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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015     *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(** * 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.