diff options
Diffstat (limited to 'theories/ZArith/auxiliary.v')
-rw-r--r-- | theories/ZArith/auxiliary.v | 118 |
1 files changed, 60 insertions, 58 deletions
diff --git a/theories/ZArith/auxiliary.v b/theories/ZArith/auxiliary.v index 28cbd1e4..726fb45a 100644 --- a/theories/ZArith/auxiliary.v +++ b/theories/ZArith/auxiliary.v @@ -6,11 +6,11 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: auxiliary.v 5920 2004-07-16 20:01:26Z herbelin $ i*) +(*i $Id: auxiliary.v 9302 2006-10-27 21:21:17Z barras $ i*) (** Binary Integers (Pierre Crégut, CNET, Lannion, France) *) -Require Export Arith. +Require Export Arith_base. Require Import BinInt. Require Import Zorder. Require Import Decidable. @@ -19,132 +19,134 @@ Require Export Compare_dec. Open Local Scope Z_scope. -(**********************************************************************) -(** Moving terms from one side to the other of an inequality *) +(***************************************************************) +(** * Moving terms from one side to the other of an inequality *) Theorem Zne_left : forall n m:Z, Zne n m -> Zne (n + - m) 0. Proof. -intros x y; unfold Zne in |- *; unfold not in |- *; intros H1 H2; apply H1; - apply Zplus_reg_l with (- y); rewrite Zplus_opp_l; - rewrite Zplus_comm; trivial with arith. + intros x y; unfold Zne in |- *; unfold not in |- *; intros H1 H2; apply H1; + apply Zplus_reg_l with (- y); rewrite Zplus_opp_l; + rewrite Zplus_comm; trivial with arith. Qed. Theorem Zegal_left : forall n m:Z, n = m -> n + - m = 0. Proof. -intros x y H; apply (Zplus_reg_l y); rewrite Zplus_permute; - rewrite Zplus_opp_r; do 2 rewrite Zplus_0_r; assumption. + intros x y H; apply (Zplus_reg_l y); rewrite Zplus_permute; + rewrite Zplus_opp_r; do 2 rewrite Zplus_0_r; assumption. Qed. Theorem Zle_left : forall n m:Z, n <= m -> 0 <= m + - n. Proof. -intros x y H; replace 0 with (x + - x). -apply Zplus_le_compat_r; trivial. -apply Zplus_opp_r. + intros x y H; replace 0 with (x + - x). + apply Zplus_le_compat_r; trivial. + apply Zplus_opp_r. Qed. Theorem Zle_left_rev : forall n m:Z, 0 <= m + - n -> n <= m. Proof. -intros x y H; apply Zplus_le_reg_r with (- x). -rewrite Zplus_opp_r; trivial. + intros x y H; apply Zplus_le_reg_r with (- x). + rewrite Zplus_opp_r; trivial. Qed. Theorem Zlt_left_rev : forall n m:Z, 0 < m + - n -> n < m. Proof. -intros x y H; apply Zplus_lt_reg_r with (- x). -rewrite Zplus_opp_r; trivial. + intros x y H; apply Zplus_lt_reg_r with (- x). + rewrite Zplus_opp_r; trivial. Qed. Theorem Zlt_left : forall n m:Z, n < m -> 0 <= m + -1 + - n. Proof. -intros x y H; apply Zle_left; apply Zsucc_le_reg; - change (Zsucc x <= Zsucc (Zpred y)) in |- *; rewrite <- Zsucc_pred; - apply Zlt_le_succ; assumption. + intros x y H; apply Zle_left; apply Zsucc_le_reg; + change (Zsucc x <= Zsucc (Zpred y)) in |- *; rewrite <- Zsucc_pred; + apply Zlt_le_succ; assumption. Qed. Theorem Zlt_left_lt : forall n m:Z, n < m -> 0 < m + - n. Proof. -intros x y H; replace 0 with (x + - x). -apply Zplus_lt_compat_r; trivial. -apply Zplus_opp_r. + intros x y H; replace 0 with (x + - x). + apply Zplus_lt_compat_r; trivial. + apply Zplus_opp_r. Qed. Theorem Zge_left : forall n m:Z, n >= m -> 0 <= n + - m. Proof. -intros x y H; apply Zle_left; apply Zge_le; assumption. + intros x y H; apply Zle_left; apply Zge_le; assumption. Qed. Theorem Zgt_left : forall n m:Z, n > m -> 0 <= n + -1 + - m. Proof. -intros x y H; apply Zlt_left; apply