refactor(Normed/Group/Quotient): streamline, multiplicativise

[YaelDillies]

• Multiplicativise using the (not so) new GroupNorm API.
• Deprecate the lemmas about the quotient norm that hold for all norms.
• Move all remaining lemmas to a single QuotientAddGroup namespace. They were currently scattered across _root_, AddSubgroup and QuotientAddGroup.
• Follow the naming convention in lemma names.

---

• depends on: [Merged by Bors] - chore: replace the lemma that dist 1 = norm by dist 1 a = ‖a‖ #21340

[github-actions]

PR summary 1278b3a994

Import changes exceeding 2%

%	File
+2.56%	Mathlib.Analysis.Normed.Group.AddCircle

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Analysis.Normed.Group.AddCircle	1445	1482	+37 (+2.56%)

Import changes for all files

Files	Import difference
34 files Mathlib.MeasureTheory.Decomposition.Lebesgue Mathlib.MeasureTheory.Function.ContinuousMapDense Mathlib.MeasureTheory.Function.ConvergenceInMeasure Mathlib.MeasureTheory.Function.L1Space.AEEqFun Mathlib.MeasureTheory.Function.L1Space.Integrable Mathlib.MeasureTheory.Function.LocallyIntegrable Mathlib.MeasureTheory.Function.LpOrder Mathlib.MeasureTheory.Function.LpSeminorm.Basic Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality Mathlib.MeasureTheory.Function.LpSeminorm.Trim Mathlib.MeasureTheory.Function.LpSpace.ContinuousCompMeasurePreserving Mathlib.MeasureTheory.Function.LpSpace.DomAct.Basic Mathlib.MeasureTheory.Function.LpSpace.DomAct.Continuous Mathlib.MeasureTheory.Function.LpSpace Mathlib.MeasureTheory.Function.SimpleFuncDenseLp Mathlib.MeasureTheory.Function.SpecialFunctions.Basic Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp Mathlib.MeasureTheory.Function.UnifTight Mathlib.MeasureTheory.Function.UniformIntegrable Mathlib.MeasureTheory.Group.Integral Mathlib.MeasureTheory.Integral.Bochner Mathlib.MeasureTheory.Integral.BoundedContinuousFunction Mathlib.MeasureTheory.Integral.IntegrableOn Mathlib.MeasureTheory.Integral.MeanInequalities Mathlib.MeasureTheory.Integral.SetToL1 Mathlib.MeasureTheory.Integral.VitaliCaratheodory Mathlib.MeasureTheory.Measure.FiniteMeasure Mathlib.MeasureTheory.Measure.SeparableMeasure Mathlib.Probability.Independence.Integrable Mathlib.Probability.Kernel.Integral Mathlib.Probability.Moments.IntegrableExpMul Mathlib.Probability.ProbabilityMassFunction.Integrals	2
72 files Mathlib.Algebra.Order.AbsoluteValue.Equivalence Mathlib.Analysis.Analytic.RadiusLiminf Mathlib.Analysis.CStarAlgebra.Multiplier Mathlib.Analysis.CStarAlgebra.lpSpace Mathlib.Analysis.Calculus.ContDiff.WithLp Mathlib.Analysis.Calculus.FDeriv.WithLp Mathlib.Analysis.Convex.SpecificFunctions.Basic Mathlib.Analysis.Convex.SpecificFunctions.Pow Mathlib.Analysis.MeanInequalitiesPow Mathlib.Analysis.MeanInequalities Mathlib.Analysis.Normed.Algebra.Norm Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt Mathlib.Analysis.Normed.Algebra.UnitizationL1 Mathlib.Analysis.Normed.Lp.LpEquiv Mathlib.Analysis.Normed.Lp.PiLp Mathlib.Analysis.Normed.Lp.ProdLp Mathlib.Analysis.Normed.Lp.lpSpace Mathlib.Analysis.Normed.Ring.IsPowMulFaithful Mathlib.Analysis.Normed.Ring.SeminormFromBounded Mathlib.Analysis.Normed.Ring.SeminormFromConst Mathlib.Analysis.Normed.Ring.Seminorm Mathlib.Analysis.NormedSpace.DualNumber Mathlib.Analysis.PSeries Mathlib.Analysis.SpecialFunctions.CompareExp Mathlib.Analysis.SpecialFunctions.Complex.Circle Mathlib.Analysis.SpecialFunctions.Complex.Log Mathlib.Analysis.SpecialFunctions.Log.Base Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog Mathlib.Analysis.SpecialFunctions.Log.Monotone Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics Mathlib.Analysis.SpecialFunctions.Pow.Complex Mathlib.Analysis.SpecialFunctions.Pow.Continuity Mathlib.Analysis.SpecialFunctions.Pow.NNReal Mathlib.Analysis.SpecialFunctions.Pow.Real Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex Mathlib.Analysis.SpecificLimits.FloorPow Mathlib.Combinatorics.Additive.Corner.Roth Mathlib.Combinatorics.SimpleGraph.Regularity.Bound Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk Mathlib.Combinatorics.SimpleGraph.Regularity.Increment Mathlib.Combinatorics.SimpleGraph.Regularity.Lemma Mathlib.Combinatorics.SimpleGraph.Triangle.Removal Mathlib.Computability.AkraBazzi.GrowsPolynomially Mathlib.Dynamics.TopologicalEntropy.CoverEntropy Mathlib.Dynamics.TopologicalEntropy.NetEntropy Mathlib.Dynamics.TopologicalEntropy.Semiconj Mathlib.LinearAlgebra.QuadraticForm.Complex Mathlib.MeasureTheory.Function.SpecialFunctions.Arctan Mathlib.MeasureTheory.Measure.Doubling Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree Mathlib.NumberTheory.ClassNumber.Finite Mathlib.NumberTheory.ClassNumber.FunctionField Mathlib.NumberTheory.Cyclotomic.Basic Mathlib.NumberTheory.ModularForms.EisensteinSeries.IsBoundedAtImInfty Mathlib.NumberTheory.ModularForms.EisensteinSeries.UniformConvergence Mathlib.NumberTheory.MulChar.Lemmas Mathlib.NumberTheory.Ostrowski Mathlib.NumberTheory.SumPrimeReciprocals Mathlib.RingTheory.Perfection Mathlib.RingTheory.Polynomial.Cyclotomic.Basic Mathlib.RingTheory.Polynomial.Cyclotomic.Expand Mathlib.RingTheory.Polynomial.Cyclotomic.Roots Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral Mathlib.RingTheory.RootsOfUnity.AlgebraicallyClosed Mathlib.RingTheory.RootsOfUnity.Complex Mathlib.Topology.Algebra.PontryaginDual Mathlib.Topology.MetricSpace.GromovHausdorff Mathlib.Topology.MetricSpace.HolderNorm Mathlib.Topology.MetricSpace.Holder Mathlib.Topology.MetricSpace.Kuratowski	3
Mathlib.Analysis.SpecialFunctions.Complex.Arg Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle	4
Mathlib.Analysis.Normed.Group.AddCircle	37

