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diff --git a/contrib/omega/OmegaLemmas.v b/contrib/omega/OmegaLemmas.v new file mode 100644 index 00000000..6f0ea2c6 --- /dev/null +++ b/contrib/omega/OmegaLemmas.v @@ -0,0 +1,269 @@ +(************************************************************************) +(* 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 *) +(************************************************************************) + +(*i $Id: OmegaLemmas.v,v 1.4.2.1 2004/07/16 19:30:12 herbelin Exp $ i*) + +Require Import ZArith_base. + +(** These are specific variants of theorems dedicated for the Omega tactic *) + +Lemma new_var : forall x:Z, exists y : Z, x = y. +intros x; exists x; trivial with arith. +Qed. + +Lemma OMEGA1 : forall x y:Z, x = y -> (0 <= x)%Z -> (0 <= y)%Z. +intros x y H; rewrite H; auto with arith. +Qed. + +Lemma OMEGA2 : forall x y:Z, (0 <= x)%Z -> (0 <= y)%Z -> (0 <= x + y)%Z. +exact Zplus_le_0_compat. +Qed. + +Lemma OMEGA3 : + forall x y k:Z, (k > 0)%Z -> x = (y * k)%Z -> x = 0%Z -> y = 0%Z. + +intros x y k H1 H2 H3; apply (Zmult_integral_l k); + [ unfold not in |- *; intros H4; absurd (k > 0)%Z; + [ rewrite H4; unfold Zgt in |- *; simpl in |- *; discriminate + | assumption ] + | rewrite <- H2; assumption ]. +Qed. + +Lemma OMEGA4 : forall x y z:Z, (x > 0)%Z -> (y > x)%Z -> (z * y + x)%Z <> 0%Z. + +unfold not in |- *; intros x y z H1 H2 H3; cut (y > 0)%Z; + [ intros H4; cut (0 <= z * y + x)%Z; + [ intros H5; generalize (Zmult_le_approx y z x H4 H2 H5); intros H6; + absurd (z * y + x > 0)%Z; + [ rewrite H3; unfold Zgt in |- *; simpl in |- *; discriminate + | apply Zle_gt_trans with x; + [ pattern x at 1 in |- *; rewrite <- (Zplus_0_l x); + apply Zplus_le_compat_r; rewrite Zmult_comm; + generalize H4; unfold Zgt in |- *; case y; + [ simpl in |- *; intros H7; discriminate H7 + | intros p H7; rewrite <- (Zmult_0_r (Zpos p)); + unfold Zle in |- *; rewrite Zcompare_mult_compat; + exact H6 + | simpl in |- *; intros p H7; discriminate H7 ] + | assumption ] ] + | rewrite H3; unfold Zle in |- *; simpl in |- *; discriminate ] + | apply Zgt_trans with x; [ assumption | assumption ] ]. +Qed. + +Lemma OMEGA5 : forall x y z:Z, x = 0%Z -> y = 0%Z -> (x + y * z)%Z = 0%Z. + +intros x y z H1 H2; rewrite H1; rewrite H2; simpl in |- *; trivial with arith. +Qed. + +Lemma OMEGA6 : forall x y z:Z, (0 <= x)%Z -> y = 0%Z -> (0 <= x + y * z)%Z. + +intros x y z H1 H2; rewrite H2; simpl in |- *; rewrite