feat(library): add constructions of categories, some changes in eq, sigma and path
in eq, add theorem for proof irrelevance and congruence for binary functions in sigma, add some support for triplets in path, comment out two unneccesary definitions in category, add Cat, slice, coslice, product and arrow categories, also add fully bundled approach
This commit is contained in:
Floris van Doorn
parent
ab7d2b1751
commit
5396e422d2
4 changed files with 280 additions and 31 deletions
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@ -6,20 +6,23 @@
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import logic.core.eq logic.core.connectives
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import data.unit data.sigma data.prod
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import algebra.function
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import logic.axioms.funext
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open eq
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inductive category [class] (ob : Type) : Type :=
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mk : Π (mor : ob → ob → Type) (comp : Π{A B C : ob}, mor B C → mor A B → mor A C)
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mk : Π (mor : ob → ob → Type) (comp : Π⦃A B C : ob⦄, mor B C → mor A B → mor A C)
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(id : Π {A : ob}, mor A A),
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(Π {A B C D : ob} {f : mor A B} {g : mor B C} {h : mor C D},
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(Π ⦃A B C D : ob⦄ {h : mor C D} {g : mor B C} {f : mor A B},
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comp h (comp g f) = comp (comp h g) f) →
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(Π {A B : ob} {f : mor A B}, comp f id = f) →
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(Π {A B : ob} {f : mor A B}, comp id f = f) →
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(Π ⦃A B : ob⦄ {f : mor A B}, comp id f = f) →
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(Π ⦃A B : ob⦄ {f : mor A B}, comp f id = f) →
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category ob
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inductive Category : Type :=
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mk : Π (A : Type), category A → Category
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namespace category
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precedence `∘` : 60
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section
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parameters {ob : Type} {Cat : category ob} {A B C D : ob}
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@ -32,16 +35,17 @@ namespace category
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rec (λ mor compose id assoc idr idl, id) Cat
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definition ID (A : ob) : mor A A := @id A
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precedence `∘` : 60
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infixr `∘` := compose
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theorem assoc : Π {A B C D : ob} {f : mor A B} {g : mor B C} {h : mor C D},
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theorem assoc : Π {A B C D : ob} {h : mor C D} {g : mor B C} {f : mor A B},
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h ∘ (g ∘ f) = (h ∘ g) ∘ f :=
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rec (λ mor comp id assoc idr idl, assoc) Cat
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theorem id_right : Π {A B : ob} {f : mor A B}, f ∘ id = f :=
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rec (λ mor comp id assoc idr idl, idr) Cat
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theorem id_left : Π {A B : ob} {f : mor A B}, id ∘ f = f :=
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rec (λ mor comp id assoc idr idl, idl) Cat
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rec (λ mor comp id assoc idl idr, idl) Cat
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theorem id_right : Π {A B : ob} {f : mor A B}, f ∘ id = f :=
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rec (λ mor comp id assoc idl idr, idr) Cat
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theorem id_compose {A : ob} : (ID A) ∘ id = id :=
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id_left
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@ -178,8 +182,16 @@ namespace category
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... = h : id_right
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end
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infixr `∘` := compose
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postfix `⁻¹` := inverse
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section
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definition objects [coercion] (C : Category) : Type
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:= Category.rec (fun c s, c) C
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definition category_instance [instance] (C : Category) : category (objects C)
