path: root/theories/Numbers/Integer/Abstract/ZAdd.v

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(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
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(*    //   *      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                     *)
(************************************************************************)

Require Export ZBase.

Module ZAddProp (Import Z : ZAxiomsMiniSig').
Include ZBaseProp Z.

(** Theorems that are either not valid on N or have different proofs
    on N and Z *)

Hint Rewrite opp_0 : nz.

Theorem add_pred_l : forall n m, P n + m == P (n + m).
Proof.
intros n m.
rewrite <- (succ_pred n) at 2.
now rewrite add_succ_l, pred_succ.
Qed.

Theorem add_pred_r : forall n m, n + P m == P (n + m).
Proof.
intros n m; rewrite 2 (add_comm n); apply add_pred_l.
Qed.

Theorem add_opp_r : forall n m, n + (- m) == n - m.
Proof.
nzinduct m.
now nzsimpl.
intro m. rewrite opp_succ, sub_succ_r, add_pred_r. now rewrite pred_inj_wd.
Qed.

Theorem sub_0_l : forall n, 0 - n == - n.
Proof.
intro n; rewrite <- add_opp_r; now rewrite add_0_l.
Qed.

Theorem sub_succ_l : forall n m, S n - m == S (n - m).
Proof.
intros n m; rewrite <- 2 add_opp_r; now rewrite add_succ_l.
Qed.

Theorem sub_pred_l : forall n m, P n - m == P (n - m).
Proof.
intros n m. rewrite <- (succ_pred n) at 2.
rewrite sub_succ_l; now rewrite pred_succ.
Qed.

Theorem sub_pred_r : forall n m, n - (P m) == S (n - m).
Proof.
intros n m. rewrite <- (succ_pred m) at 2.
rewrite sub_succ_r; now rewrite succ_pred.
Qed.

Theorem opp_pred : forall n, - (P n) == S (- n).
Proof.
intro n. rewrite <- (succ_pred n) at 2.
rewrite opp_succ. now rewrite succ_pred.
Qed.

Theorem sub_diag : forall n, n - n == 0.
Proof.
nzinduct n.
now nzsimpl.
intro n. rewrite sub_succ_r, sub_succ_l; now rewrite pred_succ.
Qed.

Theorem add_opp_diag_l : forall n, - n + n == 0.
Proof.
intro n; now rewrite add_comm, add_opp_r, sub_diag.
Qed.

Theorem add_opp_diag_r : forall n, n + (- n) == 0.
Proof.
intro n; rewrite add_comm; apply add_opp_diag_l.
Qed.

Theorem add_opp_l : forall n m, - m + n == n - m.
Proof.
intros n m; rewrite <- add_opp_r; now rewrite add_comm.
Qed.

Theorem add_sub_assoc : forall n m p, n + (m - p) == (n + m) - p.
Proof.
intros n m p; rewrite <- 2 add_opp_r; now rewrite add_assoc.
Qed.

Theorem opp_involutive : forall n, - (- n) == n.
Proof.
nzinduct n.
now nzsimpl.
intro n. rewrite opp_succ, opp_pred. now rewrite succ_inj_wd.
Qed.

Theorem opp_add_distr : forall n m, - (n + m) == - n + (- m).
Proof.
intros n m; nzinduct n.
now nzsimpl.
intro n. rewrite add_succ_l; do 2 rewrite opp_succ; rewrite add_pred_l.
now rewrite pred_inj_wd.
Qed.

Theorem opp_sub_distr : forall n m, - (n - m) == - n + m.
Proof.
intros n m; rewrite <- add_opp_r, opp_add_distr.
now rewrite opp_involutive.
Qed.

Theorem opp_inj : forall n m, - n == - m -> n == m.
Proof.
intros n m H. apply opp_wd in H. now rewrite 2 opp_involutive in H.
Qed.

Theorem opp_inj_wd : forall n m, - n == - m <-> n == m.
Proof.
intros n m; split; [apply opp_inj | intros; now f_equiv].
Qed.

Theorem eq_opp_l : forall n m, - n == m <-> n == - m.
Proof.
intros n m. now rewrite <- (opp_inj_wd (- n) m), opp_involutive.
Qed.

Theorem eq_opp_r : forall n m, n == - m <-> - n == m.
Proof.
symmetry; apply eq_opp_l.
Qed.

Theorem sub_add_distr : forall n m p, n - (m + p) == (n - m) - p.
Proof.
intros n m p; rewrite <- add_opp_r, opp_add_distr, add_assoc.
now rewrite 2 add_opp_r.
Qed.

Theorem sub_sub_distr : forall n m p, n - (m - p) == (n - m) + p.
Proof.
intros n m p; rewrite <- add_opp_r, opp_sub_distr, add_assoc.
now rewrite add_opp_r.
Qed.

Theorem sub_opp_l : forall n m, - n - m == - m - n.
Proof.
intros n m. rewrite <- 2 add_opp_r. now rewrite add_comm.
Qed.

Theorem sub_opp_r : forall n m, n - (- m) == n + m.
Proof.
intros n m; rewrite <- add_opp_r; now rewrite opp_involutive.
Qed.

