
+
+section image
+
+variable (G : Subpresheaf F) (f : F ⟶ F')
+
+/-- The image of a subpresheaf by a morphism of presheaves of types. -/
+@[simps]
+def image : Subpresheaf F' where
+  obj i := (f.app i) '' (G.obj i)
+  map := by
+    rintro Δ Δ' φ _ ⟨x, hx, rfl⟩
+    exact ⟨F.map φ x, G.map φ hx, by apply FunctorToTypes.naturality⟩
+
+lemma image_top : (⊤ : Subpresheaf F).image f = range f := by aesop
+
+lemma image_comp (g : F' ⟶ F'') :
+    G.image (f ≫ g) = (G.image f).image g := by aesop
+
+lemma range_comp (g : F' ⟶ F'') :
+    range (f ≫ g) = (range f).image g := by aesop
+
+end image
+
+section preimage
+
+/-- The preimage of a subpresheaf by a morphism of presheaves of types. -/
+@[simps]
+def preimage (G : Subpresheaf F) (p : F' ⟶ F) : Subpresheaf F' where
+  obj n := p.app n ⁻¹' (G.obj n)
+  map f := (Set.preimage_mono (G.map f)).trans (by
+    simp only [Set.preimage_preimage, FunctorToTypes.naturality _ _ p f]
+    rfl)
+
+@[simp]
+lemma preimage_id (G : Subpresheaf F) :
+    G.preimage (𝟙 F) = G := by aesop
+
+lemma preimage_comp (G : Subpresheaf F) (f : F'' ⟶ F') (g : F' ⟶ F) :
+    G.preimage (f ≫ g) = (G.preimage g).preimage f := by aesop
+
+lemma image_le_iff (G : Subpresheaf F) (f : F ⟶ F') (G' : Subpresheaf F') :
+    G.image f ≤ G' ↔ G ≤ G'.preimage f := by
+  simp [Subpresheaf.le_def]
+
+/-- Given a morphism `p : F' ⟶ F` of presheaves of types and `G : Subpresheaf F`,
+this is the morphism from the preimage of `G` by `p` to `G`. -/
+def fromPreimage (G : Subpresheaf F) (p : F' ⟶ F) :
+    (G.preimage p).toPresheaf ⟶ G.toPresheaf :=
+  lift ((G.preimage p).ι ≫ p) (by
+    rw [range_comp, range_ι, image_le_iff])
+
+@[reassoc]
+lemma fromPreimage_ι (G : Subpresheaf F) (p : F' ⟶ F) :
+    G.fromPreimage p ≫ G.ι = (G.preimage p).ι ≫ p := rfl
+
+lemma preimage_eq_top_iff (G : Subpresheaf F) (p : F' ⟶ F) :
+    G.preimage p = ⊤ ↔ range p ≤ G := by
+  rw [← image_top, image_le_iff]
+  aesop
+
+@[simp]
+lemma preimage_image_of_epi (G : Subpresheaf F) (p : F' ⟶ F) [hp : Epi p] :
+    (G.preimage p).image p = G := by
+  apply le_antisymm
+  · rw [image_le_iff]
+  · intro i x hx
+    simp only [NatTrans.epi_iff_epi_app, epi_iff_surjective] at hp
+    obtain ⟨y, rfl⟩ := hp _ x
+    exact ⟨y, hx, rfl⟩
+
+end preimage
+
+end Subpresheaf
+
+@[deprecated (since := "2025-01-25")] alias imagePresheaf := Subpresheaf.range
+@[deprecated (since := "2025-01-25")] alias imagePresheaf_id := Subpresheaf.range_id
+@[deprecated (since := "2025-01-25")] alias toImagePresheaf := Subpresheaf.toRange
+@[deprecated (since := "2025-01-25")] alias toImagePresheaf_ι := Subpresheaf.toRange_ι
+@[deprecated (since := "2025-01-25")] alias imagePresheaf_comp_le := Subpresheaf.range_comp_le
 
 end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Subpresheaf/OfSection.lean b/Mathlib/CategoryTheory/Subpresheaf/OfSection.lean
new file mode 100644
index 00000000000000..3bb6facb0c17fc
--- /dev/null
+++ b/Mathlib/CategoryTheory/Subpresheaf/OfSection.lean
@@ -0,0 +1,105 @@
+/-
+Copyright (c) 2025 Joël Riou. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Joël Riou
+-/
+import Mathlib.CategoryTheory.Subpresheaf.Image
+import Mathlib.CategoryTheory.Yoneda
+
+/-!
+# The subpresheaf generated by a section
+
+Given a presheaf of types `F : Cᵒᵖ ⥤ Type w` and a section `x : F.obj X`,
+we define `Subpresheaf.ofSection x : Subpresheaf F` as the subpresheaf
+of `F` generated by `x`.
+
+-/
+
+universe w v u
+
+namespace CategoryTheory
+
+variable {C : Type u} [Category.{v} C]
+
+namespace Subpresheaf
+
+section
+
+variable {F : Cᵒᵖ ⥤ Type w} {X : Cᵒᵖ} (x : F.obj X)
+
+/-- The subpresheaf of `F : Cᵒᵖ ⥤ Type w` that is generated
+by a section `x : F.obj X`. -/
+@[simps (config := .lemmasOnly)]
+def ofSection : Subpresheaf F where
+  obj U := setOf (fun u ↦ ∃ (f : X ⟶ U), F.map f x = u)
+  map {U V} g := by
+    rintro _ ⟨f, rfl⟩
+    exact ⟨f ≫ g, by simp⟩
+
+lemma mem_ofSection_obj : x ∈ (ofSection x).obj X := ⟨𝟙 _, by simp⟩
+
+@[simp]
+lemma ofSection_le_iff (G : Subpresheaf F) :
+    ofSection x ≤ G ↔ x ∈ G.obj X := by
+  constructor
+  · intro hx
+    exact hx _ (mem_ofSection_obj x)
+  · rintro hx U _ ⟨f, rfl⟩
+    exact G.map f hx
+
+@[simp]
+lemma ofSection_image {F' : Cᵒᵖ ⥤ Type w} (f : F ⟶ F') :
+    (ofSection x).image f = ofSection (f.app _ x) := by
+  apply le_antisymm
+  · rw [image_le_iff, ofSection_le_iff, preimage_obj, Set.mem_preimage]
+    exact ⟨𝟙 X, by simp⟩
+  · simp only [ofSection_le_iff, image_obj, Set.mem_image]
+    exact ⟨x, mem_ofSection_obj x, rfl⟩
+
+end
+
+section
+
+variable {F : Cᵒᵖ ⥤ Type v}
+
+lemma ofSection_eq_range {X : Cᵒᵖ} (x : F.obj X) :
+    ofSection x = range (yonedaEquiv.symm x) := by
+  ext U y
+  simp only [ofSection_obj, Set.mem_setOf_eq, Opposite.op_unop, range_obj, yoneda_obj_obj,
+    Set.mem_range]
+  constructor
+  · rintro ⟨f, rfl⟩
+    exact ⟨f.unop, rfl⟩
+  · rintro ⟨f, rfl⟩
+    exact ⟨f.op, rfl⟩
+
+lemma range_eq_ofSection {X : C} (f : yoneda.obj X ⟶ F) :
+    range f = ofSection (yonedaEquiv f) := by
+  rw [ofSection_eq_range, Equiv.symm_apply_apply]
+
+end
+
+section
+
+variable {F : Cᵒᵖ ⥤ Type max v w}
+
+lemma ofSection_eq_range' {X : Cᵒᵖ} (x : F.obj X) :
+    ofSection x = range ((yonedaCompUliftFunctorEquiv F X.unop).symm x) := by
+  ext U y
+  simp only [Opposite.op_unop, range_obj, Functor.comp_obj, yoneda_obj_obj, uliftFunctor_obj,
+    Set.mem_range, ULift.exists]
+  constructor
+  · rintro ⟨f, rfl⟩
+    exact ⟨f.unop, rfl⟩
+  · rintro ⟨f, rfl⟩
+    exact ⟨f.op, rfl⟩
+
+lemma range_eq_ofSection' {X : C} (f : yoneda.obj X ⋙ uliftFunctor.{w} ⟶ F) :
+    range f = ofSection ((yonedaCompUliftFunctorEquiv F X) f) := by
+  rw [ofSection_eq_range', Equiv.symm_apply_apply]
+
+end
+
+end Subpresheaf
+
+end CategoryTheory
diff --git a/Mathlib/CategoryTheory/Subpresheaf/Sieves.lean b/Mathlib/CategoryTheory/Subpresheaf/Sieves.lean
index 12ffd8506ee789..7c84f551edcd18 100644
--- a/Mathlib/CategoryTheory/Subpresheaf/Sieves.lean
+++ b/Mathlib/CategoryTheory/Subpresheaf/Sieves.lean
@@ -3,7 +3,6 @@ Copyright (c) 2022 Andrew Yang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Andrew Yang
 -/
-
 import Mathlib.CategoryTheory.Subpresheaf.Basic
 import Mathlib.CategoryTheory.Sites.IsSheafFor
 
diff --git a/Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean b/Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean
index 99b4e12ea64112..add552a2ee67aa 100644
--- a/Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean
+++ b/Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean
@@ -182,7 +182,8 @@ theorem ruzsa_triangle_inequality_mul_mul_mul (A B C : Finset G) :
   refine le_trans ?_
     (mul_le_mul (hUA _ hB') (cast_le.2 <| card_le_card <| mul_subset_mul_right hU.2)
       (zero_le _) (zero_le _))
-  #adaptation_note /-- 2024-11-01 `le_div_iff₀` is synthesizing wrong `GroupWithZero` without `@` -/
+  #adaptation_note /-- 2024-11-01
+  `le_div_iff₀` is synthesizing wrong `GroupWithZero` without `@` -/
   rw [← mul_div_right_comm, ← mul_assoc,
     @le_div_iff₀ _ (_) _ _ _ _ _ _ _ (cast_pos.2 hU.1.card_pos)]
   exact mod_cast pluennecke_petridis_inequality_mul C (mul_aux hU.1 hU.2 hUA)
diff --git a/Mathlib/Combinatorics/Enumerative/DyckWord.lean b/Mathlib/Combinatorics/Enumerative/DyckWord.lean
index 5a8e966e55c59c..a48532e748a046 100644
--- a/Mathlib/Combinatorics/Enumerative/DyckWord.lean
+++ b/Mathlib/Combinatorics/Enumerative/DyckWord.lean
@@ -97,7 +97,7 @@ lemma toList_ne_nil : p.toList ≠ [] ↔ p ≠ 0 := toList_eq_nil.ne
 instance : Unique (AddUnits DyckWord) where
   uniq p := by
     obtain ⟨a, b, h, -⟩ := p
-    obtain ⟨ha, hb⟩ := append_eq_nil.mp (toList_eq_nil.mpr h)
+    obtain ⟨ha, hb⟩ := append_eq_nil_iff.mp (toList_eq_nil.mpr h)
     congr
     · exact toList_eq_nil.mp ha
     · exact toList_eq_nil.mp hb
diff --git a/Mathlib/Combinatorics/SimpleGraph/Clique.lean b/Mathlib/Combinatorics/SimpleGraph/Clique.lean
index 7738b767a8d3a8..433d517a5cba77 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Clique.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Clique.lean
@@ -367,6 +367,10 @@ protected theorem CliqueFree.replaceVertex [DecidableEq α] (h : G.CliqueFree n)
     conv at hφ => enter [a, b]; rw [G.adj_replaceVertex_iff_of_ne _ (mt a) (mt b)]
     exact hφ
 
+@[simp]
+lemma cliqueFree_one : G.CliqueFree 1 ↔ IsEmpty α := by
+  simp [CliqueFree, isEmpty_iff]
+
 @[simp]
 theorem cliqueFree_two : G.CliqueFree 2 ↔ G = ⊥ := by
   classical
diff --git a/Mathlib/Combinatorics/SimpleGraph/Coloring.lean b/Mathlib/Combinatorics/SimpleGraph/Coloring.lean
index 54483becc09063..c0fa0618bf9ad3 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Coloring.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Coloring.lean
@@ -137,6 +137,10 @@ theorem isEmpty_of_colorable_zero (h : G.Colorable 0) : IsEmpty V := by
   obtain ⟨i, hi⟩ := h.some v
   exact Nat.not_lt_zero _ hi
 
+@[simp]
+lemma colorable_zero_iff : G.Colorable 0 ↔ IsEmpty V :=
+   ⟨G.isEmpty_of_colorable_zero, fun _ ↦ G.colorable_of_isEmpty 0⟩
+
 /-- The "tautological" coloring of a graph, using the vertices of the graph as colors. -/
 def selfColoring : G.Coloring V := Coloring.mk id fun {_ _} => G.ne_of_adj
 
diff --git a/Mathlib/Combinatorics/SimpleGraph/Matching.lean b/Mathlib/Combinatorics/SimpleGraph/Matching.lean
index e98fe155b060bd..310d9aa562bd12 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Matching.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Matching.lean
@@ -243,7 +243,7 @@ lemma IsPerfectMatching.induce_connectedComponent_isMatching (h : M.IsPerfectMat
   simpa [h.2.verts_eq_univ] using h.1.induce_connectedComponent c
 
 @[simp]
-lemma IsPerfectMatching.toSubgraph_spanningCoe_iff (h : M.spanningCoe ≤ G') :
+lemma IsPerfectMatching.toSubgraph_iff (h : M.spanningCoe ≤ G') :
     (G'.toSubgraph M.spanningCoe h).IsPerfectMatching ↔ M.IsPerfectMatching := by
   simp only [isPerfectMatching_iff, toSubgraph_adj, spanningCoe_adj]
 
@@ -256,13 +256,11 @@ section Finite
 lemma even_card_of_isPerfectMatching [Fintype V] [DecidableEq V] [DecidableRel G.Adj]
     (c : ConnectedComponent G) (hM : M.IsPerfectMatching) :
     Even (Fintype.card c.supp) := by
-  #adaptation_note
-  /--
-  After https://github.com/leanprover/lean4/pull/5020, some instances that use the chain of coercions
+  #adaptation_note /-- https://github.com/leanprover/lean4/pull/5020
+  some instances that use the chain of coercions
   `[SetLike X], X → Set α → Sort _` are
   blocked by the discrimination tree. This can be fixed by redeclaring the instance for `X`
-  using the double coercion but the proper fix seems to avoid the double coercion.
-  -/
+  using the double coercion but the proper fix seems to avoid the double coercion. -/
   letI : DecidablePred fun x ↦ x ∈ (M.induce c.supp).verts := fun a ↦ G.instDecidableMemSupp c a
   simpa using (hM.induce_connectedComponent_isMatching c).even_card
 
@@ -339,9 +337,18 @@ lemma IsCycles.other_adj_of_adj (h : G.IsCycles) (hadj : G.Adj v w) :
   obtain ⟨w', hww'⟩ := (G.neighborSet v).exists_ne_of_one_lt_ncard (by omega) w
   exact ⟨w', ⟨hww'.2.symm, hww'.1⟩⟩
 
+lemma IsCycles.induce_supp (c : G.ConnectedComponent) (h : G.IsCycles) :
+    (G.induce c.supp).spanningCoe.IsCycles := by
+  intro v ⟨w, hw⟩
+  rw [mem_neighborSet, c.adj_spanningCoe_induce_supp] at hw
+  rw [← h ⟨w, hw.2⟩]
+  congr
+  ext w'
+  simp only [mem_neighborSet, c.adj_spanningCoe_induce_supp, hw, true_and]
+
 open scoped symmDiff
 
-lemma Subgraph.IsPerfectMatching.symmDiff_spanningCoe_IsCycles
+lemma Subgraph.IsPerfectMatching.symmDiff_isCycles
     {M : Subgraph G} {M' : Subgraph G'} (hM : M.IsPerfectMatching)
     (hM' : M'.IsPerfectMatching) : (M.spanningCoe ∆ M'.spanningCoe).IsCycles := by
   intro v
@@ -368,10 +375,12 @@ for `SimpleGraph` rather than `SimpleGraph.Subgraph`.
 def IsAlternating (G G' : SimpleGraph V) :=
   ∀ ⦃v w w': V⦄, w ≠ w' → G.Adj v w → G.Adj v w' → (G'.Adj v w ↔ ¬ G'.Adj v w')
 
-lemma IsPerfectMatching.symmDiff_spanningCoe_of_isAlternating {M : Subgraph G}
-    (hM : M.IsPerfectMatching) (hG' : G'.IsAlternating M.spanningCoe) (hG'cyc : G'.IsCycles)  :
-    (SimpleGraph.toSubgraph (M.spanningCoe ∆ G')
-      (by rfl)).IsPerfectMatching := by
+lemma IsAlternating.mono {G'' : SimpleGraph V} (halt : G.IsAlternating G') (h : G'' ≤ G) :
+    G''.IsAlternating G' := fun _ _ _ hww' hvw hvw' ↦ halt hww' (h hvw) (h hvw')
+
+lemma IsPerfectMatching.symmDiff_of_isAlternating (hM : M.IsPerfectMatching)
+    (hG' : G'.IsAlternating M.spanningCoe) (hG'cyc : G'.IsCycles) :
+    (⊤ : Subgraph (M.spanningCoe ∆ G')).IsPerfectMatching := by
   rw [Subgraph.isPerfectMatching_iff]
   intro v
   simp only [toSubgraph_adj, symmDiff_def, sup_adj, sdiff_adj, Subgraph.spanningCoe_adj]
@@ -380,20 +389,33 @@ lemma IsPerfectMatching.symmDiff_spanningCoe_of_isAlternating {M : Subgraph G}
   · obtain ⟨w', hw'⟩ := hG'cyc.other_adj_of_adj h
     have hmadj :  M.Adj v w ↔ ¬M.Adj v w' := by simpa using hG' hw'.1 h hw'.2
     use w'
-    simp only [hmadj.mp hw.1, hw'.2, not_true_eq_false, and_self, not_false_eq_true, or_true,
-      true_and]
+    simp only [Subgraph.top_adj, sup_adj, sdiff_adj, Subgraph.spanningCoe_adj, hmadj.mp hw.1, hw'.2,
+      not_true_eq_false, and_self, not_false_eq_true, or_true, true_and]
     rintro y (hl | hr)
     · aesop
     · obtain ⟨w'', hw''⟩ := hG'cyc.other_adj_of_adj hr.1
       by_contra! hc
-      simp_all only [show M.Adj v y ↔ ¬M.Adj v w' from by simpa using hG' hc hr.1 hw'.2,
-        not_false_eq_true, ne_eq, iff_true, not_true_eq_false, and_false]
+      simp_all [show M.Adj v y ↔ ¬M.Adj v w' from by simpa using hG' hc hr.1 hw'.2]
   · use w
-    simp only [hw.1, h, not_false_eq_true, and_self, not_true_eq_false, or_false, true_and]
+    simp only [Subgraph.top_adj, sup_adj, sdiff_adj, Subgraph.spanningCoe_adj, hw.1, h,
+      not_false_eq_true, and_self, not_true_eq_false, or_false, true_and]
     rintro y (hl | hr)
     · exact hw.2 _ hl.1
     · have ⟨w', hw'⟩ := hG'cyc.other_adj_of_adj hr.1
-      simp_all only [show M.Adj v y ↔ ¬M.Adj v w' from by simpa using hG' hw'.1 hr.1 hw'.2, not_not,
-        ne_eq, and_false]
+      simp_all [show M.Adj v y ↔ ¬M.Adj v w' from by simpa using hG' hw'.1 hr.1 hw'.2]
+
+lemma IsPerfectMatching.isAlternating_symmDiff_left {M' : Subgraph G'}
+    (hM : M.IsPerfectMatching) (hM' : M'.IsPerfectMatching) :
+    (M.spanningCoe ∆ M'.spanningCoe).IsAlternating M.spanningCoe := by
+  intro v w w' hww' hvw hvw'
+  obtain ⟨v1, hm1, hv1⟩ := hM.1 (hM.2 v)
+  obtain ⟨v2, hm2, hv2⟩ := hM'.1 (hM'.2 v)
+  simp only [symmDiff_def] at *
+  aesop
+
+lemma IsPerfectMatching.isAlternating_symmDiff_right
+    {M' : Subgraph G'} (hM : M.IsPerfectMatching) (hM' : M'.IsPerfectMatching) :
+    (M.spanningCoe ∆ M'.spanningCoe).IsAlternating M'.spanningCoe := by
+  simpa [symmDiff_comm] using isAlternating_symmDiff_left hM' hM
 
 end SimpleGraph
diff --git a/Mathlib/Combinatorics/SimpleGraph/Path.lean b/Mathlib/Combinatorics/SimpleGraph/Path.lean
index 3211d0a9774fee..d016e6d94ad996 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Path.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Path.lean
@@ -1071,6 +1071,22 @@ theorem mem_supp_iff (C : G.ConnectedComponent) (v : V) :
     v ∈ C.supp ↔ G.connectedComponentMk v = C :=
   Iff.rfl
 
+lemma mem_supp_congr_adj {v w : V} (c : G.ConnectedComponent) (hadj : G.Adj v w) :
+    v ∈ c.supp ↔ w ∈ c.supp := by
+  simp only [ConnectedComponent.mem_supp_iff] at *
+  constructor <;> intro h <;> simp only [← h] <;> apply connectedComponentMk_eq_of_adj
+  · exact hadj.symm
+  · exact hadj
+
+lemma adj_spanningCoe_induce_supp {v w : V} (c : G.ConnectedComponent) :
+    (G.induce c.supp).spanningCoe.Adj v w ↔ v ∈ c.supp ∧ G.Adj v w := by
+  by_cases h : v ∈ c.supp
+  · refine ⟨by aesop, ?_⟩
+    intro h'
+    have : w ∈ c.supp := by rwa [c.mem_supp_congr_adj h'.2] at h
+    aesop
+  · aesop
+
 theorem connectedComponentMk_mem {v : V} : v ∈ G.connectedComponentMk v :=
   rfl
 
diff --git a/Mathlib/Combinatorics/SimpleGraph/Sum.lean b/Mathlib/Combinatorics/SimpleGraph/Sum.lean
index 25de075d43605d..aaf460a2f7c93e 100644
--- a/Mathlib/Combinatorics/SimpleGraph/Sum.lean
+++ b/Mathlib/Combinatorics/SimpleGraph/Sum.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Iván Renison
 -/
 import Mathlib.Combinatorics.SimpleGraph.Basic
+import Mathlib.Combinatorics.SimpleGraph.Coloring
 import Mathlib.Combinatorics.SimpleGraph.Maps
 
 /-!
@@ -46,18 +47,12 @@ variable {G : SimpleGraph α} {H : SimpleGraph β}
 /-- The disjoint sum is commutative up to isomorphism. `Iso.sumComm` as a graph isomorphism. -/
 @[simps!]
 def Iso.sumComm : G ⊕g H ≃g H ⊕g G := ⟨Equiv.sumComm α β, by
-  intro u v
-  cases u <;> cases v <;> simp⟩
+  rintro (u | u) (v | v) <;> simp⟩
 
 /-- The disjoint sum is associative up to isomorphism. `Iso.sumAssoc` as a graph isomorphism. -/
 @[simps!]
 def Iso.sumAssoc {I : SimpleGraph γ} : (G ⊕g H) ⊕g I ≃g G ⊕g (H ⊕g I) := ⟨Equiv.sumAssoc α β γ, by
-  intro u v
-  cases u <;> cases v <;> rename_i u v
-  · cases u <;> cases v <;> simp
-  · cases u <;> simp
-  · cases v <;> simp
-  · simp⟩
+  rintro ((u | u) | u) ((v | v) | v) <;> simp⟩
 
 /-- The embedding of `G` into `G ⊕g H`. -/
 @[simps]
@@ -73,4 +68,71 @@ def Embedding.sumInr : H ↪g G ⊕g H where
   inj' u v := by simp
   map_rel_iff' := by simp
 
+/-- Color `G ⊕g H` with colorings of `G` and `H` -/
+def Coloring.sum (cG : G.Coloring γ) (cH : H.Coloring γ) : (G ⊕g H).Coloring γ where
+  toFun := Sum.elim cG cH
+  map_rel' {u v} huv := by cases u <;> cases v <;> simp_all [cG.valid, cH.valid]
+
+/-- Get coloring of `G` from coloring of `G ⊕g H` -/
+def Coloring.sumLeft (c : (G ⊕g H).Coloring γ) : G.Coloring γ := c.comp Embedding.sumInl.toHom
+
+/-- Get coloring of `H` from coloring of `G ⊕g H` -/
+def Coloring.sumRight (c : (G ⊕g H).Coloring γ) : H.Coloring γ := c.comp Embedding.sumInr.toHom
+
+@[simp]
+theorem Coloring.sumLeft_sum (cG : G.Coloring γ) (cH : H.Coloring γ) : (cG.sum cH).sumLeft = cG :=
+  rfl
+
+@[simp]
+theorem Coloring.sumRight_sum (cG : G.Coloring γ) (cH : H.Coloring γ) : (cG.sum cH).sumRight = cH :=
+  rfl
+
+@[simp]
+theorem Coloring.sum_sumLeft_sumRight (c : (G ⊕g H).Coloring γ) : c.sumLeft.sum c.sumRight = c := by
+  ext (u | u) <;> rfl
+
+/-- Bijection between `(G ⊕g H).Coloring γ` and `G.Coloring γ × H.Coloring γ` -/
+def Coloring.sumEquiv : (G ⊕g H).Coloring γ ≃ G.Coloring γ × H.Coloring γ where
+  toFun c := ⟨c.sumLeft, c.sumRight⟩
+  invFun p := p.1.sum p.2
+  left_inv c := by simp [sum_sumLeft_sumRight c]
+  right_inv p := rfl
+
+/-- Color `G ⊕g H` with `Fin (n + m)` given a coloring of `G` with `Fin n` and a coloring of `H`
+with `Fin m` -/
+def Coloring.sumFin {n m : ℕ} (cG : G.Coloring (Fin n)) (cH : H.Coloring (Fin m)) :
+    (G ⊕g H).Coloring (Fin (max n m)) := sum
+  (G.recolorOfEmbedding (Fin.castLEEmb (n.le_max_left m)) cG)
+  (H.recolorOfEmbedding (Fin.castLEEmb (n.le_max_right m)) cH)
+
+theorem Colorable.sum_max {n m : ℕ} (hG : G.Colorable n) (hH : H.Colorable m) :
+    (G ⊕g H).Colorable (max n m) := Nonempty.intro (hG.some.sumFin hH.some)
+
+theorem Colorable.of_sum_left {n : ℕ} (h : (G ⊕g H).Colorable n) : G.Colorable n :=
+  Nonempty.intro (h.some.sumLeft)
+
+theorem Colorable.of_sum_right {n : ℕ} (h : (G ⊕g H).Colorable n) : H.Colorable n :=
+  Nonempty.intro (h.some.sumRight)
+
+@[simp]
+theorem colorable_sum {n : ℕ} : (G ⊕g H).Colorable n ↔ G.Colorable n ∧ H.Colorable n :=
+  ⟨fun cGH => ⟨cGH.of_sum_left, cGH.of_sum_right⟩,
+    fun ⟨cG, cH⟩ => by rw [← n.max_self]; exact cG.sum_max cH⟩
+
+theorem chromaticNumber_le_sum_left : G.chromaticNumber ≤ (G ⊕g H).chromaticNumber :=
+  chromaticNumber_le_of_forall_imp (fun _ h ↦ h.of_sum_left)
+
+theorem chromaticNumber_le_sum_right : H.chromaticNumber ≤ (G ⊕g H).chromaticNumber :=
+  chromaticNumber_le_of_forall_imp (fun _ h ↦ h.of_sum_right)
+
+@[simp]
+theorem chromaticNumber_sum :
+    (G ⊕g H).chromaticNumber = max G.chromaticNumber H.chromaticNumber := by
+  refine eq_max chromaticNumber_le_sum_left chromaticNumber_le_sum_right ?_
+  rintro (n | n) hG hH
+  · simp [show (none : ℕ∞) = (⊤ : ℕ∞) from rfl]
+  · let cG : G.Coloring (Fin n) := (chromaticNumber_le_iff_colorable.mp hG).some
+    let cH : H.Coloring (Fin n) := (chromaticNumber_le_iff_colorable.mp hH).some
+    exact chromaticNumber_le_iff_colorable.mpr (Nonempty.intro (cG.sum cH))
+
 end SimpleGraph
diff --git a/Mathlib/Computability/DFA.lean b/Mathlib/Computability/DFA.lean
index e55cb83d9c17c5..dd5c1c6b0348b8 100644
--- a/Mathlib/Computability/DFA.lean
+++ b/Mathlib/Computability/DFA.lean
@@ -6,7 +6,6 @@ Authors: Fox Thomson, Chris Wong
 import Mathlib.Computability.Language
 import Mathlib.Data.Countable.Small
 import Mathlib.Data.Fintype.Card
-import Mathlib.Data.List.Indexes
 import Mathlib.Tactic.NormNum
 
 /-!
@@ -185,9 +184,9 @@ theorem comap_id : M.comap id = M := rfl
 @[simp]
 theorem evalFrom_comap (f : α' → α) (s : σ) (x : List α') :
     (M.comap f).evalFrom s x = M.evalFrom s (x.map f) := by
-  induction x using List.list_reverse_induction with
-  | base => simp
-  | ind x a ih => simp [ih]
+  induction x using List.reverseRecOn with
+  | nil => simp
+  | append_singleton x a ih => simp [ih]
 
 @[simp]
 theorem eval_comap (f : α' → α) (x : List α') : (M.comap f).eval x = M.eval (x.map f) := by
@@ -226,9 +225,9 @@ theorem symm_reindex (g : σ ≃ σ') : (reindex (α := α) g).symm = reindex g.
 @[simp]
 theorem evalFrom_reindex (g : σ ≃ σ') (s : σ') (x : List α) :
     (reindex g M).evalFrom s x = g (M.evalFrom (g.symm s) x) := by
-  induction x using List.list_reverse_induction with
-  | base => simp
-  | ind x a ih => simp [ih]
+  induction x using List.reverseRecOn with
+  | nil => simp
+  | append_singleton x a ih => simp [ih]
 
 @[simp]
 theorem eval_reindex (g : σ ≃ σ') (x : List α) : (reindex g M).eval x = g (M.eval x) := by
diff --git a/Mathlib/Computability/MyhillNerode.lean b/Mathlib/Computability/MyhillNerode.lean
index 0ab520eb652381..9a9fc6bd140eee 100644
--- a/Mathlib/Computability/MyhillNerode.lean
+++ b/Mathlib/Computability/MyhillNerode.lean
@@ -85,9 +85,9 @@ theorem accepts_toDFA : L.toDFA.accepts = L := by
   ext x
   rw [DFA.mem_accepts]
   suffices L.toDFA.eval x = L.leftQuotient x by simp [this]
-  induction x using List.list_reverse_induction with
-  | base => simp
-  | ind x a ih => simp [ih, leftQuotient_append]
+  induction x using List.reverseRecOn with
+  | nil => simp
+  | append_singleton x a ih => simp [ih, leftQuotient_append]
 
 theorem IsRegular.of_finite_range_leftQuotient (h : Set.Finite (Set.range L.leftQuotient)) :
     L.IsRegular :=
diff --git a/Mathlib/Computability/PartrecCode.lean b/Mathlib/Computability/PartrecCode.lean
index 4a04dbd781f9d2..fc3cd1f4049019 100644
--- a/Mathlib/Computability/PartrecCode.lean
+++ b/Mathlib/Computability/PartrecCode.lean
@@ -32,6 +32,7 @@ of some code.
 * `Nat.Partrec.Code.exists_code`: Partial recursiveness is equivalent to being the eval of a code.
 * `Nat.Partrec.Code.evaln_prim`: `evaln` is primitive recursive.
 * `Nat.Partrec.Code.fixed_point`: Roger's fixed point theorem.
+* `Nat.Partrec.Code.fixed_point₂`: Kleene's second recursion theorem.
 
 ## References
 
@@ -979,9 +980,9 @@ theorem eval_part : Partrec₂ eval :=
     (evaln_prim.to_comp.comp ((Computable.snd.pair (fst.comp fst)).pair (snd.comp fst))).to₂).of_eq
     fun a => by simp [eval_eq_rfindOpt]
 
-/-- Roger's fixed-point theorem: Any total, computable `f` has a fixed point: That is, under the
-interpretation given by `Nat.Partrec.Code.eval`, there is a code `c` such that `c` and `f c` have
-the same evaluation.
+/-- **Roger's fixed-point theorem**: any total, computable `f` has a fixed point.
+That is, under the interpretation given by `Nat.Partrec.Code.eval`, there is a code `c`
+such that `c` and `f c` have the same evaluation.
 -/
 theorem fixed_point {f : Code → Code} (hf : Computable f) : ∃ c : Code, eval (f c) = eval c :=
   let g (x y : ℕ) : Part ℕ := eval (ofNat Code x) x >>= fun b => eval (ofNat Code b) y
@@ -1000,6 +1001,7 @@ theorem fixed_point {f : Code → Code} (hf : Computable f) : ∃ c : Code, eval
       show eval (f (curry cg (encode cF))) n = eval (curry cg (encode cF)) n by
         simp [F, g, eg', eF', Part.map_id']⟩
 
+/-- **Kleene's second recursion theorem** -/
 theorem fixed_point₂ {f : Code → ℕ →. ℕ} (hf : Partrec₂ f) : ∃ c : Code, eval c = f c :=
   let ⟨cf, ef⟩ := exists_code.1 hf
   (fixed_point (curry_prim.comp (_root_.Primrec.const cf) Primrec.encode).to_comp).imp fun c e =>
diff --git a/Mathlib/Computability/Primrec.lean b/Mathlib/Computability/Primrec.lean
index 20b710c3ba8449..a4c9369593bb28 100644
--- a/Mathlib/Computability/Primrec.lean
+++ b/Mathlib/Computability/Primrec.lean
@@ -333,7 +333,7 @@ theorem const (x : σ) : Primrec₂ fun (_ : α) (_ : β) => x :=
   Primrec.const _
 
 protected theorem pair : Primrec₂ (@Prod.mk α β) :=
-  .pair .fst .snd
+  Primrec.pair .fst .snd
 
 theorem left : Primrec₂ fun (a : α) (_ : β) => a :=
   .fst
diff --git a/Mathlib/Computability/RegularExpressions.lean b/Mathlib/Computability/RegularExpressions.lean
index 4e10a5bf7e78c1..a1922965cec30b 100644
--- a/Mathlib/Computability/RegularExpressions.lean
+++ b/Mathlib/Computability/RegularExpressions.lean
@@ -92,7 +92,7 @@ theorem comp_def (P Q : RegularExpression α) : comp P Q = P * Q :=
   rfl
 
 -- This was renamed to `matches'` during the port of Lean 4 as `matches` is a reserved word.
-#adaptation_note /-- around nightly-2024-02-25,
+#adaptation_note /-- nightly-2024-02-25
   we need to write `comp x y` in the pattern `comp P Q`, instead of `x * y`. -/
 /-- `matches' P` provides a language which contains all strings that `P` matches -/
 -- Porting note: was '@[simp] but removed based on
@@ -136,7 +136,7 @@ theorem matches'_pow (P : RegularExpression α) : ∀ n : ℕ, (P ^ n).matches'
 theorem matches'_star (P : RegularExpression α) : P.star.matches' = P.matches'∗ :=
   rfl
 
-#adaptation_note /-- around nightly-2024-02-25,
+#adaptation_note /-- nightly-2024-02-25
   we need to write `comp x y` in the pattern `comp P Q`, instead of `x * y`. -/
 /-- `matchEpsilon P` is true if and only if `P` matches the empty string -/
 def matchEpsilon : RegularExpression α → Bool
@@ -150,7 +150,7 @@ def matchEpsilon : RegularExpression α → Bool
 section DecidableEq
 variable [DecidableEq α]
 
-#adaptation_note /-- around nightly-2024-02-25,
+#adaptation_note /-- nightly-2024-02-25
   we need to write `comp x y` in the pattern `comp P Q`, instead of `x * y`. -/
 /-- `P.deriv a` matches `x` if `P` matches `a :: x`, the Brzozowski derivative of `P` with respect
   to `a` -/
@@ -231,7 +231,7 @@ theorem mul_rmatch_iff (P Q : RegularExpression α) (x : List α) :
       rw [rmatch, rmatch]
       rwa [Bool.and_eq_true_iff] at h
     · rintro ⟨t, u, h₁, h₂⟩
-      cases' List.append_eq_nil.1 h₁.symm with ht hu
+      cases' List.append_eq_nil_iff.1 h₁.symm with ht hu
       subst ht
       subst hu
       repeat rw [rmatch] at h₂
@@ -343,7 +343,7 @@ instance (P : RegularExpression α) : DecidablePred (· ∈ P.matches') := fun _
 
 end DecidableEq
 
-#adaptation_note /-- around nightly-2024-02-25,
+#adaptation_note /-- nightly-2024-02-25
   we need to write `comp x y` in the pattern `comp P Q`, instead of `x * y`. -/
 /-- Map the alphabet of a regular expression. -/
 @[simp]
@@ -361,7 +361,7 @@ protected theorem map_pow (f : α → β) (P : RegularExpression α) :
   | 0 => by unfold map; rfl
   | n + 1 => (congr_arg (· * map f P) (RegularExpression.map_pow f P n) :)
 
-#adaptation_note /-- around nightly-2024-02-25,
+#adaptation_note /-- nightly-2024-02-25
   we need to write `comp x y` in the pattern `comp P Q`, instead of `x * y`. -/
 @[simp]
 theorem map_id : ∀ P : RegularExpression α, P.map id = P
@@ -372,7 +372,7 @@ theorem map_id : ∀ P : RegularExpression α, P.map id = P
   | comp R S => by simp_rw [map, map_id]; rfl
   | star R => by simp_rw [map, map_id]
 
-#adaptation_note /-- around nightly-2024-02-25,
+#adaptation_note /-- nightly-2024-02-25
   we need to write `comp x y` in the pattern `comp P Q`, instead of `x * y`. -/
 @[simp]
 theorem map_map (g : β → γ) (f : α → β) : ∀ P : RegularExpression α, (P.map f).map g = P.map (g ∘ f)
@@ -383,7 +383,7 @@ theorem map_map (g : β → γ) (f : α → β) : ∀ P : RegularExpression α,
   | comp R S => by simp only [map, Function.comp_apply, map_map]
   | star R => by simp only [map, Function.comp_apply, map_map]
 
-#adaptation_note /-- around nightly-2024-02-25,
+#adaptation_note /-- nightly-2024-02-25
   we need to write `comp x y` in the pattern `comp R S`,
   instead of `x * y` (and the `erw` was just `rw`). -/
 /-- The language of the map is the map of the language. -/
diff --git a/Mathlib/Computability/Tape.lean b/Mathlib/Computability/Tape.lean
index cf293a33770e00..920c5c776721a1 100644
--- a/Mathlib/Computability/Tape.lean
+++ b/Mathlib/Computability/Tape.lean
@@ -377,7 +377,7 @@ theorem ListBlank.append_assoc {Γ} [Inhabited Γ] (l₁ l₂ : List Γ) (l₃ :
 element is sent to a sequence of default elements. -/
 def ListBlank.flatMap {Γ Γ'} [Inhabited Γ] [Inhabited Γ'] (l : ListBlank Γ) (f : Γ → List Γ')
     (hf : ∃ n, f default = List.replicate n default) : ListBlank Γ' := by
-  apply l.liftOn (fun l ↦ ListBlank.mk (List.flatMap l f))
+  apply l.liftOn (fun l ↦ ListBlank.mk (l.flatMap f))
   rintro l _ ⟨i, rfl⟩; cases' hf with n e; refine Quotient.sound' (Or.inl ⟨i * n, ?_⟩)
   rw [List.flatMap_append, mul_comm]; congr
   induction' i with i IH
diff --git a/Mathlib/Control/Fold.lean b/Mathlib/Control/Fold.lean
index 62e4e80dddadbb..fd6adaeacbf13e 100644
--- a/Mathlib/Control/Fold.lean
+++ b/Mathlib/Control/Fold.lean
@@ -118,10 +118,7 @@ def Foldl.ofFreeMonoid (f : β → α → β) : FreeMonoid α →* Monoid.Foldl
   map_one' := rfl
   map_mul' := by
     intros
-    #adaptation_note /-- nightly-2024-03-16: simp was
-    simp only [FreeMonoid.toList_mul, flip, unop_op, List.foldl_append, op_inj] -/
-    simp only [FreeMonoid.toList_mul, unop_op, List.foldl_append, op_inj, Function.flip_def,
-      List.foldl_append]
+    simp only [FreeMonoid.toList_mul, unop_op, List.foldl_append, op_inj, Function.flip_def]
     rfl
 
 abbrev Foldr (α : Type u) : Type u :=
@@ -280,9 +277,6 @@ theorem foldr.ofFreeMonoid_comp_of (f : β → α → α) :
 theorem foldlm.ofFreeMonoid_comp_of {m} [Monad m] [LawfulMonad m] (f : α → β → m α) :
     foldlM.ofFreeMonoid f ∘ FreeMonoid.of = foldlM.mk ∘ flip f := by
   ext1 x
-  #adaptation_note /-- nightly-2024-03-16: simp was
-  simp only [foldlM.ofFreeMonoid, flip, MonoidHom.coe_mk, OneHom.coe_mk, Function.comp_apply,
-    FreeMonoid.toList_of, List.foldlM_cons, List.foldlM_nil, bind_pure, foldlM.mk, op_inj] -/
   simp only [foldlM.ofFreeMonoid, Function.flip_def, MonoidHom.coe_mk, OneHom.coe_mk,
     Function.comp_apply, FreeMonoid.toList_of, List.foldlM_cons, List.foldlM_nil, bind_pure,
     foldlM.mk, op_inj]
@@ -292,8 +286,6 @@ theorem foldlm.ofFreeMonoid_comp_of {m} [Monad m] [LawfulMonad m] (f : α → β
 theorem foldrm.ofFreeMonoid_comp_of {m} [Monad m] [LawfulMonad m] (f : β → α → m α) :
     foldrM.ofFreeMonoid f ∘ FreeMonoid.of = foldrM.mk ∘ f := by
   ext
-  #adaptation_note /-- nightly-2024-03-16: simp was
-  simp [(· ∘ ·), foldrM.ofFreeMonoid, foldrM.mk, flip] -/
   simp [(· ∘ ·), foldrM.ofFreeMonoid, foldrM.mk, Function.flip_def]
 
 theorem toList_spec (xs : t α) : toList xs = FreeMonoid.toList (foldMap FreeMonoid.of xs) :=
@@ -306,8 +298,6 @@ theorem toList_spec (xs : t α) : toList xs = FreeMonoid.toList (foldMap FreeMon
           simp only [FreeMonoid.reverse_reverse, List.foldr_reverse, List.foldl_flip_cons_eq_append,
             List.append_nil, List.reverse_reverse]
       _ = (unop (Foldl.ofFreeMonoid (flip cons) (foldMap FreeMonoid.of xs)) []).reverse := by
-            #adaptation_note /-- nightly-2024-03-16: simp was
-            simp [flip, List.foldr_reverse, Foldl.ofFreeMonoid, unop_op] -/
             simp [Function.flip_def, List.foldr_reverse, Foldl.ofFreeMonoid, unop_op]
       _ = toList xs := by
             rw [foldMap_hom_free (Foldl.ofFreeMonoid (flip <| @cons α))]
@@ -337,8 +327,6 @@ theorem toList_map (f : α → β) (xs : t α) : toList (f <$> xs) = f <$> toLis
 @[simp]
 theorem foldl_map (g : β → γ) (f : α → γ → α) (a : α) (l : t β) :
     foldl f a (g <$> l) = foldl (fun x y => f x (g y)) a l := by
-  #adaptation_note /-- nightly-2024-03-16: simp was
-  simp only [foldl, foldMap_map, (· ∘ ·), flip] -/
   simp only [foldl, foldMap_map, Function.comp_def, Function.flip_def]
 
 @[simp]
@@ -382,8 +370,6 @@ theorem foldrm_toList (f : α → β → m β) (x : β) (xs : t α) :
 @[simp]
 theorem foldlm_map (g : β → γ) (f : α → γ → m α) (a : α) (l : t β) :
     foldlm f a (g <$> l) = foldlm (fun x y => f x (g y)) a l := by
-  #adaptation_note /-- nightly-2024-03-16: simp was
-  simp only [foldlm, foldMap_map, (· ∘ ·), flip] -/
   simp only [foldlm, foldMap_map, Function.comp_def, Function.flip_def]
 
 @[simp]
diff --git a/Mathlib/Data/Bool/Basic.lean b/Mathlib/Data/Bool/Basic.lean
index 5f5bfc9ba28c85..0857104a5b71e5 100644
--- a/Mathlib/Data/Bool/Basic.lean
+++ b/Mathlib/Data/Bool/Basic.lean
@@ -49,8 +49,8 @@ theorem or_eq_true_eq_eq_true_or_eq_true (a b : Bool) :
 
 theorem not_eq_true_eq_eq_false (a : Bool) : (not a = true) = (a = false) := by cases a <;> simp
 
-#adaptation_note /-- this is no longer a simp lemma,
-  as after nightly-2024-03-05 the LHS simplifies. -/
+#adaptation_note /-- nightly-2024-03-05
+this is no longer a simp lemma, as the LHS simplifies. -/
 theorem and_eq_false_eq_eq_false_or_eq_false (a b : Bool) :
     ((a && b) = false) = (a = false ∨ b = false) := by
   cases a <;> cases b <;> simp
diff --git a/Mathlib/Data/Complex/Abs.lean b/Mathlib/Data/Complex/Abs.lean
index 7531c38a54ff39..92b673fb0b86ab 100644
--- a/Mathlib/Data/Complex/Abs.lean
+++ b/Mathlib/Data/Complex/Abs.lean
@@ -291,10 +291,16 @@ theorem lim_abs (f : CauSeq ℂ Complex.abs) : lim (cauSeqAbs f) = Complex.abs (
     let ⟨i, hi⟩ := equiv_lim f ε ε0
     ⟨i, fun j hj => lt_of_le_of_lt (Complex.abs.abs_abv_sub_le_abv_sub _ _) (hi j hj)⟩
 
+lemma ne_zero_of_re_pos {s : ℂ} (hs : 0 < s.re) : s ≠ 0 :=
+  fun h ↦ (zero_re ▸ h ▸ hs).false
+
 lemma ne_zero_of_one_lt_re {s : ℂ} (hs : 1 < s.re) : s ≠ 0 :=
-  fun h ↦ ((zero_re ▸ h ▸ hs).trans zero_lt_one).false
+  ne_zero_of_re_pos <| zero_lt_one.trans hs
+
+lemma re_neg_ne_zero_of_re_pos {s : ℂ} (hs : 0 < s.re) : (-s).re ≠ 0 :=
+  ne_iff_lt_or_gt.mpr <| Or.inl <| neg_re s ▸ (neg_lt_zero.mpr hs)
 
 lemma re_neg_ne_zero_of_one_lt_re {s : ℂ} (hs : 1 < s.re) : (-s).re ≠ 0 :=
-  ne_iff_lt_or_gt.mpr <| Or.inl <| neg_re s ▸ by linarith
+  re_neg_ne_zero_of_re_pos <| zero_lt_one.trans hs
 
 end Complex
diff --git a/Mathlib/Data/Complex/Exponential.lean b/Mathlib/Data/Complex/Exponential.lean
index 8972434892eb7d..fd4dfde3db5b64 100644
--- a/Mathlib/Data/Complex/Exponential.lean
+++ b/Mathlib/Data/Complex/Exponential.lean
@@ -12,10 +12,9 @@ import Mathlib.Tactic.Bound.Attribute
 import Mathlib.Algebra.CharP.Defs
 
 /-!
-# Exponential, trigonometric and hyperbolic trigonometric functions
+# Exponential Function
 
-This file contains the definitions of the real and complex exponential, sine, cosine, tangent,
-hyperbolic sine, hyperbolic cosine, and hyperbolic tangent functions.
+This file contains the definitions of the real and complex exponential function.
 
 -/
 
@@ -51,40 +50,6 @@ def exp' (z : ℂ) : CauSeq ℂ Complex.abs :=
 def exp (z : ℂ) : ℂ :=
   CauSeq.lim (exp' z)
 
-/-- The complex sine function, defined via `exp` -/
-@[pp_nodot]
-def sin (z : ℂ) : ℂ :=
-  (exp (-z * I) - exp (z * I)) * I / 2
-
-/-- The complex cosine function, defined via `exp` -/
-@[pp_nodot]
-def cos (z : ℂ) : ℂ :=
-  (exp (z * I) + exp (-z * I)) / 2
-
-/-- The complex tangent function, defined as `sin z / cos z` -/
-@[pp_nodot]
-def tan (z : ℂ) : ℂ :=
-  sin z / cos z
-
-/-- The complex cotangent function, defined as `cos z / sin z` -/
-def cot (z : ℂ) : ℂ :=
-  cos z / sin z
-
-/-- The complex hyperbolic sine function, defined via `exp` -/
-@[pp_nodot]
-def sinh (z : ℂ) : ℂ :=
-  (exp z - exp (-z)) / 2
-
-/-- The complex hyperbolic cosine function, defined via `exp` -/
-@[pp_nodot]
-def cosh (z : ℂ) : ℂ :=
-  (exp z + exp (-z)) / 2
-
-/-- The complex hyperbolic tangent function, defined as `sinh z / cosh z` -/
-@[pp_nodot]
-def tanh (z : ℂ) : ℂ :=
-  sinh z / cosh z
-
 /-- scoped notation for the complex exponential function -/
 scoped notation "cexp" => Complex.exp
 
@@ -103,41 +68,6 @@ noncomputable section
 nonrec def exp (x : ℝ) : ℝ :=
   (exp x).re
 
-/-- The real sine function, defined as the real part of the complex sine -/
-@[pp_nodot]
-nonrec def sin (x : ℝ) : ℝ :=
-  (sin x).re
-
-/-- The real cosine function, defined as the real part of the complex cosine -/
-@[pp_nodot]
-nonrec def cos (x : ℝ) : ℝ :=
-  (cos x).re
-
-/-- The real tangent function, defined as the real part of the complex tangent -/
-@[pp_nodot]
-nonrec def tan (x : ℝ) : ℝ :=
-  (tan x).re
-
-/-- The real cotangent function, defined as the real part of the complex cotangent -/
-nonrec def cot (x : ℝ) : ℝ :=
-  (cot x).re
-
-/-- The real hypebolic sine function, defined as the real part of the complex hyperbolic sine -/
-@[pp_nodot]
-nonrec def sinh (x : ℝ) : ℝ :=
-  (sinh x).re
-
-/-- The real hypebolic cosine function, defined as the real part of the complex hyperbolic cosine -/
-@[pp_nodot]
-nonrec def cosh (x : ℝ) : ℝ :=
-  (cosh x).re
-
-/-- The real hypebolic tangent function, defined as the real part of
-the complex hyperbolic tangent -/
-@[pp_nodot]
-nonrec def tanh (x : ℝ) : ℝ :=
-  (tanh x).re
-
 /-- scoped notation for the real exponential function -/
 scoped notation "rexp" => Real.exp
 
@@ -248,427 +178,6 @@ theorem exp_ofReal_im (x : ℝ) : (exp x).im = 0 := by rw [← ofReal_exp_ofReal
 theorem exp_ofReal_re (x : ℝ) : (exp x).re = Real.exp x :=
   rfl
 
-theorem two_sinh : 2 * sinh x = exp x - exp (-x) :=
-  mul_div_cancel₀ _ two_ne_zero
-
-theorem two_cosh : 2 * cosh x = exp x + exp (-x) :=
-  mul_div_cancel₀ _ two_ne_zero
-
-@[simp]
-theorem sinh_zero : sinh 0 = 0 := by simp [sinh]
-
-@[simp]
-theorem sinh_neg : sinh (-x) = -sinh x := by simp [sinh, exp_neg, (neg_div _ _).symm, add_mul]
-
-private theorem sinh_add_aux {a b c d : ℂ} :
-    (a - b) * (c + d) + (a + b) * (c - d) = 2 * (a * c - b * d) := by ring
-
-theorem sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y := by
-  rw [← mul_right_inj' (two_ne_zero' ℂ), two_sinh, exp_add, neg_add, exp_add, eq_comm, mul_add, ←
-    mul_assoc, two_sinh, mul_left_comm, two_sinh, ← mul_right_inj' (two_ne_zero' ℂ), mul_add,
-    mul_left_comm, two_cosh, ← mul_assoc, two_cosh]
-  exact sinh_add_aux
-
-@[simp]
-theorem cosh_zero : cosh 0 = 1 := by simp [cosh]
-
-@[simp]
-theorem cosh_neg : cosh (-x) = cosh x := by simp [add_comm, cosh, exp_neg]
-
-private theorem cosh_add_aux {a b c d : ℂ} :
-    (a + b) * (c + d) + (a - b) * (c - d) = 2 * (a * c + b * d) := by ring
-
-theorem cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y := by
-  rw [← mul_right_inj' (two_ne_zero' ℂ), two_cosh, exp_add, neg_add, exp_add, eq_comm, mul_add, ←
-    mul_assoc, two_cosh, ← mul_assoc, two_sinh, ← mul_right_inj' (two_ne_zero' ℂ), mul_add,
-    mul_left_comm, two_cosh, mul_left_comm, two_sinh]
-  exact cosh_add_aux
-
-theorem sinh_sub : sinh (x - y) = sinh x * cosh y - cosh x * sinh y := by
-  simp [sub_eq_add_neg, sinh_add, sinh_neg, cosh_neg]
-
-theorem cosh_sub : cosh (x - y) = cosh x * cosh y - sinh x * sinh y := by
-  simp [sub_eq_add_neg, cosh_add, sinh_neg, cosh_neg]
-
-theorem sinh_conj : sinh (conj x) = conj (sinh x) := by
-  rw [sinh, ← RingHom.map_neg, exp_conj, exp_conj, ← RingHom.map_sub, sinh, map_div₀]
-  -- Porting note: not nice
-  simp [← one_add_one_eq_two]
-
-@[simp]
-theorem ofReal_sinh_ofReal_re (x : ℝ) : ((sinh x).re : ℂ) = sinh x :=
-  conj_eq_iff_re.1 <| by rw [← sinh_conj, conj_ofReal]
-
-@[simp, norm_cast]
-theorem ofReal_sinh (x : ℝ) : (Real.sinh x : ℂ) = sinh x :=
-  ofReal_sinh_ofReal_re _
-
-@[simp]
-theorem sinh_ofReal_im (x : ℝ) : (sinh x).im = 0 := by rw [← ofReal_sinh_ofReal_re, ofReal_im]
-
-theorem sinh_ofReal_re (x : ℝ) : (sinh x).re = Real.sinh x :=
-  rfl
-
-theorem cosh_conj : cosh (conj x) = conj (cosh x) := by
-  rw [cosh, ← RingHom.map_neg, exp_conj, exp_conj, ← RingHom.map_add, cosh, map_div₀]
-  -- Porting note: not nice
-  simp [← one_add_one_eq_two]
-
-theorem ofReal_cosh_ofReal_re (x : ℝ) : ((cosh x).re : ℂ) = cosh x :=
-  conj_eq_iff_re.1 <| by rw [← cosh_conj, conj_ofReal]
-
-@[simp, norm_cast]
-theorem ofReal_cosh (x : ℝ) : (Real.cosh x : ℂ) = cosh x :=
-  ofReal_cosh_ofReal_re _
-
-@[simp]
-theorem cosh_ofReal_im (x : ℝ) : (cosh x).im = 0 := by rw [← ofReal_cosh_ofReal_re, ofReal_im]
-
-@[simp]
-theorem cosh_ofReal_re (x : ℝ) : (cosh x).re = Real.cosh x :=
-  rfl
-
-theorem tanh_eq_sinh_div_cosh : tanh x = sinh x / cosh x :=
-  rfl
-
-@[simp]
-theorem tanh_zero : tanh 0 = 0 := by simp [tanh]
-
-@[simp]
-theorem tanh_neg : tanh (-x) = -tanh x := by simp [tanh, neg_div]
-
-theorem tanh_conj : tanh (conj x) = conj (tanh x) := by
-  rw [tanh, sinh_conj, cosh_conj, ← map_div₀, tanh]
-
-@[simp]
-theorem ofReal_tanh_ofReal_re (x : ℝ) : ((tanh x).re : ℂ) = tanh x :=
-  conj_eq_iff_re.1 <| by rw [← tanh_conj, conj_ofReal]
-
-@[simp, norm_cast]
-theorem ofReal_tanh (x : ℝ) : (Real.tanh x : ℂ) = tanh x :=
-  ofReal_tanh_ofReal_re _
-
-@[simp]
-theorem tanh_ofReal_im (x : ℝ) : (tanh x).im = 0 := by rw [← ofReal_tanh_ofReal_re, ofReal_im]
-
-theorem tanh_ofReal_re (x : ℝ) : (tanh x).re = Real.tanh x :=
-  rfl
-
-@[simp]
-theorem cosh_add_sinh : cosh x + sinh x = exp x := by
-  rw [← mul_right_inj' (two_ne_zero' ℂ), mul_add, two_cosh, two_sinh, add_add_sub_cancel, two_mul]
-
-@[simp]
-theorem sinh_add_cosh : sinh x + cosh x = exp x := by rw [add_comm, cosh_add_sinh]
-
-@[simp]
-theorem exp_sub_cosh : exp x - cosh x = sinh x :=
-  sub_eq_iff_eq_add.2 (sinh_add_cosh x).symm
-
-@[simp]
-theorem exp_sub_sinh : exp x - sinh x = cosh x :=
-  sub_eq_iff_eq_add.2 (cosh_add_sinh x).symm
-
-@[simp]
-theorem cosh_sub_sinh : cosh x - sinh x = exp (-x) := by
-  rw [← mul_right_inj' (two_ne_zero' ℂ), mul_sub, two_cosh, two_sinh, add_sub_sub_cancel, two_mul]
-
-@[simp]
-theorem sinh_sub_cosh : sinh x - cosh x = -exp (-x) := by rw [← neg_sub, cosh_sub_sinh]
-
-@[simp]
-theorem cosh_sq_sub_sinh_sq : cosh x ^ 2 - sinh x ^ 2 = 1 := by
-  rw [sq_sub_sq, cosh_add_sinh, cosh_sub_sinh, ← exp_add, add_neg_cancel, exp_zero]
-
-theorem cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 := by
-  rw [← cosh_sq_sub_sinh_sq x]
-  ring
-
-theorem sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 := by
-  rw [← cosh_sq_sub_sinh_sq x]
-  ring
-
-theorem cosh_two_mul : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2 := by rw [two_mul, cosh_add, sq, sq]
-
-theorem sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x := by
-  rw [two_mul, sinh_add]
-  ring
-
-theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x := by
-  have h1 : x + 2 * x = 3 * x := by ring
-  rw [← h1, cosh_add x (2 * x)]
-  simp only [cosh_two_mul, sinh_two_mul]
-  have h2 : sinh x * (2 * sinh x * cosh x) = 2 * cosh x * sinh x ^ 2 := by ring
-  rw [h2, sinh_sq]
-  ring
-
-theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x := by
-  have h1 : x + 2 * x = 3 * x := by ring
-  rw [← h1, sinh_add x (2 * x)]
-  simp only [cosh_two_mul, sinh_two_mul]
-  have h2 : cosh x * (2 * sinh x * cosh x) = 2 * sinh x * cosh x ^ 2 := by ring
-  rw [h2, cosh_sq]
-  ring
-
-@[simp]
-theorem sin_zero : sin 0 = 0 := by simp [sin]
-
-@[simp]
-theorem sin_neg : sin (-x) = -sin x := by
-  simp [sin, sub_eq_add_neg, exp_neg, (neg_div _ _).symm, add_mul]
-
-theorem two_sin : 2 * sin x = (exp (-x * I) - exp (x * I)) * I :=
-  mul_div_cancel₀ _ two_ne_zero
-
-theorem two_cos : 2 * cos x = exp (x * I) + exp (-x * I) :=
-  mul_div_cancel₀ _ two_ne_zero
-
-theorem sinh_mul_I : sinh (x * I) = sin x * I := by
-  rw [← mul_right_inj' (two_ne_zero' ℂ), two_sinh, ← mul_assoc, two_sin, mul_assoc, I_mul_I,
-    mul_neg_one, neg_sub, neg_mul_eq_neg_mul]
-
-theorem cosh_mul_I : cosh (x * I) = cos x := by
-  rw [← mul_right_inj' (two_ne_zero' ℂ), two_cosh, two_cos, neg_mul_eq_neg_mul]
-
-theorem tanh_mul_I : tanh (x * I) = tan x * I := by
-  rw [tanh_eq_sinh_div_cosh, cosh_mul_I, sinh_mul_I, mul_div_right_comm, tan]
-
-theorem cos_mul_I : cos (x * I) = cosh x := by rw [← cosh_mul_I]; ring_nf; simp
-
-theorem sin_mul_I : sin (x * I) = sinh x * I := by
-  have h : I * sin (x * I) = -sinh x := by
-    rw [mul_comm, ← sinh_mul_I]
-    ring_nf
-    simp
-  rw [← neg_neg (sinh x), ← h]
-  apply Complex.ext <;> simp
-
-theorem tan_mul_I : tan (x * I) = tanh x * I := by
-  rw [tan, sin_mul_I, cos_mul_I, mul_div_right_comm, tanh_eq_sinh_div_cosh]
-
-theorem sin_add : sin (x + y) = sin x * cos y + cos x * sin y := by
-  rw [← mul_left_inj' I_ne_zero, ← sinh_mul_I, add_mul, add_mul, mul_right_comm, ← sinh_mul_I,
-    mul_assoc, ← sinh_mul_I, ← cosh_mul_I, ← cosh_mul_I, sinh_add]
-
-@[simp]
-theorem cos_zero : cos 0 = 1 := by simp [cos]
-
-@[simp]
-theorem cos_neg : cos (-x) = cos x := by simp [cos, sub_eq_add_neg, exp_neg, add_comm]
-
-theorem cos_add : cos (x + y) = cos x * cos y - sin x * sin y := by
-  rw [← cosh_mul_I, add_mul, cosh_add, cosh_mul_I, cosh_mul_I, sinh_mul_I, sinh_mul_I,
-    mul_mul_mul_comm, I_mul_I, mul_neg_one, sub_eq_add_neg]
-
-theorem sin_sub : sin (x - y) = sin x * cos y - cos x * sin y := by
-  simp [sub_eq_add_neg, sin_add, sin_neg, cos_neg]
-
-theorem cos_sub : cos (x - y) = cos x * cos y + sin x * sin y := by
-  simp [sub_eq_add_neg, cos_add, sin_neg, cos_neg]
-
-theorem sin_add_mul_I (x y : ℂ) : sin (x + y * I) = sin x * cosh y + cos x * sinh y * I := by
-  rw [sin_add, cos_mul_I, sin_mul_I, mul_assoc]
-
-theorem sin_eq (z : ℂ) : sin z = sin z.re * cosh z.im + cos z.re * sinh z.im * I := by
-  convert sin_add_mul_I z.re z.im; exact (re_add_im z).symm
-
-theorem cos_add_mul_I (x y : ℂ) : cos (x + y * I) = cos x * cosh y - sin x * sinh y * I := by
-  rw [cos_add, cos_mul_I, sin_mul_I, mul_assoc]
-
-theorem cos_eq (z : ℂ) : cos z = cos z.re * cosh z.im - sin z.re * sinh z.im * I := by
-  convert cos_add_mul_I z.re z.im; exact (re_add_im z).symm
-
-theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) := by
-  have s1 := sin_add ((x + y) / 2) ((x - y) / 2)
-  have s2 := sin_sub ((x + y) / 2) ((x - y) / 2)
-  rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel_right, add_self_div_two] at s1
-  rw [div_sub_div_same, ← sub_add, add_sub_cancel_left, add_self_div_two] at s2
-  rw [s1, s2]
-  ring
-
-theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) := by
-  have s1 := cos_add ((x + y) / 2) ((x - y) / 2)
-  have s2 := cos_sub ((x + y) / 2) ((x - y) / 2)
-  rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel_right, add_self_div_two] at s1
-  rw [div_sub_div_same, ← sub_add, add_sub_cancel_left, add_self_div_two] at s2
-  rw [s1, s2]
-  ring
-
-theorem sin_add_sin : sin x + sin y = 2 * sin ((x + y) / 2) * cos ((x - y) / 2) := by
-  simpa using sin_sub_sin x (-y)
-
-theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) := by
-  calc
-    cos x + cos y = cos ((x + y) / 2 + (x - y) / 2) + cos ((x + y) / 2 - (x - y) / 2) := ?_
-    _ =
-        cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2) +
-          (cos ((x + y) / 2) * cos ((x - y) / 2) + sin ((x + y) / 2) * sin ((x - y) / 2)) :=
-      ?_
-    _ = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) := ?_
-
-  · congr <;> field_simp
-  · rw [cos_add, cos_sub]
-  ring
-
-theorem sin_conj : sin (conj x) = conj (sin x) := by
-  rw [← mul_left_inj' I_ne_zero, ← sinh_mul_I, ← conj_neg_I, ← RingHom.map_mul, ← RingHom.map_mul,
-    sinh_conj, mul_neg, sinh_neg, sinh_mul_I, mul_neg]
-
-@[simp]
-theorem ofReal_sin_ofReal_re (x : ℝ) : ((sin x).re : ℂ) = sin x :=
-  conj_eq_iff_re.1 <| by rw [← sin_conj, conj_ofReal]
-
-@[simp, norm_cast]
-theorem ofReal_sin (x : ℝ) : (Real.sin x : ℂ) = sin x :=
-  ofReal_sin_ofReal_re _
-
-@[simp]
-theorem sin_ofReal_im (x : ℝ) : (sin x).im = 0 := by rw [← ofReal_sin_ofReal_re, ofReal_im]
-
-theorem sin_ofReal_re (x : ℝ) : (sin x).re = Real.sin x :=
-  rfl
-
-theorem cos_conj : cos (conj x) = conj (cos x) := by
-  rw [← cosh_mul_I, ← conj_neg_I, ← RingHom.map_mul, ← cosh_mul_I, cosh_conj, mul_neg, cosh_neg]
-
-@[simp]
-theorem ofReal_cos_ofReal_re (x : ℝ) : ((cos x).re : ℂ) = cos x :=
-  conj_eq_iff_re.1 <| by rw [← cos_conj, conj_ofReal]
-
-@[simp, norm_cast]
-theorem ofReal_cos (x : ℝ) : (Real.cos x : ℂ) = cos x :=
-  ofReal_cos_ofReal_re _
-
-@[simp]
-theorem cos_ofReal_im (x : ℝ) : (cos x).im = 0 := by rw [← ofReal_cos_ofReal_re, ofReal_im]
-
-theorem cos_ofReal_re (x : ℝ) : (cos x).re = Real.cos x :=
-  rfl
-
-@[simp]
-theorem tan_zero : tan 0 = 0 := by simp [tan]
-
-theorem tan_eq_sin_div_cos : tan x = sin x / cos x :=
-  rfl
-
-theorem cot_eq_cos_div_sin : cot x = cos x / sin x :=
-  rfl
-
-theorem tan_mul_cos {x : ℂ} (hx : cos x ≠ 0) : tan x * cos x = sin x := by
-  rw [tan_eq_sin_div_cos, div_mul_cancel₀ _ hx]
-
-@[simp]
-theorem tan_neg : tan (-x) = -tan x := by simp [tan, neg_div]
-
-theorem tan_conj : tan (conj x) = conj (tan x) := by rw [tan, sin_conj, cos_conj, ← map_div₀, tan]
-
-theorem cot_conj : cot (conj x) = conj (cot x) := by rw [cot, sin_conj, cos_conj, ← map_div₀, cot]
-
-@[simp]
-theorem ofReal_tan_ofReal_re (x : ℝ) : ((tan x).re : ℂ) = tan x :=
-  conj_eq_iff_re.1 <| by rw [← tan_conj, conj_ofReal]
-
-@[simp]
-theorem ofReal_cot_ofReal_re (x : ℝ) : ((cot x).re : ℂ) = cot x :=
-  conj_eq_iff_re.1 <| by rw [← cot_conj, conj_ofReal]
-
-@[simp, norm_cast]
-theorem ofReal_tan (x : ℝ) : (Real.tan x : ℂ) = tan x :=
-  ofReal_tan_ofReal_re _
-
-@[simp, norm_cast]
-theorem ofReal_cot (x : ℝ) : (Real.cot x : ℂ) = cot x :=
-  ofReal_cot_ofReal_re _
-
-@[simp]
-theorem tan_ofReal_im (x : ℝ) : (tan x).im = 0 := by rw [← ofReal_tan_ofReal_re, ofReal_im]
-
-theorem tan_ofReal_re (x : ℝ) : (tan x).re = Real.tan x :=
-  rfl
-
-theorem cos_add_sin_I : cos x + sin x * I = exp (x * I) := by
-  rw [← cosh_add_sinh, sinh_mul_I, cosh_mul_I]
-
-theorem cos_sub_sin_I : cos x - sin x * I = exp (-x * I) := by
-  rw [neg_mul, ← cosh_sub_sinh, sinh_mul_I, cosh_mul_I]
-
-@[simp]
-theorem sin_sq_add_cos_sq : sin x ^ 2 + cos x ^ 2 = 1 :=
-  Eq.trans (by rw [cosh_mul_I, sinh_mul_I, mul_pow, I_sq, mul_neg_one, sub_neg_eq_add, add_comm])
-    (cosh_sq_sub_sinh_sq (x * I))
-
-@[simp]
-theorem cos_sq_add_sin_sq : cos x ^ 2 + sin x ^ 2 = 1 := by rw [add_comm, sin_sq_add_cos_sq]
-
-theorem cos_two_mul' : cos (2 * x) = cos x ^ 2 - sin x ^ 2 := by rw [two_mul, cos_add, ← sq, ← sq]
-
-theorem cos_two_mul : cos (2 * x) = 2 * cos x ^ 2 - 1 := by
-  rw [cos_two_mul', eq_sub_iff_add_eq.2 (sin_sq_add_cos_sq x), ← sub_add, sub_add_eq_add_sub,
-    two_mul]
-
-theorem sin_two_mul : sin (2 * x) = 2 * sin x * cos x := by
-  rw [two_mul, sin_add, two_mul, add_mul, mul_comm]
-
-theorem cos_sq : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 := by
-  simp [cos_two_mul, div_add_div_same, mul_div_cancel_left₀, two_ne_zero, -one_div]
-
-theorem cos_sq' : cos x ^ 2 = 1 - sin x ^ 2 := by rw [← sin_sq_add_cos_sq x, add_sub_cancel_left]
-
-theorem sin_sq : sin x ^ 2 = 1 - cos x ^ 2 := by rw [← sin_sq_add_cos_sq x, add_sub_cancel_right]
-
-theorem inv_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2 := by
-  rw [tan_eq_sin_div_cos, div_pow]
-  field_simp
-
-theorem tan_sq_div_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) :
-    tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2 := by
-  simp only [← tan_mul_cos hx, mul_pow, ← inv_one_add_tan_sq hx, div_eq_mul_inv, one_mul]
-
-theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x := by
-  have h1 : x + 2 * x = 3 * x := by ring
-  rw [← h1, cos_add x (2 * x)]
-  simp only [cos_two_mul, sin_two_mul, mul_add, mul_sub, mul_one, sq]
-  have h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2 := by ring
-  rw [h2, cos_sq']
-  ring
-
-theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 := by
-  have h1 : x + 2 * x = 3 * x := by ring
-  rw [← h1, sin_add x (2 * x)]
-  simp only [cos_two_mul, sin_two_mul, cos_sq']
-  have h2 : cos x * (2 * sin x * cos x) = 2 * sin x * cos x ^ 2 := by ring
-  rw [h2, cos_sq']
-  ring
-
-theorem exp_mul_I : exp (x * I) = cos x + sin x * I :=
-  (cos_add_sin_I _).symm
-
-theorem exp_add_mul_I : exp (x + y * I) = exp x * (cos y + sin y * I) := by rw [exp_add, exp_mul_I]
-
-theorem exp_eq_exp_re_mul_sin_add_cos : exp x = exp x.re * (cos x.im + sin x.im * I) := by
-  rw [← exp_add_mul_I, re_add_im]
-
-theorem exp_re : (exp x).re = Real.exp x.re * Real.cos x.im := by
-  rw [exp_eq_exp_re_mul_sin_add_cos]
-  simp [exp_ofReal_re, cos_ofReal_re]
-
-theorem exp_im : (exp x).im = Real.exp x.re * Real.sin x.im := by
-  rw [exp_eq_exp_re_mul_sin_add_cos]
-  simp [exp_ofReal_re, sin_ofReal_re]
-
-@[simp]
-theorem exp_ofReal_mul_I_re (x : ℝ) : (exp (x * I)).re = Real.cos x := by
-  simp [exp_mul_I, cos_ofReal_re]
-
-@[simp]
-theorem exp_ofReal_mul_I_im (x : ℝ) : (exp (x * I)).im = Real.sin x := by
-  simp [exp_mul_I, sin_ofReal_re]
-
-/-- **De Moivre's formula** -/
-theorem cos_add_sin_mul_I_pow (n : ℕ) (z : ℂ) :
-    (cos z + sin z * I) ^ n = cos (↑n * z) + sin (↑n * z) * I := by
-  rw [← exp_mul_I, ← exp_mul_I, ← exp_nat_mul, mul_assoc]
-
 end Complex
 
 namespace Real
@@ -716,241 +225,6 @@ nonrec theorem exp_neg : exp (-x) = (exp x)⁻¹ :=
 theorem exp_sub : exp (x - y) = exp x / exp y := by
   simp [sub_eq_add_neg, exp_add, exp_neg, div_eq_mul_inv]
 
-@[simp]
-theorem sin_zero : sin 0 = 0 := by simp [sin]
-
-@[simp]
-theorem sin_neg : sin (-x) = -sin x := by simp [sin, exp_neg, (neg_div _ _).symm, add_mul]
-
-nonrec theorem sin_add : sin (x + y) = sin x * cos y + cos x * sin y :=
-  ofReal_injective <| by simp [sin_add]
-
-@[simp]
-theorem cos_zero : cos 0 = 1 := by simp [cos]
-
-@[simp]
-theorem cos_neg : cos (-x) = cos x := by simp [cos, exp_neg]
-
-@[simp]
-theorem cos_abs : cos |x| = cos x := by
-  cases le_total x 0 <;> simp only [*, _root_.abs_of_nonneg, abs_of_nonpos, cos_neg]
-
-nonrec theorem cos_add : cos (x + y) = cos x * cos y - sin x * sin y :=
-  ofReal_injective <| by simp [cos_add]
-
-theorem sin_sub : sin (x - y) = sin x * cos y - cos x * sin y := by
-  simp [sub_eq_add_neg, sin_add, sin_neg, cos_neg]
-
-theorem cos_sub : cos (x - y) = cos x * cos y + sin x * sin y := by
-  simp [sub_eq_add_neg, cos_add, sin_neg, cos_neg]
-
-nonrec theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) :=
-  ofReal_injective <| by simp [sin_sub_sin]
-
-nonrec theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) :=
-  ofReal_injective <| by simp [cos_sub_cos]
-
-nonrec theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) :=
-  ofReal_injective <| by simp [cos_add_cos]
-
-theorem two_mul_sin_mul_sin (x y : ℝ) : 2 * sin x * sin y = cos (x - y) - cos (x + y) := by
-  simp [cos_add, cos_sub]
-  ring
-
-theorem two_mul_cos_mul_cos (x y : ℝ) : 2 * cos x * cos y = cos (x - y) + cos (x + y) := by
-  simp [cos_add, cos_sub]
-  ring
-
-theorem two_mul_sin_mul_cos (x y : ℝ) : 2 * sin x * cos y = sin (x - y) + sin (x + y) := by
-  simp [sin_add, sin_sub]
-  ring
-
-nonrec theorem tan_eq_sin_div_cos : tan x = sin x / cos x :=
-  ofReal_injective <| by simp only [ofReal_tan, tan_eq_sin_div_cos, ofReal_div, ofReal_sin,
-    ofReal_cos]
-
-nonrec theorem cot_eq_cos_div_sin : cot x = cos x / sin x :=
-  ofReal_injective <| by simp [cot_eq_cos_div_sin]
-
-theorem tan_mul_cos {x : ℝ} (hx : cos x ≠ 0) : tan x * cos x = sin x := by
-  rw [tan_eq_sin_div_cos, div_mul_cancel₀ _ hx]
-
-@[simp]
-theorem tan_zero : tan 0 = 0 := by simp [tan]
-
-@[simp]
-theorem tan_neg : tan (-x) = -tan x := by simp [tan, neg_div]
-
-@[simp]
-nonrec theorem sin_sq_add_cos_sq : sin x ^ 2 + cos x ^ 2 = 1 :=
-  ofReal_injective (by simp [sin_sq_add_cos_sq])
-
-@[simp]
-theorem cos_sq_add_sin_sq : cos x ^ 2 + sin x ^ 2 = 1 := by rw [add_comm, sin_sq_add_cos_sq]
-
-theorem sin_sq_le_one : sin x ^ 2 ≤ 1 := by
-  rw [← sin_sq_add_cos_sq x]; exact le_add_of_nonneg_right (sq_nonneg _)
-
-theorem cos_sq_le_one : cos x ^ 2 ≤ 1 := by
-  rw [← sin_sq_add_cos_sq x]; exact le_add_of_nonneg_left (sq_nonneg _)
-
-theorem abs_sin_le_one : |sin x| ≤ 1 :=
-  abs_le_one_iff_mul_self_le_one.2 <| by simp only [← sq, sin_sq_le_one]
-
-theorem abs_cos_le_one : |cos x| ≤ 1 :=
-  abs_le_one_iff_mul_self_le_one.2 <| by simp only [← sq, cos_sq_le_one]
-
-theorem sin_le_one : sin x ≤ 1 :=
-  (abs_le.1 (abs_sin_le_one _)).2
-
-theorem cos_le_one : cos x ≤ 1 :=
-  (abs_le.1 (abs_cos_le_one _)).2
-
-theorem neg_one_le_sin : -1 ≤ sin x :=
-  (abs_le.1 (abs_sin_le_one _)).1
-
-theorem neg_one_le_cos : -1 ≤ cos x :=
-  (abs_le.1 (abs_cos_le_one _)).1
-
-nonrec theorem cos_two_mul : cos (2 * x) = 2 * cos x ^ 2 - 1 :=
-  ofReal_injective <| by simp [cos_two_mul]
-
-nonrec theorem cos_two_mul' : cos (2 * x) = cos x ^ 2 - sin x ^ 2 :=
-  ofReal_injective <| by simp [cos_two_mul']
-
-nonrec theorem sin_two_mul : sin (2 * x) = 2 * sin x * cos x :=
-  ofReal_injective <| by simp [sin_two_mul]
-
-nonrec theorem cos_sq : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 :=
-  ofReal_injective <| by simp [cos_sq]
-
-theorem cos_sq' : cos x ^ 2 = 1 - sin x ^ 2 := by rw [← sin_sq_add_cos_sq x, add_sub_cancel_left]
-
-theorem sin_sq : sin x ^ 2 = 1 - cos x ^ 2 :=
-  eq_sub_iff_add_eq.2 <| sin_sq_add_cos_sq _
-
-lemma sin_sq_eq_half_sub : sin x ^ 2 = 1 / 2 - cos (2 * x) / 2 := by
-  rw [sin_sq, cos_sq, ← sub_sub, sub_half]
-
-theorem abs_sin_eq_sqrt_one_sub_cos_sq (x : ℝ) : |sin x| = √(1 - cos x ^ 2) := by
-  rw [← sin_sq, sqrt_sq_eq_abs]
-
-theorem abs_cos_eq_sqrt_one_sub_sin_sq (x : ℝ) : |cos x| = √(1 - sin x ^ 2) := by
-  rw [← cos_sq', sqrt_sq_eq_abs]
-
-theorem inv_one_add_tan_sq {x : ℝ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2 :=
-  have : Complex.cos x ≠ 0 := mt (congr_arg re) hx
-  ofReal_inj.1 <| by simpa using Complex.inv_one_add_tan_sq this
-
-theorem tan_sq_div_one_add_tan_sq {x : ℝ} (hx : cos x ≠ 0) :
-    tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2 := by
-  simp only [← tan_mul_cos hx, mul_pow, ← inv_one_add_tan_sq hx, div_eq_mul_inv, one_mul]
-
-theorem inv_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < cos x) : (√(1 + tan x ^ 2))⁻¹ = cos x := by
-  rw [← sqrt_sq hx.le, ← sqrt_inv, inv_one_add_tan_sq hx.ne']
-
-theorem tan_div_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < cos x) :
-    tan x / √(1 + tan x ^ 2) = sin x := by
-  rw [← tan_mul_cos hx.ne', ← inv_sqrt_one_add_tan_sq hx, div_eq_mul_inv]
-
-nonrec theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x := by
-  rw [← ofReal_inj]; simp [cos_three_mul]
-
-nonrec theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 := by
-  rw [← ofReal_inj]; simp [sin_three_mul]
-
-/-- The definition of `sinh` in terms of `exp`. -/
-nonrec theorem sinh_eq (x : ℝ) : sinh x = (exp x - exp (-x)) / 2 :=
-  ofReal_injective <| by simp [Complex.sinh]
-
-@[simp]
-theorem sinh_zero : sinh 0 = 0 := by simp [sinh]
-
-@[simp]
-theorem sinh_neg : sinh (-x) = -sinh x := by simp [sinh, exp_neg, (neg_div _ _).symm, add_mul]
-
-nonrec theorem sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y := by
-  rw [← ofReal_inj]; simp [sinh_add]
-
-/-- The definition of `cosh` in terms of `exp`. -/
-theorem cosh_eq (x : ℝ) : cosh x = (exp x + exp (-x)) / 2 :=
-  eq_div_of_mul_eq two_ne_zero <| by
-    rw [cosh, exp, exp, Complex.ofReal_neg, Complex.cosh, mul_two, ← Complex.add_re, ← mul_two,
-      div_mul_cancel₀ _ (two_ne_zero' ℂ), Complex.add_re]
-
-@[simp]
-theorem cosh_zero : cosh 0 = 1 := by simp [cosh]
-
-@[simp]
-theorem cosh_neg : cosh (-x) = cosh x :=
-  ofReal_inj.1 <| by simp
-
-@[simp]
-theorem cosh_abs : cosh |x| = cosh x := by
-  cases le_total x 0 <;> simp [*, _root_.abs_of_nonneg, abs_of_nonpos]
-
-nonrec theorem cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y := by
-  rw [← ofReal_inj]; simp [cosh_add]
-
-theorem sinh_sub : sinh (x - y) = sinh x * cosh y - cosh x * sinh y := by
-  simp [sub_eq_add_neg, sinh_add, sinh_neg, cosh_neg]
-
-theorem cosh_sub : cosh (x - y) = cosh x * cosh y - sinh x * sinh y := by
-  simp [sub_eq_add_neg, cosh_add, sinh_neg, cosh_neg]
-
-nonrec theorem tanh_eq_sinh_div_cosh : tanh x = sinh x / cosh x :=
-  ofReal_inj.1 <| by simp [tanh_eq_sinh_div_cosh]
-
-@[simp]
-theorem tanh_zero : tanh 0 = 0 := by simp [tanh]
-
-@[simp]
-theorem tanh_neg : tanh (-x) = -tanh x := by simp [tanh, neg_div]
-
-@[simp]
-theorem cosh_add_sinh : cosh x + sinh x = exp x := by rw [← ofReal_inj]; simp
-
-@[simp]
-theorem sinh_add_cosh : sinh x + cosh x = exp x := by rw [add_comm, cosh_add_sinh]
-
-@[simp]
-theorem exp_sub_cosh : exp x - cosh x = sinh x :=
-  sub_eq_iff_eq_add.2 (sinh_add_cosh x).symm
-
-@[simp]
-theorem exp_sub_sinh : exp x - sinh x = cosh x :=
-  sub_eq_iff_eq_add.2 (cosh_add_sinh x).symm
-
-@[simp]
-theorem cosh_sub_sinh : cosh x - sinh x = exp (-x) := by
-  rw [← ofReal_inj]
-  simp
-
-@[simp]
-theorem sinh_sub_cosh : sinh x - cosh x = -exp (-x) := by rw [← neg_sub, cosh_sub_sinh]
-
-@[simp]
-theorem cosh_sq_sub_sinh_sq (x : ℝ) : cosh x ^ 2 - sinh x ^ 2 = 1 := by rw [← ofReal_inj]; simp
-
-nonrec theorem cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 := by rw [← ofReal_inj]; simp [cosh_sq]
-
-theorem cosh_sq' : cosh x ^ 2 = 1 + sinh x ^ 2 :=
-  (cosh_sq x).trans (add_comm _ _)
-
-nonrec theorem sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 := by rw [← ofReal_inj]; simp [sinh_sq]
-
-nonrec theorem cosh_two_mul : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2 := by
-  rw [← ofReal_inj]; simp [cosh_two_mul]
-
-nonrec theorem sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x := by
-  rw [← ofReal_inj]; simp [sinh_two_mul]
-
-nonrec theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x := by
-  rw [← ofReal_inj]; simp [cosh_three_mul]
-
-nonrec theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x := by
-  rw [← ofReal_inj]; simp [sinh_three_mul]
-
 open IsAbsoluteValue Nat
 
 theorem sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i ∈ range n, x ^ i / i ! ≤ exp x :=
@@ -1055,13 +329,6 @@ theorem exp_le_one_iff {x : ℝ} : exp x ≤ 1 ↔ x ≤ 0 :=
 theorem one_le_exp_iff {x : ℝ} : 1 ≤ exp x ↔ 0 ≤ x :=
   exp_zero ▸ exp_le_exp
 
-/-- `Real.cosh` is always positive -/
-theorem cosh_pos (x : ℝ) : 0 < Real.cosh x :=
-  (cosh_eq x).symm ▸ half_pos (add_pos (exp_pos x) (exp_pos (-x)))
-
-theorem sinh_lt_cosh : sinh x < cosh x :=
-  lt_of_pow_lt_pow_left₀ 2 (cosh_pos _).le <| (cosh_sq x).symm ▸ lt_add_one _
-
 end Real
 
 namespace Complex
@@ -1334,125 +601,6 @@ theorem exp_1_approx_succ_eq {n} {a₁ b₁ : ℝ} {m : ℕ} (en : n + 1 = m) {r
 theorem exp_approx_start (x a b : ℝ) (h : |exp x - expNear 0 x a| ≤ |x| ^ 0 / Nat.factorial 0 * b) :
     |exp x - a| ≤ b := by simpa using h
 
-theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
-  calc
-    |cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
-      rw [← abs_ofReal]; simp
-    _ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
-      simp [Complex.cos, sub_div, add_div, neg_div, div_self (two_ne_zero' ℂ)]
-    _ = abs
-          (((Complex.exp (x * I) - ∑ m ∈ range 4, (x * I) ^ m / m.factorial) +
-              (Complex.exp (-x * I) - ∑ m ∈ range 4, (-x * I) ^ m / m.factorial)) / 2) :=
-      (congr_arg Complex.abs
-        (congr_arg (fun x : ℂ => x / 2)
-          (by
-            simp only [sum_range_succ, neg_mul, pow_succ, pow_zero, mul_one, range_zero, sum_empty,
-              Nat.factorial, Nat.cast_one, ne_eq, one_ne_zero, not_false_eq_true, div_self,
-              zero_add, div_one, Nat.mul_one, Nat.cast_succ, Nat.cast_mul, Nat.cast_ofNat, mul_neg,
-              neg_neg]
-            apply Complex.ext <;> simp [div_eq_mul_inv, normSq] <;> ring_nf
-            )))
-    _ ≤ abs ((Complex.exp (x * I) - ∑ m ∈ range 4, (x * I) ^ m / m.factorial) / 2) +
-          abs ((Complex.exp (-x * I) - ∑ m ∈ range 4, (-x * I) ^ m / m.factorial) / 2) := by
-      rw [add_div]; exact Complex.abs.add_le _ _
-    _ = abs (Complex.exp (x * I) - ∑ m ∈ range 4, (x * I) ^ m / m.factorial) / 2 +
-          abs (Complex.exp (-x * I) - ∑ m ∈ range 4, (-x * I) ^ m / m.factorial) / 2 := by
-      simp [map_div₀]
-    _ ≤ Complex.abs (x * I) ^ 4 * (Nat.succ 4 * ((Nat.factorial 4) * (4 : ℕ) : ℝ)⁻¹) / 2 +
-          Complex.abs (-x * I) ^ 4 * (Nat.succ 4 * ((Nat.factorial 4) * (4 : ℕ) : ℝ)⁻¹) / 2 := by
-      gcongr
-      · exact Complex.exp_bound (by simpa) (by decide)
-      · exact Complex.exp_bound (by simpa) (by decide)
-    _ ≤ |x| ^ 4 * (5 / 96) := by norm_num [Nat.factorial]
-
-theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
-  calc
-    |sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
-      rw [← abs_ofReal]; simp
-    _ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
-          (2 * x - x ^ 3 / 3 : ℝ)) / 2) := by
-      simp [Complex.sin, sub_div, add_div, neg_div, mul_div_cancel_left₀ _ (two_ne_zero' ℂ),
-        div_div, show (3 : ℂ) * 2 = 6 by norm_num]
-    _ = Complex.abs (((Complex.exp (-x * I) - ∑ m ∈ range 4, (-x * I) ^ m / m.factorial) -
-                (Complex.exp (x * I) - ∑ m ∈ range 4, (x * I) ^ m / m.factorial)) * I / 2) :=
-      (congr_arg Complex.abs
-        (congr_arg (fun x : ℂ => x / 2)
-          (by
-            simp only [sum_range_succ, neg_mul, pow_succ, pow_zero, mul_one, ofReal_sub, ofReal_mul,
-              ofReal_ofNat, ofReal_div, range_zero, sum_empty, Nat.factorial, Nat.cast_one, ne_eq,
-              one_ne_zero, not_false_eq_true, div_self, zero_add, div_one, mul_neg, neg_neg,
-              Nat.mul_one, Nat.cast_succ, Nat.cast_mul, Nat.cast_ofNat]
-            apply Complex.ext <;> simp [div_eq_mul_inv, normSq]; ring)))
-    _ ≤ abs ((Complex.exp (-x * I) - ∑ m ∈ range 4, (-x * I) ^ m / m.factorial) * I / 2) +
-          abs (-((Complex.exp (x * I) - ∑ m ∈ range 4, (x * I) ^ m / m.factorial) * I) / 2) := by
-      rw [sub_mul, sub_eq_add_neg, add_div]; exact Complex.abs.add_le _ _
-    _ = abs (Complex.exp (x * I) - ∑ m ∈ range 4, (x * I) ^ m / m.factorial) / 2 +
-          abs (Complex.exp (-x * I) - ∑ m ∈ range 4, (-x * I) ^ m / m.factorial) / 2 := by
-      simp [add_comm, map_div₀]
-    _ ≤ Complex.abs (x * I) ^ 4 * (Nat.succ 4 * (Nat.factorial 4 * (4 : ℕ) : ℝ)⁻¹) / 2 +
-          Complex.abs (-x * I) ^ 4 * (Nat.succ 4 * (Nat.factorial 4 * (4 : ℕ) : ℝ)⁻¹) / 2 := by
-      gcongr
-      · exact Complex.exp_bound (by simpa) (by decide)
-      · exact Complex.exp_bound (by simpa) (by decide)
-    _ ≤ |x| ^ 4 * (5 / 96) := by norm_num [Nat.factorial]
-
-theorem cos_pos_of_le_one {x : ℝ} (hx : |x| ≤ 1) : 0 < cos x :=
-  calc 0 < 1 - x ^ 2 / 2 - |x| ^ 4 * (5 / 96) :=
-      sub_pos.2 <|
-        lt_sub_iff_add_lt.2
-          (calc
-            |x| ^ 4 * (5 / 96) + x ^ 2 / 2 ≤ 1 * (5 / 96) + 1 / 2 := by
-                  gcongr
-                  · exact pow_le_one₀ (abs_nonneg _) hx
-                  · rw [sq, ← abs_mul_self, abs_mul]
-                    exact mul_le_one₀ hx (abs_nonneg _) hx
-            _ < 1 := by norm_num)
-    _ ≤ cos x := sub_le_comm.1 (abs_sub_le_iff.1 (cos_bound hx)).2
-
-theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x :=
-  calc 0 < x - x ^ 3 / 6 - |x| ^ 4 * (5 / 96) :=
-      sub_pos.2 <| lt_sub_iff_add_lt.2
-          (calc
-            |x| ^ 4 * (5 / 96) + x ^ 3 / 6 ≤ x * (5 / 96) + x / 6 := by
-                gcongr
-                · calc
-                    |x| ^ 4 ≤ |x| ^ 1 :=
-                      pow_le_pow_of_le_one (abs_nonneg _)
-                        (by rwa [_root_.abs_of_nonneg (le_of_lt hx0)]) (by decide)
-                    _ = x := by simp [_root_.abs_of_nonneg (le_of_lt hx0)]
-                · calc
-                    x ^ 3 ≤ x ^ 1 := pow_le_pow_of_le_one (le_of_lt hx0) hx (by decide)
-                    _ = x := pow_one _
-            _ < x := by linarith)
-    _ ≤ sin x :=
-      sub_le_comm.1 (abs_sub_le_iff.1 (sin_bound (by rwa [_root_.abs_of_nonneg (le_of_lt hx0)]))).2
-
-theorem sin_pos_of_pos_of_le_two {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 2) : 0 < sin x :=
-  have : x / 2 ≤ 1 := (div_le_iff₀ (by norm_num)).mpr (by simpa)
-  calc
-    0 < 2 * sin (x / 2) * cos (x / 2) :=
-      mul_pos (mul_pos (by norm_num) (sin_pos_of_pos_of_le_one (half_pos hx0) this))
-        (cos_pos_of_le_one (by rwa [_root_.abs_of_nonneg (le_of_lt (half_pos hx0))]))
-    _ = sin x := by rw [← sin_two_mul, two_mul, add_halves]
-
-theorem cos_one_le : cos 1 ≤ 2 / 3 :=
-  calc
-    cos 1 ≤ |(1 : ℝ)| ^ 4 * (5 / 96) + (1 - 1 ^ 2 / 2) :=
-      sub_le_iff_le_add.1 (abs_sub_le_iff.1 (cos_bound (by simp))).1
-    _ ≤ 2 / 3 := by norm_num
-
-theorem cos_one_pos : 0 < cos 1 :=
-  cos_pos_of_le_one (le_of_eq abs_one)
-
-theorem cos_two_neg : cos 2 < 0 :=
-  calc cos 2 = cos (2 * 1) := congr_arg cos (mul_one _).symm
-    _ = _ := Real.cos_two_mul 1
-    _ ≤ 2 * (2 / 3) ^ 2 - 1 := by
-      gcongr
-      · exact cos_one_pos.le
-      · apply cos_one_le
-    _ < 0 := by norm_num
-
 theorem exp_bound_div_one_sub_of_interval' {x : ℝ} (h1 : 0 < x) (h2 : x < 1) :
     Real.exp x < 1 / (1 - x) := by
   have H : 0 < 1 - (1 + x + x ^ 2) * (1 - x) := calc
@@ -1527,41 +675,13 @@ def evalExp : PositivityExt where eval {u α} _ _ e := do
     pure (.positive q(Real.exp_pos $a))
   | _, _, _ => throwError "not Real.exp"
 
-/-- Extension for the `positivity` tactic: `Real.cosh` is always positive. -/
-@[positivity Real.cosh _]
-def evalCosh : PositivityExt where eval {u α} _ _ e := do
-  match u, α, e with
-  | 0, ~q(ℝ), ~q(Real.cosh $a) =>
-    assertInstancesCommute
-    return .positive q(Real.cosh_pos $a)
-  | _, _, _ => throwError "not Real.cosh"
-
-example (x : ℝ) : 0 < x.cosh := by positivity
-
 end Mathlib.Meta.Positivity
 
 namespace Complex
 
-@[simp]
-theorem abs_cos_add_sin_mul_I (x : ℝ) : abs (cos x + sin x * I) = 1 := by
-  have := Real.sin_sq_add_cos_sq x
-  simp_all [add_comm, abs, normSq, sq, sin_ofReal_re, cos_ofReal_re, mul_re]
-
 @[simp]
 theorem abs_exp_ofReal (x : ℝ) : abs (exp x) = Real.exp x := by
   rw [← ofReal_exp]
   exact abs_of_nonneg (le_of_lt (Real.exp_pos _))
 
-@[simp]
-theorem abs_exp_ofReal_mul_I (x : ℝ) : abs (exp (x * I)) = 1 := by
-  rw [exp_mul_I, abs_cos_add_sin_mul_I]
-
-theorem abs_exp (z : ℂ) : abs (exp z) = Real.exp z.re := by
-  rw [exp_eq_exp_re_mul_sin_add_cos, map_mul, abs_exp_ofReal, abs_cos_add_sin_mul_I, mul_one]
-
-theorem abs_exp_eq_iff_re_eq {x y : ℂ} : abs (exp x) = abs (exp y) ↔ x.re = y.re := by
-  rw [abs_exp, abs_exp, Real.exp_eq_exp]
-
 end Complex
-
-set_option linter.style.longFile 1700
diff --git a/Mathlib/Data/Complex/Trigonometric.lean b/Mathlib/Data/Complex/Trigonometric.lean
new file mode 100644
index 00000000000000..ea6b53a5110384
--- /dev/null
+++ b/Mathlib/Data/Complex/Trigonometric.lean
@@ -0,0 +1,951 @@
+/-
+Copyright (c) 2018 Chris Hughes. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
+-/
+import Mathlib.Data.Complex.Exponential
+
+/-!
+# Trigonometric and hyperbolic trigonometric functions
+
+This file contains the definitions of the sine, cosine, tangent,
+hyperbolic sine, hyperbolic cosine, and hyperbolic tangent functions.
+
+-/
+
+open CauSeq Finset IsAbsoluteValue
+open scoped ComplexConjugate
+
+namespace Complex
+
+noncomputable section
+
+/-- The complex sine function, defined via `exp` -/
+@[pp_nodot]
+def sin (z : ℂ) : ℂ :=
+  (exp (-z * I) - exp (z * I)) * I / 2
+
+/-- The complex cosine function, defined via `exp` -/
+@[pp_nodot]
+def cos (z : ℂ) : ℂ :=
+  (exp (z * I) + exp (-z * I)) / 2
+
+/-- The complex tangent function, defined as `sin z / cos z` -/
+@[pp_nodot]
+def tan (z : ℂ) : ℂ :=
+  sin z / cos z
+
+/-- The complex cotangent function, defined as `cos z / sin z` -/
+def cot (z : ℂ) : ℂ :=
+  cos z / sin z
+
+/-- The complex hyperbolic sine function, defined via `exp` -/
+@[pp_nodot]
+def sinh (z : ℂ) : ℂ :=
+  (exp z - exp (-z)) / 2
+
+/-- The complex hyperbolic cosine function, defined via `exp` -/
+@[pp_nodot]
+def cosh (z : ℂ) : ℂ :=
+  (exp z + exp (-z)) / 2
+
+/-- The complex hyperbolic tangent function, defined as `sinh z / cosh z` -/
+@[pp_nodot]
+def tanh (z : ℂ) : ℂ :=
+  sinh z / cosh z
+
+end
+
+end Complex
+
+namespace Real
+
+open Complex
+
+noncomputable section
+
+/-- The real sine function, defined as the real part of the complex sine -/
+@[pp_nodot]
+nonrec def sin (x : ℝ) : ℝ :=
+  (sin x).re
+
+/-- The real cosine function, defined as the real part of the complex cosine -/
+@[pp_nodot]
+nonrec def cos (x : ℝ) : ℝ :=
+  (cos x).re
+
+/-- The real tangent function, defined as the real part of the complex tangent -/
+@[pp_nodot]
+nonrec def tan (x : ℝ) : ℝ :=
+  (tan x).re
+
+/-- The real cotangent function, defined as the real part of the complex cotangent -/
+nonrec def cot (x : ℝ) : ℝ :=
+  (cot x).re
+
+/-- The real hypebolic sine function, defined as the real part of the complex hyperbolic sine -/
+@[pp_nodot]
+nonrec def sinh (x : ℝ) : ℝ :=
+  (sinh x).re
+
+/-- The real hypebolic cosine function, defined as the real part of the complex hyperbolic cosine -/
+@[pp_nodot]
+nonrec def cosh (x : ℝ) : ℝ :=
+  (cosh x).re
+
+/-- The real hypebolic tangent function, defined as the real part of
+the complex hyperbolic tangent -/
+@[pp_nodot]
+nonrec def tanh (x : ℝ) : ℝ :=
+  (tanh x).re
+
+end
+
+end Real
+
+namespace Complex
+
+variable (x y : ℂ)
+
+theorem two_sinh : 2 * sinh x = exp x - exp (-x) :=
+  mul_div_cancel₀ _ two_ne_zero
+
+theorem two_cosh : 2 * cosh x = exp x + exp (-x) :=
+  mul_div_cancel₀ _ two_ne_zero
+
+@[simp]
+theorem sinh_zero : sinh 0 = 0 := by simp [sinh]
+
+@[simp]
+theorem sinh_neg : sinh (-x) = -sinh x := by simp [sinh, exp_neg, (neg_div _ _).symm, add_mul]
+
+private theorem sinh_add_aux {a b c d : ℂ} :
+    (a - b) * (c + d) + (a + b) * (c - d) = 2 * (a * c - b * d) := by ring
+
+theorem sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y := by
+  rw [← mul_right_inj' (two_ne_zero' ℂ), two_sinh, exp_add, neg_add, exp_add, eq_comm, mul_add, ←
+    mul_assoc, two_sinh, mul_left_comm, two_sinh, ← mul_right_inj' (two_ne_zero' ℂ), mul_add,
+    mul_left_comm, two_cosh, ← mul_assoc, two_cosh]
+  exact sinh_add_aux
+
+@[simp]
+theorem cosh_zero : cosh 0 = 1 := by simp [cosh]
+
+@[simp]
+theorem cosh_neg : cosh (-x) = cosh x := by simp [add_comm, cosh, exp_neg]
+
+private theorem cosh_add_aux {a b c d : ℂ} :
+    (a + b) * (c + d) + (a - b) * (c - d) = 2 * (a * c + b * d) := by ring
+
+theorem cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y := by
+  rw [← mul_right_inj' (two_ne_zero' ℂ), two_cosh, exp_add, neg_add, exp_add, eq_comm, mul_add, ←
+    mul_assoc, two_cosh, ← mul_assoc, two_sinh, ← mul_right_inj' (two_ne_zero' ℂ), mul_add,
+    mul_left_comm, two_cosh, mul_left_comm, two_sinh]
+  exact cosh_add_aux
+
+theorem sinh_sub : sinh (x - y) = sinh x * cosh y - cosh x * sinh y := by
+  simp [sub_eq_add_neg, sinh_add, sinh_neg, cosh_neg]
+
+theorem cosh_sub : cosh (x - y) = cosh x * cosh y - sinh x * sinh y := by
+  simp [sub_eq_add_neg, cosh_add, sinh_neg, cosh_neg]
+
+theorem sinh_conj : sinh (conj x) = conj (sinh x) := by
+  rw [sinh, ← RingHom.map_neg, exp_conj, exp_conj, ← RingHom.map_sub, sinh, map_div₀]
+  -- Porting note: not nice
+  simp [← one_add_one_eq_two]
+
+@[simp]
+theorem ofReal_sinh_ofReal_re (x : ℝ) : ((sinh x).re : ℂ) = sinh x :=
+  conj_eq_iff_re.1 <| by rw [← sinh_conj, conj_ofReal]
+
+@[simp, norm_cast]
+theorem ofReal_sinh (x : ℝ) : (Real.sinh x : ℂ) = sinh x :=
+  ofReal_sinh_ofReal_re _
+
+@[simp]
+theorem sinh_ofReal_im (x : ℝ) : (sinh x).im = 0 := by rw [← ofReal_sinh_ofReal_re, ofReal_im]
+
+theorem sinh_ofReal_re (x : ℝ) : (sinh x).re = Real.sinh x :=
+  rfl
+
+theorem cosh_conj : cosh (conj x) = conj (cosh x) := by
+  rw [cosh, ← RingHom.map_neg, exp_conj, exp_conj, ← RingHom.map_add, cosh, map_div₀]
+  -- Porting note: not nice
+  simp [← one_add_one_eq_two]
+
+theorem ofReal_cosh_ofReal_re (x : ℝ) : ((cosh x).re : ℂ) = cosh x :=
+  conj_eq_iff_re.1 <| by rw [← cosh_conj, conj_ofReal]
+
+@[simp, norm_cast]
+theorem ofReal_cosh (x : ℝ) : (Real.cosh x : ℂ) = cosh x :=
+  ofReal_cosh_ofReal_re _
+
+@[simp]
+theorem cosh_ofReal_im (x : ℝ) : (cosh x).im = 0 := by rw [← ofReal_cosh_ofReal_re, ofReal_im]
+
+@[simp]
+theorem cosh_ofReal_re (x : ℝ) : (cosh x).re = Real.cosh x :=
+  rfl
+
+theorem tanh_eq_sinh_div_cosh : tanh x = sinh x / cosh x :=
+  rfl
+
+@[simp]
+theorem tanh_zero : tanh 0 = 0 := by simp [tanh]
+
+@[simp]
+theorem tanh_neg : tanh (-x) = -tanh x := by simp [tanh, neg_div]
+
+theorem tanh_conj : tanh (conj x) = conj (tanh x) := by
+  rw [tanh, sinh_conj, cosh_conj, ← map_div₀, tanh]
+
+@[simp]
+theorem ofReal_tanh_ofReal_re (x : ℝ) : ((tanh x).re : ℂ) = tanh x :=
+  conj_eq_iff_re.1 <| by rw [← tanh_conj, conj_ofReal]
+
+@[simp, norm_cast]
+theorem ofReal_tanh (x : ℝ) : (Real.tanh x : ℂ) = tanh x :=
+  ofReal_tanh_ofReal_re _
+
+@[simp]
+theorem tanh_ofReal_im (x : ℝ) : (tanh x).im = 0 := by rw [← ofReal_tanh_ofReal_re, ofReal_im]
+
+theorem tanh_ofReal_re (x : ℝ) : (tanh x).re = Real.tanh x :=
+  rfl
+
+@[simp]
+theorem cosh_add_sinh : cosh x + sinh x = exp x := by
+  rw [← mul_right_inj' (two_ne_zero' ℂ), mul_add, two_cosh, two_sinh, add_add_sub_cancel, two_mul]
+
+@[simp]
+theorem sinh_add_cosh : sinh x + cosh x = exp x := by rw [add_comm, cosh_add_sinh]
+
+@[simp]
+theorem exp_sub_cosh : exp x - cosh x = sinh x :=
+  sub_eq_iff_eq_add.2 (sinh_add_cosh x).symm
+
+@[simp]
+theorem exp_sub_sinh : exp x - sinh x = cosh x :=
+  sub_eq_iff_eq_add.2 (cosh_add_sinh x).symm
+
+@[simp]
+theorem cosh_sub_sinh : cosh x - sinh x = exp (-x) := by
+  rw [← mul_right_inj' (two_ne_zero' ℂ), mul_sub, two_cosh, two_sinh, add_sub_sub_cancel, two_mul]
+
+@[simp]
+theorem sinh_sub_cosh : sinh x - cosh x = -exp (-x) := by rw [← neg_sub, cosh_sub_sinh]
+
+@[simp]
+theorem cosh_sq_sub_sinh_sq : cosh x ^ 2 - sinh x ^ 2 = 1 := by
+  rw [sq_sub_sq, cosh_add_sinh, cosh_sub_sinh, ← exp_add, add_neg_cancel, exp_zero]
+
+theorem cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 := by
+  rw [← cosh_sq_sub_sinh_sq x]
+  ring
+
+theorem sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 := by
+  rw [← cosh_sq_sub_sinh_sq x]
+  ring
+
+theorem cosh_two_mul : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2 := by rw [two_mul, cosh_add, sq, sq]
+
+theorem sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x := by
+  rw [two_mul, sinh_add]
+  ring
+
+theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x := by
+  have h1 : x + 2 * x = 3 * x := by ring
+  rw [← h1, cosh_add x (2 * x)]
+  simp only [cosh_two_mul, sinh_two_mul]
+  have h2 : sinh x * (2 * sinh x * cosh x) = 2 * cosh x * sinh x ^ 2 := by ring
+  rw [h2, sinh_sq]
+  ring
+
+theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x := by
+  have h1 : x + 2 * x = 3 * x := by ring
+  rw [← h1, sinh_add x (2 * x)]
+  simp only [cosh_two_mul, sinh_two_mul]
+  have h2 : cosh x * (2 * sinh x * cosh x) = 2 * sinh x * cosh x ^ 2 := by ring
+  rw [h2, cosh_sq]
+  ring
+
+@[simp]
+theorem sin_zero : sin 0 = 0 := by simp [sin]
+
+@[simp]
+theorem sin_neg : sin (-x) = -sin x := by
+  simp [sin, sub_eq_add_neg, exp_neg, (neg_div _ _).symm, add_mul]
+
+theorem two_sin : 2 * sin x = (exp (-x * I) - exp (x * I)) * I :=
+  mul_div_cancel₀ _ two_ne_zero
+
+theorem two_cos : 2 * cos x = exp (x * I) + exp (-x * I) :=
+  mul_div_cancel₀ _ two_ne_zero
+
+theorem sinh_mul_I : sinh (x * I) = sin x * I := by
+  rw [← mul_right_inj' (two_ne_zero' ℂ), two_sinh, ← mul_assoc, two_sin, mul_assoc, I_mul_I,
+    mul_neg_one, neg_sub, neg_mul_eq_neg_mul]
+
+theorem cosh_mul_I : cosh (x * I) = cos x := by
+  rw [← mul_right_inj' (two_ne_zero' ℂ), two_cosh, two_cos, neg_mul_eq_neg_mul]
+
+theorem tanh_mul_I : tanh (x * I) = tan x * I := by
+  rw [tanh_eq_sinh_div_cosh, cosh_mul_I, sinh_mul_I, mul_div_right_comm, tan]
+
+theorem cos_mul_I : cos (x * I) = cosh x := by rw [← cosh_mul_I]; ring_nf; simp
+
+theorem sin_mul_I : sin (x * I) = sinh x * I := by
+  have h : I * sin (x * I) = -sinh x := by
+    rw [mul_comm, ← sinh_mul_I]
+    ring_nf
+    simp
+  rw [← neg_neg (sinh x), ← h]
+  apply Complex.ext <;> simp
+
+theorem tan_mul_I : tan (x * I) = tanh x * I := by
+  rw [tan, sin_mul_I, cos_mul_I, mul_div_right_comm, tanh_eq_sinh_div_cosh]
+
+theorem sin_add : sin (x + y) = sin x * cos y + cos x * sin y := by
+  rw [← mul_left_inj' I_ne_zero, ← sinh_mul_I, add_mul, add_mul, mul_right_comm, ← sinh_mul_I,
+    mul_assoc, ← sinh_mul_I, ← cosh_mul_I, ← cosh_mul_I, sinh_add]
+
+@[simp]
+theorem cos_zero : cos 0 = 1 := by simp [cos]
+
+@[simp]
+theorem cos_neg : cos (-x) = cos x := by simp [cos, sub_eq_add_neg, exp_neg, add_comm]
+
+theorem cos_add : cos (x + y) = cos x * cos y - sin x * sin y := by
+  rw [← cosh_mul_I, add_mul, cosh_add, cosh_mul_I, cosh_mul_I, sinh_mul_I, sinh_mul_I,
+    mul_mul_mul_comm, I_mul_I, mul_neg_one, sub_eq_add_neg]
+
+theorem sin_sub : sin (x - y) = sin x * cos y - cos x * sin y := by
+  simp [sub_eq_add_neg, sin_add, sin_neg, cos_neg]
+
+theorem cos_sub : cos (x - y) = cos x * cos y + sin x * sin y := by
+  simp [sub_eq_add_neg, cos_add, sin_neg, cos_neg]
+
+theorem sin_add_mul_I (x y : ℂ) : sin (x + y * I) = sin x * cosh y + cos x * sinh y * I := by
+  rw [sin_add, cos_mul_I, sin_mul_I, mul_assoc]
+
+theorem sin_eq (z : ℂ) : sin z = sin z.re * cosh z.im + cos z.re * sinh z.im * I := by
+  convert sin_add_mul_I z.re z.im; exact (re_add_im z).symm
+
+theorem cos_add_mul_I (x y : ℂ) : cos (x + y * I) = cos x * cosh y - sin x * sinh y * I := by
+  rw [cos_add, cos_mul_I, sin_mul_I, mul_assoc]
+
+theorem cos_eq (z : ℂ) : cos z = cos z.re * cosh z.im - sin z.re * sinh z.im * I := by
+  convert cos_add_mul_I z.re z.im; exact (re_add_im z).symm
+
+theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) := by
+  have s1 := sin_add ((x + y) / 2) ((x - y) / 2)
+  have s2 := sin_sub ((x + y) / 2) ((x - y) / 2)
+  rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel_right, add_self_div_two] at s1
+  rw [div_sub_div_same, ← sub_add, add_sub_cancel_left, add_self_div_two] at s2
+  rw [s1, s2]
+  ring
+
+theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) := by
+  have s1 := cos_add ((x + y) / 2) ((x - y) / 2)
+  have s2 := cos_sub ((x + y) / 2) ((x - y) / 2)
+  rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel_right, add_self_div_two] at s1
+  rw [div_sub_div_same, ← sub_add, add_sub_cancel_left, add_self_div_two] at s2
+  rw [s1, s2]
+  ring
+
+theorem sin_add_sin : sin x + sin y = 2 * sin ((x + y) / 2) * cos ((x - y) / 2) := by
+  simpa using sin_sub_sin x (-y)
+
+theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) := by
+  calc
+    cos x + cos y = cos ((x + y) / 2 + (x - y) / 2) + cos ((x + y) / 2 - (x - y) / 2) := ?_
+    _ =
+        cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2) +
+          (cos ((x + y) / 2) * cos ((x - y) / 2) + sin ((x + y) / 2) * sin ((x - y) / 2)) :=
+      ?_
+    _ = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) := ?_
+
+  · congr <;> field_simp
+  · rw [cos_add, cos_sub]
+  ring
+
+theorem sin_conj : sin (conj x) = conj (sin x) := by
+  rw [← mul_left_inj' I_ne_zero, ← sinh_mul_I, ← conj_neg_I, ← RingHom.map_mul, ← RingHom.map_mul,
+    sinh_conj, mul_neg, sinh_neg, sinh_mul_I, mul_neg]
+
+@[simp]
+theorem ofReal_sin_ofReal_re (x : ℝ) : ((sin x).re : ℂ) = sin x :=
+  conj_eq_iff_re.1 <| by rw [← sin_conj, conj_ofReal]
+
+@[simp, norm_cast]
+theorem ofReal_sin (x : ℝ) : (Real.sin x : ℂ) = sin x :=
+  ofReal_sin_ofReal_re _
+
+@[simp]
+theorem sin_ofReal_im (x : ℝ) : (sin x).im = 0 := by rw [← ofReal_sin_ofReal_re, ofReal_im]
+
+theorem sin_ofReal_re (x : ℝ) : (sin x).re = Real.sin x :=
+  rfl
+
+theorem cos_conj : cos (conj x) = conj (cos x) := by
+  rw [← cosh_mul_I, ← conj_neg_I, ← RingHom.map_mul, ← cosh_mul_I, cosh_conj, mul_neg, cosh_neg]
+
+@[simp]
+theorem ofReal_cos_ofReal_re (x : ℝ) : ((cos x).re : ℂ) = cos x :=
+  conj_eq_iff_re.1 <| by rw [← cos_conj, conj_ofReal]
+
+@[simp, norm_cast]
+theorem ofReal_cos (x : ℝ) : (Real.cos x : ℂ) = cos x :=
+  ofReal_cos_ofReal_re _
+
+@[simp]
+theorem cos_ofReal_im (x : ℝ) : (cos x).im = 0 := by rw [← ofReal_cos_ofReal_re, ofReal_im]
+
+theorem cos_ofReal_re (x : ℝ) : (cos x).re = Real.cos x :=
+  rfl
+
+@[simp]
+theorem tan_zero : tan 0 = 0 := by simp [tan]
+
+theorem tan_eq_sin_div_cos : tan x = sin x / cos x :=
+  rfl
+
+theorem cot_eq_cos_div_sin : cot x = cos x / sin x :=
+  rfl
+
+theorem tan_mul_cos {x : ℂ} (hx : cos x ≠ 0) : tan x * cos x = sin x := by
+  rw [tan_eq_sin_div_cos, div_mul_cancel₀ _ hx]
+
+@[simp]
+theorem tan_neg : tan (-x) = -tan x := by simp [tan, neg_div]
+
+theorem tan_conj : tan (conj x) = conj (tan x) := by rw [tan, sin_conj, cos_conj, ← map_div₀, tan]
+
+theorem cot_conj : cot (conj x) = conj (cot x) := by rw [cot, sin_conj, cos_conj, ← map_div₀, cot]
+
+@[simp]
+theorem ofReal_tan_ofReal_re (x : ℝ) : ((tan x).re : ℂ) = tan x :=
+  conj_eq_iff_re.1 <| by rw [← tan_conj, conj_ofReal]
+
+@[simp]
+theorem ofReal_cot_ofReal_re (x : ℝ) : ((cot x).re : ℂ) = cot x :=
+  conj_eq_iff_re.1 <| by rw [← cot_conj, conj_ofReal]
+
+@[simp, norm_cast]
+theorem ofReal_tan (x : ℝ) : (Real.tan x : ℂ) = tan x :=
+  ofReal_tan_ofReal_re _
+
+@[simp, norm_cast]
+theorem ofReal_cot (x : ℝ) : (Real.cot x : ℂ) = cot x :=
+  ofReal_cot_ofReal_re _
+
+@[simp]
+theorem tan_ofReal_im (x : ℝ) : (tan x).im = 0 := by rw [← ofReal_tan_ofReal_re, ofReal_im]
+
+theorem tan_ofReal_re (x : ℝ) : (tan x).re = Real.tan x :=
+  rfl
+
+theorem cos_add_sin_I : cos x + sin x * I = exp (x * I) := by
+  rw [← cosh_add_sinh, sinh_mul_I, cosh_mul_I]
+
+theorem cos_sub_sin_I : cos x - sin x * I = exp (-x * I) := by
+  rw [neg_mul, ← cosh_sub_sinh, sinh_mul_I, cosh_mul_I]
+
+@[simp]
+theorem sin_sq_add_cos_sq : sin x ^ 2 + cos x ^ 2 = 1 :=
+  Eq.trans (by rw [cosh_mul_I, sinh_mul_I, mul_pow, I_sq, mul_neg_one, sub_neg_eq_add, add_comm])
+    (cosh_sq_sub_sinh_sq (x * I))
+
+@[simp]
+theorem cos_sq_add_sin_sq : cos x ^ 2 + sin x ^ 2 = 1 := by rw [add_comm, sin_sq_add_cos_sq]
+
+theorem cos_two_mul' : cos (2 * x) = cos x ^ 2 - sin x ^ 2 := by rw [two_mul, cos_add, ← sq, ← sq]
+
+theorem cos_two_mul : cos (2 * x) = 2 * cos x ^ 2 - 1 := by
+  rw [cos_two_mul', eq_sub_iff_add_eq.2 (sin_sq_add_cos_sq x), ← sub_add, sub_add_eq_add_sub,
+    two_mul]
+
+theorem sin_two_mul : sin (2 * x) = 2 * sin x * cos x := by
+  rw [two_mul, sin_add, two_mul, add_mul, mul_comm]
+
+theorem cos_sq : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 := by
+  simp [cos_two_mul, div_add_div_same, mul_div_cancel_left₀, two_ne_zero, -one_div]
+
+theorem cos_sq' : cos x ^ 2 = 1 - sin x ^ 2 := by rw [← sin_sq_add_cos_sq x, add_sub_cancel_left]
+
+theorem sin_sq : sin x ^ 2 = 1 - cos x ^ 2 := by rw [← sin_sq_add_cos_sq x, add_sub_cancel_right]
+
+theorem inv_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2 := by
+  rw [tan_eq_sin_div_cos, div_pow]
+  field_simp
+
+theorem tan_sq_div_one_add_tan_sq {x : ℂ} (hx : cos x ≠ 0) :
+    tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2 := by
+  simp only [← tan_mul_cos hx, mul_pow, ← inv_one_add_tan_sq hx, div_eq_mul_inv, one_mul]
+
+theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x := by
+  have h1 : x + 2 * x = 3 * x := by ring
+  rw [← h1, cos_add x (2 * x)]
+  simp only [cos_two_mul, sin_two_mul, mul_add, mul_sub, mul_one, sq]
+  have h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2 := by ring
+  rw [h2, cos_sq']
+  ring
+
+theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 := by
+  have h1 : x + 2 * x = 3 * x := by ring
+  rw [← h1, sin_add x (2 * x)]
+  simp only [cos_two_mul, sin_two_mul, cos_sq']
+  have h2 : cos x * (2 * sin x * cos x) = 2 * sin x * cos x ^ 2 := by ring
+  rw [h2, cos_sq']
+  ring
+
+theorem exp_mul_I : exp (x * I) = cos x + sin x * I :=
+  (cos_add_sin_I _).symm
+
+theorem exp_add_mul_I : exp (x + y * I) = exp x * (cos y + sin y * I) := by rw [exp_add, exp_mul_I]
+
+theorem exp_eq_exp_re_mul_sin_add_cos : exp x = exp x.re * (cos x.im + sin x.im * I) := by
+  rw [← exp_add_mul_I, re_add_im]
+
+theorem exp_re : (exp x).re = Real.exp x.re * Real.cos x.im := by
+  rw [exp_eq_exp_re_mul_sin_add_cos]
+  simp [exp_ofReal_re, cos_ofReal_re]
+
+theorem exp_im : (exp x).im = Real.exp x.re * Real.sin x.im := by
+  rw [exp_eq_exp_re_mul_sin_add_cos]
+  simp [exp_ofReal_re, sin_ofReal_re]
+
+@[simp]
+theorem exp_ofReal_mul_I_re (x : ℝ) : (exp (x * I)).re = Real.cos x := by
+  simp [exp_mul_I, cos_ofReal_re]
+
+@[simp]
+theorem exp_ofReal_mul_I_im (x : ℝ) : (exp (x * I)).im = Real.sin x := by
+  simp [exp_mul_I, sin_ofReal_re]
+
+/-- **De Moivre's formula** -/
+theorem cos_add_sin_mul_I_pow (n : ℕ) (z : ℂ) :
+    (cos z + sin z * I) ^ n = cos (↑n * z) + sin (↑n * z) * I := by
+  rw [← exp_mul_I, ← exp_mul_I, ← exp_nat_mul, mul_assoc]
+
+end Complex
+
+namespace Real
+
+open Complex
+
+variable (x y : ℝ)
+
+@[simp]
+theorem sin_zero : sin 0 = 0 := by simp [sin]
+
+@[simp]
+theorem sin_neg : sin (-x) = -sin x := by simp [sin, exp_neg, (neg_div _ _).symm, add_mul]
+
+nonrec theorem sin_add : sin (x + y) = sin x * cos y + cos x * sin y :=
+  ofReal_injective <| by simp [sin_add]
+
+@[simp]
+theorem cos_zero : cos 0 = 1 := by simp [cos]
+
+@[simp]
+theorem cos_neg : cos (-x) = cos x := by simp [cos, exp_neg]
+
+@[simp]
+theorem cos_abs : cos |x| = cos x := by
+  cases le_total x 0 <;> simp only [*, _root_.abs_of_nonneg, abs_of_nonpos, cos_neg]
+
+nonrec theorem cos_add : cos (x + y) = cos x * cos y - sin x * sin y :=
+  ofReal_injective <| by simp [cos_add]
+
+theorem sin_sub : sin (x - y) = sin x * cos y - cos x * sin y := by
+  simp [sub_eq_add_neg, sin_add, sin_neg, cos_neg]
+
+theorem cos_sub : cos (x - y) = cos x * cos y + sin x * sin y := by
+  simp [sub_eq_add_neg, cos_add, sin_neg, cos_neg]
+
+nonrec theorem sin_sub_sin : sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2) :=
+  ofReal_injective <| by simp [sin_sub_sin]
+
+nonrec theorem cos_sub_cos : cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2) :=
+  ofReal_injective <| by simp [cos_sub_cos]
+
+nonrec theorem cos_add_cos : cos x + cos y = 2 * cos ((x + y) / 2) * cos ((x - y) / 2) :=
+  ofReal_injective <| by simp [cos_add_cos]
+
+theorem two_mul_sin_mul_sin (x y : ℝ) : 2 * sin x * sin y = cos (x - y) - cos (x + y) := by
+  simp [cos_add, cos_sub]
+  ring
+
+theorem two_mul_cos_mul_cos (x y : ℝ) : 2 * cos x * cos y = cos (x - y) + cos (x + y) := by
+  simp [cos_add, cos_sub]
+  ring
+
+theorem two_mul_sin_mul_cos (x y : ℝ) : 2 * sin x * cos y = sin (x - y) + sin (x + y) := by
+  simp [sin_add, sin_sub]
+  ring
+
+nonrec theorem tan_eq_sin_div_cos : tan x = sin x / cos x :=
+  ofReal_injective <| by simp only [ofReal_tan, tan_eq_sin_div_cos, ofReal_div, ofReal_sin,
+    ofReal_cos]
+
+nonrec theorem cot_eq_cos_div_sin : cot x = cos x / sin x :=
+  ofReal_injective <| by simp [cot_eq_cos_div_sin]
+
+theorem tan_mul_cos {x : ℝ} (hx : cos x ≠ 0) : tan x * cos x = sin x := by
+  rw [tan_eq_sin_div_cos, div_mul_cancel₀ _ hx]
+
+@[simp]
+theorem tan_zero : tan 0 = 0 := by simp [tan]
+
+@[simp]
+theorem tan_neg : tan (-x) = -tan x := by simp [tan, neg_div]
+
+@[simp]
+nonrec theorem sin_sq_add_cos_sq : sin x ^ 2 + cos x ^ 2 = 1 :=
+  ofReal_injective (by simp [sin_sq_add_cos_sq])
+
+@[simp]
+theorem cos_sq_add_sin_sq : cos x ^ 2 + sin x ^ 2 = 1 := by rw [add_comm, sin_sq_add_cos_sq]
+
+theorem sin_sq_le_one : sin x ^ 2 ≤ 1 := by
+  rw [← sin_sq_add_cos_sq x]; exact le_add_of_nonneg_right (sq_nonneg _)
+
+theorem cos_sq_le_one : cos x ^ 2 ≤ 1 := by
+  rw [← sin_sq_add_cos_sq x]; exact le_add_of_nonneg_left (sq_nonneg _)
+
+theorem abs_sin_le_one : |sin x| ≤ 1 :=
+  abs_le_one_iff_mul_self_le_one.2 <| by simp only [← sq, sin_sq_le_one]
+
+theorem abs_cos_le_one : |cos x| ≤ 1 :=
+  abs_le_one_iff_mul_self_le_one.2 <| by simp only [← sq, cos_sq_le_one]
+
+theorem sin_le_one : sin x ≤ 1 :=
+  (abs_le.1 (abs_sin_le_one _)).2
+
+theorem cos_le_one : cos x ≤ 1 :=
+  (abs_le.1 (abs_cos_le_one _)).2
+
+theorem neg_one_le_sin : -1 ≤ sin x :=
+  (abs_le.1 (abs_sin_le_one _)).1
+
+theorem neg_one_le_cos : -1 ≤ cos x :=
+  (abs_le.1 (abs_cos_le_one _)).1
+
+nonrec theorem cos_two_mul : cos (2 * x) = 2 * cos x ^ 2 - 1 :=
+  ofReal_injective <| by simp [cos_two_mul]
+
+nonrec theorem cos_two_mul' : cos (2 * x) = cos x ^ 2 - sin x ^ 2 :=
+  ofReal_injective <| by simp [cos_two_mul']
+
+nonrec theorem sin_two_mul : sin (2 * x) = 2 * sin x * cos x :=
+  ofReal_injective <| by simp [sin_two_mul]
+
+nonrec theorem cos_sq : cos x ^ 2 = 1 / 2 + cos (2 * x) / 2 :=
+  ofReal_injective <| by simp [cos_sq]
+
+theorem cos_sq' : cos x ^ 2 = 1 - sin x ^ 2 := by rw [← sin_sq_add_cos_sq x, add_sub_cancel_left]
+
+theorem sin_sq : sin x ^ 2 = 1 - cos x ^ 2 :=
+  eq_sub_iff_add_eq.2 <| sin_sq_add_cos_sq _
+
+lemma sin_sq_eq_half_sub : sin x ^ 2 = 1 / 2 - cos (2 * x) / 2 := by
+  rw [sin_sq, cos_sq, ← sub_sub, sub_half]
+
+theorem abs_sin_eq_sqrt_one_sub_cos_sq (x : ℝ) : |sin x| = √(1 - cos x ^ 2) := by
+  rw [← sin_sq, sqrt_sq_eq_abs]
+
+theorem abs_cos_eq_sqrt_one_sub_sin_sq (x : ℝ) : |cos x| = √(1 - sin x ^ 2) := by
+  rw [← cos_sq', sqrt_sq_eq_abs]
+
+theorem inv_one_add_tan_sq {x : ℝ} (hx : cos x ≠ 0) : (1 + tan x ^ 2)⁻¹ = cos x ^ 2 :=
+  have : Complex.cos x ≠ 0 := mt (congr_arg re) hx
+  ofReal_inj.1 <| by simpa using Complex.inv_one_add_tan_sq this
+
+theorem tan_sq_div_one_add_tan_sq {x : ℝ} (hx : cos x ≠ 0) :
+    tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2 := by
+  simp only [← tan_mul_cos hx, mul_pow, ← inv_one_add_tan_sq hx, div_eq_mul_inv, one_mul]
+
+theorem inv_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < cos x) : (√(1 + tan x ^ 2))⁻¹ = cos x := by
+  rw [← sqrt_sq hx.le, ← sqrt_inv, inv_one_add_tan_sq hx.ne']
+
+theorem tan_div_sqrt_one_add_tan_sq {x : ℝ} (hx : 0 < cos x) :
+    tan x / √(1 + tan x ^ 2) = sin x := by
+  rw [← tan_mul_cos hx.ne', ← inv_sqrt_one_add_tan_sq hx, div_eq_mul_inv]
+
+nonrec theorem cos_three_mul : cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x := by
+  rw [← ofReal_inj]; simp [cos_three_mul]
+
+nonrec theorem sin_three_mul : sin (3 * x) = 3 * sin x - 4 * sin x ^ 3 := by
+  rw [← ofReal_inj]; simp [sin_three_mul]
+
+/-- The definition of `sinh` in terms of `exp`. -/
+nonrec theorem sinh_eq (x : ℝ) : sinh x = (exp x - exp (-x)) / 2 :=
+  ofReal_injective <| by simp [Complex.sinh]
+
+@[simp]
+theorem sinh_zero : sinh 0 = 0 := by simp [sinh]
+
+@[simp]
+theorem sinh_neg : sinh (-x) = -sinh x := by simp [sinh, exp_neg, (neg_div _ _).symm, add_mul]
+
+nonrec theorem sinh_add : sinh (x + y) = sinh x * cosh y + cosh x * sinh y := by
+  rw [← ofReal_inj]; simp [sinh_add]
+
+/-- The definition of `cosh` in terms of `exp`. -/
+theorem cosh_eq (x : ℝ) : cosh x = (exp x + exp (-x)) / 2 :=
+  eq_div_of_mul_eq two_ne_zero <| by
+    rw [cosh, exp, exp, Complex.ofReal_neg, Complex.cosh, mul_two, ← Complex.add_re, ← mul_two,
+      div_mul_cancel₀ _ (two_ne_zero' ℂ), Complex.add_re]
+
+@[simp]
+theorem cosh_zero : cosh 0 = 1 := by simp [cosh]
+
+@[simp]
+theorem cosh_neg : cosh (-x) = cosh x :=
+  ofReal_inj.1 <| by simp
+
+@[simp]
+theorem cosh_abs : cosh |x| = cosh x := by
+  cases le_total x 0 <;> simp [*, _root_.abs_of_nonneg, abs_of_nonpos]
+
+nonrec theorem cosh_add : cosh (x + y) = cosh x * cosh y + sinh x * sinh y := by
+  rw [← ofReal_inj]; simp [cosh_add]
+
+theorem sinh_sub : sinh (x - y) = sinh x * cosh y - cosh x * sinh y := by
+  simp [sub_eq_add_neg, sinh_add, sinh_neg, cosh_neg]
+
+theorem cosh_sub : cosh (x - y) = cosh x * cosh y - sinh x * sinh y := by
+  simp [sub_eq_add_neg, cosh_add, sinh_neg, cosh_neg]
+
+nonrec theorem tanh_eq_sinh_div_cosh : tanh x = sinh x / cosh x :=
+  ofReal_inj.1 <| by simp [tanh_eq_sinh_div_cosh]
+
+@[simp]
+theorem tanh_zero : tanh 0 = 0 := by simp [tanh]
+
+@[simp]
+theorem tanh_neg : tanh (-x) = -tanh x := by simp [tanh, neg_div]
+
+@[simp]
+theorem cosh_add_sinh : cosh x + sinh x = exp x := by rw [← ofReal_inj]; simp
+
+@[simp]
+theorem sinh_add_cosh : sinh x + cosh x = exp x := by rw [add_comm, cosh_add_sinh]
+
+@[simp]
+theorem exp_sub_cosh : exp x - cosh x = sinh x :=
+  sub_eq_iff_eq_add.2 (sinh_add_cosh x).symm
+
+@[simp]
+theorem exp_sub_sinh : exp x - sinh x = cosh x :=
+  sub_eq_iff_eq_add.2 (cosh_add_sinh x).symm
+
+@[simp]
+theorem cosh_sub_sinh : cosh x - sinh x = exp (-x) := by
+  rw [← ofReal_inj]
+  simp
+
+@[simp]
+theorem sinh_sub_cosh : sinh x - cosh x = -exp (-x) := by rw [← neg_sub, cosh_sub_sinh]
+
+@[simp]
+theorem cosh_sq_sub_sinh_sq (x : ℝ) : cosh x ^ 2 - sinh x ^ 2 = 1 := by rw [← ofReal_inj]; simp
+
+nonrec theorem cosh_sq : cosh x ^ 2 = sinh x ^ 2 + 1 := by rw [← ofReal_inj]; simp [cosh_sq]
+
+theorem cosh_sq' : cosh x ^ 2 = 1 + sinh x ^ 2 :=
+  (cosh_sq x).trans (add_comm _ _)
+
+nonrec theorem sinh_sq : sinh x ^ 2 = cosh x ^ 2 - 1 := by rw [← ofReal_inj]; simp [sinh_sq]
+
+nonrec theorem cosh_two_mul : cosh (2 * x) = cosh x ^ 2 + sinh x ^ 2 := by
+  rw [← ofReal_inj]; simp [cosh_two_mul]
+
+nonrec theorem sinh_two_mul : sinh (2 * x) = 2 * sinh x * cosh x := by
+  rw [← ofReal_inj]; simp [sinh_two_mul]
+
+nonrec theorem cosh_three_mul : cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x := by
+  rw [← ofReal_inj]; simp [cosh_three_mul]
+
+nonrec theorem sinh_three_mul : sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x := by
+  rw [← ofReal_inj]; simp [sinh_three_mul]
+
+open IsAbsoluteValue Nat
+
+private theorem add_one_lt_exp_of_pos {x : ℝ} (hx : 0 < x) : x + 1 < exp x :=
+  (by nlinarith : x + 1 < 1 + x + x ^ 2 / 2).trans_le (quadratic_le_exp_of_nonneg hx.le)
+
+private theorem add_one_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) : x + 1 ≤ exp x := by
+  rcases eq_or_lt_of_le hx with (rfl | h)
+  · simp
+  exact (add_one_lt_exp_of_pos h).le
+
+/-- `Real.cosh` is always positive -/
+theorem cosh_pos (x : ℝ) : 0 < Real.cosh x :=
+  (cosh_eq x).symm ▸ half_pos (add_pos (exp_pos x) (exp_pos (-x)))
+
+theorem sinh_lt_cosh : sinh x < cosh x :=
+  lt_of_pow_lt_pow_left₀ 2 (cosh_pos _).le <| (cosh_sq x).symm ▸ lt_add_one _
+
+end Real
+
+namespace Real
+
+open Complex Finset
+
+theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
+  calc
+    |cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
+      rw [← abs_ofReal]; simp
+    _ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
+      simp [Complex.cos, sub_div, add_div, neg_div, div_self (two_ne_zero' ℂ)]
+    _ = abs
+          (((Complex.exp (x * I) - ∑ m ∈ range 4, (x * I) ^ m / m.factorial) +
+              (Complex.exp (-x * I) - ∑ m ∈ range 4, (-x * I) ^ m / m.factorial)) / 2) :=
+      (congr_arg Complex.abs
+        (congr_arg (fun x : ℂ => x / 2)
+          (by
+            simp only [sum_range_succ, neg_mul, pow_succ, pow_zero, mul_one, range_zero, sum_empty,
+              Nat.factorial, Nat.cast_one, ne_eq, one_ne_zero, not_false_eq_true, div_self,
+              zero_add, div_one, Nat.mul_one, Nat.cast_succ, Nat.cast_mul, Nat.cast_ofNat, mul_neg,
+              neg_neg]
+            apply Complex.ext <;> simp [div_eq_mul_inv, normSq] <;> ring_nf
+            )))
+    _ ≤ abs ((Complex.exp (x * I) - ∑ m ∈ range 4, (x * I) ^ m / m.factorial) / 2) +
+          abs ((Complex.exp (-x * I) - ∑ m ∈ range 4, (-x * I) ^ m / m.factorial) / 2) := by
+      rw [add_div]; exact Complex.abs.add_le _ _
+    _ = abs (Complex.exp (x * I) - ∑ m ∈ range 4, (x * I) ^ m / m.factorial) / 2 +
+          abs (Complex.exp (-x * I) - ∑ m ∈ range 4, (-x * I) ^ m / m.factorial) / 2 := by
+      simp [map_div₀]
+    _ ≤ Complex.abs (x * I) ^ 4 * (Nat.succ 4 * ((Nat.factorial 4) * (4 : ℕ) : ℝ)⁻¹) / 2 +
+          Complex.abs (-x * I) ^ 4 * (Nat.succ 4 * ((Nat.factorial 4) * (4 : ℕ) : ℝ)⁻¹) / 2 := by
+      gcongr
+      · exact Complex.exp_bound (by simpa) (by decide)
+      · exact Complex.exp_bound (by simpa) (by decide)
+    _ ≤ |x| ^ 4 * (5 / 96) := by norm_num [Nat.factorial]
+
+theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
+  calc
+    |sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
+      rw [← abs_ofReal]; simp
+    _ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
+          (2 * x - x ^ 3 / 3 : ℝ)) / 2) := by
+      simp [Complex.sin, sub_div, add_div, neg_div, mul_div_cancel_left₀ _ (two_ne_zero' ℂ),
+        div_div, show (3 : ℂ) * 2 = 6 by norm_num]
+    _ = Complex.abs (((Complex.exp (-x * I) - ∑ m ∈ range 4, (-x * I) ^ m / m.factorial) -
+                (Complex.exp (x * I) - ∑ m ∈ range 4, (x * I) ^ m / m.factorial)) * I / 2) :=
+      (congr_arg Complex.abs
+        (congr_arg (fun x : ℂ => x / 2)
+          (by
+            simp only [sum_range_succ, neg_mul, pow_succ, pow_zero, mul_one, ofReal_sub, ofReal_mul,
+              ofReal_ofNat, ofReal_div, range_zero, sum_empty, Nat.factorial, Nat.cast_one, ne_eq,
+              one_ne_zero, not_false_eq_true, div_self, zero_add, div_one, mul_neg, neg_neg,
+              Nat.mul_one, Nat.cast_succ, Nat.cast_mul, Nat.cast_ofNat]
+            apply Complex.ext <;> simp [div_eq_mul_inv, normSq]; ring)))
+    _ ≤ abs ((Complex.exp (-x * I) - ∑ m ∈ range 4, (-x * I) ^ m / m.factorial) * I / 2) +
+          abs (-((Complex.exp (x * I) - ∑ m ∈ range 4, (x * I) ^ m / m.factorial) * I) / 2) := by
+      rw [sub_mul, sub_eq_add_neg, add_div]; exact Complex.abs.add_le _ _
+    _ = abs (Complex.exp (x * I) - ∑ m ∈ range 4, (x * I) ^ m / m.factorial) / 2 +
+          abs (Complex.exp (-x * I) - ∑ m ∈ range 4, (-x * I) ^ m / m.factorial) / 2 := by
+      simp [add_comm, map_div₀]
+    _ ≤ Complex.abs (x * I) ^ 4 * (Nat.succ 4 * (Nat.factorial 4 * (4 : ℕ) : ℝ)⁻¹) / 2 +
+          Complex.abs (-x * I) ^ 4 * (Nat.succ 4 * (Nat.factorial 4 * (4 : ℕ) : ℝ)⁻¹) / 2 := by
+      gcongr
+      · exact Complex.exp_bound (by simpa) (by decide)
+      · exact Complex.exp_bound (by simpa) (by decide)
+    _ ≤ |x| ^ 4 * (5 / 96) := by norm_num [Nat.factorial]
+
+theorem cos_pos_of_le_one {x : ℝ} (hx : |x| ≤ 1) : 0 < cos x :=
+  calc 0 < 1 - x ^ 2 / 2 - |x| ^ 4 * (5 / 96) :=
+      sub_pos.2 <|
+        lt_sub_iff_add_lt.2
+          (calc
+            |x| ^ 4 * (5 / 96) + x ^ 2 / 2 ≤ 1 * (5 / 96) + 1 / 2 := by
+                  gcongr
+                  · exact pow_le_one₀ (abs_nonneg _) hx
+                  · rw [sq, ← abs_mul_self, abs_mul]
+                    exact mul_le_one₀ hx (abs_nonneg _) hx
+            _ < 1 := by norm_num)
+    _ ≤ cos x := sub_le_comm.1 (abs_sub_le_iff.1 (cos_bound hx)).2
+
+theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x :=
+  calc 0 < x - x ^ 3 / 6 - |x| ^ 4 * (5 / 96) :=
+      sub_pos.2 <| lt_sub_iff_add_lt.2
+          (calc
+            |x| ^ 4 * (5 / 96) + x ^ 3 / 6 ≤ x * (5 / 96) + x / 6 := by
+                gcongr
+                · calc
+                    |x| ^ 4 ≤ |x| ^ 1 :=
+                      pow_le_pow_of_le_one (abs_nonneg _)
+                        (by rwa [_root_.abs_of_nonneg (le_of_lt hx0)]) (by decide)
+                    _ = x := by simp [_root_.abs_of_nonneg (le_of_lt hx0)]
+                · calc
+                    x ^ 3 ≤ x ^ 1 := pow_le_pow_of_le_one (le_of_lt hx0) hx (by decide)
+                    _ = x := pow_one _
+            _ < x := by linarith)
+    _ ≤ sin x :=
+      sub_le_comm.1 (abs_sub_le_iff.1 (sin_bound (by rwa [_root_.abs_of_nonneg (le_of_lt hx0)]))).2
+
+theorem sin_pos_of_pos_of_le_two {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 2) : 0 < sin x :=
+  have : x / 2 ≤ 1 := (div_le_iff₀ (by norm_num)).mpr (by simpa)
+  calc
+    0 < 2 * sin (x / 2) * cos (x / 2) :=
+      mul_pos (mul_pos (by norm_num) (sin_pos_of_pos_of_le_one (half_pos hx0) this))
+        (cos_pos_of_le_one (by rwa [_root_.abs_of_nonneg (le_of_lt (half_pos hx0))]))
+    _ = sin x := by rw [← sin_two_mul, two_mul, add_halves]
+
+theorem cos_one_le : cos 1 ≤ 2 / 3 :=
+  calc
+    cos 1 ≤ |(1 : ℝ)| ^ 4 * (5 / 96) + (1 - 1 ^ 2 / 2) :=
+      sub_le_iff_le_add.1 (abs_sub_le_iff.1 (cos_bound (by simp))).1
+    _ ≤ 2 / 3 := by norm_num
+
+theorem cos_one_pos : 0 < cos 1 :=
+  cos_pos_of_le_one (le_of_eq abs_one)
+
+theorem cos_two_neg : cos 2 < 0 :=
+  calc cos 2 = cos (2 * 1) := congr_arg cos (mul_one _).symm
+    _ = _ := Real.cos_two_mul 1
+    _ ≤ 2 * (2 / 3) ^ 2 - 1 := by
+      gcongr
+      · exact cos_one_pos.le
+      · apply cos_one_le
+    _ < 0 := by norm_num
+
+end Real
+
+namespace Mathlib.Meta.Positivity
+open Lean.Meta Qq
+
+/-- Extension for the `positivity` tactic: `Real.cosh` is always positive. -/
+@[positivity Real.cosh _]
+def evalCosh : PositivityExt where eval {u α} _ _ e := do
+  match u, α, e with
+  | 0, ~q(ℝ), ~q(Real.cosh $a) =>
+    assertInstancesCommute
+    return .positive q(Real.cosh_pos $a)
+  | _, _, _ => throwError "not Real.cosh"
+
+example (x : ℝ) : 0 < x.cosh := by positivity
+
+end Mathlib.Meta.Positivity
+
+namespace Complex
+
+@[simp]
+theorem abs_cos_add_sin_mul_I (x : ℝ) : abs (cos x + sin x * I) = 1 := by
+  have := Real.sin_sq_add_cos_sq x
+  simp_all [add_comm, abs, normSq, sq, sin_ofReal_re, cos_ofReal_re, mul_re]
+
+@[simp]
+theorem abs_exp_ofReal_mul_I (x : ℝ) : abs (exp (x * I)) = 1 := by
+  rw [exp_mul_I, abs_cos_add_sin_mul_I]
+
+theorem abs_exp (z : ℂ) : abs (exp z) = Real.exp z.re := by
+  rw [exp_eq_exp_re_mul_sin_add_cos, map_mul, abs_exp_ofReal, abs_cos_add_sin_mul_I, mul_one]
+
+theorem abs_exp_eq_iff_re_eq {x y : ℂ} : abs (exp x) = abs (exp y) ↔ x.re = y.re := by
+  rw [abs_exp, abs_exp, Real.exp_eq_exp]
+
+end Complex
diff --git a/Mathlib/Data/DFinsupp/Order.lean b/Mathlib/Data/DFinsupp/Order.lean
index 9201ee22c9791b..0ea38f0ada1209 100644
--- a/Mathlib/Data/DFinsupp/Order.lean
+++ b/Mathlib/Data/DFinsupp/Order.lean
@@ -44,14 +44,7 @@ lemma le_def : f ≤ g ↔ ∀ i, f i ≤ g i := Iff.rfl
 def orderEmbeddingToFun : (Π₀ i, α i) ↪o ∀ i, α i where
   toFun := DFunLike.coe
   inj' := DFunLike.coe_injective
-  map_rel_iff' :=
-    #adaptation_note
-    /--
-    This proof used to be `rfl`,
-    but has been temporarily broken by https://github.com/leanprover/lean4/pull/5329.
-    It can hopefully be restored after https://github.com/leanprover/lean4/pull/5359
-    -/
-    Iff.rfl
+  map_rel_iff' := Iff.rfl
 
 @[simp, norm_cast]
 lemma coe_orderEmbeddingToFun : ⇑(orderEmbeddingToFun (α := α)) = DFunLike.coe := rfl
diff --git a/Mathlib/Data/DFinsupp/Sigma.lean b/Mathlib/Data/DFinsupp/Sigma.lean
index 10d83e6df6193c..c0fd6d64a85c2f 100644
--- a/Mathlib/Data/DFinsupp/Sigma.lean
+++ b/Mathlib/Data/DFinsupp/Sigma.lean
@@ -63,8 +63,7 @@ theorem sigmaCurry_zero [∀ i j, Zero (δ i j)] :
 
 @[simp]
 theorem sigmaCurry_add [∀ i j, AddZeroClass (δ i j)] (f g : Π₀ (i : Σ _, _), δ i.1 i.2) :
-    #adaptation_note
-    /-- After https://github.com/leanprover/lean4/pull/6024
+    #adaptation_note /-- https://github.com/leanprover/lean4/pull/6024
     we needed to add the `(_ : Π₀ (i) (j), δ i j)` type annotation. -/
     sigmaCurry (f + g) = (sigmaCurry f + sigmaCurry g : Π₀ (i) (j), δ i j) := by
   ext (i j)
@@ -73,8 +72,7 @@ theorem sigmaCurry_add [∀ i j, AddZeroClass (δ i j)] (f g : Π₀ (i : Σ _,
 @[simp]
 theorem sigmaCurry_smul [Monoid γ] [∀ i j, AddMonoid (δ i j)] [∀ i j, DistribMulAction γ (δ i j)]
     (r : γ) (f : Π₀ (i : Σ _, _), δ i.1 i.2) :
-    #adaptation_note
-    /-- After https://github.com/leanprover/lean4/pull/6024
+    #adaptation_note /-- https://github.com/leanprover/lean4/pull/6024
     we needed to add the `(_ : Π₀ (i) (j), δ i j)` type annotation. -/
     sigmaCurry (r • f) = (r • sigmaCurry f : Π₀ (i) (j), δ i j) := by
   ext (i j)
diff --git a/Mathlib/Data/DFinsupp/WellFounded.lean b/Mathlib/Data/DFinsupp/WellFounded.lean
index 4b312a0571ec77..ac37b3707bd633 100644
--- a/Mathlib/Data/DFinsupp/WellFounded.lean
+++ b/Mathlib/Data/DFinsupp/WellFounded.lean
@@ -212,9 +212,7 @@ protected theorem DFinsupp.wellFoundedLT [∀ i, Zero (α i)] [∀ i, Preorder (
       refine Lex.wellFounded' ?_ (fun i ↦ IsWellFounded.wf) ?_
       · rintro i ⟨a⟩
         apply hbot
-      · #adaptation_note /-- nightly-2024-03-16: simp was
-        simp (config := { unfoldPartialApp := true }) only [Function.swap] -/
-        simp only [Function.swap_def]
+      · simp +unfoldPartialApp only [Function.swap]
         exact IsWellFounded.wf
     refine Subrelation.wf (fun h => ?_) <| InvImage.wf (mapRange e fun _ ↦ rfl) this
     have := IsStrictOrder.swap (@WellOrderingRel ι)
diff --git a/Mathlib/Data/ENNReal/Inv.lean b/Mathlib/Data/ENNReal/Inv.lean
index 9b9e842c4e348f..63249a86934640 100644
--- a/Mathlib/Data/ENNReal/Inv.lean
+++ b/Mathlib/Data/ENNReal/Inv.lean
@@ -44,7 +44,8 @@ protected theorem div_eq_inv_mul : a / b = b⁻¹ * a := by rw [div_eq_mul_inv,
   show sInf { b : ℝ≥0∞ | 1 ≤ 0 * b } = ∞ by simp
 
 @[simp] theorem inv_top : ∞⁻¹ = 0 :=
-  bot_unique <| le_of_forall_le_of_dense fun a (h : 0 < a) => sInf_le <| by simp [*, h.ne', top_mul]
+  bot_unique <| le_of_forall_gt_imp_ge_of_dense fun a (h : 0 < a) => sInf_le <| by
+    simp [*, h.ne', top_mul]
 
 theorem coe_inv_le : (↑r⁻¹ : ℝ≥0∞) ≤ (↑r)⁻¹ :=
   le_sInf fun b (hb : 1 ≤ ↑r * b) =>
@@ -444,7 +445,7 @@ instance : SMulPosMono ℝ≥0 ℝ≥0∞ where
   elim _r _ _a _b hab := mul_le_mul_right' (coe_le_coe.2 hab) _
 
 theorem le_of_forall_nnreal_lt {x y : ℝ≥0∞} (h : ∀ r : ℝ≥0, ↑r < x → ↑r ≤ y) : x ≤ y := by
-  refine le_of_forall_ge_of_dense fun r hr => ?_
+  refine le_of_forall_lt_imp_le_of_dense fun r hr => ?_
   lift r to ℝ≥0 using ne_top_of_lt hr
   exact h r hr
 
@@ -533,7 +534,7 @@ private lemma exists_lt_mul_right {a b c : ℝ≥0∞} (hc : c < a * b) : ∃ b'
   simp_rw [mul_comm a] at hc ⊢; exact exists_lt_mul_left hc
 
 lemma mul_le_of_forall_lt {a b c : ℝ≥0∞} (h : ∀ a' < a, ∀ b' < b, a' * b' ≤ c) : a * b ≤ c := by
-  refine le_of_forall_ge_of_dense fun d hd ↦ ?_
+  refine le_of_forall_lt_imp_le_of_dense fun d hd ↦ ?_
   obtain ⟨a', ha', hd⟩ := exists_lt_mul_left hd
   obtain ⟨b', hb', hd⟩ := exists_lt_mul_right hd
   exact le_trans hd.le <| h _ ha' _ hb'
diff --git a/Mathlib/Data/Fin/Basic.lean b/Mathlib/Data/Fin/Basic.lean
index 7dc978fddb1020..331cbae4643f70 100644
--- a/Mathlib/Data/Fin/Basic.lean
+++ b/Mathlib/Data/Fin/Basic.lean
@@ -240,6 +240,12 @@ This one instead uses a `NeZero n` typeclass hypothesis.
 theorem val_zero' (n : ℕ) [NeZero n] : ((0 : Fin n) : ℕ) = 0 :=
   rfl
 
+/-- Fin.mk_zero` in `Lean` only applies in `Fin (n + 1)`.
+This one instead uses a `NeZero n` typeclass hypothesis.
+-/
+@[simp]
+theorem mk_zero' (n : ℕ) [NeZero n] : (⟨0, pos_of_neZero n⟩ : Fin n) = 0 := rfl
+
 /--
 The `Fin.zero_le` in `Lean` only applies in `Fin (n+1)`.
 This one instead uses a `NeZero n` typeclass hypothesis.
@@ -565,9 +571,15 @@ theorem leftInverse_cast (eq : n = m) : LeftInverse (Fin.cast eq.symm) (Fin.cast
 theorem rightInverse_cast (eq : n = m) : RightInverse (Fin.cast eq.symm) (Fin.cast eq) :=
   fun _ => rfl
 
+@[simp]
+theorem cast_inj (eq : n = m) {a b : Fin n} : a.cast eq = b.cast eq ↔ a = b := by
+  simp [← val_inj]
+
+@[simp]
 theorem cast_lt_cast (eq : n = m) {a b : Fin n} : a.cast eq < b.cast eq ↔ a < b :=
   Iff.rfl
 
+@[simp]
 theorem cast_le_cast (eq : n = m) {a b : Fin n} : a.cast eq ≤ b.cast eq ↔ a ≤ b :=
   Iff.rfl
 
diff --git a/Mathlib/Data/Finite/Sum.lean b/Mathlib/Data/Finite/Sum.lean
index 177f8e038134a3..4c3ecc2999ef54 100644
--- a/Mathlib/Data/Finite/Sum.lean
+++ b/Mathlib/Data/Finite/Sum.lean
@@ -24,4 +24,13 @@ theorem sum_left (β) [Finite (α ⊕ β)] : Finite α :=
 theorem sum_right (α) [Finite (α ⊕ β)] : Finite β :=
   of_injective (Sum.inr : β → α ⊕ β) Sum.inr_injective
 
+instance {α β : Sort*} [Finite α] [Finite β] : Finite (α ⊕' β) :=
+  of_equiv _ ((Equiv.psumEquivSum _ _).symm.trans (Equiv.plift.psumCongr Equiv.plift))
+
+theorem psum_left {α β : Sort*} [Finite (α ⊕' β)] : Finite α :=
+  of_injective (PSum.inl : α → α ⊕' β) PSum.inl_injective
+
+theorem psum_right {α β : Sort*} [Finite (α ⊕' β)] : Finite β :=
+  of_injective (PSum.inr : β → α ⊕' β) PSum.inr_injective
+
 end Finite
diff --git a/Mathlib/Data/Finset/Lattice/Fold.lean b/Mathlib/Data/Finset/Lattice/Fold.lean
index 0a04809d8a4e61..c5ae9f5e4dcdb9 100644
--- a/Mathlib/Data/Finset/Lattice/Fold.lean
+++ b/Mathlib/Data/Finset/Lattice/Fold.lean
@@ -250,6 +250,11 @@ theorem sup_mem (s : Set α) (w₁ : ⊥ ∈ s) (w₂ : ∀ᵉ (x ∈ s) (y ∈
 protected theorem sup_eq_bot_iff (f : β → α) (S : Finset β) : S.sup f = ⊥ ↔ ∀ s ∈ S, f s = ⊥ := by
   classical induction' S using Finset.induction with a S _ hi <;> simp [*]
 
+@[simp]
+theorem sup_eq_bot_of_isEmpty [IsEmpty β] (f : β → α) (S : Finset β) : S.sup f = ⊥ := by
+  rw [Finset.sup_eq_bot_iff]
+  exact fun x _ => False.elim <| IsEmpty.false x
+
 end Sup
 
 theorem sup_eq_iSup [CompleteLattice β] (s : Finset α) (f : α → β) : s.sup f = ⨆ a ∈ s, f a :=
diff --git a/Mathlib/Data/Finsupp/Basic.lean b/Mathlib/Data/Finsupp/Basic.lean
index 6abe2e3b9c254f..8edd359b3d1c99 100644
--- a/Mathlib/Data/Finsupp/Basic.lean
+++ b/Mathlib/Data/Finsupp/Basic.lean
@@ -1466,8 +1466,8 @@ theorem extendDomain_subtypeDomain (f : α →₀ M) (hf : ∀ a ∈ f.support,
   ext a
   by_cases h : P a
   · exact dif_pos h
-  · #adaptation_note
-    /-- Prior to nightly-2024-06-18, this `rw` was done by `dsimp`. -/
+  · #adaptation_note /-- nightly-2024-06-18
+    this `rw` was done by `dsimp`. -/
     rw [extendDomain_toFun]
     dsimp
     rw [if_neg h, eq_comm, ← not_mem_support_iff]
@@ -1478,8 +1478,8 @@ theorem extendDomain_subtypeDomain (f : α →₀ M) (hf : ∀ a ∈ f.support,
 theorem extendDomain_single (a : Subtype P) (m : M) :
     (single a m).extendDomain = single a.val m := by
   ext a'
-  #adaptation_note
-  /-- Prior to nightly-2024-06-18, this `rw` was instead `dsimp only`. -/
+  #adaptation_note /-- nightly-2024-06-18
+  this `rw` was instead `dsimp only`. -/
   rw [extendDomain_toFun]
   obtain rfl | ha := eq_or_ne a.val a'
   · simp_rw [single_eq_same, dif_pos a.prop]
diff --git a/Mathlib/Data/Fintype/Basic.lean b/Mathlib/Data/Fintype/Basic.lean
index ff4691dc547893..3207bf60506dd0 100644
--- a/Mathlib/Data/Fintype/Basic.lean
+++ b/Mathlib/Data/Fintype/Basic.lean
@@ -1060,11 +1060,7 @@ variable [Fintype α] [DecidableEq β] {f : α → β}
 /-- `bijInv f` is the unique inverse to a bijection `f`. This acts
   as a computable alternative to `Function.invFun`. -/
 def bijInv (f_bij : Bijective f) (b : β) : α :=
-  Fintype.choose (fun a => f a = b)
-    (by
-      rcases f_bij.right b with ⟨a', fa_eq_b⟩
-      rw [← fa_eq_b]
-      exact ⟨a', ⟨rfl, fun a h => f_bij.left h⟩⟩)
+  Fintype.choose (fun a => f a = b) (f_bij.existsUnique b)
 
 theorem leftInverse_bijInv (f_bij : Bijective f) : LeftInverse (bijInv f_bij) f := fun a =>
   f_bij.left (choose_spec (fun a' => f a' = f a) _)
diff --git a/Mathlib/Data/List/Basic.lean b/Mathlib/Data/List/Basic.lean
index d51295c269ce17..187b3b0cb21bcc 100644
--- a/Mathlib/Data/List/Basic.lean
+++ b/Mathlib/Data/List/Basic.lean
@@ -29,6 +29,10 @@ open Nat hiding one_pos
 
 namespace List
 
+-- Renamed in lean core; to be removed with the version bump.
+alias replicate_append_replicate := append_replicate_replicate
+alias append_eq_nil_iff := append_eq_nil
+
 universe u v w
 
 variable {ι : Type*} {α : Type u} {β : Type v} {γ : Type w} {l₁ l₂ : List α}
@@ -208,7 +212,7 @@ theorem eq_replicate_length {a : α} : ∀ {l : List α}, l = replicate l.length
   | (b :: l) => by simp [eq_replicate_length, replicate_succ]
 
 theorem replicate_add (m n) (a : α) : replicate (m + n) a = replicate m a ++ replicate n a := by
-  rw [append_replicate_replicate]
+  rw [replicate_append_replicate]
 
 theorem replicate_subset_singleton (n) (a : α) : replicate n a ⊆ [a] := fun _ h =>
   mem_singleton.2 (eq_of_mem_replicate h)
@@ -379,7 +383,8 @@ theorem getLastI_eq_getLast? [Inhabited α] : ∀ l : List α, l.getLastI = l.ge
   | [_, _, _] => rfl
   | _ :: _ :: c :: l => by simp [getLastI, getLastI_eq_getLast? (c :: l)]
 
-#adaptation_note /-- 2024-07-10: removed `@[simp]` since the LHS simplifies using the simp set. -/
+#adaptation_note /-- 2024-07-10
+removed `@[simp]` since the LHS simplifies using the simp set. -/
 theorem getLast?_append_cons :
     ∀ (l₁ : List α) (a : α) (l₂ : List α), getLast? (l₁ ++ a :: l₂) = getLast? (a :: l₂)
   | [], _, _ => rfl
@@ -670,19 +675,19 @@ variable [DecidableEq α]
 
 -- indexOf_cons_eq _ rfl
 @[simp]
-theorem indexOf_cons_self (a : α) (l : List α) : indexOf a (a :: l) = 0 := by
+theorem indexOf_cons_self {a : α} {l : List α} : indexOf a (a :: l) = 0 := by
   rw [indexOf, findIdx_cons, beq_self_eq_true, cond]
 
 -- fun e => if_pos e
 theorem indexOf_cons_eq {a b : α} (l : List α) : b = a → indexOf a (b :: l) = 0
-  | e => by rw [← e]; exact indexOf_cons_self b l
+  | e => by rw [← e]; exact indexOf_cons_self
 
 -- fun n => if_neg n
 @[simp]
 theorem indexOf_cons_ne {a b : α} (l : List α) : b ≠ a → indexOf a (b :: l) = succ (indexOf a l)
   | h => by simp only [indexOf, findIdx_cons, Bool.cond_eq_ite, beq_iff_eq, h, ite_false]
 
-theorem indexOf_eq_length {a : α} {l : List α} : indexOf a l = length l ↔ a ∉ l := by
+theorem indexOf_eq_length_iff {a : α} {l : List α} : indexOf a l = length l ↔ a ∉ l := by
   induction' l with b l ih
   · exact iff_of_true rfl (not_mem_nil _)
   simp only [length, mem_cons, indexOf_cons, eq_comm]
@@ -693,9 +698,12 @@ theorem indexOf_eq_length {a : α} {l : List α} : indexOf a l = length l ↔ a
     rw [← ih]
     exact succ_inj'
 
+@[deprecated (since := "2025-01-28")]
+alias indexOf_eq_length := indexOf_eq_length_iff
+
 @[simp]
 theorem indexOf_of_not_mem {l : List α} {a : α} : a ∉ l → indexOf a l = length l :=
-  indexOf_eq_length.2
+  indexOf_eq_length_iff.2
 
 theorem indexOf_le_length {a : α} {l : List α} : indexOf a l ≤ length l := by
   induction' l with b l ih; · rfl
@@ -705,8 +713,8 @@ theorem indexOf_le_length {a : α} {l : List α} : indexOf a l ≤ length l := b
   · rw [if_neg h]; exact succ_le_succ ih
 
 theorem indexOf_lt_length_iff {a} {l : List α} : indexOf a l < length l ↔ a ∈ l :=
-  ⟨fun h => Decidable.byContradiction fun al => Nat.ne_of_lt h <| indexOf_eq_length.2 al,
-   fun al => (lt_of_le_of_ne indexOf_le_length) fun h => indexOf_eq_length.1 h al⟩
+  ⟨fun h => Decidable.byContradiction fun al => Nat.ne_of_lt h <| indexOf_eq_length_iff.2 al,
+   fun al => (lt_of_le_of_ne indexOf_le_length) fun h => indexOf_eq_length_iff.1 h al⟩
 
 @[deprecated (since := "2025-01-22")] alias indexOf_lt_length := indexOf_lt_length_iff
 
@@ -875,7 +883,7 @@ theorem flatMap_pure_eq_map (f : α → β) (l : List α) : l.flatMap (pure ∘
 @[deprecated (since := "2024-10-16")] alias bind_pure_eq_map := flatMap_pure_eq_map
 
 theorem flatMap_congr {l : List α} {f g : α → List β} (h : ∀ x ∈ l, f x = g x) :
-    List.flatMap l f = List.flatMap l g :=
+    l.flatMap f = l.flatMap g :=
   (congr_arg List.flatten <| map_congr_left h :)
 
 @[deprecated (since := "2024-10-16")] alias bind_congr := flatMap_congr
@@ -1670,11 +1678,9 @@ theorem filter_eq_foldr (p : α → Bool) (l : List α) :
     filter p l = foldr (fun a out => bif p a then a :: out else out) [] l := by
   induction l <;> simp [*, filter]; rfl
 
-#adaptation_note
-/--
+#adaptation_note /-- nightly-2024-07-27
 This has to be temporarily renamed to avoid an unintentional collision.
-The prime should be removed at nightly-2024-07-27.
--/
+The prime should be removed at nightly-2024-07-27. -/
 @[simp]
 theorem filter_subset' (l : List α) : filter p l ⊆ l :=
   (filter_sublist l).subset
diff --git a/Mathlib/Data/List/Defs.lean b/Mathlib/Data/List/Defs.lean
index 9718d8cc7eb2c8..a0351827a88e0a 100644
--- a/Mathlib/Data/List/Defs.lean
+++ b/Mathlib/Data/List/Defs.lean
@@ -488,10 +488,6 @@ theorem length_mapAccumr₂ :
 
 end MapAccumr
 
-/- #adaptation_note: this attribute should be removed after Mathlib moves to v4.15.0-rc1. -/
-set_option allowUnsafeReducibility true in
-attribute [semireducible] Fin.foldr.loop
-
 section Deprecated
 
 @[deprecated List.mem_cons (since := "2024-08-10")]
diff --git a/Mathlib/Data/List/Indexes.lean b/Mathlib/Data/List/Indexes.lean
index e9680556a8109d..e307495cd9f2cd 100644
--- a/Mathlib/Data/List/Indexes.lean
+++ b/Mathlib/Data/List/Indexes.lean
@@ -9,6 +9,14 @@ import Mathlib.Data.List.Basic
 # Lemmas about List.*Idx functions.
 