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:= Category.rec (fun c s, s) C
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end
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section
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open unit
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@ -192,19 +204,24 @@ namespace category
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parameter {ob : Type}
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definition opposite (C : category ob) : category ob :=
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category.mk (λa b, mor b a) (λ a b c f g, g ∘ f) (λ a, id) (λ a b c d f g h, symm assoc)
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(λ a b f, id_left) (λ a b f, id_right)
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(λ a b f, id_right) (λ a b f, id_left)
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precedence `∘op` : 60
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infixr `∘op` := @compose _ (opposite _) _ _ _
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parameters {C : category ob} {a b c : ob} {f : @mor ob C a b} {g : @mor ob C b c}
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parameters {C : category ob} {a b c : ob}
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theorem compose_op : f ∘op g = g ∘ f :=
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theorem compose_op {f : @mor ob C a b} {g : mor b c} : f ∘op g = g ∘ f :=
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rfl
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theorem op_op {C : category ob} : opposite (opposite C) = C :=
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rec (λ mor comp id assoc idl idr, sorry) C
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rec (λ mor comp id assoc idl idr, rfl) C
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end
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definition Opposite (C : Category) : Category :=
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Category.mk (objects C) (opposite (category_instance C))
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section
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--need extensionality
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-- definition type_cat : category Type :=
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@ -217,15 +234,18 @@ end category
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open category
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inductive functor {obC obD : Type} (C : category obC) (D : category obD) : Type :=
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mk : Π (obF : obC → obD) (morF : Π{A B : obC}, mor A B → mor (obF A) (obF B)),
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(Π {A : obC}, morF (ID A) = ID (obF A)) →
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(Π {A B C : obC} {f : mor A B} {g : mor B C}, morF (g ∘ f) = morF g ∘ morF f) →
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@functor obC obD C D
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mk : Π (obF : obC → obD) (morF : Π⦃A B : obC⦄, mor A B → mor (obF A) (obF B)),
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(Π ⦃A : obC⦄, morF (ID A) = ID (obF A)) →
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(Π ⦃A B C : obC⦄ {f : mor A B} {g : mor B C}, morF (g ∘ f) = morF g ∘ morF f) →
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functor C D
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inductive Functor (C D : Category) : Type :=
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mk : functor (category_instance C) (category_instance D) → Functor C D
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infixl `⇒`:25 := functor
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namespace functor
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section
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section basic_functor
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parameters {obC obD : Type} {C : category obC} {D : category obD}
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definition object [coercion] (F : C ⇒ D) : obC → obD := rec (λ obF morF Hid Hcomp, obF) F
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@ -238,10 +258,12 @@ namespace functor
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theorem respect_comp (F : C ⇒ D) : Π {a b c : obC} {f : mor a b} {g : mor b c},
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F (g ∘ f) = F g ∘ F f :=
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rec (λ obF morF Hid Hcomp, Hcomp) F
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end
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end basic_functor
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section category_functor
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definition compose {obC obD obE : Type} {C : category obC} {D : category obD} {E : category obE}
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(F : C ⇒ D) (G : D ⇒ E) : C ⇒ E :=
