diff --git a/src/analysis/inner_product_space/pi_L2.lean b/src/analysis/inner_product_space/pi_L2.lean
index ab8caa6dc83db..d4ae133a5b30f 100644
--- a/src/analysis/inner_product_space/pi_L2.lean
+++ b/src/analysis/inner_product_space/pi_L2.lean
@@ -701,7 +701,7 @@ sum has an orthonormal basis indexed by `fin n` and subordinate to that direct s
 /-- An `n`-dimensional `inner_product_space` equipped with a decomposition as an internal direct
 sum has an orthonormal basis indexed by `fin n` and subordinate to that direct sum. This function
 provides the mapping by which it is subordinate. -/
-def direct_sum.is_internal.subordinate_orthonormal_basis_index
+@[irreducible] def direct_sum.is_internal.subordinate_orthonormal_basis_index
   (a : fin n) (hV' : @orthogonal_family 𝕜 _ _ _ _ (λ i, V i) _ (λ i, (V i).subtypeₗᵢ)) : ι :=
 ((hV.sigma_orthonormal_basis_index_equiv hn hV').symm a).1
 
@@ -712,12 +712,10 @@ lemma direct_sum.is_internal.subordinate_orthonormal_basis_subordinate
   (hV.subordinate_orthonormal_basis hn hV' a) ∈
   V (hV.subordinate_orthonormal_basis_index hn a hV') :=
 by simpa only [direct_sum.is_internal.subordinate_orthonormal_basis,
-  orthonormal_basis.coe_reindex]
+  orthonormal_basis.coe_reindex, direct_sum.is_internal.subordinate_orthonormal_basis_index]
   using hV.collected_orthonormal_basis_mem hV' (λ i, (std_orthonormal_basis 𝕜 (V i)))
     ((hV.sigma_orthonormal_basis_index_equiv hn hV').symm a)
 
-attribute [irreducible] direct_sum.is_internal.subordinate_orthonormal_basis_index
-
 end subordinate_orthonormal_basis
 
 end finite_dimensional
diff --git a/src/analysis/inner_product_space/spectrum.lean b/src/analysis/inner_product_space/spectrum.lean
index f6d03066c8430..07ce6120ccac8 100644
--- a/src/analysis/inner_product_space/spectrum.lean
+++ b/src/analysis/inner_product_space/spectrum.lean
@@ -179,7 +179,7 @@ finite-dimensional inner product space `E`.
 
 TODO Postcompose with a permutation so that these eigenvectors are listed in increasing order of
 eigenvalue. -/
-noncomputable def eigenvector_basis : orthonormal_basis (fin n) 𝕜 E :=
+@[irreducible] noncomputable def eigenvector_basis : orthonormal_basis (fin n) 𝕜 E :=
 hT.direct_sum_is_internal.subordinate_orthonormal_basis hn
   hT.orthogonal_family_eigenspaces'
 
@@ -187,7 +187,7 @@ hT.direct_sum_is_internal.subordinate_orthonormal_basis hn
 for a self-adjoint operator `T` on `E`.
 
 TODO Postcompose with a permutation so that these eigenvalues are listed in increasing order. -/
-noncomputable def eigenvalues (i : fin n) : ℝ :=
+@[irreducible] noncomputable def eigenvalues (i : fin n) : ℝ :=
 @is_R_or_C.re 𝕜 _ $
   hT.direct_sum_is_internal.subordinate_orthonormal_basis_index hn i
     hT.orthogonal_family_eigenspaces'
@@ -198,11 +198,13 @@ begin
   let v : E := hT.eigenvector_basis hn i,
   let μ : 𝕜 := hT.direct_sum_is_internal.subordinate_orthonormal_basis_index
     hn i hT.orthogonal_family_eigenspaces',
+  simp_rw [eigenvalues],
   change has_eigenvector T (is_R_or_C.re μ) v,
   have key : has_eigenvector T μ v,
   { have H₁ : v ∈ eigenspace T μ,
-    { exact hT.direct_sum_is_internal.subordinate_orthonormal_basis_subordinate
-      hn i hT.orthogonal_family_eigenspaces' },
+    { simp_rw [v, eigenvector_basis],
+      exact hT.direct_sum_is_internal.subordinate_orthonormal_basis_subordinate
+        hn i hT.orthogonal_family_eigenspaces' },
     have H₂ : v ≠ 0 := by simpa using (hT.eigenvector_basis hn).to_basis.ne_zero i,
     exact ⟨H₁, H₂⟩ },
   have re_μ : ↑(is_R_or_C.re μ) = μ,
@@ -212,9 +214,7 @@ begin
 end
 
 lemma has_eigenvalue_eigenvalues (i : fin n) : has_eigenvalue T (hT.eigenvalues hn i) :=
-    module.End.has_eigenvalue_of_has_eigenvector (hT.has_eigenvector_eigenvector_basis hn i)
-
-attribute [irreducible] eigenvector_basis eigenvalues
+module.End.has_eigenvalue_of_has_eigenvector (hT.has_eigenvector_eigenvector_basis hn i)
 