 Some specification lemmas for `List.mapIdx`, `List.mapIdxM`, `List.foldlIdx` and `List.foldrIdx`.
+
+As of 2025-01-29, these are not used anywhere in Mathlib. Moreover, with
+`List.enum` and `List.enumFrom` being replaced by `List.zipIdx`
+in Lean's `nightly-2025-01-29` release, they now use deprecated functions and theorems.
+Rather than updating this unused material, we are deprecating it.
+Anyone wanting to restore this material is welcome to do so, but will need to update uses of
+`List.enum` and `List.enumFrom` to use `List.zipIdx` instead.
+However, note that this material will later be implemented in the Lean standard library.
 -/
 
 assert_not_exists MonoidWithZero
@@ -23,17 +31,10 @@ variable {α : Type u} {β : Type v}
 
 section MapIdx
 
+@[deprecated reverseRecOn (since := "2025-01-28")]
 theorem list_reverse_induction (p : List α → Prop) (base : p [])
-    (ind : ∀ (l : List α) (e : α), p l → p (l ++ [e])) : (∀ (l : List α), p l) := by
-  let q := fun l ↦ p (reverse l)
-  have pq : ∀ l, p (reverse l) → q l := by simp only [q, reverse_reverse]; intro; exact id
-  have qp : ∀ l, q (reverse l) → p l := by simp only [q, reverse_reverse]; intro; exact id
-  intro l
-  apply qp
-  generalize (reverse l) = l
-  induction' l with head tail ih
-  · apply pq; simp only [reverse_nil, base]
-  · apply pq; simp only [reverse_cons]; apply ind; apply qp; rw [reverse_reverse]; exact ih
+    (ind : ∀ (l : List α) (e : α), p l → p (l ++ [e])) : (∀ (l : List α), p l) :=
+  fun l => l.reverseRecOn base ind
 
 @[deprecated (since := "2024-10-15")] alias mapIdxGo_length := mapIdx_go_length
 
@@ -41,7 +42,7 @@ theorem mapIdx_append_one : ∀ {f : ℕ → α → β} {l : List α} {e : α},
     mapIdx f (l ++ [e]) = mapIdx f l ++ [f l.length e] :=
   mapIdx_concat
 
-@[local simp]
+@[deprecated "Deprecated without replacement." (since := "2025-01-29"), local simp]
 theorem map_enumFrom_eq_zipWith : ∀ (l : List α) (n : ℕ) (f : ℕ → α → β),
     map (uncurry f) (enumFrom n l) = zipWith (fun i ↦ f (i + n)) (range (length l)) l := by
   intro l
@@ -67,6 +68,7 @@ theorem map_enumFrom_eq_zipWith : ∀ (l : List α) (n : ℕ) (f : ℕ → α 
 
 @[deprecated (since := "2024-10-15")] alias mapIdx_eq_nil := mapIdx_eq_nil_iff
 
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 theorem get_mapIdx (l : List α) (f : ℕ → α → β) (i : ℕ) (h : i < l.length)
     (h' : i < (l.mapIdx f).length := h.trans_le length_mapIdx.ge) :
     (l.mapIdx f).get ⟨i, h'⟩ = f i (l.get ⟨i, h⟩) := by
@@ -159,30 +161,41 @@ section FoldrIdx
 
 -- Porting note: Changed argument order of `foldrIdxSpec` to align better with `foldrIdx`.
 /-- Specification of `foldrIdx`. -/
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 def foldrIdxSpec (f : ℕ → α → β → β) (b : β) (as : List α) (start : ℕ) : β :=
   foldr (uncurry f) b <| enumFrom start as
 
+set_option linter.deprecated false in
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 theorem foldrIdxSpec_cons (f : ℕ → α → β → β) (b a as start) :
     foldrIdxSpec f b (a :: as) start = f start a (foldrIdxSpec f b as (start + 1)) :=
   rfl
 
+set_option linter.deprecated false in
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 theorem foldrIdx_eq_foldrIdxSpec (f : ℕ → α → β → β) (b as start) :
     foldrIdx f b as start = foldrIdxSpec f b as start := by
   induction as generalizing start
   · rfl
   · simp only [foldrIdx, foldrIdxSpec_cons, *]
 
+set_option linter.deprecated false in
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 theorem foldrIdx_eq_foldr_enum (f : ℕ → α → β → β) (b : β) (as : List α) :
     foldrIdx f b as = foldr (uncurry f) b (enum as) := by
   simp only [foldrIdx, foldrIdxSpec, foldrIdx_eq_foldrIdxSpec, enum]
 
 end FoldrIdx
 
+set_option linter.deprecated false in
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 theorem indexesValues_eq_filter_enum (p : α → Prop) [DecidablePred p] (as : List α) :
     indexesValues p as = filter (p ∘ Prod.snd) (enum as) := by
   simp (config := { unfoldPartialApp := true }) [indexesValues, foldrIdx_eq_foldr_enum, uncurry,
     filter_eq_foldr, cond_eq_if]
 
+set_option linter.deprecated false in
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 theorem findIdxs_eq_map_indexesValues (p : α → Prop) [DecidablePred p] (as : List α) :
     findIdxs p as = map Prod.fst (indexesValues p as) := by
   simp (config := { unfoldPartialApp := true }) only [indexesValues_eq_filter_enum,
@@ -193,19 +206,26 @@ section FoldlIdx
 
 -- Porting note: Changed argument order of `foldlIdxSpec` to align better with `foldlIdx`.
 /-- Specification of `foldlIdx`. -/
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 def foldlIdxSpec (f : ℕ → α → β → α) (a : α) (bs : List β) (start : ℕ) : α :=
   foldl (fun a p ↦ f p.fst a p.snd) a <| enumFrom start bs
 
+set_option linter.deprecated false in
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 theorem foldlIdxSpec_cons (f : ℕ → α → β → α) (a b bs start) :
     foldlIdxSpec f a (b :: bs) start = foldlIdxSpec f (f start a b) bs (start + 1) :=
   rfl
 
+set_option linter.deprecated false in
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 theorem foldlIdx_eq_foldlIdxSpec (f : ℕ → α → β → α) (a bs start) :
     foldlIdx f a bs start = foldlIdxSpec f a bs start := by
   induction bs generalizing start a
   · rfl
   · simp [foldlIdxSpec, *]
 
+set_option linter.deprecated false in
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 theorem foldlIdx_eq_foldl_enum (f : ℕ → α → β → α) (a : α) (bs : List β) :
     foldlIdx f a bs = foldl (fun a p ↦ f p.fst a p.snd) a (enum bs) := by
   simp only [foldlIdx, foldlIdxSpec, foldlIdx_eq_foldlIdxSpec, enum]
@@ -217,11 +237,15 @@ section FoldIdxM
 -- Porting note: `foldrM_eq_foldr` now depends on `[LawfulMonad m]`
 variable {m : Type u → Type v} [Monad m]
 
+set_option linter.deprecated false in
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 theorem foldrIdxM_eq_foldrM_enum {β} (f : ℕ → α → β → m β) (b : β) (as : List α) [LawfulMonad m] :
     foldrIdxM f b as = foldrM (uncurry f) b (enum as) := by
   simp (config := { unfoldPartialApp := true }) only [foldrIdxM, foldrM_eq_foldr,
     foldrIdx_eq_foldr_enum, uncurry]
 
+set_option linter.deprecated false in
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 theorem foldlIdxM_eq_foldlM_enum [LawfulMonad m] {β} (f : ℕ → β → α → m β) (b : β) (as : List α) :
     foldlIdxM f b as = List.foldlM (fun b p ↦ f p.fst b p.snd) b (enum as) := by
   rw [foldlIdxM, foldlM_eq_foldl, foldlIdx_eq_foldl_enum]
@@ -234,15 +258,20 @@ section MapIdxM
 variable {m : Type u → Type v} [Monad m]
 
 /-- Specification of `mapIdxMAux`. -/
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 def mapIdxMAuxSpec {β} (f : ℕ → α → m β) (start : ℕ) (as : List α) : m (List β) :=
   List.traverse (uncurry f) <| enumFrom start as
 
 -- Note: `traverse` the class method would require a less universe-polymorphic
 -- `m : Type u → Type u`.
+set_option linter.deprecated false in
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 theorem mapIdxMAuxSpec_cons {β} (f : ℕ → α → m β) (start : ℕ) (a : α) (as : List α) :
     mapIdxMAuxSpec f start (a :: as) = cons <$> f start a <*> mapIdxMAuxSpec f (start + 1) as :=
   rfl
 
+set_option linter.deprecated false in
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 theorem mapIdxMGo_eq_mapIdxMAuxSpec
     [LawfulMonad m] {β} (f : ℕ → α → m β) (arr : Array β) (as : List α) :
     mapIdxM.go f as arr = (arr.toList ++ ·) <$> mapIdxMAuxSpec f arr.size as := by
@@ -266,6 +295,8 @@ theorem mapIdxMGo_eq_mapIdxMAuxSpec
       simp only [Array.push_toList, append_assoc, singleton_append, Array.size_push,
         map_eq_pure_bind]
 
+set_option linter.deprecated false in
+@[deprecated "Deprecated without replacement." (since := "2025-01-29")]
 theorem mapIdxM_eq_mmap_enum [LawfulMonad m] {β} (f : ℕ → α → m β) (as : List α) :
     as.mapIdxM f = List.traverse (uncurry f) (enum as) := by
   simp only [mapIdxM, mapIdxMGo_eq_mapIdxMAuxSpec, Array.toList_toArray,
diff --git a/Mathlib/Data/List/MinMax.lean b/Mathlib/Data/List/MinMax.lean
index 71564ef6cb8486..241a87b598d882 100644
--- a/Mathlib/Data/List/MinMax.lean
+++ b/Mathlib/Data/List/MinMax.lean
@@ -38,7 +38,7 @@ def argAux (a : Option α) (b : α) : Option α :=
 @[simp]
 theorem foldl_argAux_eq_none : l.foldl (argAux r) o = none ↔ l = [] ∧ o = none :=
   List.reverseRecOn l (by simp) fun tl hd => by
-    simp only [foldl_append, foldl_cons, argAux, foldl_nil, append_eq_nil, and_false, false_and,
+    simp only [foldl_append, foldl_cons, argAux, foldl_nil, append_eq_nil_iff, and_false, false_and,
       iff_false]
     cases foldl (argAux r) o tl
     · simp
diff --git a/Mathlib/Data/List/Monad.lean b/Mathlib/Data/List/Monad.lean
index 3132c8376d9229..0c420b1774ff69 100644
--- a/Mathlib/Data/List/Monad.lean
+++ b/Mathlib/Data/List/Monad.lean
@@ -16,9 +16,9 @@ namespace List
 variable {α : Type u}
 
 instance instMonad : Monad List.{u} where
-  pure := @List.singleton
-  bind := @List.flatMap
-  map := @List.map
+  pure x := [x]
+  bind l f := l.flatMap f
+  map f l := l.map f
 
 @[simp] theorem pure_def (a : α) : pure a = [a] := rfl
 
diff --git a/Mathlib/Data/List/Permutation.lean b/Mathlib/Data/List/Permutation.lean
index bdd3205fa2f21a..d598a0ba42d67f 100644
--- a/Mathlib/Data/List/Permutation.lean
+++ b/Mathlib/Data/List/Permutation.lean
@@ -326,7 +326,7 @@ theorem perm_permutations'Aux_comm (a b : α) (l : List α) :
           map (cons c) ((permutations'Aux a l).flatMap (permutations'Aux b)) := by
     intros a' b'
     simp only [flatMap_map, permutations'Aux]
-    show List.flatMap (permutations'Aux _ l) (fun a => ([b' :: c :: a] ++
+    show (permutations'Aux _ l).flatMap (fun a => ([b' :: c :: a] ++
       map (cons c) (permutations'Aux _ a))) ~ _
     refine (flatMap_append_perm _ (fun x => [b' :: c :: x]) _).symm.trans ?_
     rw [← map_eq_flatMap, ← map_flatMap]
diff --git a/Mathlib/Data/Matrix/Basis.lean b/Mathlib/Data/Matrix/Basis.lean
index 69638ec2666c4c..d7eeeed30a9772 100644
--- a/Mathlib/Data/Matrix/Basis.lean
+++ b/Mathlib/Data/Matrix/Basis.lean
@@ -49,6 +49,17 @@ theorem stdBasisMatrix_zero (i : m) (j : n) : stdBasisMatrix i j (0 : α) = 0 :=
   ext
   simp
 
+@[simp]
+lemma transpose_stdBasisMatrix (i : m) (j : n) (a : α) :
+    (stdBasisMatrix i j a)ᵀ = stdBasisMatrix j i a := by
+  aesop (add unsafe unfold stdBasisMatrix)
+
+@[simp]
+lemma map_stdBasisMatrix (i : m) (j : n) (a : α) {β : Type*} [Zero β]
+    {F : Type*} [FunLike F α β] [ZeroHomClass F α β] (f : F) :
+    (stdBasisMatrix i j a).map f = stdBasisMatrix i j (f a) := by
+  aesop (add unsafe unfold stdBasisMatrix)
+
 end Zero
 
 theorem stdBasisMatrix_add [AddZeroClass α] (i : m) (j : n) (a b : α) :
diff --git a/Mathlib/Data/Matrix/ConjTranspose.lean b/Mathlib/Data/Matrix/ConjTranspose.lean
index 4b856340799ca6..f43539add8aae8 100644
--- a/Mathlib/Data/Matrix/ConjTranspose.lean
+++ b/Mathlib/Data/Matrix/ConjTranspose.lean
@@ -11,6 +11,7 @@ import Mathlib.Algebra.Star.BigOperators
 import Mathlib.Algebra.Star.Module
 import Mathlib.Algebra.Star.Pi
 import Mathlib.Data.Fintype.BigOperators
+import Mathlib.Data.Matrix.Basis
 import Mathlib.Data.Matrix.Mul
 
 /-!
@@ -40,6 +41,13 @@ def conjTranspose [Star α] (M : Matrix m n α) : Matrix n m α :=
 @[inherit_doc]
 scoped postfix:1024 "ᴴ" => Matrix.conjTranspose
 
+@[simp]
+lemma conjTranspose_stdBasisMatrix [DecidableEq n] [DecidableEq m] [AddMonoid α]
+    [StarAddMonoid α] (i : m) (j : n) (a : α) :
+    (stdBasisMatrix i j a)ᴴ = stdBasisMatrix j i (star a) := by
+  show (stdBasisMatrix i j a).transpose.map starAddEquiv = stdBasisMatrix j i (star a)
+  simp
+
 section Diagonal
 
 variable [DecidableEq n]
diff --git a/Mathlib/Data/Matrix/Mul.lean b/Mathlib/Data/Matrix/Mul.lean
index 4f71ea9008fa3a..c074c0ef47fffb 100644
--- a/Mathlib/Data/Matrix/Mul.lean
+++ b/Mathlib/Data/Matrix/Mul.lean
@@ -722,12 +722,12 @@ theorem mulVec_smul [Fintype n] [Monoid R] [NonUnitalNonAssocSemiring S] [Distri
 
 @[simp]
 theorem mulVec_single [Fintype n] [DecidableEq n] [NonUnitalNonAssocSemiring R] (M : Matrix m n R)
-    (j : n) (x : R) : M *ᵥ Pi.single j x = fun i => M i j * x :=
+    (j : n) (x : R) : M *ᵥ Pi.single j x = MulOpposite.op x • Mᵀ j :=
   funext fun _ => dotProduct_single _ _ _
 
 @[simp]
 theorem single_vecMul [Fintype m] [DecidableEq m] [NonUnitalNonAssocSemiring R] (M : Matrix m n R)
-    (i : m) (x : R) : Pi.single i x ᵥ* M = fun j => x * M i j :=
+    (i : m) (x : R) : Pi.single i x ᵥ* M = x • M i :=
   funext fun _ => single_dotProduct _ _ _
 
 theorem mulVec_single_one [Fintype n] [DecidableEq n] [NonAssocSemiring R]
@@ -736,7 +736,7 @@ theorem mulVec_single_one [Fintype n] [DecidableEq n] [NonAssocSemiring R]
 
 theorem single_one_vecMul [Fintype m] [DecidableEq m] [NonAssocSemiring R]
     (i : m) (M : Matrix m n R) :
-    Pi.single i 1 ᵥ* M = M i := by simp
+    Pi.single i 1 ᵥ* M = M i := by ext; simp
 
 theorem diagonal_mulVec_single [Fintype n] [DecidableEq n] [NonUnitalNonAssocSemiring R] (v : n → R)
     (j : n) (x : R) : diagonal v *ᵥ Pi.single j x = Pi.single j (v j * x) := by
@@ -781,12 +781,28 @@ section NonAssocSemiring
 
 variable [NonAssocSemiring α]
 
-theorem mulVec_one [Fintype n] (A : Matrix m n α) : A *ᵥ 1 = fun i => ∑ j, A i j := by
+theorem mulVec_one [Fintype n] (A : Matrix m n α) : A *ᵥ 1 = ∑ j, Aᵀ j := by
   ext; simp [mulVec, dotProduct]
 
-theorem vec_one_mul [Fintype m] (A : Matrix m n α) : 1 ᵥ* A = fun j => ∑ i, A i j := by
+theorem one_vecMul [Fintype m] (A : Matrix m n α) : 1 ᵥ* A = ∑ i, A i := by
   ext; simp [vecMul, dotProduct]
 
+@[deprecated (since := "2025-01-26")] alias vec_one_mul := one_vecMul
+
+lemma ext_of_mulVec_single [DecidableEq n] [Fintype n] {M N : Matrix m n α}
+    (h : ∀ i, M *ᵥ Pi.single i 1 = N *ᵥ Pi.single i 1) :
+    M = N := by
+  ext i j
+  simp_rw [mulVec_single_one] at h
+  exact congrFun (h j) i
+
+lemma ext_of_single_vecMul [DecidableEq m] [Fintype m] {M N : Matrix m n α}
+    (h : ∀ i, Pi.single i 1 ᵥ* M = Pi.single i 1 ᵥ* N) :
+    M = N := by
+  ext i j
+  simp_rw [single_one_vecMul] at h
+  exact congrFun (h i) j
+
 variable [Fintype m] [Fintype n] [DecidableEq m]
 
 @[simp]
diff --git a/Mathlib/Data/Matrix/PEquiv.lean b/Mathlib/Data/Matrix/PEquiv.lean
index eb70fede2430c4..98cb355cc674a8 100644
--- a/Mathlib/Data/Matrix/PEquiv.lean
+++ b/Mathlib/Data/Matrix/PEquiv.lean
@@ -40,7 +40,7 @@ open Matrix
 universe u v
 
 variable {k l m n : Type*}
-variable {α : Type v}
+variable {α β : Type*}
 
 open Matrix
 
@@ -55,13 +55,28 @@ theorem toMatrix_apply [DecidableEq n] [Zero α] [One α] (f : m ≃. n) (i j) :
     toMatrix f i j = if j ∈ f i then (1 : α) else 0 :=
   rfl
 
-theorem mul_matrix_apply [Fintype m] [DecidableEq m] [Semiring α] (f : l ≃. m) (M : Matrix m n α)
+theorem toMatrix_mul_apply [Fintype m] [DecidableEq m] [Semiring α] (f : l ≃. m) (M : Matrix m n α)
     (i j) : (f.toMatrix * M :) i j = Option.casesOn (f i) 0 fun fi => M fi j := by
   dsimp [toMatrix, Matrix.mul_apply]
   cases' h : f i with fi
   · simp [h]
   · rw [Finset.sum_eq_single fi] <;> simp +contextual [h, eq_comm]
 
+@[deprecated (since := "2025-01-27")] alias mul_matrix_apply := toMatrix_mul_apply
+
+theorem mul_toMatrix_apply [Fintype m] [Semiring α] [DecidableEq n] (M : Matrix l m α) (f : m ≃. n)
+    (i j) : (M * f.toMatrix :) i j = Option.casesOn (f.symm j) 0 (M i) := by
+  dsimp [Matrix.mul_apply, toMatrix_apply]
+  cases' h : f.symm j with fj
+  · simp [h, ← f.eq_some_iff]
+  · rw [Finset.sum_eq_single fj]
+    · simp [h, ← f.eq_some_iff]
+    · rintro b - n
+      simp [h, ← f.eq_some_iff, n.symm]
+    · simp
+
+@[deprecated (since := "2025-01-27")] alias matrix_mul_apply := mul_toMatrix_apply
+
 theorem toMatrix_symm [DecidableEq m] [DecidableEq n] [Zero α] [One α] (f : m ≃. n) :
     (f.symm.toMatrix : Matrix n m α) = f.toMatrixᵀ := by
   ext
@@ -74,32 +89,53 @@ theorem toMatrix_refl [DecidableEq n] [Zero α] [One α] :
   ext
   simp [toMatrix_apply, one_apply]
 
-theorem matrix_mul_apply [Fintype m] [Semiring α] [DecidableEq n] (M : Matrix l m α) (f : m ≃. n)
-    (i j) : (M * f.toMatrix :) i j = Option.casesOn (f.symm j) 0 fun fj => M i fj := by
-  dsimp [toMatrix, Matrix.mul_apply]
-  cases' h : f.symm j with fj
-  · simp [h, ← f.eq_some_iff]
-  · rw [Finset.sum_eq_single fj]
-    · simp [h, ← f.eq_some_iff]
-    · rintro b - n
-      simp [h, ← f.eq_some_iff, n.symm]
-    · simp
+@[simp]
+theorem toMatrix_toPEquiv_apply [DecidableEq n] [Zero α] [One α] (f : m ≃ n) (i) :
+    f.toPEquiv.toMatrix i = Pi.single (f i) (1 : α) := by
+  ext
+  simp [toMatrix_apply, Pi.single_apply, eq_comm]
+
+@[simp]
+theorem transpose_toMatrix_toPEquiv_apply
+    [DecidableEq m] [DecidableEq n] [Zero α] [One α] (f : m ≃ n) (j) :
+    f.toPEquiv.toMatrixᵀ j = Pi.single (f.symm j) (1 : α) := by
+  ext
+  simp [toMatrix_apply, Pi.single_apply, eq_comm, ← Equiv.apply_eq_iff_eq_symm_apply]
 
-theorem toPEquiv_mul_matrix [Fintype m] [DecidableEq m] [Semiring α] (f : m ≃ m)
-    (M : Matrix m n α) : f.toPEquiv.toMatrix * M = M.submatrix f id := by
+theorem toMatrix_toPEquiv_mul [Fintype m] [DecidableEq m]
+    [Semiring α] (f : l ≃ m) (M : Matrix m n α) :
+    f.toPEquiv.toMatrix * M = M.submatrix f id := by
   ext i j
-  rw [mul_matrix_apply, Equiv.toPEquiv_apply, submatrix_apply, id]
+  rw [toMatrix_mul_apply, Equiv.toPEquiv_apply, submatrix_apply, id]
 
-theorem mul_toPEquiv_toMatrix {m n α : Type*} [Fintype n] [DecidableEq n] [Semiring α] (f : n ≃ n)
-    (M : Matrix m n α) : M * f.toPEquiv.toMatrix = M.submatrix id f.symm :=
+@[deprecated (since := "2025-01-27")] alias toPEquiv_mul_matrix := toMatrix_toPEquiv_mul
+
+theorem mul_toMatrix_toPEquiv [Fintype m] [DecidableEq n]
+    [Semiring α] (M : Matrix l m α) (f : m ≃ n) :
+    (M * f.toPEquiv.toMatrix) = M.submatrix id f.symm :=
   Matrix.ext fun i j => by
-    rw [PEquiv.matrix_mul_apply, ← Equiv.toPEquiv_symm, Equiv.toPEquiv_apply,
+    rw [PEquiv.mul_toMatrix_apply, ← Equiv.toPEquiv_symm, Equiv.toPEquiv_apply,
       Matrix.submatrix_apply, id]
 
+@[deprecated (since := "2025-01-27")] alias mul_toPEquiv_toMatrix := mul_toMatrix_toPEquiv
+
+lemma toMatrix_toPEquiv_mulVec [DecidableEq n] [Fintype n]
+    [NonAssocSemiring α] (σ : m ≃ n) (a : n → α) :
+    σ.toPEquiv.toMatrix *ᵥ a = a ∘ σ := by
+  ext j
+  simp [toMatrix, mulVec, dotProduct]
+
+lemma vecMul_toMatrix_toPEquiv [DecidableEq n] [Fintype m]
+    [NonAssocSemiring α] (σ : m ≃ n) (a : m → α) :
+    a ᵥ* σ.toPEquiv.toMatrix = a ∘ σ.symm := by
+  classical
+  ext j
+  simp [toMatrix, σ.apply_eq_iff_eq_symm_apply, vecMul, dotProduct]
+
 theorem toMatrix_trans [Fintype m] [DecidableEq m] [DecidableEq n] [Semiring α] (f : l ≃. m)
     (g : m ≃. n) : ((f.trans g).toMatrix : Matrix l n α) = f.toMatrix * g.toMatrix := by
   ext i j
-  rw [mul_matrix_apply]
+  rw [toMatrix_mul_apply]
   dsimp [toMatrix, PEquiv.trans]
   cases f i <;> simp
 
@@ -110,19 +146,18 @@ theorem toMatrix_bot [DecidableEq n] [Zero α] [One α] :
 
 theorem toMatrix_injective [DecidableEq n] [MonoidWithZero α] [Nontrivial α] :
     Function.Injective (@toMatrix m n α _ _ _) := by
-  classical
-    intro f g
-    refine not_imp_not.1 ?_
-    simp only [Matrix.ext_iff.symm, toMatrix_apply, PEquiv.ext_iff, not_forall, exists_imp]
-    intro i hi
-    use i
-    cases' hf : f i with fi
-    · cases' hg : g i with gi
-      · rw [hf, hg] at hi; exact (hi rfl).elim
-      · use gi
-        simp
-    · use fi
-      simp [hf.symm, Ne.symm hi]
+  intro f g
+  refine not_imp_not.1 ?_
+  simp only [Matrix.ext_iff.symm, toMatrix_apply, PEquiv.ext_iff, not_forall, exists_imp]
+  intro i hi
+  use i
+  cases' hf : f i with fi
+  · cases' hg : g i with gi
+    · rw [hf, hg] at hi; exact (hi rfl).elim
+    · use gi
+      simp
+  · use fi
+    simp [hf.symm, Ne.symm hi]
 
 theorem toMatrix_swap [DecidableEq n] [Ring α] (i j : n) :
     (Equiv.swap i j).toPEquiv.toMatrix =
@@ -152,8 +187,16 @@ theorem single_mul_single_right [Fintype n] [Fintype k] [DecidableEq n] [Decidab
   rw [← Matrix.mul_assoc, single_mul_single]
 
 /-- We can also define permutation matrices by permuting the rows of the identity matrix. -/
-theorem equiv_toPEquiv_toMatrix [DecidableEq n] [Zero α] [One α] (σ : Equiv n n) (i j : n) :
-    σ.toPEquiv.toMatrix i j = (1 : Matrix n n α) (σ i) j :=
-  if_congr Option.some_inj rfl rfl
+theorem toMatrix_toPEquiv_eq [DecidableEq n] [Zero α] [One α] (σ : Equiv.Perm n) :
+    σ.toPEquiv.toMatrix = (1 : Matrix n n α).submatrix σ id :=
+  Matrix.ext fun _ _ => if_congr Option.some_inj rfl rfl
+
+@[deprecated (since := "2025-01-27")] alias equiv_toPEquiv_toMatrix := toMatrix_toPEquiv_eq
+
+@[simp]
+lemma map_toMatrix [DecidableEq n] [NonAssocSemiring α] [NonAssocSemiring β]
+    (f : α →+* β) (σ : m ≃. n) : σ.toMatrix.map f = σ.toMatrix := by
+  ext i j
+  simp
 
 end PEquiv
diff --git a/Mathlib/Data/Matroid/Basic.lean b/Mathlib/Data/Matroid/Basic.lean
index 60a143ac60a47d..4bef987a38c329 100644
--- a/Mathlib/Data/Matroid/Basic.lean
+++ b/Mathlib/Data/Matroid/Basic.lean
@@ -217,6 +217,9 @@ namespace Matroid
 
 variable {α : Type*} {M : Matroid α}
 
+instance (M : Matroid α) : Nonempty {B // M.Base B} :=
+  nonempty_subtype.2 M.exists_base
+
 /-- Typeclass for a matroid having finite ground set. Just a wrapper for `M.E.Finite`-/
 @[mk_iff] protected class Finite (M : Matroid α) : Prop where
   /-- The ground set is finite -/
diff --git a/Mathlib/Data/Matroid/Circuit.lean b/Mathlib/Data/Matroid/Circuit.lean
index d8f568c93c09e2..d074210e882958 100644
--- a/Mathlib/Data/Matroid/Circuit.lean
+++ b/Mathlib/Data/Matroid/Circuit.lean
@@ -16,6 +16,24 @@ In matroids arising from graphs, circuits correspond to graphical cycles.
 # Main Declarations
 
 * `Matroid.Circuit M C` means that `C` is minimally dependent in `M`.
+* For an `Indep`endent set `I` whose closure contains an element `e ∉ I`,
+  `Matroid.fundCircuit M e I` is the unique circuit contained in `insert e I`.
+* `Matroid.Indep.fundCircuit_circuit` states that `Matroid.fundCircuit M e I` is indeed a circuit.
+* `Circuit.eq_fundCircuit_of_subset` states that `Matroid.fundCircuit M e I` is the
+  unique circuit contained in `insert e I`.
+* `Matroid.dep_iff_superset_circuit` states that the dependent subsets of the ground set
+  are precisely those that contain a circuit.
+* `Matroid.ext_circuit` : a matroid is determined by its collection of circuits.
+
+# Implementation Details
+
+Since `Matroid.fundCircuit M e I` is only sensible if `I` is independent and `e ∈ M.closure I \ I`,
+to avoid hypotheses being explicitly included in the definition,
+junk values need to be chosen if either hypothesis fails.
+The definition is chosen so that the junk values satisfy
+`M.fundCircuit e I = {e}` for `e ∈ I` or `e ∉ M.E` and
+`M.fundCircuit e I = insert e I` if `e ∈ M.E \ M.closure I`.
+These make the useful statement `e ∈ M.fundCircuit e I ⊆ insert e I` true unconditionally.
 -/
 
 variable {α : Type*} {M : Matroid α} {C C' I X Y R : Set α} {e f x y : α}
@@ -87,6 +105,10 @@ lemma Circuit.not_ssubset (hC : M.Circuit C) (hC' : M.Circuit C') : ¬C' ⊂ C :
 lemma Circuit.eq_of_subset_circuit (hC : M.Circuit C) (hC' : M.Circuit C') (h : C ⊆ C') : C = C' :=
   hC'.eq_of_dep_subset hC.dep h
 
+lemma Circuit.eq_of_superset_circuit (hC : M.Circuit C) (hC' : M.Circuit C') (h : C' ⊆ C) :
+    C = C' :=
+  (hC'.eq_of_subset_circuit hC h).symm
+
 lemma circuit_iff_dep_forall_diff_singleton_indep :
     M.Circuit C ↔ M.Dep C ∧ ∀ e ∈ C, M.Indep (C \ {e}) := by
   wlog hCE : C ⊆ M.E
@@ -155,8 +177,7 @@ lemma Circuit.mem_closure_diff_singleton_of_mem (hC : M.Circuit C) (heC : e ∈
 
 /-! ### Restriction -/
 
-lemma Circuit.circuit_restrict_of_subset (hC : M.Circuit C) (hCR : C ⊆ R) :
-    (M ↾ R).Circuit C := by
+lemma Circuit.circuit_restrict_of_subset (hC : M.Circuit C) (hCR : C ⊆ R) : (M ↾ R).Circuit C := by
   simp_rw [circuit_iff, restrict_dep_iff, dep_iff, and_imp] at *
   exact ⟨⟨hC.1.1, hCR⟩, fun I hI _ hIC ↦ hC.2 hI (hIC.trans hC.1.2) hIC⟩
 
@@ -166,4 +187,167 @@ lemma restrict_circuit_iff (hR : R ⊆ M.E := by aesop_mat) :
   simp_rw [circuit_iff, restrict_dep_iff, and_imp, dep_iff]
   exact fun hC hCR h ↦ ⟨⟨⟨hC,hCR.trans hR⟩,fun I hI hIC ↦ h hI.1 (hIC.trans hCR) hIC⟩,hCR⟩
 
+/-! ### Fundamental Circuits -/
+
+/-- For an independent set `I` and some `e ∈ M.closure I \ I`,
+`M.fundCircuit e I` is the unique circuit contained in `insert e I`.
+For the fact that this is a circuit, see `Matroid.Indep.fundCircuit_circuit`,
+and the fact that it is unique, see `Matroid.Circuit.eq_fundCircuit_of_subset`.
+Has the junk value `{e}` if `e ∈ I` or `e ∉ M.E`, and `insert e I` if `e ∈ M.E \ M.closure I`. -/
+def fundCircuit (M : Matroid α) (e : α) (I : Set α) : Set α :=
+  insert e (I ∩ ⋂₀ {J | J ⊆ I ∧ M.closure {e} ⊆ M.closure J})
+
+lemma fundCircuit_eq_sInter (he : e ∈ M.closure I) :
+    M.fundCircuit e I = insert e (⋂₀ {J | J ⊆ I ∧ e ∈ M.closure J}) := by
+  rw [fundCircuit]
+  simp_rw [closure_subset_closure_iff_subset_closure
+    (show {e} ⊆ M.E by simpa using mem_ground_of_mem_closure he), singleton_subset_iff]
+  rw [inter_eq_self_of_subset_right (sInter_subset_of_mem (by simpa))]
+
+lemma fundCircuit_subset_insert (M : Matroid α) (e : α) (I : Set α) :
+    M.fundCircuit e I ⊆ insert e I :=
+  insert_subset_insert inter_subset_left
+
+lemma fundCircuit_subset_ground (he : e ∈ M.E) (hI : I ⊆ M.E := by aesop_mat) :
+    M.fundCircuit e I ⊆ M.E :=
+  (M.fundCircuit_subset_insert e I).trans (insert_subset he hI)
+
+lemma mem_fundCircuit (M : Matroid α) (e : α) (I : Set α) : e ∈ fundCircuit M e I :=
+  mem_insert ..
+
+lemma fundCircuit_diff_eq_inter (M : Matroid α) (heI : e ∉ I) :
+    (M.fundCircuit e I) \ {e} = (M.fundCircuit e I) ∩ I :=
+  (subset_inter diff_subset (by simp [fundCircuit_subset_insert])).antisymm
+    (subset_diff_singleton inter_subset_left (by simp [heI]))
+
+/-- The fundamental circuit of `e` and `X` has the junk value `{e}` if `e ∈ X` -/
+lemma fundCircuit_eq_of_mem (heX : e ∈ X) : M.fundCircuit e X = {e} := by
+  suffices h : ∀ a ∈ X, (∀ t ⊆ X, M.closure {e} ⊆ M.closure t → a ∈ t) → a = e by
+    simpa [subset_antisymm_iff, fundCircuit]
+  exact fun b hbX h ↦ h _ (singleton_subset_iff.2 heX) Subset.rfl
+
+lemma fundCircuit_eq_of_not_mem_ground (heX : e ∉ M.E) : M.fundCircuit e X = {e} := by
+  suffices h : ∀ a ∈ X, (∀ t ⊆ X, M.closure {e} ⊆ M.closure t → a ∈ t) → a = e by
+    simpa [subset_antisymm_iff, fundCircuit]
+  simp_rw [← M.closure_inter_ground {e}, singleton_inter_eq_empty.2 heX]
+  exact fun a haX h ↦ by simpa using h ∅ (empty_subset X) rfl.subset
+
+lemma Indep.fundCircuit_circuit (hI : M.Indep I) (hecl : e ∈ M.closure I) (heI : e ∉ I) :
+    M.Circuit (M.fundCircuit e I) := by
+  have aux : ⋂₀ {J | J ⊆ I ∧ e ∈ M.closure J} ⊆ I := sInter_subset_of_mem (by simpa)
+  rw [fundCircuit_eq_sInter hecl]
+  refine (hI.subset aux).insert_circuit_of_forall ?_ ?_ ?_
+  · simp [show ∃ x ⊆ I, e ∈ M.closure x ∧ e ∉ x from ⟨I, by simp [hecl, heI]⟩]
+  · rw [hI.closure_sInter_eq_biInter_closure_of_forall_subset ⟨I, by simpa⟩ (by simp +contextual)]
+    simp
+  simp only [mem_sInter, mem_setOf_eq, and_imp]
+  exact fun f hf hecl ↦ (hf _ (diff_subset.trans aux) hecl).2 rfl
+
+lemma Indep.mem_fundCircuit_iff (hI : M.Indep I) (hecl : e ∈ M.closure I) (heI : e ∉ I) :
+    x ∈ M.fundCircuit e I ↔ M.Indep (insert e I \ {x}) := by
+  obtain rfl | hne := eq_or_ne x e
+  · simp [hI.diff, mem_fundCircuit]
+  suffices (∀ t ⊆ I, e ∈ M.closure t → x ∈ t) ↔ e ∉ M.closure (I \ {x}) by
+    simpa [fundCircuit_eq_sInter hecl, hne, ← insert_diff_singleton_comm hne.symm,
+      (hI.diff _).insert_indep_iff, mem_ground_of_mem_closure hecl, heI]
+  refine ⟨fun h hecl ↦ (h _ diff_subset hecl).2 rfl, fun h J hJ heJ ↦ by_contra fun hxJ ↦ h ?_⟩
+  exact M.closure_subset_closure (subset_diff_singleton hJ hxJ) heJ
+
+lemma Base.fundCircuit_circuit {B : Set α} (hB : M.Base B) (hxE : x ∈ M.E) (hxB : x ∉ B) :
+    M.Circuit (M.fundCircuit x B) :=
+  hB.indep.fundCircuit_circuit (by rwa [hB.closure_eq]) hxB
+
+/-- For `I` independent, `M.fundCircuit e I` is the only circuit contained in `insert e I`. -/
+lemma Circuit.eq_fundCircuit_of_subset (hC : M.Circuit C) (hI : M.Indep I) (hCss : C ⊆ insert e I) :
+    C = M.fundCircuit e I := by
+  obtain hCI | ⟨heC, hCeI⟩ := subset_insert_iff.1 hCss
+  · exact (hC.not_indep (hI.subset hCI)).elim
+  suffices hss : M.fundCircuit e I ⊆ C by
+    refine hC.eq_of_superset_circuit (hI.fundCircuit_circuit ?_ fun heI ↦ ?_) hss
+    · rw [hI.mem_closure_iff]
+      exact .inl (hC.dep.superset hCss (insert_subset (hC.subset_ground heC) hI.subset_ground))
+    exact hC.not_indep (hI.subset (hCss.trans (by simp [heI])))
+  rw [fundCircuit_eq_sInter <|
+    M.closure_subset_closure hCeI <| hC.mem_closure_diff_singleton_of_mem heC]
+  refine insert_subset heC <| (sInter_subset_of_mem (t := C \ {e}) ?_).trans diff_subset
+  simp [hCss, hC.mem_closure_diff_singleton_of_mem heC]
+
+lemma fundCircuit_restrict {R : Set α} (hIR : I ⊆ R) (heR : e ∈ R) (hR : R ⊆ M.E) :
+    (M ↾ R).fundCircuit e I = M.fundCircuit e I := by
+  simp_rw [fundCircuit, M.restrict_closure_eq (R := R) (X := {e}) (by simpa)]
+  refine subset_antisymm (insert_subset_insert (inter_subset_inter_right _ ?_))
+    (insert_subset_insert (inter_subset_inter_right _ ?_))
+  · refine subset_sInter fun J ⟨hJI, heJ⟩ ↦ sInter_subset_of_mem ⟨hJI, ?_⟩
+    simp only [restrict_closure_eq', union_subset_iff, subset_union_right, and_true]
+    refine (inter_subset_inter_left _ ?_).trans subset_union_left
+    rwa [inter_eq_self_of_subset_left (hJI.trans hIR)]
+  refine subset_sInter fun J ⟨hJI, heJ⟩ ↦ sInter_subset_of_mem
+    ⟨hJI, M.closure_subset_closure_of_subset_closure ?_⟩
+  rw [restrict_closure_eq _ (hJI.trans hIR) hR] at heJ
+  simp only [subset_inter_iff, inter_subset_right, and_true] at heJ
+  exact subset_trans (by simpa [M.mem_closure_of_mem' (mem_singleton e) (hR heR)]) heJ
+
+@[simp] lemma fundCircuit_restrict_univ (M : Matroid α) :
+    (M ↾ univ).fundCircuit e I = M.fundCircuit e I := by
+  have aux (A B) : M.closure A ⊆ B ∪ univ \ M.E ↔ M.closure A ⊆ B := by
+    refine ⟨fun h ↦ ?_, fun h ↦ h.trans subset_union_left⟩
+    refine (subset_inter h (M.closure_subset_ground A)).trans ?_
+    simp [union_inter_distrib_right]
+  simp [fundCircuit, aux]
+
+/-! ### Dependence -/
+
+lemma Dep.exists_circuit_subset (hX : M.Dep X) : ∃ C, C ⊆ X ∧ M.Circuit C := by
+  obtain ⟨I, hI⟩ := M.exists_basis X
+  obtain ⟨e, heX, heI⟩ := exists_of_ssubset
+    (hI.subset.ssubset_of_ne (by rintro rfl; exact hI.indep.not_dep hX))
+  exact ⟨M.fundCircuit e I, (M.fundCircuit_subset_insert e I).trans (insert_subset heX hI.subset),
+    hI.indep.fundCircuit_circuit (hI.subset_closure heX) heI⟩
+
+lemma dep_iff_superset_circuit (hX : X ⊆ M.E := by aesop_mat) :
+    M.Dep X ↔ ∃ C, C ⊆ X ∧ M.Circuit C :=
+  ⟨Dep.exists_circuit_subset, fun ⟨C, hCX, hC⟩ ↦ hC.dep.superset hCX⟩
+
+/-- A version of `Matroid.dep_iff_superset_circuit` that has the supportedness hypothesis
+as part of the equivalence, rather than a hypothesis. -/
+lemma dep_iff_superset_circuit' : M.Dep X ↔ (∃ C, C ⊆ X ∧ M.Circuit C) ∧ X ⊆ M.E :=
+  ⟨fun h ↦ ⟨h.exists_circuit_subset, h.subset_ground⟩, fun ⟨⟨C, hCX, hC⟩, h⟩ ↦ hC.dep.superset hCX⟩
+
+/-- A version of `Matroid.indep_iff_forall_subset_not_circuit` that has the supportedness hypothesis
+as part of the equivalence, rather than a hypothesis. -/
+lemma indep_iff_forall_subset_not_circuit' :
+    M.Indep I ↔ (∀ C, C ⊆ I → ¬M.Circuit C) ∧ I ⊆ M.E := by
+  simp_rw [indep_iff_not_dep, dep_iff_superset_circuit']
+  aesop
+
+lemma indep_iff_forall_subset_not_circuit (hI : I ⊆ M.E := by aesop_mat) :
+    M.Indep I ↔ ∀ C, C ⊆ I → ¬M.Circuit C := by
+  rw [indep_iff_forall_subset_not_circuit', and_iff_left hI]
+
+/-! ### Extensionality -/
+
+lemma ext_circuit {M₁ M₂ : Matroid α} (hE : M₁.E = M₂.E)
+    (h : ∀ ⦃C⦄, C ⊆ M₁.E → (M₁.Circuit C ↔ M₂.Circuit C)) : M₁ = M₂ := by
+  have h' {C} : M₁.Circuit C ↔ M₂.Circuit C :=
+    (em (C ⊆ M₁.E)).elim (h (C := C)) (fun hC ↦ iff_of_false (mt Circuit.subset_ground hC)
+      (mt Circuit.subset_ground fun hss ↦ hC (hss.trans_eq hE.symm)))
+  refine ext_indep hE fun I hI ↦ ?_
+  simp_rw [indep_iff_forall_subset_not_circuit hI, h',
+    indep_iff_forall_subset_not_circuit (hI.trans_eq hE)]
+
+/-- A stronger version of `Matroid.ext_circuit`:
+two matroids on the same ground set are equal if no circuit of one is independent in the other. -/
+lemma ext_circuit_not_indep {M₁ M₂ : Matroid α} (hE : M₁.E = M₂.E)
+    (h₁ : ∀ C, M₁.Circuit C → ¬ M₂.Indep C) (h₂ : ∀ C, M₂.Circuit C → ¬ M₁.Indep C) :
+    M₁ = M₂ := by
+  refine ext_circuit hE fun C hCE ↦ ⟨fun hC ↦ ?_, fun hC ↦ ?_⟩
+  · obtain ⟨C', hC'C, hC'⟩ := ((not_indep_iff (by rwa [← hE])).1 (h₁ C hC)).exists_circuit_subset
+    rwa [← hC.eq_of_not_indep_subset (h₂ C' hC') hC'C]
+  obtain ⟨C', hC'C, hC'⟩ := ((not_indep_iff hCE).1 (h₂ C hC)).exists_circuit_subset
+  rwa [← hC.eq_of_not_indep_subset (h₁ C' hC') hC'C]
+
+lemma ext_iff_circuit {M₁ M₂ : Matroid α} :
+    M₁ = M₂ ↔ M₁.E = M₂.E ∧ ∀ C, M₁.Circuit C ↔ M₂.Circuit C :=
+  ⟨fun h ↦ by simp [h], fun h ↦ ext_circuit h.1 fun C hC ↦ h.2 (C := C)⟩
+
 end Matroid
diff --git a/Mathlib/Data/Matroid/Rank/Cardinal.lean b/Mathlib/Data/Matroid/Rank/Cardinal.lean
index b12a9f56efadbb..b08d04f6201229 100644
--- a/Mathlib/Data/Matroid/Rank/Cardinal.lean
+++ b/Mathlib/Data/Matroid/Rank/Cardinal.lean
@@ -3,46 +3,294 @@ Copyright (c) 2025 Peter Nelson and Junyan Xu. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Peter Nelson, Junyan Xu
 -/
-import Mathlib.Data.Matroid.Restrict
+import Mathlib.Data.Matroid.Closure
+import Mathlib.Data.Matroid.Map
 import Mathlib.SetTheory.Cardinal.Arithmetic
 
 /-!
-# Invariance of cardinality of bases of a finitary matroid
+# Cardinal-valued rank
 
 In a finitary matroid, all bases have the same cardinality.
-In fact, something stronger holds: if `B` and `B'` are bases, then `#(B \ B') = #(B' \ B)`.
-This file provides proofs of these facts,
-as well as variants when each of `B` and `B'` is a `Basis` or `Basis'` of some common set `X`.
+In fact, something stronger holds: if `I` and `J` are both bases for a set `X`,
+then `#(I \ J) = #(J \ I)` and (consequently) `#I = #J`.
+This file introduces a typeclass `InvariantCardinalRank` that applies to any matroid
+such that this property holds for all `I`, `J` and `X`.
 
-Some extra assumption like `Finitary` is necessary for these to be true,
+A matroid satisfying this condition has a well-defined cardinality-valued rank function,
+both for itself and all its minors.
+
+# Main Declarations
+
+* `Matroid.InvariantCardinalRank` : a typeclass capturing the idea that a matroid and all its minors
+  have a well-behaved cardinal-valued rank function.
+* `Matroid.cRank M` is the supremum of the cardinalities of the bases of matroid `M`.
+* `Matroid.cRk M X` is the supremum of the cardinalities of the bases of a set `X` in a matroid `M`.
+* `invariantCardinalRank_of_finitary` is the instance
+   showing that `Finitary` matroids are `InvariantCardinalRank`.
+* `cRk_inter_add_cRk_union_le` states that cardinal rank is submodular.
+
+# Notes
+
+It is not the case that all matroids are `InvariantCardinalRank`,
 since the equicardinality of bases in general matroids is independent of ZFC
-(see the docstring of `Data.Matroid.Basic`).
-Lemmas like `Matroid.Base.cardinalMk_diff_comm_of_finitary`
-become true for all matroids if they are weakened by replacing `Cardinal.mk`
-with the cruder `ℕ∞`-valued `encard`; see, for example, `Matroid.Base.encard_diff_comm`.
+(see the module docstring of `Mathlib.Data.Matroid.Basic`).
+Lemmas like `Matroid.Base.cardinalMk_diff_comm` become true for all matroids
+only if they are weakened by replacing `Cardinal.mk`
+with the cruder `ℕ∞`-valued `Set.encard`; see, for example, `Matroid.Base.encard_diff_comm`.
+
+# Implementation Details
+
+Since the functions `cRank` and `cRk` are defined as suprema,
+independently of the `Matroid.InvariantCardinalRank` typeclass,
+they are well-defined for all matroids.
+However, for matroids that do not satisfy `InvariantCardinalRank`, they are badly behaved.
+For instance, in general `cRk` is not submodular,
+and its value may differ on a set `X` and the closure of `X`.
+We state and prove theorems without `InvariantCardinalRank` whenever possible,
+which sometime makes their proofs longer than they would be with the instance.
 
-TODO
+# TODO
+
+* Higgs' theorem : if the generalized continuum hypothesis holds,
+  then every matroid is `InvariantCardinalRank`.
 
-* Higg's theorem that, if the generalized continuum hypothesis holds,
-  all bases of any matroid are equicardinal.
-* API for a `Cardinal`-valued rank function.
 -/
 
-variable {α : Type*} {M : Matroid α} {I J B B' X : Set α} [M.Finitary]
+universe u v
+
+variable {α : Type u} {β : Type v} {f : α → β} {M : Matroid α} {I J B B' X Y : Set α}
 
 open Cardinal Set
 
 namespace Matroid
 
-theorem Base.cardinalMk_diff_comm_of_finitary (hB : M.Base B) (hB' : M.Base B') :
-    #(B \ B' : Set α) = #(B' \ B : Set α) := by
-  wlog hge : #(B' \ B : Set α) ≤ #(B \ B' : Set α) with aux
-  · exact (aux hB' hB (not_le.1 hge).le).symm
+section Rank
+
+variable {κ : Cardinal}
+
+/-- The rank (supremum of the cardinalities of bases) of a matroid `M` as a `Cardinal`. -/
+noncomputable def cRank (M : Matroid α) := ⨆ (B : {B // M.Base B}), #B
+
+/-- The rank (supremum of the cardinalities of bases) of a set `X` in a matroid `M`,
+as a `Cardinal`. -/
+noncomputable def cRk (M : Matroid α) (X : Set α) := (M ↾ X).cRank
+
+theorem Base.cardinalMk_le_cRank (hB : M.Base B) : #B ≤ M.cRank :=
+  le_ciSup (f := fun (B : {B // M.Base B}) ↦ #(B.1)) (bddAbove_range _) ⟨B, hB⟩
+
+theorem Basis'.cardinalMk_le_cRk (hIX : M.Basis' I X) : #I ≤ M.cRk X :=
+  (base_restrict_iff'.2 hIX).cardinalMk_le_cRank
+
+theorem Basis.cardinalMk_le_cRk (hIX : M.Basis I X) : #I ≤ M.cRk X :=
+  hIX.basis'.cardinalMk_le_cRk
+
+theorem cRank_le_iff : M.cRank ≤ κ ↔ ∀ ⦃B⦄, M.Base B → #B ≤ κ :=
+  ⟨fun h _ hB ↦ (hB.cardinalMk_le_cRank.trans h), fun h ↦ ciSup_le fun ⟨_, hB⟩ ↦ h hB⟩
+
+theorem cRk_le_iff : M.cRk X ≤ κ ↔ ∀ ⦃I⦄, M.Basis' I X → #I ≤ κ := by
+  simp_rw [cRk, cRank_le_iff, base_restrict_iff']
+
+theorem Indep.cardinalMk_le_cRk_of_subset (hI : M.Indep I) (hIX : I ⊆ X) : #I ≤ M.cRk X :=
+  let ⟨_, hJ, hIJ⟩ := hI.subset_basis'_of_subset hIX
+  (mk_le_mk_of_subset hIJ).trans hJ.cardinalMk_le_cRk
+
+theorem cRk_le_cardinalMk (M : Matroid α) (X : Set α) : M.cRk X ≤ #X :=
+  ciSup_le fun ⟨_, hI⟩ ↦ mk_le_mk_of_subset hI.subset_ground
+
+@[simp] theorem cRk_ground (M : Matroid α) : M.cRk M.E = M.cRank := by
+  rw [cRk, restrict_ground_eq_self]
+
+@[simp] theorem cRank_restrict (M : Matroid α) (X : Set α) : (M ↾ X).cRank = M.cRk X := rfl
+
+theorem cRk_mono (M : Matroid α) : Monotone M.cRk := by
+  simp only [Monotone, le_eq_subset, cRk_le_iff]
+  intro X Y hXY I hIX
+  obtain ⟨J, hJ, hIJ⟩ := hIX.indep.subset_basis'_of_subset (hIX.subset.trans hXY)
+  exact (mk_le_mk_of_subset hIJ).trans hJ.cardinalMk_le_cRk
+
+theorem cRk_le_of_subset (M : Matroid α) (hXY : X ⊆ Y) : M.cRk X ≤ M.cRk Y :=
+  M.cRk_mono hXY
+
+@[simp] theorem cRk_inter_ground (M : Matroid α) (X : Set α) : M.cRk (X ∩ M.E) = M.cRk X :=
+  (M.cRk_le_of_subset inter_subset_left).antisymm <| cRk_le_iff.2
+    fun _ h ↦ h.basis_inter_ground.cardinalMk_le_cRk
+
+theorem cRk_restrict_subset (M : Matroid α) (hYX : Y ⊆ X) : (M ↾ X).cRk Y = M.cRk Y := by
+  have aux : ∀ ⦃I⦄, M.Basis' I Y ↔ (M ↾ X).Basis' I Y := by
+    simp_rw [basis'_restrict_iff, inter_eq_self_of_subset_left hYX, iff_self_and]
+    exact fun I h ↦ h.subset.trans hYX
+  simp_rw [le_antisymm_iff, cRk_le_iff]
+  exact ⟨fun I hI ↦ (aux.2 hI).cardinalMk_le_cRk, fun I hI ↦ (aux.1 hI).cardinalMk_le_cRk⟩
+
+theorem cRk_restrict (M : Matroid α) (X Y : Set α) : (M ↾ X).cRk Y = M.cRk (X ∩ Y) := by
+  rw [← cRk_inter_ground, restrict_ground_eq, cRk_restrict_subset _ inter_subset_right,
+    inter_comm]
+
+theorem Indep.cRk_eq_cardinalMk (hI : M.Indep I) : #I = M.cRk I :=
+  (M.cRk_le_cardinalMk I).antisymm' (hI.basis_self.cardinalMk_le_cRk)
+
+@[simp] theorem cRk_map_image_lift (M : Matroid α) (hf : InjOn f M.E) (X : Set α)
+    (hX : X ⊆ M.E := by aesop_mat) : lift.{u,v} ((M.map f hf).cRk (f '' X)) = lift (M.cRk X) := by
+  nth_rw 1 [cRk, cRank, le_antisymm_iff, lift_iSup (bddAbove_range _), cRk, cRank, cRk, cRank]
+  nth_rw 2 [lift_iSup (bddAbove_range _)]
+  simp only [ciSup_le_iff (bddAbove_range _), ge_iff_le, Subtype.forall, base_restrict_iff',
+    basis'_iff_basis hX, basis'_iff_basis (show f '' X ⊆ (M.map f hf).E from image_mono hX)]
+  refine ⟨fun I hI ↦ ?_, fun I hI ↦ ?_⟩
+  · obtain ⟨I, X', hIX, rfl, hXX'⟩ := map_basis_iff'.1 hI
+    rw [mk_image_eq_of_injOn_lift _ _ (hf.mono hIX.indep.subset_ground), lift_le]
+    obtain rfl : X = X' := by rwa [hf.image_eq_image_iff hX hIX.subset_ground] at hXX'
+    exact hIX.cardinalMk_le_cRk
+  rw [← mk_image_eq_of_injOn_lift _ _ (hf.mono hI.indep.subset_ground), lift_le]
+  exact (hI.map hf).cardinalMk_le_cRk
+
+@[simp] theorem cRk_map_image {β : Type u} {f : α → β} (M : Matroid α) (hf : InjOn f M.E)
+    (X : Set α) (hX : X ⊆ M.E := by aesop_mat) : (M.map f hf).cRk (f '' X) = M.cRk X :=
+  lift_inj.1 <| M.cRk_map_image_lift ..
+
+theorem cRk_map_eq {β : Type u} {f : α → β} {X : Set β} (M : Matroid α) (hf : InjOn f M.E) :
+    (M.map f hf).cRk X = M.cRk (f ⁻¹' X) := by
+  rw [← M.cRk_inter_ground, ← M.cRk_map_image hf _, image_preimage_inter, ← map_ground _ _ hf,
+    cRk_inter_ground]
+
+@[simp] theorem cRk_comap_lift (M : Matroid β) (f : α → β) (X : Set α) :
+    lift.{v,u} ((M.comap f).cRk X) = lift (M.cRk (f '' X)) := by
+  nth_rw 1 [cRk, cRank, le_antisymm_iff, lift_iSup (bddAbove_range _), cRk, cRank, cRk, cRank]
+  nth_rw 2 [lift_iSup (bddAbove_range _)]
+  simp only [ciSup_le_iff (bddAbove_range _), ge_iff_le, Subtype.forall, base_restrict_iff',
+    comap_basis'_iff, and_imp]
+  refine ⟨fun I hI hfI hIX ↦ ?_, fun I hIX ↦ ?_⟩
+  · rw [← mk_image_eq_of_injOn_lift _ _ hfI, lift_le]
+    exact hI.cardinalMk_le_cRk
+  obtain ⟨I₀, hI₀X, rfl, hfI₀⟩ := show ∃ I₀ ⊆ X, f '' I₀ = I ∧ InjOn f I₀ by
+    obtain ⟨I₀, hI₀ss, hbij⟩ := exists_subset_bijOn (f ⁻¹' I ∩ X) f
+    refine ⟨I₀, hI₀ss.trans inter_subset_right, ?_, hbij.injOn⟩
+    rw [hbij.image_eq, image_preimage_inter, inter_eq_self_of_subset_left hIX.subset]
+  rw [mk_image_eq_of_injOn_lift _ _ hfI₀, lift_le]
+  exact Basis'.cardinalMk_le_cRk <| comap_basis'_iff.2 ⟨hIX, hfI₀, hI₀X⟩
+
+@[simp] theorem cRk_comap {β : Type u} (M : Matroid β) (f : α → β) (X : Set α) :
+    (M.comap f).cRk X = M.cRk (f '' X) :=
+  lift_inj.1 <| M.cRk_comap_lift ..
+
+end Rank
+
+section Invariant
+
+/-- A class stating that cardinality-valued rank is well-defined
+(i.e. all bases are equicardinal) for a matroid `M` and its minors.
+Notably, this holds for `Finitary` matroids; see `Matroid.invariantCardinalRank_of_finitary`.  -/
+@[mk_iff]
+class InvariantCardinalRank (M : Matroid α) : Prop where
+  forall_card_basis_diff :
+    ∀ ⦃I J X⦄, M.Basis I X → M.Basis J X → #(I \ J : Set α) = #(J \ I : Set α)
+
+variable [InvariantCardinalRank M]
+
+theorem Basis.cardinalMk_diff_comm (hIX : M.Basis I X) (hJX : M.Basis J X) :
+    #(I \ J : Set α) = #(J \ I : Set α) :=
+  InvariantCardinalRank.forall_card_basis_diff hIX hJX
+
+theorem Basis'.cardinalMk_diff_comm (hIX : M.Basis' I X) (hJX : M.Basis' J X) :
+    #(I \ J : Set α) = #(J \ I : Set α) :=
+  hIX.basis_inter_ground.cardinalMk_diff_comm hJX.basis_inter_ground
+
+theorem Base.cardinalMk_diff_comm (hB : M.Base B) (hB' : M.Base B') :
+    #(B \ B' : Set α) = #(B' \ B : Set α) :=
+  hB.basis_ground.cardinalMk_diff_comm hB'.basis_ground
+
+theorem Basis.cardinalMk_eq (hIX : M.Basis I X) (hJX : M.Basis J X) : #I = #J := by
+  rw [← diff_union_inter I J,
+    mk_union_of_disjoint (disjoint_sdiff_left.mono_right inter_subset_right),
+    hIX.cardinalMk_diff_comm hJX,
+    ← mk_union_of_disjoint (disjoint_sdiff_left.mono_right inter_subset_left),
+    inter_comm, diff_union_inter]
+
+theorem Basis'.cardinalMk_eq (hIX : M.Basis' I X) (hJX : M.Basis' J X) : #I = #J :=
+  hIX.basis_inter_ground.cardinalMk_eq hJX.basis_inter_ground
+
+theorem Base.cardinalMk_eq (hB : M.Base B) (hB' : M.Base B') : #B = #B' :=
+  hB.basis_ground.cardinalMk_eq hB'.basis_ground
+
+theorem Indep.cardinalMk_le_base (hI : M.Indep I) (hB : M.Base B) : #I ≤ #B :=
+  have ⟨_B', hB', hIB'⟩ := hI.exists_base_superset
+  hB'.cardinalMk_eq hB ▸ mk_le_mk_of_subset hIB'
+
+theorem Indep.cardinalMk_le_basis' (hI : M.Indep I) (hJ : M.Basis' J X) (hIX : I ⊆ X) :
+    #I ≤ #J :=
+  have ⟨_J', hJ', hIJ'⟩ := hI.subset_basis'_of_subset hIX
+  hJ'.cardinalMk_eq hJ ▸ mk_le_mk_of_subset hIJ'
+
+theorem Indep.cardinalMk_le_basis (hI : M.Indep I) (hJ : M.Basis J X) (hIX : I ⊆ X) :
+    #I ≤ #J :=
+  hI.cardinalMk_le_basis' hJ.basis' hIX
+
+theorem Base.cardinalMk_eq_cRank (hB : M.Base B) : #B = M.cRank := by
+  have hrw : ∀ B' : {B : Set α // M.Base B}, #B' = #B := fun B' ↦ B'.2.cardinalMk_eq hB
+  simp [cRank, hrw]
+
+/-- Restrictions of matroids with cardinal rank functions have cardinal rank functions- -/
+instance invariantCardinalRank_restrict [InvariantCardinalRank M] :
+    InvariantCardinalRank (M ↾ X) := by
+  refine ⟨fun I J Y hI hJ ↦ ?_⟩
+  rw [basis_restrict_iff'] at hI hJ
+  exact hI.1.cardinalMk_diff_comm hJ.1
+
+theorem Basis'.cardinalMk_eq_cRk (hIX : M.Basis' I X) : #I = M.cRk X := by
+  rw [cRk, (base_restrict_iff'.2 hIX).cardinalMk_eq_cRank]
+
+theorem Basis.cardinalMk_eq_cRk (hIX : M.Basis I X) : #I = M.cRk X :=
+  hIX.basis'.cardinalMk_eq_cRk
+
+@[simp] theorem cRk_closure (M : Matroid α) [InvariantCardinalRank M] (X : Set α) :
+    M.cRk (M.closure X) = M.cRk X := by
+  obtain ⟨I, hI⟩ := M.exists_basis' X
+  rw [← hI.basis_closure_right.cardinalMk_eq_cRk, ← hI.cardinalMk_eq_cRk]
+
+theorem cRk_closure_congr (hXY : M.closure X = M.closure Y) : M.cRk X = M.cRk Y := by
+  rw [← cRk_closure, hXY, cRk_closure]
+
+variable (M : Matroid α) [InvariantCardinalRank M] (e : α) (X Y : Set α)
+
+@[simp] theorem cRk_union_closure_right_eq : M.cRk (X ∪ M.closure Y) = M.cRk (X ∪ Y) :=
+  M.cRk_closure_congr (M.closure_union_closure_right_eq _ _)
+
+@[simp] theorem cRk_union_closure_left_eq : M.cRk (M.closure X ∪ Y) = M.cRk (X ∪ Y) :=
+  M.cRk_closure_congr (M.closure_union_closure_left_eq _ _)
+
+@[simp] theorem cRk_insert_closure_eq : M.cRk (insert e (M.closure X)) = M.cRk (insert e X) := by
+  rw [← union_singleton, cRk_union_closure_left_eq, union_singleton]
+
+theorem cRk_union_closure_eq : M.cRk (M.closure X ∪ M.closure Y) = M.cRk (X ∪ Y) := by
+  simp
+
+/-- The `Cardinal` rank function is submodular. -/
+theorem cRk_inter_add_cRk_union_le : M.cRk (X ∩ Y) + M.cRk (X ∪ Y) ≤ M.cRk X + M.cRk Y := by
+  obtain ⟨Ii, hIi⟩ := M.exists_basis' (X ∩ Y)
+  obtain ⟨IX, hIX, hIX'⟩ :=
+    hIi.indep.subset_basis'_of_subset (hIi.subset.trans inter_subset_left)
+  obtain ⟨IY, hIY, hIY'⟩ :=
+    hIi.indep.subset_basis'_of_subset (hIi.subset.trans inter_subset_right)
+  rw [← cRk_union_closure_eq, ← hIX.closure_eq_closure, ← hIY.closure_eq_closure,
+    cRk_union_closure_eq, ← hIi.cardinalMk_eq_cRk, ← hIX.cardinalMk_eq_cRk,
+    ← hIY.cardinalMk_eq_cRk, ← mk_union_add_mk_inter, add_comm]
+  exact add_le_add (M.cRk_le_cardinalMk _) (mk_le_mk_of_subset (subset_inter hIX' hIY'))
+
+end Invariant
+
+section Instances
+
+/-- `Finitary` matroids have a cardinality-valued rank function. -/
+instance invariantCardinalRank_of_finitary [Finitary M] : InvariantCardinalRank M := by
+  suffices aux : ∀ ⦃B B'⦄ ⦃N : Matroid α⦄, Finitary N → N.Base B → N.Base B' →
+      #(B \ B' : Set α) ≤ #(B' \ B : Set α) from
+    ⟨fun I J X hI hJ ↦ (aux (restrict_finitary X) hI.base_restrict hJ.base_restrict).antisymm
+      (aux (restrict_finitary X) hJ.base_restrict hI.base_restrict)⟩
+  intro B B' N hfin hB hB'
   by_cases h : (B' \ B).Finite
   · rw [← cast_ncard h, ← cast_ncard, hB.ncard_diff_comm hB']
-    exact (diff_finite_comm hB' hB).mp h
+    exact (hB'.diff_finite_comm hB).mp h
   rw [← Set.Infinite, ← infinite_coe_iff] at h
-  have (a : α) (ha : a ∈ B' \ B) : ∃ S : Set α, Finite S ∧ S ⊆ B ∧ ¬ M.Indep (insert a S) := by
+  have (a : α) (ha : a ∈ B' \ B) : ∃ S : Set α, Finite S ∧ S ⊆ B ∧ ¬ N.Indep (insert a S) := by
     have := (hB.insert_dep ⟨hB'.subset_ground ha.1, ha.2⟩).1
     contrapose! this
     exact Finitary.indep_of_forall_finite _ fun J hJ fin ↦ (this (J \ {a}) fin.diff.to_subtype <|
@@ -50,7 +298,7 @@ theorem Base.cardinalMk_diff_comm_of_finitary (hB : M.Base B) (hB' : M.Base B')
   choose S S_fin hSB dep using this
   let U := ⋃ a : ↥(B' \ B), S a a.2
   suffices B \ B' ⊆ U by
-    refine hge.antisymm' <| (mk_le_mk_of_subset this).trans <| (mk_iUnion_le ..).trans
+    refine (mk_le_mk_of_subset this).trans <| (mk_iUnion_le ..).trans
       <| (mul_le_max_of_aleph0_le_left (by simp)).trans ?_
     simp only [sup_le_iff, le_refl, true_and]
     exact ciSup_le' fun e ↦ (lt_aleph0_of_finite _).le.trans <| by simp
@@ -63,40 +311,35 @@ theorem Base.cardinalMk_diff_comm_of_finitary (hB : M.Base B) (hB' : M.Base B')
   refine dep a this (ind.subset <| insert_subset_insert <| .trans ?_ subset_union_right)
   exact subset_iUnion_of_subset ⟨a, this⟩ subset_rfl
 
-theorem Base.cardinalMk_eq_of_finitary (hB : M.Base B) (hB' : M.Base B') : #B = #B' := by
-  rw [← diff_union_inter B B',
-    mk_union_of_disjoint (disjoint_sdiff_left.mono_right inter_subset_right),
-    hB.cardinalMk_diff_comm_of_finitary hB',
-    ← mk_union_of_disjoint (disjoint_sdiff_left.mono_right inter_subset_left),
-    inter_comm, diff_union_inter]
-
-theorem Basis'.cardinalMk_diff_comm_of_finitary (hIX : M.Basis' I X) (hJX : M.Basis' J X) :
-    #(I \ J : Set α) = #(J \ I : Set α) := by
-  rw [← base_restrict_iff'] at hIX hJX
-  rw [hIX.cardinalMk_diff_comm_of_finitary hJX]
-
-theorem Basis.cardinalMk_diff_comm_of_finitary (hIX : M.Basis I X) (hJX : M.Basis J X) :
-    #(I \ J : Set α) = #(J \ I : Set α) :=
-  hIX.basis'.cardinalMk_diff_comm_of_finitary hJX.basis'
-
-theorem Basis'.cardinalMk_eq_of_finitary (hIX : M.Basis' I X) (hJX : M.Basis' J X) : #I = #J := by
-  rw [← base_restrict_iff'] at hIX hJX
-  rw [hIX.cardinalMk_eq_of_finitary hJX]
-
-theorem Basis.cardinalMk_eq_of_finitary (hIX : M.Basis I X) (hJX : M.Basis J X) : #I = #J :=
-  hIX.basis'.cardinalMk_eq_of_finitary hJX.basis'
+instance invariantCardinalRank_map (M : Matroid α) [InvariantCardinalRank M] (hf : InjOn f M.E) :
+    InvariantCardinalRank (M.map f hf) := by
+  refine ⟨fun I J X hI hJ ↦ ?_⟩
+  obtain ⟨I, X, hIX, rfl, rfl⟩ := map_basis_iff'.1 hI
+  obtain ⟨J, X', hJX, rfl, h'⟩ := map_basis_iff'.1 hJ
+  obtain rfl : X = X' := by
+    rwa [InjOn.image_eq_image_iff hf hIX.subset_ground hJX.subset_ground] at h'
+  have hcard := hIX.cardinalMk_diff_comm hJX
+  rwa [← lift_inj.{u,v},
+    ← mk_image_eq_of_injOn_lift _ _ (hf.mono ((hIX.indep.diff _).subset_ground)),
+    ← mk_image_eq_of_injOn_lift _ _ (hf.mono ((hJX.indep.diff _).subset_ground)),
+    lift_inj, (hf.mono hIX.indep.subset_ground).image_diff,
+    (hf.mono hJX.indep.subset_ground).image_diff, inter_comm,
+    hf.image_inter hJX.indep.subset_ground hIX.indep.subset_ground,
+    diff_inter_self_eq_diff, diff_self_inter] at hcard
 
-theorem Indep.cardinalMk_le_base_of_finitary (hI : M.Indep I) (hB : M.Base B) : #I ≤ #B :=
-  have ⟨_B', hB', hIB'⟩ := hI.exists_base_superset
-  hB'.cardinalMk_eq_of_finitary hB ▸ mk_le_mk_of_subset hIB'
+instance invariantCardinalRank_comap (M : Matroid β) [InvariantCardinalRank M] (f : α → β) :
+    InvariantCardinalRank (M.comap f) := by
+  refine ⟨fun I J X hI hJ ↦ ?_⟩
+  obtain ⟨hI, hfI, hIX⟩ := comap_basis_iff.1 hI
+  obtain ⟨hJ, hfJ, hJX⟩ := comap_basis_iff.1 hJ
+  rw [← lift_inj.{u,v}, ← mk_image_eq_of_injOn_lift _ _ (hfI.mono diff_subset),
+    ← mk_image_eq_of_injOn_lift _ _ (hfJ.mono diff_subset), lift_inj, hfI.image_diff,
+    hfJ.image_diff, ← diff_union_diff_cancel inter_subset_left (image_inter_subset f I J),
+    inter_comm, diff_inter_self_eq_diff, mk_union_of_disjoint, hI.cardinalMk_diff_comm hJ,
+    ← diff_union_diff_cancel inter_subset_left (image_inter_subset f J I), inter_comm,
+    diff_inter_self_eq_diff, mk_union_of_disjoint, inter_comm J I] <;>
+  exact disjoint_sdiff_left.mono_right (diff_subset.trans inter_subset_left)
 
-theorem Indep.cardinalMk_le_basis'_of_finitary (hI : M.Indep I) (hJ : M.Basis' J X) (hIX : I ⊆ X) :
-    #I ≤ #J :=
-  have ⟨_J', hJ', hIJ'⟩ := hI.subset_basis'_of_subset hIX
-  hJ'.cardinalMk_eq_of_finitary hJ ▸ mk_le_mk_of_subset hIJ'
-
-theorem Indep.cardinalMk_le_basis_of_finitary (hI : M.Indep I) (hJ : M.Basis J X) (hIX : I ⊆ X) :
-    #I ≤ #J :=
-  hI.cardinalMk_le_basis'_of_finitary hJ.basis' hIX
+end Instances
 
 end Matroid
diff --git a/Mathlib/Data/NNReal/Defs.lean b/Mathlib/Data/NNReal/Defs.lean
index 95519a608d3833..be4890f48ea145 100644
--- a/Mathlib/Data/NNReal/Defs.lean
+++ b/Mathlib/Data/NNReal/Defs.lean
@@ -800,7 +800,7 @@ nonrec theorem div_le_div_left {a b c : ℝ≥0} (a0 : 0 < a) (b0 : 0 < b) (c0 :
   div_le_div_iff_of_pos_left a0 b0 c0
 
 theorem le_of_forall_lt_one_mul_le {x y : ℝ≥0} (h : ∀ a < 1, a * x ≤ y) : x ≤ y :=
-  le_of_forall_ge_of_dense fun a ha => by
+  le_of_forall_lt_imp_le_of_dense fun a ha => by
     have hx : x ≠ 0 := pos_iff_ne_zero.1 (lt_of_le_of_lt (zero_le _) ha)
     have hx' : x⁻¹ ≠ 0 := by rwa [Ne, inv_eq_zero]
     have : a * x⁻¹ < 1 := by rwa [← lt_inv_iff_mul_lt hx', inv_inv]
diff --git a/Mathlib/Data/Nat/Choose/Bounds.lean b/Mathlib/Data/Nat/Choose/Bounds.lean
index d4b21ddf606d9a..e89df1b02a1add 100644
--- a/Mathlib/Data/Nat/Choose/Bounds.lean
+++ b/Mathlib/Data/Nat/Choose/Bounds.lean
@@ -50,4 +50,20 @@ theorem pow_le_choose (r n : ℕ) : ((n + 1 - r : ℕ) ^ r : α) / r ! ≤ n.cho
     exact n.pow_sub_le_descFactorial r
   exact mod_cast r.factorial_pos
 
+theorem choose_succ_le_two_pow (n k : ℕ) : (n + 1).choose k ≤ 2 ^ n := by
+  by_cases lt : n + 1 < k
+  · simp [choose_eq_zero_of_lt lt]
+  · cases' n with n
+    · cases k <;> simp_all
+    · cases' k with k
+      · rw [choose_zero_right]
+        exact Nat.one_le_two_pow
+      · rw [choose_succ_succ', two_pow_succ]
+        exact Nat.add_le_add (choose_succ_le_two_pow n k) (choose_succ_le_two_pow n (k + 1))
+
+theorem choose_le_two_pow (n k : ℕ) (p : 0 < n) : n.choose k < 2 ^ n := by
+  refine lt_of_le_of_lt ?_ (Nat.two_pow_pred_lt_two_pow p)
+  rw [← Nat.sub_add_cancel p]
+  exact choose_succ_le_two_pow (n - 1) k
+
 end Nat
diff --git a/Mathlib/Data/Nat/Log.lean b/Mathlib/Data/Nat/Log.lean
index 2d4196b2fe33ab..f576bc2d52e16d 100644
--- a/Mathlib/Data/Nat/Log.lean
+++ b/Mathlib/Data/Nat/Log.lean
@@ -24,14 +24,6 @@ assert_not_exists OrderTop
 
 namespace Nat
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/5338 we just unused argument warnings,
-but these are used in the decreasing by blocks.
-If instead we inline the `have` blocks, the unusedHavesSuffices linter triggers.
--/
-set_option linter.unusedVariables false
-
 /-! ### Floor logarithm -/
 
 
diff --git a/Mathlib/Data/PEquiv.lean b/Mathlib/Data/PEquiv.lean
index 007830364ceb2b..fa6cbb13d14d73 100644
--- a/Mathlib/Data/PEquiv.lean
+++ b/Mathlib/Data/PEquiv.lean
@@ -432,6 +432,7 @@ theorem toPEquiv_trans (f : α ≃ β) (g : β ≃ γ) :
 theorem toPEquiv_symm (f : α ≃ β) : f.symm.toPEquiv = f.toPEquiv.symm :=
   rfl
 
+@[simp]
 theorem toPEquiv_apply (f : α ≃ β) (x : α) : f.toPEquiv x = some (f x) :=
   rfl
 
diff --git a/Mathlib/Data/PFunctor/Univariate/M.lean b/Mathlib/Data/PFunctor/Univariate/M.lean
index 70b6e183f4a7e6..0d37382ac25ee4 100644
--- a/Mathlib/Data/PFunctor/Univariate/M.lean
+++ b/Mathlib/Data/PFunctor/Univariate/M.lean
@@ -161,7 +161,7 @@ end Approx
 open Approx
 
 /-- Internal definition for `M`. It is needed to avoid name clashes
-between `M.mk` and `M.cases_on` and the declarations generated for
+between `M.mk` and `M.casesOn` and the declarations generated for
 the structure -/
 structure MIntl where
   /-- An `n`-th level approximation, for each depth `n` -/
@@ -378,8 +378,7 @@ theorem casesOn_mk' {r : M F → Sort*} {a} (x : F.B a → M F)
 /-- `IsPath p x` tells us if `p` is a valid path through `x` -/
 inductive IsPath : Path F → M F → Prop
   | nil (x : M F) : IsPath [] x
-  |
-  cons (xs : Path F) {a} (x : M F) (f : F.B a → M F) (i : F.B a) :
+  | cons (xs : Path F) {a} (x : M F) (f : F.B a → M F) (i : F.B a) :
     x = M.mk ⟨a, f⟩ → IsPath xs (f i) → IsPath (⟨a, i⟩ :: xs) x
 
 theorem isPath_cons {xs : Path F} {a a'} {f : F.B a → M F} {i : F.B a'} :
diff --git a/Mathlib/Data/Real/Archimedean.lean b/Mathlib/Data/Real/Archimedean.lean
index 039acaddf3405f..3e9bf433eb681a 100644
--- a/Mathlib/Data/Real/Archimedean.lean
+++ b/Mathlib/Data/Real/Archimedean.lean
@@ -42,7 +42,7 @@ theorem of_near (f : ℕ → ℚ) (x : ℝ) (h : ∀ ε > 0, ∃ i, ∀ j ≥ i,
   ⟨isCauSeq_iff_lift.2 (CauSeq.of_near _ (const abs x) h),
     sub_eq_zero.1 <|
       abs_eq_zero.1 <|
-        (eq_of_le_of_forall_le_of_dense (abs_nonneg _)) fun _ε ε0 =>
+        (eq_of_le_of_forall_lt_imp_le_of_dense (abs_nonneg _)) fun _ε ε0 =>
           mk_near_of_forall_near <| (h _ ε0).imp fun _i h j ij => le_of_lt (h j ij)⟩
 
 theorem exists_floor (x : ℝ) : ∃ ub : ℤ, (ub : ℝ) ≤ x ∧ ∀ z : ℤ, (z : ℝ) ≤ x → z ≤ ub :=
@@ -88,7 +88,7 @@ theorem exists_isLUB (hne : s.Nonempty) (hbdd : BddAbove s) : ∃ x, IsLUB s x :
     simpa using sub_lt_iff_lt_add'.2 (lt_of_le_of_lt hy <| sub_lt_iff_lt_add.1 <| hf₂ _ k0 _ yS)
   let g : CauSeq ℚ abs := ⟨fun n => f n / n, hg⟩
   refine ⟨mk g, ⟨fun x xS => ?_, fun y h => ?_⟩⟩
-  · refine le_of_forall_ge_of_dense fun z xz => ?_
+  · refine le_of_forall_lt_imp_le_of_dense fun z xz => ?_
     cases' exists_nat_gt (x - z)⁻¹ with K hK
     refine le_mk_of_forall_le ⟨K, fun n nK => ?_⟩
     replace xz := sub_pos.2 xz
diff --git a/Mathlib/Data/Real/EReal.lean b/Mathlib/Data/Real/EReal.lean
index 17c70dbd11726e..8e308ba21e9aa2 100644
--- a/Mathlib/Data/Real/EReal.lean
+++ b/Mathlib/Data/Real/EReal.lean
@@ -1291,7 +1291,7 @@ private lemma exists_lt_add_right {a b c : EReal} (hc : c < a + b) : ∃ b' < b,
   simp_rw [add_comm a] at hc ⊢; exact exists_lt_add_left hc
 
 lemma add_le_of_forall_lt {a b c : EReal} (h : ∀ a' < a, ∀ b' < b, a' + b' ≤ c) : a + b ≤ c := by
-  refine le_of_forall_ge_of_dense fun d hd ↦ ?_
+  refine le_of_forall_lt_imp_le_of_dense fun d hd ↦ ?_
   obtain ⟨a', ha', hd⟩ := exists_lt_add_left hd
   obtain ⟨b', hb', hd⟩ := exists_lt_add_right hd
   exact hd.le.trans (h _ ha' _ hb')
diff --git a/Mathlib/Data/Rel.lean b/Mathlib/Data/Rel.lean
index ced1fb359df561..cae0b3a83e60e8 100644
--- a/Mathlib/Data/Rel.lean
+++ b/Mathlib/Data/Rel.lean
@@ -136,12 +136,10 @@ theorem inv_comp (r : Rel α β) (s : Rel β γ) : inv (r • s) = inv s • inv
 
 @[simp]
 theorem inv_bot : (⊥ : Rel α β).inv = (⊥ : Rel β α) := by
-  #adaptation_note /-- nightly-2024-03-16: simp was `simp [Bot.bot, inv, flip]` -/
   simp [Bot.bot, inv, Function.flip_def]
 
 @[simp]
 theorem inv_top : (⊤ : Rel α β).inv = (⊤ : Rel β α) := by
-  #adaptation_note /-- nightly-2024-03-16: simp was `simp [Top.top, inv, flip]` -/
   simp [Top.top, inv, Function.flip_def]
 
 /-- Image of a set under a relation -/
diff --git a/Mathlib/Data/SetLike/Basic.lean b/Mathlib/Data/SetLike/Basic.lean
index 44719c1ca0a00c..9f99e745a2ce21 100644
--- a/Mathlib/Data/SetLike/Basic.lean
+++ b/Mathlib/Data/SetLike/Basic.lean
@@ -214,4 +214,14 @@ theorem exists_of_lt : p < q → ∃ x ∈ q, x ∉ p :=
 theorem lt_iff_le_and_exists : p < q ↔ p ≤ q ∧ ∃ x ∈ q, x ∉ p := by
   rw [lt_iff_le_not_le, not_le_iff_exists]
 
+/-- membership is inherited from `Set X` -/
+abbrev instSubtypeSet {X} {p : Set X → Prop} : SetLike {s // p s} X where
+  coe := (↑)
+  coe_injective' := Subtype.val_injective
+
+/-- membership is inherited from `S` -/
+abbrev instSubtype {X S} [SetLike S X] {p : S → Prop} : SetLike {s // p s} X where
+  coe := (↑)
+  coe_injective' := SetLike.coe_injective.comp Subtype.val_injective
+
 end SetLike
diff --git a/Mathlib/Data/Sign.lean b/Mathlib/Data/Sign.lean
index 4607e88ca0d3ec..ffb6b9cb726dbb 100644
--- a/Mathlib/Data/Sign.lean
+++ b/Mathlib/Data/Sign.lean
@@ -115,11 +115,8 @@ instance : BoundedOrder SignType where
   le_top := LE.of_pos
   bot := -1
   bot_le :=
-    #adaptation_note
-    /--
-    Added `by exact` after https://github.com/leanprover/lean4/pull/6053,
-    but don't understand why it was needed.
-    -/
+    #adaptation_note /-- https://github.com/leanprover/lean4/pull/6053
+    Added `by exact`, but don't understand why it was needed. -/
     by exact LE.of_neg
 
 instance : HasDistribNeg SignType :=
diff --git a/Mathlib/Data/Subtype.lean b/Mathlib/Data/Subtype.lean
index ea0164bca8d418..ebde31a801bf53 100644
--- a/Mathlib/Data/Subtype.lean
+++ b/Mathlib/Data/Subtype.lean
@@ -155,7 +155,6 @@ def map {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ a, p a →
     Subtype p → Subtype q :=
   fun x ↦ ⟨f x, h x x.prop⟩
 
-#adaptation_note /-- nightly-2024-03-16: added to replace simp [Subtype.map] -/
 theorem map_def {p : α → Prop} {q : β → Prop} (f : α → β) (h : ∀ a, p a → q (f a)) :
     map f h = fun x ↦ ⟨f x, h x x.prop⟩ :=
   rfl
diff --git a/Mathlib/Data/Sum/Basic.lean b/Mathlib/Data/Sum/Basic.lean
index 30d502873b1397..cf572757b919fa 100644
--- a/Mathlib/Data/Sum/Basic.lean
+++ b/Mathlib/Data/Sum/Basic.lean
@@ -269,3 +269,17 @@ def in₂ (c : γ) : α ⊕ (β ⊕ γ) :=
   inr <| inr c
 
 end Sum3
+
+/-!
+### PSum
+-/
+
+namespace PSum
+
+variable {α β : Sort*}
+
+theorem inl_injective : Function.Injective (PSum.inl : α → α ⊕' β) := fun _ _ ↦ inl.inj
+
+theorem inr_injective : Function.Injective (PSum.inr : β → α ⊕' β) := fun _ _ ↦ inr.inj
+
+end PSum
diff --git a/Mathlib/Deprecated/Group.lean b/Mathlib/Deprecated/Group.lean
deleted file mode 100644
index dd913b42c9bec0..00000000000000
--- a/Mathlib/Deprecated/Group.lean
+++ /dev/null
@@ -1,391 +0,0 @@
-/-
-Copyright (c) 2019 Yury Kudryashov. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Yury Kudryashov
--/
-import Mathlib.Algebra.Group.Units.Hom
-import Mathlib.Algebra.Ring.Hom.Defs
-import Mathlib.Algebra.Group.TypeTags.Basic
-
-/-!
-# Unbundled monoid and group homomorphisms
-
-This file is deprecated, and is no longer imported by anything in mathlib other than other
-deprecated files, and test files. You should not need to import it.
-
-This file defines predicates for unbundled monoid and group homomorphisms. Instead of using
-this file, please use `MonoidHom`, defined in `Algebra.Hom.Group`, with notation `→*`, for
-morphisms between monoids or groups. For example use `φ : G →* H` to represent a group
-homomorphism between multiplicative groups, and `ψ : A →+ B` to represent a group homomorphism
-between additive groups.
-
-## Main Definitions
-
-`IsMonoidHom` (deprecated), `IsGroupHom` (deprecated)
-
-## Tags
-
-IsGroupHom, IsMonoidHom
-
--/
-
-
-universe u v
-
-variable {α : Type u} {β : Type v}
-
-/-- Predicate for maps which preserve an addition. -/
-structure IsAddHom {α β : Type*} [Add α] [Add β] (f : α → β) : Prop where
-  /-- The proposition that `f` preserves addition. -/
-  map_add : ∀ x y, f (x + y) = f x + f y
-
-/-- Predicate for maps which preserve a multiplication. -/
-@[to_additive]
-structure IsMulHom {α β : Type*} [Mul α] [Mul β] (f : α → β) : Prop where
-  /-- The proposition that `f` preserves multiplication. -/
-  map_mul : ∀ x y, f (x * y) = f x * f y
-
-namespace IsMulHom
-
-variable [Mul α] [Mul β] {γ : Type*} [Mul γ]
-
-/-- The identity map preserves multiplication. -/
-@[to_additive "The identity map preserves addition"]
-theorem id : IsMulHom (id : α → α) :=
-  { map_mul := fun _ _ => rfl }
-
-/-- The composition of maps which preserve multiplication, also preserves multiplication. -/
-@[to_additive "The composition of addition preserving maps also preserves addition"]
-theorem comp {f : α → β} {g : β → γ} (hf : IsMulHom f) (hg : IsMulHom g) : IsMulHom (g ∘ f) :=
-  { map_mul := fun x y => by simp only [Function.comp, hf.map_mul, hg.map_mul] }
-
-/-- A product of maps which preserve multiplication,
-preserves multiplication when the target is commutative. -/
-@[to_additive
-      "A sum of maps which preserves addition, preserves addition when the target
-      is commutative."]
-theorem mul {α β} [Semigroup α] [CommSemigroup β] {f g : α → β} (hf : IsMulHom f)
-    (hg : IsMulHom g) : IsMulHom fun a => f a * g a :=
-  { map_mul := fun a b => by
-      simp only [hf.map_mul, hg.map_mul, mul_comm, mul_assoc, mul_left_comm] }
-
-/-- The inverse of a map which preserves multiplication,
-preserves multiplication when the target is commutative. -/
-@[to_additive
-      "The negation of a map which preserves addition, preserves addition when
-      the target is commutative."]
-theorem inv {α β} [Mul α] [CommGroup β] {f : α → β} (hf : IsMulHom f) : IsMulHom fun a => (f a)⁻¹ :=
-  { map_mul := fun a b => (hf.map_mul a b).symm ▸ mul_inv _ _ }
-
-end IsMulHom
-
-/-- Predicate for additive monoid homomorphisms
-(deprecated -- use the bundled `MonoidHom` version). -/
-structure IsAddMonoidHom [AddZeroClass α] [AddZeroClass β] (f : α → β) extends IsAddHom f :
-  Prop where
-  /-- The proposition that `f` preserves the additive identity. -/
-  map_zero : f 0 = 0
-
-/-- Predicate for monoid homomorphisms (deprecated -- use the bundled `MonoidHom` version). -/
-@[to_additive]
-structure IsMonoidHom [MulOneClass α] [MulOneClass β] (f : α → β) extends IsMulHom f : Prop where
-  /-- The proposition that `f` preserves the multiplicative identity. -/
-  map_one : f 1 = 1
-
-namespace MonoidHom
-
-variable {M : Type*} {N : Type*} {mM : MulOneClass M} {mN : MulOneClass N}
-
-/-- Interpret a map `f : M → N` as a homomorphism `M →* N`. -/
-@[to_additive "Interpret a map `f : M → N` as a homomorphism `M →+ N`."]
-def of {f : M → N} (h : IsMonoidHom f) : M →* N where
-  toFun := f
-  map_one' := h.2
-  map_mul' := h.1.1
-
-@[to_additive (attr := simp)]
-theorem coe_of {f : M → N} (hf : IsMonoidHom f) : ⇑(MonoidHom.of hf) = f :=
-  rfl
-
-@[to_additive]
-theorem isMonoidHom_coe (f : M →* N) : IsMonoidHom (f : M → N) :=
-  { map_mul := f.map_mul
-    map_one := f.map_one }
-
-end MonoidHom
-
-namespace MulEquiv
-
-variable {M : Type*} {N : Type*} [MulOneClass M] [MulOneClass N]
-
-/-- A multiplicative isomorphism preserves multiplication (deprecated). -/
-@[to_additive "An additive isomorphism preserves addition (deprecated)."]
-theorem isMulHom (h : M ≃* N) : IsMulHom h :=
-  ⟨map_mul h⟩
-
-/-- A multiplicative bijection between two monoids is a monoid hom
-  (deprecated -- use `MulEquiv.toMonoidHom`). -/
-@[to_additive
-      "An additive bijection between two additive monoids is an additive
-      monoid hom (deprecated). "]
-theorem isMonoidHom (h : M ≃* N) : IsMonoidHom h :=
-  { map_mul := map_mul h
-    map_one := h.map_one }
-
-end MulEquiv
-
-namespace IsMonoidHom
-
-/-- A monoid homomorphism preserves multiplication. -/
-@[to_additive "An additive monoid homomorphism preserves addition."]
-theorem map_mul' [MulOneClass α] [MulOneClass β] {f : α → β} (hf : IsMonoidHom f) (x y) :
-    f (x * y) = f x * f y :=
-  hf.map_mul x y
-
-/-- The inverse of a map which preserves multiplication,
-preserves multiplication when the target is commutative. -/
-@[to_additive
-      "The negation of a map which preserves addition, preserves addition
-      when the target is commutative."]
-theorem inv {α β} [MulOneClass α] [CommGroup β] {f : α → β} (hf : IsMonoidHom f) :
-    IsMonoidHom fun a => (f a)⁻¹ :=
-  { map_one := hf.map_one.symm ▸ inv_one
-    map_mul := fun a b => (hf.map_mul a b).symm ▸ mul_inv _ _ }
-
-end IsMonoidHom
-
-/-- A map to a group preserving multiplication is a monoid homomorphism. -/
-@[to_additive "A map to an additive group preserving addition is an additive monoid
-homomorphism."]
-theorem IsMulHom.to_isMonoidHom [MulOneClass α] [Group β] {f : α → β} (hf : IsMulHom f) :
-    IsMonoidHom f :=
-  { map_one := (mul_right_eq_self (a := f 1)).1 <| by rw [← hf.map_mul, one_mul]
-    map_mul := hf.map_mul }
-
-namespace IsMonoidHom
-
-variable [MulOneClass α] [MulOneClass β] {f : α → β}
-
-/-- The identity map is a monoid homomorphism. -/
-@[to_additive "The identity map is an additive monoid homomorphism."]
-theorem id : IsMonoidHom (@id α) :=
-  { map_one := rfl
-    map_mul := fun _ _ => rfl }
-
-/-- The composite of two monoid homomorphisms is a monoid homomorphism. -/
-@[to_additive
-      "The composite of two additive monoid homomorphisms is an additive monoid
-      homomorphism."]
-theorem comp (hf : IsMonoidHom f) {γ} [MulOneClass γ] {g : β → γ} (hg : IsMonoidHom g) :
-    IsMonoidHom (g ∘ f) :=
-  { IsMulHom.comp hf.toIsMulHom hg.toIsMulHom with
-    map_one := show g _ = 1 by rw [hf.map_one, hg.map_one] }
-
-end IsMonoidHom
-
-namespace IsAddMonoidHom
-
-/-- Left multiplication in a ring is an additive monoid morphism. -/
-theorem isAddMonoidHom_mul_left {γ : Type*} [NonUnitalNonAssocSemiring γ] (x : γ) :
-    IsAddMonoidHom fun y : γ => x * y :=
-  { map_zero := mul_zero x
-    map_add := fun y z => mul_add x y z }
-
-/-- Right multiplication in a ring is an additive monoid morphism. -/
-theorem isAddMonoidHom_mul_right {γ : Type*} [NonUnitalNonAssocSemiring γ] (x : γ) :
-    IsAddMonoidHom fun y : γ => y * x :=
-  { map_zero := zero_mul x
-    map_add := fun y z => add_mul y z x }
-
-end IsAddMonoidHom
-
-/-- Predicate for additive group homomorphism (deprecated -- use bundled `MonoidHom`). -/
-structure IsAddGroupHom [AddGroup α] [AddGroup β] (f : α → β) extends IsAddHom f : Prop
-
-/-- Predicate for group homomorphisms (deprecated -- use bundled `MonoidHom`). -/
-@[to_additive]
-structure IsGroupHom [Group α] [Group β] (f : α → β) extends IsMulHom f : Prop
-
-@[to_additive]
-theorem MonoidHom.isGroupHom {G H : Type*} {_ : Group G} {_ : Group H} (f : G →* H) :
-    IsGroupHom (f : G → H) :=
-  { map_mul := f.map_mul }
-
-@[to_additive]
-theorem MulEquiv.isGroupHom {G H : Type*} {_ : Group G} {_ : Group H} (h : G ≃* H) :
-    IsGroupHom h :=
-  { map_mul := map_mul h }
-
-/-- Construct `IsGroupHom` from its only hypothesis. -/
-@[to_additive "Construct `IsAddGroupHom` from its only hypothesis."]
-theorem IsGroupHom.mk' [Group α] [Group β] {f : α → β} (hf : ∀ x y, f (x * y) = f x * f y) :
-    IsGroupHom f :=
-  { map_mul := hf }
-
-namespace IsGroupHom
-
-variable [Group α] [Group β] {f : α → β} (hf : IsGroupHom f)
-
-/-- The identity is a group homomorphism. -/
-@[to_additive "The identity is an additive group homomorphism."]
-theorem id : IsGroupHom (@id α) :=
-  { map_mul := fun _ _ => rfl }
-
-section
-include hf
-
-open IsMulHom (map_mul)
-
-theorem map_mul' : ∀ x y, f (x * y) = f x * f y :=
-  hf.toIsMulHom.map_mul
-
-/-- A group homomorphism is a monoid homomorphism. -/
-@[to_additive "An additive group homomorphism is an additive monoid homomorphism."]
-theorem to_isMonoidHom : IsMonoidHom f :=
-  hf.toIsMulHom.to_isMonoidHom
-
-/-- A group homomorphism sends 1 to 1. -/
-@[to_additive "An additive group homomorphism sends 0 to 0."]
-theorem map_one : f 1 = 1 :=
-  hf.to_isMonoidHom.map_one
-
-/-- A group homomorphism sends inverses to inverses. -/
-@[to_additive "An additive group homomorphism sends negations to negations."]
-theorem map_inv (a : α) : f a⁻¹ = (f a)⁻¹ :=
-  eq_inv_of_mul_eq_one_left <| by rw [← hf.map_mul, inv_mul_cancel, hf.map_one]
-
-@[to_additive]
-theorem map_div (a b : α) : f (a / b) = f a / f b := by
-  simp_rw [div_eq_mul_inv, hf.map_mul, hf.map_inv]
-
-/-- The composition of two group homomorphisms is a group homomorphism. -/
-@[to_additive
-      "The composition of two additive group homomorphisms is an additive
-      group homomorphism."]
-theorem comp {γ} [Group γ] {g : β → γ} (hg : IsGroupHom g) :
-    IsGroupHom (g ∘ f) :=
-  { IsMulHom.comp hf.toIsMulHom hg.toIsMulHom with }
-
-/-- A group homomorphism is injective iff its kernel is trivial. -/
-@[to_additive "An additive group homomorphism is injective if its kernel is trivial."]
-theorem injective_iff : Function.Injective f ↔ ∀ a, f a = 1 → a = 1 :=
-  ⟨fun h _ => by rw [← hf.map_one]; exact @h _ _, fun h x y hxy =>
-    eq_of_div_eq_one <| h _ <| by rwa [hf.map_div, div_eq_one]⟩
-
-end
-
-/-- The product of group homomorphisms is a group homomorphism if the target is commutative. -/
-@[to_additive
-      "The sum of two additive group homomorphisms is an additive group homomorphism
-      if the target is commutative."]
-theorem mul {α β} [Group α] [CommGroup β] {f g : α → β} (hf : IsGroupHom f) (hg : IsGroupHom g) :
-    IsGroupHom fun a => f a * g a :=
-  { map_mul := (hf.toIsMulHom.mul hg.toIsMulHom).map_mul }
-
-/-- The inverse of a group homomorphism is a group homomorphism if the target is commutative. -/
-@[to_additive
-      "The negation of an additive group homomorphism is an additive group homomorphism
-      if the target is commutative."]
-theorem inv {α β} [Group α] [CommGroup β] {f : α → β} (hf : IsGroupHom f) :
-    IsGroupHom fun a => (f a)⁻¹ :=
-  { map_mul := hf.toIsMulHom.inv.map_mul }
-
-end IsGroupHom
-
-namespace RingHom
-
-/-!
-These instances look redundant, because `Deprecated.Ring` provides `IsRingHom` for a `→+*`.
-Nevertheless these are harmless, and helpful for stripping out dependencies on `Deprecated.Ring`.
--/
-
-
-variable {R : Type*} {S : Type*}
-
-section
-
-variable [NonAssocSemiring R] [NonAssocSemiring S]
-
-theorem to_isMonoidHom (f : R →+* S) : IsMonoidHom f :=
-  { map_one := f.map_one
-    map_mul := f.map_mul }
-
-theorem to_isAddMonoidHom (f : R →+* S) : IsAddMonoidHom f :=
-  { map_zero := f.map_zero
-    map_add := f.map_add }
-
-end
-
-section
-
-variable [Ring R] [Ring S]
-
-theorem to_isAddGroupHom (f : R →+* S) : IsAddGroupHom f :=
-  { map_add := f.map_add }
-
-end
-
-end RingHom
-
-/-- Inversion is a group homomorphism if the group is commutative. -/
-@[to_additive
-      "Negation is an `AddGroup` homomorphism if the `AddGroup` is commutative."]
-theorem Inv.isGroupHom [CommGroup α] : IsGroupHom (Inv.inv : α → α) :=
-  { map_mul := mul_inv }
-
-/-- The difference of two additive group homomorphisms is an additive group
-homomorphism if the target is commutative. -/
-theorem IsAddGroupHom.sub {α β} [AddGroup α] [AddCommGroup β] {f g : α → β} (hf : IsAddGroupHom f)
-    (hg : IsAddGroupHom g) : IsAddGroupHom fun a => f a - g a := by
-  simpa only [sub_eq_add_neg] using hf.add hg.neg
-
-namespace Units
-
-variable {M : Type*} {N : Type*} [Monoid M] [Monoid N]
-
-/-- The group homomorphism on units induced by a multiplicative morphism. -/
-abbrev map' {f : M → N} (hf : IsMonoidHom f) : Mˣ →* Nˣ :=
-  map (MonoidHom.of hf)
-
-@[simp]
-theorem coe_map' {f : M → N} (hf : IsMonoidHom f) (x : Mˣ) : ↑((map' hf : Mˣ → Nˣ) x) = f x :=
-  rfl
-
-theorem coe_isMonoidHom : IsMonoidHom (↑· : Mˣ → M) :=
-  (coeHom M).isMonoidHom_coe
-
-end Units
-
-namespace IsUnit
-
-variable {M : Type*} {N : Type*} [Monoid M] [Monoid N]
-
-theorem map' {f : M → N} (hf : IsMonoidHom f) {x : M} (h : IsUnit x) : IsUnit (f x) :=
-  h.map (MonoidHom.of hf)
-
-end IsUnit
-
-theorem Additive.isAddHom [Mul α] [Mul β] {f : α → β} (hf : IsMulHom f) :
-    @IsAddHom (Additive α) (Additive β) _ _ f :=
-  { map_add := hf.map_mul }
-
-theorem Multiplicative.isMulHom [Add α] [Add β] {f : α → β} (hf : IsAddHom f) :
-    @IsMulHom (Multiplicative α) (Multiplicative β) _ _ f :=
-  { map_mul := hf.map_add }
-
--- defeq abuse
-theorem Additive.isAddMonoidHom [MulOneClass α] [MulOneClass β] {f : α → β}
-    (hf : IsMonoidHom f) : @IsAddMonoidHom (Additive α) (Additive β) _ _ f :=
-  { Additive.isAddHom hf.toIsMulHom with map_zero := hf.map_one }
-
-theorem Multiplicative.isMonoidHom [AddZeroClass α] [AddZeroClass β] {f : α → β}
-    (hf : IsAddMonoidHom f) : @IsMonoidHom (Multiplicative α) (Multiplicative β) _ _ f :=
-  { Multiplicative.isMulHom hf.toIsAddHom with map_one := hf.map_zero }
-
-theorem Additive.isAddGroupHom [Group α] [Group β] {f : α → β} (hf : IsGroupHom f) :
-    @IsAddGroupHom (Additive α) (Additive β) _ _ f :=
-  { map_add := hf.toIsMulHom.map_mul }
-
-theorem Multiplicative.isGroupHom [AddGroup α] [AddGroup β] {f : α → β} (hf : IsAddGroupHom f) :
-    @IsGroupHom (Multiplicative α) (Multiplicative β) _ _ f :=
-  { map_mul := hf.toIsAddHom.map_add }
diff --git a/Mathlib/Deprecated/Ring.lean b/Mathlib/Deprecated/Ring.lean
deleted file mode 100644
index fc34ab12c07eeb..00000000000000
--- a/Mathlib/Deprecated/Ring.lean
+++ /dev/null
@@ -1,150 +0,0 @@
-/-
-Copyright (c) 2020 Mario Carneiro. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Mario Carneiro
--/
-import Mathlib.Deprecated.Group
-
-/-!
-# Unbundled semiring and ring homomorphisms (deprecated)
-
-This file is deprecated, and is no longer imported by anything in mathlib other than other
-deprecated files, and test files. You should not need to import it.
-
-This file defines predicates for unbundled semiring and ring homomorphisms. Instead of using
-this file, please use `RingHom`, defined in `Algebra.Hom.Ring`, with notation `→+*`, for
-morphisms between semirings or rings. For example use `φ : A →+* B` to represent a
-ring homomorphism.
-
-## Main Definitions
-
-`IsSemiringHom` (deprecated), `IsRingHom` (deprecated)
-
-## Tags
-
-IsSemiringHom, IsRingHom
-
--/
-
-
-universe u v
-
-variable {α : Type u}
-
-/-- Predicate for semiring homomorphisms (deprecated -- use the bundled `RingHom` version). -/
-structure IsSemiringHom {α : Type u} {β : Type v} [Semiring α] [Semiring β] (f : α → β) : Prop where
-  /-- The proposition that `f` preserves the additive identity. -/
-  map_zero : f 0 = 0
-  /-- The proposition that `f` preserves the multiplicative identity. -/
-  map_one : f 1 = 1
-  /-- The proposition that `f` preserves addition. -/
-  map_add : ∀ x y, f (x + y) = f x + f y
-  /-- The proposition that `f` preserves multiplication. -/
-  map_mul : ∀ x y, f (x * y) = f x * f y
-
-namespace IsSemiringHom
-
-variable {β : Type v} [Semiring α] [Semiring β]
-variable {f : α → β}
-
-/-- The identity map is a semiring homomorphism. -/
-theorem id : IsSemiringHom (@id α) := by constructor <;> intros <;> rfl
-
-/-- The composition of two semiring homomorphisms is a semiring homomorphism. -/
-theorem comp (hf : IsSemiringHom f) {γ} [Semiring γ] {g : β → γ} (hg : IsSemiringHom g) :
-    IsSemiringHom (g ∘ f) :=
-  { map_zero := by simpa [map_zero hf] using map_zero hg
-    map_one := by simpa [map_one hf] using map_one hg
-    map_add := fun {x y} => by simp [map_add hf, map_add hg]
-    map_mul := fun {x y} => by simp [map_mul hf, map_mul hg] }
-
-/-- A semiring homomorphism is an additive monoid homomorphism. -/
-theorem to_isAddMonoidHom (hf : IsSemiringHom f) : IsAddMonoidHom f :=
-  { ‹IsSemiringHom f› with map_add := by apply @‹IsSemiringHom f›.map_add }
-
-/-- A semiring homomorphism is a monoid homomorphism. -/
-theorem to_isMonoidHom (hf : IsSemiringHom f) : IsMonoidHom f :=
-  { ‹IsSemiringHom f› with }
-
-end IsSemiringHom
-
-/-- Predicate for ring homomorphisms (deprecated -- use the bundled `RingHom` version). -/
-structure IsRingHom {α : Type u} {β : Type v} [Ring α] [Ring β] (f : α → β) : Prop where
-  /-- The proposition that `f` preserves the multiplicative identity. -/
-  map_one : f 1 = 1
-  /-- The proposition that `f` preserves multiplication. -/
-  map_mul : ∀ x y, f (x * y) = f x * f y
-  /-- The proposition that `f` preserves addition. -/
-  map_add : ∀ x y, f (x + y) = f x + f y
-
-namespace IsRingHom
-
-variable {β : Type v} [Ring α] [Ring β]
-
-/-- A map of rings that is a semiring homomorphism is also a ring homomorphism. -/
-theorem of_semiring {f : α → β} (H : IsSemiringHom f) : IsRingHom f :=
-  { H with }
-
-variable {f : α → β} {x y : α}
-
-/-- Ring homomorphisms map zero to zero. -/
-theorem map_zero (hf : IsRingHom f) : f 0 = 0 :=
-  calc
-    f 0 = f (0 + 0) - f 0 := by rw [hf.map_add]; simp
-    _ = 0 := by simp
-
-/-- Ring homomorphisms preserve additive inverses. -/
-theorem map_neg (hf : IsRingHom f) : f (-x) = -f x :=
-  calc
-    f (-x) = f (-x + x) - f x := by rw [hf.map_add]; simp
-    _ = -f x := by simp [hf.map_zero]
-
-/-- Ring homomorphisms preserve subtraction. -/
-theorem map_sub (hf : IsRingHom f) : f (x - y) = f x - f y := by
-  simp [sub_eq_add_neg, hf.map_add, hf.map_neg]
-
-/-- The identity map is a ring homomorphism. -/
-theorem id : IsRingHom (@id α) := by constructor <;> intros <;> rfl
-
--- see Note [no instance on morphisms]
-/-- The composition of two ring homomorphisms is a ring homomorphism. -/
-theorem comp (hf : IsRingHom f) {γ} [Ring γ] {g : β → γ} (hg : IsRingHom g) : IsRingHom (g ∘ f) :=
-  { map_add := fun x y => by simp only [Function.comp_apply, map_add hf, map_add hg]
-    map_mul := fun x y => by simp only [Function.comp_apply, map_mul hf, map_mul hg]
-    map_one := by simp only [Function.comp_apply, map_one hf, map_one hg] }
-
-/-- A ring homomorphism is also a semiring homomorphism. -/
-theorem to_isSemiringHom (hf : IsRingHom f) : IsSemiringHom f :=
-  { ‹IsRingHom f› with map_zero := map_zero hf }
-
-theorem to_isAddGroupHom (hf : IsRingHom f) : IsAddGroupHom f :=
-  { map_add := hf.map_add }
-
-end IsRingHom
-
-variable {β : Type v} {rα : Semiring α} {rβ : Semiring β}
-
-namespace RingHom
-
-section
-
-/-- Interpret `f : α → β` with `IsSemiringHom f` as a ring homomorphism. -/
-def of {f : α → β} (hf : IsSemiringHom f) : α →+* β :=
-  { MonoidHom.of hf.to_isMonoidHom, AddMonoidHom.of hf.to_isAddMonoidHom with toFun := f }
-
-@[simp]
-theorem coe_of {f : α → β} (hf : IsSemiringHom f) : ⇑(of hf) = f :=
-  rfl
-
-theorem to_isSemiringHom (f : α →+* β) : IsSemiringHom f :=
-  { map_zero := f.map_zero
-    map_one := f.map_one
-    map_add := f.map_add
-    map_mul := f.map_mul }
-
-end
-
-theorem to_isRingHom {α γ} [Ring α] [Ring γ] (g : α →+* γ) : IsRingHom g :=
-  IsRingHom.of_semiring g.to_isSemiringHom
-
-end RingHom
diff --git a/Mathlib/Deprecated/Subfield.lean b/Mathlib/Deprecated/Subfield.lean
deleted file mode 100644
index 3ce93a3a26acc4..00000000000000
--- a/Mathlib/Deprecated/Subfield.lean
+++ /dev/null
@@ -1,150 +0,0 @@
-/-
-Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Andreas Swerdlow
--/
-import Mathlib.Algebra.Field.Basic
-import Mathlib.Deprecated.Subring
-import Mathlib.Algebra.GroupWithZero.Units.Lemmas
-
-/-!
-# Unbundled subfields (deprecated)
-
-This file is deprecated, and is no longer imported by anything in mathlib other than other
-deprecated files, and test files. You should not need to import it.
-
-This file defines predicates for unbundled subfields. Instead of using this file, please use
-`Subfield`, defined in `FieldTheory.Subfield`, for subfields of fields.
-
-## Main definitions
-
-`IsSubfield (S : Set F) : Prop` : the predicate that `S` is the underlying set of a subfield
-of the field `F`. The bundled variant `Subfield F` should be used in preference to this.
-
-## Tags
-
-IsSubfield, subfield
--/
-
-
-variable {F : Type*} [Field F] (S : Set F)
-
-/-- `IsSubfield (S : Set F)` is the predicate saying that a given subset of a field is
-the set underlying a subfield. This structure is deprecated; use the bundled variant
-`Subfield F` to model subfields of a field. -/
-structure IsSubfield extends IsSubring S : Prop where
-  inv_mem : ∀ {x : F}, x ∈ S → x⁻¹ ∈ S
-
-theorem IsSubfield.div_mem {S : Set F} (hS : IsSubfield S) {x y : F} (hx : x ∈ S) (hy : y ∈ S) :
-    x / y ∈ S := by
-  rw [div_eq_mul_inv]
-  exact hS.toIsSubring.toIsSubmonoid.mul_mem hx (hS.inv_mem hy)
-
-theorem IsSubfield.pow_mem {a : F} {n : ℤ} {s : Set F} (hs : IsSubfield s) (h : a ∈ s) :
-    a ^ n ∈ s := by
-  cases' n with n n
-  · suffices a ^ (n : ℤ) ∈ s by exact this
-    rw [zpow_natCast]
-    exact hs.toIsSubring.toIsSubmonoid.pow_mem h
-  · rw [zpow_negSucc]
-    exact hs.inv_mem (hs.toIsSubring.toIsSubmonoid.pow_mem h)
-
-theorem Univ.isSubfield : IsSubfield (@Set.univ F) :=
-  { Univ.isSubmonoid, IsAddSubgroup.univ_addSubgroup with
-    inv_mem := fun _ ↦ trivial }
-
-theorem Preimage.isSubfield {K : Type*} [Field K] (f : F →+* K) {s : Set K} (hs : IsSubfield s) :
-    IsSubfield (f ⁻¹' s) :=
-  { f.isSubring_preimage hs.toIsSubring with
-    inv_mem := fun {a} (ha : f a ∈ s) ↦ show f a⁻¹ ∈ s by
-      rw [map_inv₀]
-      exact hs.inv_mem ha }
-
-theorem Image.isSubfield {K : Type*} [Field K] (f : F →+* K) {s : Set F} (hs : IsSubfield s) :
-    IsSubfield (f '' s) :=
-  { f.isSubring_image hs.toIsSubring with
-    inv_mem := fun ⟨x, xmem, ha⟩ ↦ ⟨x⁻¹, hs.inv_mem xmem, ha ▸ map_inv₀ f x⟩ }
-
-theorem Range.isSubfield {K : Type*} [Field K] (f : F →+* K) : IsSubfield (Set.range f) := by
-  rw [← Set.image_univ]
-  apply Image.isSubfield _ Univ.isSubfield
-
-namespace Field
-
-/-- `Field.closure s` is the minimal subfield that includes `s`. -/
-def closure : Set F :=
-  { x | ∃ y ∈ Ring.closure S, ∃ z ∈ Ring.closure S, y / z = x }
-
-variable {S}
-
-theorem ring_closure_subset : Ring.closure S ⊆ closure S :=
-  fun x hx ↦ ⟨x, hx, 1, Ring.closure.isSubring.toIsSubmonoid.one_mem, div_one x⟩
-
-theorem closure.isSubmonoid : IsSubmonoid (closure S) :=
-  { mul_mem := by
-      rintro _ _ ⟨p, hp, q, hq, hq0, rfl⟩ ⟨r, hr, s, hs, hs0, rfl⟩
-      exact ⟨p * r, IsSubmonoid.mul_mem Ring.closure.isSubring.toIsSubmonoid hp hr, q * s,
-        IsSubmonoid.mul_mem Ring.closure.isSubring.toIsSubmonoid hq hs,
-        (div_mul_div_comm _ _ _ _).symm⟩
-    one_mem := ring_closure_subset <| IsSubmonoid.one_mem Ring.closure.isSubring.toIsSubmonoid }
-
-theorem closure.isSubfield : IsSubfield (closure S) :=
-  { closure.isSubmonoid with
-    add_mem := by
-      intro a b ha hb
-      rcases id ha with ⟨p, hp, q, hq, rfl⟩
-      rcases id hb with ⟨r, hr, s, hs, rfl⟩
-      by_cases hq0 : q = 0
-      · rwa [hq0, div_zero, zero_add]
-      by_cases hs0 : s = 0
-      · rwa [hs0, div_zero, add_zero]
-      exact ⟨p * s + q * r,
-        IsAddSubmonoid.add_mem Ring.closure.isSubring.toIsAddSubgroup.toIsAddSubmonoid
-          (Ring.closure.isSubring.toIsSubmonoid.mul_mem hp hs)
-          (Ring.closure.isSubring.toIsSubmonoid.mul_mem hq hr),
-        q * s, Ring.closure.isSubring.toIsSubmonoid.mul_mem hq hs, (div_add_div p r hq0 hs0).symm⟩
-    zero_mem := ring_closure_subset Ring.closure.isSubring.toIsAddSubgroup.toIsAddSubmonoid.zero_mem
-    neg_mem := by
-      rintro _ ⟨p, hp, q, hq, rfl⟩
-      exact ⟨-p, Ring.closure.isSubring.toIsAddSubgroup.neg_mem hp, q, hq, neg_div q p⟩
-    inv_mem := by
-      rintro _ ⟨p, hp, q, hq, rfl⟩
-      exact ⟨q, hq, p, hp, (inv_div _ _).symm⟩ }
-
-theorem mem_closure {a : F} (ha : a ∈ S) : a ∈ closure S :=
-  ring_closure_subset <| Ring.mem_closure ha
-
-theorem subset_closure : S ⊆ closure S :=
-  fun _ ↦ mem_closure
-
-theorem closure_subset {T : Set F} (hT : IsSubfield T) (H : S ⊆ T) : closure S ⊆ T := by
-  rintro _ ⟨p, hp, q, hq, hq0, rfl⟩
-  exact hT.div_mem (Ring.closure_subset hT.toIsSubring H hp)
-    (Ring.closure_subset hT.toIsSubring H hq)
-
-theorem closure_subset_iff {s t : Set F} (ht : IsSubfield t) : closure s ⊆ t ↔ s ⊆ t :=
-  ⟨Set.Subset.trans subset_closure, closure_subset ht⟩
-
-@[gcongr]
-theorem closure_mono {s t : Set F} (H : s ⊆ t) : closure s ⊆ closure t :=
-  closure_subset closure.isSubfield <| Set.Subset.trans H subset_closure
-
-end Field
-
-theorem isSubfield_iUnion_of_directed {ι : Type*} [Nonempty ι] {s : ι → Set F}
-    (hs : ∀ i, IsSubfield (s i)) (directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :
-    IsSubfield (⋃ i, s i) :=
-  { inv_mem := fun hx ↦
-      let ⟨i, hi⟩ := Set.mem_iUnion.1 hx
-      Set.mem_iUnion.2 ⟨i, (hs i).inv_mem hi⟩
-    toIsSubring := isSubring_iUnion_of_directed (fun i ↦ (hs i).toIsSubring) directed }
-
-theorem IsSubfield.inter {S₁ S₂ : Set F} (hS₁ : IsSubfield S₁) (hS₂ : IsSubfield S₂) :
-    IsSubfield (S₁ ∩ S₂) :=
-  { IsSubring.inter hS₁.toIsSubring hS₂.toIsSubring with
-    inv_mem := fun hx ↦ ⟨hS₁.inv_mem hx.1, hS₂.inv_mem hx.2⟩ }
-
-theorem IsSubfield.iInter {ι : Sort*} {S : ι → Set F} (h : ∀ y : ι, IsSubfield (S y)) :
-    IsSubfield (Set.iInter S) :=
-  { IsSubring.iInter fun y ↦ (h y).toIsSubring with
-    inv_mem := fun hx ↦ Set.mem_iInter.2 fun y ↦ (h y).inv_mem <| Set.mem_iInter.1 hx y }
diff --git a/Mathlib/Deprecated/Subgroup.lean b/Mathlib/Deprecated/Subgroup.lean
deleted file mode 100644
index 3842d160c48acc..00000000000000
--- a/Mathlib/Deprecated/Subgroup.lean
+++ /dev/null
@@ -1,627 +0,0 @@
-/-
-Copyright (c) 2018 Johannes Hölzl. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johannes Hölzl, Mitchell Rowett, Kim Morrison, Johan Commelin, Mario Carneiro,
-  Michael Howes
--/
-import Mathlib.Algebra.Group.Subgroup.Basic
-import Mathlib.Deprecated.Submonoid
-
-/-!
-# Unbundled subgroups (deprecated)
-
-This file is deprecated, and is no longer imported by anything in mathlib other than other
-deprecated files, and test files. You should not need to import it.
-
-This file defines unbundled multiplicative and additive subgroups. Instead of using this file,
-please use `Subgroup G` and `AddSubgroup A`, defined in `Mathlib.Algebra.Group.Subgroup.Basic`.
-
-## Main definitions
-
-`IsAddSubgroup (S : Set A)` : the predicate that `S` is the underlying subset of an additive
-subgroup of `A`. The bundled variant `AddSubgroup A` should be used in preference to this.
-
-`IsSubgroup (S : Set G)` : the predicate that `S` is the underlying subset of a subgroup
-of `G`. The bundled variant `Subgroup G` should be used in preference to this.
-
-## Tags
-
-subgroup, subgroups, IsSubgroup
--/
-
-
-open Set Function
-
-variable {G : Type*} {H : Type*} {A : Type*} {a b : G}
-
-section Group
-
-variable [Group G] [AddGroup A]
-
-/-- `s` is an additive subgroup: a set containing 0 and closed under addition and negation. -/
-structure IsAddSubgroup (s : Set A) extends IsAddSubmonoid s : Prop where
-  /-- The proposition that `s` is closed under negation. -/
-  neg_mem {a} : a ∈ s → -a ∈ s
-
-/-- `s` is a subgroup: a set containing 1 and closed under multiplication and inverse. -/
-@[to_additive]
-structure IsSubgroup (s : Set G) extends IsSubmonoid s : Prop where
-  /-- The proposition that `s` is closed under inverse. -/
-  inv_mem {a} : a ∈ s → a⁻¹ ∈ s
-
-@[to_additive]
-theorem IsSubgroup.div_mem {s : Set G} (hs : IsSubgroup s) {x y : G} (hx : x ∈ s) (hy : y ∈ s) :
-    x / y ∈ s := by simpa only [div_eq_mul_inv] using hs.mul_mem hx (hs.inv_mem hy)
-
-theorem Additive.isAddSubgroup {s : Set G} (hs : IsSubgroup s) : @IsAddSubgroup (Additive G) _ s :=
-  @IsAddSubgroup.mk (Additive G) _ _ (Additive.isAddSubmonoid hs.toIsSubmonoid) hs.inv_mem
-
-theorem Additive.isAddSubgroup_iff {s : Set G} : @IsAddSubgroup (Additive G) _ s ↔ IsSubgroup s :=
-  ⟨by rintro ⟨⟨h₁, h₂⟩, h₃⟩; exact @IsSubgroup.mk G _ _ ⟨h₁, @h₂⟩ @h₃, fun h =>
-    Additive.isAddSubgroup h⟩
-
-theorem Multiplicative.isSubgroup {s : Set A} (hs : IsAddSubgroup s) :
-    @IsSubgroup (Multiplicative A) _ s :=
-  @IsSubgroup.mk (Multiplicative A) _ _ (Multiplicative.isSubmonoid hs.toIsAddSubmonoid) hs.neg_mem
-
-theorem Multiplicative.isSubgroup_iff {s : Set A} :
-    @IsSubgroup (Multiplicative A) _ s ↔ IsAddSubgroup s :=
-  ⟨by rintro ⟨⟨h₁, h₂⟩, h₃⟩; exact @IsAddSubgroup.mk A _ _ ⟨h₁, @h₂⟩ @h₃, fun h =>
-    Multiplicative.isSubgroup h⟩
-
-@[to_additive of_add_neg]
-theorem IsSubgroup.of_div (s : Set G) (one_mem : (1 : G) ∈ s)
-    (div_mem : ∀ {a b : G}, a ∈ s → b ∈ s → a * b⁻¹ ∈ s) : IsSubgroup s :=
-  have inv_mem : ∀ a, a ∈ s → a⁻¹ ∈ s := fun a ha => by
-    have : 1 * a⁻¹ ∈ s := div_mem one_mem ha
-    convert this using 1
-    rw [one_mul]
-  { inv_mem := inv_mem _
-    mul_mem := fun {a b} ha hb => by
-      have : a * b⁻¹⁻¹ ∈ s := div_mem ha (inv_mem b hb)
-      convert this
-      rw [inv_inv]
-    one_mem }
-
-theorem IsAddSubgroup.of_sub (s : Set A) (zero_mem : (0 : A) ∈ s)
-    (sub_mem : ∀ {a b : A}, a ∈ s → b ∈ s → a - b ∈ s) : IsAddSubgroup s :=
-  IsAddSubgroup.of_add_neg s zero_mem fun {x y} hx hy => by
-    simpa only [sub_eq_add_neg] using sub_mem hx hy
-
-@[to_additive]
-theorem IsSubgroup.inter {s₁ s₂ : Set G} (hs₁ : IsSubgroup s₁) (hs₂ : IsSubgroup s₂) :
-    IsSubgroup (s₁ ∩ s₂) :=
-  { IsSubmonoid.inter hs₁.toIsSubmonoid hs₂.toIsSubmonoid with
-    inv_mem := fun hx => ⟨hs₁.inv_mem hx.1, hs₂.inv_mem hx.2⟩ }
-
-@[to_additive]
-theorem IsSubgroup.iInter {ι : Sort*} {s : ι → Set G} (hs : ∀ y : ι, IsSubgroup (s y)) :
-    IsSubgroup (Set.iInter s) :=
-  { IsSubmonoid.iInter fun y => (hs y).toIsSubmonoid with
-    inv_mem := fun h =>
-      Set.mem_iInter.2 fun y => IsSubgroup.inv_mem (hs _) (Set.mem_iInter.1 h y) }
-
-@[to_additive]
-theorem isSubgroup_iUnion_of_directed {ι : Type*} [Nonempty ι] {s : ι → Set G}
-    (hs : ∀ i, IsSubgroup (s i)) (directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :
-    IsSubgroup (⋃ i, s i) :=
-  { inv_mem := fun ha =>
-      let ⟨i, hi⟩ := Set.mem_iUnion.1 ha
-      Set.mem_iUnion.2 ⟨i, (hs i).inv_mem hi⟩
-    toIsSubmonoid := isSubmonoid_iUnion_of_directed (fun i => (hs i).toIsSubmonoid) directed }
-
-end Group
-
-namespace IsSubgroup
-
-open IsSubmonoid
-
-variable [Group G] {s : Set G} (hs : IsSubgroup s)
-include hs
-
-@[to_additive]
-theorem inv_mem_iff : a⁻¹ ∈ s ↔ a ∈ s :=
-  ⟨fun h => by simpa using hs.inv_mem h, inv_mem hs⟩
-
-@[to_additive]
-theorem mul_mem_cancel_right (h : a ∈ s) : b * a ∈ s ↔ b ∈ s :=
-  ⟨fun hba => by simpa using hs.mul_mem hba (hs.inv_mem h), fun hb => hs.mul_mem hb h⟩
-
-@[to_additive]
-theorem mul_mem_cancel_left (h : a ∈ s) : a * b ∈ s ↔ b ∈ s :=
-  ⟨fun hab => by simpa using hs.mul_mem (hs.inv_mem h) hab, hs.mul_mem h⟩
-
-end IsSubgroup
-
-/-- `IsNormalAddSubgroup (s : Set A)` expresses the fact that `s` is a normal additive subgroup
-of the additive group `A`. Important: the preferred way to say this in Lean is via bundled
-subgroups `S : AddSubgroup A` and `hs : S.normal`, and not via this structure. -/
-structure IsNormalAddSubgroup [AddGroup A] (s : Set A) extends IsAddSubgroup s : Prop where
-  /-- The proposition that `s` is closed under (additive) conjugation. -/
-  normal : ∀ n ∈ s, ∀ g : A, g + n + -g ∈ s
-
-/-- `IsNormalSubgroup (s : Set G)` expresses the fact that `s` is a normal subgroup
-of the group `G`. Important: the preferred way to say this in Lean is via bundled
-subgroups `S : Subgroup G` and not via this structure. -/
-@[to_additive]
-structure IsNormalSubgroup [Group G] (s : Set G) extends IsSubgroup s : Prop where
-  /-- The proposition that `s` is closed under conjugation. -/
-  normal : ∀ n ∈ s, ∀ g : G, g * n * g⁻¹ ∈ s
-
-@[to_additive]
-theorem isNormalSubgroup_of_commGroup [CommGroup G] {s : Set G} (hs : IsSubgroup s) :
-    IsNormalSubgroup s :=
-  { hs with normal := fun n hn g => by rwa [mul_right_comm, mul_inv_cancel, one_mul] }
-
-theorem Additive.isNormalAddSubgroup [Group G] {s : Set G} (hs : IsNormalSubgroup s) :
-    @IsNormalAddSubgroup (Additive G) _ s :=
-  @IsNormalAddSubgroup.mk (Additive G) _ _ (Additive.isAddSubgroup hs.toIsSubgroup)
-    (@IsNormalSubgroup.normal _ ‹Group (Additive G)› _ hs)
-    -- Porting note: Lean needs help synthesising
-
-theorem Additive.isNormalAddSubgroup_iff [Group G] {s : Set G} :
-    @IsNormalAddSubgroup (Additive G) _ s ↔ IsNormalSubgroup s :=
-  ⟨by rintro ⟨h₁, h₂⟩; exact @IsNormalSubgroup.mk G _ _ (Additive.isAddSubgroup_iff.1 h₁) @h₂,
-    fun h => Additive.isNormalAddSubgroup h⟩
-
-theorem Multiplicative.isNormalSubgroup [AddGroup A] {s : Set A} (hs : IsNormalAddSubgroup s) :
-    @IsNormalSubgroup (Multiplicative A) _ s :=
-  @IsNormalSubgroup.mk (Multiplicative A) _ _ (Multiplicative.isSubgroup hs.toIsAddSubgroup)
-    (@IsNormalAddSubgroup.normal _ ‹AddGroup (Multiplicative A)› _ hs)
-
-theorem Multiplicative.isNormalSubgroup_iff [AddGroup A] {s : Set A} :
-    @IsNormalSubgroup (Multiplicative A) _ s ↔ IsNormalAddSubgroup s :=
-  ⟨by
-    rintro ⟨h₁, h₂⟩
-    exact @IsNormalAddSubgroup.mk A _ _ (Multiplicative.isSubgroup_iff.1 h₁) @h₂,
-    fun h => Multiplicative.isNormalSubgroup h⟩
-
-namespace IsSubgroup
-
-variable [Group G]
-
--- Normal subgroup properties
-@[to_additive]
-theorem mem_norm_comm {s : Set G} (hs : IsNormalSubgroup s) {a b : G} (hab : a * b ∈ s) :
-    b * a ∈ s := by
-  simpa using (hs.normal (a * b) hab a⁻¹)
-
-@[to_additive]
-theorem mem_norm_comm_iff {s : Set G} (hs : IsNormalSubgroup s) {a b : G} : a * b ∈ s ↔ b * a ∈ s :=
-  ⟨mem_norm_comm hs, mem_norm_comm hs⟩
-
-/-- The trivial subgroup -/
-@[to_additive "the trivial additive subgroup"]
-def trivial (G : Type*) [Group G] : Set G :=
-  {1}
-
-@[to_additive (attr := simp)]
-theorem mem_trivial {g : G} : g ∈ trivial G ↔ g = 1 :=
-  mem_singleton_iff
-
-@[to_additive]
-theorem trivial_normal : IsNormalSubgroup (trivial G) := by refine ⟨⟨⟨?_, ?_⟩, ?_⟩, ?_⟩ <;> simp
-
-@[to_additive]
-theorem eq_trivial_iff {s : Set G} (hs : IsSubgroup s) : s = trivial G ↔ ∀ x ∈ s, x = (1 : G) := by
-  simp only [Set.ext_iff, IsSubgroup.mem_trivial]
-  exact ⟨fun h x => (h x).1, fun h x => ⟨h x, fun hx => hx.symm ▸ hs.toIsSubmonoid.one_mem⟩⟩
-
-@[to_additive]
-theorem univ_subgroup : IsNormalSubgroup (@univ G) := by refine ⟨⟨⟨?_, ?_⟩, ?_⟩, ?_⟩ <;> simp
-
-/-- The underlying set of the center of a group. -/
-@[to_additive addCenter "The underlying set of the center of an additive group."]
-def center (G : Type*) [Group G] : Set G :=
-  { z | ∀ g, g * z = z * g }
-
-@[to_additive mem_add_center]
-theorem mem_center {a : G} : a ∈ center G ↔ ∀ g, g * a = a * g :=
-  Iff.rfl
-
-@[to_additive add_center_normal]
-theorem center_normal : IsNormalSubgroup (center G) :=
-  { one_mem := by simp [center]
-    mul_mem := fun ha hb g => by
-      rw [← mul_assoc, mem_center.2 ha g, mul_assoc, mem_center.2 hb g, ← mul_assoc]
-    inv_mem := fun {a} ha g =>
-      calc
-        g * a⁻¹ = a⁻¹ * (g * a) * a⁻¹ := by simp [ha g]
-        _ = a⁻¹ * g := by rw [← mul_assoc, mul_assoc]; simp
-    normal := fun n ha g h =>
-      calc
-        h * (g * n * g⁻¹) = h * n := by simp [ha g, mul_assoc]
-        _ = g * g⁻¹ * n * h := by rw [ha h]; simp
-        _ = g * n * g⁻¹ * h := by rw [mul_assoc g, ha g⁻¹, ← mul_assoc]
-         }
-
-/-- The underlying set of the normalizer of a subset `S : Set G` of a group `G`. That is,
-  the elements `g : G` such that `g * S * g⁻¹ = S`. -/
-@[to_additive addNormalizer
-      "The underlying set of the normalizer of a subset `S : Set A` of an
-      additive group `A`. That is, the elements `a : A` such that `a + S - a = S`."]
-def normalizer (s : Set G) : Set G :=
-  { g : G | ∀ n, n ∈ s ↔ g * n * g⁻¹ ∈ s }
-
-@[to_additive]
-theorem normalizer_isSubgroup (s : Set G) : IsSubgroup (normalizer s) :=
-  { one_mem := by simp [normalizer]
-    mul_mem := fun {a b}
-      (ha : ∀ n, n ∈ s ↔ a * n * a⁻¹ ∈ s) (hb : ∀ n, n ∈ s ↔ b * n * b⁻¹ ∈ s) n =>
-      by rw [mul_inv_rev, ← mul_assoc, mul_assoc a, mul_assoc a, ← ha, ← hb]
-    inv_mem := fun {a} (ha : ∀ n, n ∈ s ↔ a * n * a⁻¹ ∈ s) n => by
-      rw [ha (a⁻¹ * n * a⁻¹⁻¹)]; simp [mul_assoc] }
-
-@[to_additive subset_add_normalizer]
-theorem subset_normalizer {s : Set G} (hs : IsSubgroup s) : s ⊆ normalizer s := fun g hg n => by
-  rw [IsSubgroup.mul_mem_cancel_right hs ((IsSubgroup.inv_mem_iff hs).2 hg),
-    IsSubgroup.mul_mem_cancel_left hs hg]
-
-end IsSubgroup
-
--- Homomorphism subgroups
-namespace IsGroupHom
-
-open IsSubmonoid IsSubgroup
-
-/-- `ker f : Set G` is the underlying subset of the kernel of a map `G → H`. -/
-@[to_additive "`ker f : Set A` is the underlying subset of the kernel of a map `A → B`"]
-def ker [Group H] (f : G → H) : Set G :=
-  preimage f (trivial H)
-
-@[to_additive]
-theorem mem_ker [Group H] (f : G → H) {x : G} : x ∈ ker f ↔ f x = 1 :=
-  mem_trivial
-
-variable [Group G] [Group H]
-
-@[to_additive]
-theorem one_ker_inv {f : G → H} (hf : IsGroupHom f) {a b : G} (h : f (a * b⁻¹) = 1) :
-    f a = f b := by
-  rw [hf.map_mul, hf.map_inv] at h
-  rw [← inv_inv (f b), eq_inv_of_mul_eq_one_left h]
-
-@[to_additive]
-theorem one_ker_inv' {f : G → H} (hf : IsGroupHom f) {a b : G} (h : f (a⁻¹ * b) = 1) :
-    f a = f b := by
-  rw [hf.map_mul, hf.map_inv] at h
-  apply inv_injective
-  rw [eq_inv_of_mul_eq_one_left h]
-
-@[to_additive]
-theorem inv_ker_one {f : G → H} (hf : IsGroupHom f) {a b : G} (h : f a = f b) :
-    f (a * b⁻¹) = 1 := by
-  have : f a * (f b)⁻¹ = 1 := by rw [h, mul_inv_cancel]
-  rwa [← hf.map_inv, ← hf.map_mul] at this
-
-@[to_additive]
-theorem inv_ker_one' {f : G → H} (hf : IsGroupHom f) {a b : G} (h : f a = f b) :
-    f (a⁻¹ * b) = 1 := by
-  have : (f a)⁻¹ * f b = 1 := by rw [h, inv_mul_cancel]
-  rwa [← hf.map_inv, ← hf.map_mul] at this
-
-@[to_additive]
-theorem one_iff_ker_inv {f : G → H} (hf : IsGroupHom f) (a b : G) : f a = f b ↔ f (a * b⁻¹) = 1 :=
-  ⟨hf.inv_ker_one, hf.one_ker_inv⟩
-
-@[to_additive]
-theorem one_iff_ker_inv' {f : G → H} (hf : IsGroupHom f) (a b : G) : f a = f b ↔ f (a⁻¹ * b) = 1 :=
-  ⟨hf.inv_ker_one', hf.one_ker_inv'⟩
-
-@[to_additive]
-theorem inv_iff_ker {f : G → H} (hf : IsGroupHom f) (a b : G) : f a = f b ↔ a * b⁻¹ ∈ ker f := by
-  rw [mem_ker]; exact one_iff_ker_inv hf _ _
-
-@[to_additive]
-theorem inv_iff_ker' {f : G → H} (hf : IsGroupHom f) (a b : G) : f a = f b ↔ a⁻¹ * b ∈ ker f := by
-  rw [mem_ker]; exact one_iff_ker_inv' hf _ _
-
-@[to_additive]
-theorem image_subgroup {f : G → H} (hf : IsGroupHom f) {s : Set G} (hs : IsSubgroup s) :
-    IsSubgroup (f '' s) :=
-  { mul_mem := fun {a₁ a₂} ⟨b₁, hb₁, eq₁⟩ ⟨b₂, hb₂, eq₂⟩ =>
-      ⟨b₁ * b₂, hs.mul_mem hb₁ hb₂, by simp [eq₁, eq₂, hf.map_mul]⟩
-    one_mem := ⟨1, hs.toIsSubmonoid.one_mem, hf.map_one⟩
-    inv_mem := fun {a} ⟨b, hb, Eq⟩ =>
-      ⟨b⁻¹, hs.inv_mem hb, by
-        rw [hf.map_inv]
-        simp [*]⟩ }
-
-@[to_additive]
-theorem range_subgroup {f : G → H} (hf : IsGroupHom f) : IsSubgroup (Set.range f) :=
-  @Set.image_univ _ _ f ▸ hf.image_subgroup univ_subgroup.toIsSubgroup
-
-attribute [local simp] IsSubmonoid.one_mem IsSubgroup.inv_mem
-  IsSubmonoid.mul_mem IsNormalSubgroup.normal
-
-@[to_additive]
-theorem preimage {f : G → H} (hf : IsGroupHom f) {s : Set H} (hs : IsSubgroup s) :
-    IsSubgroup (f ⁻¹' s) where
-  one_mem := by simp [hf.map_one, hs.one_mem]
-  mul_mem := by simp_all [hf.map_mul, hs.mul_mem]
-  inv_mem := by simp_all [hf.map_inv]
-
-@[to_additive]
-theorem preimage_normal {f : G → H} (hf : IsGroupHom f) {s : Set H} (hs : IsNormalSubgroup s) :
-    IsNormalSubgroup (f ⁻¹' s) :=
-  { one_mem := by simp [hf.map_one, hs.toIsSubgroup.one_mem]
-    mul_mem := by simp +contextual [hf.map_mul, hs.toIsSubgroup.mul_mem]
-    inv_mem := by simp +contextual [hf.map_inv, hs.toIsSubgroup.inv_mem]
-    normal := by simp +contextual [hs.normal, hf.map_mul, hf.map_inv] }
-
-@[to_additive]
-theorem isNormalSubgroup_ker {f : G → H} (hf : IsGroupHom f) : IsNormalSubgroup (ker f) :=
-  hf.preimage_normal trivial_normal
-
-@[to_additive]
-theorem injective_of_trivial_ker {f : G → H} (hf : IsGroupHom f) (h : ker f = trivial G) :
-    Function.Injective f := by
-  intro a₁ a₂ hfa
-  simp [Set.ext_iff, ker, IsSubgroup.trivial] at h
-  have ha : a₁ * a₂⁻¹ = 1 := by rw [← h]; exact hf.inv_ker_one hfa
-  rw [eq_inv_of_mul_eq_one_left ha, inv_inv a₂]
-
-@[to_additive]
-theorem trivial_ker_of_injective {f : G → H} (hf : IsGroupHom f) (h : Function.Injective f) :
-    ker f = trivial G :=
-  Set.ext fun x =>
-    Iff.intro
-      (fun hx => by
-        suffices f x = f 1 by simpa using h this
-        simp [hf.map_one]; rwa [mem_ker] at hx)
-      (by simp +contextual [mem_ker, hf.map_one])
-
-@[to_additive]
-theorem injective_iff_trivial_ker {f : G → H} (hf : IsGroupHom f) :
-    Function.Injective f ↔ ker f = trivial G :=
-  ⟨hf.trivial_ker_of_injective, hf.injective_of_trivial_ker⟩
-
-@[to_additive]
-theorem trivial_ker_iff_eq_one {f : G → H} (hf : IsGroupHom f) :
-    ker f = trivial G ↔ ∀ x, f x = 1 → x = 1 := by
-  rw [Set.ext_iff]
-  simpa [ker] using ⟨fun h x hx => (h x).1 hx, fun h x => ⟨h x, fun hx => by rw [hx, hf.map_one]⟩⟩
-
-end IsGroupHom
-
-namespace AddGroup
-
-variable [AddGroup A]
-
-/-- If `A` is an additive group and `s : Set A`, then `InClosure s : Set A` is the underlying
-subset of the subgroup generated by `s`. -/
-inductive InClosure (s : Set A) : A → Prop
-  | basic {a : A} : a ∈ s → InClosure s a
-  | zero : InClosure s 0
-  | neg {a : A} : InClosure s a → InClosure s (-a)
-  | add {a b : A} : InClosure s a → InClosure s b → InClosure s (a + b)
-
-end AddGroup
-
-namespace Group
-
-open IsSubmonoid IsSubgroup
-
-variable [Group G] {s : Set G}
-
-/-- If `G` is a group and `s : Set G`, then `InClosure s : Set G` is the underlying
-subset of the subgroup generated by `s`. -/
-@[to_additive]
-inductive InClosure (s : Set G) : G → Prop
-  | basic {a : G} : a ∈ s → InClosure s a
-  | one : InClosure s 1
-  | inv {a : G} : InClosure s a → InClosure s a⁻¹
-  | mul {a b : G} : InClosure s a → InClosure s b → InClosure s (a * b)
-
-/-- `Group.closure s` is the subgroup generated by `s`, i.e. the smallest subgroup containing `s`.
--/
-@[to_additive
-  "`AddGroup.closure s` is the additive subgroup generated by `s`, i.e., the
-  smallest additive subgroup containing `s`."]
-def closure (s : Set G) : Set G :=
-  { a | InClosure s a }
-
-@[to_additive]
-theorem mem_closure {a : G} : a ∈ s → a ∈ closure s :=
-  InClosure.basic
-
-@[to_additive]
-theorem closure.isSubgroup (s : Set G) : IsSubgroup (closure s) :=
-  { one_mem := InClosure.one
-    mul_mem := InClosure.mul
-    inv_mem := InClosure.inv }
-
-@[to_additive]
-theorem subset_closure {s : Set G} : s ⊆ closure s := fun _ => mem_closure
-
-@[to_additive]
-theorem closure_subset {s t : Set G} (ht : IsSubgroup t) (h : s ⊆ t) : closure s ⊆ t := fun a ha =>
-  by induction ha <;> simp [h _, *, ht.one_mem, ht.mul_mem, IsSubgroup.inv_mem_iff]
-
-@[to_additive]
-theorem closure_subset_iff {s t : Set G} (ht : IsSubgroup t) : closure s ⊆ t ↔ s ⊆ t :=
-  ⟨fun h _ ha => h (mem_closure ha), fun h _ ha => closure_subset ht h ha⟩
-
-@[to_additive (attr := gcongr)]
-theorem closure_mono {s t : Set G} (h : s ⊆ t) : closure s ⊆ closure t :=
-  closure_subset (closure.isSubgroup _) <| Set.Subset.trans h subset_closure
-
-@[to_additive (attr := simp)]
-theorem closure_subgroup {s : Set G} (hs : IsSubgroup s) : closure s = s :=
-  Set.Subset.antisymm (closure_subset hs <| Set.Subset.refl s) subset_closure
-
-@[to_additive]
-theorem exists_list_of_mem_closure {s : Set G} {a : G} (h : a ∈ closure s) :
-    ∃ l : List G, (∀ x ∈ l, x ∈ s ∨ x⁻¹ ∈ s) ∧ l.prod = a :=
-  InClosure.recOn h
-    (fun {x} hxs => ⟨[x], List.forall_mem_singleton.2 <| Or.inl hxs, List.prod_singleton⟩)
-    ⟨[], List.forall_mem_nil _, rfl⟩
-    (fun {_} _ ⟨L, HL1, HL2⟩ =>
-      ⟨L.reverse.map Inv.inv, fun _ hx =>
-        let ⟨y, hy1, hy2⟩ := List.exists_of_mem_map hx
-        hy2 ▸ Or.imp id (by rw [inv_inv]; exact id) (HL1 _ <| List.mem_reverse.1 hy1).symm,
-        HL2 ▸
-          List.recOn L inv_one.symm fun hd tl ih => by
-            rw [List.reverse_cons, List.map_append, List.prod_append, ih, List.map_singleton,
-              List.prod_cons, List.prod_nil, mul_one, List.prod_cons, mul_inv_rev]⟩)
-    fun {x y} _ _ ⟨L1, HL1, HL2⟩ ⟨L2, HL3, HL4⟩ =>
-    ⟨L1 ++ L2, List.forall_mem_append.2 ⟨HL1, HL3⟩, by rw [List.prod_append, HL2, HL4]⟩
-
-@[to_additive]
-theorem image_closure [Group H] {f : G → H} (hf : IsGroupHom f) (s : Set G) :
-    f '' closure s = closure (f '' s) :=
-  le_antisymm
-    (by
-      rintro _ ⟨x, hx, rfl⟩
-      exact InClosure.recOn hx
-        (by intros _ ha; exact subset_closure (mem_image_of_mem f ha))
-        (by
-          rw [hf.map_one]
-          apply IsSubmonoid.one_mem (closure.isSubgroup _).toIsSubmonoid)
-        (by
-          intros _ _
-          rw [hf.map_inv]
-          apply IsSubgroup.inv_mem (closure.isSubgroup _))
-        (by
-          intros _ _ _ _ ha hb
-          rw [hf.map_mul]
-          exact (closure.isSubgroup (f '' s)).toIsSubmonoid.mul_mem ha hb))
-    (closure_subset (hf.image_subgroup <| closure.isSubgroup _) <|
-      Set.image_subset _ subset_closure)
-
-@[to_additive]
-theorem mclosure_subset {s : Set G} : Monoid.Closure s ⊆ closure s :=
-  Monoid.closure_subset (closure.isSubgroup _).toIsSubmonoid <| subset_closure
-
-@[to_additive]
-theorem mclosure_inv_subset {s : Set G} : Monoid.Closure (Inv.inv ⁻¹' s) ⊆ closure s :=
-  Monoid.closure_subset (closure.isSubgroup _).toIsSubmonoid fun x hx =>
-    inv_inv x ▸ ((closure.isSubgroup _).inv_mem <| subset_closure hx)
-
-@[to_additive]
-theorem closure_eq_mclosure {s : Set G} : closure s = Monoid.Closure (s ∪ Inv.inv ⁻¹' s) :=
-  Set.Subset.antisymm
-    (@closure_subset _ _ _ (Monoid.Closure (s ∪ Inv.inv ⁻¹' s))
-      { one_mem := (Monoid.closure.isSubmonoid _).one_mem
-        mul_mem := (Monoid.closure.isSubmonoid _).mul_mem
-        inv_mem := fun hx =>
-          Monoid.InClosure.recOn hx
-            (fun {x} hx =>
-              Or.casesOn hx
-                (fun hx =>
-                  Monoid.subset_closure <| Or.inr <| show x⁻¹⁻¹ ∈ s from (inv_inv x).symm ▸ hx)
-                fun hx => Monoid.subset_closure <| Or.inl hx)
-            ((@inv_one G _).symm ▸ IsSubmonoid.one_mem (Monoid.closure.isSubmonoid _))
-            fun {x y} _ _ ihx ihy =>
-            (mul_inv_rev x y).symm ▸ IsSubmonoid.mul_mem (Monoid.closure.isSubmonoid _) ihy ihx }
-      (Set.Subset.trans Set.subset_union_left Monoid.subset_closure))
-    (Monoid.closure_subset (closure.isSubgroup _).toIsSubmonoid <|
-      Set.union_subset subset_closure fun x hx =>
-        inv_inv x ▸ (IsSubgroup.inv_mem (closure.isSubgroup _) <| subset_closure hx))
-
-@[to_additive]
-theorem mem_closure_union_iff {G : Type*} [CommGroup G] {s t : Set G} {x : G} :
-    x ∈ closure (s ∪ t) ↔ ∃ y ∈ closure s, ∃ z ∈ closure t, y * z = x := by
-  simp only [closure_eq_mclosure, Monoid.mem_closure_union_iff, exists_prop, preimage_union]
-  constructor
-  · rintro ⟨_, ⟨ys, hys, yt, hyt, rfl⟩, _, ⟨zs, hzs, zt, hzt, rfl⟩, rfl⟩
-    refine ⟨_, ⟨_, hys, _, hzs, rfl⟩, _, ⟨_, hyt, _, hzt, rfl⟩, ?_⟩
-    rw [mul_assoc, mul_assoc, mul_left_comm zs]
-  · rintro ⟨_, ⟨ys, hys, zs, hzs, rfl⟩, _, ⟨yt, hyt, zt, hzt, rfl⟩, rfl⟩
-    refine ⟨_, ⟨ys, hys, yt, hyt, rfl⟩, _, ⟨zs, hzs, zt, hzt, rfl⟩, ?_⟩
-    rw [mul_assoc, mul_assoc, mul_left_comm yt]
-
-end Group
-
-namespace IsSubgroup
-
-variable [Group G]
-
-@[to_additive]
-theorem trivial_eq_closure : trivial G = Group.closure ∅ :=
-  Subset.antisymm (by simp [Set.subset_def, (Group.closure.isSubgroup _).one_mem])
-    (Group.closure_subset trivial_normal.toIsSubgroup <| by simp)
-
-end IsSubgroup
-
-/- The normal closure of a set s is the subgroup closure of all the conjugates of
-elements of s. It is the smallest normal subgroup containing s. -/
-namespace Group
-
-variable {s : Set G} [Group G]
-
-theorem conjugatesOf_subset {t : Set G} (ht : IsNormalSubgroup t) {a : G} (h : a ∈ t) :
-    conjugatesOf a ⊆ t := fun x hc => by
-  obtain ⟨c, w⟩ := isConj_iff.1 hc
-  have H := IsNormalSubgroup.normal ht a h c
-  rwa [← w]
-
-theorem conjugatesOfSet_subset' {s t : Set G} (ht : IsNormalSubgroup t) (h : s ⊆ t) :
-    conjugatesOfSet s ⊆ t :=
-  Set.iUnion₂_subset fun _ H => conjugatesOf_subset ht (h H)
-
-/-- The normal closure of a set s is the subgroup closure of all the conjugates of
-elements of s. It is the smallest normal subgroup containing s. -/
-def normalClosure (s : Set G) : Set G :=
-  closure (conjugatesOfSet s)
-
-theorem conjugatesOfSet_subset_normalClosure : conjugatesOfSet s ⊆ normalClosure s :=
-  subset_closure
-
-theorem subset_normalClosure : s ⊆ normalClosure s :=
-  Set.Subset.trans subset_conjugatesOfSet conjugatesOfSet_subset_normalClosure
-
-/-- The normal closure of a set is a subgroup. -/
-theorem normalClosure.isSubgroup (s : Set G) : IsSubgroup (normalClosure s) :=
-  closure.isSubgroup (conjugatesOfSet s)
-
-/-- The normal closure of s is a normal subgroup. -/
-theorem normalClosure.is_normal : IsNormalSubgroup (normalClosure s) :=
-  { normalClosure.isSubgroup _ with
-    normal := fun n h g => by
-      induction' h with x hx x hx ihx x y hx hy ihx ihy
-      · exact conjugatesOfSet_subset_normalClosure (conj_mem_conjugatesOfSet hx)
-      · simpa using (normalClosure.isSubgroup s).one_mem
-      · rw [← conj_inv]
-        exact (normalClosure.isSubgroup _).inv_mem ihx
-      · rw [← conj_mul]
-        exact (normalClosure.isSubgroup _).toIsSubmonoid.mul_mem ihx ihy }
-
-/-- The normal closure of s is the smallest normal subgroup containing s. -/
-theorem normalClosure_subset {s t : Set G} (ht : IsNormalSubgroup t) (h : s ⊆ t) :
-    normalClosure s ⊆ t := fun a w => by
-  induction w with
-  | basic hx => exact conjugatesOfSet_subset' ht h <| hx
-  | one => exact ht.toIsSubgroup.toIsSubmonoid.one_mem
-  | inv _ ihx => exact ht.toIsSubgroup.inv_mem ihx
-  | mul _ _ ihx ihy => exact ht.toIsSubgroup.toIsSubmonoid.mul_mem ihx ihy
-
-theorem normalClosure_subset_iff {s t : Set G} (ht : IsNormalSubgroup t) :
-    s ⊆ t ↔ normalClosure s ⊆ t :=
-  ⟨normalClosure_subset ht, Set.Subset.trans subset_normalClosure⟩
-
-@[gcongr]
-theorem normalClosure_mono {s t : Set G} : s ⊆ t → normalClosure s ⊆ normalClosure t := fun h =>
-  normalClosure_subset normalClosure.is_normal (Set.Subset.trans h subset_normalClosure)
-
-end Group
-
-/-- Create a bundled subgroup from a set `s` and `[IsSubgroup s]`. -/
-@[to_additive "Create a bundled additive subgroup from a set `s` and `[IsAddSubgroup s]`."]
-def Subgroup.of [Group G] {s : Set G} (h : IsSubgroup s) : Subgroup G where
-  carrier := s
-  one_mem' := h.1.1
-  mul_mem' := h.1.2
-  inv_mem' := h.2
-
-@[to_additive]
-theorem Subgroup.isSubgroup [Group G] (K : Subgroup G) : IsSubgroup (K : Set G) :=
-  { one_mem := K.one_mem'
-    mul_mem := K.mul_mem'
-    inv_mem := K.inv_mem' }
-
--- this will never fire if it's an instance
-@[to_additive]
-theorem Subgroup.of_normal [Group G] (s : Set G) (h : IsSubgroup s) (n : IsNormalSubgroup s) :
-    Subgroup.Normal (Subgroup.of h) :=
-  { conj_mem := n.normal }
diff --git a/Mathlib/Deprecated/Submonoid.lean b/Mathlib/Deprecated/Submonoid.lean
deleted file mode 100644
index 25e8af4f8368fb..00000000000000
--- a/Mathlib/Deprecated/Submonoid.lean
+++ /dev/null
@@ -1,363 +0,0 @@
-/-
-Copyright (c) 2018 Johannes Hölzl. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard
--/
-import Mathlib.Algebra.Group.Submonoid.Defs
-import Mathlib.Deprecated.Group
-import Mathlib.Data.Set.Lattice
-import Mathlib.Algebra.BigOperators.Group.Finset.Defs
-
-/-!
-# Unbundled submonoids (deprecated)
-
-This file is deprecated, and is no longer imported by anything in mathlib other than other
-deprecated files, and test files. You should not need to import it.
-
-This file defines unbundled multiplicative and additive submonoids. Instead of using this file,
-please use `Submonoid G` and `AddSubmonoid A`, defined in `GroupTheory.Submonoid.Basic`.
-
-## Main definitions
-
-`IsAddSubmonoid (S : Set M)` : the predicate that `S` is the underlying subset of an additive
-submonoid of `M`. The bundled variant `AddSubmonoid M` should be used in preference to this.
-
-`IsSubmonoid (S : Set M)` : the predicate that `S` is the underlying subset of a submonoid
-of `M`. The bundled variant `Submonoid M` should be used in preference to this.
-
-## Tags
-Submonoid, Submonoids, IsSubmonoid
--/
-
-
-variable {M : Type*} [Monoid M] {s : Set M} {A : Type*} [AddMonoid A]
-
-/-- `s` is an additive submonoid: a set containing 0 and closed under addition.
-Note that this structure is deprecated, and the bundled variant `AddSubmonoid A` should be
-preferred. -/
-structure IsAddSubmonoid (s : Set A) : Prop where
-  /-- The proposition that s contains 0. -/
-  zero_mem : (0 : A) ∈ s
-  /-- The proposition that s is closed under addition. -/
-  add_mem {a b} : a ∈ s → b ∈ s → a + b ∈ s
-
-/-- `s` is a submonoid: a set containing 1 and closed under multiplication.
-Note that this structure is deprecated, and the bundled variant `Submonoid M` should be
-preferred. -/
-@[to_additive]
-structure IsSubmonoid (s : Set M) : Prop where
-  /-- The proposition that s contains 1. -/
-  one_mem : (1 : M) ∈ s
-  /-- The proposition that s is closed under multiplication. -/
-  mul_mem {a b} : a ∈ s → b ∈ s → a * b ∈ s
-
-theorem Additive.isAddSubmonoid {s : Set M} :
-    IsSubmonoid s → @IsAddSubmonoid (Additive M) _ s
-  | ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩
-
-theorem Additive.isAddSubmonoid_iff {s : Set M} :
-    @IsAddSubmonoid (Additive M) _ s ↔ IsSubmonoid s :=
-  ⟨fun ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩, Additive.isAddSubmonoid⟩
-
-theorem Multiplicative.isSubmonoid {s : Set A} :
-    IsAddSubmonoid s → @IsSubmonoid (Multiplicative A) _ s
-  | ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩
-
-theorem Multiplicative.isSubmonoid_iff {s : Set A} :
-    @IsSubmonoid (Multiplicative A) _ s ↔ IsAddSubmonoid s :=
-  ⟨fun ⟨h₁, h₂⟩ => ⟨h₁, @h₂⟩, Multiplicative.isSubmonoid⟩
-
-/-- The intersection of two submonoids of a monoid `M` is a submonoid of `M`. -/
-@[to_additive
-      "The intersection of two `AddSubmonoid`s of an `AddMonoid` `M` is an `AddSubmonoid` of M."]
-theorem IsSubmonoid.inter {s₁ s₂ : Set M} (is₁ : IsSubmonoid s₁) (is₂ : IsSubmonoid s₂) :
-    IsSubmonoid (s₁ ∩ s₂) :=
-  { one_mem := ⟨is₁.one_mem, is₂.one_mem⟩
-    mul_mem := @fun _ _ hx hy => ⟨is₁.mul_mem hx.1 hy.1, is₂.mul_mem hx.2 hy.2⟩ }
-
-/-- The intersection of an indexed set of submonoids of a monoid `M` is a submonoid of `M`. -/
-@[to_additive
-      "The intersection of an indexed set of `AddSubmonoid`s of an `AddMonoid` `M` is
-      an `AddSubmonoid` of `M`."]
-theorem IsSubmonoid.iInter {ι : Sort*} {s : ι → Set M} (h : ∀ y : ι, IsSubmonoid (s y)) :
-    IsSubmonoid (Set.iInter s) :=
-  { one_mem := Set.mem_iInter.2 fun y => (h y).one_mem
-    mul_mem := fun h₁ h₂ =>
-      Set.mem_iInter.2 fun y => (h y).mul_mem (Set.mem_iInter.1 h₁ y) (Set.mem_iInter.1 h₂ y) }
-
-/-- The union of an indexed, directed, nonempty set of submonoids of a monoid `M` is a submonoid
-    of `M`. -/
-@[to_additive
-      "The union of an indexed, directed, nonempty set of `AddSubmonoid`s of an `AddMonoid` `M`
-      is an `AddSubmonoid` of `M`. "]
-theorem isSubmonoid_iUnion_of_directed {ι : Type*} [hι : Nonempty ι] {s : ι → Set M}
-    (hs : ∀ i, IsSubmonoid (s i)) (Directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :
-    IsSubmonoid (⋃ i, s i) :=
-  { one_mem :=
-      let ⟨i⟩ := hι
-      Set.mem_iUnion.2 ⟨i, (hs i).one_mem⟩
-    mul_mem := fun ha hb =>
-      let ⟨i, hi⟩ := Set.mem_iUnion.1 ha
-      let ⟨j, hj⟩ := Set.mem_iUnion.1 hb
-      let ⟨k, hk⟩ := Directed i j
-      Set.mem_iUnion.2 ⟨k, (hs k).mul_mem (hk.1 hi) (hk.2 hj)⟩ }
-
-section powers
-
-/-- The set of natural number powers `1, x, x², ...` of an element `x` of a monoid. -/
-@[to_additive
-      "The set of natural number multiples `0, x, 2x, ...` of an element `x` of an `AddMonoid`."]
-def powers (x : M) : Set M :=
-  { y | ∃ n : ℕ, x ^ n = y }
-
-/-- 1 is in the set of natural number powers of an element of a monoid. -/
-@[to_additive "0 is in the set of natural number multiples of an element of an `AddMonoid`."]
-theorem powers.one_mem {x : M} : (1 : M) ∈ powers x :=
-  ⟨0, pow_zero _⟩
-
-/-- An element of a monoid is in the set of that element's natural number powers. -/
-@[to_additive
-      "An element of an `AddMonoid` is in the set of that element's natural number multiples."]
-theorem powers.self_mem {x : M} : x ∈ powers x :=
-  ⟨1, pow_one _⟩
-
-/-- The set of natural number powers of an element of a monoid is closed under multiplication. -/
-@[to_additive
-      "The set of natural number multiples of an element of an `AddMonoid` is closed under
-      addition."]
-theorem powers.mul_mem {x y z : M} : y ∈ powers x → z ∈ powers x → y * z ∈ powers x :=
-  fun ⟨n₁, h₁⟩ ⟨n₂, h₂⟩ => ⟨n₁ + n₂, by simp only [pow_add, *]⟩
-
-/-- The set of natural number powers of an element of a monoid `M` is a submonoid of `M`. -/
-@[to_additive
-      "The set of natural number multiples of an element of an `AddMonoid` `M` is
-      an `AddSubmonoid` of `M`."]
-theorem powers.isSubmonoid (x : M) : IsSubmonoid (powers x) :=
-  { one_mem := powers.one_mem
-    mul_mem := powers.mul_mem }
-
-/-- A monoid is a submonoid of itself. -/
-@[to_additive "An `AddMonoid` is an `AddSubmonoid` of itself."]
-theorem Univ.isSubmonoid : IsSubmonoid (@Set.univ M) := by constructor <;> simp
-
-/-- The preimage of a submonoid under a monoid hom is a submonoid of the domain. -/
-@[to_additive
-      "The preimage of an `AddSubmonoid` under an `AddMonoid` hom is
-      an `AddSubmonoid` of the domain."]
-theorem IsSubmonoid.preimage {N : Type*} [Monoid N] {f : M → N} (hf : IsMonoidHom f) {s : Set N}
-    (hs : IsSubmonoid s) : IsSubmonoid (f ⁻¹' s) :=
-  { one_mem := show f 1 ∈ s by (rw [IsMonoidHom.map_one hf]; exact hs.one_mem)
-    mul_mem := fun {a b} (ha : f a ∈ s) (hb : f b ∈ s) =>
-      show f (a * b) ∈ s by (rw [IsMonoidHom.map_mul' hf]; exact hs.mul_mem ha hb) }
-
-/-- The image of a submonoid under a monoid hom is a submonoid of the codomain. -/
-@[to_additive
-      "The image of an `AddSubmonoid` under an `AddMonoid` hom is an `AddSubmonoid` of the
-      codomain."]
-theorem IsSubmonoid.image {γ : Type*} [Monoid γ] {f : M → γ} (hf : IsMonoidHom f) {s : Set M}
-    (hs : IsSubmonoid s) : IsSubmonoid (f '' s) :=
-  { one_mem := ⟨1, hs.one_mem, hf.map_one⟩
-    mul_mem := @fun a b ⟨x, hx⟩ ⟨y, hy⟩ =>
-      ⟨x * y, hs.mul_mem hx.1 hy.1, by rw [hf.map_mul, hx.2, hy.2]⟩ }
-
-/-- The image of a monoid hom is a submonoid of the codomain. -/
-@[to_additive "The image of an `AddMonoid` hom is an `AddSubmonoid` of the codomain."]
-theorem Range.isSubmonoid {γ : Type*} [Monoid γ] {f : M → γ} (hf : IsMonoidHom f) :
-    IsSubmonoid (Set.range f) := by
-  rw [← Set.image_univ]
-  exact Univ.isSubmonoid.image hf
-
-/-- Submonoids are closed under natural powers. -/
-@[to_additive
-      "An `AddSubmonoid` is closed under multiplication by naturals."]
-theorem IsSubmonoid.pow_mem {a : M} (hs : IsSubmonoid s) (h : a ∈ s) : ∀ {n : ℕ}, a ^ n ∈ s
-  | 0 => by
-    rw [pow_zero]
-    exact hs.one_mem
-  | n + 1 => by
-    rw [pow_succ]
-    exact hs.mul_mem (IsSubmonoid.pow_mem hs h) h
-
-/-- The set of natural number powers of an element of a `Submonoid` is a subset of the
-`Submonoid`. -/
-@[to_additive
-      "The set of natural number multiples of an element of an `AddSubmonoid` is a subset of
-      the `AddSubmonoid`."]
-theorem IsSubmonoid.powers_subset {a : M} (hs : IsSubmonoid s) (h : a ∈ s) : powers a ⊆ s :=
-  fun _ ⟨_, hx⟩ => hx ▸ hs.pow_mem h
-
-end powers
-
-namespace IsSubmonoid
-
-/-- The product of a list of elements of a submonoid is an element of the submonoid. -/
-@[to_additive
-      "The sum of a list of elements of an `AddSubmonoid` is an element of the `AddSubmonoid`."]
-theorem list_prod_mem (hs : IsSubmonoid s) : ∀ {l : List M}, (∀ x ∈ l, x ∈ s) → l.prod ∈ s
-  | [], _ => hs.one_mem
-  | a :: l, h =>
-    suffices a * l.prod ∈ s by simpa
-    have : a ∈ s ∧ ∀ x ∈ l, x ∈ s := by simpa using h
-    hs.mul_mem this.1 (list_prod_mem hs this.2)
-
-/-- The product of a multiset of elements of a submonoid of a `CommMonoid` is an element of
-the submonoid. -/
-@[to_additive
-      "The sum of a multiset of elements of an `AddSubmonoid` of an `AddCommMonoid`
-      is an element of the `AddSubmonoid`. "]
-theorem multiset_prod_mem {M} [CommMonoid M] {s : Set M} (hs : IsSubmonoid s) (m : Multiset M) :
-    (∀ a ∈ m, a ∈ s) → m.prod ∈ s := by
-  refine Quotient.inductionOn m fun l hl => ?_
-  rw [Multiset.quot_mk_to_coe, Multiset.prod_coe]
-  exact list_prod_mem hs hl
-
-/-- The product of elements of a submonoid of a `CommMonoid` indexed by a `Finset` is an element
-of the submonoid. -/
-@[to_additive
-      "The sum of elements of an `AddSubmonoid` of an `AddCommMonoid` indexed by
-      a `Finset` is an element of the `AddSubmonoid`."]
-theorem finset_prod_mem {M A} [CommMonoid M] {s : Set M} (hs : IsSubmonoid s) (f : A → M) :
-    ∀ t : Finset A, (∀ b ∈ t, f b ∈ s) → (∏ b ∈ t, f b) ∈ s
-  | ⟨m, hm⟩, _ => multiset_prod_mem hs _ (by simpa)
-
-end IsSubmonoid
-
-namespace AddMonoid
-
-/-- The inductively defined membership predicate for the submonoid generated by a subset of a
-    monoid. -/
-inductive InClosure (s : Set A) : A → Prop
-  | basic {a : A} : a ∈ s → InClosure _ a
-  | zero : InClosure _ 0
-  | add {a b : A} : InClosure _ a → InClosure _ b → InClosure _ (a + b)
-
-end AddMonoid
-
-namespace Monoid
-
-/-- The inductively defined membership predicate for the `Submonoid` generated by a subset of an
-    monoid. -/
-@[to_additive]
-inductive InClosure (s : Set M) : M → Prop
-  | basic {a : M} : a ∈ s → InClosure _ a
-  | one : InClosure _ 1
-  | mul {a b : M} : InClosure _ a → InClosure _ b → InClosure _ (a * b)
-
-/-- The inductively defined submonoid generated by a subset of a monoid. -/
-@[to_additive
-      "The inductively defined `AddSubmonoid` generated by a subset of an `AddMonoid`."]
-def Closure (s : Set M) : Set M :=
-  { a | InClosure s a }
-
-@[to_additive]
-theorem closure.isSubmonoid (s : Set M) : IsSubmonoid (Closure s) :=
-  { one_mem := InClosure.one
-    mul_mem := InClosure.mul }
-
-/-- A subset of a monoid is contained in the submonoid it generates. -/
-@[to_additive
-    "A subset of an `AddMonoid` is contained in the `AddSubmonoid` it generates."]
-theorem subset_closure {s : Set M} : s ⊆ Closure s := fun _ => InClosure.basic
-
-/-- The submonoid generated by a set is contained in any submonoid that contains the set. -/
-@[to_additive
-      "The `AddSubmonoid` generated by a set is contained in any `AddSubmonoid` that
-      contains the set."]
-theorem closure_subset {s t : Set M} (ht : IsSubmonoid t) (h : s ⊆ t) : Closure s ⊆ t := fun a ha =>
-  by induction ha <;> simp [h _, *, IsSubmonoid.one_mem, IsSubmonoid.mul_mem]
-
-/-- Given subsets `t` and `s` of a monoid `M`, if `s ⊆ t`, the submonoid of `M` generated by `s` is
-    contained in the submonoid generated by `t`. -/
-@[to_additive (attr := gcongr)
-      "Given subsets `t` and `s` of an `AddMonoid M`, if `s ⊆ t`, the `AddSubmonoid`
-      of `M` generated by `s` is contained in the `AddSubmonoid` generated by `t`."]
-theorem closure_mono {s t : Set M} (h : s ⊆ t) : Closure s ⊆ Closure t :=
-  closure_subset (closure.isSubmonoid t) <| Set.Subset.trans h subset_closure
-
-/-- The submonoid generated by an element of a monoid equals the set of natural number powers of
-    the element. -/
-@[to_additive
-      "The `AddSubmonoid` generated by an element of an `AddMonoid` equals the set of
-      natural number multiples of the element."]
-theorem closure_singleton {x : M} : Closure ({x} : Set M) = powers x :=
-  Set.eq_of_subset_of_subset
-      (closure_subset (powers.isSubmonoid x) <| Set.singleton_subset_iff.2 <| powers.self_mem) <|
-    IsSubmonoid.powers_subset (closure.isSubmonoid _) <|
-      Set.singleton_subset_iff.1 <| subset_closure
-
-/-- The image under a monoid hom of the submonoid generated by a set equals the submonoid generated
-    by the image of the set under the monoid hom. -/
-@[to_additive
-      "The image under an `AddMonoid` hom of the `AddSubmonoid` generated by a set equals
-      the `AddSubmonoid` generated by the image of the set under the `AddMonoid` hom."]
-theorem image_closure {A : Type*} [Monoid A] {f : M → A} (hf : IsMonoidHom f) (s : Set M) :
-    f '' Closure s = Closure (f '' s) :=
-  le_antisymm
-    (by
-      rintro _ ⟨x, hx, rfl⟩
-      induction' hx with z hz
-      · solve_by_elim [subset_closure, Set.mem_image_of_mem]
-      · rw [hf.map_one]
-        apply IsSubmonoid.one_mem (closure.isSubmonoid (f '' s))
-      · rw [hf.map_mul]
-        solve_by_elim [(closure.isSubmonoid _).mul_mem] )
-    (closure_subset (IsSubmonoid.image hf (closure.isSubmonoid _)) <|
-      Set.image_subset _ subset_closure)
-
-/-- Given an element `a` of the submonoid of a monoid `M` generated by a set `s`, there exists
-a list of elements of `s` whose product is `a`. -/
-@[to_additive
-      "Given an element `a` of the `AddSubmonoid` of an `AddMonoid M` generated by
-      a set `s`, there exists a list of elements of `s` whose sum is `a`."]
-theorem exists_list_of_mem_closure {s : Set M} {a : M} (h : a ∈ Closure s) :
-    ∃ l : List M, (∀ x ∈ l, x ∈ s) ∧ l.prod = a := by
-  induction h with
-  | @basic a ha => exists [a]; simp [ha]
-  | one => exists []; simp
-  | mul _ _ ha hb =>
-    rcases ha with ⟨la, ha, eqa⟩
-    rcases hb with ⟨lb, hb, eqb⟩
-    exists la ++ lb
-    simp only [List.mem_append, or_imp, List.prod_append, eqa.symm, eqb.symm, and_true]
-    exact fun a => ⟨ha a, hb a⟩
-
-/-- Given sets `s, t` of a commutative monoid `M`, `x ∈ M` is in the submonoid of `M` generated by
-    `s ∪ t` iff there exists an element of the submonoid generated by `s` and an element of the
-    submonoid generated by `t` whose product is `x`. -/
-@[to_additive
-      "Given sets `s, t` of a commutative `AddMonoid M`, `x ∈ M` is in the `AddSubmonoid`
-      of `M` generated by `s ∪ t` iff there exists an element of the `AddSubmonoid` generated by `s`
-      and an element of the `AddSubmonoid` generated by `t` whose sum is `x`."]
-theorem mem_closure_union_iff {M : Type*} [CommMonoid M] {s t : Set M} {x : M} :
-    x ∈ Closure (s ∪ t) ↔ ∃ y ∈ Closure s, ∃ z ∈ Closure t, y * z = x :=
-  ⟨fun hx =>
-    let ⟨L, HL1, HL2⟩ := exists_list_of_mem_closure hx
-    HL2 ▸
-      List.recOn L
-        (fun _ =>
-          ⟨1, (closure.isSubmonoid _).one_mem, 1, (closure.isSubmonoid _).one_mem, mul_one _⟩)
-        (fun hd tl ih HL1 =>
-          let ⟨y, hy, z, hz, hyzx⟩ := ih (List.forall_mem_of_forall_mem_cons HL1)
-          Or.casesOn (HL1 hd <| List.mem_cons_self _ _)
-            (fun hs =>
-              ⟨hd * y, (closure.isSubmonoid _).mul_mem (subset_closure hs) hy, z, hz, by
-                rw [mul_assoc, List.prod_cons, ← hyzx]⟩)
-            fun ht =>
-            ⟨y, hy, z * hd, (closure.isSubmonoid _).mul_mem hz (subset_closure ht), by
-              rw [← mul_assoc, List.prod_cons, ← hyzx, mul_comm hd]⟩)
-        HL1,
-    fun ⟨_, hy, _, hz, hyzx⟩ =>
-    hyzx ▸
-      (closure.isSubmonoid _).mul_mem (closure_mono Set.subset_union_left hy)
-        (closure_mono Set.subset_union_right hz)⟩
-
-end Monoid
-
-/-- Create a bundled submonoid from a set `s` and `[IsSubmonoid s]`. -/
-@[to_additive "Create a bundled additive submonoid from a set `s` and `[IsAddSubmonoid s]`."]
-def Submonoid.of {s : Set M} (h : IsSubmonoid s) : Submonoid M :=
-  ⟨⟨s, @fun _ _ => h.2⟩, h.1⟩
-
-@[to_additive]
-theorem Submonoid.isSubmonoid (S : Submonoid M) : IsSubmonoid (S : Set M) := by
-  exact ⟨S.2, S.1.2⟩
diff --git a/Mathlib/Deprecated/Subring.lean b/Mathlib/Deprecated/Subring.lean
deleted file mode 100644
index 05bea1fc07bb06..00000000000000
--- a/Mathlib/Deprecated/Subring.lean
+++ /dev/null
@@ -1,205 +0,0 @@
-/-
-Copyright (c) 2018 Johan Commelin. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Johan Commelin
--/
-import Mathlib.Algebra.Ring.Subring.Defs
-import Mathlib.Deprecated.Group
-import Mathlib.Deprecated.Subgroup
-
-/-!
-# Unbundled subrings (deprecated)
-
-This file is deprecated, and is no longer imported by anything in mathlib other than other
-deprecated files, and test files. You should not need to import it.
-
-This file defines predicates for unbundled subrings. Instead of using this file, please use
-`Subring`, defined in `Mathlib.Algebra.Ring.Subring.Basic`, for subrings of rings.
-
-## Main definitions
-
-`IsSubring (S : Set R) : Prop` : the predicate that `S` is the underlying set of a subring
-of the ring `R`. The bundled variant `Subring R` should be used in preference to this.
-
-## Tags
-
-IsSubring
--/
-
-
-universe u v
-
-open Group
-
-variable {R : Type u} [Ring R]
-
-/-- `S` is a subring: a set containing 1 and closed under multiplication, addition and additive
-inverse. -/
-structure IsSubring (S : Set R) extends IsAddSubgroup S, IsSubmonoid S : Prop
-
-/-- Construct a `Subring` from a set satisfying `IsSubring`. -/
-def IsSubring.subring {S : Set R} (hs : IsSubring S) : Subring R where
-  carrier := S
-  one_mem' := hs.one_mem
-  mul_mem' := hs.mul_mem
-  zero_mem' := hs.zero_mem
-  add_mem' := hs.add_mem
-  neg_mem' := hs.neg_mem
-
-namespace RingHom
-
-theorem isSubring_preimage {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set S}
-    (hs : IsSubring s) : IsSubring (f ⁻¹' s) :=
-  { IsAddGroupHom.preimage f.to_isAddGroupHom hs.toIsAddSubgroup,
-    IsSubmonoid.preimage f.to_isMonoidHom hs.toIsSubmonoid with }
-
-theorem isSubring_image {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) {s : Set R}
-    (hs : IsSubring s) : IsSubring (f '' s) :=
-  { IsAddGroupHom.image_addSubgroup f.to_isAddGroupHom hs.toIsAddSubgroup,
-    IsSubmonoid.image f.to_isMonoidHom hs.toIsSubmonoid with }
-
-theorem isSubring_set_range {R : Type u} {S : Type v} [Ring R] [Ring S] (f : R →+* S) :
-    IsSubring (Set.range f) :=
-  { IsAddGroupHom.range_addSubgroup f.to_isAddGroupHom, Range.isSubmonoid f.to_isMonoidHom with }
-
-end RingHom
-
-theorem IsSubring.inter {S₁ S₂ : Set R} (hS₁ : IsSubring S₁) (hS₂ : IsSubring S₂) :
-    IsSubring (S₁ ∩ S₂) :=
-  { IsAddSubgroup.inter hS₁.toIsAddSubgroup hS₂.toIsAddSubgroup,
-    IsSubmonoid.inter hS₁.toIsSubmonoid hS₂.toIsSubmonoid with }
-
-theorem IsSubring.iInter {ι : Sort*} {S : ι → Set R} (h : ∀ y : ι, IsSubring (S y)) :
-    IsSubring (Set.iInter S) :=
-  { IsAddSubgroup.iInter fun i ↦ (h i).toIsAddSubgroup,
-    IsSubmonoid.iInter fun i ↦ (h i).toIsSubmonoid with }
-
-theorem isSubring_iUnion_of_directed {ι : Type*} [Nonempty ι] {s : ι → Set R}
-    (h : ∀ i, IsSubring (s i)) (directed : ∀ i j, ∃ k, s i ⊆ s k ∧ s j ⊆ s k) :
-    IsSubring (⋃ i, s i) :=
-  { toIsAddSubgroup := isAddSubgroup_iUnion_of_directed (fun i ↦ (h i).toIsAddSubgroup) directed
-    toIsSubmonoid := isSubmonoid_iUnion_of_directed (fun i ↦ (h i).toIsSubmonoid) directed }
-
-namespace Ring
-
-/-- The smallest subring containing a given subset of a ring, considered as a set. This function
-is deprecated; use `Subring.closure`. -/
-def closure (s : Set R) :=
-  AddGroup.closure (Monoid.Closure s)
-
-variable {s : Set R}
-
--- attribute [local reducible] closure -- Porting note: not available in Lean4
-
-theorem exists_list_of_mem_closure {a : R} (h : a ∈ closure s) :
-    ∃ L : List (List R), (∀ l ∈ L, ∀ x ∈ l, x ∈ s ∨ x = (-1 : R)) ∧ (L.map List.prod).sum = a :=
-  AddGroup.InClosure.recOn h
-    fun {x} hx ↦ match x, Monoid.exists_list_of_mem_closure hx with
-    | _, ⟨L, h1, rfl⟩ =>
-      ⟨[L], List.forall_mem_singleton.2 fun r hr ↦ Or.inl (h1 r hr), List.sum_singleton⟩
-    ⟨[], List.forall_mem_nil _, rfl⟩
-    fun {b} _ ih ↦ match b, ih with
-    | _, ⟨L1, h1, rfl⟩ =>
-      ⟨L1.map (List.cons (-1)),
-        fun L2 h2 ↦ match L2, List.mem_map.1 h2 with
-        | _, ⟨L3, h3, rfl⟩ => List.forall_mem_cons.2 ⟨Or.inr rfl, h1 L3 h3⟩, by
-        simp only [List.map_map, Function.comp_def, List.prod_cons, neg_one_mul]
-        refine List.recOn L1 neg_zero.symm fun hd tl ih ↦ ?_
-        rw [List.map_cons, List.sum_cons, ih, List.map_cons, List.sum_cons, neg_add]⟩
-    fun {r1 r2} _ _ ih1 ih2 ↦ match r1, r2, ih1, ih2 with
-    | _, _, ⟨L1, h1, rfl⟩, ⟨L2, h2, rfl⟩ =>
-      ⟨L1 ++ L2, List.forall_mem_append.2 ⟨h1, h2⟩, by rw [List.map_append, List.sum_append]⟩
-
-@[elab_as_elim]
-protected theorem InClosure.recOn {C : R → Prop} {x : R} (hx : x ∈ closure s) (h1 : C 1)
-    (hneg1 : C (-1)) (hs : ∀ z ∈ s, ∀ n, C n → C (z * n)) (ha : ∀ {x y}, C x → C y → C (x + y)) :
-    C x := by
-  have h0 : C 0 := add_neg_cancel (1 : R) ▸ ha h1 hneg1
-  rcases exists_list_of_mem_closure hx with ⟨L, HL, rfl⟩
-  clear hx
-  induction' L with hd tl ih
-  · exact h0
-  rw [List.forall_mem_cons] at HL
-  suffices C (List.prod hd) by
-    rw [List.map_cons, List.sum_cons]
-    exact ha this (ih HL.2)
-  replace HL := HL.1
-  clear ih tl
-  -- Porting note: Expanded `rsuffices`
-  suffices ∃ L, (∀ x ∈ L, x ∈ s) ∧ (List.prod hd = List.prod L ∨ List.prod hd = -List.prod L) by
-    rcases this with ⟨L, HL', HP | HP⟩ <;> rw [HP] <;> clear HP HL
-    · induction' L with hd tl ih
-      · exact h1
-      rw [List.forall_mem_cons] at HL'
-      rw [List.prod_cons]
-      exact hs _ HL'.1 _ (ih HL'.2)
-    · induction' L with hd tl ih
-      · exact hneg1
-      rw [List.prod_cons, neg_mul_eq_mul_neg]
-      rw [List.forall_mem_cons] at HL'
-      exact hs _ HL'.1 _ (ih HL'.2)
-  induction' hd with hd tl ih
-  · exact ⟨[], List.forall_mem_nil _, Or.inl rfl⟩
-  rw [List.forall_mem_cons] at HL
-  rcases ih HL.2 with ⟨L, HL', HP | HP⟩ <;> cases' HL.1 with hhd hhd
-  · exact
-      ⟨hd::L, List.forall_mem_cons.2 ⟨hhd, HL'⟩,
-        Or.inl <| by rw [List.prod_cons, List.prod_cons, HP]⟩
-  · exact ⟨L, HL', Or.inr <| by rw [List.prod_cons, hhd, neg_one_mul, HP]⟩
-  · exact
-      ⟨hd::L, List.forall_mem_cons.2 ⟨hhd, HL'⟩,
-        Or.inr <| by rw [List.prod_cons, List.prod_cons, HP, neg_mul_eq_mul_neg]⟩
-  · exact ⟨L, HL', Or.inl <| by rw [List.prod_cons, hhd, HP, neg_one_mul, neg_neg]⟩
-
-theorem closure.isSubring : IsSubring (closure s) :=
-  { AddGroup.closure.isAddSubgroup _ with
-    one_mem := AddGroup.mem_closure <| IsSubmonoid.one_mem <| Monoid.closure.isSubmonoid _
-    mul_mem := fun {a _} ha hb ↦ AddGroup.InClosure.recOn hb
-      (fun {c} hc ↦ AddGroup.InClosure.recOn ha
-        (fun hd ↦ AddGroup.subset_closure ((Monoid.closure.isSubmonoid _).mul_mem hd hc))
-        ((zero_mul c).symm ▸ (AddGroup.closure.isAddSubgroup _).zero_mem)
-        (fun {d} _ hdc ↦ neg_mul_eq_neg_mul d c ▸ (AddGroup.closure.isAddSubgroup _).neg_mem hdc)
-        fun {d e} _ _ hdc hec ↦
-          (add_mul d e c).symm ▸ (AddGroup.closure.isAddSubgroup _).add_mem hdc hec)
-      ((mul_zero a).symm ▸ (AddGroup.closure.isAddSubgroup _).zero_mem)
-      (fun {c} _ hac ↦ neg_mul_eq_mul_neg a c ▸ (AddGroup.closure.isAddSubgroup _).neg_mem hac)
-      fun {c d} _ _ hac had ↦
-        (mul_add a c d).symm ▸ (AddGroup.closure.isAddSubgroup _).add_mem hac had }
-
-theorem mem_closure {a : R} : a ∈ s → a ∈ closure s :=
-  AddGroup.mem_closure ∘ @Monoid.subset_closure _ _ _ _
-
-theorem subset_closure : s ⊆ closure s :=
-  fun _ ↦ mem_closure
-
-theorem closure_subset {t : Set R} (ht : IsSubring t) : s ⊆ t → closure s ⊆ t :=
-  AddGroup.closure_subset ht.toIsAddSubgroup ∘ Monoid.closure_subset ht.toIsSubmonoid
-
-theorem closure_subset_iff {s t : Set R} (ht : IsSubring t) : closure s ⊆ t ↔ s ⊆ t :=
-  (AddGroup.closure_subset_iff ht.toIsAddSubgroup).trans
-    ⟨Set.Subset.trans Monoid.subset_closure, Monoid.closure_subset ht.toIsSubmonoid⟩
-
-@[gcongr]
-theorem closure_mono {s t : Set R} (H : s ⊆ t) : closure s ⊆ closure t :=
-  closure_subset closure.isSubring <| Set.Subset.trans H subset_closure
-
-theorem image_closure {S : Type*} [Ring S] (f : R →+* S) (s : Set R) :
-    f '' closure s = closure (f '' s) := by
-  refine le_antisymm ?_ (closure_subset (RingHom.isSubring_image _ closure.isSubring) <|
-    Set.image_subset _ subset_closure)
-  rintro _ ⟨x, hx, rfl⟩
-  apply AddGroup.InClosure.recOn (motive := fun {x} _ ↦ f x ∈ closure (f '' s)) hx _ <;> intros
-  · rw [f.map_zero]
-    apply closure.isSubring.zero_mem
-  · rw [f.map_neg]
-    apply closure.isSubring.neg_mem
-    assumption
-  · rw [f.map_add]
-    apply closure.isSubring.add_mem
-    assumption'
-  · apply AddGroup.mem_closure
-    rw [← Monoid.image_closure f.to_isMonoidHom]
-    apply Set.mem_image_of_mem
-    assumption
-
-end Ring
diff --git a/Mathlib/FieldTheory/KummerExtension.lean b/Mathlib/FieldTheory/KummerExtension.lean
index 50462a0390dace..352ba6623e838b 100644
--- a/Mathlib/FieldTheory/KummerExtension.lean
+++ b/Mathlib/FieldTheory/KummerExtension.lean
@@ -175,10 +175,8 @@ theorem Polynomial.separable_X_pow_sub_C_of_irreducible : (X ^ n - C a).Separabl
     AdjoinRoot.algebraMap_eq,
     X_pow_sub_C_eq_prod (hζ.map_of_injective (algebraMap K _).injective) hn
     (root_X_pow_sub_C_pow n a), separable_prod_X_sub_C_iff']
-  #adaptation_note
-  /--
-  After https://github.com/leanprover/lean4/pull/5376 we need to provide this helper instance.
-  -/
+  #adaptation_note /-- https://github.com/leanprover/lean4/pull/5376
+  we need to provide this helper instance. -/
   have : MonoidHomClass (K →+* K[n√a]) K K[n√a] := inferInstance
   exact (hζ.map_of_injective (algebraMap K K[n√a]).injective).injOn_pow_mul
     (root_X_pow_sub_C_ne_zero (lt_of_le_of_ne (show 1 ≤ n from hn) (Ne.symm hn')) _)
diff --git a/Mathlib/FieldTheory/Minpoly/Basic.lean b/Mathlib/FieldTheory/Minpoly/Basic.lean
index 0b88000978e5fe..8c19cd2535cbdd 100644
--- a/Mathlib/FieldTheory/Minpoly/Basic.lean
+++ b/Mathlib/FieldTheory/Minpoly/Basic.lean
@@ -15,7 +15,6 @@ such as irreducibility under the assumption `B` is a domain.
 -/
 