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(G : D ⇒ E) (F : C ⇒ D) : C ⇒ E :=
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functor.mk C E
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(λx, G (F x))
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(λ a b f, G (F f))
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@ -252,4 +274,178 @@ namespace functor
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G (F (g ∘ f)) = G (F g ∘ F f) : {respect_comp F}
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... = G (F g) ∘ G (F f) : respect_comp G)
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precedence `∘∘` : 60
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infixr `∘∘` := compose
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-- theorem congr_functor {obC obD : Type} {C : category obC} {D : category obD} {o : obC → obD}
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-- {m m' : Π⦃A B : obC⦄, mor A B → mor (o A) (o B)} {i i' c c'}
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-- (Hm : ∀{a b : obC} {f : mor a b}, m f = m' f)
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-- : functor.mk C D o m i c = functor.mk C D o m' i' c' :=
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-- sorry
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-- theorem congr_functor_refl {obC obD : Type} {C : category obC} {D : category obD} {o : obC → obD}
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-- {m : Π⦃A B : obC⦄, mor A B → mor (o A) (o B)} {i i' c c'}
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-- : functor.mk C D o m i c = functor.mk C D o m i' c' :=
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-- rfl
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theorem assoc {obA obB obC obD : Type} {A : category obA} {B : category obB}
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{C : category obC} {D : category obD} (H : C ⇒ D) (G : B ⇒ C) (F : A ⇒ B) :
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H ∘∘ (G ∘∘ F) = (H ∘∘ G) ∘∘ F :=
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rfl
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-- later check whether we want implicit or explicit arguments here. For the moment, define both
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definition id {ob : Type} {C : category ob} : functor C C :=
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mk C C (λa, a) (λ a b f, f) (λ a, rfl) (λ a b c f g, rfl)
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definition ID {ob : Type} (C : category ob) : functor C C := id
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definition Id {C : Category} : Functor C C := Functor.mk id
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definition iD (C : Category) : Functor C C := Functor.mk id
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theorem id_left {obC obB : Type} {B : category obB} {C : category obC} (F : B ⇒ C)
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: id ∘∘ F = F :=
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rec (λ obF morF idF compF, rfl) F
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theorem id_right {obC obB : Type} {B : category obB} {C : category obC} (F : B ⇒ C)
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: F ∘∘ id = F :=
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rec (λ obF morF idF compF, rfl) F
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end category_functor
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section Functor
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-- parameters {C D E : Category} (G : Functor D E) (F : Functor C D)
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definition Functor_functor {C D : Category} (F : Functor C D) : functor (category_instance C) (category_instance D) :=
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Functor.rec (λ x, x) F
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definition Compose {C D E : Category} (G : Functor D E) (F : Functor C D) : Functor C E :=
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Functor.mk (compose (Functor_functor G) (Functor_functor F))
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-- namespace Functor
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precedence `∘∘` : 60
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infixr `∘∘` := Compose
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-- end Functor
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definition Assoc {A B C D : Category} (H : Functor C D) (G : Functor B C) (F : Functor A B)
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: H ∘∘ (G ∘∘ F) = (H ∘∘ G) ∘∘ F :=
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rfl
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theorem Id_left {B : Category} {C : Category} (F : Functor B C)
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: Id ∘∘ F = F :=
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Functor.rec (λ f, subst (id_left f) rfl) F
