[Merged by Bors] - feat(RingTheory/MvPolynomial/Groebner): division algorithm wrt monomial orders

[AntoineChambert-Loir]

Formalize the division algorithm with respect to a monomial order

This is part of an ongoing work to formalize Groebner theory.
For the moment, only the lexicographic order is present in mathlib, but
subsequent PRs will introduce the homogeneous lexicographic order and the homogeneous
reverse lexicographic order.

---

• depends on: [Merged by Bors] - feat(Data/Finsupp/MonomialOrder): monomial orders #19453

[github-actions]

PR summary 9351e29a98

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.RingTheory.MvPolynomial.MonomialOrder (new file)	1102
Mathlib.RingTheory.MvPolynomial.Groebner (new file)	1103

---

Declarations diff

+ coeff_degree_eq_zero_iff
+ coeff_degree_ne_zero_iff
+ coeff_eq_zero_of_lt
+ coeff_mul_of_degree_add
+ degree
+ degree_C
+ degree_add_le
+ degree_add_of_lt
+ degree_add_of_ne
+ degree_eq_zero_iff_totalDegree_eq_zero
+ degree_le_iff
+ degree_lt_iff
+ degree_monomial
+ degree_monomial_le
+ degree_mul
+ degree_mul_le
+ degree_mul_of_isRegular_left
+ degree_mul_of_isRegular_right
+ degree_mul_of_nonzero_mul
+ degree_neg
+ degree_reduce_lt
+ degree_smul
+ degree_smul_le
+ degree_sub_LTerm_le
+ degree_sub_LTerm_lt
+ degree_sub_le
+ degree_sub_of_lt
+ degree_zero
+ div
+ div_set
+ eq_C_of_degree_eq_zero
+ lCoeff
+ lCoeff_add_of_lt
+ lCoeff_eq_zero_iff
+ lCoeff_is_unit_iff
+ lCoeff_monomial
+ lCoeff_mul
+ lCoeff_mul_of_isRegular_left
+ lCoeff_mul_of_isRegular_right
+ lCoeff_ne_zero_iff
+ lCoeff_sub_of_lt
+ lCoeff_zero
+ le_degree
+ reduce
+ subLTerm
+ sup_mem_of_nonempty

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:

• [Merged by Bors] - feat(Data/Finsupp/MonomialOrder): monomial orders #19453
  By Dependent Issues (🤖). Happy coding!

[jcommelin]

This looks great! Thanks for doing this 🎉

bors merge

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

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