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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
(** * Basic facts about Prop as a type *)
(** An intuitionistic theorem from topos theory [[LambekScott]]
References:
[[LambekScott]] Jim Lambek, Phil J. Scott, Introduction to higher
order categorical logic, Cambridge Studies in Advanced Mathematics
(Book 7), 1988.
*)
Theorem injection_is_involution_in_Prop
(f : Prop -> Prop)
(inj : forall A B, (f A <-> f B) -> (A <-> B))
(ext : forall A B, A <-> B -> f A <-> f B)
: forall A, f (f A) <-> A.
Proof.
intros.
enough (f (f (f A)) <-> f A) by (apply inj; assumption).
split; intro H.
- now_show (f A).
enough (f A <-> True) by firstorder.
enough (f (f A) <-> f True) by (apply inj; assumption).
split; intro H'.
+ now_show (f True).
enough (f (f (f A)) <-> f True) by firstorder.
apply ext; firstorder.
+ now_show (f (f A)).
enough (f (f A) <-> True) by firstorder.
apply inj; firstorder.
- now_show (f (f (f A))).
enough (f A <-> f (f (f A))) by firstorder.
apply ext.
split; intro H'.
+ now_show (f (f A)).
enough (f A <-> f (f A)) by firstorder.
apply ext; firstorder.
+ now_show A.
enough (f A <-> A) by firstorder.
apply inj; firstorder.
Defined.
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