 
-open scoped Classical
 open Polynomial Set Function
 
 variable {A B B' : Type*}
@@ -24,6 +23,7 @@ section MinPolyDef
 
 variable (A) [CommRing A] [Ring B] [Algebra A B]
 
+open scoped Classical in
 /-- Suppose `x : B`, where `B` is an `A`-algebra.
 
 The minimal polynomial `minpoly A x` of `x`
@@ -64,6 +64,7 @@ theorem ne_zero_iff [Nontrivial A] : minpoly A x ≠ 0 ↔ IsIntegral A x :=
 
 theorem algHom_eq (f : B →ₐ[A] B') (hf : Function.Injective f) (x : B) :
     minpoly A (f x) = minpoly A x := by
+  classical
   simp_rw [minpoly, isIntegral_algHom_iff _ hf, ← Polynomial.aeval_def, aeval_algHom,
     AlgHom.comp_apply, _root_.map_eq_zero_iff f hf]
 
diff --git a/Mathlib/FieldTheory/Minpoly/Field.lean b/Mathlib/FieldTheory/Minpoly/Field.lean
index 0652b50888c6b4..b3f90cac4e69d1 100644
--- a/Mathlib/FieldTheory/Minpoly/Field.lean
+++ b/Mathlib/FieldTheory/Minpoly/Field.lean
@@ -18,7 +18,6 @@ are irreducible, and uniquely determined by their defining property.
 -/
 
 
-open scoped Classical
 open Polynomial Set Function minpoly
 
 namespace minpoly
@@ -181,6 +180,7 @@ theorem neg {B : Type*} [CommRing B] [Algebra A B] (x : B) :
 
 section AlgHomFintype
 
+open scoped Classical in
 /-- A technical finiteness result. -/
 noncomputable def Fintype.subtypeProd {E : Type*} {X : Set E} (hX : X.Finite) {L : Type*}
     (F : E → Multiset L) : Fintype (∀ x : X, { l : L // l ∈ F x }) :=
diff --git a/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean b/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean
index 14a02db0ca5883..5c4ceecc798cce 100644
--- a/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean
+++ b/Mathlib/FieldTheory/RatFunc/AsPolynomial.lean
@@ -32,8 +32,6 @@ namespace RatFunc
 
 section Eval
 
-open scoped Classical
-
 open scoped nonZeroDivisors Polynomial
 
 open RatFunc
diff --git a/Mathlib/FieldTheory/RatFunc/Basic.lean b/Mathlib/FieldTheory/RatFunc/Basic.lean
index c5ca7de7845f57..9a1cde7fbd08bb 100644
--- a/Mathlib/FieldTheory/RatFunc/Basic.lean
+++ b/Mathlib/FieldTheory/RatFunc/Basic.lean
@@ -48,8 +48,6 @@ universe u v
 
 noncomputable section
 
-open scoped Classical
-
 open scoped nonZeroDivisors Polynomial
 
 variable {K : Type u}
@@ -320,6 +318,8 @@ open RatFunc
 
 variable {G₀ L R S F : Type*} [CommGroupWithZero G₀] [Field L] [CommRing R] [CommRing S]
 variable [FunLike F R[X] S[X]]
+
+open scoped Classical in
 /-- Lift a monoid homomorphism that maps polynomials `φ : R[X] →* S[X]`
 to a `RatFunc R →* RatFunc S`,
 on the condition that `φ` maps non zero divisors to non zero divisors,
@@ -741,6 +741,7 @@ open GCDMonoid Polynomial
 
 variable [Field K]
 
+open scoped Classical in
 /-- `RatFunc.numDenom` are numerator and denominator of a rational function over a field,
 normalized such that the denominator is monic. -/
 def numDenom (x : RatFunc K) : K[X] × K[X] :=
@@ -775,6 +776,7 @@ def numDenom (x : RatFunc K) : K[X] × K[X] :=
         rw [inv_div, mul_comm, mul_div_assoc, ← mul_assoc, inv_inv, mul_inv_cancel₀ ha',
           one_mul, inv_div])
 
+open scoped Classical in
 @[simp]
 theorem numDenom_div (p : K[X]) {q : K[X]} (hq : q ≠ 0) :
     numDenom (algebraMap _ _ p / algebraMap _ _ q) =
@@ -790,6 +792,7 @@ normalized such that the denominator is monic. -/
 def num (x : RatFunc K) : K[X] :=
   x.numDenom.1
 
+open scoped Classical in
 private theorem num_div' (p : K[X]) {q : K[X]} (hq : q ≠ 0) :
     num (algebraMap _ _ p / algebraMap _ _ q) =
       Polynomial.C (q / gcd p q).leadingCoeff⁻¹ * (p / gcd p q) := by
@@ -798,6 +801,7 @@ private theorem num_div' (p : K[X]) {q : K[X]} (hq : q ≠ 0) :
 @[simp]
 theorem num_zero : num (0 : RatFunc K) = 0 := by convert num_div' (0 : K[X]) one_ne_zero <;> simp
 
+open scoped Classical in
 @[simp]
 theorem num_div (p q : K[X]) :
     num (algebraMap _ _ p / algebraMap _ _ q) =
@@ -814,10 +818,12 @@ theorem num_algebraMap (p : K[X]) : num (algebraMap _ _ p) = p := by convert num
 
 theorem num_div_dvd (p : K[X]) {q : K[X]} (hq : q ≠ 0) :
     num (algebraMap _ _ p / algebraMap _ _ q) ∣ p := by
+  classical
   rw [num_div _ q, C_mul_dvd]
   · exact EuclideanDomain.div_dvd_of_dvd (gcd_dvd_left p q)
   · simpa only [Ne, inv_eq_zero, Polynomial.leadingCoeff_eq_zero] using right_div_gcd_ne_zero hq
 
+open scoped Classical in
 /-- A version of `num_div_dvd` with the LHS in simp normal form -/
 @[simp]
 theorem num_div_dvd' (p : K[X]) {q : K[X]} (hq : q ≠ 0) :
@@ -828,6 +834,7 @@ normalized such that it is monic. -/
 def denom (x : RatFunc K) : K[X] :=
   x.numDenom.2
 
+open scoped Classical in
 @[simp]
 theorem denom_div (p : K[X]) {q : K[X]} (hq : q ≠ 0) :
     denom (algebraMap _ _ p / algebraMap _ _ q) =
@@ -835,6 +842,7 @@ theorem denom_div (p : K[X]) {q : K[X]} (hq : q ≠ 0) :
   rw [denom, numDenom_div _ hq]
 
 theorem monic_denom (x : RatFunc K) : (denom x).Monic := by
+  classical
   induction x using RatFunc.induction_on with
   | f p q hq =>
     rw [denom_div p hq, mul_comm]
@@ -857,6 +865,7 @@ theorem denom_algebraMap (p : K[X]) : denom (algebraMap _ (RatFunc K) p) = 1 :=
 
 @[simp]
 theorem denom_div_dvd (p q : K[X]) : denom (algebraMap _ _ p / algebraMap _ _ q) ∣ q := by
+  classical
   by_cases hq : q = 0
   · simp [hq]
   rw [denom_div _ hq, C_mul_dvd]
@@ -865,6 +874,7 @@ theorem denom_div_dvd (p q : K[X]) : denom (algebraMap _ _ p / algebraMap _ _ q)
 
 @[simp]
 theorem num_div_denom (x : RatFunc K) : algebraMap _ _ (num x) / algebraMap _ _ (denom x) = x := by
+  classical
   induction' x using RatFunc.induction_on with p q hq
   have q_div_ne_zero : q / gcd p q ≠ 0 := right_div_gcd_ne_zero hq
   rw [num_div p q, denom_div p hq, RingHom.map_mul, RingHom.map_mul, mul_div_mul_left,
@@ -878,6 +888,7 @@ theorem num_div_denom (x : RatFunc K) : algebraMap _ _ (num x) / algebraMap _ _
     exact inv_ne_zero (Polynomial.leadingCoeff_ne_zero.mpr q_div_ne_zero)
 
 theorem isCoprime_num_denom (x : RatFunc K) : IsCoprime x.num x.denom := by
+  classical
   induction' x using RatFunc.induction_on with p q hq
   rw [num_div, denom_div _ hq]
   exact (isCoprime_mul_unit_left
diff --git a/Mathlib/FieldTheory/RatFunc/Defs.lean b/Mathlib/FieldTheory/RatFunc/Defs.lean
index a0e7e1e83de5be..c3fc63ae2deaa7 100644
--- a/Mathlib/FieldTheory/RatFunc/Defs.lean
+++ b/Mathlib/FieldTheory/RatFunc/Defs.lean
@@ -48,8 +48,6 @@ the codomain is not a field or even an integral domain.
 
 noncomputable section
 
-open scoped Classical
-
 open scoped nonZeroDivisors Polynomial
 
 universe u v
diff --git a/Mathlib/FieldTheory/Relrank.lean b/Mathlib/FieldTheory/Relrank.lean
index 2ed666bbdae0ab..aa621bbab5393f 100644
--- a/Mathlib/FieldTheory/Relrank.lean
+++ b/Mathlib/FieldTheory/Relrank.lean
@@ -35,12 +35,10 @@ variable {E : Type v} [Field E] {L : Type w} [Field L]
 
 variable (A B C : Subfield E)
 
-#adaptation_note
-/--
+#adaptation_note /-- nightly-2024-11-14
 This `synthInstance.maxHeartbeats` (and below) was required after nightly-2024-11-14;
 it's not exactly clear why, but we were very close to the limit previously,
-so probably we should not particularly blame changes in Lean, and instead optimize in Mathlib.
--/
+so probably we should not particularly blame changes in Lean, and instead optimize in Mathlib. -/
 set_option synthInstance.maxHeartbeats 400000 in
 /-- `Subfield.relrank A B` is defined to be `[B : A ⊓ B]` as a `Cardinal`, in particular,
 when `A ≤ B` it is `[B : A]`, the degree of the field extension `B / A`.
diff --git a/Mathlib/Geometry/Euclidean/Inversion/Basic.lean b/Mathlib/Geometry/Euclidean/Inversion/Basic.lean
index ef7c492a1d056c..7fc9dd70fae57d 100644
--- a/Mathlib/Geometry/Euclidean/Inversion/Basic.lean
+++ b/Mathlib/Geometry/Euclidean/Inversion/Basic.lean
@@ -40,7 +40,6 @@ sphere. -/
 def inversion (c : P) (R : ℝ) (x : P) : P :=
   (R / dist x c) ^ 2 • (x -ᵥ c) +ᵥ c
 
-#adaptation_note /-- nightly-2024-03-16: added to replace simp [inversion] -/
 theorem inversion_def :
     inversion = fun (c : P) (R : ℝ) (x : P) => (R / dist x c) ^ 2 • (x -ᵥ c) +ᵥ c :=
   rfl
diff --git a/Mathlib/Geometry/Euclidean/Inversion/Calculus.lean b/Mathlib/Geometry/Euclidean/Inversion/Calculus.lean
index 877daf129a72af..f948ab3a60200a 100644
--- a/Mathlib/Geometry/Euclidean/Inversion/Calculus.lean
+++ b/Mathlib/Geometry/Euclidean/Inversion/Calculus.lean
@@ -89,9 +89,7 @@ theorem hasFDerivAt_inversion (hx : x ≠ c) :
       ((R / dist x c) ^ 2 • (reflection (ℝ ∙ (x - c))ᗮ : F →L[ℝ] F)) x := by
   rcases add_left_surjective c x with ⟨x, rfl⟩
   have : HasFDerivAt (inversion c R) (?_ : F →L[ℝ] F) (c + x) := by
-    #adaptation_note /-- nightly-2024-03-16: simp was
-    simp (config := { unfoldPartialApp := true }) only [inversion] -/
-    simp only [inversion_def]
+    simp +unfoldPartialApp only [inversion]
     simp_rw [dist_eq_norm, div_pow, div_eq_mul_inv]
     have A := (hasFDerivAt_id (𝕜 := ℝ) (c + x)).sub_const c
     have B := ((hasDerivAt_inv <| by simpa using hx).comp_hasFDerivAt _ A.norm_sq).const_mul
diff --git a/Mathlib/Geometry/Manifold/AnalyticManifold.lean b/Mathlib/Geometry/Manifold/AnalyticManifold.lean
index 9b6c9a3025046a..700a30607806f2 100644
--- a/Mathlib/Geometry/Manifold/AnalyticManifold.lean
+++ b/Mathlib/Geometry/Manifold/AnalyticManifold.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Michael Lee, Geoffrey Irving
 -/
 import Mathlib.Analysis.Analytic.Constructions
-import Mathlib.Geometry.Manifold.IsManifold
+import Mathlib.Geometry.Manifold.IsManifold.Basic
 
 /-!
 # Analytic manifolds (possibly with boundary or corners)
diff --git a/Mathlib/Geometry/Manifold/ChartedSpace.lean b/Mathlib/Geometry/Manifold/ChartedSpace.lean
index 7bcb7a3778950f..c5f523f226b655 100644
--- a/Mathlib/Geometry/Manifold/ChartedSpace.lean
+++ b/Mathlib/Geometry/Manifold/ChartedSpace.lean
@@ -851,7 +851,7 @@ theorem piChartedSpace_chartAt {ι : Type*} [Finite ι] (H : ι → Type*)
 section sum
 
 variable [TopologicalSpace H] [TopologicalSpace M] [TopologicalSpace M']
-    [cm : ChartedSpace H M] [cm' : ChartedSpace H M'] [Nonempty H] [Nonempty M] [Nonempty M']
+    [cm : ChartedSpace H M] [cm' : ChartedSpace H M'] [Nonempty H]
 
 /-- The disjoint union of two charted spaces on `H` is a charted space over `H`. -/
 instance ChartedSpace.sum : ChartedSpace H (M ⊕ M') where
diff --git a/Mathlib/Geometry/Manifold/ContMDiff/Defs.lean b/Mathlib/Geometry/Manifold/ContMDiff/Defs.lean
index 4bb05df82f4ff0..ee537cfccc4d7c 100644
--- a/Mathlib/Geometry/Manifold/ContMDiff/Defs.lean
+++ b/Mathlib/Geometry/Manifold/ContMDiff/Defs.lean
@@ -3,7 +3,7 @@ Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Floris van Doorn
 -/
-import Mathlib.Geometry.Manifold.IsManifold
+import Mathlib.Geometry.Manifold.IsManifold.Basic
 import Mathlib.Geometry.Manifold.LocalInvariantProperties
 
 /-!
diff --git a/Mathlib/Geometry/Manifold/Instances/Real.lean b/Mathlib/Geometry/Manifold/Instances/Real.lean
index 78db01d88e9e28..f18b90045c1e6a 100644
--- a/Mathlib/Geometry/Manifold/Instances/Real.lean
+++ b/Mathlib/Geometry/Manifold/Instances/Real.lean
@@ -3,8 +3,8 @@ Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathlib.Geometry.Manifold.InteriorBoundary
-import Mathlib.Geometry.Manifold.IsManifold
+import Mathlib.Geometry.Manifold.IsManifold.Basic
+import Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary
 import Mathlib.Analysis.InnerProductSpace.PiL2
 
 /-!
diff --git a/Mathlib/Geometry/Manifold/IntegralCurve/ExistUnique.lean b/Mathlib/Geometry/Manifold/IntegralCurve/ExistUnique.lean
index 9a921471207e6b..3f35ee20c69bf7 100644
--- a/Mathlib/Geometry/Manifold/IntegralCurve/ExistUnique.lean
+++ b/Mathlib/Geometry/Manifold/IntegralCurve/ExistUnique.lean
@@ -6,7 +6,7 @@ Authors: Winston Yin
 import Mathlib.Analysis.ODE.Gronwall
 import Mathlib.Analysis.ODE.PicardLindelof
 import Mathlib.Geometry.Manifold.IntegralCurve.Transform
-import Mathlib.Geometry.Manifold.InteriorBoundary
+import Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary
 
 /-!
 # Existence and uniqueness of integral curves
diff --git a/Mathlib/Geometry/Manifold/IsManifold.lean b/Mathlib/Geometry/Manifold/IsManifold/Basic.lean
similarity index 99%
rename from Mathlib/Geometry/Manifold/IsManifold.lean
rename to Mathlib/Geometry/Manifold/IsManifold/Basic.lean
index 69d14ee8012e61..069cea5c3f202d 100644
--- a/Mathlib/Geometry/Manifold/IsManifold.lean
+++ b/Mathlib/Geometry/Manifold/IsManifold/Basic.lean
@@ -829,7 +829,7 @@ variable {M' : Type*} [TopologicalSpace M'] [ChartedSpace H M']
 
 /-- The disjoint union of two `C^n` manifolds modelled on `(E, H)`
 is a `C^n` manifold modeled on `(E, H)`. -/
-instance disjointUnion [Nonempty M] [Nonempty M'] [Nonempty H] : IsManifold I n (M ⊕ M') where
+instance disjointUnion [Nonempty H] : IsManifold I n (M ⊕ M') where
   compatible {e} e' he he' := by
     obtain (⟨f, hf, hef⟩ | ⟨f, hf, hef⟩) := ChartedSpace.mem_atlas_sum he
     · obtain (⟨f', hf', he'f'⟩ | ⟨f', hf', he'f'⟩) := ChartedSpace.mem_atlas_sum he'
diff --git a/Mathlib/Geometry/Manifold/InteriorBoundary.lean b/Mathlib/Geometry/Manifold/IsManifold/InteriorBoundary.lean
similarity index 98%
rename from Mathlib/Geometry/Manifold/InteriorBoundary.lean
rename to Mathlib/Geometry/Manifold/IsManifold/InteriorBoundary.lean
index 7f6c2a06f8852a..9cc24f55b41587 100644
--- a/Mathlib/Geometry/Manifold/InteriorBoundary.lean
+++ b/Mathlib/Geometry/Manifold/IsManifold/InteriorBoundary.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Michael Rothgang
 -/
 
-import Mathlib.Geometry.Manifold.IsManifold
+import Mathlib.Geometry.Manifold.IsManifold.Basic
 
 /-!
 # Interior and boundary of a manifold
@@ -250,12 +250,11 @@ end prod
 section disjointUnion
 
 variable {M' : Type*} [TopologicalSpace M'] [ChartedSpace H M'] {n : WithTop ℕ∞}
-  [hM : IsManifold I n M] [hM' : IsManifold I n M']
-  [Nonempty M] [Nonempty M'] [Nonempty H]
+  [hM : IsManifold I n M] [hM' : IsManifold I n M'] [Nonempty H]
   {E' : Type*} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type*} [TopologicalSpace H']
   {J : Type*} {J : ModelWithCorners 𝕜 E' H'}
   {N N' : Type*} [TopologicalSpace N] [TopologicalSpace N'] [ChartedSpace H' N] [ChartedSpace H' N']
-  [IsManifold J n N] [IsManifold J n N'] [Nonempty N] [Nonempty N'] [Nonempty H']
+  [IsManifold J n N] [IsManifold J n N'] [Nonempty H']
 
 open Topology
 
diff --git a/Mathlib/Geometry/Manifold/MFDeriv/Defs.lean b/Mathlib/Geometry/Manifold/MFDeriv/Defs.lean
index c269a880e9c260..8fae4d69ee6bdd 100644
--- a/Mathlib/Geometry/Manifold/MFDeriv/Defs.lean
+++ b/Mathlib/Geometry/Manifold/MFDeriv/Defs.lean
@@ -3,7 +3,7 @@ Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Floris van Doorn
 -/
-import Mathlib.Geometry.Manifold.IsManifold
+import Mathlib.Geometry.Manifold.IsManifold.Basic
 import Mathlib.Geometry.Manifold.LocalInvariantProperties
 
 /-!
diff --git a/Mathlib/Geometry/Manifold/MFDeriv/NormedSpace.lean b/Mathlib/Geometry/Manifold/MFDeriv/NormedSpace.lean
index d920ed00741905..158ebd2729aa13 100644
--- a/Mathlib/Geometry/Manifold/MFDeriv/NormedSpace.lean
+++ b/Mathlib/Geometry/Manifold/MFDeriv/NormedSpace.lean
@@ -81,12 +81,9 @@ end Module
 
 /-! ### Linear maps between normed spaces are differentiable -/
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/6024
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/6024
 we needed to add the named arguments `𝕜 := 𝕜` and `F := ((F₂ →L[𝕜] F₃) →L[𝕜] F₁ →L[𝕜] F₃))`
-to `ContinuousLinearMap.differentiable`.
--/
+to `ContinuousLinearMap.differentiable`. -/
 theorem MDifferentiableWithinAt.clm_precomp {f : M → F₁ →L[𝕜] F₂} {s : Set M} {x : M}
     (hf : MDifferentiableWithinAt I 𝓘(𝕜, F₁ →L[𝕜] F₂) f s x) :
     MDifferentiableWithinAt I 𝓘(𝕜, (F₂ →L[𝕜] F₃) →L[𝕜] (F₁ →L[𝕜] F₃))
@@ -95,12 +92,9 @@ theorem MDifferentiableWithinAt.clm_precomp {f : M → F₁ →L[𝕜] F₂} {s
     (ContinuousLinearMap.differentiable (𝕜 := 𝕜) (F := ((F₂ →L[𝕜] F₃) →L[𝕜] F₁ →L[𝕜] F₃))
       (ContinuousLinearMap.compL 𝕜 F₁ F₂ F₃).flip) hf
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/6024
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/6024
 we needed to add the named arguments `𝕜 := 𝕜` and `F := ((F₂ →L[𝕜] F₃) →L[𝕜] F₁ →L[𝕜] F₃))`
-to `ContinuousLinearMap.differentiable`.
--/
+to `ContinuousLinearMap.differentiable`. -/
 nonrec theorem MDifferentiableAt.clm_precomp {f : M → F₁ →L[𝕜] F₂} {x : M}
     (hf : MDifferentiableAt I 𝓘(𝕜, F₁ →L[𝕜] F₂) f x) :
     MDifferentiableAt I 𝓘(𝕜, (F₂ →L[𝕜] F₃) →L[𝕜] (F₁ →L[𝕜] F₃))
@@ -121,12 +115,9 @@ theorem MDifferentiable.clm_precomp
       (fun y ↦ (f y).precomp F₃ : M → (F₂ →L[𝕜] F₃) →L[𝕜] (F₁ →L[𝕜] F₃)) := fun x ↦
   (hf x).clm_precomp
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/6024
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/6024
 we needed to add the named arguments `𝕜 := 𝕜` and `F := ((F₁ →L[𝕜] F₂) →L[𝕜] (F₁ →L[𝕜] F₃))`
-to `ContinuousLinearMap.differentiable`.
--/
+to `ContinuousLinearMap.differentiable`. -/
 theorem MDifferentiableWithinAt.clm_postcomp {f : M → F₂ →L[𝕜] F₃} {s : Set M} {x : M}
     (hf : MDifferentiableWithinAt I 𝓘(𝕜, F₂ →L[𝕜] F₃) f s x) :
     MDifferentiableWithinAt I 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] (F₁ →L[𝕜] F₃))
@@ -136,12 +127,9 @@ theorem MDifferentiableWithinAt.clm_postcomp {f : M → F₂ →L[𝕜] F₃} {s
     (ContinuousLinearMap.differentiable (𝕜 := 𝕜) (F := ((F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₃))
       (ContinuousLinearMap.compL 𝕜 F₁ F₂ F₃)) hf
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/6024
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/6024
 we needed to add the named arguments `𝕜 := 𝕜` and `F := ((F₁ →L[𝕜] F₂) →L[𝕜] (F₁ →L[𝕜] F₃))`
-to `ContinuousLinearMap.differentiable`.
--/
+to `ContinuousLinearMap.differentiable`. -/
 theorem MDifferentiableAt.clm_postcomp {f : M → F₂ →L[𝕜] F₃} {x : M}
     (hf : MDifferentiableAt I 𝓘(𝕜, F₂ →L[𝕜] F₃) f x) :
     MDifferentiableAt I 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] (F₁ →L[𝕜] F₃))
diff --git a/Mathlib/Geometry/Manifold/Metrizable.lean b/Mathlib/Geometry/Manifold/Metrizable.lean
index 5427ae259b734b..fa2cb6d5dd8920 100644
--- a/Mathlib/Geometry/Manifold/Metrizable.lean
+++ b/Mathlib/Geometry/Manifold/Metrizable.lean
@@ -3,7 +3,7 @@ Copyright (c) 2022 Yury Kudryashov. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yury Kudryashov
 -/
-import Mathlib.Geometry.Manifold.IsManifold
+import Mathlib.Geometry.Manifold.IsManifold.Basic
 import Mathlib.Topology.Compactness.Paracompact
 import Mathlib.Topology.Metrizable.Urysohn
 
diff --git a/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean b/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean
index e05dffc7df37db..a4c26869dcdfeb 100644
--- a/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean
+++ b/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/HasColimits.lean
@@ -227,9 +227,10 @@ theorem coequalizer_π_stalk_isLocalHom (x : Y) :
   constructor
   rintro a ha
   rcases TopCat.Presheaf.germ_exist _ _ a with ⟨U, hU, s, rfl⟩
-  rw [← CommRingCat.forget_map_apply, PresheafedSpace.stalkMap_germ_apply
-    (coequalizer.π (C := SheafedSpace _) f.toShHom g.toShHom) U _ hU] at ha
-  rw [CommRingCat.forget_map_apply]
+  rw [← CommRingCat.forget_map_apply, forget_map_eq_coe,
+    PresheafedSpace.stalkMap_germ_apply
+      (coequalizer.π (C := SheafedSpace _) f.toShHom g.toShHom) U _ hU] at ha
+  rw [coe_toHasForget_instFunLike]
   let V := imageBasicOpen f g U s
   have hV : (coequalizer.π f.toShHom g.toShHom).base ⁻¹'
       ((coequalizer.π f.toShHom g.toShHom).base '' V.1) = V.1 :=
diff --git a/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean b/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean
index e96b86ffdeceb6..5120aff056674d 100644
--- a/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean
+++ b/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean
@@ -113,7 +113,7 @@ lemma residueFieldMap_comp {Z : LocallyRingedSpace.{u}} (g : Y ⟶ Z) (x : X) :
   simp only [comp_toShHom, SheafedSpace.comp_base, Function.comp_apply, residueFieldMap,
     CommRingCat.hom_comp, TopCat.comp_app]
   simp_rw [stalkMap_comp]
-  apply IsLocalRing.ResidueField.map_comp
+  apply IsLocalRing.ResidueField.map_comp (Hom.stalkMap g (f.base x)).hom (Hom.stalkMap f x).hom
 
 @[reassoc]
 lemma evaluation_naturality {V : Opens Y} (x : (Opens.map f.base).obj V) :
diff --git a/Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean b/Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
index c3fefa900b4902..912cb224dd41a2 100644
--- a/Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
+++ b/Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean
@@ -192,14 +192,16 @@ theorem snd_invApp_t_app' (i j k : D.J) (U : Opens (pullback (D.f i j) (D.f i k)
     rw [this, Set.image_comp, Set.image_comp, Set.preimage_image_eq]
     swap
     · refine Function.HasLeftInverse.injective ⟨(D.t i k).base, fun x => ?_⟩
-      rw [← comp_apply, ← comp_base, D.t_inv, id_base, id_apply]
+      rw [← CategoryTheory.comp_apply, ← comp_base, D.t_inv, id_base, CategoryTheory.id_apply]
     refine congr_arg (_ '' ·) ?_
     refine congr_fun ?_ _
     refine Set.image_eq_preimage_of_inverse ?_ ?_
     · intro x
-      rw [← comp_apply, ← comp_base, IsIso.inv_hom_id, id_base, id_apply]
+      rw [← CategoryTheory.comp_apply, ← comp_base, IsIso.inv_hom_id, id_base,
+        CategoryTheory.id_apply]
     · intro x
-      rw [← comp_apply, ← comp_base, IsIso.hom_inv_id, id_base, id_apply]
+      rw [← CategoryTheory.comp_apply, ← comp_base, IsIso.hom_inv_id, id_base,
+        CategoryTheory.id_apply]
   · rw [← IsIso.eq_inv_comp, IsOpenImmersion.inv_invApp, Category.assoc,
       (D.t' k i j).c.naturality_assoc]
     simp_rw [← Category.assoc]
diff --git a/Mathlib/GroupTheory/CommutingProbability.lean b/Mathlib/GroupTheory/CommutingProbability.lean
index 606360fa0cce66..815d7acb04b4e7 100644
--- a/Mathlib/GroupTheory/CommutingProbability.lean
+++ b/Mathlib/GroupTheory/CommutingProbability.lean
@@ -26,8 +26,6 @@ assert_not_exists Ideal TwoSidedIdeal
 
 noncomputable section
 
-open scoped Classical
-
 open Fintype
 
 variable (M : Type*) [Mul M]
@@ -79,6 +77,7 @@ variable {M}
 
 theorem commProb_eq_one_iff [h : Nonempty M] :
     commProb M = 1 ↔ Std.Commutative ((· * ·) : M → M → M) := by
+  classical
   haveI := Fintype.ofFinite M
   rw [commProb, ← Set.coe_setOf, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card]
   rw [div_eq_one_iff_eq, ← Nat.cast_pow, Nat.cast_inj, sq, ← card_prod,
diff --git a/Mathlib/GroupTheory/Coxeter/Length.lean b/Mathlib/GroupTheory/Coxeter/Length.lean
index aff28c1d3b6853..c987abb55aa891 100644
--- a/Mathlib/GroupTheory/Coxeter/Length.lean
+++ b/Mathlib/GroupTheory/Coxeter/Length.lean
@@ -50,7 +50,7 @@ assert_not_exists TwoSidedIdeal
 
 namespace CoxeterSystem
 
-open List Matrix Function Classical
+open List Matrix Function
 
 variable {B : Type*}
 variable {W : Type*} [Group W]
@@ -65,6 +65,7 @@ private theorem exists_word_with_prod (w : W) : ∃ n ω, ω.length = n ∧ π 
   rcases cs.wordProd_surjective w with ⟨ω, rfl⟩
   use ω.length, ω
 
+open scoped Classical in
 /-- The length of `w`; i.e., the minimum number of simple reflections that
 must be multiplied to form `w`. -/
 noncomputable def length (w : W) : ℕ := Nat.find (cs.exists_word_with_prod w)
@@ -72,9 +73,11 @@ noncomputable def length (w : W) : ℕ := Nat.find (cs.exists_word_with_prod w)
 local prefix:100 "ℓ" => cs.length
 
 theorem exists_reduced_word (w : W) : ∃ ω, ω.length = ℓ w ∧ w = π ω := by
+  classical
   have := Nat.find_spec (cs.exists_word_with_prod w)
   tauto
 
+open scoped Classical in
 theorem length_wordProd_le (ω : List B) : ℓ (π ω) ≤ ω.length :=
   Nat.find_min' (cs.exists_word_with_prod (π ω)) ⟨ω, by tauto⟩
 
diff --git a/Mathlib/GroupTheory/Exponent.lean b/Mathlib/GroupTheory/Exponent.lean
index 8adffc6d68bd07..0aa68664c8997c 100644
--- a/Mathlib/GroupTheory/Exponent.lean
+++ b/Mathlib/GroupTheory/Exponent.lean
@@ -47,8 +47,6 @@ universe u
 
 variable {G : Type u}
 
-open scoped Classical
-
 namespace Monoid
 
 section Monoid
@@ -63,6 +61,7 @@ variable (G) [Monoid G]
 def ExponentExists :=
   ∃ n, 0 < n ∧ ∀ g : G, g ^ n = 1
 
+open scoped Classical in
 /-- The exponent of a group is the smallest positive integer `n` such that `g ^ n = 1` for all
   `g ∈ G` if it exists, otherwise it is zero by convention. -/
 @[to_additive
@@ -132,6 +131,7 @@ theorem exponent_eq_zero_iff_forall : exponent G = 0 ↔ ∀ n > 0, ∃ g : G, g
 
 @[to_additive exponent_nsmul_eq_zero]
 theorem pow_exponent_eq_one (g : G) : g ^ exponent G = 1 := by
+  classical
   by_cases h : ExponentExists G
   · simp_rw [exponent, dif_pos h]
     exact (Nat.find_spec h).2 g
@@ -150,6 +150,7 @@ theorem exponent_pos_of_exists (n : ℕ) (hpos : 0 < n) (hG : ∀ g : G, g ^ n =
 
 @[to_additive]
 theorem exponent_min' (n : ℕ) (hpos : 0 < n) (hG : ∀ g : G, g ^ n = 1) : exponent G ≤ n := by
+  classical
   rw [exponent, dif_pos]
   · apply Nat.find_min'
     exact ⟨hpos, hG⟩
@@ -467,6 +468,7 @@ theorem exponent_eq_iSup_orderOf (h : ∀ g : G, 0 < orderOf g) :
     simp_rw [Set.mem_range, exists_exists_eq_and]
     exact ⟨g, hx⟩
 
+open scoped Classical in
 @[to_additive]
 theorem exponent_eq_iSup_orderOf' :
     exponent G = if ∃ g : G, orderOf g = 0 then 0 else ⨆ g : G, orderOf g := by
@@ -541,6 +543,7 @@ open Finset Monoid
 @[to_additive]
 theorem Monoid.exponent_pi_eq_zero {ι : Type*} {M : ι → Type*} [∀ i, Monoid (M i)] {j : ι}
     (hj : exponent (M j) = 0) : exponent ((i : ι) → M i) = 0 := by
+  classical
   rw [@exponent_eq_zero_iff, ExponentExists] at hj ⊢
   push_neg at hj ⊢
   peel hj with n hn _
diff --git a/Mathlib/GroupTheory/FreeAbelianGroup.lean b/Mathlib/GroupTheory/FreeAbelianGroup.lean
index 030548d59fea55..60054af51dd65b 100644
--- a/Mathlib/GroupTheory/FreeAbelianGroup.lean
+++ b/Mathlib/GroupTheory/FreeAbelianGroup.lean
@@ -129,8 +129,7 @@ end lift
 
 section
 
-open scoped Classical
-
+open scoped Classical in
 theorem of_injective : Function.Injective (of : α → FreeAbelianGroup α) :=
   fun x y hoxy ↦ Classical.by_contradiction fun hxy : x ≠ y ↦
     let f : FreeAbelianGroup α →+ ℤ := lift fun z ↦ if x = z then (1 : ℤ) else 0
diff --git a/Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean b/Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean
index 477433e03185a5..95d573ec602a6f 100644
--- a/Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean
+++ b/Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean
@@ -161,8 +161,7 @@ theorem support_nsmul (k : ℕ) (h : k ≠ 0) (a : FreeAbelianGroup X) :
   apply support_zsmul k _ a
   exact mod_cast h
 
-open scoped Classical
-
+open scoped Classical in
 theorem support_add (a b : FreeAbelianGroup X) : support (a + b) ⊆ a.support ∪ b.support := by
   simp only [support, AddMonoidHom.map_add]
   apply Finsupp.support_add
diff --git a/Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean b/Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean
index 2343d928c711c1..ce18362bd2a60f 100644
--- a/Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean
+++ b/Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean
@@ -50,8 +50,6 @@ free group, free groupoid, Nielsen-Schreier
 
 noncomputable section
 
-open scoped Classical
-
 universe v u
 
 /- Porting note: ./././Mathport/Syntax/Translate/Command.lean:229:11:unsupported:
@@ -95,12 +93,8 @@ theorem ext_functor {G} [Groupoid.{v} G] [IsFreeGroupoid G] {X : Type v} [Group
   let ⟨_, _, u⟩ := @unique_lift G _ _ X _ fun (a b : Generators G) (e : a ⟶ b) => g.map (of e)
   _root_.trans (u _ h) (u _ fun _ _ _ => rfl).symm
 
-#adaptation_note
-/--
-The new unused variable linter in
-https://github.com/leanprover/lean4/pull/5338
-flags `{ e // _ }`.
--/
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/5338
+The new unused variable linter flags `{ e // _ }`. -/
 set_option linter.unusedVariables false in
 /-- An action groupoid over a free group is free. More generally, one could show that the groupoid
 of elements over a free groupoid is free, but this version is easier to prove and suffices for our
@@ -210,6 +204,7 @@ def functorOfMonoidHom {X} [Monoid X] (f : End (root' T) →* X) :
     rw [comp_as_mul, ← f.map_mul]
     simp only [IsIso.inv_hom_id_assoc, loopOfHom, End.mul_def, Category.assoc]
 
+open scoped Classical in
 /-- Given a free groupoid and an arborescence of its generating quiver, the vertex
     group at the root is freely generated by loops coming from generating arrows
     in the complement of the tree. -/
diff --git a/Mathlib/GroupTheory/GroupAction/Defs.lean b/Mathlib/GroupTheory/GroupAction/Defs.lean
index d2cba75e644f68..fe04a1f0932dfb 100644
--- a/Mathlib/GroupTheory/GroupAction/Defs.lean
+++ b/Mathlib/GroupTheory/GroupAction/Defs.lean
@@ -5,8 +5,8 @@ Authors: Chris Hughes
 -/
 import Mathlib.Algebra.Group.Pointwise.Set.Basic
 import Mathlib.Algebra.Group.Subgroup.Defs
-import Mathlib.Algebra.Group.Submonoid.Operations
 import Mathlib.Algebra.GroupWithZero.Action.Defs
+import Mathlib.Algebra.Group.Submonoid.MulAction
 
 /-!
 # Definition of `orbit`, `fixedPoints` and `stabilizer`
diff --git a/Mathlib/GroupTheory/MonoidLocalization/Basic.lean b/Mathlib/GroupTheory/MonoidLocalization/Basic.lean
index 69ac2dcd677060..c1d5b153c65e4b 100644
--- a/Mathlib/GroupTheory/MonoidLocalization/Basic.lean
+++ b/Mathlib/GroupTheory/MonoidLocalization/Basic.lean
@@ -6,6 +6,7 @@ Authors: Amelia Livingston
 import Mathlib.Algebra.Group.Submonoid.Defs
 import Mathlib.GroupTheory.Congruence.Hom
 import Mathlib.GroupTheory.OreLocalization.Basic
+import Mathlib.Algebra.Group.Submonoid.Operations
 
 /-!
 # Localizations of commutative monoids
diff --git a/Mathlib/GroupTheory/Nilpotent.lean b/Mathlib/GroupTheory/Nilpotent.lean
index 2b1d6262d8668a..8c188f330ea885 100644
--- a/Mathlib/GroupTheory/Nilpotent.lean
+++ b/Mathlib/GroupTheory/Nilpotent.lean
@@ -355,23 +355,24 @@ theorem nilpotent_iff_lowerCentralSeries : IsNilpotent G ↔ ∃ n, lowerCentral
 
 section Classical
 
-open scoped Classical
-
 variable [hG : IsNilpotent G]
 variable (G)
 
+open scoped Classical in
 /-- The nilpotency class of a nilpotent group is the smallest natural `n` such that
 the `n`'th term of the upper central series is `G`. -/
 noncomputable def Group.nilpotencyClass : ℕ := Nat.find (IsNilpotent.nilpotent G)
 
 variable {G}
 
+open scoped Classical in
 @[simp]
 theorem upperCentralSeries_nilpotencyClass : upperCentralSeries G (Group.nilpotencyClass G) = ⊤ :=
   Nat.find_spec (IsNilpotent.nilpotent G)
 
 theorem upperCentralSeries_eq_top_iff_nilpotencyClass_le {n : ℕ} :
     upperCentralSeries G n = ⊤ ↔ Group.nilpotencyClass G ≤ n := by
+  classical
   constructor
   · intro h
     exact Nat.find_le h
@@ -379,6 +380,7 @@ theorem upperCentralSeries_eq_top_iff_nilpotencyClass_le {n : ℕ} :
     rw [eq_top_iff, ← upperCentralSeries_nilpotencyClass]
     exact upperCentralSeries_mono _ h
 
+open scoped Classical in
 /-- The nilpotency class of a nilpotent `G` is equal to the smallest `n` for which an ascending
 central series reaches `G` in its `n`'th term. -/
 theorem least_ascending_central_series_length_eq_nilpotencyClass :
@@ -391,6 +393,7 @@ theorem least_ascending_central_series_length_eq_nilpotencyClass :
     rw [← top_le_iff, ← hn]
     exact ascending_central_series_le_upper H hH n
 
+open scoped Classical in
 /-- The nilpotency class of a nilpotent `G` is equal to the smallest `n` for which the descending
 central series reaches `⊥` in its `n`'th term. -/
 theorem least_descending_central_series_length_eq_nilpotencyClass :
@@ -409,6 +412,7 @@ theorem least_descending_central_series_length_eq_nilpotencyClass :
     rw [tsub_self]
     exact hH.1
 
+open scoped Classical in
 /-- The nilpotency class of a nilpotent `G` is equal to the length of the lower central series. -/
 theorem lowerCentralSeries_length_eq_nilpotencyClass :
     Nat.find (nilpotent_iff_lowerCentralSeries.mp hG) = Group.nilpotencyClass (G := G) := by
@@ -423,11 +427,13 @@ theorem lowerCentralSeries_length_eq_nilpotencyClass :
 @[simp]
 theorem lowerCentralSeries_nilpotencyClass :
     lowerCentralSeries G (Group.nilpotencyClass G) = ⊥ := by
+  classical
   rw [← lowerCentralSeries_length_eq_nilpotencyClass]
   exact Nat.find_spec (nilpotent_iff_lowerCentralSeries.mp hG)
 
 theorem lowerCentralSeries_eq_bot_iff_nilpotencyClass_le {n : ℕ} :
     lowerCentralSeries G n = ⊥ ↔ Group.nilpotencyClass G ≤ n := by
+  classical
   constructor
   · intro h
     rw [← lowerCentralSeries_length_eq_nilpotencyClass]
diff --git a/Mathlib/GroupTheory/OrderOfElement.lean b/Mathlib/GroupTheory/OrderOfElement.lean
index a53461a680fb88..4a207ce12d1ef9 100644
--- a/Mathlib/GroupTheory/OrderOfElement.lean
+++ b/Mathlib/GroupTheory/OrderOfElement.lean
@@ -10,6 +10,7 @@ import Mathlib.Algebra.Order.Group.Action
 import Mathlib.Algebra.Order.Ring.Abs
 import Mathlib.Dynamics.PeriodicPts.Lemmas
 import Mathlib.GroupTheory.Index
+import Mathlib.NumberTheory.Divisors
 import Mathlib.Order.Interval.Set.Infinite
 import Mathlib.Tactic.Positivity
 
@@ -689,28 +690,12 @@ variable [Monoid G] {x : G} {n : ℕ}
 
 @[to_additive]
 theorem sum_card_orderOf_eq_card_pow_eq_one [Fintype G] [DecidableEq G] (hn : n ≠ 0) :
-    (∑ m ∈ (Finset.range n.succ).filter (· ∣ n),
+    (∑ m ∈ divisors n,
         (Finset.univ.filter fun x : G => orderOf x = m).card) =
-      (Finset.univ.filter fun x : G => x ^ n = 1).card :=
-  calc
-    (∑ m ∈ (Finset.range n.succ).filter (· ∣ n),
-          (Finset.univ.filter fun x : G => orderOf x = m).card) = _ :=
-      (Finset.card_biUnion
-          (by
-            intros
-            apply Finset.disjoint_filter.2
-            rintro _ _ rfl; assumption)).symm
-    _ = _ :=
-      congr_arg Finset.card
-        (Finset.ext
-          (by
-            intro x
-            suffices orderOf x ≤ n ∧ orderOf x ∣ n ↔ x ^ n = 1 by simpa [Nat.lt_succ_iff]
-            exact
-              ⟨fun h => by
-                let ⟨m, hm⟩ := h.2
-                rw [hm, pow_mul, pow_orderOf_eq_one, one_pow], fun h =>
-                ⟨orderOf_le_of_pow_eq_one hn.bot_lt h, orderOf_dvd_of_pow_eq_one h⟩⟩))
+      (Finset.univ.filter fun x : G => x ^ n = 1).card := by
+  refine (Finset.card_biUnion ?_).symm.trans ?_
+  · simp +contextual [disjoint_iff, Finset.ext_iff]
+  · congr; ext; simp [hn, orderOf_dvd_iff_pow_eq_one]
 
 @[to_additive]
 theorem orderOf_le_card_univ [Fintype G] : orderOf x ≤ Fintype.card G :=
diff --git a/Mathlib/GroupTheory/OreLocalization/Basic.lean b/Mathlib/GroupTheory/OreLocalization/Basic.lean
index 24b43521f550bb..356e809eb79d9b 100644
--- a/Mathlib/GroupTheory/OreLocalization/Basic.lean
+++ b/Mathlib/GroupTheory/OreLocalization/Basic.lean
@@ -3,9 +3,10 @@ Copyright (c) 2022 Jakob von Raumer. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
 -/
-import Mathlib.Algebra.Group.Submonoid.Operations
 import Mathlib.GroupTheory.OreLocalization.OreSet
 import Mathlib.Tactic.Common
+import Mathlib.Algebra.Group.Submonoid.MulAction
+import Mathlib.Algebra.Group.Units.Defs
 
 /-!
 
diff --git a/Mathlib/GroupTheory/Perm/Cycle/Concrete.lean b/Mathlib/GroupTheory/Perm/Cycle/Concrete.lean
index 4f238742343a89..233d79ecb80e8a 100644
--- a/Mathlib/GroupTheory/Perm/Cycle/Concrete.lean
+++ b/Mathlib/GroupTheory/Perm/Cycle/Concrete.lean
@@ -39,8 +39,6 @@ The forward direction of `Equiv.Perm.isoCycle'` uses `Fintype.choose` of the uni
 result, relying on the `Fintype` instance of a `Cycle.nodup` subtype.
 It is unclear if this works faster than the `Equiv.Perm.toCycle`, which relies
 on recursion over `Finset.univ`.
-Running `#eval` on even a simple noncyclic permutation `c[(1 : Fin 7), 2, 3] * c[0, 5]`
-to show it takes a long time. TODO: is this because computing the cycle factors is slow?
 
 -/
 
@@ -469,9 +467,35 @@ set_option linter.unusedTactic false in
 notation3 (prettyPrint := false) "c["(l", "* => foldr (h t => List.cons h t) List.nil)"]" =>
   Cycle.formPerm (Cycle.ofList l) (Iff.mpr Cycle.nodup_coe_iff (by decide))
 
-unsafe instance repr_perm [Repr α] : Repr (Perm α) :=
-  ⟨fun f _ => repr (Multiset.pmap (fun (g : Perm α) (hg : g.IsCycle) => isoCycle ⟨g, hg⟩)
-    (Perm.cycleFactorsFinset f).val -- toCycle is faster?
-    fun _ hg => (mem_cycleFactorsFinset_iff.mp (Finset.mem_def.mpr hg)).left)⟩
+/-- Represents a permutation as product of disjoint cycles:
+```
+#eval (c[0, 1, 2, 3] : Perm (Fin 4))
+-- c[0, 1, 2, 3]
+
+#eval (c[3, 1] * c[0, 2] : Perm (Fin 4))
+-- c[0, 2] * c[1, 3]
+
+#eval (c[1, 2, 3] * c[0, 1, 2] : Perm (Fin 4))
+-- c[0, 2] * c[1, 3]
+
+#eval (c[1, 2, 3] * c[0, 1, 2] * c[3, 1] * c[0, 2] : Perm (Fin 4))
+-- 1
+```
+-/
+unsafe instance instRepr [Repr α] : Repr (Perm α) where
+  reprPrec f prec :=
+    -- Obtain a list of formats which represents disjoint cycles.
+    letI l := Quot.unquot <| Multiset.map repr <| Multiset.pmap toCycle
+      (Perm.cycleFactorsFinset f).val
+      fun _ hg => (mem_cycleFactorsFinset_iff.mp (Finset.mem_def.mpr hg)).left
+    -- And intercalate `*`s.
+    match l with
+    | []  => "1"
+    | [f] => f
+    | l   =>
+      -- multiple terms, use `*` precedence
+      (if prec ≥ 70 then Lean.Format.paren else id)
+      (Lean.Format.fill
+        (Lean.Format.joinSep l (" *" ++ Lean.Format.line)))
 
 end Equiv.Perm
diff --git a/Mathlib/GroupTheory/Perm/Cycle/Type.lean b/Mathlib/GroupTheory/Perm/Cycle/Type.lean
index e9a7dd8bc94db6..ebbd77c65b3711 100644
--- a/Mathlib/GroupTheory/Perm/Cycle/Type.lean
+++ b/Mathlib/GroupTheory/Perm/Cycle/Type.lean
@@ -406,8 +406,8 @@ theorem zero_eq : vectorsProdEqOne G 0 = {Vector.nil} :=
   Set.eq_singleton_iff_unique_mem.mpr ⟨Eq.refl (1 : G), fun v _ => v.eq_nil⟩
 
 theorem one_eq : vectorsProdEqOne G 1 = {Vector.nil.cons 1} := by
-  simp_rw [Set.eq_singleton_iff_unique_mem, mem_iff, Vector.toList_singleton, List.prod_singleton,
-    Vector.head_cons, true_and]
+  simp_rw [Set.eq_singleton_iff_unique_mem, mem_iff, List.Vector.toList_singleton,
+    List.prod_singleton, List.Vector.head_cons, true_and]
   exact fun v hv => v.cons_head_tail.symm.trans (congr_arg₂ Vector.cons hv v.tail.eq_nil)
 
 instance zeroUnique : Unique (vectorsProdEqOne G 0) := by
diff --git a/Mathlib/GroupTheory/Perm/Fin.lean b/Mathlib/GroupTheory/Perm/Fin.lean
index 6d0fa972e68ea1..ce47c8d3ed178c 100644
--- a/Mathlib/GroupTheory/Perm/Fin.lean
+++ b/Mathlib/GroupTheory/Perm/Fin.lean
@@ -142,13 +142,13 @@ def cycleRange {n : ℕ} (i : Fin n) : Perm (Fin n) :=
         ext
         simp))
 
-theorem cycleRange_of_gt {n : ℕ} {i j : Fin n.succ} (h : i < j) : cycleRange i j = j := by
+theorem cycleRange_of_gt {n : ℕ} {i j : Fin n} (h : i < j) : cycleRange i j = j := by
   rw [cycleRange, ofLeftInverse'_eq_ofInjective,
     ← Function.Embedding.toEquivRange_eq_ofInjective, ← viaFintypeEmbedding,
     viaFintypeEmbedding_apply_not_mem_range]
   simpa
 
-theorem cycleRange_of_le {n : ℕ} {i j : Fin n.succ} (h : j ≤ i) :
+theorem cycleRange_of_le {n : ℕ} [NeZero n] {i j : Fin n} (h : j ≤ i) :
     cycleRange i j = if j = i then 0 else j + 1 := by
   cases n
   · subsingleton
@@ -165,8 +165,10 @@ theorem cycleRange_of_le {n : ℕ} {i j : Fin n.succ} (h : j ≤ i) :
   · rw [Fin.val_add_one_of_lt]
     exact lt_of_lt_of_le (lt_of_le_of_ne h (mt (congr_arg _) heq)) (le_last i)
 
-theorem coe_cycleRange_of_le {n : ℕ} {i j : Fin n.succ} (h : j ≤ i) :
+theorem coe_cycleRange_of_le {n : ℕ} {i j : Fin n} (h : j ≤ i) :
     (cycleRange i j : ℕ) = if j = i then 0 else (j : ℕ) + 1 := by
+  cases' n with n
+  · exact absurd le_rfl i.pos.not_le
   rw [cycleRange_of_le h]
   split_ifs with h'
   · rfl
@@ -176,20 +178,20 @@ theorem coe_cycleRange_of_le {n : ℕ} {i j : Fin n.succ} (h : j ≤ i) :
         (j : ℕ) < i := Fin.lt_iff_val_lt_val.mp (lt_of_le_of_ne h h')
         _ ≤ n := Nat.lt_succ_iff.mp i.2)
 
-theorem cycleRange_of_lt {n : ℕ} {i j : Fin n.succ} (h : j < i) : cycleRange i j = j + 1 := by
+theorem cycleRange_of_lt {n : ℕ} [NeZero n] {i j : Fin n} (h : j < i) : cycleRange i j = j + 1 := by
   rw [cycleRange_of_le h.le, if_neg h.ne]
 
-theorem coe_cycleRange_of_lt {n : ℕ} {i j : Fin n.succ} (h : j < i) :
+theorem coe_cycleRange_of_lt {n : ℕ} {i j : Fin n} (h : j < i) :
     (cycleRange i j : ℕ) = j + 1 := by rw [coe_cycleRange_of_le h.le, if_neg h.ne]
 
-theorem cycleRange_of_eq {n : ℕ} {i j : Fin n.succ} (h : j = i) : cycleRange i j = 0 := by
+theorem cycleRange_of_eq {n : ℕ} [NeZero n] {i j : Fin n} (h : j = i) : cycleRange i j = 0 := by
   rw [cycleRange_of_le h.le, if_pos h]
 
 @[simp]
-theorem cycleRange_self {n : ℕ} (i : Fin n.succ) : cycleRange i i = 0 :=
+theorem cycleRange_self {n : ℕ} [NeZero n] (i : Fin n) : cycleRange i i = 0 :=
   cycleRange_of_eq rfl
 
-theorem cycleRange_apply {n : ℕ} (i j : Fin n.succ) :
+theorem cycleRange_apply {n : ℕ} [NeZero n] (i j : Fin n) :
     cycleRange i j = if j < i then j + 1 else if j = i then 0 else j := by
   split_ifs with h₁ h₂
   · exact cycleRange_of_lt h₁
@@ -197,11 +199,11 @@ theorem cycleRange_apply {n : ℕ} (i j : Fin n.succ) :
   · exact cycleRange_of_gt (lt_of_le_of_ne (le_of_not_gt h₁) (Ne.symm h₂))
 
 @[simp]
-theorem cycleRange_zero (n : ℕ) : cycleRange (0 : Fin n.succ) = 1 := by
+theorem cycleRange_zero (n : ℕ) [NeZero n] : cycleRange (0 : Fin n) = 1 := by
   ext j
-  refine Fin.cases ?_ (fun j => ?_) j
+  rcases (Fin.zero_le' j).eq_or_lt with rfl | hj
   · simp
-  · rw [cycleRange_of_gt (Fin.succ_pos j), one_apply]
+  · rw [cycleRange_of_gt hj, one_apply]
 
 @[simp]
 theorem cycleRange_last (n : ℕ) : cycleRange (last n) = finRotate (n + 1) := by
@@ -209,10 +211,12 @@ theorem cycleRange_last (n : ℕ) : cycleRange (last n) = finRotate (n + 1) := b
   rw [coe_cycleRange_of_le (le_last _), coe_finRotate]
 
 @[simp]
-theorem cycleRange_zero' {n : ℕ} (h : 0 < n) : cycleRange ⟨0, h⟩ = 1 := by
-  cases' n with n
-  · cases h
-  exact cycleRange_zero n
+theorem cycleRange_mk_zero {n : ℕ} (h : 0 < n) : cycleRange ⟨0, h⟩ = 1 :=
+  have : NeZero n := .of_pos h
+  cycleRange_zero n
+
+@[deprecated (since := "2025-01-28")]
+alias cycleRange_zero' := cycleRange_mk_zero
 
 @[simp]
 theorem sign_cycleRange {n : ℕ} (i : Fin n) : Perm.sign (cycleRange i) = (-1) ^ (i : ℕ) := by
@@ -248,7 +252,7 @@ theorem cycleRange_succAbove {n : ℕ} (i : Fin (n + 1)) (j : Fin n) :
   · rw [Fin.succAbove_of_le_castSucc _ _ h, Fin.cycleRange_of_gt (Fin.le_castSucc_iff.mp h)]
 
 @[simp]
-theorem cycleRange_symm_zero {n : ℕ} (i : Fin (n + 1)) : i.cycleRange.symm 0 = i :=
+theorem cycleRange_symm_zero {n : ℕ} [NeZero n] (i : Fin n) : i.cycleRange.symm 0 = i :=
   i.cycleRange.injective (by simp)
 
 @[simp]
@@ -256,14 +260,15 @@ theorem cycleRange_symm_succ {n : ℕ} (i : Fin (n + 1)) (j : Fin n) :
     i.cycleRange.symm j.succ = i.succAbove j :=
   i.cycleRange.injective (by simp)
 
-theorem isCycle_cycleRange {n : ℕ} {i : Fin (n + 1)} (h0 : i ≠ 0) : IsCycle (cycleRange i) := by
+theorem isCycle_cycleRange {n : ℕ} [NeZero n] {i : Fin n} (h0 : i ≠ 0) :
+    IsCycle (cycleRange i) := by
   cases' i with i hi
   cases i
   · exact (h0 rfl).elim
   exact isCycle_finRotate.extendDomain _
 
 @[simp]
-theorem cycleType_cycleRange {n : ℕ} {i : Fin (n + 1)} (h0 : i ≠ 0) :
+theorem cycleType_cycleRange {n : ℕ} [NeZero n] {i : Fin n} (h0 : i ≠ 0) :
     cycleType (cycleRange i) = {(i + 1 : ℕ)} := by
   cases' i with i hi
   cases i
diff --git a/Mathlib/GroupTheory/Perm/ViaEmbedding.lean b/Mathlib/GroupTheory/Perm/ViaEmbedding.lean
index 4d0a88429acabe..7193c769d6b76b 100644
--- a/Mathlib/GroupTheory/Perm/ViaEmbedding.lean
+++ b/Mathlib/GroupTheory/Perm/ViaEmbedding.lean
@@ -19,18 +19,20 @@ namespace Perm
 
 variable (e : Perm α) (ι : α ↪ β)
 
-open scoped Classical
-
+open scoped Classical in
 /-- Noncomputable version of `Equiv.Perm.viaFintypeEmbedding` that does not assume `Fintype` -/
 noncomputable def viaEmbedding : Perm β :=
   extendDomain e (ofInjective ι.1 ι.2)
 
+open scoped Classical in
 theorem viaEmbedding_apply (x : α) : e.viaEmbedding ι (ι x) = ι (e x) :=
   extendDomain_apply_image e (ofInjective ι.1 ι.2) x
 
+open scoped Classical in
 theorem viaEmbedding_apply_of_not_mem (x : β) (hx : x ∉ Set.range ι) : e.viaEmbedding ι x = x :=
   extendDomain_apply_not_subtype e (ofInjective ι.1 ι.2) hx
 
+open scoped Classical in
 /-- `viaEmbedding` as a group homomorphism -/
 noncomputable def viaEmbeddingHom : Perm α →* Perm β :=
   extendDomainHom (ofInjective ι.1 ι.2)
@@ -38,6 +40,7 @@ noncomputable def viaEmbeddingHom : Perm α →* Perm β :=
 theorem viaEmbeddingHom_apply : viaEmbeddingHom ι e = viaEmbedding e ι :=
   rfl
 
+open scoped Classical in
 theorem viaEmbeddingHom_injective : Function.Injective (viaEmbeddingHom ι) :=
   extendDomainHom_injective (ofInjective ι.1 ι.2)
 
diff --git a/Mathlib/GroupTheory/QuotientGroup/Finite.lean b/Mathlib/GroupTheory/QuotientGroup/Finite.lean
index d21f2e376e3738..c27b649ddef251 100644
--- a/Mathlib/GroupTheory/QuotientGroup/Finite.lean
+++ b/Mathlib/GroupTheory/QuotientGroup/Finite.lean
@@ -20,13 +20,12 @@ universe u v w x
 
 namespace Group
 
-open scoped Classical
-
 open QuotientGroup Subgroup
 
 variable {F G H : Type u} [Group F] [Group G] [Group H] [Fintype F] [Fintype H]
 variable (f : F →* G) (g : G →* H)
 
+open scoped Classical in
 /-- If `F` and `H` are finite such that `ker(G →* H) ≤ im(F →* G)`, then `G` is finite. -/
 @[to_additive "If `F` and `H` are finite such that `ker(G →+ H) ≤ im(F →+ G)`, then `G` is finite."]
 noncomputable def fintypeOfKerLeRange (h : g.ker ≤ f.range) : Fintype G :=
diff --git a/Mathlib/GroupTheory/Schreier.lean b/Mathlib/GroupTheory/Schreier.lean
index d7bbfc3d608794..b0d77367ab543b 100644
--- a/Mathlib/GroupTheory/Schreier.lean
+++ b/Mathlib/GroupTheory/Schreier.lean
@@ -29,12 +29,11 @@ section CommGroup
 
 open Subgroup
 
-open scoped Classical
-
 variable (G : Type*) [CommGroup G] [Group.FG G]
 
 @[to_additive]
 theorem card_dvd_exponent_pow_rank : Nat.card G ∣ Monoid.exponent G ^ Group.rank G := by
+  classical
   obtain ⟨S, hS1, hS2⟩ := Group.rank_spec G
   rw [← hS1, ← Fintype.card_coe, ← Finset.card_univ, ← Finset.prod_const]
   let f : (∀ g : S, zpowers (g : G)) →* G := noncommPiCoprod fun s t _ x y _ _ => mul_comm x _
diff --git a/Mathlib/GroupTheory/SchurZassenhaus.lean b/Mathlib/GroupTheory/SchurZassenhaus.lean
index b36b24b2c9fa80..0611210466d315 100644
--- a/Mathlib/GroupTheory/SchurZassenhaus.lean
+++ b/Mathlib/GroupTheory/SchurZassenhaus.lean
@@ -117,8 +117,6 @@ private theorem exists_right_complement'_of_coprime_aux (hH : Nat.Coprime (Nat.c
 
 end SchurZassenhausAbelian
 
-open scoped Classical
-
 universe u
 
 namespace SchurZassenhausInduction
diff --git a/Mathlib/GroupTheory/SpecificGroups/Cyclic.lean b/Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
index 44e9501b1c31a5..d000af6ee0f64a 100644
--- a/Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
+++ b/Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
@@ -456,8 +456,8 @@ private theorem card_orderOf_eq_totient_aux₁ {d : ℕ} (hd : d ∣ Fintype.car
   have h2 :
     (∑ m ∈ d.divisors, #{a : α | orderOf a = m}) =
       ∑ m ∈ d.divisors, φ m := by
-    rw [← filter_dvd_eq_divisors hd0, sum_card_orderOf_eq_card_pow_eq_one hd0,
-      filter_dvd_eq_divisors hd0, sum_totient, ← ha, card_pow_eq_one_eq_orderOf_aux hn a]
+    rw [sum_card_orderOf_eq_card_pow_eq_one hd0, sum_totient,
+      ← ha, card_pow_eq_one_eq_orderOf_aux hn a]
   simpa [← cons_self_properDivisors hd0, ← h1] using h2
 
 @[to_additive]
@@ -472,7 +472,7 @@ theorem card_orderOf_eq_totient_aux₂ {d : ℕ} (hd : d ∣ Fintype.card α) :
   apply lt_irrefl c
   calc
     c = ∑ m ∈ c.divisors, #{a : α | orderOf a = m} := by
-      simp only [← filter_dvd_eq_divisors hc0.ne', sum_card_orderOf_eq_card_pow_eq_one hc0.ne']
+      simp only [sum_card_orderOf_eq_card_pow_eq_one hc0.ne']
       apply congr_arg card
       simp [c]
     _ = ∑ m ∈ c.divisors.erase d, #{a : α | orderOf a = m} := by
diff --git a/Mathlib/GroupTheory/Subgroup/Center.lean b/Mathlib/GroupTheory/Subgroup/Center.lean
index 6dd4dbfa414472..b8dc94177cf81a 100644
--- a/Mathlib/GroupTheory/Subgroup/Center.lean
+++ b/Mathlib/GroupTheory/Subgroup/Center.lean
@@ -40,6 +40,16 @@ theorem center_toSubmonoid : (center G).toSubmonoid = Submonoid.center G :=
 instance center.isCommutative : (center G).IsCommutative :=
   ⟨⟨fun a b => Subtype.ext (b.2.comm a).symm⟩⟩
 
+variable {G} in
+/-- The center of isomorphic groups are isomorphic. -/
+@[to_additive (attr := simps!) "The center of isomorphic additive groups are isomorphic."]
+def centerCongr {H} [Group H] (e : G ≃* H) : center G ≃* center H := Submonoid.centerCongr e
+
+/-- The center of a group is isomorphic to the center of its opposite. -/
+@[to_additive (attr := simps!)
+"The center of an additive group is isomorphic to the center of its opposite."]
+def centerToMulOpposite : center G ≃* center Gᵐᵒᵖ := Submonoid.centerToMulOpposite
+
 /-- For a group with zero, the center of the units is the same as the units of the center. -/
 @[simps! apply_val_coe symm_apply_coe_val]
 def centerUnitsEquivUnitsCenter (G₀ : Type*) [GroupWithZero G₀] :
diff --git a/Mathlib/GroupTheory/Submonoid/Center.lean b/Mathlib/GroupTheory/Submonoid/Center.lean
index 44028df37b8689..addceff4248907 100644
--- a/Mathlib/GroupTheory/Submonoid/Center.lean
+++ b/Mathlib/GroupTheory/Submonoid/Center.lean
@@ -126,3 +126,61 @@ def unitsCenterToCenterUnits [Monoid M] : (Submonoid.center M)ˣ →* Submonoid.
 theorem unitsCenterToCenterUnits_injective [Monoid M] :
     Function.Injective (unitsCenterToCenterUnits M) :=
   fun _a _b h => Units.ext <| Subtype.ext <| congr_arg (Units.val ∘ Subtype.val) h
+
+section congr
+
+variable {M} {N : Type*}
+
+@[to_additive] theorem _root_.MulEquivClass.apply_mem_center {F} [EquivLike F M N] [Mul M] [Mul N]
+    [MulEquivClass F M N] (e : F) {x : M} (hx : x ∈ Set.center M) : e x ∈ Set.center N := by
+  let e := MulEquivClass.toMulEquiv e
+  show e x ∈ Set.center N
+  constructor <;>
+  (intros; apply e.symm.injective;
+    simp only [map_mul, e.symm_apply_apply, (isMulCentral_iff _).mp hx])
+
+@[to_additive] theorem _root_.MulEquivClass.apply_mem_center_iff {F} [EquivLike F M N]
+    [Mul M] [Mul N] [MulEquivClass F M N] (e : F) {x : M} :
+    e x ∈ Set.center N ↔ x ∈ Set.center M :=
+  ⟨(by simpa using MulEquivClass.apply_mem_center (MulEquivClass.toMulEquiv e).symm ·),
+    MulEquivClass.apply_mem_center e⟩
+
+/-- The center of isomorphic magmas are isomorphic. -/
+@[to_additive (attr := simps) "The center of isomorphic additive magmas are isomorphic."]
+def Subsemigroup.centerCongr [Mul M] [Mul N] (e : M ≃* N) : center M ≃* center N where
+  toFun r := ⟨e r, MulEquivClass.apply_mem_center e r.2⟩
+  invFun s := ⟨e.symm s, MulEquivClass.apply_mem_center e.symm s.2⟩
+  left_inv _ := Subtype.ext (e.left_inv _)
+  right_inv _ := Subtype.ext (e.right_inv _)
+  map_mul' _ _ := Subtype.ext (map_mul ..)
+
+/-- The center of isomorphic monoids are isomorphic. -/
+@[to_additive (attr := simps!) "The center of isomorphic additive monoids are isomorphic."]
+def Submonoid.centerCongr [MulOneClass M] [MulOneClass N] (e : M ≃* N) : center M ≃* center N :=
+  Subsemigroup.centerCongr e
+
+@[to_additive] theorem MulOpposite.op_mem_center_iff [Mul M] {x : M} :
+    op x ∈ Set.center Mᵐᵒᵖ ↔ x ∈ Set.center M := by
+  simp_rw [Set.mem_center_iff, isMulCentral_iff, MulOpposite.forall, ← op_mul, op_inj]; aesop
+
+@[to_additive] theorem MulOpposite.unop_mem_center_iff [Mul M] {x : Mᵐᵒᵖ} :
+    unop x ∈ Set.center M ↔ x ∈ Set.center Mᵐᵒᵖ :=
+  op_mem_center_iff.symm
+
+/-- The center of a magma is isomorphic to the center of its opposite. -/
+@[to_additive (attr := simps)
+"The center of an additive magma is isomorphic to the center of its opposite."]
+def Subsemigroup.centerToMulOpposite [Mul M] : center M ≃* center Mᵐᵒᵖ where
+  toFun r := ⟨_, MulOpposite.op_mem_center_iff.mpr r.2⟩
+  invFun r := ⟨_, MulOpposite.unop_mem_center_iff.mpr r.2⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_mul' r _ := Subtype.ext (congr_arg MulOpposite.op <| r.2.1 _)
+
+/-- The center of a monoid is isomorphic to the center of its opposite. -/
+@[to_additive (attr := simps!)
+"The center of an additive monoid is isomorphic to the center of its opposite. "]
+def Submonoid.centerToMulOpposite [MulOneClass M] : center M ≃* center Mᵐᵒᵖ :=
+  Subsemigroup.centerToMulOpposite
+
+end congr
diff --git a/Mathlib/InformationTheory/Hamming.lean b/Mathlib/InformationTheory/Hamming.lean
index 8ad9b91993a9b9..47a11f3d846157 100644
--- a/Mathlib/InformationTheory/Hamming.lean
+++ b/Mathlib/InformationTheory/Hamming.lean
@@ -244,6 +244,8 @@ instance [Zero α] [∀ i, Zero (β i)] [∀ i, SMulWithZero α (β i)] : SMulWi
 instance [∀ i, AddMonoid (β i)] : AddMonoid (Hamming β) :=
   Pi.addMonoid
 
+instance [∀ i, AddGroup (β i)] : AddGroup (Hamming β) := Pi.addGroup
+
 instance [∀ i, AddCommMonoid (β i)] : AddCommMonoid (Hamming β) :=
   Pi.addCommMonoid
 
@@ -396,15 +398,15 @@ instance [∀ i, Zero (β i)] : Norm (Hamming β) :=
 theorem norm_eq_hammingNorm [∀ i, Zero (β i)] (x : Hamming β) : ‖x‖ = hammingNorm (ofHamming x) :=
   rfl
 
--- Porting note: merged `SeminormedAddCommGroup` and `NormedAddCommGroup` instances
+instance [∀ i, AddGroup (β i)] : NormedAddGroup (Hamming β) where
+  dist_eq := by push_cast; exact mod_cast hammingDist_eq_hammingNorm
 
 instance [∀ i, AddCommGroup (β i)] : NormedAddCommGroup (Hamming β) where
   dist_eq := by push_cast; exact mod_cast hammingDist_eq_hammingNorm
 
 @[simp, push_cast]
-theorem nnnorm_eq_hammingNorm [∀ i, AddCommGroup (β i)] (x : Hamming β) :
-    ‖x‖₊ = hammingNorm (ofHamming x) :=
-  rfl
+theorem nnnorm_eq_hammingNorm [∀ i, AddGroup (β i)] (x : Hamming β) :
+    ‖x‖₊ = hammingNorm (ofHamming x) := rfl
 
 end
 
diff --git a/Mathlib/Lean/Meta/CongrTheorems.lean b/Mathlib/Lean/Meta/CongrTheorems.lean
index a8dafe249c20f2..fa7daf4c35b9d0 100644
--- a/Mathlib/Lean/Meta/CongrTheorems.lean
+++ b/Mathlib/Lean/Meta/CongrTheorems.lean
@@ -105,6 +105,11 @@ class FastIsEmpty (α : Sort u) : Prop where
 protected theorem FastSubsingleton.elim {α : Sort u} [h : FastSubsingleton α] : (a b : α) → a = b :=
   h.inst.allEq
 
+protected theorem FastSubsingleton.helim {α β : Sort u} [FastSubsingleton α]
+    (h₂ : α = β) (a : α) (b : β) : HEq a b := by
+  have : Subsingleton α := FastSubsingleton.inst
+  exact Subsingleton.helim h₂ a b
+
 instance (priority := 100) {α : Type u} [inst : FastIsEmpty α] : FastSubsingleton α where
   inst := have := inst.inst; inferInstance
 
diff --git a/Mathlib/LinearAlgebra/BilinearForm/Properties.lean b/Mathlib/LinearAlgebra/BilinearForm/Properties.lean
index 1c1eeeec108bc9..8ae2bfd5616d2b 100644
--- a/Mathlib/LinearAlgebra/BilinearForm/Properties.lean
+++ b/Mathlib/LinearAlgebra/BilinearForm/Properties.lean
@@ -275,8 +275,8 @@ noncomputable def dualBasis (B : BilinForm K V) (hB : B.Nondegenerate) (b : Basi
 theorem dualBasis_repr_apply
     (B : BilinForm K V) (hB : B.Nondegenerate) (b : Basis ι K V) (x i) :
     (B.dualBasis hB b).repr x i = B x (b i) := by
-  #adaptation_note
-  /-- Before https://github.com/leanprover/lean4/pull/4814, we did not need the `@` in front of `toDual_def` in the `rw`.
+  #adaptation_note /-- https://github.com/leanprover/lean4/pull/4814
+  we did not need the `@` in front of `toDual_def` in the `rw`.
   I'm confused! -/
   rw [dualBasis, Basis.map_repr, LinearEquiv.symm_symm, LinearEquiv.trans_apply,
     Basis.dualBasis_repr, @toDual_def]
diff --git a/Mathlib/LinearAlgebra/Dual.lean b/Mathlib/LinearAlgebra/Dual.lean
index a6c69a3ba7efa0..295a03c7af209f 100644
--- a/Mathlib/LinearAlgebra/Dual.lean
+++ b/Mathlib/LinearAlgebra/Dual.lean
@@ -1712,9 +1712,7 @@ theorem finiteDimensional_quot_dualCoannihilator_iff {W : Submodule K (Dual K V)
     FiniteDimensional K (V ⧸ W.dualCoannihilator) ↔ FiniteDimensional K W :=
   ⟨fun _ ↦ FiniteDimensional.of_injective _ W.flip_quotDualCoannihilatorToDual_injective,
     fun _ ↦ by
-      #adaptation_note
-      /--
-      After https://github.com/leanprover/lean4/pull/4119
+      #adaptation_note /-- https://github.com/leanprover/lean4/pull/4119
       the `Free K W` instance isn't found unless we use `set_option maxSynthPendingDepth 2`, or add
       explicit instances:
       ```
@@ -1896,4 +1894,4 @@ noncomputable def dualDistribEquiv : Dual R M ⊗[R] Dual R N ≃ₗ[R] Dual R (
 
 end TensorProduct
 
-set_option linter.style.longFile 2100
+set_option linter.style.longFile 2000
diff --git a/Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean b/Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean
index fcb07a9d441a26..aa94d3765c6685 100644
--- a/Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean
+++ b/Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean
@@ -219,7 +219,7 @@ theorem inf_iSup_genEigenspace [FiniteDimensional K V] (h : ∀ x ∈ p, f x ∈
       rw [Algebra.algebraMap_eq_smul_one, ← End.genEigenspace_nat]
       apply aux μ' hμ'
     · have := this.supIndep' (m.support.erase μ)
-      apply Finset.supIndep_antimono_fun _ this
+      apply this.antitone_fun
       intro μ' hμ'
       rw [Algebra.algebraMap_eq_smul_one, ← End.genEigenspace_nat]
       apply aux μ' hμ'
diff --git a/Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean b/Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean
index 8befb3f75b370b..8ae8358bfca550 100644
--- a/Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean
@@ -39,7 +39,7 @@ theorem PiToModule.fromMatrix_apply [DecidableEq ι] (A : Matrix ι ι R) (w : 
 theorem PiToModule.fromMatrix_apply_single_one [DecidableEq ι] (A : Matrix ι ι R) (j : ι) :
     PiToModule.fromMatrix R b A (Pi.single j 1) = ∑ i : ι, A i j • b i := by
   rw [PiToModule.fromMatrix_apply, Fintype.linearCombination_apply, Matrix.mulVec_single]
-  simp_rw [mul_one]
+  simp_rw [MulOpposite.op_one, one_smul, transpose_apply]
 
 /-- The endomorphisms of `M` acts on `(ι → R) →ₗ[R] M`, and takes the projection
 to a `(ι → R) →ₗ[R] M`. -/
diff --git a/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Defs.lean b/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Defs.lean
index 6a59c9e32abf27..e61efb9e17587f 100644
--- a/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Defs.lean
+++ b/Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup/Defs.lean
@@ -130,6 +130,7 @@ end CoeLemmas
 variable {S T : Type*} [CommRing S] [CommRing T]
 
 /-- A ring homomorphism ``f : R →+* S`` induces a homomorphism ``GLₙ(f) : GLₙ(R) →* GLₙ(S)``. -/
+@[simps! apply_val]
 def map (f : R →+* S) : GL n R →* GL n S := Units.map <| (RingHom.mapMatrix f).toMonoidHom
 
 @[simp]
diff --git a/Mathlib/LinearAlgebra/Matrix/Hermitian.lean b/Mathlib/LinearAlgebra/Matrix/Hermitian.lean
index 89a34b681a4a6a..c7e39b7cc0e1d0 100644
--- a/Mathlib/LinearAlgebra/Matrix/Hermitian.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Hermitian.lean
@@ -279,7 +279,7 @@ theorem isHermitian_iff_isSymmetric [Fintype n] [DecidableEq n] {A : Matrix n n
   · intro h
     ext i j
     simpa only [(Pi.single_star i 1).symm, ← star_mulVec, mul_one, dotProduct_single,
-      single_vecMul, star_one, one_mul] using h (Pi.single i 1) (Pi.single j 1)
+      single_one_vecMul, star_one] using h (Pi.single i 1) (Pi.single j 1)
 
 end RCLike
 
diff --git a/Mathlib/LinearAlgebra/Matrix/Permutation.lean b/Mathlib/LinearAlgebra/Matrix/Permutation.lean
index 96c0d8f5fc7712..990b92bc1ee055 100644
--- a/Mathlib/LinearAlgebra/Matrix/Permutation.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Permutation.lean
@@ -39,7 +39,7 @@ namespace Matrix
 /-- The determinant of a permutation matrix equals its sign. -/
 @[simp]
 theorem det_permutation [CommRing R] : det (σ.permMatrix R) = Perm.sign σ := by
-  rw [← Matrix.mul_one (σ.permMatrix R), PEquiv.toPEquiv_mul_matrix,
+  rw [← Matrix.mul_one (σ.permMatrix R), PEquiv.toMatrix_toPEquiv_mul,
     det_permute, det_one, mul_one]
 
 /-- The trace of a permutation matrix equals the number of fixed points. -/
diff --git a/Mathlib/LinearAlgebra/Matrix/PosDef.lean b/Mathlib/LinearAlgebra/Matrix/PosDef.lean
index 5e46c1956c392a..7776201ccffd13 100644
--- a/Mathlib/LinearAlgebra/Matrix/PosDef.lean
+++ b/Mathlib/LinearAlgebra/Matrix/PosDef.lean
@@ -363,8 +363,8 @@ theorem _root_.Matrix.posDef_diagonal_iff
     PosDef (diagonal d) ↔ ∀ i, 0 < d i := by
   refine ⟨fun h i => ?_, .diagonal⟩
   have := h.2 (Pi.single i 1)
-  simp only [mulVec_single, mul_one, dotProduct_diagonal', Pi.star_apply, Pi.single_eq_same,
-    star_one, one_mul, Function.ne_iff, Pi.zero_apply] at this
+  simp_rw [mulVec_single_one, ← Pi.single_star, star_one, single_dotProduct, one_mul,
+    transpose_apply, diagonal_apply_eq, Function.ne_iff] at this
   exact this ⟨i, by simp⟩
 
 protected theorem one [StarOrderedRing R] [DecidableEq n] [NoZeroDivisors R] :
diff --git a/Mathlib/LinearAlgebra/Matrix/Spectrum.lean b/Mathlib/LinearAlgebra/Matrix/Spectrum.lean
index 3367f16ef2d733..ef6287892c69eb 100644
--- a/Mathlib/LinearAlgebra/Matrix/Spectrum.lean
+++ b/Mathlib/LinearAlgebra/Matrix/Spectrum.lean
@@ -74,6 +74,11 @@ lemma eigenvectorUnitary_coe {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n
       (EuclideanSpace.basisFun n 𝕜).toBasis.toMatrix (hA.eigenvectorBasis).toBasis :=
   rfl
 
+@[simp]
+theorem eigenvectorUnitary_transpose_apply (j : n) :
+    (eigenvectorUnitary hA)ᵀ j = ⇑(hA.eigenvectorBasis j) :=
+  rfl
+
 @[simp]
 theorem eigenvectorUnitary_apply (i j : n) :
     eigenvectorUnitary hA i j = ⇑(hA.eigenvectorBasis j) i :=
@@ -81,7 +86,7 @@ theorem eigenvectorUnitary_apply (i j : n) :
 
 theorem eigenvectorUnitary_mulVec (j : n) :
     eigenvectorUnitary hA *ᵥ Pi.single j 1 = ⇑(hA.eigenvectorBasis j) := by
-  simp only [mulVec_single, eigenvectorUnitary_apply, mul_one]
+  simp_rw [mulVec_single_one, eigenvectorUnitary_transpose_apply]
 
 theorem star_eigenvectorUnitary_mulVec (j : n) :
     (star (eigenvectorUnitary hA : Matrix n n 𝕜)) *ᵥ ⇑(hA.eigenvectorBasis j) = Pi.single j 1 := by
diff --git a/Mathlib/LinearAlgebra/Matrix/ToLin.lean b/Mathlib/LinearAlgebra/Matrix/ToLin.lean
index 1bce52c084cc20..506f38bc302e36 100644
--- a/Mathlib/LinearAlgebra/Matrix/ToLin.lean
+++ b/Mathlib/LinearAlgebra/Matrix/ToLin.lean
@@ -249,7 +249,7 @@ variable [Fintype n]
 @[simp]
 theorem Matrix.mulVecLin_one [DecidableEq n] :
     Matrix.mulVecLin (1 : Matrix n n R) = LinearMap.id := by
-  ext; simp [Matrix.one_apply, Pi.single_apply]
+  ext; simp [Matrix.one_apply, Pi.single_apply, eq_comm]
 
 @[simp]
 theorem Matrix.mulVecLin_mul [Fintype m] (M : Matrix l m R) (N : Matrix m n R) :
diff --git a/Mathlib/LinearAlgebra/Orientation.lean b/Mathlib/LinearAlgebra/Orientation.lean
index 549ff52629c7bb..004d248871f558 100644
--- a/Mathlib/LinearAlgebra/Orientation.lean
+++ b/Mathlib/LinearAlgebra/Orientation.lean
@@ -337,8 +337,8 @@ theorem eq_or_eq_neg [FiniteDimensional R M] (x₁ x₂ : Orientation R M ι)
   -- Porting note: this needs to be made explicit for the simp below
   have orientation_neg_neg :
       ∀ f : Basis ι R M, - -Basis.orientation f = Basis.orientation f := by
-    #adaptation_note
-    /-- `set_option maxSynthPendingDepth 2` required after https://github.com/leanprover/lean4/pull/4119 -/
+    #adaptation_note /-- https://github.com/leanprover/lean4/pull/4119
+    `set_option maxSynthPendingDepth 2` required -/
     set_option maxSynthPendingDepth 2 in simp
   rcases e.orientation_eq_or_eq_neg x₁ with (h₁ | h₁) <;>
     rcases e.orientation_eq_or_eq_neg x₂ with (h₂ | h₂) <;> simp [h₁, h₂, orientation_neg_neg]
diff --git a/Mathlib/LinearAlgebra/Pi.lean b/Mathlib/LinearAlgebra/Pi.lean
index 38d4beb411b38d..c638022cfd82a2 100644
--- a/Mathlib/LinearAlgebra/Pi.lean
+++ b/Mathlib/LinearAlgebra/Pi.lean
@@ -11,6 +11,7 @@ import Mathlib.Algebra.Module.Submodule.Ker
 import Mathlib.Algebra.Module.Submodule.Range
 import Mathlib.Algebra.Module.Equiv.Basic
 import Mathlib.Logic.Equiv.Fin
+import Mathlib.LinearAlgebra.Prod
 
 /-!
 # Pi types of modules
@@ -571,6 +572,17 @@ noncomputable def Function.ExtendByZero.linearMap : (ι → R) →ₗ[R] η →
 
 end Extend
 
+variable (R) in
+/-- `Fin.consEquiv` as a continuous linear equivalence.  -/
+@[simps]
+def Fin.consLinearEquiv
+    {n : ℕ} (M : Fin n.succ → Type*) [Semiring R] [∀ i, AddCommMonoid (M i)] [∀ i, Module R (M i)] :
+    (M 0 × Π i, M (Fin.succ i)) ≃ₗ[R] (Π i, M i) where
+  __ := Fin.consEquiv M
+  map_add' x y := funext <| Fin.cases rfl (by simp)
+  map_smul' c x := funext <| Fin.cases rfl (by simp)
+
+
 /-! ### Bundled versions of `Matrix.vecCons` and `Matrix.vecEmpty`
 
 The idea of these definitions is to be able to define a map as `x ↦ ![f₁ x, f₂ x, f₃ x]`, where
@@ -604,14 +616,8 @@ theorem LinearMap.vecEmpty_apply (m : M) : (LinearMap.vecEmpty : M →ₗ[R] Fin
 
 /-- A linear map into `Fin n.succ → M₃` can be built out of a map into `M₃` and a map into
 `Fin n → M₃`. -/
-def LinearMap.vecCons {n} (f : M →ₗ[R] M₂) (g : M →ₗ[R] Fin n → M₂) : M →ₗ[R] Fin n.succ → M₂ where
-  toFun m := Matrix.vecCons (f m) (g m)
-  map_add' x y := by
-    simp only []
-    rw [f.map_add, g.map_add, Matrix.cons_add_cons (f x)]
-  map_smul' c x := by
-    simp only []
-    rw [f.map_smul, g.map_smul, RingHom.id_apply, Matrix.smul_cons c (f x)]
+def LinearMap.vecCons {n} (f : M →ₗ[R] M₂) (g : M →ₗ[R] Fin n → M₂) : M →ₗ[R] Fin n.succ → M₂ :=
+  Fin.consLinearEquiv R (fun _ : Fin n.succ => M₂) ∘ₗ f.prod g
 
 @[simp]
 theorem LinearMap.vecCons_apply {n} (f : M →ₗ[R] M₂) (g : M →ₗ[R] Fin n → M₂) (m : M) :
diff --git a/Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean b/Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean
index 7a26465a9f79c0..6854ebb704c2fe 100644
--- a/Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean
+++ b/Mathlib/LinearAlgebra/QuadraticForm/TensorProduct.lean
@@ -27,7 +27,7 @@ section CommRing
 variable [CommRing R] [CommRing A]
 variable [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup N₁] [AddCommGroup N₂]
 variable [Algebra R A] [Module R M₁] [Module A M₁] [Module R N₁] [Module A N₁]
-variable [SMulCommClass R A M₁] [IsScalarTower R A M₁] [SMulCommClass R A N₁] [IsScalarTower R A N₁]
+variable [SMulCommClass R A M₁] [IsScalarTower R A M₁] [IsScalarTower R A N₁]
 variable [Module R M₂] [Module R N₂]
 
 section InvertibleTwo
@@ -144,13 +144,13 @@ end QuadraticForm
 
 end InvertibleTwo
 
-/-- If two quadratic forms from `A ⊗[R] M₂` agree on elements of the form `1 ⊗ m`, they are equal.
+/-- If two quadratic maps from `A ⊗[R] M₂` agree on elements of the form `1 ⊗ m`, they are equal.
 
-In other words, if a base change exists for a quadratic form, it is unique.
+In other words, if a base change exists for a quadratic map, it is unique.
 
 Note that unlike `QuadraticForm.baseChange`, this does not need `Invertible (2 : R)`. -/
 @[ext]
-theorem baseChange_ext ⦃Q₁ Q₂ : QuadraticForm A (A ⊗[R] M₂)⦄
+theorem baseChange_ext ⦃Q₁ Q₂ : QuadraticMap A (A ⊗[R] M₂) N₁⦄
     (h : ∀ m, Q₁ (1 ⊗ₜ m) = Q₂ (1 ⊗ₜ m)) :
     Q₁ = Q₂ := by
   replace h (a m) : Q₁ (a ⊗ₜ m) = Q₂ (a ⊗ₜ m) := by
@@ -166,6 +166,6 @@ theorem baseChange_ext ⦃Q₁ Q₂ : QuadraticForm A (A ⊗[R] M₂)⦄
       rw [← TensorProduct.tmul_add, h, h, h]
     replace := congr($this x y)
     dsimp [polar] at this
-    linear_combination this + hx + hy
+    linear_combination (norm := module) this + hx + hy
 
 end CommRing
diff --git a/Mathlib/LinearAlgebra/Semisimple.lean b/Mathlib/LinearAlgebra/Semisimple.lean
index feb4bde80dc93f..4135e45fe83c2e 100644
--- a/Mathlib/LinearAlgebra/Semisimple.lean
+++ b/Mathlib/LinearAlgebra/Semisimple.lean
@@ -274,13 +274,10 @@ theorem IsSemisimple.of_mem_adjoin_pair {a : End K M} (ha : a ∈ Algebra.adjoin
     (AdjoinRoot.powerBasis' <| minpoly.monic <| Algebra.IsIntegral.isIntegral f).finite
   have : Module.Finite R S :=
     (AdjoinRoot.powerBasis' <| (minpoly.monic <| Algebra.IsIntegral.isIntegral g).map _).finite
-  #adaptation_note
-  /--
-  After https://github.com/leanprover/lean4/pull/4119 we either need
-  to specify the `(S := R)` argument, or use `set_option maxSynthPendingDepth 2 in`.
+  #adaptation_note /-- https://github.com/leanprover/lean4/pull/4119
+  we either need to specify the `(S := R)` argument, or use `set_option maxSynthPendingDepth 2 in`.
 
-  In either case this step is too slow!
-  -/
+  In either case this step is too slow! -/
   set_option maxSynthPendingDepth 2 in
   have : IsScalarTower K R S := .of_algebraMap_eq fun _ ↦ rfl
   have : Module.Finite K S := .trans R S
diff --git a/Mathlib/Logic/Function/Defs.lean b/Mathlib/Logic/Function/Defs.lean
index 0caf5f335d524b..8196c2377e17e6 100644
--- a/Mathlib/Logic/Function/Defs.lean
+++ b/Mathlib/Logic/Function/Defs.lean
@@ -48,7 +48,6 @@ scoped infixl:2 " on " => onFun
 
 abbrev swap {φ : α → β → Sort u₃} (f : ∀ x y, φ x y) : ∀ y x, φ x y := fun y x => f x y
 
-#adaptation_note /-- nightly-2024-03-16: added to replace simp [Function.swap] -/
 theorem swap_def {φ : α → β → Sort u₃} (f : ∀ x y, φ x y) : swap f = fun y x => f x y := rfl
 
 -- Porting note: removed, it was never used
diff --git a/Mathlib/Logic/IsEmpty.lean b/Mathlib/Logic/IsEmpty.lean
index fa48bd9fc85c3f..d6c0c753066073 100644
--- a/Mathlib/Logic/IsEmpty.lean
+++ b/Mathlib/Logic/IsEmpty.lean
@@ -213,10 +213,22 @@ variable {α β : Type*} (R : α → β → Prop)
 theorem leftTotal_empty [IsEmpty α] : LeftTotal R := by
   simp only [LeftTotal, IsEmpty.forall_iff]
 
+theorem leftTotal_iff_isEmpty_left [IsEmpty β] : LeftTotal R ↔ IsEmpty α := by
+  simp only [LeftTotal, IsEmpty.exists_iff, isEmpty_iff]
+
 @[simp]
 theorem rightTotal_empty [IsEmpty β] : RightTotal R := by
   simp only [RightTotal, IsEmpty.forall_iff]
 
+theorem rightTotal_iff_isEmpty_right [IsEmpty α] : RightTotal R ↔ IsEmpty β := by
+  simp only [RightTotal, IsEmpty.exists_iff, isEmpty_iff, imp_self]
+
 @[simp]
 theorem biTotal_empty [IsEmpty α] [IsEmpty β] : BiTotal R :=
   ⟨leftTotal_empty R, rightTotal_empty R⟩
+
+theorem biTotal_iff_isEmpty_right [IsEmpty α] : BiTotal R ↔ IsEmpty β := by
+  simp only [BiTotal, leftTotal_empty, rightTotal_iff_isEmpty_right, true_and]
+
+theorem biTotal_iff_isEmpty_left [IsEmpty β] : BiTotal R ↔ IsEmpty α := by
+  simp only [BiTotal, leftTotal_iff_isEmpty_left, rightTotal_empty, and_true]
diff --git a/Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean b/Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean
index 07c0bcd00e44e1..c3611c1f706ca8 100644
--- a/Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean
+++ b/Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean
@@ -232,25 +232,28 @@ theorem measurable_nnnorm : Measurable (nnnorm : α → ℝ≥0) :=
   continuous_nnnorm.measurable
 
 @[measurability, fun_prop]
-theorem Measurable.nnnorm {f : β → α} (hf : Measurable f) : Measurable fun a => ‖f a‖₊ :=
+protected theorem Measurable.nnnorm {f : β → α} (hf : Measurable f) : Measurable fun a => ‖f a‖₊ :=
   measurable_nnnorm.comp hf
 
 @[measurability, fun_prop]
-theorem AEMeasurable.nnnorm {f : β → α} {μ : Measure β} (hf : AEMeasurable f μ) :
+protected lemma AEMeasurable.nnnorm {f : β → α} {μ : Measure β} (hf : AEMeasurable f μ) :
     AEMeasurable (fun a => ‖f a‖₊) μ :=
   measurable_nnnorm.comp_aemeasurable hf
 
 @[measurability]
-theorem measurable_ennnorm : Measurable fun x : α => (‖x‖₊ : ℝ≥0∞) :=
-  measurable_nnnorm.coe_nnreal_ennreal
+lemma measurable_enorm : Measurable (enorm : α → ℝ≥0∞) := continuous_enorm.measurable
 
 @[measurability, fun_prop]
-theorem Measurable.ennnorm {f : β → α} (hf : Measurable f) : Measurable fun a => (‖f a‖₊ : ℝ≥0∞) :=
+protected lemma Measurable.enorm {f : β → α} (hf : Measurable f) : Measurable (‖f ·‖ₑ) :=
   hf.nnnorm.coe_nnreal_ennreal
 
 @[measurability, fun_prop]
-theorem AEMeasurable.ennnorm {f : β → α} {μ : Measure β} (hf : AEMeasurable f μ) :
-    AEMeasurable (fun a => (‖f a‖₊ : ℝ≥0∞)) μ :=
-  measurable_ennnorm.comp_aemeasurable hf
+protected lemma AEMeasurable.enorm {f : β → α} {μ : Measure β} (hf : AEMeasurable f μ) :
+    AEMeasurable (‖f ·‖ₑ) μ :=
+  measurable_enorm.comp_aemeasurable hf
+
+@[deprecated (since := "2025-01-21")] alias measurable_ennnorm := measurable_enorm
+@[deprecated (since := "2025-01-21")] alias Measurable.ennnorm := Measurable.enorm
+@[deprecated (since := "2025-01-21")] alias AEMeasurable.ennnorm := AEMeasurable.enorm
 
 end NormedAddCommGroup
diff --git a/Mathlib/MeasureTheory/Constructions/Pi.lean b/Mathlib/MeasureTheory/Constructions/Pi.lean
index d2ad209b30457b..e81c5ea04ae2fd 100644
--- a/Mathlib/MeasureTheory/Constructions/Pi.lean
+++ b/Mathlib/MeasureTheory/Constructions/Pi.lean
@@ -54,7 +54,7 @@ noncomputable section
 
 open Function Set MeasureTheory.OuterMeasure Filter MeasurableSpace Encodable
 
-open scoped Classical Topology ENNReal
+open scoped Topology ENNReal
 
 universe u v
 
@@ -168,6 +168,7 @@ open List MeasurableEquiv
 
 variable [Encodable ι]
 
+open scoped Classical in
 /-- The product measure on an encodable finite type, defined by mapping `Measure.tprod` along the
   equivalence `MeasurableEquiv.piMeasurableEquivTProd`.
   The definition `MeasureTheory.Measure.pi` should be used instead of this one. -/
@@ -176,6 +177,7 @@ def pi' : Measure (∀ i, α i) :=
 
 theorem pi'_pi [∀ i, SigmaFinite (μ i)] (s : ∀ i, Set (α i)) :
     pi' μ (pi univ s) = ∏ i, μ i (s i) := by
+  classical
   rw [pi']
   rw [← MeasurableEquiv.piMeasurableEquivTProd_symm_apply, MeasurableEquiv.map_apply,
     MeasurableEquiv.piMeasurableEquivTProd_symm_apply, elim_preimage_pi, tprod_tprod _ μ, ←
@@ -342,6 +344,7 @@ theorem pi_empty_univ {α : Type*} [Fintype α] [IsEmpty α] {β : α → Type*}
 
 theorem pi_eval_preimage_null {i : ι} {s : Set (α i)} (hs : μ i s = 0) :
     Measure.pi μ (eval i ⁻¹' s) = 0 := by
+  classical
   -- WLOG, `s` is measurable
   rcases exists_measurable_superset_of_null hs with ⟨t, hst, _, hμt⟩
   suffices Measure.pi μ (eval i ⁻¹' t) = 0 from measure_mono_null (preimage_mono hst) this
diff --git a/Mathlib/MeasureTheory/Covering/Besicovitch.lean b/Mathlib/MeasureTheory/Covering/Besicovitch.lean
index 9f8b24ead46f20..b2868a890cc06d 100644
--- a/Mathlib/MeasureTheory/Covering/Besicovitch.lean
+++ b/Mathlib/MeasureTheory/Covering/Besicovitch.lean
@@ -102,7 +102,7 @@ universe u
 
 open Metric Set Filter Fin MeasureTheory TopologicalSpace
 
-open scoped Topology Classical ENNReal MeasureTheory NNReal
+open scoped Topology ENNReal MeasureTheory NNReal
 
 /-!
 ### Satellite configurations
@@ -528,6 +528,7 @@ theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMea
     (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
     ∃ t : Finset α, ↑t ⊆ s ∧ μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
       (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) := by
+  classical
   -- exclude the trivial case where `μ s = 0`.
   rcases le_or_lt (μ s) 0 with (hμs | hμs)
   · have : μ s = 0 := le_bot_iff.1 hμs
@@ -667,6 +668,7 @@ theorem exists_disjoint_closedBall_covering_ae_of_finiteMeasure_aux (μ : Measur
     ∃ t : Set (α × ℝ), t.Countable ∧ (∀ p ∈ t, p.1 ∈ s) ∧ (∀ p ∈ t, p.2 ∈ f p.1) ∧
       μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ t), closedBall p.1 p.2) = 0 ∧
         t.PairwiseDisjoint fun p => closedBall p.1 p.2 := by
+  classical
   rcases HasBesicovitchCovering.no_satelliteConfig (α := α) with ⟨N, τ, hτ, hN⟩
   /- Introduce a property `P` on finsets saying that we have a nice disjoint covering of a
       subset of `s` by admissible balls. -/
@@ -862,6 +864,7 @@ theorem exists_closedBall_covering_tsum_measure_le (μ : Measure α) [SFinite μ
     disjoint subfamilies. Making sure that they are all included in a neighborhood `v` of `s'` of
     measure at most `ε / (2 N)`, the sum of their measures is at most `ε / 2`,
     completing the proof. -/
+  classical
   obtain ⟨u, su, u_open, μu⟩ : ∃ U, U ⊇ s ∧ IsOpen U ∧ μ U ≤ μ s + ε / 2 :=
     Set.exists_isOpen_le_add _ _
       (by
diff --git a/Mathlib/MeasureTheory/Covering/Differentiation.lean b/Mathlib/MeasureTheory/Covering/Differentiation.lean
index 672bfe1a4f090c..f54de11ae1ded1 100644
--- a/Mathlib/MeasureTheory/Covering/Differentiation.lean
+++ b/Mathlib/MeasureTheory/Covering/Differentiation.lean
@@ -770,10 +770,10 @@ theorem ae_tendsto_lintegral_div {f : α → ℝ≥0∞} (hf : AEMeasurable f μ
   apply lintegral_congr_ae
   exact ae_restrict_of_ae hf.ae_eq_mk
 
-theorem ae_tendsto_lintegral_nnnorm_sub_div'_of_integrable {f : α → E} (hf : Integrable f μ)
+theorem ae_tendsto_lintegral_enorm_sub_div'_of_integrable {f : α → E} (hf : Integrable f μ)
     (h'f : StronglyMeasurable f) :
-    ∀ᵐ x ∂μ, Tendsto (fun a => (∫⁻ y in a, ‖f y - f x‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 0) := by
-  /- For every `c`, then `(∫⁻ y in a, ‖f y - c‖₊ ∂μ) / μ a` tends almost everywhere to `‖f x - c‖`.
+    ∀ᵐ x ∂μ, Tendsto (fun a => (∫⁻ y in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0) := by
+  /- For every `c`, then `(∫⁻ y in a, ‖f y - c‖ₑ ∂μ) / μ a` tends almost everywhere to `‖f x - c‖`.
     We apply this to a countable set of `c` which is dense in the range of `f`, to deduce the
     desired convergence.
     A minor technical inconvenience is that constants are not integrable, so to apply previous
@@ -784,37 +784,31 @@ theorem ae_tendsto_lintegral_nnnorm_sub_div'_of_integrable {f : α → E} (hf :
   have main :
     ∀ᵐ x ∂μ,
       ∀ᵉ (n : ℕ) (c ∈ t),
-        Tendsto (fun a => (∫⁻ y in a, ‖f y - (A.set n).indicator (fun _ => c) y‖₊ ∂μ) / μ a)
-          (v.filterAt x) (𝓝 ‖f x - (A.set n).indicator (fun _ => c) x‖₊) := by
+        Tendsto (fun a => (∫⁻ y in a, ‖f y - (A.set n).indicator (fun _ => c) y‖ₑ ∂μ) / μ a)
+          (v.filterAt x) (𝓝 ‖f x - (A.set n).indicator (fun _ => c) x‖ₑ) := by
     #adaptation_note /-- 2024-04-23
     The next two lines were previously just `simp_rw [ae_all_iff, ae_ball_iff t_count]`. -/
     simp_rw [ae_all_iff]
     intro x; rw [ae_ball_iff t_count]; revert x
     intro n c _
     apply ae_tendsto_lintegral_div'
-    · refine (h'f.sub ?_).ennnorm
+    · refine (h'f.sub ?_).enorm
       exact stronglyMeasurable_const.indicator (IsOpen.measurableSet (A.set_mem n))
     · apply ne_of_lt
       calc
-        (∫⁻ y, ↑‖f y - (A.set n).indicator (fun _ : α => c) y‖₊ ∂μ) ≤
-            ∫⁻ y, ‖f y‖₊ + ‖(A.set n).indicator (fun _ : α => c) y‖₊ ∂μ := by
-          apply lintegral_mono
-          intro x
-          dsimp
-          rw [← ENNReal.coe_add]
-          exact ENNReal.coe_le_coe.2 (nnnorm_sub_le _ _)
-        _ = (∫⁻ y, ‖f y‖₊ ∂μ) + ∫⁻ y, ‖(A.set n).indicator (fun _ : α => c) y‖₊ ∂μ :=
-          (lintegral_add_left h'f.ennnorm _)
+        ∫⁻ y, ‖f y - (A.set n).indicator (fun _ : α => c) y‖ₑ ∂μ
+          ≤ ∫⁻ y, ‖f y‖ₑ + ‖(A.set n).indicator (fun _ : α => c) y‖ₑ ∂μ :=
+          lintegral_mono fun x ↦ enorm_sub_le
+        _ = ∫⁻ y, ‖f y‖ₑ ∂μ + ∫⁻ y, ‖(A.set n).indicator (fun _ : α => c) y‖ₑ ∂μ :=
+          lintegral_add_left h'f.enorm _
         _ < ∞ + ∞ :=
           haveI I : Integrable ((A.set n).indicator fun _ : α => c) μ := by
             simp only [integrable_indicator_iff (IsOpen.measurableSet (A.set_mem n)),
               integrableOn_const, A.finite n, or_true]
           ENNReal.add_lt_add hf.2 I.2
   filter_upwards [main, v.ae_eventually_measure_pos] with x hx h'x
-  have M :
-    ∀ c ∈ t, Tendsto (fun a => (∫⁻ y in a, ‖f y - c‖₊ ∂μ) / μ a)
-      (v.filterAt x) (𝓝 ‖f x - c‖₊) := by
-    intro c hc
+  have M c (hc : c ∈ t) :
+      Tendsto (fun a => (∫⁻ y in a, ‖f y - c‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 ‖f x - c‖ₑ) := by
     obtain ⟨n, xn⟩ : ∃ n, x ∈ A.set n := by simpa [← A.spanning] using mem_univ x
     specialize hx n c hc
     simp only [xn, indicator_of_mem] at hx
@@ -825,18 +819,18 @@ theorem ae_tendsto_lintegral_nnnorm_sub_div'_of_integrable {f : α → E} (hf :
     apply setLIntegral_congr_fun h'a
     filter_upwards with y hy using (by simp only [ha hy, indicator_of_mem])
   apply ENNReal.tendsto_nhds_zero.2 fun ε εpos => ?_
-  obtain ⟨c, ct, xc⟩ : ∃ c ∈ t, (‖f x - c‖₊ : ℝ≥0∞) < ε / 2 := by
-    simp_rw [← edist_eq_coe_nnnorm_sub]
+  obtain ⟨c, ct, xc⟩ : ∃ c ∈ t, ‖f x - c‖ₑ < ε / 2 := by
+    simp_rw [← edist_eq_enorm_sub]
     have : f x ∈ closure t := ht (mem_range_self _)
     exact EMetric.mem_closure_iff.1 this (ε / 2) (ENNReal.half_pos (ne_of_gt εpos))
   filter_upwards [(tendsto_order.1 (M c ct)).2 (ε / 2) xc, h'x, v.eventually_measure_lt_top x] with
     a ha h'a h''a
   apply ENNReal.div_le_of_le_mul
   calc
-    (∫⁻ y in a, ‖f y - f x‖₊ ∂μ) ≤ ∫⁻ y in a, ‖f y - c‖₊ + ‖f x - c‖₊ ∂μ := by
+    (∫⁻ y in a, ‖f y - f x‖ₑ ∂μ) ≤ ∫⁻ y in a, ‖f y - c‖ₑ + ‖f x - c‖ₑ ∂μ := by
       apply lintegral_mono fun x => ?_
-      simpa only [← edist_eq_coe_nnnorm_sub] using edist_triangle_right _ _ _
-    _ = (∫⁻ y in a, ‖f y - c‖₊ ∂μ) + ∫⁻ _ in a, ‖f x - c‖₊ ∂μ :=
+      simpa only [← edist_eq_enorm_sub] using edist_triangle_right _ _ _
+    _ = (∫⁻ y in a, ‖f y - c‖ₑ ∂μ) + ∫⁻ _ in a, ‖f x - c‖ₑ ∂μ :=
       (lintegral_add_right _ measurable_const)
     _ ≤ ε / 2 * μ a + ε / 2 * μ a := by
       gcongr
@@ -846,10 +840,14 @@ theorem ae_tendsto_lintegral_nnnorm_sub_div'_of_integrable {f : α → E} (hf :
         gcongr
     _ = ε * μ a := by rw [← add_mul, ENNReal.add_halves]
 
-theorem ae_tendsto_lintegral_nnnorm_sub_div_of_integrable {f : α → E} (hf : Integrable f μ) :
-    ∀ᵐ x ∂μ, Tendsto (fun a => (∫⁻ y in a, ‖f y - f x‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 0) := by
+@[deprecated (since := "2025-01-22")]
+alias ae_tendsto_lintegral_nnnorm_sub_div'_of_integrable :=
+  ae_tendsto_lintegral_enorm_sub_div'_of_integrable
+
+theorem ae_tendsto_lintegral_enorm_sub_div_of_integrable {f : α → E} (hf : Integrable f μ) :
+    ∀ᵐ x ∂μ, Tendsto (fun a => (∫⁻ y in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0) := by
   have I : Integrable (hf.1.mk f) μ := hf.congr hf.1.ae_eq_mk
-  filter_upwards [v.ae_tendsto_lintegral_nnnorm_sub_div'_of_integrable I hf.1.stronglyMeasurable_mk,
+  filter_upwards [v.ae_tendsto_lintegral_enorm_sub_div'_of_integrable I hf.1.stronglyMeasurable_mk,
     hf.1.ae_eq_mk] with x hx h'x
   apply hx.congr _
   intro a
@@ -859,14 +857,18 @@ theorem ae_tendsto_lintegral_nnnorm_sub_div_of_integrable {f : α → E} (hf : I
   filter_upwards [hf.1.ae_eq_mk] with y hy
   rw [hy, h'x]
 
-theorem ae_tendsto_lintegral_nnnorm_sub_div {f : α → E} (hf : LocallyIntegrable f μ) :
-    ∀ᵐ x ∂μ, Tendsto (fun a => (∫⁻ y in a, ‖f y - f x‖₊ ∂μ) / μ a) (v.filterAt x) (𝓝 0) := by
+@[deprecated (since := "2025-01-22")]
+alias ae_tendsto_lintegral_nnnorm_sub_div_of_integrable :=
+  ae_tendsto_lintegral_enorm_sub_div_of_integrable
+
+theorem ae_tendsto_lintegral_enorm_sub_div {f : α → E} (hf : LocallyIntegrable f μ) :
+    ∀ᵐ x ∂μ, Tendsto (fun a => (∫⁻ y in a, ‖f y - f x‖ₑ ∂μ) / μ a) (v.filterAt x) (𝓝 0) := by
   rcases hf.exists_nat_integrableOn with ⟨u, u_open, u_univ, hu⟩
   have : ∀ n, ∀ᵐ x ∂μ,
-      Tendsto (fun a => (∫⁻ y in a, ‖(u n).indicator f y - (u n).indicator f x‖₊ ∂μ) / μ a)
+      Tendsto (fun a => (∫⁻ y in a, ‖(u n).indicator f y - (u n).indicator f x‖ₑ ∂μ) / μ a)
       (v.filterAt x) (𝓝 0) := by
     intro n
-    apply ae_tendsto_lintegral_nnnorm_sub_div_of_integrable
+    apply ae_tendsto_lintegral_enorm_sub_div_of_integrable
     exact (integrable_indicator_iff (u_open n).measurableSet).2 (hu n)
   filter_upwards [ae_all_iff.2 this] with x hx
   obtain ⟨n, hn⟩ : ∃ n, x ∈ u n := by simpa only [← u_univ, mem_iUnion] using mem_univ x
@@ -877,11 +879,14 @@ theorem ae_tendsto_lintegral_nnnorm_sub_div {f : α → E} (hf : LocallyIntegrab
   refine setLIntegral_congr_fun h'a (Eventually.of_forall (fun y hy ↦ ?_))
   rw [indicator_of_mem (ha hy) f, indicator_of_mem hn f]
 
