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Require Import TestSuite.admit.
(* File reduced by coq-bug-finder from 279 lines to 219 lines. *)
Set Implicit Arguments.
Set Universe Polymorphism.
Definition admit {T} : T.
Admitted.
Module Export Overture.
Reserved Notation "g 'o' f" (at level 40, left associativity).
Inductive paths {A : Type} (a : A) : A -> Type :=
idpath : paths a a.
Arguments idpath {A a} , [A] a.
Notation "x = y :> A" := (@paths A x y) : type_scope.
Notation "x = y" := (x = y :>_) : type_scope.
Delimit Scope path_scope with path.
Local Open Scope path_scope.
Definition ap {A B:Type} (f:A -> B) {x y:A} (p:x = y) : f x = f y
:= match p with idpath => idpath end.
Definition apD10 {A} {B:A->Type} {f g : forall x, B x} (h:f=g)
: forall x, f x = g x
:= fun x => match h with idpath => idpath end.
Class IsEquiv {A B : Type} (f : A -> B) := BuildIsEquiv { equiv_inv : B -> A }.
Delimit Scope equiv_scope with equiv.
Local Open Scope equiv_scope.
Notation "f ^-1" := (@equiv_inv _ _ f _) (at level 3) : equiv_scope.
Class Funext.
Axiom isequiv_apD10 : `{Funext} -> forall (A : Type) (P : A -> Type) f g, IsEquiv (@apD10 A P f g) .
#[export] Existing Instance isequiv_apD10.
Definition path_forall `{Funext} {A : Type} {P : A -> Type} (f g : forall x : A, P x) :
(forall x, f x = g x) -> f = g
:=
(@apD10 A P f g)^-1.
End Overture.
Module Export Core.
Set Implicit Arguments.
Delimit Scope morphism_scope with morphism.
Delimit Scope category_scope with category.
Delimit Scope object_scope with object.
Record PreCategory :=
{
object :> Type;
morphism : object -> object -> Type;
compose : forall s d d',
morphism d d'
-> morphism s d
-> morphism s d'
where "f 'o' g" := (compose f g);
associativity : forall x1 x2 x3 x4
(m1 : morphism x1 x2)
(m2 : morphism x2 x3)
(m3 : morphism x3 x4),
(m3 o m2) o m1 = m3 o (m2 o m1)
}.
Bind Scope category_scope with PreCategory.
Arguments compose [!C%_category s%_object d%_object d'%_object] m1%_morphism m2%_morphism : rename.
Infix "o" := compose : morphism_scope.
End Core.
Local Open Scope morphism_scope.
Record Functor (C D : PreCategory) :=
{
object_of :> C -> D;
morphism_of : forall s d, morphism C s d
-> morphism D (object_of s) (object_of d)
}.
Inductive Unit : Set :=
tt : Unit.
Definition indiscrete_category (X : Type) : PreCategory
:= @Build_PreCategory X
(fun _ _ => Unit)
(fun _ _ _ _ _ => tt)
(fun _ _ _ _ _ _ _ => idpath).
Record NaturalTransformation C D (F G : Functor C D) := { components_of :> forall c, morphism D (F c) (G c) }.
Section path_natural_transformation.
Context `{Funext}.
Variable C : PreCategory.
Variable D : PreCategory.
Variables F G : Functor C D.
Section path.
Variables T U : NaturalTransformation F G.
Lemma path'_natural_transformation
: components_of T = components_of U
-> T = U.
admit.
Defined.
Lemma path_natural_transformation
: (forall x, T x = U x)
-> T = U.
Proof.
intros.
apply path'_natural_transformation.
apply path_forall; assumption.
Qed.
End path.
End path_natural_transformation.
Ltac path_natural_transformation :=
repeat match goal with
| _ => intro
| _ => apply path_natural_transformation; simpl
end.
Definition comma_category A B C (S : Functor A C) (T : Functor B C)
: PreCategory.
admit.
Defined.
Definition compose C D (F F' F'' : Functor C D)
(T' : NaturalTransformation F' F'') (T : NaturalTransformation F F')
: NaturalTransformation F F''
:= Build_NaturalTransformation F F''
(fun c => T' c o T c).
Infix "o" := compose : natural_transformation_scope.
Local Open Scope natural_transformation_scope.
Definition associativity `{fs : Funext}
C D F G H I
(V : @NaturalTransformation C D F G)
(U : @NaturalTransformation C D G H)
(T : @NaturalTransformation C D H I)
: (T o U) o V = T o (U o V).
Proof.
path_natural_transformation.
apply associativity.
Qed.
Definition functor_category `{Funext} (C D : PreCategory) : PreCategory
:= @Build_PreCategory (Functor C D)
(@NaturalTransformation C D)
(@compose C D)
(@associativity _ C D).
Notation "C -> D" := (functor_category C D) : category_scope.
Definition compose_functor `{Funext} (C D E : PreCategory) : object ((C -> D) -> ((D -> E) -> (C -> E))).
admit.
Defined.
Definition pullback_along `{Funext} (C C' D : PreCategory) (p : Functor C C')
: object ((C' -> D) -> (C -> D))
:= Eval hnf in compose_functor _ _ _ p.
Definition IsColimit `{Funext} C D (F : Functor D C)
(x : object
(@comma_category (indiscrete_category Unit)
(@functor_category H (indiscrete_category Unit) C)
(@functor_category H D C)
admit
(@pullback_along H D (indiscrete_category Unit) C
admit))) : Type
:= admit.
Generalizable All Variables.
Axiom fs : Funext.
#[export] Existing Instance fs.
Section bar.
Variable D : PreCategory.
Context `(has_colimits
: forall F : Functor D C,
@IsColimit _ C D F (colimits F)).
(* Error: Unsatisfied constraints: Top.3773 <= Set
(maybe a bugged tactic). *)
End bar.
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