draft(Module): Module length is additive in short exact sequences

[Raph-DG]

In this PR, we define the module length to be the krull dimension of the lattice of submodules and prove that it is additive in short exact sequences. Relies on #22069, #22036 and #21869

---

• depends on: feat(Set): Added some simple set lemmas #22036
• depends on: feat(LinearMap): Added lemmas relating kernels of linear maps to Submodule.map #22069
• depends on: feat(Order): Trimmed length of a RelSeries  #21869

[github-actions]

[lint-style (comment with "bot fix style" to have the bot commit all style suggestions)] _{reported by reviewdog 🐶}

Suggested change

	· let n' : Fin (rs.length) := {val := n, isLt := by rw[o] ; exact lt_add_one n}
	· let n' : Fin (rs.length) := {val := n, isLt := by rw[o]; exact lt_add_one n}

[github-actions]

PR summary 5cdfd92a0c

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Order.TrimmedLength (new file)	1218
Mathlib.Algebra.Module.Length (new file)	1222

---

Declarations diff

+ Module.length
+ RelSeries.length_eq_trimmedLength_iff
+ RelSeries.moduleLength_ge_trimmedLength
+ RelSeries.trim
+ RelSeries.trimmedLength
+ RelSeries.trimmedLength_additive
+ RelSeries.trimmedLength_eraseLast_le
+ RelSeries.trimmedLength_eraseLast_of_eq
+ RelSeries.trimmedLength_eraseLast_of_lt
+ RelSeries.trimmedLength_exists_le
+ RelSeries.trimmedLength_le_length
+ add_iSup
+ coe_unbotD_iSup
+ iSup_add
+ iSup_le_add
+ instance (rs : RelSeries (α := α) (· ≤ ·)) :
+ ker_inf_lt_ker_inf
+ ker_inter_mono_of_map_eq
+ map_lt_map_of_ker_inf_eq
+ map_lt_map_or
+ map_mono_of_ker_inter_eq
+ ne_map_or_ne_kernel_inter_of_lt
+ range_inter_ssubset_iff_preimage_ssubset
+ rel_of_le
+ ssubset_iff_exists

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:

• feat(Set): Added some simple set lemmas #22036
• feat(LinearMap): Added lemmas relating kernels of linear maps to Submodule.map #22069
• feat(Order): Trimmed length of a RelSeries  #21869
  By Dependent Issues (🤖). Happy coding!