+@[deprecated (since := "2025-01-22")]
+alias ae_tendsto_lintegral_nnnorm_sub_div := ae_tendsto_lintegral_enorm_sub_div
+
 /-- *Lebesgue differentiation theorem*: for almost every point `x`, the
 average of `‖f y - f x‖` on `a` tends to `0` as `a` shrinks to `x` along a Vitali family. -/
 theorem ae_tendsto_average_norm_sub {f : α → E} (hf : LocallyIntegrable f μ) :
     ∀ᵐ x ∂μ, Tendsto (fun a => ⨍ y in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0) := by
-  filter_upwards [v.ae_tendsto_lintegral_nnnorm_sub_div hf] with x hx
+  filter_upwards [v.ae_tendsto_lintegral_enorm_sub_div hf] with x hx
   have := (ENNReal.tendsto_toReal ENNReal.zero_ne_top).comp hx
   simp only [ENNReal.zero_toReal] at this
   apply Tendsto.congr' _ this
@@ -891,6 +896,7 @@ theorem ae_tendsto_average_norm_sub {f : α → E} (hf : LocallyIntegrable f μ)
   have A : IntegrableOn (fun y => (‖f y - f x‖₊ : ℝ)) a μ := by
     simp_rw [coe_nnnorm]
     exact (h''a.sub (integrableOn_const.2 (Or.inr h'a))).norm
+  dsimp [enorm]
   rw [lintegral_coe_eq_integral _ A, ENNReal.toReal_ofReal (by positivity)]
   simp only [coe_nnnorm, smul_eq_mul]
 
diff --git a/Mathlib/MeasureTheory/Covering/Vitali.lean b/Mathlib/MeasureTheory/Covering/Vitali.lean
index 5b2934b6db6d82..61f42f43718a68 100644
--- a/Mathlib/MeasureTheory/Covering/Vitali.lean
+++ b/Mathlib/MeasureTheory/Covering/Vitali.lean
@@ -310,7 +310,7 @@ theorem exists_disjoint_covering_ae
         linarith [Idist_v b ⟨bu, hbx⟩]
   -- we will show that, in `ball x (R x)`, almost all `s` is covered by the family `u`.
   refine ⟨_ ∩ ball x (R x), inter_mem_nhdsWithin _ (ball_mem_nhds _ (hR0 _)),
-    nonpos_iff_eq_zero.mp (le_of_forall_le_of_dense fun ε εpos => ?_)⟩
+    nonpos_iff_eq_zero.mp (le_of_forall_gt_imp_ge_of_dense fun ε εpos => ?_)⟩
   -- the elements of `v` are disjoint and all contained in a finite volume ball, hence the sum
   -- of their measures is finite.
   have I : (∑' a : v, μ (B a)) < ∞ := by
diff --git a/Mathlib/MeasureTheory/Decomposition/IntegralRNDeriv.lean b/Mathlib/MeasureTheory/Decomposition/IntegralRNDeriv.lean
new file mode 100644
index 00000000000000..e2ddf8630cf97b
--- /dev/null
+++ b/Mathlib/MeasureTheory/Decomposition/IntegralRNDeriv.lean
@@ -0,0 +1,99 @@
+/-
+Copyright (c) 2025 Rémy Degenne. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Rémy Degenne, Lorenzo Luccioli
+-/
+import Mathlib.Analysis.Convex.Continuous
+import Mathlib.Analysis.Convex.Integral
+import Mathlib.MeasureTheory.Decomposition.RadonNikodym
+
+/-!
+# Integrals of functions of Radon-Nikodym derivatives
+
+## Main statements
+
+* `mul_le_integral_rnDeriv_of_ac`: for a convex continuous function `f` on `[0, ∞)`, if `μ`
+  is absolutely continuous with respect to `ν`, then
+  `(ν univ).toReal * f ((μ univ).toReal / (ν univ).toReal) ≤ ∫ x, f (μ.rnDeriv ν x).toReal ∂ν`.
+
+-/
+
+
+open Set
+
+namespace MeasureTheory
+
+variable {α : Type*} {mα : MeasurableSpace α} {μ ν : Measure α} {f : ℝ → ℝ}
+
+/-- For a convex continuous function `f` on `[0, ∞)`, if `μ` is absolutely continuous
+with respect to a probability measure `ν`, then
+`f (μ univ).toReal ≤ ∫ x, f (μ.rnDeriv ν x).toReal ∂ν`. -/
+lemma le_integral_rnDeriv_of_ac [IsFiniteMeasure μ] [IsProbabilityMeasure ν]
+    (hf_cvx : ConvexOn ℝ (Ici 0) f) (hf_cont : ContinuousWithinAt f (Ici 0) 0)
+    (hf_int : Integrable (fun x ↦ f (μ.rnDeriv ν x).toReal) ν) (hμν : μ ≪ ν) :
+    f (μ univ).toReal ≤ ∫ x, f (μ.rnDeriv ν x).toReal ∂ν := by
+  have hf_cont' : ContinuousOn f (Ici 0) := by
+    intro x hx
+    rcases eq_or_lt_of_le (α := ℝ) (hx : 0 ≤ x) with rfl | hx_pos
+    · exact hf_cont
+    · have h := hf_cvx.continuousOn_interior x
+      simp only [nonempty_Iio, interior_Ici', mem_Ioi] at h
+      rw [continuousWithinAt_iff_continuousAt (Ioi_mem_nhds hx_pos)] at h
+      exact (h hx_pos).continuousWithinAt
+  calc f (μ univ).toReal
+    = f (∫ x, (μ.rnDeriv ν x).toReal ∂ν) := by rw [Measure.integral_toReal_rnDeriv hμν]
+  _ ≤ ∫ x, f (μ.rnDeriv ν x).toReal ∂ν := by
+    rw [← average_eq_integral, ← average_eq_integral]
+    exact ConvexOn.map_average_le hf_cvx hf_cont' isClosed_Ici (by simp)
+      Measure.integrable_toReal_rnDeriv hf_int
+
+/-- For a convex continuous function `f` on `[0, ∞)`, if `μ` is absolutely continuous
+with respect to `ν`, then
+`(ν univ).toReal * f ((μ univ).toReal / (ν univ).toReal) ≤ ∫ x, f (μ.rnDeriv ν x).toReal ∂ν`. -/
+lemma mul_le_integral_rnDeriv_of_ac [IsFiniteMeasure μ] [IsFiniteMeasure ν]
+    (hf_cvx : ConvexOn ℝ (Ici 0) f) (hf_cont : ContinuousWithinAt f (Ici 0) 0)
+    (hf_int : Integrable (fun x ↦ f (μ.rnDeriv ν x).toReal) ν) (hμν : μ ≪ ν) :
+    (ν univ).toReal * f ((μ univ).toReal / (ν univ).toReal)
+      ≤ ∫ x, f (μ.rnDeriv ν x).toReal ∂ν := by
+  by_cases hν : ν = 0
+  · simp [hν]
+  have : NeZero ν := ⟨hν⟩
+  let μ' := (ν univ)⁻¹ • μ
+  let ν' := (ν univ)⁻¹ • ν
+  have : IsFiniteMeasure μ' := μ.smul_finite (by simp [hν])
+  have hμν' : μ' ≪ ν' := hμν.smul _
+  have h_rnDeriv_eq : μ'.rnDeriv ν' =ᵐ[ν] μ.rnDeriv ν := by
+    have h1' : μ'.rnDeriv ν' =ᵐ[ν'] (ν univ)⁻¹ • μ.rnDeriv ν' :=
+      Measure.rnDeriv_smul_left_of_ne_top' (μ := ν') (ν := μ) (by simp [hν])
+    have h1 : μ'.rnDeriv ν' =ᵐ[ν] (ν univ)⁻¹ • μ.rnDeriv ν' := by
+      rwa [Measure.ae_smul_measure_eq] at h1'
+      simp
+    have h2 : μ.rnDeriv ν' =ᵐ[ν] (ν univ)⁻¹⁻¹ • μ.rnDeriv ν :=
+      Measure.rnDeriv_smul_right_of_ne_top' (μ := ν) (ν := μ) (by simp) (by simp [hν])
+    filter_upwards [h1, h2] with x h1 h2
+    rw [h1, Pi.smul_apply, smul_eq_mul, h2]
+    simp only [inv_inv, Pi.smul_apply, smul_eq_mul]
+    rw [← mul_assoc, ENNReal.inv_mul_cancel, one_mul]
+    · simp [hν]
+    · simp
+  have h_eq : ∫ x, f (μ'.rnDeriv ν' x).toReal ∂ν'
+      = (ν univ).toReal⁻¹ * ∫ x, f ((μ.rnDeriv ν x).toReal) ∂ν := by
+    rw [integral_smul_measure, smul_eq_mul, ENNReal.toReal_inv]
+    congr 1
+    refine integral_congr_ae ?_
+    filter_upwards [h_rnDeriv_eq] with x hx
+    rw [hx]
+  have h : f (μ' univ).toReal ≤ ∫ x, f (μ'.rnDeriv ν' x).toReal ∂ν' :=
+    le_integral_rnDeriv_of_ac hf_cvx hf_cont ?_ hμν'
+  swap
+  · refine Integrable.smul_measure ?_ (by simp [hν])
+    refine (integrable_congr ?_).mpr hf_int
+    filter_upwards [h_rnDeriv_eq] with x hx
+    rw [hx]
+  rw [h_eq, mul_comm, ← div_le_iff₀, div_eq_inv_mul, inv_inv] at h
+  · convert h
+    · simp only [div_eq_inv_mul, Measure.smul_apply, smul_eq_mul, ENNReal.toReal_mul,
+      ENNReal.toReal_inv, μ']
+  · simp [ENNReal.toReal_pos_iff, hν]
+
+end MeasureTheory
diff --git a/Mathlib/MeasureTheory/Decomposition/Jordan.lean b/Mathlib/MeasureTheory/Decomposition/Jordan.lean
index ce9122ea87126f..e4c8c69908a4c9 100644
--- a/Mathlib/MeasureTheory/Decomposition/Jordan.lean
+++ b/Mathlib/MeasureTheory/Decomposition/Jordan.lean
@@ -43,7 +43,7 @@ Jordan decomposition theorem
 
 noncomputable section
 
-open scoped Classical MeasureTheory ENNReal NNReal
+open scoped MeasureTheory ENNReal NNReal
 
 variable {α : Type*} [MeasurableSpace α]
 
@@ -185,7 +185,6 @@ end JordanDecomposition
 
 namespace SignedMeasure
 
-open scoped Classical
 open JordanDecomposition Measure Set VectorMeasure
 
 variable {s : SignedMeasure α}
diff --git a/Mathlib/MeasureTheory/Decomposition/Lebesgue.lean b/Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
index f1cececd23a181..ca9899a429ecca 100644
--- a/Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
+++ b/Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
@@ -5,7 +5,7 @@ Authors: Kexing Ying
 -/
 import Mathlib.MeasureTheory.Decomposition.UnsignedHahn
 import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
-import Mathlib.MeasureTheory.Function.L1Space
+import Mathlib.MeasureTheory.Function.L1Space.Integrable
 import Mathlib.MeasureTheory.Measure.Sub
 
 /-!
diff --git a/Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean b/Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean
index d0c1bf8f1d9423..f8f7b82e720718 100644
--- a/Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean
+++ b/Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean
@@ -43,7 +43,7 @@ Radon-Nikodym theorem
 
 noncomputable section
 
-open scoped Classical MeasureTheory NNReal ENNReal
+open scoped MeasureTheory NNReal ENNReal
 
 variable {α β : Type*} {m : MeasurableSpace α}
 
diff --git a/Mathlib/MeasureTheory/Decomposition/SignedHahn.lean b/Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
index 0a690c953d507f..003de465b8ccb2 100644
--- a/Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
+++ b/Mathlib/MeasureTheory/Decomposition/SignedHahn.lean
@@ -37,7 +37,7 @@ Hahn decomposition theorem
 
 noncomputable section
 
-open scoped Classical NNReal ENNReal MeasureTheory
+open scoped NNReal ENNReal MeasureTheory
 
 variable {α β : Type*} [MeasurableSpace α]
 variable {M : Type*} [AddCommMonoid M] [TopologicalSpace M] [OrderedAddCommMonoid M]
@@ -99,6 +99,7 @@ private theorem existsNatOneDivLTMeasure_of_not_negative (hi : ¬s ≤[i] 0) :
   let ⟨n, hn⟩ := exists_nat_one_div_lt hj
   ⟨n, k, hj₂, hj₁, hn⟩
 
+open scoped Classical in
 /-- Given the set `i`, if `i` is not negative, `findExistsOneDivLT s i` is the
 least natural number `n` such that `ExistsOneDivLT s i n`, otherwise, it returns 0. -/
 private def findExistsOneDivLT (s : SignedMeasure α) (i : Set α) : ℕ :=
@@ -111,9 +112,11 @@ private theorem findExistsOneDivLT_spec (hi : ¬s ≤[i] 0) :
 
 private theorem findExistsOneDivLT_min (hi : ¬s ≤[i] 0) {m : ℕ}
     (hm : m < findExistsOneDivLT s i) : ¬ExistsOneDivLT s i m := by
+  classical
   rw [findExistsOneDivLT, dif_pos hi] at hm
   exact Nat.find_min _ hm
 
+open scoped Classical in
 /-- Given the set `i`, if `i` is not negative, `someExistsOneDivLT` chooses the set
 `k` from `ExistsOneDivLT s i (findExistsOneDivLT s i)`, otherwise, it returns the
 empty set. -/
@@ -222,6 +225,7 @@ private theorem restrictNonposSeq_disjoint : Pairwise (Disjoint on restrictNonpo
 private theorem exists_subset_restrict_nonpos' (hi₁ : MeasurableSet i) (hi₂ : s i < 0)
     (hn : ¬∀ n : ℕ, ¬s ≤[i \ ⋃ l < n, restrictNonposSeq s i l] 0) :
     ∃ j : Set α, MeasurableSet j ∧ j ⊆ i ∧ s ≤[j] 0 ∧ s j < 0 := by
+  classical
   by_cases h : s ≤[i] 0
   · exact ⟨i, hi₁, Set.Subset.refl _, h, hi₂⟩
   push_neg at hn
diff --git a/Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean b/Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean
index 354c4e16c02103..c20637659cb297 100644
--- a/Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean
+++ b/Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean
@@ -42,7 +42,7 @@ Lebesgue decomposition theorem
 
 noncomputable section
 
-open scoped Classical MeasureTheory NNReal ENNReal
+open scoped MeasureTheory NNReal ENNReal
 
 open Set
 
diff --git a/Mathlib/MeasureTheory/Function/AEMeasurableOrder.lean b/Mathlib/MeasureTheory/Function/AEMeasurableOrder.lean
index 0f4dba7c8de330..a26a30a2fad26d 100644
--- a/Mathlib/MeasureTheory/Function/AEMeasurableOrder.lean
+++ b/Mathlib/MeasureTheory/Function/AEMeasurableOrder.lean
@@ -23,7 +23,6 @@ as possible.
 
 open MeasureTheory Set TopologicalSpace
 
-open scoped Classical
 open ENNReal NNReal
 
 /-- If a function `f : α → β` is such that the level sets `{f < p}` and `{q < f}` have measurable
@@ -36,6 +35,7 @@ theorem MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets {α : Type
     (h : ∀ p ∈ s, ∀ q ∈ s, p < q → ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
       { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ μ (u ∩ v) = 0) :
     AEMeasurable f μ := by
+  classical
   haveI : Encodable s := s_count.toEncodable
   have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
       { x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
diff --git a/Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean b/Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean
index e5fb77b18b62e5..98df7b831fb53b 100644
--- a/Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean
+++ b/Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean
@@ -69,7 +69,7 @@ conditional expectation, conditional expected value
 -/
 
 open TopologicalSpace MeasureTheory.Lp Filter
-open scoped Classical ENNReal Topology MeasureTheory
+open scoped ENNReal Topology MeasureTheory
 
 namespace MeasureTheory
   -- 𝕜 for ℝ or ℂ
@@ -80,6 +80,7 @@ variable {α β E 𝕜 : Type*} [RCLike 𝕜] {m m₀ : MeasurableSpace α} {μ
 section NormedAddCommGroup
 variable [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
 
+open scoped Classical in
 variable (m) in
 /-- Conditional expectation of a function. It is defined as 0 if any one of the following conditions
 is true:
@@ -108,6 +109,7 @@ theorem condExp_of_not_sigmaFinite (hm : m ≤ m₀) (hμm_not : ¬SigmaFinite (
 
 @[deprecated (since := "2025-01-21")] alias condexp_of_not_sigmaFinite := condExp_of_not_sigmaFinite
 
+open scoped Classical in
 theorem condExp_of_sigmaFinite (hm : m ≤ m₀) [hμm : SigmaFinite (μ.trim hm)] :
     μ[f|m] =
       if Integrable f μ then
@@ -320,6 +322,7 @@ theorem condExp_add (hf : Integrable f μ) (hg : Integrable g μ) (m : Measurabl
 theorem condExp_finset_sum {ι : Type*} {s : Finset ι} {f : ι → α → E}
     (hf : ∀ i ∈ s, Integrable (f i) μ) (m : MeasurableSpace α) :
     μ[∑ i ∈ s, f i|m] =ᵐ[μ] ∑ i ∈ s, μ[f i|m] := by
+  classical
   induction' s using Finset.induction_on with i s his heq hf
   · rw [Finset.sum_empty, Finset.sum_empty, condExp_zero]
   · rw [Finset.sum_insert his, Finset.sum_insert his]
diff --git a/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean b/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean
index 1871e8db5807f7..5bb82053bb1764 100644
--- a/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean
+++ b/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean
@@ -132,15 +132,15 @@ theorem norm_condExpIndL1Fin_le (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (x
     ‖condExpIndL1Fin hm hs hμs x‖ ≤ (μ s).toReal * ‖x‖ := by
   rw [L1.norm_eq_integral_norm, ← ENNReal.toReal_ofReal (norm_nonneg x), ← ENNReal.toReal_mul,
     ← ENNReal.ofReal_le_iff_le_toReal (ENNReal.mul_ne_top hμs ENNReal.ofReal_ne_top),
-    ofReal_integral_norm_eq_lintegral_nnnorm]
+    ofReal_integral_norm_eq_lintegral_enorm]
   swap; · rw [← memℒp_one_iff_integrable]; exact Lp.memℒp _
   have h_eq :
-    ∫⁻ a, ‖condExpIndL1Fin hm hs hμs x a‖₊ ∂μ = ∫⁻ a, ‖condExpIndSMul hm hs hμs x a‖₊ ∂μ := by
+    ∫⁻ a, ‖condExpIndL1Fin hm hs hμs x a‖ₑ ∂μ = ∫⁻ a, ‖condExpIndSMul hm hs hμs x a‖ₑ ∂μ := by
     refine lintegral_congr_ae ?_
     refine (condExpIndL1Fin_ae_eq_condExpIndSMul hm hs hμs x).mono fun z hz => ?_
     dsimp only
     rw [hz]
-  rw [h_eq, ofReal_norm_eq_coe_nnnorm]
+  rw [h_eq, ofReal_norm_eq_enorm]
   exact lintegral_nnnorm_condExpIndSMul_le hm hs hμs x
 
 @[deprecated (since := "2025-01-21")] alias norm_condexpIndL1Fin_le := norm_condExpIndL1Fin_le
@@ -170,11 +170,10 @@ alias condexpIndL1Fin_disjoint_union := condExpIndL1Fin_disjoint_union
 
 end CondexpIndL1Fin
 
-open scoped Classical
-
 section CondexpIndL1
 
 
+open scoped Classical in
 /-- Conditional expectation of the indicator of a set, as a function in L1. Its value for sets
 which are not both measurable and of finite measure is not used: we set it to 0. -/
 def condExpIndL1 {m m0 : MeasurableSpace α} (hm : m ≤ m0) (μ : Measure α) (s : Set α)
diff --git a/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean b/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean
index b6524435ba77e5..c51e17e6232436 100644
--- a/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean
+++ b/Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean
@@ -194,7 +194,7 @@ theorem lintegral_nnnorm_condExpL2_le (hs : MeasurableSet[m] s) (hμs : μ s ≠
     dsimp only
     simp_rw [hx]
   rw [lintegral_congr_ae hg_nnnorm_eq.symm]
-  refine lintegral_nnnorm_le_of_forall_fin_meas_integral_eq
+  refine lintegral_enorm_le_of_forall_fin_meas_integral_eq
     hm (Lp.stronglyMeasurable f) ?_ ?_ ?_ ?_ hs hμs
   · exact integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim hμs
   · exact hg_meas
@@ -225,7 +225,7 @@ theorem condExpL2_ae_eq_zero_of_ae_eq_zero (hs : MeasurableSet[m] s) (hμs : μ
     dsimp only
     rw [hx]
     simp
-  · exact (Lp.stronglyMeasurable _).ennnorm
+  · exact (Lp.stronglyMeasurable _).enorm
 
 @[deprecated (since := "2025-01-21")]
 alias condexpL2_ae_eq_zero_of_ae_eq_zero := condExpL2_ae_eq_zero_of_ae_eq_zero
@@ -383,7 +383,7 @@ theorem setLIntegral_nnnorm_condExpL2_indicator_le (hm : m ≤ m0) (hs : Measura
     _ = (∫⁻ a in t, ‖(condExpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs 1) : α → ℝ) a‖₊ ∂μ) * ‖x‖₊ := by
       simp_rw [nnnorm_smul, ENNReal.coe_mul]
       rw [lintegral_mul_const, lpMeas_coe]
-      exact (Lp.stronglyMeasurable _).ennnorm
+      exact (Lp.stronglyMeasurable _).enorm
     _ ≤ μ (s ∩ t) * ‖x‖₊ :=
       mul_le_mul_right' (lintegral_nnnorm_condExpL2_indicator_le_real hs hμs ht hμt) _
 
@@ -488,7 +488,7 @@ theorem setLIntegral_nnnorm_condExpIndSMul_le (hm : m ≤ m0) (hs : MeasurableSe
     _ = (∫⁻ a in t, ‖(condExpL2 ℝ ℝ hm (indicatorConstLp 2 hs hμs 1) : α → ℝ) a‖₊ ∂μ) * ‖x‖₊ := by
       simp_rw [nnnorm_smul, ENNReal.coe_mul]
       rw [lintegral_mul_const, lpMeas_coe]
-      exact (Lp.stronglyMeasurable _).ennnorm
+      exact (Lp.stronglyMeasurable _).enorm
     _ ≤ μ (s ∩ t) * ‖x‖₊ :=
       mul_le_mul_right' (lintegral_nnnorm_condExpL2_indicator_le_real hs hμs ht hμt) _
 
diff --git a/Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean b/Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
index 41c370cc5b4b63..3f4e72c29f95a1 100644
--- a/Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
+++ b/Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean
@@ -71,12 +71,12 @@ theorem eLpNorm_one_condExp_le_eLpNorm (f : α → ℝ) : eLpNorm (μ[f|m]) 1 μ
           (ae_of_all μ (fun x => neg_le_abs (f x) : ∀ x, -f x ≤ |f x|)))] with x hx₁ hx₂
       exact abs_le_abs hx₁ hx₂
     _ = eLpNorm f 1 μ := by
-      rw [eLpNorm_one_eq_lintegral_nnnorm, eLpNorm_one_eq_lintegral_nnnorm,
-        ← ENNReal.toReal_eq_toReal (hasFiniteIntegral_iff_nnnorm.mp integrable_condExp.2).ne
-          (hasFiniteIntegral_iff_nnnorm.mp hf.2).ne,
-        ← integral_norm_eq_lintegral_nnnorm
+      rw [eLpNorm_one_eq_lintegral_enorm, eLpNorm_one_eq_lintegral_enorm,
+        ← ENNReal.toReal_eq_toReal (hasFiniteIntegral_iff_enorm.mp integrable_condExp.2).ne
+          (hasFiniteIntegral_iff_enorm.mp hf.2).ne,
+        ← integral_norm_eq_lintegral_enorm
           (stronglyMeasurable_condExp.mono hm).aestronglyMeasurable,
-        ← integral_norm_eq_lintegral_nnnorm hf.1]
+        ← integral_norm_eq_lintegral_enorm hf.1]
       simp_rw [Real.norm_eq_abs]
       rw (config := {occs := .pos [2]}) [← integral_condExp hm]
       refine integral_congr_ae ?_
@@ -106,9 +106,9 @@ theorem integral_abs_condExp_le (f : α → ℝ) : ∫ x, |(μ[f|m]) x| ∂μ 
       Algebra.id.smul_eq_mul, mul_zero]
     positivity
   rw [integral_eq_lintegral_of_nonneg_ae, integral_eq_lintegral_of_nonneg_ae]
-  · apply ENNReal.toReal_mono <;> simp_rw [← Real.norm_eq_abs, ofReal_norm_eq_coe_nnnorm]
+  · apply ENNReal.toReal_mono <;> simp_rw [← Real.norm_eq_abs, ofReal_norm_eq_enorm]
     · exact hfint.2.ne
-    · rw [← eLpNorm_one_eq_lintegral_nnnorm, ← eLpNorm_one_eq_lintegral_nnnorm]
+    · rw [← eLpNorm_one_eq_lintegral_enorm, ← eLpNorm_one_eq_lintegral_enorm]
       exact eLpNorm_one_condExp_le_eLpNorm _
   · filter_upwards with x using abs_nonneg _
   · simp_rw [← Real.norm_eq_abs]
@@ -237,7 +237,7 @@ alias Integrable.uniformIntegrable_condexp := Integrable.uniformIntegrable_condE
 section PullOut
 
 -- TODO: this section could be generalized beyond multiplication, to any bounded bilinear map.
-/-- Auxiliary lemma for `condexp_mul_of_stronglyMeasurable_left`. -/
+/-- Auxiliary lemma for `condExp_mul_of_stronglyMeasurable_left`. -/
 theorem condExp_stronglyMeasurable_simpleFunc_mul (hm : m ≤ m0) (f : @SimpleFunc α m ℝ) {g : α → ℝ}
     (hg : Integrable g μ) : μ[(f * g : α → ℝ)|m] =ᵐ[μ] f * μ[g|m] := by
   have : ∀ (s c) (f : α → ℝ), Set.indicator s (Function.const α c) * f = s.indicator (c • f) := by
diff --git a/Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean b/Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean
index 033ce021ee4bed..490cbe24cc8175 100644
--- a/Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean
+++ b/Mathlib/MeasureTheory/Function/ConditionalExpectation/Unique.lean
@@ -204,18 +204,22 @@ theorem integral_norm_le_of_forall_fin_meas_integral_eq (hm : m ≤ m0) {f g : 
 
 /-- Let `m` be a sub-σ-algebra of `m0`, `f` an `m0`-measurable function and `g` an `m`-measurable
 function, such that their integrals coincide on `m`-measurable sets with finite measure.
-Then `∫⁻ x in s, ‖g x‖₊ ∂μ ≤ ∫⁻ x in s, ‖f x‖₊ ∂μ` on all `m`-measurable sets with finite
+Then `∫⁻ x in s, ‖g x‖ₑ ∂μ ≤ ∫⁻ x in s, ‖f x‖ₑ ∂μ` on all `m`-measurable sets with finite
 measure. -/
-theorem lintegral_nnnorm_le_of_forall_fin_meas_integral_eq (hm : m ≤ m0) {f g : α → ℝ}
+theorem lintegral_enorm_le_of_forall_fin_meas_integral_eq (hm : m ≤ m0) {f g : α → ℝ}
     (hf : StronglyMeasurable f) (hfi : IntegrableOn f s μ) (hg : StronglyMeasurable[m] g)
     (hgi : IntegrableOn g s μ)
     (hgf : ∀ t, MeasurableSet[m] t → μ t < ∞ → ∫ x in t, g x ∂μ = ∫ x in t, f x ∂μ)
-    (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) : (∫⁻ x in s, ‖g x‖₊ ∂μ) ≤ ∫⁻ x in s, ‖f x‖₊ ∂μ := by
-  rw [← ofReal_integral_norm_eq_lintegral_nnnorm hfi, ←
-    ofReal_integral_norm_eq_lintegral_nnnorm hgi, ENNReal.ofReal_le_ofReal_iff]
+    (hs : MeasurableSet[m] s) (hμs : μ s ≠ ∞) : (∫⁻ x in s, ‖g x‖ₑ ∂μ) ≤ ∫⁻ x in s, ‖f x‖ₑ ∂μ := by
+  rw [← ofReal_integral_norm_eq_lintegral_enorm hfi, ←
+    ofReal_integral_norm_eq_lintegral_enorm hgi, ENNReal.ofReal_le_ofReal_iff]
   · exact integral_norm_le_of_forall_fin_meas_integral_eq hm hf hfi hg hgi hgf hs hμs
   · positivity
 
+@[deprecated (since := "2025-01-21")]
+alias lintegral_nnnorm_le_of_forall_fin_meas_integral_eq :=
+  lintegral_enorm_le_of_forall_fin_meas_integral_eq
+
 end IntegralNormLE
 
 end MeasureTheory
diff --git a/Mathlib/MeasureTheory/Function/ContinuousMapDense.lean b/Mathlib/MeasureTheory/Function/ContinuousMapDense.lean
index 633e66b2ab991a..8110b019c55bb8 100644
--- a/Mathlib/MeasureTheory/Function/ContinuousMapDense.lean
+++ b/Mathlib/MeasureTheory/Function/ContinuousMapDense.lean
@@ -117,11 +117,8 @@ theorem exists_continuous_eLpNorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠
   have gc_mem : Memℒp (fun x => g x • c) p μ := by
     refine Memℒp.smul_of_top_left (memℒp_top_const _) ?_
     refine ⟨g.continuous.aestronglyMeasurable, ?_⟩
-    have : eLpNorm (v.indicator fun _x => (1 : ℝ)) p μ < ⊤ := by
-      refine (eLpNorm_indicator_const_le _ _).trans_lt ?_
-      simp only [lt_top_iff_ne_top, hμv.ne, nnnorm_one, ENNReal.coe_one, one_div, one_mul, Ne,
-        ENNReal.rpow_eq_top_iff, inv_lt_zero, false_and, or_false, not_and, not_lt,
-        ENNReal.toReal_nonneg, imp_true_iff]
+    have : eLpNorm (v.indicator fun _x => (1 : ℝ)) p μ < ⊤ :=
+      (eLpNorm_indicator_const_le _ _).trans_lt <| by simp [lt_top_iff_ne_top, hμv.ne]
     refine (eLpNorm_mono fun x => ?_).trans_lt this
     by_cases hx : x ∈ v
     · simp only [hx, abs_of_nonneg (hg_range x).1, (hg_range x).2, Real.norm_eq_abs,
@@ -219,8 +216,8 @@ theorem Integrable.exists_hasCompactSupport_lintegral_sub_le
     [R1Space α] [WeaklyLocallyCompactSpace α] [μ.Regular]
     {f : α → E} (hf : Integrable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
     ∃ g : α → E,
-      HasCompactSupport g ∧ (∫⁻ x, ‖f x - g x‖₊ ∂μ) ≤ ε ∧ Continuous g ∧ Integrable g μ := by
-  simp only [← memℒp_one_iff_integrable, ← eLpNorm_one_eq_lintegral_nnnorm] at hf ⊢
+      HasCompactSupport g ∧ ∫⁻ x, ‖f x - g x‖ₑ ∂μ ≤ ε ∧ Continuous g ∧ Integrable g μ := by
+  simp only [← memℒp_one_iff_integrable, ← eLpNorm_one_eq_lintegral_enorm] at hf ⊢
   exact hf.exists_hasCompactSupport_eLpNorm_sub_le ENNReal.one_ne_top hε
 
 /-- In a locally compact space, any integrable function can be approximated by compactly supported
@@ -230,7 +227,7 @@ theorem Integrable.exists_hasCompactSupport_integral_sub_le
     {f : α → E} (hf : Integrable f μ) {ε : ℝ} (hε : 0 < ε) :
     ∃ g : α → E, HasCompactSupport g ∧ (∫ x, ‖f x - g x‖ ∂μ) ≤ ε ∧
       Continuous g ∧ Integrable g μ := by
-  simp only [← memℒp_one_iff_integrable, ← eLpNorm_one_eq_lintegral_nnnorm, ← ENNReal.ofReal_one]
+  simp only [← memℒp_one_iff_integrable, ← eLpNorm_one_eq_lintegral_enorm, ← ENNReal.ofReal_one]
     at hf ⊢
   simpa using hf.exists_hasCompactSupport_integral_rpow_sub_le zero_lt_one hε
 
@@ -309,8 +306,8 @@ theorem Memℒp.exists_boundedContinuous_integral_rpow_sub_le [μ.WeaklyRegular]
 version in terms of `∫⁻`. -/
 theorem Integrable.exists_boundedContinuous_lintegral_sub_le [μ.WeaklyRegular] {f : α → E}
     (hf : Integrable f μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
-    ∃ g : α →ᵇ E, (∫⁻ x, ‖f x - g x‖₊ ∂μ) ≤ ε ∧ Integrable g μ := by
-  simp only [← memℒp_one_iff_integrable, ← eLpNorm_one_eq_lintegral_nnnorm] at hf ⊢
+    ∃ g : α →ᵇ E, ∫⁻ x, ‖f x - g x‖ₑ ∂μ ≤ ε ∧ Integrable g μ := by
+  simp only [← memℒp_one_iff_integrable, ← eLpNorm_one_eq_lintegral_enorm] at hf ⊢
   exact hf.exists_boundedContinuous_eLpNorm_sub_le ENNReal.one_ne_top hε
 
 /-- Any integrable function can be approximated by bounded continuous functions,
@@ -318,7 +315,7 @@ version in terms of `∫`. -/
 theorem Integrable.exists_boundedContinuous_integral_sub_le [μ.WeaklyRegular] {f : α → E}
     (hf : Integrable f μ) {ε : ℝ} (hε : 0 < ε) :
     ∃ g : α →ᵇ E, (∫ x, ‖f x - g x‖ ∂μ) ≤ ε ∧ Integrable g μ := by
-  simp only [← memℒp_one_iff_integrable, ← eLpNorm_one_eq_lintegral_nnnorm, ← ENNReal.ofReal_one]
+  simp only [← memℒp_one_iff_integrable, ← eLpNorm_one_eq_lintegral_enorm, ← ENNReal.ofReal_one]
     at hf ⊢
   simpa using hf.exists_boundedContinuous_integral_rpow_sub_le zero_lt_one hε
 
diff --git a/Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean b/Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
index f9a7daf12fa481..e8eaafe2a7ed05 100644
--- a/Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
+++ b/Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
@@ -270,7 +270,7 @@ theorem tendstoInMeasure_of_tendsto_eLpNorm_of_stronglyMeasurable (hp_ne_zero :
   · rw [← ENNReal.ofReal_rpow_of_pos hε]
     convert mul_meas_ge_le_pow_eLpNorm' μ hp_ne_zero hp_ne_top ((hf n).sub hg).aestronglyMeasurable
         (ENNReal.ofReal ε)
-    rw [dist_eq_norm, ← ENNReal.ofReal_le_ofReal_iff (norm_nonneg _), ofReal_norm_eq_coe_nnnorm]
+    rw [dist_eq_norm, ← ENNReal.ofReal_le_ofReal_iff (norm_nonneg _), ofReal_norm_eq_enorm]
     exact Iff.rfl
   · rw [Ne, ENNReal.ofReal_eq_zero, not_le]
     exact Or.inl (Real.rpow_pos_of_pos hε _)
diff --git a/Mathlib/MeasureTheory/Function/Egorov.lean b/Mathlib/MeasureTheory/Function/Egorov.lean
index 9e40665728a348..104398ecddfc9d 100644
--- a/Mathlib/MeasureTheory/Function/Egorov.lean
+++ b/Mathlib/MeasureTheory/Function/Egorov.lean
@@ -23,7 +23,6 @@ convergence in measure.
 
 noncomputable section
 
-open scoped Classical
 open MeasureTheory NNReal ENNReal Topology
 
 namespace MeasureTheory
diff --git a/Mathlib/MeasureTheory/Function/L1Space/AEEqFun.lean b/Mathlib/MeasureTheory/Function/L1Space/AEEqFun.lean
new file mode 100644
index 00000000000000..b63aa09f634b42
--- /dev/null
+++ b/Mathlib/MeasureTheory/Function/L1Space/AEEqFun.lean
@@ -0,0 +1,231 @@
+/-
+Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Zhouhang Zhou
+-/
+import Mathlib.MeasureTheory.Function.L1Space.Integrable
+
+/-!
+# `L¹` space
+
+In this file we establish an API between `Integrable` and the space `L¹` of equivalence
+classes of integrable functions, already defined as a special case of `L^p` spaces for `p = 1`.
+
+## Notation
+
+* `α →₁[μ] β` is the type of `L¹` space, where `α` is a `MeasureSpace` and `β` is a
+  `NormedAddCommGroup`. `f : α →ₘ β` is a "function" in `L¹`.
+  In comments, `[f]` is also used to denote an `L¹` function.
+
+  `₁` can be typed as `\1`.
+
+## Tags
+
+function space, l1
+
+-/
+
+noncomputable section
+
+open EMetric ENNReal Filter MeasureTheory NNReal Set
+
+variable {α β : Type*} {m : MeasurableSpace α} {μ ν : Measure α}
+variable [NormedAddCommGroup β]
+
+namespace MeasureTheory
+
+namespace AEEqFun
+
+section
+
+/-- A class of almost everywhere equal functions is `Integrable` if its function representative
+is integrable. -/
+def Integrable (f : α →ₘ[μ] β) : Prop :=
+  MeasureTheory.Integrable f μ
+
+theorem integrable_mk {f : α → β} (hf : AEStronglyMeasurable f μ) :
+    Integrable (mk f hf : α →ₘ[μ] β) ↔ MeasureTheory.Integrable f μ := by
+  simp only [Integrable]
+  apply integrable_congr
+  exact coeFn_mk f hf
+
+theorem integrable_coeFn {f : α →ₘ[μ] β} : MeasureTheory.Integrable f μ ↔ Integrable f := by
+  rw [← integrable_mk, mk_coeFn]
+
+theorem integrable_zero : Integrable (0 : α →ₘ[μ] β) :=
+  (MeasureTheory.integrable_zero α β μ).congr (coeFn_mk _ _).symm
+
+end
+
+section
+
+theorem Integrable.neg {f : α →ₘ[μ] β} : Integrable f → Integrable (-f) :=
+  induction_on f fun _f hfm hfi => (integrable_mk _).2 ((integrable_mk hfm).1 hfi).neg
+
+section
+
+theorem integrable_iff_mem_L1 {f : α →ₘ[μ] β} : Integrable f ↔ f ∈ (α →₁[μ] β) := by
+  rw [← integrable_coeFn, ← memℒp_one_iff_integrable, Lp.mem_Lp_iff_memℒp]
+
+theorem Integrable.add {f g : α →ₘ[μ] β} : Integrable f → Integrable g → Integrable (f + g) := by
+  refine induction_on₂ f g fun f hf g hg hfi hgi => ?_
+  simp only [integrable_mk, mk_add_mk] at hfi hgi ⊢
+  exact hfi.add hgi
+
+theorem Integrable.sub {f g : α →ₘ[μ] β} (hf : Integrable f) (hg : Integrable g) :
+    Integrable (f - g) :=
+  (sub_eq_add_neg f g).symm ▸ hf.add hg.neg
+
+end
+
+section BoundedSMul
+
+variable {𝕜 : Type*} [NormedRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β]
+
+theorem Integrable.smul {c : 𝕜} {f : α →ₘ[μ] β} : Integrable f → Integrable (c • f) :=
+  induction_on f fun _f hfm hfi => (integrable_mk _).2 <|
+    by simpa using ((integrable_mk hfm).1 hfi).smul c
+
+end BoundedSMul
+
+end
+
+end AEEqFun
+
+namespace L1
+
+
+theorem integrable_coeFn (f : α →₁[μ] β) : Integrable f μ := by
+  rw [← memℒp_one_iff_integrable]
+  exact Lp.memℒp f
+
+theorem hasFiniteIntegral_coeFn (f : α →₁[μ] β) : HasFiniteIntegral f μ :=
+  (integrable_coeFn f).hasFiniteIntegral
+
+theorem stronglyMeasurable_coeFn (f : α →₁[μ] β) : StronglyMeasurable f :=
+  Lp.stronglyMeasurable f
+
+theorem measurable_coeFn [MeasurableSpace β] [BorelSpace β] (f : α →₁[μ] β) : Measurable f :=
+  (Lp.stronglyMeasurable f).measurable
+
+theorem aestronglyMeasurable_coeFn (f : α →₁[μ] β) : AEStronglyMeasurable f μ :=
+  Lp.aestronglyMeasurable f
+
+theorem aemeasurable_coeFn [MeasurableSpace β] [BorelSpace β] (f : α →₁[μ] β) : AEMeasurable f μ :=
+  (Lp.stronglyMeasurable f).measurable.aemeasurable
+
+theorem edist_def (f g : α →₁[μ] β) : edist f g = ∫⁻ a, edist (f a) (g a) ∂μ := by
+  simp only [Lp.edist_def, eLpNorm, one_ne_zero, eLpNorm'_eq_lintegral_enorm, Pi.sub_apply,
+    one_toReal, ENNReal.rpow_one, ne_eq, not_false_eq_true, div_self, ite_false]
+  simp [edist_eq_enorm_sub]
+
+theorem dist_def (f g : α →₁[μ] β) : dist f g = (∫⁻ a, edist (f a) (g a) ∂μ).toReal := by
+  simp_rw [dist_edist, edist_def]
+
+theorem norm_def (f : α →₁[μ] β) : ‖f‖ = (∫⁻ a, ‖f a‖ₑ ∂μ).toReal := by
+  simp [Lp.norm_def, eLpNorm, eLpNorm'_eq_lintegral_enorm]
+
+/-- Computing the norm of a difference between two L¹-functions. Note that this is not a
+  special case of `norm_def` since `(f - g) x` and `f x - g x` are not equal
+  (but only a.e.-equal). -/
+theorem norm_sub_eq_lintegral (f g : α →₁[μ] β) : ‖f - g‖ = (∫⁻ x, ‖f x - g x‖ₑ ∂μ).toReal := by
+  rw [norm_def]
+  congr 1
+  rw [lintegral_congr_ae]
+  filter_upwards [Lp.coeFn_sub f g] with _ ha
+  simp only [ha, Pi.sub_apply]
+
+theorem ofReal_norm_eq_lintegral (f : α →₁[μ] β) : ENNReal.ofReal ‖f‖ = ∫⁻ x, ‖f x‖ₑ ∂μ := by
+  rw [norm_def, ENNReal.ofReal_toReal]
+  exact ne_of_lt (hasFiniteIntegral_coeFn f)
+
+/-- Computing the norm of a difference between two L¹-functions. Note that this is not a
+  special case of `ofReal_norm_eq_lintegral` since `(f - g) x` and `f x - g x` are not equal
+  (but only a.e.-equal). -/
+theorem ofReal_norm_sub_eq_lintegral (f g : α →₁[μ] β) :
+    ENNReal.ofReal ‖f - g‖ = ∫⁻ x, ‖f x - g x‖ₑ ∂μ := by
+  simp_rw [ofReal_norm_eq_lintegral, ← edist_zero_eq_enorm]
+  apply lintegral_congr_ae
+  filter_upwards [Lp.coeFn_sub f g] with _ ha
+  simp only [ha, Pi.sub_apply]
+
+end L1
+
+namespace Integrable
+
+
+/-- Construct the equivalence class `[f]` of an integrable function `f`, as a member of the
+space `Lp β 1 μ`. -/
+def toL1 (f : α → β) (hf : Integrable f μ) : α →₁[μ] β :=
+  (memℒp_one_iff_integrable.2 hf).toLp f
+
+@[simp]
+theorem toL1_coeFn (f : α →₁[μ] β) (hf : Integrable f μ) : hf.toL1 f = f := by
+  simp [Integrable.toL1]
+
+theorem coeFn_toL1 {f : α → β} (hf : Integrable f μ) : hf.toL1 f =ᵐ[μ] f :=
+  AEEqFun.coeFn_mk _ _
+
+@[simp]
+theorem toL1_zero (h : Integrable (0 : α → β) μ) : h.toL1 0 = 0 :=
+  rfl
+
+@[simp]
+theorem toL1_eq_mk (f : α → β) (hf : Integrable f μ) :
+    (hf.toL1 f : α →ₘ[μ] β) = AEEqFun.mk f hf.aestronglyMeasurable :=
+  rfl
+
+@[simp]
+theorem toL1_eq_toL1_iff (f g : α → β) (hf : Integrable f μ) (hg : Integrable g μ) :
+    toL1 f hf = toL1 g hg ↔ f =ᵐ[μ] g :=
+  Memℒp.toLp_eq_toLp_iff _ _
+
+theorem toL1_add (f g : α → β) (hf : Integrable f μ) (hg : Integrable g μ) :
+    toL1 (f + g) (hf.add hg) = toL1 f hf + toL1 g hg :=
+  rfl
+
+theorem toL1_neg (f : α → β) (hf : Integrable f μ) : toL1 (-f) (Integrable.neg hf) = -toL1 f hf :=
+  rfl
+
+theorem toL1_sub (f g : α → β) (hf : Integrable f μ) (hg : Integrable g μ) :
+    toL1 (f - g) (hf.sub hg) = toL1 f hf - toL1 g hg :=
+  rfl
+
+theorem norm_toL1 (f : α → β) (hf : Integrable f μ) :
+    ‖hf.toL1 f‖ = (∫⁻ a, edist (f a) 0 ∂μ).toReal := by
+  simp [toL1, Lp.norm_toLp, eLpNorm, eLpNorm'_eq_lintegral_enorm]
+
+theorem enorm_toL1 {f : α → β} (hf : Integrable f μ) : ‖hf.toL1 f‖ₑ = ∫⁻ a, ‖f a‖ₑ ∂μ := by
+  simpa [Integrable.toL1, eLpNorm, eLpNorm', enorm] using ENNReal.coe_toNNReal hf.2.ne
+
+@[deprecated (since := "2025-01-20")] alias nnnorm_toL1 := enorm_toL1
+
+theorem norm_toL1_eq_lintegral_norm (f : α → β) (hf : Integrable f μ) :
+    ‖hf.toL1 f‖ = ENNReal.toReal (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) := by
+  rw [norm_toL1, lintegral_norm_eq_lintegral_edist]
+
+@[simp]
+theorem edist_toL1_toL1 (f g : α → β) (hf : Integrable f μ) (hg : Integrable g μ) :
+    edist (hf.toL1 f) (hg.toL1 g) = ∫⁻ a, edist (f a) (g a) ∂μ := by
+  simp only [toL1, Lp.edist_toLp_toLp, eLpNorm, one_ne_zero, eLpNorm'_eq_lintegral_enorm,
+    Pi.sub_apply, one_toReal, ENNReal.rpow_one, ne_eq, not_false_eq_true, div_self, ite_false]
+  simp [edist_eq_enorm_sub]
+
+theorem edist_toL1_zero (f : α → β) (hf : Integrable f μ) :
+    edist (hf.toL1 f) 0 = ∫⁻ a, edist (f a) 0 ∂μ := by
+  simp only [edist_zero_right, Lp.enorm_def, toL1_eq_mk, eLpNorm_aeeqFun]
+  apply eLpNorm_one_eq_lintegral_enorm
+
+variable {𝕜 : Type*} [NormedRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β]
+
+theorem toL1_smul (f : α → β) (hf : Integrable f μ) (k : 𝕜) :
+    toL1 (fun a => k • f a) (hf.smul k) = k • toL1 f hf :=
+  rfl
+
+theorem toL1_smul' (f : α → β) (hf : Integrable f μ) (k : 𝕜) :
+    toL1 (k • f) (hf.smul k) = k • toL1 f hf :=
+  rfl
+
+end Integrable
+
+end MeasureTheory
diff --git a/Mathlib/MeasureTheory/Function/L1Space/HasFiniteIntegral.lean b/Mathlib/MeasureTheory/Function/L1Space/HasFiniteIntegral.lean
new file mode 100644
index 00000000000000..8aee3621a62b03
--- /dev/null
+++ b/Mathlib/MeasureTheory/Function/L1Space/HasFiniteIntegral.lean
@@ -0,0 +1,398 @@
+/-
+Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Zhouhang Zhou
+-/
+import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
+import Mathlib.MeasureTheory.Measure.WithDensity
+
+/-!
+# Function with finite integral
+
+In this file we define the predicate `HasFiniteIntegral`, which is then used to define the
+predicate `Integrable` in the corresponding file.
+
+## Main definition
+
+* Let `f : α → β` be a function, where `α` is a `MeasureSpace` and `β` a `NormedAddCommGroup`.
+  Then `HasFiniteIntegral f` means `∫⁻ a, ‖f a‖ₑ < ∞`.
+
+## Tags
+
+finite integral
+
+-/
+
+
+noncomputable section
+
+open Topology ENNReal MeasureTheory NNReal
+
+open Set Filter TopologicalSpace ENNReal EMetric MeasureTheory
+
+variable {α β γ ε : Type*} {m : MeasurableSpace α} {μ ν : Measure α}
+variable [NormedAddCommGroup β] [NormedAddCommGroup γ] [ENorm ε]
+
+namespace MeasureTheory
+
+/-! ### Some results about the Lebesgue integral involving a normed group -/
+
+lemma lintegral_enorm_eq_lintegral_edist (f : α → β) :
+    ∫⁻ a, ‖f a‖ₑ ∂μ = ∫⁻ a, edist (f a) 0 ∂μ := by simp only [edist_zero_eq_enorm]
+
+@[deprecated (since := "2025-01-20")]
+alias lintegral_nnnorm_eq_lintegral_edist := lintegral_enorm_eq_lintegral_edist
+
+theorem lintegral_norm_eq_lintegral_edist (f : α → β) :
+    ∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ = ∫⁻ a, edist (f a) 0 ∂μ := by
+  simp only [ofReal_norm_eq_enorm, edist_zero_eq_enorm]
+
+theorem lintegral_edist_triangle {f g h : α → β} (hf : AEStronglyMeasurable f μ)
+    (hh : AEStronglyMeasurable h μ) :
+    (∫⁻ a, edist (f a) (g a) ∂μ) ≤ (∫⁻ a, edist (f a) (h a) ∂μ) + ∫⁻ a, edist (g a) (h a) ∂μ := by
+  rw [← lintegral_add_left' (hf.edist hh)]
+  refine lintegral_mono fun a => ?_
+  apply edist_triangle_right
+
+-- Yaël: Why do the following four lemmas even exist?
+theorem lintegral_enorm_zero : ∫⁻ _ : α, ‖(0 : β)‖ₑ ∂μ = 0 := by simp
+
+theorem lintegral_enorm_add_left {f : α → β} (hf : AEStronglyMeasurable f μ) (g : α → γ) :
+    ∫⁻ a, ‖f a‖ₑ + ‖g a‖ₑ ∂μ = ∫⁻ a, ‖f a‖ₑ ∂μ + ∫⁻ a, ‖g a‖ₑ ∂μ :=
+  lintegral_add_left' hf.enorm _
+
+theorem lintegral_enorm_add_right (f : α → β) {g : α → γ} (hg : AEStronglyMeasurable g μ) :
+    ∫⁻ a, ‖f a‖ₑ + ‖g a‖ₑ ∂μ = ∫⁻ a, ‖f a‖ₑ ∂μ + ∫⁻ a, ‖g a‖ₑ ∂μ :=
+  lintegral_add_right' _ hg.enorm
+
+theorem lintegral_enorm_neg {f : α → β} : ∫⁻ a, ‖(-f) a‖ₑ ∂μ = ∫⁻ a, ‖f a‖ₑ ∂μ := by simp
+
+@[deprecated (since := "2025-01-21")] alias lintegral_nnnorm_zero := lintegral_enorm_zero
+@[deprecated (since := "2025-01-21")] alias lintegral_nnnorm_add_left := lintegral_enorm_add_left
+@[deprecated (since := "2025-01-21")] alias lintegral_nnnorm_add_right := lintegral_enorm_add_right
+@[deprecated (since := "2025-01-21")] alias lintegral_nnnorm_neg := lintegral_enorm_neg
+
+/-! ### The predicate `HasFiniteIntegral` -/
+
+
+/-- `HasFiniteIntegral f μ` means that the integral `∫⁻ a, ‖f a‖ ∂μ` is finite.
+  `HasFiniteIntegral f` means `HasFiniteIntegral f volume`. -/
+def HasFiniteIntegral {_ : MeasurableSpace α} (f : α → ε)
+    (μ : Measure α := by volume_tac) : Prop :=
+  ∫⁻ a, ‖f a‖ₑ ∂μ < ∞
+
+theorem hasFiniteIntegral_def {_ : MeasurableSpace α} (f : α → ε) (μ : Measure α) :
+    HasFiniteIntegral f μ ↔ (∫⁻ a, ‖f a‖ₑ ∂μ < ∞) :=
+  Iff.rfl
+
+theorem hasFiniteIntegral_iff_enorm {f : α → β} : HasFiniteIntegral f μ ↔ ∫⁻ a, ‖f a‖ₑ ∂μ < ∞ := by
+  simp only [HasFiniteIntegral, ofReal_norm_eq_enorm, enorm_eq_nnnorm]
+
+@[deprecated (since := "2025-01-20")]
+alias hasFiniteIntegral_iff_nnnorm := hasFiniteIntegral_iff_enorm
+
+theorem hasFiniteIntegral_iff_norm (f : α → β) :
+    HasFiniteIntegral f μ ↔ (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) < ∞ := by
+  simp only [hasFiniteIntegral_iff_enorm, ofReal_norm_eq_enorm]
+
+theorem hasFiniteIntegral_iff_edist (f : α → β) :
+    HasFiniteIntegral f μ ↔ (∫⁻ a, edist (f a) 0 ∂μ) < ∞ := by
+  simp only [hasFiniteIntegral_iff_norm, edist_dist, dist_zero_right]
+
+theorem hasFiniteIntegral_iff_ofReal {f : α → ℝ} (h : 0 ≤ᵐ[μ] f) :
+    HasFiniteIntegral f μ ↔ (∫⁻ a, ENNReal.ofReal (f a) ∂μ) < ∞ := by
+  rw [hasFiniteIntegral_iff_enorm, lintegral_enorm_of_ae_nonneg h]
+
+theorem hasFiniteIntegral_iff_ofNNReal {f : α → ℝ≥0} :
+    HasFiniteIntegral (fun x => (f x : ℝ)) μ ↔ (∫⁻ a, f a ∂μ) < ∞ := by
+  simp [hasFiniteIntegral_iff_norm]
+
+theorem HasFiniteIntegral.mono {f : α → β} {g : α → γ} (hg : HasFiniteIntegral g μ)
+    (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ ‖g a‖) : HasFiniteIntegral f μ := by
+  simp only [hasFiniteIntegral_iff_norm] at *
+  calc
+    (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) ≤ ∫⁻ a : α, ENNReal.ofReal ‖g a‖ ∂μ :=
+      lintegral_mono_ae (h.mono fun a h => ofReal_le_ofReal h)
+    _ < ∞ := hg
+
+theorem HasFiniteIntegral.mono' {f : α → β} {g : α → ℝ} (hg : HasFiniteIntegral g μ)
+    (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ g a) : HasFiniteIntegral f μ :=
+  hg.mono <| h.mono fun _x hx => le_trans hx (le_abs_self _)
+
+theorem HasFiniteIntegral.congr' {f : α → β} {g : α → γ} (hf : HasFiniteIntegral f μ)
+    (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) : HasFiniteIntegral g μ :=
+  hf.mono <| EventuallyEq.le <| EventuallyEq.symm h
+
+theorem hasFiniteIntegral_congr' {f : α → β} {g : α → γ} (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) :
+    HasFiniteIntegral f μ ↔ HasFiniteIntegral g μ :=
+  ⟨fun hf => hf.congr' h, fun hg => hg.congr' <| EventuallyEq.symm h⟩
+
+theorem HasFiniteIntegral.congr {f g : α → β} (hf : HasFiniteIntegral f μ) (h : f =ᵐ[μ] g) :
+    HasFiniteIntegral g μ :=
+  hf.congr' <| h.fun_comp norm
+
+theorem hasFiniteIntegral_congr {f g : α → β} (h : f =ᵐ[μ] g) :
+    HasFiniteIntegral f μ ↔ HasFiniteIntegral g μ :=
+  hasFiniteIntegral_congr' <| h.fun_comp norm
+
+theorem hasFiniteIntegral_const_iff {c : β} :
+    HasFiniteIntegral (fun _ : α => c) μ ↔ c = 0 ∨ IsFiniteMeasure μ := by
+  simp [hasFiniteIntegral_iff_enorm, lintegral_const, lt_top_iff_ne_top, ENNReal.mul_eq_top,
+    or_iff_not_imp_left, isFiniteMeasure_iff]
+
+lemma hasFiniteIntegral_const_iff_isFiniteMeasure {c : β} (hc : c ≠ 0) :
+    HasFiniteIntegral (fun _ ↦ c) μ ↔ IsFiniteMeasure μ := by
+  simp [hasFiniteIntegral_const_iff, hc, isFiniteMeasure_iff]
+
+theorem hasFiniteIntegral_const [IsFiniteMeasure μ] (c : β) :
+    HasFiniteIntegral (fun _ : α => c) μ :=
+  hasFiniteIntegral_const_iff.2 <| .inr ‹_›
+
+theorem HasFiniteIntegral.of_mem_Icc [IsFiniteMeasure μ] (a b : ℝ) {X : α → ℝ}
+    (h : ∀ᵐ ω ∂μ, X ω ∈ Set.Icc a b) :
+    HasFiniteIntegral X μ := by
+  apply (hasFiniteIntegral_const (max ‖a‖ ‖b‖)).mono'
+  filter_upwards [h.mono fun ω h ↦ h.1, h.mono fun ω h ↦ h.2] with ω using abs_le_max_abs_abs
+
+theorem hasFiniteIntegral_of_bounded [IsFiniteMeasure μ] {f : α → β} {C : ℝ}
+    (hC : ∀ᵐ a ∂μ, ‖f a‖ ≤ C) : HasFiniteIntegral f μ :=
+  (hasFiniteIntegral_const C).mono' hC
+
+theorem HasFiniteIntegral.of_finite [Finite α] [IsFiniteMeasure μ] {f : α → β} :
+    HasFiniteIntegral f μ :=
+  let ⟨_⟩ := nonempty_fintype α
+  hasFiniteIntegral_of_bounded <| ae_of_all μ <| norm_le_pi_norm f
+
+theorem HasFiniteIntegral.mono_measure {f : α → β} (h : HasFiniteIntegral f ν) (hμ : μ ≤ ν) :
+    HasFiniteIntegral f μ :=
+  lt_of_le_of_lt (lintegral_mono' hμ le_rfl) h
+
+theorem HasFiniteIntegral.add_measure {f : α → β} (hμ : HasFiniteIntegral f μ)
+    (hν : HasFiniteIntegral f ν) : HasFiniteIntegral f (μ + ν) := by
+  simp only [HasFiniteIntegral, lintegral_add_measure] at *
+  exact add_lt_top.2 ⟨hμ, hν⟩
+
+theorem HasFiniteIntegral.left_of_add_measure {f : α → β} (h : HasFiniteIntegral f (μ + ν)) :
+    HasFiniteIntegral f μ :=
+  h.mono_measure <| Measure.le_add_right <| le_rfl
+
+theorem HasFiniteIntegral.right_of_add_measure {f : α → β} (h : HasFiniteIntegral f (μ + ν)) :
+    HasFiniteIntegral f ν :=
+  h.mono_measure <| Measure.le_add_left <| le_rfl
+
+@[simp]
+theorem hasFiniteIntegral_add_measure {f : α → β} :
+    HasFiniteIntegral f (μ + ν) ↔ HasFiniteIntegral f μ ∧ HasFiniteIntegral f ν :=
+  ⟨fun h => ⟨h.left_of_add_measure, h.right_of_add_measure⟩, fun h => h.1.add_measure h.2⟩
+
+theorem HasFiniteIntegral.smul_measure {f : α → β} (h : HasFiniteIntegral f μ) {c : ℝ≥0∞}
+    (hc : c ≠ ∞) : HasFiniteIntegral f (c • μ) := by
+  simp only [HasFiniteIntegral, lintegral_smul_measure] at *
+  exact mul_lt_top hc.lt_top h
+
+@[simp]
+theorem hasFiniteIntegral_zero_measure {m : MeasurableSpace α} (f : α → β) :
+    HasFiniteIntegral f (0 : Measure α) := by
+  simp only [HasFiniteIntegral, lintegral_zero_measure, zero_lt_top]
+
+variable (α β μ)
+
+@[simp]
+theorem hasFiniteIntegral_zero : HasFiniteIntegral (fun _ : α => (0 : β)) μ := by
+  simp [hasFiniteIntegral_iff_enorm]
+
+variable {α β μ}
+
+theorem HasFiniteIntegral.neg {f : α → β} (hfi : HasFiniteIntegral f μ) :
+    HasFiniteIntegral (-f) μ := by simpa [hasFiniteIntegral_iff_enorm] using hfi
+
+@[simp]
+theorem hasFiniteIntegral_neg_iff {f : α → β} : HasFiniteIntegral (-f) μ ↔ HasFiniteIntegral f μ :=
+  ⟨fun h => neg_neg f ▸ h.neg, HasFiniteIntegral.neg⟩
+
+theorem HasFiniteIntegral.norm {f : α → β} (hfi : HasFiniteIntegral f μ) :
+    HasFiniteIntegral (fun a => ‖f a‖) μ := by simpa [hasFiniteIntegral_iff_enorm] using hfi
+
+theorem hasFiniteIntegral_norm_iff (f : α → β) :
+    HasFiniteIntegral (fun a => ‖f a‖) μ ↔ HasFiniteIntegral f μ :=
+  hasFiniteIntegral_congr' <| Eventually.of_forall fun x => norm_norm (f x)
+
+theorem hasFiniteIntegral_toReal_of_lintegral_ne_top {f : α → ℝ≥0∞} (hf : ∫⁻ x, f x ∂μ ≠ ∞) :
+    HasFiniteIntegral (fun x ↦ (f x).toReal) μ := by
+  have h x : ‖(f x).toReal‖ₑ = .ofReal (f x).toReal := by
+    rw [Real.enorm_of_nonneg ENNReal.toReal_nonneg]
+  simp_rw [hasFiniteIntegral_iff_enorm, h]
+  refine lt_of_le_of_lt (lintegral_mono fun x => ?_) (lt_top_iff_ne_top.2 hf)
+  by_cases hfx : f x = ∞
+  · simp [hfx]
+  · lift f x to ℝ≥0 using hfx with fx h
+    simp [← h, ← NNReal.coe_le_coe]
+
+lemma hasFiniteIntegral_toReal_iff {f : α → ℝ≥0∞} (hf : ∀ᵐ x ∂μ, f x ≠ ∞) :
+    HasFiniteIntegral (fun x ↦ (f x).toReal) μ ↔ ∫⁻ x, f x ∂μ ≠ ∞ := by
+  have : ∀ᵐ x ∂μ, .ofReal (f x).toReal = f x := by filter_upwards [hf] with x hx; simp [hx]
+  simp [hasFiniteIntegral_iff_enorm, Real.enorm_of_nonneg ENNReal.toReal_nonneg,
+    lintegral_congr_ae this, lt_top_iff_ne_top]
+
+theorem isFiniteMeasure_withDensity_ofReal {f : α → ℝ} (hfi : HasFiniteIntegral f μ) :
+    IsFiniteMeasure (μ.withDensity fun x => ENNReal.ofReal <| f x) := by
+  refine isFiniteMeasure_withDensity ((lintegral_mono fun x => ?_).trans_lt hfi).ne
+  exact Real.ofReal_le_enorm (f x)
+
+section DominatedConvergence
+
+variable {F : ℕ → α → β} {f : α → β} {bound : α → ℝ}
+
+theorem all_ae_ofReal_F_le_bound (h : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) :
+    ∀ n, ∀ᵐ a ∂μ, ENNReal.ofReal ‖F n a‖ ≤ ENNReal.ofReal (bound a) := fun n =>
+  (h n).mono fun _ h => ENNReal.ofReal_le_ofReal h
+
+theorem all_ae_tendsto_ofReal_norm (h : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop <| 𝓝 <| f a) :
+    ∀ᵐ a ∂μ, Tendsto (fun n => ENNReal.ofReal ‖F n a‖) atTop <| 𝓝 <| ENNReal.ofReal ‖f a‖ :=
+  h.mono fun _ h => tendsto_ofReal <| Tendsto.comp (Continuous.tendsto continuous_norm _) h
+
+theorem all_ae_ofReal_f_le_bound (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
+    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :
+    ∀ᵐ a ∂μ, ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a) := by
+  have F_le_bound := all_ae_ofReal_F_le_bound h_bound
+  rw [← ae_all_iff] at F_le_bound
+  apply F_le_bound.mp ((all_ae_tendsto_ofReal_norm h_lim).mono _)
+  intro a tendsto_norm F_le_bound
+  exact le_of_tendsto' tendsto_norm F_le_bound
+
+theorem hasFiniteIntegral_of_dominated_convergence {F : ℕ → α → β} {f : α → β} {bound : α → ℝ}
+    (bound_hasFiniteIntegral : HasFiniteIntegral bound μ)
+    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
+    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) : HasFiniteIntegral f μ := by
+  /- `‖F n a‖ ≤ bound a` and `‖F n a‖ --> ‖f a‖` implies `‖f a‖ ≤ bound a`,
+    and so `∫ ‖f‖ ≤ ∫ bound < ∞` since `bound` is has_finite_integral -/
+  rw [hasFiniteIntegral_iff_norm]
+  calc
+    (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) ≤ ∫⁻ a, ENNReal.ofReal (bound a) ∂μ :=
+      lintegral_mono_ae <| all_ae_ofReal_f_le_bound h_bound h_lim
+    _ < ∞ := by
+      rw [← hasFiniteIntegral_iff_ofReal]
+      · exact bound_hasFiniteIntegral
+      exact (h_bound 0).mono fun a h => le_trans (norm_nonneg _) h
+
+theorem tendsto_lintegral_norm_of_dominated_convergence {F : ℕ → α → β} {f : α → β} {bound : α → ℝ}
+    (F_measurable : ∀ n, AEStronglyMeasurable (F n) μ)
+    (bound_hasFiniteIntegral : HasFiniteIntegral bound μ)
+    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
+    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :
+    Tendsto (fun n => ∫⁻ a, ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 0) := by
+  have f_measurable : AEStronglyMeasurable f μ :=
+    aestronglyMeasurable_of_tendsto_ae _ F_measurable h_lim
+  let b a := 2 * ENNReal.ofReal (bound a)
+  /- `‖F n a‖ ≤ bound a` and `F n a --> f a` implies `‖f a‖ ≤ bound a`, and thus by the
+    triangle inequality, have `‖F n a - f a‖ ≤ 2 * (bound a)`. -/
+  have hb : ∀ n, ∀ᵐ a ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a := by
+    intro n
+    filter_upwards [all_ae_ofReal_F_le_bound h_bound n,
+      all_ae_ofReal_f_le_bound h_bound h_lim] with a h₁ h₂
+    calc
+      ENNReal.ofReal ‖F n a - f a‖ ≤ ENNReal.ofReal ‖F n a‖ + ENNReal.ofReal ‖f a‖ := by
+        rw [← ENNReal.ofReal_add]
+        · apply ofReal_le_ofReal
+          apply norm_sub_le
+        · exact norm_nonneg _
+        · exact norm_nonneg _
+      _ ≤ ENNReal.ofReal (bound a) + ENNReal.ofReal (bound a) := add_le_add h₁ h₂
+      _ = b a := by rw [← two_mul]
+  -- On the other hand, `F n a --> f a` implies that `‖F n a - f a‖ --> 0`
+  have h : ∀ᵐ a ∂μ, Tendsto (fun n => ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 0) := by
+    rw [← ENNReal.ofReal_zero]
+    refine h_lim.mono fun a h => (continuous_ofReal.tendsto _).comp ?_
+    rwa [← tendsto_iff_norm_sub_tendsto_zero]
+  /- Therefore, by the dominated convergence theorem for nonnegative integration, have
+    ` ∫ ‖f a - F n a‖ --> 0 ` -/
+  suffices Tendsto (fun n => ∫⁻ a, ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 (∫⁻ _ : α, 0 ∂μ)) by
+    rwa [lintegral_zero] at this
+  -- Using the dominated convergence theorem.
+  refine tendsto_lintegral_of_dominated_convergence' _ ?_ hb ?_ ?_
+  -- Show `fun a => ‖f a - F n a‖` is almost everywhere measurable for all `n`
+  · exact fun n =>
+      measurable_ofReal.comp_aemeasurable ((F_measurable n).sub f_measurable).norm.aemeasurable
+  -- Show `2 * bound` `HasFiniteIntegral`
+  · rw [hasFiniteIntegral_iff_ofReal] at bound_hasFiniteIntegral
+    · calc
+        ∫⁻ a, b a ∂μ = 2 * ∫⁻ a, ENNReal.ofReal (bound a) ∂μ := by
+          rw [lintegral_const_mul']
+          exact coe_ne_top
+        _ ≠ ∞ := mul_ne_top coe_ne_top bound_hasFiniteIntegral.ne
+    filter_upwards [h_bound 0] with _ h using le_trans (norm_nonneg _) h
+  -- Show `‖f a - F n a‖ --> 0`
+  · exact h
+
+end DominatedConvergence
+
+section PosPart
+
+/-! Lemmas used for defining the positive part of an `L¹` function -/
+
+
+theorem HasFiniteIntegral.max_zero {f : α → ℝ} (hf : HasFiniteIntegral f μ) :
+    HasFiniteIntegral (fun a => max (f a) 0) μ :=
+  hf.mono <| Eventually.of_forall fun x => by simp [abs_le, le_abs_self]
+
+theorem HasFiniteIntegral.min_zero {f : α → ℝ} (hf : HasFiniteIntegral f μ) :
+    HasFiniteIntegral (fun a => min (f a) 0) μ :=
+  hf.mono <| Eventually.of_forall fun x => by simpa [abs_le] using neg_abs_le _
+
+end PosPart
+
+section NormedSpace
+
+variable {𝕜 : Type*}
+
+theorem HasFiniteIntegral.smul [NormedAddCommGroup 𝕜] [SMulZeroClass 𝕜 β] [BoundedSMul 𝕜 β] (c : 𝕜)
+    {f : α → β} : HasFiniteIntegral f μ → HasFiniteIntegral (c • f) μ := by
+  simp only [HasFiniteIntegral]; intro hfi
+  calc
+    ∫⁻ a : α, ‖c • f a‖ₑ ∂μ ≤ ∫⁻ a : α, ‖c‖ₑ * ‖f a‖ₑ ∂μ := lintegral_mono fun i ↦ enorm_smul_le
+    _ < ∞ := by
+      rw [lintegral_const_mul']
+      exacts [mul_lt_top coe_lt_top hfi, coe_ne_top]
+
+theorem hasFiniteIntegral_smul_iff [NormedRing 𝕜] [MulActionWithZero 𝕜 β] [BoundedSMul 𝕜 β] {c : 𝕜}
+    (hc : IsUnit c) (f : α → β) : HasFiniteIntegral (c • f) μ ↔ HasFiniteIntegral f μ := by
+  obtain ⟨c, rfl⟩ := hc
+  constructor
+  · intro h
+    simpa only [smul_smul, Units.inv_mul, one_smul] using h.smul ((c⁻¹ : 𝕜ˣ) : 𝕜)
+  exact HasFiniteIntegral.smul _
+
+theorem HasFiniteIntegral.const_mul [NormedRing 𝕜] {f : α → 𝕜} (h : HasFiniteIntegral f μ) (c : 𝕜) :
+    HasFiniteIntegral (fun x => c * f x) μ :=
+  h.smul c
+
+theorem HasFiniteIntegral.mul_const [NormedRing 𝕜] {f : α → 𝕜} (h : HasFiniteIntegral f μ) (c : 𝕜) :
+    HasFiniteIntegral (fun x => f x * c) μ :=
+  h.smul (MulOpposite.op c)
+
+section count
+
+variable [MeasurableSingletonClass α] {f : α → β}
+
+/-- A function has finite integral for the counting measure iff its norm is summable. -/
+lemma hasFiniteIntegral_count_iff :
+    HasFiniteIntegral f Measure.count ↔ Summable (‖f ·‖) := by
+  simp only [hasFiniteIntegral_iff_enorm, enorm, lintegral_count, lt_top_iff_ne_top,
+    tsum_coe_ne_top_iff_summable, ← summable_coe, coe_nnnorm]
+
+end count
+
+section restrict
+
+variable {E : Type*} [NormedAddCommGroup E] {f : α → E}
+
+lemma HasFiniteIntegral.restrict (h : HasFiniteIntegral f μ) {s : Set α} :
+    HasFiniteIntegral f (μ.restrict s) := by
+  refine lt_of_le_of_lt ?_ h
+  simpa [Measure.restrict_univ] using lintegral_mono_set (subset_univ s)
+
+end restrict
+
+end NormedSpace
+
+end MeasureTheory
diff --git a/Mathlib/MeasureTheory/Function/L1Space.lean b/Mathlib/MeasureTheory/Function/L1Space/Integrable.lean
similarity index 61%
rename from Mathlib/MeasureTheory/Function/L1Space.lean
rename to Mathlib/MeasureTheory/Function/L1Space/Integrable.lean
index 2624f72f454f23..37cfdc0b9cc29f 100644
--- a/Mathlib/MeasureTheory/Function/L1Space.lean
+++ b/Mathlib/MeasureTheory/Function/L1Space/Integrable.lean
@@ -3,35 +3,23 @@ Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Zhouhang Zhou
 -/
+import Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral
 import Mathlib.MeasureTheory.Function.LpOrder
 import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas
 
 /-!
-# Integrable functions and `L¹` space
+# Integrable functions
 
-In the first part of this file, the predicate `Integrable` is defined and basic properties of
+In this file, the predicate `Integrable` is defined and basic properties of
 integrable functions are proved.
 
 Such a predicate is already available under the name `Memℒp 1`. We give a direct definition which
-is easier to use, and show that it is equivalent to `Memℒp 1`
+is easier to use, and show that it is equivalent to `Memℒp 1`.
 
-In the second part, we establish an API between `Integrable` and the space `L¹` of equivalence
-classes of integrable functions, already defined as a special case of `L^p` spaces for `p = 1`.
+## Main definition
 
-## Notation
-
-* `α →₁[μ] β` is the type of `L¹` space, where `α` is a `MeasureSpace` and `β` is a
-  `NormedAddCommGroup` with a `SecondCountableTopology`. `f : α →ₘ β` is a "function" in `L¹`.
-  In comments, `[f]` is also used to denote an `L¹` function.
-
-  `₁` can be typed as `\1`.
-
-## Main definitions
-
-* Let `f : α → β` be a function, where `α` is a `MeasureSpace` and `β` a `NormedAddCommGroup`.
-  Then `HasFiniteIntegral f` means `(∫⁻ a, ‖f a‖₊) < ∞`.
-
-* If `β` is moreover a `MeasurableSpace` then `f` is called `Integrable` if
+* Let `f : α → β` be a function, where `α` is a `MeasureSpace` and `β` a `NormedAddCommGroup`
+  which also a `MeasurableSpace`. Then `f` is called `Integrable` if
   `f` is `Measurable` and `HasFiniteIntegral f` holds.
 
 ## Implementation notes
@@ -41,367 +29,20 @@ To prove something for an arbitrary integrable function, a useful theorem is
 
 ## Tags
 
-integrable, function space, l1
+integrable
 
 -/
 
 
 noncomputable section
 
-open scoped Classical
-open Topology ENNReal MeasureTheory NNReal
-
-open Set Filter TopologicalSpace ENNReal EMetric MeasureTheory
+open EMetric ENNReal Filter MeasureTheory NNReal Set
 
 variable {α β γ δ ε : Type*} {m : MeasurableSpace α} {μ ν : Measure α} [MeasurableSpace δ]
 variable [NormedAddCommGroup β] [NormedAddCommGroup γ] [ENorm ε]
 
 namespace MeasureTheory
 
-/-! ### Some results about the Lebesgue integral involving a normed group -/
-
-
-theorem lintegral_nnnorm_eq_lintegral_edist (f : α → β) :
-    ∫⁻ a, ‖f a‖₊ ∂μ = ∫⁻ a, edist (f a) 0 ∂μ := by simp only [edist_eq_coe_nnnorm]
-
-theorem lintegral_norm_eq_lintegral_edist (f : α → β) :
-    ∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ = ∫⁻ a, edist (f a) 0 ∂μ := by
-  simp only [ofReal_norm_eq_coe_nnnorm, edist_eq_coe_nnnorm]
-
-theorem lintegral_edist_triangle {f g h : α → β} (hf : AEStronglyMeasurable f μ)
-    (hh : AEStronglyMeasurable h μ) :
-    (∫⁻ a, edist (f a) (g a) ∂μ) ≤ (∫⁻ a, edist (f a) (h a) ∂μ) + ∫⁻ a, edist (g a) (h a) ∂μ := by
-  rw [← lintegral_add_left' (hf.edist hh)]
-  refine lintegral_mono fun a => ?_
-  apply edist_triangle_right
-
-theorem lintegral_nnnorm_zero : (∫⁻ _ : α, ‖(0 : β)‖₊ ∂μ) = 0 := by simp
-
-theorem lintegral_nnnorm_add_left {f : α → β} (hf : AEStronglyMeasurable f μ) (g : α → γ) :
-    ∫⁻ a, ‖f a‖₊ + ‖g a‖₊ ∂μ = (∫⁻ a, ‖f a‖₊ ∂μ) + ∫⁻ a, ‖g a‖₊ ∂μ :=
-  lintegral_add_left' hf.ennnorm _
-
-theorem lintegral_nnnorm_add_right (f : α → β) {g : α → γ} (hg : AEStronglyMeasurable g μ) :
-    ∫⁻ a, ‖f a‖₊ + ‖g a‖₊ ∂μ = (∫⁻ a, ‖f a‖₊ ∂μ) + ∫⁻ a, ‖g a‖₊ ∂μ :=
-  lintegral_add_right' _ hg.ennnorm
-
-theorem lintegral_nnnorm_neg {f : α → β} : (∫⁻ a, ‖(-f) a‖₊ ∂μ) = ∫⁻ a, ‖f a‖₊ ∂μ := by
-  simp only [Pi.neg_apply, nnnorm_neg]
-
-/-! ### The predicate `HasFiniteIntegral` -/
-
-
-/-- `HasFiniteIntegral f μ` means that the integral `∫⁻ a, ‖f a‖ ∂μ` is finite.
-  `HasFiniteIntegral f` means `HasFiniteIntegral f volume`. -/
-def HasFiniteIntegral {_ : MeasurableSpace α} (f : α → ε)
-    (μ : Measure α := by volume_tac) : Prop :=
-  (∫⁻ a, ‖f a‖ₑ ∂μ) < ∞
-
-theorem hasFiniteIntegral_def {_ : MeasurableSpace α} (f : α → ε) (μ : Measure α) :
-    HasFiniteIntegral f μ ↔ ((∫⁻ a, ‖f a‖ₑ ∂μ) < ∞) :=
-  Iff.rfl
-
-theorem hasFiniteIntegral_iff_nnnorm {f : α → β} :
-    HasFiniteIntegral f μ ↔ (∫⁻ a, ‖f a‖₊ ∂μ) < ∞ := by
-  simp only [HasFiniteIntegral, ofReal_norm_eq_coe_nnnorm, enorm_eq_nnnorm]
-
-theorem hasFiniteIntegral_iff_norm (f : α → β) :
-    HasFiniteIntegral f μ ↔ (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) < ∞ := by
-  simp only [hasFiniteIntegral_iff_nnnorm, ofReal_norm_eq_coe_nnnorm]
-
-theorem hasFiniteIntegral_iff_edist (f : α → β) :
-    HasFiniteIntegral f μ ↔ (∫⁻ a, edist (f a) 0 ∂μ) < ∞ := by
-  simp only [hasFiniteIntegral_iff_norm, edist_dist, dist_zero_right]
-
-theorem hasFiniteIntegral_iff_ofReal {f : α → ℝ} (h : 0 ≤ᵐ[μ] f) :
-    HasFiniteIntegral f μ ↔ (∫⁻ a, ENNReal.ofReal (f a) ∂μ) < ∞ := by
-  rw [hasFiniteIntegral_iff_nnnorm, lintegral_nnnorm_eq_of_ae_nonneg h]
-
-theorem hasFiniteIntegral_iff_ofNNReal {f : α → ℝ≥0} :
-    HasFiniteIntegral (fun x => (f x : ℝ)) μ ↔ (∫⁻ a, f a ∂μ) < ∞ := by
-  simp [hasFiniteIntegral_iff_norm]
-
-theorem HasFiniteIntegral.mono {f : α → β} {g : α → γ} (hg : HasFiniteIntegral g μ)
-    (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ ‖g a‖) : HasFiniteIntegral f μ := by
-  simp only [hasFiniteIntegral_iff_norm] at *
-  calc
-    (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) ≤ ∫⁻ a : α, ENNReal.ofReal ‖g a‖ ∂μ :=
-      lintegral_mono_ae (h.mono fun a h => ofReal_le_ofReal h)
-    _ < ∞ := hg
-
-theorem HasFiniteIntegral.mono' {f : α → β} {g : α → ℝ} (hg : HasFiniteIntegral g μ)
-    (h : ∀ᵐ a ∂μ, ‖f a‖ ≤ g a) : HasFiniteIntegral f μ :=
-  hg.mono <| h.mono fun _x hx => le_trans hx (le_abs_self _)
-
-theorem HasFiniteIntegral.congr' {f : α → β} {g : α → γ} (hf : HasFiniteIntegral f μ)
-    (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) : HasFiniteIntegral g μ :=
-  hf.mono <| EventuallyEq.le <| EventuallyEq.symm h
-
-theorem hasFiniteIntegral_congr' {f : α → β} {g : α → γ} (h : ∀ᵐ a ∂μ, ‖f a‖ = ‖g a‖) :
-    HasFiniteIntegral f μ ↔ HasFiniteIntegral g μ :=
-  ⟨fun hf => hf.congr' h, fun hg => hg.congr' <| EventuallyEq.symm h⟩
-
-theorem HasFiniteIntegral.congr {f g : α → β} (hf : HasFiniteIntegral f μ) (h : f =ᵐ[μ] g) :
-    HasFiniteIntegral g μ :=
-  hf.congr' <| h.fun_comp norm
-
-theorem hasFiniteIntegral_congr {f g : α → β} (h : f =ᵐ[μ] g) :
-    HasFiniteIntegral f μ ↔ HasFiniteIntegral g μ :=
-  hasFiniteIntegral_congr' <| h.fun_comp norm
-
-theorem hasFiniteIntegral_const_iff {c : β} :
-    HasFiniteIntegral (fun _ : α => c) μ ↔ c = 0 ∨ IsFiniteMeasure μ := by
-  simp [hasFiniteIntegral_iff_nnnorm, lintegral_const, lt_top_iff_ne_top, ENNReal.mul_eq_top,
-    or_iff_not_imp_left, isFiniteMeasure_iff]
-
-lemma hasFiniteIntegral_const_iff_isFiniteMeasure {c : β} (hc : c ≠ 0) :
-    HasFiniteIntegral (fun _ ↦ c) μ ↔ IsFiniteMeasure μ := by
-  simp [hasFiniteIntegral_const_iff, hc, isFiniteMeasure_iff]
-
-theorem hasFiniteIntegral_const [IsFiniteMeasure μ] (c : β) :
-    HasFiniteIntegral (fun _ : α => c) μ :=
-  hasFiniteIntegral_const_iff.2 <| .inr ‹_›
-
-theorem HasFiniteIntegral.of_mem_Icc [IsFiniteMeasure μ] (a b : ℝ) {X : α → ℝ}
-    (h : ∀ᵐ ω ∂μ, X ω ∈ Set.Icc a b) :
-    HasFiniteIntegral X μ := by
-  apply (hasFiniteIntegral_const (max ‖a‖ ‖b‖)).mono'
-  filter_upwards [h.mono fun ω h ↦ h.1, h.mono fun ω h ↦ h.2] with ω using abs_le_max_abs_abs
-
-theorem hasFiniteIntegral_of_bounded [IsFiniteMeasure μ] {f : α → β} {C : ℝ}
-    (hC : ∀ᵐ a ∂μ, ‖f a‖ ≤ C) : HasFiniteIntegral f μ :=
-  (hasFiniteIntegral_const C).mono' hC
-
-theorem HasFiniteIntegral.of_finite [Finite α] [IsFiniteMeasure μ] {f : α → β} :
-    HasFiniteIntegral f μ :=
-  let ⟨_⟩ := nonempty_fintype α
-  hasFiniteIntegral_of_bounded <| ae_of_all μ <| norm_le_pi_norm f
-
-theorem HasFiniteIntegral.mono_measure {f : α → β} (h : HasFiniteIntegral f ν) (hμ : μ ≤ ν) :
-    HasFiniteIntegral f μ :=
-  lt_of_le_of_lt (lintegral_mono' hμ le_rfl) h
-
-theorem HasFiniteIntegral.add_measure {f : α → β} (hμ : HasFiniteIntegral f μ)
-    (hν : HasFiniteIntegral f ν) : HasFiniteIntegral f (μ + ν) := by
-  simp only [HasFiniteIntegral, lintegral_add_measure] at *
-  exact add_lt_top.2 ⟨hμ, hν⟩
-
-theorem HasFiniteIntegral.left_of_add_measure {f : α → β} (h : HasFiniteIntegral f (μ + ν)) :
-    HasFiniteIntegral f μ :=
-  h.mono_measure <| Measure.le_add_right <| le_rfl
-
-theorem HasFiniteIntegral.right_of_add_measure {f : α → β} (h : HasFiniteIntegral f (μ + ν)) :
-    HasFiniteIntegral f ν :=
-  h.mono_measure <| Measure.le_add_left <| le_rfl
-
-@[simp]
-theorem hasFiniteIntegral_add_measure {f : α → β} :
-    HasFiniteIntegral f (μ + ν) ↔ HasFiniteIntegral f μ ∧ HasFiniteIntegral f ν :=
-  ⟨fun h => ⟨h.left_of_add_measure, h.right_of_add_measure⟩, fun h => h.1.add_measure h.2⟩
-
-theorem HasFiniteIntegral.smul_measure {f : α → β} (h : HasFiniteIntegral f μ) {c : ℝ≥0∞}
-    (hc : c ≠ ∞) : HasFiniteIntegral f (c • μ) := by
-  simp only [HasFiniteIntegral, lintegral_smul_measure] at *
-  exact mul_lt_top hc.lt_top h
-
-@[simp]
-theorem hasFiniteIntegral_zero_measure {m : MeasurableSpace α} (f : α → β) :
-    HasFiniteIntegral f (0 : Measure α) := by
-  simp only [HasFiniteIntegral, lintegral_zero_measure, zero_lt_top]
-
-variable (α β μ)
-
-@[simp]
-theorem hasFiniteIntegral_zero : HasFiniteIntegral (fun _ : α => (0 : β)) μ := by
-  simp [hasFiniteIntegral_iff_nnnorm]
-
-variable {α β μ}
-
-theorem HasFiniteIntegral.neg {f : α → β} (hfi : HasFiniteIntegral f μ) :
-    HasFiniteIntegral (-f) μ := by simpa [hasFiniteIntegral_iff_nnnorm] using hfi
-
-@[simp]
-theorem hasFiniteIntegral_neg_iff {f : α → β} : HasFiniteIntegral (-f) μ ↔ HasFiniteIntegral f μ :=
-  ⟨fun h => neg_neg f ▸ h.neg, HasFiniteIntegral.neg⟩
-
-theorem HasFiniteIntegral.norm {f : α → β} (hfi : HasFiniteIntegral f μ) :
-    HasFiniteIntegral (fun a => ‖f a‖) μ := by
-  have eq : (fun a => (nnnorm ‖f a‖ : ℝ≥0∞)) = fun a => (‖f a‖₊ : ℝ≥0∞) := by
-    funext
-    rw [nnnorm_norm]
-  rwa [hasFiniteIntegral_iff_nnnorm, eq]
-
-theorem hasFiniteIntegral_norm_iff (f : α → β) :
-    HasFiniteIntegral (fun a => ‖f a‖) μ ↔ HasFiniteIntegral f μ :=
-  hasFiniteIntegral_congr' <| Eventually.of_forall fun x => norm_norm (f x)
-
-theorem hasFiniteIntegral_toReal_of_lintegral_ne_top {f : α → ℝ≥0∞} (hf : ∫⁻ x, f x ∂μ ≠ ∞) :
-    HasFiniteIntegral (fun x ↦ (f x).toReal) μ := by
-  have h x : (‖(f x).toReal‖₊ : ℝ≥0∞) = ENNReal.ofNNReal ⟨(f x).toReal, ENNReal.toReal_nonneg⟩ := by
-    rw [Real.nnnorm_of_nonneg]
-  simp_rw [hasFiniteIntegral_iff_nnnorm, h]
-  refine lt_of_le_of_lt (lintegral_mono fun x => ?_) (lt_top_iff_ne_top.2 hf)
-  by_cases hfx : f x = ∞
-  · simp [hfx]
-  · lift f x to ℝ≥0 using hfx with fx h
-    simp [← h, ← NNReal.coe_le_coe]
-
-lemma hasFiniteIntegral_toReal_iff {f : α → ℝ≥0∞} (hf_ne_top : ∀ᵐ x ∂μ, f x ≠ ∞) :
-    HasFiniteIntegral (fun x ↦ (f x).toReal) μ ↔ ∫⁻ x, f x ∂μ ≠ ∞ := by
-  have h_eq x : (‖(f x).toReal‖₊ : ℝ≥0∞)
-      = ENNReal.ofNNReal ⟨(f x).toReal, ENNReal.toReal_nonneg⟩ := by
-    rw [Real.nnnorm_of_nonneg]
-  simp_rw [hasFiniteIntegral_iff_nnnorm, h_eq, lt_top_iff_ne_top]
-  convert Iff.rfl using 2
-  refine lintegral_congr_ae ?_
-  filter_upwards [hf_ne_top] with x hfx
-  lift f x to ℝ≥0 using hfx with fx h
-  simp only [coe_toReal, ENNReal.coe_inj]
-  rfl
-
-theorem isFiniteMeasure_withDensity_ofReal {f : α → ℝ} (hfi : HasFiniteIntegral f μ) :
-    IsFiniteMeasure (μ.withDensity fun x => ENNReal.ofReal <| f x) := by
-  refine isFiniteMeasure_withDensity ((lintegral_mono fun x => ?_).trans_lt hfi).ne
-  exact Real.ofReal_le_ennnorm (f x)
-
-section DominatedConvergence
-
-variable {F : ℕ → α → β} {f : α → β} {bound : α → ℝ}
-
-theorem all_ae_ofReal_F_le_bound (h : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) :
-    ∀ n, ∀ᵐ a ∂μ, ENNReal.ofReal ‖F n a‖ ≤ ENNReal.ofReal (bound a) := fun n =>
-  (h n).mono fun _ h => ENNReal.ofReal_le_ofReal h
-
-theorem all_ae_tendsto_ofReal_norm (h : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop <| 𝓝 <| f a) :
-    ∀ᵐ a ∂μ, Tendsto (fun n => ENNReal.ofReal ‖F n a‖) atTop <| 𝓝 <| ENNReal.ofReal ‖f a‖ :=
-  h.mono fun _ h => tendsto_ofReal <| Tendsto.comp (Continuous.tendsto continuous_norm _) h
-
-theorem all_ae_ofReal_f_le_bound (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
-    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :
-    ∀ᵐ a ∂μ, ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a) := by
-  have F_le_bound := all_ae_ofReal_F_le_bound h_bound
-  rw [← ae_all_iff] at F_le_bound
-  apply F_le_bound.mp ((all_ae_tendsto_ofReal_norm h_lim).mono _)
-  intro a tendsto_norm F_le_bound
-  exact le_of_tendsto' tendsto_norm F_le_bound
-
-theorem hasFiniteIntegral_of_dominated_convergence {F : ℕ → α → β} {f : α → β} {bound : α → ℝ}
-    (bound_hasFiniteIntegral : HasFiniteIntegral bound μ)
-    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
-    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) : HasFiniteIntegral f μ := by
-  /- `‖F n a‖ ≤ bound a` and `‖F n a‖ --> ‖f a‖` implies `‖f a‖ ≤ bound a`,
-    and so `∫ ‖f‖ ≤ ∫ bound < ∞` since `bound` is has_finite_integral -/
-  rw [hasFiniteIntegral_iff_norm]
-  calc
-    (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) ≤ ∫⁻ a, ENNReal.ofReal (bound a) ∂μ :=
-      lintegral_mono_ae <| all_ae_ofReal_f_le_bound h_bound h_lim
-    _ < ∞ := by
-      rw [← hasFiniteIntegral_iff_ofReal]
-      · exact bound_hasFiniteIntegral
-      exact (h_bound 0).mono fun a h => le_trans (norm_nonneg _) h
-
-theorem tendsto_lintegral_norm_of_dominated_convergence {F : ℕ → α → β} {f : α → β} {bound : α → ℝ}
-    (F_measurable : ∀ n, AEStronglyMeasurable (F n) μ)
-    (bound_hasFiniteIntegral : HasFiniteIntegral bound μ)
-    (h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
-    (h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :
-    Tendsto (fun n => ∫⁻ a, ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 0) := by
-  have f_measurable : AEStronglyMeasurable f μ :=
-    aestronglyMeasurable_of_tendsto_ae _ F_measurable h_lim
-  let b a := 2 * ENNReal.ofReal (bound a)
-  /- `‖F n a‖ ≤ bound a` and `F n a --> f a` implies `‖f a‖ ≤ bound a`, and thus by the
-    triangle inequality, have `‖F n a - f a‖ ≤ 2 * (bound a)`. -/
-  have hb : ∀ n, ∀ᵐ a ∂μ, ENNReal.ofReal ‖F n a - f a‖ ≤ b a := by
-    intro n
-    filter_upwards [all_ae_ofReal_F_le_bound h_bound n,
-      all_ae_ofReal_f_le_bound h_bound h_lim] with a h₁ h₂
-    calc
-      ENNReal.ofReal ‖F n a - f a‖ ≤ ENNReal.ofReal ‖F n a‖ + ENNReal.ofReal ‖f a‖ := by
-        rw [← ENNReal.ofReal_add]
-        · apply ofReal_le_ofReal
-          apply norm_sub_le
-        · exact norm_nonneg _
-        · exact norm_nonneg _
-      _ ≤ ENNReal.ofReal (bound a) + ENNReal.ofReal (bound a) := add_le_add h₁ h₂
-      _ = b a := by rw [← two_mul]
-  -- On the other hand, `F n a --> f a` implies that `‖F n a - f a‖ --> 0`
-  have h : ∀ᵐ a ∂μ, Tendsto (fun n => ENNReal.ofReal ‖F n a - f a‖) atTop (𝓝 0) := by
-    rw [← ENNReal.ofReal_zero]
-    refine h_lim.mono fun a h => (continuous_ofReal.tendsto _).comp ?_
-    rwa [← tendsto_iff_norm_sub_tendsto_zero]
-  /- Therefore, by the dominated convergence theorem for nonnegative integration, have
-    ` ∫ ‖f a - F n a‖ --> 0 ` -/
-  suffices Tendsto (fun n => ∫⁻ a, ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 (∫⁻ _ : α, 0 ∂μ)) by
-    rwa [lintegral_zero] at this
-  -- Using the dominated convergence theorem.
-  refine tendsto_lintegral_of_dominated_convergence' _ ?_ hb ?_ ?_
-  -- Show `fun a => ‖f a - F n a‖` is almost everywhere measurable for all `n`
-  · exact fun n =>
-      measurable_ofReal.comp_aemeasurable ((F_measurable n).sub f_measurable).norm.aemeasurable
-  -- Show `2 * bound` `HasFiniteIntegral`
-  · rw [hasFiniteIntegral_iff_ofReal] at bound_hasFiniteIntegral
-    · calc
-        ∫⁻ a, b a ∂μ = 2 * ∫⁻ a, ENNReal.ofReal (bound a) ∂μ := by
-          rw [lintegral_const_mul']
-          exact coe_ne_top
-        _ ≠ ∞ := mul_ne_top coe_ne_top bound_hasFiniteIntegral.ne
-    filter_upwards [h_bound 0] with _ h using le_trans (norm_nonneg _) h
-  -- Show `‖f a - F n a‖ --> 0`
-  · exact h
-
-end DominatedConvergence
-
-section PosPart
-
-/-! Lemmas used for defining the positive part of an `L¹` function -/
-
-
-theorem HasFiniteIntegral.max_zero {f : α → ℝ} (hf : HasFiniteIntegral f μ) :
-    HasFiniteIntegral (fun a => max (f a) 0) μ :=
-  hf.mono <| Eventually.of_forall fun x => by simp [abs_le, le_abs_self]
-
-theorem HasFiniteIntegral.min_zero {f : α → ℝ} (hf : HasFiniteIntegral f μ) :
-    HasFiniteIntegral (fun a => min (f a) 0) μ :=
-  hf.mono <| Eventually.of_forall fun x => by simpa [abs_le] using neg_abs_le _
-
-end PosPart
-
-section NormedSpace
-
-variable {𝕜 : Type*}
-
-theorem HasFiniteIntegral.smul [NormedAddCommGroup 𝕜] [SMulZeroClass 𝕜 β] [BoundedSMul 𝕜 β] (c : 𝕜)
-    {f : α → β} : HasFiniteIntegral f μ → HasFiniteIntegral (c • f) μ := by
-  simp only [HasFiniteIntegral]; intro hfi
-  calc
-    (∫⁻ a : α, ‖c • f a‖₊ ∂μ) ≤ ∫⁻ a : α, ‖c‖₊ * ‖f a‖₊ ∂μ := by
-      refine lintegral_mono ?_
-      intro i
-      -- After https://github.com/leanprover/lean4/pull/2734, we need to do beta reduction `exact mod_cast`
-      beta_reduce
-      exact mod_cast (nnnorm_smul_le c (f i))
-    _ < ∞ := by
-      rw [lintegral_const_mul']
-      exacts [mul_lt_top coe_lt_top hfi, coe_ne_top]
-
-theorem hasFiniteIntegral_smul_iff [NormedRing 𝕜] [MulActionWithZero 𝕜 β] [BoundedSMul 𝕜 β] {c : 𝕜}
-    (hc : IsUnit c) (f : α → β) : HasFiniteIntegral (c • f) μ ↔ HasFiniteIntegral f μ := by
-  obtain ⟨c, rfl⟩ := hc
-  constructor
-  · intro h
-    simpa only [smul_smul, Units.inv_mul, one_smul] using h.smul ((c⁻¹ : 𝕜ˣ) : 𝕜)
-  exact HasFiniteIntegral.smul _
-
-theorem HasFiniteIntegral.const_mul [NormedRing 𝕜] {f : α → 𝕜} (h : HasFiniteIntegral f μ) (c : 𝕜) :
-    HasFiniteIntegral (fun x => c * f x) μ :=
-  h.smul c
-
-theorem HasFiniteIntegral.mul_const [NormedRing 𝕜] {f : α → 𝕜} (h : HasFiniteIntegral f μ) (c : 𝕜) :
-    HasFiniteIntegral (fun x => f x * c) μ :=
-  h.smul (MulOpposite.op c)
-
-end NormedSpace
-
 /-! ### The predicate `Integrable` -/
 
 variable [TopologicalSpace ε]
@@ -418,7 +59,7 @@ def Integrable {α} {_ : MeasurableSpace α} (f : α → ε)
 scoped notation "Integrable[" mα "]" => @Integrable _ _ _ _ mα
 
 theorem memℒp_one_iff_integrable {f : α → β} : Memℒp f 1 μ ↔ Integrable f μ := by
-  simp_rw [Integrable, hasFiniteIntegral_iff_nnnorm, Memℒp, eLpNorm_one_eq_lintegral_nnnorm]
+  simp_rw [Integrable, hasFiniteIntegral_iff_enorm, Memℒp, eLpNorm_one_eq_lintegral_enorm]
 
 theorem Integrable.aestronglyMeasurable {f : α → β} (hf : Integrable f μ) :
     AEStronglyMeasurable f μ :=
@@ -480,7 +121,6 @@ lemma Integrable.of_finite [Finite α] [MeasurableSingletonClass α] [IsFiniteMe
 @[deprecated Integrable.of_finite (since := "2024-10-05"), nolint deprecatedNoSince]
 lemma Integrable.of_isEmpty [IsEmpty α] {f : α → β} : Integrable f μ := .of_finite
 
-
 theorem Memℒp.integrable_norm_rpow {f : α → β} {p : ℝ≥0∞} (hf : Memℒp f p μ) (hp_ne_zero : p ≠ 0)
     (hp_ne_top : p ≠ ∞) : Integrable (fun x : α => ‖f x‖ ^ p.toReal) μ := by
   rw [← memℒp_one_iff_integrable]
@@ -537,6 +177,7 @@ theorem integrable_zero_measure {_ : MeasurableSpace α} {f : α → β} :
 
 theorem integrable_finset_sum_measure {ι} {m : MeasurableSpace α} {f : α → β} {μ : ι → Measure α}
     {s : Finset ι} : Integrable f (∑ i ∈ s, μ i) ↔ ∀ i ∈ s, Integrable f (μ i) := by
+  classical
   induction s using Finset.induction_on <;> simp [*]
 
 theorem Integrable.smul_measure {f : α → β} (h : Integrable f μ) {c : ℝ≥0∞} (hc : c ≠ ∞) :
@@ -567,6 +208,7 @@ theorem Integrable.to_average {f : α → β} (h : Integrable f μ) : Integrable
   · apply h.smul_measure
     simpa
 
+open scoped Classical in
 theorem integrable_average [IsFiniteMeasure μ] {f : α → β} :
     Integrable f ((μ univ)⁻¹ • μ) ↔ Integrable f μ :=
   (eq_or_ne μ 0).by_cases (fun h => by simp [h]) fun h =>
@@ -625,12 +267,8 @@ variable {α β μ}
 theorem Integrable.add' {f g : α → β} (hf : Integrable f μ) (hg : Integrable g μ) :
     HasFiniteIntegral (f + g) μ :=
   calc
-    (∫⁻ a, ‖f a + g a‖₊ ∂μ) ≤ ∫⁻ a, ‖f a‖₊ + ‖g a‖₊ ∂μ :=
-      lintegral_mono fun a => by
-        -- After https://github.com/leanprover/lean4/pull/2734, we need to do beta reduction before `exact mod_cast`
-        beta_reduce
-        exact mod_cast nnnorm_add_le _ _
-    _ = _ := lintegral_nnnorm_add_left hf.aestronglyMeasurable _
+    ∫⁻ a, ‖f a + g a‖ₑ ∂μ ≤ ∫⁻ a, ‖f a‖ₑ + ‖g a‖ₑ ∂μ := lintegral_mono fun _ ↦ enorm_add_le _ _
+    _ = _ := lintegral_enorm_add_left hf.aestronglyMeasurable _
     _ < ∞ := add_lt_top.2 ⟨hf.hasFiniteIntegral, hg.hasFiniteIntegral⟩
 
 @[fun_prop]
@@ -789,7 +427,7 @@ theorem Integrable.bdd_mul {F : Type*} [NormedDivisionRing F] {f g : α → F} (
 vector-valued function by a scalar function with finite essential supremum is integrable. -/
 theorem Integrable.essSup_smul {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 𝕜 β] {f : α → β}
     (hf : Integrable f μ) {g : α → 𝕜} (g_aestronglyMeasurable : AEStronglyMeasurable g μ)
-    (ess_sup_g : essSup (fun x => (‖g x‖₊ : ℝ≥0∞)) μ ≠ ∞) :
+    (ess_sup_g : essSup (‖g ·‖ₑ) μ ≠ ∞) :
     Integrable (fun x : α => g x • f x) μ := by
   rw [← memℒp_one_iff_integrable] at *
   refine ⟨g_aestronglyMeasurable.smul hf.1, ?_⟩
@@ -804,8 +442,7 @@ theorem Integrable.essSup_smul {𝕜 : Type*} [NormedField 𝕜] [NormedSpace 
 scalar-valued function by a vector-value function with finite essential supremum is integrable. -/
 theorem Integrable.smul_essSup {𝕜 : Type*} [NormedRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β]
     {f : α → 𝕜} (hf : Integrable f μ) {g : α → β}
-    (g_aestronglyMeasurable : AEStronglyMeasurable g μ)
-    (ess_sup_g : essSup (fun x => (‖g x‖₊ : ℝ≥0∞)) μ ≠ ∞) :
+    (g_aestronglyMeasurable : AEStronglyMeasurable g μ) (ess_sup_g : essSup (‖g ·‖ₑ) μ ≠ ∞) :
     Integrable (fun x : α => f x • g x) μ := by
   rw [← memℒp_one_iff_integrable] at *
   refine ⟨hf.1.smul g_aestronglyMeasurable, ?_⟩
@@ -864,8 +501,8 @@ theorem Memℒp.integrable {q : ℝ≥0∞} (hq1 : 1 ≤ q) {f : α → β} [IsF
 where `‖f x‖ ≥ ε` is finite for all positive `ε`. -/
 theorem Integrable.measure_norm_ge_lt_top {f : α → β} (hf : Integrable f μ) {ε : ℝ} (hε : 0 < ε) :
     μ { x | ε ≤ ‖f x‖ } < ∞ := by
-  rw [show { x | ε ≤ ‖f x‖ } = { x | ENNReal.ofReal ε ≤ ‖f x‖₊ } by
-      simp only [ENNReal.ofReal, Real.toNNReal_le_iff_le_coe, ENNReal.coe_le_coe, coe_nnnorm]]
+  rw [show { x | ε ≤ ‖f x‖ } = { x | .ofReal ε ≤ ‖f x‖ₑ } by
+      simp [ENNReal.ofReal, Real.toNNReal_le_iff_le_coe]]
   refine (meas_ge_le_mul_pow_eLpNorm μ one_ne_zero ENNReal.one_ne_top hf.1 ?_).trans_lt ?_
   · simpa only [Ne, ENNReal.ofReal_eq_zero, not_le] using hε
   apply ENNReal.mul_lt_top
@@ -944,12 +581,6 @@ section count
 
 variable [MeasurableSingletonClass α] {f : α → β}
 
-/-- A function has finite integral for the counting measure iff its norm is summable. -/
-lemma hasFiniteIntegral_count_iff :
-    HasFiniteIntegral f Measure.count ↔ Summable (‖f ·‖) := by
-  simp only [hasFiniteIntegral_iff_nnnorm, lintegral_count, lt_top_iff_ne_top,
-    ENNReal.tsum_coe_ne_top_iff_summable,  ← NNReal.summable_coe, coe_nnnorm]
-
 /-- A function is integrable for the counting measure iff its norm is summable. -/
 lemma integrable_count_iff :
     Integrable f Measure.count ↔ Summable (‖f ·‖) := by
@@ -985,16 +616,11 @@ theorem integrable_withDensity_iff_integrable_coe_smul {f : α → ℝ≥0} (hf
     {g : α → E} :
     Integrable g (μ.withDensity fun x => f x) ↔ Integrable (fun x => (f x : ℝ) • g x) μ := by
   by_cases H : AEStronglyMeasurable (fun x : α => (f x : ℝ) • g x) μ
-  · simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, hasFiniteIntegral_iff_nnnorm, H]
+  · simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, hasFiniteIntegral_iff_enorm, H,
+      true_and]
     rw [lintegral_withDensity_eq_lintegral_mul₀' hf.coe_nnreal_ennreal.aemeasurable]
-    · rw [iff_iff_eq]
-      congr
-      ext1 x
-      simp only [nnnorm_smul, NNReal.nnnorm_eq, coe_mul, Pi.mul_apply]
-    · rw [aemeasurable_withDensity_ennreal_iff hf]
-      convert H.ennnorm using 1
-      ext1 x
-      simp only [nnnorm_smul, NNReal.nnnorm_eq, coe_mul]
+    · simp [enorm_smul]
+    · simpa [aemeasurable_withDensity_ennreal_iff hf, enorm_smul] using H.enorm
   · simp only [Integrable, aestronglyMeasurable_withDensity_iff hf, H, false_and]
 
 theorem integrable_withDensity_iff_integrable_smul {f : α → ℝ≥0} (hf : Measurable f) {g : α → E} :
@@ -1093,9 +719,8 @@ noncomputable def withDensitySMulLI {f : α → ℝ≥0} (f_meas : Measurable f)
     congr 1
     apply lintegral_congr_ae
     filter_upwards [(memℒ1_smul_of_L1_withDensity f_meas u).coeFn_toLp] with x hx
-    rw [hx, Pi.mul_apply]
-    change (‖(f x : ℝ) • u x‖₊ : ℝ≥0∞) = (f x : ℝ≥0∞) * (‖u x‖₊ : ℝ≥0∞)
-    simp only [nnnorm_smul, NNReal.nnnorm_eq, ENNReal.coe_mul]
+    rw [hx]
+    simp [NNReal.smul_def, enorm_smul]
 
 @[simp]
 theorem withDensitySMulLI_apply {f : α → ℝ≥0} (f_meas : Measurable f)
@@ -1110,7 +735,7 @@ section ENNReal
 
 theorem mem_ℒ1_toReal_of_lintegral_ne_top {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ)
     (hfi : ∫⁻ x, f x ∂μ ≠ ∞) : Memℒp (fun x ↦ (f x).toReal) 1 μ := by
-  rw [Memℒp, eLpNorm_one_eq_lintegral_nnnorm]
+  rw [Memℒp, eLpNorm_one_eq_lintegral_enorm]
   exact ⟨(AEMeasurable.ennreal_toReal hfm).aestronglyMeasurable,
     hasFiniteIntegral_toReal_of_lintegral_ne_top hfi⟩
 
@@ -1193,8 +818,8 @@ variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
 theorem integrable_smul_const {f : α → 𝕜} {c : E} (hc : c ≠ 0) :
     Integrable (fun x => f x • c) μ ↔ Integrable f μ := by
   simp_rw [Integrable, aestronglyMeasurable_smul_const_iff (f := f) hc, and_congr_right_iff,
-    hasFiniteIntegral_iff_nnnorm, nnnorm_smul, ENNReal.coe_mul]
-  intro _; rw [lintegral_mul_const' _ _ ENNReal.coe_ne_top, ENNReal.mul_lt_top_iff]
+    hasFiniteIntegral_iff_enorm, enorm_smul]
+  intro _; rw [lintegral_mul_const' _ _ enorm_ne_top, ENNReal.mul_lt_top_iff]
   have : ∀ x : ℝ≥0∞, x = 0 → x < ∞ := by simp
   simp [hc, or_iff_left_of_imp (this _)]
 
@@ -1295,15 +920,13 @@ theorem Integrable.trim (hm : m ≤ m0) (hf_int : Integrable f μ') (hf : Strong
   refine ⟨hf.aestronglyMeasurable, ?_⟩
   rw [HasFiniteIntegral, lintegral_trim hm _]
   · exact hf_int.2
-  · exact @StronglyMeasurable.ennnorm _ m _ _ f hf
+  · exact @StronglyMeasurable.enorm _ m _ _ f hf
 
 theorem integrable_of_integrable_trim (hm : m ≤ m0) (hf_int : Integrable f (μ'.trim hm)) :
     Integrable f μ' := by
   obtain ⟨hf_meas_ae, hf⟩ := hf_int
   refine ⟨aestronglyMeasurable_of_aestronglyMeasurable_trim hm hf_meas_ae, ?_⟩
-  rw [HasFiniteIntegral] at hf ⊢
-  rwa [lintegral_trim_ae hm _] at hf
-  exact AEStronglyMeasurable.ennnorm hf_meas_ae
+  simpa [HasFiniteIntegral, lintegral_trim_ae hm hf_meas_ae.enorm] using hf
 
 end Trim
 
@@ -1313,228 +936,22 @@ variable {E : Type*} {m0 : MeasurableSpace α} [NormedAddCommGroup E]
 
 theorem integrable_of_forall_fin_meas_le' {μ : Measure α} (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
     (C : ℝ≥0∞) (hC : C < ∞) {f : α → E} (hf_meas : AEStronglyMeasurable f μ)
-    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → (∫⁻ x in s, ‖f x‖₊ ∂μ) ≤ C) : Integrable f μ :=
+    (hf : ∀ s, MeasurableSet[m] s → μ s ≠ ∞ → ∫⁻ x in s, ‖f x‖ₑ ∂μ ≤ C) : Integrable f μ :=
   ⟨hf_meas, (lintegral_le_of_forall_fin_meas_trim_le hm C hf).trans_lt hC⟩
 
 theorem integrable_of_forall_fin_meas_le [SigmaFinite μ] (C : ℝ≥0∞) (hC : C < ∞) {f : α → E}
     (hf_meas : AEStronglyMeasurable[m] f μ)
-    (hf : ∀ s : Set α, MeasurableSet[m] s → μ s ≠ ∞ → (∫⁻ x in s, ‖f x‖₊ ∂μ) ≤ C) :
+    (hf : ∀ s : Set α, MeasurableSet[m] s → μ s ≠ ∞ → ∫⁻ x in s, ‖f x‖ₑ ∂μ ≤ C) :
     Integrable f μ :=
   have : SigmaFinite (μ.trim le_rfl) := by rwa [@trim_eq_self _ m]
   integrable_of_forall_fin_meas_le' le_rfl C hC hf_meas hf
 
 end SigmaFinite
 
-/-! ### The predicate `Integrable` on measurable functions modulo a.e.-equality -/
-
-
-namespace AEEqFun
-
-section
-
-/-- A class of almost everywhere equal functions is `Integrable` if its function representative
-is integrable. -/
-def Integrable (f : α →ₘ[μ] β) : Prop :=
-  MeasureTheory.Integrable f μ
-
-theorem integrable_mk {f : α → β} (hf : AEStronglyMeasurable f μ) :
-    Integrable (mk f hf : α →ₘ[μ] β) ↔ MeasureTheory.Integrable f μ := by
-  simp only [Integrable]
-  apply integrable_congr
-  exact coeFn_mk f hf
-
-theorem integrable_coeFn {f : α →ₘ[μ] β} : MeasureTheory.Integrable f μ ↔ Integrable f := by
-  rw [← integrable_mk, mk_coeFn]
-
-theorem integrable_zero : Integrable (0 : α →ₘ[μ] β) :=
-  (MeasureTheory.integrable_zero α β μ).congr (coeFn_mk _ _).symm
-
-end
-
-section
-
-theorem Integrable.neg {f : α →ₘ[μ] β} : Integrable f → Integrable (-f) :=
-  induction_on f fun _f hfm hfi => (integrable_mk _).2 ((integrable_mk hfm).1 hfi).neg
-
-section
-
-theorem integrable_iff_mem_L1 {f : α →ₘ[μ] β} : Integrable f ↔ f ∈ (α →₁[μ] β) := by
-  rw [← integrable_coeFn, ← memℒp_one_iff_integrable, Lp.mem_Lp_iff_memℒp]
-
-theorem Integrable.add {f g : α →ₘ[μ] β} : Integrable f → Integrable g → Integrable (f + g) := by
-  refine induction_on₂ f g fun f hf g hg hfi hgi => ?_
-  simp only [integrable_mk, mk_add_mk] at hfi hgi ⊢
-  exact hfi.add hgi
-
-theorem Integrable.sub {f g : α →ₘ[μ] β} (hf : Integrable f) (hg : Integrable g) :
-    Integrable (f - g) :=
-  (sub_eq_add_neg f g).symm ▸ hf.add hg.neg
-
-end
-
-section BoundedSMul
-
-variable {𝕜 : Type*} [NormedRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β]
-
-theorem Integrable.smul {c : 𝕜} {f : α →ₘ[μ] β} : Integrable f → Integrable (c • f) :=
-  induction_on f fun _f hfm hfi => (integrable_mk _).2 <|
-    by simpa using ((integrable_mk hfm).1 hfi).smul c
-
-end BoundedSMul
-
-end
-
-end AEEqFun
-
-namespace L1
-
-
-theorem integrable_coeFn (f : α →₁[μ] β) : Integrable f μ := by
-  rw [← memℒp_one_iff_integrable]
-  exact Lp.memℒp f
-
-theorem hasFiniteIntegral_coeFn (f : α →₁[μ] β) : HasFiniteIntegral f μ :=
-  (integrable_coeFn f).hasFiniteIntegral
-
-theorem stronglyMeasurable_coeFn (f : α →₁[μ] β) : StronglyMeasurable f :=
-  Lp.stronglyMeasurable f
-
-theorem measurable_coeFn [MeasurableSpace β] [BorelSpace β] (f : α →₁[μ] β) : Measurable f :=
-  (Lp.stronglyMeasurable f).measurable
-
-theorem aestronglyMeasurable_coeFn (f : α →₁[μ] β) : AEStronglyMeasurable f μ :=
-  Lp.aestronglyMeasurable f
-
-theorem aemeasurable_coeFn [MeasurableSpace β] [BorelSpace β] (f : α →₁[μ] β) : AEMeasurable f μ :=
-  (Lp.stronglyMeasurable f).measurable.aemeasurable
-
-theorem edist_def (f g : α →₁[μ] β) : edist f g = ∫⁻ a, edist (f a) (g a) ∂μ := by
-  simp only [Lp.edist_def, eLpNorm, one_ne_zero, eLpNorm'_eq_lintegral_nnnorm, Pi.sub_apply,
-    one_toReal, ENNReal.rpow_one, ne_eq, not_false_eq_true, div_self, ite_false]
-  simp [edist_eq_coe_nnnorm_sub]
-
-theorem dist_def (f g : α →₁[μ] β) : dist f g = (∫⁻ a, edist (f a) (g a) ∂μ).toReal := by
-  simp_rw [dist_edist, edist_def]
-
-theorem norm_def (f : α →₁[μ] β) : ‖f‖ = (∫⁻ a, ‖f a‖₊ ∂μ).toReal := by
-  simp [Lp.norm_def, eLpNorm, eLpNorm'_eq_lintegral_nnnorm]
-
-/-- Computing the norm of a difference between two L¹-functions. Note that this is not a
-  special case of `norm_def` since `(f - g) x` and `f x - g x` are not equal
-  (but only a.e.-equal). -/
-theorem norm_sub_eq_lintegral (f g : α →₁[μ] β) :
-    ‖f - g‖ = (∫⁻ x, (‖f x - g x‖₊ : ℝ≥0∞) ∂μ).toReal := by
-  rw [norm_def]
-  congr 1
-  rw [lintegral_congr_ae]
-  filter_upwards [Lp.coeFn_sub f g] with _ ha
-  simp only [ha, Pi.sub_apply]
-
-theorem ofReal_norm_eq_lintegral (f : α →₁[μ] β) :
-    ENNReal.ofReal ‖f‖ = ∫⁻ x, (‖f x‖₊ : ℝ≥0∞) ∂μ := by
-  rw [norm_def, ENNReal.ofReal_toReal]
-  exact ne_of_lt (hasFiniteIntegral_coeFn f)
-
-/-- Computing the norm of a difference between two L¹-functions. Note that this is not a
-  special case of `ofReal_norm_eq_lintegral` since `(f - g) x` and `f x - g x` are not equal
-  (but only a.e.-equal). -/
-theorem ofReal_norm_sub_eq_lintegral (f g : α →₁[μ] β) :
-    ENNReal.ofReal ‖f - g‖ = ∫⁻ x, (‖f x - g x‖₊ : ℝ≥0∞) ∂μ := by
-  simp_rw [ofReal_norm_eq_lintegral, ← edist_eq_coe_nnnorm]
-  apply lintegral_congr_ae
-  filter_upwards [Lp.coeFn_sub f g] with _ ha
-  simp only [ha, Pi.sub_apply]
-
-end L1
-
-namespace Integrable
-
-
-/-- Construct the equivalence class `[f]` of an integrable function `f`, as a member of the
-space `L1 β 1 μ`. -/
-def toL1 (f : α → β) (hf : Integrable f μ) : α →₁[μ] β :=
-  (memℒp_one_iff_integrable.2 hf).toLp f
-
-@[simp]
-theorem toL1_coeFn (f : α →₁[μ] β) (hf : Integrable f μ) : hf.toL1 f = f := by
-  simp [Integrable.toL1]
-
-theorem coeFn_toL1 {f : α → β} (hf : Integrable f μ) : hf.toL1 f =ᵐ[μ] f :=
-  AEEqFun.coeFn_mk _ _
-
-@[simp]
-theorem toL1_zero (h : Integrable (0 : α → β) μ) : h.toL1 0 = 0 :=
-  rfl
-
-@[simp]
-theorem toL1_eq_mk (f : α → β) (hf : Integrable f μ) :
-    (hf.toL1 f : α →ₘ[μ] β) = AEEqFun.mk f hf.aestronglyMeasurable :=
-  rfl
-
-@[simp]
-theorem toL1_eq_toL1_iff (f g : α → β) (hf : Integrable f μ) (hg : Integrable g μ) :
-    toL1 f hf = toL1 g hg ↔ f =ᵐ[μ] g :=
-  Memℒp.toLp_eq_toLp_iff _ _
-
-theorem toL1_add (f g : α → β) (hf : Integrable f μ) (hg : Integrable g μ) :
-    toL1 (f + g) (hf.add hg) = toL1 f hf + toL1 g hg :=
-  rfl
-
-theorem toL1_neg (f : α → β) (hf : Integrable f μ) : toL1 (-f) (Integrable.neg hf) = -toL1 f hf :=
-  rfl
-
-theorem toL1_sub (f g : α → β) (hf : Integrable f μ) (hg : Integrable g μ) :
-    toL1 (f - g) (hf.sub hg) = toL1 f hf - toL1 g hg :=
-  rfl
-
-theorem norm_toL1 (f : α → β) (hf : Integrable f μ) :
-    ‖hf.toL1 f‖ = ENNReal.toReal (∫⁻ a, edist (f a) 0 ∂μ) := by
-  simp only [toL1, Lp.norm_toLp, eLpNorm, one_ne_zero, eLpNorm'_eq_lintegral_nnnorm, one_toReal,
-    ENNReal.rpow_one, ne_eq, not_false_eq_true, div_self, ite_false]
-  simp [edist_eq_coe_nnnorm]
-
-theorem nnnorm_toL1 {f : α → β} (hf : Integrable f μ) :
-    (‖hf.toL1 f‖₊ : ℝ≥0∞) = ∫⁻ a, ‖f a‖₊ ∂μ := by
-  simpa [Integrable.toL1, eLpNorm, eLpNorm'] using ENNReal.coe_toNNReal hf.2.ne
-
-theorem norm_toL1_eq_lintegral_norm (f : α → β) (hf : Integrable f μ) :
-    ‖hf.toL1 f‖ = ENNReal.toReal (∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) := by
-  rw [norm_toL1, lintegral_norm_eq_lintegral_edist]
-
-@[simp]
-theorem edist_toL1_toL1 (f g : α → β) (hf : Integrable f μ) (hg : Integrable g μ) :
-    edist (hf.toL1 f) (hg.toL1 g) = ∫⁻ a, edist (f a) (g a) ∂μ := by
-  simp only [toL1, Lp.edist_toLp_toLp, eLpNorm, one_ne_zero, eLpNorm'_eq_lintegral_nnnorm,
-    Pi.sub_apply, one_toReal, ENNReal.rpow_one, ne_eq, not_false_eq_true, div_self, ite_false]
-  simp [edist_eq_coe_nnnorm_sub]
-
-theorem edist_toL1_zero (f : α → β) (hf : Integrable f μ) :
-    edist (hf.toL1 f) 0 = ∫⁻ a, edist (f a) 0 ∂μ := by
-  simp only [edist_zero_right, Lp.nnnorm_coe_ennreal, toL1_eq_mk, eLpNorm_aeeqFun]
-  apply eLpNorm_one_eq_lintegral_nnnorm
-
-variable {𝕜 : Type*} [NormedRing 𝕜] [Module 𝕜 β] [BoundedSMul 𝕜 β]
-
-theorem toL1_smul (f : α → β) (hf : Integrable f μ) (k : 𝕜) :
-    toL1 (fun a => k • f a) (hf.smul k) = k • toL1 f hf :=
-  rfl
-
-theorem toL1_smul' (f : α → β) (hf : Integrable f μ) (k : 𝕜) :
-    toL1 (k • f) (hf.smul k) = k • toL1 f hf :=
-  rfl
-
-end Integrable
-
 section restrict
 
 variable {E : Type*} [NormedAddCommGroup E] {f : α → E}
 
-lemma HasFiniteIntegral.restrict (h : HasFiniteIntegral f μ) {s : Set α} :
-    HasFiniteIntegral f (μ.restrict s) := by
-  refine lt_of_le_of_lt ?_ h
-  convert lintegral_mono_set (μ := μ) (s := s) (t := univ) (f := fun x ↦ ↑‖f x‖₊) (subset_univ s)
-  exact Measure.restrict_univ.symm
-
 /-- One should usually use `MeasureTheory.Integrable.IntegrableOn` instead. -/
 lemma Integrable.restrict (hf : Integrable f μ) {s : Set α} : Integrable f (μ.restrict s) :=
   hf.mono_measure Measure.restrict_le_self
@@ -1592,5 +1009,3 @@ lemma integrable_prod {f : α → E × F} :
   ⟨fun h ↦ ⟨h.fst, h.snd⟩, fun h ↦ h.1.prod_mk h.2⟩
 
 end MeasureTheory
-
-set_option linter.style.longFile 1700
diff --git a/Mathlib/MeasureTheory/Function/L2Space.lean b/Mathlib/MeasureTheory/Function/L2Space.lean
index 3594c9e9b9d387..878c21c3d7fa32 100644
--- a/Mathlib/MeasureTheory/Function/L2Space.lean
+++ b/Mathlib/MeasureTheory/Function/L2Space.lean
@@ -156,7 +156,7 @@ theorem integral_inner_eq_sq_eLpNorm (f : α →₂[μ] E) :
   ext1 x
   have h_two : (2 : ℝ) = ((2 : ℕ) : ℝ) := by simp
   rw [← Real.rpow_natCast _ 2, ← h_two, ←
-    ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) zero_le_two, ofReal_norm_eq_coe_nnnorm]
+    ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) zero_le_two, ofReal_norm_eq_enorm]
   norm_cast
 
 @[deprecated (since := "2024-07-27")]
@@ -169,7 +169,7 @@ private theorem norm_sq_eq_inner' (f : α →₂[μ] E) : ‖f‖ ^ 2 = RCLike.r
   · rw [← ENNReal.rpow_natCast, eLpNorm_eq_eLpNorm' two_ne_zero ENNReal.two_ne_top, eLpNorm', ←
       ENNReal.rpow_mul, one_div, h_two]
     simp [enorm_eq_nnnorm]
-  · refine (lintegral_rpow_nnnorm_lt_top_of_eLpNorm'_lt_top zero_lt_two ?_).ne
+  · refine (lintegral_rpow_enorm_lt_top_of_eLpNorm'_lt_top zero_lt_two ?_).ne
     rw [← h_two, ← eLpNorm_eq_eLpNorm' two_ne_zero ENNReal.two_ne_top]
     exact Lp.eLpNorm_lt_top f
 
diff --git a/Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean b/Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
index a2019826e26f5a..34188e7455aaaf 100644
--- a/Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
+++ b/Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
@@ -70,18 +70,22 @@ this quantity is finite -/
 def eLpNorm' {_ : MeasurableSpace α} (f : α → ε) (q : ℝ) (μ : Measure α) : ℝ≥0∞ :=
   (∫⁻ a, ‖f a‖ₑ ^ q ∂μ) ^ (1 / q)
 
-lemma eLpNorm'_eq_lintegral_nnnorm {_ : MeasurableSpace α} (f : α → F) (q : ℝ) (μ : Measure α) :
-    eLpNorm' f q μ = (∫⁻ a, ‖f a‖₊ ^ q ∂μ) ^ (1 / q) :=
+lemma eLpNorm'_eq_lintegral_enorm {_ : MeasurableSpace α} (f : α → F) (q : ℝ) (μ : Measure α) :
+    eLpNorm' f q μ = (∫⁻ a, ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) :=
   rfl
 
+@[deprecated (since := "2025-01-17")]
+alias eLpNorm'_eq_lintegral_nnnorm := eLpNorm'_eq_lintegral_enorm
+
 /-- seminorm for `ℒ∞`, equal to the essential supremum of `‖f‖`. -/
 def eLpNormEssSup {_ : MeasurableSpace α} (f : α → ε) (μ : Measure α) :=
   essSup (fun x => ‖f x‖ₑ) μ
 
-lemma eLpNormEssSup_eq_essSup_nnnorm {_ : MeasurableSpace α} (f : α → F) (μ : Measure α) :
-    eLpNormEssSup f μ = essSup (fun x => (‖f x‖₊ : ℝ≥0∞)) μ :=
-  rfl
+lemma eLpNormEssSup_eq_essSup_enorm {_ : MeasurableSpace α} (f : α → F) (μ : Measure α) :
+    eLpNormEssSup f μ = essSup (‖f ·‖ₑ) μ := rfl
 
+@[deprecated (since := "2025-01-17")]
+alias eLpNormEssSup_eq_essSup_nnnorm := eLpNormEssSup_eq_essSup_enorm
 
 /-- `ℒp` seminorm, equal to `0` for `p=0`, to `(∫ ‖f a‖^p ∂μ) ^ (1/p)` for `0 < p < ∞` and to
 `essSup ‖f‖ μ` for `p = ∞`. -/
@@ -102,25 +106,31 @@ lemma eLpNorm_nnreal_eq_eLpNorm' {f : α → F} {p : ℝ≥0} (hp : p ≠ 0) :
 
 @[deprecated (since := "2024-07-27")] alias snorm_nnreal_eq_snorm' := eLpNorm_nnreal_eq_eLpNorm'
 
-theorem eLpNorm_eq_lintegral_rpow_nnnorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
-    eLpNorm f p μ = (∫⁻ x, (‖f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) ^ (1 / p.toReal) := by
-  rw [eLpNorm_eq_eLpNorm' hp_ne_zero hp_ne_top, eLpNorm'_eq_lintegral_nnnorm]
+theorem eLpNorm_eq_lintegral_rpow_enorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {f : α → F} :
+    eLpNorm f p μ = (∫⁻ x, ‖f x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal) := by
+  rw [eLpNorm_eq_eLpNorm' hp_ne_zero hp_ne_top, eLpNorm'_eq_lintegral_enorm]
 
 @[deprecated (since := "2024-07-27")]
-alias snorm_eq_lintegral_rpow_nnnorm := eLpNorm_eq_lintegral_rpow_nnnorm
+alias snorm_eq_lintegral_rpow_nnnorm := eLpNorm_eq_lintegral_rpow_enorm
+
+@[deprecated (since := "2025-01-17")]
+alias eLpNorm_eq_lintegral_rpow_nnnorm := eLpNorm_eq_lintegral_rpow_enorm
 
 lemma eLpNorm_nnreal_eq_lintegral {f : α → F} {p : ℝ≥0} (hp : p ≠ 0) :
-    eLpNorm f p μ = (∫⁻ x, ‖f x‖₊ ^ (p : ℝ) ∂μ) ^ (1 / (p : ℝ)) :=
+    eLpNorm f p μ = (∫⁻ x, ‖f x‖ₑ ^ (p : ℝ) ∂μ) ^ (1 / (p : ℝ)) :=
   eLpNorm_nnreal_eq_eLpNorm' hp
 
 @[deprecated (since := "2024-07-27")] alias snorm_nnreal_eq_lintegral := eLpNorm_nnreal_eq_lintegral
 
-theorem eLpNorm_one_eq_lintegral_nnnorm {f : α → F} : eLpNorm f 1 μ = ∫⁻ x, ‖f x‖₊ ∂μ := by
-  simp_rw [eLpNorm_eq_lintegral_rpow_nnnorm one_ne_zero ENNReal.coe_ne_top, ENNReal.one_toReal,
+theorem eLpNorm_one_eq_lintegral_enorm {f : α → F} : eLpNorm f 1 μ = ∫⁻ x, ‖f x‖ₑ ∂μ := by
+  simp_rw [eLpNorm_eq_lintegral_rpow_enorm one_ne_zero ENNReal.coe_ne_top, ENNReal.one_toReal,
     one_div_one, ENNReal.rpow_one]
 
+@[deprecated (since := "2025-01-17")]
+alias eLpNorm_one_eq_lintegral_nnnorm := eLpNorm_one_eq_lintegral_enorm
+
 @[deprecated (since := "2024-07-27")]
-alias snorm_one_eq_lintegral_nnnorm := eLpNorm_one_eq_lintegral_nnnorm
+alias snorm_one_eq_lintegral_nnnorm := eLpNorm_one_eq_lintegral_enorm
 
 @[simp]
 theorem eLpNorm_exponent_top {f : α → F} : eLpNorm f ∞ μ = eLpNormEssSup f μ := by simp [eLpNorm]
@@ -138,18 +148,21 @@ theorem Memℒp.aestronglyMeasurable {f : α → E} {p : ℝ≥0∞} (h : Memℒ
     AEStronglyMeasurable f μ :=
   h.1
 
-theorem lintegral_rpow_nnnorm_eq_rpow_eLpNorm' {f : α → F} (hq0_lt : 0 < q) :
-    ∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ = eLpNorm' f q μ ^ q := by
-  rw [eLpNorm'_eq_lintegral_nnnorm, ← ENNReal.rpow_mul, one_div, inv_mul_cancel₀, ENNReal.rpow_one]
-  exact (ne_of_lt hq0_lt).symm
+theorem lintegral_rpow_enorm_eq_rpow_eLpNorm' {f : α → F} (hq0_lt : 0 < q) :
+    ∫⁻ a, ‖f a‖ₑ ^ q ∂μ = eLpNorm' f q μ ^ q := by
+  rw [eLpNorm'_eq_lintegral_enorm, ← ENNReal.rpow_mul, one_div, inv_mul_cancel₀, ENNReal.rpow_one]
+  exact hq0_lt.ne'
+
+@[deprecated (since := "2025-01-17")]
+alias lintegral_rpow_nnnorm_eq_rpow_eLpNorm' := lintegral_rpow_enorm_eq_rpow_eLpNorm'
 
 @[deprecated (since := "2024-07-27")]
-alias lintegral_rpow_nnnorm_eq_rpow_snorm' := lintegral_rpow_nnnorm_eq_rpow_eLpNorm'
+alias lintegral_rpow_nnnorm_eq_rpow_snorm' := lintegral_rpow_enorm_eq_rpow_eLpNorm'
 
 lemma eLpNorm_nnreal_pow_eq_lintegral {f : α → F} {p : ℝ≥0} (hp : p ≠ 0) :
-    eLpNorm f p μ ^ (p : ℝ) = ∫⁻ x, ‖f x‖₊ ^ (p : ℝ) ∂μ := by
+    eLpNorm f p μ ^ (p : ℝ) = ∫⁻ x, ‖f x‖ₑ ^ (p : ℝ) ∂μ := by
   simp [eLpNorm_eq_eLpNorm' (by exact_mod_cast hp) ENNReal.coe_ne_top,
-    lintegral_rpow_nnnorm_eq_rpow_eLpNorm' ((NNReal.coe_pos.trans pos_iff_ne_zero).mpr hp)]
+    lintegral_rpow_enorm_eq_rpow_eLpNorm' ((NNReal.coe_pos.trans pos_iff_ne_zero).mpr hp)]
 
 @[deprecated (since := "2024-07-27")]
 alias snorm_nnreal_pow_eq_lintegral := eLpNorm_nnreal_pow_eq_lintegral
@@ -170,31 +183,39 @@ theorem Memℒp.eLpNorm_ne_top {f : α → E} (hfp : Memℒp f p μ) : eLpNorm f
 @[deprecated (since := "2024-07-27")]
 alias Memℒp.snorm_ne_top := Memℒp.eLpNorm_ne_top
 
-theorem lintegral_rpow_nnnorm_lt_top_of_eLpNorm'_lt_top {f : α → F} (hq0_lt : 0 < q)
-    (hfq : eLpNorm' f q μ < ∞) : (∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) < ∞ := by
-  rw [lintegral_rpow_nnnorm_eq_rpow_eLpNorm' hq0_lt]
+theorem lintegral_rpow_enorm_lt_top_of_eLpNorm'_lt_top {f : α → F} (hq0_lt : 0 < q)
+    (hfq : eLpNorm' f q μ < ∞) : ∫⁻ a, ‖f a‖ₑ ^ q ∂μ < ∞ := by
+  rw [lintegral_rpow_enorm_eq_rpow_eLpNorm' hq0_lt]
   exact ENNReal.rpow_lt_top_of_nonneg (le_of_lt hq0_lt) (ne_of_lt hfq)
 
 @[deprecated (since := "2024-07-27")]
 alias lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top :=
-  lintegral_rpow_nnnorm_lt_top_of_eLpNorm'_lt_top
+  lintegral_rpow_enorm_lt_top_of_eLpNorm'_lt_top
+
+@[deprecated (since := "2025-01-17")]
+alias lintegral_rpow_nnnorm_lt_top_of_eLpNorm'_lt_top' :=
+  lintegral_rpow_enorm_lt_top_of_eLpNorm'_lt_top
 
-theorem lintegral_rpow_nnnorm_lt_top_of_eLpNorm_lt_top {f : α → F} (hp_ne_zero : p ≠ 0)
-    (hp_ne_top : p ≠ ∞) (hfp : eLpNorm f p μ < ∞) : (∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) < ∞ := by
-  apply lintegral_rpow_nnnorm_lt_top_of_eLpNorm'_lt_top
+theorem lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top {f : α → F} (hp_ne_zero : p ≠ 0)
+    (hp_ne_top : p ≠ ∞) (hfp : eLpNorm f p μ < ∞) : ∫⁻ a, ‖f a‖ₑ ^ p.toReal ∂μ < ∞ := by
+  apply lintegral_rpow_enorm_lt_top_of_eLpNorm'_lt_top
   · exact ENNReal.toReal_pos hp_ne_zero hp_ne_top
   · simpa [eLpNorm_eq_eLpNorm' hp_ne_zero hp_ne_top] using hfp
 
 @[deprecated (since := "2024-07-27")]
-alias lintegral_rpow_nnnorm_lt_top_of_snorm_lt_top := lintegral_rpow_nnnorm_lt_top_of_eLpNorm_lt_top
+alias lintegral_rpow_nnnorm_lt_top_of_snorm_lt_top := lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top
+
+@[deprecated (since := "2025-01-17")]
+alias lintegral_rpow_nnnorm_lt_top_of_eLpNorm_lt_top :=
+  lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top
 
 theorem eLpNorm_lt_top_iff_lintegral_rpow_nnnorm_lt_top {f : α → F} (hp_ne_zero : p ≠ 0)
-    (hp_ne_top : p ≠ ∞) : eLpNorm f p μ < ∞ ↔ (∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) < ∞ :=
-  ⟨lintegral_rpow_nnnorm_lt_top_of_eLpNorm_lt_top hp_ne_zero hp_ne_top, by
+    (hp_ne_top : p ≠ ∞) : eLpNorm f p μ < ∞ ↔ ∫⁻ a, (‖f a‖ₑ) ^ p.toReal ∂μ < ∞ :=
+  ⟨lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top hp_ne_zero hp_ne_top, by
     intro h
     have hp' := ENNReal.toReal_pos hp_ne_zero hp_ne_top
     have : 0 < 1 / p.toReal := div_pos zero_lt_one hp'
-    simpa [eLpNorm_eq_lintegral_rpow_nnnorm hp_ne_zero hp_ne_top] using
+    simpa [eLpNorm_eq_lintegral_rpow_enorm hp_ne_zero hp_ne_top] using
       ENNReal.rpow_lt_top_of_nonneg (le_of_lt this) (ne_of_lt h)⟩
 
 @[deprecated (since := "2024-07-27")]
@@ -224,7 +245,7 @@ theorem memℒp_zero_iff_aestronglyMeasurable {f : α → E} :
 
 @[simp]
 theorem eLpNorm'_zero (hp0_lt : 0 < q) : eLpNorm' (0 : α → F) q μ = 0 := by
-  simp [eLpNorm'_eq_lintegral_nnnorm, hp0_lt]
+  simp [eLpNorm'_eq_lintegral_enorm, hp0_lt]
 
 @[deprecated (since := "2024-07-27")]
 alias snorm'_zero := eLpNorm'_zero
@@ -233,16 +254,14 @@ alias snorm'_zero := eLpNorm'_zero
 theorem eLpNorm'_zero' (hq0_ne : q ≠ 0) (hμ : μ ≠ 0) : eLpNorm' (0 : α → F) q μ = 0 := by
   rcases le_or_lt 0 q with hq0 | hq_neg
   · exact eLpNorm'_zero (lt_of_le_of_ne hq0 hq0_ne.symm)
-  · simp [eLpNorm'_eq_lintegral_nnnorm, ENNReal.rpow_eq_zero_iff, hμ, hq_neg]
+  · simp [eLpNorm'_eq_lintegral_enorm, ENNReal.rpow_eq_zero_iff, hμ, hq_neg]
 
 @[deprecated (since := "2024-07-27")]
 alias snorm'_zero' := eLpNorm'_zero'
 
 @[simp]
 theorem eLpNormEssSup_zero : eLpNormEssSup (0 : α → F) μ = 0 := by
-  simp_rw [eLpNormEssSup_eq_essSup_nnnorm, Pi.zero_apply, nnnorm_zero, ENNReal.coe_zero,
-    ← ENNReal.bot_eq_zero]
-  exact essSup_const_bot
+  simp [eLpNormEssSup, ← bot_eq_zero', essSup_const_bot]
 
 @[deprecated (since := "2024-07-27")]
 alias snormEssSup_zero := eLpNormEssSup_zero
@@ -320,7 +339,7 @@ section Neg
 
 @[simp]
 theorem eLpNorm'_neg (f : α → F) (q : ℝ) (μ : Measure α) : eLpNorm' (-f) q μ = eLpNorm' f q μ := by
-  simp [eLpNorm'_eq_lintegral_nnnorm]
+  simp [eLpNorm'_eq_lintegral_enorm]
 
 @[deprecated (since := "2024-07-27")]
 alias snorm'_neg := eLpNorm'_neg
@@ -330,7 +349,7 @@ theorem eLpNorm_neg (f : α → F) (p : ℝ≥0∞) (μ : Measure α) : eLpNorm
   by_cases h0 : p = 0
   · simp [h0]
   by_cases h_top : p = ∞
-  · simp [h_top, eLpNormEssSup_eq_essSup_nnnorm]
+  · simp [h_top, eLpNormEssSup_eq_essSup_enorm]
   simp [eLpNorm_eq_eLpNorm' h0 h_top]
 
 lemma eLpNorm_sub_comm (f g : α → E) (p : ℝ≥0∞) (μ : Measure α) :
@@ -350,8 +369,8 @@ end Neg
 section Const
 
 theorem eLpNorm'_const (c : F) (hq_pos : 0 < q) :
-    eLpNorm' (fun _ : α => c) q μ = (‖c‖₊ : ℝ≥0∞) * μ Set.univ ^ (1 / q) := by
-  rw [eLpNorm'_eq_lintegral_nnnorm, lintegral_const,
+    eLpNorm' (fun _ : α => c) q μ = ‖c‖ₑ * μ Set.univ ^ (1 / q) := by
+  rw [eLpNorm'_eq_lintegral_enorm, lintegral_const,
     ENNReal.mul_rpow_of_nonneg _ _ (by simp [hq_pos.le] : 0 ≤ 1 / q)]
   congr
   rw [← ENNReal.rpow_mul]
@@ -362,37 +381,33 @@ theorem eLpNorm'_const (c : F) (hq_pos : 0 < q) :
 alias snorm'_const := eLpNorm'_const
 
 theorem eLpNorm'_const' [IsFiniteMeasure μ] (c : F) (hc_ne_zero : c ≠ 0) (hq_ne_zero : q ≠ 0) :
-    eLpNorm' (fun _ : α => c) q μ = (‖c‖₊ : ℝ≥0∞) * μ Set.univ ^ (1 / q) := by
-  rw [eLpNorm'_eq_lintegral_nnnorm, lintegral_const,
+    eLpNorm' (fun _ : α => c) q μ = ‖c‖ₑ * μ Set.univ ^ (1 / q) := by
+  rw [eLpNorm'_eq_lintegral_enorm, lintegral_const,
     ENNReal.mul_rpow_of_ne_top _ (measure_ne_top μ Set.univ)]
   · congr
     rw [← ENNReal.rpow_mul]
     suffices hp_cancel : q * (1 / q) = 1 by rw [hp_cancel, ENNReal.rpow_one]
     rw [one_div, mul_inv_cancel₀ hq_ne_zero]
   · rw [Ne, ENNReal.rpow_eq_top_iff, not_or, not_and_or, not_and_or]
-    constructor
-    · left
-      rwa [ENNReal.coe_eq_zero, nnnorm_eq_zero]
-    · exact Or.inl ENNReal.coe_ne_top
+    simp [hc_ne_zero]
 
 @[deprecated (since := "2024-07-27")]
 alias snorm'_const' := eLpNorm'_const'
 
-theorem eLpNormEssSup_const (c : F) (hμ : μ ≠ 0) :
-    eLpNormEssSup (fun _ : α => c) μ = (‖c‖₊ : ℝ≥0∞) := by
-  rw [eLpNormEssSup_eq_essSup_nnnorm, essSup_const _ hμ]
+theorem eLpNormEssSup_const (c : F) (hμ : μ ≠ 0) : eLpNormEssSup (fun _ : α => c) μ = ‖c‖ₑ := by
+  rw [eLpNormEssSup_eq_essSup_enorm, essSup_const _ hμ]
 
 @[deprecated (since := "2024-07-27")]
 alias snormEssSup_const := eLpNormEssSup_const
 
 theorem eLpNorm'_const_of_isProbabilityMeasure (c : F) (hq_pos : 0 < q) [IsProbabilityMeasure μ] :
-    eLpNorm' (fun _ : α => c) q μ = (‖c‖₊ : ℝ≥0∞) := by simp [eLpNorm'_const c hq_pos, measure_univ]
+    eLpNorm' (fun _ : α => c) q μ = ‖c‖ₑ := by simp [eLpNorm'_const c hq_pos, measure_univ]
 
 @[deprecated (since := "2024-07-27")]
 alias snorm'_const_of_isProbabilityMeasure := eLpNorm'_const_of_isProbabilityMeasure
 
 theorem eLpNorm_const (c : F) (h0 : p ≠ 0) (hμ : μ ≠ 0) :
-    eLpNorm (fun _ : α => c) p μ = (‖c‖₊ : ℝ≥0∞) * μ Set.univ ^ (1 / ENNReal.toReal p) := by
+    eLpNorm (fun _ : α => c) p μ = ‖c‖ₑ * μ Set.univ ^ (1 / ENNReal.toReal p) := by
   by_cases h_top : p = ∞
   · simp [h_top, eLpNormEssSup_const c hμ]
   simp [eLpNorm_eq_eLpNorm' h0 h_top, eLpNorm'_const, ENNReal.toReal_pos h0 h_top]
@@ -401,7 +416,7 @@ theorem eLpNorm_const (c : F) (h0 : p ≠ 0) (hμ : μ ≠ 0) :
 alias snorm_const := eLpNorm_const
 
 theorem eLpNorm_const' (c : F) (h0 : p ≠ 0) (h_top : p ≠ ∞) :
-    eLpNorm (fun _ : α => c) p μ = (‖c‖₊ : ℝ≥0∞) * μ Set.univ ^ (1 / ENNReal.toReal p) := by
+    eLpNorm (fun _ : α => c) p μ = ‖c‖ₑ * μ Set.univ ^ (1 / ENNReal.toReal p) := by
   simp [eLpNorm_eq_eLpNorm' h0 h_top, eLpNorm'_const, ENNReal.toReal_pos h0 h_top]
 
 @[deprecated (since := "2024-07-27")]
@@ -416,14 +431,11 @@ theorem eLpNorm_const_lt_top_iff {p : ℝ≥0∞} {c : F} (hp_ne_zero : p ≠ 0)
   by_cases hc : c = 0
   · simp only [hc, true_or, eq_self_iff_true, ENNReal.zero_lt_top, eLpNorm_zero']
   rw [eLpNorm_const' c hp_ne_zero hp_ne_top]
-  by_cases hμ_top : μ Set.univ = ∞
+  obtain hμ_top | hμ_top := eq_or_ne (μ .univ) ∞
   · simp [hc, hμ_top, hp]
   rw [ENNReal.mul_lt_top_iff]
-  simp only [true_and, one_div, ENNReal.rpow_eq_zero_iff, hμ, false_or, or_false,
-    ENNReal.coe_lt_top, nnnorm_eq_zero, ENNReal.coe_eq_zero,
-    MeasureTheory.Measure.measure_univ_eq_zero, hp, inv_lt_zero, hc, false_and,
-    inv_pos, or_self_iff, hμ_top, Ne.lt_top hμ_top, iff_true]
-  exact ENNReal.rpow_lt_top_of_nonneg (inv_nonneg.mpr hp.le) hμ_top
+  simpa [hμ, hc, hμ_top, hμ_top.lt_top] using
+    ENNReal.rpow_lt_top_of_nonneg (inv_nonneg.mpr hp.le) hμ_top
 
 @[deprecated (since := "2024-07-27")]
 alias snorm_const_lt_top_iff := eLpNorm_const_lt_top_iff
@@ -439,12 +451,8 @@ theorem memℒp_const (c : E) [IsFiniteMeasure μ] : Memℒp (fun _ : α => c) p
   refine ENNReal.rpow_lt_top_of_nonneg ?_ (measure_ne_top μ Set.univ)
   simp
 
-theorem memℒp_top_const (c : E) : Memℒp (fun _ : α => c) ∞ μ := by
-  refine ⟨aestronglyMeasurable_const, ?_⟩
-  by_cases h : μ = 0
-  · simp only [h, eLpNorm_measure_zero, ENNReal.zero_lt_top]
-  · rw [eLpNorm_const _ ENNReal.top_ne_zero h]
-    simp only [ENNReal.top_toReal, div_zero, ENNReal.rpow_zero, mul_one, ENNReal.coe_lt_top]
+theorem memℒp_top_const (c : E) : Memℒp (fun _ : α => c) ∞ μ :=
+  ⟨aestronglyMeasurable_const, by by_cases h : μ = 0 <;> simp [eLpNorm_const _, h]⟩
 
 theorem memℒp_const_iff {p : ℝ≥0∞} {c : E} (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) :
     Memℒp (fun _ : α => c) p μ ↔ c = 0 ∨ μ Set.univ < ∞ := by
@@ -457,9 +465,10 @@ variable {f : α → F}
 
 lemma eLpNorm'_mono_nnnorm_ae {f : α → F} {g : α → G} (hq : 0 ≤ q) (h : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ ‖g x‖₊) :
     eLpNorm' f q μ ≤ eLpNorm' g q μ := by
-  simp only [eLpNorm'_eq_lintegral_nnnorm]
+  simp only [eLpNorm'_eq_lintegral_enorm]
   gcongr ?_ ^ (1/q)
   refine lintegral_mono_ae (h.mono fun x hx => ?_)
+  dsimp [enorm]
   gcongr
 
 @[deprecated (since := "2024-07-27")]
@@ -474,9 +483,8 @@ alias snorm'_mono_ae := eLpNorm'_mono_ae
 
 theorem eLpNorm'_congr_nnnorm_ae {f g : α → F} (hfg : ∀ᵐ x ∂μ, ‖f x‖₊ = ‖g x‖₊) :
     eLpNorm' f q μ = eLpNorm' g q μ := by
-  have : (fun x => (‖f x‖₊ : ℝ≥0∞) ^ q) =ᵐ[μ] fun x => (‖g x‖₊ : ℝ≥0∞) ^ q :=
-    hfg.mono fun x hx => by simp_rw [hx]
-  simp only [eLpNorm'_eq_lintegral_nnnorm, lintegral_congr_ae this]
+  have : (‖f ·‖ₑ ^ q) =ᵐ[μ] (‖g ·‖ₑ ^ q) := hfg.mono fun x hx ↦ by simp [enorm, hx]
+  simp only [eLpNorm'_eq_lintegral_enorm, lintegral_congr_ae this]
 
 @[deprecated (since := "2024-07-27")]
 alias snorm'_congr_nnnorm_ae := eLpNorm'_congr_nnnorm_ae
@@ -591,7 +599,7 @@ theorem eLpNorm_le_of_ae_nnnorm_bound {f : α → F} {C : ℝ≥0} (hfC : ∀ᵐ
   · simp [hp]
   have : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ ‖(C : ℝ)‖₊ := hfC.mono fun x hx => hx.trans_eq C.nnnorm_eq.symm
   refine (eLpNorm_mono_ae this).trans_eq ?_
-  rw [eLpNorm_const _ hp (NeZero.ne μ), C.nnnorm_eq, one_div, ENNReal.smul_def, smul_eq_mul]
+  rw [eLpNorm_const _ hp (NeZero.ne μ), C.enorm_eq, one_div, ENNReal.smul_def, smul_eq_mul]
 
 @[deprecated (since := "2024-07-27")]
 alias snorm_le_of_ae_nnnorm_bound := eLpNorm_le_of_ae_nnnorm_bound
@@ -630,7 +638,7 @@ alias snorm_indicator_sub_indicator := eLpNorm_indicator_sub_indicator
 
 @[simp]
 theorem eLpNorm'_norm {f : α → F} :
-    eLpNorm' (fun a => ‖f a‖) q μ = eLpNorm' f q μ := by simp [eLpNorm'_eq_lintegral_nnnorm]
+    eLpNorm' (fun a => ‖f a‖) q μ = eLpNorm' f q μ := by simp [eLpNorm'_eq_lintegral_enorm]
 
 @[deprecated (since := "2024-07-27")]
 alias snorm'_norm := eLpNorm'_norm
@@ -644,8 +652,8 @@ alias snorm_norm := eLpNorm_norm
 
 theorem eLpNorm'_norm_rpow (f : α → F) (p q : ℝ) (hq_pos : 0 < q) :
     eLpNorm' (fun x => ‖f x‖ ^ q) p μ = eLpNorm' f (p * q) μ ^ q := by
-  simp_rw [eLpNorm'_eq_lintegral_nnnorm, ← ENNReal.rpow_mul, ← one_div_mul_one_div, one_div,
-    mul_assoc, inv_mul_cancel₀ hq_pos.ne.symm, mul_one, ← ofReal_norm_eq_coe_nnnorm,
+  simp_rw [eLpNorm', ← ENNReal.rpow_mul, ← one_div_mul_one_div, one_div,
+    mul_assoc, inv_mul_cancel₀ hq_pos.ne.symm, mul_one, ← ofReal_norm_eq_enorm,
     Real.norm_eq_abs, abs_eq_self.mpr (Real.rpow_nonneg (norm_nonneg _) _), mul_comm p,
     ← ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hq_pos.le, ENNReal.rpow_mul]
 
