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Diffstat (limited to 'theories/Numbers/Integer/Abstract/ZAdd.v')
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diff --git a/theories/Numbers/Integer/Abstract/ZAdd.v b/theories/Numbers/Integer/Abstract/ZAdd.v new file mode 100644 index 00000000..df941d90 --- /dev/null +++ b/theories/Numbers/Integer/Abstract/ZAdd.v @@ -0,0 +1,345 @@ +(************************************************************************) +(* 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 *) +(************************************************************************) +(* Evgeny Makarov, INRIA, 2007 *) +(************************************************************************) + +(*i $Id: ZAdd.v 11040 2008-06-03 00:04:16Z letouzey $ i*) + +Require Export ZBase. + +Module ZAddPropFunct (Import ZAxiomsMod : ZAxiomsSig). +Module Export ZBasePropMod := ZBasePropFunct ZAxiomsMod. +Open Local Scope IntScope. + +Theorem Zadd_wd : + forall n1 n2 : Z, n1 == n2 -> forall m1 m2 : Z, m1 == m2 -> n1 + m1 == n2 + m2. +Proof NZadd_wd. + +Theorem Zadd_0_l : forall n : Z, 0 + n == n. +Proof NZadd_0_l. + +Theorem Zadd_succ_l : forall n m : Z, (S n) + m == S (n + m). +Proof NZadd_succ_l. + +Theorem Zsub_0_r : forall n : Z, n - 0 == n. +Proof NZsub_0_r. + +Theorem Zsub_succ_r : forall n m : Z, n - (S m) == P (n - m). +Proof NZsub_succ_r. + +Theorem Zopp_0 : - 0 == 0. +Proof Zopp_0. + +Theorem Zopp_succ : forall n : Z, - (S n) == P (- n). +Proof Zopp_succ. + +(* Theorems that are valid for both natural numbers and integers *) + +Theorem Zadd_0_r : forall n : Z, n + 0 == n. +Proof NZadd_0_r. + +Theorem Zadd_succ_r : forall n m : Z, n + S m == S (n + m). +Proof NZadd_succ_r. + +Theorem Zadd_comm : forall n m : Z, n + m == m + n. +Proof NZadd_comm. + +Theorem Zadd_assoc : forall n m p : Z, n + (m + p) == (n + m) + p. +Proof NZadd_assoc. + +Theorem Zadd_shuffle1 : forall n m p q : Z, (n + m) + (p + q) == (n + p) + (m + q). +Proof NZadd_shuffle1. + +Theorem Zadd_shuffle2 : forall n m p q : Z, (n + m) + (p + q) == (n + q) + (m + p). +Proof NZadd_shuffle2. + +Theorem Zadd_1_l : forall n : Z, 1 + n == S n. +Proof NZadd_1_l. + +Theorem Zadd_1_r : forall n : Z, n + 1 == S n. +Proof NZadd_1_r. + +Theorem Zadd_cancel_l : forall n m p : Z, p + n == p + m <-> n == m. +Proof NZadd_cancel_l. + +Theorem Zadd_cancel_r : forall n m p : Z, n + p == m + p <-> n == m. +Proof NZadd_cancel_r. + +(* Theorems that are either not valid on N or have different proofs on N and Z *) + +Theorem Zadd_pred_l : forall n m : Z, P n + m == P (n + m). +Proof. +intros n m. +rewrite <- (Zsucc_pred n) at 2. +rewrite Zadd_succ_l. now rewrite Zpred_succ. +Qed. + +Theorem Zadd_pred_r : forall n m : Z, n + P m == P (n + m). +Proof. +intros n m; rewrite (Zadd_comm n (P m)), (Zadd_comm n m); +apply Zadd_pred_l. +Qed. + +Theorem Zadd_opp_r : forall n m : Z, n + (- m) == n - m. +Proof. +NZinduct m. +rewrite Zopp_0; rewrite Zsub_0_r; now rewrite Zadd_0_r. +intro m. rewrite Zopp_succ, Zsub_succ_r, Zadd_pred_r; now rewrite Zpred_inj_wd. +Qed. + +Theorem Zsub_0_l : forall n : Z, 0 - n == - n. +Proof. +intro n; rewrite <- Zadd_opp_r; now rewrite Zadd_0_l. +Qed. + +Theorem Zsub_succ_l : forall n m : Z, S n - m == S (n - m). +Proof. +intros n m; do 2 rewrite <- Zadd_opp_r; now rewrite Zadd_succ_l. +Qed. + +Theorem