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theorem Id_right {B : Category} {C : Category} (F : Functor B C)
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: F ∘∘ Id = F :=
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Functor.rec (λ f, subst (id_right f) rfl) F
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end Functor
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end functor
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open functor
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inductive natural_transformation {obC obD : Type} {C : category obC} {D : category obD}
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(F G : functor C D) : Type :=
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mk : Π (η : Π(a : obC), mor (object F a) (object G a)), Π{a b : obC} (f : mor a b), morphism G f ∘ η a = η b ∘ morphism F f
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→ natural_transformation F G
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inductive Natural_transformation {C D : Category} (F G : Functor C D) : Type :=
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mk : natural_transformation (Functor_functor F) (Functor_functor G) → Natural_transformation F G
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namespace natural_tranformation
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-- todo
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end natural_tranformation
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open functor
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namespace category
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section cat_Cat
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definition Cat : category Category :=
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mk (λ a b, Functor a b) (λ a b c g f, Compose g f) (λ a, Id)
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(λ a b c d h g f, Assoc h g f) (λ a b f, Id_left f) (λ a b f, Id_right f)
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end cat_Cat
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section slice
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open sigma
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definition slice {ob : Type} (C : category ob) (c : ob) : category (Σ(b : ob), mor b c) :=
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mk (λa b, Σ(g : mor (dpr1 a) (dpr1 b)), dpr2 b ∘ g = dpr2 a)
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(λ a b c g f, dpair (dpr1 g ∘ dpr1 f)
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(calc
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dpr2 c ∘ (dpr1 g ∘ dpr1 f) = (dpr2 c ∘ dpr1 g) ∘ dpr1 f : assoc
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... = dpr2 b ∘ dpr1 f : {dpr2 g}
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... = dpr2 a : {dpr2 f}))
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(λ a, dpair id id_right)
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(λ a b c d h g f, dpair_eq assoc proof_irrel)
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(λ a b f, sigma.equal id_left proof_irrel)
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(λ a b f, sigma.equal id_right proof_irrel)
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--replacing proof_irrel with rfl confuses the unifier
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end slice
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section coslice
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open sigma
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definition coslice {ob : Type} (C : category ob) (c : ob) : category (Σ(b : ob), mor c b) :=
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mk (λa b, Σ(g : mor (dpr1 a) (dpr1 b)), g ∘ dpr2 a = dpr2 b)
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(λ a b c g f, dpair (dpr1 g ∘ dpr1 f)
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(calc
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(dpr1 g ∘ dpr1 f) ∘ dpr2 a = dpr1 g ∘ (dpr1 f ∘ dpr2 a): symm assoc
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... = dpr1 g ∘ dpr2 b : {dpr2 f}
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... = dpr2 c : {dpr2 g}))
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(λ a, dpair id id_left)
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(λ a b c d h g f, dpair_eq assoc proof_irrel)
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(λ a b f, sigma.equal id_left proof_irrel)
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(λ a b f, sigma.equal id_right proof_irrel)
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-- theorem slice_coslice_opp {ob : Type} (C : category ob) (c : ob) :
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-- coslice C c = opposite (slice (opposite C) c) :=
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-- sorry
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end coslice
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section product
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open prod
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definition product {obC obD : Type} (C : category obC) (D : category obD)