@@ -658,22 +666,17 @@ theorem eLpNorm_norm_rpow (f : α → F) (hq_pos : 0 < q) :
   · simp [h0, ENNReal.zero_rpow_of_pos hq_pos]
   by_cases hp_top : p = ∞
   · simp only [hp_top, eLpNorm_exponent_top, ENNReal.top_mul', hq_pos.not_le,
-      ENNReal.ofReal_eq_zero, if_false, eLpNorm_exponent_top, eLpNormEssSup_eq_essSup_nnnorm]
-    have h_rpow :
-      essSup (fun x : α => (‖‖f x‖ ^ q‖₊ : ℝ≥0∞)) μ =
-        essSup (fun x : α => (‖f x‖₊ : ℝ≥0∞) ^ q) μ := by
+      ENNReal.ofReal_eq_zero, if_false, eLpNorm_exponent_top, eLpNormEssSup_eq_essSup_enorm]
+    have h_rpow : essSup (‖‖f ·‖ ^ q‖ₑ) μ = essSup (‖f ·‖ₑ ^ q) μ := by
       congr
       ext1 x
-      conv_rhs => rw [← nnnorm_norm]
-      rw [← ENNReal.coe_rpow_of_nonneg _ hq_pos.le, ENNReal.coe_inj]
-      ext
-      push_cast
-      rw [Real.norm_rpow_of_nonneg (norm_nonneg _)]
+      conv_rhs => rw [← enorm_norm]
+      rw [← Real.enorm_rpow_of_nonneg (norm_nonneg _) hq_pos.le]
     rw [h_rpow]
     have h_rpow_mono := ENNReal.strictMono_rpow_of_pos hq_pos
     have h_rpow_surj := (ENNReal.rpow_left_bijective hq_pos.ne.symm).2
     let iso := h_rpow_mono.orderIsoOfSurjective _ h_rpow_surj
-    exact (iso.essSup_apply (fun x => (‖f x‖₊ : ℝ≥0∞)) μ).symm
+    exact (iso.essSup_apply (fun x => ‖f x‖ₑ) μ).symm
   rw [eLpNorm_eq_eLpNorm' h0 hp_top, eLpNorm_eq_eLpNorm' _ _]
   swap
   · refine mul_ne_zero h0 ?_
@@ -774,15 +777,14 @@ lemma eLpNorm_indicator_eq_eLpNorm_restrict (hs : MeasurableSet s) :
   by_cases hp_zero : p = 0
   · simp only [hp_zero, eLpNorm_exponent_zero]
   by_cases hp_top : p = ∞
-  · simp_rw [hp_top, eLpNorm_exponent_top, eLpNormEssSup_eq_essSup_nnnorm,
-       nnnorm_indicator_eq_indicator_nnnorm, ENNReal.coe_indicator,
-       ENNReal.essSup_indicator_eq_essSup_restrict hs]
-  simp_rw [eLpNorm_eq_lintegral_rpow_nnnorm hp_zero hp_top]
-  suffices (∫⁻ x, (‖s.indicator f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) =
-      ∫⁻ x in s, (‖f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ by rw [this]
+  · simp_rw [hp_top, eLpNorm_exponent_top, eLpNormEssSup_eq_essSup_enorm,
+       enorm_indicator_eq_indicator_enorm, ENNReal.essSup_indicator_eq_essSup_restrict hs]
+  simp_rw [eLpNorm_eq_lintegral_rpow_enorm hp_zero hp_top]
+  suffices (∫⁻ x, (‖s.indicator f x‖ₑ) ^ p.toReal ∂μ) =
+      ∫⁻ x in s, ‖f x‖ₑ ^ p.toReal ∂μ by rw [this]
   rw [← lintegral_indicator hs]
   congr
-  simp_rw [nnnorm_indicator_eq_indicator_nnnorm, ENNReal.coe_indicator]
+  simp_rw [enorm_indicator_eq_indicator_enorm]
   have h_zero : (fun x => x ^ p.toReal) (0 : ℝ≥0∞) = 0 := by
     simp [ENNReal.toReal_pos hp_zero hp_top]
   -- Porting note: The implicit argument should be specified because the elaborator can't deal with
@@ -819,14 +821,14 @@ lemma eLpNormEssSup_indicator_le (s : Set α) (f : α → G) :
   exact Set.indicator_le_self s _ x
 
 lemma eLpNormEssSup_indicator_const_le (s : Set α) (c : G) :
-    eLpNormEssSup (s.indicator fun _ : α => c) μ ≤ ‖c‖₊ := by
+    eLpNormEssSup (s.indicator fun _ : α => c) μ ≤ ‖c‖ₑ := by
   by_cases hμ0 : μ = 0
   · rw [hμ0, eLpNormEssSup_measure_zero]
     exact zero_le _
   · exact (eLpNormEssSup_indicator_le s fun _ => c).trans (eLpNormEssSup_const c hμ0).le
 
 lemma eLpNormEssSup_indicator_const_eq (s : Set α) (c : G) (hμs : μ s ≠ 0) :
-    eLpNormEssSup (s.indicator fun _ : α => c) μ = ‖c‖₊ := by
+    eLpNormEssSup (s.indicator fun _ : α => c) μ = ‖c‖ₑ := by
   refine le_antisymm (eLpNormEssSup_indicator_const_le s c) ?_
   by_contra! h
   have h' := ae_iff.mp (ae_lt_of_essSup_lt h)
@@ -835,33 +837,33 @@ lemma eLpNormEssSup_indicator_const_eq (s : Set α) (c : G) (hμs : μ s ≠ 0)
   rw [Set.mem_setOf_eq, Set.indicator_of_mem hx_mem, enorm_eq_nnnorm]
 
 lemma eLpNorm_indicator_const₀ (hs : NullMeasurableSet s μ) (hp : p ≠ 0) (hp_top : p ≠ ∞) :
-    eLpNorm (s.indicator fun _ => c) p μ = ‖c‖₊ * μ s ^ (1 / p.toReal) :=
+    eLpNorm (s.indicator fun _ => c) p μ = ‖c‖ₑ * μ s ^ (1 / p.toReal) :=
   have hp_pos : 0 < p.toReal := ENNReal.toReal_pos hp hp_top
   calc
     eLpNorm (s.indicator fun _ => c) p μ
-      = (∫⁻ x, ((‖(s.indicator fun _ ↦ c) x‖₊ : ℝ≥0∞) ^ p.toReal) ∂μ) ^ (1 / p.toReal) :=
-          eLpNorm_eq_lintegral_rpow_nnnorm hp hp_top
-    _ = (∫⁻ x, (s.indicator fun _ ↦ (‖c‖₊ : ℝ≥0∞) ^ p.toReal) x ∂μ) ^ (1 / p.toReal) := by
+      = (∫⁻ x, (‖(s.indicator fun _ ↦ c) x‖ₑ ^ p.toReal) ∂μ) ^ (1 / p.toReal) :=
+          eLpNorm_eq_lintegral_rpow_enorm hp hp_top
+    _ = (∫⁻ x, (s.indicator fun _ ↦ ‖c‖ₑ ^ p.toReal) x ∂μ) ^ (1 / p.toReal) := by
       congr 2
-      refine (Set.comp_indicator_const c (fun x ↦ (‖x‖₊ : ℝ≥0∞) ^ p.toReal) ?_)
+      refine (Set.comp_indicator_const c (fun x ↦ (‖x‖ₑ) ^ p.toReal) ?_)
       simp [hp_pos]
-    _ = ‖c‖₊ * μ s ^ (1 / p.toReal) := by
+    _ = ‖c‖ₑ * μ s ^ (1 / p.toReal) := by
       rw [lintegral_indicator_const₀ hs, ENNReal.mul_rpow_of_nonneg, ← ENNReal.rpow_mul,
         mul_one_div_cancel hp_pos.ne', ENNReal.rpow_one]
       positivity
 
 lemma eLpNorm_indicator_const (hs : MeasurableSet s) (hp : p ≠ 0) (hp_top : p ≠ ∞) :
-    eLpNorm (s.indicator fun _ => c) p μ = ‖c‖₊ * μ s ^ (1 / p.toReal) :=
+    eLpNorm (s.indicator fun _ => c) p μ = ‖c‖ₑ * μ s ^ (1 / p.toReal) :=
   eLpNorm_indicator_const₀ hs.nullMeasurableSet hp hp_top
 
 lemma eLpNorm_indicator_const' (hs : MeasurableSet s) (hμs : μ s ≠ 0) (hp : p ≠ 0) :
-    eLpNorm (s.indicator fun _ => c) p μ = ‖c‖₊ * μ s ^ (1 / p.toReal) := by
+    eLpNorm (s.indicator fun _ => c) p μ = ‖c‖ₑ * μ s ^ (1 / p.toReal) := by
   by_cases hp_top : p = ∞
   · simp [hp_top, eLpNormEssSup_indicator_const_eq s c hμs]
   · exact eLpNorm_indicator_const hs hp hp_top
 
 lemma eLpNorm_indicator_const_le (c : G) (p : ℝ≥0∞) :
-    eLpNorm (s.indicator fun _ => c) p μ ≤ ‖c‖₊ * μ s ^ (1 / p.toReal) := by
+    eLpNorm (s.indicator fun _ => c) p μ ≤ ‖c‖ₑ * μ s ^ (1 / p.toReal) := by
   obtain rfl | hp := eq_or_ne p 0
   · simp only [eLpNorm_exponent_zero, zero_le']
   obtain rfl | h'p := eq_or_ne p ∞
@@ -872,9 +874,9 @@ lemma eLpNorm_indicator_const_le (c : G) (p : ℝ≥0∞) :
   calc
     eLpNorm (s.indicator fun _ => c) p μ ≤ eLpNorm (t.indicator fun _ => c) p μ :=
       eLpNorm_mono (norm_indicator_le_of_subset (subset_toMeasurable _ _) _)
-    _ = ‖c‖₊ * μ t ^ (1 / p.toReal) :=
+    _ = ‖c‖ₑ * μ t ^ (1 / p.toReal) :=
       eLpNorm_indicator_const (measurableSet_toMeasurable ..) hp h'p
-    _ = ‖c‖₊ * μ s ^ (1 / p.toReal) := by rw [measure_toMeasurable]
+    _ = ‖c‖ₑ * μ s ^ (1 / p.toReal) := by rw [measure_toMeasurable]
 
 lemma Memℒp.indicator (hs : MeasurableSet s) (hf : Memℒp f p μ) : Memℒp (s.indicator f) p μ :=
   ⟨hf.aestronglyMeasurable.indicator hs, lt_of_le_of_lt (eLpNorm_indicator_le f) hf.eLpNorm_lt_top⟩
@@ -916,18 +918,16 @@ protected lemma Memℒp.piecewise [DecidablePred (· ∈ s)] {g} (hs : Measurabl
   rw [eLpNorm_lt_top_iff_lintegral_rpow_nnnorm_lt_top hp_zero hp_top, ← lintegral_add_compl _ hs,
     ENNReal.add_lt_top]
   constructor
-  · have h : ∀ᵐ x ∂μ, x ∈ s →
-        (‖Set.piecewise s f g x‖₊ : ℝ≥0∞) ^ p.toReal = (‖f x‖₊ : ℝ≥0∞) ^ p.toReal := by
+  · have h : ∀ᵐ x ∂μ, x ∈ s → ‖Set.piecewise s f g x‖ₑ ^ p.toReal = ‖f x‖ₑ ^ p.toReal := by
       filter_upwards with a ha using by simp [ha]
     rw [setLIntegral_congr_fun hs h]
-    exact lintegral_rpow_nnnorm_lt_top_of_eLpNorm_lt_top hp_zero hp_top hf.2
-  · have h : ∀ᵐ x ∂μ, x ∈ sᶜ →
-        (‖Set.piecewise s f g x‖₊ : ℝ≥0∞) ^ p.toReal = (‖g x‖₊ : ℝ≥0∞) ^ p.toReal := by
+    exact lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top hp_zero hp_top hf.2
+  · have h : ∀ᵐ x ∂μ, x ∈ sᶜ → ‖Set.piecewise s f g x‖ₑ ^ p.toReal = ‖g x‖ₑ ^ p.toReal := by
       filter_upwards with a ha
       have ha' : a ∉ s := ha
       simp [ha']
     rw [setLIntegral_congr_fun hs.compl h]
-    exact lintegral_rpow_nnnorm_lt_top_of_eLpNorm_lt_top hp_zero hp_top hg.2
+    exact lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top hp_zero hp_top hg.2
 
 end Indicator
 
@@ -938,12 +938,9 @@ theorem eLpNorm_restrict_eq_of_support_subset {s : Set α} {f : α → F} (hsf :
   by_cases hp0 : p = 0
   · simp [hp0]
   by_cases hp_top : p = ∞
-  · simp only [hp_top, eLpNorm_exponent_top, eLpNormEssSup_eq_essSup_nnnorm]
-    apply ENNReal.essSup_restrict_eq_of_support_subset
-    apply Function.support_subset_iff.2 (fun x hx ↦ ?_)
-    simp only [ne_eq, ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx
-    exact Function.support_subset_iff.1 hsf x hx
-  · simp_rw [eLpNorm_eq_eLpNorm' hp0 hp_top, eLpNorm'_eq_lintegral_nnnorm]
+  · simp only [hp_top, eLpNorm_exponent_top, eLpNormEssSup_eq_essSup_enorm]
+    exact ENNReal.essSup_restrict_eq_of_support_subset fun x hx ↦ hsf <| enorm_ne_zero.1 hx
+  · simp_rw [eLpNorm_eq_eLpNorm' hp0 hp_top, eLpNorm'_eq_lintegral_enorm]
     congr 1
     apply setLIntegral_eq_of_support_subset
     have : ¬(p.toReal ≤ 0) := by simpa only [not_le] using ENNReal.toReal_pos hp0 hp_top
@@ -1047,7 +1044,7 @@ theorem Memℒp.smul_measure {f : α → E} {c : ℝ≥0∞} (hf : Memℒp f p 
 
 theorem eLpNorm_one_add_measure (f : α → F) (μ ν : Measure α) :
     eLpNorm f 1 (μ + ν) = eLpNorm f 1 μ + eLpNorm f 1 ν := by
-  simp_rw [eLpNorm_one_eq_lintegral_nnnorm]
+  simp_rw [eLpNorm_one_eq_lintegral_enorm]
   rw [lintegral_add_measure _ μ ν]
 
 @[deprecated (since := "2024-07-27")]
@@ -1067,11 +1064,11 @@ theorem eLpNorm_le_add_measure_left (f : α → F) (μ ν : Measure α) {p : ℝ
 @[deprecated (since := "2024-07-27")]
 alias snorm_le_add_measure_left := eLpNorm_le_add_measure_left
 
-lemma eLpNormEssSup_eq_iSup (hμ : ∀ a, μ {a} ≠ 0) (f : α → E) : eLpNormEssSup f μ = ⨆ a, ↑‖f a‖₊ :=
+lemma eLpNormEssSup_eq_iSup (hμ : ∀ a, μ {a} ≠ 0) (f : α → E) : eLpNormEssSup f μ = ⨆ a, ‖f a‖ₑ :=
   essSup_eq_iSup hμ _
 
 @[simp] lemma eLpNormEssSup_count [MeasurableSingletonClass α] (f : α → E) :
-    eLpNormEssSup f .count = ⨆ a, ↑‖f a‖₊ := essSup_count _
+    eLpNormEssSup f .count = ⨆ a, ‖f a‖ₑ := essSup_count _
 
 theorem Memℒp.left_of_add_measure {f : α → E} (h : Memℒp f p (μ + ν)) : Memℒp f p μ :=
   h.mono_measure <| Measure.le_add_right <| le_refl _
@@ -1101,22 +1098,12 @@ alias snorm'_eq_zero_of_ae_zero' := eLpNorm'_eq_zero_of_ae_zero'
 
 theorem ae_eq_zero_of_eLpNorm'_eq_zero {f : α → E} (hq0 : 0 ≤ q) (hf : AEStronglyMeasurable f μ)
     (h : eLpNorm' f q μ = 0) : f =ᵐ[μ] 0 := by
-  rw [eLpNorm'_eq_lintegral_nnnorm, ENNReal.rpow_eq_zero_iff] at h
-  cases h with
-  | inl h =>
-    rw [lintegral_eq_zero_iff' (hf.ennnorm.pow_const q)] at h
-    refine h.left.mono fun x hx => ?_
-    rw [Pi.zero_apply, ENNReal.rpow_eq_zero_iff] at hx
-    cases hx with
-    | inl hx =>
-      cases' hx with hx _
-      rwa [← ENNReal.coe_zero, ENNReal.coe_inj, nnnorm_eq_zero] at hx
-    | inr hx =>
-      exact absurd hx.left ENNReal.coe_ne_top
-  | inr h =>
-    exfalso
-    rw [one_div, inv_lt_zero] at h
-    exact hq0.not_lt h.right
+  simp only [eLpNorm'_eq_lintegral_enorm, lintegral_eq_zero_iff' (hf.enorm.pow_const q), one_div,
+    ENNReal.rpow_eq_zero_iff, inv_pos, inv_neg'', hq0.not_lt, and_false, or_false] at h
+  refine h.left.mono fun x hx ↦ ?_
+  simp only [Pi.zero_apply, ENNReal.rpow_eq_zero_iff, enorm_eq_zero, enorm_ne_top, false_and,
+    or_false] at hx
+  exact hx.1
 
 @[deprecated (since := "2024-07-27")]
 alias ae_eq_zero_of_snorm'_eq_zero := ae_eq_zero_of_eLpNorm'_eq_zero
@@ -1129,15 +1116,15 @@ theorem eLpNorm'_eq_zero_iff (hq0_lt : 0 < q) {f : α → E} (hf : AEStronglyMea
 alias snorm'_eq_zero_iff := eLpNorm'_eq_zero_iff
 
 theorem coe_nnnorm_ae_le_eLpNormEssSup {_ : MeasurableSpace α} (f : α → F) (μ : Measure α) :
-    ∀ᵐ x ∂μ, (‖f x‖₊ : ℝ≥0∞) ≤ eLpNormEssSup f μ :=
-  ENNReal.ae_le_essSup fun x => (‖f x‖₊ : ℝ≥0∞)
+    ∀ᵐ x ∂μ, ‖f x‖ₑ ≤ eLpNormEssSup f μ :=
+  ENNReal.ae_le_essSup fun x => ‖f x‖ₑ
 
 @[deprecated (since := "2024-07-27")]
 alias coe_nnnorm_ae_le_snormEssSup := coe_nnnorm_ae_le_eLpNormEssSup
 
 @[simp]
 theorem eLpNormEssSup_eq_zero_iff {f : α → F} : eLpNormEssSup f μ = 0 ↔ f =ᵐ[μ] 0 := by
-  simp [EventuallyEq, eLpNormEssSup_eq_essSup_nnnorm]
+  simp [EventuallyEq, eLpNormEssSup_eq_essSup_enorm]
 
 @[deprecated (since := "2024-07-27")]
 alias snormEssSup_eq_zero_iff := eLpNormEssSup_eq_zero_iff
@@ -1156,7 +1143,7 @@ theorem eLpNorm_eq_zero_of_ae_zero {f : α → E} (hf : f =ᵐ[μ] 0) : eLpNorm
   rw [← eLpNorm_zero (p := p) (μ := μ) (α := α) (F := E)]
   exact eLpNorm_congr_ae hf
 
-theorem ae_le_eLpNormEssSup {f : α → F} : ∀ᵐ y ∂μ, ‖f y‖₊ ≤ eLpNormEssSup f μ :=
+theorem ae_le_eLpNormEssSup {f : α → F} : ∀ᵐ y ∂μ, ‖f y‖ₑ ≤ eLpNormEssSup f μ :=
   ae_le_essSup
 
 @[deprecated (since := "2024-07-27")]
@@ -1168,7 +1155,7 @@ lemma eLpNormEssSup_lt_top_iff_isBoundedUnder :
     simp_rw [← ENNReal.coe_le_coe, ENNReal.coe_toNNReal h.ne]; exact ae_le_eLpNormEssSup⟩
   mpr := by rintro ⟨C, hC⟩; exact eLpNormEssSup_lt_top_of_ae_nnnorm_bound (C := C) hC
 
-theorem meas_eLpNormEssSup_lt {f : α → F} : μ { y | eLpNormEssSup f μ < ‖f y‖₊ } = 0 :=
+theorem meas_eLpNormEssSup_lt {f : α → F} : μ { y | eLpNormEssSup f μ < ‖f y‖ₑ } = 0 :=
   meas_essSup_lt
 
 @[deprecated (since := "2024-07-27")]
@@ -1182,7 +1169,7 @@ lemma eLpNorm_lt_top_of_finite [Finite α] [IsFiniteMeasure μ] : eLpNorm f p μ
     exact .le_of_finite
   rw [eLpNorm_lt_top_iff_lintegral_rpow_nnnorm_lt_top hp₀ hp]
   refine IsFiniteMeasure.lintegral_lt_top_of_bounded_to_ennreal μ ?_
-  simp_rw [← ENNReal.coe_rpow_of_nonneg _ ENNReal.toReal_nonneg]
+  simp_rw [enorm, ← ENNReal.coe_rpow_of_nonneg _ ENNReal.toReal_nonneg]
   norm_cast
   exact Finite.exists_le _
 
@@ -1205,7 +1192,7 @@ variable {β : Type*} {mβ : MeasurableSpace β} {f : α → β} {g : β → E}
 
 theorem eLpNormEssSup_map_measure (hg : AEStronglyMeasurable g (Measure.map f μ))
     (hf : AEMeasurable f μ) : eLpNormEssSup g (Measure.map f μ) = eLpNormEssSup (g ∘ f) μ :=
-  essSup_map_measure hg.ennnorm hf
+  essSup_map_measure hg.enorm hf
 
 @[deprecated (since := "2024-07-27")]
 alias snormEssSup_map_measure := eLpNormEssSup_map_measure
@@ -1217,8 +1204,8 @@ theorem eLpNorm_map_measure (hg : AEStronglyMeasurable g (Measure.map f μ))
   by_cases hp_top : p = ∞
   · simp_rw [hp_top, eLpNorm_exponent_top]
     exact eLpNormEssSup_map_measure hg hf
-  simp_rw [eLpNorm_eq_lintegral_rpow_nnnorm hp_zero hp_top]
-  rw [lintegral_map' (hg.ennnorm.pow_const p.toReal) hf]
+  simp_rw [eLpNorm_eq_lintegral_rpow_enorm hp_zero hp_top]
+  rw [lintegral_map' (hg.enorm.pow_const p.toReal) hf]
   rfl
 
 @[deprecated (since := "2024-07-27")]
@@ -1267,7 +1254,7 @@ theorem _root_.MeasurableEmbedding.eLpNorm_map_measure {g : β → F} (hf : Meas
   by_cases hp : p = ∞
   · simp_rw [hp, eLpNorm_exponent_top]
     exact hf.essSup_map_measure
-  · simp_rw [eLpNorm_eq_lintegral_rpow_nnnorm hp_zero hp]
+  · simp_rw [eLpNorm_eq_lintegral_rpow_enorm hp_zero hp]
     rw [hf.lintegral_map]
     rfl
 
@@ -1289,10 +1276,10 @@ section Monotonicity
 theorem eLpNorm'_le_nnreal_smul_eLpNorm'_of_ae_le_mul {f : α → F} {g : α → G} {c : ℝ≥0}
     (h : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊) {p : ℝ} (hp : 0 < p) :
     eLpNorm' f p μ ≤ c • eLpNorm' g p μ := by
-  simp_rw [eLpNorm'_eq_lintegral_nnnorm]
+  simp_rw [eLpNorm'_eq_lintegral_enorm]
   rw [← ENNReal.rpow_le_rpow_iff hp, ENNReal.smul_def, smul_eq_mul,
     ENNReal.mul_rpow_of_nonneg _ _ hp.le]
-  simp_rw [← ENNReal.rpow_mul, one_div, inv_mul_cancel₀ hp.ne.symm, ENNReal.rpow_one,
+  simp_rw [← ENNReal.rpow_mul, one_div, inv_mul_cancel₀ hp.ne.symm, ENNReal.rpow_one, enorm,
     ← ENNReal.coe_rpow_of_nonneg _ hp.le, ← lintegral_const_mul' _ _ ENNReal.coe_ne_top,
     ← ENNReal.coe_mul]
   apply lintegral_mono_ae
@@ -1305,10 +1292,10 @@ alias snorm'_le_nnreal_smul_snorm'_of_ae_le_mul := eLpNorm'_le_nnreal_smul_eLpNo
 theorem eLpNormEssSup_le_nnreal_smul_eLpNormEssSup_of_ae_le_mul {f : α → F} {g : α → G} {c : ℝ≥0}
     (h : ∀ᵐ x ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊) : eLpNormEssSup f μ ≤ c • eLpNormEssSup g μ :=
   calc
-    essSup (fun x => (‖f x‖₊ : ℝ≥0∞)) μ ≤ essSup (fun x => (↑(c * ‖g x‖₊) : ℝ≥0∞)) μ :=
+    essSup (‖f ·‖ₑ) μ ≤ essSup (fun x => (↑(c * ‖g x‖₊) : ℝ≥0∞)) μ :=
       essSup_mono_ae <| h.mono fun _ hx => ENNReal.coe_le_coe.mpr hx
-    _ = essSup (fun x => (c * ‖g x‖₊ : ℝ≥0∞)) μ := by simp_rw [ENNReal.coe_mul]
-    _ = c • essSup (fun x => (‖g x‖₊ : ℝ≥0∞)) μ := ENNReal.essSup_const_mul
+    _ = essSup (c * ‖g ·‖ₑ) μ := by simp_rw [ENNReal.coe_mul, enorm]
+    _ = c • essSup (‖g ·‖ₑ) μ := ENNReal.essSup_const_mul
 
 @[deprecated (since := "2024-07-27")]
 alias snormEssSup_le_nnreal_smul_snormEssSup_of_ae_le_mul :=
@@ -1384,20 +1371,20 @@ section BoundedSMul
 variable {𝕜 : Type*} [NormedRing 𝕜] [MulActionWithZero 𝕜 E] [MulActionWithZero 𝕜 F]
 variable [BoundedSMul 𝕜 E] [BoundedSMul 𝕜 F] {c : 𝕜} {f : α → F}
 
-theorem eLpNorm'_const_smul_le (hq : 0 < q) : eLpNorm' (c • f) q μ ≤ ‖c‖₊ • eLpNorm' f q μ :=
+theorem eLpNorm'_const_smul_le (hq : 0 < q) : eLpNorm' (c • f) q μ ≤ ‖c‖ₑ * eLpNorm' f q μ :=
   eLpNorm'_le_nnreal_smul_eLpNorm'_of_ae_le_mul (Eventually.of_forall fun _ => nnnorm_smul_le ..) hq
 
 @[deprecated (since := "2024-07-27")]
 alias snorm'_const_smul_le := eLpNorm'_const_smul_le
 
-theorem eLpNormEssSup_const_smul_le : eLpNormEssSup (c • f) μ ≤ ‖c‖₊ • eLpNormEssSup f μ :=
+theorem eLpNormEssSup_const_smul_le : eLpNormEssSup (c • f) μ ≤ ‖c‖ₑ * eLpNormEssSup f μ :=
   eLpNormEssSup_le_nnreal_smul_eLpNormEssSup_of_ae_le_mul
     (Eventually.of_forall fun _ => by simp [nnnorm_smul_le])
 
 @[deprecated (since := "2024-07-27")]
 alias snormEssSup_const_smul_le := eLpNormEssSup_const_smul_le
 
-theorem eLpNorm_const_smul_le : eLpNorm (c • f) p μ ≤ ‖c‖₊ • eLpNorm f p μ :=
+theorem eLpNorm_const_smul_le : eLpNorm (c • f) p μ ≤ ‖c‖ₑ * eLpNorm f p μ :=
   eLpNorm_le_nnreal_smul_eLpNorm_of_ae_le_mul
     (Eventually.of_forall fun _ => by simp [nnnorm_smul_le]) _
 
@@ -1425,29 +1412,31 @@ variable {𝕜 : Type*} [NormedDivisionRing 𝕜] [MulActionWithZero 𝕜 E] [Mo
 variable [BoundedSMul 𝕜 E] [BoundedSMul 𝕜 F]
 
 theorem eLpNorm'_const_smul {f : α → F} (c : 𝕜) (hq_pos : 0 < q) :
-    eLpNorm' (c • f) q μ = ‖c‖₊ • eLpNorm' f q μ := by
+    eLpNorm' (c • f) q μ = ‖c‖ₑ * eLpNorm' f q μ := by
   obtain rfl | hc := eq_or_ne c 0
-  · simp [eLpNorm'_eq_lintegral_nnnorm, hq_pos]
-  refine le_antisymm (eLpNorm'_const_smul_le hq_pos) ?_
-  simpa [hc, le_inv_smul_iff_of_pos] using eLpNorm'_const_smul_le (c := c⁻¹) (f := c • f) hq_pos
+  · simp [eLpNorm'_eq_lintegral_enorm, hq_pos]
+  refine le_antisymm (eLpNorm'_const_smul_le hq_pos) <| ENNReal.mul_le_of_le_div' ?_
+  simpa [enorm_inv, hc, ENNReal.div_eq_inv_mul]
+    using eLpNorm'_const_smul_le (c := c⁻¹) (f := c • f) hq_pos
 
 @[deprecated (since := "2024-07-27")]
 alias snorm'_const_smul := eLpNorm'_const_smul
 
 theorem eLpNormEssSup_const_smul (c : 𝕜) (f : α → F) :
-    eLpNormEssSup (c • f) μ = (‖c‖₊ : ℝ≥0∞) * eLpNormEssSup f μ := by
-  simp_rw [eLpNormEssSup_eq_essSup_nnnorm, Pi.smul_apply, nnnorm_smul, ENNReal.coe_mul,
+    eLpNormEssSup (c • f) μ = ‖c‖ₑ * eLpNormEssSup f μ := by
+  simp_rw [eLpNormEssSup_eq_essSup_enorm, Pi.smul_apply, enorm_smul,
     ENNReal.essSup_const_mul]
 
 @[deprecated (since := "2024-07-27")]
 alias snormEssSup_const_smul := eLpNormEssSup_const_smul
 
 theorem eLpNorm_const_smul (c : 𝕜) (f : α → F) (p : ℝ≥0∞) (μ : Measure α):
-    eLpNorm (c • f) p μ = (‖c‖₊ : ℝ≥0∞) * eLpNorm f p μ := by
+    eLpNorm (c • f) p μ = ‖c‖ₑ * eLpNorm f p μ := by
   obtain rfl | hc := eq_or_ne c 0
   · simp
-  refine le_antisymm eLpNorm_const_smul_le ?_
-  simpa [hc, le_inv_smul_iff_of_pos] using eLpNorm_const_smul_le (c := c⁻¹) (f := c • f)
+  refine le_antisymm eLpNorm_const_smul_le <| ENNReal.mul_le_of_le_div' ?_
+  simpa [enorm_inv, hc, ENNReal.div_eq_inv_mul]
+    using eLpNorm_const_smul_le (c := c⁻¹) (f := c • f)
 
 @[deprecated (since := "2024-07-27")]
 alias snorm_const_smul := eLpNorm_const_smul
@@ -1461,7 +1450,7 @@ end NormedSpace
 theorem le_eLpNorm_of_bddBelow (hp : p ≠ 0) (hp' : p ≠ ∞) {f : α → F} (C : ℝ≥0) {s : Set α}
     (hs : MeasurableSet s) (hf : ∀ᵐ x ∂μ, x ∈ s → C ≤ ‖f x‖₊) :
     C • μ s ^ (1 / p.toReal) ≤ eLpNorm f p μ := by
-  rw [ENNReal.smul_def, smul_eq_mul, eLpNorm_eq_lintegral_rpow_nnnorm hp hp',
+  rw [ENNReal.smul_def, smul_eq_mul, eLpNorm_eq_lintegral_rpow_enorm hp hp',
     one_div, ENNReal.le_rpow_inv_iff (ENNReal.toReal_pos hp hp'),
     ENNReal.mul_rpow_of_nonneg _ _ ENNReal.toReal_nonneg, ← ENNReal.rpow_mul,
     inv_mul_cancel₀ (ENNReal.toReal_pos hp hp').ne.symm, ENNReal.rpow_one, ← setLIntegral_const,
@@ -1469,9 +1458,9 @@ theorem le_eLpNorm_of_bddBelow (hp : p ≠ 0) (hp' : p ≠ ∞) {f : α → F} (
   refine lintegral_mono_ae ?_
   filter_upwards [hf] with x hx
   by_cases hxs : x ∈ s
-  · simp only [Set.indicator_of_mem hxs] at hx ⊢
+  · simp only [Set.indicator_of_mem, hxs, true_implies] at hx ⊢
     gcongr
-    exact hx hxs
+    rwa [coe_le_enorm]
   · simp [Set.indicator_of_not_mem hxs]
 
 @[deprecated (since := "2024-07-27")]
@@ -1511,7 +1500,7 @@ variable [MeasurableSpace E] [OpensMeasurableSpace E] {R : ℝ≥0}
 
 theorem ae_bdd_liminf_atTop_rpow_of_eLpNorm_bdd {p : ℝ≥0∞} {f : ℕ → α → E}
     (hfmeas : ∀ n, Measurable (f n)) (hbdd : ∀ n, eLpNorm (f n) p μ ≤ R) :
-    ∀ᵐ x ∂μ, liminf (fun n => ((‖f n x‖₊ : ℝ≥0∞) ^ p.toReal : ℝ≥0∞)) atTop < ∞ := by
+    ∀ᵐ x ∂μ, liminf (fun n => ((‖f n x‖ₑ) ^ p.toReal : ℝ≥0∞)) atTop < ∞ := by
   by_cases hp0 : p.toReal = 0
   · simp only [hp0, ENNReal.rpow_zero]
     filter_upwards with _
@@ -1526,7 +1515,7 @@ theorem ae_bdd_liminf_atTop_rpow_of_eLpNorm_bdd {p : ℝ≥0∞} {f : ℕ → α
           (lt_of_le_of_lt ?_
             (ENNReal.rpow_lt_top_of_nonneg ENNReal.toReal_nonneg ENNReal.coe_ne_top :
               (R : ℝ≥0∞) ^ p.toReal < ∞))).ne
-  simp_rw [eLpNorm_eq_lintegral_rpow_nnnorm hp hp', one_div] at hbdd
+  simp_rw [eLpNorm_eq_lintegral_rpow_enorm hp hp', one_div] at hbdd
   simp_rw [liminf_eq, eventually_atTop]
   exact
     sSup_le fun b ⟨a, ha⟩ =>
@@ -1537,11 +1526,11 @@ alias ae_bdd_liminf_atTop_rpow_of_snorm_bdd := ae_bdd_liminf_atTop_rpow_of_eLpNo
 
 theorem ae_bdd_liminf_atTop_of_eLpNorm_bdd {p : ℝ≥0∞} (hp : p ≠ 0) {f : ℕ → α → E}
     (hfmeas : ∀ n, Measurable (f n)) (hbdd : ∀ n, eLpNorm (f n) p μ ≤ R) :
-    ∀ᵐ x ∂μ, liminf (fun n => (‖f n x‖₊ : ℝ≥0∞)) atTop < ∞ := by
+    ∀ᵐ x ∂μ, liminf (fun n => (‖f n x‖ₑ)) atTop < ∞ := by
   by_cases hp' : p = ∞
   · subst hp'
     simp_rw [eLpNorm_exponent_top] at hbdd
-    have : ∀ n, ∀ᵐ x ∂μ, (‖f n x‖₊ : ℝ≥0∞) < R + 1 := fun n =>
+    have : ∀ n, ∀ᵐ x ∂μ, (‖f n x‖ₑ) < R + 1 := fun n =>
       ae_lt_of_essSup_lt
         (lt_of_le_of_lt (hbdd n) <| ENNReal.lt_add_right ENNReal.coe_ne_top one_ne_zero)
     rw [← ae_all_iff] at this
@@ -1551,15 +1540,15 @@ theorem ae_bdd_liminf_atTop_of_eLpNorm_bdd {p : ℝ≥0∞} (hp : p ≠ 0) {f :
   filter_upwards [ae_bdd_liminf_atTop_rpow_of_eLpNorm_bdd hfmeas hbdd] with x hx
   have hppos : 0 < p.toReal := ENNReal.toReal_pos hp hp'
   have :
-    liminf (fun n => (‖f n x‖₊ : ℝ≥0∞) ^ p.toReal) atTop =
-      liminf (fun n => (‖f n x‖₊ : ℝ≥0∞)) atTop ^ p.toReal := by
+    liminf (fun n => (‖f n x‖ₑ) ^ p.toReal) atTop =
+      liminf (fun n => (‖f n x‖ₑ)) atTop ^ p.toReal := by
     change
-      liminf (fun n => ENNReal.orderIsoRpow p.toReal hppos (‖f n x‖₊ : ℝ≥0∞)) atTop =
-        ENNReal.orderIsoRpow p.toReal hppos (liminf (fun n => (‖f n x‖₊ : ℝ≥0∞)) atTop)
+      liminf (fun n => ENNReal.orderIsoRpow p.toReal hppos (‖f n x‖ₑ)) atTop =
+        ENNReal.orderIsoRpow p.toReal hppos (liminf (fun n => (‖f n x‖ₑ)) atTop)
     refine (OrderIso.liminf_apply (ENNReal.orderIsoRpow p.toReal _) ?_ ?_ ?_ ?_).symm <;>
       isBoundedDefault
   rw [this] at hx
-  rw [← ENNReal.rpow_one (liminf (fun n => ‖f n x‖₊) atTop), ← mul_inv_cancel₀ hppos.ne.symm,
+  rw [← ENNReal.rpow_one (liminf (‖f · x‖ₑ) atTop), ← mul_inv_cancel₀ hppos.ne.symm,
     ENNReal.rpow_mul]
   exact ENNReal.rpow_lt_top_of_nonneg (inv_nonneg.2 hppos.le) hx.ne
 
@@ -1587,13 +1576,13 @@ theorem Memℒp.exists_eLpNorm_indicator_compl_lt {β : Type*} [NormedAddCommGro
   rcases eq_or_ne p 0 with rfl | hp₀
   · use ∅; simp [pos_iff_ne_zero.2 hε] -- first take care of `p = 0`
   · obtain ⟨s, hsm, hs, hε⟩ :
-        ∃ s, MeasurableSet s ∧ μ s < ∞ ∧ ∫⁻ a in sᶜ, (‖f a‖₊) ^ p.toReal ∂μ < ε ^ p.toReal := by
+        ∃ s, MeasurableSet s ∧ μ s < ∞ ∧ ∫⁻ a in sᶜ, (‖f a‖ₑ) ^ p.toReal ∂μ < ε ^ p.toReal := by
       apply exists_setLintegral_compl_lt
       · exact ((eLpNorm_lt_top_iff_lintegral_rpow_nnnorm_lt_top hp₀ hp_top).1 hf.2).ne
       · simp [*]
     refine ⟨s, hsm, hs, ?_⟩
     rwa [eLpNorm_indicator_eq_eLpNorm_restrict hsm.compl,
-      eLpNorm_eq_lintegral_rpow_nnnorm hp₀ hp_top, one_div, ENNReal.rpow_inv_lt_iff]
+      eLpNorm_eq_lintegral_rpow_enorm hp₀ hp_top, one_div, ENNReal.rpow_inv_lt_iff]
     simp [ENNReal.toReal_pos, *]
 
 @[deprecated (since := "2024-07-27")]
diff --git a/Mathlib/MeasureTheory/Function/LpSeminorm/ChebyshevMarkov.lean b/Mathlib/MeasureTheory/Function/LpSeminorm/ChebyshevMarkov.lean
index 6248e4a2c59258..968d77d53489ec 100644
--- a/Mathlib/MeasureTheory/Function/LpSeminorm/ChebyshevMarkov.lean
+++ b/Mathlib/MeasureTheory/Function/LpSeminorm/ChebyshevMarkov.lean
@@ -21,9 +21,9 @@ variable {α E : Type*} {m0 : MeasurableSpace α} [NormedAddCommGroup E]
 theorem pow_mul_meas_ge_le_eLpNorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
     (hf : AEStronglyMeasurable f μ) (ε : ℝ≥0∞) :
     (ε * μ { x | ε ≤ (‖f x‖₊ : ℝ≥0∞) ^ p.toReal }) ^ (1 / p.toReal) ≤ eLpNorm f p μ := by
-  rw [eLpNorm_eq_lintegral_rpow_nnnorm hp_ne_zero hp_ne_top]
+  rw [eLpNorm_eq_lintegral_rpow_enorm hp_ne_zero hp_ne_top]
   gcongr
-  exact mul_meas_ge_le_lintegral₀ (hf.ennnorm.pow_const _) ε
+  exact mul_meas_ge_le_lintegral₀ (hf.enorm.pow_const _) ε
 
 @[deprecated (since := "2024-07-27")]
 alias pow_mul_meas_ge_le_snorm := pow_mul_meas_ge_le_eLpNorm
diff --git a/Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean b/Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean
index 649bc7c74f6ed3..e56b6c50caf6d7 100644
--- a/Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean
+++ b/Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean
@@ -30,16 +30,16 @@ theorem eLpNorm'_le_eLpNorm'_mul_rpow_measure_univ {p q : ℝ} (hp0_lt : 0 < p)
   · rw [hpq_eq, sub_self, ENNReal.rpow_zero, mul_one]
   have hpq : p < q := lt_of_le_of_ne hpq hpq_eq
   let g := fun _ : α => (1 : ℝ≥0∞)
-  have h_rw : (∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ p ∂μ) = ∫⁻ a, ((‖f a‖₊ : ℝ≥0∞) * g a) ^ p ∂μ :=
+  have h_rw : (∫⁻ a, ‖f a‖ₑ ^ p ∂μ) = ∫⁻ a, (‖f a‖ₑ * g a) ^ p ∂μ :=
     lintegral_congr fun a => by simp [g]
-  repeat' rw [eLpNorm'_eq_lintegral_nnnorm]
+  repeat' rw [eLpNorm'_eq_lintegral_enorm]
   rw [h_rw]
   let r := p * q / (q - p)
   have hpqr : 1 / p = 1 / q + 1 / r := by field_simp [r, hp0_lt.ne', hq0_lt.ne']
   calc
     (∫⁻ a : α, (↑‖f a‖₊ * g a) ^ p ∂μ) ^ (1 / p) ≤
         (∫⁻ a : α, ↑‖f a‖₊ ^ q ∂μ) ^ (1 / q) * (∫⁻ a : α, g a ^ r ∂μ) ^ (1 / r) :=
-      ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr hp0_lt hpq hpqr μ hf.ennnorm aemeasurable_const
+      ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr hp0_lt hpq hpqr μ hf.enorm aemeasurable_const
     _ = (∫⁻ a : α, ↑‖f a‖₊ ^ q ∂μ) ^ (1 / q) * μ Set.univ ^ (1 / p - 1 / q) := by
       rw [hpqr]; simp [r, g]
 
@@ -48,7 +48,7 @@ alias snorm'_le_snorm'_mul_rpow_measure_univ := eLpNorm'_le_eLpNorm'_mul_rpow_me
 
 theorem eLpNorm'_le_eLpNormEssSup_mul_rpow_measure_univ {q : ℝ} (hq_pos : 0 < q) :
     eLpNorm' f q μ ≤ eLpNormEssSup f μ * μ Set.univ ^ (1 / q) := by
-  have h_le : (∫⁻ a : α, (‖f a‖₊ : ℝ≥0∞) ^ q ∂μ) ≤ ∫⁻ _ : α, eLpNormEssSup f μ ^ q ∂μ := by
+  have h_le : (∫⁻ a : α, ‖f a‖ₑ ^ q ∂μ) ≤ ∫⁻ _ : α, eLpNormEssSup f μ ^ q ∂μ := by
     refine lintegral_mono_ae ?_
     have h_nnnorm_le_eLpNorm_ess_sup := coe_nnnorm_ae_le_eLpNormEssSup f μ
     exact h_nnnorm_le_eLpNorm_ess_sup.mono fun x hx => by gcongr
@@ -198,7 +198,7 @@ theorem eLpNorm_le_eLpNorm_top_mul_eLpNorm (p : ℝ≥0∞) (f : α → E) {g :
     exact ha
   by_cases hp_zero : p = 0
   · simp only [hp_zero, eLpNorm_exponent_zero, mul_zero, le_zero_iff]
-  simp_rw [eLpNorm_eq_lintegral_rpow_nnnorm hp_zero hp_top, eLpNorm_exponent_top, eLpNormEssSup]
+  simp_rw [eLpNorm_eq_lintegral_rpow_enorm hp_zero hp_top, eLpNorm_exponent_top, eLpNormEssSup]
   calc
     (∫⁻ x, (‖b (f x) (g x)‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) ^ (1 / p.toReal) ≤
         (∫⁻ x, (‖f x‖₊ : ℝ≥0∞) ^ p.toReal * (‖g x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) ^ (1 / p.toReal) := by
@@ -255,7 +255,7 @@ theorem eLpNorm'_le_eLpNorm'_mul_eLpNorm' {p q r : ℝ} (hf : AEStronglyMeasurab
           h.mono fun a ha => (ENNReal.rpow_le_rpow_iff hp0_lt).mpr <| ENNReal.coe_le_coe.mpr <| ha
     _ ≤ _ := ?_
   simp_rw [eLpNorm', ENNReal.coe_mul]
-  exact ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr hp0_lt hpq hpqr μ hf.ennnorm hg.ennnorm
+  exact ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr hp0_lt hpq hpqr μ hf.enorm hg.enorm
 
 @[deprecated (since := "2024-07-27")]
 alias snorm'_le_snorm'_mul_snorm' := eLpNorm'_le_eLpNorm'_mul_eLpNorm'
diff --git a/Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean b/Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean
index bde7165f033fba..74b789bb4680a2 100644
--- a/Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean
+++ b/Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean
@@ -28,7 +28,7 @@ theorem eLpNorm'_add_le (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasura
         (∫⁻ a, ((fun a => (‖f a‖₊ : ℝ≥0∞)) + fun a => (‖g a‖₊ : ℝ≥0∞)) a ^ q ∂μ) ^ (1 / q) := by
       gcongr with a
       simp only [Pi.add_apply, ← ENNReal.coe_add, ENNReal.coe_le_coe, nnnorm_add_le]
-    _ ≤ eLpNorm' f q μ + eLpNorm' g q μ := ENNReal.lintegral_Lp_add_le hf.ennnorm hg.ennnorm hq1
+    _ ≤ eLpNorm' f q μ + eLpNorm' g q μ := ENNReal.lintegral_Lp_add_le hf.enorm hg.enorm hq1
 
 @[deprecated (since := "2024-07-27")]
 alias snorm'_add_le := eLpNorm'_add_le
@@ -41,7 +41,7 @@ theorem eLpNorm'_add_le_of_le_one (hf : AEStronglyMeasurable f μ) (hq0 : 0 ≤
       gcongr with a
       simp only [Pi.add_apply, ← ENNReal.coe_add, ENNReal.coe_le_coe, nnnorm_add_le]
     _ ≤ (2 : ℝ≥0∞) ^ (1 / q - 1) * (eLpNorm' f q μ + eLpNorm' g q μ) :=
-      ENNReal.lintegral_Lp_add_le_of_le_one hf.ennnorm hq0 hq1
+      ENNReal.lintegral_Lp_add_le_of_le_one hf.enorm hq0 hq1
 
 @[deprecated (since := "2024-07-27")]
 alias snorm'_add_le_of_le_one := eLpNorm'_add_le_of_le_one
diff --git a/Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean b/Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean
index b6776ff95e87ac..0cb5cd60fe951f 100644
--- a/Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean
+++ b/Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean
@@ -51,7 +51,7 @@ theorem essSup_trim (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : Measurable[m]
 
 theorem eLpNormEssSup_trim (hm : m ≤ m0) {f : α → E} (hf : StronglyMeasurable[m] f) :
     eLpNormEssSup f (μ.trim hm) = eLpNormEssSup f μ :=
-  essSup_trim _ (@StronglyMeasurable.ennnorm _ m _ _ _ hf)
+  essSup_trim _ (@StronglyMeasurable.enorm _ m _ _ _ hf)
 
 @[deprecated (since := "2024-07-27")]
 alias snormEssSup_trim := eLpNormEssSup_trim
diff --git a/Mathlib/MeasureTheory/Function/LpSpace.lean b/Mathlib/MeasureTheory/Function/LpSpace.lean
index 7ff9c55c9cd7d8..64caa8bea97f1f 100644
--- a/Mathlib/MeasureTheory/Function/LpSpace.lean
+++ b/Mathlib/MeasureTheory/Function/LpSpace.lean
@@ -248,10 +248,12 @@ theorem nnnorm_def (f : Lp E p μ) : ‖f‖₊ = ENNReal.toNNReal (eLpNorm f p
 protected theorem coe_nnnorm (f : Lp E p μ) : (‖f‖₊ : ℝ) = ‖f‖ :=
   rfl
 
-@[simp, norm_cast]
-theorem nnnorm_coe_ennreal (f : Lp E p μ) : (‖f‖₊ : ℝ≥0∞) = eLpNorm f p μ :=
+@[simp]
+theorem enorm_def (f : Lp E p μ) : ‖f‖ₑ = eLpNorm f p μ :=
   ENNReal.coe_toNNReal <| Lp.eLpNorm_ne_top f
 
+@[deprecated (since := "2025-01-20")] alias nnnorm_coe_ennreal := enorm_def
+
 @[simp]
 lemma norm_toLp (f : α → E) (hf : Memℒp f p μ) : ‖hf.toLp f‖ = ENNReal.toReal (eLpNorm f p μ) := by
   rw [norm_def, eLpNorm_congr_ae (Memℒp.coeFn_toLp hf)]
@@ -261,8 +263,10 @@ theorem nnnorm_toLp (f : α → E) (hf : Memℒp f p μ) :
     ‖hf.toLp f‖₊ = ENNReal.toNNReal (eLpNorm f p μ) :=
   NNReal.eq <| norm_toLp f hf
 
-theorem coe_nnnorm_toLp {f : α → E} (hf : Memℒp f p μ) : (‖hf.toLp f‖₊ : ℝ≥0∞) = eLpNorm f p μ := by
-  rw [nnnorm_toLp f hf, ENNReal.coe_toNNReal hf.2.ne]
+lemma enorm_toLp {f : α → E} (hf : Memℒp f p μ) : ‖hf.toLp f‖ₑ = eLpNorm f p μ := by
+  simp [enorm, nnnorm_toLp f hf, ENNReal.coe_toNNReal hf.2.ne]
+
+@[deprecated (since := "2025-01-20")] alias coe_nnnorm_toLp := enorm_toLp
 
 theorem dist_def (f g : Lp E p μ) : dist f g = (eLpNorm (⇑f - ⇑g) p μ).toReal := by
   simp_rw [dist, norm_def]
@@ -402,8 +406,8 @@ instance instNormedAddCommGroup [hp : Fact (1 ≤ p)] : NormedAddCommGroup (Lp E
         map_zero' := norm_zero
         neg' := by simp
         add_le' := fun f g => by
-          suffices (‖f + g‖₊ : ℝ≥0∞) ≤ ‖f‖₊ + ‖g‖₊ from mod_cast this
-          simp only [Lp.nnnorm_coe_ennreal]
+          suffices ‖f + g‖ₑ ≤ ‖f‖ₑ + ‖g‖ₑ by simpa [← ENNReal.coe_add, enorm] using this
+          simp only [Lp.enorm_def]
           exact (eLpNorm_congr_ae (AEEqFun.coeFn_add _ _)).trans_le
             (eLpNorm_add_le (Lp.aestronglyMeasurable _) (Lp.aestronglyMeasurable _) hp.out)
         eq_zero_of_map_eq_zero' := fun _ =>
@@ -426,9 +430,7 @@ variable [BoundedSMul 𝕜 E] [BoundedSMul 𝕜' E]
 
 theorem const_smul_mem_Lp (c : 𝕜) (f : Lp E p μ) : c • (f : α →ₘ[μ] E) ∈ Lp E p μ := by
   rw [mem_Lp_iff_eLpNorm_lt_top, eLpNorm_congr_ae (AEEqFun.coeFn_smul _ _)]
-  refine eLpNorm_const_smul_le.trans_lt ?_
-  rw [ENNReal.smul_def, smul_eq_mul, ENNReal.mul_lt_top_iff]
-  exact Or.inl ⟨ENNReal.coe_lt_top, f.prop⟩
+  exact eLpNorm_const_smul_le.trans_lt <| ENNReal.mul_lt_top ENNReal.coe_lt_top f.prop
 
 variable (E p μ 𝕜)
 
@@ -461,10 +463,9 @@ instance instIsScalarTower [SMul 𝕜 𝕜'] [IsScalarTower 𝕜 𝕜' E] : IsSc
 instance instBoundedSMul [Fact (1 ≤ p)] : BoundedSMul 𝕜 (Lp E p μ) :=
   -- TODO: add `BoundedSMul.of_nnnorm_smul_le`
   BoundedSMul.of_norm_smul_le fun r f => by
-    suffices (‖r • f‖₊ : ℝ≥0∞) ≤ ‖r‖₊ * ‖f‖₊ from mod_cast this
-    rw [nnnorm_def, nnnorm_def, ENNReal.coe_toNNReal (Lp.eLpNorm_ne_top _),
-      eLpNorm_congr_ae (coeFn_smul _ _), ENNReal.coe_toNNReal (Lp.eLpNorm_ne_top _)]
-    exact eLpNorm_const_smul_le
+    suffices ‖r • f‖ₑ ≤ ‖r‖ₑ * ‖f‖ₑ by simpa [← ENNReal.coe_mul, enorm] using this
+    simpa only [eLpNorm_congr_ae (coeFn_smul _ _), enorm_def]
+      using eLpNorm_const_smul_le (c := r) (f := f) (p := p)
 
 end BoundedSMul
 
@@ -536,8 +537,7 @@ theorem exists_eLpNorm_indicator_le (hp : p ≠ ∞) (c : E) {ε : ℝ≥0∞} (
   have hp₀ : 0 < p := bot_lt_iff_ne_bot.2 h'p
   have hp₀' : 0 ≤ 1 / p.toReal := div_nonneg zero_le_one ENNReal.toReal_nonneg
   have hp₀'' : 0 < p.toReal := ENNReal.toReal_pos hp₀.ne' hp
-  obtain ⟨η, hη_pos, hη_le⟩ :
-      ∃ η : ℝ≥0, 0 < η ∧ (‖c‖₊ : ℝ≥0∞) * (η : ℝ≥0∞) ^ (1 / p.toReal) ≤ ε := by
+  obtain ⟨η, hη_pos, hη_le⟩ : ∃ η : ℝ≥0, 0 < η ∧ ‖c‖ₑ * (η : ℝ≥0∞) ^ (1 / p.toReal) ≤ ε := by
     have :
       Filter.Tendsto (fun x : ℝ≥0 => ((‖c‖₊ * x ^ (1 / p.toReal) : ℝ≥0) : ℝ≥0∞)) (𝓝 0)
         (𝓝 (0 : ℝ≥0)) := by
@@ -548,8 +548,7 @@ theorem exists_eLpNorm_indicator_le (hp : p ≠ ∞) (c : E) {ε : ℝ≥0∞} (
     obtain ⟨δ, hδ, hδε'⟩ := NNReal.nhds_zero_basis.eventually_iff.mp (this.eventually_le_const hε')
     obtain ⟨η, hη, hηδ⟩ := exists_between hδ
     refine ⟨η, hη, ?_⟩
-    rw [← ENNReal.coe_rpow_of_nonneg _ hp₀', ← ENNReal.coe_mul]
-    exact hδε' hηδ
+    simpa only [← ENNReal.coe_rpow_of_nonneg _ hp₀', enorm, ← ENNReal.coe_mul] using hδε' hηδ
   refine ⟨η, hη_pos, fun s hs => ?_⟩
   refine (eLpNorm_indicator_const_le _ _).trans (le_trans ?_ hη_le)
   exact mul_le_mul_left' (ENNReal.rpow_le_rpow hs hp₀') _
@@ -614,13 +613,13 @@ theorem norm_indicatorConstLp (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) :
     ‖indicatorConstLp p hs hμs c‖ = ‖c‖ * (μ s).toReal ^ (1 / p.toReal) := by
   rw [Lp.norm_def, eLpNorm_congr_ae indicatorConstLp_coeFn,
     eLpNorm_indicator_const hs hp_ne_zero hp_ne_top, ENNReal.toReal_mul, ENNReal.toReal_rpow,
-    ENNReal.coe_toReal, coe_nnnorm]
+    toReal_enorm]
 
 theorem norm_indicatorConstLp_top (hμs_ne_zero : μ s ≠ 0) :
     ‖indicatorConstLp ∞ hs hμs c‖ = ‖c‖ := by
   rw [Lp.norm_def, eLpNorm_congr_ae indicatorConstLp_coeFn,
     eLpNorm_indicator_const' hs hμs_ne_zero ENNReal.top_ne_zero, ENNReal.top_toReal,
-    _root_.div_zero, ENNReal.rpow_zero, mul_one, ENNReal.coe_toReal, coe_nnnorm]
+    _root_.div_zero, ENNReal.rpow_zero, mul_one, toReal_enorm]
 
 theorem norm_indicatorConstLp' (hp_pos : p ≠ 0) (hμs_pos : μ s ≠ 0) :
     ‖indicatorConstLp p hs hμs c‖ = ‖c‖ * (μ s).toReal ^ (1 / p.toReal) := by
@@ -634,29 +633,33 @@ theorem norm_indicatorConstLp_le :
   rw [indicatorConstLp, Lp.norm_toLp]
   refine ENNReal.toReal_le_of_le_ofReal (by positivity) ?_
   refine (eLpNorm_indicator_const_le _ _).trans_eq ?_
-  rw [← coe_nnnorm, ENNReal.ofReal_mul (NNReal.coe_nonneg _), ENNReal.ofReal_coe_nnreal,
-    ENNReal.toReal_rpow, ENNReal.ofReal_toReal]
+  rw [ENNReal.ofReal_mul (norm_nonneg _), ofReal_norm, ENNReal.toReal_rpow, ENNReal.ofReal_toReal]
   exact ENNReal.rpow_ne_top_of_nonneg (by positivity) hμs
 
 theorem nnnorm_indicatorConstLp_le :
     ‖indicatorConstLp p hs hμs c‖₊ ≤ ‖c‖₊ * (μ s).toNNReal ^ (1 / p.toReal) :=
   norm_indicatorConstLp_le
 
-theorem ennnorm_indicatorConstLp_le :
-    (‖indicatorConstLp p hs hμs c‖₊ : ℝ≥0∞) ≤ ‖c‖₊ * (μ s) ^ (1 / p.toReal) := by
-  refine (ENNReal.coe_le_coe.mpr nnnorm_indicatorConstLp_le).trans_eq ?_
-  simp [ENNReal.coe_rpow_of_nonneg, ENNReal.coe_toNNReal hμs]
+theorem enorm_indicatorConstLp_le :
+    ‖indicatorConstLp p hs hμs c‖ₑ ≤ ‖c‖ₑ * μ s ^ (1 / p.toReal) := by
+  simpa [ENNReal.coe_rpow_of_nonneg, ENNReal.coe_toNNReal hμs, Lp.enorm_def, ← enorm_eq_nnnorm]
+    using ENNReal.coe_le_coe.2 <| nnnorm_indicatorConstLp_le (c := c) (hμs := hμs)
+
+@[deprecated (since := "2025-01-20")] alias ennnorm_indicatorConstLp_le := enorm_indicatorConstLp_le
 
-theorem edist_indicatorConstLp_eq_nnnorm {t : Set α} {ht : MeasurableSet t} {hμt : μ t ≠ ∞} :
+theorem edist_indicatorConstLp_eq_enorm {t : Set α} {ht : MeasurableSet t} {hμt : μ t ≠ ∞} :
     edist (indicatorConstLp p hs hμs c) (indicatorConstLp p ht hμt c) =
-      ‖indicatorConstLp p (hs.symmDiff ht) (measure_symmDiff_ne_top hμs hμt) c‖₊ := by
+      ‖indicatorConstLp p (hs.symmDiff ht) (measure_symmDiff_ne_top hμs hμt) c‖ₑ := by
   unfold indicatorConstLp
-  rw [Lp.edist_toLp_toLp, eLpNorm_indicator_sub_indicator, Lp.coe_nnnorm_toLp]
+  rw [Lp.edist_toLp_toLp, eLpNorm_indicator_sub_indicator, Lp.enorm_toLp]
+
+@[deprecated (since := "2025-01-20")]
+alias edist_indicatorConstLp_eq_nnnorm := edist_indicatorConstLp_eq_enorm
 
 theorem dist_indicatorConstLp_eq_norm {t : Set α} {ht : MeasurableSet t} {hμt : μ t ≠ ∞} :
     dist (indicatorConstLp p hs hμs c) (indicatorConstLp p ht hμt c) =
       ‖indicatorConstLp p (hs.symmDiff ht) (measure_symmDiff_ne_top hμs hμt) c‖ := by
-  rw [Lp.dist_edist, edist_indicatorConstLp_eq_nnnorm, ENNReal.coe_toReal, Lp.coe_nnnorm]
+  simp [Lp.dist_edist, edist_indicatorConstLp_eq_enorm, enorm, ENNReal.coe_toReal, Lp.coe_nnnorm]
 
 /-- A family of `indicatorConstLp` functions tends to an `indicatorConstLp`,
 if the underlying sets tend to the set in the sense of the measure of the symmetric difference. -/
@@ -741,12 +744,12 @@ theorem Lp.norm_const [NeZero μ] (hp_zero : p ≠ 0) :
     ‖Lp.const p μ c‖ = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal) := by
   have := NeZero.ne μ
   rw [← Memℒp.toLp_const, Lp.norm_toLp, eLpNorm_const] <;> try assumption
-  rw [ENNReal.toReal_mul, ENNReal.coe_toReal, ← ENNReal.toReal_rpow, coe_nnnorm]
+  rw [ENNReal.toReal_mul, toReal_enorm, ← ENNReal.toReal_rpow]
 
 theorem Lp.norm_const' (hp_zero : p ≠ 0) (hp_top : p ≠ ∞) :
     ‖Lp.const p μ c‖ = ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal) := by
   rw [← Memℒp.toLp_const, Lp.norm_toLp, eLpNorm_const'] <;> try assumption
-  rw [ENNReal.toReal_mul, ENNReal.coe_toReal, ← ENNReal.toReal_rpow, coe_nnnorm]
+  rw [ENNReal.toReal_mul, toReal_enorm, ← ENNReal.toReal_rpow]
 
 theorem Lp.norm_const_le : ‖Lp.const p μ c‖ ≤ ‖c‖ * (μ Set.univ).toReal ^ (1 / p.toReal) := by
   rw [← indicatorConstLp_univ]
@@ -1150,18 +1153,15 @@ namespace MeasureTheory
 namespace Lp
 
 theorem eLpNorm'_lim_eq_lintegral_liminf {ι} [Nonempty ι] [LinearOrder ι] {f : ι → α → G} {p : ℝ}
-    (hp_nonneg : 0 ≤ p) {f_lim : α → G}
-    (h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
-    eLpNorm' f_lim p μ = (∫⁻ a, atTop.liminf fun m => (‖f m a‖₊ : ℝ≥0∞) ^ p ∂μ) ^ (1 / p) := by
-  suffices h_no_pow :
-      (∫⁻ a, (‖f_lim a‖₊ : ℝ≥0∞) ^ p ∂μ) = ∫⁻ a, atTop.liminf fun m => (‖f m a‖₊ : ℝ≥0∞) ^ p ∂μ by
-    rw [eLpNorm'_eq_lintegral_nnnorm, h_no_pow]
+    {f_lim : α → G} (h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
+    eLpNorm' f_lim p μ = (∫⁻ a, atTop.liminf (‖f · a‖ₑ ^ p) ∂μ) ^ (1 / p) := by
+  suffices h_no_pow : (∫⁻ a, ‖f_lim a‖ₑ ^ p ∂μ) = ∫⁻ a, atTop.liminf fun m => ‖f m a‖ₑ ^ p ∂μ by
+    rw [eLpNorm'_eq_lintegral_enorm, h_no_pow]
   refine lintegral_congr_ae (h_lim.mono fun a ha => ?_)
   dsimp only
   rw [Tendsto.liminf_eq]
-  simp_rw [← ENNReal.coe_rpow_of_nonneg _ hp_nonneg, ENNReal.tendsto_coe]
-  refine ((NNReal.continuous_rpow_const hp_nonneg).tendsto ‖f_lim a‖₊).comp ?_
-  exact (continuous_nnnorm.tendsto (f_lim a)).comp ha
+  refine (ENNReal.continuous_rpow_const.tendsto ‖f_lim a‖₊).comp ?_
+  exact (continuous_enorm.tendsto (f_lim a)).comp ha
 
 @[deprecated (since := "2024-07-27")]
 alias snorm'_lim_eq_lintegral_liminf := eLpNorm'_lim_eq_lintegral_liminf
@@ -1170,9 +1170,9 @@ theorem eLpNorm'_lim_le_liminf_eLpNorm' {E} [NormedAddCommGroup E] {f : ℕ →
     (hp_pos : 0 < p) (hf : ∀ n, AEStronglyMeasurable (f n) μ) {f_lim : α → E}
     (h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
     eLpNorm' f_lim p μ ≤ atTop.liminf fun n => eLpNorm' (f n) p μ := by
-  rw [eLpNorm'_lim_eq_lintegral_liminf hp_pos.le h_lim]
+  rw [eLpNorm'_lim_eq_lintegral_liminf h_lim]
   rw [one_div, ← ENNReal.le_rpow_inv_iff (by simp [hp_pos] : 0 < p⁻¹), inv_inv]
-  refine (lintegral_liminf_le' fun m => (hf m).ennnorm.pow_const _).trans_eq ?_
+  refine (lintegral_liminf_le' fun m => (hf m).enorm.pow_const _).trans_eq ?_
   have h_pow_liminf :
     atTop.liminf (fun n ↦ eLpNorm' (f n) p μ) ^ p
       = atTop.liminf fun n ↦ eLpNorm' (f n) p μ ^ p := by
@@ -1181,7 +1181,7 @@ theorem eLpNorm'_lim_le_liminf_eLpNorm' {E} [NormedAddCommGroup E] {f : ℕ →
     refine (h_rpow_mono.orderIsoOfSurjective _ h_rpow_surj).liminf_apply ?_ ?_ ?_ ?_
     all_goals isBoundedDefault
   rw [h_pow_liminf]
-  simp_rw [eLpNorm'_eq_lintegral_nnnorm, ← ENNReal.rpow_mul, one_div,
+  simp_rw [eLpNorm'_eq_lintegral_enorm, ← ENNReal.rpow_mul, one_div,
     inv_mul_cancel₀ hp_pos.ne.symm, ENNReal.rpow_one]
 
 @[deprecated (since := "2024-07-27")]
@@ -1189,13 +1189,12 @@ alias snorm'_lim_le_liminf_snorm' := eLpNorm'_lim_le_liminf_eLpNorm'
 
 theorem eLpNorm_exponent_top_lim_eq_essSup_liminf {ι} [Nonempty ι] [LinearOrder ι] {f : ι → α → G}
     {f_lim : α → G} (h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
-    eLpNorm f_lim ∞ μ = essSup (fun x => atTop.liminf fun m => (‖f m x‖₊ : ℝ≥0∞)) μ := by
-  rw [eLpNorm_exponent_top, eLpNormEssSup_eq_essSup_nnnorm]
+    eLpNorm f_lim ∞ μ = essSup (fun x => atTop.liminf fun m => ‖f m x‖ₑ) μ := by
+  rw [eLpNorm_exponent_top, eLpNormEssSup_eq_essSup_enorm]
   refine essSup_congr_ae (h_lim.mono fun x hx => ?_)
   dsimp only
   apply (Tendsto.liminf_eq ..).symm
-  rw [ENNReal.tendsto_coe]
-  exact (continuous_nnnorm.tendsto (f_lim x)).comp hx
+  exact (continuous_enorm.tendsto (f_lim x)).comp hx
 
 @[deprecated (since := "2024-07-27")]
 alias snorm_exponent_top_lim_eq_essSup_liminf := eLpNorm_exponent_top_lim_eq_essSup_liminf
@@ -1206,7 +1205,7 @@ theorem eLpNorm_exponent_top_lim_le_liminf_eLpNorm_exponent_top {ι} [Nonempty 
     eLpNorm f_lim ∞ μ ≤ atTop.liminf fun n => eLpNorm (f n) ∞ μ := by
   rw [eLpNorm_exponent_top_lim_eq_essSup_liminf h_lim]
   simp_rw [eLpNorm_exponent_top, eLpNormEssSup]
-  exact ENNReal.essSup_liminf_le fun n => fun x => (‖f n x‖₊ : ℝ≥0∞)
+  exact ENNReal.essSup_liminf_le _
 
 @[deprecated (since := "2024-07-27")]
 alias snorm_exponent_top_lim_le_liminf_snorm_exponent_top :=
@@ -1335,35 +1334,31 @@ private theorem eLpNorm'_sum_norm_sub_le_tsum_of_cauchy_eLpNorm' {f : ℕ → α
 alias snorm'_sum_norm_sub_le_tsum_of_cauchy_snorm' :=
   eLpNorm'_sum_norm_sub_le_tsum_of_cauchy_eLpNorm'
 
-private theorem lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum
+private theorem lintegral_rpow_sum_enorm_sub_le_rpow_tsum
     {f : ℕ → α → E} {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞} (n : ℕ)
     (hn : eLpNorm' (fun x => ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' i, B i) :
-    (∫⁻ a, (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖₊ : ℝ≥0∞) ^ p ∂μ) ≤
-      (∑' i, B i) ^ p := by
+    (∫⁻ a, (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ≤ (∑' i, B i) ^ p := by
   have hp_pos : 0 < p := zero_lt_one.trans_le hp1
   rw [← inv_inv p, @ENNReal.le_rpow_inv_iff _ _ p⁻¹ (by simp [hp_pos]), inv_inv p]
-  simp_rw [eLpNorm'_eq_lintegral_nnnorm, one_div] at hn
+  simp_rw [eLpNorm'_eq_lintegral_enorm, one_div] at hn
   have h_nnnorm_nonneg :
-    (fun a => (‖∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖₊ : ℝ≥0∞) ^ p) = fun a =>
-      (∑ i ∈ Finset.range (n + 1), (‖f (i + 1) a - f i a‖₊ : ℝ≥0∞)) ^ p := by
+    (fun a => ‖∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ₑ ^ p) = fun a =>
+      (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p := by
     ext1 a
     congr
-    simp_rw [← ofReal_norm_eq_coe_nnnorm]
+    simp_rw [← ofReal_norm_eq_enorm]
     rw [← ENNReal.ofReal_sum_of_nonneg]
     · rw [Real.norm_of_nonneg _]
       exact Finset.sum_nonneg fun x _ => norm_nonneg _
     · exact fun x _ => norm_nonneg _
   rwa [h_nnnorm_nonneg] at hn
 
-private theorem lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum {f : ℕ → α → E}
+private theorem lintegral_rpow_tsum_coe_enorm_sub_le_tsum {f : ℕ → α → E}
     (hf : ∀ n, AEStronglyMeasurable (f n) μ) {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞}
-    (h :
-      ∀ n,
-        (∫⁻ a, (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖₊ : ℝ≥0∞) ^ p ∂μ) ≤
-          (∑' i, B i) ^ p) :
-    (∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖₊ : ℝ≥0∞) ^ p ∂μ) ^ (1 / p) ≤ ∑' i, B i := by
+    (h : ∀ n, ∫⁻ a, (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' i, B i) ^ p) :
+    (∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' i, B i := by
   have hp_pos : 0 < p := zero_lt_one.trans_le hp1
-  suffices h_pow : (∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖₊ : ℝ≥0∞) ^ p ∂μ) ≤ (∑' i, B i) ^ p by
+  suffices h_pow : (∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ≤ (∑' i, B i) ^ p by
       rwa [one_div, ← ENNReal.le_rpow_inv_iff (by simp [hp_pos] : 0 < p⁻¹), inv_inv]
   have h_tsum_1 :
     ∀ g : ℕ → ℝ≥0∞, ∑' i, g i = atTop.liminf fun n => ∑ i ∈ Finset.range (n + 1), g i := by
@@ -1372,35 +1367,29 @@ private theorem lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum {f : ℕ → α → E
   simp_rw [h_tsum_1 _]
   rw [← h_tsum_1]
   have h_liminf_pow :
-    (∫⁻ a, (atTop.liminf
-      fun n => ∑ i ∈ Finset.range (n + 1), (‖f (i + 1) a - f i a‖₊ : ℝ≥0∞)) ^ p ∂μ) =
-      ∫⁻ a, atTop.liminf
-        fun n => (∑ i ∈ Finset.range (n + 1), (‖f (i + 1) a - f i a‖₊ : ℝ≥0∞)) ^ p ∂μ := by
+    ∫⁻ a, (atTop.liminf fun n => ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ =
+      ∫⁻ a, atTop.liminf fun n => (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ := by
     refine lintegral_congr fun x => ?_
     have h_rpow_mono := ENNReal.strictMono_rpow_of_pos (zero_lt_one.trans_le hp1)
     have h_rpow_surj := (ENNReal.rpow_left_bijective hp_pos.ne.symm).2
     refine (h_rpow_mono.orderIsoOfSurjective _ h_rpow_surj).liminf_apply ?_ ?_ ?_ ?_
     all_goals isBoundedDefault
   rw [h_liminf_pow]
-  refine (lintegral_liminf_le' ?_).trans ?_
-  · exact fun n =>
-      (Finset.aemeasurable_sum (Finset.range (n + 1)) fun i _ =>
-            ((hf (i + 1)).sub (hf i)).ennnorm).pow_const
-        _
-  · exact liminf_le_of_frequently_le' (Frequently.of_forall h)
-
-private theorem tsum_nnnorm_sub_ae_lt_top {f : ℕ → α → E} (hf : ∀ n, AEStronglyMeasurable (f n) μ)
+  refine (lintegral_liminf_le' fun n ↦ ?_).trans <| liminf_le_of_frequently_le' <| .of_forall h
+  exact ((Finset.range _).aemeasurable_sum fun i _ ↦ ((hf _).sub (hf i)).enorm).pow_const _
+
+private theorem tsum_enorm_sub_ae_lt_top {f : ℕ → α → E} (hf : ∀ n, AEStronglyMeasurable (f n) μ)
     {p : ℝ} (hp1 : 1 ≤ p) {B : ℕ → ℝ≥0∞} (hB : ∑' i, B i ≠ ∞)
-    (h : (∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖₊ : ℝ≥0∞) ^ p ∂μ) ^ (1 / p) ≤ ∑' i, B i) :
-    ∀ᵐ x ∂μ, (∑' i, ‖f (i + 1) x - f i x‖₊ : ℝ≥0∞) < ∞ := by
+    (h : (∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' i, B i) :
+    ∀ᵐ x ∂μ, ∑' i, ‖f (i + 1) x - f i x‖ₑ < ∞ := by
   have hp_pos : 0 < p := zero_lt_one.trans_le hp1
-  have h_integral : (∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖₊ : ℝ≥0∞) ^ p ∂μ) < ∞ := by
+  have h_integral : ∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ < ∞ := by
     have h_tsum_lt_top : (∑' i, B i) ^ p < ∞ := ENNReal.rpow_lt_top_of_nonneg hp_pos.le hB
     refine lt_of_le_of_lt ?_ h_tsum_lt_top
     rwa [one_div, ← ENNReal.le_rpow_inv_iff (by simp [hp_pos] : 0 < p⁻¹), inv_inv] at h
-  have rpow_ae_lt_top : ∀ᵐ x ∂μ, (∑' i, ‖f (i + 1) x - f i x‖₊ : ℝ≥0∞) ^ p < ∞ := by
+  have rpow_ae_lt_top : ∀ᵐ x ∂μ, (∑' i, ‖f (i + 1) x - f i x‖ₑ) ^ p < ∞ := by
     refine ae_lt_top' (AEMeasurable.pow_const ?_ _) h_integral.ne
-    exact AEMeasurable.ennreal_tsum fun n => ((hf (n + 1)).sub (hf n)).ennnorm
+    exact AEMeasurable.ennreal_tsum fun n => ((hf (n + 1)).sub (hf n)).enorm
   refine rpow_ae_lt_top.mono fun x hx => ?_
   rwa [← ENNReal.lt_rpow_inv_iff hp_pos,
     ENNReal.top_rpow_of_pos (by simp [hp_pos] : 0 < p⁻¹)] at hx
@@ -1413,15 +1402,13 @@ theorem ae_tendsto_of_cauchy_eLpNorm' [CompleteSpace E] {f : ℕ → α → E} {
     have h1 :
       ∀ n, eLpNorm' (fun x => ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) x - f i x‖) p μ ≤ ∑' i, B i :=
       eLpNorm'_sum_norm_sub_le_tsum_of_cauchy_eLpNorm' hf hp1 h_cau
-    have h2 :
-      ∀ n,
-        (∫⁻ a, (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖₊ : ℝ≥0∞) ^ p ∂μ) ≤
-          (∑' i, B i) ^ p :=
-      fun n => lintegral_rpow_sum_coe_nnnorm_sub_le_rpow_tsum hp1 n (h1 n)
-    have h3 : (∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖₊ : ℝ≥0∞) ^ p ∂μ) ^ (1 / p) ≤ ∑' i, B i :=
-      lintegral_rpow_tsum_coe_nnnorm_sub_le_tsum hf hp1 h2
-    have h4 : ∀ᵐ x ∂μ, (∑' i, ‖f (i + 1) x - f i x‖₊ : ℝ≥0∞) < ∞ :=
-      tsum_nnnorm_sub_ae_lt_top hf hp1 hB h3
+    have h2 n :
+        ∫⁻ a, (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ ≤ (∑' i, B i) ^ p :=
+      lintegral_rpow_sum_enorm_sub_le_rpow_tsum hp1 n (h1 n)
+    have h3 : (∫⁻ a, (∑' i, ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' i, B i :=
+      lintegral_rpow_tsum_coe_enorm_sub_le_tsum hf hp1 h2
+    have h4 : ∀ᵐ x ∂μ, ∑' i, ‖f (i + 1) x - f i x‖ₑ < ∞ :=
+      tsum_enorm_sub_ae_lt_top hf hp1 hB h3
     exact h4.mono fun x hx => .of_nnnorm <| ENNReal.tsum_coe_ne_top_iff_summable.mp hx.ne
   have h :
     ∀ᵐ x ∂μ, ∃ l : E,
@@ -1453,7 +1440,7 @@ theorem ae_tendsto_of_cauchy_eLpNorm [CompleteSpace E] {f : ℕ → α → E}
     ∀ᵐ x ∂μ, ∃ l : E, atTop.Tendsto (fun n => f n x) (𝓝 l) := by
   by_cases hp_top : p = ∞
   · simp_rw [hp_top] at *
-    have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → (‖(f n - f m) x‖₊ : ℝ≥0∞) < B N := by
+    have h_cau_ae : ∀ᵐ x ∂μ, ∀ N n m, N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N := by
       simp_rw [ae_all_iff]
       exact fun N n m hnN hmN => ae_lt_of_essSup_lt (h_cau N n m hnN hmN)
     simp_rw [eLpNorm_exponent_top, eLpNormEssSup] at h_cau
@@ -1463,7 +1450,7 @@ theorem ae_tendsto_of_cauchy_eLpNorm [CompleteSpace E] {f : ℕ → α → E}
       specialize hx N n m hnN hmN
       rw [_root_.dist_eq_norm,
         ← ENNReal.ofReal_le_iff_le_toReal (ENNReal.ne_top_of_tsum_ne_top hB N),
-        ofReal_norm_eq_coe_nnnorm]
+        ofReal_norm_eq_enorm]
       exact hx.le
     · rw [← ENNReal.zero_toReal]
       exact
@@ -1745,27 +1732,35 @@ namespace MeasureTheory
 
 namespace Lp
 
-theorem pow_mul_meas_ge_le_norm (f : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) (ε : ℝ≥0∞) :
-    (ε * μ { x | ε ≤ (‖f x‖₊ : ℝ≥0∞) ^ p.toReal }) ^ (1 / p.toReal) ≤ ENNReal.ofReal ‖f‖ :=
+/-- A version of **Markov's inequality** with elements of Lp. -/
+lemma pow_mul_meas_ge_le_enorm (f : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) (ε : ℝ≥0∞) :
+    (ε * μ {x | ε ≤ ‖f x‖ₑ ^ p.toReal}) ^ (1 / p.toReal) ≤ ENNReal.ofReal ‖f‖ :=
   (ENNReal.ofReal_toReal (eLpNorm_ne_top f)).symm ▸
     pow_mul_meas_ge_le_eLpNorm μ hp_ne_zero hp_ne_top (Lp.aestronglyMeasurable f) ε
 
-theorem mul_meas_ge_le_pow_norm (f : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) (ε : ℝ≥0∞) :
-    ε * μ { x | ε ≤ (‖f x‖₊ : ℝ≥0∞) ^ p.toReal } ≤ ENNReal.ofReal ‖f‖ ^ p.toReal :=
+/-- A version of **Markov's inequality** with elements of Lp. -/
+lemma mul_meas_ge_le_pow_enorm (f : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) (ε : ℝ≥0∞) :
+    ε * μ {x | ε ≤ ‖f x‖ₑ ^ p.toReal} ≤ ENNReal.ofReal ‖f‖ ^ p.toReal :=
   (ENNReal.ofReal_toReal (eLpNorm_ne_top f)).symm ▸
     mul_meas_ge_le_pow_eLpNorm μ hp_ne_zero hp_ne_top (Lp.aestronglyMeasurable f) ε
 
-/-- A version of Markov's inequality with elements of Lp. -/
-theorem mul_meas_ge_le_pow_norm' (f : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
-    (ε : ℝ≥0∞) : ε ^ p.toReal * μ { x | ε ≤ ‖f x‖₊ } ≤ ENNReal.ofReal ‖f‖ ^ p.toReal :=
+/-- A version of **Markov's inequality** with elements of Lp. -/
+theorem mul_meas_ge_le_pow_enorm' (f : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
+    (ε : ℝ≥0∞) : ε ^ p.toReal * μ {x | ε ≤ ‖f x‖₊ } ≤ ENNReal.ofReal ‖f‖ ^ p.toReal :=
   (ENNReal.ofReal_toReal (eLpNorm_ne_top f)).symm ▸
     mul_meas_ge_le_pow_eLpNorm' μ hp_ne_zero hp_ne_top (Lp.aestronglyMeasurable f) ε
 
-theorem meas_ge_le_mul_pow_norm (f : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {ε : ℝ≥0∞}
-    (hε : ε ≠ 0) : μ { x | ε ≤ ‖f x‖₊ } ≤ ε⁻¹ ^ p.toReal * ENNReal.ofReal ‖f‖ ^ p.toReal :=
+/-- A version of **Markov's inequality** with elements of Lp. -/
+theorem meas_ge_le_mul_pow_enorm (f : Lp E p μ) (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞) {ε : ℝ≥0∞}
+    (hε : ε ≠ 0) : μ {x | ε ≤ ‖f x‖₊} ≤ ε⁻¹ ^ p.toReal * ENNReal.ofReal ‖f‖ ^ p.toReal :=
   (ENNReal.ofReal_toReal (eLpNorm_ne_top f)).symm ▸
     meas_ge_le_mul_pow_eLpNorm μ hp_ne_zero hp_ne_top (Lp.aestronglyMeasurable f) hε
 
+@[deprecated (since := "2025-01-20")] alias pow_mul_meas_ge_le_norm := pow_mul_meas_ge_le_enorm
+@[deprecated (since := "2025-01-20")] alias mul_meas_ge_le_pow_norm := mul_meas_ge_le_pow_enorm
+@[deprecated (since := "2025-01-20")] alias mul_meas_ge_le_pow_norm' := mul_meas_ge_le_pow_enorm'
+@[deprecated (since := "2025-01-20")] alias meas_ge_le_mul_pow_norm := meas_ge_le_mul_pow_enorm
+
 end Lp
 
 end MeasureTheory
diff --git a/Mathlib/MeasureTheory/Function/SimpleFunc.lean b/Mathlib/MeasureTheory/Function/SimpleFunc.lean
index 3d030c368a939f..80239aa069a9a6 100644
--- a/Mathlib/MeasureTheory/Function/SimpleFunc.lean
+++ b/Mathlib/MeasureTheory/Function/SimpleFunc.lean
@@ -28,7 +28,6 @@ open Filter ENNReal
 
 open Function (support)
 
-open scoped Classical
 open Topology NNReal ENNReal MeasureTheory
 
 namespace MeasureTheory
@@ -177,6 +176,7 @@ theorem sum_range_measure_preimage_singleton (f : α →ₛ β) (μ : Measure α
     (∑ y ∈ f.range, μ (f ⁻¹' {y})) = μ univ := by
   rw [f.sum_measure_preimage_singleton, coe_range, preimage_range]
 
+open scoped Classical in
 /-- If-then-else as a `SimpleFunc`. -/
 def piecewise (s : Set α) (hs : MeasurableSet s) (f g : α →ₛ β) : α →ₛ β :=
   ⟨s.piecewise f g, fun _ =>
@@ -184,15 +184,18 @@ def piecewise (s : Set α) (hs : MeasurableSet s) (f g : α →ₛ β) : α →
     f.measurable.piecewise hs g.measurable trivial,
     (f.finite_range.union g.finite_range).subset range_ite_subset⟩
 
+open scoped Classical in
 @[simp]
 theorem coe_piecewise {s : Set α} (hs : MeasurableSet s) (f g : α →ₛ β) :
     ⇑(piecewise s hs f g) = s.piecewise f g :=
   rfl
 
+open scoped Classical in
 theorem piecewise_apply {s : Set α} (hs : MeasurableSet s) (f g : α →ₛ β) (a) :
     piecewise s hs f g a = if a ∈ s then f a else g a :=
   rfl
 
+open scoped Classical in
 @[simp]
 theorem piecewise_compl {s : Set α} (hs : MeasurableSet sᶜ) (f g : α →ₛ β) :
     piecewise sᶜ hs f g = piecewise s hs.of_compl g f :=
@@ -206,6 +209,7 @@ theorem piecewise_univ (f g : α →ₛ β) : piecewise univ MeasurableSet.univ
 theorem piecewise_empty (f g : α →ₛ β) : piecewise ∅ MeasurableSet.empty f g = g :=
   coe_injective <| by simp
 
+open scoped Classical in
 @[simp]
 theorem piecewise_same (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) :
     piecewise s hs f f = f :=
@@ -215,6 +219,7 @@ theorem support_indicator [Zero β] {s : Set α} (hs : MeasurableSet s) (f : α
     Function.support (f.piecewise s hs (SimpleFunc.const α 0)) = s ∩ Function.support f :=
   Set.support_indicator
 
+open scoped Classical in
 theorem range_indicator {s : Set α} (hs : MeasurableSet s) (hs_nonempty : s.Nonempty)
     (hs_ne_univ : s ≠ univ) (x y : β) :
     (piecewise s hs (const α x) (const α y)).range = {x, y} := by
@@ -261,12 +266,14 @@ theorem range_map [DecidableEq γ] (g : β → γ) (f : α →ₛ β) : (f.map g
 theorem map_const (g : β → γ) (b : β) : (const α b).map g = const α (g b) :=
   rfl
 
+open scoped Classical in
 theorem map_preimage (f : α →ₛ β) (g : β → γ) (s : Set γ) :
     f.map g ⁻¹' s = f ⁻¹' ↑{b ∈ f.range | g b ∈ s} := by
   simp only [coe_range, sep_mem_eq, coe_map, Finset.coe_filter,
     ← mem_preimage, inter_comm, preimage_inter_range, ← Finset.mem_coe]
   exact preimage_comp
 
+open scoped Classical in
 theorem map_preimage_singleton (f : α →ₛ β) (g : β → γ) (c : γ) :
     f.map g ⁻¹' {c} = f ⁻¹' ↑{b ∈ f.range | g b = c} :=
   map_preimage _ _ _
@@ -560,6 +567,7 @@ instance instPreorder : Preorder (α →ₛ β) := Preorder.lift (⇑)
 @[gcongr] protected alias ⟨_, GCongr.coe_le_coe⟩ := coe_le_coe
 @[gcongr] protected alias ⟨_, GCongr.coe_lt_coe⟩ := coe_lt_coe
 
+open scoped Classical in
 @[gcongr]
 lemma piecewise_mono (hf : ∀ a ∈ s, f₁ a ≤ f₂ a) (hg : ∀ a ∉ s, g₁ a ≤ g₂ a) :
     piecewise s hs f₁ g₁ ≤ piecewise s hs f₂ g₂ := Set.piecewise_mono hf hg
@@ -600,6 +608,7 @@ instance instBoundedOrder [LE β] [BoundedOrder β] : BoundedOrder (α →ₛ β
 
 theorem finset_sup_apply [SemilatticeSup β] [OrderBot β] {f : γ → α →ₛ β} (s : Finset γ) (a : α) :
     s.sup f a = s.sup fun c => f c a := by
+  classical
   refine Finset.induction_on s rfl ?_
   intro a s _ ih
   rw [Finset.sup_insert, Finset.sup_insert, sup_apply, ih]
@@ -608,6 +617,7 @@ section Restrict
 
 variable [Zero β]
 
+open scoped Classical in
 /-- Restrict a simple function `f : α →ₛ β` to a set `s`. If `s` is measurable,
 then `f.restrict s a = if a ∈ s then f a else 0`, otherwise `f.restrict s = const α 0`. -/
 def restrict (f : α →ₛ β) (s : Set α) : α →ₛ β :=
@@ -620,6 +630,7 @@ theorem restrict_of_not_measurable {f : α →ₛ β} {s : Set α} (hs : ¬Measu
 @[simp]
 theorem coe_restrict (f : α →ₛ β) {s : Set α} (hs : MeasurableSet s) :
     ⇑(restrict f s) = indicator s f := by
+  classical
   rw [restrict, dif_pos hs, coe_piecewise, coe_zero, piecewise_eq_indicator]
 
 @[simp]
@@ -628,6 +639,7 @@ theorem restrict_univ (f : α →ₛ β) : restrict f univ = f := by simp [restr
 @[simp]
 theorem restrict_empty (f : α →ₛ β) : restrict f ∅ = 0 := by simp [restrict]
 
+open scoped Classical in
 theorem map_restrict_of_zero [Zero γ] {g : β → γ} (hg : g 0 = 0) (f : α →ₛ β) (s : Set α) :
     (f.restrict s).map g = (f.map g).restrict s :=
   ext fun x =>
@@ -657,6 +669,7 @@ theorem mem_restrict_range {r : β} {s : Set α} {f : α →ₛ β} (hs : Measur
     r ∈ (restrict f s).range ↔ r = 0 ∧ s ≠ univ ∨ r ∈ f '' s := by
   rw [← Finset.mem_coe, coe_range, coe_restrict _ hs, mem_range_indicator]
 
+open scoped Classical in
 theorem mem_image_of_mem_range_restrict {r : β} {s : Set α} {f : α →ₛ β}
     (hr : r ∈ (restrict f s).range) (h0 : r ≠ 0) : r ∈ f '' s :=
   if hs : MeasurableSet s then by simpa [mem_restrict_range hs, h0, -mem_range] using hr
@@ -664,6 +677,7 @@ theorem mem_image_of_mem_range_restrict {r : β} {s : Set α} {f : α →ₛ β}
     rw [restrict_of_not_measurable hs] at hr
     exact (h0 <| eq_zero_of_mem_range_zero hr).elim
 
+open scoped Classical in
 @[gcongr, mono]
 theorem restrict_mono [Preorder β] (s : Set α) {f g : α →ₛ β} (H : f ≤ g) :
     f.restrict s ≤ g.restrict s :=
@@ -685,6 +699,7 @@ of the set `{i k | k ≤ n ∧ i k ≤ f a}`, see `approx_apply` and `iSup_appro
 def approx (i : ℕ → β) (f : α → β) (n : ℕ) : α →ₛ β :=
   (Finset.range n).sup fun k => restrict (const α (i k)) { a : α | i k ≤ f a }
 
+open scoped Classical in
 theorem approx_apply [TopologicalSpace β] [OrderClosedTopology β] [MeasurableSpace β]
     [OpensMeasurableSpace β] {i : ℕ → β} {f : α → β} {n : ℕ} (a : α) (hf : Measurable f) :
     (approx i f n : α →ₛ β) a = (Finset.range n).sup fun k => if i k ≤ f a then i k else 0 := by
@@ -904,6 +919,7 @@ theorem lintegral_sum {m : MeasurableSpace α} {ι} (f : α →ₛ ℝ≥0∞) (
     ENNReal.tsum_mul_left]
   apply ENNReal.tsum_comm
 
+open scoped Classical in
 theorem restrict_lintegral (f : α →ₛ ℝ≥0∞) {s : Set α} (hs : MeasurableSet s) :
     (restrict f s).lintegral μ = ∑ r ∈ f.range, r * μ (f ⁻¹' {r} ∩ s) :=
   calc
@@ -1006,6 +1022,7 @@ section FinMeasSupp
 
 open Finset Function
 
+open scoped Classical in
 theorem support_eq [MeasurableSpace α] [Zero β] (f : α →ₛ β) :
     support f = ⋃ y ∈ {y ∈ f.range | y ≠ 0}, f ⁻¹' {y} :=
   Set.ext fun x => by
@@ -1020,6 +1037,7 @@ theorem measurableSet_support [MeasurableSpace α] (f : α →ₛ β) : Measurab
 
 lemma measure_support_lt_top (f : α →ₛ β) (hf : ∀ y, y ≠ 0 → μ (f ⁻¹' {y}) < ∞) :
     μ (support f) < ∞ := by
+  classical
   rw [support_eq]
   refine (measure_biUnion_finset_le _ _).trans_lt (ENNReal.sum_lt_top.mpr fun y hy => ?_)
   rw [Finset.mem_filter] at hy
@@ -1034,6 +1052,7 @@ theorem finMeasSupp_iff_support : f.FinMeasSupp μ ↔ μ (support f) < ∞ :=
   Iff.rfl
 
 theorem finMeasSupp_iff : f.FinMeasSupp μ ↔ ∀ y, y ≠ 0 → μ (f ⁻¹' {y}) < ∞ := by
+  classical
   constructor
   · refine fun h y hy => lt_of_le_of_lt (measure_mono ?_) h
     exact fun x hx (H : f x = 0) => hy <| H ▸ Eq.symm hx
@@ -1124,6 +1143,7 @@ protected theorem induction {α γ} [MeasurableSpace α] [AddMonoid γ] {P : Sim
         P (SimpleFunc.piecewise s hs (SimpleFunc.const _ c) (SimpleFunc.const _ 0)))
     (h_add : ∀ ⦃f g : SimpleFunc α γ⦄, Disjoint (support f) (support g) → P f → P g → P (f + g))
     (f : SimpleFunc α γ) : P f := by
+  classical
   generalize h : f.range \ {0} = s
   rw [← Finset.coe_inj, Finset.coe_sdiff, Finset.coe_singleton, SimpleFunc.coe_range] at h
   induction s using Finset.induction generalizing f with
diff --git a/Mathlib/MeasureTheory/Function/SimpleFuncDense.lean b/Mathlib/MeasureTheory/Function/SimpleFuncDense.lean
index d8993de25b2648..a764aa7f10af51 100644
--- a/Mathlib/MeasureTheory/Function/SimpleFuncDense.lean
+++ b/Mathlib/MeasureTheory/Function/SimpleFuncDense.lean
@@ -34,7 +34,6 @@ by a sequence of simple functions.
 
 open Set Function Filter TopologicalSpace ENNReal EMetric Finset
 
-open scoped Classical
 open Topology ENNReal MeasureTheory
 
 variable {α β ι E F 𝕜 : Type*}
diff --git a/Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean b/Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
index 7ad4405faa9c1a..a7d019b6e5ebea 100644
--- a/Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
+++ b/Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
@@ -3,7 +3,7 @@ Copyright (c) 2022 Zhouhang Zhou. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
 -/
-import Mathlib.MeasureTheory.Function.L1Space
+import Mathlib.MeasureTheory.Function.L1Space.AEEqFun
 import Mathlib.MeasureTheory.Function.SimpleFuncDense
 
 /-!
@@ -45,7 +45,7 @@ noncomputable section
 
 open Set Function Filter TopologicalSpace ENNReal EMetric Finset
 
-open scoped Classical Topology ENNReal MeasureTheory
+open scoped Topology ENNReal MeasureTheory
 
 variable {α β ι E F 𝕜 : Type*}
 
@@ -73,16 +73,14 @@ theorem nnnorm_approxOn_le [OpensMeasurableSpace E] {f : β → E} (hf : Measura
 theorem norm_approxOn_y₀_le [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E}
     {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (x : β) (n : ℕ) :
     ‖approxOn f hf s y₀ h₀ n x - y₀‖ ≤ ‖f x - y₀‖ + ‖f x - y₀‖ := by
-  have := edist_approxOn_y0_le hf h₀ x n
-  repeat rw [edist_comm y₀, edist_eq_coe_nnnorm_sub] at this
-  exact mod_cast this
+  simpa [enorm, edist_eq_enorm_sub, ← ENNReal.coe_add, norm_sub_rev]
+    using edist_approxOn_y0_le hf h₀ x n
 
 theorem norm_approxOn_zero_le [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f) {s : Set E}
     (h₀ : (0 : E) ∈ s) [SeparableSpace s] (x : β) (n : ℕ) :
     ‖approxOn f hf s 0 h₀ n x‖ ≤ ‖f x‖ + ‖f x‖ := by
-  have := edist_approxOn_y0_le hf h₀ x n
-  simp only [edist_comm (0 : E), edist_eq_coe_nnnorm] at this
-  exact mod_cast this
+  simpa [enorm, edist_eq_enorm_sub, ← ENNReal.coe_add, norm_sub_rev]
+    using edist_approxOn_y0_le hf h₀ x n
 
 theorem tendsto_approxOn_Lp_eLpNorm [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f)
     {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] (hp_ne_top : p ≠ ∞) {μ : Measure β}
@@ -91,35 +89,28 @@ theorem tendsto_approxOn_Lp_eLpNorm [OpensMeasurableSpace E] {f : β → E} (hf
   by_cases hp_zero : p = 0
   · simpa only [hp_zero, eLpNorm_exponent_zero] using tendsto_const_nhds
   have hp : 0 < p.toReal := toReal_pos hp_zero hp_ne_top
-  suffices
-      Tendsto (fun n => ∫⁻ x, (‖approxOn f hf s y₀ h₀ n x - f x‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) atTop
-        (𝓝 0) by
-    simp only [eLpNorm_eq_lintegral_rpow_nnnorm hp_zero hp_ne_top]
+  suffices Tendsto (fun n => ∫⁻ x, ‖approxOn f hf s y₀ h₀ n x - f x‖ₑ ^ p.toReal ∂μ) atTop (𝓝 0) by
+    simp only [eLpNorm_eq_lintegral_rpow_enorm hp_zero hp_ne_top]
     convert continuous_rpow_const.continuousAt.tendsto.comp this
     simp [zero_rpow_of_pos (_root_.inv_pos.mpr hp)]
   -- We simply check the conditions of the Dominated Convergence Theorem:
   -- (1) The function "`p`-th power of distance between `f` and the approximation" is measurable
-  have hF_meas :
-    ∀ n, Measurable fun x => (‖approxOn f hf s y₀ h₀ n x - f x‖₊ : ℝ≥0∞) ^ p.toReal := by
-    simpa only [← edist_eq_coe_nnnorm_sub] using fun n =>
+  have hF_meas n : Measurable fun x => ‖approxOn f hf s y₀ h₀ n x - f x‖ₑ ^ p.toReal := by
+    simpa only [← edist_eq_enorm_sub] using
       (approxOn f hf s y₀ h₀ n).measurable_bind (fun y x => edist y (f x) ^ p.toReal) fun y =>
         (measurable_edist_right.comp hf).pow_const p.toReal
   -- (2) The functions "`p`-th power of distance between `f` and the approximation" are uniformly
   -- bounded, at any given point, by `fun x => ‖f x - y₀‖ ^ p.toReal`
-  have h_bound :
-    ∀ n, (fun x => (‖approxOn f hf s y₀ h₀ n x - f x‖₊ : ℝ≥0∞) ^ p.toReal) ≤ᵐ[μ] fun x =>
-        (‖f x - y₀‖₊ : ℝ≥0∞) ^ p.toReal :=
-    fun n =>
-    Eventually.of_forall fun x =>
-      rpow_le_rpow (coe_mono (nnnorm_approxOn_le hf h₀ x n)) toReal_nonneg
+  have h_bound n :
+    (fun x ↦ ‖approxOn f hf s y₀ h₀ n x - f x‖ₑ ^ p.toReal) ≤ᵐ[μ] (‖f · - y₀‖ₑ ^ p.toReal) :=
+    .of_forall fun x => rpow_le_rpow (coe_mono (nnnorm_approxOn_le hf h₀ x n)) toReal_nonneg
   -- (3) The bounding function `fun x => ‖f x - y₀‖ ^ p.toReal` has finite integral
-  have h_fin : (∫⁻ a : β, (‖f a - y₀‖₊ : ℝ≥0∞) ^ p.toReal ∂μ) ≠ ⊤ :=
-    (lintegral_rpow_nnnorm_lt_top_of_eLpNorm_lt_top hp_zero hp_ne_top hi).ne
+  have h_fin : (∫⁻ a : β, ‖f a - y₀‖ₑ ^ p.toReal ∂μ) ≠ ⊤ :=
+    (lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top hp_zero hp_ne_top hi).ne
   -- (4) The functions "`p`-th power of distance between `f` and the approximation" tend pointwise
   -- to zero
   have h_lim :
-    ∀ᵐ a : β ∂μ,
-      Tendsto (fun n => (‖approxOn f hf s y₀ h₀ n a - f a‖₊ : ℝ≥0∞) ^ p.toReal) atTop (𝓝 0) := by
+    ∀ᵐ a : β ∂μ, Tendsto (‖approxOn f hf s y₀ h₀ · a - f a‖ₑ ^ p.toReal) atTop (𝓝 0) := by
     filter_upwards [hμ] with a ha
     have : Tendsto (fun n => (approxOn f hf s y₀ h₀ n) a - f a) atTop (𝓝 (f a - f a)) :=
       (tendsto_approxOn hf h₀ ha).sub tendsto_const_nhds
@@ -222,13 +213,15 @@ section Integrable
 variable [MeasurableSpace β]
 variable [MeasurableSpace E] [NormedAddCommGroup E]
 
-theorem tendsto_approxOn_L1_nnnorm [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f)
+theorem tendsto_approxOn_L1_enorm [OpensMeasurableSpace E] {f : β → E} (hf : Measurable f)
     {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s] {μ : Measure β}
     (hμ : ∀ᵐ x ∂μ, f x ∈ closure s) (hi : HasFiniteIntegral (fun x => f x - y₀) μ) :
-    Tendsto (fun n => ∫⁻ x, ‖approxOn f hf s y₀ h₀ n x - f x‖₊ ∂μ) atTop (𝓝 0) := by
-  simpa [eLpNorm_one_eq_lintegral_nnnorm] using
+    Tendsto (fun n => ∫⁻ x, ‖approxOn f hf s y₀ h₀ n x - f x‖ₑ ∂μ) atTop (𝓝 0) := by
+  simpa [eLpNorm_one_eq_lintegral_enorm] using
     tendsto_approxOn_Lp_eLpNorm hf h₀ one_ne_top hμ
-      (by simpa [eLpNorm_one_eq_lintegral_nnnorm] using hi)
+      (by simpa [eLpNorm_one_eq_lintegral_enorm] using hi)
+
+@[deprecated (since := "2025-01-21")] alias tendsto_approxOn_L1_nnnorm := tendsto_approxOn_L1_enorm
 
 theorem integrable_approxOn [BorelSpace E] {f : β → E} {μ : Measure β} (fmeas : Measurable f)
     (hf : Integrable f μ) {s : Set E} {y₀ : E} (h₀ : y₀ ∈ s) [SeparableSpace s]
@@ -236,14 +229,17 @@ theorem integrable_approxOn [BorelSpace E] {f : β → E} {μ : Measure β} (fme
   rw [← memℒp_one_iff_integrable] at hf hi₀ ⊢
   exact memℒp_approxOn fmeas hf h₀ hi₀ n
 
-theorem tendsto_approxOn_range_L1_nnnorm [OpensMeasurableSpace E] {f : β → E} {μ : Measure β}
+theorem tendsto_approxOn_range_L1_enorm [OpensMeasurableSpace E] {f : β → E} {μ : Measure β}
     [SeparableSpace (range f ∪ {0} : Set E)] (fmeas : Measurable f) (hf : Integrable f μ) :
-    Tendsto (fun n => ∫⁻ x, ‖approxOn f fmeas (range f ∪ {0}) 0 (by simp) n x - f x‖₊ ∂μ) atTop
+    Tendsto (fun n => ∫⁻ x, ‖approxOn f fmeas (range f ∪ {0}) 0 (by simp) n x - f x‖ₑ ∂μ) atTop
       (𝓝 0) := by
-  apply tendsto_approxOn_L1_nnnorm fmeas
+  apply tendsto_approxOn_L1_enorm fmeas
   · filter_upwards with x using subset_closure (by simp)
   · simpa using hf.2
 
+@[deprecated (since := "2025-01-21")]
+alias tendsto_approxOn_range_L1_nnnorm := tendsto_approxOn_range_L1_enorm
+
 theorem integrable_approxOn_range [BorelSpace E] {f : β → E} {μ : Measure β} (fmeas : Measurable f)
     [SeparableSpace (range f ∪ {0} : Set E)] (hf : Integrable f μ) (n : ℕ) :
     Integrable (approxOn f fmeas (range f ∪ {0}) 0 (by simp) n) μ :=
@@ -278,10 +274,9 @@ theorem memℒp_top (f : α →ₛ E) (μ : Measure α) : Memℒp f ∞ μ :=
   memℒp_top_of_bound f.aestronglyMeasurable C <| Eventually.of_forall hfC
 
 protected theorem eLpNorm'_eq {p : ℝ} (f : α →ₛ F) (μ : Measure α) :
-    eLpNorm' f p μ = (∑ y ∈ f.range, (‖y‖₊ : ℝ≥0∞) ^ p * μ (f ⁻¹' {y})) ^ (1 / p) := by
-  have h_map : (fun a => (‖f a‖₊ : ℝ≥0∞) ^ p) = f.map fun a : F => (‖a‖₊ : ℝ≥0∞) ^ p := by
-    simp; rfl
-  rw [eLpNorm'_eq_lintegral_nnnorm, h_map, lintegral_eq_lintegral, map_lintegral]
+    eLpNorm' f p μ = (∑ y ∈ f.range, ‖y‖ₑ ^ p * μ (f ⁻¹' {y})) ^ (1 / p) := by
+  have h_map : (‖f ·‖ₑ ^ p) = f.map (‖·‖ₑ ^ p) := by simp; rfl
+  rw [eLpNorm'_eq_lintegral_enorm, h_map, lintegral_eq_lintegral, map_lintegral]
 
 @[deprecated (since := "2024-07-27")]
 protected alias snorm'_eq := SimpleFunc.eLpNorm'_eq
diff --git a/Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean b/Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
index 3b54bf1d8f7dba..3ea95e6147099a 100644
--- a/Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
+++ b/Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
@@ -812,10 +812,12 @@ protected theorem nnnorm {_ : MeasurableSpace α} {β : Type*} [SeminormedAddCom
   continuous_nnnorm.comp_stronglyMeasurable hf
 
 @[measurability]
-protected theorem ennnorm {_ : MeasurableSpace α} {β : Type*} [SeminormedAddCommGroup β]
-    {f : α → β} (hf : StronglyMeasurable f) : Measurable fun a => (‖f a‖₊ : ℝ≥0∞) :=
+protected theorem enorm {_ : MeasurableSpace α} {β : Type*} [SeminormedAddCommGroup β]
+    {f : α → β} (hf : StronglyMeasurable f) : Measurable (‖f ·‖ₑ) :=
   (ENNReal.continuous_coe.comp_stronglyMeasurable hf.nnnorm).measurable
 
+@[deprecated (since := "2025-01-21")] alias ennnorm := StronglyMeasurable.enorm
+
 @[measurability]
 protected theorem real_toNNReal {_ : MeasurableSpace α} {f : α → ℝ} (hf : StronglyMeasurable f) :
     StronglyMeasurable fun x => (f x).toNNReal :=
@@ -1438,10 +1440,12 @@ protected theorem nnnorm {β : Type*} [SeminormedAddCommGroup β] {f : α → β
   continuous_nnnorm.comp_aestronglyMeasurable hf
 
 @[measurability]
-protected theorem ennnorm {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
-    (hf : AEStronglyMeasurable f μ) : AEMeasurable (fun a => (‖f a‖₊ : ℝ≥0∞)) μ :=
+protected theorem enorm {β : Type*} [SeminormedAddCommGroup β] {f : α → β}
+    (hf : AEStronglyMeasurable f μ) : AEMeasurable (‖f ·‖ₑ) μ :=
   (ENNReal.continuous_coe.comp_aestronglyMeasurable hf.nnnorm).aemeasurable
 
+@[deprecated (since := "2025-01-20")] alias ennnorm := AEStronglyMeasurable.enorm
+
 @[aesop safe 20 apply (rule_sets := [Measurable])]
 protected theorem edist {β : Type*} [SeminormedAddCommGroup β] {f g : α → β}
     (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) :
diff --git a/Mathlib/MeasureTheory/Function/UnifTight.lean b/Mathlib/MeasureTheory/Function/UnifTight.lean
index 6c36077b3a716f..1b98552847b15d 100644
--- a/Mathlib/MeasureTheory/Function/UnifTight.lean
+++ b/Mathlib/MeasureTheory/Function/UnifTight.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Igor Khavkine
 -/
 import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
-import Mathlib.MeasureTheory.Function.L1Space
 import Mathlib.MeasureTheory.Function.UniformIntegrable
 
 /-!
diff --git a/Mathlib/MeasureTheory/Function/UniformIntegrable.lean b/Mathlib/MeasureTheory/Function/UniformIntegrable.lean
index 172756a4d9dbd6..8ba93267f9ee78 100644
--- a/Mathlib/MeasureTheory/Function/UniformIntegrable.lean
+++ b/Mathlib/MeasureTheory/Function/UniformIntegrable.lean
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kexing Ying
 -/
 import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
-import Mathlib.MeasureTheory.Function.L1Space
+import Mathlib.MeasureTheory.Function.L1Space.Integrable
 
 /-!
 # Uniform integrability
@@ -45,7 +45,7 @@ uniform integrable, uniformly absolutely continuous integral, Vitali convergence
 
 noncomputable section
 
-open scoped Classical MeasureTheory NNReal ENNReal Topology
+open scoped MeasureTheory NNReal ENNReal Topology
 
 namespace MeasureTheory
 
@@ -126,6 +126,7 @@ protected theorem sub (hf : UnifIntegrable f p μ) (hg : UnifIntegrable g p μ)
 
 protected theorem ae_eq (hf : UnifIntegrable f p μ) (hfg : ∀ n, f n =ᵐ[μ] g n) :
     UnifIntegrable g p μ := by
+  classical
   intro ε hε
   obtain ⟨δ, hδ_pos, hfδ⟩ := hf hε
   refine ⟨δ, hδ_pos, fun n s hs hμs => (le_of_eq <| eLpNorm_congr_ae ?_).trans (hfδ n s hs hμs)⟩
@@ -224,12 +225,12 @@ theorem Memℒp.integral_indicator_norm_ge_le (hf : Memℒp f 1 μ) (hmeas : Str
 which does not require measurability. -/
 theorem Memℒp.integral_indicator_norm_ge_nonneg_le_of_meas (hf : Memℒp f 1 μ)
     (hmeas : StronglyMeasurable f) {ε : ℝ} (hε : 0 < ε) :
-    ∃ M : ℝ, 0 ≤ M ∧ (∫⁻ x, ‖{ x | M ≤ ‖f x‖₊ }.indicator f x‖₊ ∂μ) ≤ ENNReal.ofReal ε :=
+    ∃ M : ℝ, 0 ≤ M ∧ (∫⁻ x, ‖{ x | M ≤ ‖f x‖₊ }.indicator f x‖ₑ ∂μ) ≤ ENNReal.ofReal ε :=
   let ⟨M, hM⟩ := hf.integral_indicator_norm_ge_le hmeas hε
   ⟨max M 0, le_max_right _ _, by simpa⟩
 
 theorem Memℒp.integral_indicator_norm_ge_nonneg_le (hf : Memℒp f 1 μ) {ε : ℝ} (hε : 0 < ε) :
-    ∃ M : ℝ, 0 ≤ M ∧ (∫⁻ x, ‖{ x | M ≤ ‖f x‖₊ }.indicator f x‖₊ ∂μ) ≤ ENNReal.ofReal ε := by
+    ∃ M : ℝ, 0 ≤ M ∧ (∫⁻ x, ‖{ x | M ≤ ‖f x‖₊ }.indicator f x‖ₑ ∂μ) ≤ ENNReal.ofReal ε := by
   have hf_mk : Memℒp (hf.1.mk f) 1 μ := (memℒp_congr_ae hf.1.ae_eq_mk).mp hf
   obtain ⟨M, hM_pos, hfM⟩ :=
     hf_mk.integral_indicator_norm_ge_nonneg_le_of_meas hf.1.stronglyMeasurable_mk hε
@@ -281,27 +282,25 @@ theorem Memℒp.eLpNorm_indicator_norm_ge_le (hf : Memℒp f p μ) (hmeas : Stro
   obtain ⟨M, hM', hM⟩ := Memℒp.integral_indicator_norm_ge_nonneg_le
     (μ := μ) (hf.norm_rpow hp_ne_zero hp_ne_top) (Real.rpow_pos_of_pos hε p.toReal)
   refine ⟨M ^ (1 / p.toReal), ?_⟩
-  rw [eLpNorm_eq_lintegral_rpow_nnnorm hp_ne_zero hp_ne_top, ← ENNReal.rpow_one (ENNReal.ofReal ε)]
+  rw [eLpNorm_eq_lintegral_rpow_enorm hp_ne_zero hp_ne_top, ← ENNReal.rpow_one (ENNReal.ofReal ε)]
   conv_rhs => rw [← mul_one_div_cancel (ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm]
   rw [ENNReal.rpow_mul,
     ENNReal.rpow_le_rpow_iff (one_div_pos.2 <| ENNReal.toReal_pos hp_ne_zero hp_ne_top),
     ENNReal.ofReal_rpow_of_pos hε]
-  convert hM
-  rename_i x
-  rw [← ENNReal.coe_rpow_of_nonneg _ ENNReal.toReal_nonneg, nnnorm_indicator_eq_indicator_nnnorm,
-    nnnorm_indicator_eq_indicator_nnnorm]
+  convert hM using 3 with x
+  rw [enorm_indicator_eq_indicator_enorm, enorm_indicator_eq_indicator_enorm]
   have hiff : M ^ (1 / p.toReal) ≤ ‖f x‖₊ ↔ M ≤ ‖‖f x‖ ^ p.toReal‖₊ := by
     rw [coe_nnnorm, coe_nnnorm, Real.norm_rpow_of_nonneg (norm_nonneg _), norm_norm,
       ← Real.rpow_le_rpow_iff hM' (Real.rpow_nonneg (norm_nonneg _) _)
         (one_div_pos.2 <| ENNReal.toReal_pos hp_ne_zero hp_ne_top), ← Real.rpow_mul (norm_nonneg _),
       mul_one_div_cancel (ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm, Real.rpow_one]
   by_cases hx : x ∈ { x : α | M ^ (1 / p.toReal) ≤ ‖f x‖₊ }
-  · rw [Set.indicator_of_mem hx, Set.indicator_of_mem, Real.nnnorm_of_nonneg]
-    · rfl
+  · rw [Set.indicator_of_mem hx, Set.indicator_of_mem, Real.enorm_of_nonneg (by positivity),
+      ← ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) ENNReal.toReal_nonneg, ofReal_norm]
     rw [Set.mem_setOf_eq]
     rwa [← hiff]
   · rw [Set.indicator_of_not_mem hx, Set.indicator_of_not_mem]
-    · simp [(ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm]
+    · simp [ENNReal.toReal_pos hp_ne_zero hp_ne_top]
     · rw [Set.mem_setOf_eq]
       rwa [← hiff]
 
@@ -499,7 +498,7 @@ theorem eLpNorm_sub_le_of_dist_bdd (μ : Measure α)
   refine le_trans (eLpNorm_mono this) ?_
   rw [eLpNorm_indicator_const hs hp hp']
   refine mul_le_mul_right' (le_of_eq ?_) _
-  rw [← ofReal_norm_eq_coe_nnnorm, Real.norm_eq_abs, abs_of_nonneg hc]
+  rw [← ofReal_norm_eq_enorm, Real.norm_eq_abs, abs_of_nonneg hc]
 
 @[deprecated (since := "2024-07-27")]
 alias snorm_sub_le_of_dist_bdd := eLpNorm_sub_le_of_dist_bdd
@@ -920,33 +919,23 @@ theorem uniformIntegrable_average
     refine ⟨δ, hδ₁, fun n s hs hle => ?_⟩
     simp_rw [Finset.smul_sum, Finset.indicator_sum]
     refine le_trans (eLpNorm_sum_le (fun i _ => ((hf₁ i).const_smul _).indicator hs) hp) ?_
-    have : ∀ i, s.indicator ((n : ℝ) ⁻¹ • f i) = (↑n : ℝ)⁻¹ • s.indicator (f i) :=
-      fun i ↦ indicator_const_smul _ _ _
-    simp_rw [this, eLpNorm_const_smul, ← Finset.mul_sum, nnnorm_inv, Real.nnnorm_natCast]
-    by_cases hn : (↑(↑n : ℝ≥0)⁻¹ : ℝ≥0∞) = 0
-    · simp only [hn, zero_mul, zero_le]
-    refine le_trans ?_ (?_ : ↑(↑n : ℝ≥0)⁻¹ * n • ENNReal.ofReal ε ≤ ENNReal.ofReal ε)
-    · refine (ENNReal.mul_le_mul_left hn ENNReal.coe_ne_top).2 ?_
-      conv_rhs => rw [← Finset.card_range n]
-      exact Finset.sum_le_card_nsmul _ _ _ fun i _ => hδ₂ _ _ hs hle
-    · simp only [ENNReal.coe_eq_zero, inv_eq_zero, Nat.cast_eq_zero] at hn
-      rw [nsmul_eq_mul, ← mul_assoc, ENNReal.coe_inv, ENNReal.coe_natCast,
-        ENNReal.inv_mul_cancel _ (ENNReal.natCast_ne_top _), one_mul]
-      all_goals simpa only [Ne, Nat.cast_eq_zero]
+    have this i : s.indicator ((n : ℝ) ⁻¹ • f i) = (↑n : ℝ)⁻¹ • s.indicator (f i) :=
+      indicator_const_smul _ _ _
+    obtain rfl | hn := eq_or_ne n 0
+    · simp
+    simp_rw [this, eLpNorm_const_smul, ← Finset.mul_sum]
+    rw [enorm_inv (by positivity), Real.enorm_natCast, ← ENNReal.div_eq_inv_mul]
+    refine ENNReal.div_le_of_le_mul' ?_
+    simpa using Finset.sum_le_card_nsmul (.range n) _ _ fun i _ => hδ₂ _ _ hs hle
   · obtain ⟨C, hC⟩ := hf₃
     simp_rw [Finset.smul_sum]
     refine ⟨C, fun n => (eLpNorm_sum_le (fun i _ => (hf₁ i).const_smul _) hp).trans ?_⟩
-    simp_rw [eLpNorm_const_smul, ← Finset.mul_sum, nnnorm_inv, Real.nnnorm_natCast]
-    by_cases hn : (↑(↑n : ℝ≥0)⁻¹ : ℝ≥0∞) = 0
-    · simp only [hn, zero_mul, zero_le]
-    refine le_trans ?_ (?_ : ↑(↑n : ℝ≥0)⁻¹ * (n • C : ℝ≥0∞) ≤ C)
-    · refine (ENNReal.mul_le_mul_left hn ENNReal.coe_ne_top).2 ?_
-      conv_rhs => rw [← Finset.card_range n]
-      exact Finset.sum_le_card_nsmul _ _ _ fun i _ => hC i
-    · simp only [ENNReal.coe_eq_zero, inv_eq_zero, Nat.cast_eq_zero] at hn
-      rw [nsmul_eq_mul, ← mul_assoc, ENNReal.coe_inv, ENNReal.coe_natCast,
-        ENNReal.inv_mul_cancel _ (ENNReal.natCast_ne_top _), one_mul]
-      all_goals simpa only [Ne, Nat.cast_eq_zero]
+    obtain rfl | hn := eq_or_ne n 0
+    · simp
+    simp_rw [eLpNorm_const_smul, ← Finset.mul_sum]
+    rw [enorm_inv (by positivity), Real.enorm_natCast, ← ENNReal.div_eq_inv_mul]
+    refine ENNReal.div_le_of_le_mul' ?_
+    simpa using Finset.sum_le_card_nsmul (.range n) _ _ fun i _ => hC i
 
 /-- The averaging of a uniformly integrable real-valued sequence is also uniformly integrable. -/
 theorem uniformIntegrable_average_real (hp : 1 ≤ p) {f : ℕ → α → ℝ} (hf : UniformIntegrable f p μ) :
diff --git a/Mathlib/MeasureTheory/Group/Measure.lean b/Mathlib/MeasureTheory/Group/Measure.lean
index 0dc9999c175a6f..62c46156f922a0 100644
--- a/Mathlib/MeasureTheory/Group/Measure.lean
+++ b/Mathlib/MeasureTheory/Group/Measure.lean
@@ -764,9 +764,7 @@ nonrec theorem _root_.MulEquiv.isHaarMeasure_map [BorelSpace G] [TopologicalGrou
     [TopologicalGroup H] (e : G ≃* H) (he : Continuous e) (hesymm : Continuous e.symm) :
     IsHaarMeasure (Measure.map e μ) :=
   let f : G ≃ₜ H := .mk e
-  #adaptation_note
-  /--
-  After https://github.com/leanprover/lean4/pull/6024
+  #adaptation_note /-- https://github.com/leanprover/lean4/pull/6024
   we needed to write `e.toMonoidHom` instead of just `e`, to avoid unification issues.
   -/
   isHaarMeasure_map μ e.toMonoidHom he e.surjective f.isClosedEmbedding.tendsto_cocompact
diff --git a/Mathlib/MeasureTheory/Integral/Bochner.lean b/Mathlib/MeasureTheory/Integral/Bochner.lean
index 51627a057494c3..a9ff72df2468ee 100644
--- a/Mathlib/MeasureTheory/Integral/Bochner.lean
+++ b/Mathlib/MeasureTheory/Integral/Bochner.lean
@@ -868,12 +868,14 @@ theorem norm_integral_le_lintegral_norm (f : α → G) :
     · rw [integral_undef hf, norm_zero]; exact toReal_nonneg
   · simp [integral, hG]
 
-theorem ennnorm_integral_le_lintegral_ennnorm (f : α → G) :
-    (‖∫ a, f a ∂μ‖₊ : ℝ≥0∞) ≤ ∫⁻ a, ‖f a‖₊ ∂μ := by
-  simp_rw [← ofReal_norm_eq_coe_nnnorm]
+theorem enorm_integral_le_lintegral_enorm (f : α → G) : ‖∫ a, f a ∂μ‖ₑ ≤ ∫⁻ a, ‖f a‖ₑ ∂μ := by
+  simp_rw [← ofReal_norm_eq_enorm]
   apply ENNReal.ofReal_le_of_le_toReal
   exact norm_integral_le_lintegral_norm f
 
+@[deprecated (since := "2025-01-21")]
+alias ennnorm_integral_le_lintegral_ennnorm := enorm_integral_le_lintegral_enorm
+
 theorem integral_eq_zero_of_ae {f : α → G} (hf : f =ᵐ[μ] 0) : ∫ a, f a ∂μ = 0 := by
   simp [integral_congr_ae hf, integral_zero]
 
@@ -887,7 +889,7 @@ theorem HasFiniteIntegral.tendsto_setIntegral_nhds_zero {ι} {f : α → G}
     ENNReal.coe_zero]
   exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
     (tendsto_setLIntegral_zero (ne_of_lt hf) hs) (fun i => zero_le _)
-    fun i => ennnorm_integral_le_lintegral_ennnorm _
+    fun i => enorm_integral_le_lintegral_enorm _
 
 @[deprecated (since := "2024-04-17")]
 alias HasFiniteIntegral.tendsto_set_integral_nhds_zero :=
@@ -907,7 +909,7 @@ alias Integrable.tendsto_set_integral_nhds_zero :=
 /-- If `F i → f` in `L1`, then `∫ x, F i x ∂μ → ∫ x, f x ∂μ`. -/
 theorem tendsto_integral_of_L1 {ι} (f : α → G) (hfi : Integrable f μ) {F : ι → α → G} {l : Filter ι}
     (hFi : ∀ᶠ i in l, Integrable (F i) μ)
-    (hF : Tendsto (fun i => ∫⁻ x, ‖F i x - f x‖₊ ∂μ) l (𝓝 0)) :
+    (hF : Tendsto (fun i => ∫⁻ x, ‖F i x - f x‖ₑ ∂μ) l (𝓝 0)) :
     Tendsto (fun i => ∫ x, F i x ∂μ) l (𝓝 <| ∫ x, f x ∂μ) := by
   by_cases hG : CompleteSpace G
   · simp only [integral, hG, L1.integral]
@@ -919,18 +921,18 @@ lemma tendsto_integral_of_L1' {ι} (f : α → G) (hfi : Integrable f μ) {F : 
     (hFi : ∀ᶠ i in l, Integrable (F i) μ) (hF : Tendsto (fun i ↦ eLpNorm (F i - f) 1 μ) l (𝓝 0)) :
     Tendsto (fun i ↦ ∫ x, F i x ∂μ) l (𝓝 (∫ x, f x ∂μ)) := by
   refine tendsto_integral_of_L1 f hfi hFi ?_
-  simp_rw [eLpNorm_one_eq_lintegral_nnnorm, Pi.sub_apply] at hF
+  simp_rw [eLpNorm_one_eq_lintegral_enorm, Pi.sub_apply] at hF
   exact hF
 
 /-- If `F i → f` in `L1`, then `∫ x in s, F i x ∂μ → ∫ x in s, f x ∂μ`. -/
 lemma tendsto_setIntegral_of_L1 {ι} (f : α → G) (hfi : Integrable f μ) {F : ι → α → G}
     {l : Filter ι}
-    (hFi : ∀ᶠ i in l, Integrable (F i) μ) (hF : Tendsto (fun i ↦ ∫⁻ x, ‖F i x - f x‖₊ ∂μ) l (𝓝 0))
+    (hFi : ∀ᶠ i in l, Integrable (F i) μ) (hF : Tendsto (fun i ↦ ∫⁻ x, ‖F i x - f x‖ₑ ∂μ) l (𝓝 0))
     (s : Set α) :
     Tendsto (fun i ↦ ∫ x in s, F i x ∂μ) l (𝓝 (∫ x in s, f x ∂μ)) := by
   refine tendsto_integral_of_L1 f hfi.restrict ?_ ?_
   · filter_upwards [hFi] with i hi using hi.restrict
-  · simp_rw [← eLpNorm_one_eq_lintegral_nnnorm] at hF ⊢
+  · simp_rw [← eLpNorm_one_eq_lintegral_enorm] at hF ⊢
     exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hF (fun _ ↦ zero_le')
       (fun _ ↦ eLpNorm_mono_measure _ Measure.restrict_le_self)
 
@@ -944,7 +946,7 @@ lemma tendsto_setIntegral_of_L1' {ι} (f : α → G) (hfi : Integrable f μ) {F
     (s : Set α) :
     Tendsto (fun i ↦ ∫ x in s, F i x ∂μ) l (𝓝 (∫ x in s, f x ∂μ)) := by
   refine tendsto_setIntegral_of_L1 f hfi hFi ?_ s
-  simp_rw [eLpNorm_one_eq_lintegral_nnnorm, Pi.sub_apply] at hF
+  simp_rw [eLpNorm_one_eq_lintegral_enorm, Pi.sub_apply] at hF
   exact hF
 
 @[deprecated (since := "2024-04-17")]
@@ -1048,16 +1050,22 @@ theorem integral_eq_lintegral_of_nonneg_ae {f : α → ℝ} (hf : 0 ≤ᵐ[μ] f
       rw [Real.norm_eq_abs, abs_of_nonneg h]
     rw [this, hfi]; rfl
 
-theorem integral_norm_eq_lintegral_nnnorm {P : Type*} [NormedAddCommGroup P] {f : α → P}
-    (hf : AEStronglyMeasurable f μ) : ∫ x, ‖f x‖ ∂μ = ENNReal.toReal (∫⁻ x, ‖f x‖₊ ∂μ) := by
+theorem integral_norm_eq_lintegral_enorm {P : Type*} [NormedAddCommGroup P] {f : α → P}
+    (hf : AEStronglyMeasurable f μ) : ∫ x, ‖f x‖ ∂μ = (∫⁻ x, ‖f x‖ₑ ∂μ).toReal := by
   rw [integral_eq_lintegral_of_nonneg_ae _ hf.norm]
-  · simp_rw [ofReal_norm_eq_coe_nnnorm]
+  · simp_rw [ofReal_norm_eq_enorm]
   · filter_upwards; simp_rw [Pi.zero_apply, norm_nonneg, imp_true_iff]
 
-theorem ofReal_integral_norm_eq_lintegral_nnnorm {P : Type*} [NormedAddCommGroup P] {f : α → P}
-    (hf : Integrable f μ) : ENNReal.ofReal (∫ x, ‖f x‖ ∂μ) = ∫⁻ x, ‖f x‖₊ ∂μ := by
-  rw [integral_norm_eq_lintegral_nnnorm hf.aestronglyMeasurable, ENNReal.ofReal_toReal]
-  exact lt_top_iff_ne_top.mp (hasFiniteIntegral_iff_nnnorm.mpr hf.2)
+@[deprecated (since := "2025-01-21")]
+alias integral_norm_eq_lintegral_nnnorm := integral_norm_eq_lintegral_enorm
+
+theorem ofReal_integral_norm_eq_lintegral_enorm {P : Type*} [NormedAddCommGroup P] {f : α → P}
+    (hf : Integrable f μ) : ENNReal.ofReal (∫ x, ‖f x‖ ∂μ) = ∫⁻ x, ‖f x‖ₑ ∂μ := by
+  rw [integral_norm_eq_lintegral_enorm hf.aestronglyMeasurable, ENNReal.ofReal_toReal]
+  exact lt_top_iff_ne_top.mp (hasFiniteIntegral_iff_enorm.mpr hf.2)
+
+@[deprecated (since := "2025-01-21")]
+alias ofReal_integral_norm_eq_lintegral_nnnorm := ofReal_integral_norm_eq_lintegral_enorm
 
 theorem integral_eq_integral_pos_part_sub_integral_neg_part {f : α → ℝ} (hf : Integrable f μ) :
     ∫ a, f a ∂μ = ∫ a, (Real.toNNReal (f a) : ℝ) ∂μ - ∫ a, (Real.toNNReal (-f a) : ℝ) ∂μ := by
@@ -1079,13 +1087,13 @@ theorem lintegral_coe_eq_integral (f : α → ℝ≥0) (hfi : Integrable (fun x
   rw [← lt_top_iff_ne_top]
   convert hfi.hasFiniteIntegral
   -- Porting note: `convert` no longer unfolds `HasFiniteIntegral`
-  simp_rw [hasFiniteIntegral_iff_nnnorm, NNReal.nnnorm_eq]
+  simp_rw [hasFiniteIntegral_iff_enorm, NNReal.enorm_eq]
 
 theorem ofReal_integral_eq_lintegral_ofReal {f : α → ℝ} (hfi : Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) :
     ENNReal.ofReal (∫ x, f x ∂μ) = ∫⁻ x, ENNReal.ofReal (f x) ∂μ := by
   have : f =ᵐ[μ] (‖f ·‖) := f_nn.mono fun _x hx ↦ (abs_of_nonneg hx).symm
-  simp_rw [integral_congr_ae this, ofReal_integral_norm_eq_lintegral_nnnorm hfi,
-    ← ofReal_norm_eq_coe_nnnorm]
+  simp_rw [integral_congr_ae this, ofReal_integral_norm_eq_lintegral_enorm hfi,
+    ← ofReal_norm_eq_enorm]
   exact lintegral_congr_ae (this.symm.fun_comp ENNReal.ofReal)
 
 theorem integral_toReal {f : α → ℝ≥0∞} (hfm : AEMeasurable f μ) (hf : ∀ᵐ x ∂μ, f x < ∞) :
@@ -1249,7 +1257,7 @@ lemma tendsto_of_integral_tendsto_of_monotone {μ : Measure α} {f : ℕ → α
   have h := tendsto_of_lintegral_tendsto_of_monotone ?_ h_tendsto h_mono h_bound ?_
   rotate_left
   · exact (hF_int.1.aemeasurable.sub (hf_int 0).1.aemeasurable).ennreal_ofReal
-  · exact ((lintegral_ofReal_le_lintegral_nnnorm _).trans_lt (hF_int.sub (hf_int 0)).2).ne
+  · exact ((lintegral_ofReal_le_lintegral_enorm _).trans_lt (hF_int.sub (hf_int 0)).2).ne
   filter_upwards [h, hf_mono, hf_bound] with a ha ha_mono ha_bound
   have h1 : (fun i ↦ f i a) = fun i ↦ (f' i a).toReal + f 0 a := by
     unfold f'
@@ -1296,11 +1304,11 @@ section NormedAddCommGroup
 variable {H : Type*} [NormedAddCommGroup H]
 
 theorem L1.norm_eq_integral_norm (f : α →₁[μ] H) : ‖f‖ = ∫ a, ‖f a‖ ∂μ := by
-  simp only [eLpNorm, eLpNorm'_eq_lintegral_nnnorm, ENNReal.one_toReal, ENNReal.rpow_one,
+  simp only [eLpNorm, eLpNorm'_eq_lintegral_enorm, ENNReal.one_toReal, ENNReal.rpow_one,
     Lp.norm_def, if_false, ENNReal.one_ne_top, one_ne_zero, _root_.div_one]
   rw [integral_eq_lintegral_of_nonneg_ae (Eventually.of_forall (by simp [norm_nonneg]))
       (Lp.aestronglyMeasurable f).norm]
-  simp [ofReal_norm_eq_coe_nnnorm]
+  simp [ofReal_norm_eq_enorm]
 
 theorem L1.dist_eq_integral_dist (f g : α →₁[μ] H) : dist f g = ∫ a, dist (f a) (g a) ∂μ := by
   simp only [dist_eq_norm, L1.norm_eq_integral_norm]
@@ -1314,14 +1322,14 @@ theorem L1.norm_of_fun_eq_integral_norm {f : α → H} (hf : Integrable f μ) :
 theorem Memℒp.eLpNorm_eq_integral_rpow_norm {f : α → H} {p : ℝ≥0∞} (hp1 : p ≠ 0) (hp2 : p ≠ ∞)
     (hf : Memℒp f p μ) :
     eLpNorm f p μ = ENNReal.ofReal ((∫ a, ‖f a‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹) := by
-  have A : ∫⁻ a : α, ENNReal.ofReal (‖f a‖ ^ p.toReal) ∂μ = ∫⁻ a : α, ‖f a‖₊ ^ p.toReal ∂μ := by
-    simp_rw [← ofReal_rpow_of_nonneg (norm_nonneg _) toReal_nonneg, ofReal_norm_eq_coe_nnnorm]
-  simp only [eLpNorm_eq_lintegral_rpow_nnnorm hp1 hp2, one_div]
+  have A : ∫⁻ a : α, ENNReal.ofReal (‖f a‖ ^ p.toReal) ∂μ = ∫⁻ a : α, ‖f a‖ₑ ^ p.toReal ∂μ := by
+    simp_rw [← ofReal_rpow_of_nonneg (norm_nonneg _) toReal_nonneg, ofReal_norm_eq_enorm]
+  simp only [eLpNorm_eq_lintegral_rpow_enorm hp1 hp2, one_div]
   rw [integral_eq_lintegral_of_nonneg_ae]; rotate_left
   · exact ae_of_all _ fun x => by positivity
   · exact (hf.aestronglyMeasurable.norm.aemeasurable.pow_const _).aestronglyMeasurable
   rw [A, ← ofReal_rpow_of_nonneg toReal_nonneg (inv_nonneg.2 toReal_nonneg), ofReal_toReal]
-  exact (lintegral_rpow_nnnorm_lt_top_of_eLpNorm_lt_top hp1 hp2 hf.2).ne
+  exact (lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top hp1 hp2 hf.2).ne
 
 @[deprecated (since := "2024-07-27")]
 alias Memℒp.snorm_eq_integral_rpow_norm := Memℒp.eLpNorm_eq_integral_rpow_norm
@@ -1428,8 +1436,8 @@ theorem tendsto_integral_norm_approxOn_sub
     (fmeas : Measurable f) (hf : Integrable f μ) [SeparableSpace (range f ∪ {0} : Set E)] :
     Tendsto (fun n ↦ ∫ x, ‖SimpleFunc.approxOn f fmeas (range f ∪ {0}) 0 (by simp) n x - f x‖ ∂μ)
       atTop (𝓝 0) := by
-  convert (tendsto_toReal zero_ne_top).comp (tendsto_approxOn_range_L1_nnnorm fmeas hf) with n
-  rw [integral_norm_eq_lintegral_nnnorm]
+  convert (tendsto_toReal zero_ne_top).comp (tendsto_approxOn_range_L1_enorm fmeas hf) with n
+  rw [integral_norm_eq_lintegral_enorm]
   · simp
   · apply (SimpleFunc.aestronglyMeasurable _).sub
     apply (stronglyMeasurable_iff_measurable_separable.2 ⟨fmeas, ?_⟩ ).aestronglyMeasurable
@@ -1484,9 +1492,9 @@ theorem integral_finset_sum_measure {ι} {m : MeasurableSpace α} {f : α → G}
 
 theorem nndist_integral_add_measure_le_lintegral
     {f : α → G} (h₁ : Integrable f μ) (h₂ : Integrable f ν) :
-    (nndist (∫ x, f x ∂μ) (∫ x, f x ∂(μ + ν)) : ℝ≥0∞) ≤ ∫⁻ x, ‖f x‖₊ ∂ν := by
+    (nndist (∫ x, f x ∂μ) (∫ x, f x ∂(μ + ν)) : ℝ≥0∞) ≤ ∫⁻ x, ‖f x‖ₑ ∂ν := by
   rw [integral_add_measure h₁ h₂, nndist_comm, nndist_eq_nnnorm, add_sub_cancel_left]
-  exact ennnorm_integral_le_lintegral_ennnorm _
+  exact enorm_integral_le_lintegral_enorm _
 
 theorem hasSum_integral_measure {ι} {m : MeasurableSpace α} {f : α → G} {μ : ι → Measure α}
     (hf : Integrable f (Measure.sum μ)) :
@@ -1495,11 +1503,11 @@ theorem hasSum_integral_measure {ι} {m : MeasurableSpace α} {f : α → G} {μ
   simp only [HasSum, ← integral_finset_sum_measure fun i _ => hfi i]
   refine Metric.nhds_basis_ball.tendsto_right_iff.mpr fun ε ε0 => ?_
   lift ε to ℝ≥0 using ε0.le
-  have hf_lt : (∫⁻ x, ‖f x‖₊ ∂Measure.sum μ) < ∞ := hf.2
-  have hmem : ∀ᶠ y in 𝓝 (∫⁻ x, ‖f x‖₊ ∂Measure.sum μ), (∫⁻ x, ‖f x‖₊ ∂Measure.sum μ) < y + ε := by
+  have hf_lt : (∫⁻ x, ‖f x‖ₑ ∂Measure.sum μ) < ∞ := hf.2
+  have hmem : ∀ᶠ y in 𝓝 (∫⁻ x, ‖f x‖ₑ ∂Measure.sum μ), (∫⁻ x, ‖f x‖ₑ ∂Measure.sum μ) < y + ε := by
     refine tendsto_id.add tendsto_const_nhds (lt_mem_nhds (α := ℝ≥0∞) <| ENNReal.lt_add_right ?_ ?_)
     exacts [hf_lt.ne, ENNReal.coe_ne_zero.2 (NNReal.coe_ne_zero.1 ε0.ne')]
-  refine ((hasSum_lintegral_measure (fun x => ‖f x‖₊) μ).eventually hmem).mono fun s hs => ?_
+  refine ((hasSum_lintegral_measure (fun x => ‖f x‖ₑ) μ).eventually hmem).mono fun s hs => ?_
   obtain ⟨ν, hν⟩ : ∃ ν, (∑ i ∈ s, μ i) + ν = Measure.sum μ := by
     refine ⟨Measure.sum fun i : ↥(sᶜ : Set ι) => μ i, ?_⟩
     simpa only [← Measure.sum_coe_finset] using Measure.sum_add_sum_compl (s : Set ι) μ
@@ -1677,26 +1685,26 @@ theorem integral_mul_norm_le_Lp_mul_Lq {E} [NormedAddCommGroup E] {f g : α →
   rw [ENNReal.toReal_rpow, ENNReal.toReal_rpow, ← ENNReal.toReal_mul]
   -- replace norms by nnnorm
   have h_left : ∫⁻ a, ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ =
-      ∫⁻ a, ((fun x => (‖f x‖₊ : ℝ≥0∞)) * fun x => (‖g x‖₊ : ℝ≥0∞)) a ∂μ := by
-    simp_rw [Pi.mul_apply, ← ofReal_norm_eq_coe_nnnorm, ENNReal.ofReal_mul (norm_nonneg _)]
-  have h_right_f : ∫⁻ a, ENNReal.ofReal (‖f a‖ ^ p) ∂μ = ∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ p ∂μ := by
+      ∫⁻ a, ((‖f ·‖ₑ) * (‖g ·‖ₑ)) a ∂μ := by
+    simp_rw [Pi.mul_apply, ← ofReal_norm_eq_enorm, ENNReal.ofReal_mul (norm_nonneg _)]
+  have h_right_f : ∫⁻ a, .ofReal (‖f a‖ ^ p) ∂μ = ∫⁻ a, ‖f a‖ₑ ^ p ∂μ := by
     refine lintegral_congr fun x => ?_
-    rw [← ofReal_norm_eq_coe_nnnorm, ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hpq.nonneg]
-  have h_right_g : ∫⁻ a, ENNReal.ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ a, (‖g a‖₊ : ℝ≥0∞) ^ q ∂μ := by
+    rw [← ofReal_norm_eq_enorm, ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hpq.nonneg]
+  have h_right_g : ∫⁻ a, .ofReal (‖g a‖ ^ q) ∂μ = ∫⁻ a, ‖g a‖ₑ ^ q ∂μ := by
     refine lintegral_congr fun x => ?_
-    rw [← ofReal_norm_eq_coe_nnnorm, ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hpq.symm.nonneg]
+    rw [← ofReal_norm_eq_enorm, ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hpq.symm.nonneg]
   rw [h_left, h_right_f, h_right_g]
   -- we can now apply `ENNReal.lintegral_mul_le_Lp_mul_Lq` (up to the `toReal` application)
   refine ENNReal.toReal_mono ?_ ?_
   · refine ENNReal.mul_ne_top ?_ ?_
     · convert hf.eLpNorm_ne_top
-      rw [eLpNorm_eq_lintegral_rpow_nnnorm]
+      rw [eLpNorm_eq_lintegral_rpow_enorm]
       · rw [ENNReal.toReal_ofReal hpq.nonneg]
       · rw [Ne, ENNReal.ofReal_eq_zero, not_le]
         exact hpq.pos
       · exact ENNReal.coe_ne_top
     · convert hg.eLpNorm_ne_top
-      rw [eLpNorm_eq_lintegral_rpow_nnnorm]
+      rw [eLpNorm_eq_lintegral_rpow_enorm]
       · rw [ENNReal.toReal_ofReal hpq.symm.nonneg]
       · rw [Ne, ENNReal.ofReal_eq_zero, not_le]
         exact hpq.symm.pos
@@ -1838,12 +1846,12 @@ theorem integral_trim (hm : m ≤ m0) {f : β → G} (hf : StronglyMeasurable[m]
     integral_trim_simpleFunc hm (f_seq n) (hf_seq_int n)
   have h_lim_1 : atTop.Tendsto (fun n => ∫ x, f_seq n x ∂μ) (𝓝 (∫ x, f x ∂μ)) := by
     refine tendsto_integral_of_L1 f hf_int (Eventually.of_forall hf_seq_int) ?_
-    exact SimpleFunc.tendsto_approxOn_range_L1_nnnorm (hf.mono hm).measurable hf_int
+    exact SimpleFunc.tendsto_approxOn_range_L1_enorm (hf.mono hm).measurable hf_int
   have h_lim_2 : atTop.Tendsto (fun n => ∫ x, f_seq n x ∂μ) (𝓝 (∫ x, f x ∂μ.trim hm)) := by
     simp_rw [hf_seq_eq]
     refine @tendsto_integral_of_L1 β G _ _ m (μ.trim hm) _ f (hf_int.trim hm hf) _ _
       (Eventually.of_forall hf_seq_int_m) ?_
-    exact @SimpleFunc.tendsto_approxOn_range_L1_nnnorm β G m _ _ _ f _ _ hf.measurable
+    exact @SimpleFunc.tendsto_approxOn_range_L1_enorm β G m _ _ _ f _ _ hf.measurable
       (hf_int.trim hm hf)
   exact tendsto_nhds_unique h_lim_1 h_lim_2
 
diff --git a/Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean b/Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean
index d76ba1fe92b5b3..6eb67e7491eb74 100644
--- a/Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean
+++ b/Mathlib/MeasureTheory/Integral/BoundedContinuousFunction.lean
@@ -47,7 +47,7 @@ theorem lintegral_lt_top_of_nnreal (f : X →ᵇ ℝ≥0) : ∫⁻ x, f x ∂μ
 theorem integrable_of_nnreal [OpensMeasurableSpace X] (f : X →ᵇ ℝ≥0) :
     Integrable (((↑) : ℝ≥0 → ℝ) ∘ ⇑f) μ := by
   refine ⟨(NNReal.continuous_coe.comp f.continuous).measurable.aestronglyMeasurable, ?_⟩
-  simp only [hasFiniteIntegral_iff_nnnorm, Function.comp_apply, NNReal.nnnorm_eq]
+  simp only [hasFiniteIntegral_iff_enorm, Function.comp_apply, NNReal.enorm_eq]
   exact lintegral_lt_top_of_nnreal _ f
 
 theorem integral_eq_integral_nnrealPart_sub [OpensMeasurableSpace X] (f : X →ᵇ ℝ) :
diff --git a/Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean b/Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean
index 827108c784d82d..95aead3dd63b1f 100644
--- a/Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean
+++ b/Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean
@@ -56,7 +56,7 @@ divergence theorem, Bochner integral
 
 open Set Finset TopologicalSpace Function BoxIntegral MeasureTheory Filter
 
-open scoped Classical Topology Interval
+open scoped Topology Interval
 
 universe u
 
diff --git a/Mathlib/MeasureTheory/Integral/DominatedConvergence.lean b/Mathlib/MeasureTheory/Integral/DominatedConvergence.lean
index 3a7713df35ddae..063908dfd80a56 100644
--- a/Mathlib/MeasureTheory/Integral/DominatedConvergence.lean
+++ b/Mathlib/MeasureTheory/Integral/DominatedConvergence.lean
@@ -102,11 +102,11 @@ theorem hasSum_integral_of_dominated_convergence {ι} [Countable ι] {F : ι →
       _ ≤ ∑' n, bound n a := sum_le_tsum _ (fun n _ => ha0 n) has
 
 theorem integral_tsum {ι} [Countable ι] {f : ι → α → G} (hf : ∀ i, AEStronglyMeasurable (f i) μ)
-    (hf' : ∑' i, ∫⁻ a : α, ‖f i a‖₊ ∂μ ≠ ∞) :
+    (hf' : ∑' i, ∫⁻ a : α, ‖f i a‖ₑ ∂μ ≠ ∞) :
     ∫ a : α, ∑' i, f i a ∂μ = ∑' i, ∫ a : α, f i a ∂μ := by
   by_cases hG : CompleteSpace G; swap
   · simp [integral, hG]
-  have hf'' : ∀ i, AEMeasurable (fun x => (‖f i x‖₊ : ℝ≥0∞)) μ := fun i => (hf i).ennnorm
+  have hf'' i : AEMeasurable (‖f i ·‖ₑ) μ := (hf i).enorm
   have hhh : ∀ᵐ a : α ∂μ, Summable fun n => (‖f n a‖₊ : ℝ) := by
     rw [← lintegral_tsum hf''] at hf'
     refine (ae_lt_top' (AEMeasurable.ennreal_tsum hf'') hf').mono ?_
@@ -124,7 +124,7 @@ theorem integral_tsum {ι} [Countable ι] {f : ι → α → G} (hf : ∀ i, AES
     apply AEMeasurable.nnreal_tsum
     exact fun i => (hf i).nnnorm.aemeasurable
   · dsimp [HasFiniteIntegral]
-    have : ∫⁻ a, ∑' n, ‖f n a‖₊ ∂μ < ⊤ := by rwa [lintegral_tsum hf'', lt_top_iff_ne_top]
+    have : ∫⁻ a, ∑' n, ‖f n a‖ₑ ∂μ < ⊤ := by rwa [lintegral_tsum hf'', lt_top_iff_ne_top]
     convert this using 1
     apply lintegral_congr_ae
     simp_rw [← coe_nnnorm, ← NNReal.coe_tsum, enorm_eq_nnnorm, NNReal.nnnorm_eq]
@@ -140,14 +140,13 @@ lemma hasSum_integral_of_summable_integral_norm {ι} [Countable ι] {F : ι →
   · simp [integral, hE, hasSum_zero]
   rw [integral_tsum (fun i ↦ (hF_int i).1)]
   · exact (hF_sum.of_norm_bounded _ fun i ↦ norm_integral_le_integral_norm _).hasSum
-  have (i : ι) : ∫⁻ (a : α), ‖F i a‖₊ ∂μ = ‖(∫ a : α, ‖F i a‖ ∂μ)‖₊ := by
+  have (i : ι) : ∫⁻ a, ‖F i a‖ₑ ∂μ = ‖∫ a, ‖F i a‖ ∂μ‖ₑ := by
+    dsimp [enorm]
     rw [lintegral_coe_eq_integral _ (hF_int i).norm, coe_nnreal_eq, coe_nnnorm,
       Real.norm_of_nonneg (integral_nonneg (fun a ↦ norm_nonneg (F i a)))]
     simp only [coe_nnnorm]
-  rw [funext this, ← ENNReal.coe_tsum]
-  · apply coe_ne_top
-  · simp_rw [← NNReal.summable_coe, coe_nnnorm]
-    exact hF_sum.abs
+  rw [funext this]
+  exact ENNReal.tsum_coe_ne_top_iff_summable.2 <| NNReal.summable_coe.1 hF_sum.abs
 
 lemma integral_tsum_of_summable_integral_norm {ι} [Countable ι] {F : ι → α → E}
     (hF_int : ∀ i : ι, Integrable (F i) μ) (hF_sum : Summable fun i ↦ ∫ a, ‖F i a‖ ∂μ) :
diff --git a/Mathlib/MeasureTheory/Integral/FundThmCalculus.lean b/Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
index a88490002c8be2..788e73736ccf9f 100644
--- a/Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
+++ b/Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
@@ -141,10 +141,9 @@ integral, fundamental theorem of calculus, FTC-1, FTC-2, change of variables in
 
 noncomputable section
 
-open scoped Classical
 open MeasureTheory Set Filter Function
 
-open scoped Classical Topology Filter ENNReal Interval NNReal
+open scoped Topology Filter ENNReal Interval NNReal
 
 variable {ι 𝕜 E A : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
 
@@ -1223,7 +1222,7 @@ theorem integrableOn_deriv_right_of_nonneg (hcont : ContinuousOn g (Icc a b))
   let F : ℝ → ℝ := (↑) ∘ f
   have intF : IntegrableOn F (Ioo a b) := by
     refine ⟨f.measurable.coe_nnreal_real.aestronglyMeasurable, ?_⟩
-    simpa only [F, hasFiniteIntegral_iff_nnnorm, comp_apply, NNReal.nnnorm_eq] using fint
+    simpa only [F, hasFiniteIntegral_iff_enorm, comp_apply, NNReal.enorm_eq] using fint
   have A : ∫⁻ x : ℝ in Ioo a b, f x = ENNReal.ofReal (∫ x in Ioo a b, F x) :=
     lintegral_coe_eq_integral _ intF
   rw [A] at hf
diff --git a/Mathlib/MeasureTheory/Integral/IntegrableOn.lean b/Mathlib/MeasureTheory/Integral/IntegrableOn.lean
index 9093ea80babe21..d5c7a91ba8e6ae 100644
--- a/Mathlib/MeasureTheory/Integral/IntegrableOn.lean
+++ b/Mathlib/MeasureTheory/Integral/IntegrableOn.lean
@@ -3,7 +3,7 @@ Copyright (c) 2021 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Zhouhang Zhou, Yury Kudryashov
 -/
-import Mathlib.MeasureTheory.Function.L1Space
+import Mathlib.MeasureTheory.Function.L1Space.Integrable
 import Mathlib.Analysis.NormedSpace.IndicatorFunction
 
 /-! # Functions integrable on a set and at a filter
@@ -232,8 +232,8 @@ theorem MeasurePreserving.integrableOn_image [MeasurableSpace β] {e : α → β
 
 theorem integrable_indicator_iff (hs : MeasurableSet s) :
     Integrable (indicator s f) μ ↔ IntegrableOn f s μ := by
-  simp_rw [IntegrableOn, Integrable, hasFiniteIntegral_iff_nnnorm,
-    nnnorm_indicator_eq_indicator_nnnorm, ENNReal.coe_indicator, lintegral_indicator hs,
+  simp_rw [IntegrableOn, Integrable, hasFiniteIntegral_iff_enorm,
+    enorm_indicator_eq_indicator_enorm, lintegral_indicator hs,
     aestronglyMeasurable_indicator_iff hs]
 
 theorem IntegrableOn.integrable_indicator (h : IntegrableOn f s μ) (hs : MeasurableSet s) :
@@ -335,7 +335,7 @@ theorem integrableOn_Lp_of_measure_ne_top {E} [NormedAddCommGroup E] {p : ℝ≥
 theorem Integrable.lintegral_lt_top {f : α → ℝ} (hf : Integrable f μ) :
     (∫⁻ x, ENNReal.ofReal (f x) ∂μ) < ∞ :=
   calc
-    (∫⁻ x, ENNReal.ofReal (f x) ∂μ) ≤ ∫⁻ x, ↑‖f x‖₊ ∂μ := lintegral_ofReal_le_lintegral_nnnorm f
+    (∫⁻ x, ENNReal.ofReal (f x) ∂μ) ≤ ∫⁻ x, ↑‖f x‖₊ ∂μ := lintegral_ofReal_le_lintegral_enorm f
     _ < ∞ := hf.2
 
 theorem IntegrableOn.setLIntegral_lt_top {f : α → ℝ} {s : Set α} (hf : IntegrableOn f s μ) :
diff --git a/Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean b/Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
index 99c22b52d22c31..e0363619f99bbb 100644
--- a/Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
+++ b/Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
@@ -396,46 +396,62 @@ section Integrable
 
 variable {α ι E : Type*} [MeasurableSpace α] {μ : Measure α} {l : Filter ι} [NormedAddCommGroup E]
 
-theorem AECover.integrable_of_lintegral_nnnorm_bounded [l.NeBot] [l.IsCountablyGenerated]
+theorem AECover.integrable_of_lintegral_enorm_bounded [l.NeBot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ)
-    (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ ENNReal.ofReal I) : Integrable f μ := by
+    (hbounded : ∀ᶠ i in l, ∫⁻ x in φ i, ‖f x‖ₑ ∂μ ≤ ENNReal.ofReal I) : Integrable f μ := by
   refine ⟨hfm, (le_of_tendsto ?_ hbounded).trans_lt ENNReal.ofReal_lt_top⟩
-  exact hφ.lintegral_tendsto_of_countably_generated hfm.ennnorm
+  exact hφ.lintegral_tendsto_of_countably_generated hfm.enorm
 
-theorem AECover.integrable_of_lintegral_nnnorm_tendsto [l.NeBot] [l.IsCountablyGenerated]
+@[deprecated (since := "2025-01-22")]
+alias AECover.integrable_of_lintegral_nnnorm_bounded :=
+ AECover.integrable_of_lintegral_enorm_bounded
+
+theorem AECover.integrable_of_lintegral_enorm_tendsto [l.NeBot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfm : AEStronglyMeasurable f μ)
-    (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 <| ENNReal.ofReal I)) :
+    (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖ₑ ∂μ) l (𝓝 <| .ofReal I)) :
     Integrable f μ := by
-  refine hφ.integrable_of_lintegral_nnnorm_bounded (max 1 (I + 1)) hfm ?_
+  refine hφ.integrable_of_lintegral_enorm_bounded (max 1 (I + 1)) hfm ?_
   refine htendsto.eventually (ge_mem_nhds ?_)
   refine (ENNReal.ofReal_lt_ofReal_iff (lt_max_of_lt_left zero_lt_one)).2 ?_
   exact lt_max_of_lt_right (lt_add_one I)
 
-theorem AECover.integrable_of_lintegral_nnnorm_bounded' [l.NeBot] [l.IsCountablyGenerated]
+@[deprecated (since := "2025-01-22")]
+alias AECover.integrable_of_lintegral_nnnorm_tendsto :=
+ AECover.integrable_of_lintegral_enorm_tendsto
+
+theorem AECover.integrable_of_lintegral_enorm_bounded' [l.NeBot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ)
-    (hbounded : ∀ᶠ i in l, (∫⁻ x in φ i, ‖f x‖₊ ∂μ) ≤ I) : Integrable f μ :=
-  hφ.integrable_of_lintegral_nnnorm_bounded I hfm
+    (hbounded : ∀ᶠ i in l, ∫⁻ x in φ i, ‖f x‖ₑ ∂μ ≤ I) : Integrable f μ :=
+  hφ.integrable_of_lintegral_enorm_bounded I hfm
     (by simpa only [ENNReal.ofReal_coe_nnreal] using hbounded)
 
-theorem AECover.integrable_of_lintegral_nnnorm_tendsto' [l.NeBot] [l.IsCountablyGenerated]
+@[deprecated (since := "2025-01-22")]
+alias AECover.integrable_of_lintegral_nnnorm_bounded' :=
+ AECover.integrable_of_lintegral_enorm_bounded'
+
+theorem AECover.integrable_of_lintegral_enorm_tendsto' [l.NeBot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ≥0) (hfm : AEStronglyMeasurable f μ)
-    (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖₊ ∂μ) l (𝓝 I)) : Integrable f μ :=
-  hφ.integrable_of_lintegral_nnnorm_tendsto I hfm
+    (htendsto : Tendsto (fun i => ∫⁻ x in φ i, ‖f x‖ₑ ∂μ) l (𝓝 I)) : Integrable f μ :=
+  hφ.integrable_of_lintegral_enorm_tendsto I hfm
     (by simpa only [ENNReal.ofReal_coe_nnreal] using htendsto)
 
+@[deprecated (since := "2025-01-22")]
+alias AECover.integrable_of_lintegral_nnnorm_tendsto' :=
+ AECover.integrable_of_lintegral_enorm_tendsto'
+
 theorem AECover.integrable_of_integral_norm_bounded [l.NeBot] [l.IsCountablyGenerated]
     {φ : ι → Set α} (hφ : AECover μ l φ) {f : α → E} (I : ℝ) (hfi : ∀ i, IntegrableOn f (φ i) μ)
     (hbounded : ∀ᶠ i in l, (∫ x in φ i, ‖f x‖ ∂μ) ≤ I) : Integrable f μ := by
   have hfm : AEStronglyMeasurable f μ :=
     hφ.aestronglyMeasurable fun i => (hfi i).aestronglyMeasurable
-  refine hφ.integrable_of_lintegral_nnnorm_bounded I hfm ?_
+  refine hφ.integrable_of_lintegral_enorm_bounded I hfm ?_
   conv at hbounded in integral _ _ =>
     rw [integral_eq_lintegral_of_nonneg_ae (ae_of_all _ fun x => @norm_nonneg E _ (f x))
         hfm.norm.restrict]
   conv at hbounded in ENNReal.ofReal _ =>
     rw [← coe_nnnorm, ENNReal.ofReal_coe_nnreal]
   refine hbounded.mono fun i hi => ?_
-  rw [← ENNReal.ofReal_toReal <| ne_top_of_lt <| hasFiniteIntegral_iff_nnnorm.mp (hfi i).2]
+  rw [← ENNReal.ofReal_toReal <| ne_top_of_lt <| hasFiniteIntegral_iff_enorm.mp (hfi i).2]
   apply ENNReal.ofReal_le_ofReal hi
 
 theorem AECover.integrable_of_integral_norm_tendsto [l.NeBot] [l.IsCountablyGenerated]
@@ -958,23 +974,27 @@ theorem _root_.HasCompactSupport.integral_Iic_deriv_eq (hf : ContDiff ℝ 1 f)
   exact h2f.filter_mono _root_.atBot_le_cocompact |>.tendsto
 
 open UniformSpace in
-lemma _root_.HasCompactSupport.ennnorm_le_lintegral_Ici_deriv
+lemma _root_.HasCompactSupport.enorm_le_lintegral_Ici_deriv
     {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
     {f : ℝ → F} (hf : ContDiff ℝ 1 f) (h'f : HasCompactSupport f) (x : ℝ) :
-    (‖f x‖₊ : ℝ≥0∞) ≤ ∫⁻ y in Iic x, ‖deriv f y‖₊ := by
+    ‖f x‖ₑ ≤ ∫⁻ y in Iic x, ‖deriv f y‖ₑ := by
   let I : F →L[ℝ] Completion F := Completion.toComplL
   let f' : ℝ → Completion F := I ∘ f
   have hf' : ContDiff ℝ 1 f' := hf.continuousLinearMap_comp I
   have h'f' : HasCompactSupport f' := h'f.comp_left rfl
-  have : (‖f' x‖₊ : ℝ≥0∞) ≤ ∫⁻ y in Iic x, ‖deriv f' y‖₊ := by
+  have : ‖f' x‖ₑ ≤ ∫⁻ y in Iic x, ‖deriv f' y‖ₑ := by
     rw [← HasCompactSupport.integral_Iic_deriv_eq hf' h'f' x]
-    exact ennnorm_integral_le_lintegral_ennnorm _
+    exact enorm_integral_le_lintegral_enorm _
   convert this with y
-  · simp [f', I, Completion.nnnorm_coe]
+  · simp [f', I, Completion.enorm_coe]
   · rw [fderiv_comp_deriv _ I.differentiableAt (hf.differentiable le_rfl _)]
     simp only [ContinuousLinearMap.fderiv]
     simp [I]
 