Zgt_lt; assumption. + intros x y H; apply Zlt_left; apply Zgt_lt; assumption. Qed. Theorem Zgt_left_gt : forall n m:Z, n > m -> n + - m > 0. Proof. -intros x y H; replace 0 with (y + - y). -apply Zplus_gt_compat_r; trivial. -apply Zplus_opp_r. + intros x y H; replace 0 with (y + - y). + apply Zplus_gt_compat_r; trivial. + apply Zplus_opp_r. Qed. Theorem Zgt_left_rev : forall n m:Z, n + - m > 0 -> n > m. Proof. -intros x y H; apply Zplus_gt_reg_r with (- y). -rewrite Zplus_opp_r; trivial. + intros x y H; apply Zplus_gt_reg_r with (- y). + rewrite Zplus_opp_r; trivial. Qed. (**********************************************************************) -(** Factorization lemmas *) +(** * Factorization lemmas *) Theorem Zred_factor0 : forall n:Z, n = n * 1. -intro x; rewrite (Zmult_1_r x); reflexivity. + intro x; rewrite (Zmult_1_r x); reflexivity. Qed. Theorem Zred_factor1 : forall n:Z, n + n = n * 2. Proof. -exact Zplus_diag_eq_mult_2. + exact Zplus_diag_eq_mult_2. Qed. Theorem Zred_factor2 : forall n m:Z, n + n * m = n * (1 + m). - -intros x y; pattern x at 1 in |- *; rewrite <- (Zmult_1_r x); - rewrite <- Zmult_plus_distr_r; trivial with arith. +Proof. + intros x y; pattern x at 1 in |- *; rewrite <- (Zmult_1_r x); + rewrite <- Zmult_plus_distr_r; trivial with arith. Qed. Theorem Zred_factor3 : forall n m:Z, n * m + n = n * (1 + m). - -intros x y; pattern x at 2 in |- *; rewrite <- (Zmult_1_r x); - rewrite <- Zmult_plus_distr_r; rewrite Zplus_comm; - trivial with arith. +Proof. + intros x y; pattern x at 2 in |- *; rewrite <- (Zmult_1_r x); + rewrite <- Zmult_plus_distr_r; rewrite Zplus_comm; + trivial with arith. Qed. + Theorem Zred_factor4 : forall n m p:Z, n * m + n * p = n * (m + p). -intros x y z; symmetry in |- *; apply Zmult_plus_distr_r. +Proof. + intros x y z; symmetry in |- *; apply Zmult_plus_distr_r. Qed. Theorem Zred_factor5 : forall n m:Z, n * 0 + m = m. - -intros x y; rewrite <- Zmult_0_r_reverse; auto with arith. +Proof. + intros x y; rewrite <- Zmult_0_r_reverse; auto with arith. Qed. Theorem Zred_factor6 : forall n:Z, n = n + 0. - -intro; rewrite Zplus_0_r; trivial with arith. +Proof. + intro; rewrite Zplus_0_r; trivial with arith. Qed. Theorem Zle_mult_approx : - forall n m p:Z, n > 0 -> p > 0 -> 0 <= m -> 0 <= m * n + p. - -intros x y z H1 H2 H3; apply Zle_trans with (m := y * x); - [ apply Zmult_gt_0_le_0_compat; assumption - | pattern (y * x) at 1 in |- *; rewrite <- Zplus_0_r; - apply Zplus_le_compat_l; apply Zlt_le_weak; apply Zgt_lt; - assumption ]. + forall n m p:Z, n > 0 -> p > 0 -> 0 <= m -> 0 <= m * n + p. +Proof. + intros x y z H1 H2 H3; apply Zle_trans with (m := y * x); + [ apply Zmult_gt_0_le_0_compat; assumption + | pattern (y * x) at 1 in |- *; rewrite <- Zplus_0_r; + apply Zplus_le_compat_l; apply Zlt_le_weak; apply Zgt_lt; + assumption ]. Qed. Theorem Zmult_le_approx : - forall n m p:Z, n > 0 -> n > p -> 0 <= m * n + p -> 0 <= m. - -intros x y z H1 H2 H3; apply Zlt_succ_le; apply Zmult_gt_0_lt_0_reg_r with x; - [ assumption - | apply Zle_lt_trans with (1 := H3); rewrite <- Zmult_succ_l_reverse; - apply Zplus_lt_compat_l; apply Zgt_lt; assumption ]. - + forall n m p:Z, n > 0 -> n > p -> 0 <= m * n + p -> 0 <= m. +Proof. + intros x y z H1 H2 H3; apply Zlt_succ_le; apply Zmult_gt_0_lt_0_reg_r with x; + [ assumption + | apply Zle_lt_trans with (1 := H3); rewrite <- Zmult_succ_l_reverse; + apply Zplus_lt_compat_l; apply Zgt_lt; assumption ]. Qed. + |