(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \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)         *)
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(* Certified Haskell Prelude.
 * Author: Matthieu Sozeau
 * Institution: LRI, CNRS UMR 8623 - UniversitÃcopyright Paris Sud
 *              91405 Orsay, France *)

Require Import Coq.Program.Program.

Set Implicit Arguments.
Unset Strict Implicit.

Require Import Coq.Classes.SetoidTactics.

Goal not True == not (not False) -> ((not True -> True)) \/ True.
  intros.
  clrewrite H.
  clrewrite <- H.
  right ; auto.
Defined.

Definition reduced_thm := Eval compute in Unnamed_thm.

(* Print reduced_thm. *)

Lemma foo [ Setoid a R ] : True. (* forall x y, R x y -> x -> y. *)
Proof.
  intros.
  Print respect2.
  pose setoid_morphism.
  pose (respect2 (b0:=b)).
  simpl in b0.
  unfold binary_respectful in b0.
  pose (arrow_morphism R).
  pose (respect2 (b0:=b1)).
  unfold binary_respectful in b2.

  pose (eq_morphism (A:=a)).
  pose (respect2 (b0:=b3)).
  unfold binary_respectful in b4.
  exact I.
Qed.

Goal forall A B C (H : A <-> B) (H' : B <-> C), A /\ B <-> B /\ C.
  intros.
  Set Printing All.
  Print iff_morphism.
  clrewrite H.
  clrewrite H'.
  reflexivity.
Defined.

Goal forall A B C (H : A <-> B) (H' : B <-> C), A /\ B <-> B /\ C.
  intros.
  rewrite H.
  rewrite H'.
  reflexivity.
Defined.

Require Import Setoid_tac.
Require Import Setoid_Prop.

(* Print Unnamed_thm0. *)
(* Print Unnamed_thm1. *)