+@[deprecated (since := "2025-01-22")]
+alias _root_.HasCompactSupport.ennnorm_le_lintegral_Ici_deriv :=
+  HasCompactSupport.enorm_le_lintegral_Ici_deriv
+
 end IicFTC
 
 section UnivFTC
diff --git a/Mathlib/MeasureTheory/Integral/IntervalIntegral.lean b/Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
index bfb8d81c0bd4a7..338687b657efa8 100644
--- a/Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
+++ b/Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
@@ -49,10 +49,9 @@ integral
 
 noncomputable section
 
-open scoped Classical
 open MeasureTheory Set Filter Function
 
-open scoped Classical Topology Filter ENNReal Interval NNReal
+open scoped Topology Filter ENNReal Interval NNReal
 
 variable {ι 𝕜 E F A : Type*} [NormedAddCommGroup E]
 
diff --git a/Mathlib/MeasureTheory/Integral/Lebesgue.lean b/Mathlib/MeasureTheory/Integral/Lebesgue.lean
index b0cbd7bec0648f..171fca1f4b81a8 100644
--- a/Mathlib/MeasureTheory/Integral/Lebesgue.lean
+++ b/Mathlib/MeasureTheory/Integral/Lebesgue.lean
@@ -323,22 +323,31 @@ theorem setLIntegral_congr_fun {f g : α → ℝ≥0∞} {s : Set α} (hs : Meas
 @[deprecated (since := "2024-06-29")]
 alias set_lintegral_congr_fun := setLIntegral_congr_fun
 
-theorem lintegral_ofReal_le_lintegral_nnnorm (f : α → ℝ) :
-    ∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ x, ‖f x‖₊ ∂μ := by
-  simp_rw [← ofReal_norm_eq_coe_nnnorm]
+theorem lintegral_ofReal_le_lintegral_enorm (f : α → ℝ) :
+    ∫⁻ x, ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ x, ‖f x‖ₑ ∂μ := by
+  simp_rw [← ofReal_norm_eq_enorm]
   refine lintegral_mono fun x => ENNReal.ofReal_le_ofReal ?_
   rw [Real.norm_eq_abs]
   exact le_abs_self (f x)
 
-theorem lintegral_nnnorm_eq_of_ae_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ᵐ[μ] f) :
-    ∫⁻ x, ‖f x‖₊ ∂μ = ∫⁻ x, ENNReal.ofReal (f x) ∂μ := by
+@[deprecated (since := "2025-01-17")]
+alias lintegral_ofReal_le_lintegral_nnnorm := lintegral_ofReal_le_lintegral_enorm
+
+theorem lintegral_enorm_of_ae_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ᵐ[μ] f) :
+    ∫⁻ x, ‖f x‖ₑ ∂μ = ∫⁻ x, .ofReal (f x) ∂μ := by
   apply lintegral_congr_ae
   filter_upwards [h_nonneg] with x hx
-  rw [Real.nnnorm_of_nonneg hx, ENNReal.ofReal_eq_coe_nnreal hx]
+  rw [Real.enorm_eq_ofReal hx]
+
+@[deprecated (since := "2025-01-17")]
+alias lintegral_nnnorm_eq_of_ae_nonneg := lintegral_enorm_of_ae_nonneg
+
+theorem lintegral_enorm_of_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ f) :
+    ∫⁻ x, ‖f x‖ₑ ∂μ = ∫⁻ x, .ofReal (f x) ∂μ :=
+  lintegral_enorm_of_ae_nonneg <| .of_forall h_nonneg
 
-theorem lintegral_nnnorm_eq_of_nonneg {f : α → ℝ} (h_nonneg : 0 ≤ f) :
-    ∫⁻ x, ‖f x‖₊ ∂μ = ∫⁻ x, ENNReal.ofReal (f x) ∂μ :=
-  lintegral_nnnorm_eq_of_ae_nonneg (Filter.Eventually.of_forall h_nonneg)
+@[deprecated (since := "2025-01-17")]
+alias lintegral_nnnorm_eq_of_nonneg := lintegral_enorm_of_nonneg
 
 /-- **Monotone convergence theorem** -- sometimes called **Beppo-Levi convergence**.
 See `lintegral_iSup_directed` for a more general form. -/
diff --git a/Mathlib/MeasureTheory/Integral/Prod.lean b/Mathlib/MeasureTheory/Integral/Prod.lean
index 886d5b00bc9104..193e5df6d8fb36 100644
--- a/Mathlib/MeasureTheory/Integral/Prod.lean
+++ b/Mathlib/MeasureTheory/Integral/Prod.lean
@@ -35,7 +35,7 @@ product measure, Fubini's theorem, Fubini-Tonelli theorem
 
 noncomputable section
 
-open scoped Classical Topology ENNReal MeasureTheory
+open scoped Topology ENNReal MeasureTheory
 
 open Set Function Real ENNReal
 
@@ -59,7 +59,7 @@ along one of the variables (using either the Lebesgue or Bochner integral) is me
 theorem measurableSet_integrable [SFinite ν] ⦃f : α → β → E⦄
     (hf : StronglyMeasurable (uncurry f)) : MeasurableSet {x | Integrable (f x) ν} := by
   simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and]
-  exact measurableSet_lt (Measurable.lintegral_prod_right hf.ennnorm) measurable_const
+  exact measurableSet_lt (Measurable.lintegral_prod_right hf.enorm) measurable_const
 
 section
 
@@ -70,6 +70,7 @@ variable [NormedSpace ℝ E]
   This version has `f` in curried form. -/
 theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SFinite ν] ⦃f : α → β → E⦄
     (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂ν := by
+  classical
   by_cases hE : CompleteSpace E; swap; · simp [integral, hE, stronglyMeasurable_const]
   borelize E
   haveI : SeparableSpace (range (uncurry f) ∪ {0} : Set E) :=
@@ -148,7 +149,7 @@ variable [SFinite ν]
 theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
     (h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ := by
   refine ⟨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, ?_⟩
-  simp_rw [hasFiniteIntegral_iff_nnnorm, ennnorm_eq_ofReal toReal_nonneg]
+  simp_rw [hasFiniteIntegral_iff_enorm, enorm_eq_ofReal toReal_nonneg]
   convert h2s.lt_top using 1
   rw [prod_apply hs]
   apply lintegral_congr_ae
@@ -216,11 +217,11 @@ theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasu
     HasFiniteIntegral f (μ.prod ν) ↔
       (∀ᵐ x ∂μ, HasFiniteIntegral (fun y => f (x, y)) ν) ∧
         HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂ν) μ := by
-  simp only [hasFiniteIntegral_iff_nnnorm, lintegral_prod_of_measurable _ h1f.ennnorm]
+  simp only [hasFiniteIntegral_iff_enorm, lintegral_prod_of_measurable _ h1f.enorm]
   have (x) : ∀ᵐ y ∂ν, 0 ≤ ‖f (x, y)‖ := by filter_upwards with y using norm_nonneg _
   simp_rw [integral_eq_lintegral_of_nonneg_ae (this _)
       (h1f.norm.comp_measurable measurable_prod_mk_left).aestronglyMeasurable,
-    ennnorm_eq_ofReal toReal_nonneg, ofReal_norm_eq_coe_nnnorm]
+    enorm_eq_ofReal toReal_nonneg, ofReal_norm_eq_enorm]
   -- this fact is probably too specialized to be its own lemma
   have : ∀ {p q r : Prop} (_ : r → p), (r ↔ p ∧ q) ↔ p → (r ↔ q) := fun {p q r} h1 => by
     rw [← and_congr_right_iff, and_iff_right_of_imp h1]
@@ -228,7 +229,7 @@ theorem hasFiniteIntegral_prod_iff ⦃f : α × β → E⦄ (h1f : StronglyMeasu
   · intro h2f; rw [lintegral_congr_ae]
     filter_upwards [h2f] with x hx
     rw [ofReal_toReal]; rw [← lt_top_iff_ne_top]; exact hx
-  · intro h2f; refine ae_lt_top ?_ h2f.ne; exact h1f.ennnorm.lintegral_prod_right'
+  · intro h2f; refine ae_lt_top ?_ h2f.ne; exact h1f.enorm.lintegral_prod_right'
 
 theorem hasFiniteIntegral_prod_iff' ⦃f : α × β → E⦄ (h1f : AEStronglyMeasurable f (μ.prod ν)) :
     HasFiniteIntegral f (μ.prod ν) ↔
@@ -385,17 +386,15 @@ theorem continuous_integral_integral :
   refine
     tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_prod_left
       (Eventually.of_forall fun h => (L1.integrable_coeFn h).integral_prod_left) ?_
-  simp_rw [←
-    lintegral_fn_integral_sub (fun x => (‖x‖₊ : ℝ≥0∞)) (L1.integrable_coeFn _)
-      (L1.integrable_coeFn g)]
+  simp_rw [← lintegral_fn_integral_sub _ (L1.integrable_coeFn _) (L1.integrable_coeFn g)]
   apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _ (fun i => zero_le _) _
-  · exact fun i => ∫⁻ x, ∫⁻ y, ‖i (x, y) - g (x, y)‖₊ ∂ν ∂μ
-  swap; · exact fun i => lintegral_mono fun x => ennnorm_integral_le_lintegral_ennnorm _
+  · exact fun i => ∫⁻ x, ∫⁻ y, ‖i (x, y) - g (x, y)‖ₑ ∂ν ∂μ
+  swap; · exact fun i => lintegral_mono fun x => enorm_integral_le_lintegral_enorm _
   show
-    Tendsto (fun i : α × β →₁[μ.prod ν] E => ∫⁻ x, ∫⁻ y : β, ‖i (x, y) - g (x, y)‖₊ ∂ν ∂μ) (𝓝 g)
+    Tendsto (fun i : α × β →₁[μ.prod ν] E => ∫⁻ x, ∫⁻ y : β, ‖i (x, y) - g (x, y)‖ₑ ∂ν ∂μ) (𝓝 g)
       (𝓝 0)
-  have : ∀ i : α × β →₁[μ.prod ν] E, Measurable fun z => (‖i z - g z‖₊ : ℝ≥0∞) := fun i =>
-    ((Lp.stronglyMeasurable i).sub (Lp.stronglyMeasurable g)).ennnorm
+  have this (i : α × β →₁[μ.prod ν] E) : Measurable fun z => ‖i z - g z‖ₑ :=
+    ((Lp.stronglyMeasurable i).sub (Lp.stronglyMeasurable g)).enorm
   -- Porting note: was
   -- simp_rw [← lintegral_prod_of_measurable _ (this _), ← L1.ofReal_norm_sub_eq_lintegral, ←
   --   ofReal_zero]
diff --git a/Mathlib/MeasureTheory/Integral/SetIntegral.lean b/Mathlib/MeasureTheory/Integral/SetIntegral.lean
index f97aa2469f9e89..05f685c4df4b5b 100644
--- a/Mathlib/MeasureTheory/Integral/SetIntegral.lean
+++ b/Mathlib/MeasureTheory/Integral/SetIntegral.lean
@@ -54,7 +54,7 @@ assert_not_exists InnerProductSpace
 noncomputable section
 
 open Filter Function MeasureTheory RCLike Set TopologicalSpace Topology
-open scoped Classical ENNReal NNReal
+open scoped ENNReal NNReal
 
 variable {X Y E F : Type*}
 
@@ -141,6 +141,7 @@ theorem integral_finset_biUnion {ι : Type*} (t : Finset ι) {s : ι → Set X}
     (hs : ∀ i ∈ t, MeasurableSet (s i)) (h's : Set.Pairwise (↑t) (Disjoint on s))
     (hf : ∀ i ∈ t, IntegrableOn f (s i) μ) :
     ∫ x in ⋃ i ∈ t, s i, f x ∂μ = ∑ i ∈ t, ∫ x in s i, f x ∂μ := by
+  classical
   induction' t using Finset.induction_on with a t hat IH hs h's
   · simp
   · simp only [Finset.coe_insert, Finset.forall_mem_insert, Set.pairwise_insert,
@@ -211,8 +212,7 @@ theorem ofReal_setIntegral_one_of_measure_ne_top {X : Type*} {m : MeasurableSpac
     ENNReal.ofReal (∫ _ in s, (1 : ℝ) ∂μ) = ENNReal.ofReal (∫ _ in s, ‖(1 : ℝ)‖ ∂μ) := by
       simp only [norm_one]
     _ = ∫⁻ _ in s, 1 ∂μ := by
-      rw [ofReal_integral_norm_eq_lintegral_nnnorm (integrableOn_const.2 (Or.inr hs.lt_top))]
-      simp only [nnnorm_one, ENNReal.coe_one]
+      simpa [ofReal_integral_norm_eq_lintegral_enorm (integrableOn_const.2 (.inr hs.lt_top))]
     _ = μ s := setLIntegral_one _
 
 @[deprecated (since := "2024-04-17")]
@@ -244,7 +244,7 @@ theorem tendsto_setIntegral_of_monotone
   have hSm : MeasurableSet S := MeasurableSet.iUnion_of_monotone h_mono hsm
   have hsub {i} : s i ⊆ S := subset_iUnion s i
   rw [← withDensity_apply _ hSm] at hfi'
-  set ν := μ.withDensity fun x => ‖f x‖₊ with hν
+  set ν := μ.withDensity (‖f ·‖ₑ) with hν
   refine Metric.nhds_basis_closedBall.tendsto_right_iff.2 fun ε ε0 => ?_
   lift ε to ℝ≥0 using ε0.le
   have : ∀ᶠ i in atTop, ν (s i) ∈ Icc (ν S - ε) (ν S + ε) :=
@@ -252,7 +252,7 @@ theorem tendsto_setIntegral_of_monotone
   filter_upwards [this] with i hi
   rw [mem_closedBall_iff_norm', ← integral_diff (hsm i) hfi hsub, ← coe_nnnorm, NNReal.coe_le_coe, ←
     ENNReal.coe_le_coe]
-  refine (ennnorm_integral_le_lintegral_ennnorm _).trans ?_
+  refine (enorm_integral_le_lintegral_enorm _).trans ?_
   rw [← withDensity_apply _ (hSm.diff (hsm _)), ← hν, measure_diff hsub (hsm _).nullMeasurableSet]
   exacts [tsub_le_iff_tsub_le.mp hi.1,
     (hi.2.trans_lt <| ENNReal.add_lt_top.2 ⟨hfi', ENNReal.coe_lt_top⟩).ne]
@@ -661,7 +661,7 @@ theorem setIntegral_gt_gt {R : ℝ} {f : X → ℝ} (hR : 0 ≤ R)
     (μ {x | ↑R < f x}).toReal * R < ∫ x in {x | ↑R < f x}, f x ∂μ := by
   have : IntegrableOn (fun _ => R) {x | ↑R < f x} μ := by
     refine ⟨aestronglyMeasurable_const, lt_of_le_of_lt ?_ hfint.2⟩
-    refine setLIntegral_mono_ae hfint.1.ennnorm <| ae_of_all _ fun x hx => ?_
+    refine setLIntegral_mono_ae hfint.1.enorm <| ae_of_all _ fun x hx => ?_
     simp only [ENNReal.coe_le_coe, Real.nnnorm_of_nonneg hR, enorm_eq_nnnorm,
       Real.nnnorm_of_nonneg (hR.trans <| le_of_lt hx), Subtype.mk_le_mk]
     exact le_of_lt hx
@@ -1333,6 +1333,8 @@ end ContinuousMap
 theorem integral_ofReal {f : X → ℝ} : ∫ x, (f x : 𝕜) ∂μ = ↑(∫ x, f x ∂μ) :=
   (@RCLike.ofRealLI 𝕜 _).integral_comp_comm f
 
+theorem integral_complex_ofReal {f : X → ℝ} : ∫ x, (f x : ℂ) ∂μ = ∫ x, f x ∂μ := integral_ofReal
+
 theorem integral_re {f : X → 𝕜} (hf : Integrable f μ) :
     ∫ x, RCLike.re (f x) ∂μ = RCLike.re (∫ x, f x ∂μ) :=
   (@RCLike.reCLM 𝕜 _).integral_comp_comm hf
@@ -1416,10 +1418,7 @@ theorem integral_withDensity_eq_integral_smul {f : X → ℝ≥0} (f_meas : Meas
     · rfl
     · exact integral_nonneg fun x => NNReal.coe_nonneg _
     · refine ⟨f_meas.coe_nnreal_real.aemeasurable.aestronglyMeasurable, ?_⟩
-      rw [withDensity_apply _ s_meas] at hs
-      rw [hasFiniteIntegral_iff_nnnorm]
-      convert hs with x
-      simp only [NNReal.nnnorm_eq]
+      simpa [withDensity_apply _ s_meas, hasFiniteIntegral_iff_enorm] using hs
   · intro u u' _ u_int u'_int h h'
     change
       (∫ x : X, u x + u' x ∂μ.withDensity fun x : X => ↑(f x)) = ∫ x : X, f x • (u x + u' x) ∂μ
diff --git a/Mathlib/MeasureTheory/Integral/SetToL1.lean b/Mathlib/MeasureTheory/Integral/SetToL1.lean
index 9c3a4587da7c6d..2fdacefa4379d0 100644
--- a/Mathlib/MeasureTheory/Integral/SetToL1.lean
+++ b/Mathlib/MeasureTheory/Integral/SetToL1.lean
@@ -607,13 +607,13 @@ namespace SimpleFunc
 
 theorem norm_eq_sum_mul (f : α →₁ₛ[μ] G) :
     ‖f‖ = ∑ x ∈ (toSimpleFunc f).range, (μ (toSimpleFunc f ⁻¹' {x})).toReal * ‖x‖ := by
-  rw [norm_toSimpleFunc, eLpNorm_one_eq_lintegral_nnnorm]
-  have h_eq := SimpleFunc.map_apply (fun x => (‖x‖₊ : ℝ≥0∞)) (toSimpleFunc f)
+  rw [norm_toSimpleFunc, eLpNorm_one_eq_lintegral_enorm]
+  have h_eq := SimpleFunc.map_apply (‖·‖ₑ) (toSimpleFunc f)
   simp_rw [← h_eq]
   rw [SimpleFunc.lintegral_eq_lintegral, SimpleFunc.map_lintegral, ENNReal.toReal_sum]
   · congr
     ext1 x
-    rw [ENNReal.toReal_mul, mul_comm, ← ofReal_norm_eq_coe_nnnorm,
+    rw [ENNReal.toReal_mul, mul_comm, ← ofReal_norm_eq_enorm,
       ENNReal.toReal_ofReal (norm_nonneg _)]
   · intro x _
     by_cases hx0 : x = 0
@@ -1350,14 +1350,14 @@ theorem continuous_setToFun (hT : DominatedFinMeasAdditive μ T C) :
 /-- If `F i → f` in `L1`, then `setToFun μ T hT (F i) → setToFun μ T hT f`. -/
 theorem tendsto_setToFun_of_L1 (hT : DominatedFinMeasAdditive μ T C) {ι} (f : α → E)
     (hfi : Integrable f μ) {fs : ι → α → E} {l : Filter ι} (hfsi : ∀ᶠ i in l, Integrable (fs i) μ)
-    (hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖₊ ∂μ) l (𝓝 0)) :
+    (hfs : Tendsto (fun i => ∫⁻ x, ‖fs i x - f x‖ₑ ∂μ) l (𝓝 0)) :
     Tendsto (fun i => setToFun μ T hT (fs i)) l (𝓝 <| setToFun μ T hT f) := by
   classical
     let f_lp := hfi.toL1 f
     let F_lp i := if hFi : Integrable (fs i) μ then hFi.toL1 (fs i) else 0
     have tendsto_L1 : Tendsto F_lp l (𝓝 f_lp) := by
       rw [Lp.tendsto_Lp_iff_tendsto_ℒp']
-      simp_rw [eLpNorm_one_eq_lintegral_nnnorm, Pi.sub_apply]
+      simp_rw [eLpNorm_one_eq_lintegral_enorm, Pi.sub_apply]
       refine (tendsto_congr' ?_).mp hfs
       filter_upwards [hfsi] with i hi
       refine lintegral_congr_ae ?_
@@ -1380,7 +1380,7 @@ theorem tendsto_setToFun_approxOn_of_measurable (hT : DominatedFinMeasAdditive 
       (𝓝 <| setToFun μ T hT f) :=
   tendsto_setToFun_of_L1 hT _ hfi
     (Eventually.of_forall (SimpleFunc.integrable_approxOn hfm hfi h₀ h₀i))
-    (SimpleFunc.tendsto_approxOn_L1_nnnorm hfm _ hs (hfi.sub h₀i).2)
+    (SimpleFunc.tendsto_approxOn_L1_enorm hfm _ hs (hfi.sub h₀i).2)
 
 theorem tendsto_setToFun_approxOn_of_measurable_of_range_subset
     (hT : DominatedFinMeasAdditive μ T C) [MeasurableSpace E] [BorelSpace E] {f : α → E}
@@ -1569,7 +1569,7 @@ theorem tendsto_setToFun_of_dominated_convergence (hT : DominatedFinMeasAdditive
   rw [← Integrable.toL1_sub]
   refine ((fs_int n).sub f_int).coeFn_toL1.mono fun x hx => ?_
   dsimp only
-  rw [hx, ofReal_norm_eq_coe_nnnorm, Pi.sub_apply]
+  rw [hx, ofReal_norm_eq_enorm, Pi.sub_apply]
 
 /-- Lebesgue dominated convergence theorem for filters with a countable basis -/
 theorem tendsto_setToFun_filter_of_dominated_convergence (hT : DominatedFinMeasAdditive μ T C) {ι}
diff --git a/Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean b/Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean
index b7600589ddd7c8..14e6554bcb7bdf 100644
--- a/Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean
+++ b/Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean
@@ -504,9 +504,7 @@ def arrowProdEquivProdArrow (α β γ : Type*) [MeasurableSpace α] [MeasurableS
   __ := Equiv.arrowProdEquivProdArrow α β γ
   measurable_toFun _ h := by
     simp_rw [Equiv.arrowProdEquivProdArrow, coe_fn_mk]
-    #adaptation_note
-    /--
-    After https://github.com/leanprover/lean4/pull/6024
+    #adaptation_note /-- https://github.com/leanprover/lean4/pull/6024
     we need provide the type hints `(a : γ → α × β)`, to avoid unification issues.
     -/
     exact MeasurableSet.preimage h (Measurable.prod_mk
diff --git a/Mathlib/MeasureTheory/MeasurableSpace/Pi.lean b/Mathlib/MeasureTheory/MeasurableSpace/Pi.lean
index 82e1418f467fcf..fc311ca2b168c0 100644
--- a/Mathlib/MeasureTheory/MeasurableSpace/Pi.lean
+++ b/Mathlib/MeasureTheory/MeasurableSpace/Pi.lean
@@ -20,7 +20,6 @@ generate the σ-algebra on the indexed product of `α i`s.
 noncomputable section
 
 open Function Set Filter MeasurableSpace Encodable
-open scoped Classical
 
 variable {ι : Type*} {α : ι → Type*}
 
@@ -63,6 +62,7 @@ theorem IsCountablySpanning.pi {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsCoun
 theorem generateFrom_pi_eq {C : ∀ i, Set (Set (α i))} (hC : ∀ i, IsCountablySpanning (C i)) :
     (@MeasurableSpace.pi _ _ fun i => generateFrom (C i)) =
     generateFrom (pi univ '' pi univ C) := by
+  classical
   cases nonempty_encodable ι
   apply le_antisymm
   · refine iSup_le ?_; intro i; rw [comap_generateFrom]
diff --git a/Mathlib/MeasureTheory/Measure/AddContent.lean b/Mathlib/MeasureTheory/Measure/AddContent.lean
index 2332cc948c41d4..3475421c5788ae 100644
--- a/Mathlib/MeasureTheory/Measure/AddContent.lean
+++ b/Mathlib/MeasureTheory/Measure/AddContent.lean
@@ -109,27 +109,27 @@ lemma addContent_union' (hs : s ∈ C) (ht : t ∈ C) (hst : s ∪ t ∈ C) (h_d
 
 section IsSetSemiring
 
-lemma addContent_eq_add_diffFinset₀_of_subset (hC : IsSetSemiring C)
+lemma addContent_eq_add_disjointOfDiffUnion_of_subset (hC : IsSetSemiring C)
     (hs : s ∈ C) (hI : ↑I ⊆ C) (hI_ss : ∀ t ∈ I, t ⊆ s)
     (h_dis : PairwiseDisjoint (I : Set (Set α)) id) :
-    m s = ∑ i ∈ I, m i + ∑ i ∈ hC.diffFinset₀ hs hI, m i := by
+    m s = ∑ i ∈ I, m i + ∑ i ∈ hC.disjointOfDiffUnion hs hI, m i := by
   classical
-  conv_lhs => rw [← hC.sUnion_union_diffFinset₀_of_subset hs hI hI_ss]
+  conv_lhs => rw [← hC.sUnion_union_disjointOfDiffUnion_of_subset hs hI hI_ss]
   rw [addContent_sUnion]
   · rw [sum_union]
-    exact hC.disjoint_diffFinset₀ hs hI
+    exact hC.disjoint_disjointOfDiffUnion hs hI
   · rw [coe_union]
-    exact Set.union_subset hI (hC.diffFinset₀_subset hs hI)
+    exact Set.union_subset hI (hC.disjointOfDiffUnion_subset hs hI)
   · rw [coe_union]
-    exact hC.pairwiseDisjoint_union_diffFinset₀ hs hI h_dis
-  · rwa [hC.sUnion_union_diffFinset₀_of_subset hs hI hI_ss]
+    exact hC.pairwiseDisjoint_union_disjointOfDiffUnion hs hI h_dis
+  · rwa [hC.sUnion_union_disjointOfDiffUnion_of_subset hs hI hI_ss]
 
 lemma sum_addContent_le_of_subset (hC : IsSetSemiring C)
     (h_ss : ↑I ⊆ C) (h_dis : PairwiseDisjoint (I : Set (Set α)) id)
     (ht : t ∈ C) (hJt : ∀ s ∈ I, s ⊆ t) :
     ∑ u ∈ I, m u ≤ m t := by
   classical
-  rw [addContent_eq_add_diffFinset₀_of_subset hC ht h_ss hJt h_dis]
+  rw [addContent_eq_add_disjointOfDiffUnion_of_subset hC ht h_ss hJt h_dis]
   exact le_add_right le_rfl
 
 lemma addContent_mono (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C)
diff --git a/Mathlib/MeasureTheory/Measure/Haar/Quotient.lean b/Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
index a782f7e429aeba..7915e996c9a658 100644
--- a/Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
+++ b/Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
@@ -393,7 +393,7 @@ lemma QuotientGroup.integral_eq_integral_automorphize {E : Type*} [NormedAddComm
   rw [integral_tsum]
   · exact fun i ↦ (hf₁.1.comp_quasiMeasurePreserving
       (measurePreserving_smul i μ).quasiMeasurePreserving).restrict
-  · rw [← h𝓕.lintegral_eq_tsum'' (fun x ↦ ‖f x‖₊)]
+  · rw [← h𝓕.lintegral_eq_tsum'' (‖f ·‖ₑ)]
     exact ne_of_lt hf₁.2
 
 /-- This is the **Unfolding Trick**: Given a subgroup `Γ` of a group `G`, the integral of a
@@ -403,7 +403,7 @@ lemma QuotientGroup.integral_eq_integral_automorphize {E : Type*} [NormedAddComm
 lemma QuotientGroup.integral_mul_eq_integral_automorphize_mul {K : Type*} [NormedField K]
     [NormedSpace ℝ K] [μ.IsMulRightInvariant] {f : G → K}
     (f_ℒ_1 : Integrable f μ) {g : G ⧸ Γ → K} (hg : AEStronglyMeasurable g μ_𝓕)
-    (g_ℒ_infinity : essSup (fun x ↦ ↑‖g x‖₊) μ_𝓕 ≠ ∞)
+    (g_ℒ_infinity : essSup (fun x ↦ ↑‖g x‖ₑ) μ_𝓕 ≠ ∞)
     (F_ae_measurable : AEStronglyMeasurable (QuotientGroup.automorphize f) μ_𝓕) :
     ∫ x : G, g (x : G ⧸ Γ) * (f x) ∂μ
       = ∫ x : G ⧸ Γ, g x * (QuotientGroup.automorphize f x) ∂μ_𝓕 := by
@@ -418,8 +418,7 @@ lemma QuotientGroup.integral_mul_eq_integral_automorphize_mul {K : Type*} [Norme
     have : AEStronglyMeasurable (fun (x : G) ↦ g (x : (G ⧸ Γ))) μ :=
       (hg.mono_ac h𝓕.absolutelyContinuous_map).comp_measurable meas_π
     refine Integrable.essSup_smul f_ℒ_1 this ?_
-    have hg' : AEStronglyMeasurable (fun x ↦ (‖g x‖₊ : ℝ≥0∞)) μ_𝓕 :=
-      (ENNReal.continuous_coe.comp continuous_nnnorm).comp_aestronglyMeasurable hg
+    have hg' : AEStronglyMeasurable (‖g ·‖ₑ) μ_𝓕 := continuous_enorm.comp_aestronglyMeasurable hg
     rw [← essSup_comp_quotientGroup_mk h𝓕 hg'.aemeasurable]
     exact g_ℒ_infinity
   have H₂ : AEStronglyMeasurable (QuotientGroup.automorphize ((g ∘ π) * f)) μ_𝓕 := by
@@ -447,7 +446,7 @@ local notation "μ_𝓕" => Measure.map (@QuotientAddGroup.mk G' _ Γ') (μ'.res
 lemma QuotientAddGroup.integral_mul_eq_integral_automorphize_mul {K : Type*} [NormedField K]
     [NormedSpace ℝ K] [μ'.IsAddRightInvariant] {f : G' → K}
     (f_ℒ_1 : Integrable f μ') {g : G' ⧸ Γ' → K} (hg : AEStronglyMeasurable g μ_𝓕)
-    (g_ℒ_infinity : essSup (fun x ↦ (‖g x‖₊ : ℝ≥0∞)) μ_𝓕 ≠ ∞)
+    (g_ℒ_infinity : essSup (‖g ·‖ₑ) μ_𝓕 ≠ ∞)
     (F_ae_measurable : AEStronglyMeasurable (QuotientAddGroup.automorphize f) μ_𝓕)
     (h𝓕 : IsAddFundamentalDomain Γ'.op 𝓕' μ') :
     ∫ x : G', g (x : G' ⧸ Γ') * (f x) ∂μ'
@@ -463,8 +462,7 @@ lemma QuotientAddGroup.integral_mul_eq_integral_automorphize_mul {K : Type*} [No
     have : AEStronglyMeasurable (fun (x : G') ↦ g (x : (G' ⧸ Γ'))) μ' :=
       (hg.mono_ac h𝓕.absolutelyContinuous_map).comp_measurable meas_π
     refine Integrable.essSup_smul f_ℒ_1 this ?_
-    have hg' : AEStronglyMeasurable (fun x ↦ (‖g x‖₊ : ℝ≥0∞)) μ_𝓕 :=
-      (ENNReal.continuous_coe.comp continuous_nnnorm).comp_aestronglyMeasurable hg
+    have hg' : AEStronglyMeasurable (‖g ·‖ₑ) μ_𝓕 := continuous_enorm.comp_aestronglyMeasurable hg
     rw [← essSup_comp_quotientAddGroup_mk h𝓕 hg'.aemeasurable]
     exact g_ℒ_infinity
   have H₂ : AEStronglyMeasurable (QuotientAddGroup.automorphize ((g ∘ π) * f)) μ_𝓕 := by
diff --git a/Mathlib/MeasureTheory/Measure/Hausdorff.lean b/Mathlib/MeasureTheory/Measure/Hausdorff.lean
index 2f8b288b9314bd..5bd12b8bc56832 100644
--- a/Mathlib/MeasureTheory/Measure/Hausdorff.lean
+++ b/Mathlib/MeasureTheory/Measure/Hausdorff.lean
@@ -501,7 +501,7 @@ theorem mkMetric_le_liminf_tsum {β : Type*} {ι : β → Type*} [∀ n, Countab
   haveI : ∀ n, Encodable (ι n) := fun n => Encodable.ofCountable _
   simp only [mkMetric_apply]
   refine iSup₂_le fun ε hε => ?_
-  refine le_of_forall_le_of_dense fun c hc => ?_
+  refine le_of_forall_gt_imp_ge_of_dense fun c hc => ?_
   rcases ((frequently_lt_of_liminf_lt (by isBoundedDefault) hc).and_eventually
         ((hr.eventually (gt_mem_nhds hε)).and (ht.and hst))).exists with
     ⟨n, hn, hrn, htn, hstn⟩
diff --git a/Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean b/Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
index aa7a1f50934429..2027a22e7dfebb 100644
--- a/Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
+++ b/Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
@@ -563,7 +563,7 @@ theorem _root_.AlternatingMap.measure_parallelepiped (ω : G [⋀^Fin n]→ₗ[
   conv_rhs => rw [ω.eq_smul_basis_det (finBasisOfFinrankEq ℝ G _i.out)]
   simp only [addHaar_parallelepiped, AlternatingMap.measure, coe_nnreal_smul_apply,
     AlternatingMap.smul_apply, Algebra.id.smul_eq_mul, abs_mul, ENNReal.ofReal_mul (abs_nonneg _),
-    Real.ennnorm_eq_ofReal_abs]
+    ← Real.enorm_eq_ofReal_abs, enorm]
 
 instance (ω : G [⋀^Fin n]→ₗ[ℝ] ℝ) : IsAddLeftInvariant ω.measure := by
   rw [AlternatingMap.measure]; infer_instance
diff --git a/Mathlib/MeasureTheory/Measure/Prod.lean b/Mathlib/MeasureTheory/Measure/Prod.lean
index 6248de656ceb81..e20bbe27b66ac1 100644
--- a/Mathlib/MeasureTheory/Measure/Prod.lean
+++ b/Mathlib/MeasureTheory/Measure/Prod.lean
@@ -851,6 +851,13 @@ instance fst.instIsProbabilityMeasure [IsProbabilityMeasure ρ] : IsProbabilityM
     rw [fst_univ]
     exact measure_univ
 
+instance fst.instIsZeroOrProbabilityMeasure [IsZeroOrProbabilityMeasure ρ] :
+    IsZeroOrProbabilityMeasure ρ.fst := by
+  rcases eq_zero_or_isProbabilityMeasure ρ with h | h
+  · simp only [h, fst_zero]
+    infer_instance
+  · infer_instance
+
 @[simp]
 lemma fst_prod [IsProbabilityMeasure ν] : (μ.prod ν).fst = μ := by
   ext1 s hs
@@ -909,6 +916,13 @@ instance snd.instIsProbabilityMeasure [IsProbabilityMeasure ρ] : IsProbabilityM
     rw [snd_univ]
     exact measure_univ
 
+instance snd.instIsZeroOrProbabilityMeasure [IsZeroOrProbabilityMeasure ρ] :
+    IsZeroOrProbabilityMeasure ρ.snd := by
+  rcases eq_zero_or_isProbabilityMeasure ρ with h | h
+  · simp only [h, snd_zero]
+    infer_instance
+  · infer_instance
+
 @[simp]
 lemma snd_prod [IsProbabilityMeasure μ] : (μ.prod ν).snd = ν := by
   ext1 s hs
diff --git a/Mathlib/MeasureTheory/SetSemiring.lean b/Mathlib/MeasureTheory/SetSemiring.lean
index 503f9ca6b3e8d5..976dcb2c87dc3e 100644
--- a/Mathlib/MeasureTheory/SetSemiring.lean
+++ b/Mathlib/MeasureTheory/SetSemiring.lean
@@ -6,6 +6,7 @@ Authors: Rémy Degenne, Peter Pfaffelhuber
 import Mathlib.Data.Nat.Lattice
 import Mathlib.Data.Set.Accumulate
 import Mathlib.Data.Set.Pairwise.Lattice
+import Mathlib.Order.CompleteLattice
 import Mathlib.MeasureTheory.PiSystem
 
 /-! # Semirings and rings of sets
@@ -24,19 +25,27 @@ A ring of sets is a set of sets containing `∅`, stable by union, set differenc
 ## Main definitions
 
 * `MeasureTheory.IsSetSemiring C`: property of being a semi-ring of sets.
-* `MeasureTheory.IsSetSemiring.diffFinset hs ht`: for `s, t` in a semi-ring `C`
+* `MeasureTheory.IsSetSemiring.disjointOfDiff hs ht`: for `s, t` in a semi-ring `C`
   (with `hC : IsSetSemiring C`) with `hs : s ∈ C`, `ht : t ∈ C`, this is a `Finset` of
-  pairwise disjoint sets such that `s \ t = ⋃₀ hC.diffFinset hs ht`.
-* `MeasureTheory.IsSetSemiring.diffFinset₀ hs hI`: for `hs : s ∈ C` and a finset `I` of sets in `C`
-  (with `hI : ↑I ⊆ C`), this is a `Finset` of pairwise disjoint sets such that
-  `s \ ⋃₀ I = ⋃₀ hC.diffFinset₀ hs hI`.
+  pairwise disjoint sets such that `s \ t = ⋃₀ hC.disjointOfDiff hs ht`.
+* `MeasureTheory.IsSetSemiring.disjointOfDiffUnion hs hI`: for `hs : s ∈ C` and a finset
+  `I` of sets in `C` (with `hI : ↑I ⊆ C`), this is a `Finset` of pairwise disjoint sets such that
+  `s \ ⋃₀ I = ⋃₀ hC.disjointOfDiffUnion hs hI`.
+* `MeasureTheory.IsSetSemiring.disjointOfUnion hJ`: for `hJ ⊆ C`, this is a
+  `Finset` of pairwise disjoint sets such that `⋃₀ J = ⋃₀ hC.disjointOfUnion hJ`.
 
 * `MeasureTheory.IsSetRing`: property of being a ring of sets.
 
 ## Main statements
 
 * `MeasureTheory.IsSetSemiring.exists_disjoint_finset_diff_eq`: the existence of the `Finset` given
-  by the definition `IsSetSemiring.diffFinset₀` (see above).
+  by the definition `IsSetSemiring.disjointOfDiffUnion` (see above).
+* `MeasureTheory.IsSetSemiring.disjointOfUnion_props`: In a `hC : IsSetSemiring C`,
+  for a `J : Finset (Set α)` with `J ⊆ C`, there is
+  for every `x in J` some `K x ⊆ C` finite, such that
+  * `⋃ x ∈ J, K x` are pairwise disjoint and do not contain ∅,
+  * `⋃ s ∈ K x, s ⊆ x`,
+  * `⋃ x ∈ J, x = ⋃ x ∈ J, ⋃ s ∈ K x, s`.
 
 -/
 
@@ -58,84 +67,86 @@ namespace IsSetSemiring
 
 lemma isPiSystem (hC : IsSetSemiring C) : IsPiSystem C := fun s hs t ht _ ↦ hC.inter_mem s hs t ht
 
-section diffFinset
+section disjointOfDiff
 
 open scoped Classical in
 /-- In a semi-ring of sets `C`, for all sets `s, t ∈ C`, `s \ t` is equal to a disjoint union of
 finitely many sets in `C`. The finite set of sets in the union is not unique, but this definition
 gives an arbitrary `Finset (Set α)` that satisfies the equality.
 
-We remove the empty set to ensure that `t ∉ hC.diffFinset hs ht` even if `t = ∅`. -/
-noncomputable def diffFinset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
+We remove the empty set to ensure that `t ∉ hC.disjointOfDiff hs ht` even if `t = ∅`. -/
+noncomputable def disjointOfDiff (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
     Finset (Set α) :=
   (hC.diff_eq_sUnion' s hs t ht).choose \ {∅}
 
-lemma empty_not_mem_diffFinset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
-    ∅ ∉ hC.diffFinset hs ht := by
+lemma empty_nmem_disjointOfDiff (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
+    ∅ ∉ hC.disjointOfDiff hs ht := by
   classical
-  simp only [diffFinset, mem_sdiff, Finset.mem_singleton, eq_self_iff_true, not_true,
-    and_false, not_false_iff]
+  simp only [disjointOfDiff, mem_sdiff, Finset.mem_singleton, eq_self_iff_true,
+    not_true, and_false, not_false_iff]
 
-lemma diffFinset_subset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
-    ↑(hC.diffFinset hs ht) ⊆ C := by
+lemma subset_disjointOfDiff (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
+    ↑(hC.disjointOfDiff hs ht) ⊆ C := by
   classical
-  simp only [diffFinset, coe_sdiff, coe_singleton, diff_singleton_subset_iff]
+  simp only [disjointOfDiff, coe_sdiff, coe_singleton, diff_singleton_subset_iff]
   exact (hC.diff_eq_sUnion' s hs t ht).choose_spec.1.trans (Set.subset_insert _ _)
 
-lemma pairwiseDisjoint_diffFinset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
-    PairwiseDisjoint (hC.diffFinset hs ht : Set (Set α)) id := by
+lemma pairwiseDisjoint_disjointOfDiff (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
+    PairwiseDisjoint (hC.disjointOfDiff hs ht : Set (Set α)) id := by
   classical
-  simp only [diffFinset, coe_sdiff, coe_singleton]
+  simp only [disjointOfDiff, coe_sdiff, coe_singleton]
   exact Set.PairwiseDisjoint.subset (hC.diff_eq_sUnion' s hs t ht).choose_spec.2.1
       diff_subset
 
-lemma sUnion_diffFinset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
-    ⋃₀ hC.diffFinset hs ht = s \ t := by
+lemma sUnion_disjointOfDiff (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
+    ⋃₀ hC.disjointOfDiff hs ht = s \ t := by
   classical
   rw [(hC.diff_eq_sUnion' s hs t ht).choose_spec.2.2]
-  simp only [diffFinset, coe_sdiff, coe_singleton, diff_singleton_subset_iff]
+  simp only [disjointOfDiff, coe_sdiff, coe_singleton, diff_singleton_subset_iff]
   rw [sUnion_diff_singleton_empty]
 
-lemma not_mem_diffFinset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
-    t ∉ hC.diffFinset hs ht := by
+lemma nmem_disjointOfDiff (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
+    t ∉ hC.disjointOfDiff hs ht := by
   intro hs_mem
   suffices t ⊆ s \ t by
     have h := @disjoint_sdiff_self_right _ t s _
     specialize h le_rfl this
     simp only [Set.bot_eq_empty, Set.le_eq_subset, subset_empty_iff] at h
-    refine hC.empty_not_mem_diffFinset hs ht ?_
+    refine hC.empty_nmem_disjointOfDiff hs ht ?_
     rwa [← h]
-  rw [← hC.sUnion_diffFinset hs ht]
+  rw [← hC.sUnion_disjointOfDiff hs ht]
   exact subset_sUnion_of_mem hs_mem
 
-lemma sUnion_insert_diffFinset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) (hst : t ⊆ s) :
-    ⋃₀ insert t (hC.diffFinset hs ht) = s := by
-  conv_rhs => rw [← union_diff_cancel hst, ← hC.sUnion_diffFinset hs ht]
+lemma sUnion_insert_disjointOfDiff (hC : IsSetSemiring C) (hs : s ∈ C)
+    (ht : t ∈ C) (hst : t ⊆ s) :
+    ⋃₀ insert t (hC.disjointOfDiff hs ht) = s := by
+  conv_rhs => rw [← union_diff_cancel hst, ← hC.sUnion_disjointOfDiff hs ht]
   simp only [mem_coe, sUnion_insert]
 
-lemma disjoint_sUnion_diffFinset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
-    Disjoint t (⋃₀ hC.diffFinset hs ht) := by
-  rw [hC.sUnion_diffFinset]
+lemma disjoint_sUnion_disjointOfDiff (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
+    Disjoint t (⋃₀ hC.disjointOfDiff hs ht) := by
+  rw [hC.sUnion_disjointOfDiff]
   exact disjoint_sdiff_right
 
-lemma pairwiseDisjoint_insert_diffFinset (hC : IsSetSemiring C) (hs : s ∈ C) (ht : t ∈ C) :
-    PairwiseDisjoint (insert t (hC.diffFinset hs ht) : Set (Set α)) id := by
-  have h := hC.pairwiseDisjoint_diffFinset hs ht
-  refine PairwiseDisjoint.insert_of_not_mem h (hC.not_mem_diffFinset hs ht) fun u hu ↦ ?_
+lemma pairwiseDisjoint_insert_disjointOfDiff (hC : IsSetSemiring C) (hs : s ∈ C)
+    (ht : t ∈ C) :
+    PairwiseDisjoint (insert t (hC.disjointOfDiff hs ht) : Set (Set α)) id := by
+  have h := hC.pairwiseDisjoint_disjointOfDiff hs ht
+  refine PairwiseDisjoint.insert_of_not_mem h (hC.nmem_disjointOfDiff hs ht) fun u hu ↦ ?_
   simp_rw [id]
-  refine Disjoint.mono_right ?_ (hC.disjoint_sUnion_diffFinset hs ht)
+  refine Disjoint.mono_right ?_ (hC.disjoint_sUnion_disjointOfDiff hs ht)
   simp only [Set.le_eq_subset]
   exact subset_sUnion_of_mem hu
 
-end diffFinset
+end disjointOfDiff
 
-section diffFinset₀
+section disjointOfDiffUnion
 
 variable {I : Finset (Set α)}
 
 /-- In a semiring of sets `C`, for all set `s ∈ C` and finite set of sets `I ⊆ C`, there is a
 finite set of sets in `C` whose union is `s \ ⋃₀ I`.
-See `IsSetSemiring.diffFinset₀` for a definition that gives such a set. -/
+See `IsSetSemiring.disjointOfDiffUnion` for a definition that gives such a set. -/
 lemma exists_disjoint_finset_diff_eq (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
     ∃ J : Finset (Set α), ↑J ⊆ C ∧ PairwiseDisjoint (J : Set (Set α)) id ∧
       s \ ⋃₀ I = ⋃₀ J := by
@@ -151,14 +162,14 @@ lemma exists_disjoint_finset_diff_eq (hC : IsSetSemiring C) (hs : s ∈ C) (hI :
   rw [coe_insert] at hI
   have ht : t ∈ C := hI (Set.mem_insert _ _)
   obtain ⟨J, h_ss, h_dis, h_eq⟩ := h ((Set.subset_insert _ _).trans hI)
-  let Ju : ∀ u ∈ C, Finset (Set α) := fun u hu ↦ hC.diffFinset hu ht
+  let Ju : ∀ u ∈ C, Finset (Set α) := fun u hu ↦ hC.disjointOfDiff hu ht
   have hJu_subset : ∀ (u) (hu : u ∈ C), ↑(Ju u hu) ⊆ C := by
     intro u hu x hx
-    exact hC.diffFinset_subset hu ht hx
+    exact hC.subset_disjointOfDiff hu ht hx
   have hJu_disj : ∀ (u) (hu : u ∈ C), (Ju u hu : Set (Set α)).PairwiseDisjoint id := fun u hu ↦
-    hC.pairwiseDisjoint_diffFinset hu ht
+    hC.pairwiseDisjoint_disjointOfDiff hu ht
   have hJu_sUnion : ∀ (u) (hu : u ∈ C), ⋃₀ (Ju u hu : Set (Set α)) = u \ t :=
-    fun u hu ↦ hC.sUnion_diffFinset hu ht
+    fun u hu ↦ hC.sUnion_disjointOfDiff hu ht
   have hJu_disj' : ∀ (u) (hu : u ∈ C) (v) (hv : v ∈ C) (_h_dis : Disjoint u v),
       Disjoint (⋃₀ (Ju u hu : Set (Set α))) (⋃₀ ↑(Ju v hv)) := by
     intro u hu v hv huv_disj
@@ -202,80 +213,228 @@ lemma exists_disjoint_finset_diff_eq (hC : IsSetSemiring C) (hs : s ∈ C) (hI :
 
 open scoped Classical in
 /-- In a semiring of sets `C`, for all set `s ∈ C` and finite set of sets `I ⊆ C`,
-`diffFinset₀` is a finite set of sets in `C` such that `s \ ⋃₀ I = ⋃₀ (hC.diffFinset₀ hs I hI)`.
-`diffFinset` is a special case of `diffFinset₀` where `I` is a singleton. -/
-noncomputable def diffFinset₀ (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) : Finset (Set α) :=
+`disjointOfDiffUnion` is a finite set of sets in `C` such that
+`s \ ⋃₀ I = ⋃₀ (hC.disjointOfDiffUnion hs I hI)`.
+`disjointOfDiff` is a special case of `disjointOfDiffUnion` where `I` is a
+singleton. -/
+noncomputable def disjointOfDiffUnion (hC : IsSetSemiring C) (hs : s ∈ C)
+  (hI : ↑I ⊆ C) : Finset (Set α) :=
   (hC.exists_disjoint_finset_diff_eq hs hI).choose \ {∅}
 
-lemma empty_not_mem_diffFinset₀ (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
-    ∅ ∉ hC.diffFinset₀ hs hI := by
+lemma empty_nmem_disjointOfDiffUnion (hC : IsSetSemiring C) (hs : s ∈ C)
+    (hI : ↑I ⊆ C) :
+    ∅ ∉ hC.disjointOfDiffUnion hs hI := by
   classical
-  simp only [diffFinset₀, mem_sdiff, Finset.mem_singleton, eq_self_iff_true, not_true,
-    and_false, not_false_iff]
+  simp only [disjointOfDiffUnion, mem_sdiff, Finset.mem_singleton, eq_self_iff_true,
+    not_true, and_false, not_false_iff]
 
-lemma diffFinset₀_subset (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
-    ↑(hC.diffFinset₀ hs hI) ⊆ C := by
+lemma disjointOfDiffUnion_subset (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
+    ↑(hC.disjointOfDiffUnion hs hI) ⊆ C := by
   classical
-  simp only [diffFinset₀, coe_sdiff, coe_singleton, diff_singleton_subset_iff]
+  simp only [disjointOfDiffUnion, coe_sdiff, coe_singleton, diff_singleton_subset_iff]
   exact (hC.exists_disjoint_finset_diff_eq hs hI).choose_spec.1.trans (Set.subset_insert _ _)
 
-lemma pairwiseDisjoint_diffFinset₀ (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
-    PairwiseDisjoint (hC.diffFinset₀ hs hI : Set (Set α)) id := by
+lemma pairwiseDisjoint_disjointOfDiffUnion (hC : IsSetSemiring C) (hs : s ∈ C)
+    (hI : ↑I ⊆ C) : PairwiseDisjoint (hC.disjointOfDiffUnion hs hI : Set (Set α)) id := by
   classical
-  simp only [diffFinset₀, coe_sdiff, coe_singleton]
+  simp only [disjointOfDiffUnion, coe_sdiff, coe_singleton]
   exact Set.PairwiseDisjoint.subset
     (hC.exists_disjoint_finset_diff_eq hs hI).choose_spec.2.1 diff_subset
 
-lemma diff_sUnion_eq_sUnion_diffFinset₀ (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
-    s \ ⋃₀ I = ⋃₀ hC.diffFinset₀ hs hI := by
+lemma diff_sUnion_eq_sUnion_disjointOfDiffUnion (hC : IsSetSemiring C) (hs : s ∈ C)
+    (hI : ↑I ⊆ C) : s \ ⋃₀ I = ⋃₀ hC.disjointOfDiffUnion hs hI := by
   classical
   rw [(hC.exists_disjoint_finset_diff_eq hs hI).choose_spec.2.2]
-  simp only [diffFinset₀, coe_sdiff, coe_singleton, diff_singleton_subset_iff]
+  simp only [disjointOfDiffUnion, coe_sdiff, coe_singleton, diff_singleton_subset_iff]
   rw [sUnion_diff_singleton_empty]
 
-lemma sUnion_diffFinset₀_subset (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
-    ⋃₀ (hC.diffFinset₀ hs hI : Set (Set α)) ⊆ s := by
-  rw [← hC.diff_sUnion_eq_sUnion_diffFinset₀]
+lemma sUnion_disjointOfDiffUnion_subset (hC : IsSetSemiring C) (hs : s ∈ C)
+    (hI : ↑I ⊆ C) : ⋃₀ (hC.disjointOfDiffUnion hs hI : Set (Set α)) ⊆ s := by
+  rw [← hC.diff_sUnion_eq_sUnion_disjointOfDiffUnion]
   exact diff_subset
 
-lemma disjoint_sUnion_diffFinset₀ (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
-    Disjoint (⋃₀ (I : Set (Set α))) (⋃₀ hC.diffFinset₀ hs hI) := by
-  rw [← hC.diff_sUnion_eq_sUnion_diffFinset₀]; exact Set.disjoint_sdiff_right
-
-lemma disjoint_diffFinset₀ (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
-    Disjoint I (hC.diffFinset₀ hs hI) := by
+lemma subset_of_diffUnion_disjointOfDiffUnion (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C)
+    (t : Set α) (ht : t ∈ (hC.disjointOfDiffUnion hs hI : Set (Set α))) :
+    t ⊆ s \ ⋃₀ I := by
+  revert t ht
+  rw [← sUnion_subset_iff, hC.diff_sUnion_eq_sUnion_disjointOfDiffUnion hs hI]
+
+lemma subset_of_mem_disjointOfDiffUnion (hC : IsSetSemiring C) {I : Finset (Set α)}
+    (hs : s ∈ C) (hI : ↑I ⊆ C) (t : Set α)
+    (ht : t ∈ (hC.disjointOfDiffUnion hs hI : Set (Set α))) :
+    t ⊆ s := by
+  apply le_trans <| hC.subset_of_diffUnion_disjointOfDiffUnion hs hI t ht
+  exact sdiff_le (a := s) (b := ⋃₀ I)
+
+lemma disjoint_sUnion_disjointOfDiffUnion (hC : IsSetSemiring C) (hs : s ∈ C)
+    (hI : ↑I ⊆ C) :
+    Disjoint (⋃₀ (I : Set (Set α))) (⋃₀ hC.disjointOfDiffUnion hs hI) := by
+  rw [← hC.diff_sUnion_eq_sUnion_disjointOfDiffUnion]; exact Set.disjoint_sdiff_right
+
+lemma disjoint_disjointOfDiffUnion (hC : IsSetSemiring C) (hs : s ∈ C) (hI : ↑I ⊆ C) :
+    Disjoint I (hC.disjointOfDiffUnion hs hI) := by
   by_contra h
   rw [Finset.not_disjoint_iff] at h
-  obtain ⟨u, huI, hu_diffFinset₀⟩ := h
-  have h_disj : u ≤ ⊥ := hC.disjoint_sUnion_diffFinset₀ hs hI (subset_sUnion_of_mem huI)
-    (subset_sUnion_of_mem hu_diffFinset₀)
+  obtain ⟨u, huI, hu_disjointOfDiffUnion⟩ := h
+  have h_disj : u ≤ ⊥ :=
+    hC.disjoint_sUnion_disjointOfDiffUnion hs hI (subset_sUnion_of_mem huI)
+    (subset_sUnion_of_mem hu_disjointOfDiffUnion)
   simp only [Set.bot_eq_empty, Set.le_eq_subset, subset_empty_iff] at h_disj
-  refine hC.empty_not_mem_diffFinset₀ hs hI ?_
-  rwa [h_disj] at hu_diffFinset₀
+  refine hC.empty_nmem_disjointOfDiffUnion hs hI ?_
+  rwa [h_disj] at hu_disjointOfDiffUnion
 
-lemma pairwiseDisjoint_union_diffFinset₀ (hC : IsSetSemiring C) (hs : s ∈ C)
+lemma pairwiseDisjoint_union_disjointOfDiffUnion (hC : IsSetSemiring C) (hs : s ∈ C)
     (hI : ↑I ⊆ C) (h_dis : PairwiseDisjoint (I : Set (Set α)) id) :
-    PairwiseDisjoint (I ∪ hC.diffFinset₀ hs hI : Set (Set α)) id := by
+    PairwiseDisjoint (I ∪ hC.disjointOfDiffUnion hs hI : Set (Set α)) id := by
   rw [pairwiseDisjoint_union]
-  refine ⟨h_dis, hC.pairwiseDisjoint_diffFinset₀ hs hI, fun u hu v hv _ ↦ ?_⟩
+  refine ⟨h_dis, hC.pairwiseDisjoint_disjointOfDiffUnion hs hI, fun u hu v hv _ ↦ ?_⟩
   simp_rw [id]
   exact disjoint_of_subset (subset_sUnion_of_mem hu) (subset_sUnion_of_mem hv)
-    (hC.disjoint_sUnion_diffFinset₀ hs hI)
+    (hC.disjoint_sUnion_disjointOfDiffUnion hs hI)
 
-lemma sUnion_union_sUnion_diffFinset₀_of_subset (hC : IsSetSemiring C) (hs : s ∈ C)
-    (hI : ↑I ⊆ C) (hI_ss : ∀ t ∈ I, t ⊆ s) :
-    ⋃₀ I ∪ ⋃₀ hC.diffFinset₀ hs hI = s := by
+lemma sUnion_union_sUnion_disjointOfDiffUnion_of_subset (hC : IsSetSemiring C)
+    (hs : s ∈ C) (hI : ↑I ⊆ C) (hI_ss : ∀ t ∈ I, t ⊆ s) :
+    ⋃₀ I ∪ ⋃₀ hC.disjointOfDiffUnion hs hI = s := by
   conv_rhs => rw [← union_diff_cancel (Set.sUnion_subset hI_ss : ⋃₀ ↑I ⊆ s),
-    hC.diff_sUnion_eq_sUnion_diffFinset₀ hs hI]
+    hC.diff_sUnion_eq_sUnion_disjointOfDiffUnion hs hI]
 
-lemma sUnion_union_diffFinset₀_of_subset (hC : IsSetSemiring C) (hs : s ∈ C)
+lemma sUnion_union_disjointOfDiffUnion_of_subset (hC : IsSetSemiring C) (hs : s ∈ C)
     (hI : ↑I ⊆ C) (hI_ss : ∀ t ∈ I, t ⊆ s) [DecidableEq (Set α)] :
-    ⋃₀ ↑(I ∪ hC.diffFinset₀ hs hI) = s := by
-  conv_rhs => rw [← sUnion_union_sUnion_diffFinset₀_of_subset hC hs hI hI_ss]
+    ⋃₀ ↑(I ∪ hC.disjointOfDiffUnion hs hI) = s := by
+  conv_rhs => rw [← sUnion_union_sUnion_disjointOfDiffUnion_of_subset hC hs hI hI_ss]
   simp_rw [coe_union]
   rw [sUnion_union]
 
-end diffFinset₀
+end disjointOfDiffUnion
+
+section disjointOfUnion
+
+
+variable {j : Set α} {J : Finset (Set α)}
+
+open Set MeasureTheory Order
+
+theorem disjointOfUnion_props (hC : IsSetSemiring C) (h1 : ↑J ⊆ C) :
+    ∃ K : Set α → Finset (Set α),
+      PairwiseDisjoint J K
+      ∧ (∀ i ∈ J, ↑(K i) ⊆ C)
+      ∧ PairwiseDisjoint (⋃ x ∈ J, (K x : Set (Set α))) id
+      ∧ (∀ j ∈ J, ⋃₀ K j ⊆ j)
+      ∧ (∀ j ∈ J, ∅ ∉ K j)
+      ∧ ⋃₀ J = ⋃₀ (⋃ x ∈ J, (K x : Set (Set α))) := by
+  classical
+  induction J using Finset.cons_induction with
+  | empty => simp
+  | cons s J hJ hind =>
+    rw [cons_eq_insert, coe_insert, Set.insert_subset_iff] at h1
+    obtain ⟨K, hK0, ⟨hK1, hK2, hK3, hK4, hK5⟩⟩ := hind h1.2
+    let K1 : Set α → Finset (Set α) := fun (t : Set α) ↦
+      if t = s then (hC.disjointOfDiffUnion h1.1 h1.2) else K t
+    have hK1s : K1 s = hC.disjointOfDiffUnion h1.1 h1.2 := by simp [K1]
+    have hK1_of_ne t (ht : t ≠ s) : K1 t = K t := by simp [K1, ht]
+    use K1
+    simp only [cons_eq_insert, disjiUnion_eq_biUnion, Finset.biUnion_insert, coe_union, coe_biUnion,
+      mem_coe, Set.union_subset_iff, iUnion_subset_iff, Finset.mem_insert, sUnion_subset_iff,
+      forall_eq_or_imp, coe_insert, sUnion_insert, exists_and_left, exists_prop]
+    -- two simplification rules for induction hypothesis
+    have ht1' : ∀ x ∈ J, K1 x = K x := fun x hx ↦ hK1_of_ne _ (fun h_eq ↦ hJ (h_eq ▸ hx))
+    have ht2 : (⋃ x ∈ J, (K1 x : Set (Set α))) = ⋃ x ∈ J, ((K x : Set (Set α))) := by
+      apply iUnion₂_congr
+      intros x hx
+      exact_mod_cast hK1_of_ne _ (ne_of_mem_of_not_mem hx hJ)
+    simp only [hK1s]
+    refine ⟨?_, ⟨hC.disjointOfDiffUnion_subset h1.1 h1.2, ?_⟩, ?_,
+      ⟨hC.subset_of_mem_disjointOfDiffUnion h1.1 h1.2, ?_⟩, ?_, ?_⟩
+    · apply Set.Pairwise.insert
+      · intro j hj i hi hij
+        rw [Function.onFun, ht1' j hj, ht1' i hi]
+        exact hK0 hj hi hij
+      · intro i hi _
+        have h7 : Disjoint ↑(hC.disjointOfDiffUnion h1.1 h1.2) (K i : Set (Set α)) := by
+          refine disjoint_of_sSup_disjoint_of_le_of_le
+            (hC.subset_of_diffUnion_disjointOfDiffUnion h1.1 h1.2) ?_
+            (@disjoint_sdiff_left _ (⋃₀ J) s) (Or.inl
+              (hC.empty_nmem_disjointOfDiffUnion h1.1 h1.2))
+          simp only [mem_coe, Set.le_eq_subset]
+          apply sUnion_subset_iff.mp
+          exact (hK3 i hi).trans (subset_sUnion_of_mem hi)
+        have h8 : Function.onFun Disjoint K1 s i := by
+          refine Finset.disjoint_iff_inter_eq_empty.mpr ?_
+          rw [ht1' i hi, hK1s]
+          rw [Set.disjoint_iff_inter_eq_empty] at h7
+          exact_mod_cast h7
+        exact ⟨h8, Disjoint.symm h8⟩
+    · intros i hi
+      rw [ht1' i hi]
+      exact hK1 i hi
+    · simp only [iUnion_iUnion_eq_or_left]
+      refine pairwiseDisjoint_union.mpr ⟨?_, ?_, ?_⟩
+      · rw [hK1s]
+        exact hC.pairwiseDisjoint_disjointOfDiffUnion h1.1 h1.2
+      · simpa [ht2]
+      · simp only [mem_coe, mem_iUnion, exists_prop, ne_eq, id_eq, forall_exists_index, and_imp]
+        intros i hi j x hx h3 h4
+        obtain ki : i ⊆ s \ ⋃₀ J := hC.subset_of_diffUnion_disjointOfDiffUnion h1.1 h1.2 _
+          (hK1s ▸ hi)
+        obtain hx2 : j ⊆ x := subset_trans (subset_sUnion_of_mem (ht1' x hx ▸ h3)) (hK3 x hx)
+        obtain kj : j ⊆ ⋃₀ J := hx2.trans <| subset_sUnion_of_mem hx
+        exact disjoint_of_subset ki kj disjoint_sdiff_left
+    · intros a ha
+      simp_rw [hK1_of_ne _ (ne_of_mem_of_not_mem ha hJ)]
+      change ∀ t' ∈ (K a : Set (Set α)), t' ⊆ a
+      rw [← sUnion_subset_iff]
+      exact hK3 a ha
+    · refine ⟨hC.empty_nmem_disjointOfDiffUnion h1.1 h1.2, ?_⟩
+      intros a ha
+      rw [ht1' a ha]
+      exact hK4 a ha
+    · simp only [iUnion_iUnion_eq_or_left, ht2, sUnion_union, ht2, K1]
+      simp_rw [apply_ite, hK5,
+      ← hC.diff_sUnion_eq_sUnion_disjointOfDiffUnion h1.1 h1.2, hK5]
+      simp only [↓reduceIte, diff_union_self]
+
+/-- For some `hJ : J ⊆ C` and `j : Set α`, where `hC : IsSetSemiring C`, this is
+a `Finset (Set α)` such that `K j := hC.disjointOfUnion hJ` are disjoint
+and `⋃₀ K j ⊆ j`, for `j ∈ J`.
+Using these we write `⋃₀ J` as a disjoint union `⋃₀ J = ⋃₀ ⋃ x ∈ J, (K x)`.
+See `MeasureTheory.IsSetSemiring.disjointOfUnion_props`.-/
+noncomputable def disjointOfUnion (hC : IsSetSemiring C) (hJ : ↑J ⊆ C) (j : Set α) :=
+  (hC.disjointOfUnion_props hJ).choose j
+
+lemma pairwiseDisjoint_disjointOfUnion (hC : IsSetSemiring C) (hJ : ↑J ⊆ C) :
+    PairwiseDisjoint J (hC.disjointOfUnion hJ) :=
+  (Exists.choose_spec (hC.disjointOfUnion_props hJ)).1
+
+lemma disjointOfUnion_subset (hC : IsSetSemiring C) (hJ : ↑J ⊆ C) (hj : j ∈ J) :
+    (disjointOfUnion hC hJ j : Set (Set α)) ⊆ C :=
+  (Exists.choose_spec (hC.disjointOfUnion_props hJ)).2.1 _ hj
+
+lemma pairwiseDisjoint_biUnion_disjointOfUnion (hC : IsSetSemiring C) (hJ : ↑J ⊆ C) :
+    PairwiseDisjoint (⋃ x ∈ J, (hC.disjointOfUnion hJ x : Set (Set α))) id :=
+  (Exists.choose_spec (hC.disjointOfUnion_props hJ)).2.2.1
+
+lemma pairwiseDisjoint_disjointOfUnion_of_mem (hC : IsSetSemiring C) (hJ : ↑J ⊆ C) (hj : j ∈ J) :
+    PairwiseDisjoint (hC.disjointOfUnion hJ j : Set (Set α)) id := by
+  apply PairwiseDisjoint.subset (hC.pairwiseDisjoint_biUnion_disjointOfUnion hJ)
+  exact subset_iUnion₂_of_subset j hj fun ⦃a⦄ a ↦ a
+
+lemma disjointOfUnion_subset_of_mem (hC : IsSetSemiring C) (hJ : ↑J ⊆ C) (hj : j ∈ J) :
+    ⋃₀ hC.disjointOfUnion hJ j ⊆ j :=
+  (Exists.choose_spec (hC.disjointOfUnion_props hJ)).2.2.2.1 j hj
+
+lemma subset_of_mem_disjointOfUnion (hC : IsSetSemiring C) (hJ : ↑J ⊆ C) (hj : j ∈ J) {x : Set α}
+    (hx : x ∈ (hC.disjointOfUnion hJ) j) : x ⊆ j :=
+  sUnion_subset_iff.mp (hC.disjointOfUnion_subset_of_mem hJ hj) x hx
+
+lemma empty_nmem_disjointOfUnion (hC : IsSetSemiring C) (hJ : ↑J ⊆ C) (hj : j ∈ J) :
+    ∅ ∉ hC.disjointOfUnion hJ j :=
+  (Exists.choose_spec (hC.disjointOfUnion_props hJ)).2.2.2.2.1 j hj
+
+lemma sUnion_disjointOfUnion (hC : IsSetSemiring C) (hJ : ↑J ⊆ C) :
+    ⋃₀ ⋃ x ∈ J, (hC.disjointOfUnion hJ x : Set (Set α)) = ⋃₀ J
+    := (Exists.choose_spec (hC.disjointOfUnion_props hJ)).2.2.2.2.2.symm
+
+end disjointOfUnion
 
 end IsSetSemiring
 
diff --git a/Mathlib/ModelTheory/PartialEquiv.lean b/Mathlib/ModelTheory/PartialEquiv.lean
index 6a0442f49b37a2..2c595a9111371a 100644
--- a/Mathlib/ModelTheory/PartialEquiv.lean
+++ b/Mathlib/ModelTheory/PartialEquiv.lean
@@ -347,7 +347,8 @@ lemma partialEquivLimit_comp_inclusion {i : ι} :
 
 theorem le_partialEquivLimit (i : ι) : S i ≤ partialEquivLimit S :=
   ⟨le_iSup (f := fun i ↦ (S i).dom) _, by
-    #adaptation_note /-- After https://github.com/leanprover/lean4/pull/5020, these two `simp` calls cannot be combined. -/
+    #adaptation_note /-- https://github.com/leanprover/lean4/pull/5020
+    these two `simp` calls cannot be combined. -/
     simp only [partialEquivLimit_comp_inclusion]
     simp only [cod_partialEquivLimit, dom_partialEquivLimit, ← Embedding.comp_assoc,
       subtype_comp_inclusion]⟩
diff --git a/Mathlib/NumberTheory/AbelSummation.lean b/Mathlib/NumberTheory/AbelSummation.lean
index 202136f5c40add..3594e9a1f4abea 100644
--- a/Mathlib/NumberTheory/AbelSummation.lean
+++ b/Mathlib/NumberTheory/AbelSummation.lean
@@ -51,8 +51,8 @@ namespace abelSummationProof
 
 open intervalIntegral IntervalIntegrable
 
-private theorem sumlocc (n : ℕ) :
-    ∀ᵐ t, t ∈ Set.Icc (n : ℝ) (n + 1) → ∑ k ∈ Icc 0 ⌊t⌋₊, c k = ∑ k ∈ Icc 0 n, c k := by
+private theorem sumlocc {m : ℕ} (n : ℕ) :
+    ∀ᵐ t, t ∈ Set.Icc (n : ℝ) (n + 1) → ∑ k ∈ Icc m ⌊t⌋₊, c k = ∑ k ∈ Icc m n, c k := by
   filter_upwards [Ico_ae_eq_Icc] with t h ht
   rw [Nat.floor_eq_on_Ico _ _ (h.mpr ht)]
 
@@ -83,11 +83,12 @@ private theorem ineqofmemIco' {k : ℕ} (hk : k ∈ Ico (⌊a⌋₊ + 1) ⌊b⌋
     a ≤ k ∧ k + 1 ≤ b :=
   ineqofmemIco (by rwa [← Finset.coe_Ico])
 
-private theorem integrablemulsum (ha : 0 ≤ a) {g : ℝ → 𝕜} (hg_int : IntegrableOn g (Set.Icc a b)) :
-    IntegrableOn (fun t ↦ g t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k) (Set.Icc a b) := by
+theorem _root_.integrableOn_mul_sum_Icc {m : ℕ} (ha : 0 ≤ a) {g : ℝ → 𝕜}
+    (hg_int : IntegrableOn g (Set.Icc a b)) :
+    IntegrableOn (fun t ↦ g t * ∑ k ∈ Icc m ⌊t⌋₊, c k) (Set.Icc a b) := by
   obtain hab | hab := le_or_gt a b
   · obtain hb | hb := eq_or_lt_of_le (Nat.floor_le_floor hab)
-    · have : ∀ᵐ t, t ∈ Set.Icc a b → ∑ k ∈ Icc 0 ⌊a⌋₊, c k = ∑ k ∈ Icc 0 ⌊t⌋₊, c k := by
+    · have : ∀ᵐ t, t ∈ Set.Icc a b → ∑ k ∈ Icc m ⌊a⌋₊, c k = ∑ k ∈ Icc m ⌊t⌋₊, c k := by
         filter_upwards [sumlocc c ⌊a⌋₊] with t ht₁ ht₂
         rw [ht₁ ⟨(Nat.floor_le ha).trans ht₂.1, hb ▸ ht₂.2.trans (Nat.lt_floor_add_one b).le⟩]
       rw [← ae_restrict_iff' measurableSet_Icc] at this
@@ -95,7 +96,7 @@ private theorem integrablemulsum (ha : 0 ≤ a) {g : ℝ → 𝕜} (hg_int : Int
         (hg_int.mul_const _) ((Filter.EventuallyEq.refl _ g).mul this)
     · have h_locint {t₁ t₂ : ℝ} {n : ℕ} (h : t₁ ≤ t₂) (h₁ : n ≤ t₁) (h₂ : t₂ ≤ n + 1)
           (h₃ : a ≤ t₁) (h₄ : t₂ ≤ b) :
-          IntervalIntegrable (fun t ↦ g t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k) volume t₁ t₂ := by
+          IntervalIntegrable (fun t ↦ g t * ∑ k ∈ Icc m ⌊t⌋₊, c k) volume t₁ t₂ := by
         rw [intervalIntegrable_iff_integrableOn_Icc_of_le h]
         exact (IntegrableOn.mono_set (hg_int.mul_const _) (Set.Icc_subset_Icc h₃ h₄)).congr
           <| ae_restrict_of_ae_restrict_of_subset (Set.Icc_subset_Icc h₁ h₂)
@@ -154,15 +155,15 @@ theorem _root_.sum_mul_eq_sub_sub_integral_mul (ha : 0 ≤ a) (hab : a ≤ b)
   -- (Note we have 5 goals, but the 1st and 3rd are identical. TODO: find a non-hacky way of dealing
   -- with both at once.)
   · rw [intervalIntegrable_iff_integrableOn_Icc_of_le aux6]
-    exact (integrablemulsum c ha hf_int).mono_set (Set.Icc_subset_Icc_right aux5)
+    exact (integrableOn_mul_sum_Icc c ha hf_int).mono_set (Set.Icc_subset_Icc_right aux5)
   · rw [intervalIntegrable_iff_integrableOn_Icc_of_le aux5]
-    exact (integrablemulsum c ha hf_int).mono_set (Set.Icc_subset_Icc_left aux6)
+    exact (integrableOn_mul_sum_Icc c ha hf_int).mono_set (Set.Icc_subset_Icc_left aux6)
   · rw [intervalIntegrable_iff_integrableOn_Icc_of_le aux6]
-    exact (integrablemulsum c ha hf_int).mono_set (Set.Icc_subset_Icc_right aux5)
+    exact (integrableOn_mul_sum_Icc c ha hf_int).mono_set (Set.Icc_subset_Icc_right aux5)
   · rw [intervalIntegrable_iff_integrableOn_Icc_of_le aux3]
-    exact (integrablemulsum c ha hf_int).mono_set (Set.Icc_subset_Icc_right aux4)
+    exact (integrableOn_mul_sum_Icc c ha hf_int).mono_set (Set.Icc_subset_Icc_right aux4)
   · exact fun k hk ↦ (intervalIntegrable_iff_integrableOn_Icc_of_le (mod_cast k.le_succ)).mpr
-      <| (integrablemulsum c ha hf_int).mono_set
+      <| (integrableOn_mul_sum_Icc c ha hf_int).mono_set
         <| (Set.Icc_subset_Icc_iff (mod_cast k.le_succ)).mpr <| mod_cast (ineqofmemIco hk)
 
 /-- A version of `sum_mul_eq_sub_sub_integral_mul` where the endpoints are `Nat`. -/
@@ -209,7 +210,7 @@ theorem sum_mul_eq_sub_integral_mul₀ (hc : c 0 = 0) (b : ℝ)
       f b * (∑ k ∈ Icc 0 ⌊b⌋₊, c k) - ∫ t in Set.Ioc 1 b, deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k := by
   obtain hb | hb := le_or_gt 1 b
   · have : 1 ≤ ⌊b⌋₊ := (Nat.one_le_floor_iff _).mpr hb
-    nth_rewrite 1 [Icc_eq_cons_Ioc (by omega), sum_cons, ← Nat.Icc_succ_left,
+    nth_rewrite 1 [Icc_eq_cons_Ioc (Nat.zero_le _), sum_cons, ← Nat.Icc_succ_left,
       Icc_eq_cons_Ioc (by omega), sum_cons]
     rw [Nat.succ_eq_add_one, zero_add, ← Nat.floor_one (α := ℝ),
       sum_mul_eq_sub_sub_integral_mul c zero_le_one hb hf_diff hf_int, Nat.floor_one, Nat.cast_one,
@@ -235,20 +236,22 @@ section limit
 
 open Filter Topology abelSummationProof intervalIntegral
 
-private theorem locallyintegrablemulsum (ha : 0 ≤ a) {g : ℝ → 𝕜} (hg : IntegrableOn g (Set.Ici a)) :
-    LocallyIntegrableOn (fun t ↦ g t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k) (Set.Ici a) := by
+theorem locallyIntegrableOn_mul_sum_Icc {m : ℕ} (ha : 0 ≤ a) {g : ℝ → 𝕜}
+    (hg : LocallyIntegrableOn g (Set.Ici a)) :
+    LocallyIntegrableOn (fun t ↦ g t * ∑ k ∈ Icc m ⌊t⌋₊, c k) (Set.Ici a) := by
   refine (locallyIntegrableOn_iff isLocallyClosed_Ici).mpr fun K hK₁ hK₂ ↦ ?_
   by_cases hK₃ : K.Nonempty
   · have h_inf : a ≤ sInf K := (hK₁ (hK₂.sInf_mem hK₃))
     refine IntegrableOn.mono_set ?_ (Bornology.IsBounded.subset_Icc_sInf_sSup hK₂.isBounded)
-    refine integrablemulsum _ (ha.trans h_inf) (hg.mono_set ?_)
+    refine integrableOn_mul_sum_Icc _ (ha.trans h_inf) ?_
+    refine hg.integrableOn_compact_subset ?_ isCompact_Icc
     exact (Set.Icc_subset_Ici_iff (Real.sInf_le_sSup _ hK₂.bddBelow hK₂.bddAbove)).mpr h_inf
   · rw [Set.not_nonempty_iff_eq_empty.mp hK₃]
     exact integrableOn_empty
 
 theorem tendsto_sum_mul_atTop_nhds_one_sub_integral
     (hf_diff : ∀ t ∈ Set.Ici 0, DifferentiableAt ℝ f t)
-    (hf_int : IntegrableOn (deriv f) (Set.Ici 0)) {l : 𝕜}
+    (hf_int : LocallyIntegrableOn (deriv f) (Set.Ici 0)) {l : 𝕜}
     (h_lim : Tendsto (fun n : ℕ ↦ f n * ∑ k ∈ Icc 0 n, c k) atTop (𝓝 l))
     {g : ℝ → 𝕜} (hg_dom : (fun t ↦ deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k) =O[atTop] g)
     (hg_int : IntegrableAtFilter g atTop) :
@@ -259,14 +262,15 @@ theorem tendsto_sum_mul_atTop_nhds_one_sub_integral
     refine Tendsto.congr (fun _ ↦ by rw [← integral_of_le (Nat.cast_nonneg _)]) ?_
     refine intervalIntegral_tendsto_integral_Ioi _ ?_ tendsto_natCast_atTop_atTop
     exact integrableOn_Ici_iff_integrableOn_Ioi.mp
-      <| (locallyintegrablemulsum c le_rfl hf_int).integrableOn_of_isBigO_atTop hg_dom hg_int
+      <| (locallyIntegrableOn_mul_sum_Icc c le_rfl hf_int).integrableOn_of_isBigO_atTop
+        hg_dom hg_int
   refine (h_lim.sub h_lim').congr (fun _ ↦ ?_)
-  rw [sum_mul_eq_sub_integral_mul' _ _ (fun t ht ↦ hf_diff _ ht.1)
-    (hf_int.mono_set Set.Icc_subset_Ici_self)]
+  rw [sum_mul_eq_sub_integral_mul' _ _ (fun t ht ↦ hf_diff _ ht.1)]
+  exact hf_int.integrableOn_compact_subset Set.Icc_subset_Ici_self isCompact_Icc
 
 theorem tendsto_sum_mul_atTop_nhds_one_sub_integral₀ (hc : c 0 = 0)
     (hf_diff : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ f t)
-    (hf_int : IntegrableOn (deriv f) (Set.Ici 1)) {l : 𝕜}
+    (hf_int : LocallyIntegrableOn (deriv f) (Set.Ici 1)) {l : 𝕜}
     (h_lim: Tendsto (fun n : ℕ ↦ f n * ∑ k ∈ Icc 0 n, c k) atTop (𝓝 l))
     {g : ℝ → ℝ} (hg_dom : (fun t ↦ deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k) =O[atTop] g)
     (hg_int : IntegrableAtFilter g atTop) :
@@ -280,10 +284,11 @@ theorem tendsto_sum_mul_atTop_nhds_one_sub_integral₀ (hc : c 0 = 0)
       atTop (𝓝 (∫ t in Set.Ioi 1, deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c k)) := by
     refine Tendsto.congr' h (intervalIntegral_tendsto_integral_Ioi _ ?_ tendsto_natCast_atTop_atTop)
     exact integrableOn_Ici_iff_integrableOn_Ioi.mp
-      <| (locallyintegrablemulsum c zero_le_one hf_int).integrableOn_of_isBigO_atTop hg_dom hg_int
+      <| (locallyIntegrableOn_mul_sum_Icc c zero_le_one hf_int).integrableOn_of_isBigO_atTop
+        hg_dom hg_int
   refine (h_lim.sub h_lim').congr (fun _ ↦ ?_)
-  rw [sum_mul_eq_sub_integral_mul₀' _ hc _ (fun t ht ↦ hf_diff _ ht.1)
-    (hf_int.mono_set Set.Icc_subset_Ici_self)]
+  rw [sum_mul_eq_sub_integral_mul₀' _ hc _ (fun t ht ↦ hf_diff _ ht.1)]
+  exact hf_int.integrableOn_compact_subset Set.Icc_subset_Ici_self isCompact_Icc
 
 end limit
 
@@ -293,7 +298,7 @@ open Filter abelSummationProof
 
 private theorem summable_mul_of_bigO_atTop_aux (m : ℕ)
     (h_bdd : (fun n : ℕ ↦ ‖f n‖ * ∑ k ∈ Icc 0 n, ‖c k‖) =O[atTop] fun _ ↦ (1 : ℝ))
-    (hf_int : IntegrableOn (deriv (fun t ↦ ‖f t‖)) (Set.Ici (m : ℝ)))
+    (hf_int : LocallyIntegrableOn (deriv (fun t ↦ ‖f t‖)) (Set.Ici (m : ℝ)))
     (hf : ∀ n : ℕ, ∑ k ∈ Icc 0 n, ‖f k‖ * ‖c k‖ =
       ‖f n‖ * ∑ k ∈ Icc 0 n, ‖c k‖ -
         ∫ (t : ℝ) in Set.Ioc ↑m ↑n, deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖)
@@ -325,26 +330,27 @@ private theorem summable_mul_of_bigO_atTop_aux (m : ℕ)
       · exact add_le_add_left
           (le_trans (neg_le_abs _) (Real.norm_eq_abs _ ▸ norm_integral_le_integral_norm _)) _
       · refine add_le_add_left (setIntegral_mono_set ?_ ?_ Set.Ioc_subset_Ioi_self.eventuallyLE) C₁
-        · refine integrableOn_Ici_iff_integrableOn_Ioi.mp <|
+        · exact integrableOn_Ici_iff_integrableOn_Ioi.mp <|
             (integrable_norm_iff h_mes.aestronglyMeasurable).mpr <|
-              (locallyintegrablemulsum _ m.cast_nonneg hf_int).integrableOn_of_isBigO_atTop hg₁ hg₂
+              (locallyIntegrableOn_mul_sum_Icc _ m.cast_nonneg hf_int).integrableOn_of_isBigO_atTop
+                hg₁ hg₂
         · filter_upwards with t using norm_nonneg _
 
 theorem summable_mul_of_bigO_atTop
     (hf_diff : ∀ t ∈ Set.Ici 0, DifferentiableAt ℝ (fun x ↦ ‖f x‖) t)
-    (hf_int : IntegrableOn (deriv (fun t ↦ ‖f t‖)) (Set.Ici 0))
+    (hf_int : LocallyIntegrableOn (deriv (fun t ↦ ‖f t‖)) (Set.Ici 0))
     (h_bdd : (fun n : ℕ ↦ ‖f n‖ * ∑ k ∈ Icc 0 n, ‖c k‖) =O[atTop] fun _ ↦ (1 : ℝ))
     {g : ℝ → ℝ} (hg₁ : (fun t ↦ deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 0 ⌊t⌋₊, ‖c k‖) =O[atTop] g)
     (hg₂ : IntegrableAtFilter g atTop) :
     Summable (fun n : ℕ ↦ f n * c n) := by
   refine summable_mul_of_bigO_atTop_aux c 0 h_bdd (by rwa [Nat.cast_zero]) (fun n ↦ ?_) hg₁ hg₂
   exact_mod_cast sum_mul_eq_sub_integral_mul' _ _ (fun _ ht ↦ hf_diff _ ht.1)
-    (hf_int.mono_set Set.Icc_subset_Ici_self)
+    (hf_int.integrableOn_compact_subset Set.Icc_subset_Ici_self isCompact_Icc)
 
 /-- A version of `summable_mul_of_bigO_atTop` that can be useful to avoid difficulties near zero. -/
 theorem summable_mul_of_bigO_atTop'
     (hf_diff : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x ↦ ‖f x‖) t)
-    (hf_int : IntegrableOn (deriv (fun t ↦ ‖f t‖)) (Set.Ici 1))
+    (hf_int : LocallyIntegrableOn (deriv (fun t ↦ ‖f t‖)) (Set.Ici 1))
     (h_bdd : (fun n : ℕ ↦ ‖f n‖ * ∑ k ∈ Icc 1 n, ‖c k‖) =O[atTop] fun _ ↦ (1 : ℝ))
     {g : ℝ → ℝ} (hg₁ : (fun t ↦ deriv (fun t ↦ ‖f t‖) t * ∑ k ∈ Icc 1 ⌊t⌋₊, ‖c k‖) =O[atTop] g)
     (hg₂ : IntegrableAtFilter g atTop) :
@@ -358,7 +364,8 @@ theorem summable_mul_of_bigO_atTop'
   refine Summable.congr_atTop (summable_mul_of_bigO_atTop_aux (fun n ↦ if n = 0 then 0 else c n) 1
     h_bdd (by rwa [Nat.cast_one]) (fun n ↦ ?_) hg₁ hg₂) ?_
   · exact_mod_cast sum_mul_eq_sub_integral_mul₀' _ (by simp only [reduceIte, norm_zero]) n
-      (fun _ ht ↦ hf_diff _ ht.1) (hf_int.mono_set Set.Icc_subset_Ici_self)
+      (fun _ ht ↦ hf_diff _ ht.1)
+      (hf_int.integrableOn_compact_subset Set.Icc_subset_Ici_self isCompact_Icc)
   · filter_upwards [eventually_ne_atTop 0] with k hk
     simp_rw [if_neg hk]
 
diff --git a/Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean b/Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean
index 7e288de6b9c304..bff8147059e585 100644
--- a/Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean
+++ b/Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean
@@ -98,8 +98,8 @@ theorem exists_approx_aux (n : ℕ) (h : abv.IsAdmissible) :
     · rwa [Finset.length_toList]
     · ext
       simpa [(Finset.nodup_toList _).getElem_inj_iff] using h
-    · #adaptation_note
-      /-- This proof was nicer prior to https://github.com/leanprover/lean4/pull/4400.
+    · #adaptation_note /-- https://github.com/leanprover/lean4/pull/4400
+      This proof was nicer before.
       Please feel welcome to improve it, by avoiding use of `List.get` in favour of `GetElem`. -/
       have : ∀ i h, t ((Finset.univ.filter fun x ↦ t x = s).toList.get ⟨i, h⟩) = s := fun i h ↦
         (Finset.mem_filter.mp (Finset.mem_toList.mp (List.get_mem _ ⟨i, h⟩))).2
diff --git a/Mathlib/NumberTheory/ClassNumber/FunctionField.lean b/Mathlib/NumberTheory/ClassNumber/FunctionField.lean
index 04ead5e6912634..6c6aecd30fe2d0 100644
--- a/Mathlib/NumberTheory/ClassNumber/FunctionField.lean
+++ b/Mathlib/NumberTheory/ClassNumber/FunctionField.lean
@@ -29,12 +29,11 @@ variable [Algebra Fq[X] F] [Algebra (RatFunc Fq) F]
 variable [IsScalarTower Fq[X] (RatFunc Fq) F]
 variable [FunctionField Fq F] [Algebra.IsSeparable (RatFunc Fq) F]
 
-open scoped Classical
-
 namespace RingOfIntegers
 
 open FunctionField
 
+open scoped Classical in
 noncomputable instance : Fintype (ClassGroup (ringOfIntegers Fq F)) :=
   ClassGroup.fintypeOfAdmissibleOfFinite (RatFunc Fq) F
     (Polynomial.cardPowDegreeIsAdmissible :
diff --git a/Mathlib/NumberTheory/Cyclotomic/Rat.lean b/Mathlib/NumberTheory/Cyclotomic/Rat.lean
index dad9eca2e69658..6f56aadeff7a6b 100644
--- a/Mathlib/NumberTheory/Cyclotomic/Rat.lean
+++ b/Mathlib/NumberTheory/Cyclotomic/Rat.lean
@@ -214,12 +214,10 @@ theorem integralPowerBasis_gen [hcycl : IsCyclotomicExtension {p ^ k} ℚ K]
     simp only [adjoinEquivRingOfIntegers_apply, IsIntegralClosure.algebraMap_lift]
     rfl
 
-#adaptation_note
-/--
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/5338
 We name `hcycl` so it can be used as a named argument,
 but since https://github.com/leanprover/lean4/pull/5338, this is considered unused,
-so we need to disable the linter.
--/
+so we need to disable the linter. -/
 set_option linter.unusedVariables false in
 @[simp]
 theorem integralPowerBasis_dim [hcycl : IsCyclotomicExtension {p ^ k} ℚ K]
diff --git a/Mathlib/NumberTheory/Dioph.lean b/Mathlib/NumberTheory/Dioph.lean
index 6c7c378d55eb41..2b065a6f7dd2e5 100644
--- a/Mathlib/NumberTheory/Dioph.lean
+++ b/Mathlib/NumberTheory/Dioph.lean
@@ -26,7 +26,7 @@ there exists `t : ℕ^β` with `p (v, t) = 0`.
 * `Dioph`: a predicate stating that a set is Diophantine, i.e. a set `S ⊆ ℕ^α` is
   Diophantine if there exists a polynomial on `α ⊕ β` such that `v ∈ S` iff there
   exists `t : ℕ^β` with `p (v, t) = 0`.
-* `dioph_fn`: a predicate on a function stating that it is Diophantine in the sense that its graph
+* `DiophFn`: a predicate on a function stating that it is Diophantine in the sense that its graph
   is Diophantine as a set.
 
 ## Main statements
diff --git a/Mathlib/NumberTheory/LSeries/Basic.lean b/Mathlib/NumberTheory/LSeries/Basic.lean
index 3faf700efb4030..3f0a95887411db 100644
--- a/Mathlib/NumberTheory/LSeries/Basic.lean
+++ b/Mathlib/NumberTheory/LSeries/Basic.lean
@@ -68,6 +68,12 @@ lemma term_def (f : ℕ → ℂ) (s : ℂ) (n : ℕ) :
     term f s n = if n = 0 then 0 else f n / n ^ s :=
   rfl
 
+/-- An alternate spelling of `term_def` for the case `f 0 = 0`. -/
+lemma term_def₀ {f : ℕ → ℂ} (hf : f 0 = 0) (s : ℂ) (n : ℕ) :
+    LSeries.term f s n = f n * (n : ℂ) ^ (- s) := by
+  rw [LSeries.term]
+  split_ifs with h <;> simp [h, hf, cpow_neg, div_eq_inv_mul, mul_comm]
+
 @[simp]
 lemma term_zero (f : ℕ → ℂ) (s : ℂ) : term f s 0 = 0 := rfl
 
diff --git a/Mathlib/NumberTheory/LSeries/DirichletContinuation.lean b/Mathlib/NumberTheory/LSeries/DirichletContinuation.lean
index 4b59c880997f0a..46c73ec9805a35 100644
--- a/Mathlib/NumberTheory/LSeries/DirichletContinuation.lean
+++ b/Mathlib/NumberTheory/LSeries/DirichletContinuation.lean
@@ -38,7 +38,7 @@ All definitions and theorems are in the `DirichletCharacter` namespace.
   `completedLFunction χ s = N ^ (s - 1 / 2) * rootNumber χ * completedLFunction χ⁻¹ s`.
 -/
 
-open HurwitzZeta Complex Finset Classical ZMod Filter
+open HurwitzZeta Complex Finset ZMod Filter
 
 open scoped Real Topology
 
@@ -198,6 +198,7 @@ section gammaFactor
 
 omit [NeZero N] -- not required for these declarations
 
+open scoped Classical in
 /-- The Archimedean Gamma factor: `Gammaℝ s` if `χ` is even, and `Gammaℝ (s + 1)` otherwise. -/
 noncomputable def gammaFactor (χ : DirichletCharacter ℂ N) (s : ℂ) :=
   if χ.Even then Gammaℝ s else Gammaℝ (s + 1)
@@ -258,6 +259,7 @@ lemma LFunction_eq_completed_div_gammaFactor (χ : DirichletCharacter ℂ N) (s
   · exact LFunction_eq_completed_div_gammaFactor_even hχ.to_fun _ (h.imp_right χ.map_zero')
   · apply LFunction_eq_completed_div_gammaFactor_odd hχ.to_fun
 
+open scoped Classical in
 /--
 Global root number of `χ` (for `χ` primitive; junk otherwise). Defined as
 `gaussSum χ stdAddChar / I ^ a / N ^ (1 / 2)`, where `a = 0` if even, `a = 1` if odd. (The factor
@@ -277,6 +279,7 @@ namespace IsPrimitive
 /-- **Functional equation** for primitive Dirichlet L-functions. -/
 theorem completedLFunction_one_sub {χ : DirichletCharacter ℂ N} (hχ : IsPrimitive χ) (s : ℂ) :
     completedLFunction χ (1 - s) = N ^ (s - 1 / 2) * rootNumber χ * completedLFunction χ⁻¹ s := by
+  classical
   -- First handle special case of Riemann zeta
   rcases eq_or_ne N 1 with rfl | hN
   · simp only [completedLFunction_modOne_eq, completedRiemannZeta_one_sub, Nat.cast_one, one_cpow,
diff --git a/Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean b/Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean
index 9782d336636d6d..ceccbcc2a6eab6 100644
--- a/Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean
+++ b/Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean
@@ -46,7 +46,7 @@ multiples of `1 / s` and `1 / (1 - s)`.
 -/
 noncomputable section
 
-open Complex Filter Topology Asymptotics Real Set Classical MeasureTheory
+open Complex Filter Topology Asymptotics Real Set MeasureTheory
 
 namespace HurwitzZeta
 
diff --git a/Mathlib/NumberTheory/LSeries/Nonvanishing.lean b/Mathlib/NumberTheory/LSeries/Nonvanishing.lean
index 24e61f8fb24586..c9d59f72fadfd9 100644
--- a/Mathlib/NumberTheory/LSeries/Nonvanishing.lean
+++ b/Mathlib/NumberTheory/LSeries/Nonvanishing.lean
@@ -374,11 +374,8 @@ private lemma LFunction_ne_zero_of_not_quadratic_or_ne_one {t : ℝ} (h : χ ^ 2
   -- go via absolute value to translate into a statement over `ℝ`
   replace H := (H₀.trans H).norm_right
   simp only [norm_eq_abs, abs_ofReal] at H
-  #adaptation_note
-  /--
-  After https://github.com/leanprover/lean4/pull/6024
-  we needed to add `(F' := ℝ)` to `H.of_norm_right`.
-  -/
+  #adaptation_note /-- https://github.com/leanprover/lean4/pull/6024
+  we needed to add `(F' := ℝ)` to `H.of_norm_right`. -/
   exact isLittleO_irrefl (.of_forall (fun _ ↦ one_ne_zero)) <|
     (H.of_norm_right (F' := ℝ)).trans_isLittleO <| isLittleO_id_one.mono nhdsWithin_le_nhds
 
diff --git a/Mathlib/NumberTheory/LSeries/SumCoeff.lean b/Mathlib/NumberTheory/LSeries/SumCoeff.lean
new file mode 100644
index 00000000000000..3c70a12320fde4
--- /dev/null
+++ b/Mathlib/NumberTheory/LSeries/SumCoeff.lean
@@ -0,0 +1,155 @@
+/-
+Copyright (c) 2025 Xavier Roblot. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Xavier Roblot
+-/
+import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
+import Mathlib.Analysis.InnerProductSpace.Calculus
+import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
+import Mathlib.NumberTheory.AbelSummation
+import Mathlib.NumberTheory.LSeries.Basic
+
+/-!
+# Partial sums of coefficients of L-series
+
+We prove several results involving partial sums of coefficients (or norm of coefficients) of
+L-series.
+
+## Main results
+
+* `LSeriesSummable_of_sum_norm_bigO`: for `f : ℕ → ℂ`, if the partial sums
+  `∑ k ∈ Icc 1 n, ‖f k‖` are `O(n ^ r)` for some real `0 ≤ r`, then the L-series `LSeries f`
+  converges at `s : ℂ` for all `s` such that `r < s.re`.
+
+* `LSeries_eq_mul_integral` : for `f : ℕ → ℂ`, if the partial sums `∑ k ∈ Icc 1 n, f k` are
+  `O(n ^ r)` for some real `0 ≤ r` and the L-series `LSeries f` converges at `s : ℂ` with
+  `r < s.re`, then `LSeries f s = s * ∫ t in Set.Ioi 1, (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * t ^ (-(s + 1))`.
+
+-/
+
+open Finset Filter MeasureTheory Topology Complex Asymptotics
+
+section summable
+
+variable {f : ℕ → ℂ} {r : ℝ} {s : ℂ}
+
+private theorem LSeriesSummable_of_sum_norm_bigO_aux (hf : f 0 = 0)
+    (hO : (fun n ↦ ∑ k ∈ Icc 1 n, ‖f k‖) =O[atTop] fun n ↦ (n : ℝ) ^ r)
+    (hr : 0 ≤ r) (hs : r < s.re) :
+    LSeriesSummable f s := by
+  have h₁ : -s ≠ 0 := neg_ne_zero.mpr <| ne_zero_of_re_pos (hr.trans_lt hs)
+  have h₂ : (-s).re + r ≤ 0 := by
+    rw [neg_re, neg_add_nonpos_iff]
+    exact hs.le
+  have h₃ (t : ℝ) (ht : t ∈ Set.Ici 1) : DifferentiableAt ℝ (fun x : ℝ ↦ ‖(x : ℂ) ^ (-s)‖) t :=
+    have ht' : t ≠ 0 := (zero_lt_one.trans_le ht).ne'
+    (differentiableAt_id.ofReal_cpow_const ht' h₁).norm ℝ <|
+      (cpow_ne_zero_iff_of_exponent_ne_zero h₁).mpr <| ofReal_ne_zero.mpr ht'
+  have h₄ : (deriv fun t : ℝ ↦ ‖(t : ℂ) ^ (-s)‖) =ᶠ[atTop] fun t ↦ -s.re * t ^ (-(s.re +1)) := by
+    filter_upwards [eventually_gt_atTop 0] with t ht
+    rw [deriv_norm_ofReal_cpow _ ht, neg_re, neg_add']
+  simp_rw [LSeriesSummable, funext (LSeries.term_def₀ hf s), mul_comm (f _)]
+  refine summable_mul_of_bigO_atTop' (f := fun t ↦ (t : ℂ) ^ (-s))
+    (g := fun t ↦ t ^ (-(s.re + 1) + r)) _ h₃ ?_ ?_ ?_ ?_
+  · refine (integrableOn_Ici_iff_integrableOn_Ioi.mpr
+      (integrableOn_Ioi_deriv_norm_ofReal_cpow zero_lt_one ?_)).locallyIntegrableOn
+    exact neg_re _ ▸ neg_nonpos.mpr  <| hr.trans hs.le
+  · refine (IsBigO.mul_atTop_rpow_natCast_of_isBigO_rpow _ _ _ ?_ hO h₂).congr_right  (by simp)
+    exact (norm_ofReal_cpow_eventually_eq_atTop _).isBigO.natCast_atTop
+  · refine h₄.isBigO.of_const_mul_right.mul_atTop_rpow_of_isBigO_rpow _ r _ ?_ le_rfl
+    exact (hO.comp_tendsto tendsto_nat_floor_atTop).trans <|
+      isEquivalent_nat_floor.isBigO.rpow hr (eventually_ge_atTop 0)
+  · rwa [integrableAtFilter_rpow_atTop_iff, neg_add_lt_iff_lt_add, add_neg_cancel_right]
+
+/-- If the partial sums `∑ k ∈ Icc 1 n, ‖f k‖` are `O(n ^ r)` for some real `0 ≤ r`, then the
+L-series `LSeries f` converges at `s : ℂ` for all `s` such that `r < s.re`. -/
+theorem LSeriesSummable_of_sum_norm_bigO
+    (hO : (fun n ↦ ∑ k ∈ Icc 1 n, ‖f k‖) =O[atTop] fun n ↦ (n : ℝ) ^ r)
+    (hr : 0 ≤ r) (hs : r < s.re) :
+    LSeriesSummable f s := by
+  have h₁ : (fun n ↦ if n = 0 then 0 else f n) =ᶠ[atTop] f := by
+    filter_upwards [eventually_ne_atTop 0] with n hn using by simp_rw [if_neg hn]
+  refine (LSeriesSummable_of_sum_norm_bigO_aux (if_pos rfl) ?_ hr hs).congr' _ h₁
+  refine hO.congr' (Eventually.of_forall fun _ ↦ Finset.sum_congr rfl fun _ h ↦ ?_) EventuallyEq.rfl
+  rw [if_neg (zero_lt_one.trans_le (mem_Icc.mp h).1).ne']
+
+/-- If `f` takes nonnegative real values and the partial sums `∑ k ∈ Icc 1 n, f k` are `O(n ^ r)`
+for some real `0 ≤ r`, then the L-series `LSeries f` converges at `s : ℂ` for all `s`
+such that `r < s.re`. -/
+theorem LSeriesSummable_of_sum_norm_bigO_and_nonneg
+    {f : ℕ → ℝ} (hO : (fun n ↦ ∑ k ∈ Icc 1 n, f k) =O[atTop] fun n ↦ (n : ℝ) ^ r)
+    (hf : ∀ n, 0 ≤ f n) (hr : 0 ≤ r) (hs : r < s.re) :
+    LSeriesSummable (fun n ↦ f n) s :=
+  LSeriesSummable_of_sum_norm_bigO (by simpa [_root_.abs_of_nonneg (hf _)]) hr hs
+
+end summable
+
+section integralrepresentation
+
+private theorem LSeries_eq_mul_integral_aux {f : ℕ → ℂ} (hf : f 0 = 0) {r : ℝ} (hr : 0 ≤ r) {s : ℂ}
+    (hs : r < s.re) (hS : LSeriesSummable f s)
+    (hO : (fun n ↦ ∑ k ∈ Icc 1 n, f k) =O[atTop] fun n ↦ (n : ℝ) ^ r) :
+    LSeries f s = s * ∫ t in Set.Ioi (1 : ℝ), (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * t ^ (- (s + 1)) := by
+  have h₁ : (-s - 1).re + r < -1 := by
+    rwa [sub_re, one_re, neg_re, neg_sub_left, neg_add_lt_iff_lt_add, add_neg_cancel_comm]
+  have h₂ : s ≠ 0 := ne_zero_of_re_pos (hr.trans_lt hs)
+  have h₃ (t : ℝ) (ht : t ∈ Set.Ici 1) : DifferentiableAt ℝ (fun x : ℝ ↦ (x : ℂ) ^ (-s)) t :=
+    differentiableAt_id.ofReal_cpow_const (zero_lt_one.trans_le ht).ne' (neg_ne_zero.mpr h₂)
+  have h₄ : ∀ n, ∑ k ∈ Icc 0 n, f k = ∑ k ∈ Icc 1 n, f k := fun n ↦ by
+    rw [← Nat.Icc_insert_succ_left n.zero_le, sum_insert (by aesop), hf, zero_add, zero_add]
+  simp_rw [← h₄] at hO
+  rw [← integral_mul_left]
+  refine tendsto_nhds_unique ((tendsto_add_atTop_iff_nat 1).mpr hS.hasSum.tendsto_sum_nat) ?_
+  simp_rw [Nat.range_succ_eq_Icc_zero, LSeries.term_def₀ hf, mul_comm (f _)]
+  convert tendsto_sum_mul_atTop_nhds_one_sub_integral₀ (f := fun x ↦ (x : ℂ) ^ (-s)) (l := 0)
+    ?_ hf h₃ ?_ ?_ ?_ (integrableAtFilter_rpow_atTop_iff.mpr h₁)
+  · rw [zero_sub, ← integral_neg]
+    refine setIntegral_congr_fun measurableSet_Ioi fun t ht ↦ ?_
+    rw [deriv_ofReal_cpow_const (zero_lt_one.trans ht).ne', h₄]
+    · ring_nf
+    · exact neg_ne_zero.mpr <| ne_zero_of_re_pos (hr.trans_lt hs)
+  · refine (integrableOn_Ici_iff_integrableOn_Ioi.mpr <|
+      integrableOn_Ioi_deriv_ofReal_cpow zero_lt_one
+        (by simpa using hr.trans_lt hs)).locallyIntegrableOn
+  · have hlim : Tendsto (fun n : ℕ ↦ (n : ℝ) ^ (-(s.re - r))) atTop (𝓝 0) :=
+      (tendsto_rpow_neg_atTop (by rwa [sub_pos])).comp tendsto_natCast_atTop_atTop
+    refine (IsBigO.mul_atTop_rpow_natCast_of_isBigO_rpow (-s.re) _ _ ?_ hO ?_).trans_tendsto hlim
+    · exact isBigO_norm_left.mp <| (norm_ofReal_cpow_eventually_eq_atTop _).isBigO.natCast_atTop
+    · linarith
+  · refine .mul_atTop_rpow_of_isBigO_rpow (-(s + 1).re) r _ ?_ ?_ (by rw [← neg_re, neg_add'])
+    · simpa [- neg_add_rev, neg_add'] using isBigO_deriv_ofReal_cpow_const_atTop _
+    · exact (hO.comp_tendsto tendsto_nat_floor_atTop).trans <|
+        isEquivalent_nat_floor.isBigO.rpow hr (eventually_ge_atTop 0)
+
+/-- If the partial sums `∑ k ∈ Icc 1 n, f k` are `O(n ^ r)` for some real `0 ≤ r` and the
+L-series `LSeries f` converges at `s : ℂ` with `r < s.re`, then
+`LSeries f s = s * ∫ t in Set.Ioi 1, (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * t ^ (-(s + 1))`. -/
+theorem LSeries_eq_mul_integral (f : ℕ → ℂ) {r : ℝ} (hr : 0 ≤ r) {s : ℂ} (hs : r < s.re)
+    (hS : LSeriesSummable f s)
+    (hO : (fun n ↦ ∑ k ∈ Icc 1 n, f k) =O[atTop] fun n ↦ (n : ℝ) ^ r) :
+    LSeries f s = s * ∫ t in Set.Ioi (1 : ℝ), (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * t ^ (-(s + 1)) := by
+  rw [← LSeriesSummable_congr' s (f := fun n ↦ if n = 0 then 0 else f n)
+    (by filter_upwards [eventually_ne_atTop 0] with n h using if_neg h)] at hS
+  have (n) : ∑ k ∈ Icc 1 n, (if k = 0 then 0 else f k) = ∑ k ∈ Icc 1 n, f k :=
+    Finset.sum_congr rfl fun k hk ↦ by rw [if_neg (zero_lt_one.trans_le (mem_Icc.mp hk).1).ne']
+  rw [← LSeries_congr _ (fun _ ↦ if_neg _), LSeries_eq_mul_integral_aux (if_pos rfl) hr hs hS] <;>
+  simp_all
+
+/-- A version of `LSeries_eq_mul_integral` where we use the stronger condition that the partial sums
+`∑ k ∈ Icc 1 n, ‖f k‖` are `O(n ^ r)` to deduce the integral representation. -/
+theorem LSeries_eq_mul_integral' (f : ℕ → ℂ) {r : ℝ} (hr : 0 ≤ r) {s : ℂ} (hs : r < s.re)
+    (hO : (fun n ↦ ∑ k ∈ Icc 1 n, ‖f k‖) =O[atTop] fun n ↦ (n : ℝ) ^ r) :
+    LSeries f s = s * ∫ t in Set.Ioi (1 : ℝ), (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * t ^ (-(s + 1)) :=
+  LSeries_eq_mul_integral _ hr hs (LSeriesSummable_of_sum_norm_bigO hO hr hs) <|
+    (isBigO_of_le _ fun _ ↦ (norm_sum_le _ _).trans <| Real.le_norm_self _).trans hO
+
+/-- If `f` takes nonnegative real values and the partial sums `∑ k ∈ Icc 1 n, f k` are `O(n ^ r)`
+for some real `0 ≤ r`, then for `s : ℂ` with `r < s.re`, we have
+`LSeries f s = s * ∫ t in Set.Ioi 1, (∑ k ∈ Icc 1 ⌊t⌋₊, f k) * t ^ (-(s + 1))`. -/
+theorem LSeries_eq_mul_integral_of_nonneg (f : ℕ → ℝ) {r : ℝ} (hr : 0 ≤ r) {s : ℂ} (hs : r < s.re)
+    (hO : (fun n ↦ ∑ k ∈ Icc 1 n, f k) =O[atTop] fun n ↦ (n : ℝ) ^ r) (hf : ∀ n, 0 ≤ f n) :
+    LSeries (fun n ↦ f n) s =
+      s * ∫ t in Set.Ioi (1 : ℝ), (∑ k ∈ Icc 1 ⌊t⌋₊, (f k : ℂ)) * t ^ (-(s + 1)) :=
+  LSeries_eq_mul_integral' _ hr hs <| hO.congr_left fun _ ↦ by simp [_root_.abs_of_nonneg (hf _)]
+
+end integralrepresentation
diff --git a/Mathlib/NumberTheory/LSeries/ZMod.lean b/Mathlib/NumberTheory/LSeries/ZMod.lean
index e25eb91d295544..67ee56461069a2 100644
--- a/Mathlib/NumberTheory/LSeries/ZMod.lean
+++ b/Mathlib/NumberTheory/LSeries/ZMod.lean
@@ -55,7 +55,7 @@ Results for completed L-functions:
   the functional equation relating `completedLFunction Φ (1 - s)` to `completedLFunction (𝓕 Φ) s`.
 -/
 
-open HurwitzZeta Complex ZMod Finset Classical Topology Filter Set
+open HurwitzZeta Complex ZMod Finset Topology Filter Set
 
 open scoped Real
 
diff --git a/Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean b/Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
index f408526e3fb67b..d609a2dcc3395b 100644
--- a/Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
+++ b/Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
@@ -508,10 +508,10 @@ open NumberTheorySymbols jacobiSym
 private def fastJacobiSymAux (a b : ℕ) (flip : Bool) (ha0 : a > 0) : ℤ :=
   if ha4 : a % 4 = 0 then
     fastJacobiSymAux (a / 4) b flip
-      (a.div_pos (Nat.le_of_dvd ha0 (Nat.dvd_of_mod_eq_zero ha4)) (by decide))
+      (Nat.div_pos (Nat.le_of_dvd ha0 (Nat.dvd_of_mod_eq_zero ha4)) (by decide))
   else if ha2 : a % 2 = 0 then
     fastJacobiSymAux (a / 2) b (xor (b % 8 = 3 ∨ b % 8 = 5) flip)
-      (a.div_pos (Nat.le_of_dvd ha0 (Nat.dvd_of_mod_eq_zero ha2)) (by decide))
+      (Nat.div_pos (Nat.le_of_dvd ha0 (Nat.dvd_of_mod_eq_zero ha2)) (by decide))
   else if ha1 : a = 1 then
     if flip then -1 else 1
   else if hba : b % a = 0 then
diff --git a/Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean b/Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean
index b106e2e0d0de90..321736dea55cba 100644
--- a/Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean
+++ b/Mathlib/NumberTheory/ModularForms/JacobiTheta/Bounds.lean
@@ -34,7 +34,7 @@ hence Dirichlet L-functions, etc).
   `∞`.
 -/
 
-open Set Filter Topology Asymptotics Real Classical
+open Set Filter Topology Asymptotics Real
 
 noncomputable section
 
diff --git a/Mathlib/NumberTheory/ModularForms/QExpansion.lean b/Mathlib/NumberTheory/ModularForms/QExpansion.lean
index 27d4e3389e4bb1..beb1790478637b 100644
--- a/Mathlib/NumberTheory/ModularForms/QExpansion.lean
+++ b/Mathlib/NumberTheory/ModularForms/QExpansion.lean
@@ -149,7 +149,7 @@ lemma qExpansionFormalMultilinearSeries_apply_norm (m : ℕ) :
 
 lemma qExpansionFormalMultilinearSeries_radius [NeZero n] [ModularFormClass F Γ(n) k] :
     1 ≤ (qExpansionFormalMultilinearSeries n f).radius := by
-  refine le_of_forall_ge_of_dense fun r hr ↦ ?_
+  refine le_of_forall_lt_imp_le_of_dense fun r hr ↦ ?_
   lift r to NNReal using hr.ne_top
   apply FormalMultilinearSeries.le_radius_of_summable
   simp only [qExpansionFormalMultilinearSeries_apply_norm]
@@ -161,8 +161,8 @@ lemma qExpansionFormalMultilinearSeries_radius [NeZero n] [ModularFormClass F Γ
 lemma hasFPowerSeries_cuspFunction [NeZero n] [ModularFormClass F Γ(n) k] :
     HasFPowerSeriesOnBall (cuspFunction n f) (qExpansionFormalMultilinearSeries n f) 0 1 := by
   refine ⟨qExpansionFormalMultilinearSeries_radius n f, zero_lt_one, fun hy ↦ ?_⟩
-  rw [EMetric.mem_ball, edist_zero_right, ENNReal.coe_lt_one_iff, ← NNReal.coe_lt_one,
-    coe_nnnorm, norm_eq_abs] at hy
+  rw [EMetric.mem_ball, edist_zero_right, enorm_eq_nnnorm, ENNReal.coe_lt_one_iff,
+    ← NNReal.coe_lt_one, coe_nnnorm, norm_eq_abs] at hy
   simpa [qExpansionFormalMultilinearSeries] using hasSum_qExpansion_of_abs_lt n f hy
 
 end ModularFormClass
diff --git a/Mathlib/NumberTheory/NumberField/AdeleRing.lean b/Mathlib/NumberTheory/NumberField/AdeleRing.lean
index 927332bf8bdebf..fcbefa15560a6b 100644
--- a/Mathlib/NumberTheory/NumberField/AdeleRing.lean
+++ b/Mathlib/NumberTheory/NumberField/AdeleRing.lean
@@ -39,8 +39,6 @@ namespace NumberField
 
 open InfinitePlace AbsoluteValue.Completion InfinitePlace.Completion DedekindDomain IsDedekindDomain
 
-open scoped Classical
-
 /-! ## The infinite adele ring
 
 The infinite adele ring is the finite product of completions of a number field over its
@@ -76,6 +74,7 @@ theorem algebraMap_apply (x : K) (v : InfinitePlace K) :
 instance locallyCompactSpace [NumberField K] : LocallyCompactSpace (InfiniteAdeleRing K) :=
   Pi.locallyCompactSpace_of_finite
 
+open scoped Classical in
 /-- The ring isomorphism between the infinite adele ring of a number field and the
 space `ℝ ^ r₁ × ℂ ^ r₂`, where `(r₁, r₂)` is the signature of the number field. -/
 abbrev ringEquiv_mixedSpace :
diff --git a/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean b/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean
index 46dd10425cc974..8d446e73f8ab60 100644
--- a/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean
+++ b/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean
@@ -309,10 +309,9 @@ end commMap
 
 noncomputable section norm
 
-open scoped Classical
-
 variable {K}
 
+open scoped Classical in
 /-- The norm at the infinite place `w` of an element of the mixed space. --/
 def normAtPlace (w : InfinitePlace K) : (mixedSpace K) →*₀ ℝ where
   toFun x := if hw : IsReal w then ‖x.1 ⟨w, hw⟩‖ else ‖x.2 ⟨w, not_isReal_iff_isComplex.mp hw⟩‖
@@ -380,6 +379,7 @@ theorem exists_normAtPlace_ne_zero_iff {x : mixedSpace K} :
 
 variable [NumberField K]
 
+open scoped Classical in
 theorem nnnorm_eq_sup_normAtPlace (x : mixedSpace K) :
     ‖x‖₊ = univ.sup fun w ↦ ⟨normAtPlace w x, normAtPlace_nonneg w x⟩ := by
   have :
@@ -395,6 +395,7 @@ theorem nnnorm_eq_sup_normAtPlace (x : mixedSpace K) :
   · ext w
     simp [normAtPlace_apply_isComplex w.prop]
 
+open scoped Classical in
 theorem norm_eq_sup'_normAtPlace (x : mixedSpace K) :
     ‖x‖ = univ.sup' univ_nonempty fun w ↦ normAtPlace w x := by
   rw [← coe_nnnorm, nnnorm_eq_sup_normAtPlace, ← sup'_eq_sup univ_nonempty, ← NNReal.val_eq_coe,
@@ -460,8 +461,6 @@ end norm
 
 noncomputable section stdBasis
 
-open scoped Classical
-
 open Complex MeasureTheory MeasureTheory.Measure ZSpan Matrix ComplexConjugate
 
 variable [NumberField K]
@@ -469,6 +468,7 @@ variable [NumberField K]
 /-- The type indexing the basis `stdBasis`. -/
 abbrev index := {w : InfinitePlace K // IsReal w} ⊕ ({w : InfinitePlace K // IsComplex w}) × (Fin 2)
 
+open scoped Classical in
 /-- The `ℝ`-basis of the mixed space of `K` formed by the vector equal to `1` at `w` and `0`
 elsewhere for `IsReal w` and by the couple of vectors equal to `1` (resp. `I`) at `w` and `0`
 elsewhere for `IsComplex w`. -/
@@ -494,6 +494,7 @@ theorem stdBasis_apply_ofIsComplex_snd (x : mixedSpace K)
 
 variable (K)
 
+open scoped Classical in
 theorem fundamentalDomain_stdBasis :
     fundamentalDomain (stdBasis K) =
         (Set.univ.pi fun _ => Set.Ico 0 1) ×ˢ
@@ -501,6 +502,7 @@ theorem fundamentalDomain_stdBasis :
   ext
   simp [stdBasis, mem_fundamentalDomain, Complex.measurableEquivPi]
 
+open scoped Classical in
 theorem volume_fundamentalDomain_stdBasis :
     volume (fundamentalDomain (stdBasis K)) = 1 := by
   rw [fundamentalDomain_stdBasis, volume_eq_prod, prod_prod, volume_pi, volume_pi, pi_pi, pi_pi,
@@ -508,6 +510,7 @@ theorem volume_fundamentalDomain_stdBasis :
     sub_zero, ENNReal.ofReal_one, prod_const_one, prod_const_one, prod_const_one, one_mul]
   exact (MeasurableSet.pi Set.countable_univ (fun _ _ => measurableSet_Ico)).nullMeasurableSet
 
+open scoped Classical in
 /-- The `Equiv` between `index K` and `K →+* ℂ` defined by sending a real infinite place `w` to
 the unique corresponding embedding `w.embedding`, and the pair `⟨w, 0⟩` (resp. `⟨w, 1⟩`) for a
 complex infinite place `w` to `w.embedding` (resp. `conjugate w.embedding`). -/
@@ -544,6 +547,7 @@ theorem indexEquiv_apply_ofIsComplex_snd (w : {w : InfinitePlace K // IsComplex
 
 variable (K)
 
+open scoped Classical in
 /-- The matrix that gives the representation on `stdBasis` of the image by `commMap` of an
 element `x` of `(K →+* ℂ) → ℂ` fixed by the map `x_φ ↦ conj x_(conjugate φ)`,
 see `stdBasis_repr_eq_matrixToStdBasis_mul`. -/
@@ -551,6 +555,7 @@ def matrixToStdBasis : Matrix (index K) (index K) ℂ :=
   fromBlocks (diagonal fun _ => 1) 0 0 <| reindex (Equiv.prodComm _ _) (Equiv.prodComm _ _)
     (blockDiagonal (fun _ => (2 : ℂ)⁻¹ • !![1, 1; - I, I]))
 
+open scoped Classical in
 theorem det_matrixToStdBasis :
     (matrixToStdBasis K).det = (2⁻¹ * I) ^ nrComplexPlaces K :=
   calc
@@ -563,6 +568,7 @@ theorem det_matrixToStdBasis :
   _ = (2⁻¹ * Complex.I) ^ Fintype.card {w : InfinitePlace K // IsComplex w} := by
       rw [prod_const, Fintype.card]
 
+open scoped Classical in
 /-- Let `x : (K →+* ℂ) → ℂ` such that `x_φ = conj x_(conj φ)` for all `φ : K →+* ℂ`, then the
 representation of `commMap K x` on `stdBasis` is given (up to reindexing) by the product of
 `matrixToStdBasis` by `x`. -/
diff --git a/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean b/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
index 0eb57e9a67069c..fe204bdeade808 100644
--- a/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
+++ b/Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
@@ -74,8 +74,6 @@ theorem convexBodyLT_convex : Convex ℝ (convexBodyLT K f) :=
 
 open Fintype MeasureTheory MeasureTheory.Measure ENNReal
 
-open scoped Classical
-
 variable [NumberField K]
 
 /-- The fudge factor that appears in the formula for the volume of `convexBodyLT`. -/
@@ -88,6 +86,7 @@ theorem convexBodyLTFactor_ne_zero : convexBodyLTFactor K ≠ 0 :=
 theorem one_le_convexBodyLTFactor : 1 ≤ convexBodyLTFactor K :=
   one_le_mul (one_le_pow₀ one_le_two) (one_le_pow₀ (one_le_two.trans Real.two_le_pi))
 
+open scoped Classical in
 /-- The volume of `(ConvexBodyLt K f)` where `convexBodyLT K f` is the set of points `x`
 such that `‖x w‖ < f w` for all infinite places `w`. -/
 theorem convexBodyLT_volume :
@@ -122,6 +121,7 @@ but one and choose the value at the remaining place so that we can apply
 `exists_ne_zero_mem_ringOfIntegers_lt`. -/
 theorem adjust_f {w₁ : InfinitePlace K} (B : ℝ≥0) (hf : ∀ w, w ≠ w₁ → f w ≠ 0) :
     ∃ g : InfinitePlace K → ℝ≥0, (∀ w, w ≠ w₁ → g w = f w) ∧ ∏ w, (g w) ^ mult w = B := by
+  classical
   let S := ∏ w ∈ Finset.univ.erase w₁, (f w) ^ mult w
   refine ⟨Function.update f w₁ ((B * S⁻¹) ^ (mult w₁ : ℝ)⁻¹), ?_, ?_⟩
   · exact fun w hw => Function.update_of_ne hw _ f
@@ -139,10 +139,9 @@ section convexBodyLT'
 
 open  Metric ENNReal NNReal
 
-open scoped Classical
-
 variable (f : InfinitePlace K → ℝ≥0) (w₀ : {w : InfinitePlace K // IsComplex w})
 
+open scoped Classical in
 /-- A version of `convexBodyLT` with an additional condition at a fixed complex place. This is
 needed to ensure the element constructed is not real, see for example
 `exists_primitive_element_lt_of_isComplex`.
@@ -191,8 +190,6 @@ theorem convexBodyLT'_convex : Convex ℝ (convexBodyLT' K f w₀) := by
 
 open MeasureTheory MeasureTheory.Measure
 
-open scoped Classical
-
 variable [NumberField K]
 
 /-- The fudge factor that appears in the formula for the volume of `convexBodyLT'`. -/
@@ -205,6 +202,7 @@ theorem convexBodyLT'Factor_ne_zero : convexBodyLT'Factor K ≠ 0 :=
 theorem one_le_convexBodyLT'Factor : 1 ≤ convexBodyLT'Factor K :=
   one_le_mul (one_le_pow₀ one_le_two) (one_le_pow₀ (one_le_two.trans Real.two_le_pi))
 
+open scoped Classical in
 theorem convexBodyLT'_volume :
     volume (convexBodyLT' K f w₀) = convexBodyLT'Factor K * ∏ w, (f w) ^ (mult w) := by
   have vol_box : ∀ B : ℝ≥0, volume {x : ℂ | |x.re| < 1 ∧ |x.im| < B^2} = 4*B^2 := by
@@ -258,7 +256,7 @@ section convexBodySum
 
 open ENNReal MeasureTheory Fintype
 
-open scoped Real Classical NNReal
+open scoped Real NNReal
 
 variable [NumberField K] (B : ℝ)
 variable {K}
@@ -270,6 +268,7 @@ noncomputable abbrev convexBodySumFun (x : mixedSpace K) : ℝ := ∑ w, mult w
 theorem convexBodySumFun_apply (x : mixedSpace K) :
     convexBodySumFun x = ∑ w,  mult w * normAtPlace w x := rfl
 
+open scoped Classical in
 theorem convexBodySumFun_apply' (x : mixedSpace K) :
     convexBodySumFun x = ∑ w, ‖x.1 w‖ + 2 * ∑ w, ‖x.2 w‖ := by
   simp_rw [convexBodySumFun_apply, ← Finset.sum_add_sum_compl {w | IsReal w}.toFinset,
@@ -310,6 +309,7 @@ theorem convexBodySumFun_eq_zero_iff (x : mixedSpace K) :
       (mem_nonZeroDivisors_iff_ne_zero.mpr <| Nat.cast_ne_zero.mpr mult_ne_zero)]
   simp_rw [Finset.mem_univ, true_implies]
 
+open scoped Classical in
 theorem norm_le_convexBodySumFun (x : mixedSpace K) : ‖x‖ ≤ convexBodySumFun x := by
   rw [norm_eq_sup'_normAtPlace]
   refine (Finset.sup'_le_iff _ _).mpr fun w _ ↦ ?_
@@ -333,6 +333,7 @@ theorem convexBodySumFun_continuous :
   `∑ w real, ‖x w‖ + 2 * ∑ w complex, ‖x w‖ ≤ B`. -/
 abbrev convexBodySum : Set (mixedSpace K)  := { x | convexBodySumFun x ≤ B }
 
+open scoped Classical in
 theorem convexBodySum_volume_eq_zero_of_le_zero {B} (hB : B ≤ 0) :
     volume (convexBodySum K B) = 0 := by
   obtain hB | hB := lt_or_eq_of_le hB
@@ -363,12 +364,14 @@ theorem convexBodySum_convex : Convex ℝ (convexBodySum K B) := by
   exact (abs_eq_self.mpr h).symm
 
 theorem convexBodySum_isBounded : Bornology.IsBounded (convexBodySum K B) := by
+  classical
   refine Metric.isBounded_iff.mpr ⟨B + B, fun x hx y hy => ?_⟩
   refine le_trans (norm_sub_le x y) (add_le_add ?_ ?_)
   · exact le_trans (norm_le_convexBodySumFun x) hx
   · exact le_trans (norm_le_convexBodySumFun y) hy
 
 theorem convexBodySum_compact : IsCompact (convexBodySum K B) := by
+  classical
   rw [Metric.isCompact_iff_isClosed_bounded]
   refine ⟨?_, convexBodySum_isBounded K B⟩
   convert IsClosed.preimage (convexBodySumFun_continuous K) (isClosed_Icc : IsClosed (Set.Icc 0 B))
@@ -385,6 +388,7 @@ theorem convexBodySumFactor_ne_zero : convexBodySumFactor K ≠ 0 := by
     (pow_ne_zero _ (div_ne_zero NNReal.pi_ne_zero two_ne_zero))
 
 open MeasureTheory MeasureTheory.Measure Real in
+open scoped Classical in
 theorem convexBodySum_volume :
     volume (convexBodySum K B) = (convexBodySumFactor K) * (.ofReal B) ^ (finrank ℚ K) := by
   obtain hB | hB := le_or_lt B 0
@@ -440,13 +444,13 @@ end convexBodySum
 
 section minkowski
 
-open scoped Classical
 open MeasureTheory MeasureTheory.Measure Module ZSpan Real Submodule
 
 open scoped ENNReal NNReal nonZeroDivisors IntermediateField
 
 variable [NumberField K] (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ)
 
+open scoped Classical in
 /-- The bound that appears in **Minkowski Convex Body theorem**, see
 `MeasureTheory.exists_ne_zero_mem_lattice_of_measure_mul_two_pow_lt_measure`. See
 `NumberField.mixedEmbedding.volume_fundamentalDomain_idealLatticeBasis_eq` and
@@ -456,6 +460,7 @@ noncomputable def minkowskiBound : ℝ≥0∞ :=
   volume (fundamentalDomain (fractionalIdealLatticeBasis K I)) *
     (2 : ℝ≥0∞) ^ (finrank ℝ (mixedSpace K))
 
+open scoped Classical in
 theorem volume_fundamentalDomain_fractionalIdealLatticeBasis :
     volume (fundamentalDomain (fractionalIdealLatticeBasis K I)) =
       .ofReal (FractionalIdeal.absNorm I.1) *  volume (fundamentalDomain (latticeBasis K)) := by
@@ -471,17 +476,20 @@ theorem volume_fundamentalDomain_fractionalIdealLatticeBasis :
     rw [mixedEmbedding.det_basisOfFractionalIdeal_eq_norm]
 
 theorem minkowskiBound_lt_top : minkowskiBound K I < ⊤ := by
+  classical
   refine ENNReal.mul_lt_top ?_ ?_
   · exact (fundamentalDomain_isBounded _).measure_lt_top
   · exact ENNReal.pow_lt_top (lt_top_iff_ne_top.mpr ENNReal.two_ne_top) _
 
 theorem minkowskiBound_pos : 0 < minkowskiBound K I := by
+  classical
   refine zero_lt_iff.mpr (mul_ne_zero ?_ ?_)
   · exact ZSpan.measure_fundamentalDomain_ne_zero _
   · exact ENNReal.pow_ne_zero two_ne_zero _
 
 variable {f : InfinitePlace K → ℝ≥0} (I : (FractionalIdeal (𝓞 K)⁰ K)ˣ)
 
+open scoped Classical in
 /-- Let `I` be a fractional ideal of `K`. Assume that `f : InfinitePlace K → ℝ≥0` is such that
 `minkowskiBound K I < volume (convexBodyLT K f)` where `convexBodyLT K f` is the set of
 points `x` such that `‖x w‖ < f w` for all infinite places `w` (see `convexBodyLT_volume` for
@@ -499,6 +507,7 @@ theorem exists_ne_zero_mem_ideal_lt (h : minkowskiBound K I < volume (convexBody
   obtain ⟨a, ha, rfl⟩ := hx
   exact ⟨a, ha, by simpa using h_nz, (convexBodyLT_mem K f).mp h_mem⟩
 
+open scoped Classical in
 /-- A version of `exists_ne_zero_mem_ideal_lt` where the absolute value of the real part of `a` is
 smaller than `1` at some fixed complex place. This is useful to ensure that `a` is not real. -/
 theorem exists_ne_zero_mem_ideal_lt' (w₀ : {w : InfinitePlace K // IsComplex w})
@@ -515,6 +524,7 @@ theorem exists_ne_zero_mem_ideal_lt' (w₀ : {w : InfinitePlace K // IsComplex w
   obtain ⟨a, ha, rfl⟩ := hx
   exact ⟨a, ha, by simpa using h_nz, (convexBodyLT'_mem K f w₀).mp h_mem⟩
 
+open scoped Classical in
 /-- A version of `exists_ne_zero_mem_ideal_lt` for the ring of integers of `K`. -/
 theorem exists_ne_zero_mem_ringOfIntegers_lt (h : minkowskiBound K ↑1 < volume (convexBodyLT K f)) :
     ∃ a : 𝓞 K, a ≠ 0 ∧ ∀ w : InfinitePlace K, w a < f w := by
@@ -522,6 +532,7 @@ theorem exists_ne_zero_mem_ringOfIntegers_lt (h : minkowskiBound K ↑1 < volume
   obtain ⟨a, rfl⟩ := (FractionalIdeal.mem_one_iff _).mp h_mem
   exact ⟨a, RingOfIntegers.coe_ne_zero_iff.mp h_nz, h_bd⟩
 
+open scoped Classical in
 /-- A version of `exists_ne_zero_mem_ideal_lt'` for the ring of integers of `K`. -/
 theorem exists_ne_zero_mem_ringOfIntegers_lt' (w₀ : {w : InfinitePlace K // IsComplex w})
     (h : minkowskiBound K ↑1 < volume (convexBodyLT' K f w₀)) :
@@ -535,6 +546,7 @@ theorem exists_primitive_element_lt_of_isReal {w₀ : InfinitePlace K} (hw₀ :
     (hB : minkowskiBound K ↑1 < convexBodyLTFactor K * B) :
     ∃ a : 𝓞 K, ℚ⟮(a : K)⟯ = ⊤ ∧
       ∀ w : InfinitePlace K, w a < max B 1 := by
+  classical
   have : minkowskiBound K ↑1 < volume (convexBodyLT K (fun w ↦ if w = w₀ then B else 1)) := by
     rw [convexBodyLT_volume, ← Finset.prod_erase_mul _ _ (Finset.mem_univ w₀)]
     simp_rw [ite_pow, one_pow]
@@ -551,6 +563,7 @@ theorem exists_primitive_element_lt_of_isComplex {w₀ : InfinitePlace K} (hw₀
     {B : ℝ≥0} (hB : minkowskiBound K ↑1 < convexBodyLT'Factor K * B) :
     ∃ a : 𝓞 K, ℚ⟮(a : K)⟯ = ⊤ ∧
       ∀ w : InfinitePlace K, w a < Real.sqrt (1 + B ^ 2) := by
+  classical
   have : minkowskiBound K ↑1 <
       volume (convexBodyLT' K (fun w ↦ if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩) := by
     rw [convexBodyLT'_volume, ← Finset.prod_erase_mul _ _ (Finset.mem_univ w₀)]
@@ -577,6 +590,7 @@ theorem exists_primitive_element_lt_of_isComplex {w₀ : InfinitePlace K} (hw₀
       rw [NNReal.coe_one, Real.le_sqrt' zero_lt_one, one_pow]
       norm_num
 
+open scoped Classical in
 /-- Let `I` be a fractional ideal of `K`. Assume that `B : ℝ` is such that
 `minkowskiBound K I < volume (convexBodySum K B)` where `convexBodySum K B` is the set of points
 `x` such that `∑ w real, ‖x w‖ + 2 * ∑ w complex, ‖x w‖ ≤ B` (see `convexBodySum_volume` for
@@ -615,6 +629,7 @@ theorem exists_ne_zero_mem_ideal_of_norm_le {B : ℝ}
   · rw [← Nat.cast_sum, sum_mult_eq, Nat.cast_pos]
     exact finrank_pos
 
+open scoped Classical in
 theorem exists_ne_zero_mem_ringOfIntegers_of_norm_le {B : ℝ}
     (h : (minkowskiBound K ↑1) ≤ volume (convexBodySum K B)) :
     ∃ a : 𝓞 K, a ≠ 0 ∧ |Algebra.norm ℚ (a : K)| ≤ (B / finrank ℚ K) ^ finrank ℚ K := by
diff --git a/Mathlib/NumberTheory/NumberField/Embeddings.lean b/Mathlib/NumberTheory/NumberField/Embeddings.lean
index b14ba38413d3d6..d92da2e2f64b27 100644
--- a/Mathlib/NumberTheory/NumberField/Embeddings.lean
+++ b/Mathlib/NumberTheory/NumberField/Embeddings.lean
@@ -30,7 +30,7 @@ for `x ∈ K`, we have `Π_w ‖x‖_w = |norm(x)|` where the product is over th
 number field, embeddings, places, infinite places
 -/
 
-open scoped Classical Finset
+open scoped Finset
 
 namespace NumberField.Embeddings
 
@@ -97,6 +97,7 @@ variable (K A)
 /-- Let `B` be a real number. The set of algebraic integers in `K` whose conjugates are all
 smaller in norm than `B` is finite. -/
 theorem finite_of_norm_le (B : ℝ) : {x : K | IsIntegral ℤ x ∧ ∀ φ : K →+* A, ‖φ x‖ ≤ B}.Finite := by
+  classical
   let C := Nat.ceil (max B 1 ^ finrank ℚ K * (finrank ℚ K).choose (finrank ℚ K / 2))
   have := bUnion_roots_finite (algebraMap ℤ K) (finrank ℚ K) (finite_Icc (-C : ℤ) C)
   refine this.subset fun x hx => ?_; simp_rw [mem_iUnion]
@@ -422,6 +423,7 @@ lemma isComplex_mk_iff {φ : K →+* ℂ} :
 theorem not_isReal_of_mk_isComplex {φ : K →+* ℂ} (h : IsComplex (mk φ)) :
     ¬ ComplexEmbedding.IsReal φ := by rwa [← isComplex_mk_iff]
 
+open scoped Classical in
 /-- The multiplicity of an infinite place, that is the number of distinct complex embeddings that
 define it, see `card_filter_mk_eq`. -/
 noncomputable def mult (w : InfinitePlace K) : ℕ := if (IsReal w) then 1 else 2
@@ -437,6 +439,7 @@ theorem one_le_mult {w : InfinitePlace K} : (1 : ℝ) ≤ mult w := by
   rw [← Nat.cast_one, Nat.cast_le]
   exact mult_pos
 
+open scoped Classical in
 theorem card_filter_mk_eq [NumberField K] (w : InfinitePlace K) : #{φ | mk φ = w} = mult w := by
   conv_lhs =>
     congr; congr; ext
@@ -450,11 +453,13 @@ theorem card_filter_mk_eq [NumberField K] (w : InfinitePlace K) : #{φ | mk φ =
   · refine Finset.card_pair ?_
     rwa [Ne, eq_comm, ← ComplexEmbedding.isReal_iff, ← isReal_iff]
 
+open scoped Classical in
 noncomputable instance NumberField.InfinitePlace.fintype [NumberField K] :
     Fintype (InfinitePlace K) := Set.fintypeRange _
 
 theorem sum_mult_eq [NumberField K] :
     ∑ w : InfinitePlace K, mult w = Module.finrank ℚ K := by
+  classical
   rw [← Embeddings.card K ℂ, Fintype.card, Finset.card_eq_sum_ones, ← Finset.univ.sum_fiberwise
     (fun φ => InfinitePlace.mk φ)]
   exact Finset.sum_congr rfl
@@ -490,6 +495,7 @@ variable [NumberField K]
 `‖·‖_w` is the normalized absolute value for `w`. -/
 theorem prod_eq_abs_norm (x : K) :
     ∏ w : InfinitePlace K, w x ^ mult w = abs (Algebra.norm ℚ x) := by
+  classical
   convert (congr_arg Complex.abs (@Algebra.norm_eq_prod_embeddings ℚ _ _ _ _ ℂ _ _ _ _ _ x)).symm
   · rw [map_prod, ← Fintype.prod_equiv RingHom.equivRatAlgHom (fun f => Complex.abs (f x))
       (fun φ => Complex.abs (φ x)) fun _ => by simp [RingHom.equivRatAlgHom_apply]; rfl]
@@ -555,25 +561,30 @@ section NumberField
 
 variable [NumberField K]
 
+open scoped Classical in
 /-- The number of infinite real places of the number field `K`. -/
 noncomputable abbrev nrRealPlaces := card { w : InfinitePlace K // IsReal w }
 
 @[deprecated (since := "2024-10-24")] alias NrRealPlaces := nrRealPlaces
 
+open scoped Classical in
 /-- The number of infinite complex places of the number field `K`. -/
 noncomputable abbrev nrComplexPlaces := card { w : InfinitePlace K // IsComplex w }
 
 @[deprecated (since := "2024-10-24")] alias NrComplexPlaces := nrComplexPlaces
 
+open scoped Classical in
 theorem card_real_embeddings :
     card { φ : K →+* ℂ // ComplexEmbedding.IsReal φ } = nrRealPlaces K := Fintype.card_congr mkReal
 
 theorem card_eq_nrRealPlaces_add_nrComplexPlaces :
     Fintype.card (InfinitePlace K) = nrRealPlaces K + nrComplexPlaces K := by
+  classical
   convert Fintype.card_subtype_or_disjoint (IsReal (K := K)) (IsComplex (K := K))
     (disjoint_isReal_isComplex K) using 1
   exact (Fintype.card_of_subtype _ (fun w ↦ ⟨fun _ ↦ isReal_or_isComplex w, fun _ ↦ by simp⟩)).symm
 
+open scoped Classical in
 theorem card_complex_embeddings :
     card { φ : K →+* ℂ // ¬ComplexEmbedding.IsReal φ } = 2 * nrComplexPlaces K := by
   suffices ∀ w : { w : InfinitePlace K // IsComplex w },
@@ -593,6 +604,7 @@ theorem card_complex_embeddings :
 
 theorem card_add_two_mul_card_eq_rank :
     nrRealPlaces K + 2 * nrComplexPlaces K = finrank ℚ K := by
+  classical
   rw [← card_real_embeddings, ← card_complex_embeddings, Fintype.card_subtype_compl,
     ← Embeddings.card K ℂ, Nat.add_sub_of_le]
   exact Fintype.card_subtype_le _
@@ -838,6 +850,7 @@ variable (k w)
 
 lemma nat_card_stabilizer_eq_one_or_two :
     Nat.card (Stab w) = 1 ∨ Nat.card (Stab w) = 2 := by
+  classical
   rw [← SetLike.coe_sort_coe, ← mk_embedding w]
   by_cases h : ∃ σ, ComplexEmbedding.IsConj (k := k) (embedding w) σ
   · obtain ⟨σ, hσ⟩ := h
@@ -869,6 +882,7 @@ lemma not_isUnramified_iff_card_stabilizer_eq_two [IsGalois k K] :
   rw [isUnramified_iff_card_stabilizer_eq_one]
   obtain (e|e) := nat_card_stabilizer_eq_one_or_two k w <;> rw [e] <;> decide
 
+open scoped Classical in
 lemma card_stabilizer [IsGalois k K] :
     Nat.card (Stab w) = if IsUnramified k w then 1 else 2 := by
   split
@@ -934,6 +948,7 @@ variable (k K)
 variable [NumberField K]
 
 open Finset in
+open scoped Classical in
 lemma card_isUnramified [NumberField k] [IsGalois k K] :
     #{w : InfinitePlace K | w.IsUnramified k} =
       #{w : InfinitePlace k | w.IsUnramifiedIn K} * finrank k K := by
@@ -956,6 +971,7 @@ lemma card_isUnramified [NumberField k] [IsGalois k K] :
   · simp [isUnramifiedIn_comap]
 
 open Finset in
+open scoped Classical in
 lemma card_isUnramified_compl [NumberField k] [IsGalois k K] :
     #({w : InfinitePlace K | w.IsUnramified k} : Finset _)ᶜ =
       #({w : InfinitePlace k | w.IsUnramifiedIn K} : Finset _)ᶜ * (finrank k K / 2) := by
@@ -977,6 +993,7 @@ lemma card_isUnramified_compl [NumberField k] [IsGalois k K] :
       rwa [← isUnramifiedIn_comap]
   · simp [isUnramifiedIn_comap]
 
+open scoped Classical in
 lemma card_eq_card_isUnramifiedIn [NumberField k] [IsGalois k K] :
     Fintype.card (InfinitePlace K) =
       #{w : InfinitePlace k | w.IsUnramifiedIn K} * finrank k K +
@@ -1038,6 +1055,7 @@ open Module in
 lemma IsUnramifiedAtInfinitePlaces.card_infinitePlace [NumberField k] [NumberField K]
     [IsGalois k K] [IsUnramifiedAtInfinitePlaces k K] :
     Fintype.card (InfinitePlace K) = Fintype.card (InfinitePlace k) * finrank k K := by
+  classical
   rw [InfinitePlace.card_eq_card_isUnramifiedIn (k := k) (K := K), Finset.filter_true_of_mem,
     Finset.card_univ, Finset.card_eq_zero.mpr, zero_mul, add_zero]
   · exact Finset.compl_univ
diff --git a/Mathlib/NumberTheory/NumberField/FinitePlaces.lean b/Mathlib/NumberTheory/NumberField/FinitePlaces.lean
index d030b4900a54c4..5d5cf658791b0c 100644
--- a/Mathlib/NumberTheory/NumberField/FinitePlaces.lean
+++ b/Mathlib/NumberTheory/NumberField/FinitePlaces.lean
@@ -25,7 +25,7 @@ into a completion of `K` associated to a non-zero prime ideal of `𝓞 K`.
 number field, places, finite places
 -/
 
-open Ideal IsDedekindDomain HeightOneSpectrum NumberField WithZeroMulInt
+open Ideal IsDedekindDomain HeightOneSpectrum WithZeroMulInt
 
 namespace NumberField
 
@@ -33,9 +33,8 @@ section absoluteValue
 
 variable {K : Type*} [Field K] [NumberField K] (v : HeightOneSpectrum (𝓞 K))
 
-/-- The norm of a maximal ideal as an element of `ℝ≥0` is `> 1`  -/
-lemma one_lt_norm : 1 < (absNorm v.asIdeal : NNReal) := by
-  norm_cast
+/-- The norm of a maximal ideal is `> 1`  -/
+lemma one_lt_norm : 1 < absNorm v.asIdeal := by
   by_contra! h
   apply IsPrime.ne_top v.isPrime
   rw [← absNorm_eq_one_iff]
@@ -44,7 +43,12 @@ lemma one_lt_norm : 1 < (absNorm v.asIdeal : NNReal) := by
     exact (v.asIdeal.fintypeQuotientOfFreeOfNeBot v.ne_bot).finite
   omega
 
-private lemma norm_ne_zero : (absNorm v.asIdeal : NNReal) ≠ 0 := ne_zero_of_lt (one_lt_norm v)
+/-- The norm of a maximal ideal as an element of `ℝ≥0` is `> 1`  -/
+lemma one_lt_norm_nnreal : 1 < (absNorm v.asIdeal : NNReal) := mod_cast one_lt_norm v
+
+/-- The norm of a maximal ideal as an element of `ℝ≥0` is `≠ 0`  -/
+lemma norm_ne_zero : (absNorm v.asIdeal : NNReal) ≠ 0 :=
+  ne_zero_of_lt (one_lt_norm_nnreal v)
 
 /-- The `v`-adic absolute value on `K` defined as the norm of `v` raised to negative `v`-adic
 valuation.-/
@@ -57,7 +61,7 @@ noncomputable def vadicAbv : AbsoluteValue K ℝ where
     -- the triangle inequality is implied by the ultrametric one
     apply le_trans _ <| max_le_add_of_nonneg (zero_le ((toNNReal (norm_ne_zero v)) (v.valuation x)))
       (zero_le ((toNNReal (norm_ne_zero v)) (v.valuation y)))
-    have h_mono := (toNNReal_strictMono (one_lt_norm v)).monotone
+    have h_mono := (toNNReal_strictMono (one_lt_norm_nnreal v)).monotone
     rw [← h_mono.map_max] --max goes inside withZeroMultIntToNNReal
     exact h_mono (v.valuation.map_add x y)
 
@@ -80,7 +84,7 @@ noncomputable instance instRankOneValuedAdicCompletion :
     map_one' := rfl
     map_mul' := MonoidWithZeroHom.map_mul (toNNReal (norm_ne_zero v))
   }
-  strictMono' := toNNReal_strictMono (one_lt_norm v)
+  strictMono' := toNNReal_strictMono (one_lt_norm_nnreal v)
   nontrivial' := by
     rcases Submodule.exists_mem_ne_zero_of_ne_bot v.ne_bot with ⟨x, hx1, hx2⟩
     use (x : K)
@@ -133,19 +137,19 @@ open FinitePlace
 
 /-- The `v`-adic norm of an integer is at most 1. -/
 theorem norm_le_one (x : 𝓞 K) : ‖embedding v x‖ ≤ 1 := by
-  rw [norm_def', NNReal.coe_le_one, toNNReal_le_one_iff (one_lt_norm v)]
+  rw [norm_def', NNReal.coe_le_one, toNNReal_le_one_iff (one_lt_norm_nnreal v)]
   exact valuation_le_one v x
 
 /-- The `v`-adic norm of an integer is 1 if and only if it is not in the ideal. -/
 theorem norm_eq_one_iff_not_mem (x : 𝓞 K) : ‖(embedding v) x‖ = 1 ↔ x ∉ v.asIdeal := by
   rw [norm_def_int, NNReal.coe_eq_one, toNNReal_eq_one_iff (v.intValuationDef x)
-    (norm_ne_zero v) (one_lt_norm v).ne', ← dvd_span_singleton,
+    (norm_ne_zero v) (one_lt_norm_nnreal v).ne', ← dvd_span_singleton,
     ← intValuation_lt_one_iff_dvd, not_lt]
   exact (intValuation_le_one v x).ge_iff_eq.symm
 
 /-- The `v`-adic norm of an integer is less than 1 if and only if it is in the ideal. -/
 theorem norm_lt_one_iff_mem (x : 𝓞 K) : ‖embedding v x‖ < 1 ↔ x ∈ v.asIdeal := by
-  rw [norm_def_int, NNReal.coe_lt_one, toNNReal_lt_one_iff (one_lt_norm v),
+  rw [norm_def_int, NNReal.coe_lt_one, toNNReal_lt_one_iff (one_lt_norm_nnreal v),
     intValuation_lt_one_iff_dvd]
   exact dvd_span_singleton
 
@@ -265,9 +269,9 @@ lemma embedding_mul_absNorm (v : HeightOneSpectrum (𝓞 K)) {x : 𝓞 K} (h_x_n
     WithZeroMulInt.toNNReal_neg_apply _
       (v.valuation.ne_zero_iff.mpr (RingOfIntegers.coe_ne_zero_iff.mpr h_x_nezero))]
   push_cast
-  rw [← zpow_natCast, ← zpow_add₀ <| mod_cast (zero_lt_one.trans (one_lt_norm v)).ne']
+  rw [← zpow_natCast, ← zpow_add₀ <| mod_cast (zero_lt_one.trans (one_lt_norm_nnreal v)).ne']
   norm_cast
-  rw [zpow_eq_one_iff_right₀ (Nat.cast_nonneg' _) (mod_cast (one_lt_norm v).ne')]
+  rw [zpow_eq_one_iff_right₀ (Nat.cast_nonneg' _) (mod_cast (one_lt_norm_nnreal v).ne')]
   simp [valuation_eq_intValuationDef, intValuationDef_if_neg, h_x_nezero]
 
 end IsDedekindDomain.HeightOneSpectrum
diff --git a/Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean b/Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean
index 1d17123834ee72..88da50e57f12cc 100644
--- a/Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean
+++ b/Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean
@@ -400,9 +400,7 @@ theorem logEmbeddingQuot_injective :
     Function.comp_apply, EmbeddingLike.apply_eq_iff_eq] at h
   exact (EmbeddingLike.apply_eq_iff_eq _).mp <| (QuotientGroup.kerLift_injective _).eq_iff.mp h
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/4119
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/4119
 the `Module ℤ (Additive ((𝓞 K)ˣ ⧸ NumberField.Units.torsion K))` instance required below isn't found
 unless we use `set_option maxSynthPendingDepth 2`, or add
 explicit instances:
diff --git a/Mathlib/NumberTheory/NumberField/Units/Regulator.lean b/Mathlib/NumberTheory/NumberField/Units/Regulator.lean
index cd7cb76b5bf075..31347cb3073d0e 100644
--- a/Mathlib/NumberTheory/NumberField/Units/Regulator.lean
+++ b/Mathlib/NumberTheory/NumberField/Units/Regulator.lean
@@ -32,21 +32,22 @@ namespace NumberField.Units
 
 variable (K : Type*) [Field K]
 
-open MeasureTheory Classical NumberField.InfinitePlace
+open MeasureTheory NumberField.InfinitePlace
   NumberField NumberField.Units.dirichletUnitTheorem
 
 variable [NumberField K]
 
+open scoped Classical in
 /-- The regulator of a number field `K`. -/
 def regulator : ℝ := ZLattice.covolume (unitLattice K)
 
+open scoped Classical in
 theorem regulator_ne_zero : regulator K ≠ 0 := ZLattice.covolume_ne_zero (unitLattice K) volume
 
+open scoped Classical in
 theorem regulator_pos : 0 < regulator K := ZLattice.covolume_pos (unitLattice K) volume
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/4119
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/4119
 the `Module ℤ (Additive ((𝓞 K)ˣ ⧸ NumberField.Units.torsion K))` instance required below isn't found
 unless we use `set_option maxSynthPendingDepth 2`, or add
 explicit instances:
@@ -56,6 +57,7 @@ local instance : CommGroup (𝓞 K)ˣ := inferInstance
 -/
 set_option maxSynthPendingDepth 2 -- Note this is active for the remainder of the file.
 