Zplus_0_r; assumption. +Qed. + +Lemma OMEGA7 : + forall x y z t:Z, + (z > 0)%Z -> + (t > 0)%Z -> (0 <= x)%Z -> (0 <= y)%Z -> (0 <= x * z + y * t)%Z. + +intros x y z t H1 H2 H3 H4; rewrite <- (Zplus_0_l 0); apply Zplus_le_compat; + apply Zmult_gt_0_le_0_compat; assumption. +Qed. + +Lemma OMEGA8 : + forall x y:Z, (0 <= x)%Z -> (0 <= y)%Z -> x = (- y)%Z -> x = 0%Z. + +intros x y H1 H2 H3; elim (Zle_lt_or_eq 0 x H1); + [ intros H4; absurd (0 < x)%Z; + [ change (0 >= x)%Z in |- *; apply Zle_ge; apply Zplus_le_reg_l with y; + rewrite H3; rewrite Zplus_opp_r; rewrite Zplus_0_r; + assumption + | assumption ] + | intros H4; rewrite H4; trivial with arith ]. +Qed. + +Lemma OMEGA9 : + forall x y z t:Z, y = 0%Z -> x = z -> (y + (- x + z) * t)%Z = 0%Z. + +intros x y z t H1 H2; rewrite H2; rewrite Zplus_opp_l; rewrite Zmult_0_l; + rewrite Zplus_0_r; assumption. +Qed. + +Lemma OMEGA10 : + forall v c1 c2 l1 l2 k1 k2:Z, + ((v * c1 + l1) * k1 + (v * c2 + l2) * k2)%Z = + (v * (c1 * k1 + c2 * k2) + (l1 * k1 + l2 * k2))%Z. + +intros; repeat rewrite Zmult_plus_distr_l || rewrite Zmult_plus_distr_r; + repeat rewrite Zmult_assoc; repeat elim Zplus_assoc; + rewrite (Zplus_permute (l1 * k1) (v * c2 * k2)); trivial with arith. +Qed. + +Lemma OMEGA11 : + forall v1 c1 l1 l2 k1:Z, + ((v1 * c1 + l1) * k1 + l2)%Z = (v1 * (c1 * k1) + (l1 * k1 + l2))%Z. + +intros; repeat rewrite Zmult_plus_distr_l || rewrite Zmult_plus_distr_r; + repeat rewrite Zmult_assoc; repeat elim Zplus_assoc; + trivial with arith. +Qed. + +Lemma OMEGA12 : + forall v2 c2 l1 l2 k2:Z, + (l1 + (v2 * c2 + l2) * k2)%Z = (v2 * (c2 * k2) + (l1 + l2 * k2))%Z. + +intros; repeat rewrite Zmult_plus_distr_l || rewrite Zmult_plus_distr_r; + repeat rewrite Zmult_assoc; repeat elim Zplus_assoc; + rewrite Zplus_permute; trivial with arith. +Qed. + +Lemma OMEGA13 : + forall (v l1 l2:Z) (x:positive), + (v * Zpos x + l1 + (v * Zneg x + l2))%Z = (l1 + l2)%Z. + +intros; rewrite Zplus_assoc; rewrite (Zplus_comm (v * Zpos x) l1); + rewrite (Zplus_assoc_reverse l1); rewrite <- Zmult_plus_distr_r; + rewrite <- Zopp_neg; rewrite (Zplus_comm (- Zneg x) (Zneg x)); + rewrite Zplus_opp_r; rewrite Zmult_0_r; rewrite Zplus_0_r; + trivial with arith. +Qed. + +Lemma OMEGA14 : + forall (v l1 l2:Z) (x:positive), + (v * Zneg x + l1 + (v * Zpos x + l2))%Z = (l1 + l2)%Z. + +intros; rewrite Zplus_assoc; rewrite (Zplus_comm (v * Zneg x) l1); + rewrite (Zplus_assoc_reverse l1); rewrite <- Zmult_plus_distr_r; + rewrite <- Zopp_neg; rewrite