feat(RingTheory): homotopy of Koszul complex and cocomplex

[Thmoas-Guan]

Homotopy between smul id and zero for Koszul complex and cocomplex, this gives results on annihilator of homology.

This PR have some proofs generated by LLM.

Co-authored-by: Wang Jingting wangjt2020@163.com
Co-authored-by: Zichen Wang zichenwang25@stu.pku.edu.cn

---

• depends on: feat(RingTheory): definition of Koszul complex #40929
• depends on: feat(RingTheory): definition of Koszul cocomplex #40930

[github-actions]

PR summary 8653ea14d5

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.RingTheory.KoszulComplex.Cocomplex (new file)	1984
Mathlib.RingTheory.KoszulComplex.Complex (new file)	1985
Mathlib.RingTheory.KoszulComplex.Homotopy (new file)	2128

---

Declarations diff (regex)

+ GradedAlgebra.linearGMul
+ GradedAlgebra.linearGMul_eq_mul
+ contraction_wedge_cartan_formula
+ contraction_wedge_zero_degree
+ d_def
+ free
+ isZero_X_ofList_of_length_le
+ isZero_X_ofList_of_length_lt
+ koszulCocomplex
+ koszulCocomplex.d
+ koszulCocomplex.d_apply_ιMulti
+ koszulCocomplex.homotopySMulIdZero
+ koszulCocomplex.mem_annihilator_homology
+ koszulCocomplex.ofList_ideal_le_mem_annihilator_homology
+ koszulCocomplex.scalar_homotopy_comm
+ koszulCocomplex.scalar_homotopy_comm_zero
+ koszulComplex
+ koszulComplex.d
+ koszulComplex.dAlternating
+ koszulComplex.dAlternating_apply
+ koszulComplex.d_comp_d
+ koszulComplex.d_def
+ koszulComplex.homotopySMulIdZero
+ koszulComplex.mem_annihilator_homology
+ koszulComplex.ofList_ideal_annihilator_homology
+ koszulComplex.range_le_annihilator_homology
+ koszulComplex.scalar_homotopy_comm
+ koszulComplex.scalar_homotopy_comm_zero
+ map_dAlternating_apply
+ map_d_comm
+ map_id_refl
+ obj_d'
+ of_d'
+ removeNth_comp
++ isZero_X_of_span_eq_top
++ isoOfEquiv
++ map
++ map_comp
++ map_f
++ map_id
++ ofList
- of.d

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

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

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 8653ea1).

• +57 new declarations
• −4 removed declarations

+AlgebraicTopology.NormalizedMooreComplex.obj_d'
-ChainComplex.of.d
-ChainComplex.of.d.congr_simp
+ChainComplex.of_d'
+Fin.removeNth_comp
+GradedAlgebra.linearGMul
+GradedAlgebra.linearGMul.congr_simp
+GradedAlgebra.linearGMul_eq_mul
+contraction_wedge_cartan_formula
+contraction_wedge_zero_degree
+groupHomology.chainsFunctor_obj
-groupHomology.chainsFunctor_obj_X_carrier
-groupHomology.chainsFunctor_obj_d
+koszulCocomplex
+koszulCocomplex.d
+koszulCocomplex.d_apply_ιMulti
+koszulCocomplex.d_def
+koszulCocomplex.free
+koszulCocomplex.homotopySMulIdZero
+koszulCocomplex.isZero_X_ofList_of_length_le
+koszulCocomplex.isZero_X_of_span_eq_top
+koszulCocomplex.isoOfEquiv
+koszulCocomplex.map
+koszulCocomplex.map.congr_simp
+koszulCocomplex.map_comp
+koszulCocomplex.map_comp_assoc
+koszulCocomplex.map_f
+koszulCocomplex.map_id
+koszulCocomplex.map_id_assoc
+koszulCocomplex.map_id_refl
+koszulCocomplex.map_id_refl_assoc
+koszulCocomplex.mem_annihilator_homology
+koszulCocomplex.ofList
+koszulCocomplex.ofList_ideal_le_mem_annihilator_homology
+koszulCocomplex.scalar_homotopy_comm
+koszulCocomplex.scalar_homotopy_comm_zero
+koszulComplex
+koszulComplex.d
+koszulComplex.dAlternating
+koszulComplex.dAlternating_apply
+koszulComplex.d_comp_d
+koszulComplex.d_def
+koszulComplex.homotopySMulIdZero
+koszulComplex.isZero_X_ofList_of_length_lt
+koszulComplex.isZero_X_of_span_eq_top
+koszulComplex.isoOfEquiv
+koszulComplex.map
+koszulComplex.map.congr_simp
+koszulComplex.map_comp
+koszulComplex.map_comp_assoc
+koszulComplex.map_dAlternating_apply
+koszulComplex.map_d_comm
+koszulComplex.map_f
+koszulComplex.map_id
+koszulComplex.map_id_assoc
+koszulComplex.mem_annihilator_homology
+koszulComplex.ofList
+koszulComplex.ofList_ideal_annihilator_homology
+koszulComplex.range_le_annihilator_homology
+koszulComplex.scalar_homotopy_comm
+koszulComplex.scalar_homotopy_comm_zero

---

Decrease in strong tech debt: (relative, absolute) = (1.63, 0.00)

Current number	Change	Type (strong)
backward.defeqAttrib.useBackward	4224	-4
backward.isDefEq.respectTransparency	4680	1

Increase in weak tech debt: (relative, absolute) = (3.00, 0.00)

Current number	Change	Type (weak)
exposed public sections	5064	3

Current commit 8653ea14d5
Reference commit 98b840bb29

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.py 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).