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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Import Arith.
Require Export Ring.
Set Implicit Arguments.
Ltac isnatcst t :=
let t := eval hnf in t in
match t with
O => true
| S ?p => isnatcst p
| _ => false
end.
Ltac natcst t :=
match isnatcst t with
true => t
| _ => NotConstant
end.
Ltac natprering :=
match goal with
|- context C [S ?p] =>
match p with
O => fail 1 (* avoid replacing 1 with 1+0 ! *)
| S _ => fail 1
| _ => fold (1+p)%nat; natprering
end
| _ => idtac
end.
Lemma natSRth : semi_ring_theory O (S O) plus mult (@eq nat).
Proof.
constructor. exact plus_0_l. exact plus_comm. exact plus_assoc.
exact mult_1_l. exact mult_0_l. exact mult_comm. exact mult_assoc.
exact mult_plus_distr_r.
Qed.
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_ok : forall n m:nat, nateq n m = true -> n = m.
Proof.
simple induction n; simple induction m; simpl; intros; try discriminate.
trivial.
rewrite (H n1 H1).
trivial.
Qed.
Add Ring natr : natSRth
(decidable nateq_ok, constants [natcst], preprocess [natprering]).
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