diff options
Diffstat (limited to 'theories/Numbers/Cyclic')
| -rw-r--r-- | theories/Numbers/Cyclic/Abstract/CyclicAxioms.v | 46 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/Abstract/NZCyclic.v | 14 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v | 50 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v | 115 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v | 30 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v | 272 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v | 80 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v | 186 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v | 88 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v | 393 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v | 26 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v | 4 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/Int31/Cyclic31.v | 609 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/Int31/Int31.v | 18 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/Int31/Ring31.v | 4 | ||||
| -rw-r--r-- | theories/Numbers/Cyclic/ZModulo/ZModulo.v | 191 |
16 files changed, 1047 insertions, 1079 deletions
diff --git a/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v b/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v index 59656eed..9a8a7691 100644 --- a/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v +++ b/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -111,7 +111,7 @@ Module ZnZ. (* Conversion functions with Z *) spec_to_Z : forall x, 0 <= [| x |] < wB; spec_of_pos : forall p, - Zpos p = (Z_of_N (fst (of_pos p)))*wB + [|(snd (of_pos p))|]; + Zpos p = (Z.of_N (fst (of_pos p)))*wB + [|(snd (of_pos p))|]; spec_zdigits : [| zdigits |] = Zpos digits; spec_more_than_1_digit: 1 < Zpos digits; @@ -284,11 +284,11 @@ Module ZnZ. generalize (spec_of_pos p). case (of_pos p); intros n w1; simpl. case n; simpl Npos; auto with zarith. - intros p1 Hp1; contradict Hp; apply Zle_not_lt. + intros p1 Hp1; contradict Hp; apply Z.le_ngt. replace (base digits) with (1 * base digits + 0) by ring. rewrite Hp1. - apply Zplus_le_compat. - apply Zmult_le_compat; auto with zarith. + apply Z.add_le_mono. + apply Z.mul_le_mono_nonneg; auto with zarith. case p1; simpl; intros; red; simpl; intros; discriminate. unfold base; auto with zarith. case (spec_to_Z w1); auto with zarith. @@ -305,7 +305,7 @@ Module ZnZ. Proof. intros p; case p; simpl; try rewrite spec_0; auto. intros; rewrite of_pos_correct; auto with zarith. - intros p1 (H1, _); contradict H1; apply Zlt_not_le; red; simpl; auto. + intros p1 (H1, _); contradict H1; apply Z.lt_nge; red; simpl; auto. Qed. End Of_Z. @@ -346,46 +346,46 @@ Ltac zify := unfold eq in *; autorewrite with cyclic. Lemma add_0_l : forall x, 0 + x == x. Proof. -intros. zify. rewrite Zplus_0_l. +intros. zify. rewrite Z.add_0_l. apply Zmod_small. apply ZnZ.spec_to_Z. Qed. Lemma add_comm : forall x y, x + y == y + x. Proof. -intros. zify. now rewrite Zplus_comm. +intros. zify. now rewrite Z.add_comm. Qed. Lemma add_assoc : forall x y z, x + (y + z) == x + y + z. Proof. -intros. zify. now rewrite Zplus_mod_idemp_r, Zplus_mod_idemp_l, Zplus_assoc. +intros. zify. now rewrite Zplus_mod_idemp_r, Zplus_mod_idemp_l, Z.add_assoc. Qed. Lemma mul_1_l : forall x, 1 * x == x. Proof. -intros. zify. rewrite Zmult_1_l. +intros. zify. rewrite Z.mul_1_l. apply Zmod_small. apply ZnZ.spec_to_Z. Qed. Lemma mul_comm : forall x y, x * y == y * x. Proof. -intros. zify. now rewrite Zmult_comm. +intros. zify. now rewrite Z.mul_comm. Qed. Lemma mul_assoc : forall x y z, x * (y * z) == x * y * z. Proof. -intros. zify. now rewrite Zmult_mod_idemp_r, Zmult_mod_idemp_l, Zmult_assoc. +intros. zify. now rewrite Zmult_mod_idemp_r, Zmult_mod_idemp_l, Z.mul_assoc. Qed. Lemma mul_add_distr_r : forall x y z, (x+y)*z == x*z + y*z. Proof. -intros. zify. now rewrite <- Zplus_mod, Zmult_mod_idemp_l, Zmult_plus_distr_l. +intros. zify. now rewrite <- Zplus_mod, Zmult_mod_idemp_l, Z.mul_add_distr_r. Qed. Lemma add_opp_r : forall x y, x + - y == x-y. Proof. -intros. zify. rewrite <- Zminus_mod_idemp_r. unfold Zminus. -destruct (Z_eq_dec ([|y|] mod wB) 0) as [EQ|NEQ]. -rewrite Z_mod_zero_opp_full, EQ, 2 Zplus_0_r; auto. +intros. zify. rewrite <- Zminus_mod_idemp_r. unfold Z.sub. +destruct (Z.eq_dec ([|y|] mod wB) 0) as [EQ|NEQ]. +rewrite Z_mod_zero_opp_full, EQ, 2 Z.add_0_r; auto. rewrite Z_mod_nz_opp_full by auto. rewrite <- Zplus_mod_idemp_r, <- Zminus_mod_idemp_l. rewrite Z_mod_same_full. simpl. now rewrite Zplus_mod_idemp_r. @@ -393,7 +393,7 @@ Qed. Lemma add_opp_diag_r : forall x, x + - x == 0. Proof. -intros. red. rewrite add_opp_r. zify. now rewrite Zminus_diag, Zmod_0_l. +intros. red. rewrite add_opp_r. zify. now rewrite Z.sub_diag, Zmod_0_l. Qed. Lemma CyclicRing : ring_theory 0 1 ZnZ.add ZnZ.mul ZnZ.sub ZnZ.opp eq. @@ -413,19 +413,9 @@ Lemma eqb_eq : forall x y, eqb x y = true <-> x == y. Proof. intros. unfold eqb, eq. rewrite ZnZ.spec_compare. - case Zcompare_spec; intuition; try discriminate. + case Z.compare_spec; intuition; try discriminate. Qed. -(* POUR HUGO: -Lemma eqb_eq : forall x y, eqb x y = true <-> x == y. -Proof. - intros. unfold eqb, eq. generalize (ZnZ.spec_compare x y). - case (ZnZ.compare x y); intuition; try discriminate. - (* BUG ?! using destruct instead of case won't work: - it gives 3 subcases, but ZnZ.compare x y is still there in them! *) -Qed. -*) - Lemma eqb_correct : forall x y, eqb x y = true -> x==y. Proof. now apply eqb_eq. Qed. diff --git a/theories/Numbers/Cyclic/Abstract/NZCyclic.v b/theories/Numbers/Cyclic/Abstract/NZCyclic.v index c52cbe10..1d5b78ec 100644 --- a/theories/Numbers/Cyclic/Abstract/NZCyclic.v +++ b/theories/Numbers/Cyclic/Abstract/NZCyclic.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -69,7 +69,7 @@ Program Instance mul_wd : Proper (eq ==> eq ==> eq) mul. Theorem gt_wB_1 : 1 < wB. Proof. -unfold base. apply Zpower_gt_1; unfold Zlt; auto with zarith. +unfold base. apply Zpower_gt_1; unfold Z.lt; auto with zarith. Qed. Theorem gt_wB_0 : 0 < wB. @@ -161,20 +161,20 @@ End Induction. Theorem add_0_l : forall n, 0 + n == n. Proof. intro n. zify. -rewrite Zplus_0_l. apply Zmod_small. apply ZnZ.spec_to_Z. +rewrite Z.add_0_l. apply Zmod_small. apply ZnZ.spec_to_Z. Qed. Theorem add_succ_l : forall n m, (S n) + m == S (n + m). Proof. intros n m. zify. rewrite succ_mod_wB. repeat rewrite Zplus_mod_idemp_l; try apply gt_wB_0. -rewrite <- (Zplus_assoc ([| n |] mod wB) 1 [| m |]). rewrite Zplus_mod_idemp_l. -rewrite (Zplus_comm 1 [| m |]); now rewrite Zplus_assoc. +rewrite <- (Z.add_assoc ([| n |] mod wB) 1 [| m |]). rewrite Zplus_mod_idemp_l. +rewrite (Z.add_comm 1 [| m |]); now rewrite Z.add_assoc. Qed. Theorem sub_0_r : forall n, n - 0 == n. Proof. -intro n. zify. rewrite Zminus_0_r. apply NZ_to_Z_mod. +intro n. zify. rewrite Z.sub_0_r. apply NZ_to_Z_mod. Qed. Theorem sub_succ_r : forall n m, n - (S m) == P (n - m). @@ -192,7 +192,7 @@ Qed. Theorem mul_succ_l : forall n m, (S n) * m == n * m + m. Proof. intros n m. zify. rewrite Zplus_mod_idemp_l, Zmult_mod_idemp_l. -now rewrite Zmult_plus_distr_l, Zmult_1_l. +now rewrite Z.mul_add_distr_r, Z.mul_1_l. Qed. Definition t := t. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v index deb216dd..35d8b595 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -182,7 +182,7 @@ Section DoubleAdd. destruct x as [ |xh xl];simpl. apply spec_ww_1. generalize (spec_w_succ_c xl);destruct (w_succ_c xl) as [l|l]; intro H;unfold interp_carry in H. simpl;rewrite H;ring. - rewrite <- Zplus_assoc;rewrite <- H;rewrite Zmult_1_l. + rewrite <- Z.add_assoc;rewrite <- H;rewrite Z.mul_1_l. assert ([|l|] = 0). generalize (spec_to_Z xl)(spec_to_Z l);omega. rewrite H0;generalize (spec_w_succ_c xh);destruct (w_succ_c xh) as [h|h]; intro H1;unfold interp_carry in H1. @@ -195,19 +195,19 @@ Section DoubleAdd. Lemma spec_ww_add_c : forall x y, [+[ww_add_c x y]] = [[x]] + [[y]]. Proof. destruct x as [ |xh xl];simpl;trivial. - destruct y as [ |yh yl];simpl. rewrite Zplus_0_r;trivial. + destruct y as [ |yh yl];simpl. rewrite Z.add_0_r;trivial. replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|])) with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|])). 