+open scoped Classical in
 theorem regulator_eq_det' (e : {w : InfinitePlace K // w ≠ w₀} ≃ Fin (rank K)) :
     regulator K = |(Matrix.of fun i ↦
       logEmbedding K (Additive.ofMul (fundSystem K (e i)))).det| := by
@@ -63,6 +65,7 @@ theorem regulator_eq_det' (e : {w : InfinitePlace K // w ≠ w₀} ≃ Fin (rank
     (((basisModTorsion K).map (logEmbeddingEquiv K)).reindex e.symm), Basis.coe_reindex,
     Function.comp_def, Basis.map_apply, ← fundSystem_mk, Equiv.symm_symm, logEmbeddingEquiv_apply]
 
+open scoped Classical in
 /-- Let `u : Fin (rank K) → (𝓞 K)ˣ` be a family of units and let `w₁` and `w₂` be two infinite
 places. Then, the two square matrices with entries `(mult w * log w (u i))_i, {w ≠ w_i}`, `i = 1,2`,
 have the same determinant in absolute value. -/
@@ -98,6 +101,7 @@ theorem abs_det_eq_abs_det (u : Fin (rank K) → (𝓞 K)ˣ)
       Units.norm, Rat.cast_one, Real.log_one]
     exact fun _ _ ↦ pow_ne_zero _ <| (map_ne_zero _).mpr (coe_ne_zero _)
 
+open scoped Classical in
 /-- For any infinite place `w'`, the regulator is equal to the absolute value of the determinant
 of the matrix `(mult w * log w (fundSystem K i)))_i, {w ≠ w'}`. -/
 theorem regulator_eq_det (w' : InfinitePlace K) (e : {w // w ≠ w'} ≃ Fin (rank K)) :
diff --git a/Mathlib/NumberTheory/Padics/RingHoms.lean b/Mathlib/NumberTheory/Padics/RingHoms.lean
index f16855f13878b8..f6b9067b3e3c91 100644
--- a/Mathlib/NumberTheory/Padics/RingHoms.lean
+++ b/Mathlib/NumberTheory/Padics/RingHoms.lean
@@ -41,7 +41,6 @@ The constructions of the ring homomorphisms go through an auxiliary constructor
 
 noncomputable section
 
-open scoped Classical
 open Nat IsLocalRing Padic
 
 namespace PadicInt
@@ -293,6 +292,7 @@ def residueField : IsLocalRing.ResidueField ℤ_[p] ≃+* ZMod p := by
   exact_mod_cast (@PadicInt.ker_toZMod p _) ▸ RingHom.quotientKerEquivOfSurjective
     (ZMod.ringHom_surjective PadicInt.toZMod)
 
+open scoped Classical in
 /-- `appr n x` gives a value `v : ℕ` such that `x` and `↑v : ℤ_p` are congruent mod `p^n`.
 See `appr_spec`. -/
 -- Porting note: removing irreducible solves a lot of problems
diff --git a/Mathlib/NumberTheory/RamificationInertia/Basic.lean b/Mathlib/NumberTheory/RamificationInertia/Basic.lean
index 24f96fd9081fbb..137fee8752dbb4 100644
--- a/Mathlib/NumberTheory/RamificationInertia/Basic.lean
+++ b/Mathlib/NumberTheory/RamificationInertia/Basic.lean
@@ -675,25 +675,27 @@ end FactLeComap
 
 section FactorsMap
 
-open scoped Classical
-
 /-! ## Properties of the factors of `p.map (algebraMap R S)` -/
 
 
 variable [IsDedekindDomain S] [Algebra R S]
 
+open scoped Classical in
 theorem Factors.ne_bot (P : (factors (map (algebraMap R S) p)).toFinset) : (P : Ideal S) ≠ ⊥ :=
   (prime_of_factor _ (Multiset.mem_toFinset.mp P.2)).ne_zero
 
+open scoped Classical in
 instance Factors.isPrime (P : (factors (map (algebraMap R S) p)).toFinset) :
     IsPrime (P : Ideal S) :=
   Ideal.isPrime_of_prime (prime_of_factor _ (Multiset.mem_toFinset.mp P.2))
 
+open scoped Classical in
 theorem Factors.ramificationIdx_ne_zero (P : (factors (map (algebraMap R S) p)).toFinset) :
     ramificationIdx (algebraMap R S) p P ≠ 0 :=
   IsDedekindDomain.ramificationIdx_ne_zero (ne_zero_of_mem_factors (Multiset.mem_toFinset.mp P.2))
     (Factors.isPrime p P) (Ideal.le_of_dvd (dvd_of_mem_factors (Multiset.mem_toFinset.mp P.2)))
 
+open scoped Classical in
 instance Factors.fact_ramificationIdx_neZero (P : (factors (map (algebraMap R S) p)).toFinset) :
     NeZero (ramificationIdx (algebraMap R S) p P) :=
   ⟨Factors.ramificationIdx_ne_zero p P⟩
@@ -702,14 +704,17 @@ set_option synthInstance.checkSynthOrder false
 -- Porting note: this is okay since, as noted above, in this file the value of `f` can be inferred
 attribute [local instance] Quotient.algebraQuotientOfRamificationIdxNeZero
 
+open scoped Classical in
 instance Factors.isScalarTower (P : (factors (map (algebraMap R S) p)).toFinset) :
     IsScalarTower R (R ⧸ p) (S ⧸ (P : Ideal S)) :=
   IsScalarTower.of_algebraMap_eq' rfl
 
+open scoped Classical in
 instance Factors.liesOver [p.IsMaximal] (P : (factors (map (algebraMap R S) p)).toFinset) :
     P.1.LiesOver p :=
   ⟨(comap_eq_of_scalar_tower_quotient (algebraMap (R ⧸ p) (S ⧸ P.1)).injective).symm⟩
 
+open scoped Classical in
 theorem Factors.finrank_pow_ramificationIdx [p.IsMaximal]
     (P : (factors (map (algebraMap R S) p)).toFinset) :
     finrank (R ⧸ p) (S ⧸ (P : Ideal S) ^ ramificationIdx (algebraMap R S) p P) =
@@ -717,6 +722,7 @@ theorem Factors.finrank_pow_ramificationIdx [p.IsMaximal]
   rw [finrank_prime_pow_ramificationIdx, inertiaDeg_algebraMap]
   exacts [Factors.ne_bot p P, NeZero.ne _]
 
+open scoped Classical in
 instance Factors.finiteDimensional_quotient_pow [Module.Finite R S] [p.IsMaximal]
     (P : (factors (map (algebraMap R S) p)).toFinset) :
     FiniteDimensional (R ⧸ p) (S ⧸ (P : Ideal S) ^ ramificationIdx (algebraMap R S) p P) := by
@@ -726,6 +732,7 @@ instance Factors.finiteDimensional_quotient_pow [Module.Finite R S] [p.IsMaximal
 
 universe w
 
+open scoped Classical in
 /-- **Chinese remainder theorem** for a ring of integers: if the prime ideal `p : Ideal R`
 factors in `S` as `∏ i, P i ^ e i`, then `S ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`. -/
 noncomputable def Factors.piQuotientEquiv (p : Ideal R) (hp : map (algebraMap R S) p ≠ ⊥) :
@@ -752,6 +759,7 @@ theorem Factors.piQuotientEquiv_map (p : Ideal R) (hp : map (algebraMap R S) p 
 
 variable (S)
 
+open scoped Classical in
 /-- **Chinese remainder theorem** for a ring of integers: if the prime ideal `p : Ideal R`
 factors in `S` as `∏ i, P i ^ e i`,
 then `S ⧸ I` factors `R ⧸ I`-linearly as `Π i, R ⧸ (P i ^ e i)`. -/
@@ -767,6 +775,7 @@ noncomputable def Factors.piQuotientLinearEquiv (p : Ideal R) (hp : map (algebra
         RingHomCompTriple.comp_apply, Pi.mul_apply, Pi.algebraMap_apply]
       congr }
 
+open scoped Classical in
 /-- The **fundamental identity** of ramification index `e` and inertia degree `f`:
 for `P` ranging over the primes lying over `p`, `∑ P, e P * f P = [Frac(S) : Frac(R)]`;
 here `S` is a finite `R`-module (and thus `Frac(S) : Frac(R)` is a finite extension) and `p`
diff --git a/Mathlib/Order/Antisymmetrization.lean b/Mathlib/Order/Antisymmetrization.lean
index a64b7b3da17a81..1ec2f0a7747118 100644
--- a/Mathlib/Order/Antisymmetrization.lean
+++ b/Mathlib/Order/Antisymmetrization.lean
@@ -40,7 +40,7 @@ def AntisymmRel (a b : α) : Prop :=
   r a b ∧ r b a
 
 theorem antisymmRel_swap : AntisymmRel (swap r) = AntisymmRel r :=
-  funext fun _ => funext fun _ => propext and_comm
+  funext₂ fun _ _ ↦ propext and_comm
 
 @[refl]
 theorem antisymmRel_refl [IsRefl α r] (a : α) : AntisymmRel r a a :=
diff --git a/Mathlib/Order/Basic.lean b/Mathlib/Order/Basic.lean
index 6b4dfb21914633..11f4a94f61cf10 100644
--- a/Mathlib/Order/Basic.lean
+++ b/Mathlib/Order/Basic.lean
@@ -1263,32 +1263,47 @@ instance [∀ i, Preorder (π i)] [∀ i, DenselyOrdered (π i)] :
           ⟨le_update_iff.2 ⟨ha.le, fun _ _ ↦ le_rfl⟩, i, by rwa [update_self]⟩,
           update_le_iff.2 ⟨hb.le, fun _ _ ↦ hab _⟩, i, by rwa [update_self]⟩⟩
 
-theorem le_of_forall_le_of_dense [LinearOrder α] [DenselyOrdered α] {a₁ a₂ : α}
-    (h : ∀ a, a₂ < a → a₁ ≤ a) : a₁ ≤ a₂ :=
+section LinearOrder
+variable [LinearOrder α] [DenselyOrdered α] {a₁ a₂ : α}
+
+theorem le_of_forall_gt_imp_ge_of_dense (h : ∀ a, a₂ < a → a₁ ≤ a) : a₁ ≤ a₂ :=
   le_of_not_gt fun ha ↦
     let ⟨a, ha₁, ha₂⟩ := exists_between ha
     lt_irrefl a <| lt_of_lt_of_le ‹a < a₁› (h _ ‹a₂ < a›)
 
-theorem eq_of_le_of_forall_le_of_dense [LinearOrder α] [DenselyOrdered α] {a₁ a₂ : α} (h₁ : a₂ ≤ a₁)
-    (h₂ : ∀ a, a₂ < a → a₁ ≤ a) : a₁ = a₂ :=
-  le_antisymm (le_of_forall_le_of_dense h₂) h₁
+lemma forall_gt_imp_ge_iff_le_of_dense : (∀ a, a₂ < a → a₁ ≤ a) ↔ a₁ ≤ a₂ :=
+  ⟨le_of_forall_gt_imp_ge_of_dense, fun ha _a ha₂ ↦ ha.trans ha₂.le⟩
+
+lemma eq_of_le_of_forall_lt_imp_le_of_dense (h₁ : a₂ ≤ a₁) (h₂ : ∀ a, a₂ < a → a₁ ≤ a) : a₁ = a₂ :=
+  le_antisymm (le_of_forall_gt_imp_ge_of_dense h₂) h₁
 
-theorem le_of_forall_ge_of_dense [LinearOrder α] [DenselyOrdered α] {a₁ a₂ : α}
-    (h : ∀ a₃ < a₁, a₃ ≤ a₂) : a₁ ≤ a₂ :=
+theorem le_of_forall_lt_imp_le_of_dense (h : ∀ a < a₁, a ≤ a₂) : a₁ ≤ a₂ :=
   le_of_not_gt fun ha ↦
     let ⟨a, ha₁, ha₂⟩ := exists_between ha
     lt_irrefl a <| lt_of_le_of_lt (h _ ‹a < a₁›) ‹a₂ < a›
 
-theorem eq_of_le_of_forall_ge_of_dense [LinearOrder α] [DenselyOrdered α] {a₁ a₂ : α} (h₁ : a₂ ≤ a₁)
-    (h₂ : ∀ a₃ < a₁, a₃ ≤ a₂) : a₁ = a₂ :=
-  (le_of_forall_ge_of_dense h₂).antisymm h₁
+lemma forall_lt_imp_le_iff_le_of_dense : (∀ a < a₁, a ≤ a₂) ↔ a₁ ≤ a₂ :=
+  ⟨le_of_forall_lt_imp_le_of_dense, fun ha _a ha₁ ↦ ha₁.le.trans ha⟩
+
+theorem eq_of_le_of_forall_gt_imp_ge_of_dense (h₁ : a₂ ≤ a₁) (h₂ : ∀ a < a₁, a ≤ a₂) : a₁ = a₂ :=
+  (le_of_forall_lt_imp_le_of_dense h₂).antisymm h₁
+
+@[deprecated (since := "2025-01-21")]
+alias le_of_forall_le_of_dense := le_of_forall_gt_imp_ge_of_dense
+
+@[deprecated (since := "2025-01-21")]
+alias le_of_forall_ge_of_dense := le_of_forall_lt_imp_le_of_dense
+
+@[deprecated (since := "2025-01-21")] alias forall_lt_le_iff := forall_lt_imp_le_iff_le_of_dense
+@[deprecated (since := "2025-01-21")] alias forall_gt_ge_iff := forall_gt_imp_ge_iff_le_of_dense
+
+@[deprecated (since := "2025-01-21")]
+alias eq_of_le_of_forall_le_of_dense := eq_of_le_of_forall_lt_imp_le_of_dense
 
-theorem forall_lt_le_iff [LinearOrder α] [DenselyOrdered α] {a b : α} : (∀ c < a, c ≤ b) ↔ a ≤ b :=
-  ⟨le_of_forall_ge_of_dense, fun hab _c hca ↦ hca.le.trans hab⟩
+@[deprecated (since := "2025-01-21")]
+alias eq_of_le_of_forall_ge_of_dense := eq_of_le_of_forall_gt_imp_ge_of_dense
 
-theorem forall_gt_ge_iff [LinearOrder α] [DenselyOrdered α] {a b : α} :
-    (∀ c, a < c → b ≤ c) ↔ b ≤ a :=
-  forall_lt_le_iff (α := αᵒᵈ)
+end LinearOrder
 
 theorem dense_or_discrete [LinearOrder α] (a₁ a₂ : α) :
     (∃ a, a₁ < a ∧ a < a₂) ∨ (∀ a, a₁ < a → a₂ ≤ a) ∧ ∀ a < a₂, a ≤ a₁ :=
diff --git a/Mathlib/Order/Bounds/Basic.lean b/Mathlib/Order/Bounds/Basic.lean
index ff8f494fcda960..a161ef7fab20c1 100644
--- a/Mathlib/Order/Bounds/Basic.lean
+++ b/Mathlib/Order/Bounds/Basic.lean
@@ -458,7 +458,7 @@ theorem exists_glb_Ioi (i : γ) : ∃ j, IsGLB (Ioi i) j :=
 variable [DenselyOrdered γ]
 
 theorem isLUB_Iio {a : γ} : IsLUB (Iio a) a :=
-  ⟨fun _ hx => le_of_lt hx, fun _ hy => le_of_forall_ge_of_dense hy⟩
+  ⟨fun _ hx => le_of_lt hx, fun _ hy => le_of_forall_lt_imp_le_of_dense hy⟩
 
 theorem isGLB_Ioi {a : γ} : IsGLB (Ioi a) a :=
   @isLUB_Iio γᵒᵈ _ _ a
diff --git a/Mathlib/Order/CompleteLattice.lean b/Mathlib/Order/CompleteLattice.lean
index 6dfb80d1a72f2e..5e469fe3ff642b 100644
--- a/Mathlib/Order/CompleteLattice.lean
+++ b/Mathlib/Order/CompleteLattice.lean
@@ -1739,6 +1739,18 @@ theorem disjoint_sSup_right {a : Set α} {b : α} (d : Disjoint b (sSup a)) {i}
     Disjoint b i :=
   disjoint_iff_inf_le.mpr (iSup₂_le_iff.mp (iSup_inf_le_inf_sSup.trans d.le_bot) i hi :)
 
+lemma disjoint_of_sSup_disjoint_of_le_of_le {a b : α} {c d : Set α} (hs : ∀ e ∈ c, e ≤ a)
+    (ht : ∀ e ∈ d, e ≤ b) (hd : Disjoint a b) (he : ⊥ ∉ c ∨ ⊥ ∉ d) : Disjoint c d := by
+  rw [disjoint_iff_forall_ne]
+  intros x hx y hy
+  rw [Disjoint.ne_iff]
+  · aesop
+  · exact Disjoint.mono (hs x hx) (ht y hy) hd
+
+lemma disjoint_of_sSup_disjoint {a b : Set α} (hd : Disjoint (sSup a) (sSup b))
+    (he : ⊥ ∉ a ∨ ⊥ ∉ b) : Disjoint a b :=
+  disjoint_of_sSup_disjoint_of_le_of_le (fun _ hc ↦ le_sSup hc) (fun _ hc ↦ le_sSup hc) hd he
+
 end CompleteLattice
 
 -- See note [reducible non-instances]
diff --git a/Mathlib/Order/CompleteLattice/SetLike.lean b/Mathlib/Order/CompleteLattice/SetLike.lean
new file mode 100644
index 00000000000000..2e5426d136ae59
--- /dev/null
+++ b/Mathlib/Order/CompleteLattice/SetLike.lean
@@ -0,0 +1,56 @@
+/-
+Copyright (c) 2024 Sven Manthe. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Sven Manthe
+-/
+import Mathlib.Order.CompleteSublattice
+
+/-!
+# `SetLike` instance for elements of `CompleteSublattice (Set X)`
+
+This file provides lemmas for the `SetLike` instance for elements of `CompleteSublattice (Set X)`
+-/
+
+attribute [local instance] SetLike.instSubtypeSet
+
+namespace Sublattice
+
+variable {X : Type*} {L : Sublattice (Set X)}
+
+variable {S T : L} {x : X}
+
+@[ext] lemma ext_mem (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T := SetLike.ext h
+
+lemma mem_subtype : x ∈ L.subtype T ↔ x ∈ T := Iff.rfl
+
+@[simp] lemma setLike_mem_inf : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T := by simp [← mem_subtype]
+@[simp] lemma setLike_mem_sup : x ∈ S ⊔ T ↔ x ∈ S ∨ x ∈ T := by simp [← mem_subtype]
+
+@[simp] lemma setLike_mem_coe : x ∈ T.val ↔ x ∈ T := Iff.rfl
+@[simp] lemma setLike_mem_mk (U : Set X) (h : U ∈ L) : x ∈ (⟨U, h⟩ : L) ↔ x ∈ U := Iff.rfl
+
+end Sublattice
+
+namespace CompleteSublattice
+
+variable {X : Type*} {L : CompleteSublattice (Set X)}
+
+variable {S T : L} {𝒮 : Set L} {I : Sort*} {f : I → L} {x : X}
+
+@[ext] lemma ext (h : ∀ x, x ∈ S ↔ x ∈ T) : S = T := SetLike.ext h
+
+lemma mem_subtype : x ∈ L.subtype T ↔ x ∈ T := Iff.rfl
+
+@[simp] lemma mem_inf : x ∈ S ⊓ T ↔ x ∈ S ∧ x ∈ T := by simp [← mem_subtype]
+@[simp] lemma mem_sInf : x ∈ sInf 𝒮 ↔ ∀ T ∈ 𝒮, x ∈ T := by simp [← mem_subtype]
+@[simp] lemma mem_iInf : x ∈ ⨅ i : I, f i ↔ ∀ i : I, x ∈ f i := by simp [← mem_subtype]
+@[simp] lemma mem_top : x ∈ (⊤ : L) := by simp [← mem_subtype]
+
+@[simp] lemma mem_sup : x ∈ S ⊔ T ↔ x ∈ S ∨ x ∈ T := by simp [← mem_subtype]
+@[simp] lemma mem_sSup : x ∈ sSup 𝒮 ↔ ∃ T ∈ 𝒮, x ∈ T := by simp [← mem_subtype]
+@[simp] lemma mem_iSup : x ∈ ⨆ i : I, f i ↔ ∃ i : I, x ∈ f i := by simp [← mem_subtype]
+@[simp] lemma not_mem_bot : ¬ x ∈ (⊥ : L) := by simp [← mem_subtype]
+
+@[simp] lemma mem_mk (U : Set X) (h : U ∈ L) : x ∈ (⟨U, h⟩ : L) ↔ x ∈ U := Iff.rfl
+
+end CompleteSublattice
diff --git a/Mathlib/Order/DirectedInverseSystem.lean b/Mathlib/Order/DirectedInverseSystem.lean
index 943b33a61c2285..fe714d8d388630 100644
--- a/Mathlib/Order/DirectedInverseSystem.lean
+++ b/Mathlib/Order/DirectedInverseSystem.lean
@@ -144,15 +144,13 @@ theorem mk_injective (h : ∀ i j hij, Function.Injective (f i j hij)) (i) :
 
 section map₀
 
-open Classical (arbitrary)
-
 variable [Nonempty ι] (ih : ∀ i, F i)
 
 /-- "Nullary map" to construct an element in the direct limit. -/
-noncomputable def map₀ : DirectLimit F f := ⟦⟨arbitrary ι, ih _⟩⟧
+noncomputable def map₀ : DirectLimit F f := ⟦⟨Classical.arbitrary ι, ih _⟩⟧
 
 theorem map₀_def (compat : ∀ i j h, f i j h (ih i) = ih j) (i) : map₀ f ih = ⟦⟨i, ih i⟩⟧ :=
-  have ⟨j, hcj, hij⟩ := exists_ge_ge (arbitrary ι) i
+  have ⟨j, hcj, hij⟩ := exists_ge_ge (Classical.arbitrary ι) i
   Quotient.sound ⟨j, hcj, hij, (compat ..).trans (compat ..).symm⟩
 
 end map₀
diff --git a/Mathlib/Order/Filter/SmallSets.lean b/Mathlib/Order/Filter/SmallSets.lean
index 1e40a2da092eb3..28c02d2e8dc150 100644
--- a/Mathlib/Order/Filter/SmallSets.lean
+++ b/Mathlib/Order/Filter/SmallSets.lean
@@ -54,6 +54,21 @@ theorem hasBasis_smallSets (l : Filter α) :
     HasBasis l.smallSets (fun t : Set α => t ∈ l) powerset :=
   l.basis_sets.smallSets
 
+theorem Eventually.exists_mem_basis_of_smallSets {p : ι → Prop} {s : ι → Set α} {P : Set α → Prop}
+    (h₁ : ∀ᶠ t in l.smallSets, P t) (h₂ : HasBasis l p s) : ∃ i, p i ∧ P (s i) :=
+  (h₂.smallSets.eventually_iff.mp h₁).imp fun _i ⟨hpi, hi⟩ ↦ ⟨hpi, hi Subset.rfl⟩
+
+theorem Frequently.smallSets_of_forall_mem_basis {p : ι → Prop} {s : ι → Set α} {P : Set α → Prop}
+    (h₁ : ∀ i, p i → P (s i)) (h₂ : HasBasis l p s) : ∃ᶠ t in l.smallSets, P t :=
+  h₂.smallSets.frequently_iff.mpr fun _ hi => ⟨_, Subset.rfl, h₁ _ hi⟩
+
+theorem Eventually.exists_mem_of_smallSets {p : Set α → Prop}
+    (h : ∀ᶠ t in l.smallSets, p t) : ∃ s ∈ l, p s :=
+  h.exists_mem_basis_of_smallSets l.basis_sets
+
+/-! No `Frequently.smallSets_of_forall_mem (h : ∀ s ∈ l, p s) : ∃ᶠ t in l.smallSets, p t` as
+`Filter.frequently_smallSets_mem : ∃ᶠ t in l.smallSets, t ∈ l` is preferred. -/
+
 /-- `g` converges to `f.smallSets` if for all `s ∈ f`, eventually we have `g x ⊆ s`. -/
 theorem tendsto_smallSets_iff {f : α → Set β} :
     Tendsto f la lb.smallSets ↔ ∀ t ∈ lb, ∀ᶠ x in la, f x ⊆ t :=
diff --git a/Mathlib/Order/Fin/Basic.lean b/Mathlib/Order/Fin/Basic.lean
index 174adbcde047ab..8f534eba16e5a9 100644
--- a/Mathlib/Order/Fin/Basic.lean
+++ b/Mathlib/Order/Fin/Basic.lean
@@ -71,7 +71,7 @@ instance instLattice      : Lattice (Fin n)      := inferInstance
 
 lemma top_eq_last (n : ℕ) : ⊤ = Fin.last n := rfl
 
-lemma bot_eq_zero (n : ℕ) : ⊥ = (0 : Fin (n + 1)) := rfl
+lemma bot_eq_zero (n : ℕ) [NeZero n] : ⊥ = (0 : Fin n) := rfl
 
 @[simp] theorem rev_bot [NeZero n] : rev (⊥ : Fin n) = ⊤ := rfl
 @[simp] theorem rev_top [NeZero n] : rev (⊤ : Fin n) = ⊥ := rev_rev _
@@ -79,6 +79,26 @@ lemma bot_eq_zero (n : ℕ) : ⊥ = (0 : Fin (n + 1)) := rfl
 theorem rev_zero_eq_top (n : ℕ) [NeZero n] : rev (0 : Fin n) = ⊤ := rfl
 theorem rev_last_eq_bot (n : ℕ) : rev (last n) = ⊥ := by rw [rev_last, bot_eq_zero]
 
+@[simp]
+theorem succ_top (n : ℕ) [NeZero n] : (⊤ : Fin n).succ = ⊤ := by
+  rw [← rev_zero_eq_top, ← rev_zero_eq_top, ← rev_castSucc, castSucc_zero']
+
+@[simp]
+theorem val_top (n : ℕ) [NeZero n] : ((⊤ : Fin n) : ℕ) = n - 1 := rfl
+
+@[simp]
+theorem zero_eq_top {n : ℕ} [NeZero n] : (0 : Fin n) = ⊤ ↔ n = 1 := by
+  rw [← bot_eq_zero, subsingleton_iff_bot_eq_top, subsingleton_iff_le_one, LE.le.le_iff_eq]
+  exact pos_of_neZero n
+
+@[simp]
+theorem top_eq_zero {n : ℕ} [NeZero n] : (⊤ : Fin n) = 0 ↔ n = 1 :=
+  eq_comm.trans zero_eq_top
+
+@[simp]
+theorem cast_top {m n : ℕ} [NeZero m] [NeZero n] (h : m = n) : (⊤ : Fin m).cast h = ⊤ := by
+  simp [← val_inj, h]
+
 section ToFin
 variable {α : Type*} [Preorder α] {f : α → Fin (n + 1)}
 
@@ -117,6 +137,16 @@ lemma strictAnti_iff_succ_lt : StrictAnti f ↔ ∀ i : Fin n, f i.succ < f (cas
 lemma antitone_iff_succ_le : Antitone f ↔ ∀ i : Fin n, f i.succ ≤ f (castSucc i) :=
   antitone_iff_forall_lt.trans <| liftFun_iff_succ (· ≥ ·)
 
+lemma orderHom_injective_iff {α : Type*} [PartialOrder α] {n : ℕ} (f : Fin (n + 1) →o α) :
+    Function.Injective f ↔ ∀ (i : Fin n), f i.castSucc ≠ f i.succ := by
+  constructor
+  · intro hf i hi
+    have := hf hi
+    simp [Fin.ext_iff] at this
+  · intro hf
+    refine (strictMono_iff_lt_succ (f := f).2 fun i ↦ ?_).injective
+    exact lt_of_le_of_ne (f.monotone (Fin.castSucc_le_succ i)) (hf i)
+
 end FromFin
 
 /-! #### Monotonicity -/
@@ -225,6 +255,8 @@ def succOrderEmb (n : ℕ) : Fin n ↪o Fin (n + 1) := .ofStrictMono succ strict
 
 @[simp, norm_cast] lemma coe_succOrderEmb : ⇑(succOrderEmb n) = Fin.succ := rfl
 
+@[simp] lemma succOrderEmb_toEmbedding : (succOrderEmb n).toEmbedding = succEmb n := rfl
+
 /-- `Fin.castLE` as an `OrderEmbedding`.
 
 `castLEEmb h i` embeds `i` into a larger `Fin` type. -/
diff --git a/Mathlib/Order/Interval/Finset/Nat.lean b/Mathlib/Order/Interval/Finset/Nat.lean
index d446eb5f357437..56f30d50f82363 100644
--- a/Mathlib/Order/Interval/Finset/Nat.lean
+++ b/Mathlib/Order/Interval/Finset/Nat.lean
@@ -65,6 +65,9 @@ lemma range_eq_Icc_zero_sub_one (n : ℕ) (hn : n ≠ 0) : range n = Icc 0 (n -
 theorem _root_.Finset.range_eq_Ico : range = Ico 0 :=
   Ico_zero_eq_range.symm
 
+theorem range_succ_eq_Icc_zero (n : ℕ) : range (n + 1) = Icc 0 n := by
+  rw [range_eq_Icc_zero_sub_one _ (Nat.add_one_ne_zero _), Nat.add_sub_cancel_right]
+
 @[simp] lemma card_Icc : #(Icc a b) = b + 1 - a := List.length_range' ..
 @[simp] lemma card_Ico : #(Ico a b) = b - a := List.length_range' ..
 @[simp] lemma card_Ioc : #(Ioc a b) = b - a := List.length_range' ..
diff --git a/Mathlib/Order/RelIso/Set.lean b/Mathlib/Order/RelIso/Set.lean
index d52b7837ed95bc..90cb227e3edb90 100644
--- a/Mathlib/Order/RelIso/Set.lean
+++ b/Mathlib/Order/RelIso/Set.lean
@@ -3,6 +3,7 @@ Copyright (c) 2017 Mario Carneiro. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Mario Carneiro
 -/
+import Mathlib.Order.Directed
 import Mathlib.Order.RelIso.Basic
 import Mathlib.Logic.Embedding.Set
 import Mathlib.Logic.Equiv.Set
@@ -35,6 +36,14 @@ theorem map_sup [SemilatticeSup α] [LinearOrder β] [FunLike F β α]
     a (m ⊔ n) = a m ⊔ a n :=
   map_inf (α := αᵒᵈ) (β := βᵒᵈ) _ _ _
 
+theorem directed [FunLike F α β] [RelHomClass F r s] {ι : Sort*} {a : ι → α} {f : F}
+    (ha : Directed r a) : Directed s (f ∘ a) :=
+  ha.mono_comp _ fun _ _ h ↦ map_rel f h
+
+theorem directedOn [FunLike F α β] [RelHomClass F r s] {f : F}
+    {t : Set α} (hs : DirectedOn r t) : DirectedOn s (f '' t) :=
+  hs.mono_comp fun _ _ h ↦ map_rel f h
+
 end RelHomClass
 
 namespace RelIso
diff --git a/Mathlib/Order/SupIndep.lean b/Mathlib/Order/SupIndep.lean
index 68a3a6a1f2487f..4895b49790087a 100644
--- a/Mathlib/Order/SupIndep.lean
+++ b/Mathlib/Order/SupIndep.lean
@@ -5,7 +5,6 @@ Authors: Aaron Anderson, Kevin Buzzard, Yaël Dillies, Eric Wieser
 -/
 import Mathlib.Data.Finset.Sigma
 import Mathlib.Data.Finset.Pairwise
-import Mathlib.Data.Finset.Powerset
 import Mathlib.Data.Fintype.Basic
 import Mathlib.Order.CompleteLatticeIntervals
 
@@ -58,13 +57,27 @@ def SupIndep (s : Finset ι) (f : ι → α) : Prop :=
 
 variable {s t : Finset ι} {f : ι → α} {i : ι}
 
-instance [DecidableEq ι] [DecidableEq α] : Decidable (SupIndep s f) := by
-  refine @Finset.decidableForallOfDecidableSubsets _ _ _ (?_)
-  rintro t -
-  refine @Finset.decidableDforallFinset _ _ _ (?_)
-  rintro i -
-  have : Decidable (Disjoint (f i) (sup t f)) := decidable_of_iff' (_ = ⊥) disjoint_iff
-  infer_instance
+/-- The RHS looks like the definition of `iSupIndep`. -/
+theorem supIndep_iff_disjoint_erase [DecidableEq ι] :
+    s.SupIndep f ↔ ∀ i ∈ s, Disjoint (f i) ((s.erase i).sup f) :=
+  ⟨fun hs _ hi => hs (erase_subset _ _) hi (not_mem_erase _ _), fun hs _ ht i hi hit =>
+    (hs i hi).mono_right (sup_mono fun _ hj => mem_erase.2 ⟨ne_of_mem_of_not_mem hj hit, ht hj⟩)⟩
+
+/-- If both the index type and the lattice have decidable equality,
+then the `SupIndep` predicate is decidable.
+
+TODO: speedup the definition and drop the `[DecidableEq ι]` assumption
+by iterating over the pairs `(a, t)` such that `s = Finset.cons a t _`
+using something like `List.eraseIdx`
+or by generating both `f i` and `(s.erase i).sup f` in one loop over `s`.
+Yet another possible optimization is to precompute partial suprema of `f`
+over the inits and tails of the list representing `s`,
+store them in 2 `Array`s,
+then compute each `sup` in 1 operation. -/
+instance [DecidableEq ι] [DecidableEq α] : Decidable (SupIndep s f) :=
+  have : ∀ i, Decidable (Disjoint (f i) ((s.erase i).sup f)) := fun _ ↦
+    decidable_of_iff _ disjoint_iff.symm
+  decidable_of_iff _ supIndep_iff_disjoint_erase.symm
 
 theorem SupIndep.subset (ht : t.SupIndep f) (h : s ⊆ t) : s.SupIndep f := fun _ hu _ hi =>
   ht (hu.trans h) (h hi)
@@ -73,6 +86,7 @@ theorem SupIndep.subset (ht : t.SupIndep f) (h : s ⊆ t) : s.SupIndep f := fun
 theorem supIndep_empty (f : ι → α) : (∅ : Finset ι).SupIndep f := fun _ _ a ha =>
   (not_mem_empty a ha).elim
 
+@[simp]
 theorem supIndep_singleton (i : ι) (f : ι → α) : ({i} : Finset ι).SupIndep f :=
   fun s hs j hji hj => by
     rw [eq_empty_of_ssubset_singleton ⟨hs, fun h => hj (h hji)⟩, sup_empty]
@@ -82,53 +96,28 @@ theorem SupIndep.pairwiseDisjoint (hs : s.SupIndep f) : (s : Set ι).PairwiseDis
   fun _ ha _ hb hab =>
     sup_singleton.subst <| hs (singleton_subset_iff.2 hb) ha <| not_mem_singleton.2 hab
 
+@[deprecated (since := "2025-01-17")] alias sup_indep.pairwise_disjoint := SupIndep.pairwiseDisjoint
+
 theorem SupIndep.le_sup_iff (hs : s.SupIndep f) (hts : t ⊆ s) (hi : i ∈ s) (hf : ∀ i, f i ≠ ⊥) :
     f i ≤ t.sup f ↔ i ∈ t := by
   refine ⟨fun h => ?_, le_sup⟩
   by_contra hit
   exact hf i (disjoint_self.1 <| (hs hts hi hit).mono_right h)
 
-/-- The RHS looks like the definition of `iSupIndep`. -/
-theorem supIndep_iff_disjoint_erase [DecidableEq ι] :
-    s.SupIndep f ↔ ∀ i ∈ s, Disjoint (f i) ((s.erase i).sup f) :=
-  ⟨fun hs _ hi => hs (erase_subset _ _) hi (not_mem_erase _ _), fun hs _ ht i hi hit =>
-    (hs i hi).mono_right (sup_mono fun _ hj => mem_erase.2 ⟨ne_of_mem_of_not_mem hj hit, ht hj⟩)⟩
+theorem SupIndep.antitone_fun {g : ι → α} (hle : ∀ x ∈ s, f x ≤ g x) (h : s.SupIndep g) :
+    s.SupIndep f := fun _t hts i his hit ↦
+  (h hts his hit).mono (hle i his) <| Finset.sup_mono_fun fun x hx ↦ hle x <| hts hx
 
-theorem supIndep_antimono_fun {g : ι → α} (h : ∀ x ∈ s, f x ≤ g x) (h : s.SupIndep g) :
-    s.SupIndep f := by
-  classical
-  induction s using Finset.induction_on with
-  | empty => apply Finset.supIndep_empty
-  | @insert i s his IH =>
-  rename_i hle
-  rw [Finset.supIndep_iff_disjoint_erase] at h ⊢
-  intro j hj
-  simp_all only [Finset.mem_insert, or_true, implies_true, true_implies, forall_eq_or_imp,
-    Finset.erase_insert_eq_erase, not_false_eq_true, Finset.erase_eq_of_not_mem]
-  obtain rfl | hj := hj
-  · simp only [Finset.erase_insert_eq_erase]
-    apply h.left.mono hle.left
-    apply (Finset.sup_mono _).trans (Finset.sup_mono_fun hle.right)
-    exact Finset.erase_subset _ _
-  · apply (h.right j hj).mono (hle.right j hj) (Finset.sup_mono_fun _)
-    intro k hk
-    simp only [Finset.mem_erase, ne_eq, Finset.mem_insert] at hk
-    obtain ⟨-, rfl | hk⟩ := hk
-    · exact hle.left
-    · exact hle.right k hk
-
-theorem SupIndep.image [DecidableEq ι] {s : Finset ι'} {g : ι' → ι} (hs : s.SupIndep (f ∘ g)) :
-    (s.image g).SupIndep f := by
+@[deprecated (since := "2025-01-17")]
+alias supIndep_antimono_fun := SupIndep.antitone_fun
+
+protected theorem SupIndep.image [DecidableEq ι] {s : Finset ι'} {g : ι' → ι}
+    (hs : s.SupIndep (f ∘ g)) : (s.image g).SupIndep f := by
   intro t ht i hi hit
-  rw [mem_image] at hi
-  obtain ⟨i, hi, rfl⟩ := hi
-  haveI : DecidableEq ι' := Classical.decEq _
-  suffices hts : t ⊆ (s.erase i).image g by
-    refine (supIndep_iff_disjoint_erase.1 hs i hi).mono_right ((sup_mono hts).trans ?_)
-    rw [sup_image]
-  rintro j hjt
-  obtain ⟨j, hj, rfl⟩ := mem_image.1 (ht hjt)
-  exact mem_image_of_mem _ (mem_erase.2 ⟨ne_of_apply_ne g (ne_of_mem_of_not_mem hjt hit), hj⟩)
+  rcases subset_image_iff.mp ht with ⟨t, hts, rfl⟩
+  rcases mem_image.mp hi with ⟨i, his, rfl⟩
+  rw [sup_image]
+  exact hs hts his (hit <| mem_image_of_mem _ ·)
 
 theorem supIndep_map {s : Finset ι'} {g : ι' ↪ ι} : (s.map g).SupIndep f ↔ s.SupIndep (f ∘ g) := by
   refine ⟨fun hs t ht i hi hit => ?_, fun hs => ?_⟩
@@ -140,24 +129,11 @@ theorem supIndep_map {s : Finset ι'} {g : ι' ↪ ι} : (s.map g).SupIndep f 
 
 @[simp]
 theorem supIndep_pair [DecidableEq ι] {i j : ι} (hij : i ≠ j) :
-    ({i, j} : Finset ι).SupIndep f ↔ Disjoint (f i) (f j) :=
-  ⟨fun h => h.pairwiseDisjoint (by simp) (by simp) hij,
-   fun h => by
-    rw [supIndep_iff_disjoint_erase]
-    intro k hk
-    rw [Finset.mem_insert, Finset.mem_singleton] at hk
-    obtain rfl | rfl := hk
-    · convert h using 1
-      rw [Finset.erase_insert, Finset.sup_singleton]
-      simpa using hij
-    · convert h.symm using 1
-      have : ({i, k} : Finset ι).erase k = {i} := by
-        ext
-        rw [mem_erase, mem_insert, mem_singleton, mem_singleton, and_or_left, Ne,
-          not_and_self_iff, or_false, and_iff_right_of_imp]
-        rintro rfl
-        exact hij
-      rw [this, Finset.sup_singleton]⟩
+    ({i, j} : Finset ι).SupIndep f ↔ Disjoint (f i) (f j) := by
+  suffices Disjoint (f i) (f j) → Disjoint (f j) ((Finset.erase {i, j} j).sup f) by
+    simpa [supIndep_iff_disjoint_erase, hij]
+  rw [pair_comm]
+  simp [hij.symm, disjoint_comm]
 
 theorem supIndep_univ_bool (f : Bool → α) :
     (Finset.univ : Finset Bool).SupIndep f ↔ Disjoint (f false) (f true) :=
@@ -167,29 +143,14 @@ theorem supIndep_univ_bool (f : Bool → α) :
 @[simp]
 theorem supIndep_univ_fin_two (f : Fin 2 → α) :
     (Finset.univ : Finset (Fin 2)).SupIndep f ↔ Disjoint (f 0) (f 1) :=
-  haveI : (0 : Fin 2) ≠ 1 := by simp
+  have : (0 : Fin 2) ≠ 1 := by simp
   supIndep_pair this
 
-theorem SupIndep.attach (hs : s.SupIndep f) : s.attach.SupIndep fun a => f a := by
-  intro t _ i _ hi
-  classical
-    have : (fun (a : { x // x ∈ s }) => f ↑a) = f ∘ (fun a : { x // x ∈ s } => ↑a) := rfl
-    rw [this, ← Finset.sup_image]
-    refine hs (image_subset_iff.2 fun (j : { x // x ∈ s }) _ => j.2) i.2 fun hi' => hi ?_
-    rw [mem_image] at hi'
-    obtain ⟨j, hj, hji⟩ := hi'
-    rwa [Subtype.ext hji] at hj
-
 @[simp]
 theorem supIndep_attach : (s.attach.SupIndep fun a => f a) ↔ s.SupIndep f := by
-  refine ⟨fun h t ht i his hit => ?_, SupIndep.attach⟩
-  classical
-  convert h (filter_subset (fun (i : { x // x ∈ s }) => (i : ι) ∈ t) _) (mem_attach _ ⟨i, ‹_›⟩)
-    fun hi => hit <| by simpa using hi using 1
-  refine eq_of_forall_ge_iff ?_
-  simp only [Finset.sup_le_iff, mem_filter, mem_attach, true_and, Function.comp_apply,
-    Subtype.forall, Subtype.coe_mk]
-  exact fun a => forall_congr' fun j => ⟨fun h _ => h, fun h hj => h (ht hj) hj⟩
+  simpa [Finset.attach_map_val] using (supIndep_map (s := s.attach) (g := .subtype _)).symm
+
+alias ⟨_, SupIndep.attach⟩ := supIndep_attach
 
 end Lattice
 
@@ -201,11 +162,10 @@ theorem supIndep_iff_pairwiseDisjoint : s.SupIndep f ↔ (s : Set ι).PairwiseDi
   ⟨SupIndep.pairwiseDisjoint, fun hs _ ht _ hi hit =>
     Finset.disjoint_sup_right.2 fun _ hj => hs hi (ht hj) (ne_of_mem_of_not_mem hj hit).symm⟩
 
-alias ⟨sup_indep.pairwise_disjoint, _root_.Set.PairwiseDisjoint.supIndep⟩ :=
-  supIndep_iff_pairwiseDisjoint
+alias ⟨_, _root_.Set.PairwiseDisjoint.supIndep⟩ := supIndep_iff_pairwiseDisjoint
 
 /-- Bind operation for `SupIndep`. -/
-theorem SupIndep.sup [DecidableEq ι] {s : Finset ι'} {g : ι' → Finset ι} {f : ι → α}
+protected theorem SupIndep.sup [DecidableEq ι] {s : Finset ι'} {g : ι' → Finset ι} {f : ι → α}
     (hs : s.SupIndep fun i => (g i).sup f) (hg : ∀ i' ∈ s, (g i').SupIndep f) :
     (s.sup g).SupIndep f := by
   simp_rw [supIndep_iff_pairwiseDisjoint] at hs hg ⊢
@@ -213,15 +173,15 @@ theorem SupIndep.sup [DecidableEq ι] {s : Finset ι'} {g : ι' → Finset ι} {
   exact hs.biUnion_finset hg
 
 /-- Bind operation for `SupIndep`. -/
-theorem SupIndep.biUnion [DecidableEq ι] {s : Finset ι'} {g : ι' → Finset ι} {f : ι → α}
+protected theorem SupIndep.biUnion [DecidableEq ι] {s : Finset ι'} {g : ι' → Finset ι} {f : ι → α}
     (hs : s.SupIndep fun i => (g i).sup f) (hg : ∀ i' ∈ s, (g i').SupIndep f) :
     (s.biUnion g).SupIndep f := by
   rw [← sup_eq_biUnion]
   exact hs.sup hg
 
 /-- Bind operation for `SupIndep`. -/
-theorem SupIndep.sigma {β : ι → Type*} {s : Finset ι} {g : ∀ i, Finset (β i)} {f : Sigma β → α}
-    (hs : s.SupIndep fun i => (g i).sup fun b => f ⟨i, b⟩)
+protected theorem SupIndep.sigma {β : ι → Type*} {s : Finset ι} {g : ∀ i, Finset (β i)}
+    {f : Sigma β → α} (hs : s.SupIndep fun i => (g i).sup fun b => f ⟨i, b⟩)
     (hg : ∀ i ∈ s, (g i).SupIndep fun b => f ⟨i, b⟩) : (s.sigma g).SupIndep f := by
   rintro t ht ⟨i, b⟩ hi hit
   rw [Finset.disjoint_sup_right]
@@ -235,7 +195,7 @@ theorem SupIndep.sigma {β : ι → Type*} {s : Finset ι} {g : ∀ i, Finset (
     · convert le_sup (α := α) hi.2; simp
     · convert le_sup (α := α) hj.2; simp
 
-theorem SupIndep.product {s : Finset ι} {t : Finset ι'} {f : ι × ι' → α}
+protected theorem SupIndep.product {s : Finset ι} {t : Finset ι'} {f : ι × ι' → α}
     (hs : s.SupIndep fun i => t.sup fun i' => f (i, i'))
     (ht : t.SupIndep fun i' => s.sup fun i => f (i, i')) : (s ×ˢ t).SupIndep f := by
   rintro u hu ⟨i, i'⟩ hi hiu
diff --git a/Mathlib/Order/WithBot.lean b/Mathlib/Order/WithBot.lean
index 63a1594ad82a06..8cbe3710d04299 100644
--- a/Mathlib/Order/WithBot.lean
+++ b/Mathlib/Order/WithBot.lean
@@ -956,7 +956,7 @@ lemma le_of_forall_lt_iff_le [LinearOrder α] [DenselyOrdered α] [NoMinOrder α
   | coe x =>
     rw [le_coe_iff]
     rintro y rfl
-    exact le_of_forall_le_of_dense (by exact_mod_cast h)
+    exact le_of_forall_gt_imp_ge_of_dense (by exact_mod_cast h)
 
 lemma ge_of_forall_gt_iff_ge [LinearOrder α] [DenselyOrdered α] [NoMinOrder α]
     {x y : WithBot α} : (∀ z : α, z < x → z ≤ y) ↔ x ≤ y := by
@@ -966,7 +966,7 @@ lemma ge_of_forall_gt_iff_ge [LinearOrder α] [DenselyOrdered α] [NoMinOrder α
   | coe y =>
     rw [le_coe_iff]
     rintro h x rfl
-    exact le_of_forall_ge_of_dense (by exact_mod_cast h)
+    exact le_of_forall_lt_imp_le_of_dense (by exact_mod_cast h)
 
 section LE
 
diff --git a/Mathlib/Probability/Density.lean b/Mathlib/Probability/Density.lean
index a89d87c31a7920..3a90248fdc4734 100644
--- a/Mathlib/Probability/Density.lean
+++ b/Mathlib/Probability/Density.lean
@@ -289,15 +289,14 @@ theorem integral_mul_eq_integral [HasPDF X ℙ] : ∫ x, x * (pdf X ℙ volume x
     _ = _ := integral_pdf_smul measurable_id.aestronglyMeasurable
 
 theorem hasFiniteIntegral_mul {f : ℝ → ℝ} {g : ℝ → ℝ≥0∞} (hg : pdf X ℙ =ᵐ[volume] g)
-    (hgi : ∫⁻ x, ‖f x‖₊ * g x ≠ ∞) :
+    (hgi : ∫⁻ x, ‖f x‖ₑ * g x ≠ ∞) :
     HasFiniteIntegral fun x => f x * (pdf X ℙ volume x).toReal := by
-  rw [hasFiniteIntegral_iff_nnnorm]
-  have : (fun x => ↑‖f x‖₊ * g x) =ᵐ[volume] fun x => ‖f x * (pdf X ℙ volume x).toReal‖₊ := by
-    refine ae_eq_trans (Filter.EventuallyEq.mul (ae_eq_refl fun x => (‖f x‖₊ : ℝ≥0∞))
-      (ae_eq_trans hg.symm ofReal_toReal_ae_eq.symm)) ?_
-    simp_rw [← smul_eq_mul, nnnorm_smul, ENNReal.coe_mul, smul_eq_mul]
-    refine Filter.EventuallyEq.mul (ae_eq_refl _) ?_
-    simp only [Real.ennnorm_eq_ofReal ENNReal.toReal_nonneg, ae_eq_refl]
+  rw [hasFiniteIntegral_iff_enorm]
+  have : (fun x => ‖f x‖ₑ * g x) =ᵐ[volume] fun x => ‖f x * (pdf X ℙ volume x).toReal‖ₑ := by
+    refine ae_eq_trans ((ae_eq_refl _).mul (ae_eq_trans hg.symm ofReal_toReal_ae_eq.symm)) ?_
+    simp_rw [← smul_eq_mul, enorm_smul, smul_eq_mul]
+    refine .mul (ae_eq_refl _) ?_
+    simp only [Real.enorm_eq_ofReal ENNReal.toReal_nonneg, ae_eq_refl]
   rwa [lt_top_iff_ne_top, ← lintegral_congr_ae this]
 
 end Real
diff --git a/Mathlib/Probability/Distributions/Uniform.lean b/Mathlib/Probability/Distributions/Uniform.lean
index 3efd4efda464d5..4c50fb841441c2 100644
--- a/Mathlib/Probability/Distributions/Uniform.lean
+++ b/Mathlib/Probability/Distributions/Uniform.lean
@@ -153,11 +153,11 @@ theorem mul_pdf_integrable (hcs : IsCompact s) (huX : IsUniform X s ℙ) :
       (measurable_pdf X ℙ).aemeasurable.ennreal_toReal.aestronglyMeasurable
   refine hasFiniteIntegral_mul (pdf_eq hcs.measurableSet huX) ?_
   set ind := (volume s)⁻¹ • (1 : ℝ → ℝ≥0∞)
-  have : ∀ x, ↑‖x‖₊ * s.indicator ind x = s.indicator (fun x => ‖x‖₊ * ind x) x := fun x =>
+  have : ∀ x, ‖x‖ₑ * s.indicator ind x = s.indicator (fun x => ‖x‖ₑ * ind x) x := fun x =>
     (s.indicator_mul_right (fun x => ↑‖x‖₊) ind).symm
   simp only [ind, this, lintegral_indicator hcs.measurableSet, mul_one, Algebra.id.smul_eq_mul,
     Pi.one_apply, Pi.smul_apply]
-  rw [lintegral_mul_const _ measurable_nnnorm.coe_nnreal_ennreal]
+  rw [lintegral_mul_const _ measurable_enorm]
   exact ENNReal.mul_ne_top (setLIntegral_lt_top_of_isCompact hnt.2 hcs continuous_nnnorm).ne
     (ENNReal.inv_lt_top.2 (pos_iff_ne_zero.mpr hnt.1)).ne
 
@@ -209,12 +209,13 @@ namespace PMF
 
 variable {α : Type*}
 
-open scoped Classical NNReal ENNReal
+open scoped NNReal ENNReal
 
 section UniformOfFinset
 
 /-- Uniform distribution taking the same non-zero probability on the nonempty finset `s` -/
 def uniformOfFinset (s : Finset α) (hs : s.Nonempty) : PMF α := by
+  classical
   refine ofFinset (fun a => if a ∈ s then s.card⁻¹ else 0) s ?_ ?_
   · simp only [Finset.sum_ite_mem, Finset.inter_self, Finset.sum_const, nsmul_eq_mul]
     have : (s.card : ℝ≥0∞) ≠ 0 := by
@@ -225,6 +226,7 @@ def uniformOfFinset (s : Finset α) (hs : s.Nonempty) : PMF α := by
 
 variable {s : Finset α} (hs : s.Nonempty) {a : α}
 
+open scoped Classical in
 @[simp]
 theorem uniformOfFinset_apply (a : α) :
     uniformOfFinset s hs a = if a ∈ s then (s.card : ℝ≥0∞)⁻¹ else 0 :=
@@ -249,6 +251,7 @@ section Measure
 
 variable (t : Set α)
 
+open scoped Classical in
 @[simp]
 theorem toOuterMeasure_uniformOfFinset_apply :
     (uniformOfFinset s hs).toOuterMeasure t = (s.filter (· ∈ t)).card / s.card :=
@@ -266,6 +269,7 @@ theorem toOuterMeasure_uniformOfFinset_apply :
     _ = (s.filter (· ∈ t)).card / s.card := by
         simp only [div_eq_mul_inv, Finset.sum_const, nsmul_eq_mul]
 
+open scoped Classical in
 @[simp]
 theorem toMeasure_uniformOfFinset_apply [MeasurableSpace α] (ht : MeasurableSet t) :
     (uniformOfFinset s hs).toMeasure t = (s.filter (· ∈ t)).card / s.card :=
@@ -298,11 +302,13 @@ section Measure
 
 variable (s : Set α)
 
+open scoped Classical in
 theorem toOuterMeasure_uniformOfFintype_apply :
     (uniformOfFintype α).toOuterMeasure s = Fintype.card s / Fintype.card α := by
   rw [uniformOfFintype, toOuterMeasure_uniformOfFinset_apply,Fintype.card_ofFinset]
   rfl
 
+open scoped Classical in
 theorem toMeasure_uniformOfFintype_apply [MeasurableSpace α] (hs : MeasurableSet s) :
     (uniformOfFintype α).toMeasure s = Fintype.card s / Fintype.card α := by
   simp [uniformOfFintype, hs]
@@ -313,6 +319,7 @@ end UniformOfFintype
 
 section OfMultiset
 
+open scoped Classical in
 /-- Given a non-empty multiset `s` we construct the `PMF` which sends `a` to the fraction of
   elements in `s` that are `a`. -/
 def ofMultiset (s : Multiset α) (hs : s ≠ 0) : PMF α :=
@@ -334,14 +341,17 @@ def ofMultiset (s : Multiset α) (hs : s ≠ 0) : PMF α :=
 
 variable {s : Multiset α} (hs : s ≠ 0)
 
+open scoped Classical in
 @[simp]
 theorem ofMultiset_apply (a : α) : ofMultiset s hs a = s.count a / (Multiset.card s) :=
   rfl
 
+open scoped Classical in
 @[simp]
 theorem support_ofMultiset : (ofMultiset s hs).support = s.toFinset :=
   Set.ext (by simp [mem_support_iff, hs])
 
+open scoped Classical in
 theorem mem_support_ofMultiset_iff (a : α) : a ∈ (ofMultiset s hs).support ↔ a ∈ s.toFinset := by
   simp
 
@@ -353,6 +363,7 @@ section Measure
 
 variable (t : Set α)
 
+open scoped Classical in
 @[simp]
 theorem toOuterMeasure_ofMultiset_apply :
     (ofMultiset s hs).toOuterMeasure t =
@@ -361,6 +372,7 @@ theorem toOuterMeasure_ofMultiset_apply :
   refine tsum_congr fun x => ?_
   by_cases hx : x ∈ t <;> simp [Set.indicator, hx, div_eq_mul_inv]
 
+open scoped Classical in
 @[simp]
 theorem toMeasure_ofMultiset_apply [MeasurableSpace α] (ht : MeasurableSet t) :
     (ofMultiset s hs).toMeasure t = (∑' x, (s.filter (· ∈ t)).count x : ℝ≥0∞) / (Multiset.card s) :=
diff --git a/Mathlib/Probability/Independence/Integrable.lean b/Mathlib/Probability/Independence/Integrable.lean
index c33b372d816a2c..0d06f7f91d98f2 100644
--- a/Mathlib/Probability/Independence/Integrable.lean
+++ b/Mathlib/Probability/Independence/Integrable.lean
@@ -3,7 +3,7 @@ Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathlib.MeasureTheory.Function.L1Space
+import Mathlib.MeasureTheory.Function.L1Space.Integrable
 import Mathlib.Probability.Independence.Basic
 
 /-!
diff --git a/Mathlib/Probability/Integration.lean b/Mathlib/Probability/Integration.lean
index 77a6aa46b89e1e..7d987ea9c09355 100644
--- a/Mathlib/Probability/Integration.lean
+++ b/Mathlib/Probability/Integration.lean
@@ -153,19 +153,16 @@ theorem lintegral_prod_eq_prod_lintegral_of_indepFun {ι : Type*} [DecidableEq 
 theorem IndepFun.integrable_mul {β : Type*} [MeasurableSpace β] {X Y : Ω → β}
     [NormedDivisionRing β] [BorelSpace β] (hXY : IndepFun X Y μ) (hX : Integrable X μ)
     (hY : Integrable Y μ) : Integrable (X * Y) μ := by
-  let nX : Ω → ENNReal := fun a => ‖X a‖₊
-  let nY : Ω → ENNReal := fun a => ‖Y a‖₊
-  have hXY' : IndepFun (fun a => ‖X a‖₊) (fun a => ‖Y a‖₊) μ :=
-    hXY.comp measurable_nnnorm measurable_nnnorm
-  have hXY'' : IndepFun nX nY μ :=
-    hXY'.comp measurable_coe_nnreal_ennreal measurable_coe_nnreal_ennreal
-  have hnX : AEMeasurable nX μ := hX.1.aemeasurable.nnnorm.coe_nnreal_ennreal
-  have hnY : AEMeasurable nY μ := hY.1.aemeasurable.nnnorm.coe_nnreal_ennreal
+  let nX : Ω → ℝ≥0∞ := fun a => ‖X a‖ₑ
+  let nY : Ω → ℝ≥0∞ := fun a => ‖Y a‖ₑ
+  have hXY' : IndepFun nX nY μ := hXY.comp measurable_enorm measurable_enorm
+  have hnX : AEMeasurable nX μ := hX.1.aemeasurable.enorm
+  have hnY : AEMeasurable nY μ := hY.1.aemeasurable.enorm
   have hmul : ∫⁻ a, nX a * nY a ∂μ = (∫⁻ a, nX a ∂μ) * ∫⁻ a, nY a ∂μ :=
-    lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun' hnX hnY hXY''
+    lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun' hnX hnY hXY'
   refine ⟨hX.1.mul hY.1, ?_⟩
   simp only [nX, nY] at hmul
-  simp_rw [hasFiniteIntegral_iff_nnnorm, Pi.mul_apply, nnnorm_mul, ENNReal.coe_mul, hmul]
+  simp_rw [hasFiniteIntegral_iff_enorm, Pi.mul_apply, enorm_mul, hmul]
   exact ENNReal.mul_lt_top hX.2 hY.2
 
 /-- If the product of two independent real-valued random variables is integrable and
@@ -175,18 +172,16 @@ theorem IndepFun.integrable_left_of_integrable_mul {β : Type*} [MeasurableSpace
     (hX : AEStronglyMeasurable X μ) (hY : AEStronglyMeasurable Y μ) (h'Y : ¬Y =ᵐ[μ] 0) :
     Integrable X μ := by
   refine ⟨hX, ?_⟩
-  have I : (∫⁻ ω, ‖Y ω‖₊ ∂μ) ≠ 0 := fun H ↦ by
-    have I : (fun ω => ‖Y ω‖₊ : Ω → ℝ≥0∞) =ᵐ[μ] 0 := (lintegral_eq_zero_iff' hY.ennnorm).1 H
+  have I : (∫⁻ ω, ‖Y ω‖ₑ ∂μ) ≠ 0 := fun H ↦ by
+    have I : (fun ω => ‖Y ω‖ₑ : Ω → ℝ≥0∞) =ᵐ[μ] 0 := (lintegral_eq_zero_iff' hY.enorm).1 H
     apply h'Y
     filter_upwards [I] with ω hω
     simpa using hω
-  refine hasFiniteIntegral_iff_nnnorm.mpr <| lt_top_iff_ne_top.2 fun H => ?_
-  have J : IndepFun (fun ω => ‖X ω‖₊ : Ω → ℝ≥0∞) (fun ω => ‖Y ω‖₊ : Ω → ℝ≥0∞) μ := by
-    have M : Measurable fun x : β => (‖x‖₊ : ℝ≥0∞) := measurable_nnnorm.coe_nnreal_ennreal
-    apply IndepFun.comp hXY M M
-  have A : (∫⁻ ω, ‖X ω * Y ω‖₊ ∂μ) < ∞ := h'XY.2
-  simp only [nnnorm_mul, ENNReal.coe_mul] at A
-  rw [lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun'' hX.ennnorm hY.ennnorm J, H] at A
+  refine hasFiniteIntegral_iff_enorm.mpr <| lt_top_iff_ne_top.2 fun H => ?_
+  have J : IndepFun (‖X ·‖ₑ) (‖Y ·‖ₑ) μ := hXY.comp measurable_enorm measurable_enorm
+  have A : ∫⁻ ω, ‖X ω * Y ω‖ₑ ∂μ < ∞ := h'XY.2
+  simp only [enorm_mul, ENNReal.coe_mul] at A
+  rw [lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun'' hX.enorm hY.enorm J, H] at A
   simp only [ENNReal.top_mul I, lt_self_iff_false] at A
 
 /-- If the product of two independent real-valued random variables is integrable and the
@@ -196,19 +191,17 @@ theorem IndepFun.integrable_right_of_integrable_mul {β : Type*} [MeasurableSpac
     (hX : AEStronglyMeasurable X μ) (hY : AEStronglyMeasurable Y μ) (h'X : ¬X =ᵐ[μ] 0) :
     Integrable Y μ := by
   refine ⟨hY, ?_⟩
-  have I : (∫⁻ ω, ‖X ω‖₊ ∂μ) ≠ 0 := fun H ↦ by
-    have I : (fun ω => ‖X ω‖₊ : Ω → ℝ≥0∞) =ᵐ[μ] 0 := (lintegral_eq_zero_iff' hX.ennnorm).1 H
+  have I : ∫⁻ ω, ‖X ω‖ₑ ∂μ ≠ 0 := fun H ↦ by
+    have I : ((‖X ·‖ₑ) : Ω → ℝ≥0∞) =ᵐ[μ] 0 := (lintegral_eq_zero_iff' hX.enorm).1 H
     apply h'X
     filter_upwards [I] with ω hω
     simpa using hω
   refine lt_top_iff_ne_top.2 fun H => ?_
-  have J : IndepFun (fun ω => ‖X ω‖₊ : Ω → ℝ≥0∞) (fun ω => ‖Y ω‖₊ : Ω → ℝ≥0∞) μ := by
-    have M : Measurable fun x : β => (‖x‖₊ : ℝ≥0∞) := measurable_nnnorm.coe_nnreal_ennreal
-    apply IndepFun.comp hXY M M
-  have A : (∫⁻ ω, ‖X ω * Y ω‖₊ ∂μ) < ∞ := h'XY.2
-  simp only [nnnorm_mul, ENNReal.coe_mul] at A
-  simp_rw [enorm_eq_nnnorm] at H
-  rw [lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun'' hX.ennnorm hY.ennnorm J, H] at A
+  have J : IndepFun (fun ω => ‖X ω‖ₑ : Ω → ℝ≥0∞) (fun ω => ‖Y ω‖ₑ : Ω → ℝ≥0∞) μ :=
+    IndepFun.comp hXY measurable_enorm measurable_enorm
+  have A : ∫⁻ ω, ‖X ω * Y ω‖ₑ ∂μ < ∞ := h'XY.2
+  simp only [enorm_mul, ENNReal.coe_mul] at A
+  rw [lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun'' hX.enorm hY.enorm J, H] at A
   simp only [ENNReal.mul_top I, lt_self_iff_false] at A
 
 /-- The (Bochner) integral of the product of two independent, nonnegative random
diff --git a/Mathlib/Probability/Kernel/Composition/Basic.lean b/Mathlib/Probability/Kernel/Composition/Basic.lean
index c5cdd7bb8284a8..3a6ccc33841e1d 100644
--- a/Mathlib/Probability/Kernel/Composition/Basic.lean
+++ b/Mathlib/Probability/Kernel/Composition/Basic.lean
@@ -158,8 +158,7 @@ theorem measurable_compProdFun (κ : Kernel α β) [IsSFiniteKernel κ] (η : Ke
 @[deprecated (since := "2024-08-30")]
 alias measurable_compProdFun_of_finite := measurable_compProdFun
 
-open scoped Classical
-
+open scoped Classical in
 /-- Composition-Product of kernels. For s-finite kernels, it satisfies
 `∫⁻ bc, f bc ∂(compProd κ η a) = ∫⁻ b, ∫⁻ c, f (b, c) ∂(η (a, b)) ∂(κ a)`
 (see `ProbabilityTheory.Kernel.lintegral_compProd`).
@@ -259,6 +258,7 @@ lemma compProd_preimage_fst {s : Set β} (hs : MeasurableSet s) (κ : Kernel α
 lemma compProd_deterministic_apply [MeasurableSingletonClass γ] {f : α × β → γ} (hf : Measurable f)
     {s : Set (β × γ)} (hs : MeasurableSet s) (κ : Kernel α β) [IsSFiniteKernel κ] (x : α) :
     (κ ⊗ₖ deterministic f hf) x s = κ x {b | (b, f (x, b)) ∈ s} := by
+  classical
   simp only [deterministic_apply, measurableSet_setOf, Set.mem_setOf_eq, Measure.dirac_apply,
     Set.mem_setOf_eq, Set.indicator_apply, Pi.one_apply, compProd_apply hs]
   let t := {b | (b, f (x, b)) ∈ s}
diff --git a/Mathlib/Probability/Kernel/Composition/CompNotation.lean b/Mathlib/Probability/Kernel/Composition/CompNotation.lean
new file mode 100644
index 00000000000000..a206f6de862820
--- /dev/null
+++ b/Mathlib/Probability/Kernel/Composition/CompNotation.lean
@@ -0,0 +1,56 @@
+/-
+Copyright (c) 2025 Rémy Degenne. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Rémy Degenne, Lorenzo Luccioli
+-/
+import Mathlib.Probability.Kernel.Basic
+
+/-!
+# Notation for the compostition of a measure and a kernel
+
+This operation, for which we introduce the notation `∘ₘ`, takes `μ : Measure α` and
+`κ : Kernel α β` and creates `κ ∘ₘ μ : Measure β`. The integral of a function against `κ ∘ₘ μ` is
+`∫⁻ x, f x ∂(κ ∘ₘ μ) = ∫⁻ a, ∫⁻ b, f b ∂(κ a) ∂μ`.
+
+This file does not define composition but only introduces notation for
+`MeasureTheory.Measure.bind μ κ`.
+
+## Notations
+
+* `κ ∘ₘ μ = MeasureTheory.Measure.bind μ κ`, for `κ` a kernel.
+-/
+
+/- This file is only for lemmas that are direct specializations of `Measure.bind` to kernels,
+anything more involved shoud go elsewhere (for example the `MeasureComp` file). -/
+assert_not_exists ProbabilityTheory.Kernel.compProd
+
+open ProbabilityTheory
+
+namespace MeasureTheory.Measure
+
+variable {α β : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
+  {μ : Measure α} {κ : Kernel α β}
+
+/-- Composition of a measure and a kernel.
+
+Notation for `MeasureTheory.Measure.bind` -/
+scoped[ProbabilityTheory] notation3 κ " ∘ₘ " μ:100 => MeasureTheory.Measure.bind μ κ
+
+@[simp]
+lemma comp_apply_univ [IsMarkovKernel κ] : (κ ∘ₘ μ) Set.univ = μ Set.univ := by
+  simp [bind_apply .univ κ.measurable]
+
+lemma deterministic_comp_eq_map {f : α → β} (hf : Measurable f) :
+    Kernel.deterministic f hf ∘ₘ μ = μ.map f :=
+  Measure.bind_dirac_eq_map μ hf
+
+@[simp]
+lemma id_comp : Kernel.id ∘ₘ μ = μ := by rw [Kernel.id, deterministic_comp_eq_map, Measure.map_id]
+
+lemma swap_comp {μ : Measure (α × β)} : (Kernel.swap α β) ∘ₘ μ = μ.map Prod.swap :=
+  deterministic_comp_eq_map measurable_swap
+
+@[simp]
+lemma const_comp {ν : Measure β} : (Kernel.const α ν) ∘ₘ μ = μ Set.univ • ν := μ.bind_const
+
+end MeasureTheory.Measure
diff --git a/Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean b/Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean
index a9ab89e8cf8f4d..3b0cddf917b806 100644
--- a/Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean
+++ b/Mathlib/Probability/Kernel/Composition/IntegralCompProd.lean
@@ -45,7 +45,7 @@ namespace ProbabilityTheory
 theorem hasFiniteIntegral_prod_mk_left (a : α) {s : Set (β × γ)} (h2s : (κ ⊗ₖ η) a s ≠ ∞) :
     HasFiniteIntegral (fun b => (η (a, b) (Prod.mk b ⁻¹' s)).toReal) (κ a) := by
   let t := toMeasurable ((κ ⊗ₖ η) a) s
-  simp_rw [hasFiniteIntegral_iff_nnnorm, ennnorm_eq_ofReal toReal_nonneg]
+  simp_rw [hasFiniteIntegral_iff_enorm, enorm_eq_ofReal toReal_nonneg]
   calc
     ∫⁻ b, ENNReal.ofReal (η (a, b) (Prod.mk b ⁻¹' s)).toReal ∂κ a
     _ ≤ ∫⁻ b, η (a, b) (Prod.mk b ⁻¹' t) ∂κ a := by
@@ -82,19 +82,19 @@ theorem hasFiniteIntegral_compProd_iff ⦃f : β × γ → E⦄ (h1f : StronglyM
     HasFiniteIntegral f ((κ ⊗ₖ η) a) ↔
       (∀ᵐ x ∂κ a, HasFiniteIntegral (fun y => f (x, y)) (η (a, x))) ∧
         HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) := by
-  simp only [hasFiniteIntegral_iff_nnnorm]
-  rw [Kernel.lintegral_compProd _ _ _ h1f.ennnorm]
+  simp only [hasFiniteIntegral_iff_enorm]
+  rw [lintegral_compProd _ _ _ h1f.enorm]
   have : ∀ x, ∀ᵐ y ∂η (a, x), 0 ≤ ‖f (x, y)‖ := fun x => Eventually.of_forall fun y => norm_nonneg _
   simp_rw [integral_eq_lintegral_of_nonneg_ae (this _)
       (h1f.norm.comp_measurable measurable_prod_mk_left).aestronglyMeasurable,
-    ennnorm_eq_ofReal toReal_nonneg, ofReal_norm_eq_coe_nnnorm]
+    enorm_eq_ofReal toReal_nonneg, ofReal_norm_eq_enorm]
   have : ∀ {p q r : Prop} (_ : r → p), (r ↔ p ∧ q) ↔ p → (r ↔ q) := fun {p q r} h1 => by
     rw [← and_congr_right_iff, and_iff_right_of_imp h1]
   rw [this]
   · intro h2f; rw [lintegral_congr_ae]
     filter_upwards [h2f] with x hx
     rw [ofReal_toReal]; rw [← lt_top_iff_ne_top]; exact hx
-  · intro h2f; refine ae_lt_top ?_ h2f.ne; exact h1f.ennnorm.lintegral_kernel_prod_right''
+  · intro h2f; refine ae_lt_top ?_ h2f.ne; exact h1f.enorm.lintegral_kernel_prod_right''
 
 theorem hasFiniteIntegral_compProd_iff' ⦃f : β × γ → E⦄
     (h1f : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
@@ -200,21 +200,18 @@ theorem Kernel.continuous_integral_integral :
   refine
     tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_compProd
       (Eventually.of_forall fun h => (L1.integrable_coeFn h).integral_compProd) ?_
-  simp_rw [←
-    Kernel.lintegral_fn_integral_sub (fun x => (‖x‖₊ : ℝ≥0∞)) (L1.integrable_coeFn _)
-      (L1.integrable_coeFn g)]
+  simp_rw [← lintegral_fn_integral_sub (‖·‖ₑ) (L1.integrable_coeFn _) (L1.integrable_coeFn g)]
   apply tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _ (fun i => zero_le _) _
-  · exact fun i => ∫⁻ x, ∫⁻ y, ‖i (x, y) - g (x, y)‖₊ ∂η (a, x) ∂κ a
-  swap; · exact fun i => lintegral_mono fun x => ennnorm_integral_le_lintegral_ennnorm _
+  · exact fun i => ∫⁻ x, ∫⁻ y, ‖i (x, y) - g (x, y)‖ₑ ∂η (a, x) ∂κ a
+  swap; · exact fun i => lintegral_mono fun x => enorm_integral_le_lintegral_enorm _
   show
     Tendsto
-      (fun i : β × γ →₁[(κ ⊗ₖ η) a] E => ∫⁻ x, ∫⁻ y : γ, ‖i (x, y) - g (x, y)‖₊ ∂η (a, x) ∂κ a)
+      (fun i : β × γ →₁[(κ ⊗ₖ η) a] E => ∫⁻ x, ∫⁻ y : γ, ‖i (x, y) - g (x, y)‖ₑ ∂η (a, x) ∂κ a)
       (𝓝 g) (𝓝 0)
-  have : ∀ i : (MeasureTheory.Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))),
-      Measurable fun z => (‖i z - g z‖₊ : ℝ≥0∞) := fun i =>
-    ((Lp.stronglyMeasurable i).sub (Lp.stronglyMeasurable g)).ennnorm
-  simp_rw [← Kernel.lintegral_compProd _ _ _ (this _), ← L1.ofReal_norm_sub_eq_lintegral, ←
-    ofReal_zero]
+  have this (i : Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))) :
+      Measurable fun z => ‖i z - g z‖ₑ :=
+    ((Lp.stronglyMeasurable i).sub (Lp.stronglyMeasurable g)).enorm
+  simp_rw [← lintegral_compProd _ _ _ (this _), ← L1.ofReal_norm_sub_eq_lintegral, ← ofReal_zero]
   refine (continuous_ofReal.tendsto 0).comp ?_
   rw [← tendsto_iff_norm_sub_tendsto_zero]
   exact tendsto_id
diff --git a/Mathlib/Probability/Kernel/Composition/Lemmas.lean b/Mathlib/Probability/Kernel/Composition/Lemmas.lean
new file mode 100644
index 00000000000000..f4afde59594803
--- /dev/null
+++ b/Mathlib/Probability/Kernel/Composition/Lemmas.lean
@@ -0,0 +1,103 @@
+/-
+Copyright (c) 2025 Rémy Degenne. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Rémy Degenne, Lorenzo Luccioli
+-/
+import Mathlib.Probability.Kernel.Composition.MeasureComp
+import Mathlib.Probability.Kernel.Composition.ParallelComp
+
+/-!
+# Lemmas relating different ways to compose measures and kernels
+
+This file contains lemmas about the composition of measures and kernels that do not fit in any of
+the other files in this directory, because they involve several types of compositions/products.
+
+## Main statements
+
+* `parallelComp_comp_parallelComp`: `(η ∥ₖ η') ∘ₖ (κ ∥ₖ κ') = (η ∘ₖ κ) ∥ₖ (η' ∘ₖ κ')`
+
+-/
+
+open MeasureTheory ProbabilityTheory
+
+variable {α β γ δ : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
+  {mγ : MeasurableSpace γ} {mδ : MeasurableSpace δ}
+  {μ : Measure α} {ν : Measure β} {κ : Kernel α β}
+
+namespace MeasureTheory.Measure
+
+lemma compProd_eq_parallelComp_comp_copy_comp [SFinite μ] [IsSFiniteKernel κ] :
+    μ ⊗ₘ κ = (Kernel.id ∥ₖ κ) ∘ₘ Kernel.copy α ∘ₘ μ := by
+  rw [compProd_eq_comp_prod, ← Kernel.parallelComp_comp_copy, Measure.comp_assoc]
+
+lemma prod_comp_right [SFinite ν] {κ : Kernel β γ} [IsSFiniteKernel κ] :
+    μ.prod (κ ∘ₘ ν) = (Kernel.id ∥ₖ κ) ∘ₘ (μ.prod ν) := by
+  ext s hs
+  rw [Measure.prod_apply hs, Measure.bind_apply hs (Kernel.measurable _)]
+  simp_rw [Measure.bind_apply (measurable_prod_mk_left hs) (Kernel.measurable _)]
+  rw [MeasureTheory.lintegral_prod]
+  swap; · exact (Kernel.measurable_coe _ hs).aemeasurable
+  congr with a
+  congr with b
+  rw [Kernel.parallelComp_apply, Kernel.id_apply, Measure.prod_apply hs, lintegral_dirac']
+  exact measurable_measure_prod_mk_left hs
+
+lemma prod_comp_left [SFinite μ] [SFinite ν] {κ : Kernel α γ} [IsSFiniteKernel κ] :
+    (κ ∘ₘ μ).prod ν = (κ ∥ₖ Kernel.id) ∘ₘ (μ.prod ν) := by
+  have h1 : (κ ∘ₘ μ).prod ν = (ν.prod (κ ∘ₘ μ)).map Prod.swap := by rw [Measure.prod_swap]
+  have h2 : (κ ∥ₖ Kernel.id) ∘ₘ (μ.prod ν) = ((Kernel.id ∥ₖ κ) ∘ₘ (ν.prod μ)).map Prod.swap := by
+    calc (κ ∥ₖ Kernel.id) ∘ₘ (μ.prod ν)
+    _ = (κ ∥ₖ Kernel.id) ∘ₘ ((ν.prod μ).map Prod.swap) := by rw [Measure.prod_swap]
+    _ = (κ ∥ₖ Kernel.id) ∘ₘ ((Kernel.swap _ _) ∘ₘ (ν.prod μ)) := by
+      rw [Kernel.swap, Measure.deterministic_comp_eq_map]
+    _ = (Kernel.swap _ _) ∘ₘ ((Kernel.id ∥ₖ κ) ∘ₘ (ν.prod μ)) := by
+      rw [Measure.comp_assoc, Measure.comp_assoc, Kernel.swap_parallelComp]
+    _ = ((Kernel.id ∥ₖ κ) ∘ₘ (ν.prod μ)).map Prod.swap := by
+      rw [Kernel.swap, Measure.deterministic_comp_eq_map]
+  rw [← Measure.prod_comp_right, ← h1] at h2
+  exact h2.symm
+
+end MeasureTheory.Measure
+
+namespace ProbabilityTheory.Kernel
+
+variable {α' β' γ' : Type*} {mα' : MeasurableSpace α'} {mβ' : MeasurableSpace β'}
+  {mγ' : MeasurableSpace γ'}
+  [IsSFiniteKernel κ]
+
+lemma parallelComp_id_left_comp_parallelComp
+    {η : Kernel α' γ} [IsSFiniteKernel η] {ξ : Kernel γ δ} [IsSFiniteKernel ξ] :
+    (Kernel.id ∥ₖ ξ) ∘ₖ (κ ∥ₖ η) = κ ∥ₖ (ξ ∘ₖ η) := by
+  ext a
+  rw [parallelComp_apply, comp_apply, comp_apply, parallelComp_apply, Measure.prod_comp_right]
+
+lemma parallelComp_id_right_comp_parallelComp {η : Kernel α' γ} [IsSFiniteKernel η]
+    {ξ : Kernel γ δ} [IsSFiniteKernel ξ] :
+    (ξ ∥ₖ Kernel.id) ∘ₖ (η ∥ₖ κ) = (ξ ∘ₖ η) ∥ₖ κ := by
+  ext a
+  rw [parallelComp_apply, comp_apply, comp_apply, parallelComp_apply, Measure.prod_comp_left]
+
+lemma parallelComp_comp_parallelComp {η : Kernel β γ} [IsSFiniteKernel η]
+    {κ' : Kernel α' β'} [IsSFiniteKernel κ'] {η' : Kernel β' γ'} [IsSFiniteKernel η'] :
+    (η ∥ₖ η') ∘ₖ (κ ∥ₖ κ') = (η ∘ₖ κ) ∥ₖ (η' ∘ₖ κ') := by
+  rw [← parallelComp_id_left_comp_parallelComp, ← parallelComp_id_right_comp_parallelComp,
+    ← comp_assoc, parallelComp_id_left_comp_parallelComp, comp_id]
+
+lemma parallelComp_comp_prod {η : Kernel β γ} [IsSFiniteKernel η]
+    {κ' : Kernel α β'} [IsSFiniteKernel κ'] {η' : Kernel β' γ'} [IsSFiniteKernel η'] :
+    (η ∥ₖ η') ∘ₖ (κ ×ₖ κ') = (η ∘ₖ κ) ×ₖ (η' ∘ₖ κ') := by
+  rw [← parallelComp_comp_copy, ← comp_assoc, parallelComp_comp_parallelComp,
+    ← parallelComp_comp_copy]
+
+lemma parallelComp_comm {η : Kernel γ δ} [IsSFiniteKernel η] :
+    (Kernel.id ∥ₖ κ) ∘ₖ (η ∥ₖ Kernel.id) = (η ∥ₖ Kernel.id) ∘ₖ (Kernel.id ∥ₖ κ) := by
+  rw [parallelComp_id_left_comp_parallelComp, parallelComp_id_right_comp_parallelComp,
+    comp_id, comp_id]
+
+end ProbabilityTheory.Kernel
+
+lemma MeasureTheory.Measure.parallelComp_comp_compProd [SFinite μ]
+    [IsSFiniteKernel κ] {η : Kernel β γ} [IsSFiniteKernel η] :
+    (Kernel.id ∥ₖ η) ∘ₘ (μ ⊗ₘ κ) = μ ⊗ₘ (η ∘ₖ κ) := by
+  rw [Measure.compProd_eq_comp_prod, Measure.compProd_eq_comp_prod, Measure.comp_assoc,
+    Kernel.parallelComp_comp_prod, Kernel.id_comp]
diff --git a/Mathlib/Probability/Kernel/Composition/MeasureComp.lean b/Mathlib/Probability/Kernel/Composition/MeasureComp.lean
new file mode 100644
index 00000000000000..80b35782496b39
--- /dev/null
+++ b/Mathlib/Probability/Kernel/Composition/MeasureComp.lean
@@ -0,0 +1,112 @@
+/-
+Copyright (c) 2025 Rémy Degenne. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Rémy Degenne, Lorenzo Luccioli
+-/
+import Mathlib.Probability.Kernel.Composition.CompNotation
+import Mathlib.Probability.Kernel.Composition.MeasureCompProd
+
+/-!
+# Lemmas about the composition of a measure and a kernel
+
+Basic lemmas about the composition `κ ∘ₘ μ` of a kernel `κ` and a measure `μ`.
+
+-/
+
+open scoped ENNReal
+
+open ProbabilityTheory MeasureTheory
+
+namespace MeasureTheory.Measure
+
+variable {α β γ : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ}
+  {μ ν : Measure α} {κ η : Kernel α β}
+
+lemma comp_assoc {η : Kernel β γ} : η ∘ₘ (κ ∘ₘ μ) = (η ∘ₖ κ) ∘ₘ μ :=
+  Measure.bind_bind κ.measurable η.measurable
+
+@[simp]
+lemma snd_compProd (μ : Measure α) [SFinite μ] (κ : Kernel α β) [IsSFiniteKernel κ] :
+    (μ ⊗ₘ κ).snd = κ ∘ₘ μ := by
+  ext s hs
+  rw [bind_apply hs κ.measurable, snd_apply hs, compProd_apply]
+  · rfl
+  · exact measurable_snd hs
+
+instance [SFinite μ] [IsSFiniteKernel κ] : SFinite (κ ∘ₘ μ) := by
+  rw [← snd_compProd]; infer_instance
+
+instance [IsFiniteMeasure μ] [IsFiniteKernel κ] : IsFiniteMeasure (κ ∘ₘ μ) := by
+  rw [← snd_compProd]; infer_instance
+
+instance [IsProbabilityMeasure μ] [IsMarkovKernel κ] : IsProbabilityMeasure (κ ∘ₘ μ) := by
+  rw [← snd_compProd]; infer_instance
+
+instance [IsZeroOrProbabilityMeasure μ] [IsZeroOrMarkovKernel κ] :
+    IsZeroOrProbabilityMeasure (κ ∘ₘ μ) := by
+  rw [← snd_compProd]; infer_instance
+
+lemma map_comp (μ : Measure α) (κ : Kernel α β) {f : β → γ} (hf : Measurable f) :
+    (κ ∘ₘ μ).map f = (κ.map f) ∘ₘ μ := by
+  ext s hs
+  rw [Measure.map_apply hf hs, Measure.bind_apply (hf hs) κ.measurable,
+    Measure.bind_apply hs (Kernel.measurable _)]
+  simp_rw [Kernel.map_apply' _ hf _ hs]
+
+section CompProd
+
+lemma compProd_eq_comp_prod (μ : Measure α) [SFinite μ] (κ : Kernel α β) [IsSFiniteKernel κ] :
+    μ ⊗ₘ κ = (Kernel.id ×ₖ κ) ∘ₘ μ := by
+  rw [compProd, Kernel.compProd_prodMkLeft_eq_comp]
+  rfl
+
+lemma compProd_id_eq_copy_comp [SFinite μ] : μ ⊗ₘ Kernel.id = Kernel.copy α ∘ₘ μ := by
+  rw [compProd_id, Kernel.copy, deterministic_comp_eq_map]
+
+end CompProd
+
+section AddSMul
+
+@[simp]
+lemma comp_add [SFinite μ] [SFinite ν] [IsSFiniteKernel κ] :
+    κ ∘ₘ (μ + ν) = κ ∘ₘ μ + κ ∘ₘ ν := by
+  simp_rw [← snd_compProd, compProd_add_left]
+  simp
+
+lemma add_comp [SFinite μ] [IsSFiniteKernel κ] [IsSFiniteKernel η] :
+    (κ + η) ∘ₘ μ = κ ∘ₘ μ + η ∘ₘ μ := by
+  simp_rw [← snd_compProd, compProd_add_right]
+  simp
+
+/-- Same as `add_comp` except that it uses `⇑κ + ⇑η` instead of `⇑(κ + η)` in order to have
+a simp-normal form on the left of the equality. -/
+@[simp]
+lemma add_comp' [SFinite μ] [IsSFiniteKernel κ] [IsSFiniteKernel η] :
+    (⇑κ + ⇑η) ∘ₘ μ = κ ∘ₘ μ + η ∘ₘ μ := by rw [← Kernel.coe_add, add_comp]
+
+lemma comp_smul (a : ℝ≥0∞) : κ ∘ₘ (a • μ) = a • (κ ∘ₘ μ) := by
+  ext s hs
+  simp only [bind_apply hs κ.measurable, lintegral_smul_measure, smul_apply, smul_eq_mul]
+
+end AddSMul
+
+section AbsolutelyContinuous
+
+variable [SFinite μ] [SFinite ν] [IsSFiniteKernel κ] [IsSFiniteKernel η]
+
+lemma AbsolutelyContinuous.comp (hμν : μ ≪ ν) (hκη : ∀ᵐ a ∂μ, κ a ≪ η a) :
+    κ ∘ₘ μ ≪ η ∘ₘ ν := by
+  simp_rw [← snd_compProd, Measure.snd]
+  exact (hμν.compProd hκη).map measurable_snd
+
+lemma AbsolutelyContinuous.comp_right (hμν : μ ≪ ν) (κ : Kernel α γ) [IsSFiniteKernel κ]  :
+    κ ∘ₘ μ ≪ κ ∘ₘ ν :=
+  hμν.comp (ae_of_all μ fun _ _ a ↦ a)
+
+lemma AbsolutelyContinuous.comp_left (μ : Measure α) [SFinite μ] (hκη : ∀ᵐ a ∂μ, κ a ≪ η a) :
+    κ ∘ₘ μ ≪ η ∘ₘ μ :=
+  μ.absolutelyContinuous_refl.comp hκη
+
+end AbsolutelyContinuous
+
+end MeasureTheory.Measure
diff --git a/Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean b/Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean
index 64933b8111dc2f..d39c73a222f5cc 100644
--- a/Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean
+++ b/Mathlib/Probability/Kernel/Composition/MeasureCompProd.lean
@@ -81,6 +81,17 @@ lemma _root_.ProbabilityTheory.Kernel.compProd_apply_eq_compProd_sectR {γ : Typ
   simp_rw [Kernel.compProd_apply hs, compProd_apply hs, Kernel.sectR_apply]
   rfl
 
+lemma compProd_id [SFinite μ] : μ ⊗ₘ Kernel.id = μ.map (fun x ↦ (x, x)) := by
+  ext s hs
+  rw [compProd_apply hs, map_apply (measurable_id.prod measurable_id) hs]
+  have h_meas a : MeasurableSet (Prod.mk a ⁻¹' s) := measurable_prod_mk_left hs
+  simp_rw [Kernel.id_apply, dirac_apply' _ (h_meas _)]
+  calc ∫⁻ a, (Prod.mk a ⁻¹' s).indicator 1 a ∂μ
+  _ = ∫⁻ a, ((fun x ↦ (x, x)) ⁻¹' s).indicator 1 a ∂μ := rfl
+  _ = μ ((fun x ↦ (x, x)) ⁻¹' s) := by
+    rw [lintegral_indicator_one]
+    exact (measurable_id.prod measurable_id) hs
+
 lemma compProd_congr [IsSFiniteKernel κ] [IsSFiniteKernel η]
     (h : κ =ᵐ[μ] η) : μ ⊗ₘ κ = μ ⊗ₘ η := by
   by_cases hμ : SFinite μ
@@ -207,6 +218,17 @@ instance [IsFiniteMeasure μ] [IsFiniteKernel κ] : IsFiniteMeasure (μ ⊗ₘ 
 instance [IsProbabilityMeasure μ] [IsMarkovKernel κ] : IsProbabilityMeasure (μ ⊗ₘ κ) := by
   rw [compProd]; infer_instance
 
+instance [IsZeroOrProbabilityMeasure μ] [IsZeroOrMarkovKernel κ] :
+    IsZeroOrProbabilityMeasure (μ ⊗ₘ κ) := by
+  rw [compProd]
+  rcases eq_zero_or_isProbabilityMeasure μ with rfl | h
+  · simp only [Kernel.const_zero, Kernel.compProd_zero_left, Kernel.zero_apply]
+    infer_instance
+  rcases eq_zero_or_isMarkovKernel κ with rfl | hκ
+  · simp only [Kernel.prodMkLeft_zero, Kernel.compProd_zero_right, Kernel.zero_apply]
+    infer_instance
+  · infer_instance
+
 section AbsolutelyContinuous
 
 lemma AbsolutelyContinuous.compProd_left [SFinite ν] (hμν : μ ≪ ν) (κ : Kernel α β) :
diff --git a/Mathlib/Probability/Kernel/CondDistrib.lean b/Mathlib/Probability/Kernel/CondDistrib.lean
index 8868754b1be920..a0933088f0f3dd 100644
--- a/Mathlib/Probability/Kernel/CondDistrib.lean
+++ b/Mathlib/Probability/Kernel/CondDistrib.lean
@@ -321,7 +321,7 @@ theorem _root_.MeasureTheory.Integrable.comp_snd_map_prod_mk
   by_cases hX : AEMeasurable X μ
   · have hf := hf_int.1.comp_snd_map_prod_mk X (mΩ := mΩ) (mβ := mβ)
     refine ⟨hf, ?_⟩
-    rw [hasFiniteIntegral_iff_nnnorm, lintegral_map' hf.ennnorm (hX.prod_mk aemeasurable_id)]
+    rw [hasFiniteIntegral_iff_enorm, lintegral_map' hf.enorm (hX.prod_mk aemeasurable_id)]
     exact hf_int.2
   · rw [Measure.map_of_not_aemeasurable]
     · simp
diff --git a/Mathlib/Probability/Kernel/Disintegration/CondCDF.lean b/Mathlib/Probability/Kernel/Disintegration/CondCDF.lean
index 7a63f0c847a347..25bf36467a9843 100644
--- a/Mathlib/Probability/Kernel/Disintegration/CondCDF.lean
+++ b/Mathlib/Probability/Kernel/Disintegration/CondCDF.lean
@@ -198,7 +198,7 @@ lemma integrable_preCDF (ρ : Measure (α × ℝ)) [IsFiniteMeasure ρ] (x : ℚ
     Integrable (fun a ↦ (preCDF ρ x a).toReal) ρ.fst := by
   refine integrable_of_forall_fin_meas_le _ (measure_lt_top ρ.fst univ) ?_ fun t _ _ ↦ ?_
   · exact measurable_preCDF.ennreal_toReal.aestronglyMeasurable
-  · simp_rw [← ofReal_norm_eq_coe_nnnorm, Real.norm_of_nonneg ENNReal.toReal_nonneg]
+  · simp_rw [← ofReal_norm_eq_enorm, Real.norm_of_nonneg ENNReal.toReal_nonneg]
     rw [← lintegral_one]
     refine (setLIntegral_le_lintegral _ _).trans (lintegral_mono_ae ?_)
     filter_upwards [preCDF_le_one ρ] with a ha using ENNReal.ofReal_toReal_le.trans (ha _)
diff --git a/Mathlib/Probability/Kernel/MeasurableIntegral.lean b/Mathlib/Probability/Kernel/MeasurableIntegral.lean
index d3eb71066d3ede..f226af6adbb258 100644
--- a/Mathlib/Probability/Kernel/MeasurableIntegral.lean
+++ b/Mathlib/Probability/Kernel/MeasurableIntegral.lean
@@ -32,7 +32,7 @@ theorem ProbabilityTheory.measurableSet_kernel_integrable ⦃f : α → β → E
     (hf : StronglyMeasurable (uncurry f)) :
     MeasurableSet {x | Integrable (f x) (κ x)} := by
   simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and]
-  exact measurableSet_lt (Measurable.lintegral_kernel_prod_right hf.ennnorm) measurable_const
+  exact measurableSet_lt (Measurable.lintegral_kernel_prod_right hf.enorm) measurable_const
 
 open ProbabilityTheory.Kernel
 
diff --git a/Mathlib/Probability/Kernel/Proper.lean b/Mathlib/Probability/Kernel/Proper.lean
index b7cb185ae7528f..724d6cff74ea3a 100644
--- a/Mathlib/Probability/Kernel/Proper.lean
+++ b/Mathlib/Probability/Kernel/Proper.lean
@@ -3,7 +3,7 @@ Copyright (c) 2024 Yaël Dillies, Kalle Kytölä, Kin Yau James Wong. All rights
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies, Kalle Kytölä, Kin Yau James Wong
 -/
-import Mathlib.Probability.Kernel.Basic
+import Mathlib.Probability.Kernel.Composition.CompNotation
 
 /-!
 # Proper kernels
@@ -66,14 +66,17 @@ alias ⟨IsProper.restrict_eq_indicator_smul, IsProper.of_restrict_eq_indicator_
 alias ⟨IsProper.inter_eq_indicator_mul, IsProper.of_inter_eq_indicator_mul⟩ :=
   isProper_iff_inter_eq_indicator_mul
 
-lemma IsProper.setLIntegral_eq_bind (hπ : IsProper π) (h𝓑𝓧 : 𝓑 ≤ 𝓧) {μ : Measure[𝓧] X}
+lemma IsProper.setLIntegral_eq_comp (hπ : IsProper π) (h𝓑𝓧 : 𝓑 ≤ 𝓧) {μ : Measure[𝓧] X}
     (hA : MeasurableSet[𝓧] A) (hB : MeasurableSet[𝓑] B) :
-    ∫⁻ a in B, π a A ∂μ = μ.bind π (A ∩ B) := by
+    ∫⁻ a in B, π a A ∂μ = (π ∘ₘ μ) (A ∩ B) := by
   rw [Measure.bind_apply (by measurability) (π.measurable.mono h𝓑𝓧 le_rfl)]
   simp only [hπ.inter_eq_indicator_mul h𝓑𝓧 hA hB, ← indicator_mul_const, Pi.one_apply, one_mul]
   rw [← lintegral_indicator (h𝓑𝓧 _ hB)]
   rfl
 
+@[deprecated (since := "2025-01-24")]
+alias IsProper.setLIntegral_eq_bind := IsProper.setLIntegral_eq_comp
+
 /-- Auxiliary lemma for `IsProper.lintegral_mul` and
 `IsProper.setLIntegral_eq_indicator_mul_lintegral`. -/
 private lemma IsProper.lintegral_indicator_mul_indicator (hπ : IsProper π) (h𝓑𝓧 : 𝓑 ≤ 𝓧)
diff --git a/Mathlib/Probability/Martingale/Convergence.lean b/Mathlib/Probability/Martingale/Convergence.lean
index 78e152f66e7705..00ffa32b2454bd 100644
--- a/Mathlib/Probability/Martingale/Convergence.lean
+++ b/Mathlib/Probability/Martingale/Convergence.lean
@@ -154,7 +154,7 @@ theorem Submartingale.upcrossings_ae_lt_top' [IsFiniteMeasure μ] (hf : Submarti
   rw [mul_comm, ← ENNReal.le_div_iff_mul_le] at this
   · refine (lt_of_le_of_lt this (ENNReal.div_lt_top ?_ ?_)).ne
     · have hR' : ∀ n, ∫⁻ ω, ‖f n ω - a‖₊ ∂μ ≤ R + ‖a‖₊ * μ Set.univ := by
-        simp_rw [eLpNorm_one_eq_lintegral_nnnorm] at hbdd
+        simp_rw [eLpNorm_one_eq_lintegral_enorm] at hbdd
         intro n
         refine (lintegral_mono ?_ : ∫⁻ ω, ‖f n ω - a‖₊ ∂μ ≤ ∫⁻ ω, ‖f n ω‖₊ + ‖a‖₊ ∂μ).trans ?_
         · intro ω
diff --git a/Mathlib/Probability/Moments/Basic.lean b/Mathlib/Probability/Moments/Basic.lean
index ec9aaa1d554221..edcfe1a187ce9e 100644
--- a/Mathlib/Probability/Moments/Basic.lean
+++ b/Mathlib/Probability/Moments/Basic.lean
@@ -184,6 +184,12 @@ theorem mgf_pos [IsProbabilityMeasure μ] (h_int_X : Integrable (fun ω => exp (
     0 < mgf X μ t :=
   mgf_pos' (IsProbabilityMeasure.ne_zero μ) h_int_X
 
+lemma mgf_pos_iff [hμ : NeZero μ] :
+    0 < mgf X μ t ↔ Integrable (fun ω ↦ exp (t * X ω)) μ := by
+  refine ⟨fun h ↦ ?_, fun h ↦ mgf_pos' hμ.out h⟩
+  contrapose! h with h
+  simp [mgf_undef h]
+
 lemma exp_cgf_of_neZero [hμ : NeZero μ] (hX : Integrable (fun ω ↦ exp (t * X ω)) μ) :
     exp (cgf X μ t) = mgf X μ t := by rw [cgf, exp_log (mgf_pos' hμ.out hX)]
 
@@ -196,6 +202,11 @@ lemma mgf_id_map (hX : AEMeasurable X μ) : mgf id (μ.map X) = mgf X μ := by
   · rfl
   · exact (measurable_const_mul _).exp.aestronglyMeasurable
 
+lemma mgf_congr_identDistrib {Ω' : Type*} {mΩ' : MeasurableSpace Ω'} {μ' : Measure Ω'}
+    {Y : Ω' → ℝ} (h : IdentDistrib X Y μ μ') :
+    mgf X μ = mgf Y μ' := by
+  rw [← mgf_id_map h.aemeasurable_fst, ← mgf_id_map h.aemeasurable_snd, h.map_eq]
+
 theorem mgf_neg : mgf (-X) μ t = mgf X μ (-t) := by simp_rw [mgf, Pi.neg_apply, mul_neg, neg_mul]
 
 theorem cgf_neg : cgf (-X) μ t = cgf X μ (-t) := by simp_rw [cgf, mgf_neg]
diff --git a/Mathlib/Probability/Moments/ComplexMGF.lean b/Mathlib/Probability/Moments/ComplexMGF.lean
new file mode 100644
index 00000000000000..1849cec292373a
--- /dev/null
+++ b/Mathlib/Probability/Moments/ComplexMGF.lean
@@ -0,0 +1,298 @@
+/-
+Copyright (c) 2025 Rémy Degenne. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Rémy Degenne
+-/
+import Mathlib.Analysis.Analytic.IsolatedZeros
+import Mathlib.Analysis.Calculus.ParametricIntegral
+import Mathlib.Analysis.Complex.CauchyIntegral
+import Mathlib.Probability.Moments.Basic
+import Mathlib.Probability.Moments.IntegrableExpMul
+
+/-!
+# The complex-valued moment generating function
+
+The moment generating function (mgf) is `t : ℝ ↦ μ[fun ω ↦ rexp (t * X ω)]`. It can be extended to
+a complex function `z : ℂ ↦ μ[fun ω ↦ cexp (z * X ω)]`, which we call `complexMGF X μ`.
+That function is holomorphic on the vertical strip with base the interior of the interval
+of definition of the mgf.
+On the vertical line that goes through 0, `complexMGF X μ` is equal to the characteristic function.
+This allows us to link properties of the characteristic function and the mgf (mostly deducing
+properties of the mgf from those of the characteristic function).
+
+## Main definitions
+
+* `complexMGF X μ`: the function `z : ℂ ↦ μ[fun ω ↦ cexp (z * X ω)]`.
+
+## Main results
+
+* `complexMGF_ofReal`: for `x : ℝ`, `complexMGF X μ x = mgf X μ x`.
+
+* `hasDerivAt_complexMGF`: for all `z : ℂ` such that the real part `z.re` belongs to the interior
+  of the interval of definition of the mgf, `complexMGF X μ` is differentiable at `z`
+  with derivative `μ[X * exp (z * X)]`.
+* `differentiableOn_complexMGF`: `complexMGF X μ` is holomorphic on the vertical strip
+  `{z | z.re ∈ interior (integrableExpSet X μ)}`.
+* `analyticOn_complexMGF`: `complexMGF X μ` is analytic on the vertical strip
+  `{z | z.re ∈ interior (integrableExpSet X μ)}`.
+
+* `eqOn_complexMGF_of_mgf`: if two random variables have the same moment generating function,
+  then they have the same `complexMGF` on the vertical strip
+  `{z | z.re ∈ interior (integrableExpSet X μ)}`.
+  Once we know that equal `mgf` implies equal distributions, we will be able to show that
+  the `complexMGF` are equal everywhere, not only on the strip.
+  This lemma will be used in the proof of the equality of distributions.
+
+## TODO
+
+Once we have a definition for the characteristic function, we will be able to prove the following.
+
+* `x : ℝ ↦ complexMGF X μ (I * x)` is equal to the characteristic function of
+  the random variable `X`.
+* As a consequence, if two random variables have same `mgf`, then they have the same
+  characteristic function and the same distribution.
+
+-/
+
+
+open MeasureTheory Filter Finset Real Complex
+
+open scoped MeasureTheory ProbabilityTheory ENNReal NNReal Topology
+
+namespace ProbabilityTheory
+
+variable {Ω ι : Type*} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : Measure Ω} {t u v : ℝ} {z ε : ℂ}
+
+/-- Complex extension of the moment generating function. -/
+noncomputable
+def complexMGF (X : Ω → ℝ) (μ : Measure Ω) (z : ℂ) : ℂ := μ[fun ω ↦ cexp (z * X ω)]
+
+lemma complexMGF_undef (hX : AEMeasurable X μ) (h : ¬ Integrable (fun ω ↦ rexp (z.re * X ω)) μ) :
+    complexMGF X μ z = 0 := by
+  rw [complexMGF, integral_undef]
+  rw [← integrable_norm_iff (AEMeasurable.aestronglyMeasurable <| by fun_prop)]
+  simpa [Complex.norm_eq_abs, Complex.abs_exp] using h
+
+lemma complexMGF_id_map (hX : AEMeasurable X μ) : complexMGF id (μ.map X) = complexMGF X μ := by
+  ext t
+  rw [complexMGF, integral_map hX]
+  · rfl
+  · exact AEMeasurable.aestronglyMeasurable <| by fun_prop
+
+lemma complexMGF_congr_identDistrib {Ω' : Type*} {mΩ' : MeasurableSpace Ω'} {μ' : Measure Ω'}
+    {Y : Ω' → ℝ}  (h : IdentDistrib X Y μ μ') :
+    complexMGF X μ = complexMGF Y μ' := by
+  rw [← complexMGF_id_map h.aemeasurable_fst, ← complexMGF_id_map h.aemeasurable_snd, h.map_eq]
+
+lemma abs_complexMGF_le_mgf : abs (complexMGF X μ z) ≤ mgf X μ z.re := by
+  rw [complexMGF, ← re_add_im z]
+  simp_rw [add_mul, Complex.exp_add, re_add_im, ← Complex.norm_eq_abs]
+  calc ‖∫ ω, cexp (z.re * X ω) * cexp (z.im * I * X ω) ∂μ‖
+  _ ≤ ∫ ω, ‖cexp (z.re * X ω) * cexp (z.im * I * X ω)‖ ∂μ := norm_integral_le_integral_norm _
+  _ = ∫ ω, rexp (z.re * X ω) ∂μ := by simp [Complex.abs_exp]
+
+lemma complexMGF_ofReal (x : ℝ) : complexMGF X μ x = mgf X μ x := by
+  rw [complexMGF, mgf, ← integral_complex_ofReal]
+  norm_cast
+
+lemma re_complexMGF_ofReal (x : ℝ) : (complexMGF X μ x).re = mgf X μ x := by
+  simp [complexMGF_ofReal]
+
+lemma re_complexMGF_ofReal' : (fun x : ℝ ↦ (complexMGF X μ x).re) = mgf X μ := by
+  ext x
+  exact re_complexMGF_ofReal x
+
+section Analytic
+
+/-- For `z : ℂ` with `z.re ∈ interior (integrableExpSet X μ)`, the derivative of the function
+`z' ↦ μ[X ^ n * cexp (z' * X)]` at `z` is `μ[X ^ (n + 1) * cexp (z * X)]`. -/
+lemma hasDerivAt_integral_pow_mul_exp (hz : z.re ∈ interior (integrableExpSet X μ)) (n : ℕ) :
+    HasDerivAt (fun z ↦ μ[fun ω ↦ X ω ^ n * cexp (z * X ω)])
+        μ[fun ω ↦ X ω ^ (n + 1) * cexp (z * X ω)] z := by
+  have hX : AEMeasurable X μ := aemeasurable_of_mem_interior_integrableExpSet hz
+  have hz' := hz
+  rw [mem_interior_iff_mem_nhds, mem_nhds_iff_exists_Ioo_subset] at hz'
+  obtain ⟨l, u, hlu, h_subset⟩ := hz'
+  let t := ((z.re - l) ⊓ (u - z.re)) / 2
+  have h_pos : 0 < (z.re - l) ⊓ (u - z.re) := by simp [hlu.1, hlu.2]
+  have ht : 0 < t := half_pos h_pos
+  refine (hasDerivAt_integral_of_dominated_loc_of_deriv_le
+    (bound := fun ω ↦ |X ω| ^ (n + 1) * rexp (z.re * X ω + t/2 * |X ω|))
+    (F := fun z ω ↦ X ω ^ n * cexp (z * X ω))
+    (F' := fun z ω ↦ X ω ^ (n + 1) * cexp (z * X ω)) (half_pos ht) ?_ ?_ ?_ ?_ ?_ ?_).2
+  · exact .of_forall fun z ↦ AEMeasurable.aestronglyMeasurable (by fun_prop)
+  · exact integrable_pow_mul_cexp_of_re_mem_interior_integrableExpSet hz n
+  · exact AEMeasurable.aestronglyMeasurable (by fun_prop)
+  · refine ae_of_all _ fun ω ε hε ↦ ?_
+    simp only [norm_mul, norm_pow, norm_real, Real.norm_eq_abs, Complex.norm_eq_abs]
+    rw [Complex.abs_ofReal, Complex.abs_exp]
+    simp only [mul_re, ofReal_re, ofReal_im, mul_zero, sub_zero]
+    gcongr
+    have : ε = z + (ε - z) := by simp
+    rw [this, add_re, add_mul]
+    gcongr _ + ?_
+    refine (le_abs_self _).trans ?_
+    rw [abs_mul]
+    gcongr
+    refine (abs_re_le_abs _).trans ?_
+    simp only [Metric.mem_ball, dist_eq_norm, Complex.norm_eq_abs] at hε
+    exact hε.le
+  · refine integrable_pow_abs_mul_exp_add_of_integrable_exp_mul ?_ ?_ ?_ ?_ (t := t) (n + 1)
+    · exact h_subset (add_half_inf_sub_mem_Ioo hlu)
+    · exact h_subset (sub_half_inf_sub_mem_Ioo hlu)
+    · positivity
+    · exact lt_of_lt_of_le (by simp [ht]) (le_abs_self _)
+  · refine ae_of_all _ fun ω ε hε ↦ ?_
+    simp only
+    simp_rw [pow_succ, mul_assoc]
+    refine HasDerivAt.const_mul _ ?_
+    simp_rw [← smul_eq_mul, Complex.exp_eq_exp_ℂ]
+    convert hasDerivAt_exp_smul_const (X ω : ℂ) ε using 1
+    rw [smul_eq_mul, mul_comm]
+
+/-- For all `z : ℂ` with `z.re ∈ interior (integrableExpSet X μ)`,
+`complexMGF X μ` is differentiable at `z` with derivative `μ[X * exp (z * X)]`. -/
+theorem hasDerivAt_complexMGF (hz : z.re ∈ interior (integrableExpSet X μ)) :
+    HasDerivAt (complexMGF X μ) μ[fun ω ↦ X ω * cexp (z * X ω)] z := by
+  convert hasDerivAt_integral_pow_mul_exp hz 0
+  · simp [complexMGF]
+  · simp
+
+/-- `complexMGF X μ` is holomorphic on the vertical strip
+`{z | z.re ∈ interior (integrableExpSet X μ)}`. -/
+theorem differentiableOn_complexMGF :
+    DifferentiableOn ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet X μ)} := by
+  intro z hz
+  have h := hasDerivAt_complexMGF hz
+  rw [hasDerivAt_iff_hasFDerivAt] at h
+  exact h.hasFDerivWithinAt.differentiableWithinAt
+
+theorem analyticOnNhd_complexMGF :
+    AnalyticOnNhd ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet X μ)} :=
+  differentiableOn_complexMGF.analyticOnNhd (isOpen_interior.preimage Complex.continuous_re)
+
+/-- `complexMGF X μ` is analytic on the vertical strip
+  `{z | z.re ∈ interior (integrableExpSet X μ)}`. -/
+theorem analyticOn_complexMGF :
+    AnalyticOn ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet X μ)} :=
+  analyticOnNhd_complexMGF.analyticOn
+
+/-- `complexMGF X μ` is analytic at any point `z` with `z.re ∈ interior (integrableExpSet X μ)`. -/
+lemma analyticAt_complexMGF (hz : z.re ∈ interior (integrableExpSet X μ)) :
+    AnalyticAt ℂ (complexMGF X μ) z :=
+  analyticOnNhd_complexMGF z hz
+
+end Analytic
+
+section Deriv
+
+/-! ### Iterated derivatives of `complexMGF` -/
+
+lemma hasDerivAt_iteratedDeriv_complexMGF (hz : z.re ∈ interior (integrableExpSet X μ)) (n : ℕ) :
+    HasDerivAt (iteratedDeriv n (complexMGF X μ)) μ[fun ω ↦ X ω ^ (n + 1) * cexp (z * X ω)] z := by
+  induction n generalizing z with
+  | zero => simp [hasDerivAt_complexMGF hz]
+  | succ n hn =>
+    rw [iteratedDeriv_succ]
+    have : deriv (iteratedDeriv n (complexMGF X μ))
+        =ᶠ[𝓝 z] fun z ↦ μ[fun ω ↦ X ω ^ (n + 1) * cexp (z * X ω)] := by
+      have h_mem : ∀ᶠ y in 𝓝 z, y.re ∈ interior (integrableExpSet X μ) := by
+        refine IsOpen.eventually_mem ?_ hz
+        exact isOpen_interior.preimage Complex.continuous_re
+      filter_upwards [h_mem] with y hy using HasDerivAt.deriv (hn hy)
+    rw [EventuallyEq.hasDerivAt_iff this]
+    exact hasDerivAt_integral_pow_mul_exp hz (n + 1)
+
+/-- For `z : ℂ` with `z.re ∈ interior (integrableExpSet X μ)`, the n-th derivative of the function
+`complexMGF X μ` at `z` is `μ[X ^ n * cexp (z * X)]`. -/
+lemma iteratedDeriv_complexMGF (hz : z.re ∈ interior (integrableExpSet X μ)) (n : ℕ) :
+    iteratedDeriv n (complexMGF X μ) z = μ[fun ω ↦ X ω ^ n * cexp (z * X ω)] := by
+  induction n generalizing z with
+  | zero => simp [complexMGF]
+  | succ n hn =>
+    rw [iteratedDeriv_succ]
+    exact (hasDerivAt_iteratedDeriv_complexMGF hz n).deriv
+
+end Deriv
+
+section EqOfMGF
+
+/-! We prove that if two random variables have the same `mgf`, then
+they also have the same `complexMGF`.-/
+
+variable {Ω' : Type*} {mΩ' : MeasurableSpace Ω'} {Y : Ω' → ℝ} {μ' : Measure Ω'}
+
+/-- If two random variables have the same moment generating function then they have
+the same `integrableExpSet`. -/
+lemma integrableExpSet_eq_of_mgf' (hXY : mgf X μ = mgf Y μ') (hμμ' : μ = 0 ↔ μ' = 0) :
+    integrableExpSet X μ = integrableExpSet Y μ' := by
+  ext t
+  simp only [integrableExpSet, Set.mem_setOf_eq]
+  by_cases hμ : μ = 0
+  · simp [hμ, hμμ'.mp hμ]
+  have : NeZero μ := ⟨hμ⟩
+  have : NeZero μ' := ⟨(not_iff_not.mpr hμμ').mp hμ⟩
+  rw [← mgf_pos_iff, ← mgf_pos_iff, hXY]
+
+/-- If two random variables have the same moment generating function then they have
+the same `integrableExpSet`. -/
+lemma integrableExpSet_eq_of_mgf [IsProbabilityMeasure μ]
+    (hXY : mgf X μ = mgf Y μ') :
+    integrableExpSet X μ = integrableExpSet Y μ' := by
+  refine integrableExpSet_eq_of_mgf' hXY ?_
+  simp only [IsProbabilityMeasure.ne_zero, false_iff]
+  suffices mgf Y μ' 0 ≠ 0 by
+    intro h_contra
+    simp [h_contra] at this
+  rw [← hXY]
+  exact (mgf_pos (by simp)).ne'
+
+/-- If two random variables have the same moment generating function then they have
+the same `complexMGF` on the vertical strip `{z | z.re ∈ interior (integrableExpSet X μ)}`.
+
+TODO: once we know that equal `mgf` implies equal distributions, we will be able to show that
+the `complexMGF` are equal everywhere, not only on the strip.
+This lemma will be used in the proof of the equality of distributions. -/
+lemma eqOn_complexMGF_of_mgf' (hXY : mgf X μ = mgf Y μ') (hμμ' : μ = 0 ↔ μ' = 0) :
+    Set.EqOn (complexMGF X μ) (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet X μ)} := by
+  by_cases h_empty : interior (integrableExpSet X μ) = ∅
+  · simp [h_empty]
+  rw [← ne_eq, ← Set.nonempty_iff_ne_empty] at h_empty
+  obtain ⟨t, ht⟩ := h_empty
+  have hX : AnalyticOnNhd ℂ (complexMGF X μ) {z | z.re ∈ interior (integrableExpSet X μ)} :=
+    analyticOnNhd_complexMGF
+  have hY : AnalyticOnNhd ℂ (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet Y μ')} :=
+    analyticOnNhd_complexMGF
+  rw [integrableExpSet_eq_of_mgf' hXY hμμ'] at hX ht ⊢
+  refine AnalyticOnNhd.eqOn_of_preconnected_of_frequently_eq hX hY
+    (convex_integrableExpSet.interior.linear_preimage reLm).isPreconnected
+    (z₀ := (t : ℂ)) (by simp [ht]) ?_
+  have h_real : ∃ᶠ (x : ℝ) in 𝓝[≠] t, complexMGF X μ x = complexMGF Y μ' x := by
+    refine .of_forall fun y ↦ ?_
+    rw [complexMGF_ofReal, complexMGF_ofReal, hXY]
+  rw [frequently_iff_seq_forall] at h_real ⊢
+  obtain ⟨xs, hx_tendsto, hx_eq⟩ := h_real
+  refine ⟨fun n ↦ xs n, ?_, fun n ↦ ?_⟩
+  · rw [tendsto_nhdsWithin_iff] at hx_tendsto ⊢
+    constructor
+    · rw [tendsto_ofReal_iff]
+      exact hx_tendsto.1
+    · simpa using hx_tendsto.2
+  · simp [hx_eq]
+
+/-- If two random variables have the same moment generating function then they have
+the same `complexMGF` on the vertical strip `{z | z.re ∈ interior (integrableExpSet X μ)}`. -/
+lemma eqOn_complexMGF_of_mgf [IsProbabilityMeasure μ]
+    (hXY : mgf X μ = mgf Y μ') :
+    Set.EqOn (complexMGF X μ) (complexMGF Y μ') {z | z.re ∈ interior (integrableExpSet X μ)} := by
+  refine eqOn_complexMGF_of_mgf' hXY ?_
+  simp only [IsProbabilityMeasure.ne_zero, false_iff]
+  suffices mgf Y μ' 0 ≠ 0 by
+    intro h_contra
+    simp [h_contra] at this
+  rw [← hXY]
+  exact (mgf_pos (by simp)).ne'
+
+end EqOfMGF
+
+end ProbabilityTheory
diff --git a/Mathlib/Probability/Moments/IntegrableExpMul.lean b/Mathlib/Probability/Moments/IntegrableExpMul.lean
index f2135d63a157ed..74d8ee95ca9403 100644
--- a/Mathlib/Probability/Moments/IntegrableExpMul.lean
+++ b/Mathlib/Probability/Moments/IntegrableExpMul.lean
@@ -3,7 +3,7 @@ Copyright (c) 2024 Rémy Degenne. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Rémy Degenne
 -/
-import Mathlib.MeasureTheory.Function.L1Space
+import Mathlib.MeasureTheory.Function.L1Space.Integrable
 import Mathlib.MeasureTheory.Order.Group.Lattice
 
 /-!
@@ -539,6 +539,63 @@ lemma memℒp_of_mem_interior_integrableExpSet (h : 0 ∈ interior (integrableEx
   simp only [norm_eq_abs, ENNReal.coe_toReal]
   exact integrable_rpow_abs_of_mem_interior_integrableExpSet h p.2
 
+section Complex
+
+open Complex
+
+variable {z : ℂ}
+
+lemma integrable_cexp_mul_of_re_mem_integrableExpSet (hX : AEMeasurable X μ)
+    (hz : z.re ∈ integrableExpSet X μ) :
+    Integrable (fun ω ↦ cexp (z * X ω)) μ := by
+  rw [← integrable_norm_iff]
+  · simpa [Complex.norm_eq_abs, Complex.abs_exp] using hz
+  · exact AEMeasurable.aestronglyMeasurable (by fun_prop)
+
+lemma integrable_cexp_mul_of_re_mem_interior_integrableExpSet
+    (hz : z.re ∈ interior (integrableExpSet X μ)) :
+    Integrable (fun ω ↦ cexp (z * X ω)) μ :=
+  integrable_cexp_mul_of_re_mem_integrableExpSet
+    (aemeasurable_of_mem_interior_integrableExpSet hz) (interior_subset hz)
+
+lemma integrable_rpow_abs_mul_cexp_of_re_mem_interior_integrableExpSet
+    (hz : z.re ∈ interior (integrableExpSet X μ)) {p : ℝ} (hp : 0 ≤ p) :
+    Integrable (fun ω ↦ (|X ω| ^ p : ℝ) * cexp (z * X ω)) μ := by
+  have hX : AEMeasurable X μ := aemeasurable_of_mem_interior_integrableExpSet hz
+  rw [← integrable_norm_iff]
+  swap; · exact AEMeasurable.aestronglyMeasurable (by fun_prop)
+  simpa [abs_rpow_of_nonneg (abs_nonneg _), Complex.abs_exp]
+    using integrable_rpow_abs_mul_exp_of_mem_interior_integrableExpSet hz hp
+
+lemma integrable_pow_abs_mul_cexp_of_re_mem_interior_integrableExpSet
+    (hz : z.re ∈ interior (integrableExpSet X μ)) (n : ℕ) :
+    Integrable (fun ω ↦ |X ω| ^ n * cexp (z * X ω)) μ := by
+  convert integrable_rpow_abs_mul_cexp_of_re_mem_interior_integrableExpSet hz (Nat.cast_nonneg n)
+  simp
+
+lemma integrable_rpow_mul_cexp_of_re_mem_interior_integrableExpSet
+    (hz : z.re ∈ interior (integrableExpSet X μ)) {p : ℝ} (hp : 0 ≤ p) :
+    Integrable (fun ω ↦ (X ω ^ p : ℝ) * cexp (z * X ω)) μ := by
+  have hX : AEMeasurable X μ := aemeasurable_of_mem_interior_integrableExpSet hz
+  rw [← integrable_norm_iff]
+  swap; · exact AEMeasurable.aestronglyMeasurable (by fun_prop)
+  simp only [norm_mul, norm_real, Real.norm_eq_abs, Complex.norm_eq_abs, Complex.abs_exp, mul_re,
+    ofReal_re, ofReal_im, mul_zero, sub_zero, Complex.abs_ofReal]
+  refine (integrable_rpow_abs_mul_exp_of_mem_interior_integrableExpSet hz hp).mono ?_ ?_
+  · exact AEMeasurable.aestronglyMeasurable (by fun_prop)
+  refine ae_of_all _ fun ω ↦ ?_
+  simp only [norm_mul, Real.norm_eq_abs, abs_abs, Real.abs_exp]
+  gcongr
+  exact abs_rpow_le_abs_rpow _ _
+
+lemma integrable_pow_mul_cexp_of_re_mem_interior_integrableExpSet
+    (hz : z.re ∈ interior (integrableExpSet X μ)) (n : ℕ) :
+    Integrable (fun ω ↦ X ω ^ n * cexp (z * X ω)) μ := by
+  convert integrable_rpow_mul_cexp_of_re_mem_interior_integrableExpSet hz (Nat.cast_nonneg n)
+  simp
+
+end Complex
+
 end IntegrableExpSet
 
 end FiniteMoments
diff --git a/Mathlib/Probability/Moments/MGFAnalytic.lean b/Mathlib/Probability/Moments/MGFAnalytic.lean
new file mode 100644
index 00000000000000..57f5d8242d2e64
--- /dev/null
+++ b/Mathlib/Probability/Moments/MGFAnalytic.lean
@@ -0,0 +1,124 @@
+/-
+Copyright (c) 2025 Rémy Degenne. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Rémy Degenne
+-/
+import Mathlib.Probability.Moments.ComplexMGF
+import Mathlib.Analysis.SpecialFunctions.Complex.Analytic
+
+/-!
+# The moment generating function is analytic
+
+The moment generating function `mgf X μ` of a random variable `X` with respect to a measure `μ`
+is analytic on the interior of `integrableExpSet X μ`, the interval on which it is defined.
+
+## Main results
+
+* `analyticOn_mgf`: the moment generating function is analytic on the interior of the interval
+  on which it is defined.
+* `iteratedDeriv_mgf`: the n-th derivative of the mgf at `t` is `μ[X ^ n * exp (t * X)]`.
+
+-/
+
+
+open MeasureTheory Filter Finset Real
+
+open scoped MeasureTheory ProbabilityTheory ENNReal NNReal Topology Nat
+
+namespace ProbabilityTheory
+
+variable {Ω ι : Type*} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : Measure Ω} {t u v : ℝ}
+
+section Deriv
+
+/-- For `t : ℝ` with `t ∈ interior (integrableExpSet X μ)`, the derivative of the function
+`x ↦ μ[X ^ n * rexp (x * X)]` at `t` is `μ[X ^ (n + 1) * rexp (t * X)]`. -/
+lemma hasDerivAt_integral_pow_mul_exp_real (ht : t ∈ interior (integrableExpSet X μ)) (n : ℕ) :
+    HasDerivAt (fun t ↦ μ[fun ω ↦ X ω ^ n * rexp (t * X ω)])
+      μ[fun ω ↦ X ω ^ (n + 1) * rexp (t * X ω)] t := by
+  have h_re_of_mem n t (ht' : t ∈ interior (integrableExpSet X μ)) :
+      (∫ ω, X ω ^ n * Complex.exp (t * X ω) ∂μ).re = ∫ ω, X ω ^ n * rexp (t * X ω) ∂μ := by
+    rw [← RCLike.re_eq_complex_re, ← integral_re]
+    · norm_cast
+    · refine integrable_pow_mul_cexp_of_re_mem_interior_integrableExpSet ?_ n
+      simpa using ht'
+  have h_re n : ∀ᶠ t' : ℝ in 𝓝 t, (∫ ω, X ω ^ n * Complex.exp (t' * X ω) ∂μ).re
+      = ∫ ω, X ω ^ n * rexp (t' * X ω) ∂μ := by
+    filter_upwards [isOpen_interior.eventually_mem ht] with t ht' using h_re_of_mem n t ht'
+  rw [← EventuallyEq.hasDerivAt_iff (h_re _), ← h_re_of_mem _ t ht]
+  exact (hasDerivAt_integral_pow_mul_exp (by simp [ht]) n).real_of_complex
+
+/-- For `t ∈ interior (integrableExpSet X μ)`, the derivative of `mgf X μ` at `t` is
+`μ[X * exp (t * X)]`. -/
+lemma hasDerivAt_mgf (h : t ∈ interior (integrableExpSet X μ)) :
+    HasDerivAt (mgf X μ) (μ[fun ω ↦ X ω * exp (t * X ω)]) t := by
+  convert hasDerivAt_integral_pow_mul_exp_real h 0
+  · simp [mgf]
+  · simp
+
+lemma hasDerivAt_iteratedDeriv_mgf (ht : t ∈ interior (integrableExpSet X μ)) (n : ℕ) :
+    HasDerivAt (iteratedDeriv n (mgf X μ)) μ[fun ω ↦ X ω ^ (n + 1) * exp (t * X ω)] t := by
+  induction n generalizing t with
+  | zero => simp [hasDerivAt_mgf ht]
+  | succ n hn =>
+    rw [iteratedDeriv_succ]
+    have : deriv (iteratedDeriv n (mgf X μ))
+        =ᶠ[𝓝 t] fun t ↦ μ[fun ω ↦ X ω ^ (n + 1) * exp (t * X ω)] := by
+      have h_mem : ∀ᶠ y in 𝓝 t, y ∈ interior (integrableExpSet X μ) :=
+        isOpen_interior.eventually_mem ht
+      filter_upwards [h_mem] with y hy using HasDerivAt.deriv (hn hy)
+    rw [EventuallyEq.hasDerivAt_iff this]
+    exact hasDerivAt_integral_pow_mul_exp_real ht (n + 1)
+
+/-- For `t ∈ interior (integrableExpSet X μ)`, the n-th derivative of `mgf X μ` at `t` is
+`μ[X ^ n * exp (t * X)]`. -/
+lemma iteratedDeriv_mgf (ht : t ∈ interior (integrableExpSet X μ)) (n : ℕ) :
+    iteratedDeriv n (mgf X μ) t = μ[fun ω ↦ X ω ^ n * exp (t * X ω)] := by
+  induction n generalizing t with
+  | zero => simp [mgf]
+  | succ n hn =>
+    rw [iteratedDeriv_succ]
+    exact (hasDerivAt_iteratedDeriv_mgf ht n).deriv
+
+/-- The derivatives of the moment generating function at zero are the moments. -/
+lemma iteratedDeriv_mgf_zero (h : 0 ∈ interior (integrableExpSet X μ)) (n : ℕ) :
+    iteratedDeriv n (mgf X μ) 0 = μ[X ^ n] := by
+  simp [iteratedDeriv_mgf h n]
+
+/-- For `t ∈ interior (integrableExpSet X μ)`, the derivative of `mgf X μ` at `t` is
+`μ[X * exp (t * X)]`. -/
+lemma deriv_mgf (h : t ∈ interior (integrableExpSet X μ)) :
+    deriv (mgf X μ) t = μ[fun ω ↦ X ω * exp (t * X ω)] :=
+  (hasDerivAt_mgf h).deriv
+
+lemma deriv_mgf_zero (h : 0 ∈ interior (integrableExpSet X μ)) : deriv (mgf X μ) 0 = μ[X] := by
+  simp [deriv_mgf h]
+
+end Deriv
+
+section Analytic
+
+/-- The moment generating function is analytic at every `t ∈ interior (integrableExpSet X μ)`. -/
+lemma analyticAt_mgf (ht : t ∈ interior (integrableExpSet X μ)) :
+    AnalyticAt ℝ (mgf X μ) t := by
+  rw [← re_complexMGF_ofReal']
+  exact (analyticAt_complexMGF (by simp [ht])).re_ofReal
+
+lemma analyticOnNhd_mgf : AnalyticOnNhd ℝ (mgf X μ) (interior (integrableExpSet X μ)) :=
+  fun _ hx ↦ analyticAt_mgf hx
+
+/-- The moment generating function is analytic on the interior of the interval on which it is
+defined. -/
+lemma analyticOn_mgf : AnalyticOn ℝ (mgf X μ) (interior (integrableExpSet X μ)) :=
+  analyticOnNhd_mgf.analyticOn
+
+lemma hasFPowerSeriesAt_mgf (hv : v ∈ interior (integrableExpSet X μ)) :
+    HasFPowerSeriesAt (mgf X μ)
+      (FormalMultilinearSeries.ofScalars ℝ
+        (fun n ↦ (μ[fun ω ↦ X ω ^ n * exp (v * X ω)] : ℝ) / n !)) v := by
+  convert (analyticAt_mgf hv).hasFPowerSeriesAt
+  rw [iteratedDeriv_mgf hv]
+
+end Analytic
+
+end ProbabilityTheory
diff --git a/Mathlib/Probability/Process/Filtration.lean b/Mathlib/Probability/Process/Filtration.lean
index 82b1ace233b965..1c4556148126c7 100644
--- a/Mathlib/Probability/Process/Filtration.lean
+++ b/Mathlib/Probability/Process/Filtration.lean
@@ -33,7 +33,7 @@ filtration, stochastic process
 
 open Filter Order TopologicalSpace
 
-open scoped Classical MeasureTheory NNReal ENNReal Topology
+open scoped MeasureTheory NNReal ENNReal Topology
 
 namespace MeasureTheory
 
@@ -133,6 +133,7 @@ theorem sSup_def (s : Set (Filtration ι m)) (i : ι) :
     sSup s i = sSup ((fun f : Filtration ι m => f i) '' s) :=
   rfl
 
+open scoped Classical in
 noncomputable instance : InfSet (Filtration ι m) :=
   ⟨fun s =>
     { seq := fun i => if Set.Nonempty s then sInf ((fun f : Filtration ι m => f i) '' s) else m
@@ -151,6 +152,7 @@ noncomputable instance : InfSet (Filtration ι m) :=
         obtain ⟨f, hf_mem⟩ := h_nonempty
         exact le_trans (sInf_le ⟨f, hf_mem, rfl⟩) (f.le i) }⟩
 
+open scoped Classical in
 theorem sInf_def (s : Set (Filtration ι m)) (i : ι) :
     sInf s i = if Set.Nonempty s then sInf ((fun f : Filtration ι m => f i) '' s) else m :=
   rfl
@@ -295,6 +297,7 @@ section Limit
 variable {E : Type*} [Zero E] [TopologicalSpace E] {ℱ : Filtration ι m} {f : ι → Ω → E}
   {μ : Measure Ω}
 
+open scoped Classical in
 /-- Given a process `f` and a filtration `ℱ`, if `f` converges to some `g` almost everywhere and
 `g` is `⨆ n, ℱ n`-measurable, then `limitProcess f ℱ μ` chooses said `g`, else it returns 0.
 
diff --git a/Mathlib/Probability/Process/HittingTime.lean b/Mathlib/Probability/Process/HittingTime.lean
index 5ac75975d64832..7610ba2b87de1d 100644
--- a/Mathlib/Probability/Process/HittingTime.lean
+++ b/Mathlib/Probability/Process/HittingTime.lean
@@ -53,7 +53,6 @@ noncomputable def hitting [Preorder ι] [InfSet ι] (u : ι → Ω → β)
   fun x => if ∃ j ∈ Set.Icc n m, u j x ∈ s
     then sInf (Set.Icc n m ∩ {i : ι | u i x ∈ s}) else m
 
-#adaptation_note /-- nightly-2024-03-16: added to replace simp [hitting] -/
 open scoped Classical in
 theorem hitting_def [Preorder ι] [InfSet ι] (u : ι → Ω → β) (s : Set β) (n m : ι) :
     hitting u s n m =
diff --git a/Mathlib/Probability/Variance.lean b/Mathlib/Probability/Variance.lean
index 5c3c9f6c42be48..886ceef0e7ebe4 100644
--- a/Mathlib/Probability/Variance.lean
+++ b/Mathlib/Probability/Variance.lean
@@ -49,7 +49,7 @@ namespace ProbabilityTheory
 /-- The `ℝ≥0∞`-valued variance of a real-valued random variable defined as the Lebesgue integral of
 `(X - 𝔼[X])^2`. -/
 def evariance {Ω : Type*} {_ : MeasurableSpace Ω} (X : Ω → ℝ) (μ : Measure Ω) : ℝ≥0∞ :=
-  ∫⁻ ω, (‖X ω - μ[X]‖₊ : ℝ≥0∞) ^ 2 ∂μ
+  ∫⁻ ω, ‖X ω - μ[X]‖ₑ ^ 2 ∂μ
 
 /-- The `ℝ`-valued variance of a real-valued random variable defined by applying `ENNReal.toReal`
 to `evariance`. -/
@@ -82,7 +82,7 @@ variable {Ω : Type*} {m : MeasurableSpace Ω} {X : Ω → ℝ} {μ : Measure Ω
 theorem _root_.MeasureTheory.Memℒp.evariance_lt_top [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
     evariance X μ < ∞ := by
   have := ENNReal.pow_lt_top (hX.sub <| memℒp_const <| μ[X]).2 2
-  rw [eLpNorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, ← ENNReal.rpow_two] at this
+  rw [eLpNorm_eq_lintegral_rpow_enorm two_ne_zero ENNReal.two_ne_top, ← ENNReal.rpow_two] at this
   simp only [ENNReal.toReal_ofNat, Pi.sub_apply, ENNReal.one_toReal, one_div] at this
   rw [← ENNReal.rpow_mul, inv_mul_cancel₀ (two_ne_zero : (2 : ℝ) ≠ 0), ENNReal.rpow_one] at this
   simp_rw [ENNReal.rpow_two] at this
@@ -94,7 +94,7 @@ theorem evariance_eq_top [IsFiniteMeasure μ] (hXm : AEStronglyMeasurable X μ)
   rw [← Ne, ← lt_top_iff_ne_top] at h
   have : Memℒp (fun ω => X ω - μ[X]) 2 μ := by
     refine ⟨hXm.sub aestronglyMeasurable_const, ?_⟩
-    rw [eLpNorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top]
+    rw [eLpNorm_eq_lintegral_rpow_enorm two_ne_zero ENNReal.two_ne_top]
     simp only [ENNReal.toReal_ofNat, ENNReal.one_toReal, ENNReal.rpow_two, Ne]
     exact ENNReal.rpow_lt_top_of_nonneg (by linarith) h.ne
   refine hX ?_
@@ -117,11 +117,8 @@ theorem _root_.MeasureTheory.Memℒp.ofReal_variance_eq [IsFiniteMeasure μ] (hX
 theorem evariance_eq_lintegral_ofReal (X : Ω → ℝ) (μ : Measure Ω) :
     evariance X μ = ∫⁻ ω, ENNReal.ofReal ((X ω - μ[X]) ^ 2) ∂μ := by
   rw [evariance]
-  congr
-  ext1 ω
-  rw [pow_two, ← ENNReal.coe_mul, ← nnnorm_mul, ← pow_two]
-  congr
-  exact (Real.toNNReal_eq_nnnorm_of_nonneg <| sq_nonneg _).symm
+  congr with ω
+  rw [pow_two, ← enorm_mul, ← pow_two, Real.enorm_of_nonneg (sq_nonneg _)]
 
 theorem _root_.MeasureTheory.Memℒp.variance_eq_of_integral_eq_zero (hX : Memℒp X 2 μ)
     (hXint : μ[X] = 0) : variance X μ = μ[X ^ (2 : Nat)] := by
@@ -152,25 +149,16 @@ theorem evariance_eq_zero_iff (hX : AEMeasurable X μ) :
     evariance X μ = 0 ↔ X =ᵐ[μ] fun _ => μ[X] := by
   rw [evariance, lintegral_eq_zero_iff']
   constructor <;> intro hX <;> filter_upwards [hX] with ω hω
-  · simpa only [Pi.zero_apply, sq_eq_zero_iff, ENNReal.coe_eq_zero, nnnorm_eq_zero, sub_eq_zero]
-      using hω
-  · rw [hω]
-    simp
-  · exact (hX.sub_const _).ennnorm.pow_const _ -- TODO `measurability` and `fun_prop` fail
+  · simpa [sub_eq_zero] using hω
+  · simp [hω]
+  · exact (hX.sub_const _).enorm.pow_const _ -- TODO `measurability` and `fun_prop` fail
 
 theorem evariance_mul (c : ℝ) (X : Ω → ℝ) (μ : Measure Ω) :
     evariance (fun ω => c * X ω) μ = ENNReal.ofReal (c ^ 2) * evariance X μ := by
   rw [evariance, evariance, ← lintegral_const_mul' _ _ ENNReal.ofReal_lt_top.ne]
-  congr
-  ext1 ω
-  rw [ENNReal.ofReal, ← ENNReal.coe_pow, ← ENNReal.coe_pow, ← ENNReal.coe_mul]
-  congr
-  rw [← sq_abs, ← Real.rpow_two, Real.toNNReal_rpow_of_nonneg (abs_nonneg _), NNReal.rpow_two,
-    ← mul_pow, Real.toNNReal_mul_nnnorm _ (abs_nonneg _)]
-  conv_rhs => rw [← nnnorm_norm, norm_mul, norm_abs_eq_norm, ← norm_mul, nnnorm_norm, mul_sub]
-  congr
-  rw [mul_comm]
-  simp_rw [← smul_eq_mul, ← integral_smul_const, smul_eq_mul, mul_comm]
+  congr with ω
+  rw [integral_mul_left, ← mul_sub, enorm_mul, mul_pow, ← enorm_pow,
+    Real.enorm_of_nonneg (sq_nonneg _)]
 
 @[simp]
 theorem variance_zero (μ : Measure Ω) : variance 0 μ = 0 := by
@@ -228,20 +216,20 @@ theorem variance_le_expectation_sq [IsProbabilityMeasure μ] {X : Ω → ℝ}
   · exact (AEMeasurable.pow_const (hm.aemeasurable.sub_const _) _).aestronglyMeasurable
 
 theorem evariance_def' [IsProbabilityMeasure μ] {X : Ω → ℝ} (hX : AEStronglyMeasurable X μ) :
-    evariance X μ = (∫⁻ ω, (‖X ω‖₊ ^ 2 :) ∂μ) - ENNReal.ofReal (μ[X] ^ 2) := by
+    evariance X μ = (∫⁻ ω, ‖X ω‖ₑ ^ 2 ∂μ) - ENNReal.ofReal (μ[X] ^ 2) := by
   by_cases hℒ : Memℒp X 2 μ
   · rw [← hℒ.ofReal_variance_eq, variance_def' hℒ, ENNReal.ofReal_sub _ (sq_nonneg _)]
     congr
+    simp_rw [← enorm_pow, enorm]
     rw [lintegral_coe_eq_integral]
-    · congr 2 with ω
-      simp only [Pi.pow_apply, NNReal.coe_pow, coe_nnnorm, Real.norm_eq_abs, Even.pow_abs even_two]
-    · exact hℒ.abs.integrable_sq
+    · simp
+    · simpa using hℒ.abs.integrable_sq
   · symm
     rw [evariance_eq_top hX hℒ, ENNReal.sub_eq_top_iff]
     refine ⟨?_, ENNReal.ofReal_ne_top⟩
     rw [Memℒp, not_and] at hℒ
     specialize hℒ hX
-    simp only [eLpNorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, not_lt, top_le_iff,
+    simp only [eLpNorm_eq_lintegral_rpow_enorm two_ne_zero ENNReal.two_ne_top, not_lt, top_le_iff,
       ENNReal.toReal_ofNat, one_div, ENNReal.rpow_eq_top_iff, inv_lt_zero, inv_pos, and_true,
       or_iff_not_imp_left, not_and_or, zero_lt_two] at hℒ
     exact mod_cast hℒ fun _ => zero_le_two
@@ -255,7 +243,7 @@ theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AEStronglyMeasurable
   · congr
     simp only [Pi.sub_apply, ENNReal.coe_le_coe, ← Real.norm_eq_abs, ← coe_nnnorm,
       NNReal.coe_le_coe, ENNReal.ofReal_coe_nnreal]
-  · rw [eLpNorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top]
+  · rw [eLpNorm_eq_lintegral_rpow_enorm two_ne_zero ENNReal.two_ne_top]
     simp only [show ENNReal.ofNNReal (c ^ 2) = (ENNReal.ofNNReal c) ^ 2 by norm_cast,
       ENNReal.toReal_ofNat, one_div, Pi.sub_apply]
     rw [div_eq_mul_inv, ENNReal.inv_pow, mul_comm, ENNReal.rpow_two]
@@ -263,6 +251,7 @@ theorem meas_ge_le_evariance_div_sq {X : Ω → ℝ} (hX : AEStronglyMeasurable
     simp_rw [← ENNReal.rpow_mul, inv_mul_cancel₀ (two_ne_zero : (2 : ℝ) ≠ 0), ENNReal.rpow_two,
       ENNReal.rpow_one, evariance]
 
+
 /-- **Chebyshev's inequality**: one can control the deviation probability of a real random variable
 from its expectation in terms of the variance. -/
 theorem meas_ge_le_variance_div_sq [IsFiniteMeasure μ] {X : Ω → ℝ} (hX : Memℒp X 2 μ) {c : ℝ}
diff --git a/Mathlib/RepresentationTheory/Character.lean b/Mathlib/RepresentationTheory/Character.lean
index ada933887ae231..6dc6591865ab82 100644
--- a/Mathlib/RepresentationTheory/Character.lean
+++ b/Mathlib/RepresentationTheory/Character.lean
@@ -115,7 +115,7 @@ theorem char_orthonormal (V W : FDRep k G) [Simple V] [Simple W] :
     rw [char_iso (FDRep.dualTensorIsoLinHom W.ρ V)]
   -- The average over the group of the character of a representation equals the dimension of the
   -- space of invariants.
-  rw [average_char_eq_finrank_invariants, ← FDRep.endMulEquiv_comp_ρ (of _),
+  rw [average_char_eq_finrank_invariants, ← FDRep.endRingEquiv_comp_ρ (of _),
       FDRep.of_ρ (linHom W.ρ V.ρ)]
   -- The space of invariants of `Hom(W, V)` is the subspace of `G`-equivariant linear maps,
   -- `Hom_G(W, V)`.
diff --git a/Mathlib/RepresentationTheory/FDRep.lean b/Mathlib/RepresentationTheory/FDRep.lean
index dd5870c18de169..dae66e0f2aa478 100644
--- a/Mathlib/RepresentationTheory/FDRep.lean
+++ b/Mathlib/RepresentationTheory/FDRep.lean
@@ -82,15 +82,16 @@ instance (V W : FDRep k G) : FiniteDimensional k (V ⟶ W) :=
 
 /-- The monoid homomorphism corresponding to the action of `G` onto `V : FDRep k G`. -/
 def ρ (V : FDRep k G) : G →* V →ₗ[k] V :=
-  (ModuleCat.endMulEquiv _).toMonoidHom.comp (Action.ρ V)
+  (ModuleCat.endRingEquiv _).toMonoidHom.comp (Action.ρ V)
 
 @[simp]
-lemma endMulEquiv_symm_comp_ρ (V : FDRep k G) :
-    (MonoidHomClass.toMonoidHom (ModuleCat.endMulEquiv V.V.obj).symm).comp (ρ V) = Action.ρ V := rfl
+lemma endRingEquiv_symm_comp_ρ (V : FDRep k G) :
+    (MonoidHomClass.toMonoidHom (ModuleCat.endRingEquiv V.V.obj).symm).comp (ρ V) = Action.ρ V :=
+  rfl
 
 @[simp]
-lemma endMulEquiv_comp_ρ (V : FDRep k G) :
-    (MonoidHomClass.toMonoidHom (ModuleCat.endMulEquiv V.V.obj)).comp (Action.ρ V) = ρ V := rfl
+lemma endRingEquiv_comp_ρ (V : FDRep k G) :
+    (MonoidHomClass.toMonoidHom (ModuleCat.endRingEquiv V.V.obj)).comp (Action.ρ V) = ρ V := rfl
 
 @[simp]
 lemma hom_action_ρ (V : FDRep k G) (g : G) : (Action.ρ V g).hom = ρ V g := rfl
@@ -111,7 +112,7 @@ theorem Iso.conj_ρ {V W : FDRep k G} (i : V ≅ W) (g : G) :
 @[simps ρ]
 def of {V : Type u} [AddCommGroup V] [Module k V] [FiniteDimensional k V]
     (ρ : Representation k G V) : FDRep k G :=
-  ⟨FGModuleCat.of k V, ρ ≫ MonCat.ofHom (ModuleCat.endMulEquiv _).symm.toMonoidHom⟩
+  ⟨FGModuleCat.of k V, ρ ≫ MonCat.ofHom (ModuleCat.endRingEquiv _).symm.toMonoidHom⟩
 
 instance : HasForget₂ (FDRep k G) (Rep k G) where
   forget₂ := (forget₂ (FGModuleCat k) (ModuleCat k)).mapAction (MonCat.of G)
diff --git a/Mathlib/RepresentationTheory/GroupCohomology/Basic.lean b/Mathlib/RepresentationTheory/GroupCohomology/Basic.lean
index 756771dc9da34d..c620c7d2c03c88 100644
--- a/Mathlib/RepresentationTheory/GroupCohomology/Basic.lean
+++ b/Mathlib/RepresentationTheory/GroupCohomology/Basic.lean
@@ -146,8 +146,7 @@ and the homogeneous `linearYonedaObjResolution`. -/
   -- https://github.com/leanprover-community/mathlib4/issues/5164
   change d n A f g = diagonalHomEquiv (n + 1) A
     ((resolution k G).d (n + 1) n ≫ (diagonalHomEquiv n A).symm f) g
-  rw [diagonalHomEquiv_apply, Action.comp_hom, ModuleCat.hom_comp, LinearMap.comp_apply,
-    resolution.d_eq]
+  rw [diagonalHomEquiv_apply, Action.comp_hom, ConcreteCategory.comp_apply, resolution.d_eq]
   erw [resolution.d_of (Fin.partialProd g)]
   simp only [map_sum, ← Finsupp.smul_single_one _ ((-1 : k) ^ _)]
   -- This used to be `rw`, but we need `erw` after https://github.com/leanprover/lean4/pull/2644
@@ -188,8 +187,9 @@ noncomputable abbrev inhomogeneousCochains : CochainComplex (ModuleCat k) ℕ :=
       ((linearYonedaObjResolution A).d_comp_d n (n + 1) (n + 2)))
     simp only [ModuleCat.hom_comp, LinearMap.comp_apply] at this
     dsimp only
-    simp only [d_eq, LinearEquiv.toModuleIso_inv_hom, LinearEquiv.toModuleIso_hom_hom,
-      ModuleCat.hom_comp, LinearMap.comp_apply, LinearEquiv.coe_coe, ModuleCat.hom_zero]
+    simp only [d_eq, LinearEquiv.toModuleIso_inv, LinearEquiv.toModuleIso_hom,
+      ModuleCat.hom_comp, LinearMap.comp_apply, LinearEquiv.coe_coe, ModuleCat.hom_zero,
+      ModuleCat.ofHom_comp]
     /- Porting note: I can see I need to rewrite `LinearEquiv.coe_coe` twice to at
       least reduce the need for `symm_apply_apply` to be an `erw`. However, even `erw` refuses to
       rewrite the second `coe_coe`... -/
@@ -213,11 +213,11 @@ def inhomogeneousCochainsIso : inhomogeneousCochains A ≅ linearYonedaObjResolu
   ext
   simp only [ChainComplex.linearYonedaObj_X, linearYoneda_obj_obj_carrier, CochainComplex.of_x,
     linearYoneda_obj_obj_isAddCommGroup, linearYoneda_obj_obj_isModule, Iso.symm_hom,
-    ChainComplex.linearYonedaObj_d, ModuleCat.hom_comp, linearYoneda_obj_map_hom,
-    Quiver.Hom.unop_op, LinearEquiv.toModuleIso_inv_hom, LinearMap.coe_comp, Function.comp_apply,
-    Linear.leftComp_apply, inhomogeneousCochains.d_def, d_eq, LinearEquiv.toModuleIso_hom_hom,
-    ModuleCat.ofHom_comp, Category.assoc, LinearEquiv.comp_coe, LinearEquiv.self_trans_symm,
-    LinearEquiv.refl_toLinearMap, LinearMap.id_comp, LinearEquiv.coe_coe]
+    LinearEquiv.toModuleIso_inv, ChainComplex.linearYonedaObj_d, ModuleCat.hom_comp,
+    ModuleCat.hom_ofHom, LinearMap.coe_comp, Function.comp_apply, Linear.leftComp_apply,
+    inhomogeneousCochains.d_def, d_eq, LinearEquiv.toModuleIso_hom, ModuleCat.ofHom_comp,
+    Category.assoc, LinearEquiv.comp_coe, LinearEquiv.self_trans_symm, LinearEquiv.refl_toLinearMap,
+    LinearMap.id_comp, LinearEquiv.coe_coe]
   rfl
 
 /-- The `n`-cocycles `Zⁿ(G, A)` of a `k`-linear `G`-representation `A`, i.e. the kernel of the
diff --git a/Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean b/Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean
index 1c8844cd09b486..ccb8715cbcaa61 100644
--- a/Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean
+++ b/Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean
@@ -205,7 +205,7 @@ theorem dTwo_comp_dOne : dTwo A ∘ₗ dOne A = 0 := by
   show (ModuleCat.ofHom (dOne A) ≫ ModuleCat.ofHom (dTwo A)).hom = _
   have h1 := congr_arg ModuleCat.ofHom (dOne_comp_eq A)
   have h2 := congr_arg ModuleCat.ofHom (dTwo_comp_eq A)
-  simp only [ModuleCat.ofHom_comp, ModuleCat.ofHom_comp, ← LinearEquiv.toModuleIso_hom_hom] at h1 h2
+  simp only [ModuleCat.ofHom_comp, ModuleCat.ofHom_comp, ← LinearEquiv.toModuleIso_hom] at h1 h2
   simp only [(Iso.eq_inv_comp _).2 h2, (Iso.eq_inv_comp _).2 h1, ModuleCat.ofHom_hom,
     ModuleCat.hom_ofHom, Category.assoc, Iso.hom_inv_id_assoc, HomologicalComplex.d_comp_d_assoc,
     zero_comp, comp_zero, ModuleCat.hom_zero]
diff --git a/Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean b/Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean
index b53ab4e4d4058c..ad038c15186b45 100644
--- a/Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean
+++ b/Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean
@@ -494,7 +494,7 @@ theorem d_eq (n : ℕ) : ((groupCohomology.resolution k G).d (n + 1) n).hom =
 /- Porting note: want to rewrite `LinearMap.smul_apply` but simp/simp_rw won't do it; I need erw,
 so using Finset.sum_congr to get rid of the binder -/
   refine Finset.sum_congr rfl fun _ _ => ?_
-  simp only [ModuleCat.hom_smul, SimplexCategory.len_mk]
+  simp only [ModuleCat.hom_smul, SimplexCategory.len_mk, ModuleCat.hom_ofHom]
   erw [LinearMap.smul_apply]
   rw [Finsupp.lmapDomain_apply, Finsupp.mapDomain_single, Finsupp.smul_single', mul_one]
   rfl
@@ -560,7 +560,8 @@ theorem forget₂ToModuleCatHomotopyEquiv_f_0_eq :
         LinearMap.id fun i => rfl,
       LinearMap.id_comp]
     rfl
-  · congr
+  · rw [ModuleCat.hom_comp]
+    congr
     · ext x
       dsimp (config := { unfoldPartialApp := true }) [HomotopyEquiv.ofIso,
         Finsupp.LinearEquiv.finsuppUnique]
diff --git a/Mathlib/RepresentationTheory/Rep.lean b/Mathlib/RepresentationTheory/Rep.lean
index 68e82a8670b12a..5e1201f82dfade 100644
--- a/Mathlib/RepresentationTheory/Rep.lean
+++ b/Mathlib/RepresentationTheory/Rep.lean
@@ -61,11 +61,11 @@ instance (V : Rep k G) : Module k V := by
 -/
 def ρ (V : Rep k G) : Representation k G V :=
 -- Porting note: was `V.ρ`
-  (ModuleCat.endMulEquiv V.V).toMonoidHom.comp (Action.ρ V)
+  (ModuleCat.endRingEquiv V.V).toMonoidHom.comp (Action.ρ V)
 
 /-- Lift an unbundled representation to `Rep`. -/
 def of {V : Type u} [AddCommGroup V] [Module k V] (ρ : G →* V →ₗ[k] V) : Rep k G :=
-  ⟨ModuleCat.of k V, MonCat.ofHom ((ModuleCat.endMulEquiv _).symm.toMonoidHom.comp ρ) ⟩
+  ⟨ModuleCat.of k V, MonCat.ofHom ((ModuleCat.endRingEquiv _).symm.toMonoidHom.comp ρ) ⟩
 
 @[simp]
 theorem coe_of {V : Type u} [AddCommGroup V] [Module k V] (ρ : G →* V →ₗ[k] V) :
@@ -77,7 +77,7 @@ theorem of_ρ {V : Type u} [AddCommGroup V] [Module k V] (ρ : G →* V →ₗ[k
   rfl
 
 theorem Action_ρ_eq_ρ {A : Rep k G} :
-    Action.ρ A = (ModuleCat.endMulEquiv _).symm.toMonoidHom.comp A.ρ :=
+    Action.ρ A = (ModuleCat.endRingEquiv _).symm.toMonoidHom.comp A.ρ :=
   rfl
 
 @[simp]
@@ -296,7 +296,7 @@ noncomputable def leftRegularHom (A : Rep k G) (x : A) : Rep.ofMulAction k G G 
 theorem leftRegularHom_apply {A : Rep k G} (x : A) :
     (leftRegularHom A x).hom (Finsupp.single 1 1) = x := by
   -- This used to be `rw`, but we need `erw` after https://github.com/leanprover/lean4/pull/2644
-  erw [leftRegularHom_hom_hom, Finsupp.lift_apply, Finsupp.sum_single_index, one_smul,
+  erw [leftRegularHom_hom, Finsupp.lift_apply, Finsupp.sum_single_index, one_smul,
     A.ρ.map_one, LinearMap.one_apply]
   rw [zero_smul]
 
@@ -314,9 +314,10 @@ noncomputable def leftRegularHomEquiv (A : Rep k G) : (Rep.ofMulAction k G G ⟶
       f.hom ((ofMulAction k G G).ρ x (Finsupp.single (1 : G) (1 : k))) =
         A.ρ x (f.hom (Finsupp.single (1 : G) (1 : k))) :=
       LinearMap.ext_iff.1 (ModuleCat.hom_ext_iff.mp (f.comm x)) (Finsupp.single 1 1)
-    simp only [leftRegularHom_hom_hom, LinearMap.comp_apply, Finsupp.lsingle_apply,
+    simp only [leftRegularHom_hom, LinearMap.comp_apply, Finsupp.lsingle_apply,
       Finsupp.lift_apply, ← this, coe_of, of_ρ, Representation.ofMulAction_single x (1 : G) (1 : k),
-      smul_eq_mul, mul_one, zero_smul, Finsupp.sum_single_index, one_smul]
+      smul_eq_mul, mul_one, zero_smul, Finsupp.sum_single_index, one_smul,
+      ConcreteCategory.hom_ofHom]
     -- Mismatched `Zero k` instances
     rfl
   right_inv x := leftRegularHom_apply x
@@ -324,7 +325,7 @@ noncomputable def leftRegularHomEquiv (A : Rep k G) : (Rep.ofMulAction k G G ⟶
 theorem leftRegularHomEquiv_symm_single {A : Rep k G} (x : A) (g : G) :
     ((leftRegularHomEquiv A).symm x).hom (Finsupp.single g 1) = A.ρ g x := by
   -- This used to be `rw`, but we need `erw` after https://github.com/leanprover/lean4/pull/2644
-  erw [leftRegularHomEquiv_symm_apply, leftRegularHom_hom_hom, Finsupp.lift_apply,
+  erw [leftRegularHomEquiv_symm_apply, leftRegularHom_hom, Finsupp.lift_apply,
     Finsupp.sum_single_index, one_smul]
   rw [zero_smul]
 
diff --git a/Mathlib/RingTheory/Adjoin/Tower.lean b/Mathlib/RingTheory/Adjoin/Tower.lean
index 5e0b6870afb102..f0bec2c20ba932 100644
--- a/Mathlib/RingTheory/Adjoin/Tower.lean
+++ b/Mathlib/RingTheory/Adjoin/Tower.lean
@@ -141,7 +141,8 @@ variable [Algebra A B] [Algebra B C] [Algebra A C] [IsScalarTower A B C]
 A is noetherian, and C is algebra-finite over A, and C is module-finite over B,
 then B is algebra-finite over A.
 