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: category (obC × obD) :=
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mk (λa b, mor (pr1 a) (pr1 b) × mor (pr2 a) (pr2 b))
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(λ a b c g f, (pr1 g ∘ pr1 f , pr2 g ∘ pr2 f) )
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(λ a, (id,id))
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(λ a b c d h g f, pair_eq assoc assoc )
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(λ a b f, prod.equal id_left id_left )
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(λ a b f, prod.equal id_right id_right)
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end product
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section arrow
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open sigma eq_ops
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definition arrow {ob : Type} (C : category ob) : category (Σ(a b : ob), mor a b) :=
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mk (λa b, Σ(g : mor (dpr1 a) (dpr1 b)) (h : mor (dpr2' a) (dpr2' b)),
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dpr3 b ∘ g = h ∘ dpr3 a)
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(λ a b c g f, dpair (dpr1 g ∘ dpr1 f) (dpair (dpr2' g ∘ dpr2' f)
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(calc
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dpr3 c ∘ (dpr1 g ∘ dpr1 f) = (dpr3 c ∘ dpr1 g) ∘ dpr1 f : assoc
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... = (dpr2' g ∘ dpr3 b) ∘ dpr1 f : {dpr3 g}
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... = dpr2' g ∘ (dpr3 b ∘ dpr1 f) : symm assoc
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... = dpr2' g ∘ (dpr2' f ∘ dpr3 a) : {dpr3 f}
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... = (dpr2' g ∘ dpr2' f) ∘ dpr3 a : assoc)))
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(λ a, dpair id (dpair id (id_right ⬝ (symm id_left))))
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(λ a b c d h g f, dtrip_eq2 assoc assoc proof_irrel)
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(λ a b f, trip.equal2 id_left id_left proof_irrel)
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(λ a b f, trip.equal2 id_right id_right proof_irrel)
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-- (λ a b c d h g f, dpair_eq assoc sorry)
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-- (λ a b f, sigma.equal id_left sorry)
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-- (λ a b f, sigma.equal id_right sorry)
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end arrow
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-- definition foo {obC obD : Type} (C : category obC) (D : category obD)
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-- : category (obC × obD) :=
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-- mk (λa b, sorry)
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-- (λ a b c g f, sorry)
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-- (λ a, sorry)
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-- (λ a b c d h g f, sorry)
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-- (λ a b f, sorry)
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-- (λ a b f, sorry)
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end category
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@ -27,19 +27,59 @@ section
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theorem dpair_eq {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂} (H₁ : a₁ = a₂) (H₂ : eq.rec_on H₁ b₁ = b₂) :
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dpair a₁ b₁ = dpair a₂ b₂ :=
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eq.rec_on H₁
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(λ (b₂ : B a₁) (H₁ : a₁ = a₁) (H₂ : eq.rec_on H₁ b₁ = b₂),
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calc
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dpair a₁ b₁ = dpair a₁ (eq.rec_on H₁ b₁) : {(eq.rec_on_id H₁ b₁)⁻¹}
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... = dpair a₁ b₂ : {H₂})
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b₂ H₁ H₂
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congr_arg2 dpair H₁ H₂
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protected theorem equal {p₁ p₂ : Σx : A, B x} :
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∀(H₁ : dpr1 p₁ = dpr1 p₂) (H₂ : eq.rec_on H₁ (dpr2 p₁) = (dpr2 p₂)), p₁ = p₂ :=
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∀(H₁ : dpr1 p₁ = dpr1 p₂) (H₂ : eq.rec_on H₁ (dpr2 p₁) = dpr2 p₂), p₁ = p₂ :=
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destruct p₁ (take a₁ b₁, destruct p₂ (take a₂ b₂ H₁ H₂, dpair_eq H₁ H₂))
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protected theorem is_inhabited [instance] (H₁ : inhabited A) (H₂ : inhabited (B (default A))) :
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inhabited (sigma B) :=
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inhabited.destruct H₁ (λa, inhabited.destruct H₂ (λb, inhabited.mk (dpair (default A) b)))
