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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* $Id$ *)
(* Instantiation of the Ring tactic for the naturals of Arith $*)
Require Import Bool.
Require Export LegacyRing.
Require Export Arith.
Require Import Eqdep_dec.
Open Local Scope nat_scope.
Unboxed Fixpoint nateq (n m:nat) {struct m} : bool :=
match n, m with
| O, O => true
| S n', S m' => nateq n' m'
| _, _ => false
end.
Lemma nateq_prop : forall n m:nat, Is_true (nateq n m) -> n = m.
Proof.
simple induction n; simple induction m; intros; try contradiction.
trivial.
unfold Is_true in H1.
rewrite (H n1 H1).
trivial.
Qed.
Hint Resolve nateq_prop: arithring.
Definition NatTheory : Semi_Ring_Theory plus mult 1 0 nateq.
split; intros; auto with arith arithring.
(* apply (fun n m p:nat => plus_reg_l m p n) with (n := n).
trivial.*)
Defined.
Add Legacy Semi Ring nat plus mult 1 0 nateq NatTheory [ 0 S ].
Goal forall n:nat, S n = 1 + n.
intro; reflexivity.
Save S_to_plus_one.
(* Replace all occurrences of (S exp) by (plus (S O) exp), except when
exp is already O and only for those occurrences than can be reached by going
down plus and mult operations *)
Ltac rewrite_S_to_plus_term t :=
match constr:t with
| 1 => constr:1
| (S ?X1) =>
let t1 := rewrite_S_to_plus_term X1 in
constr:(1 + t1)
| (?X1 + ?X2) =>
let t1 := rewrite_S_to_plus_term X1
with t2 := rewrite_S_to_plus_term X2 in
constr:(t1 + t2)
| (?X1 * ?X2) =>
let t1 := rewrite_S_to_plus_term X1
with t2 := rewrite_S_to_plus_term X2 in
constr:(t1 * t2)
| _ => constr:t
end.
(* Apply S_to_plus on both sides of an equality *)
Ltac rewrite_S_to_plus :=
match goal with
| |- (?X1 = ?X2) =>
try
let t1 :=
(**) (**)
rewrite_S_to_plus_term X1
with t2 := rewrite_S_to_plus_term X2 in
change (t1 = t2) in |- *
| |- (?X1 = ?X2) =>
try
let t1 :=
(**) (**)
rewrite_S_to_plus_term X1
with t2 := rewrite_S_to_plus_term X2 in
change (t1 = t2) in |- *
end.
Ltac ring_nat := rewrite_S_to_plus; ring.
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