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(* -*- mode: coq; coq-prog-args: ("-emacs" "-indices-matter") -*- *)
(* NOTE: This bug is only triggered with -load-vernac-source / in interactive mode. *)
(* File reduced by coq-bug-finder from 139 lines to 124 lines. *)
Set Universe Polymorphism.
Reserved Notation "g 'o' f" (at level 40, left associativity).
Generalizable All Variables.
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.
Definition transport {A : Type} (P : A -> Type) {x y : A} (p : x = y) (u : P x) : P y :=
match p with idpath => u end.
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.
Definition ap11 {A B} {f g:A->B} (h:f=g) {x y:A} (p:x=y) : f x = g y.
admit.
Defined.
Class IsEquiv {A B : Type} (f : A -> B) := {}.
Class Contr_internal (A : Type) := BuildContr {
center : A ;
contr : (forall y : A, center = y)
}.
Arguments center A {_}.
Inductive trunc_index : Type :=
| minus_two : trunc_index
| trunc_S : trunc_index -> trunc_index.
Fixpoint nat_to_trunc_index (n : nat) : trunc_index
:= match n with
| 0 => trunc_S (trunc_S minus_two)
| S n' => trunc_S (nat_to_trunc_index n')
end.
Coercion nat_to_trunc_index : nat >-> trunc_index.
Fixpoint IsTrunc_internal (n : trunc_index) (A : Type) : Type :=
match n with
| minus_two => Contr_internal A
| trunc_S n' => forall (x y : A), IsTrunc_internal n' (x = y)
end.
Class IsTrunc (n : trunc_index) (A : Type) : Type :=
Trunc_is_trunc : IsTrunc_internal n A.
Instance istrunc_paths (A : Type) n `{H : IsTrunc (trunc_S n) A} (x y : A)
: IsTrunc n (x = y)
:= H x y.
Notation Contr := (IsTrunc minus_two).
Notation IsHSet := (IsTrunc 0).
Class Funext :=
{ isequiv_apD10 :> forall (A : Type) (P : A -> Type) f g, IsEquiv (@apD10 A P f g) }.
Global Instance contr_forall `{Funext} `{P : A -> Type} `{forall a, Contr (P a)}
: Contr (forall a, P a) | 100.
admit.
Defined.
Hint Extern 0 => progress change Contr_internal with Contr in * : typeclass_instances.
Global Instance trunc_forall `{Funext} `{P : A -> Type} `{forall a, IsTrunc n (P a)}
: IsTrunc n (forall a, P a) | 100.
Proof.
generalize dependent P.
induction n as [ | n' IH]; [ | admit ]; simpl; intros P ?.
exact _.
Defined.
Set Implicit Arguments.
Record PreCategory :=
{ object :> Type;
morphism : object -> object -> Type;
identity : forall x, morphism x x;
compose : forall s d d', morphism d d' -> morphism s d -> morphism s d';
trunc_morphism : forall s d, IsHSet (morphism s d) }.
Existing Instance trunc_morphism.
Infix "o" := (@compose _ _ _ _) : morphism_scope.
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);
composition_of : forall s d d'
(m1 : morphism C s d) (m2: morphism C d d'),
morphism_of _ _ (m2 o m1)
= (morphism_of _ _ m2) o (morphism_of _ _ m1);
identity_of : forall x, morphism_of _ _ (@identity _ x)
= @identity _ (object_of x) }.
Definition Sect {A B : Type} (s : A -> B) (r : B -> A) :=
forall x : A, r (s x) = x.
Section path_functor.
Context `{Funext}.
Variable C : PreCategory.
Variable D : PreCategory.
Local Notation path_functor'_T F G
:= { HO : object_of F = object_of G
| transport (fun GO => forall s d, morphism C s d -> morphism D (GO s) (GO d))
HO
(morphism_of F)
= morphism_of G }
(only parsing).
Definition path_functor'_sig (F G : Functor C D) : path_functor'_T F G -> F = G.
Proof.
intros [H' H''].
destruct F, G; simpl in *.
induction H'. (* while destruct H' works *) destruct H''.
apply ap11; [ apply ap | ];
apply center; abstract exact _.
Set Printing Universes.
(* Fail Defined.*)
(* The command has indeed failed with message:
=> Error: path_functor'_sig_subproof already exists. *)
Defined.
(* Anomaly: Backtrack.backto 55: a state with no vcs_backup. Please report. *)
End path_functor.
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