---

Declarations diff

+ coe_fract
+ exists_norm_mk'_lt
+ exists_norm_mul_lt
+ groupSeminorm
+ instNorm
+ instNormedCommGroup
+ instSeminormedCommGroup
+ instance : NormedAddCommGroup (AddCircle p) := QuotientAddGroup.instNormedAddCommGroup _
+ le_infDist
+ le_norm_iff
+ nhds_one_hasBasis
+ norm_eq_groupSeminorm
+ norm_eq_infDist
+ norm_lt_iff
+ norm_mk
+ norm_mk'
+ norm_mk'_eq_zero
+ norm_mk'_eq_zero_iff_mem_closure
+ norm_mk'_le_norm
+ norm_mk_eq_zero
+ norm_mk_eq_zero_iff_mem_closure
+ norm_mk_le_norm
- QuotientAddGroup.norm_eq_infDist
- QuotientAddGroup.norm_lt_iff
- QuotientAddGroup.norm_mk
- instance : NormedAddCommGroup (AddCircle p)

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

---

Decrease in tech debt: (relative, absolute) = (2.00, 0.00)

Current number	Change	Type
1401	-2	erw

Current commit 1278b3a994
Reference commit cd5fbfb76d

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• [Merged by Bors] - chore: replace the lemma that dist 1 = norm by dist 1 a = ‖a‖ #21340
  By Dependent Issues (🤖). Happy coding!