diff options
Diffstat (limited to 'theories/Numbers/Integer/Abstract')
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZAdd.v | 345 | ||||
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZAddOrder.v | 373 | ||||
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZAxioms.v | 65 | ||||
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZBase.v | 86 | ||||
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZDomain.v | 69 | ||||
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZLt.v | 432 | ||||
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZMul.v | 115 | ||||
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZMulOrder.v | 343 |
8 files changed, 1828 insertions, 0 deletions
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. + diff --git a/theories/Numbers/Integer/Abstract/ZAddOrder.v b/theories/Numbers/Integer/Abstract/ZAddOrder.v new file mode 100644 index 00000000..101ea634 --- /dev/null +++ b/theories/Numbers/Integer/Abstract/ZAddOrder.v @@ -0,0 +1,373 @@ +(************************************************************************) +(* 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: ZAddOrder.v 11040 2008-06-03 00:04:16Z letouzey $ i*) + +Require Export ZLt. + +Module ZAddOrderPropFunct (Import ZAxiomsMod : ZAxiomsSig). +Module Export ZOrderPropMod := ZOrderPropFunct ZAxiomsMod. +Open Local Scope IntScope. + +(* Theorems that are true on both natural numbers and integers *) + +Theorem Zadd_lt_mono_l : forall n m p : Z, n < m <-> p + n < p + m. +Proof NZadd_lt_mono_l. + +Theorem Zadd_lt_mono_r : forall n m p : Z, n < m <-> n + p < m + p. +Proof NZadd_lt_mono_r. + +Theorem Zadd_lt_mono : forall n m p q : Z, n < m -> p < q -> n + p < m + q. +Proof NZadd_lt_mono. + +Theorem Zadd_le_mono_l : forall n m p : Z, n <= m <-> p + n <= p + m. +Proof NZadd_le_mono_l. + +Theorem Zadd_le_mono_r : forall n m p : Z, n <= m <-> n + p <= m + p. +Proof NZadd_le_mono_r. + +Theorem Zadd_le_mono : forall n m p q : Z, n <= m -> p <= q -> n + p <= m + q. +Proof NZadd_le_mono. + +Theorem Zadd_lt_le_mono : forall n m p q : Z, n < m -> p <= q -> n + p < m + q. +Proof NZadd_lt_le_mono. + +Theorem Zadd_le_lt_mono : forall n m p q : Z, n <= m -> p < q -> n + p < m + q. +Proof NZadd_le_lt_mono. + +Theorem Zadd_pos_pos : forall n m : Z, 0 < n -> 0 < m -> 0 < n + m. +Proof NZadd_pos_pos. + +Theorem Zadd_pos_nonneg : forall n m : Z, 0 < n -> 0 <= m -> 0 < n + m. +Proof NZadd_pos_nonneg. + +Theorem Zadd_nonneg_pos : forall n m : Z, 0 <= n -> 0 < m -> 0 < n + m. +Proof NZadd_nonneg_pos. + +Theorem Zadd_nonneg_nonneg : forall n m : Z, 0 <= n -> 0 <= m -> 0 <= n + m. +Proof NZadd_nonneg_nonneg. + +Theorem Zlt_add_pos_l : forall n m : Z, 0 < n -> m < n + m. +Proof NZlt_add_pos_l. + +Theorem Zlt_add_pos_r : forall n m : Z, 0 < n -> m < m + n. +Proof NZlt_add_pos_r. + +Theorem Zle_lt_add_lt : forall n m p q : Z, n <= m -> p + m < q + n -> p < q. +Proof NZle_lt_add_lt. + +Theorem Zlt_le_add_lt : forall n m p q : Z, n < m -> p + m <= q + n -> p < q. +Proof NZlt_le_add_lt. + +Theorem Zle_le_add_le : forall n m p q : Z, n <= m -> p + m <= q + n -> p <= q. +Proof NZle_le_add_le. + +Theorem Zadd_lt_cases : forall n m p q : Z, n + m < p + q -> n < p \/ m < q. +Proof NZadd_lt_cases. + +Theorem Zadd_le_cases : forall n m p q : Z, n + m <= p + q -> n <= p \/ m <= q. +Proof NZadd_le_cases. + +Theorem Zadd_neg_cases : forall n m : Z, n + m < 0 -> n < 0 \/ m < 0. +Proof NZadd_neg_cases. + +Theorem Zadd_pos_cases : forall n m : Z, 0 < n + m -> 0 < n \/ 0 < m. +Proof NZadd_pos_cases. + +Theorem Zadd_nonpos_cases : forall n m : Z, n + m <= 0 -> n <= 0 \/ m <= 0. +Proof NZadd_nonpos_cases. + +Theorem Zadd_nonneg_cases : forall n m : Z, 0 <= n + m -> 0 <= n \/ 0 <= m. +Proof NZadd_nonneg_cases. + +(* Theorems that are either not valid on N or have different proofs on N and Z *) + +Theorem Zadd_neg_neg : forall n m : Z, n < 0 -> m < 0 -> n + m < 0. +Proof. +intros n m H1 H2. rewrite <- (Zadd_0_l 0). now apply Zadd_lt_mono. +Qed. + +Theorem Zadd_neg_nonpos : forall n m : Z, n < 0 -> m <= 0 -> n + m < 0. +Proof. +intros n m H1 H2. rewrite <- (Zadd_0_l 0). now apply Zadd_lt_le_mono. +Qed. + +Theorem Zadd_nonpos_neg : forall n m : Z, n <= 0 -> m < 0 -> n + m < 0. +Proof. +intros n m H1 H2. rewrite <- (Zadd_0_l 0). now apply Zadd_le_lt_mono. +Qed. + +Theorem Zadd_nonpos_nonpos : forall n m : Z, n <= 0 -> m <= 0 -> n + m <= 0. +Proof. +intros n m H1 H2. rewrite <- (Zadd_0_l 0). now apply Zadd_le_mono. +Qed. + +(** Sub and order *) + +Theorem Zlt_0_sub : forall n m : Z, 0 < m - n <-> n < m. +Proof. +intros n m. stepl (0 + n < m - n + n) by symmetry; apply Zadd_lt_mono_r. +rewrite Zadd_0_l; now rewrite Zsub_simpl_r. +Qed. + +Notation Zsub_pos := Zlt_0_sub (only parsing). + +Theorem Zle_0_sub : forall n m : Z, 0 <= m - n <-> n <= m. +Proof. +intros n m; stepl (0 + n <= m - n + n) by symmetry; apply Zadd_le_mono_r. +rewrite Zadd_0_l; now rewrite Zsub_simpl_r. +Qed. + +Notation Zsub_nonneg := Zle_0_sub (only parsing). + +Theorem Zlt_sub_0 : forall n m : Z, n - m < 0 <-> n < m. +Proof. +intros n m. stepl (n - m + m < 0 + m) by symmetry; apply Zadd_lt_mono_r. +rewrite Zadd_0_l; now rewrite Zsub_simpl_r. +Qed. + +Notation Zsub_neg := Zlt_sub_0 (only parsing). + +Theorem Zle_sub_0 : forall n m : Z, n - m <= 0 <-> n <= m. +Proof. +intros n m. stepl (n - m + m <= 0 + m) by symmetry; apply Zadd_le_mono_r. +rewrite Zadd_0_l; now rewrite Zsub_simpl_r. +Qed. + +Notation Zsub_nonpos := Zle_sub_0 (only parsing). + +Theorem Zopp_lt_mono : forall n m : Z, n < m <-> - m < - n. +Proof. +intros n m. stepr (m + - m < m + - n) by symmetry; apply Zadd_lt_mono_l. +do 2 rewrite Zadd_opp_r. rewrite Zsub_diag. symmetry; apply Zlt_0_sub. +Qed. + +Theorem Zopp_le_mono : forall n m : Z, n <= m <-> - m <= - n. +Proof. +intros n m. stepr (m + - m <= m + - n) by symmetry; apply Zadd_le_mono_l. +do 2 rewrite