ab94e71e37
We solve them by creating a fresh metavariable ?w
?u := max ?w v
Remark: this is a precise solution.
91 lines
3 KiB
Text
91 lines
3 KiB
Text
universe variables u v u1 u2 v1 v2
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set_option pp.universes true
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open smt_tactic
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meta def blast : tactic unit := using_smt $ intros >> add_lemmas_from_facts >> repeat_at_most 3 ematch
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notation `♮` := by blast
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structure semigroup_morphism { α β : Type u } ( s : semigroup α ) ( t: semigroup β ) :=
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(map: α → β)
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(multiplicative : ∀ x y : α, map(x * y) = map(x) * map(y))
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attribute [simp] semigroup_morphism.multiplicative
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instance semigroup_morphism_to_map { α β : Type u } { s : semigroup α } { t: semigroup β } : has_coe_to_fun (semigroup_morphism s t) :=
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{ F := λ f, Π x : α, β,
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coe := semigroup_morphism.map }
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@[reducible] definition semigroup_identity { α : Type u } ( s: semigroup α ) : semigroup_morphism s s := ⟨ id, ♮ ⟩
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@[reducible] definition semigroup_morphism_composition
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{ α β γ : Type u } { s: semigroup α } { t: semigroup β } { u: semigroup γ}
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( f: semigroup_morphism s t ) ( g: semigroup_morphism t u ) : semigroup_morphism s u :=
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{
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map := λ x, g (f x),
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multiplicative := begin blast, simp end
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}
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@[reducible] definition semigroup_product { α β : Type u } ( s : semigroup α ) ( t: semigroup β ) : semigroup (α × β) := {
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mul := λ p q, (p^.fst * q^.fst, p^.snd * q^.snd),
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mul_assoc := begin
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intros,
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simp [@mul.equations._eqn_1 (α × β)],
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dsimp,
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simp
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end
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}
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definition semigroup_morphism_product
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{ α β γ δ : Type u }
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{ s_f : semigroup α } { s_g: semigroup β } { t_f : semigroup γ} { t_g: semigroup δ }
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( f : semigroup_morphism s_f t_f ) ( g : semigroup_morphism s_g t_g )
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: semigroup_morphism (semigroup_product s_f s_g) (semigroup_product t_f t_g) := {
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map := λ p, (f p.1, g p.2),
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multiplicative :=
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begin
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-- cf https://groups.google.com/d/msg/lean-user/bVs5FdjClp4/tfHiVjLIBAAJ
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intros,
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unfold mul has_mul.mul,
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dsimp,
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simp
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end
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}
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structure Category :=
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(Obj : Type u)
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(Hom : Obj → Obj → Type v)
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structure Functor (C : Category.{ u1 v1 }) (D : Category.{ u2 v2 }) :=
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(onObjects : C^.Obj → D^.Obj)
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(onMorphisms : Π { X Y : C^.Obj },
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C^.Hom X Y → D^.Hom (onObjects X) (onObjects Y))
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@[reducible] definition ProductCategory (C : Category) (D : Category) :
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Category :=
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{
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Obj := C^.Obj × D^.Obj,
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Hom := (λ X Y : C^.Obj × D^.Obj, C^.Hom (X^.fst) (Y^.fst) × D^.Hom (X^.snd) (Y^.snd))
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}
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namespace ProductCategory
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notation C `×` D := ProductCategory C D
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end ProductCategory
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structure PreMonoidalCategory
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extends carrier : Category :=
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(tensor : Functor (carrier × carrier) carrier)
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definition CategoryOfSemigroups : Category :=
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{
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Obj := Σ α : Type u, semigroup α,
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Hom := λ s t, semigroup_morphism s.2 t.2
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}
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definition PreMonoidalCategoryOfSemigroups : PreMonoidalCategory := {
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CategoryOfSemigroups with
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tensor := {
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onObjects := λ p, sigma.mk (p.1.1 × p.2.1) (semigroup_product p.1.2 p.2.2),
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onMorphisms := λ s t f, semigroup_morphism_product f.1 f.2
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}
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}
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