diff options
Diffstat (limited to 'theories/omega/OmegaLemmas.v')
| -rw-r--r-- | theories/omega/OmegaLemmas.v | 307 |
1 files changed, 307 insertions, 0 deletions
diff --git a/theories/omega/OmegaLemmas.v b/theories/omega/OmegaLemmas.v new file mode 100644 index 0000000000..d2378569fc --- /dev/null +++ b/theories/omega/OmegaLemmas.v @@ -0,0 +1,307 @@ +(************************************************************************) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2019 *) +(* <O___,, * (see CREDITS file for the list of authors) *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) +(************************************************************************) + +Require Import BinInt Znat. +Local Open Scope Z_scope. + +(** Factorization lemmas *) + +Theorem Zred_factor0 n : n = n * 1. +Proof. + now Z.nzsimpl. +Qed. + +Theorem Zred_factor1 n : n + n = n * 2. +Proof. + rewrite Z.mul_comm. apply Z.add_diag. +Qed. + +Theorem Zred_factor2 n m : n + n * m = n * (1 + m). +Proof. + rewrite Z.mul_add_distr_l; now Z.nzsimpl. +Qed. + +Theorem Zred_factor3 n m : n * m + n = n * (1 + m). +Proof. + now Z.nzsimpl. +Qed. + +Theorem Zred_factor4 n m p : n * m + n * p = n * (m + p). +Proof. + symmetry; apply Z.mul_add_distr_l. +Qed. + +Theorem Zred_factor5 n m : n * 0 + m = m. +Proof. + now Z.nzsimpl. +Qed. + +Theorem Zred_factor6 n : n = n + 0. +Proof. + now Z.nzsimpl. +Qed. + +(** Other specific variants of theorems dedicated for the Omega tactic *) + +Lemma new_var : forall x : Z, exists y : Z, x = y. +Proof. +intros x; now exists x. +Qed. + +Lemma OMEGA1 x y : x = y -> 0 <= x -> 0 <= y. +Proof. +now intros ->. +Qed. + +Lemma OMEGA2 x y : 0 <= x -> 0 <= y -> 0 <= x + y. +Proof. +Z.order_pos. +Qed. + +Lemma OMEGA3 x y k : k > 0 -> x = y * k -> x = 0 -> y = 0. +Proof. +intros LT -> EQ. apply Z.mul_eq_0 in EQ. destruct EQ; now subst. +Qed. + +Lemma OMEGA4 x y z : x > 0 -> y > x -> z * y + x <> 0. +Proof. +Z.swap_greater. intros Hx Hxy. +rewrite Z.add_move_0_l, <- Z.mul_opp_l. +destruct (Z.lt_trichotomy (-z) 1) as [LT|[->|GT]]. +- intro. revert LT. apply Z.le_ngt, (Z.le_succ_l 0). + apply Z.mul_pos_cancel_r with y; Z.order. +- Z.nzsimpl. Z.order. +- rewrite (Z.mul_lt_mono_pos_r y), Z.mul_1_l in GT; Z.order. +Qed. + +Lemma OMEGA5 x y z : x = 0 -> y = 0 -> x + y * z = 0. +Proof. +now intros -> ->. +Qed. + +Lemma OMEGA6 x y z : 0 <= x -> y = 0 -> 0 <= x + y * z. +Proof. +intros H ->. now Z.nzsimpl. +Qed. + +Lemma OMEGA7 x y z t : + z > 0 -> t > 0 -> 0 <= x -> 0 <= y -> 0 <= x * z + y * t. +Proof. +intros. Z.swap_greater. Z.order_pos. +Qed. + +Lemma OMEGA8 x y : 0 <= x -> 0 <= y -> x = - y -> x = 0. +Proof. +intros H1 H2 H3. rewrite <- Z.opp_nonpos_nonneg in H2. Z.order. +Qed. + +Lemma OMEGA9 x y z t : y = 0 -> x = z -> y + (- x + z) * t = 0. +Proof. +intros. subst. now rewrite Z.add_opp_diag_l. +Qed. + +Lemma OMEGA10 v c1 c2 l1 l2 k1 k2 : + (v * c1 + l1) * k1 + (v * c2 + l2) * k2 = + v * (c1 * k1 + c2 * k2) + (l1 * k1 + l2 * k2). +Proof. +rewrite ?Z.mul_add_distr_r, ?Z.mul_add_distr_l, ?Z.mul_assoc. +rewrite <- !Z.add_assoc. f_equal. apply Z.add_shuffle3. +Qed. + +Lemma OMEGA11 v1 c1 l1 l2 k1 : + (v1 * c1 + l1) * k1 + l2 = v1 * (c1 * k1) + (l1 * k1 + l2). +Proof. +rewrite ?Z.mul_add_distr_r, ?Z.mul_add_distr_l, ?Z.mul_assoc. +now rewrite Z.add_assoc. +Qed. + +Lemma OMEGA12 v2 c2 l1 l2 k2 : + l1 + (v2 * c2 + l2) * k2 = v2 * (c2 * k2) + (l1 + l2 * k2). +Proof. +rewrite ?Z.mul_add_distr_r, ?Z.mul_add_distr_l, ?Z.mul_assoc. +apply Z.add_shuffle3. +Qed. + +Lemma OMEGA13 (v l1 l2 : Z) (x : positive) : + v * Zpos x + l1 + (v * Zneg x + l2) = l1 + l2. +Proof. + rewrite Z.add_shuffle1. + rewrite <- Z.mul_add_distr_l, <- Pos2Z.opp_neg, Z.add_opp_diag_r. + now Z.nzsimpl. +Qed. + +Lemma OMEGA14 (v l1 l2 : Z) (x : positive) : + v * Zneg x + l1 + (v * Zpos x + l2) = l1 + l2. +Proof. + rewrite Z.add_shuffle1. + rewrite <- Z.mul_add_distr_l, <- Pos2Z.opp_neg, Z.add_opp_diag_r. + now Z.nzsimpl. +Qed. + +Lemma OMEGA15 v c1 c2 l1 l2 k2 : + v * c1 + l1 + (v * c2 + l2) * k2 = v * (c1 + c2 * k2) + (l1 + l2 * k2). +Proof. + rewrite ?Z.mul_add_distr_r, ?Z.mul_add_distr_l, ?Z.mul_assoc. + apply Z.add_shuffle1. +Qed. + +Lemma OMEGA16 v c l k : (v * c + l) * k = v * (c * k) + l * k. +Proof. + now rewrite ?Z.mul_add_distr_r, ?Z.mul_add_distr_l, ?Z.mul_assoc. +Qed. + +Lemma OMEGA17 x y z : Zne x 0 -> y = 0 -> Zne (x + y * z) 0. +Proof. + unfold Zne, not. intros NE EQ. subst. now Z.nzsimpl. +Qed. + +Lemma OMEGA18 x y k : x = y * k -> Zne x 0 -> Zne y 0. +Proof. + unfold Zne, not. intros. subst; auto. +Qed. + +Lemma OMEGA19 x : Zne x 0 -> 0 <= x + -1 \/ 0 <= x * -1 + -1. +Proof. + unfold Zne. intros Hx. apply Z.lt_gt_cases in Hx. + destruct Hx as [LT|GT]. + - right. change (-1) with (-(1)). + rewrite Z.mul_opp_r, <- Z.opp_add_distr. Z.nzsimpl. + rewrite Z.opp_nonneg_nonpos. now apply Z.le_succ_l. + - left. now apply Z.lt_le_pred. +Qed. + +Lemma OMEGA20 x y z : Zne x 0 -> y = 0 -> Zne (x + y * z) 0. +Proof. + unfold Zne, not. intros H1 H2 H3; apply H1; rewrite H2 in H3; + simpl in H3; rewrite Z.add_0_r in H3; trivial with arith. +Qed. + +Definition fast_Zplus_comm (x y : Z) (P : Z -> Prop) + (H : P (y + x)) := eq_ind_r P H (Z.add_comm x y). + +Definition fast_Zplus_assoc_reverse (n m p : Z) (P : Z -> Prop) + (H : P (n + (m + p))) := eq_ind_r P H (Zplus_assoc_reverse n m p). + +Definition fast_Zplus_assoc (n m p : Z) (P : Z -> Prop) + (H : P (n + m + p)) := eq_ind_r P H (Z.add_assoc n m p). + +Definition fast_Zplus_permute (n m p : Z) (P : Z -> Prop) + (H : P (m + (n + p))) := eq_ind_r P H (Z.add_shuffle3 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))) := + 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))) := + 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))) := + 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))) := + 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)) := 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)) := 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)) := eq_ind_r P H (OMEGA14 v l1 l2 x). +Definition fast_Zred_factor0 (x : Z) (P : Z -> Prop) + (H : P (x * 1)) := eq_ind_r P H (Zred_factor0 x). + +Definition fast_Zopp_eq_mult_neg_1 (x : Z) (P : Z -> Prop) + (H : P (x * -1)) := eq_ind_r P H (Z.opp_eq_mul_m1 x). + +Definition fast_Zmult_comm (x y : Z) (P : Z -> Prop) + (H : P (y * x)) := eq_ind_r P H (Z.mul_comm x y). + +Definition fast_Zopp_plus_distr (x y : Z) (P : Z -> Prop) + (H : P (- x + - y)) := eq_ind_r P H (Z.opp_add_distr x