Zplus_opp_r; rewrite Zmult_0_r; + rewrite Zplus_0_r; trivial with arith. +Qed. +Lemma OMEGA15 : + forall v c1 c2 l1 l2 k2:Z, + (v * c1 + l1 + (v * c2 + l2) * k2)%Z = + (v * (c1 + c2 * k2) + (l1 + l2 * k2))%Z. + +intros; repeat rewrite Zmult_plus_distr_l || rewrite Zmult_plus_distr_r; + repeat rewrite Zmult_assoc; repeat elim Zplus_assoc; + rewrite (Zplus_permute l1 (v * c2 * k2)); trivial with arith. +Qed. + +Lemma OMEGA16 : + forall v c l k:Z, ((v * c + l) * k)%Z = (v * (c * k) + l * k)%Z. + +intros; repeat rewrite Zmult_plus_distr_l || rewrite Zmult_plus_distr_r; + repeat rewrite Zmult_assoc; repeat elim Zplus_assoc; + trivial with arith. +Qed. + +Lemma OMEGA17 : forall x y z:Z, Zne x 0 -> y = 0%Z -> Zne (x + y * z) 0. + +unfold Zne, not in |- *; intros x y z H1 H2 H3; apply H1; + apply Zplus_reg_l with (y * z)%Z; rewrite Zplus_comm; + rewrite H3; rewrite H2; auto with arith. +Qed. + +Lemma OMEGA18 : forall x y k:Z, x = (y * k)%Z -> Zne x 0 -> Zne y 0. + +unfold Zne, not in |- *; intros x y k H1 H2 H3; apply H2; rewrite H1; + rewrite H3; auto with arith. +Qed. + +Lemma OMEGA19 : + forall x:Z, Zne x 0 -> (0 <= x + -1)%Z \/ (0 <= x * -1 + -1)%Z. + +unfold Zne in |- *; intros x H; elim (Zle_or_lt 0 x); + [ intros H1; elim Zle_lt_or_eq with (1 := H1); + [ intros H2; left; change (0 <= Zpred x)%Z in |- *; apply Zsucc_le_reg; + rewrite <- Zsucc_pred; apply Zlt_le_succ; assumption + | intros H2; absurd (x = 0%Z); auto with arith ] + | intros H1; right; rewrite <- Zopp_eq_mult_neg_1; rewrite Zplus_comm; + apply Zle_left; apply Zsucc_le_reg; simpl in |- *; + apply Zlt_le_succ; auto with arith ]. +Qed. + +Lemma OMEGA20 : forall x y z:Z, Zne x 0 -> y = 0%Z -> Zne (x + y * z) 0. + +unfold Zne, not in |- *; intros x y z H1 H2 H3; apply H1; rewrite H2 in H3; + simpl in H3; rewrite Zplus_0_r in H3; trivial with arith. +Qed. + +Definition fast_Zplus_sym (x y:Z) (P:Z -> Prop) (H:P (y + x)%Z) := + eq_ind_r P H (Zplus_comm x y). + +Definition fast_Zplus_assoc_r (n m p:Z) (P:Z -> Prop) + (H:P (n + (m + p))%Z) := eq_ind_r P H (Zplus_assoc_reverse n m p). + +Definition fast_Zplus_assoc_l (n m p:Z) (P:Z -> Prop) + (H:P (n + m + p)%Z) := eq_ind_r P H (Zplus_assoc n m p). + +Definition fast_Zplus_permute (n m p:Z) (P:Z -> Prop) + (H:P (m + (n + p))%Z) := eq_ind_r P H (Zplus_permute n m p). + +Definition fast_OMEGA10 (v c1 c2 l1 l2 k1 k2:Z) (P:Z -> Prop) + (H:P (v * (c1 * k1 + c2 * k2) + (l1 * k1 + l2 * k2))%Z) := + eq_ind_r P H (OMEGA10 v c1 c2 l1 l2 k1 