Zsub_pred_l : forall n m : Z, P n - m == P (n - m). +Proof. +intros n m. rewrite <- (Zsucc_pred n) at 2. +rewrite Zsub_succ_l; now rewrite Zpred_succ. +Qed. + +Theorem Zsub_pred_r : forall n m : Z, n - (P m) == S (n - m). +Proof. +intros n m. rewrite <- (Zsucc_pred m) at 2. +rewrite Zsub_succ_r; now rewrite Zsucc_pred. +Qed. + +Theorem Zopp_pred : forall n : Z, - (P n) == S (- n). +Proof. +intro n. rewrite <- (Zsucc_pred n) at 2. +rewrite Zopp_succ. now rewrite Zsucc_pred. +Qed. + +Theorem Zsub_diag : forall n : Z, n - n == 0. +Proof. +NZinduct n. +now rewrite Zsub_0_r. +intro n. rewrite Zsub_succ_r, Zsub_succ_l; now rewrite Zpred_succ. +Qed. + +Theorem Zadd_opp_diag_l : forall n : Z, - n + n == 0. +Proof. +intro n; now rewrite Zadd_comm, Zadd_opp_r, Zsub_diag. +Qed. + +Theorem Zadd_opp_diag_r : forall n : Z, n + (- n) == 0. +Proof. +intro n; rewrite Zadd_comm; apply Zadd_opp_diag_l. +Qed. + +Theorem Zadd_opp_l : forall n m : Z, - m + n == n - m. +Proof. +intros n m; rewrite <- Zadd_opp_r; now rewrite Zadd_comm. +Qed. + +Theorem Zadd_sub_assoc : forall n m p : Z, n + (m - p) == (n + m) - p. +Proof. +intros n m p; do 2 rewrite <- Zadd_opp_r; now rewrite Zadd_assoc. +Qed. + +Theorem Zopp_involutive : forall n : Z, - (- n) == n. +Proof. +NZinduct n. +now do 2 rewrite Zopp_0. +intro n. rewrite Zopp_succ, Zopp_pred; now rewrite Zsucc_inj_wd. +Qed. + +Theorem Zopp_add_distr : forall n m : Z, - (n + m) == - n + (- m). +Proof. +intros n m; NZinduct n. +rewrite Zopp_0; now do 2 rewrite Zadd_0_l. +intro n. rewrite Zadd_succ_l; do 2 rewrite Zopp_succ; rewrite Zadd_pred_l. +now rewrite Zpred_inj_wd. +Qed. + +Theorem Zopp_sub_distr : forall n m : Z, - (n - m) == - n + m. +Proof. +intros n m; rewrite <- Zadd_opp_r, Zopp_add_distr. +now rewrite Zopp_involutive. +Qed. + +Theorem Zopp_inj : forall n m : Z, - n == - m -> n == m. +Proof. +intros n m H. apply Zopp_wd in H. now do 2 rewrite Zopp_involutive in H. +Qed. + +Theorem Zopp_inj_wd : forall n m : Z, - n == - m <-> n == m. +Proof. +intros n m; split; [apply Zopp_inj | apply Zopp_wd]. +Qed. + +Theorem Zeq_opp_l : forall n m : Z, - n == m <-> n == - m. +Proof. +intros n m. now rewrite <- (Zopp_inj_wd (- n) m), Zopp_involutive. +Qed. + +Theorem Zeq_opp_r : forall n m : Z, n == - m <-> - n == m. +Proof. +symmetry; apply Zeq_opp_l. +Qed. + +Theorem Zsub_add_distr : forall n m p : Z, n - (m + p) == (n - m) - p. +Proof. +intros n m p; rewrite <- Zadd_opp_r, Zopp_add_distr, Zadd_assoc. +now do 2 rewrite Zadd_opp_r. +Qed. + +Theorem Zsub_sub_distr : forall n m p : Z, n - (m - p) == (n - m) + p. +Proof. +intros n m p; rewrite <- Zadd_opp_r, Zopp_sub_distr, Zadd_assoc. +now rewrite Zadd_opp_r. +Qed. + +Theorem sub_opp_l : forall n m : Z, - n - m == - m - n. +Proof. +intros n m. do 2 rewrite <- Zadd_opp_r. now rewrite Zadd_comm. +Qed. + +Theorem Zsub_opp_r : forall n m : Z, n - (- m) == n + m. +Proof. +intros n m; rewrite <- Zadd_opp_r; now rewrite Zopp_involutive. +Qed. + +Theorem Zadd_sub_swap : forall n m p : Z, n + m - p == n - p + m. +Proof. +intros n m p. rewrite <- Zadd_sub_assoc, <- (Zadd_opp_r n p), <- Zadd_assoc. +now rewrite Zadd_opp_l. +Qed. + +Theorem Zsub_cancel_l : forall n m p : Z, n - m == n - p <-> m == p. +Proof. +intros n m p. rewrite <- (Zadd_cancel_l (n - m) (n - p) (- n)). +do 2 rewrite Zadd_sub_assoc. rewrite