# Zulip Chat Archive

## Stream: general

### Topic: subtype.* diamonds

#### Yury G. Kudryashov (Feb 13 2020 at 06:23):

With current definitions Lean fails to unify `group.to_monoid subtype.group`

with `subtype.monoid group.to_monoid`

. This diff makes it work:

diff --git a/src/group_theory/subgroup.lean b/src/group_theory/subgroup.lean index d54cd8eba..ac0b1dbcb 100644 --- a/src/group_theory/subgroup.lean +++ b/src/group_theory/subgroup.lean @@ -50,11 +50,13 @@ theorem multiplicative.is_subgroup_iff @[to_additive add_group] instance subtype.group {s : set α} [is_subgroup s] : group s := -by subtype_instance +{ inv := λ x, ⟨(x:α)⁻¹, is_subgroup.inv_mem x.2⟩, + mul_left_inv := λ x, subtype.eq $ mul_left_inv x.1, + .. subtype.monoid } @[to_additive add_comm_group] instance subtype.comm_group {α : Type*} [comm_group α] {s : set α} [is_subgroup s] : comm_group s := -by subtype_instance +{ .. subtype.group, .. subtype.comm_monoid } @[simp, to_additive] lemma is_subgroup.coe_inv {s : set α} [is_subgroup s] (a : s) : ((a⁻¹ : s) : α) = a⁻¹ := rfl diff --git a/src/group_theory/submonoid.lean b/src/group_theory/submonoid.lean index e9fa5895e..c644e06d9 100644 --- a/src/group_theory/submonoid.lean +++ b/src/group_theory/submonoid.lean @@ -221,12 +221,17 @@ end is_submonoid /-- Submonoids are themselves monoids. -/ @[to_additive add_monoid "An `add_submonoid` is itself an `add_monoid`."] instance subtype.monoid {s : set α} [is_submonoid s] : monoid s := -by subtype_instance +{ one := ⟨1, is_submonoid.one_mem s⟩, + mul := λ x y, ⟨x * y, is_submonoid.mul_mem x.2 y.2⟩, + mul_one := λ x, subtype.eq $ mul_one x.1, + one_mul := λ x, subtype.eq $ one_mul x.1, + mul_assoc := λ x y z, subtype.eq $ mul_assoc x.1 y.1 z.1 } /-- Submonoids of commutative monoids are themselves commutative monoids. -/ @[to_additive add_comm_monoid "An `add_submonoid` of a commutative `add_monoid` is itself a commutative `add_monoid`. "] instance subtype.comm_monoid {α} [comm_monoid α] {s : set α} [is_submonoid s] : comm_monoid s := -by subtype_instance +{ mul_comm := λ x y, subtype.eq $ mul_comm x.1 y.1, + .. subtype.monoid } /-- Submonoids inherit the 1 of the monoid. -/ @[simp, to_additive "An `add_submonoid` inherits the 0 of the `add_monoid`. "]

However I understand that a proper fix should modify the `subtype_instance`

tactic, and I fail to understand how it works.

#### Yury G. Kudryashov (Feb 13 2020 at 06:25):

Motivated by https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/action.20of.20subgroup.20inference.20issues

#### Yury G. Kudryashov (Feb 13 2020 at 07:04):

Header says that `subtype_instance`

was written by @Simon Hudon . Simon, could you please explain how it works (or even better fix it)?

#### Yury G. Kudryashov (Feb 13 2020 at 07:05):

E.g., I see that `subtype.monoid`

has some rewrite on a `rfl`

in the definition of `one`

and I can't understand where it comes from.

#### Simon Hudon (Feb 13 2020 at 07:15):

I don't know that there's a simple solution to make those instances commute. You maybe have to prove that they are equal

#### Johan Commelin (Feb 13 2020 at 07:19):

But Yury gave a diff that makes them defeq, right?

#### Yury G. Kudryashov (Feb 13 2020 at 07:31):

The following code works with definition from the diff:

import group_theory.group_action variables {M G α : Type} [monoid M] [group G] instance subset_has_scalar [has_scalar M α] (s : set M) : has_scalar s α := ⟨λ s b, s.1 • b⟩ instance submonoid_mul_action [mul_action M α] (s : set M) [is_submonoid s] : mul_action s α := ⟨λ x, (one_smul M x : (1 : s).1 • x = x), λ x y, @mul_smul M _ _ _ x.1 y.1⟩ variables [mul_action G α] (s : set G) [is_subgroup s] #check mul_action.orbit_rel s α

#### Yury G. Kudryashov (Feb 13 2020 at 07:32):

It fails with current definitions, and proving equality wouldn't help.

#### Simon Hudon (Feb 13 2020 at 07:38):

Sorry, it's not straightforward and I won't have time to look into it in the near future

#### Simon Hudon (Feb 13 2020 at 07:38):

You may have to go with your version until someone can adapt the tactics

#### Yury G. Kudryashov (Feb 13 2020 at 07:38):

OK.

#### Yury G. Kudryashov (Feb 13 2020 at 08:10):

Last updated: May 11 2021 at 13:22 UTC