2:ring. generalize (spec_w_add_c xl yl);destruct (w_add_c xl yl) as [l|l]; intros H;unfold interp_carry in H;rewrite <- H. generalize (spec_w_add_c xh yh);destruct (w_add_c xh yh) as [h|h]; intros H1;unfold interp_carry in *;rewrite <- H1. trivial. - repeat rewrite Zmult_1_l;rewrite spec_w_WW;rewrite wwB_wBwB; ring. - rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + repeat rewrite Z.mul_1_l;rewrite spec_w_WW;rewrite wwB_wBwB; ring. + rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. generalize (spec_w_add_carry_c xh yh);destruct (w_add_carry_c xh yh) as [h|h]; intros H1;unfold interp_carry in *;rewrite <- H1. simpl;ring. - repeat rewrite Zmult_1_l;rewrite wwB_wBwB;rewrite spec_w_WW;ring. + repeat rewrite Z.mul_1_l;rewrite wwB_wBwB;rewrite spec_w_WW;ring. Qed. Section Cont. @@ -221,23 +221,23 @@ Section DoubleAdd. destruct x as [ |xh xl];simpl;trivial. apply spec_f0;trivial. destruct y as [ |yh yl];simpl. - apply spec_f0;simpl;rewrite Zplus_0_r;trivial. + apply spec_f0;simpl;rewrite Z.add_0_r;trivial. generalize (spec_w_add_c xl yl);destruct (w_add_c xl yl) as [l|l]; intros H;unfold interp_carry in H. generalize (spec_w_add_c xh yh);destruct (w_add_c xh yh) as [h|h]; intros H1;unfold interp_carry in *. apply spec_f0. simpl;rewrite H;rewrite H1;ring. apply spec_f1. simpl;rewrite spec_w_WW;rewrite H. - rewrite Zplus_assoc;rewrite wwB_wBwB. rewrite Zpower_2; rewrite <- Zmult_plus_distr_l. - rewrite Zmult_1_l in H1;rewrite H1;ring. + rewrite Z.add_assoc;rewrite wwB_wBwB. rewrite Z.pow_2_r; rewrite <- Z.mul_add_distr_r. + rewrite Z.mul_1_l in H1;rewrite H1;ring. generalize (spec_w_add_carry_c xh yh);destruct (w_add_carry_c xh yh) as [h|h]; intros H1;unfold interp_carry in *. - apply spec_f0;simpl;rewrite H1. rewrite Zmult_plus_distr_l. - rewrite <- Zplus_assoc;rewrite H;ring. + apply spec_f0;simpl;rewrite H1. rewrite Z.mul_add_distr_r. + rewrite <- Z.add_assoc;rewrite H;ring. apply spec_f1. simpl;rewrite spec_w_WW;rewrite wwB_wBwB. - rewrite Zplus_assoc; rewrite Zpower_2; rewrite <- Zmult_plus_distr_l. - rewrite Zmult_1_l in H1;rewrite H1. rewrite Zmult_plus_distr_l. - rewrite <- Zplus_assoc;rewrite H;ring. + rewrite Z.add_assoc; rewrite Z.pow_2_r; rewrite <- Z.mul_add_distr_r. + rewrite Z.mul_1_l in H1;rewrite H1. rewrite Z.mul_add_distr_r. + rewrite <- Z.add_assoc;rewrite H;ring. Qed. End Cont. @@ -248,19 +248,19 @@ Section DoubleAdd. destruct x as [ |xh xl];intro y;simpl. exact (spec_ww_succ_c y). destruct y as [ |yh yl];simpl. - rewrite Zplus_0_r;exact (spec_ww_succ_c (WW xh xl)). + rewrite Z.add_0_r;exact (spec_ww_succ_c (WW xh xl)). replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|]) + 1) with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|]+1)). 2:ring. generalize (spec_w_add_carry_c xl yl);destruct (w_add_carry_c xl yl) as [l|l];intros H;unfold interp_carry in H;rewrite <- H. generalize (spec_w_add_c xh yh);destruct (w_add_c xh yh) as [h|h]; intros H1;unfold interp_carry in H1;rewrite <- H1. trivial. - unfold interp_carry;repeat rewrite Zmult_1_l;simpl;rewrite wwB_wBwB;ring. - rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + unfold interp_carry;repeat rewrite Z.mul_1_l;simpl;rewrite wwB_wBwB;ring. + rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. generalize (spec_w_add_carry_c xh yh);destruct (w_add_carry_c xh yh) as [h|h];intros H1;unfold interp_carry in H1;rewrite <- H1. trivial. unfold interp_carry;rewrite spec_w_WW; - repeat rewrite Zmult_1_l;simpl;rewrite wwB_wBwB;ring. + repeat rewrite Z.mul_1_l;simpl;rewrite wwB_wBwB;ring. Qed. Lemma spec_ww_succ : forall x, [[ww_succ x]] = ([[x]] + 1) mod wwB. @@ -268,14 +268,14 @@ Section DoubleAdd. destruct x as [ |xh xl];simpl. rewrite spec_ww_1;rewrite Zmod_small;trivial. split;[intro;discriminate|apply wwB_pos]. - rewrite <- Zplus_assoc;generalize (spec_w_succ_c xl); + rewrite <- Z.add_assoc;generalize (spec_w_succ_c xl); destruct (w_succ_c xl) as[l|l];intro H;unfold interp_carry in H;rewrite <-H. rewrite Zmod_small;trivial. rewrite wwB_wBwB;apply beta_mult;apply spec_to_Z. assert ([|l|] = 0). clear spec_ww_1 spec_w_1 spec_w_0. assert (H1:= spec_to_Z l); assert (H2:= spec_to_Z xl); omega. - rewrite H0;rewrite Zplus_0_r;rewrite <- Zmult_plus_distr_l;rewrite wwB_wBwB. - rewrite Zpower_2; rewrite Zmult_mod_distr_r;try apply lt_0_wB. + rewrite H0;rewrite Z.add_0_r;rewrite <- Z.mul_add_distr_r;rewrite wwB_wBwB. + rewrite Z.pow_2_r; rewrite Zmult_mod_distr_r;try apply lt_0_wB. rewrite spec_w_W0;rewrite spec_w_succ;trivial. Qed. @@ -284,7 +284,7 @@ Section DoubleAdd. destruct x as [ |xh xl];intros y;simpl. rewrite Zmod_small;trivial. apply spec_ww_to_Z;trivial. destruct y as [ |yh yl]. - change [[W0]] with 0;rewrite Zplus_0_r. + change [[W0]] with 0;rewrite Z.add_0_r. rewrite Zmod_small;trivial. exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh xl)). simpl. replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|])) @@ -292,7 +292,7 @@ Section DoubleAdd. generalize (spec_w_add_c xl yl);destruct (w_add_c xl yl) as [l|l]; unfold interp_carry;intros H;simpl;rewrite <- H. rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add;trivial. - rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. rewrite(mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add_carry;trivial. Qed. @@ -302,13 +302,13 @@ Section DoubleAdd. destruct x as [ |xh xl];intros y;simpl. exact (spec_ww_succ y). destruct y as [ |yh yl]. - change [[W0]] with 0;rewrite Zplus_0_r. exact (spec_ww_succ (WW xh xl)). + change [[W0]] with 0;rewrite Z.add_0_r. exact (spec_ww_succ (WW xh xl)). simpl;replace ([|xh|] * wB + [|xl|] + ([|yh|] * wB + [|yl|]) + 1) with (([|xh|]+[|yh|])*wB + ([|xl|]+[|yl|]+1)). 2:ring. generalize (spec_w_add_carry_c xl yl);destruct (w_add_carry_c xl yl) as [l|l];unfold interp_carry;intros H;rewrite <- H;simpl ww_to_Z. rewrite(mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add;trivial. - rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. rewrite(mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_w_add_carry;trivial. Qed. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v index e6c5a0e0..ed69a8f5 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -161,13 +161,13 @@ Section DoubleBase. Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. Variable spec_to_Z : forall x, 0 <= [|x|] < wB. Variable spec_w_compare : forall x y, - w_compare x y = Zcompare [|x|] [|y|]. + w_compare x y = Z.compare [|x|] [|y|]. Lemma wwB_wBwB : wwB = wB^2. Proof. - unfold base, ww_digits;rewrite Zpower_2; rewrite (Zpos_xO w_digits). + unfold base, ww_digits;rewrite Z.pow_2_r; rewrite (Pos2Z.inj_xO w_digits). replace (2 * Zpos w_digits) with (Zpos w_digits + Zpos w_digits). - apply Zpower_exp; unfold Zge;simpl;intros;discriminate. + apply Zpower_exp; unfold Z.ge;simpl;intros;discriminate. ring. Qed. @@ -179,28 +179,28 @@ Section DoubleBase. Lemma lt_0_wB : 0 < wB. Proof. - unfold base;apply Zpower_gt_0. unfold Zlt;reflexivity. - unfold Zle;intros H;discriminate H. + unfold base;apply Z.pow_pos_nonneg. unfold Z.lt;reflexivity. + unfold Z.le;intros H;discriminate H. Qed. Lemma lt_0_wwB : 0 < wwB. - Proof. rewrite wwB_wBwB; rewrite Zpower_2; apply Zmult_lt_0_compat;apply lt_0_wB. Qed. + Proof. rewrite wwB_wBwB; rewrite Z.pow_2_r; apply Z.mul_pos_pos;apply lt_0_wB. Qed. Lemma wB_pos: 1 < wB. Proof. - unfold base;apply Zlt_le_trans with (2^1). unfold Zlt;reflexivity. - apply Zpower_le_monotone. unfold Zlt;reflexivity. - split;unfold Zle;intros H. discriminate H. + unfold base;apply Z.lt_le_trans with (2^1). unfold Z.lt;reflexivity. + apply Zpower_le_monotone. unfold Z.lt;reflexivity. + split;unfold Z.le;intros H. discriminate H. clear spec_w_0W w_0W spec_w_Bm1 spec_to_Z spec_w_WW w_WW. destruct w_digits; discriminate H. Qed. Lemma wwB_pos: 1 < wwB. Proof. - assert (H:= wB_pos);rewrite wwB_wBwB;rewrite <-(Zmult_1_r 1). - rewrite Zpower_2. - apply Zmult_lt_compat2;(split;[unfold Zlt;reflexivity|trivial]). - apply Zlt_le_weak;trivial. + assert (H:= wB_pos);rewrite wwB_wBwB;rewrite <-(Z.mul_1_r 1). + rewrite Z.pow_2_r. + apply Zmult_lt_compat2;(split;[unfold Z.lt;reflexivity|trivial]). + apply Z.lt_le_incl;trivial. Qed. Theorem