Zadd_opp_diag_l; do 2 rewrite Zsub_0_l. +apply Zopp_inj_wd. +Qed. + +Theorem Zsub_cancel_r : forall n m p : Z, n - p == m - p <-> n == m. +Proof. +intros n m p. +stepl (n - p + p == m - p + p) by apply Zadd_cancel_r. +now do 2 rewrite <- Zsub_sub_distr, Zsub_diag, Zsub_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 Zadd_move_l : forall n m p : Z, n + m == p <-> m == p - n. +Proof. +intros n m p. +stepl (n + m - n == p - n) by apply Zsub_cancel_r. +now rewrite Zadd_comm, <- Zadd_sub_assoc, Zsub_diag, Zadd_0_r. +Qed. + +Theorem Zadd_move_r : forall n m p : Z, n + m == p <-> n == p - m. +Proof. +intros n m p; rewrite Zadd_comm; now apply Zadd_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 Zsub_move_l : forall n m p : Z, n - m == p <-> - m == p - n. +Proof. +intros n m p; rewrite <- (Zadd_opp_r n m); apply Zadd_move_l. +Qed. + +Theorem Zsub_move_r : forall n m p : Z, n - m == p <-> n == p + m. +Proof. +intros n m p; rewrite <- (Zadd_opp_r n m). now rewrite Zadd_move_r, Zsub_opp_r. +Qed. + +Theorem Zadd_move_0_l : forall n m : Z, n + m == 0 <-> m == - n. +Proof. +intros n m; now rewrite Zadd_move_l, Zsub_0_l. +Qed. + +Theorem Zadd_move_0_r : forall n m : Z, n + m == 0 <-> n == - m. +Proof. +intros n m; now rewrite Zadd_move_r, Zsub_0_l. +Qed. + +Theorem Zsub_move_0_l : forall n m : Z, n - m == 0 <-> - m == - n. +Proof. +intros n m. now rewrite Zsub_move_l, Zsub_0_l. +Qed. + +Theorem Zsub_move_0_r : forall n m : Z, n - m == 0 <-> n == m. +Proof. +intros n m. now rewrite Zsub_move_r, Zadd_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 Zadd_simpl_l : forall n m : Z, n + m - n == m. +Proof. +intros; now rewrite Zadd_sub_swap, Zsub_diag, Zadd_0_l. +Qed. + +Theorem Zadd_simpl_r : forall n m : Z, n + m - m == n. +Proof. +intros; now rewrite <- Zadd_sub_assoc, Zsub_diag, Zadd_0_r. +Qed. + +Theorem Zsub_simpl_l : forall n m : Z, - n - m + n == - m. +Proof. +intros; now rewrite <- Zadd_sub_swap, Zadd_opp_diag_l, Zsub_0_l. +Qed. + +Theorem Zsub_simpl_r : forall n m : Z, n - m + m == n. +Proof. +intros; now rewrite <- Zsub_sub_distr, Zsub_diag, Zsub_0_r. +Qed. + +(* Now we have two sums or differences; the name includes the two operators +and the position of the terms being canceled *) + +Theorem Zadd_add_simpl_l_l : forall n m p : Z, (n + m) - (n + p) == m - p. +Proof. +intros n m p. now rewrite (Zadd_comm n m), <- Zadd_sub_assoc, +Zsub_add_distr, Zsub_diag, Zsub_0_l, Zadd_opp_r. +Qed. + +Theorem Zadd_add_simpl_l_r : forall n m p : Z, (n + m) - (p + n) == m - p. +Proof. +intros n m p. rewrite (Zadd_comm p n); apply Zadd_add_simpl_l_l. +Qed. + +Theorem Zadd_add_simpl_r_l : forall n m p : Z, (n + m) - (m + p) == n - p. +Proof. +intros n m p. rewrite (Zadd_comm n m); apply Zadd_add_simpl_l_l. +Qed. + +Theorem Zadd_add_simpl_r_r : forall n m p : Z, (n + m) - (p + m) == n - p. +Proof. +intros n m p. rewrite (Zadd_comm p m); apply Zadd_add_simpl_r_l. +Qed. + +Theorem Zsub_add_simpl_r_l : forall n m p : Z, (n - m) + (m + p) == n + p. +Proof. +intros n m p. now rewrite <- Zsub_sub_distr, Zsub_add_distr, Zsub_diag, +Zsub_0_l, Zsub_opp_r. +Qed. + +Theorem Zsub_add_simpl_r_r : forall n m p : Z, (n - m) + (p + m) == n + p. +Proof. +intros n m p. rewrite (Zadd_comm p m); apply Zsub_add_simpl_r_l. +Qed. + +(* Of course, there are many other variants *) + +End ZAddPropFunct. + |