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Set Implicit Arguments.
Set Printing Universes.
Set Asymmetric Patterns.
Inductive paths {A : Type} (a : A) : A -> Type :=
idpath : paths a a.
Arguments idpath {A a} , [A] a.
Arguments paths_ind [A] a P f y p.
Arguments paths_rec [A] a P f y p.
Arguments paths_rect [A] a P f y p.
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.
Arguments ap {A B} f {x y} p : simpl nomatch.
Definition Sect {A B : Type} (s : A -> B) (r : B -> A) :=
forall x : A, r (s x) = x.
(** A typeclass that includes the data making [f] into an adjoint equivalence. *)
Class IsEquiv {A B : Type} (f : A -> B) := BuildIsEquiv {
equiv_inv : B -> A ;
eisretr : Sect equiv_inv f;
eissect : Sect f equiv_inv;
eisadj : forall x : A, eisretr (f x) = ap f (eissect x)
}.
Arguments eisretr {A B} f {_} _.
Arguments eissect {A B} f {_} _.
Arguments eisadj {A B} f {_} _.
Inductive type_eq (A : Type) : Type -> Type :=
| type_eq_refl : type_eq A A
| type_eq_impossible : False -> forall B : Type, type_eq A B.
Definition type_eq_sym {A B} (p : type_eq A B) : type_eq B A
:= match p in (type_eq _ B) return (type_eq B A) with
| type_eq_refl => type_eq_refl _
| type_eq_impossible f B => type_eq_impossible _ f A
end.
Definition type_eq_sym_type_eq_sym {A B} (p : type_eq A B) : type_eq_sym (type_eq_sym p) = p
:= match p as p return type_eq_sym (type_eq_sym p) = p with
| type_eq_refl => idpath
| type_eq_impossible f _ => idpath
end.
Module Type LiftT.
Section local.
Let type_cast_up_type : Type.
Proof.
let U0 := constr:(Type) in
let U1 := constr:(Type) in
let unif := constr:(U0 : U1) in
exact (forall T : U0, { T' : U1 & type_eq T' T }).
Defined.
Axiom type_cast_up : type_cast_up_type.
End local.
Definition Lift (T : Type) := projT1 (type_cast_up T).
Definition lift {T} : T -> Lift T
:= match projT2 (type_cast_up T) in (type_eq _ T') return T' -> Lift T with
| type_eq_refl => fun x => x
| type_eq_impossible bad _ => match bad with end
end.
Section equiv.
Definition lower' {T} : Lift T -> T
:= match projT2 (type_cast_up T) in (type_eq _ T') return Lift T -> T' with
| type_eq_refl => fun x => x
| type_eq_impossible bad _ => match bad with end
end.
Definition lift_lower {T} (x : Lift T) : lift (lower' x) = x.
Proof.
unfold lower', lift.
destruct (projT2 (type_cast_up T)) as [|[]].
reflexivity.
Defined.
Definition lower_lift {T} (x : T) : lower' (lift x) = x.
Proof.
unfold lower', lift, Lift in *.
destruct (type_cast_up T) as [T' p]; simpl.
let y := match goal with |- ?y => constr:(y) end in
let P := match (eval pattern p in y) with ?f p => constr:(f) end in
apply (@transport _ P _ _ (type_eq_sym_type_eq_sym p)); simpl in *.
generalize (type_eq_sym p); intro p'; clear p.
destruct p' as [|[]]; simpl.
reflexivity.
Defined.
Global Instance isequiv_lift A : IsEquiv (@lift A).
Proof.
refine (@BuildIsEquiv
_ _
lift lower'
lift_lower
lower_lift
_).
compute.
intro x.
destruct (type_cast_up A) as [T' p].
let y := match goal with |- ?y => constr:(y) end in
let P := match (eval pattern p in y) with ?f p => constr:(f) end in
apply (@transport _ P _ _ (type_eq_sym_type_eq_sym p)); simpl in *.
generalize (type_eq_sym p); intro p'; clear p.
destruct p' as [|[]]; simpl.
reflexivity.
Defined.
End equiv.
Definition lower {A} := (@equiv_inv _ _ (@lift A) _).
End LiftT.
Module Lift : LiftT.
Section local.
Let type_cast_up_type : Type.
Proof.
let U0 := constr:(Type) in
let U1 := constr:(Type) in
let unif := constr:(U0 : U1) in
exact (forall T : U0, { T' : U1 & type_eq T' T }).
Defined.
Definition type_cast_up : type_cast_up_type
:= fun T => existT (fun T' => type_eq T' T) T (type_eq_refl _).
End local.
Definition Lift (T : Type) := projT1 (type_cast_up T).
Definition lift {T} : T -> Lift T
:= match projT2 (type_cast_up T) in (type_eq _ T') return T' -> Lift T with
| type_eq_refl => fun x => x
| type_eq_impossible bad _ => match bad with end
end.
Section equiv.
Definition lower' {T} : Lift T -> T
:= match projT2 (type_cast_up T) in (type_eq _ T') return Lift T -> T' with
| type_eq_refl => fun x => x
| type_eq_impossible bad _ => match bad with end
end.
Definition lift_lower {T} (x : Lift T) : lift (lower' x) = x.
Proof.
unfold lower', lift.
destruct (projT2 (type_cast_up T)) as [|[]].
reflexivity.
Defined.
Definition lower_lift {T} (x : T) : lower' (lift x) = x.
Proof.
unfold lower', lift, Lift in *.
destruct (type_cast_up T) as [T' p]; simpl.
let y := match goal with |- ?y => constr:(y) end in
let P := match (eval pattern p in y) with ?f p => constr:(f) end in
apply (@transport _ P _ _ (type_eq_sym_type_eq_sym p)); simpl in *.
generalize (type_eq_sym p); intro p'; clear p.
destruct p' as [|[]]; simpl.
reflexivity.
Defined.
Global Instance isequiv_lift A : IsEquiv (@lift A).
Proof.
refine (@BuildIsEquiv
_ _
lift lower'
lift_lower
lower_lift
_).
compute.
intro x.
destruct (type_cast_up A) as [T' p].
let y := match goal with |- ?y => constr:(y) end in
let P := match (eval pattern p in y) with ?f p => constr:(f) end in
apply (@transport _ P _ _ (type_eq_sym_type_eq_sym p)); simpl in *.
generalize (type_eq_sym p); intro p'; clear p.
destruct p' as [|[]]; simpl.
reflexivity.
Defined.
End equiv.
Definition lower {A} := (@equiv_inv _ _ (@lift A) _).
End Lift.
(* Toplevel input, characters 15-24:
Anomaly: Invalid argument: enforce_eq_instances called with instances of different lengths.
Please report. *)
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