Zadd_opp_r. rewrite Zsub_diag. symmetry; apply Zle_0_sub. +Qed. + +Theorem Zopp_pos_neg : forall n : Z, 0 < - n <-> n < 0. +Proof. +intro n; rewrite (Zopp_lt_mono n 0); now rewrite Zopp_0. +Qed. + +Theorem Zopp_neg_pos : forall n : Z, - n < 0 <-> 0 < n. +Proof. +intro n. rewrite (Zopp_lt_mono 0 n). now rewrite Zopp_0. +Qed. + +Theorem Zopp_nonneg_nonpos : forall n : Z, 0 <= - n <-> n <= 0. +Proof. +intro n; rewrite (Zopp_le_mono n 0); now rewrite Zopp_0. +Qed. + +Theorem Zopp_nonpos_nonneg : forall n : Z, - n <= 0 <-> 0 <= n. +Proof. +intro n. rewrite (Zopp_le_mono 0 n). now rewrite Zopp_0. +Qed. + +Theorem Zsub_lt_mono_l : forall n m p : Z, n < m <-> p - m < p - n. +Proof. +intros n m p. do 2 rewrite <- Zadd_opp_r. rewrite <- Zadd_lt_mono_l. +apply Zopp_lt_mono. +Qed. + +Theorem Zsub_lt_mono_r : forall n m p : Z, n < m <-> n - p < m - p. +Proof. +intros n m p; do 2 rewrite <- Zadd_opp_r; apply Zadd_lt_mono_r. +Qed. + +Theorem Zsub_lt_mono : forall n m p q : Z, n < m -> q < p -> n - p < m - q. +Proof. +intros n m p q H1 H2. +apply NZlt_trans with (m - p); +[now apply -> Zsub_lt_mono_r | now apply -> Zsub_lt_mono_l]. +Qed. + +Theorem Zsub_le_mono_l : forall n m p : Z, n <= m <-> p - m <= p - n. +Proof. +intros n m p; do 2 rewrite <- Zadd_opp_r; rewrite <- Zadd_le_mono_l; +apply Zopp_le_mono. +Qed. + +Theorem Zsub_le_mono_r : forall n m p : Z, n <= m <-> n - p <= m - p. +Proof. +intros n m p; do 2 rewrite <- Zadd_opp_r; apply Zadd_le_mono_r. +Qed. + +Theorem Zsub_le_mono : forall n m p q : Z, n <= m -> q <= p -> n - p <= m - q. +Proof. +intros n m p q H1 H2. +apply NZle_trans with (m - p); +[now apply -> Zsub_le_mono_r | now apply -> Zsub_le_mono_l]. +Qed. + +Theorem Zsub_lt_le_mono : forall n m p q : Z, n < m -> q <= p -> n - p < m - q. +Proof. +intros n m p q H1 H2. +apply NZlt_le_trans with (m - p); +[now apply -> Zsub_lt_mono_r | now apply -> Zsub_le_mono_l]. +Qed. + +Theorem Zsub_le_lt_mono : forall n m p q : Z, n <= m -> q < p -> n - p < m - q. +Proof. +intros n m p q H1 H2. +apply NZle_lt_trans with (m - p); +[now apply -> Zsub_le_mono_r | now apply -> Zsub_lt_mono_l]. +Qed. + +Theorem Zle_lt_sub_lt : forall n m p q : Z, n <= m -> p - n < q - m -> p < q. +Proof. +intros n m p q H1 H2. apply (Zle_lt_add_lt (- m) (- n)); +[now apply -> Zopp_le_mono | now do 2 rewrite Zadd_opp_r]. +Qed. + +Theorem Zlt_le_sub_lt : forall n m p q : Z, n < m -> p - n <= q - m -> p < q. +Proof. +intros n m p q H1 H2. apply (Zlt_le_add_lt (- m) (- n)); +[now apply -> Zopp_lt_mono | now do 2 rewrite Zadd_opp_r]. +Qed. + +Theorem Zle_le_sub_lt : forall n m p q : Z, n <= m -> p - n <= q - m -> p <= q. +Proof. +intros n m p q H1 H2. apply (Zle_le_add_le (- m) (- n)); +[now apply -> Zopp_le_mono | now do 2 rewrite Zadd_opp_r]. +Qed. + +Theorem Zlt_add_lt_sub_r : forall n m p : Z, n + p < m <-> n < m - p. +Proof. +intros n m p. stepl (n + p - p < m - p) by symmetry; apply Zsub_lt_mono_r. +now rewrite Zadd_simpl_r. +Qed. + +Theorem Zle_add_le_sub_r : forall n m p : Z, n + p <= m <-> n <= m - p. +Proof. +intros n m p. stepl (n + p - p <= m - p) by symmetry; apply Zsub_le_mono_r. +now rewrite Zadd_simpl_r. +Qed. + +Theorem Zlt_add_lt_sub_l : forall n m p : Z, n + p < m <-> p < m - n. +Proof. +intros n m p. rewrite Zadd_comm; apply Zlt_add_lt_sub_r. +Qed. + +Theorem Zle_add_le_sub_l : forall n m p : Z, n + p <= m <-> p <= m - n. +Proof. +intros n m p. rewrite Zadd_comm; apply Zle_add_le_sub_r. +Qed. + +Theorem Zlt_sub_lt_add_r : forall n m p : Z, n - p < m <-> n < m + p. +Proof. +intros n m p. stepl (n - p + p < m + p) by symmetry; apply Zadd_lt_mono_r. +now rewrite Zsub_simpl_r. +Qed. + +Theorem Zle_sub_le_add_r : forall n m p : Z, n - p <= m <-> n <= m + p. +Proof. +intros n m p. stepl (n - p + p <= m + p) by symmetry; apply Zadd_le_mono_r. +now rewrite Zsub_simpl_r. +Qed. + +Theorem Zlt_sub_lt_add_l : forall n m p : Z, n - m < p <-> n < m + p. +Proof. +intros n m p. rewrite Zadd_comm; apply Zlt_sub_lt_add_r. +Qed. + +Theorem Zle_sub_le_add_l : forall n m p : Z, n - m <= p <-> n <= m + p. +Proof. +intros n m p. rewrite Zadd_comm; apply Zle_sub_le_add_r. +Qed. + +Theorem Zlt_sub_lt_add : forall n m p q : Z, n - m < p - q <-> n + q < m + p. +Proof. +intros n m p q. rewrite Zlt_sub_lt_add_l. rewrite Zadd_sub_assoc. +now rewrite <- Zlt_add_lt_sub_r. +Qed. + +Theorem Zle_sub_le_add : forall n m p q : Z, n - m <= p - q <-> n + q <= m + p. +Proof. +intros n m p q. rewrite Zle_sub_le_add_l. rewrite Zadd_sub_assoc. +now rewrite <- Zle_add_le_sub_r. +Qed. + +Theorem Zlt_sub_pos : forall n m : Z, 0 < m <-> n - m < n. +Proof. +intros n m. stepr (n - m < n - 0) by now rewrite Zsub_0_r. apply Zsub_lt_mono_l. +Qed. + +Theorem Zle_sub_nonneg : forall n m : Z, 0 <= m <-> n - m <= n. +Proof. +intros n m. stepr (n - m <= n - 0) by now rewrite Zsub_0_r. apply Zsub_le_mono_l. +Qed. + +Theorem Zsub_lt_cases : forall n m p q : Z, n - m < p - q -> n < m \/ q < p. +Proof. +intros n m p q H. rewrite Zlt_sub_lt_add in H. now apply Zadd_lt_cases. +Qed. + +Theorem Zsub_le_cases : forall n m p q : Z, n - m <= p - q -> n <= m \/ q <= p. +Proof. +intros n m p q H. rewrite Zle_sub_le_add in H. now apply Zadd_le_cases. +Qed. + +Theorem Zsub_neg_cases : forall n m : Z, n - m < 0 -> n < 0 \/ 0 < m. +Proof. +intros n m H; rewrite <- Zadd_opp_r in H. +setoid_replace (0 < m) with (- m < 0) using relation iff by (symmetry; apply Zopp_neg_pos). +now apply Zadd_neg_cases. +Qed. + +Theorem Zsub_pos_cases : forall n m : Z, 0 < n - m -> 0 < n \/ m < 0. +Proof. +intros n m H; rewrite <- Zadd_opp_r in H. +setoid_replace (m < 0) with (0 < - m) using relation iff by (symmetry; apply Zopp_pos_neg). +now apply Zadd_pos_cases. +Qed. + +Theorem Zsub_nonpos_cases : forall n m : Z, n - m <= 0 -> n <= 0 \/ 0 <= m. +Proof. +intros n m H; rewrite <- Zadd_opp_r in H. +setoid_replace (0 <= m) with (- m <= 0) using relation iff by (symmetry; apply Zopp_nonpos_nonneg). +now apply Zadd_nonpos_cases. +Qed. + +Theorem Zsub_nonneg_cases : forall n m : Z, 0 <= n - m -> 0 <= n \/ m <= 0. +Proof. +intros n m H; rewrite <- Zadd_opp_r in H. +setoid_replace (m <= 0) with (0 <= - m) using relation iff by (symmetry; apply Zopp_nonneg_nonpos). +now apply Zadd_nonneg_cases. +Qed. + +Section PosNeg. + +Variable P : Z -> Prop. +Hypothesis P_wd : predicate_wd Zeq P. + +Add Morphism P with signature Zeq ==> iff as P_morph. Proof. exact P_wd. Qed. + +Theorem Z0_pos_neg : + P 0 -> (forall n : Z, 0 < n -> P n /\ P (- n)) -> forall n : Z, P n. +Proof. +intros H1 H2 n. destruct (Zlt_trichotomy n 0) as [H3 | [H3 | H3]]. +apply <- Zopp_pos_neg in H3. apply H2 in H3. destruct H3 as [_ H3]. +now rewrite Zopp_involutive in H3. +now rewrite H3. +apply H2 in H3; now destruct H3. +Qed. + +End PosNeg. + +Ltac Z0_pos_neg n := induction_maker n ltac:(apply Z0_pos_neg). + +End ZAddOrderPropFunct. + + diff --git a/theories/Numbers/Integer/Abstract/ZAxioms.v b/theories/Numbers/Integer/Abstract/ZAxioms.v new file mode 100644 index 00000000..c4a4b6b8 --- /dev/null +++ b/theories/Numbers/Integer/Abstract/ZAxioms.v @@ -0,0 +1,65 @@ +(************************************************************************) +(* 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: ZAxioms.v 11040 2008-06-03 00:04:16Z letouzey $ i*) + +Require Export NZAxioms. + +Set Implicit Arguments. + +Module Type ZAxiomsSig. +Declare Module Export NZOrdAxiomsMod : NZOrdAxiomsSig. + +Delimit Scope IntScope with Int. +Notation Z := NZ. +Notation Zeq := NZeq. +Notation Z0 := NZ0. +Notation Z1 := (NZsucc NZ0). +Notation S := NZsucc. +Notation P := NZpred. +Notation Zadd := NZadd. +Notation Zmul := NZmul. +Notation Zsub := NZsub. +Notation Zlt := NZlt. +Notation Zle := NZle. +Notation Zmin := NZmin. +Notation Zmax := NZmax. +Notation "x == y" := (NZeq x y) (at level 70) : IntScope. +Notation "x ~= y" := (~ NZeq x y) (at level 70) : IntScope. +Notation "0" := NZ0 : IntScope. +Notation "1" := (NZsucc NZ0) : IntScope. +Notation "x + y" := (NZadd x y) : IntScope. +Notation "x - y" := (NZsub x y) : IntScope. +Notation "x * y" := (NZmul x y) : IntScope. +Notation "x < y" := (NZlt x y) : IntScope. +Notation "x <= y" := (NZle x y) : IntScope. +Notation "x > y" := (NZlt y x) (only parsing) : IntScope. +Notation "x >= y" := (NZle y x) (only parsing) : IntScope. + +Parameter Zopp : Z -> Z. + +(*Notation "- 1" := (Zopp 1) : IntScope. +Check (-1).*) + +Add Morphism Zopp with signature Zeq ==> Zeq as Zopp_wd. + +Notation "- x" := (Zopp x) (at level 35, right associativity) : IntScope. +Notation "- 1" := (Zopp (NZsucc NZ0)) : IntScope. + +Open Local Scope IntScope. + +(* Integers are obtained by postulating that every number has a predecessor *) +Axiom Zsucc_pred : forall n : Z, S (P n) == n. + +Axiom Zopp_0 : - 0 == 0. +Axiom Zopp_succ : forall n : Z, - (S n) == P (- n). + +End ZAxiomsSig. + diff --git a/theories/Numbers/Integer/Abstract/ZBase.v b/theories/Numbers/Integer/Abstract/ZBase.v new file mode 100644 index 00000000..29e18548 --- /dev/null +++ b/theories/Numbers/Integer/Abstract/ZBase.v @@ -0,0 +1,86 @@ +(************************************************************************) +(* 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: ZBase.v 11040 2008-06-03 00:04:16Z letouzey $ i*) + +Require Export Decidable. +Require Export ZAxioms. +Require Import NZMulOrder. + +Module ZBasePropFunct (Import ZAxiomsMod : ZAxiomsSig). + +(* Note: writing "Export" instead of "Import" on the previous line leads to +some warnings about hiding repeated declarations and results in the loss of +notations in Zadd and later *) + +Open Local Scope IntScope. + +Module Export NZMulOrderMod := NZMulOrderPropFunct NZOrdAxiomsMod. + +Theorem Zsucc_wd : forall n1 n2 : Z, n1 == n2 -> S n1 == S n2. +Proof NZsucc_wd. + +Theorem Zpred_wd : forall n1 n2 : Z, n1 == n2 -> P n1 == P n2. +Proof NZpred_wd. + +Theorem Zpred_succ : forall n : Z, P (S n) == n. +Proof NZpred_succ. + +Theorem Zeq_refl : forall n : Z, n == n. +Proof (proj1 NZeq_equiv). + +Theorem Zeq_symm : forall n m : Z, n == m -> m == n. +Proof (proj2 (proj2 NZeq_equiv)). + +Theorem Zeq_trans : forall n m p : Z, n == m -> m == p -> n == p. +Proof (proj1 (proj2 NZeq_equiv)). + +Theorem Zneq_symm : forall n m : Z, n ~= m -> m ~= n. +Proof NZneq_symm. + +Theorem Zsucc_inj : forall n1 n2 : Z, S n1 == S n2 -> n1 == n2. +Proof NZsucc_inj. + +Theorem Zsucc_inj_wd : forall n1 n2 : Z, S n1 == S n2 <-> n1 == n2. +Proof NZsucc_inj_wd. + +Theorem Zsucc_inj_wd_neg : forall n m : Z, S n ~= S m <-> n ~= m. +Proof NZsucc_inj_wd_neg. + +(* Decidability and stability of equality was proved only in NZOrder, but +since it does not mention order, we'll put it here *) + +Theorem Zeq_dec : forall n m : Z, decidable (n == m). +Proof NZeq_dec. + +Theorem Zeq_dne : forall n m : Z, ~ ~ n == m <-> n == m. +Proof NZeq_dne. + +Theorem Zcentral_induction : +forall A : Z -> Prop, predicate_wd Zeq A -> + forall z : Z, A z -> + (forall n : Z, A n <-> A (S n)) -> + forall n : Z, A n. +Proof NZcentral_induction. + +(* Theorems that are true for integers but not for natural numbers *) + +Theorem Zpred_inj : forall n m : Z, P n == P m -> n == m. +Proof. +intros n m H. apply NZsucc_wd in H. now do 2 rewrite Zsucc_pred in H. +Qed. + +Theorem Zpred_inj_wd : forall n1 n2 : Z, P n1 == P n2 <-> n1 == n2. +Proof. +intros n1 n2; split; [apply Zpred_inj | apply NZpred_wd]. +Qed. + +End ZBasePropFunct. + diff --git a/theories/Numbers/Integer/Abstract/ZDomain.v b/theories/Numbers/Integer/Abstract/ZDomain.v new file mode 100644 index 00000000..15beb2b9 --- /dev/null +++ b/theories/Numbers/Integer/Abstract/ZDomain.v @@ -0,0 +1,69 @@ +(************************************************************************) +(* 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: ZDomain.v 