feat(Algebra): Define associated graded module

[Thmoas-Guan]

Define the associated graded module to a filtered module.

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• [ ] depends on: #26858

migrated from #20940

[github-actions[bot]]

PR summary 16dc83e76d

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.RingTheory.FilteredAlgebra.AssociatedGraded (new file)	849

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Declarations diff

+ AssociatedGraded + Filtration.pow_lift + Filtration.pow_mem + GradedPiece + GradedPiece.HEq_mul_assoc + GradedPiece.HEq_mul_one + GradedPiece.HEq_mul_smul + GradedPiece.HEq_one_mul + GradedPiece.HEq_one_smul + GradedPiece.add_mul + GradedPiece.add_smul + GradedPiece.gnpow + GradedPiece.gnpow_succ' + GradedPiece.gnpow_zero' + GradedPiece.intCast + GradedPiece.intCast_negSucc_ofNat + GradedPiece.intCast_ofNat + GradedPiece.mk_smul + GradedPiece.mul_add + GradedPiece.mul_zero + GradedPiece.natCast + GradedPiece.natCast_succ + GradedPiece.natCast_zero + GradedPiece.smul_add + GradedPiece.smul_zero + GradedPiece.zero_mul + GradedPiece.zero_smul + HEq_eq_mk_coe_eq + HEq_eq_mk_eq + HEq_rfl + IsModuleFiltration.hSMul + IsRingFiltration.hMul + ext + fst_smul + gnpow_def + gradedMul_def + gradedSMul_def + hMul + hSMul + hasGMul + hasGMul.gradedMul + hasGMul.mk_int + hasGMul.mul_equiv_mul + hasGMul_AddSubgroup + hasGSMul + hasGSMul.gradedSMul + hasGSMul.mul_equiv_mul + hasGSMul_AddSubgroup + hasGSMul_int + induction_on + instance (i : ι) (j : ιM) [IsModuleFiltration F F_lt FM FM_lt] : + instance : GradedMonoid.GSMul (GradedPiece F F_lt) (GradedPiece FM FM_lt) + instance [AddMonoid ι] [PartialOrder ι] [AddSubgroupClass σ R] + instance [IsRingFiltration F F_lt] : GradedMonoid.GOne (GradedPiece F F_lt) + instance [IsRingFiltration F F_lt] {i j : ι} : + instance [Preorder ι] [IsFiltration F F_lt] (i : ι) : Setoid (AddSubgroup.ofClass (F i)) + instance [hasGMul F F_lt] : DirectSum.GRing (GradedPiece F F_lt) + instance [hasGMul F F_lt] : DirectSum.GSemiring (GradedPiece F F_lt) + instance [hasGMul F F_lt] : DirectSum.Gmodule (GradedPiece F F_lt) (GradedPiece FM FM_lt) + instance [hasGMul F F_lt] : GradedMonoid.GMonoid (GradedPiece F F_lt) + instance [hasGMul F F_lt] : GradedMonoid.GMul (GradedPiece F F_lt) + instance [hasGMul F F_lt] [DecidableEq ι] : Ring (AssociatedGraded F F_lt) + instance [hasGMul F F_lt] [DecidableEq ι] [DecidableEq ιM] : + mk_apply_of_mem + mk_apply_of_not_mem + mk_eq + mk_injective + mk_mul + of + of_apply + of_eq_of_ne + of_eq_self + of_injective + sum_support_of + sum_univ_of + support_of + support_of_subset ++ mk

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.

---

No changes to technical debt.

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:

• leanprover-community/mathlib4#26858 By Dependent Issues (🤖). Happy coding!

[mathlib4-merge-conflict-bot]

This pull request has conflicts, please merge master and resolve them.

[mathlib4-merge-conflict-bot]

This pull request has conflicts, please merge master and resolve them.

[Thmoas-Guan]

I handed this PR out to another collaborator, the new PR is #33220.