[Merged by Bors] - refactor(algebra/{group,group_with_zero/basic): Generalize lemmas to division monoids

[YaelDillies]

Generalize group and group_with_zero lemmas to division_monoid. We do not actually delete the original lemmas but make them one-liners from the new ones. The next PR will then delete the old lemmas and perform the renames in all files.

Lemmas are renamed because

• one of the group or group_with_zero name has to go
• the new API should have a consistent naming convention

Pre-emptive lemma renames

group lemma, group_with_zero lemma → new division_monoid/division_comm_monoid lemma

• inv_eq_of_mul_eq_one, inv_eq_of_mul_left_eq_one → inv_eq_of_mul_eq_one_left
• -, inv_eq_of_mul_right_eq_one → inv_eq_of_mul_eq_one_right
• eq_inv_of_mul_eq_one, eq_inv_of_mul_left_eq_one → eq_inv_of_mul_eq_one_left
• -, eq_inv_of_mul_right_eq_one → eq_inv_of_mul_eq_one_right
• -, eq_one_div_of_mul_eq_one → eq_one_div_of_mul_eq_one_right
• inv_div', inv_div → inv_div
• div_one', div_one → div_one
• eq_of_div_eq_one', eq_of_div_eq_one → eq_of_div_eq_one
• div_eq_inv_mul', div_eq_inv_mul → div_eq_inv_mul
• div_right_comm', div_right_comm → div_right_comm
• div_mul_eq_mul_div', div_mul_eq_mul_div → div_mul_eq_mul_div
• mul_comm_div', div_mul_eq_mul_div_comm → mul_comm_div
• mul_inv, mul_inv₀ → mul_inv
• inv_eq_one, inv_eq_one₀ → inv_eq_one
• one_eq_inv, one_eq_inv₀ → one_eq_inv
• div_div_div_comm, div_div_div_comm₀ → div_div_div_comm
• div_mul_div_comm, div_mul_div_comm₀ → div_mul_div_comm
• mul_div_comm', - → mul_div_mul_comm
• one_inv, inv_one → inv_one
• div_div, div_div_eq_div_mul → div_div
• mul_div_left_comm, mul_div_comm → mul_div_left_comm
• div_div_assoc_swap, div_div_eq_mul_div → div_div_eq_mul_div

Other changes

• Generalize left_inverse_inv and right_inverse_inv to has_involutive_inv, and div_eq_mul_one_div to div_inv_monoid.
• Arguments order
• Arguments implicitness

---

I found that splitting the refactor in this unusual way allowed me to have a cleaner diff and be systematically sure that I wasn't missing lemmas. The only problem is that I have to unsimp the old lemmas.

Here's what to look out for if you want to check for yourself:

• All lemma renames are either because of the group/group_with_zero disparity or naming consistency within the new API.
• simp and field_simp are copied correctly.
• Every eligible group/group_with_zero lemma in the two files has been one-lined (some in other files haven't, but I will take care of them in another PR.
• Every new division_monoid lemma is used to one-line an old lemma.
• Arguments are explicit for unconditional equalities and implicit for implications and iffs.
• Everything is additivized.

[jcommelin]

I think your experiment is good. This diff looks clean!

bors merge

[bors]

Pull request successfully merged into master.

Build succeeded:

• Build mathlib
• Lint mathlib
• Lint style
• Run tests