(************************************************************************)
(*         *   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)         *)
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(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.                  *)
Inductive foo : Type := K1 : foo | K2 : foo -> foo -> foo | K3 : (nat -> foo) -> foo.
move=> x y; elim: {2}(_ == _) / eqP.
match goal with |- (y = x -> is_true ((y == x) && true) -> is_true (y == x)) => move=> -> // | _ => fail end.
match goal with |- (y <> x -> is_true ((y == x) && false) -> is_true (y == x)) => move=> _; rewrite andbC // | _ => fail end.
Qed.
move=> x y; elim: {2}_ / eqP.
match goal with |- (y = x -> is_true ((y == x) && true) -> is_true (y == x)) => move=> -> // | _ => fail end.
match goal with |- (y <> x -> is_true ((y == x) && false) -> is_true (y == x)) => move=> _; rewrite andbC // | _ => fail end.
Qed.
move=> x y; elim E: {2}_ / eqP.
match goal with _ : y = x |- (is_true ((y == x) && true) -> is_true (y == x)) => rewrite E; reflexivity | _ => fail end.
match goal with _ : y <> x |- (is_true ((y == x) && false) -> is_true (y == x)) => rewrite E => /= H; exact H  | _ => fail end.
Qed.
Inductive spec : nat -> nat -> nat -> Prop :=
| specK : forall a b c, a = 0 -> b = 2 -> c = 4 -> spec a b c.

Lemma testP8 : (1+1) * 4 = 2 + (1+1) + (2 + 2).