action of subgroup inference issues

Stream: new members

[Anne Baanen]

I have a group and want to talk about the action of a certain subgroup, but run into issues with typeclass inference. The following is the specific example that I want to get working:

import group_theory.group_action

-- My plan is to define a `mul_action` instance on any submonoid, then use inference to get it for a specific set.
variables {α β : Type} [monoid α]
instance subset_has_scalar (s : set α) [has_scalar α β] : has_scalar s β := ⟨λ s b, s.1 • b⟩
instance submonoid_mul_action (s : set α) [mul_action α β] [is_submonoid s] : mul_action s β :=
⟨λ x, (one_smul α x : (1 : s).1 • x = x), λ x y, @mul_smul α _ _ _ x.1 y.1⟩

variables {α β} [group α] [mul_action α β] (s : set α) [is_subgroup s] -- these are concrete values in my situation
#check mul_action.orbit_rel s β -- error : failed to synthesise instance for `mul_action ↥s β`

The problem seems to be a mismatch between the monoid instances inferred. With pp.all on, the submonoid_mul_action instance looks like:

    @mul_action.{0 0} (@coe_sort.{1 2} (set.{0} ?M_1) (@set.has_coe_to_sort.{0} ?M_1) s) ?M_2
      (@subtype.monoid.{0} ?M_1 ?M_3 s _inst_3)

and Lean is trying to infer:

@mul_action.{0 0} (@coe_sort.{1 2} (set.{0} α) (@set.has_coe_to_sort.{0} α) s) β
    (@group.to_monoid.{0} (@coe_sort.{1 2} (set.{0} α) (@set.has_coe_to_sort.{0} α) s)
       (@subtype.group.{0} α _inst_2 s _inst_4))

How do I convince Lean that group.to_monoid (subtype.group ...) should coincide with subtype.monoid? (It definitely does in the specific example.)

Or is there another way to set it up so that I can write mul_action.orbit_rel s β without an error, without manually writing each instance on s?

[Yury G. Kudryashov]

Could you please try to redefine subtype.group so that it will use {.. subtype.monoid}?

[Yury G. Kudryashov]

It's in group_theory/subgroup

[Yury G. Kudryashov]

I'm not sure if this will help

[Kevin Buzzard]

Just to comment that one of the mul_actions uses _inst_3 and the other uses _inst_2 and _inst_4 so they are almost certainly not defeq. You can probably see in the local context what is going on.

[Anne Baanen]

Could you please try to redefine subtype.group so that it will use {.. subtype.monoid}?

That doesn't seem to help, unfortunately: still the same error

[Anne Baanen]

Adding an extra instance declaration with an explicit monoid s parameter works, but it seems ugly to have two instances with exactly the same definition:

instance subgroup_mul_action {α β} [group α] [mul_action α β] (s : set α) [is_subgroup s] : @mul_action s β (group.to_monoid _):=
⟨one_smul α, λ x y, @mul_smul α _ _ _ x.1 y.1⟩

[Yury G. Kudryashov]

Redefined subtype.* without using subtype_instance; compiling.

[Yury G. Kudryashov]

These definitions make 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`. "]

[Yury G. Kudryashov]

See also https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/subtype.2E*.20diamonds

[Yury G. Kudryashov]

See #1981