10934 2008-05-15 21:58:20Z letouzey $ i*) + +Require Export NumPrelude. + +Module Type ZDomainSignature. + +Parameter Inline Z : Set. +Parameter Inline Zeq : Z -> Z -> Prop. +Parameter Inline e : Z -> Z -> bool. + +Axiom eq_equiv_e : forall x y : Z, Zeq x y <-> e x y. +Axiom eq_equiv : equiv Z Zeq. + +Add Relation Z Zeq + reflexivity proved by (proj1 eq_equiv) + symmetry proved by (proj2 (proj2 eq_equiv)) + transitivity proved by (proj1 (proj2 eq_equiv)) +as eq_rel. + +Delimit Scope IntScope with Int. +Bind Scope IntScope with Z. +Notation "x == y" := (Zeq x y) (at level 70) : IntScope. +Notation "x # y" := (~ Zeq x y) (at level 70) : IntScope. + +End ZDomainSignature. + +Module ZDomainProperties (Import ZDomainModule : ZDomainSignature). +Open Local Scope IntScope. + +Add Morphism e with signature Zeq ==> Zeq ==> eq_bool as e_wd. +Proof. +intros x x' Exx' y y' Eyy'. +case_eq (e x y); case_eq (e x' y'); intros H1 H2; trivial. +assert (x == y); [apply <- eq_equiv_e; now rewrite H2 | +assert (x' == y'); [rewrite <- Exx'; now rewrite <- Eyy' | +rewrite <- H1; assert (H3 : e x' y'); [now apply -> eq_equiv_e | now inversion H3]]]. +assert (x' == y'); [apply <- eq_equiv_e; now rewrite H1 | +assert (x == y); [rewrite Exx'; now rewrite Eyy' | +rewrite <- H2; assert (H3 : e x y); [now apply -> eq_equiv_e | now inversion H3]]]. +Qed. + +Theorem neq_symm : forall n m, n # m -> m # n. +Proof. +intros n m H1 H2; symmetry in H2; false_hyp H2 H1. +Qed. + +Theorem ZE_stepl : forall x y z : Z, x == y -> x == z -> z == y. +Proof. +intros x y z H1 H2; now rewrite <- H1. +Qed. + +Declare Left Step ZE_stepl. + +(* The right step lemma is just transitivity of Zeq *) +Declare Right Step (proj1 (proj2 eq_equiv)). + +End ZDomainProperties. + + diff --git a/theories/Numbers/Integer/Abstract/ZLt.v b/theories/Numbers/Integer/Abstract/ZLt.v new file mode 100644 index 00000000..2a88a535 --- /dev/null +++ b/theories/Numbers/Integer/Abstract/ZLt.v @@ -0,0 +1,432 @@ +(************************************************************************) +(* 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: ZLt.v 11040 2008-06-03 00:04:16Z letouzey $ i*) + +Require Export ZMul. + +Module ZOrderPropFunct (Import ZAxiomsMod : ZAxiomsSig). +Module Export ZMulPropMod := ZMulPropFunct ZAxiomsMod. +Open Local Scope IntScope. + +(* Axioms *) + +Theorem Zlt_wd : + forall n1 n2 : Z, n1 == n2 -> forall m1 m2 : Z, m1 == m2 -> (n1 < m1 <-> n2 < m2). +Proof NZlt_wd. + +Theorem Zle_wd : + forall n1 n2 : Z, n1 == n2 -> forall m1 m2 : Z, m1 == m2 -> (n1 <= m1 <-> n2 <= m2). +Proof NZle_wd. + +Theorem Zmin_wd : + forall n1 n2 : Z, n1 == n2 -> forall m1 m2 : Z, m1 == m2 -> Zmin n1 m1 == Zmin n2 m2. +Proof NZmin_wd. + +Theorem Zmax_wd : + forall n1 n2 : Z, n1 == n2 -> forall m1 m2 : Z, m1 == m2 -> Zmax n1 m1 == Zmax n2 m2. +Proof NZmax_wd. + +Theorem Zlt_eq_cases : forall n m : Z, n <= m <-> n < m \/ n == m. +Proof NZlt_eq_cases. + +Theorem Zlt_irrefl : forall n : Z, ~ n < n. +Proof NZlt_irrefl. + +Theorem Zlt_succ_r : forall n m : Z, n < S m <-> n <= m. +Proof NZlt_succ_r. + +Theorem Zmin_l : forall n m : Z, n <= m -> Zmin n m == n. +Proof NZmin_l. + +Theorem Zmin_r : forall n m : Z, m <= n -> Zmin n m == m. +Proof NZmin_r. + +Theorem Zmax_l : forall n m : Z, m <= n -> Zmax n m == n. +Proof NZmax_l. + +Theorem Zmax_r : forall n m : Z, n <= m -> Zmax n m == m. +Proof NZmax_r. + +(* Renaming theorems from NZOrder.v *) + +Theorem Zlt_le_incl : forall n m : Z, n < m -> n <= m. +Proof NZlt_le_incl. + +Theorem Zlt_neq : forall n m : Z, n < m -> n ~= m. +Proof NZlt_neq. + +Theorem Zle_neq : forall n m : Z, n < m <-> n <= m /\ n ~= m. +Proof NZle_neq. + +Theorem Zle_refl : forall n : Z, n <= n. +Proof NZle_refl. + +Theorem Zlt_succ_diag_r : forall n : Z, n < S n. +Proof NZlt_succ_diag_r. + +Theorem Zle_succ_diag_r : forall n : Z, n <= S n. +Proof NZle_succ_diag_r. + +Theorem Zlt_0_1 : 0 < 1. +Proof NZlt_0_1. + +Theorem Zle_0_1 : 0 <= 1. +Proof NZle_0_1. + +Theorem Zlt_lt_succ_r : forall n m : Z, n < m -> n < S m. +Proof NZlt_lt_succ_r. + +Theorem Zle_le_succ_r : forall n m : Z, n <= m -> n <= S m. +Proof NZle_le_succ_r. + +Theorem Zle_succ_r : forall n m : Z, n <= S m <-> n <= m \/ n == S m. +Proof NZle_succ_r. + +Theorem Zneq_succ_diag_l : forall n : Z, S n ~= n. +Proof NZneq_succ_diag_l. + +Theorem Zneq_succ_diag_r : forall n : Z, n ~= S n. +Proof NZneq_succ_diag_r. + +Theorem Znlt_succ_diag_l : forall n : Z, ~ S n < n. +Proof NZnlt_succ_diag_l. + +Theorem Znle_succ_diag_l : forall n : Z, ~ S n <= n. +Proof NZnle_succ_diag_l. + +Theorem Zle_succ_l : forall n m : Z, S n <= m <-> n < m. +Proof NZle_succ_l. + +Theorem Zlt_succ_l : forall n m : Z, S n < m -> n < m. +Proof NZlt_succ_l. + +Theorem Zsucc_lt_mono : forall n m : Z, n < m <-> S n < S m. +Proof NZsucc_lt_mono. + +Theorem Zsucc_le_mono : forall n m : Z, n <= m <-> S n <= S m. +Proof NZsucc_le_mono. + +Theorem Zlt_asymm : forall n m, n < m -> ~ m < n. +Proof NZlt_asymm. + +Notation Zlt_ngt := Zlt_asymm (only parsing). + +Theorem Zlt_trans : forall n m p : Z, n < m -> m < p -> n < p. +Proof NZlt_trans. + +Theorem Zle_trans : forall n m p : Z, n <= m -> m <= p -> n <= p. +Proof NZle_trans. + +Theorem Zle_lt_trans : forall n m p : Z, n <= m -> m < p -> n < p. +Proof NZle_lt_trans. + +Theorem Zlt_le_trans : forall n m p : Z, n < m -> m <= p -> n < p. +Proof NZlt_le_trans. + +Theorem Zle_antisymm : forall n m : Z, n <= m -> m <= n -> n == m. +Proof NZle_antisymm. + +Theorem Zlt_1_l : forall n m : Z, 0 < n -> n < m -> 1 < m. +Proof NZlt_1_l. + +(** Trichotomy, decidability, and double negation elimination *) + +Theorem Zlt_trichotomy : forall n m : Z, n < m \/ n == m \/ m < n. +Proof NZlt_trichotomy. + +Notation Zlt_eq_gt_cases := Zlt_trichotomy (only parsing). + +Theorem Zlt_gt_cases : forall n m : Z, n ~= m <-> n < m \/ n > m. +Proof NZlt_gt_cases. + +Theorem