k2). + +Definition fast_OMEGA11 (v1 c1 l1 l2 k1:Z) (P:Z -> Prop) + (H:P (v1 * (c1 * k1) + (l1 * k1 + l2))%Z) := + eq_ind_r P H (OMEGA11 v1 c1 l1 l2 k1). +Definition fast_OMEGA12 (v2 c2 l1 l2 k2:Z) (P:Z -> Prop) + (H:P (v2 * (c2 * k2) + (l1 + l2 * k2))%Z) := + eq_ind_r P H (OMEGA12 v2 c2 l1 l2 k2). + +Definition fast_OMEGA15 (v c1 c2 l1 l2 k2:Z) (P:Z -> Prop) + (H:P (v * (c1 + c2 * k2) + (l1 + l2 * k2))%Z) := + eq_ind_r P H (OMEGA15 v c1 c2 l1 l2 k2). +Definition fast_OMEGA16 (v c l k:Z) (P:Z -> Prop) + (H:P (v * (c * k) + l * k)%Z) := eq_ind_r P H (OMEGA16 v c l k). + +Definition fast_OMEGA13 (v l1 l2:Z) (x:positive) (P:Z -> Prop) + (H:P (l1 + l2)%Z) := eq_ind_r P H (OMEGA13 v l1 l2 x). + +Definition fast_OMEGA14 (v l1 l2:Z) (x:positive) (P:Z -> Prop) + (H:P (l1 + l2)%Z) := eq_ind_r P H (OMEGA14 v l1 l2 x). +Definition fast_Zred_factor0 (x:Z) (P:Z -> Prop) (H:P (x * 1)%Z) := + eq_ind_r P H (Zred_factor0 x). + +Definition fast_Zopp_one (x:Z) (P:Z -> Prop) (H:P (x * -1)%Z) := + eq_ind_r P H (Zopp_eq_mult_neg_1 x). + +Definition fast_Zmult_sym (x y:Z) (P:Z -> Prop) (H:P (y * x)%Z) := + eq_ind_r P H (Zmult_comm x y). + +Definition fast_Zopp_Zplus (x y:Z) (P:Z -> Prop) (H:P (- x + - y)%Z) := + eq_ind_r P H (Zopp_plus_distr x y). + +Definition fast_Zopp_Zopp (x:Z) (P:Z -> Prop) (H:P x) := + eq_ind_r P H (Zopp_involutive x). + +Definition fast_Zopp_Zmult_r (x y:Z) (P:Z -> Prop) + (H:P (x * - y)%Z) := eq_ind_r P H (Zopp_mult_distr_r x y). + +Definition fast_Zmult_plus_distr (n m p:Z) (P:Z -> Prop) + (H:P (n * p + m * p)%Z) := eq_ind_r P H (Zmult_plus_distr_l n m p). +Definition fast_Zmult_Zopp_left (x y:Z) (P:Z -> Prop) + (H:P (x * - y)%Z) := eq_ind_r P H (Zmult_opp_comm x y). + +Definition fast_Zmult_assoc_r (n m p:Z) (P:Z -> Prop) + (H:P (n * (m * p))%Z) := eq_ind_r P H (Zmult_assoc_reverse n m p). + +Definition fast_Zred_factor1 (x:Z) (P:Z -> Prop) (H:P (x * 2)%Z) := + eq_ind_r P H (Zred_factor1 x). + +Definition fast_Zred_factor2 (x y:Z) (P:Z -> Prop) + (H:P (x * (1 + y))%Z) := eq_ind_r P H (Zred_factor2 x y). +Definition fast_Zred_factor3 (x y:Z) (P:Z -> Prop) + (H:P (x * (1 + y))%Z) := eq_ind_r P H (Zred_factor3 x y). + +Definition fast_Zred_factor4 (x y z:Z) (P:Z -> Prop) + (H:P (x * (y + z))%Z) := eq_ind_r P H (Zred_factor4 x y z). + +Definition fast_Zred_factor5 (x y:Z) (P:Z -> Prop) + (H:P y) := eq_ind_r P H (Zred_factor5 x y). + +Definition fast_Zred_factor6 (x:Z) (P:Z -> Prop) (H:P (x + 0)%Z) := + eq_ind_r P H (Zred_factor6 x). |