 @[simp] lemma apply_eigenvector_basis (i : fin n) :
   T (hT.eigenvector_basis hn i) = (hT.eigenvalues hn i : 𝕜) • hT.eigenvector_basis hn i :=
diff --git a/src/data/complex/exponential.lean b/src/data/complex/exponential.lean
index 21c7ea846b9e9..f41c5375fa3cc 100644
--- a/src/data/complex/exponential.lean
+++ b/src/data/complex/exponential.lean
@@ -337,7 +337,7 @@ the complex exponential function -/
 ⟨λ n, ∑ m in range n, z ^ m / m!, is_cau_exp z⟩
 
 /-- The complex exponential function, defined via its Taylor series -/
-@[pp_nodot] def exp (z : ℂ) : ℂ := lim (exp' z)
+@[irreducible, pp_nodot] def exp (z : ℂ) : ℂ := 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
@@ -392,8 +392,9 @@ namespace complex
 variables (x y : ℂ)
 
 @[simp] lemma exp_zero : exp 0 = 1 :=
-lim_eq_of_equiv_const $
-  λ ε ε0, ⟨1, λ j hj, begin
+begin
+  rw exp,
+  refine lim_eq_of_equiv_const (λ ε ε0, ⟨1, λ j hj, _⟩),
   convert ε0,
   cases j,
   { exact absurd hj (not_le_of_gt zero_lt_one) },
@@ -403,33 +404,29 @@ lim_eq_of_equiv_const $
     { rw ← ih dec_trivial,
       simp only [sum_range_succ, pow_succ],
       simp } }
-end⟩
+end
 
 lemma exp_add : exp (x + y) = exp x * exp y :=
-show lim (⟨_, is_cau_exp (x + y)⟩ : cau_seq ℂ abs) =
-  lim (show cau_seq ℂ abs, from ⟨_, is_cau_exp x⟩)
-  * lim (show cau_seq ℂ abs, from ⟨_, is_cau_exp y⟩),
-from
-have hj : ∀ j : ℕ, ∑ m in range j, (x + y) ^ m / m! =
-    ∑ i in range j, ∑ k in range (i + 1), x ^ k / k! * (y ^ (i - k) / (i - k)!),
-  from assume j,
-    finset.sum_congr rfl (λ m hm, begin
-      rw [add_pow, div_eq_mul_inv, sum_mul],
-      refine finset.sum_congr rfl (λ i hi, _),
-      have h₁ : (m.choose i : ℂ) ≠ 0 := nat.cast_ne_zero.2
-        (pos_iff_ne_zero.1 (nat.choose_pos (nat.le_of_lt_succ (mem_range.1 hi)))),
-      have h₂ := nat.choose_mul_factorial_mul_factorial (nat.le_of_lt_succ $ finset.mem_range.1 hi),
-      rw [← h₂, nat.cast_mul, nat.cast_mul, mul_inv, mul_inv],
-      simp only [mul_left_comm (m.choose i : ℂ), mul_assoc, mul_left_comm (m.choose i : ℂ)⁻¹,
-        mul_comm (m.choose i : ℂ)],
-      rw inv_mul_cancel h₁,
-      simp [div_eq_mul_inv, mul_comm, mul_assoc, mul_left_comm]
-    end),
-by rw lim_mul_lim;
-  exact eq.symm (lim_eq_lim_of_equiv (by dsimp; simp only [hj];
-    exact cauchy_product (is_cau_abs_exp x) (is_cau_exp y)))
-
-attribute [irreducible] complex.exp
+begin
+  have hj : ∀ j : ℕ, ∑ m in range j, (x + y) ^ m / m! =
+      ∑ i in range j, ∑ k in range (i + 1), x ^ k / k! * (y ^ (i - k) / (i - k)!),
+  { assume j,
+    refine finset.sum_congr rfl (λ m hm, _),
+    rw [add_pow, div_eq_mul_inv, sum_mul],
+    refine finset.sum_congr rfl (λ i hi, _),
+    have h₁ : (m.choose i : ℂ) ≠ 0 := nat.cast_ne_zero.2
+      (pos_iff_ne_zero.1 (nat.choose_pos (nat.le_of_lt_succ (mem_range.1 hi)))),
+    have h₂ := nat.choose_mul_factorial_mul_factorial (nat.le_of_lt_succ $ finset.mem_range.1 hi),
+    rw [← h₂, nat.cast_mul, nat.cast_mul, mul_inv, mul_inv],
+    simp only [mul_left_comm (m.choose i : ℂ), mul_assoc, mul_left_comm (m.choose i : ℂ)⁻¹,
+      mul_comm (m.choose i : ℂ)],
+    rw inv_mul_cancel h₁,
+    simp [div_eq_mul_inv, mul_comm, mul_assoc, mul_left_comm] },
+  simp_rw [exp, exp', lim_mul_lim],
+  apply (lim_eq_lim_of_equiv _).symm,
+  simp only [hj],
+  exact cauchy_product (is_cau_abs_exp x) (is_cau_exp y)
+end
 