y). + +Definition fast_Zopp_mult_distr_r (x y : Z) (P : Z -> Prop) + (H : P (x * - y)) := eq_ind_r P H (Zopp_mult_distr_r x y). + +Definition fast_Zmult_plus_distr_l (n m p : Z) (P : Z -> Prop) + (H : P (n * p + m * p)) := eq_ind_r P H (Z.mul_add_distr_r n m p). +Definition fast_Zmult_assoc_reverse (n m p : Z) (P : Z -> Prop) + (H : P (n * (m * p))) := eq_ind_r P H (Zmult_assoc_reverse n m p). + +Definition fast_Zred_factor1 (x : Z) (P : Z -> Prop) + (H : P (x * 2)) := eq_ind_r P H (Zred_factor1 x). + +Definition fast_Zred_factor2 (x y : Z) (P : Z -> Prop) + (H : P (x * (1 + y))) := eq_ind_r P H (Zred_factor2 x y). + +Definition fast_Zred_factor3 (x y : Z) (P : Z -> Prop) + (H : P (x * (1 + y))) := 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))) := 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)) := eq_ind_r P H (Zred_factor6 x). + +Theorem intro_Z : + forall n:nat, exists y : Z, Z.of_nat n = y /\ 0 <= y * 1 + 0. +Proof. + intros n; exists (Z.of_nat n); split; trivial. + rewrite Z.mul_1_r, Z.add_0_r. apply Nat2Z.is_nonneg. +Qed. + +Register fast_Zplus_assoc_reverse as plugins.omega.fast_Zplus_assoc_reverse. +Register fast_Zplus_assoc as plugins.omega.fast_Zplus_assoc. +Register fast_Zmult_assoc_reverse as plugins.omega.fast_Zmult_assoc_reverse. +Register fast_Zplus_permute as plugins.omega.fast_Zplus_permute. +Register fast_Zplus_comm as plugins.omega.fast_Zplus_comm. +Register fast_Zmult_comm as plugins.omega.fast_Zmult_comm. + +Register OMEGA1 as plugins.omega.OMEGA1. +Register OMEGA2 as plugins.omega.OMEGA2. +Register OMEGA3 as plugins.omega.OMEGA3. +Register OMEGA4 as plugins.omega.OMEGA4. +Register OMEGA5 as plugins.omega.OMEGA5. +Register OMEGA6 as plugins.omega.OMEGA6. +Register OMEGA7 as plugins.omega.OMEGA7. +Register OMEGA8 as plugins.omega.OMEGA8. +Register OMEGA9 as plugins.omega.OMEGA9. +Register fast_OMEGA10 as plugins.omega.fast_OMEGA10. +Register fast_OMEGA11 as plugins.omega.fast_OMEGA11. +Register fast_OMEGA12 as plugins.omega.fast_OMEGA12. +Register fast_OMEGA13 as plugins.omega.fast_OMEGA13. +Register fast_OMEGA14 as plugins.omega.fast_OMEGA14. +Register fast_OMEGA15 as plugins.omega.fast_OMEGA15. +Register fast_OMEGA16 as plugins.omega.fast_OMEGA16. +Register OMEGA17 as plugins.omega.OMEGA17. +Register OMEGA18 as plugins.omega.OMEGA18. +Register OMEGA19 as plugins.omega.OMEGA19. +Register OMEGA20 as plugins.omega.OMEGA20. + +Register fast_Zred_factor0 as plugins.omega.fast_Zred_factor0. +Register fast_Zred_factor1 as plugins.omega.fast_Zred_factor1. +Register fast_Zred_factor2 as plugins.omega.fast_Zred_factor2. +Register fast_Zred_factor3 as plugins.omega.fast_Zred_factor3. +Register fast_Zred_factor4 as plugins.omega.fast_Zred_factor4. +Register fast_Zred_factor5 as plugins.omega.fast_Zred_factor5. +Register fast_Zred_factor6 as plugins.omega.fast_Zred_factor6. + +Register fast_Zmult_plus_distr_l as plugins.omega.fast_Zmult_plus_distr_l. +Register fast_Zopp_plus_distr as plugins.omega.fast_Zopp_plus_distr. +Register fast_Zopp_mult_distr_r as plugins.omega.fast_Zopp_mult_distr_r. +Register fast_Zopp_eq_mult_neg_1 as plugins.omega.fast_Zopp_eq_mult_neg_1. + +Register new_var as plugins.omega.new_var. +Register intro_Z as plugins.omega.intro_Z. |