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end
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section trip_quad
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parameters {A : Type} {B : A → Type} {C : Πa, B a → Type} {D : Πa b, C a b → Type}
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definition dtrip (a : A) (b : B a) (c : C a b) := dpair a (dpair b c)
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definition dquad (a : A) (b : B a) (c : C a b) (d : D a b c) := dpair a (dpair b (dpair c d))
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definition dpr1' (x : Σ a, B a) := dpr1 x
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definition dpr2' (x : Σ a b, C a b) := dpr1 (dpr2 x)
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definition dpr3 (x : Σ a b, C a b) := dpr2 (dpr2 x)
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definition dpr3' (x : Σ a b c, D a b c) := dpr1 (dpr2 (dpr2 x))
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definition dpr4 (x : Σ a b c, D a b c) := dpr2 (dpr2 (dpr2 x))
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-- theorem dtrip_eq {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂} {c₁ : C a₁ b₁} {c₂ : C a₂ b₂}
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-- (H₁ : a₁ = a₂) (H₂ : eq.rec_on H₁ b₁ = b₂) (H₃ : eq.rec_on (congr_arg2 C H₁ H₂) c₁ = c₂) :
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-- dtrip a₁ b₁ c₁ = dtrip a₂ b₂ c₂ :=
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-- eq.rec_on H₁
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-- (λ (b₂ : B a₁) (c₂ : C a₁ b₂) (H₁ : a₁ = a₁) (H₂ : eq.rec_on H₁ b₁ = b₂)
|
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-- (x : (eq (eq.rec_on (congr_arg2 C H₁ H₂) c₁) c₂)), sorry
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-- -- have H : dpair b₁ c₁ = dpair b₂ c₂, from
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-- -- dpair_eq H₂ (eq.subst (eq.rec_on_id H₁ c₁) H₃),
|
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-- -- eq.subst H rfl
|
||||
-- )
|
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-- b₂ c₂ H₁ H₂ H₃
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end trip_quad
|
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theorem dtrip_eq2 {A B : Type} {C : A → B → Type} {a₁ a₂ : A} {b₁ b₂ : B}
|
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{c₁ : C a₁ b₁} {c₂ : C a₂ b₂} (H₁ : a₁ = a₂) (H₂ : b₁ = b₂)
|
||||
(H₃ : eq.rec_on H₂ (eq.rec_on H₁ c₁) = c₂) :
|
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dtrip a₁ b₁ c₁ = dtrip a₂ b₂ c₂ :=
|
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eq.rec_on H₁
|
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(λ (b₂ : B) (c₂ : C a₁ b₂) (H₁ : a₁ = a₁) (H₂ : b₁ = b₂)
|
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(H₃ : eq.rec_on H₂ (eq.rec_on H₁ c₁) = c₂),
|
||||
have H : dpair b₁ c₁ = dpair b₂ c₂, from
|
||||
dpair_eq H₂ (eq.subst (eq.rec_on_id H₁ c₁) H₃),
|
||||
eq.subst H rfl
|
||||
)
|
||||
b₂ c₂ H₁ H₂ H₃
|
||||
|
||||
theorem trip.equal2 {A B : Type} {C : A → B → Type} {p₁ p₂ : Σa b, C a b} :
|
||||
∀(H₁ : dpr1 p₁ = dpr1 p₂) (H₂ : dpr2' p₁ = dpr2' p₂)
|
||||
(H₃ : eq.rec_on H₂ (eq.rec_on H₁ (dpr3 p₁)) = dpr3 p₂), p₁ = p₂ :=
|
||||
destruct p₁ (take a₁ q₁, destruct q₁ (take b₁ c₁, destruct p₂ (take a₂ q₂, destruct q₂
|
||||
(take b₂ c₂ H₁ H₂ H₃, dtrip_eq2 H₁ H₂ H₃))))
|
||||
|
||||
end sigma
|
||||
|
|
|
|||
|
|
@ -40,9 +40,9 @@ infixl `@`:75 := concat
|
|||
postfix `^`:100 := inverse
|
||||
|
||||
-- In Coq, these are not needed, because concat and inv are kept transparent
|
||||
definition inv_1 {A : Type} (x : A) : (idpath x)^ ≈ idpath x := idp
|
||||
-- definition inv_1 {A : Type} (x : A) : (idpath x)^ ≈ idpath x := idp
|
||||
|
||||
definition concat_11 {A : Type} (x : A) : idpath x @ idpath x ≈ idpath x := idp
|
||||
-- definition concat_11 {A : Type} (x : A) : idpath x @ idpath x ≈ idpath x := idp
|
||||
|
||||
|
||||
-- The 1-dimensional groupoid structure
|
||||
|
|
|
|||
|
|
@ -71,6 +71,16 @@ H ▸ rfl
|
|||
theorem congr_arg {A : Type} {B : Type} {a b : A} (f : A → B) (H : a = b) : f a = f b :=
|
||||
H ▸ rfl
|
||||
|
||||
theorem congr_arg2 {A : Type} {B : A → Type} {C : Type} {a₁ a₂ : A}
|
||||
{b₁ : B a₁} {b₂ : B a₂} (f : Πa, B a → C) (H₁ : a₁ = a₂) (H₂ : eq.rec_on H₁ b₁ = b₂) :
|
||||
f a₁ b₁ = f a₂ b₂ :=
|
||||
eq.rec_on H₁
|
||||
(λ (b₂ : B a₁) (H₁ : a₁ = a₁) (H₂ : eq.rec_on H₁ b₁ = b₂),
|
||||
calc
|
||||
f a₁ b₁ = f a₁ (eq.rec_on H₁ b₁) : {(eq.rec_on_id H₁ b₁)⁻¹}
|
||||
... = f a₁ b₂ : {H₂})
|
||||
b₂ H₁ H₂
|
||||
|
||||
theorem congr {A : Type} {B : Type} {f g : A → B} {a b : A} (H1 : f = g) (H2 : a = b) :
|
||||
f a = g b :=
|
||||
H1 ▸ H2 ▸ rfl
|
||||
|
|
@ -99,6 +109,9 @@ assume Ha, H2 ▸ (H1 Ha)
|
|||
theorem eq_imp_trans {a b c : Prop} (H1 : a = b) (H2 : b → c) : a → c :=
|
||||
assume Ha, H2 (H1 ▸ Ha)
|
||||
|
||||
-- proof irrelevance is built in the kernel
|
||||
theorem proof_irrel {a : Prop} {H1 H2 : a} : H1 = H2 := rfl
|
||||
|
||||
-- ne
|
||||
-- --
|
||||
|
||||
|
|
|
|||
Loading…
Reference in a new issue