 lemma exp_list_sum (l : list ℂ) : exp l.sum = (l.map exp).prod :=
 @monoid_hom.map_list_prod (multiplicative ℂ) ℂ _ _ ⟨exp, exp_zero, exp_add⟩ l
diff --git a/src/ring_theory/fractional_ideal.lean b/src/ring_theory/fractional_ideal.lean
index 458ac481a7a8f..7a28d8888942c 100644
--- a/src/ring_theory/fractional_ideal.lean
+++ b/src/ring_theory/fractional_ideal.lean
@@ -1094,14 +1094,15 @@ variables (S)
 def span_singleton (x : P) : fractional_ideal S P :=
 ⟨span R {x}, is_fractional_span_singleton x⟩
 
-local attribute [semireducible] span_singleton
+-- local attribute [semireducible] span_singleton
 
 @[simp] lemma coe_span_singleton (x : P) :
-  (span_singleton S x : submodule R P) = span R {x} := rfl
+  (span_singleton S x : submodule R P) = span R {x} :=
+by { rw span_singleton, refl }
 
 @[simp] lemma mem_span_singleton {x y : P} :
   x ∈ span_singleton S y ↔ ∃ (z : R), z • y = x :=
-submodule.mem_span_singleton
+by { rw span_singleton, exact submodule.mem_span_singleton }
 
 lemma mem_span_singleton_self (x : P) :
   x ∈ span_singleton S x :=
@@ -1111,12 +1112,13 @@ variables {S}
 
 lemma span_singleton_eq_span_singleton [no_zero_smul_divisors R P] {x y : P} :
   span_singleton S x = span_singleton S y ↔ ∃ z : Rˣ, z • x = y :=
-by { rw [← submodule.span_singleton_eq_span_singleton], exact subtype.mk_eq_mk }
+by { rw [← submodule.span_singleton_eq_span_singleton, span_singleton, span_singleton],
+  exact subtype.mk_eq_mk }
 
 lemma eq_span_singleton_of_principal (I : fractional_ideal S P)
   [is_principal (I : submodule R P)] :
   I = span_singleton S (generator (I : submodule R P)) :=
-coe_to_submodule_injective (span_singleton_generator ↑I).symm
+by { rw span_singleton, exact coe_to_submodule_injective (span_singleton_generator ↑I).symm }
 
 lemma is_principal_iff (I : fractional_ideal S P) :
   is_principal (I : submodule R P) ↔ ∃ x, I = span_singleton S x :=
@@ -1301,7 +1303,7 @@ begin
   obtain ⟨a, aI, -, ha⟩ := exists_eq_span_singleton_mul I,
   use (algebra_map R K a)⁻¹ * algebra_map R K (generator aI),
   suffices : I = span_singleton R⁰ ((algebra_map R K a)⁻¹ * algebra_map R K (generator aI)),
-  { exact congr_arg subtype.val this },
+  { rw span_singleton at this, exact congr_arg subtype.val this },
   conv_lhs { rw [ha, ←span_singleton_generator aI] },
   rw [ideal.submodule_span_eq, coe_ideal_span_singleton (generator aI),
       span_singleton_mul_span_singleton]