wB_div_2: 2 * (wB / 2) = wB. @@ -208,22 +208,22 @@ Section DoubleBase. clear spec_w_0 w_0 spec_w_1 w_1 spec_w_Bm1 w_Bm1 spec_w_WW spec_w_0W spec_to_Z;unfold base. assert (2 ^ Zpos w_digits = 2 * (2 ^ (Zpos w_digits - 1))). - pattern 2 at 2; rewrite <- Zpower_1_r. + pattern 2 at 2; rewrite <- Z.pow_1_r. rewrite <- Zpower_exp; auto with zarith. f_equal; auto with zarith. case w_digits; compute; intros; discriminate. rewrite H; f_equal; auto with zarith. - rewrite Zmult_comm; apply Z_div_mult; auto with zarith. + rewrite Z.mul_comm; apply Z_div_mult; auto with zarith. Qed. Theorem wwB_div_2 : wwB / 2 = wB / 2 * wB. Proof. clear spec_w_0 w_0 spec_w_1 w_1 spec_w_Bm1 w_Bm1 spec_w_WW spec_w_0W spec_to_Z. - rewrite wwB_wBwB; rewrite Zpower_2. + rewrite wwB_wBwB; rewrite Z.pow_2_r. pattern wB at 1; rewrite <- wB_div_2; auto. - rewrite <- Zmult_assoc. - repeat (rewrite (Zmult_comm 2); rewrite Z_div_mult); auto with zarith. + rewrite <- Z.mul_assoc. + repeat (rewrite (Z.mul_comm 2); rewrite Z_div_mult); auto with zarith. Qed. Lemma mod_wwB : forall z x, @@ -231,15 +231,15 @@ Section DoubleBase. Proof. intros z x. rewrite Zplus_mod. - pattern wwB at 1;rewrite wwB_wBwB; rewrite Zpower_2. + pattern wwB at 1;rewrite wwB_wBwB; rewrite Z.pow_2_r. rewrite Zmult_mod_distr_r;try apply lt_0_wB. rewrite (Zmod_small [|x|]). apply Zmod_small;rewrite wwB_wBwB;apply beta_mult;try apply spec_to_Z. - apply Z_mod_lt;apply Zlt_gt;apply lt_0_wB. + apply Z_mod_lt;apply Z.lt_gt;apply lt_0_wB. destruct (spec_to_Z x);split;trivial. change [|x|] with (0*wB+[|x|]). rewrite wwB_wBwB. - rewrite Zpower_2;rewrite <- (Zplus_0_r (wB*wB));apply beta_lex_inv. - apply lt_0_wB. apply spec_to_Z. split;[apply Zle_refl | apply lt_0_wB]. + rewrite Z.pow_2_r;rewrite <- (Z.add_0_r (wB*wB));apply beta_lex_inv. + apply lt_0_wB. apply spec_to_Z. split;[apply Z.le_refl | apply lt_0_wB]. Qed. Lemma wB_div : forall x y, ([|x|] * wB + [|y|]) / wB = [|x|]. @@ -265,29 +265,29 @@ Section DoubleBase. clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1. unfold base;apply Zpower_lt_monotone;auto with zarith. assert (0 < Zpos w_digits). compute;reflexivity. - unfold ww_digits;rewrite Zpos_xO;auto with zarith. + unfold ww_digits;rewrite Pos2Z.inj_xO;auto with zarith. Qed. Lemma w_to_Z_wwB : forall x, x < wB -> x < wwB. Proof. - intros x H;apply Zlt_trans with wB;trivial;apply lt_wB_wwB. + intros x H;apply Z.lt_trans with wB;trivial;apply lt_wB_wwB. Qed. Lemma spec_ww_to_Z : forall x, 0 <= [[x]] < wwB. Proof. clear spec_w_0 spec_w_1 spec_w_Bm1 w_0 w_1 w_Bm1. destruct x as [ |h l];simpl. - split;[apply Zle_refl|apply lt_0_wwB]. + split;[apply Z.le_refl|apply lt_0_wwB]. assert (H:=spec_to_Z h);assert (L:=spec_to_Z l);split. - apply Zplus_le_0_compat;auto with zarith. - rewrite <- (Zplus_0_r wwB);rewrite wwB_wBwB; rewrite Zpower_2; + apply Z.add_nonneg_nonneg;auto with zarith. + rewrite <- (Z.add_0_r wwB);rewrite wwB_wBwB; rewrite Z.pow_2_r; apply beta_lex_inv;auto with zarith. Qed. Lemma double_wB_wwB : forall n, double_wB n * double_wB n = double_wB (S n). Proof. intros n;unfold double_wB;simpl. - unfold base. rewrite Pshiftl_nat_S, (Zpos_xO (_ << _)). + unfold base. rewrite Pshiftl_nat_S, (Pos2Z.inj_xO (_ << _)). replace (2 * Zpos (w_digits << n)) with (Zpos (w_digits << n) + Zpos (w_digits << n)) by ring. symmetry; apply Zpower_exp;intro;discriminate. @@ -306,14 +306,14 @@ Section DoubleBase. intros n; elim n; clear n; auto. unfold double_wB, "<<"; auto with zarith. intros n H1; rewrite <- double_wB_wwB. - apply Zle_trans with (wB * 1). - rewrite Zmult_1_r; apply Zle_refl. - apply Zmult_le_compat; auto with zarith. - apply Zle_trans with wB; auto with zarith. - unfold base. - rewrite <- (Zpower_0_r 2). - apply Zpower_le_monotone2; auto with zarith. + apply Z.le_trans with (wB * 1). + rewrite Z.mul_1_r; apply Z.le_refl. unfold base; auto with zarith. + apply Z.mul_le_mono_nonneg; auto with zarith. + apply Z.le_trans with wB; auto with zarith. + unfold base. + rewrite <- (Z.pow_0_r 2). + apply Z.pow_le_mono_r; auto with zarith. Qed. Lemma spec_double_to_Z : @@ -326,9 +326,9 @@ Section DoubleBase. unfold double_wB,base;split;auto with zarith. assert (U0:= IHn w0);assert (U1:= IHn w1). split;auto with zarith. - apply Zlt_le_trans with ((double_wB n - 1) * double_wB n + double_wB n). + apply Z.lt_le_trans with ((double_wB n - 1) * double_wB n + double_wB n). assert (double_to_Z n w0*double_wB n <= (double_wB n - 1)*double_wB n). - apply Zmult_le_compat_r;auto with zarith. + apply Z.mul_le_mono_nonneg_r;auto with zarith. auto with zarith. rewrite <- double_wB_wwB. replace ((double_wB n - 1) * double_wB n + double_wB n) with (double_wB n * double_wB n); @@ -342,22 +342,19 @@ Section DoubleBase. clear spec_w_1 spec_w_Bm1. intros n; elim n; auto; clear n. intros n Hrec x; case x; clear x; auto. - intros xx yy H1; simpl in H1. - assert (F1: [!n | xx!] = 0). - case (Zle_lt_or_eq 0 ([!n | xx!])); auto. - case (spec_double_to_Z n xx); auto. - intros F2. - assert (F3 := double_wB_more_digits n). - assert (F4: 0 <= [!n | yy!]). - case (spec_double_to_Z n yy); auto. + intros xx yy; simpl. + destruct (spec_double_to_Z n xx) as [F1 _]. Z.le_elim F1. + - (* 0 < [!n | xx!] *) + intros; exfalso. + assert (F3 := double_wB_more_digits n). + destruct (spec_double_to_Z n yy) as [F4 _]. assert (F5: 1 * wB <= [!n | xx!] * double_wB n); auto with zarith. - apply Zmult_le_compat; auto with zarith. + apply Z.mul_le_mono_nonneg; auto with zarith. unfold base; auto with zarith. - simpl get_low; simpl double_to_Z. - generalize H1; clear H1. - rewrite F1; rewrite Zmult_0_l; rewrite Zplus_0_l. - intros H1; apply Hrec; auto. + - (* 0 = [!n | xx!] *) + rewrite <- F1; rewrite Z.mul_0_l, Z.add_0_l. + intros; apply Hrec; auto. Qed. Lemma spec_double_WW : forall n (h l : word w n), @@ -399,36 +396,36 @@ Section DoubleBase. Ltac comp2ord := match goal with | |- Lt = (?x ?= ?y) => symmetry; change (x < y) - | |- Gt = (?x ?= ?y) => symmetry; change (x > y); apply Zlt_gt + | |- Gt = (?x ?= ?y) => symmetry; change (x > y); apply Z.lt_gt end. Lemma spec_ww_compare : forall x y, - ww_compare x y = Zcompare [[x]] [[y]]. + ww_compare x y = Z.compare [[x]] [[y]]. Proof. destruct x as [ |xh xl];destruct y as [ |yh yl];simpl;trivial. (* 1st case *) rewrite 2 spec_w_compare, spec_w_0. - destruct (Zcompare_spec 0 [|yh|]) as [H|H|H]. + destruct (Z.compare_spec 0 [|yh|]) as [H|H|H]. rewrite <- H;simpl. reflexivity. symmetry. change (0 < [|yh|]*wB+[|yl|]). change 0 with (0*wB+0). rewrite <- spec_w_0 at 2. apply wB_lex_inv;trivial. - absurd (0 <= [|yh|]). apply Zlt_not_le; trivial. + absurd (0 <= [|yh|]). apply Z.lt_nge; trivial. destruct (spec_to_Z yh);trivial. (* 2nd case *) rewrite 2 spec_w_compare, spec_w_0. - destruct (Zcompare_spec [|xh|] 0) as [H|H|H]. + destruct (Z.compare_spec [|xh|] 0) as [H|H|H]. rewrite H;simpl;reflexivity. - absurd (0 <= [|xh|]). apply Zlt_not_le; trivial. + absurd (0 <= [|xh|]). apply Z.lt_nge; trivial. destruct (spec_to_Z xh);trivial. comp2ord. change 0 with (0*wB+0). rewrite <- spec_w_0 at 2. apply wB_lex_inv;trivial. (* 3rd case *) rewrite 2 spec_w_compare. - destruct (Zcompare_spec [|xh|] [|yh|]) as [H|H|H]. + destruct (Z.compare_spec [|xh|] [|yh|]) as [H|H|H]. rewrite H. - symmetry. apply Zcompare_plus_compat. + symmetry. apply Z.add_compare_mono_l. comp2ord. apply wB_lex_inv;trivial. comp2ord. apply wB_lex_inv;trivial. Qed. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v index 00a84052..35fe948e 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -390,21 +390,21 @@ Section Z_2nZ. Proof. refine (spec_ww_to_Z w_digits w_to_Z _);auto. Qed. Let spec_ww_of_pos : forall p, - Zpos p = (Z_of_N (fst (ww_of_pos p)))*wwB + [|(snd (ww_of_pos p))|]. + Zpos p = (Z.of_N (fst (ww_of_pos p)))*wwB + [|(snd (ww_of_pos p))|]. Proof. unfold ww_of_pos;intros. rewrite (ZnZ.spec_of_pos p). unfold w_of_pos. case (ZnZ.of_pos p); intros. simpl. destruct n; simpl ZnZ.to_Z. simpl;unfold w_to_Z,w_0; rewrite ZnZ.spec_0;trivial. - unfold Z_of_N. + unfold Z.of_N. rewrite (ZnZ.spec_of_pos p0). case (ZnZ.of_pos p0); intros. simpl. - unfold fst, snd,Z_of_N, to_Z, wB, w_digits, w_to_Z, w_WW. + unfold fst, snd,Z.of_N, to_Z, wB, w_digits, w_to_Z, w_WW. rewrite ZnZ.spec_WW. replace wwB with (wB*wB). unfold wB,w_to_Z,w_digits;destruct n;ring. - symmetry. rewrite <- Zpower_2; exact (wwB_wBwB w_digits). + symmetry. rewrite <- Z.pow_2_r; exact (wwB_wBwB w_digits). Qed. Let spec_ww_0 : [|W0|] = 0. @@ -417,7 +417,7 @@ Section Z_2nZ. Proof. refine (spec_ww_Bm1 w_Bm1 w_digits w_to_Z _);auto. Qed. Let spec_ww_compare : - forall x y, compare x y = Zcompare [|x|] [|y|]. + forall x y, compare x y = Z.compare [|x|] [|y|]. Proof. refine (spec_ww_compare w_0 w_digits w_to_Z w_compare _ _ _);auto. Qed. @@ -575,9 +575,9 @@ Section Z_2nZ. unfold w_add_c; case ZnZ.add_c; unfold interp_carry; simpl ww_to_Z. intros w0 Hw0; simpl; unfold w_to_Z; rewrite Hw0. unfold w_0; rewrite ZnZ.spec_0; simpl; auto with zarith. - intros w0; rewrite Zmult_1_l; simpl. + intros w0; rewrite Z.mul_1_l; simpl. unfold w_to_Z, w_1; rewrite ZnZ.spec_1; auto with zarith. - rewrite Zmult_1_l; auto. + rewrite Z.mul_1_l; auto. Qed. Let spec_low: forall x, @@ -585,7 +585,7 @@ Section Z_2nZ. intros x; case x; simpl low. unfold ww_to_Z, w_to_Z, w_0; rewrite ZnZ.spec_0; simpl; auto. intros xh xl; simpl. - rewrite Zplus_comm; rewrite Z_mod_plus; auto with zarith. + rewrite Z.add_comm; rewrite Z_mod_plus; auto with zarith. rewrite Zmod_small; auto with zarith. unfold wB, base; auto with zarith. Qed. @@ -597,7 +597,7 @@ Section Z_2nZ. rewrite spec_add2. unfold w_to_Z, w_zdigits, w_digits. rewrite ZnZ.spec_zdigits; auto. - rewrite Zpos_xO; auto with zarith. + rewrite Pos2Z.inj_xO; auto with zarith. Qed. @@ -605,7 +605,7 @@ Section Z_2nZ. Proof. refine (spec_ww_head00 w_0 w_0W w_compare w_head0 w_add2 w_zdigits _ww_zdigits - w_to_Z _ _ _ (refl_equal _ww_digits) _ _ _ _); auto. + w_to_Z _ _ _ (eq_refl _ww_digits) _ _ _ _); auto. exact ZnZ.spec_head00. exact ZnZ.spec_zdigits. Qed. @@ -623,7 +623,7 @@ Section Z_2nZ. Proof. refine (spec_ww_tail00 w_0 w_0W w_compare w_tail0 w_add2 w_zdigits _ww_zdigits - w_to_Z _ _ _ (refl_equal _ww_digits) _ _ _ _); wwauto. + w_to_Z _ _ _ (eq_refl _ww_digits) _ _ _ _); wwauto. exact ZnZ.spec_tail00. exact ZnZ.spec_zdigits. Qed. @@ -749,7 +749,7 @@ refine | false => [|x|] mod 2 = 1 end. Proof. - refine (@spec_ww_is_even t w_is_even w_0 w_1 w_Bm1 w_digits _ _ _ _ _); auto. + refine (@spec_ww_is_even t w_is_even w_digits _ _ ). exact ZnZ.spec_is_even. Qed. @@ -798,7 +798,7 @@ refine exact ZnZ.spec_zdigits. unfold w_to_Z, w_zdigits. rewrite ZnZ.spec_zdigits. - rewrite <- Zpos_xO; exact spec_ww_digits. + rewrite <- Pos2Z.inj_xO; exact spec_ww_digits. Qed. Global Instance mk_zn2z_specs_karatsuba : ZnZ.Specs mk_zn2z_ops_karatsuba. @@ -811,7 +811,7 @@ refine exact ZnZ.spec_zdigits. unfold w_to_Z, w_zdigits. rewrite ZnZ.spec_zdigits. - rewrite <- Zpos_xO; exact spec_ww_digits. + rewrite <- Pos2Z.inj_xO; exact spec_ww_digits. Qed. End Z_2nZ. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v index 0cb6848e..8525b0e1 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -80,7 +80,7 @@ Section POS_MOD. Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. Variable spec_ww_compare : forall x y, - ww_compare x y = Zcompare [[x]] [[y]]. + ww_compare x y = Z.compare [[x]] [[y]]. Variable spec_ww_sub: forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. @@ -100,7 +100,7 @@ Section POS_MOD. unfold ww_pos_mod; case w1. simpl; rewrite Zmod_small; split; auto with zarith. intros xh xl; rewrite spec_ww_compare. - case Zcompare_spec; + case Z.compare_spec; rewrite spec_w_0W; rewrite spec_zdigits; fold wB; intros H1. rewrite H1; simpl ww_to_Z. @@ -117,19 +117,19 @@ Section POS_MOD. rewrite spec_low. apply Zmod_small; auto with zarith. case (spec_to_w_Z p); intros HHH1 HHH2; split; auto with zarith. - apply Zlt_le_trans with (1 := H1). + apply Z.lt_le_trans with (1 := H1). unfold base; apply Zpower2_le_lin; auto with zarith. rewrite HH0. rewrite Zplus_mod; auto with zarith. unfold base. rewrite <- (F0 (Zpos w_digits) [[p]]). rewrite Zpower_exp; auto with zarith. - rewrite Zmult_assoc. + rewrite Z.mul_assoc. rewrite Z_mod_mult; auto with zarith. autorewrite with w_rewrite rm10. rewrite Zmod_mod; auto with zarith. rewrite spec_ww_compare. - case Zcompare_spec; rewrite spec_ww_zdigits; + case Z.compare_spec; rewrite spec_ww_zdigits; rewrite spec_zdigits; intros H2. replace (2^[[p]]) with wwB. rewrite Zmod_small; auto with zarith. @@ -143,52 +143,52 @@ Section POS_MOD. rewrite <- Zmod_div_mod; auto with zarith. rewrite Zmod_small; auto with zarith. split; auto with zarith. - apply Zlt_le_trans with (Zpos w_digits); auto with zarith. + apply Z.lt_le_trans with (Zpos w_digits); auto with zarith. unfold base; apply Zpower2_le_lin; auto with zarith. exists wB; unfold base; rewrite <- Zpower_exp; auto with zarith. rewrite spec_ww_digits; - apply f_equal with (f := Zpower 2); rewrite Zpos_xO; auto with zarith. + apply f_equal with (f := Z.pow 2); rewrite Pos2Z.inj_xO; auto with zarith. simpl ww_to_Z; autorewrite with w_rewrite. rewrite spec_pos_mod; rewrite HH0. pattern [|xh|] at 2; rewrite Z_div_mod_eq with (b := 2 ^ ([[p]] - Zpos w_digits)); auto with zarith. - rewrite (fun x => (Zmult_comm (2 ^ x))); rewrite Zmult_plus_distr_l. - unfold base; rewrite <- Zmult_assoc; rewrite <- Zpower_exp; + rewrite (fun x => (Z.mul_comm (2 ^ x))); rewrite Z.mul_add_distr_r. + unfold base; rewrite <- Z.mul_assoc; rewrite <- Zpower_exp; auto with zarith. rewrite F0; auto with zarith. - rewrite <- Zplus_assoc; rewrite Zplus_mod; auto with zarith. + rewrite <- Z.add_assoc; rewrite Zplus_mod; auto with zarith. rewrite Z_mod_mult; auto with zarith. autorewrite with rm10. rewrite Zmod_mod; auto with zarith. - apply sym_equal; apply Zmod_small; auto with zarith. + symmetry; apply Zmod_small; auto with zarith. case (spec_to_Z xh); intros U1 U2. case (spec_to_Z xl); intros U3 U4. split; auto with zarith. - apply Zplus_le_0_compat; auto with zarith. - apply Zmult_le_0_compat; auto with zarith. + apply Z.add_nonneg_nonneg; auto with zarith. + apply Z.mul_nonneg_nonneg; auto with zarith. match goal with |- 0 <= ?X mod ?Y => case (Z_mod_lt X Y); auto with zarith end. match goal with |- ?X mod ?Y * ?U + ?Z < ?T => - apply Zle_lt_trans with ((Y - 1) * U + Z ); + apply Z.le_lt_trans with ((Y - 1) * U + Z ); [case (Z_mod_lt X Y); auto with zarith | idtac] end. match goal with |- ?X * ?U + ?Y < ?Z => - apply Zle_lt_trans with (X * U + (U - 1)) + apply Z.le_lt_trans with (X * U + (U - 1)) end. - apply Zplus_le_compat_l; auto with zarith. + apply Z.add_le_mono_l; auto with zarith. case (spec_to_Z xl); unfold base; auto with zarith. - rewrite Zmult_minus_distr_r; rewrite <- Zpower_exp; auto with zarith. + rewrite Z.mul_sub_distr_r; rewrite <- Zpower_exp; auto with zarith. rewrite F0; auto with zarith. rewrite Zmod_small; auto with zarith. case (spec_to_w_Z (WW xh xl)); intros U1 U2. split; auto with zarith. - apply Zlt_le_trans with (1:= U2). + apply Z.lt_le_trans with (1:= U2). unfold base; rewrite spec_ww_digits. apply Zpower_le_monotone; auto with zarith. split; auto with zarith. - rewrite Zpos_xO; auto with zarith. + rewrite Pos2Z.inj_xO; auto with zarith. Qed. End POS_MOD. @@ -260,7 +260,7 @@ Section DoubleDiv32. Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. Variable spec_compare : - forall x y, w_compare x y = Zcompare [|x|] [|y|]. + forall x y, w_compare x y = Z.compare [|x|] [|y|]. Variable spec_w_add_c : forall x y, [+|w_add_c x y|] = [|x|] + [|y|]. Variable spec_w_add_carry_c : forall x y, [+|w_add_carry_c x y|] = [|x|] + [|y|] + 1. @@ -290,14 +290,14 @@ Section DoubleDiv32. assert (H:= spec_ww_to_Z w_digits w_to_Z spec_to_Z x). Theorem wB_div2: forall x, wB/2 <= x -> wB <= 2 * x. - intros x H; rewrite <- wB_div_2; apply Zmult_le_compat_l; auto with zarith. + intros x H; rewrite <- wB_div_2; apply Z.mul_le_mono_nonneg_l; auto with zarith. Qed. Lemma Zmult_lt_0_reg_r_2 : forall n m : Z, 0 <= n -> 0 < m * n -> 0 < m. Proof. - intros n m H1 H2;apply Zmult_lt_0_reg_r with n;trivial. - destruct (Zle_lt_or_eq _ _ H1);trivial. - subst;rewrite Zmult_0_r in H2;discriminate H2. + intros n m H1 H2;apply Z.mul_pos_cancel_r with n;trivial. + Z.le_elim H1; trivial. + subst;rewrite Z.mul_0_r in H2;discriminate H2. Qed. Theorem spec_w_div32 : forall a1 a2 a3 b1 b2, @@ -311,7 +311,7 @@ Section DoubleDiv32. intros a1 a2 a3 b1 b2 Hle Hlt. assert (U:= lt_0_wB w_digits); assert (U1:= lt_0_wwB w_digits). Spec_w_to_Z a1;Spec_w_to_Z a2;Spec_w_to_Z a3;Spec_w_to_Z b1;Spec_w_to_Z b2. - rewrite wwB_wBwB; rewrite Zpower_2; rewrite Zmult_assoc;rewrite <- Zmult_plus_distr_l. + rewrite wwB_wBwB; rewrite Z.pow_2_r; rewrite Z.mul_assoc;rewrite <- Z.mul_add_distr_r. change (w_div32 a1 a2 a3 b1 b2) with match w_compare a1 b1 with | Lt => @@ -332,7 +332,7 @@ Section DoubleDiv32. (WW (w_sub a2 b2) a3) (WW b1 b2) | Gt => (w_0, W0) (* cas absurde *) end. - rewrite spec_compare. case Zcompare_spec; intro Hcmp. + rewrite spec_compare. case Z.compare_spec; intro Hcmp. simpl in Hlt. rewrite Hcmp in Hlt;assert ([|a2|] < [|b2|]). omega. assert ([[WW (w_sub a2 b2) a3]] = ([|a2|]-[|b2|])*wB + [|a3|] + wwB). @@ -351,17 +351,17 @@ Section DoubleDiv32. rewrite H0;intros r. repeat (rewrite spec_ww_add;eauto || rewrite spec_w_Bm1 || rewrite spec_w_Bm2); - simpl ww_to_Z;try rewrite Zmult_1_l;intros H1. + simpl ww_to_Z;try rewrite Z.mul_1_l;intros H1. assert (0<= ([[r]] + ([|b1|] * wB + [|b2|])) - wwB < [|b1|] * wB + [|b2|]). Spec_ww_to_Z r;split;zarith. rewrite H1. assert (H12:= wB_div2 Hle). assert (wwB <= 2 * [|b1|] * wB). - rewrite wwB_wBwB; rewrite Zpower_2; zarith. + rewrite wwB_wBwB; rewrite Z.pow_2_r; zarith. assert (-wwB < ([|a2|] - [|b2|]) * wB + [|a3|] < 0). - split. apply Zlt_le_trans with (([|a2|] - [|b2|]) * wB);zarith. + split. apply Z.lt_le_trans with (([|a2|] - [|b2|]) * wB);zarith. rewrite wwB_wBwB;replace (-(wB^2)) with (-wB*wB);[zarith | ring]. - apply Zmult_lt_compat_r;zarith. - apply Zle_lt_trans with (([|a2|] - [|b2|]) * wB + (wB -1));zarith. + apply Z.mul_lt_mono_pos_r;zarith. + apply Z.le_lt_trans with (([|a2|] - [|b2|]) * wB + (wB -1));zarith. replace ( ([|a2|] - [|b2|]) * wB + (wB - 1)) with (([|a2|] - [|b2|] + 1) * wB + - 1);[zarith | ring]. assert (([|a2|] - [|b2|] + 1) * wB <= 0);zarith. @@ -376,13 +376,13 @@ Section DoubleDiv32. Spec_ww_to_Z (WW b1 b2). simpl in HH4;zarith. rewrite H0;intros r;repeat (rewrite spec_w_Bm1 || rewrite spec_w_Bm2); - simpl ww_to_Z;try rewrite Zmult_1_l;intros H1. + simpl ww_to_Z;try rewrite Z.mul_1_l;intros H1. assert ([[r]]=([|a2|]-[|b2|])*wB+[|a3|]+([|b1|]*wB+[|b2|])). zarith. split. rewrite H2;rewrite Hcmp;ring. split. Spec_ww_to_Z r;zarith. rewrite H2. assert (([|a2|] - [|b2|]) * wB + [|a3|] < 0);zarith. - apply Zle_lt_trans with (([|a2|] - [|b2|]) * wB + (wB -1));zarith. + apply Z.le_lt_trans with (([|a2|] - [|b2|]) * wB + (wB -1));zarith. replace ( ([|a2|] - [|b2|]) * wB + (wB - 1)) with (([|a2|] - [|b2|] + 1) * wB + - 1);[zarith|ring]. assert (([|a2|] - [|b2|] + 1) * wB <= 0);zarith. @@ -400,7 +400,7 @@ Section DoubleDiv32. rewrite H1. split. ring. split. rewrite <- H1;destruct (spec_ww_to_Z w_digits w_to_Z spec_to_Z r1);trivial. - apply Zle_lt_trans with ([|r|] * wB + [|a3|]). + apply Z.le_lt_trans with ([|r|] * wB + [|a3|]). assert ( 0 <= [|q|] * [|b2|]);zarith. apply beta_lex_inv;zarith. assert ([[r1]] = [|r|] * wB + [|a3|] - [|q|] * [|b2|] + wwB). @@ -418,10 +418,10 @@ Section DoubleDiv32. intros r2;repeat (rewrite spec_pred || rewrite spec_ww_add;eauto); simpl ww_to_Z;intros H7. assert (0 < [|q|] - 1). - assert (1 <= [|q|]). zarith. - destruct (Zle_lt_or_eq _ _ H6);zarith. - rewrite <- H8 in H2;rewrite H2 in H7. - assert (0 < [|b1|]*wB). apply Zmult_lt_0_compat;zarith. + assert (H6 : 1 <= [|q|]) by zarith. + Z.le_elim H6;zarith. + rewrite <- H6 in H2;rewrite H2 in H7. + assert (0 < [|b1|]*wB). apply Z.mul_pos_pos;zarith. Spec_ww_to_Z r2. zarith. rewrite (Zmod_small ([|q|] -1));zarith. rewrite (Zmod_small ([|q|] -1 -1));zarith. @@ -439,7 +439,7 @@ Section DoubleDiv32. < wwB). split;try omega. replace (2*([|b1|]*wB+[|b2|])) with ((2*[|b1|])*wB+2*[|b2|]). 2:ring. assert (H12:= wB_div2 Hle). assert (wwB <= 2 * [|b1|] * wB). - rewrite wwB_wBwB; rewrite Zpower_2; zarith. omega. + rewrite wwB_wBwB; rewrite Z.pow_2_r; zarith. omega. rewrite <- (Zmod_unique ([[r2]] + ([|b1|] * wB + [|b2|])) wwB @@ -534,13 +534,13 @@ Section DoubleDiv21. 0 <= [[r]] < [|b1|] * wB + [|b2|]. Variable spec_ww_1 : [[ww_1]] = 1. Variable spec_ww_compare : forall x y, - ww_compare x y = Zcompare [[x]] [[y]]. + ww_compare x y = Z.compare [[x]] [[y]]. Variable spec_ww_sub : forall x y, [[ww_sub x y]] = ([[x]] - [[y]]) mod wwB. Theorem wwB_div: wwB = 2 * (wwB / 2). Proof. - rewrite wwB_div_2; rewrite Zmult_assoc; rewrite wB_div_2; auto. - rewrite <- Zpower_2; apply wwB_wBwB. + rewrite wwB_div_2; rewrite Z.mul_assoc; rewrite wB_div_2; auto. + rewrite <- Z.pow_2_r; apply wwB_wBwB. Qed. Ltac Spec_w_to_Z x := @@ -562,7 +562,7 @@ Section DoubleDiv21. Spec_ww_to_Z b; assert (Eq: 0 < [[b]]). Spec_ww_to_Z a1;omega. generalize Hlt H ;clear Hlt H;case a1. intros H1 H2;simpl in H1;Spec_ww_to_Z a2. - rewrite spec_ww_compare. case Zcompare_spec; + rewrite spec_ww_compare. case Z.compare_spec; simpl;try rewrite spec_ww_1;autorewrite with rm10; intros;zarith. rewrite spec_ww_sub;simpl. rewrite Zmod_small;zarith. split. ring. @@ -570,32 +570,32 @@ Section DoubleDiv21. rewrite wwB_div;zarith. intros a1h a1l. Spec_w_to_Z a1h;Spec_w_to_Z a1l. Spec_ww_to_Z a2. destruct a2 as [ |a3 a4]; - (destruct b as [ |b1 b2];[unfold Zle in Eq;discriminate Eq|idtac]); + (destruct b as [ |b1 b2];[unfold Z.le in Eq;discriminate Eq|idtac]); try (Spec_w_to_Z a3; Spec_w_to_Z a4); Spec_w_to_Z b1; Spec_w_to_Z b2; intros Hlt H; match goal with |-context [w_div32 ?X ?Y ?Z ?T ?U] => generalize (@spec_w_div32 X Y Z T U); case (w_div32 X Y Z T U); intros q1 r H0 end; (assert (Eq1: wB / 2 <= [|b1|]);[ apply (@beta_lex (wB / 2) 0 [|b1|] [|b2|] wB); auto with zarith; - autorewrite with rm10;repeat rewrite (Zmult_comm wB); + autorewrite with rm10;repeat rewrite (Z.mul_comm wB); rewrite <- wwB_div_2; trivial | generalize (H0 Eq1 Hlt);clear H0;destruct r as [ |r1 r2];simpl; - try rewrite spec_w_0; try rewrite spec_w_0W;repeat rewrite Zplus_0_r; + try rewrite spec_w_0; try rewrite spec_w_0W;repeat rewrite Z.add_0_r; intros (H1,H2) ]). - split;[rewrite wwB_wBwB; rewrite Zpower_2 | trivial]. - rewrite Zmult_assoc;rewrite Zmult_plus_distr_l;rewrite <- Zmult_assoc; - rewrite <- Zpower_2; rewrite <- wwB_wBwB;rewrite H1;ring. + split;[rewrite wwB_wBwB; rewrite Z.pow_2_r | trivial]. + rewrite Z.mul_assoc;rewrite Z.mul_add_distr_r;rewrite <- Z.mul_assoc; + rewrite <- Z.pow_2_r; rewrite <- wwB_wBwB;rewrite H1;ring. destruct H2 as (H2,H3);match goal with |-context [w_div32 ?X ?Y ?Z ?T ?U] => generalize (@spec_w_div32 X Y Z T U); case (w_div32 X Y Z T U); intros q r H0;generalize (H0 Eq1 H3);clear H0;intros (H4,H5) end. split;[rewrite wwB_wBwB | trivial]. - rewrite Zpower_2. - rewrite Zmult_assoc;rewrite Zmult_plus_distr_l;rewrite <- Zmult_assoc; - rewrite <- Zpower_2. + rewrite Z.pow_2_r. + rewrite Z.mul_assoc;rewrite Z.mul_add_distr_r;rewrite <- Z.mul_assoc; + rewrite <- Z.pow_2_r. rewrite <- wwB_wBwB;rewrite H1. - rewrite spec_w_0 in H4;rewrite Zplus_0_r in H4. - repeat rewrite Zmult_plus_distr_l. rewrite <- (Zmult_assoc [|r1|]). - rewrite <- Zpower_2; rewrite <- wwB_wBwB;rewrite H4;simpl;ring. + rewrite spec_w_0 in H4;rewrite Z.add_0_r in H4. + repeat rewrite Z.mul_add_distr_r. rewrite <- (Z.mul_assoc [|r1|]). + rewrite <- Z.pow_2_r; rewrite <- wwB_wBwB;rewrite H4;simpl;ring. split;[rewrite wwB_wBwB | split;zarith]. replace (([|a1h|] * wB + [|a1l|]) * wB^2 + ([|a3|] * wB + [|a4|])) with (([|a1h|] * wwB + [|a1l|] * wB + [|a3|])*wB+ [|a4|]). @@ -793,7 +793,7 @@ Section DoubleDivGt. Variable spec_w_WW : forall h l, [[w_WW h l]] = [|h|] * wB + [|l|]. Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. Variable spec_compare : - forall x y, w_compare x y = Zcompare [|x|] [|y|]. + forall x y, w_compare x y = Z.compare [|x|] [|y|]. Variable spec_eq0 : forall x, w_eq0 x = true -> [|x|] = 0. Variable spec_opp_c : forall x, [-|w_opp_c x|] = -[|x|]. @@ -893,42 +893,42 @@ Section DoubleDivGt. end in [[WW ah al]]=[[q]]*[[WW bh bl]]+[[r]] /\ 0 <=[[r]]< [[WW bh bl]]). assert (Hh := spec_head0 Hpos). lazy zeta. - rewrite spec_compare; case Zcompare_spec; + rewrite spec_compare; case Z.compare_spec; rewrite spec_w_0; intros HH. - generalize Hh; rewrite HH; simpl Zpower; - rewrite Zmult_1_l; intros (HH1, HH2); clear HH. + generalize Hh; rewrite HH; simpl Z.pow; + rewrite Z.mul_1_l; intros (HH1, HH2); clear HH. assert (wwB <= 2*[[WW bh bl]]). - apply Zle_trans with (2*[|bh|]*wB). - rewrite wwB_wBwB; rewrite Zpower_2; apply Zmult_le_compat_r; zarith. - rewrite <- wB_div_2; apply Zmult_le_compat_l; zarith. - simpl ww_to_Z;rewrite Zmult_plus_distr_r;rewrite Zmult_assoc. + apply Z.le_trans with (2*[|bh|]*wB). + rewrite wwB_wBwB; rewrite Z.pow_2_r; apply Z.mul_le_mono_nonneg_r; zarith. + rewrite <- wB_div_2; apply Z.mul_le_mono_nonneg_l; zarith. + simpl ww_to_Z;rewrite Z.mul_add_distr_l;rewrite Z.mul_assoc. Spec_w_to_Z bl;zarith. Spec_ww_to_Z (WW ah al). rewrite spec_ww_sub;eauto. - simpl;rewrite spec_ww_1;rewrite Zmult_1_l;simpl. + simpl;rewrite spec_ww_1;rewrite Z.mul_1_l;simpl. simpl ww_to_Z in Hgt, H, HH;rewrite Zmod_small;split;zarith. case (spec_to_Z (w_head0 bh)); auto with zarith. assert ([|w_head0 bh|] < Zpos w_digits). destruct (Z_lt_ge_dec [|w_head0 bh|] (Zpos w_digits));trivial. exfalso. assert (2 ^ [|w_head0 bh|] * [|bh|] >= wB);auto with zarith. - apply Zle_ge; replace wB with (wB * 1);try ring. - Spec_w_to_Z bh;apply Zmult_le_compat;zarith. + apply Z.le_ge; replace wB with (wB * 1);try ring. + Spec_w_to_Z bh;apply Z.mul_le_mono_nonneg;zarith. unfold base;apply Zpower_le_monotone;zarith. assert (HHHH : 0 < [|w_head0 bh|] < Zpos w_digits); auto with zarith. - assert (Hb:= Zlt_le_weak _ _ H). + assert (Hb:= Z.lt_le_incl _ _ H). generalize (spec_add_mul_div w_0 ah Hb) (spec_add_mul_div ah al Hb) (spec_add_mul_div al w_0 Hb) (spec_add_mul_div bh bl Hb) (spec_add_mul_div bl w_0 Hb); - rewrite spec_w_0; repeat rewrite Zmult_0_l;repeat rewrite Zplus_0_l; - rewrite Zdiv_0_l;repeat rewrite Zplus_0_r. + rewrite spec_w_0; repeat rewrite Z.mul_0_l;repeat rewrite Z.add_0_l; + rewrite Zdiv_0_l;repeat rewrite Z.add_0_r. Spec_w_to_Z ah;Spec_w_to_Z bh. unfold base;repeat rewrite Zmod_shift_r;zarith. assert (H3:=to_Z_div_minus_p ah HHHH);assert(H4:=to_Z_div_minus_p al HHHH); assert (H5:=to_Z_div_minus_p bl HHHH). - rewrite Zmult_comm in Hh. + rewrite Z.mul_comm in Hh. assert (2^[|w_head0 bh|] < wB). unfold base;apply Zpower_lt_monotone;zarith. unfold base in H0;rewrite Zmod_small;zarith. fold wB; rewrite (Zmod_small ([|bh|] * 2 ^ [|w_head0 bh|]));zarith. @@ -943,15 +943,15 @@ Section DoubleDivGt. (w_add_mul_div (w_head0 bh) al w_0) (w_add_mul_div (w_head0 bh) bh bl) (w_add_mul_div (w_head0 bh) bl w_0)) as (q,r). - rewrite V1;rewrite V2. rewrite Zmult_plus_distr_l. - rewrite <- (Zplus_assoc ([|bh|] * 2 ^ [|w_head0 bh|] * wB)). + rewrite V1;rewrite V2. rewrite Z.mul_add_distr_r. + rewrite <- (Z.add_assoc ([|bh|] * 2 ^ [|w_head0 bh|] * wB)). unfold base;rewrite <- shift_unshift_mod;zarith. fold wB. replace ([|bh|] * 2 ^ [|w_head0 bh|] * wB + [|bl|] * 2 ^ [|w_head0 bh|]) with ([[WW bh bl]] * 2^[|w_head0 bh|]). 2:simpl;ring. - fold wwB. rewrite wwB_wBwB. rewrite Zpower_2. rewrite U1;rewrite U2;rewrite U3. - rewrite Zmult_assoc. rewrite Zmult_plus_distr_l. - rewrite (Zplus_assoc ([|ah|] / 2^(Zpos(w_digits) - [|w_head0 bh|])*wB * wB)). - rewrite <- Zmult_plus_distr_l. rewrite <- Zplus_assoc. + fold wwB. rewrite wwB_wBwB. rewrite Z.pow_2_r. rewrite U1;rewrite U2;rewrite U3. + rewrite Z.mul_assoc. rewrite Z.mul_add_distr_r. + rewrite (Z.add_assoc ([|ah|] / 2^(Zpos(w_digits) - [|w_head0 bh|])*wB * wB)). + rewrite <- Z.mul_add_distr_r. rewrite <- Z.add_assoc. unfold base;repeat rewrite <- shift_unshift_mod;zarith. fold wB. replace ([|ah|] * 2 ^ [|w_head0 bh|] * wB + [|al|] * 2 ^ [|w_head0 bh|]) with ([[WW ah al]] * 2^[|w_head0 bh|]). 2:simpl;ring. @@ -962,42 +962,42 @@ Section DoubleDivGt. unfold base. replace (2^Zpos (w_digits)) with (2^(Zpos (w_digits) - 1)*2). rewrite Z_div_mult;zarith. rewrite <- Zpower_exp;zarith. - apply Zlt_le_trans with wB;zarith. + apply Z.lt_le_trans with wB;zarith. unfold base;apply Zpower_le_monotone;zarith. pattern 2 at 2;replace 2 with (2^1);trivial. rewrite <- Zpower_exp;zarith. ring_simplify (Zpos (w_digits) - 1 + 1);trivial. change [[WW w_0 q]] with ([|w_0|]*wB+[|q|]);rewrite spec_w_0;rewrite - Zmult_0_l;rewrite Zplus_0_l. + Z.mul_0_l;rewrite Z.add_0_l. replace [[ww_add_mul_div (ww_sub w_0 w_WW w_opp_c w_opp_carry w_sub_c w_opp w_sub w_sub_carry _ww_zdigits (w_0W (w_head0 bh))) W0 r]] with ([[r]]/2^[|w_head0 bh|]). - assert (0 < 2^[|w_head0 bh|]). apply Zpower_gt_0;zarith. + assert (0 < 2^[|w_head0 bh|]). apply Z.pow_pos_nonneg;zarith. split. rewrite <- (Z_div_mult [[WW ah al]] (2^[|w_head0 bh|]));zarith. - rewrite H1;rewrite Zmult_assoc;apply Z_div_plus_l;trivial. + rewrite H1;rewrite Z.mul_assoc;apply Z_div_plus_l;trivial. split;[apply Zdiv_le_lower_bound| apply Zdiv_lt_upper_bound];zarith. rewrite spec_ww_add_mul_div. rewrite spec_ww_sub; auto with zarith. rewrite spec_ww_digits_. change (Zpos (xO (w_digits))) with (2*Zpos (w_digits));zarith. - simpl ww_to_Z;rewrite Zmult_0_l;rewrite Zplus_0_l. + simpl ww_to_Z;rewrite Z.mul_0_l;rewrite Z.add_0_l. rewrite spec_w_0W. rewrite (fun x y => Zmod_small (x-y)); auto with zarith. ring_simplify (2 * Zpos w_digits - (2 * Zpos w_digits - [|w_head0 bh|])). rewrite Zmod_small;zarith. split;[apply Zdiv_le_lower_bound| apply Zdiv_lt_upper_bound];zarith. Spec_ww_to_Z r. - apply Zlt_le_trans with wwB;zarith. - rewrite <- (Zmult_1_r wwB);apply Zmult_le_compat;zarith. + apply Z.lt_le_trans with wwB;zarith. + rewrite <- (Z.mul_1_r wwB);apply Z.mul_le_mono_nonneg;zarith. split; auto with zarith. - apply Zle_lt_trans with (2 * Zpos w_digits); auto with zarith. - unfold base, ww_digits; rewrite (Zpos_xO w_digits). + apply Z.le_lt_trans with (2 * Zpos w_digits); auto with zarith. + unfold base, ww_digits; rewrite (Pos2Z.inj_xO w_digits). apply Zpower2_lt_lin; auto with zarith. rewrite spec_ww_sub; auto with zarith. rewrite spec_ww_digits_; rewrite spec_w_0W. rewrite Zmod_small;zarith. - rewrite Zpos_xO; split; auto with zarith. - apply Zle_lt_trans with (2 * Zpos w_digits); auto with zarith. - unfold base, ww_digits; rewrite (Zpos_xO w_digits). + rewrite Pos2Z.inj_xO; split; auto with zarith. + apply Z.le_lt_trans with (2 * Zpos w_digits); auto with zarith. + unfold base, ww_digits; rewrite (Pos2Z.inj_xO w_digits). apply Zpower2_lt_lin; auto with zarith. Qed. @@ -1037,9 +1037,9 @@ Section DoubleDivGt. assert (H2:=spec_div_gt Hgt Hpos);destruct (w_div_gt al bl). repeat rewrite spec_w_0W;simpl;rewrite spec_w_0;simpl;trivial. clear H. - rewrite spec_compare; case Zcompare_spec; intros Hcmp. + rewrite spec_compare; case Z.compare_spec; intros Hcmp. rewrite spec_w_0 in Hcmp. change [[WW bh bl]] with ([|bh|]*wB+[|bl|]). - rewrite <- Hcmp;rewrite Zmult_0_l;rewrite Zplus_0_l. + rewrite <- Hcmp;rewrite Z.mul_0_l;rewrite Z.add_0_l. simpl in Hpos;rewrite <- Hcmp in Hpos;simpl in Hpos. assert (H2:= @spec_double_divn1 w w_digits w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub w_to_Z spec_to_Z spec_w_zdigits spec_w_0 spec_w_WW spec_head0 @@ -1079,7 +1079,7 @@ Section DoubleDivGt. rewrite spec_mod_gt;trivial. assert (H:=spec_div_gt Hgt Hpos). destruct (w_div_gt a b) as (q,r);simpl. - rewrite Zmult_comm in H;destruct H. + rewrite Z.mul_comm in H;destruct H. symmetry;apply Zmod_unique with [|q|];trivial. Qed. @@ -1132,7 +1132,7 @@ Section DoubleDivGt. rewrite spec_w_0W;rewrite spec_w_mod_gt_eq;trivial. destruct (w_div_gt al bl);simpl;rewrite spec_w_0W;trivial. clear H. - rewrite spec_compare; case Zcompare_spec; intros H2. + rewrite spec_compare; case Z.compare_spec; intros H2. rewrite (@spec_double_modn1_aux w w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub w_to_Z spec_w_0 spec_compare 1 (WW ah al) bl). destruct (double_divn1 w_zdigits w_0 w_WW w_head0 w_add_mul_div w_div21 w_compare w_sub 1 @@ -1149,7 +1149,7 @@ Section DoubleDivGt. rewrite (spec_ww_mod_gt_eq a b Hgt Hpos). destruct (ww_div_gt a b)as(q,r);destruct H. apply Zmod_unique with[[q]];simpl;trivial. - rewrite Zmult_comm;trivial. + rewrite Z.mul_comm;trivial. Qed. Lemma Zis_gcd_mod : forall a b d, @@ -1206,13 +1206,13 @@ Section DoubleDivGt. | Gt => W0 (* absurde *) end). rewrite spec_compare, spec_w_0. - case