Theorem add_sub_swap : forall n m p, n + m - p == n - p + m.
Proof.
intros n m p. rewrite <- add_sub_assoc, <- (add_opp_r n p), <- add_assoc.
now rewrite add_opp_l.
Qed.

Theorem sub_cancel_l : forall n m p, n - m == n - p <-> m == p.
Proof.
intros n m p. rewrite <- (add_cancel_l (n - m) (n - p) (- n)).
rewrite 2 add_sub_assoc. rewrite add_opp_diag_l; rewrite 2 sub_0_l.
apply opp_inj_wd.
Qed.

Theorem sub_cancel_r : forall n m p, n - p == m - p <-> n == m.
Proof.
intros n m p.
stepl (n - p + p == m - p + p) by apply add_cancel_r.
now do 2 rewrite <- sub_sub_distr, sub_diag, sub_0_r.
Qed.

(** The next several theorems are devoted to moving terms from one
    side of an equation to the other. The name contains the operation
    in the original equation ([add] or [sub]) and the indication
    whether the left or right term is moved. *)

Theorem add_move_l : forall n m p, n + m == p <-> m == p - n.
Proof.
intros n m p.
stepl (n + m - n == p - n) by apply sub_cancel_r.
now rewrite add_comm, <- add_sub_assoc, sub_diag, add_0_r.
Qed.

Theorem add_move_r : forall n m p, n + m == p <-> n == p - m.
Proof.
intros n m p; rewrite add_comm; now apply add_move_l.
Qed.

(** The two theorems above do not allow rewriting subformulas of the
    form [n - m == p] to [n == p + m] since subtraction is in the
    right-hand side of the equation. Hence the following two
    theorems. *)

Theorem sub_move_l : forall n m p, n - m == p <-> - m == p - n.
Proof.
intros n m p; rewrite <- (add_opp_r n m); apply add_move_l.
Qed.

Theorem sub_move_r : forall n m p, n - m == p <-> n == p + m.
Proof.
intros n m p; rewrite <- (add_opp_r n m). now rewrite add_move_r, sub_opp_r.
Qed.

Theorem add_move_0_l : forall n m, n + m == 0 <-> m == - n.
Proof.
intros n m; now rewrite add_move_l, sub_0_l.
Qed.

Theorem add_move_0_r : forall n m, n + m == 0 <-> n == - m.
Proof.
intros n m; now rewrite add_move_r, sub_0_l.
Qed.

Theorem sub_move_0_l : forall n m, n - m == 0 <-> - m == - n.
Proof.
intros n m. now rewrite sub_move_l, sub_0_l.
Qed.

Theorem sub_move_0_r : forall n m, n - m == 0 <-> n == m.
Proof.
intros n m. now rewrite sub_move_r, add_0_l.
Qed.

(** The following section is devoted to cancellation of like
    terms. The name includes the first operator and the position of
    the term being canceled. *)

Theorem add_simpl_l : forall n m, n + m - n == m.
Proof.
intros; now rewrite add_sub_swap, sub_diag, add_0_l.
Qed.

Theorem add_simpl_r : forall n m, n + m - m == n.
Proof.
intros; now rewrite <- add_sub_assoc, sub_diag, add_0_r.
Qed.

Theorem sub_simpl_l : forall n m, - n - m + n == - m.
Proof.
intros; now rewrite <- add_sub_swap, add_opp_diag_l, sub_0_l.
Qed.

Theorem sub_simpl_r : forall n m, n - m + m == n.
Proof.
intros; now rewrite <- sub_sub_distr, sub_diag, sub_0_r.
Qed.

Theorem sub_add : forall n m, m - n + n == m.
Proof.
 intros. now rewrite <- add_sub_swap, add_simpl_r.
Qed.

(** Now we have two sums or differences; the name includes the two
    operators and the position of the terms being canceled *)

Theorem add_add_simpl_l_l : forall n m p, (n + m) - (n + p) == m - p.
Proof.
intros n m p. now rewrite (add_comm n m), <- add_sub_assoc,
sub_add_distr, sub_diag, sub_0_l, add_opp_r.
Qed.

Theorem add_add_simpl_l_r : forall n m p, (n + m) - (p + n) == m - p.
Proof.
intros n m p. rewrite (add_comm p n); apply add_add_simpl_l_l.
Qed.

Theorem add_add_simpl_r_l : forall n m p, (n + m) - (m + p) == n - p.
Proof.
intros n m p. rewrite (add_comm n m); apply add_add_simpl_l_l.
Qed.

Theorem add_add_simpl_r_r : forall n m p, (n + m) - (p + m) == n - p.
Proof.
intros n m p. rewrite (add_comm p m); apply add_add_simpl_r_l.
Qed.

Theorem sub_add_simpl_r_l : forall n m p, (n - m) + (m + p) == n + p.
Proof.
intros n m p. now rewrite <- sub_sub_distr, sub_add_distr, sub_diag,
sub_0_l, sub_opp_r.
Qed.

Theorem sub_add_simpl_r_r : forall n m p, (n - m) + (p + m) == n + p.
Proof.
intros n m p. rewrite (add_comm p m); apply sub_add_simpl_r_l.
Qed.

(** Of course, there are many other variants *)

End ZAddProp.