Zle_gt_cases : forall n m : Z, n <= m \/ n > m. +Proof NZle_gt_cases. + +Theorem Zlt_ge_cases : forall n m : Z, n < m \/ n >= m. +Proof NZlt_ge_cases. + +Theorem Zle_ge_cases : forall n m : Z, n <= m \/ n >= m. +Proof NZle_ge_cases. + +(** Instances of the previous theorems for m == 0 *) + +Theorem Zneg_pos_cases : forall n : Z, n ~= 0 <-> n < 0 \/ n > 0. +Proof. +intro; apply Zlt_gt_cases. +Qed. + +Theorem Znonpos_pos_cases : forall n : Z, n <= 0 \/ n > 0. +Proof. +intro; apply Zle_gt_cases. +Qed. + +Theorem Zneg_nonneg_cases : forall n : Z, n < 0 \/ n >= 0. +Proof. +intro; apply Zlt_ge_cases. +Qed. + +Theorem Znonpos_nonneg_cases : forall n : Z, n <= 0 \/ n >= 0. +Proof. +intro; apply Zle_ge_cases. +Qed. + +Theorem Zle_ngt : forall n m : Z, n <= m <-> ~ n > m. +Proof NZle_ngt. + +Theorem Znlt_ge : forall n m : Z, ~ n < m <-> n >= m. +Proof NZnlt_ge. + +Theorem Zlt_dec : forall n m : Z, decidable (n < m). +Proof NZlt_dec. + +Theorem Zlt_dne : forall n m, ~ ~ n < m <-> n < m. +Proof NZlt_dne. + +Theorem Znle_gt : forall n m : Z, ~ n <= m <-> n > m. +Proof NZnle_gt. + +Theorem Zlt_nge : forall n m : Z, n < m <-> ~ n >= m. +Proof NZlt_nge. + +Theorem Zle_dec : forall n m : Z, decidable (n <= m). +Proof NZle_dec. + +Theorem Zle_dne : forall n m : Z, ~ ~ n <= m <-> n <= m. +Proof NZle_dne. + +Theorem Znlt_succ_r : forall n m : Z, ~ m < S n <-> n < m. +Proof NZnlt_succ_r. + +Theorem Zlt_exists_pred : + forall z n : Z, z < n -> exists k : Z, n == S k /\ z <= k. +Proof NZlt_exists_pred. + +Theorem Zlt_succ_iter_r : + forall (n : nat) (m : Z), m < NZsucc_iter (Datatypes.S n) m. +Proof NZlt_succ_iter_r. + +Theorem Zneq_succ_iter_l : + forall (n : nat) (m : Z), NZsucc_iter (Datatypes.S n) m ~= m. +Proof NZneq_succ_iter_l. + +(** Stronger variant of induction with assumptions n >= 0 (n < 0) +in the induction step *) + +Theorem Zright_induction : + forall A : Z -> Prop, predicate_wd Zeq A -> + forall z : Z, A z -> + (forall n : Z, z <= n -> A n -> A (S n)) -> + forall n : Z, z <= n -> A n. +Proof NZright_induction. + +Theorem Zleft_induction : + forall A : Z -> Prop, predicate_wd Zeq A -> + forall z : Z, A z -> + (forall n : Z, n < z -> A (S n) -> A n) -> + forall n : Z, n <= z -> A n. +Proof NZleft_induction. + +Theorem Zright_induction' : + forall A : Z -> Prop, predicate_wd Zeq A -> + forall z : Z, + (forall n : Z, n <= z -> A n) -> + (forall n : Z, z <= n -> A n -> A (S n)) -> + forall n : Z, A n. +Proof NZright_induction'. + +Theorem Zleft_induction' : + forall A : Z -> Prop, predicate_wd Zeq A -> + forall z : Z, + (forall n : Z, z <= n -> A n) -> + (forall n : Z, n < z -> A (S n) -> A n) -> + forall n : Z, A n. +Proof NZleft_induction'. + +Theorem Zstrong_right_induction : + forall A : Z -> Prop, predicate_wd Zeq A -> + forall z : Z, + (forall n : Z, z <= n -> (forall m : Z, z <= m -> m < n -> A m) -> A n) -> + forall n : Z, z <= n -> A n. +Proof NZstrong_right_induction. + +Theorem Zstrong_left_induction : + forall A : Z -> Prop, predicate_wd Zeq A -> + forall z : Z, + (forall n : Z, n <= z -> (forall m : Z, m <= z -> S n <= m -> A m) -> A n) -> + forall n : Z, n <= z -> A n. +Proof NZstrong_left_induction. + +Theorem Zstrong_right_induction' : + forall A : Z -> Prop, predicate_wd Zeq A -> + forall z : Z, + (forall n : Z, n <= z -> A n) -> + (forall n : Z, z <= n -> (forall m : Z, z <= m -> m < n -> A m) -> A n) -> + forall n : Z, A n. +Proof NZstrong_right_induction'. + +Theorem Zstrong_left_induction' : + forall A : Z -> Prop, predicate_wd Zeq A -> + forall z : Z, + (forall n : Z, z <= n -> A n) -> + (forall n : Z, n <= z -> (forall m : Z, m <= z -> S n <= m -> A m) -> A n) -> + forall n : Z, A n. +Proof NZstrong_left_induction'. + +Theorem Zorder_induction : + forall A : Z -> Prop, predicate_wd Zeq A -> + forall z : Z, A z -> + (forall n : Z, z <= n -> A n -> A (S n)) -> + (forall n : Z, n < z -> A (S n) -> A n) -> + forall n : Z, A n. +Proof NZorder_induction. + +Theorem Zorder_induction' : + forall A : Z -> Prop, predicate_wd Zeq A -> + forall z : Z, A z -> + (forall n : Z, z <= n -> A n -> A (S n)) -> + (forall n : Z, n <= z -> A n -> A (P n)) -> + forall n : Z, A n. +Proof NZorder_induction'. + +Theorem Zorder_induction_0 : + forall A : Z -> Prop, predicate_wd Zeq A -> + A 0 -> + (forall n : Z, 0 <= n -> A n -> A (S n)) -> + (forall n : Z, n < 0 -> A (S n) -> A n) -> + forall n : Z, A n. +Proof NZorder_induction_0. + +Theorem Zorder_induction'_0 : + forall A : Z -> Prop, predicate_wd Zeq A -> + A 0 -> + (forall n : Z, 0 <= n -> A n -> A (S n)) -> + (forall n : Z, n <= 0 -> A n -> A (P n)) -> + forall n : Z, A n. +Proof NZorder_induction'_0. + +Ltac Zinduct n := induction_maker n ltac:(apply Zorder_induction_0). + +(** Elimintation principle for < *) + +Theorem Zlt_ind : + forall A : Z -> Prop, predicate_wd Zeq A -> + forall n : Z, A (S n) -> + (forall m : Z, n < m -> A m -> A (S m)) -> forall m : Z, n < m -> A m. +Proof NZlt_ind. + +(** Elimintation principle for <= *) + +Theorem Zle_ind : + forall A : Z -> Prop, predicate_wd Zeq A -> + forall n : Z, A n -> + (forall m : Z, n <= m -> A m -> A (S m)) -> forall m : Z, n <= m -> A m. +Proof NZle_ind. + +(** Well-founded relations *) + +Theorem Zlt_wf : forall z : Z, well_founded (fun n m : Z => z <= n /\ n < m). +Proof NZlt_wf. + +Theorem Zgt_wf : forall z : Z, well_founded (fun n m : Z => m < n /\ n <= z). +Proof NZgt_wf. + +(* Theorems that are either not valid on N or have different proofs on N and Z *) + +Theorem Zlt_pred_l : forall n : Z, P n < n. +Proof. +intro n; rewrite <- (Zsucc_pred n) at 2; apply Zlt_succ_diag_r. +Qed. + +Theorem Zle_pred_l : forall n : Z, P n <= n. +Proof. +intro; apply Zlt_le_incl; apply Zlt_pred_l. +Qed. + +Theorem Zlt_le_pred : forall n m : Z, n < m <-> n <= P m. +Proof. +intros n m; rewrite <- (Zsucc_pred m); rewrite Zpred_succ. apply Zlt_succ_r. +Qed. + +Theorem