-References: Atiyah--Macdonald Proposition 7.8; Stacks 00IS; Altman--Kleiman 16.17. -/
+References: Atiyah--Macdonald Proposition 7.8; Altman--Kleiman 16.17. -/
+@[stacks 00IS]
 theorem fg_of_fg_of_fg [IsNoetherianRing A] (hAC : (⊤ : Subalgebra A C).FG)
     (hBC : (⊤ : Submodule B C).FG) (hBCi : Function.Injective (algebraMap B C)) :
     (⊤ : Subalgebra A B).FG :=
diff --git a/Mathlib/RingTheory/Artinian/Ring.lean b/Mathlib/RingTheory/Artinian/Ring.lean
index 4889c2cb83fb23..455b601e23f615 100644
--- a/Mathlib/RingTheory/Artinian/Ring.lean
+++ b/Mathlib/RingTheory/Artinian/Ring.lean
@@ -91,6 +91,26 @@ theorem isField_of_isReduced_of_isLocalRing [IsReduced R] [IsLocalRing R] : IsFi
   (IsArtinianRing.equivPi R).trans (RingEquiv.piUnique _) |>.toMulEquiv.isField
     _ (Ideal.Quotient.field _).toIsField
 
+section IsUnit
+
+open nonZeroDivisors
+
+/-- If an element of an artinian ring is not a zero divisor then it is a unit. -/
+theorem isUnit_of_mem_nonZeroDivisors {a : R} (ha : a ∈ R⁰) : IsUnit a :=
+  IsUnit.isUnit_iff_mulLeft_bijective.mpr <|
+    IsArtinian.bijective_of_injective_endomorphism (LinearMap.mulLeft R a)
+      fun _ _ ↦ (mul_cancel_left_mem_nonZeroDivisors ha).mp
+
+/-- In an artinian ring, an element is a unit iff it is a non-zero-divisor.
+See also `isUnit_iff_mem_nonZeroDivisors_of_finite`.-/
+theorem isUnit_iff_mem_nonZeroDivisors {a : R} : IsUnit a ↔ a ∈ R⁰ :=
+  ⟨IsUnit.mem_nonZeroDivisors, isUnit_of_mem_nonZeroDivisors⟩
+
+theorem isUnit_submonoid_eq : IsUnit.submonoid R = R⁰ := by
+  ext; simp [IsUnit.mem_submonoid_iff, isUnit_iff_mem_nonZeroDivisors]
+
+end IsUnit
+
 section Localization
 
 variable (S : Submonoid R) (L : Type*) [CommSemiring L] [Algebra R L] [IsLocalization S L]
diff --git a/Mathlib/RingTheory/Coalgebra/TensorProduct.lean b/Mathlib/RingTheory/Coalgebra/TensorProduct.lean
index dafc8270e1007d..e8efc2bbe44ae3 100644
--- a/Mathlib/RingTheory/Coalgebra/TensorProduct.lean
+++ b/Mathlib/RingTheory/Coalgebra/TensorProduct.lean
@@ -57,7 +57,7 @@ noncomputable instance TensorProduct.instCoalgebra : Coalgebra R (M ⊗[R] N) :=
       map_comp_comul := by
         rw [CoalgebraCat.ofComonObjCoalgebraStruct_comul]
         simp [-Mon_.monMonoidalStruct_tensorObj_X,
-          ModuleCat.MonoidalCategory.instMonoidalCategoryStruct_tensorHom_hom,
+          ModuleCat.MonoidalCategory.instMonoidalCategoryStruct_tensorHom,
           ModuleCat.hom_comp, ModuleCat.of, ModuleCat.ofHom,
           ModuleCat.MonoidalCategory.tensorμ_eq_tensorTensorTensorComm] }
 
diff --git a/Mathlib/RingTheory/DedekindDomain/AdicValuation.lean b/Mathlib/RingTheory/DedekindDomain/AdicValuation.lean
index 110ec9cd323fbc..d7d5139cc0b3e8 100644
--- a/Mathlib/RingTheory/DedekindDomain/AdicValuation.lean
+++ b/Mathlib/RingTheory/DedekindDomain/AdicValuation.lean
@@ -403,6 +403,9 @@ def adicCompletion :=
 instance : Field (v.adicCompletion K) := inferInstanceAs <|
   Field (@UniformSpace.Completion K v.adicValued.toUniformSpace)
 
+instance : Algebra K (v.adicCompletion K) :=
+  RingHom.toAlgebra (@UniformSpace.Completion.coeRingHom K _ v.adicValued.toUniformSpace _ _)
+
 instance : Inhabited (v.adicCompletion K) :=
   ⟨0⟩
 
diff --git a/Mathlib/RingTheory/Etale/Basic.lean b/Mathlib/RingTheory/Etale/Basic.lean
index 1f11834fdfbaa8..1b85886b995e0f 100644
--- a/Mathlib/RingTheory/Etale/Basic.lean
+++ b/Mathlib/RingTheory/Etale/Basic.lean
@@ -167,11 +167,9 @@ section
 variable (R : Type u) [CommRing R]
 variable (A : Type u) [CommRing A] [Algebra R A]
 
-/-- An `R`-algebra `A` is étale if it is formally étale and of finite presentation.
-
-Note that the definition <https://stacks.math.columbia.edu/tag/00U1> in the stacks project is
-different, but <https://stacks.math.columbia.edu/tag/00UR> shows that it is equivalent
-to the definition here. -/
+/-- An `R`-algebra `A` is étale if it is formally étale and of finite presentation. -/
+@[stacks 00U1 "Note that this is a different definition from this Stacks entry, but
+<https://stacks.math.columbia.edu/tag/00UR> shows that it is equivalent to the definition here."]
 class Etale : Prop where
   formallyEtale : FormallyEtale R A := by infer_instance
   finitePresentation : FinitePresentation R A := by infer_instance
diff --git a/Mathlib/RingTheory/FinitePresentation.lean b/Mathlib/RingTheory/FinitePresentation.lean
index 06867a417b8d1c..dbc3a551101b0b 100644
--- a/Mathlib/RingTheory/FinitePresentation.lean
+++ b/Mathlib/RingTheory/FinitePresentation.lean
@@ -212,8 +212,8 @@ theorem trans [Algebra A B] [IsScalarTower R A B] [FinitePresentation R A]
 
 open MvPolynomial
 
--- We follow the proof of https://stacks.math.columbia.edu/tag/0561
 -- TODO: extract out helper lemmas and tidy proof.
+@[stacks 0561]
 theorem of_restrict_scalars_finitePresentation [Algebra A B] [IsScalarTower R A B]
     [FinitePresentation.{w₁, w₃} R B] [FiniteType R A] :
     FinitePresentation.{w₂, w₃} A B := by
diff --git a/Mathlib/RingTheory/Finiteness/Nakayama.lean b/Mathlib/RingTheory/Finiteness/Nakayama.lean
index a5ae263188b4da..d17b83e9cc8881 100644
--- a/Mathlib/RingTheory/Finiteness/Nakayama.lean
+++ b/Mathlib/RingTheory/Finiteness/Nakayama.lean
@@ -26,8 +26,8 @@ variable {R : Type*} {M : Type*} [Semiring R] [AddCommMonoid M] [Module R M]
 
 open Set
 
-/-- **Nakayama's Lemma**. Atiyah-Macdonald 2.5, Eisenbud 4.7, Matsumura 2.2,
-[Stacks 00DV](https://stacks.math.columbia.edu/tag/00DV) -/
+/-- **Nakayama's Lemma**. Atiyah-Macdonald 2.5, Eisenbud 4.7, Matsumura 2.2. -/
+@[stacks 00DV]
 theorem exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
     [AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
     ∃ r : R, r - 1 ∈ I ∧ ∀ n ∈ N, r • n = (0 : M) := by
diff --git a/Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean b/Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean
index 5db79f945e9dd8..bbf91f8c56ddb2 100644
--- a/Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean
+++ b/Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean
@@ -714,7 +714,7 @@ def awayMapₐ : Away 𝒜 f →ₐ[𝒜 0] Away 𝒜 x where
 /-- This is a convenient constructor for `Away 𝒜 f` when `f` is homogeneous.
 `Away.mk 𝒜 hf n x hx` is the fraction `x / f ^ n`. -/
 protected def Away.mk {d : ι} (hf : f ∈ 𝒜 d) (n : ℕ) (x : A) (hx : x ∈ 𝒜 (n • d)) : Away 𝒜 f :=
-  .mk ⟨n • d, ⟨x, hx⟩, ⟨f ^ n, SetLike.pow_mem_graded n hf⟩, ⟨n, rfl⟩⟩
+  HomogeneousLocalization.mk ⟨n • d, ⟨x, hx⟩, ⟨f ^ n, SetLike.pow_mem_graded n hf⟩, ⟨n, rfl⟩⟩
 
 @[simp]
 lemma Away.val_mk {d : ι} (n : ℕ) (hf : f ∈ 𝒜 d) (x : A) (hx : x ∈ 𝒜 (n • d)) :
diff --git a/Mathlib/RingTheory/Ideal/Operations.lean b/Mathlib/RingTheory/Ideal/Operations.lean
index ee0467668b791b..de11be3c0f851c 100644
--- a/Mathlib/RingTheory/Ideal/Operations.lean
+++ b/Mathlib/RingTheory/Ideal/Operations.lean
@@ -356,22 +356,18 @@ theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι
 theorem mul_le_right : I * J ≤ I :=
   Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr
 
-#adaptation_note
-/--
-On nightly-2024-11-12, we had to add `nolint simpNF` to the following lemma,
+#adaptation_note /-- nightly-2024-11-12
+we had to add `nolint simpNF` to the following lemma,
 as otherwise we get a deterministic timeout in typeclass inference.
-This should be investigated.
--/
+This should be investigated. -/
 @[simp, nolint simpNF]
 theorem sup_mul_right_self : I ⊔ I * J = I :=
   sup_eq_left.2 Ideal.mul_le_right
 
-#adaptation_note
-/--
-On nightly-2024-11-12, we had to add `nolint simpNF` to the following lemma,
+#adaptation_note /-- nightly-2024-11-12
+we had to add `nolint simpNF` to the following lemma,
 as otherwise we get a deterministic timeout in typeclass inference.
-This should be investigated.
--/
+This should be investigated. -/
 @[simp, nolint simpNF]
 theorem mul_right_self_sup : I * J ⊔ I = I :=
   sup_eq_right.2 Ideal.mul_le_right
@@ -1035,7 +1031,8 @@ theorem subset_union_prime' {R : Type u} [CommRing R] {s : Finset ι} {f : ι 
       exact hs <| Or.inr <| Set.mem_biUnion hjt <|
         add_sub_cancel_left r s ▸ (f j).sub_mem hj <| hr j hjt
 
-/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/
+/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Matsumura Ex.1.6. -/
+@[stacks 00DS]
 theorem subset_union_prime {R : Type u} [CommRing R] {s : Finset ι} {f : ι → Ideal R} (a b : ι)
     (hp : ∀ i ∈ s, i ≠ a → i ≠ b → IsPrime (f i)) {I : Ideal R} :
     ((I : Set R) ⊆ ⋃ i ∈ (↑s : Set ι), f i) ↔ ∃ i ∈ s, I ≤ f i :=
diff --git a/Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean b/Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean
index 0f9874f3e5ae84..c8e7297bf65f1e 100644
--- a/Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean
+++ b/Mathlib/RingTheory/IntegralClosure/Algebra/Basic.lean
@@ -41,6 +41,11 @@ theorem Algebra.IsIntegral.of_injective (f : A →ₐ[R] B) (hf : Function.Injec
     [Algebra.IsIntegral R B] : Algebra.IsIntegral R A :=
   ⟨fun _ ↦ (isIntegral_algHom_iff f hf).mp (isIntegral _)⟩
 
+/-- Homomorphic image of an integral algebra is an integral algebra. -/
+theorem Algebra.IsIntegral.of_surjective [Algebra.IsIntegral R A]
+    (f : A →ₐ[R] B) (hf : Function.Surjective f) : Algebra.IsIntegral R B :=
+  isIntegral_def.mpr fun b ↦ let ⟨a, ha⟩ := hf b; ha ▸ (isIntegral_def.mp ‹_› a).map f
+
 theorem AlgEquiv.isIntegral_iff (e : A ≃ₐ[R] B) : Algebra.IsIntegral R A ↔ Algebra.IsIntegral R B :=
   ⟨fun h ↦ h.of_injective e.symm e.symm.injective, fun h ↦ h.of_injective e e.injective⟩
 
@@ -205,8 +210,15 @@ def integralClosure : Subalgebra R A where
   mul_mem' := IsIntegral.mul
   algebraMap_mem' _ := isIntegral_algebraMap
 
-end
-
-theorem mem_integralClosure_iff (R A : Type*) [CommRing R] [CommRing A] [Algebra R A] {a : A} :
-    a ∈ integralClosure R A ↔ IsIntegral R a :=
+theorem mem_integralClosure_iff {a : A} : a ∈ integralClosure R A ↔ IsIntegral R a :=
   Iff.rfl
+
+variable {R} {A B : Type*} [Ring A] [Algebra R A] [Ring B] [Algebra R B]
+
+/-- Product of two integral algebras is an integral algebra. -/
+instance Algebra.IsIntegral.prod [Algebra.IsIntegral R A] [Algebra.IsIntegral R B] :
+    Algebra.IsIntegral R (A × B) :=
+  Algebra.isIntegral_def.mpr fun x ↦
+    (Algebra.isIntegral_def.mp ‹_› x.1).pair (Algebra.isIntegral_def.mp ‹_› x.2)
+
+end
diff --git a/Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean b/Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean
index 361784bfb1bc20..5e75b8a8f6cd12 100644
--- a/Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean
+++ b/Mathlib/RingTheory/IntegralClosure/IsIntegral/Basic.lean
@@ -201,8 +201,37 @@ theorem fg_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsI
           (his a <| Set.mem_insert a s).fg_adjoin_singleton)
     his
 
+theorem Algebra.finite_adjoin_of_finite_of_isIntegral {s : Set A} (hf : s.Finite)
+    (hi : ∀ x ∈ s, IsIntegral R x) : Module.Finite R (adjoin R s) :=
+  Module.Finite.iff_fg.mpr <| fg_adjoin_of_finite hf hi
+
+theorem Algebra.finite_adjoin_simple_of_isIntegral {x : A} (hi : IsIntegral R x) :
+    Module.Finite R (adjoin R {x}) :=
+  Module.Finite.iff_fg.mpr hi.fg_adjoin_singleton
+
 theorem isNoetherian_adjoin_finset [IsNoetherianRing R] (s : Finset A)
     (hs : ∀ x ∈ s, IsIntegral R x) : IsNoetherian R (Algebra.adjoin R (s : Set A)) :=
   isNoetherian_of_fg_of_noetherian _ (fg_adjoin_of_finite s.finite_toSet hs)
 
 end
+
+section Prod
+
+variable {R A B : Type*}
+variable [CommRing R] [Ring A] [Ring B] [Algebra R A] [Algebra R B]
+
+/-- An element of a product algebra is integral if each component is integral. -/
+theorem IsIntegral.pair {x : A × B} (hx₁ : IsIntegral R x.1) (hx₂ : IsIntegral R x.2) :
+    IsIntegral R x := by
+  obtain ⟨p₁, ⟨hp₁Monic, hp₁Eval⟩⟩ := hx₁
+  obtain ⟨p₂, ⟨hp₂Monic, hp₂Eval⟩⟩ := hx₂
+  refine ⟨p₁ * p₂, ⟨hp₁Monic.mul hp₂Monic, ?_⟩⟩
+  rw [← aeval_def] at *
+  rw [aeval_prod_apply, aeval_mul, hp₁Eval, zero_mul, aeval_mul, hp₂Eval, mul_zero,
+    Prod.zero_eq_mk]
+
+/-- An element of a product algebra is integral iff each component is integral. -/
+theorem IsIntegral.pair_iff {x : A × B} : IsIntegral R x ↔ IsIntegral R x.1 ∧ IsIntegral R x.2 :=
+  ⟨fun h ↦ ⟨h.map (AlgHom.fst R A B), h.map (AlgHom.snd R A B)⟩, fun h ↦ h.1.pair h.2⟩
+
+end Prod
diff --git a/Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean b/Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean
index c62e33c4ab1b2e..ff4bf59739f942 100644
--- a/Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean
+++ b/Mathlib/RingTheory/IntegralClosure/IsIntegralClosure/Basic.lean
@@ -132,7 +132,7 @@ theorem le_integralClosure_iff_isIntegral {S : Subalgebra R A} :
       Algebra.isIntegral_def.symm
 
 theorem Algebra.IsIntegral.adjoin {S : Set A} (hS : ∀ x ∈ S, IsIntegral R x) :
-    Algebra.IsIntegral R (Algebra.adjoin R S) :=
+    Algebra.IsIntegral R (adjoin R S) :=
   le_integralClosure_iff_isIntegral.mp <| adjoin_le hS
 
 theorem integralClosure_eq_top_iff : integralClosure R A = ⊤ ↔ Algebra.IsIntegral R A := by
diff --git a/Mathlib/RingTheory/Jacobson/Ring.lean b/Mathlib/RingTheory/Jacobson/Ring.lean
index 68129d02d2fcbc..aca96b49168c82 100644
--- a/Mathlib/RingTheory/Jacobson/Ring.lean
+++ b/Mathlib/RingTheory/Jacobson/Ring.lean
@@ -702,11 +702,8 @@ lemma finite_of_finite_type_of_isJacobsonRing (R S : Type*) [CommRing R] [Field
   obtain ⟨ι, hι, f, hf⟩ := Algebra.FiniteType.iff_quotient_mvPolynomial'.mp ‹_›
   have : (algebraMap R S).IsIntegral := by
     rw [← f.comp_algebraMap]
-    #adaptation_note
-    /--
-    After https://github.com/leanprover/lean4/pull/6024
-    we needed to write `f.toRingHom` instead of just `f`, to avoid unification issues.
-    -/
+    #adaptation_note /-- https://github.com/leanprover/lean4/pull/6024
+    we needed to write `f.toRingHom` instead of just `f`, to avoid unification issues. -/
     exact MvPolynomial.comp_C_integral_of_surjective_of_isJacobsonRing f.toRingHom hf
   have : Algebra.IsIntegral R S := Algebra.isIntegral_def.mpr this
   exact Algebra.IsIntegral.finite
diff --git a/Mathlib/RingTheory/LaurentSeries.lean b/Mathlib/RingTheory/LaurentSeries.lean
index cf8a5776479281..27e5f145a98c1c 100644
--- a/Mathlib/RingTheory/LaurentSeries.lean
+++ b/Mathlib/RingTheory/LaurentSeries.lean
@@ -14,6 +14,7 @@ import Mathlib.RingTheory.Localization.FractionRing
 import Mathlib.Topology.UniformSpace.Cauchy
 import Mathlib.Algebra.Group.Int.TypeTags
 
+
 /-!
 # Laurent Series
 
@@ -60,8 +61,8 @@ such that the `X`-adic valuation `v` satisfies `v (f - Q) < γ`.
 `LaurentSeries.exists_powerSeries_of_memIntegers` show that an element in the completion of
 `RatFunc K` is in the unit ball if and only if it comes from a power series through the isomorphism
 `LaurentSeriesRingEquiv`.
-* `LaurentSeries.powerSeriesRingEquiv` is the ring isomorphism between `K⟦X⟧` and the unit ball
-inside the `X`-adic completion of `RatFunc K`.
+* `LaurentSeries.powerSeriesAlgEquiv` is the `K`-algebra isomorphism between `K⟦X⟧`
+and the unit ball inside the `X`-adic completion of `RatFunc K`.
 
 ## Implementation details
 
@@ -75,18 +76,13 @@ that it is complete and contains `RatFunc K` as a dense subspace. The isomorphis
 equivalence, expressing the mathematical idea that the completion "is unique". It is
 `LaurentSeries.comparePkg`.
 * For applications to `K⟦X⟧` it is actually more handy to use the *inverse* of the above
-equivalence: `LaurentSeries.LaurentSeriesRingEquiv` is the *topological, ring equivalence*
-`K⸨X⸩ ≃+* RatFuncAdicCompl K`.
+equivalence: `LaurentSeries.LaurentSeriesAlgEquiv` is the *topological, algebra equivalence*
+`K⸨X⸩ ≃ₐ[K] RatFuncAdicCompl K`.
 * In order to compare `K⟦X⟧` with the valuation subring in the `X`-adic completion of
 `RatFunc K` we consider its alias `LaurentSeries.powerSeries_as_subring` as a subring of `K⸨X⸩`,
 that is itself clearly isomorphic (via the inverse of `LaurentSeries.powerSeriesEquivSubring`)
 to `K⟦X⟧`.
 
-## To Do
-* The `AdicCompletion` construction is currently done for ideals in rings and does not take into
-account the relation with algebra structures on the ring, hence it does not yield a `K`-algebra
-structure on the `X`-adic completion of `K⸨X⸩`. Once this will be available, we should update
-`LaurentSeries.LaurentSeriesRingEquiv` to an algebra equivalence.
 -/
 universe u
 
@@ -1061,6 +1057,11 @@ equivalence: it goes from `K⸨X⸩` to `RatFuncAdicCompl K` -/
 abbrev LaurentSeriesRingEquiv : K⸨X⸩ ≃+* RatFuncAdicCompl K :=
   (ratfuncAdicComplRingEquiv K).symm
 
+@[simp]
+lemma LaurentSeriesRingEquiv_def (f : K⟦X⟧) :
+    (LaurentSeriesRingEquiv K) f = (LaurentSeriesPkg K).compare ratfuncAdicComplPkg (f : K⸨X⸩) :=
+  rfl
+
 @[simp]
 theorem ratfuncAdicComplRingEquiv_apply (x : RatFuncAdicCompl K) :
     ratfuncAdicComplRingEquiv K x = ratfuncAdicComplPkg.compare (LaurentSeriesPkg K) x := rfl
@@ -1071,6 +1072,17 @@ theorem coe_X_compare :
   rw [PowerSeries.coe_X, ← RatFunc.coe_X, ← LaurentSeries_coe, ← compare_coe]
   rfl
 
+theorem algebraMap_apply (a : K) : algebraMap K K⸨X⸩ a = HahnSeries.C a := by
+  simp [RingHom.algebraMap_toAlgebra]
+
+instance : Algebra K (RatFuncAdicCompl K) :=
+  RingHom.toAlgebra ((LaurentSeriesRingEquiv K).toRingHom.comp HahnSeries.C)
+
+/-- The algebra equivalence between `K⸨X⸩` and the `X`-adic completion of `RatFunc X` -/
+def LaurentSeriesAlgEquiv : K⸨X⸩ ≃ₐ[K] RatFuncAdicCompl K :=
+  AlgEquiv.ofRingEquiv (f := LaurentSeriesRingEquiv K)
+    (fun a ↦ by simp [RingHom.algebraMap_toAlgebra])
+
 open Filter WithZero
 
 open scoped WithZeroTopology Topology Multiplicative
@@ -1132,22 +1144,30 @@ section PowerSeries
 /-- In order to compare `K⟦X⟧` with the valuation subring in the `X`-adic completion of
 `RatFunc K` we consider its alias as a subring of `K⸨X⸩`. -/
 abbrev powerSeries_as_subring : Subring K⸨X⸩ :=
-  RingHom.range (HahnSeries.ofPowerSeries ℤ K)
+--  RingHom.range (HahnSeries.ofPowerSeries ℤ K)
+  Subring.map (HahnSeries.ofPowerSeries ℤ K) ⊤
 
 /-- The ring `K⟦X⟧` is isomorphic to the subring `powerSeries_as_subring K` -/
-abbrev powerSeriesEquivSubring : K⟦X⟧ ≃+* powerSeries_as_subring K := by
-  rw [powerSeries_as_subring, RingHom.range_eq_map]
-  exact ((Subring.topEquiv).symm).trans (Subring.equivMapOfInjective ⊤ (ofPowerSeries ℤ K)
+abbrev powerSeriesEquivSubring : K⟦X⟧ ≃+* powerSeries_as_subring K :=
+  ((Subring.topEquiv).symm).trans (Subring.equivMapOfInjective ⊤ (ofPowerSeries ℤ K)
     ofPowerSeries_injective)
 
+lemma powerSeriesEquivSubring_apply (f : K⟦X⟧) :
+    powerSeriesEquivSubring K f =
+      ⟨HahnSeries.ofPowerSeries ℤ K f, Subring.mem_map.mpr ⟨f, trivial, rfl⟩⟩ :=
+  rfl
+
+lemma powerSeriesEquivSubring_coe_apply (f : K⟦X⟧) :
+    (powerSeriesEquivSubring K f : K⸨X⸩) = ofPowerSeries ℤ K f :=
+  rfl
+
 /- Through the isomorphism `LaurentSeriesRingEquiv`, power series land in the unit ball inside the
 completion of `RatFunc K`. -/
 theorem mem_integers_of_powerSeries (F : K⟦X⟧) :
     (LaurentSeriesRingEquiv K) F ∈ (idealX K).adicCompletionIntegers (RatFunc K) := by
-  have : (LaurentSeriesRingEquiv K) F =
-    (LaurentSeriesPkg K).compare ratfuncAdicComplPkg (F : K⸨X⸩) := rfl
   simp only [Subring.mem_map, exists_prop, ValuationSubring.mem_toSubring,
-    mem_adicCompletionIntegers, this,  valuation_compare, val_le_one_iff_eq_coe]
+    mem_adicCompletionIntegers, LaurentSeriesRingEquiv_def,
+    valuation_compare, val_le_one_iff_eq_coe]
   exact ⟨F, rfl⟩
 
 /- Conversely, all elements in the unit ball inside the completion of `RatFunc K` come from a power
@@ -1166,13 +1186,13 @@ theorem powerSeries_ext_subring :
     Subring.map (LaurentSeriesRingEquiv K).toRingHom (powerSeries_as_subring K) =
       ((idealX K).adicCompletionIntegers (RatFunc K)).toSubring := by
   ext x
-  refine ⟨fun ⟨f, ⟨F, coe_F⟩, hF⟩ ↦ ?_, fun H ↦ ?_⟩
+  refine ⟨fun ⟨f, ⟨F, _, coe_F⟩, hF⟩ ↦ ?_, fun H ↦ ?_⟩
   · simp only [ValuationSubring.mem_toSubring, ← hF, ← coe_F]
     apply mem_integers_of_powerSeries
   · obtain ⟨F, hF⟩ := exists_powerSeries_of_memIntegers K H
     simp only [Equiv.toFun_as_coe, UniformEquiv.coe_toEquiv, exists_exists_eq_and,
       UniformEquiv.coe_symm_toEquiv, Subring.mem_map, Equiv.invFun_as_coe]
-    exact ⟨F, ⟨F, rfl⟩, hF⟩
+    exact ⟨F, ⟨F, trivial, rfl⟩, hF⟩
 
 /-- The ring isomorphism between `K⟦X⟧` and the unit ball inside the `X`-adic completion of
 `RatFunc K`. -/
@@ -1180,6 +1200,43 @@ abbrev powerSeriesRingEquiv : K⟦X⟧ ≃+* (idealX K).adicCompletionIntegers (
   ((powerSeriesEquivSubring K).trans (LaurentSeriesRingEquiv K).subringMap).trans
     <| RingEquiv.subringCongr (powerSeries_ext_subring K)
 
+lemma powerSeriesRingEquiv_coe_apply (f : K⟦X⟧) :
+    powerSeriesRingEquiv K f = LaurentSeriesRingEquiv K (f : K⸨X⸩) :=
+  rfl
+
+lemma LaurentSeriesRingEquiv_mem_valuationSubring (f : K⟦X⟧) :
+    LaurentSeriesRingEquiv K f ∈ Valued.v.valuationSubring := by
+  simp only [Valuation.mem_valuationSubring_iff]
+  rw [LaurentSeriesRingEquiv_def, valuation_compare, val_le_one_iff_eq_coe]
+  use f
+
+lemma algebraMap_C_mem_adicCompletionIntegers (x : K) :
+    ((LaurentSeriesRingEquiv K).toRingHom.comp HahnSeries.C) x ∈
+      adicCompletionIntegers (RatFunc K) (idealX K) := by
+  have : HahnSeries.C x = ofPowerSeries ℤ K (PowerSeries.C K x) := by
+    simp [C_apply, ofPowerSeries_C]
+  simp only [RingHom.comp_apply, RingEquiv.toRingHom_eq_coe, RingHom.coe_coe, this]
+  apply LaurentSeriesRingEquiv_mem_valuationSubring
+
+instance : Algebra K ((idealX K).adicCompletionIntegers (RatFunc K)) :=
+  RingHom.toAlgebra <|
+    ((LaurentSeriesRingEquiv K).toRingHom.comp HahnSeries.C).codRestrict _
+      (algebraMap_C_mem_adicCompletionIntegers K)
+
+instance : IsScalarTower K ((idealX K).adicCompletionIntegers (RatFunc K))
+    ((idealX K).adicCompletion (RatFunc K)) :=
+  IsScalarTower.of_algebraMap_eq (fun _ ↦ by rfl)
+
+/-- The algebra isomorphism between `K⟦X⟧` and the unit ball inside the `X`-adic completion of
+`RatFunc K`. -/
+def powerSeriesAlgEquiv : K⟦X⟧ ≃ₐ[K] (idealX K).adicCompletionIntegers (RatFunc K) := by
+  apply AlgEquiv.ofRingEquiv (f := powerSeriesRingEquiv K)
+  intro a
+  rw [PowerSeries.algebraMap_eq, RingHom.algebraMap_toAlgebra, ← Subtype.coe_inj,
+    powerSeriesRingEquiv_coe_apply,
+    RingHom.codRestrict_apply _ _ (algebraMap_C_mem_adicCompletionIntegers K)]
+  simp
+
 end PowerSeries
 
 end Comparison
diff --git a/Mathlib/RingTheory/LocalRing/RingHom/Basic.lean b/Mathlib/RingTheory/LocalRing/RingHom/Basic.lean
index 2c5a8113251ff2..e2b6096cd016dd 100644
--- a/Mathlib/RingTheory/LocalRing/RingHom/Basic.lean
+++ b/Mathlib/RingTheory/LocalRing/RingHom/Basic.lean
@@ -86,8 +86,8 @@ section
 variable [CommSemiring R] [IsLocalRing R] [CommSemiring S] [IsLocalRing S]
 
 /-- A ring homomorphism between local rings is a local ring hom iff it reflects units,
-i.e. any preimage of a unit is still a unit. https://stacks.math.columbia.edu/tag/07BJ
--/
+i.e. any preimage of a unit is still a unit. -/
+@[stacks 07BJ]
 theorem local_hom_TFAE (f : R →+* S) :
     List.TFAE
       [IsLocalHom f, f '' (maximalIdeal R).1 ⊆ maximalIdeal S,
diff --git a/Mathlib/RingTheory/Localization/Basic.lean b/Mathlib/RingTheory/Localization/Basic.lean
index 22ac6c09fc61ed..8b0b1e1417be31 100644
--- a/Mathlib/RingTheory/Localization/Basic.lean
+++ b/Mathlib/RingTheory/Localization/Basic.lean
@@ -296,6 +296,11 @@ theorem isLocalization_iff_of_isLocalization [IsLocalization M S] [IsLocalizatio
   ⟨fun _ ↦ isLocalization_of_algEquiv N (algEquiv M S P),
     fun _ ↦ isLocalization_of_algEquiv M (algEquiv N S P)⟩
 
+theorem iff_of_le_of_exists_dvd (N : Submonoid R) (h₁ : M ≤ N) (h₂ : ∀ n ∈ N, ∃ m ∈ M, n ∣ m) :
+    IsLocalization M S ↔ IsLocalization N S :=
+  have : IsLocalization N (Localization M) := of_le_of_exists_dvd _ _ h₁ h₂
+  isLocalization_iff_of_isLocalization _ _ (Localization M)
+
 end
 
 variable (M)
diff --git a/Mathlib/RingTheory/Localization/Pi.lean b/Mathlib/RingTheory/Localization/Pi.lean
new file mode 100644
index 00000000000000..28a3911c43cd75
--- /dev/null
+++ b/Mathlib/RingTheory/Localization/Pi.lean
@@ -0,0 +1,79 @@
+/-
+Copyright (c) 2024 Madison Crim. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Madison Crim
+-/
+import Mathlib.Algebra.Algebra.Pi
+import Mathlib.Algebra.BigOperators.Pi
+import Mathlib.Algebra.Divisibility.Prod
+import Mathlib.Algebra.Group.Submonoid.BigOperators
+import Mathlib.Algebra.Group.Subgroup.Basic
+import Mathlib.RingTheory.Localization.Basic
+import Mathlib.Algebra.Group.Pi.Units
+
+/-!
+# Localizing a product of commutative rings
+
+## Main Result
+
+ * `bijective_lift_piRingHom_algebraMap_comp_piEvalRingHom`: the canonical map from a
+    localization of a finite product of rings `R i `at a monoid `M` to the direct product of
+    localizations `R i` at the projection of `M` onto each corresponding factor is bijective.
+
+## Implementation notes
+
+See `Mathlib/RingTheory/Localization/Defs.lean` for a design overview.
+
+## Tags
+localization, commutative ring
+-/
+
+namespace IsLocalization
+
+variable {ι : Type*} (R S : ι → Type*)
+  [Π i, CommSemiring (R i)] [Π i, CommSemiring (S i)] [Π i, Algebra (R i) (S i)]
+
+/-- If `S i` is a localization of `R i` at the submonoid `M i` for each `i`,
+then `Π i, S i` is a localization of `Π i, R i` at the product submonoid. -/
+instance (M : Π i, Submonoid (R i)) [∀ i, IsLocalization (M i) (S i)] :
+    IsLocalization (.pi .univ M) (Π i, S i) where
+  map_units' m := Pi.isUnit_iff.mpr fun i ↦ map_units _ ⟨m.1 i, m.2 i ⟨⟩⟩
+  surj' z := by
+    choose rm h using fun i ↦ surj (M := M i) (z i)
+    exact ⟨(fun i ↦ (rm i).1, ⟨_, fun i _ ↦ (rm i).2.2⟩), funext h⟩
+  exists_of_eq {x y} eq := by
+    choose c hc using fun i ↦ exists_of_eq (M := M i) (congr_fun eq i)
+    exact ⟨⟨_, fun i _ ↦ (c i).2⟩, funext hc⟩
+
+variable (S' : Type*) [CommSemiring S'] [Algebra (Π i, R i) S'] (M : Submonoid (Π i, R i))
+
+theorem iff_map_piEvalRingHom [Finite ι] :
+    IsLocalization M S' ↔ IsLocalization (.pi .univ fun i ↦ M.map (Pi.evalRingHom R i)) S' :=
+  iff_of_le_of_exists_dvd M _ (fun m hm i _ ↦ ⟨m, hm, rfl⟩) fun n hn ↦ by
+    choose m mem eq using hn
+    have := Fintype.ofFinite ι
+    refine ⟨∏ i, m i ⟨⟩, prod_mem fun i _ ↦ mem i _, pi_dvd_iff.mpr fun i ↦ ?_⟩
+    rw [Fintype.prod_apply]
+    exact (eq i ⟨⟩).symm.dvd.trans (Finset.dvd_prod_of_mem _ <| Finset.mem_univ _)
+
+variable [∀ i, IsLocalization (M.map (Pi.evalRingHom R i)) (S i)]
+
+/-- Let `M` be a submonoid of a direct product of commutative rings `R i`, and let `M' i` denote
+the projection of `M` onto each corresponding factor. Given a ring homomorphism from the direct
+product `Π i, R i` to the product of the localizations of each `R i` at `M' i`, every `y : M`
+maps to a unit under this homomorphism. -/
+lemma isUnit_piRingHom_algebraMap_comp_piEvalRingHom (y : M) :
+    IsUnit ((Pi.ringHom fun i ↦ (algebraMap (R i) (S i)).comp (Pi.evalRingHom R i)) y) :=
+  Pi.isUnit_iff.mpr fun i ↦ map_units _ (⟨y.1 i, y, y.2, rfl⟩ : M.map (Pi.evalRingHom R i))
+
+/-- Let `M` be a submonoid of a direct product of commutative rings `R i`, and let `M' i` denote
+the projection of `M` onto each factor. Then the canonical map from the localization of the direct
+product `Π i, R i` at `M` to the direct product of the localizations of each `R i` at `M' i`
+is bijective. -/
+theorem bijective_lift_piRingHom_algebraMap_comp_piEvalRingHom [IsLocalization M S'] [Finite ι] :
+    Function.Bijective (lift (S := S') (isUnit_piRingHom_algebraMap_comp_piEvalRingHom R S M)) :=
+  have := (iff_map_piEvalRingHom R (Π i, S i) M).mpr inferInstance
+  (ringEquivOfRingEquiv (M := M) (T := M) _ _ (.refl _) <|
+    Submonoid.map_equiv_eq_comap_symm _ _).bijective
+
+end IsLocalization
diff --git a/Mathlib/RingTheory/Nakayama.lean b/Mathlib/RingTheory/Nakayama.lean
index ad4d01141b8f1f..55c8610c70b934 100644
--- a/Mathlib/RingTheory/Nakayama.lean
+++ b/Mathlib/RingTheory/Nakayama.lean
@@ -16,13 +16,13 @@ This file contains some alternative statements of Nakayama's Lemma as found in
 ## Main statements
 
 * `Submodule.eq_smul_of_le_smul_of_le_jacobson` - A version of (2) in
-  [Stacks: Nakayama's Lemma](https://stacks.math.columbia.edu/tag/00DV).,
+  [Stacks: Nakayama's Lemma](https://stacks.math.columbia.edu/tag/00DV),
   generalising to the Jacobson of any ideal.
 * `Submodule.eq_bot_of_le_smul_of_le_jacobson_bot` - Statement (2) in
   [Stacks: Nakayama's Lemma](https://stacks.math.columbia.edu/tag/00DV).
 
 * `Submodule.sup_smul_eq_sup_smul_of_le_smul_of_le_jacobson` - A version of (4) in
-  [Stacks: Nakayama's Lemma](https://stacks.math.columbia.edu/tag/00DV).,
+  [Stacks: Nakayama's Lemma](https://stacks.math.columbia.edu/tag/00DV),
   generalising to the Jacobson of any ideal.
 * `Submodule.smul_le_of_le_smul_of_le_jacobson_bot` - Statement (4) in
   [Stacks: Nakayama's Lemma](https://stacks.math.columbia.edu/tag/00DV).
diff --git a/Mathlib/RingTheory/NonUnitalSubring/Basic.lean b/Mathlib/RingTheory/NonUnitalSubring/Basic.lean
index 35474da03eabb6..ecf3da187d103a 100644
--- a/Mathlib/RingTheory/NonUnitalSubring/Basic.lean
+++ b/Mathlib/RingTheory/NonUnitalSubring/Basic.lean
@@ -342,6 +342,17 @@ instance center.instNonUnitalCommRing : NonUnitalCommRing (center R) :=
   { NonUnitalSubsemiring.center.instNonUnitalCommSemiring R,
     inferInstanceAs <| NonUnitalNonAssocRing (center R) with }
 
+variable {R}
+
+/-- The center of isomorphic (not necessarily unital or associative) rings are isomorphic. -/
+@[simps!] def centerCongr {S} [NonUnitalNonAssocRing S] (e : R ≃+* S) : center R ≃+* center S :=
+  NonUnitalSubsemiring.centerCongr e
+
+/-- The center of a (not necessarily uintal or associative) ring
+is isomorphic to the center of its opposite. -/
+@[simps!] def centerToMulOpposite : center R ≃+* center Rᵐᵒᵖ :=
+  NonUnitalSubsemiring.centerToMulOpposite
+
 end NonUnitalNonAssocRing
 
 section NonUnitalRing
diff --git a/Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean b/Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
index 61c5c358fc66c0..76b1cd54c68799 100644
--- a/Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
+++ b/Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
@@ -10,8 +10,10 @@ import Mathlib.Algebra.Group.Subsemigroup.Operations
 import Mathlib.Algebra.GroupWithZero.Center
 import Mathlib.Algebra.Ring.Center
 import Mathlib.Algebra.Ring.Centralizer
+import Mathlib.Algebra.Ring.Opposite
 import Mathlib.Algebra.Ring.Prod
 import Mathlib.Data.Set.Finite.Range
+import Mathlib.GroupTheory.Submonoid.Center
 import Mathlib.GroupTheory.Subsemigroup.Centralizer
 import Mathlib.RingTheory.NonUnitalSubsemiring.Defs
 
@@ -245,6 +247,19 @@ lemma _root_.Set.mem_center_iff_addMonoidHom (a : R) :
   rw [Set.mem_center_iff, isMulCentral_iff]
   simp [DFunLike.ext_iff]
 
+variable {R}
+
+/-- The center of isomorphic (not necessarily unital or associative) semirings are isomorphic. -/
+@[simps!] def centerCongr [NonUnitalNonAssocSemiring S] (e : R ≃+* S) : center R ≃+* center S where
+  __ := Subsemigroup.centerCongr e
+  map_add' _ _ := Subtype.ext <| by exact map_add e ..
+
+/-- The center of a (not necessarily unital or associative) semiring
+is isomorphic to the center of its opposite. -/
+@[simps!] def centerToMulOpposite : center R ≃+* center Rᵐᵒᵖ where
+  __ := Subsemigroup.centerToMulOpposite
+  map_add' _ _ := rfl
+
 end NonUnitalNonAssocSemiring
 
 section NonUnitalSemiring
diff --git a/Mathlib/RingTheory/Polynomial/Basic.lean b/Mathlib/RingTheory/Polynomial/Basic.lean
index 682fe26b642a96..27ce39be124cc5 100644
--- a/Mathlib/RingTheory/Polynomial/Basic.lean
+++ b/Mathlib/RingTheory/Polynomial/Basic.lean
@@ -860,7 +860,7 @@ end MvPolynomial
 
 end Prime
 
-/-- Hilbert basis theorem: a polynomial ring over a noetherian ring is a noetherian ring. -/
+/-- **Hilbert basis theorem**: a polynomial ring over a Noetherian ring is a Noetherian ring. -/
 protected theorem Polynomial.isNoetherianRing [inst : IsNoetherianRing R] : IsNoetherianRing R[X] :=
   isNoetherianRing_iff.2
     ⟨fun I : Ideal R[X] =>
diff --git a/Mathlib/RingTheory/RingHomProperties.lean b/Mathlib/RingTheory/RingHomProperties.lean
index c2ac311deb3894..ae523c83039b31 100644
--- a/Mathlib/RingTheory/RingHomProperties.lean
+++ b/Mathlib/RingTheory/RingHomProperties.lean
@@ -48,14 +48,13 @@ theorem RespectsIso.cancel_left_isIso (hP : RespectsIso @P) {R S T : CommRingCat
     (g : S ⟶ T) [IsIso f] : P (g.hom.comp f.hom) ↔ P g.hom :=
   ⟨fun H => by
     convert hP.2 (f ≫ g).hom (asIso f).symm.commRingCatIsoToRingEquiv H
-    exact (IsIso.inv_hom_id_assoc _ _).symm, hP.2 g.hom (asIso f).commRingCatIsoToRingEquiv⟩
+    simp [← CommRingCat.hom_comp], hP.2 g.hom (asIso f).commRingCatIsoToRingEquiv⟩
 
 theorem RespectsIso.cancel_right_isIso (hP : RespectsIso @P) {R S T : CommRingCat} (f : R ⟶ S)
     (g : S ⟶ T) [IsIso g] : P (g.hom.comp f.hom) ↔ P f.hom :=
   ⟨fun H => by
     convert hP.1 (f ≫ g).hom (asIso g).symm.commRingCatIsoToRingEquiv H
-    change f = f ≫ g ≫ inv g
-    simp, hP.1 f.hom (asIso g).commRingCatIsoToRingEquiv⟩
+    simp [← CommRingCat.hom_comp], hP.1 f.hom (asIso g).commRingCatIsoToRingEquiv⟩
 
 theorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type u}
     (R' S' : Type u) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']
diff --git a/Mathlib/RingTheory/Smooth/Basic.lean b/Mathlib/RingTheory/Smooth/Basic.lean
index 76b6df39845a51..9ccb330ef6a71d 100644
--- a/Mathlib/RingTheory/Smooth/Basic.lean
+++ b/Mathlib/RingTheory/Smooth/Basic.lean
@@ -340,12 +340,9 @@ section
 variable (R : Type u) [CommSemiring R]
 variable (A : Type u) [Semiring A] [Algebra R A]
 
-/-- An `R` algebra `A` is smooth if it is formally smooth and of finite presentation.
-
-In the stacks project, the definition of smooth is completely different, and tag
-<https://stacks.math.columbia.edu/tag/00TN> proves that their definition is equivalent
-to this.
--/
+/-- An `R` algebra `A` is smooth if it is formally smooth and of finite presentation. -/
+@[stacks 00T2 "In the stacks project, the definition of smooth is completely different, and tag
+<https://stacks.math.columbia.edu/tag/00TN> proves that their definition is equivalent to this."]
 class Smooth [CommSemiring R] (A : Type u) [Semiring A] [Algebra R A] : Prop where
   formallySmooth : FormallySmooth R A := by infer_instance
   finitePresentation : FinitePresentation R A := by infer_instance
diff --git a/Mathlib/RingTheory/Spectrum/Maximal/Localization.lean b/Mathlib/RingTheory/Spectrum/Maximal/Localization.lean
index d0723e00ec9b57..b41a0bebd9a729 100644
--- a/Mathlib/RingTheory/Spectrum/Maximal/Localization.lean
+++ b/Mathlib/RingTheory/Spectrum/Maximal/Localization.lean
@@ -15,8 +15,6 @@ Localization results.
 
 noncomputable section
 
-open scoped Classical
-
 variable (R S P : Type*) [CommSemiring R] [CommSemiring S] [CommSemiring P]
 
 namespace MaximalSpectrum
@@ -121,6 +119,7 @@ variable {ι} (R : ι → Type*) [∀ i, CommSemiring (R i)] [∀ i, Nontrivial
 
 theorem toPiLocalization_not_surjective_of_infinite [Infinite ι] :
     ¬ Function.Surjective (toPiLocalization (Π i, R i)) := fun surj ↦ by
+  classical
   have ⟨J, max, nmem⟩ := PrimeSpectrum.exists_maximal_nmem_range_sigmaToPi_of_infinite R
   obtain ⟨r, hr⟩ := surj (Function.update 0 ⟨J, max⟩ 1)
   have : r = 0 := funext fun i ↦ toPiLocalization_injective _ <| funext fun I ↦ by
@@ -172,6 +171,7 @@ localizations at maximal ideals. -/
 def piLocalizationToMaximal : PiLocalization R →+* MaximalSpectrum.PiLocalization R :=
   Pi.ringHom fun I ↦ Pi.evalRingHom _ I.toPrimeSpectrum
 
+open scoped Classical in
 theorem piLocalizationToMaximal_surjective : Function.Surjective (piLocalizationToMaximal R) :=
   fun r ↦ ⟨fun I ↦ if h : I.1.IsMaximal then r ⟨_, h⟩ else 0, funext fun _ ↦ dif_pos _⟩
 
@@ -197,6 +197,7 @@ variable {S}
 
 theorem isMaximal_of_toPiLocalization_surjective (surj : Function.Surjective (toPiLocalization R))
     (I : PrimeSpectrum R) : I.1.IsMaximal := by
+  classical
   have ⟨J, max, le⟩ := I.1.exists_le_maximal I.2.ne_top
   obtain ⟨r, hr⟩ := surj (Function.update 0 ⟨J, max.isPrime⟩ 1)
   by_contra h
diff --git a/Mathlib/RingTheory/Spectrum/Prime/IsOpenComapC.lean b/Mathlib/RingTheory/Spectrum/Prime/IsOpenComapC.lean
index 4a6c542cd56932..2117b0ff267734 100644
--- a/Mathlib/RingTheory/Spectrum/Prime/IsOpenComapC.lean
+++ b/Mathlib/RingTheory/Spectrum/Prime/IsOpenComapC.lean
@@ -59,11 +59,8 @@ theorem imageOfDf_eq_comap_C_compl_zeroLocus :
   · rintro ⟨xli, complement, rfl⟩
     exact comap_C_mem_imageOfDf complement
 
-/-- The morphism `C⁺ : Spec R[x] → Spec R` is open.
-Stacks Project "Lemma 00FB", first part.
-
-https://stacks.math.columbia.edu/tag/00FB
--/
+/-- The morphism `C⁺ : Spec R[x] → Spec R` is open. -/
+@[stacks 00FB "First part"]
 theorem isOpenMap_comap_C : IsOpenMap (PrimeSpectrum.comap (C : R →+* R[X])) := by
   rintro U ⟨s, z⟩
   rw [← compl_compl U, ← z, ← iUnion_of_singleton_coe s, zeroLocus_iUnion, compl_iInter,
diff --git a/Mathlib/RingTheory/Spectrum/Prime/Topology.lean b/Mathlib/RingTheory/Spectrum/Prime/Topology.lean
index 52f45bb2373677..c846ca364238f1 100644
--- a/Mathlib/RingTheory/Spectrum/Prime/Topology.lean
+++ b/Mathlib/RingTheory/Spectrum/Prime/Topology.lean
@@ -976,11 +976,9 @@ section CommSemiring
 variable [CommSemiring R]
 
 open PrimeSpectrum in
-/--
-[Stacks: Lemma 00ES (3)](https://stacks.math.columbia.edu/tag/00ES)
-Zero loci of minimal prime ideals of `R` are irreducible components in `Spec R` and any
-irreducible component is a zero locus of some minimal prime ideal.
--/
+/-- Zero loci of minimal prime ideals of `R` are irreducible components in `Spec R` and any
+irreducible component is a zero locus of some minimal prime ideal. -/
+@[stacks 00ES]
 protected def minimalPrimes.equivIrreducibleComponents :
     minimalPrimes R ≃o (irreducibleComponents <| PrimeSpectrum R)ᵒᵈ := by
   let e : {p : Ideal R | p.IsPrime ∧ ⊥ ≤ p} ≃o PrimeSpectrum R :=
diff --git a/Mathlib/RingTheory/TensorProduct/Basic.lean b/Mathlib/RingTheory/TensorProduct/Basic.lean
index 23244caec6a64f..d1e5a3d2fd7c12 100644
--- a/Mathlib/RingTheory/TensorProduct/Basic.lean
+++ b/Mathlib/RingTheory/TensorProduct/Basic.lean
@@ -636,11 +636,9 @@ tensors can be directly applied by the caller (without needing `TensorProduct.on
 def algHomOfLinearMapTensorProduct (f : A ⊗[R] B →ₗ[S] C)
     (h_mul : ∀ (a₁ a₂ : A) (b₁ b₂ : B), f ((a₁ * a₂) ⊗ₜ (b₁ * b₂)) = f (a₁ ⊗ₜ b₁) * f (a₂ ⊗ₜ b₂))
     (h_one : f (1 ⊗ₜ[R] 1) = 1) : A ⊗[R] B →ₐ[S] C :=
-  #adaptation_note
-  /--
-  After https://github.com/leanprover/lean4/pull/4119 we either need to specify
-  the `(R := S) (A := A ⊗[R] B)` arguments, or use `set_option maxSynthPendingDepth 2 in`.
-  -/
+  #adaptation_note /-- https://github.com/leanprover/lean4/pull/4119
+  we either need to specify the `(R := S) (A := A ⊗[R] B)` arguments,
+  or use `set_option maxSynthPendingDepth 2 in`. -/
   AlgHom.ofLinearMap f h_one <| (f.map_mul_iff (R := S) (A := A ⊗[R] B)).2 <| by
     -- these instances are needed by the statement of `ext`, but not by the current definition.
     letI : Algebra R C := RestrictScalars.algebra R S C
diff --git a/Mathlib/RingTheory/Unramified/Basic.lean b/Mathlib/RingTheory/Unramified/Basic.lean
index f24642133de7fa..8d6c5320108197 100644
--- a/Mathlib/RingTheory/Unramified/Basic.lean
+++ b/Mathlib/RingTheory/Unramified/Basic.lean
@@ -282,12 +282,9 @@ section
 variable (R : Type*) [CommRing R]
 variable (A : Type*) [CommRing A] [Algebra R A]
 
-/-- An `R`-algebra `A` is unramified if it is formally unramified and of finite type.
-
-Note that the Stacks project has a different definition of unramified, and tag
-<https://stacks.math.columbia.edu/tag/00UU> shows that their definition is the
-same as this one.
--/
+/-- An `R`-algebra `A` is unramified if it is formally unramified and of finite type. -/
+@[stacks 00UT "Note that the Stacks project has a different definition of unramified, and tag
+<https://stacks.math.columbia.edu/tag/00UU> shows that their definition is the same as this one."]
 class Unramified : Prop where
   formallyUnramified : FormallyUnramified R A := by infer_instance
   finiteType : FiniteType R A := by infer_instance
diff --git a/Mathlib/RingTheory/Valuation/Basic.lean b/Mathlib/RingTheory/Valuation/Basic.lean
index 2d1fe63a3a6d8b..6a320028e373a0 100644
--- a/Mathlib/RingTheory/Valuation/Basic.lean
+++ b/Mathlib/RingTheory/Valuation/Basic.lean
@@ -253,20 +253,6 @@ lemma map_apply (f : Γ₀ →*₀ Γ'₀) (hf : Monotone f) (v : Valuation R Γ
 def IsEquiv (v₁ : Valuation R Γ₀) (v₂ : Valuation R Γ'₀) : Prop :=
   ∀ r s, v₁ r ≤ v₁ s ↔ v₂ r ≤ v₂ s
 
-/-- An ordered monoid isomorphism `Γ₀ ≃ Γ'₀` induces an equivalence
-`Valuation R Γ₀ ≃ Valuation R Γ'₀`. -/
-def congr (f : Γ₀ ≃*o Γ'₀) : Valuation R Γ₀ ≃ Valuation R Γ'₀ where
-  toFun := map f f.toOrderIso.monotone
-  invFun := map f.symm f.toOrderIso.symm.monotone
-  left_inv ν := by ext; simp
-  right_inv ν := by ext; simp
-
-end Monoid
-
-section Group
-
-variable [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x y : R}
-
 @[simp]
 theorem map_neg (x : R) : v (-x) = v x :=
   v.toMonoidWithZeroHom.toMonoidHom.map_neg x
@@ -274,12 +260,6 @@ theorem map_neg (x : R) : v (-x) = v x :=
 theorem map_sub_swap (x y : R) : v (x - y) = v (y - x) :=
   v.toMonoidWithZeroHom.toMonoidHom.map_sub_swap x y
 
-theorem map_inv {R : Type*} [DivisionRing R] (v : Valuation R Γ₀) : ∀ x, v x⁻¹ = (v x)⁻¹ :=
-  map_inv₀ _
-
-theorem map_div {R : Type*} [DivisionRing R] (v : Valuation R Γ₀) : ∀ x y, v (x / y) = v x / v y :=
-  map_div₀ _
-
 theorem map_sub (x y : R) : v (x - y) ≤ max (v x) (v y) :=
   calc
     v (x - y) = v (x + -y) := by rw [sub_eq_add_neg]
@@ -290,6 +270,8 @@ theorem map_sub_le {x y g} (hx : v x ≤ g) (hy : v y ≤ g) : v (x - y) ≤ g :
   rw [sub_eq_add_neg]
   exact v.map_add_le hx (le_trans (le_of_eq (v.map_neg y)) hy)
 
+variable {x y : R}
+
 theorem map_add_of_distinct_val (h : v x ≠ v y) : v (x + y) = max (v x) (v y) := by
   suffices ¬v (x + y) < max (v x) (v y) from
     or_iff_not_imp_right.1 (le_iff_eq_or_lt.1 (v.map_add x y)) this
@@ -339,6 +321,26 @@ theorem map_one_sub_of_lt (h : v x < 1) : v (1 - x) = 1 := by
   rw [sub_eq_add_neg 1 x]
   simpa only [v.map_one, v.map_neg] using v.map_add_eq_of_lt_left h
 
+/-- An ordered monoid isomorphism `Γ₀ ≃ Γ'₀` induces an equivalence
+`Valuation R Γ₀ ≃ Valuation R Γ'₀`. -/
+def congr (f : Γ₀ ≃*o Γ'₀) : Valuation R Γ₀ ≃ Valuation R Γ'₀ where
+  toFun := map f f.toOrderIso.monotone
+  invFun := map f.symm f.toOrderIso.symm.monotone
+  left_inv ν := by ext; simp
+  right_inv ν := by ext; simp
+
+end Monoid
+
+section Group
+
+variable [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x y : R}
+
+theorem map_inv {R : Type*} [DivisionRing R] (v : Valuation R Γ₀) : ∀ x, v x⁻¹ = (v x)⁻¹ :=
+  map_inv₀ _
+
+theorem map_div {R : Type*} [DivisionRing R] (v : Valuation R Γ₀) : ∀ x y, v (x / y) = v x / v y :=
+  map_div₀ _
+
 theorem one_lt_val_iff (v : Valuation K Γ₀) {x : K} (h : x ≠ 0) : 1 < v x ↔ v x⁻¹ < 1 := by
   simp [inv_lt_one₀ (v.pos_iff.2 h)]
 
@@ -782,20 +784,6 @@ lemma map_apply (f : Γ₀ →+ Γ'₀) (ht : f ⊤ = ⊤) (hf : Monotone f) (v
 def IsEquiv (v₁ : AddValuation R Γ₀) (v₂ : AddValuation R Γ'₀) : Prop :=
   Valuation.IsEquiv v₁ v₂
 
-end Monoid
-
-section Group
-
-variable [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) {x y : R}
-
-@[simp]
-theorem map_inv (v : AddValuation K Γ₀) {x : K} : v x⁻¹ = - (v x) :=
-  map_inv₀ (toValuation v) x
-
-@[simp]
-theorem map_div (v : AddValuation K Γ₀) {x y : K} : v (x / y) = v x - v y :=
-  map_div₀ (toValuation v) x y
-
 @[simp]
 theorem map_neg (x : R) : v (-x) = v x :=
   Valuation.map_neg v x
@@ -809,6 +797,8 @@ theorem map_sub (x y : R) : min (v x) (v y) ≤ v (x - y) :=
 theorem map_le_sub {x y : R} {g : Γ₀} (hx : g ≤ v x) (hy : g ≤ v y) : g ≤ v (x - y) :=
   Valuation.map_sub_le v hx hy
 
+variable {x y : R}
+
 theorem map_add_of_distinct_val (h : v x ≠ v y) : v (x + y) = @Min.min Γ₀ _ (v x) (v y) :=
   Valuation.map_add_of_distinct_val v h
 
@@ -831,6 +821,20 @@ theorem map_sub_eq_of_lt_right {x y : R} (hx : v y < v x) :
 theorem map_eq_of_lt_sub (h : v x < v (y - x)) : v y = v x :=
   Valuation.map_eq_of_sub_lt v h
 
+end Monoid
+
+section Group
+
+variable [LinearOrderedAddCommGroupWithTop Γ₀] [Ring R] (v : AddValuation R Γ₀) {x y : R}
+
+@[simp]
+theorem map_inv (v : AddValuation K Γ₀) {x : K} : v x⁻¹ = - (v x) :=
+  map_inv₀ (toValuation v) x
+
+@[simp]
+theorem map_div (v : AddValuation K Γ₀) {x y : K} : v (x / y) = v x - v y :=
+  map_div₀ (toValuation v) x y
+
 end Group
 
 end Basic
diff --git a/Mathlib/SetTheory/Descriptive/Tree.lean b/Mathlib/SetTheory/Descriptive/Tree.lean
new file mode 100644
index 00000000000000..43a7f2f810ce28
--- /dev/null
+++ b/Mathlib/SetTheory/Descriptive/Tree.lean
@@ -0,0 +1,46 @@
+/-
+Copyright (c) 2024 Sven Manthe. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Sven Manthe
+-/
+import Mathlib.Order.CompleteLattice.SetLike
+
+/-!
+# Trees in the sense of descriptive set theory
+
+This file defines trees of depth `ω` in the sense of descriptive set theory as sets of finite
+sequences that are stable under taking prefixes.
+
+## Main declarations
+
+* `tree A`: a (possibly infinite) tree of depth at most `ω` with nodes in `A`
+-/
+
+namespace Descriptive
+
+/-- A tree is a set of finite sequences, implemented as `List A`, that is stable under
+  taking prefixes. For the definition we use the equivalent property `x ++ [a] ∈ T → x ∈ T`,
+  which is more convenient to check. We define `tree A` as a complete sublattice of
+  `Set (List A)`, which coerces to the type of trees on `A`. -/
+def tree (A : Type*) : CompleteSublattice (Set (List A)) :=
+  CompleteSublattice.mk' {T | ∀ ⦃x : List A⦄ ⦃a : A⦄, x ++ [a] ∈ T → x ∈ T}
+    (by rintro S hS x a ⟨t, ht, hx⟩; use t, ht, hS ht hx)
+    (by rintro S hS x a h T hT; exact hS hT <| h T hT)
+
+instance (A : Type*) : SetLike (tree A) (List A) := SetLike.instSubtypeSet
+
+variable {A : Type*} {T : tree A}
+
+lemma mem_of_append {x y : List A} (h : x ++ y ∈ T) : x ∈ T := by
+  induction' y with y ys ih generalizing x
+  · simpa using h
+  · exact T.prop (ih (by simpa))
+
+lemma mem_of_prefix {x y : List A} (h' : x <+: y) (h : y ∈ T) : x ∈ T := by
+  obtain ⟨_, rfl⟩ := h'; exact mem_of_append h
+
+@[simp] lemma tree_eq_bot : T = ⊥ ↔ [] ∉ T where
+  mp := by rintro rfl; simp
+  mpr h := by ext x; simpa using fun h' ↦ h <| mem_of_prefix x.nil_prefix h'
+
+end Descriptive
diff --git a/Mathlib/SetTheory/Game/Impartial.lean b/Mathlib/SetTheory/Game/Impartial.lean
index e06e5beb85b350..2d6cde0fec9aa1 100644
--- a/Mathlib/SetTheory/Game/Impartial.lean
+++ b/Mathlib/SetTheory/Game/Impartial.lean
@@ -103,16 +103,11 @@ termination_by G
 variable (G : PGame) [Impartial G]
 
 theorem nonpos : ¬0 < G := by
-  intro h
-  have h' := neg_lt_neg_iff.2 h
-  rw [neg_zero, lt_congr_left (Equiv.symm (neg_equiv_self G))] at h'
-  exact (h.trans h').false
+  apply (lt_asymm · ?_)
+  rwa [← neg_lt_neg_iff, neg_zero, ← lt_congr_right (neg_equiv_self G)]
 
 theorem nonneg : ¬G < 0 := by
-  intro h
-  have h' := neg_lt_neg_iff.2 h
-  rw [neg_zero, lt_congr_right (Equiv.symm (neg_equiv_self G))] at h'
-  exact (h.trans h').false
+  simpa using nonpos (-G)
 
 /-- In an impartial game, either the first player always wins, or the second player always wins. -/
 theorem equiv_or_fuzzy_zero : (G ≈ 0) ∨ G ‖ 0 := by
diff --git a/Mathlib/SetTheory/Game/PGame.lean b/Mathlib/SetTheory/Game/PGame.lean
index 8186fa7fea3ebe..ac6caee7a31665 100644
--- a/Mathlib/SetTheory/Game/PGame.lean
+++ b/Mathlib/SetTheory/Game/PGame.lean
@@ -1363,6 +1363,26 @@ theorem moveRight_neg_symm {x : PGame} (i) :
 theorem moveRight_neg_symm' {x : PGame} (i) :
     x.moveRight i = -(-x).moveLeft (toLeftMovesNeg i) := by simp
 
+@[simp]
+theorem forall_leftMoves_neg {x : PGame} {p : (-x).LeftMoves → Prop} :
+    (∀ i : (-x).LeftMoves, p i) ↔ (∀ i : x.RightMoves, p (toLeftMovesNeg i)) :=
+  toLeftMovesNeg.forall_congr_right.symm
+
+@[simp]
+theorem exists_leftMoves_neg {x : PGame} {p : (-x).LeftMoves → Prop} :
+    (∃ i : (-x).LeftMoves, p i) ↔ (∃ i : x.RightMoves, p (toLeftMovesNeg i)) :=
+  toLeftMovesNeg.exists_congr_right.symm
+
+@[simp]
+theorem forall_rightMoves_neg {x : PGame} {p : (-x).RightMoves → Prop} :
+    (∀ i : (-x).RightMoves, p i) ↔ (∀ i : x.LeftMoves, p (toRightMovesNeg i)) :=
+  toRightMovesNeg.forall_congr_right.symm
+
+@[simp]
+theorem exists_rightMoves_neg {x : PGame} {p : (-x).RightMoves → Prop} :
+    (∃ i : (-x).RightMoves, p i) ↔ (∃ i : x.LeftMoves, p (toRightMovesNeg i)) :=
+  toRightMovesNeg.exists_congr_right.symm
+
 theorem leftMoves_neg_cases {x : PGame} (k) {P : (-x).LeftMoves → Prop}
     (h : ∀ i, P <| toLeftMovesNeg i) :
     P k := by
diff --git a/Mathlib/SetTheory/Ordinal/Exponential.lean b/Mathlib/SetTheory/Ordinal/Exponential.lean
index 59b0607db2df3b..c58e66d7ac7e64 100644
--- a/Mathlib/SetTheory/Ordinal/Exponential.lean
+++ b/Mathlib/SetTheory/Ordinal/Exponential.lean
@@ -21,16 +21,24 @@ universe u v w
 
 namespace Ordinal
 
-/-- The ordinal exponential, defined by transfinite recursion. -/
-instance pow : Pow Ordinal Ordinal :=
-  ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩
+/-- The ordinal exponential, defined by transfinite recursion.
 
-theorem opow_def (a b : Ordinal) :
-    a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b :=
-  rfl
+We call this `opow` in theorems in order to disambiguate from other exponentials. -/
+instance instPow : Pow Ordinal Ordinal :=
+  ⟨fun a b ↦ if a = 0 then 1 - b else
+    limitRecOn b 1 (fun _ x ↦ x * a) fun o _ f ↦ ⨆ x : Iio o, f x.1 x.2⟩
+
+private theorem opow_of_ne_zero {a b : Ordinal} (h : a ≠ 0) : a ^ b =
+    limitRecOn b 1 (fun _ x ↦ x * a) fun o _ f ↦ ⨆ x : Iio o, f x.1 x.2 :=
+  if_neg h
 
--- Porting note: `if_pos rfl` → `if_true`
-theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a := by simp only [opow_def, if_true]
+/-- `0 ^ a = 1` if `a = 0` and `0 ^ a = 0` otherwise. -/
+theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a :=
+  if_pos rfl
+
+theorem zero_opow_le (a : Ordinal) : (0 : Ordinal) ^ a ≤ 1 := by
+  rw [zero_opow']
+  exact sub_le_self 1 a
 
 @[simp]
 theorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0 := by
@@ -38,29 +46,34 @@ theorem zero_opow {a : Ordinal} (a0 : a ≠ 0) : (0 : Ordinal) ^ a = 0 := by
 
 @[simp]
 theorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1 := by
-  by_cases h : a = 0
-  · simp only [opow_def, if_pos h, sub_zero]
-  · simp only [opow_def, if_neg h, limitRecOn_zero]
+  obtain rfl | h := eq_or_ne a 0
+  · rw [zero_opow', Ordinal.sub_zero]
+  · rw [opow_of_ne_zero h, limitRecOn_zero]
 
 @[simp]
-theorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a :=
-  if h : a = 0 then by subst a; simp only [zero_opow (succ_ne_zero _), mul_zero]
-  else by simp only [opow_def, limitRecOn_succ, if_neg h]
+theorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a := by
+  obtain rfl | h := eq_or_ne a 0
+  · rw [zero_opow (succ_ne_zero b), mul_zero]
+  · rw [opow_of_ne_zero h, opow_of_ne_zero h, limitRecOn_succ]
 
-theorem opow_limit {a b : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :
-    a ^ b = bsup.{u, u} b fun c _ => a ^ c := by
-  simp only [opow_def, if_neg a0]; rw [limitRecOn_limit _ _ _ _ h]
+theorem opow_limit {a b : Ordinal} (ha : a ≠ 0) (hb : IsLimit b) :
+    a ^ b = ⨆ x : Iio b, a ^ x.1 := by
+  simp_rw [opow_of_ne_zero ha, limitRecOn_limit _ _ _ _ hb]
 
 theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :
-    a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c := by rw [opow_limit a0 h, bsup_le_iff]
+    a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c := by
+  rw [opow_limit a0 h, Ordinal.iSup_le_iff, Subtype.forall]
+  rfl
 
 theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :
     a < b ^ c ↔ ∃ c' < c, a < b ^ c' := by
-  rw [← not_iff_not, not_exists]; simp only [not_lt, opow_le_of_limit b0 h, exists_prop, not_and]
+  rw [← not_iff_not, not_exists]
+  simp only [not_lt, opow_le_of_limit b0 h, exists_prop, not_and]
 
 @[simp]
 theorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a := by
-  rw [← succ_zero, opow_succ]; simp only [opow_zero, one_mul]
+  rw [← succ_zero, opow_succ]
+  simp only [opow_zero, one_mul]
 
 @[simp]
 theorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1 := by
@@ -317,8 +330,7 @@ theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :
 
 theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x ≠ 0) : b ^ log b x ≤ x := by
   rcases eq_or_ne b 0 with (rfl | b0)
-  · rw [zero_opow']
-    exact (sub_le_self _ _).trans (one_le_iff_ne_zero.2 hx)
+  · exact (zero_opow_le _).trans (one_le_iff_ne_zero.2 hx)
   rcases lt_or_eq_of_le (one_le_iff_ne_zero.2 b0) with (hb | rfl)
   · refine le_of_not_lt fun h => (lt_succ (log b x)).not_le ?_
     have := @csInf_le' _ _ { o | x < b ^ o } _ h
diff --git a/Mathlib/SetTheory/Ordinal/Notation.lean b/Mathlib/SetTheory/Ordinal/Notation.lean
index 9d81bb8ff98acd..acc0ddcba26a4c 100644
--- a/Mathlib/SetTheory/Ordinal/Notation.lean
+++ b/Mathlib/SetTheory/Ordinal/Notation.lean
@@ -749,11 +749,11 @@ instance nf_opow (o₁ o₂) [NF o₁] [NF o₂] : NF (o₁ ^ o₂) := by
   haveI := (nf_repr_split' e₂).1
   cases' a with a0 n a'
   · cases' m with m
-    · by_cases o₂ = 0 <;> simp only [(· ^ ·), Pow.pow, pow, opow, opowAux2, *] <;> decide
+    · by_cases o₂ = 0 <;> simp only [(· ^ ·), Pow.pow, opow, opowAux2, *] <;> decide
     · by_cases m = 0
-      · simp only [(· ^ ·), Pow.pow, pow, opow, opowAux2, *, zero_def]
+      · simp only [(· ^ ·), Pow.pow, opow, opowAux2, *, zero_def]
         decide
-      · simp only [(· ^ ·), Pow.pow, pow, opow, opowAux2, mulNat_eq_mul, ofNat, *]
+      · simp only [(· ^ ·), Pow.pow, opow, opowAux2, mulNat_eq_mul, ofNat, *]
         infer_instance
   · simp only [(· ^ ·), Pow.pow, opow, opowAux2, e₁, split_eq_scale_split' e₂, mulNat_eq_mul]
     have := na.fst
diff --git a/Mathlib/Tactic/CC/Addition.lean b/Mathlib/Tactic/CC/Addition.lean
index 4ad4dc22d38f02..2fb9ddbc68213f 100644
--- a/Mathlib/Tactic/CC/Addition.lean
+++ b/Mathlib/Tactic/CC/Addition.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura, Miyahara Kō
 -/
 import Mathlib.Data.Option.Defs
-import Mathlib.Lean.Expr.Basic
 import Mathlib.Tactic.CC.MkProof
 
 /-!
@@ -199,12 +198,11 @@ def pushSubsingletonEq (a b : Expr) : CCM Unit := do
   let B ← normalize (← inferType b)
   -- TODO(Leo): check if the following test is a performance bottleneck
   if ← pureIsDefEq A B then
-    -- TODO(Leo): to improve performance we can create the following proof lazily
-    let proof ← mkAppM ``Subsingleton.elim #[a, b]
+    let proof ← mkAppM ``FastSubsingleton.elim #[a, b]
     pushEq a b proof
   else
     let some AEqB ← getEqProof A B | failure
-    let proof ← mkAppM ``Subsingleton.helim #[AEqB, a, b]
+    let proof ← mkAppM ``FastSubsingleton.helim #[AEqB, a, b]
     pushHEq a b proof
 
 /-- Given the equivalent expressions `oldRoot` and `newRoot` the root of `oldRoot` is
@@ -940,9 +938,7 @@ partial def applySimpleEqvs (e : Expr) : CCM Unit := do
 to the todo list or register `e` as the canonical form of itself. -/
 partial def processSubsingletonElem (e : Expr) : CCM Unit := do
   let type ← inferType e
-  -- TODO: this is likely to become a bottleneck. See e.g.
-  -- https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/convert.20is.20often.20slow/near/433830798
-  let ss ← synthInstance? (← mkAppM ``Subsingleton #[type])
+  let ss ← synthInstance? (← mkAppM ``FastSubsingleton #[type])
   if ss.isNone then return -- type is not a subsingleton
   let type ← normalize type
   -- Make sure type has been internalized
diff --git a/Mathlib/Tactic/CC/Datatypes.lean b/Mathlib/Tactic/CC/Datatypes.lean
index 6dc3e10fd6aa5c..11aac8d38934ed 100644
--- a/Mathlib/Tactic/CC/Datatypes.lean
+++ b/Mathlib/Tactic/CC/Datatypes.lean
@@ -3,9 +3,9 @@ Copyright (c) 2016 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura, Miyahara Kō
 -/
-import Lean.Meta.Tactic.Rfl
 import Batteries.Data.RBMap.Basic
 import Mathlib.Lean.Meta.Basic
+import Mathlib.Lean.Meta.CongrTheorems
 import Mathlib.Data.Ordering.Basic
 
 /-!
@@ -428,7 +428,8 @@ Note that this only works for two-argument relations: `ModEq n` and `ModEq m` ar
 same. -/
 abbrev SymmCongruences := Std.HashMap SymmCongruencesKey (List (Expr × Name))
 
-/-- Stores the root representatives of subsingletons. -/
+/-- Stores the root representatives of subsingletons, this uses `FastSingleton` instead of
+`Subsingleton`. -/
 abbrev SubsingletonReprs := RBExprMap Expr
 
 /-- Stores the root representatives of `.instImplicit` arguments. -/
diff --git a/Mathlib/Tactic/CC/MkProof.lean b/Mathlib/Tactic/CC/MkProof.lean
index 5f17a75846cc5a..25140ed57c1d82 100644
--- a/Mathlib/Tactic/CC/MkProof.lean
+++ b/Mathlib/Tactic/CC/MkProof.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Leonardo de Moura, Miyahara Kō
 -/
 import Batteries.Data.RBMap.Alter
-import Mathlib.Logic.Basic
 import Mathlib.Tactic.CC.Datatypes
 import Mathlib.Tactic.CC.Lemmas
 import Mathlib.Tactic.Relation.Rfl
diff --git a/Mathlib/Tactic/ComputeDegree.lean b/Mathlib/Tactic/ComputeDegree.lean
index f94993bf7de959..f82b8e01897655 100644
--- a/Mathlib/Tactic/ComputeDegree.lean
+++ b/Mathlib/Tactic/ComputeDegree.lean
@@ -440,14 +440,14 @@ def miscomputedDegree? (deg : Expr) : List Expr → List MessageData
 *  `degree f ≤ d`,
 *  `coeff f d = r`, if `d` is the degree of `f`.
 
-The tactic may leave goals of the form `d' = d` `d' ≤ d`, or `r ≠ 0`, where `d'` in `ℕ` or
+The tactic may leave goals of the form `d' = d`, `d' ≤ d`, or `r ≠ 0`, where `d'` in `ℕ` or
 `WithBot ℕ` is the tactic's guess of the degree, and `r` is the coefficient's guess of the
 leading coefficient of `f`.
 
 `compute_degree` applies `norm_num` to the left-hand side of all side goals, trying to close them.
 
 The variant `compute_degree!` first applies `compute_degree`.
-Then it uses `norm_num` on all the whole remaining goals and tries `assumption`.
+Then it uses `norm_num` on all the remaining goals and tries `assumption`.
 -/
 syntax (name := computeDegree) "compute_degree" "!"? : tactic
 
diff --git a/Mathlib/Tactic/CongrExclamation.lean b/Mathlib/Tactic/CongrExclamation.lean
index 3bdb8ef60024a9..dfa8e21eef0fe2 100644
--- a/Mathlib/Tactic/CongrExclamation.lean
+++ b/Mathlib/Tactic/CongrExclamation.lean
@@ -425,11 +425,6 @@ def Lean.MVarId.congrImplies?' (mvarId : MVarId) : MetaM (Option (List MVarId))
       | throwError "unexpected number of goals"
     return [mvarId₁, mvarId₂]
 
-protected theorem FastSubsingleton.helim {α β : Sort u} [FastSubsingleton α]
-    (h₂ : α = β) (a : α) (b : β) : HEq a b := by
-  have : Subsingleton α := FastSubsingleton.inst
-  exact Subsingleton.helim h₂ a b
-
 /--
 Try to apply `Subsingleton.helim` if the goal is a `HEq`. Tries synthesizing a `Subsingleton`
 instance for both the LHS and the RHS.
diff --git a/Mathlib/Tactic/FunProp.lean b/Mathlib/Tactic/FunProp.lean
index 1e9f65989489ad..a52e272d5b9561 100644
--- a/Mathlib/Tactic/FunProp.lean
+++ b/Mathlib/Tactic/FunProp.lean
@@ -37,7 +37,7 @@ example (y : ℝ) (hy : y ≠ 0) : ContinuousAt (fun x : ℝ => 1/x) y := by fun
 **Basic debugging:**
 The most common issue is that a function is missing the appropriate theorem. For example:
 ```lean
-import Mathlib.Data.Complex.Exponential
+import Mathlib.Data.Complex.Trigonometric
 example : Continuous (fun x : ℝ => x * Real.sin x) := by fun_prop
 ```
 Fails with the error:
diff --git a/Mathlib/Tactic/Linter/TextBased.lean b/Mathlib/Tactic/Linter/TextBased.lean
index c0df2b14e30dad..5786f3360de338 100644
--- a/Mathlib/Tactic/Linter/TextBased.lean
+++ b/Mathlib/Tactic/Linter/TextBased.lean
@@ -51,8 +51,8 @@ deriving BEq
 /-- Possible errors that text-based linters can report. -/
 -- We collect these in one inductive type to centralise error reporting.
 inductive StyleError where
-  /-- The bare string "Adaptation note" (or variants thereof): instead, the
-  #adaptation_note command should be used. -/
+  /-- The bare string "Adaptation note" (or variants thereof):
+  instead, the #adaptation_note command should be used. -/
   | adaptationNote
   /-- A line ends with windows line endings (\r\n) instead of unix ones (\n). -/
   | windowsLineEnding
diff --git a/Mathlib/Tactic/NormNum/Basic.lean b/Mathlib/Tactic/NormNum/Basic.lean
index a1d9e39f9a55bd..91ce6bebb6980a 100644
--- a/Mathlib/Tactic/NormNum/Basic.lean
+++ b/Mathlib/Tactic/NormNum/Basic.lean
@@ -25,14 +25,6 @@ See other files in this directory for many more plugins.
 
 universe u
 
-#adaptation_note
-/--
-Since https://github.com/leanprover/lean4/pull/5338,
-the unused variable linter can not see usages of variables in
-`haveI' : ⋯ =Q ⋯ := ⟨⟩` clauses, so generates many false positives.
--/
-set_option linter.unusedVariables false
-
 namespace Mathlib
 open Lean
 open Meta
diff --git a/Mathlib/Tactic/NormNum/DivMod.lean b/Mathlib/Tactic/NormNum/DivMod.lean
index 90f6938e16c6ba..c07a80e7e9788d 100644
--- a/Mathlib/Tactic/NormNum/DivMod.lean
+++ b/Mathlib/Tactic/NormNum/DivMod.lean
@@ -13,14 +13,6 @@ This file adds support for the `%`, `/`, and `∣` (divisibility) operators on `
 to the `norm_num` tactic.
 -/
 
-#adaptation_note
-/--
-Since https://github.com/leanprover/lean4/pull/5338,
-the unused variable linter can not see usages of variables in
-`haveI' : ⋯ =Q ⋯ := ⟨⟩` clauses, so generates many false positives.
--/
-set_option linter.unusedVariables false
-
 namespace Mathlib
 open Lean
 open Meta
diff --git a/Mathlib/Tactic/NormNum/Eq.lean b/Mathlib/Tactic/NormNum/Eq.lean
index d467ab0571729b..277f0479a4a029 100644
--- a/Mathlib/Tactic/NormNum/Eq.lean
+++ b/Mathlib/Tactic/NormNum/Eq.lean
@@ -11,14 +11,6 @@ import Mathlib.Tactic.NormNum.Inv
 
 variable {α : Type*}
 
-#adaptation_note
-/--
-Since https://github.com/leanprover/lean4/pull/5338,
-the unused variable linter can not see usages of variables in
-`haveI' : ⋯ =Q ⋯ := ⟨⟩` clauses, so generates many false positives.
--/
-set_option linter.unusedVariables false
-
 open Lean Meta Qq
 
 namespace Mathlib.Meta.NormNum
diff --git a/Mathlib/Tactic/NormNum/GCD.lean b/Mathlib/Tactic/NormNum/GCD.lean
index c9bcc4f4c2c986..cb0a2aa2d169a4 100644
--- a/Mathlib/Tactic/NormNum/GCD.lean
+++ b/Mathlib/Tactic/NormNum/GCD.lean
@@ -16,14 +16,6 @@ Note that `Nat.coprime` is reducible and defined in terms of `Nat.gcd`, so the `
 also indirectly provides a `Nat.coprime` extension.
 -/
 
-#adaptation_note
-/--
-Since https://github.com/leanprover/lean4/pull/5338,
-the unused variable linter can not see usages of variables in
-`haveI' : ⋯ =Q ⋯ := ⟨⟩` clauses, so generates many false positives.
--/
-set_option linter.unusedVariables false
-
 namespace Tactic
 
 namespace NormNum
diff --git a/Mathlib/Tactic/NormNum/Ineq.lean b/Mathlib/Tactic/NormNum/Ineq.lean
index 44013a8b76cedb..0c27395a582bef 100644
--- a/Mathlib/Tactic/NormNum/Ineq.lean
+++ b/Mathlib/Tactic/NormNum/Ineq.lean
@@ -13,14 +13,6 @@ import Mathlib.Algebra.Order.Ring.Cast
 # `norm_num` extensions for inequalities.
 -/
 
-#adaptation_note
-/--
-Since https://github.com/leanprover/lean4/pull/5338,
-the unused variable linter can not see usages of variables in
-`haveI' : ⋯ =Q ⋯ := ⟨⟩` clauses, so generates many false positives.
--/
-set_option linter.unusedVariables false
-
 open Lean Meta Qq
 
 namespace Mathlib.Meta.NormNum
diff --git a/Mathlib/Tactic/NormNum/Inv.lean b/Mathlib/Tactic/NormNum/Inv.lean
index c722b8cf4c5870..82c34adb85e5a1 100644
--- a/Mathlib/Tactic/NormNum/Inv.lean
+++ b/Mathlib/Tactic/NormNum/Inv.lean
@@ -13,14 +13,6 @@ import Mathlib.Algebra.Field.Basic
 
 variable {u : Lean.Level}
 
-#adaptation_note
-/--
-Since https://github.com/leanprover/lean4/pull/5338,
-the unused variable linter can not see usages of variables in
-`haveI' : ⋯ =Q ⋯ := ⟨⟩` clauses, so generates many false positives.
--/
-set_option linter.unusedVariables false
-
 namespace Mathlib.Meta.NormNum
 
 open Lean.Meta Qq
diff --git a/Mathlib/Tactic/NormNum/LegendreSymbol.lean b/Mathlib/Tactic/NormNum/LegendreSymbol.lean
index 166fc060cfcdca..dfb3def6ba7c4e 100644
--- a/Mathlib/Tactic/NormNum/LegendreSymbol.lean
+++ b/Mathlib/Tactic/NormNum/LegendreSymbol.lean
@@ -356,14 +356,6 @@ namespace NormNum
 
 open Lean Elab Tactic Qq Mathlib.Meta.NormNum
 
-#adaptation_note
-/--
-Since https://github.com/leanprover/lean4/pull/5338,
-the unused variable linter can not see usages of variables in
-`haveI' : ⋯ =Q ⋯ := ⟨⟩` clauses, so generates many false positives.
--/
-set_option linter.unusedVariables false
-
 /-- This is the `norm_num` plug-in that evaluates Jacobi symbols. -/
 @[norm_num jacobiSym _ _]
 def evalJacobiSym : NormNumExt where eval {u α} e := do
diff --git a/Mathlib/Tactic/NormNum/Pow.lean b/Mathlib/Tactic/NormNum/Pow.lean
index 467a97ad68b3f0..72ced934fd53d0 100644
--- a/Mathlib/Tactic/NormNum/Pow.lean
+++ b/Mathlib/Tactic/NormNum/Pow.lean
@@ -10,14 +10,6 @@ import Mathlib.Tactic.NormNum.Basic
 ## `norm_num` plugin for `^`.
 -/
 
-#adaptation_note
-/--
-Since https://github.com/leanprover/lean4/pull/5338,
-the unused variable linter can not see usages of variables in
-`haveI' : ⋯ =Q ⋯ := ⟨⟩` clauses, so generates many false positives.
--/
-set_option linter.unusedVariables false
-
 namespace Mathlib
 open Lean
 open Meta
@@ -235,9 +227,7 @@ theorem isRat_zpow_neg {α : Type*} [DivisionRing α] {a : α} {b : ℤ} {nb : 
     IsRat (a^b) num den := by
   rwa [pb.out, Int.cast_negOfNat, zpow_neg, zpow_natCast]
 
-#adaptation_note
-/--
-Prior to https://github.com/leanprover/lean4/pull/4096,
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/4096
 the repeated
 ```
 have h : $e =Q (HPow.hPow (γ := $α) $a $b) := ⟨⟩
diff --git a/Mathlib/Topology/Algebra/Group/Basic.lean b/Mathlib/Topology/Algebra/Group/Basic.lean
index 28b8320d761a63..649dc2de97a8bb 100644
--- a/Mathlib/Topology/Algebra/Group/Basic.lean
+++ b/Mathlib/Topology/Algebra/Group/Basic.lean
@@ -813,10 +813,11 @@ theorem continuous_of_continuousAt_one₂ {H M : Type*} [CommMonoid M] [Topologi
 
 @[to_additive]
 lemma TopologicalGroup.isInducing_iff_nhds_one
-    {H : Type*} [Group H] [TopologicalSpace H] [TopologicalGroup H] {f : G →* H} :
+    {H : Type*} [Group H] [TopologicalSpace H] [TopologicalGroup H] {F : Type*}
+    [FunLike F G H] [MonoidHomClass F G H] {f : F} :
     Topology.IsInducing f ↔ 𝓝 (1 : G) = (𝓝 (1 : H)).comap f := by
   rw [Topology.isInducing_iff_nhds]
-  refine ⟨(f.map_one ▸ · 1), fun hf x ↦ ?_⟩
+  refine ⟨(map_one f ▸ · 1), fun hf x ↦ ?_⟩
   rw [← nhds_translation_mul_inv, ← nhds_translation_mul_inv (f x), Filter.comap_comap, hf,
     Filter.comap_comap]
   congr 1
diff --git a/Mathlib/Topology/Algebra/Indicator.lean b/Mathlib/Topology/Algebra/Indicator.lean
index 189c9d6f29038b..87a704835566a7 100644
--- a/Mathlib/Topology/Algebra/Indicator.lean
+++ b/Mathlib/Topology/Algebra/Indicator.lean
@@ -5,6 +5,7 @@ Authors: PFR contributors
 -/
 import Mathlib.Algebra.Group.Indicator
 import Mathlib.Topology.ContinuousOn
+import Mathlib.Topology.Clopen
 
 /-!
 # Continuity of indicator functions
@@ -36,3 +37,8 @@ theorem ContinuousOn.continuousAt_mulIndicator (hf : ContinuousOn f (interior s)
       ⟨interior s, hs, Set.eqOn_mulIndicator.symm.mono interior_subset⟩
   · exact ContinuousAt.congr continuousAt_const <| Filter.eventuallyEq_iff_exists_mem.mpr
       ⟨sᶜ, mem_interior_iff_mem_nhds.mp h, Set.eqOn_mulIndicator'.symm⟩
+
+@[to_additive]
+lemma IsClopen.continuous_mulIndicator (hs : IsClopen s) (hf : Continuous f) :
+    Continuous (s.mulIndicator f) :=
+  hf.mulIndicator (by simp [isClopen_iff_frontier_eq_empty.mp hs])
diff --git a/Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean b/Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean
index afef581ea8ffb4..70a2cc032d21ed 100644
--- a/Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean
+++ b/Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean
@@ -55,6 +55,32 @@ theorem cauchySeq_prod_of_tendsto_cofinite_one {f : α → G} (hf : Tendsto f co
   intro i hi
   simpa using Finset.disjoint_left.mp ht hi
 
+/-- Let `G` be a nonarchimedean abelian group, and let `f : ℕ → G` be a function
+such that the quotients `f (n + 1) / f n` tend to one. Then the function is a Cauchy sequence. -/
+@[to_additive "Let `G` be a nonarchimedean additive abelian group, and let `f : ℕ → G` be a
+function such that the differences `f (n + 1) - f n` tend to zero.
+Then the function is a Cauchy sequence."]
+lemma cauchySeq_of_tendsto_div_nhds_one {f : ℕ → G}
+    (hf : Tendsto (fun n ↦ f (n + 1) / f n) atTop (𝓝 1)) :
+    CauchySeq f := by
+  suffices Tendsto (fun p : ℕ × ℕ ↦ f p.2 / f p.1) atTop (𝓝 1) by simpa [CauchySeq,
+      cauchy_map_iff, prod_atTop_atTop_eq, uniformity_eq_comap_nhds_one G, atTop_neBot]
+  rw [tendsto_atTop']
+  intro s hs
+  obtain ⟨t, ht⟩ := is_nonarchimedean s hs
+  obtain ⟨N, hN⟩ : ∃ N : ℕ, ∀ b, N ≤ b → f (b + 1) / f b ∈ t := by
+      simpa using tendsto_def.mp hf t t.mem_nhds_one
+  refine ⟨(N, N), ?_⟩
+  rintro ⟨M, M'⟩ ⟨(hMN : N ≤ M), (hMN' : N ≤ M')⟩
+  apply ht
+  wlog h : M ≤ M' generalizing M M'
+  · simpa [inv_div] using t.inv_mem <| this _ _ hMN' hMN (le_of_not_ge h)
+  obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le h
+  clear h hMN'
+  induction k with
+  | zero => simpa using one_mem t
+  | succ k ih => simpa using t.mul_mem (hN _ (by omega : N ≤ M + k)) ih
+
 /-- Let `G` be a complete nonarchimedean multiplicative abelian group, and let `f : α → G` be a
 function that tends to one on the filter of cofinite sets. Then `f` is unconditionally
 multipliable. -/
diff --git a/Mathlib/Topology/Algebra/Module/Equiv.lean b/Mathlib/Topology/Algebra/Module/Equiv.lean
index 9df3e155b17c7e..faaa18273a54cc 100644
--- a/Mathlib/Topology/Algebra/Module/Equiv.lean
+++ b/Mathlib/Topology/Algebra/Module/Equiv.lean
@@ -852,6 +852,28 @@ def piFinTwo (M : Fin 2 → Type*) [∀ i, AddCommMonoid (M i)] [∀ i, Module R
 def finTwoArrow : (Fin 2 → M) ≃L[R] M × M :=
   { piFinTwo R fun _ => M with toLinearEquiv := LinearEquiv.finTwoArrow R M }
 
+section
+variable {n : ℕ} {R : Type*} {M : Fin n.succ → Type*} {N : Type*}
+variable [Semiring R]
+variable [∀ i, AddCommMonoid (M i)] [∀ i, Module R (M i)] [∀ i, TopologicalSpace (M i)]
+
+variable (R M) in
+/-- `Fin.consEquiv` as a continuous linear equivalence.  -/
+@[simps!]
+def _root_.Fin.consEquivL : (M 0 × Π i, M (Fin.succ i)) ≃L[R] (Π i, M i) where
+  __ := Fin.consLinearEquiv R M
+  continuous_toFun := continuous_id.fst.finCons continuous_id.snd
+  continuous_invFun := .prod_mk (continuous_apply 0) (by continuity)
+
+/-- `Fin.cons` in the codomain of continuous linear maps. -/
+abbrev _root_.ContinuousLinearMap.finCons
+    [AddCommMonoid N] [Module R N] [TopologicalSpace N]
+    (f : N →L[R] M 0) (fs : N →L[R] Π i, M (Fin.succ i)) :
+    N →L[R] Π i, M i :=
+  Fin.consEquivL R M ∘L f.prod fs
+
+end
+
 end
 
 end ContinuousLinearEquiv
diff --git a/Mathlib/Topology/Algebra/Monoid.lean b/Mathlib/Topology/Algebra/Monoid.lean
index f136b5fd395686..025c3e373cc410 100644
--- a/Mathlib/Topology/Algebra/Monoid.lean
+++ b/Mathlib/Topology/Algebra/Monoid.lean
@@ -9,6 +9,7 @@ import Mathlib.Topology.Algebra.MulAction
 import Mathlib.Algebra.BigOperators.Pi
 import Mathlib.Algebra.Group.ULift
 import Mathlib.Topology.ContinuousMap.Defs
+import Mathlib.Algebra.Group.Submonoid.Basic
 
 /-!
 # Theory of topological monoids
@@ -567,6 +568,12 @@ theorem exists_open_nhds_one_mul_subset {U : Set M} (hU : U ∈ 𝓝 (1 : M)) :
     ∃ V : Set M, IsOpen V ∧ (1 : M) ∈ V ∧ V * V ⊆ U := by
   simpa only [mul_subset_iff] using exists_open_nhds_one_split hU
 
+@[to_additive]
+theorem Filter.HasBasis.mul_self {p : ι → Prop} {s : ι → Set M} (h : (𝓝 1).HasBasis p s) :
+    (𝓝 1).HasBasis p fun i => s i * s i := by
+  rw [← nhds_mul_nhds_one, ← map₂_mul, ← map_uncurry_prod]
+  simpa only [← image_mul_prod] using h.prod_self.map _
+
 end MulOneClass
 
 section ContinuousMul
diff --git a/Mathlib/Topology/Algebra/Ring/Basic.lean b/Mathlib/Topology/Algebra/Ring/Basic.lean
index 613f1181d73b99..84c683b719cfe3 100644
--- a/Mathlib/Topology/Algebra/Ring/Basic.lean
+++ b/Mathlib/Topology/Algebra/Ring/Basic.lean
@@ -183,7 +183,8 @@ instance [NonUnitalNonAssocRing α] [NonUnitalNonAssocRing β] [TopologicalRing
 
 end
 
-#adaptation_note /-- nightly-2024-04-08, needed to help `Pi.instTopologicalSemiring` -/
+#adaptation_note /-- nightly-2024-04-08
+needed to help `Pi.instTopologicalSemiring` -/
 instance {β : Type*} {C : β → Type*} [∀ b, TopologicalSpace (C b)]
     [∀ b, NonUnitalNonAssocSemiring (C b)] [∀ b, TopologicalSemiring (C b)] :
     ContinuousAdd ((b : β) → C b) :=
diff --git a/Mathlib/Topology/Category/TopCat/Opens.lean b/Mathlib/Topology/Category/TopCat/Opens.lean
index 4e4930958fd0e1..cd24b21e34cb9b 100644
--- a/Mathlib/Topology/Category/TopCat/Opens.lean
+++ b/Mathlib/Topology/Category/TopCat/Opens.lean
@@ -154,16 +154,13 @@ theorem map_id_obj (U : Opens X) : (map (𝟙 X)).obj U = U :=
   let ⟨_, _⟩ := U
   rfl
 
-@[simp 1100]
+@[simp]
 theorem map_id_obj' (U) (p) : (map (𝟙 X)).obj ⟨U, p⟩ = ⟨U, p⟩ :=
   rfl
 
-@[simp 1100]
-theorem map_id_obj_unop (U : (Opens X)ᵒᵖ) : (map (𝟙 X)).obj (unop U) = unop U :=
-  let ⟨_, _⟩ := U.unop
-  rfl
+theorem map_id_obj_unop (U : (Opens X)ᵒᵖ) : (map (𝟙 X)).obj (unop U) = unop U := by
+  simp
 
-@[simp 1100]
 theorem op_map_id_obj (U : (Opens X)ᵒᵖ) : (map (𝟙 X)).op.obj U = U := by simp
 
 @[simp]
diff --git a/Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean b/Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean
index 09080f8e326cda..0939e3bde7b5ed 100644
--- a/Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean
+++ b/Mathlib/Topology/Compactness/CompactlyGeneratedSpace.lean
@@ -94,12 +94,9 @@ instance (X : Type v) [t : TopologicalSpace X] [DiscreteTopology X] :
     rw [DiscreteTopology.eq_bot (t := t)]
     exact bot_le
 
-#adaptation_note
-/--
-The new unused variable linter in
-https://github.com/leanprover/lean4/pull/5338
-flags `[tY : TopologicalSpace Y]`, but we want to use this as a named argument.
--/
+#adaptation_note /-- https://github.com/leanprover/lean4/pull/5338
+The new unused variable linter flags `[tY : TopologicalSpace Y]`,
+but we want to use this as a named argument. -/
 set_option linter.unusedVariables false in
 /-- Let `f : X → Y`. Suppose that to prove that `f` is continuous, it suffices to show that
 for every compact Hausdorff space `K` and every continuous map `g : K → X`, `f ∘ g` is continuous.
diff --git a/Mathlib/Topology/EMetricSpace/Defs.lean b/Mathlib/Topology/EMetricSpace/Defs.lean
index e3582222d36109..0c9f840cb9455f 100644
--- a/Mathlib/Topology/EMetricSpace/Defs.lean
+++ b/Mathlib/Topology/EMetricSpace/Defs.lean
@@ -596,7 +596,7 @@ theorem edist_pos {x y : γ} : 0 < edist x y ↔ x ≠ y := by simp [← not_le]
 
 /-- Two points coincide if their distance is `< ε` for all positive ε -/
 theorem eq_of_forall_edist_le {x y : γ} (h : ∀ ε > 0, edist x y ≤ ε) : x = y :=
-  eq_of_edist_eq_zero (eq_of_le_of_forall_le_of_dense bot_le h)
+  eq_of_edist_eq_zero (eq_of_le_of_forall_lt_imp_le_of_dense bot_le h)
 
 /-- Auxiliary function to replace the uniformity on an emetric space with
 a uniformity which is equal to the original one, but maybe not defeq.
diff --git a/Mathlib/Topology/Instances/Matrix.lean b/Mathlib/Topology/Instances/Matrix.lean
index 13decc88c4c0a3..c465b090b65452 100644
--- a/Mathlib/Topology/Instances/Matrix.lean
+++ b/Mathlib/Topology/Instances/Matrix.lean
@@ -73,16 +73,17 @@ theorem Continuous.matrix_elem {A : X → Matrix m n R} (hA : Continuous A) (i :
     Continuous fun x => A x i j :=
   (continuous_apply_apply i j).comp hA
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_map [TopologicalSpace S] {A : X → Matrix m n S} {f : S → R}
     (hA : Continuous A) (hf : Continuous f) : Continuous fun x => (A x).map f :=
   continuous_matrix fun _ _ => hf.comp <| hA.matrix_elem _ _
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_transpose {A : X → Matrix m n R} (hA : Continuous A) :
     Continuous fun x => (A x)ᵀ :=
   continuous_matrix fun i j => hA.matrix_elem j i
 
+@[continuity, fun_prop]
 theorem Continuous.matrix_conjTranspose [Star R] [ContinuousStar R] {A : X → Matrix m n R}
     (hA : Continuous A) : Continuous fun x => (A x)ᴴ :=
   hA.matrix_transpose.matrix_map continuous_star
@@ -90,22 +91,22 @@ theorem Continuous.matrix_conjTranspose [Star R] [ContinuousStar R] {A : X → M
 instance [Star R] [ContinuousStar R] : ContinuousStar (Matrix m m R) :=
   ⟨continuous_id.matrix_conjTranspose⟩
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_col {ι : Type*} {A : X → n → R} (hA : Continuous A) :
     Continuous fun x => col ι (A x) :=
   continuous_matrix fun i _ => (continuous_apply i).comp hA
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_row {ι : Type*} {A : X → n → R} (hA : Continuous A) :
     Continuous fun x => row ι (A x) :=
   continuous_matrix fun _ _ => (continuous_apply _).comp hA
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_diagonal [Zero R] [DecidableEq n] {A : X → n → R} (hA : Continuous A) :
     Continuous fun x => diagonal (A x) :=
   continuous_matrix fun i _ => ((continuous_apply i).comp hA).if_const _ continuous_zero
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_dotProduct [Fintype n] [Mul R] [AddCommMonoid R] [ContinuousAdd R]
     [ContinuousMul R] {A : X → n → R} {B : X → n → R} (hA : Continuous A) (hB : Continuous B) :
     Continuous fun x => dotProduct (A x) (B x) :=
@@ -113,7 +114,7 @@ theorem Continuous.matrix_dotProduct [Fintype n] [Mul R] [AddCommMonoid R] [Cont
     ((continuous_apply i).comp hA).mul ((continuous_apply i).comp hB)
 
 /-- For square matrices the usual `continuous_mul` can be used. -/
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_mul [Fintype n] [Mul R] [AddCommMonoid R] [ContinuousAdd R]
     [ContinuousMul R] {A : X → Matrix m n R} {B : X → Matrix n p R} (hA : Continuous A)
     (hB : Continuous B) : Continuous fun x => A x * B x :=
@@ -130,34 +131,34 @@ instance [Fintype n] [NonUnitalNonAssocSemiring R] [TopologicalSemiring R] :
 instance Matrix.topologicalRing [Fintype n] [NonUnitalNonAssocRing R] [TopologicalRing R] :
     TopologicalRing (Matrix n n R) where
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_vecMulVec [Mul R] [ContinuousMul R] {A : X → m → R} {B : X → n → R}
     (hA : Continuous A) (hB : Continuous B) : Continuous fun x => vecMulVec (A x) (B x) :=
   continuous_matrix fun _ _ => ((continuous_apply _).comp hA).mul ((continuous_apply _).comp hB)
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_mulVec [NonUnitalNonAssocSemiring R] [ContinuousAdd R] [ContinuousMul R]
     [Fintype n] {A : X → Matrix m n R} {B : X → n → R} (hA : Continuous A) (hB : Continuous B) :
     Continuous fun x => A x *ᵥ B x :=
   continuous_pi fun i => ((continuous_apply i).comp hA).matrix_dotProduct hB
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_vecMul [NonUnitalNonAssocSemiring R] [ContinuousAdd R] [ContinuousMul R]
     [Fintype m] {A : X → m → R} {B : X → Matrix m n R} (hA : Continuous A) (hB : Continuous B) :
     Continuous fun x => A x ᵥ* B x :=
   continuous_pi fun _i => hA.matrix_dotProduct <| continuous_pi fun _j => hB.matrix_elem _ _
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_submatrix {A : X → Matrix l n R} (hA : Continuous A) (e₁ : m → l)
     (e₂ : p → n) : Continuous fun x => (A x).submatrix e₁ e₂ :=
   continuous_matrix fun _i _j => hA.matrix_elem _ _
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_reindex {A : X → Matrix l n R} (hA : Continuous A) (e₁ : l ≃ m)
     (e₂ : n ≃ p) : Continuous fun x => reindex e₁ e₂ (A x) :=
   hA.matrix_submatrix _ _
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_diag {A : X → Matrix n n R} (hA : Continuous A) :
     Continuous fun x => Matrix.diag (A x) :=
   continuous_pi fun _ => hA.matrix_elem _ _
@@ -166,19 +167,19 @@ theorem Continuous.matrix_diag {A : X → Matrix n n R} (hA : Continuous A) :
 theorem continuous_matrix_diag : Continuous (Matrix.diag : Matrix n n R → n → R) :=
   show Continuous fun x : Matrix n n R => Matrix.diag x from continuous_id.matrix_diag
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_trace [Fintype n] [AddCommMonoid R] [ContinuousAdd R]
     {A : X → Matrix n n R} (hA : Continuous A) : Continuous fun x => trace (A x) :=
   continuous_finset_sum _ fun _ _ => hA.matrix_elem _ _
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_det [Fintype n] [DecidableEq n] [CommRing R] [TopologicalRing R]
     {A : X → Matrix n n R} (hA : Continuous A) : Continuous fun x => (A x).det := by
   simp_rw [Matrix.det_apply]
   refine continuous_finset_sum _ fun l _ => Continuous.const_smul ?_ _
   exact continuous_finset_prod _ fun l _ => hA.matrix_elem _ _
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_updateCol [DecidableEq n] (i : n) {A : X → Matrix m n R}
     {B : X → m → R} (hA : Continuous A) (hB : Continuous B) :
     Continuous fun x => (A x).updateCol i (B x) :=
@@ -189,18 +190,18 @@ theorem Continuous.matrix_updateCol [DecidableEq n] (i : n) {A : X → Matrix m
 @[deprecated (since := "2024-12-11")]
 alias Continuous.matrix_updateColumn := Continuous.matrix_updateCol
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_updateRow [DecidableEq m] (i : m) {A : X → Matrix m n R} {B : X → n → R}
     (hA : Continuous A) (hB : Continuous B) : Continuous fun x => (A x).updateRow i (B x) :=
   hA.update i hB
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_cramer [Fintype n] [DecidableEq n] [CommRing R] [TopologicalRing R]
     {A : X → Matrix n n R} {B : X → n → R} (hA : Continuous A) (hB : Continuous B) :
     Continuous fun x => cramer (A x) (B x) :=
   continuous_pi fun _ => (hA.matrix_updateCol _ hB).matrix_det
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_adjugate [Fintype n] [DecidableEq n] [CommRing R] [TopologicalRing R]
     {A : X → Matrix n n R} (hA : Continuous A) : Continuous fun x => (A x).adjugate :=
   continuous_matrix fun _j k =>
@@ -215,7 +216,7 @@ theorem continuousAt_matrix_inv [Fintype n] [DecidableEq n] [CommRing R] [Topolo
 -- lemmas about functions in `Data/Matrix/Block.lean`
 section BlockMatrices
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_fromBlocks {A : X → Matrix n l R} {B : X → Matrix n m R}
     {C : X → Matrix p l R} {D : X → Matrix p m R} (hA : Continuous A) (hB : Continuous B)
     (hC : Continuous C) (hD : Continuous D) :
@@ -223,18 +224,18 @@ theorem Continuous.matrix_fromBlocks {A : X → Matrix n l R} {B : X → Matrix
   continuous_matrix <| by
     rintro (i | i) (j | j) <;> refine Continuous.matrix_elem ?_ i j <;> assumption
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_blockDiagonal [Zero R] [DecidableEq p] {A : X → p → Matrix m n R}
     (hA : Continuous A) : Continuous fun x => blockDiagonal (A x) :=
   continuous_matrix fun ⟨i₁, i₂⟩ ⟨j₁, _j₂⟩ =>
     (((continuous_apply i₂).comp hA).matrix_elem i₁ j₁).if_const _ continuous_zero
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_blockDiag {A : X → Matrix (m × p) (n × p) R} (hA : Continuous A) :
     Continuous fun x => blockDiag (A x) :=
   continuous_pi fun _i => continuous_matrix fun _j _k => hA.matrix_elem _ _
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_blockDiagonal' [Zero R] [DecidableEq l]
     {A : X → ∀ i, Matrix (m' i) (n' i) R} (hA : Continuous A) :
     Continuous fun x => blockDiagonal' (A x) :=
@@ -245,7 +246,7 @@ theorem Continuous.matrix_blockDiagonal' [Zero R] [DecidableEq l]
       exact ((continuous_apply i₁).comp hA).matrix_elem i₂ j₂
     · exact continuous_const
 
-@[continuity]
+@[continuity, fun_prop]
 theorem Continuous.matrix_blockDiag' {A : X → Matrix (Σi, m' i) (Σi, n' i) R} (hA : Continuous A) :
     Continuous fun x => blockDiag' (A x) :=
   continuous_pi fun _i => continuous_matrix fun _j _k => hA.matrix_elem _ _
diff --git a/Mathlib/Topology/List.lean b/Mathlib/Topology/List.lean
index be04e98105dbd3..10a7e8b0991c05 100644
--- a/Mathlib/Topology/List.lean
+++ b/Mathlib/Topology/List.lean
@@ -56,8 +56,6 @@ theorem nhds_list (as : List α) : 𝓝 as = traverse 𝓝 as := by
     have : List.Forall₂ (fun a s => IsOpen s ∧ a ∈ s) u v := by
       refine List.Forall₂.flip ?_
       replace hv := hv.flip
-      #adaptation_note /-- nightly-2024-03-16: simp was
-      simp only [List.forall₂_and_left, flip] at hv ⊢ -/
       simp only [List.forall₂_and_left, Function.flip_def] at hv ⊢
       exact ⟨hv.1, hu.flip⟩
     refine mem_of_superset ?_ hvs
diff --git a/Mathlib/Topology/MetricSpace/Defs.lean b/Mathlib/Topology/MetricSpace/Defs.lean
index e678bdcfedb6b9..0ada8be44573bc 100644
--- a/Mathlib/Topology/MetricSpace/Defs.lean
+++ b/Mathlib/Topology/MetricSpace/Defs.lean
@@ -78,7 +78,7 @@ theorem dist_pos {x y : γ} : 0 < dist x y ↔ x ≠ y := by
   simpa only [not_le] using not_congr dist_le_zero
 
 theorem eq_of_forall_dist_le {x y : γ} (h : ∀ ε > 0, dist x y ≤ ε) : x = y :=
-  eq_of_dist_eq_zero (eq_of_le_of_forall_le_of_dense dist_nonneg h)
+  eq_of_dist_eq_zero (eq_of_le_of_forall_lt_imp_le_of_dense dist_nonneg h)
 
 /-- Deduce the equality of points from the vanishing of the nonnegative distance -/
 theorem eq_of_nndist_eq_zero {x y : γ} : nndist x y = 0 → x = y := by
diff --git a/Mathlib/Topology/MetricSpace/GromovHausdorff.lean b/Mathlib/Topology/MetricSpace/GromovHausdorff.lean
index a564dcfd1a9098..caf9feed00c8f0 100644
--- a/Mathlib/Topology/MetricSpace/GromovHausdorff.lean
+++ b/Mathlib/Topology/MetricSpace/GromovHausdorff.lean
@@ -311,7 +311,7 @@ theorem hausdorffDist_optimal {X : Type u} [MetricSpace X] [CompactSpace X] [Non
     let Fb := candidatesBOfCandidates F Fgood
     have : hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD Fb :=
       hausdorffDist_optimal_le_HD _ _ (candidatesBOfCandidates_mem F Fgood)
-    refine le_trans this (le_of_forall_le_of_dense fun r hr => ?_)
+    refine le_trans this (le_of_forall_gt_imp_ge_of_dense fun r hr => ?_)
     have I1 : ∀ x : X, (⨅ y, Fb (inl x, inr y)) ≤ r := by
       intro x
       have : f (inl x) ∈ (p : Set _) := Φrange ▸ (mem_range_self _)
diff --git a/Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean b/Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
index f0124bb1d3151b..5eedc8d457f845 100644
--- a/Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
+++ b/Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
@@ -480,7 +480,7 @@ the Hausdorff distance in the optimal coupling, although we only prove here the
 we need. -/
 theorem hausdorffDist_optimal_le_HD {f} (h : f ∈ candidatesB X Y) :
     hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) ≤ HD f := by
-  refine le_trans (le_of_forall_le_of_dense fun r hr => ?_) (HD_optimalGHDist_le X Y f h)
+  refine le_trans (le_of_forall_gt_imp_ge_of_dense fun r hr => ?_) (HD_optimalGHDist_le X Y f h)
   have A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r := by
     rintro _ ⟨z, rfl⟩
     have I1 : (⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y)) < r :=
diff --git a/Mathlib/Topology/MetricSpace/HausdorffDimension.lean b/Mathlib/Topology/MetricSpace/HausdorffDimension.lean
index ec335783eba45e..0235fc0d813222 100644
--- a/Mathlib/Topology/MetricSpace/HausdorffDimension.lean
+++ b/Mathlib/Topology/MetricSpace/HausdorffDimension.lean
@@ -232,7 +232,7 @@ theorem bsupr_limsup_dimH (s : Set X) : ⨆ x ∈ s, limsup dimH (𝓝[s] x).sma
   refine le_antisymm (iSup₂_le fun x _ => ?_) ?_
   · refine limsup_le_of_le isCobounded_le_of_bot ?_
     exact eventually_smallSets.2 ⟨s, self_mem_nhdsWithin, fun t => dimH_mono⟩
-  · refine le_of_forall_ge_of_dense fun r hr => ?_
+  · refine le_of_forall_lt_imp_le_of_dense fun r hr => ?_
     rcases exists_mem_nhdsWithin_lt_dimH_of_lt_dimH hr with ⟨x, hxs, hxr⟩
     refine le_iSup₂_of_le x hxs ?_; rw [limsup_eq]; refine le_sInf fun b hb => ?_
     rcases eventually_smallSets.1 hb with ⟨t, htx, ht⟩
diff --git a/Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean b/Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
index 6b6cd86da53a26..9f284fb00b8813 100644
--- a/Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
+++ b/Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean
@@ -91,7 +91,7 @@ lemma exists_isCompact_closedBall [WeaklyLocallyCompactSpace α] (x : α) :
 
 theorem biInter_gt_closedBall (x : α) (r : ℝ) : ⋂ r' > r, closedBall x r' = closedBall x r := by
   ext
-  simp [forall_gt_ge_iff]
+  simp [forall_gt_imp_ge_iff_le_of_dense]
 
 theorem biInter_gt_ball (x : α) (r : ℝ) : ⋂ r' > r, ball x r' = closedBall x r := by
   ext
@@ -100,7 +100,7 @@ theorem biInter_gt_ball (x : α) (r : ℝ) : ⋂ r' > r, ball x r' = closedBall
 theorem biUnion_lt_ball (x : α) (r : ℝ) : ⋃ r' < r, ball x r' = ball x r := by
   ext
   rw [← not_iff_not]
-  simp [forall_lt_le_iff]
+  simp [forall_lt_imp_le_iff_le_of_dense]
 
 theorem biUnion_lt_closedBall (x : α) (r : ℝ) : ⋃ r' < r, closedBall x r' = ball x r := by
   ext
diff --git a/Mathlib/Topology/NoetherianSpace.lean b/Mathlib/Topology/NoetherianSpace.lean
index 931e8748cbfcef..e26ec3b5c4b6d7 100644
--- a/Mathlib/Topology/NoetherianSpace.lean
+++ b/Mathlib/Topology/NoetherianSpace.lean
@@ -70,7 +70,7 @@ protected theorem _root_.Topology.IsInducing.noetherianSpace [NoetherianSpace α
 @[deprecated (since := "2024-10-28")]
 alias _root_.Inducing.noetherianSpace := IsInducing.noetherianSpace
 
-/-- [Stacks: Lemma 0052 (1)](https://stacks.math.columbia.edu/tag/0052)-/
+@[stacks 0052 "(1)"]
 instance NoetherianSpace.set [NoetherianSpace α] (s : Set α) : NoetherianSpace s :=
   IsInducing.subtypeVal.noetherianSpace
 
@@ -191,7 +191,7 @@ theorem NoetherianSpace.exists_finset_irreducible [NoetherianSpace α] (s : Clos
   simpa [Set.exists_finite_iff_finset, Finset.sup_id_eq_sSup]
     using NoetherianSpace.exists_finite_set_closeds_irreducible s
 
-/-- [Stacks: Lemma 0052 (2)](https://stacks.math.columbia.edu/tag/0052) -/
+@[stacks 0052 "(2)"]
 theorem NoetherianSpace.finite_irreducibleComponents [NoetherianSpace α] :
     (irreducibleComponents α).Finite := by
   obtain ⟨S : Set (Set α), hSf, hSc, hSi, hSU⟩ :=
@@ -201,7 +201,7 @@ theorem NoetherianSpace.finite_irreducibleComponents [NoetherianSpace α] :
   rcases isIrreducible_iff_sUnion_isClosed.1 hs.1 S hSc (hSU ▸ Set.subset_univ _) with ⟨t, htS, ht⟩
   rwa [ht.antisymm (hs.2 (hSi _ htS) ht)]
 
-/-- [Stacks: Lemma 0052 (3)](https://stacks.math.columbia.edu/tag/0052) -/
+@[stacks 0052 "(3)"]
 theorem NoetherianSpace.exists_open_ne_empty_le_irreducibleComponent [NoetherianSpace α]
     (Z : Set α) (H : Z ∈ irreducibleComponents α) :
     ∃ o : Set α, IsOpen o ∧ o ≠ ∅ ∧ o ≤ Z := by
diff --git a/Mathlib/Topology/PartialHomeomorph.lean b/Mathlib/Topology/PartialHomeomorph.lean
index a178c181fd1cdd..c0004b4df3cbab 100644
--- a/Mathlib/Topology/PartialHomeomorph.lean
+++ b/Mathlib/Topology/PartialHomeomorph.lean
@@ -1326,15 +1326,14 @@ theorem subtypeRestr_symm_eqOn_of_le {U V : Opens X} (hU : Nonempty U) (hV : Non
 end subtypeRestr
 
 variable {X X' Z : Type*} [TopologicalSpace X] [TopologicalSpace X'] [TopologicalSpace Z]
-    [Nonempty X] [Nonempty Z] {f : X → X'}
+  [Nonempty Z] {f : X → X'}
 
 /-- Extend a partial homeomorphism `e : X → Z` to `X' → Z`, using an open embedding `ι : X → X'`.
 On `ι(X)`, the extension is specified by `e`; its value elsewhere is arbitrary (and uninteresting).
 -/
-noncomputable def lift_openEmbedding
-    (e : PartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
+noncomputable def lift_openEmbedding (e : PartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
     PartialHomeomorph X' Z where
-  toFun := extend f e (Classical.arbitrary (α := X' → Z))
+  toFun := extend f e (fun _ ↦ (Classical.arbitrary Z))
   invFun := f ∘ e.invFun
   source := f '' e.source
   target := e.target
@@ -1352,7 +1351,9 @@ noncomputable def lift_openEmbedding
   open_source := hf.isOpenMap _ e.open_source
   open_target := e.open_target
   continuousOn_toFun := by
-    set F := (extend f (↑e) (Classical.arbitrary (X' → Z))) with F_eq
+    by_cases Nonempty X; swap
+    · intro x hx; simp_all
+    set F := (extend f e (fun _ ↦ (Classical.arbitrary Z))) with F_eq
     have heq : EqOn F (e ∘ (hf.toPartialHomeomorph).symm) (f '' e.source) := by
       intro x ⟨x₀, hx₀, hxx₀⟩
       rw [← hxx₀, F_eq, hf.injective.extend_apply e, comp_apply, hf.toPartialHomeomorph_left_inv]
@@ -1360,15 +1361,15 @@ noncomputable def lift_openEmbedding
       apply e.continuousOn_toFun.comp; swap
       · intro x' ⟨x, hx, hx'x⟩
         rw [← hx'x, hf.toPartialHomeomorph_left_inv]; exact hx
-      have : ContinuousOn (hf.toPartialHomeomorph).symm (f '' univ) := by
-        apply (hf.toPartialHomeomorph).continuousOn_invFun
+      have : ContinuousOn (hf.toPartialHomeomorph).symm (f '' univ) :=
+        (hf.toPartialHomeomorph).continuousOn_invFun
       apply this.mono; gcongr; exact fun ⦃a⦄ a ↦ trivial
-    apply ContinuousOn.congr this heq
+    exact ContinuousOn.congr this heq
   continuousOn_invFun := hf.continuous.comp_continuousOn e.continuousOn_invFun
 
 @[simp]
 lemma lift_openEmbedding_toFun (e : PartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
-    (e.lift_openEmbedding hf) = extend f e (Classical.arbitrary (α := (X' → Z))) := rfl
+    (e.lift_openEmbedding hf) = extend f e (fun _ ↦ (Classical.arbitrary Z)) := rfl
 
 lemma lift_openEmbedding_apply (e : PartialHomeomorph X Z) (hf : IsOpenEmbedding f) {x : X} :
     (lift_openEmbedding e hf) (f x) = e x := by
@@ -1404,8 +1405,8 @@ lemma lift_openEmbedding_trans_apply
 @[simp]
 lemma lift_openEmbedding_trans (e e' : PartialHomeomorph X Z) (hf : IsOpenEmbedding f) :
     (e.lift_openEmbedding hf).symm.trans (e'.lift_openEmbedding hf) = e.symm.trans e' := by
-  ext x
-  · exact e.lift_openEmbedding_trans_apply e' hf x
+  ext z
+  · exact e.lift_openEmbedding_trans_apply e' hf z
   · simp [hf.injective.extend_apply e]
   · simp_rw [PartialHomeomorph.trans_source, e.lift_openEmbedding_symm_source, e.symm_source,
       e.lift_openEmbedding_symm, e'.lift_openEmbedding_source]
diff --git a/Mathlib/Topology/SeparatedMap.lean b/Mathlib/Topology/SeparatedMap.lean
index 61b7da07704282..5773590ffc3172 100644
--- a/Mathlib/Topology/SeparatedMap.lean
+++ b/Mathlib/Topology/SeparatedMap.lean
@@ -183,25 +183,11 @@ section eqLocus
 variable {f : X → Y} {g₁ g₂ : A → X} (h₁ : Continuous g₁) (h₂ : Continuous g₂)
 include h₁ h₂
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/5338,
-the unused variable linter flags `g` here,
-but it is used in a type ascription to direct `fun_prop`.
--/
-set_option linter.unusedVariables false in
 theorem IsSeparatedMap.isClosed_eqLocus (sep : IsSeparatedMap f) (he : f ∘ g₁ = f ∘ g₂) :
     IsClosed {a | g₁ a = g₂ a} :=
   let g : A → f.Pullback f := fun a ↦ ⟨⟨g₁ a, g₂ a⟩, congr_fun he a⟩
   (isSeparatedMap_iff_isClosed_diagonal.mp sep).preimage (by fun_prop : Continuous g)
 
-#adaptation_note
-/--
-After https://github.com/leanprover/lean4/pull/5338,
-the unused variable linter flags `g` here,
-but it is used in a type ascription to direct `fun_prop`.
--/
-set_option linter.unusedVariables false in
 theorem IsLocallyInjective.isOpen_eqLocus (inj : IsLocallyInjective f) (he : f ∘ g₁ = f ∘ g₂) :
     IsOpen {a | g₁ a = g₂ a} :=
   let g : A → f.Pullback f := fun a ↦ ⟨⟨g₁ a, g₂ a⟩, congr_fun he a⟩
diff --git a/Mathlib/Topology/Separation/Regular.lean b/Mathlib/Topology/Separation/Regular.lean
index 3be8602edb54b5..ae49d2084217e1 100644
--- a/Mathlib/Topology/Separation/Regular.lean
+++ b/Mathlib/Topology/Separation/Regular.lean
@@ -3,8 +3,9 @@ Copyright (c) 2017 Johannes Hölzl. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johannes Hölzl, Mario Carneiro
 -/
-import Mathlib.Topology.Separation.Hausdorff
+import Mathlib.Tactic.StacksAttribute
 import Mathlib.Topology.Compactness.Lindelof
+import Mathlib.Topology.Separation.Hausdorff
 
 /-!
 # Regular, normal, T₃, T₄ and T₅ spaces
@@ -637,8 +638,8 @@ theorem connectedComponent_eq_iInter_isClopen [T2Space X] [CompactSpace X] (x :
           ⟨s ∩ v, H2, mem_inter H.2.1 h1⟩
 
 /-- `ConnectedComponents X` is Hausdorff when `X` is Hausdorff and compact -/
+@[stacks 0900 "The Stacks entry proves profiniteness."]
 instance ConnectedComponents.t2 [T2Space X] [CompactSpace X] : T2Space (ConnectedComponents X) := by
-  -- Proof follows that of: https://stacks.math.columbia.edu/tag/0900
   -- Fix 2 distinct connected components, with points a and b
   refine ⟨ConnectedComponents.surjective_coe.forall₂.2 fun a b ne => ?_⟩
   rw [ConnectedComponents.coe_ne_coe] at ne
diff --git a/Mathlib/Topology/UniformSpace/Dini.lean b/Mathlib/Topology/UniformSpace/Dini.lean
index aff1384ce1d4fd..97b9a70caecb89 100644
--- a/Mathlib/Topology/UniformSpace/Dini.lean
+++ b/Mathlib/Topology/UniformSpace/Dini.lean
@@ -42,7 +42,7 @@ variable {F : ι → α → G} {f : α → G}
 
 namespace Monotone
 
-/-- **Dini's theorem** If `F n` is a monotone increasing collection of continuous functions
+/-- **Dini's theorem**: if `F n` is a monotone increasing collection of continuous functions
 converging pointwise to a continuous function `f`, then `F n` converges locally uniformly to `f`. -/
 lemma tendstoLocallyUniformly_of_forall_tendsto
     (hF_cont : ∀ i, Continuous (F i)) (hF_mono : Monotone F) (hf : Continuous f)
@@ -64,7 +64,7 @@ lemma tendstoLocallyUniformly_of_forall_tendsto
   simp only [abs_of_nonpos (sub_nonpos_of_le (F_le_f _ _)), neg_sub, sub_le_sub_iff_left]
   exact hF_mono hnm z
 
-/-- **Dini's theorem** If `F n` is a monotone increasing collection of continuous functions on a
+/-- **Dini's theorem**: if `F n` is a monotone increasing collection of continuous functions on a
 set `s` converging pointwise to a continuous function `f`, then `F n` converges locally uniformly
 to `f`. -/
 lemma tendstoLocallyUniformlyOn_of_forall_tendsto {s : Set α}
@@ -75,7 +75,7 @@ lemma tendstoLocallyUniformlyOn_of_forall_tendsto {s : Set α}
   exact tendstoLocallyUniformly_of_forall_tendsto (hF_cont · |>.restrict)
     (fun _ _ h x ↦ hF_mono _ x.2 h) hf.restrict (fun x ↦ h_tendsto x x.2)
 
-/-- **Dini's theorem** If `F n` is a monotone increasing collection of continuous functions on a
+/-- **Dini's theorem**: if `F n` is a monotone increasing collection of continuous functions on a
 compact space converging pointwise to a continuous function `f`, then `F n` converges uniformly to
 `f`. -/
 lemma tendstoUniformly_of_forall_tendsto [CompactSpace α] (hF_cont : ∀ i, Continuous (F i))
@@ -84,7 +84,7 @@ lemma tendstoUniformly_of_forall_tendsto [CompactSpace α] (hF_cont : ∀ i, Con
   tendstoLocallyUniformly_iff_tendstoUniformly_of_compactSpace.mp <|
     tendstoLocallyUniformly_of_forall_tendsto hF_cont hF_mono hf h_tendsto
 
-/-- **Dini's theorem** If `F n` is a monotone increasing collection of continuous functions on a
+/-- **Dini's theorem**: if `F n` is a monotone increasing collection of continuous functions on a
 compact set `s` converging pointwise to a continuous function `f`, then `F n` converges uniformly
 to `f`. -/
 lemma tendstoUniformlyOn_of_forall_tendsto {s : Set α} (hs : IsCompact s)
@@ -98,7 +98,7 @@ end Monotone
 
 namespace Antitone
 
-/-- **Dini's theorem** If `F n` is a monotone decreasing collection of continuous functions on a
+/-- **Dini's theorem**: if `F n` is a monotone decreasing collection of continuous functions on a
 converging pointwise to a continuous function `f`, then `F n` converges locally uniformly to `f`. -/
 lemma tendstoLocallyUniformly_of_forall_tendsto
     (hF_cont : ∀ i, Continuous (F i)) (hF_anti : Antitone F) (hf : Continuous f)
@@ -106,7 +106,7 @@ lemma tendstoLocallyUniformly_of_forall_tendsto
     TendstoLocallyUniformly F f atTop :=
   Monotone.tendstoLocallyUniformly_of_forall_tendsto (G := Gᵒᵈ) hF_cont hF_anti hf h_tendsto
 
-/-- **Dini's theorem** If `F n` is a monotone decreasing collection of continuous functions on a
+/-- **Dini's theorem**: if `F n` is a monotone decreasing collection of continuous functions on a
 set `s` converging pointwise to a continuous function `f`, then `F n` converges locally uniformly
 to `f`. -/
 lemma tendstoLocallyUniformlyOn_of_forall_tendsto {s : Set α}
@@ -115,7 +115,7 @@ lemma tendstoLocallyUniformlyOn_of_forall_tendsto {s : Set α}
     TendstoLocallyUniformlyOn F f atTop s :=
   Monotone.tendstoLocallyUniformlyOn_of_forall_tendsto (G := Gᵒᵈ) hF_cont hF_anti hf h_tendsto
 
-/-- **Dini's theorem** If `F n` is a monotone decreasing collection of continuous functions on a
+/-- **Dini's theorem**: if `F n` is a monotone decreasing collection of continuous functions on a
 compact space converging pointwise to a continuous function `f`, then `F n` converges uniformly
 to `f`. -/
 lemma tendstoUniformly_of_forall_tendsto [CompactSpace α] (hF_cont : ∀ i, Continuous (F i))
@@ -123,7 +123,7 @@ lemma tendstoUniformly_of_forall_tendsto [CompactSpace α] (hF_cont : ∀ i, Con
     TendstoUniformly F f atTop :=
   Monotone.tendstoUniformly_of_forall_tendsto (G := Gᵒᵈ) hF_cont hF_anti hf h_tendsto
 
-/-- **Dini's theorem** If `F n` is a monotone decreasing collection of continuous functions on a
+/-- **Dini's theorem**: if `F n` is a monotone decreasing collection of continuous functions on a
 compact set `s` converging pointwise to a continuous `f`, then `F n` converges uniformly to `f`. -/
 lemma tendstoUniformlyOn_of_forall_tendsto {s : Set α} (hs : IsCompact s)
     (hF_cont : ∀ i, ContinuousOn (F i) s) (hF_anti : ∀ x ∈ s, Antitone (F · x))
@@ -139,7 +139,7 @@ namespace ContinuousMap
 
 variable {F : ι → C(α, G)} {f : C(α, G)}
 
-/-- **Dini's theorem** If `F n` is a monotone increasing collection of continuous functions
+/-- **Dini's theorem**: if `F n` is a monotone increasing collection of continuous functions
 converging pointwise to a continuous function `f`, then `F n` converges to `f` in the
 compact-open topology. -/
 lemma tendsto_of_monotone_of_pointwise (hF_mono : Monotone F)
@@ -148,7 +148,7 @@ lemma tendsto_of_monotone_of_pointwise (hF_mono : Monotone F)
   tendsto_of_tendstoLocallyUniformly <|
     hF_mono.tendstoLocallyUniformly_of_forall_tendsto (F · |>.continuous) f.continuous h_tendsto
 
-/-- **Dini's theorem** If `F n` is a monotone decreasing collection of continuous functions
+/-- **Dini's theorem**: if `F n` is a monotone decreasing collection of continuous functions
 converging pointwise to a continuous function `f`, then `F n` converges to `f` in the
 compact-open topology. -/
 lemma tendsto_of_antitone_of_pointwise (hF_anti : Antitone F)
diff --git a/MathlibTest/Header.lean b/MathlibTest/Header.lean
index dcfe5788d4f319..6553027d5e3fc4 100644
--- a/MathlibTest/Header.lean
+++ b/MathlibTest/Header.lean
@@ -9,7 +9,7 @@ import Lake
 import Mathlib.Tactic.Linter.Header
 import /- -/ Mathlib.Tactic -- the `TextBased` linter does not flag this `broadImport`
 import Mathlib.Tactic.Have
-import Mathlib.Deprecated.Subfield
+import Mathlib.Deprecated.MinMax
 
 /--
 warning: In the past, importing 'Lake' in mathlib has led to dramatic slow-downs of the linter (see e.g. https://github.com/leanprover-community/mathlib4/pull/13779). Please consider carefully if this import is useful and make sure to benchmark it. If this is fine, feel free to silence this linter.
diff --git a/MathlibTest/MoveAdd.lean b/MathlibTest/MoveAdd.lean
index 74d7a9fd44d709..e091e08e058a7a 100644
--- a/MathlibTest/MoveAdd.lean
+++ b/MathlibTest/MoveAdd.lean
@@ -108,13 +108,12 @@ example {a b c d e : Prop} (h : a ∨ b ∨ c ∨ d ∨ e) : a ∨ c ∨ e ∨ b
 
 end left_assoc
 
-#adaptation_note /-- nightly-2024-03-11
-This test is now failing with `unknown free variable '_fvar.36787'` -/
--- example (k : ℕ) (h0 : 0 + 2 = 9 + 0) (h9 : k + 2 = k + 9) : k + 2 = 9 + k := by
---   induction' k with k _ih
---   · exact h0
---   · move_add [9]
---     exact h9
+example (k : ℕ) (h0 : 0 + 2 = 9 + 0) (h9 : k + 2 = k + 9) : k + 2 = 9 + k := by
+  induction k with
+  | zero => exact h0
+  | succ n _ =>
+    move_add [9]
+    exact h9
 
 -- Testing internals of the tactic `move_add`.
 section tactic
diff --git a/MathlibTest/Perm.lean b/MathlibTest/Perm.lean
new file mode 100644
index 00000000000000..72aaf2dee0de92
--- /dev/null
+++ b/MathlibTest/Perm.lean
@@ -0,0 +1,14 @@
+
+import Mathlib.GroupTheory.Perm.Cycle.Concrete
+import Mathlib.Tactic.DeriveFintype
+
+open Equiv
+
+#guard with_decl_name% ex₁ unsafe (reprStr (1 : Perm (Fin 4))) == "1"
+#guard with_decl_name% ex₂ unsafe (reprStr (c[0, 1] : Perm (Fin 4))) == "c[0, 1]"
+#guard with_decl_name% ex₃
+  unsafe (reprStr (c[0, 1] * c[2, 3] : Perm (Fin 4))) == "c[0, 1] * c[2, 3]"
+#guard with_decl_name% ex₄
+  unsafe (reprPrec (c[0, 1] * c[2, 3] : Perm (Fin 4)) 70).pretty == "(c[0, 1] * c[2, 3])"
+#guard with_decl_name% ex₅
+  unsafe (reprPrec (c[0, 1] * c[1, 2] : Perm (Fin 4)) 70).pretty == "c[0, 1, 2]"
diff --git a/MathlibTest/Recall.lean b/MathlibTest/Recall.lean
index a665edc1a8f344..01092a624dc923 100644
--- a/MathlibTest/Recall.lean
+++ b/MathlibTest/Recall.lean
@@ -1,7 +1,7 @@
 import Mathlib.Tactic.Recall
 import Mathlib.Analysis.Calculus.Deriv.Basic
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
-import Mathlib.Data.Complex.Exponential
+import Mathlib.Data.Complex.Trigonometric
 
 set_option linter.style.setOption false
 -- Remark: When the test is run by make/CI, this option is not set, so we set it here.
diff --git a/MathlibTest/cc.lean b/MathlibTest/cc.lean
index 71962960104261..1263e28bba6fa8 100644
--- a/MathlibTest/cc.lean
+++ b/MathlibTest/cc.lean
@@ -250,7 +250,7 @@ axiom C : (a : A) → B a → Type
 axiom D : (a : A) → (ba : B a) → C a ba → Type
 axiom E : (a : A) → (ba : B a) → (cba : C a ba) → D a ba cba → Type
 axiom F : (a : A) → (ba : B a) → (cba : C a ba) → (dcba : D a ba cba) → E a ba cba dcba → Type
-axiom C_ss : ∀ a ba, Subsingleton (C a ba)
+axiom C_ss : ∀ a ba, Lean.Meta.FastSubsingleton (C a ba)
 axiom a1 : A
 axiom a2 : A
 axiom a3 : A
diff --git a/MathlibTest/positivity.lean b/MathlibTest/positivity.lean
index 80b80dab4bb9ed..a9b0c4816cbd62 100644
--- a/MathlibTest/positivity.lean
+++ b/MathlibTest/positivity.lean
@@ -1,5 +1,5 @@
 import Mathlib.Tactic.Positivity
-import Mathlib.Data.Complex.Exponential
+import Mathlib.Data.Complex.Trigonometric
 import Mathlib.Data.Real.Sqrt
 import Mathlib.Analysis.Normed.Group.Basic
 import Mathlib.Analysis.SpecialFunctions.Pow.Real
diff --git a/docs/1000.yaml b/docs/1000.yaml
index fc2fe3e686f5ea..ca2e502ee182b5 100644
--- a/docs/1000.yaml
+++ b/docs/1000.yaml
@@ -9,10 +9,7 @@
 # and by `make_site.py` in the leanprover-community.github.io repo.
 #
 # Current TODOs/unresolved questions:
-# - add infrastructure for updating the information upstream when formalisations are added
 # - perhaps add a wikidata version of the stacks attribute, and generate this file automatically
-#
-# TODO: display a less ambiguous title for e.g. "Liouville's theorem"
 
 Q11518:
   title: Pythagorean theorem
@@ -272,7 +269,7 @@ Q257387:
   comment: "mathlib4 pull request at https://github.com/leanprover-community/mathlib4/pull/20722"
 
 Q258374:
-  title: Carathéodory's theorem
+  title: Carathéodory's theorem (measure theory)
   decl: MeasureTheory.OuterMeasure.caratheodory
   comment: Hard to say what exactly is meant.
 
@@ -280,7 +277,7 @@ Q260928:
   title: Jordan curve theorem
 
 Q266291:
-  title: Bloch's theorem
+  title: Bloch's theorem (complex variables)
 
 Q268031:
   title: Abel's binomial theorem
@@ -383,6 +380,7 @@ Q384142:
 
 Q386292:
   title: Prime number theorem
+  comment: "Ongoing formalisation of various versions in https://github.com/AlexKontorovich/PrimeNumberTheoremAnd"
   # also in 100.yaml
 
 Q388525:
@@ -464,8 +462,8 @@ Q504843:
   title: Mycielski's theorem
 
 Q505798:
-  title: Lagrange's theorem
-  # group theorem result; exist additive and multiplicative version
+  title: Lagrange's theorem (group theory)
+  # exist additive and multiplicative version
   decl: Subgroup.card_subgroup_dvd_card
 
 Q510197:
@@ -526,7 +524,7 @@ Q574902:
   title: Tarski's indefinability theorem
 
 Q576478:
-  title: Liouville's theorem
+  title: Liouville's theorem (complex analysis)
   decl: Differentiable.apply_eq_apply_of_bounded
 
 Q578555:
@@ -554,6 +552,9 @@ Q594571:
 
 Q595466:
   title: Dini's theorem
+  decl: Monotone.tendstoLocallyUniformly_of_forall_tendsto
+  authors: Jireh Loreaux
+  date: 2024
 
 Q599876:
   title: Cochran's theorem
@@ -620,7 +621,7 @@ Q649977:
   title: Shirshov–Cohn theorem
 
 Q650738:
-  title: Folk theorem
+  title: Folk theorem (game theory)
 
 Q651593:
   title: Norton's theorem
@@ -634,7 +635,9 @@ Q656198:
 
 Q656645:
   title: Hilbert's basis theorem
-  # search mathlib or zulip; mathlib is close to this!
+  decl: Polynomial.isNoetherianRing
+  authors: Kenny Lau
+  date: 2019
 
 Q656772:
   title: Cayley–Hamilton theorem
@@ -656,7 +659,7 @@ Q657903:
   title: Hilbert–Waring theorem
 
 Q660799:
-  title: Darboux's theorem
+  title: Darboux's theorem (analysis)
   # "all derivatives have the intermediate value property"
   # TODO should exist in mathlib, add decl!
 
@@ -818,6 +821,9 @@ Q794269:
 
 Q810431:
   title: Basel problem
+  decl: hasSum_zeta_two
+  authors: David Loeffler
+  date: 2022
 
 Q828284:
   title: Parallel axis theorem
@@ -844,7 +850,7 @@ Q837506:
   comment: "will enter mathlib in 2024"
 
 Q837551:
-  title: Frobenius theorem
+  title: Frobenius theorem (differential topology)
 
 Q841893:
   title: Desargues's theorem
@@ -911,7 +917,7 @@ Q859122:
   title: Cauchy–Hadamard theorem
 
 Q865665:
-  title: Birkhoff's theorem
+  title: Birkhoff's theorem (relativity)
 
 Q866116:
   title: Hahn–Banach theorem
@@ -936,7 +942,7 @@ Q890875:
   title: Bohr–van Leeuwen theorem
 
 Q897769:
-  title: König's theorem
+  title: Kőnig's theorem (graph theory)
 
 Q899002:
   title: Pascal's theorem
@@ -1024,12 +1030,13 @@ Q943246:
   decl: littleWedderburn
 
 Q944297:
-  title: Open mapping theorem
-  # TODO: this entry points to Wikipedia's disambiguation page => wikipedia should be fixed
-  # Banach open mapping theorem/Banach-Schauder theorem in functional analysis is in mathlib (ContinuousLinearMap.isOpenMap)
+  title: Open mapping theorem (functional analysis)
+  decl: ContinuousLinearMap.isOpenMap
+  authors: Sébastien Gouëzel
+  date: 2019
 
 Q948664:
-  title: Kneser's addition theorem
+  title: Kneser's addition theorem (combinatorics)
   authors: Mantas Bakšys, Yaël Dillies
   url: https://github.com/YaelDillies/LeanCamCombi/blob/master/LeanCamCombi/Kneser/Kneser.lean
   date: 2022
@@ -1050,11 +1057,11 @@ Q966837:
   title: Cramér–Wold theorem
 
 Q967972:
-  title: Open mapping theorem
+  title: Open mapping theorem (complex analysis)
   decl: AnalyticOnNhd.is_constant_or_isOpen
 
 Q973359:
-  title: Rado's theorem
+  title: Radó's theorem (harmonic functions)
 
 Q974405:
   title: Hille–Yosida theorem
@@ -1161,7 +1168,7 @@ Q1057919:
 
 Q1058662:
   title: Darboux's theorem
-  # symplectic geometry result
+  # symplectic geometry
 
 Q1059151:
   title: FWL theorem
@@ -1213,7 +1220,7 @@ Q1076274:
   title: Choquet–Bishop–de Leeuw theorem
 
 Q1077462:
-  title: König's theorem
+  title: König's theorem (set theory)
 
 Q1077741:
   title: Hairy ball theorem
@@ -1268,7 +1275,7 @@ Q1135706:
   title: Harnack's theorem
 
 Q1136043:
-  title: Hurwitz's theorem
+  title: Hurwitz's theorem (complex analysis)
 
 Q1136262:
   title: Optical theorem
@@ -1285,12 +1292,11 @@ Q1137206:
   authors: Moritz Doll
 
 Q1139041:
-  title: Cauchy's theorem
-  # for finite groups
+  title: Cauchy's theorem (group theory)
   decl: exists_prime_orderOf_dvd_card
 
 Q1139430:
-  title: Hurwitz's theorem
+  title: Hurwitz's theorem (normed division algebras)
 
 Q1139524:
   title: Rationality theorem
@@ -1302,7 +1308,7 @@ Q1140200:
   title: PCP theorem
 
 Q1141747:
-  title: Carnot's theorem
+  title: Carnot's theorem (inradius, circumradius)
 
 Q1143540:
   title: Heckscher–Ohlin theorem
@@ -1421,7 +1427,7 @@ Q1217677:
   authors: Yury G. Kudryashov (first) and Benjamin Davidson (second)
 
 Q1227061:
-  title: Khinchin's theorem
+  title: Khinchin's theorem on Diophantine approximations
 
 Q1227702:
   title: Dirichlet's unit theorem
@@ -1475,7 +1481,7 @@ Q1317367:
   title: Japanese theorem for concyclic polygons
 
 Q1322892:
-  title: Dirac's theorems
+  title: Dirac's theorem on chordal graphs
 
 Q1330788:
   title: Weierstrass factorization theorem
@@ -1500,7 +1506,7 @@ Q1357684:
   title: Riesz representation theorem
 
 Q1361031:
-  title: Frobenius theorem
+  title: Frobenius theorem (real division algebras)
 
 Q1361393:
   title: Luzin's theorem
@@ -1564,6 +1570,9 @@ Q1455794:
 
 Q1457052:
   title: Well-ordering theorem
+  decl: exists_wellOrder
+  authors: Mario Carneiro, Chris Hughes, Violeta Hernández
+  date: 2017
 
 Q1471282:
   title: Radon's theorem
@@ -1646,7 +1655,7 @@ Q1621180:
   title: Orlicz–Pettis theorem
 
 Q1630588:
-  title: Hurwitz's theorem
+  title: Hurwitz's theorem (number theory)
 
 Q1631749:
   title: Lester's theorem
@@ -1729,6 +1738,12 @@ Q1765521:
 
 Q1766814:
   title: Smn theorem
+  decls:
+    - Nat.Partrec.Code.smn
+    - Nat.Partrec'.part_iff
+  authors: Mario Carneiro
+  date: 2018
+  comment: "This theorem is trivial when using Mathlib's computability definition, but this definition is equivalent to the usual one, as shown in `Nat.Partrec'.part_iff`."
 
 Q1785610:
   title: Poncelet's closure theorem
@@ -1781,6 +1796,9 @@ Q1930577:
 
 Q1933521:
   title: Kleene's recursion theorem
+  decl: Nat.Partrec.Code.fixed_point₂
+  authors: Mario Carneiro
+  date: 2018
 
 Q1938251:
   title: Hasse–Minkowski theorem
@@ -1811,7 +1829,7 @@ Q2000090:
   title: Art gallery theorem
 
 Q2008549:
-  title: Hilbert's theorem
+  title: Hilbert's theorem (differential geometry)
 
 Q2013213:
   title: Reuschle's theorem
@@ -1991,7 +2009,7 @@ Q2345282:
   title: Shannon's theorem
 
 Q2347139:
-  title: Carnot's theorem
+  title: Carnot's theorem (thermodynamics)
 
 Q2360531:
   title: Jordan–Schur theorem
@@ -2028,7 +2046,7 @@ Q2449984X:
   title: Pivot theorem
 
 Q2471737:
-  title: Carathéodory's theorem
+  title: Carathéodory's theorem (convex hull)
   decl: convexHull_eq_union
 
 Q2473965:
@@ -2054,7 +2072,7 @@ Q2518048:
   title: Kodaira vanishing theorem
 
 Q2524990:
-  title: Jackson's theorem
+  title: Jackson's theorem (queueing theory)
 
 Q2525646:
   title: Jordan–Hölder theorem
@@ -2109,7 +2127,7 @@ Q2800071:
   title: Perfect graph theorem
 
 Q2862403:
-  title: Bombieri's theorem
+  title: Schneider–Lang theorem
 
 Q2893695:
   title: Unmixedness theorem
@@ -2175,7 +2193,11 @@ Q3345678:
   title: Moore–Aronszajn theorem
 
 Q3424029:
-  title: Chasles's theorems
+  # TODO: refer to the individual entries instead
+  # The list in wikipedia has already been changed. This will will change automatically
+  # when the 1000+ theorem project synchronises its data with wikipedia and this file is
+  # synchronised with the 1000+ theorem's data.
+  title: Chasles's theorem (disambiguation)
 
 Q3443426:
   title: Bondy's theorem
@@ -2223,10 +2245,10 @@ Q3527009:
   title: Baker's theorem
 
 Q3527011:
-  title: Beck's theorem
+  title: Beck's theorem (geometry)
 
 Q3527015:
-  title: Bernstein's theorem
+  title: Bernstein's theorem on monotone functions
 
 Q3527017:
   title: Beurling–Lax theorem
@@ -2235,7 +2257,7 @@ Q3527019:
   title: Birch's theorem
 
 Q3527034:
-  title: Cauchy's theorem
+  title: Cauchy's theorem (geometry)
 
 Q3527040:
   title: Chomsky–Schützenberger representation theorem
@@ -2473,7 +2495,7 @@ Q4378868:
   date: 2022
 
 Q4378889:
-  title: Carathéodory's theorem
+  title: Carathéodory's theorem (conformal mapping)
 
 Q4408070:
   title: De Finetti's theorem
@@ -2497,8 +2519,7 @@ Q4454973:
   title: Lindelöf's theorem
 
 Q4454976:
-  title: Liouville's theorem
-  # theorem in conformal mappings
+  title: Liouville's theorem (conformal mappings)
 
 Q4454989:
   title: Meusnier's theorem
@@ -2585,7 +2606,7 @@ Q4734844:
   title: Alperin–Brauer–Gorenstein theorem
 
 Q4736422:
-  title: Denjoy theorem
+  title: Denjoy–Luzin theorem
 
 Q4746948:
   title: Amitsur–Levitzki theorem
@@ -2664,6 +2685,9 @@ Q4877965:
 
 Q4878586:
   title: Beck's monadicity theorem
+  decl: CategoryTheory.Monad.monadicOfCreatesGSplitCoequalizers
+  authors: Bhavik Mehta
+  date: 2020
 
 Q4880958:
   title: Behnke–Stein theorem
@@ -2678,7 +2702,7 @@ Q4891633:
   title: Berger–Kazdan comparison theorem
 
 Q4894565:
-  title: Bernstein's theorem
+  title: Bernstein's theorem (approximation theory)
 
 Q4914101:
   title: Bing's recognition theorem
@@ -2986,7 +3010,7 @@ Q5494134:
   title: Fraňková–Helly selection theorem
 
 Q5498822:
-  title: Birkhoff's theorem
+  title: Birkhoff's ergodic theorem
 
 Q5499906:
   title: Shirshov–Witt theorem
@@ -3057,10 +3081,10 @@ Q5609384:
   title: Grinberg's theorem
 
 Q5610188:
-  title: Gromov's compactness theorem
+  title: Gromov's compactness theorem (geometry)
 
 Q5610190:
-  title: Gromov's compactness theorem
+  title: Gromov's compactness theorem (topology)
 
 Q5610718:
   title: Grothendieck's connectedness theorem
@@ -3174,8 +3198,7 @@ Q6402928:
   title: Lagrange reversion theorem
 
 Q6403282:
-  title: Lagrange's theorem
-  # number theory
+  title: Lagrange's theorem (number theory)
 
 Q6407842:
   title: Killing–Hopf theorem
@@ -3653,8 +3676,7 @@ Q7818977:
   title: Tomita's theorem
 
 Q7820874:
-  title: Tonelli's theorem
-  decl: MeasureTheory.lintegral_prod
+  title: Tonelli's theorem (functional analysis)
 
 Q7824894:
   title: Topkis's theorem
@@ -3718,7 +3740,7 @@ Q7966545:
   title: Walter theorem
 
 Q7978735:
-  title: Weber's theorem
+  title: Weber's theorem (Algebraic curves)
 
 Q7980241:
   title: Weinberg–Witten theorem
@@ -3905,7 +3927,7 @@ Q18205730:
   title: Byers–Yang theorem
 
 Q18206032:
-  title: Cartan's theorem
+  title: Closed subgroup theorem
 
 Q18206266:
   title: Euclid–Euler theorem
@@ -3946,7 +3968,7 @@ Q25304301:
   title: Bregman–Minc inequality
 
 Q25345219:
-  title: BBD decomposition theorem
+  title: Decomposition theorem of Beilinson, Bernstein and Deligne
 
 Q25378690:
   title: Ionescu-Tulcea theorem
@@ -3958,7 +3980,7 @@ Q28194853:
   title: Lovelock's theorem
 
 Q28458131:
-  title: Glivenko's theorem
+  title: Glivenko's theorem (probability theory)
 
 Q31838822:
   title: Kirchberger's theorem
diff --git a/lake-manifest.json b/lake-manifest.json
index bc1e91750309a4..f3bfaf39ccd3aa 100644
--- a/lake-manifest.json
+++ b/lake-manifest.json
@@ -65,7 +65,7 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "62398edfce9977467f273b8d104f1591aae16843",
+   "rev": "b198ae51d27902453b27d01fc0ac07eeadcbe2e6",
    "name": "batteries",
    "manifestFile": "lake-manifest.json",
    "inputRev": "fin-find",
diff --git a/scripts/lint-style.lean b/scripts/lint-style.lean
index b4b7d93b9f2839..7f1b2cdd34467c 100644
--- a/scripts/lint-style.lean
+++ b/scripts/lint-style.lean
@@ -29,6 +29,12 @@ def findImports (path : System.FilePath) : IO (Array Lean.Name) := do
   return (← Lean.parseImports' (← IO.FS.readFile path) path.toString)
     |>.map (fun imp ↦ imp.module) |>.erase `Init
 
+/-- Additional imports generated by `mk_all`. -/
+def explicitImports : Array Lean.Name := #[`Batteries, `Std]
+
+/-- Remove the additional imports generated by `mk_all` so that only mathlib modules remain. -/
+def eraseExplicitImports (names : Array Lean.Name) : Array Lean.Name :=
+  explicitImports.foldl Array.erase names
 
 /-- Check that `Mathlib.Init` is transitively imported in all of Mathlib.
 
@@ -39,7 +45,7 @@ Return `true` iff there was an error.
 def checkInitImports : IO Bool := do
   -- Find any file in the Mathlib directory which does not contain any Mathlib import.
   -- We simply parse `Mathlib.lean`, as CI ensures this file is up to date.
-  let allModuleNames := (← findImports "Mathlib.lean").erase `Batteries
+  let allModuleNames := eraseExplicitImports (← findImports "Mathlib.lean")
   let mut modulesWithoutMathlibImports := #[]
   let mut importsHeaderLinter := #[]
   for module in allModuleNames do
@@ -107,8 +113,9 @@ def lintStyleCli (args : Cli.Parsed) : IO UInt32 := do
   let mut allModuleNames := #[]
   for s in ["Archive.lean", "Counterexamples.lean", "Mathlib.lean"] do
     allModuleNames := allModuleNames.append (← findImports s)
-  -- Note: since "Batteries" is added explicitly to "Mathlib.lean", we remove it here manually.
-  allModuleNames := allModuleNames.erase `Batteries
+  -- Note: since "Batteries" and "Std" are added explicitly to "Mathlib.lean", we remove them here
+  -- manually.
+  allModuleNames := eraseExplicitImports allModuleNames
 
   -- Read the `nolints` file, with manual exceptions for the linter.
   -- NB. We pass these lints to `lintModules` explicitly to prevent cache invalidation bugs:
diff --git a/scripts/mk_all.lean b/scripts/mk_all.lean
index 00bacbc8eaefb8..ae79948610542f 100644
--- a/scripts/mk_all.lean
+++ b/scripts/mk_all.lean
@@ -55,9 +55,9 @@ def mkAllCLI (args : Parsed) : IO UInt32 := do
   for d in libs.reverse do  -- reverse to create `Mathlib/Tactic.lean` before `Mathlib.lean`
     let fileName := addExtension d "lean"
     let mut allFiles ← getAllModulesSorted git d
-    -- mathlib exception: manually import Batteries in `Mathlib.lean`
+    -- mathlib exception: manually import Std and Batteries in `Mathlib.lean`
     if d == "Mathlib" then
-      allFiles := #["Batteries"] ++ allFiles
+      allFiles := #["Std", "Batteries"] ++ allFiles
     let fileContent := ("\n".intercalate (allFiles.map ("import " ++ ·)).toList).push '\n'
     if !(← pathExists fileName) then
       if check then
diff --git a/scripts/nolints_prime_decls.txt b/scripts/nolints_prime_decls.txt
index be856e67cdf248..fd259826572ac5 100644
--- a/scripts/nolints_prime_decls.txt
+++ b/scripts/nolints_prime_decls.txt
@@ -1117,7 +1117,7 @@ DoubleCentralizer.nnnorm_def'
 DoubleCentralizer.norm_def'
 dvd_antisymm'
 dvd_geom_sum₂_iff_of_dvd_sub'
-edist_eq_coe_nnnorm'
+edist_zero_eq_enorm'
 EllipticCurve.coe_inv_map_Δ'
 EllipticCurve.coe_inv_variableChange_Δ'
 EllipticCurve.coe_map_Δ'
@@ -2823,7 +2823,7 @@ MeasureTheory.lintegral_mono_fn'
 MeasureTheory.lintegral_mono_set'
 MeasureTheory.lintegral_mul_const'
 MeasureTheory.lintegral_mul_const''
-MeasureTheory.lintegral_rpow_nnnorm_eq_rpow_eLpNorm'
+MeasureTheory.lintegral_rpow_enorm_eq_rpow_eLpNorm'
 MeasureTheory.lintegral_singleton'
 MeasureTheory.lintegral_sub'
 MeasureTheory.lintegral_sub_le'
@@ -3494,7 +3494,7 @@ Num.of_to_nat'
 Num.succ_ofInt'
 odd_add_one_self'
 odd_add_self_one'
-ofReal_norm_eq_coe_nnnorm'
+ofReal_norm_eq_enorm'
 OmegaCompletePartialOrder.const_continuous'
 OmegaCompletePartialOrder.ContinuousHom.bind_continuous'
 OmegaCompletePartialOrder.ContinuousHom.forall_forall_merge'
diff --git a/scripts/noshake.json b/scripts/noshake.json
index 5a6590b022253c..2732e7a608dbe0 100644
--- a/scripts/noshake.json
+++ b/scripts/noshake.json
@@ -303,8 +303,10 @@
   "Mathlib.Tactic.CategoryTheory.BicategoryCoherence":
   ["Mathlib.CategoryTheory.Bicategory.Coherence",
    "Mathlib.Tactic.CategoryTheory.BicategoricalComp"],
+  "Mathlib.Tactic.CC.Datatypes":
+  ["Mathlib.Lean.Meta.CongrTheorems", "Mathlib.Lean.Meta.Datatypes"],
   "Mathlib.Tactic.CC.MkProof":
-  ["Mathlib.Logic.Basic", "Mathlib.Tactic.CC.Lemmas"],
+  ["Mathlib.Tactic.CC.Lemmas"],
   "Mathlib.Tactic.ByContra": ["Batteries.Tactic.Init"],
   "Mathlib.Tactic.Bound.Init": ["Aesop.Frontend.Command"],
   "Mathlib.Tactic.Bound.Attribute":
@@ -340,6 +342,8 @@
   "Mathlib.Probability.Notation":
   ["Mathlib.MeasureTheory.Decomposition.Lebesgue",
    "Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic"],
+  "Mathlib.Probability.Kernel.Proper":
+  ["Mathlib.Probability.Kernel.Composition.CompNotation"],
   "Mathlib.Order.RelClasses": ["Batteries.WF"],
   "Mathlib.Order.Filter.ListTraverse":
   ["Mathlib.Control.Traversable.Instances"],
diff --git a/scripts/technical-debt-metrics.sh b/scripts/technical-debt-metrics.sh
index efe28e4c7e85d6..0aeb3052699316 100755
--- a/scripts/technical-debt-metrics.sh
+++ b/scripts/technical-debt-metrics.sh
@@ -73,11 +73,14 @@ computeDiff () {
 # The script uses the fact that a line represents a technical debt if and only if the text before
 # the first `|` is a number.  This is then used for comparison and formatting.
 tdc () {
+# We perform word-splitting "by hand" in the "middle" entries.
+# See also the comment on the `read` line in the for-loop that follows the definition of this array.
 titlesPathsAndRegexes=(
   "porting notes"                  "*"      "Porting note"
   "backwards compatibility flags"  "*"      "set_option.*backward"
   "skipAssignedInstances flags"    "*"      "set_option tactic.skipAssignedInstances"
-  "adaptation notes"               "*"      "adaptation_note"
+  "adaptation notes"               ":^Mathlib/Tactic/AdaptationNote.lean :^Mathlib/Tactic/Linter"
+                                            "^[· ]*#adaptation_note"
   "disabled simpNF lints"          "*"      "nolint simpNF"
   "erw"                            "*"      "erw \["
   "maxHeartBeats modifications"    ":^MathlibTest" "^ *set_option .*maxHeartbeats"
@@ -87,13 +90,15 @@ for i in ${!titlesPathsAndRegexes[@]}; do
   # loop on every 3rd entry and name that entry and the following two
   if (( i % 3 == 0 )); then
     title="${titlesPathsAndRegexes[$i]}"
-    pathspec="${titlesPathsAndRegexes[$(( i + 1 ))]}"
+    # Here we perform word-splitting: `pathspec` is an array whose entries are the "words" in
+    # the string `"${titlesPathsAndRegexes[$(( i + 1 ))]}"`.
+    read -r -a pathspec <<< "${titlesPathsAndRegexes[$(( i + 1 ))]}"
     regex="${titlesPathsAndRegexes[$(( i + 2 ))]}"
     if [ "${title}" == "porting notes" ]
     then fl="-i"  # just for porting notes we ignore the case in the regex
     else fl="--"
     fi
-    printf '%s|%s\n' "$(git grep "${fl}" "${regex}" -- "${pathspec}" | wc -l)" "${title}"
+    printf '%s|%s\n' "$(git grep "${fl}" "${regex}" -- ":^scripts" "${pathspec[@]}" | wc -l)" "${title}"
   fi
 done