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(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
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(*                      Evgeny Makarov, INRIA, 2007                     *)
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Require Import NZAxioms NZBase.

Module Type NZAddProp (Import NZ : NZAxiomsSig')(Import NZBase : NZBaseProp NZ).

Hint Rewrite
 pred_succ add_0_l add_succ_l mul_0_l mul_succ_l sub_0_r sub_succ_r : nz.
Hint Rewrite one_succ two_succ : nz'.
Ltac nzsimpl := autorewrite with nz.
Ltac nzsimpl' := autorewrite with nz nz'.

Theorem add_0_r : forall n, n + 0 == n.
Proof.
nzinduct n. now nzsimpl.
intro. nzsimpl. now rewrite succ_inj_wd.
Qed.

Theorem add_succ_r : forall n m, n + S m == S (n + m).
Proof.
intros n m; nzinduct n. now nzsimpl.
intro. nzsimpl. now rewrite succ_inj_wd.
Qed.

Theorem add_succ_comm : forall n m, S n + m == n + S m.
Proof.
intros n m. now rewrite add_succ_r, add_succ_l.
Qed.

Hint Rewrite add_0_r add_succ_r : nz.

Theorem add_comm : forall n m, n + m == m + n.
Proof.
intros n m; nzinduct n. now nzsimpl.
intro. nzsimpl. now rewrite succ_inj_wd.
Qed.

Theorem add_1_l : forall n, 1 + n == S n.
Proof.
intro n; now nzsimpl'.
Qed.

Theorem add_1_r : forall n, n + 1 == S n.
Proof.
intro n; now nzsimpl'.
Qed.

Hint Rewrite add_1_l add_1_r : nz.

Theorem add_assoc : forall n m p, n + (m + p) == (n + m) + p.
Proof.
intros n m p; nzinduct n. now nzsimpl.
intro. nzsimpl. now rewrite succ_inj_wd.
Qed.

Theorem add_cancel_l : forall n m p, p + n == p + m <-> n == m.
Proof.
intros n m p; nzinduct p. now nzsimpl.
intro p. nzsimpl. now rewrite succ_inj_wd.
Qed.

Theorem add_cancel_r : forall n m p, n + p == m + p <-> n == m.
Proof.
intros n m p. rewrite (add_comm n p), (add_comm m p). apply add_cancel_l.
Qed.

Theorem add_shuffle0 : forall n m p, n+m+p == n+p+m.
Proof.
intros n m p. rewrite <- 2 add_assoc, add_cancel_l. apply add_comm.
Qed.

Theorem add_shuffle1 : forall n m p q, (n + m) + (p + q) == (n + p) + (m + q).
Proof.
intros n m p q. rewrite 2 add_assoc, add_cancel_r. apply add_shuffle0.
Qed.

Theorem add_shuffle2 : forall n m p q, (n + m) + (p + q) == (n + q) + (m + p).
Proof.
intros n m p q. rewrite (add_comm p). apply add_shuffle1.
Qed.

Theorem add_shuffle3 : forall n m p, n + (m + p) == m + (n + p).
Proof.
intros n m p. now rewrite add_comm, <- add_assoc, (add_comm p).
Qed.

Theorem sub_1_r : forall n, n - 1 == P n.
Proof.
intro n; now nzsimpl'.
Qed.

Hint Rewrite sub_1_r : nz.

End NZAddProp.