Znle_pred_r : forall n : Z, ~ n <= P n. +Proof. +intro; rewrite <- Zlt_le_pred; apply Zlt_irrefl. +Qed. + +Theorem Zlt_pred_le : forall n m : Z, P n < m <-> n <= m. +Proof. +intros n m; rewrite <- (Zsucc_pred n) at 2. +symmetry; apply Zle_succ_l. +Qed. + +Theorem Zlt_lt_pred : forall n m : Z, n < m -> P n < m. +Proof. +intros; apply <- Zlt_pred_le; now apply Zlt_le_incl. +Qed. + +Theorem Zle_le_pred : forall n m : Z, n <= m -> P n <= m. +Proof. +intros; apply Zlt_le_incl; now apply <- Zlt_pred_le. +Qed. + +Theorem Zlt_pred_lt : forall n m : Z, n < P m -> n < m. +Proof. +intros n m H; apply Zlt_trans with (P m); [assumption | apply Zlt_pred_l]. +Qed. + +Theorem Zle_pred_lt : forall n m : Z, n <= P m -> n <= m. +Proof. +intros; apply Zlt_le_incl; now apply <- Zlt_le_pred. +Qed. + +Theorem Zpred_lt_mono : forall n m : Z, n < m <-> P n < P m. +Proof. +intros; rewrite Zlt_le_pred; symmetry; apply Zlt_pred_le. +Qed. + +Theorem Zpred_le_mono : forall n m : Z, n <= m <-> P n <= P m. +Proof. +intros; rewrite <- Zlt_pred_le; now rewrite Zlt_le_pred. +Qed. + +Theorem Zlt_succ_lt_pred : forall n m : Z, S n < m <-> n < P m. +Proof. +intros n m; now rewrite (Zpred_lt_mono (S n) m), Zpred_succ. +Qed. + +Theorem Zle_succ_le_pred : forall n m : Z, S n <= m <-> n <= P m. +Proof. +intros n m; now rewrite (Zpred_le_mono (S n) m), Zpred_succ. +Qed. + +Theorem Zlt_pred_lt_succ : forall n m : Z, P n < m <-> n < S m. +Proof. +intros; rewrite Zlt_pred_le; symmetry; apply Zlt_succ_r. +Qed. + +Theorem Zle_pred_lt_succ : forall n m : Z, P n <= m <-> n <= S m. +Proof. +intros n m; now rewrite (Zpred_le_mono n (S m)), Zpred_succ. +Qed. + +Theorem Zneq_pred_l : forall n : Z, P n ~= n. +Proof. +intro; apply Zlt_neq; apply Zlt_pred_l. +Qed. + +Theorem Zlt_n1_r : forall n m : Z, n < m -> m < 0 -> n < -1. +Proof. +intros n m H1 H2. apply -> Zlt_le_pred in H2. +setoid_replace (P 0) with (-1) in H2. now apply NZlt_le_trans with m. +apply <- Zeq_opp_r. now rewrite Zopp_pred, Zopp_0. +Qed. + +End ZOrderPropFunct. + diff --git a/theories/Numbers/Integer/Abstract/ZMul.v b/theories/Numbers/Integer/Abstract/ZMul.v new file mode 100644 index 00000000..c48d1b4c --- /dev/null +++ b/theories/Numbers/Integer/Abstract/ZMul.v @@ -0,0 +1,115 @@ +(************************************************************************) +(* 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: ZMul.v 11040 2008-06-03 00:04:16Z letouzey $ i*) + +Require Export ZAdd. + +Module ZMulPropFunct (Import ZAxiomsMod : ZAxiomsSig). +Module Export ZAddPropMod := ZAddPropFunct ZAxiomsMod. +Open Local Scope IntScope. + +Theorem Zmul_wd : + forall n1 n2 : Z, n1 == n2 -> forall m1 m2 : Z, m1 == m2 -> n1 * m1 == n2 * m2. +Proof NZmul_wd. + +Theorem Zmul_0_l : forall n : Z, 0 * n == 0. +Proof NZmul_0_l. + +Theorem Zmul_succ_l : forall n m : Z, (S n) * m == n * m + m. +Proof NZmul_succ_l. + +(* Theorems that are valid for both natural numbers and integers *) + +Theorem Zmul_0_r : forall n : Z, n * 0 == 0. +Proof NZmul_0_r. + +Theorem Zmul_succ_r : forall n m : Z, n * (S m) == n * m + n. +Proof NZmul_succ_r. + +Theorem Zmul_comm : forall n m : Z, n * m == m * n. +Proof NZmul_comm. + +Theorem Zmul_add_distr_r : forall n m p : Z, (n + m) * p == n * p + m * p. +Proof NZmul_add_distr_r. + +Theorem Zmul_add_distr_l : forall n m p : Z, n * (m + p) == n * m + n * p. +Proof NZmul_add_distr_l. + +(* A note on naming: right (correspondingly, left) distributivity happens +when the sum is multiplied by a number on the right (left), not when the +sum itself is the right (left) factor in the product (see planetmath.org +and mathworld.wolfram.com). In the old library BinInt, distributivity over +subtraction was named correctly, but distributivity over addition was named +incorrectly. The names in Isabelle/HOL library are also incorrect. *) + +Theorem Zmul_assoc : forall n m p : Z, n * (m * p) == (n * m) * p. +Proof NZmul_assoc. + +Theorem Zmul_1_l : forall n : Z, 1 * n == n. +Proof NZmul_1_l. + +Theorem Zmul_1_r : forall n : Z, n * 1 == n. +Proof NZmul_1_r. + +(* The following two theorems are true in an ordered ring, +but since they don't mention order, we'll put them here *) + +Theorem Zeq_mul_0 : forall n m : Z, n * m == 0 <-> n == 0 \/ m == 0. +Proof NZeq_mul_0. + +Theorem Zneq_mul_0 : forall n m : Z, n ~= 0 /\ m ~= 0 <-> n * m ~= 0. +Proof NZneq_mul_0. + +(* Theorems that are either not valid on N or have different proofs on N and Z *) + +Theorem Zmul_pred_r : forall n m : Z, n * (P m) == n * m - n. +Proof. +intros n m. +rewrite <- (Zsucc_pred m) at 2. +now rewrite Zmul_succ_r, <- Zadd_sub_assoc, Zsub_diag, Zadd_0_r. +Qed. + +Theorem Zmul_pred_l : forall n m : Z, (P n) * m == n * m - m. +Proof. +intros n m; rewrite (Zmul_comm (P n) m), (Zmul_comm n m). apply Zmul_pred_r. +Qed. + +Theorem Zmul_opp_l : forall n m : Z, (- n) * m == - (n * m). +Proof. +intros n m. apply -> Zadd_move_0_r. +now rewrite <- Zmul_add_distr_r, Zadd_opp_diag_l, Zmul_0_l. +Qed. + +Theorem Zmul_opp_r : forall n m : Z, n * (- m) == - (n * m). +Proof. +intros n m; rewrite (Zmul_comm n (- m)), (Zmul_comm n m); apply Zmul_opp_l. +Qed. + +Theorem Zmul_opp_opp : forall n m : Z, (- n) * (- m) == n * m. +Proof. +intros n m; now rewrite Zmul_opp_l, Zmul_opp_r, Zopp_involutive. +Qed. + +Theorem Zmul_sub_distr_l : forall n m p : Z, n * (m - p) == n * m - n * p. +Proof. +intros n m p. do 2 rewrite <- Zadd_opp_r. rewrite Zmul_add_distr_l. +now rewrite Zmul_opp_r. +Qed. + +Theorem Zmul_sub_distr_r : forall n m p : Z, (n - m) * p == n * p - m * p. +Proof. +intros n m p; rewrite (Zmul_comm (n - m) p), (Zmul_comm n p), (Zmul_comm m p); +now apply Zmul_sub_distr_l. +Qed. + +End ZMulPropFunct. + + diff --git a/theories/Numbers/Integer/Abstract/ZMulOrder.v b/theories/Numbers/Integer/Abstract/ZMulOrder.v new file mode 100644 index 00000000..e3f1d9aa --- /dev/null +++ b/theories/Numbers/Integer/Abstract/ZMulOrder.v @@ -0,0 +1,343 @@ +(************************************************************************) +(* 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: ZMulOrder.v 11040 2008-06-03 00:04:16Z letouzey $ i*) + +Require Export ZAddOrder. + +Module ZMulOrderPropFunct (Import ZAxiomsMod : ZAxiomsSig). +Module Export ZAddOrderPropMod := ZAddOrderPropFunct ZAxiomsMod. +Open Local Scope IntScope. + +Theorem Zmul_lt_pred : + forall p q n m : Z, S p == q -> (p * n < p * m <-> q * n + m < q * m + n). +Proof NZmul_lt_pred. + +Theorem Zmul_lt_mono_pos_l : forall p n m : Z, 0 < p -> (n < m <-> p * n < p * m). +Proof NZmul_lt_mono_pos_l. + +Theorem Zmul_lt_mono_pos_r : forall p n m : Z, 0 < p -> (n < m <-> n * p < m * p). +Proof NZmul_lt_mono_pos_r. + +Theorem Zmul_lt_mono_neg_l : forall p n m : Z, p < 0 -> (n < m <-> p * m < p * n). +Proof NZmul_lt_mono_neg_l. + +Theorem Zmul_lt_mono_neg_r : forall p n m : Z, p < 0 -> (n < m <-> m * p < n * p). +Proof NZmul_lt_mono_neg_r. + +Theorem Zmul_le_mono_nonneg_l : forall n m p : Z, 0 <= p -> n <= m -> p * n <= p * m. +Proof NZmul_le_mono_nonneg_l. + +Theorem Zmul_le_mono_nonpos_l : forall n m p : Z, p <= 0 -> n <= m -> p * m <= p * n. +Proof NZmul_le_mono_nonpos_l. + +Theorem Zmul_le_mono_nonneg_r : forall n m p : Z, 0 <= p -> n <= m -> n * p <= m * p. +Proof NZmul_le_mono_nonneg_r. + +Theorem Zmul_le_mono_nonpos_r : forall n m p : Z, p <= 0 -> n <= m -> m * p <= n * p. +Proof NZmul_le_mono_nonpos_r. + +Theorem Zmul_cancel_l : forall n m p : Z, p ~= 0 -> (p * n == p * m <-> n == m). +Proof NZmul_cancel_l. + +Theorem Zmul_cancel_r : forall n m p : Z, p ~= 0 -> (n * p == m * p <-> n == m). +Proof NZmul_cancel_r. + +Theorem Zmul_id_l : forall n m : Z, m ~= 0 -> (n * m == m <-> n == 1). +Proof NZmul_id_l. + +Theorem Zmul_id_r : forall n m : Z, n ~= 0 -> (n * m == n <-> m == 1). +Proof NZmul_id_r. + +Theorem Zmul_le_mono_pos_l : forall n m p : Z, 0 < p -> (n <= m <-> p * n <= p * m). +Proof NZmul_le_mono_pos_l. + +Theorem Zmul_le_mono_pos_r : forall n m p : Z, 0 < p -> (n <= m <-> n * p <= m * p). +Proof NZmul_le_mono_pos_r. + +Theorem Zmul_le_mono_neg_l : forall n m p : Z, p < 0 -> (n <= m <-> p * m <= p * n). +Proof NZmul_le_mono_neg_l. + +Theorem Zmul_le_mono_neg_r : forall n m p : Z, p < 0 -> (n <= m <-> m * p <= n * p). +Proof NZmul_le_mono_neg_r. + +Theorem Zmul_lt_mono_nonneg : + forall n m p q : Z, 0 <= n -> n < m -> 0 <= p -> p < q -> n * p < m * q. +Proof NZmul_lt_mono_nonneg. + +Theorem Zmul_lt_mono_nonpos : + forall n m p q : Z, m <= 0 -> n < m -> q <= 0 -> p < q -> m * q < n * p. +Proof. +intros n m p q H1 H2 H3 H4. +apply Zle_lt_trans with (m * p). +apply Zmul_le_mono_nonpos_l; [assumption | now apply Zlt_le_incl]. +apply -> Zmul_lt_mono_neg_r; [assumption | now apply Zlt_le_trans with q]. +Qed. + +Theorem Zmul_le_mono_nonneg : + forall n m p q : Z, 0 <= n -> n <= m -> 0 <= p -> p <= q -> n * p <= m * q. +Proof NZmul_le_mono_nonneg. + +Theorem Zmul_le_mono_nonpos : + forall n m p q : Z, m <= 0 -> n <= m -> q <= 0 -> p <= q -> m * q <= n * p. +Proof. +intros n m p q H1 H2 H3 H4. +apply Zle_trans with (m * p). +now apply Zmul_le_mono_nonpos_l. +apply Zmul_le_mono_nonpos_r; [now apply Zle_trans with q | assumption]. +Qed. + +Theorem Zmul_pos_pos : forall n m : Z, 0 < n -> 0 < m -> 0 < n * m. +Proof NZmul_pos_pos. + +Theorem Zmul_neg_neg : forall n m : Z, n < 0 -> m < 0 -> 0 < n * m. +Proof NZmul_neg_neg. + +Theorem Zmul_pos_neg : forall n m : Z, 0 < n -> m < 0 -> n * m < 0. +Proof NZmul_pos_neg. + +Theorem Zmul_neg_pos : forall n m : Z, n < 0 -> 0 < m -> n * m < 0. +Proof NZmul_neg_pos. + +Theorem Zmul_nonneg_nonneg : forall n m : Z, 0 <= n -> 0 <= m -> 0 <= n * m. +Proof. +intros n m H1 H2. +rewrite <- (Zmul_0_l m). now apply Zmul_le_mono_nonneg_r. +Qed. + +Theorem Zmul_nonpos_nonpos : forall n m : Z, n <= 0 -> m <= 0 -> 0 <= n * m. +Proof. +intros n m H1 H2. +rewrite <- (Zmul_0_l m). now apply Zmul_le_mono_nonpos_r. +Qed. + +Theorem Zmul_nonneg_nonpos : forall n m : Z, 0 <= n -> m <= 0 -> n * m <= 0. +Proof. +intros n m H1 H2. +rewrite <- (Zmul_0_l m). now apply Zmul_le_mono_nonpos_r. +Qed. + +Theorem Zmul_nonpos_nonneg : forall n m : Z, n <= 0 -> 0 <= m -> n * m <= 0. +Proof. +intros; rewrite Zmul_comm; now apply Zmul_nonneg_nonpos. +Qed. + +Theorem Zlt_1_mul_pos : forall n m : Z, 1 < n -> 0 < m -> 1 < n * m. +Proof NZlt_1_mul_pos. + +Theorem Zeq_mul_0 : forall n m : Z, n * m == 0 <-> n == 0 \/ m == 0. +Proof NZeq_mul_0. + +Theorem Zneq_mul_0 : forall n m : Z, n ~= 0 /\ m ~= 0 <-> n * m ~= 0. +Proof NZneq_mul_0. + +Theorem Zeq_square_0 : forall n : Z, n * n == 0 <-> n == 0. +Proof NZeq_square_0. + +Theorem Zeq_mul_0_l : forall n m : Z, n * m == 0 -> m ~= 0 -> n == 0. +Proof NZeq_mul_0_l. + +Theorem Zeq_mul_0_r : forall n m : Z, n * m == 0 -> n ~= 0 -> m == 0. +Proof NZeq_mul_0_r. + +Theorem Zlt_0_mul : forall n m : Z, 0 < n * m <-> 0 < n /\ 0 < m \/ m < 0 /\ n < 0. +Proof NZlt_0_mul. + +Notation Zmul_pos := Zlt_0_mul (only parsing). + +Theorem Zlt_mul_0 : + forall n m : Z, n * m < 0 <-> n < 0 /\ m > 0 \/ n > 0 /\ m < 0. +Proof. +intros n m; split; [intro H | intros [[H1 H2] | [H1 H2]]]. +destruct (Zlt_trichotomy n 0) as [H1 | [H1 | H1]]; +[| rewrite H1 in H; rewrite Zmul_0_l in H; false_hyp H Zlt_irrefl |]; +(destruct (Zlt_trichotomy m 0) as [H2 | [H2 | H2]]; +[| rewrite H2 in H; rewrite Zmul_0_r in H; false_hyp H Zlt_irrefl |]); +try (left; now split); try (right; now split). +assert (H3 : n * m > 0) by now apply Zmul_neg_neg. +elimtype False; now apply (Zlt_asymm (n * m) 0). +assert (H3 : n * m > 0) by now apply Zmul_pos_pos. +elimtype False; now apply (Zlt_asymm (n * m) 0). +now apply Zmul_neg_pos. now apply Zmul_pos_neg. +Qed. + +Notation Zmul_neg := Zlt_mul_0 (only parsing). + +Theorem Zle_0_mul : + forall n m : Z, 0 <= n * m -> 0 <= n /\ 0 <= m \/ n <= 0 /\ m <= 0. +Proof. +assert (R : forall n : Z, 0 == n <-> n == 0) by (intros; split; apply Zeq_symm). +intros n m. repeat rewrite Zlt_eq_cases. repeat rewrite R. +rewrite Zlt_0_mul, Zeq_mul_0. +pose proof (Zlt_trichotomy n 0); pose proof (Zlt_trichotomy m 0). tauto. +Qed. + +Notation Zmul_nonneg := Zle_0_mul (only parsing). + +Theorem Zle_mul_0 : + forall n m : Z, n * m <= 0 -> 0 <= n /\ m <= 0 \/ n <= 0 /\ 0 <= m. +Proof. +assert (R : forall n : Z, 0 == n <-> n == 0) by (intros; split; apply Zeq_symm). +intros n m. repeat rewrite Zlt_eq_cases. repeat rewrite R. +rewrite Zlt_mul_0, Zeq_mul_0. +pose proof (Zlt_trichotomy n 0); pose proof (Zlt_trichotomy m 0). tauto. +Qed. + +Notation Zmul_nonpos := Zle_mul_0 (only parsing). + +Theorem Zle_0_square : forall n : Z, 0 <= n * n. +Proof. +intro n; destruct (Zneg_nonneg_cases n). +apply Zlt_le_incl; now apply Zmul_neg_neg. +now apply Zmul_nonneg_nonneg. +Qed. + +Notation Zsquare_nonneg := Zle_0_square (only parsing). + +Theorem Znlt_square_0 : forall n : Z, ~ n * n < 0. +Proof. +intros n H. apply -> Zlt_nge in H. apply H. apply Zsquare_nonneg. +Qed. + +Theorem Zsquare_lt_mono_nonneg : forall n m : Z, 0 <= n -> n < m -> n * n < m * m. +Proof NZsquare_lt_mono_nonneg. + +Theorem Zsquare_lt_mono_nonpos : forall n m : Z, n <= 0 -> m < n -> n * n < m * m. +Proof. +intros n m H1 H2. now apply Zmul_lt_mono_nonpos. +Qed. + +Theorem Zsquare_le_mono_nonneg : forall n m : Z, 0 <= n -> n <= m -> n * n <= m * m. +Proof NZsquare_le_mono_nonneg. + +Theorem Zsquare_le_mono_nonpos : forall n m : Z, n <= 0 -> m <= n -> n * n <= m * m. +Proof. +intros n m H1 H2. now apply Zmul_le_mono_nonpos. +Qed. + +Theorem Zsquare_lt_simpl_nonneg : forall n m : Z, 0 <= m -> n * n < m * m -> n < m. +Proof NZsquare_lt_simpl_nonneg. + +Theorem Zsquare_le_simpl_nonneg : forall n m : Z, 0 <= m -> n * n <= m * m -> n <= m. +Proof NZsquare_le_simpl_nonneg. + +Theorem Zsquare_lt_simpl_nonpos : forall n m : Z, m <= 0 -> n * n < m * m -> m < n. +Proof. +intros n m H1 H2. destruct (Zle_gt_cases n 0). +destruct (NZlt_ge_cases m n). +assumption. assert (F : m * m <= n * n) by now apply Zsquare_le_mono_nonpos. +apply -> NZle_ngt in F. false_hyp H2 F. +now apply Zle_lt_trans with 0. +Qed. + +Theorem Zsquare_le_simpl_nonpos : forall n m : NZ, m <= 0 -> n * n <= m * m -> m <= n. +Proof. +intros n m H1 H2. destruct (NZle_gt_cases n 0). +destruct (NZle_gt_cases m n). +assumption. assert (F : m * m < n * n) by now apply Zsquare_lt_mono_nonpos. +apply -> NZlt_nge in F. false_hyp H2 F. +apply Zlt_le_incl; now apply NZle_lt_trans with 0. +Qed. + +Theorem Zmul_2_mono_l : forall n m : Z, n < m -> 1 + (1 + 1) * n < (1 + 1) * m. +Proof NZmul_2_mono_l. + +Theorem Zlt_1_mul_neg : forall n m : Z, n < -1 -> m < 0 -> 1 < n * m. +Proof. +intros n m H1 H2. apply -> (NZmul_lt_mono_neg_r m) in H1. +apply <- Zopp_pos_neg in H2. rewrite Zmul_opp_l, Zmul_1_l in H1. +now apply Zlt_1_l with (- m). +assumption. +Qed. + +Theorem Zlt_mul_n1_neg : forall n m : Z, 1 < n -> m < 0 -> n * m < -1. +Proof. +intros n m H1 H2. apply -> (NZmul_lt_mono_neg_r m) in H1. +rewrite Zmul_1_l in H1. now apply Zlt_n1_r with m. +assumption. +Qed. + +Theorem Zlt_mul_n1_pos : forall n m : Z, n < -1 -> 0 < m -> n * m < -1. +Proof. +intros n m H1 H2. apply -> (NZmul_lt_mono_pos_r m) in H1. +rewrite Zmul_opp_l, Zmul_1_l in H1. +apply <- Zopp_neg_pos in H2. now apply Zlt_n1_r with (- m). +assumption. +Qed. + +Theorem Zlt_1_mul_l : forall n m : Z, 1 < n -> n * m < -1 \/ n * m == 0 \/ 1 < n * m. +Proof. +intros n m H; destruct (Zlt_trichotomy m 0) as [H1 | [H1 | H1]]. +left. now apply Zlt_mul_n1_neg. +right; left; now rewrite H1, Zmul_0_r. +right; right; now apply Zlt_1_mul_pos. +Qed. + +Theorem Zlt_n1_mul_r : forall n m : Z, n < -1 -> n * m < -1 \/ n * m == 0 \/ 1 < n * m. +Proof. +intros n m H; destruct (Zlt_trichotomy m 0) as [H1 | [H1 | H1]]. +right; right. now apply Zlt_1_mul_neg. +right; left; now rewrite H1, Zmul_0_r. +left. now apply Zlt_mul_n1_pos. +Qed. + +Theorem Zeq_mul_1 : forall n m : Z, n * m == 1 -> n == 1 \/ n == -1. +Proof. +assert (F : ~ 1 < -1). +intro H. +assert (H1 : -1 < 0). apply <- Zopp_neg_pos. apply Zlt_succ_diag_r. +assert (H2 : 1 < 0) by now apply Zlt_trans with (-1). false_hyp H2 Znlt_succ_diag_l. +Z0_pos_neg n. +intros m H; rewrite Zmul_0_l in H; false_hyp H Zneq_succ_diag_r. +intros n H; split; apply <- Zle_succ_l in H; le_elim H. +intros m H1; apply (Zlt_1_mul_l n m) in H. +rewrite H1 in H; destruct H as [H | [H | H]]. +false_hyp H F. false_hyp H Zneq_succ_diag_l. false_hyp H Zlt_irrefl. +intros; now left. +intros m H1; apply (Zlt_1_mul_l n m) in H. rewrite Zmul_opp_l in H1; +apply -> Zeq_opp_l in H1. rewrite H1 in H; destruct H as [H | [H | H]]. +false_hyp H Zlt_irrefl. apply -> Zeq_opp_l in H. rewrite Zopp_0 in H. +false_hyp H Zneq_succ_diag_l. false_hyp H F. +intros; right; symmetry; now apply Zopp_wd. +Qed. + +Theorem Zlt_mul_diag_l : forall n m : Z, n < 0 -> (1 < m <-> n * m < n). +Proof. +intros n m H. stepr (n * m < n * 1) by now rewrite Zmul_1_r. +now apply Zmul_lt_mono_neg_l. +Qed. + +Theorem Zlt_mul_diag_r : forall n m : Z, 0 < n -> (1 < m <-> n < n * m). +Proof. +intros n m H. stepr (n * 1 < n * m) by now rewrite Zmul_1_r. +now apply Zmul_lt_mono_pos_l. +Qed. + +Theorem Zle_mul_diag_l : forall n m : Z, n < 0 -> (1 <= m <-> n * m <= n). +Proof. +intros n m H. stepr (n * m <= n * 1) by now rewrite Zmul_1_r. +now apply Zmul_le_mono_neg_l. +Qed. + +Theorem Zle_mul_diag_r : forall n m : Z, 0 < n -> (1 <= m <-> n <= n * m). +Proof. +intros n m H. stepr (n * 1 <= n * m) by now rewrite Zmul_1_r. +now apply Zmul_le_mono_pos_l. +Qed. + +Theorem Zlt_mul_r : forall n m p : Z, 0 < n -> 1 < p -> n < m -> n < m * p. +Proof. +intros. stepl (n * 1) by now rewrite Zmul_1_r. +apply Zmul_lt_mono_nonneg. +now apply Zlt_le_incl. assumption. apply Zle_0_1. assumption. +Qed. + +End ZMulOrderPropFunct. + |