Zcompare_spec; intros Hbh. + case Z.compare_spec; intros Hbh. simpl ww_to_Z in *. rewrite <- Hbh. - rewrite Zmult_0_l;rewrite Zplus_0_l. + rewrite Z.mul_0_l;rewrite Z.add_0_l. rewrite spec_compare, spec_w_0. - case Zcompare_spec; intros Hbl. + case Z.compare_spec; intros Hbl. rewrite <- Hbl;apply Zis_gcd_0. - simpl;rewrite spec_w_0;rewrite Zmult_0_l;rewrite Zplus_0_l. + simpl;rewrite spec_w_0;rewrite Z.mul_0_l;rewrite Z.add_0_l. apply Zis_gcd_mod;zarith. change ([|ah|] * wB + [|al|]) with (double_to_Z w_digits w_to_Z 1 (WW ah al)). rewrite <- (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW w_head0 w_add_mul_div @@ -1220,19 +1220,19 @@ Section DoubleDivGt. spec_div21 spec_compare spec_sub 1 (WW ah al) bl Hbl). apply spec_gcd_gt. rewrite (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW); trivial. - apply Zlt_gt;match goal with | |- ?x mod ?y < ?y => + apply Z.lt_gt;match goal with | |- ?x mod ?y < ?y => destruct (Z_mod_lt x y);zarith end. Spec_w_to_Z bl;exfalso;omega. assert (H:= spec_ww_mod_gt_aux _ _ _ Hgt Hbh). assert (H2 : 0 < [[WW bh bl]]). - simpl;Spec_w_to_Z bl. apply Zlt_le_trans with ([|bh|]*wB);zarith. - apply Zmult_lt_0_compat;zarith. + simpl;Spec_w_to_Z bl. apply Z.lt_le_trans with ([|bh|]*wB);zarith. + apply Z.mul_pos_pos;zarith. apply Zis_gcd_mod;trivial. rewrite <- H. simpl in *;destruct (ww_mod_gt_aux ah al bh bl) as [ |mh ml]. simpl;apply Zis_gcd_0;zarith. - rewrite spec_compare, spec_w_0; case Zcompare_spec; intros Hmh. + rewrite spec_compare, spec_w_0; case Z.compare_spec; intros Hmh. simpl;rewrite <- Hmh;simpl. - rewrite spec_compare, spec_w_0; case Zcompare_spec; intros Hml. + rewrite spec_compare, spec_w_0; case Z.compare_spec; intros Hml. rewrite <- Hml;simpl;apply Zis_gcd_0. simpl; rewrite spec_w_0; simpl. apply Zis_gcd_mod;zarith. @@ -1242,38 +1242,38 @@ Section DoubleDivGt. spec_div21 spec_compare spec_sub 1 (WW bh bl) ml Hml). apply spec_gcd_gt. rewrite (@spec_double_modn1 w w_digits w_zdigits w_0 w_WW); trivial. - apply Zlt_gt;match goal with | |- ?x mod ?y < ?y => + apply Z.lt_gt;match goal with | |- ?x mod ?y < ?y => destruct (Z_mod_lt x y);zarith end. Spec_w_to_Z ml;exfalso;omega. assert ([[WW bh bl]] > [[WW mh ml]]). - rewrite H;simpl; apply Zlt_gt;match goal with | |- ?x mod ?y < ?y => + rewrite H;simpl; apply Z.lt_gt;match goal with | |- ?x mod ?y < ?y => destruct (Z_mod_lt x y);zarith end. assert (H1:= spec_ww_mod_gt_aux _ _ _ H0 Hmh). assert (H3 : 0 < [[WW mh ml]]). - simpl;Spec_w_to_Z ml. apply Zlt_le_trans with ([|mh|]*wB);zarith. - apply Zmult_lt_0_compat;zarith. + simpl;Spec_w_to_Z ml. apply Z.lt_le_trans with ([|mh|]*wB);zarith. + apply Z.mul_pos_pos;zarith. apply Zis_gcd_mod;zarith. simpl in *;rewrite <- H1. destruct (ww_mod_gt_aux bh bl mh ml) as [ |rh rl]. simpl; apply Zis_gcd_0. simpl;apply Hcont. simpl in H1;rewrite H1. - apply Zlt_gt;match goal with | |- ?x mod ?y < ?y => + apply Z.lt_gt;match goal with | |- ?x mod ?y < ?y => destruct (Z_mod_lt x y);zarith end. - apply Zle_trans with (2^n/2). + apply Z.le_trans with (2^n/2). apply Zdiv_le_lower_bound;zarith. - apply Zle_trans with ([|bh|] * wB + [|bl|]);zarith. - assert (H3' := Z_div_mod_eq [[WW bh bl]] [[WW mh ml]] (Zlt_gt _ _ H3)). - assert (H4' : 0 <= [[WW bh bl]]/[[WW mh ml]]). - apply Zge_le;apply Z_div_ge0;zarith. simpl in *;rewrite H1. + apply Z.le_trans with ([|bh|] * wB + [|bl|]);zarith. + assert (H3' := Z_div_mod_eq [[WW bh bl]] [[WW mh ml]] (Z.lt_gt _ _ H3)). + assert (H4 : 0 <= [[WW bh bl]]/[[WW mh ml]]). + apply Z.ge_le;apply Z_div_ge0;zarith. simpl in *;rewrite H1. pattern ([|bh|] * wB + [|bl|]) at 2;rewrite H3'. - destruct (Zle_lt_or_eq _ _ H4'). + Z.le_elim H4. assert (H6' : [[WW bh bl]] mod [[WW mh ml]] = [[WW bh bl]] - [[WW mh ml]] * ([[WW bh bl]]/[[WW mh ml]])). simpl;pattern ([|bh|] * wB + [|bl|]) at 2;rewrite H3';ring. simpl in H6'. assert ([[WW mh ml]] <= [[WW mh ml]] * ([[WW bh bl]]/[[WW mh ml]])). - simpl;pattern ([|mh|]*wB+[|ml|]) at 1;rewrite <- Zmult_1_r;zarith. + simpl;pattern ([|mh|]*wB+[|ml|]) at 1;rewrite <- Z.mul_1_r;zarith. simpl in *;assert (H8 := Z_mod_lt [[WW bh bl]] [[WW mh ml]]);simpl in H8; zarith. assert (H8 := Z_mod_lt [[WW bh bl]] [[WW mh ml]]);simpl in *;zarith. - rewrite <- H4 in H3';rewrite Zmult_0_r in H3';simpl in H3';zarith. + rewrite <- H4 in H3';rewrite Z.mul_0_r in H3';simpl in H3';zarith. pattern n at 1;replace n with (n-1+1);try ring. rewrite Zpower_exp;zarith. change (2^1) with 2. rewrite Z_div_mult;zarith. @@ -1295,27 +1295,27 @@ Section DoubleDivGt. [[ww_gcd_gt_aux p cont ah al bh bl]]. Proof. induction p;intros cont n Hcont ah al bh bl Hgt Hs;simpl ww_gcd_gt_aux. - assert (0 < Zpos p). unfold Zlt;reflexivity. + assert (0 < Zpos p). unfold Z.lt;reflexivity. apply spec_ww_gcd_gt_aux_body with (n := Zpos (xI p) + n); - trivial;rewrite Zpos_xI. + trivial;rewrite Pos2Z.inj_xI. intros. apply IHp with (n := Zpos p + n);zarith. intros. apply IHp with (n := n );zarith. - apply Zle_trans with (2 ^ (2* Zpos p + 1+ n -1));zarith. - apply Zpower_le_monotone2;zarith. - assert (0 < Zpos p). unfold Zlt;reflexivity. + apply Z.le_trans with (2 ^ (2* Zpos p + 1+ n -1));zarith. + apply Z.pow_le_mono_r;zarith. + assert (0 < Zpos p). unfold Z.lt;reflexivity. apply spec_ww_gcd_gt_aux_body with (n := Zpos (xO p) + n );trivial. - rewrite (Zpos_xO p). + rewrite (Pos2Z.inj_xO p). intros. apply IHp with (n := Zpos p + n - 1);zarith. intros. apply IHp with (n := n -1 );zarith. intros;apply Hcont;zarith. - apply Zle_trans with (2^(n-1));zarith. - apply Zpower_le_monotone2;zarith. - apply Zle_trans with (2 ^ (Zpos p + n -1));zarith. - apply Zpower_le_monotone2;zarith. - apply Zle_trans with (2 ^ (2*Zpos p + n -1));zarith. - apply Zpower_le_monotone2;zarith. + apply Z.le_trans with (2^(n-1));zarith. + apply Z.pow_le_mono_r;zarith. + apply Z.le_trans with (2 ^ (Zpos p + n -1));zarith. + apply Z.pow_le_mono_r;zarith. + apply Z.le_trans with (2 ^ (2*Zpos p + n -1));zarith. + apply Z.pow_le_mono_r;zarith. apply spec_ww_gcd_gt_aux_body with (n := n+1);trivial. - rewrite Zplus_comm;trivial. + rewrite Z.add_comm;trivial. ring_simplify (n + 1 - 1);trivial. Qed. @@ -1353,7 +1353,7 @@ Section DoubleDiv. Variable spec_to_Z : forall x, 0 <= [|x|] < wB. Variable spec_ww_1 : [[ww_1]] = 1. Variable spec_ww_compare : forall x y, - ww_compare x y = Zcompare [[x]] [[y]]. + ww_compare x y = Z.compare [[x]] [[y]]. Variable spec_ww_div_gt : forall a b, [[a]] > [[b]] -> 0 < [[b]] -> let (q,r) := ww_div_gt a b in [[a]] = [[q]] * [[b]] + [[r]] /\ @@ -1375,7 +1375,7 @@ Section DoubleDiv. 0 <= [[r]] < [[b]]. Proof. intros a b Hpos;unfold ww_div. - rewrite spec_ww_compare; case Zcompare_spec; intros. + rewrite spec_ww_compare; case Z.compare_spec; intros. simpl;rewrite spec_ww_1;split;zarith. simpl;split;[ring|Spec_ww_to_Z a;zarith]. apply spec_ww_div_gt;auto with zarith. @@ -1385,7 +1385,7 @@ Section DoubleDiv. [[ww_mod a b]] = [[a]] mod [[b]]. Proof. intros a b Hpos;unfold ww_mod. - rewrite spec_ww_compare; case Zcompare_spec; intros. + rewrite spec_ww_compare; case Z.compare_spec; intros. simpl;apply Zmod_unique with 1;try rewrite H;zarith. Spec_ww_to_Z a;symmetry;apply Zmod_small;zarith. apply spec_ww_mod_gt;auto with zarith. @@ -1406,7 +1406,7 @@ Section DoubleDiv. Variable spec_w_0 : [|w_0|] = 0. Variable spec_w_1 : [|w_1|] = 1. Variable spec_compare : - forall x y, w_compare x y = Zcompare [|x|] [|y|]. + forall x y, w_compare x y = Z.compare [|x|] [|y|]. Variable spec_eq0 : forall x, w_eq0 x = true -> [|x|] = 0. Variable spec_gcd_gt : forall a b, [|a|] > [|b|] -> Zis_gcd [|a|] [|b|] [|w_gcd_gt a b|]. @@ -1439,7 +1439,7 @@ Section DoubleDiv. assert (H1:= beta_lex _ _ _ _ _ Hle (spec_to_Z yl) H). Spec_w_to_Z yh;zarith. unfold gcd_cont; rewrite spec_compare, spec_w_1. - case Zcompare_spec; intros Hcmpy. + case Z.compare_spec; intros Hcmpy. simpl;rewrite H;simpl; rewrite spec_ww_1;rewrite <- Hcmpy;apply Zis_gcd_mod;zarith. rewrite <- (Zmod_unique ([|xh|]*wB+[|xl|]) 1 ([|xh|]*wB+[|xl|]) 0);zarith. @@ -1485,7 +1485,7 @@ Section DoubleDiv. Spec_w_to_Z bh;assert ([|bh|] = 0);zarith. rewrite H1 in Hgt;simpl in Hgt. rewrite H1;simpl;auto. clear H. apply spec_gcd_gt_fix with (n:= 0);trivial. - rewrite Zplus_0_r;rewrite spec_ww_digits_. + rewrite Z.add_0_r;rewrite spec_ww_digits_. change (2 ^ Zpos (xO w_digits)) with wwB. Spec_ww_to_Z (WW bh bl);zarith. Qed. @@ -1498,7 +1498,7 @@ Section DoubleDiv. | Eq => a | Lt => ww_gcd_gt b a end). - rewrite spec_ww_compare; case Zcompare_spec; intros Hcmp. + rewrite spec_ww_compare; case Z.compare_spec; intros Hcmp. Spec_ww_to_Z b;rewrite Hcmp. apply Zis_gcd_for_euclid with 1;zarith. ring_simplify ([[b]] - 1 * [[b]]). apply Zis_gcd_0;zarith. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v index 062282f2..5cb7405a 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -62,7 +62,7 @@ Section GENDIVN1. [|a1|] *wB+ [|a2|] = [|q|] * [|b|] + [|r|] /\ 0 <= [|r|] < [|b|]. Variable spec_compare : - forall x y, w_compare x y = Zcompare [|x|] [|y|]. + forall x y, w_compare x y = Z.compare [|x|] [|y|]. Variable spec_sub: forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB. @@ -107,8 +107,8 @@ Section GENDIVN1. destruct H4;split;trivial. rewrite spec_double_WW;trivial. rewrite <- double_wB_wwB. - rewrite Zmult_assoc;rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. - rewrite H0;rewrite Zmult_plus_distr_l;rewrite <- Zplus_assoc. + rewrite Z.mul_assoc;rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. + rewrite H0;rewrite Z.mul_add_distr_r;rewrite <- Z.add_assoc. rewrite H4;ring. Qed. @@ -160,7 +160,7 @@ Section GENDIVN1. Lemma p_lt_double_digits : forall n, [|p|] <= Zpos (w_digits << n). Proof. induction n;simpl. trivial. - case (spec_to_Z p); rewrite Pshiftl_nat_S, Zpos_xO;auto with zarith. + case (spec_to_Z p); rewrite Pshiftl_nat_S, Pos2Z.inj_xO;auto with zarith. Qed. Lemma spec_double_divn1_p : forall n r h l, @@ -225,11 +225,11 @@ Section GENDIVN1. replace (2 ^ (Zpos (w_digits << (S n)) - [|p|])) with (2^(Zpos (w_digits << n) - [|p|])*2^Zpos (w_digits << n)). rewrite Zdiv_mult_cancel_r;auto with zarith. - rewrite Zmult_plus_distr_l with (p:= 2^[|p|]). + rewrite Z.mul_add_distr_r with (p:= 2^[|p|]). pattern ([!n|hl!] * 2^[|p|]) at 2; rewrite (shift_unshift_mod (Zpos(w_digits << n))([|p|])([!n|hl!])); auto with zarith. - rewrite Zplus_assoc. + rewrite Z.add_assoc. replace ([!n|hh!] * 2^Zpos (w_digits << n)* 2^[|p|] + ([!n|hl!] / 2^(Zpos (w_digits << n)-[|p|])* @@ -238,7 +238,7 @@ Section GENDIVN1. (([!n|hh!] *2^[|p|] + double_to_Z w_digits w_to_Z n hl / 2^(Zpos (w_digits << n)-[|p|])) * 2^Zpos(w_digits << n));try (ring;fail). - rewrite <- Zplus_assoc. + rewrite <- Z.add_assoc. rewrite <- (Zmod_shift_r ([|p|]));auto with zarith. replace (2 ^ Zpos (w_digits << n) * 2 ^ Zpos (w_digits << n)) with @@ -246,12 +246,12 @@ Section GENDIVN1. rewrite (Zmod_shift_r (Zpos (w_digits << n)));auto with zarith. replace (2 ^ (Zpos (w_digits << n) + Zpos (w_digits << n))) with (2^Zpos(w_digits << n) *2^Zpos(w_digits << n)). - rewrite (Zmult_comm (([!n|hh!] * 2 ^ [|p|] + + rewrite (Z.mul_comm (([!n|hh!] * 2 ^ [|p|] + [!n|hl!] / 2 ^ (Zpos (w_digits << n) - [|p|])))). rewrite Zmult_mod_distr_l;auto with zarith. ring. rewrite Zpower_exp;auto with zarith. - assert (0 < Zpos (w_digits << n)). unfold Zlt;reflexivity. + assert (0 < Zpos (w_digits << n)). unfold Z.lt;reflexivity. auto with zarith. apply Z_mod_lt;auto with zarith. rewrite Zpower_exp;auto with zarith. @@ -320,7 +320,7 @@ Section GENDIVN1. replace (Zpos w_digits - Zpos w_digits) with 0;try ring. simpl. rewrite <- (Zdiv_unique [|x|] 1 [|x|] 0);auto with zarith. assert (U2 := spec_double_digits n). - assert (U3 : 0 < Zpos w_digits). exact (refl_equal Lt). + assert (U3 : 0 < Zpos w_digits). exact (eq_refl Lt). destruct x;unfold high;fold high. unfold double_to_Z,zn2z_to_Z;rewrite spec_0. rewrite Zdiv_0_l;trivial. @@ -365,30 +365,30 @@ Section GENDIVN1. intros n a b H. unfold double_divn1. case (spec_head0 H); intros H0 H1. case (spec_to_Z (w_head0 b)); intros HH1 HH2. - rewrite spec_compare; case Zcompare_spec; + rewrite spec_compare; case Z.compare_spec; rewrite spec_0; intros H2; auto with zarith. assert (Hv1: wB/2 <= [|b|]). - generalize H0; rewrite H2; rewrite Zpower_0_r; - rewrite Zmult_1_l; auto. + generalize H0; rewrite H2; rewrite Z.pow_0_r; + rewrite Z.mul_1_l; auto. assert (Hv2: [|w_0|] < [|b|]). rewrite spec_0; auto. generalize (spec_double_divn1_0 Hv1 n a Hv2). - rewrite spec_0;rewrite Zmult_0_l; rewrite Zplus_0_l; auto. + rewrite spec_0;rewrite Z.mul_0_l; rewrite Z.add_0_l; auto. contradict H2; auto with zarith. assert (HHHH : 0 < [|w_head0 b|]); auto with zarith. assert ([|w_head0 b|] < Zpos w_digits). - case (Zle_or_lt (Zpos w_digits) [|w_head0 b|]); auto; intros HH. + case (Z.le_gt_cases (Zpos w_digits) [|w_head0 b|]); auto; intros HH. assert (2 ^ [|w_head0 b|] < wB). - apply Zle_lt_trans with (2 ^ [|w_head0 b|] * [|b|]);auto with zarith. + apply Z.le_lt_trans with (2 ^ [|w_head0 b|] * [|b|]);auto with zarith. replace (2 ^ [|w_head0 b|]) with (2^[|w_head0 b|] * 1);try (ring;fail). - apply Zmult_le_compat;auto with zarith. + apply Z.mul_le_mono_nonneg;auto with zarith. assert (wB <= 2^[|w_head0 b|]). unfold base;apply Zpower_le_monotone;auto with zarith. omega. assert ([|w_add_mul_div (w_head0 b) b w_0|] = 2 ^ [|w_head0 b|] * [|b|]). rewrite (spec_add_mul_div b w_0); auto with zarith. rewrite spec_0;rewrite Zdiv_0_l; try omega. - rewrite Zplus_0_r; rewrite Zmult_comm. + rewrite Z.add_0_r; rewrite Z.mul_comm. rewrite Zmod_small; auto with zarith. assert (H5 := spec_to_Z (high n a)). assert @@ -396,21 +396,21 @@ Section GENDIVN1. <[|w_add_mul_div (w_head0 b) b w_0|]). rewrite H4. rewrite spec_add_mul_div;auto with zarith. - rewrite spec_0;rewrite Zmult_0_l;rewrite Zplus_0_l. + rewrite spec_0;rewrite Z.mul_0_l;rewrite Z.add_0_l. assert (([|high n a|]/2^(Zpos w_digits - [|w_head0 b|])) < wB). apply Zdiv_lt_upper_bound;auto with zarith. - apply Zlt_le_trans with wB;auto with zarith. + apply Z.lt_le_trans with wB;auto with zarith. pattern wB at 1;replace wB with (wB*1);try ring. - apply Zmult_le_compat;auto with zarith. - assert (H6 := Zpower_gt_0 2 (Zpos w_digits - [|w_head0 b|])); + apply Z.mul_le_mono_nonneg;auto with zarith. + assert (H6 := Z.pow_pos_nonneg 2 (Zpos w_digits - [|w_head0 b|])); auto with zarith. rewrite Zmod_small;auto with zarith. apply Zdiv_lt_upper_bound;auto with zarith. - apply Zlt_le_trans with wB;auto with zarith. - apply Zle_trans with (2 ^ [|w_head0 b|] * [|b|] * 2). + apply Z.lt_le_trans with wB;auto with zarith. + apply Z.le_trans with (2 ^ [|w_head0 b|] * [|b|] * 2). rewrite <- wB_div_2; try omega. - apply Zmult_le_compat;auto with zarith. - pattern 2 at 1;rewrite <- Zpower_1_r. + apply Z.mul_le_mono_nonneg;auto with zarith. + pattern 2 at 1;rewrite <- Z.pow_1_r. apply Zpower_le_monotone;split;auto with zarith. rewrite <- H4 in H0. assert (Hb3: [|w_head0 b|] <= Zpos w_digits); auto with zarith. @@ -420,9 +420,9 @@ Section GENDIVN1. (double_0 w_0 n)) as (q,r). assert (U:= spec_double_digits n). rewrite spec_double_0 in H7;trivial;rewrite Zdiv_0_l in H7. - rewrite Zplus_0_r in H7. + rewrite Z.add_0_r in H7. rewrite spec_add_mul_div in H7;auto with zarith. - rewrite spec_0 in H7;rewrite Zmult_0_l in H7;rewrite Zplus_0_l in H7. + rewrite spec_0 in H7;rewrite Z.mul_0_l in H7;rewrite Z.add_0_l in H7. assert (([|high n a|] / 2 ^ (Zpos w_digits - [|w_head0 b|])) mod wB = [!n|a!] / 2^(Zpos (w_digits << n) - [|w_head0 b|])). rewrite Zmod_small;auto with zarith. @@ -431,29 +431,29 @@ Section GENDIVN1. replace (Zpos (w_digits << n) - Zpos w_digits + (Zpos w_digits - [|w_head0 b|])) with (Zpos (w_digits << n) - [|w_head0 b|]);trivial;ring. - assert (H8 := Zpower_gt_0 2 (Zpos w_digits - [|w_head0 b|]));auto with zarith. + assert (H8 := Z.pow_pos_nonneg 2 (Zpos w_digits - [|w_head0 b|]));auto with zarith. split;auto with zarith. - apply Zle_lt_trans with ([|high n a|]);auto with zarith. + apply Z.le_lt_trans with ([|high n a|]);auto with zarith. apply Zdiv_le_upper_bound;auto with zarith. - pattern ([|high n a|]) at 1;rewrite <- Zmult_1_r. - apply Zmult_le_compat;auto with zarith. + pattern ([|high n a|]) at 1;rewrite <- Z.mul_1_r. + apply Z.mul_le_mono_nonneg;auto with zarith. rewrite H8 in H7;unfold double_wB,base in H7. rewrite <- shift_unshift_mod in H7;auto with zarith. rewrite H4 in H7. assert ([|w_add_mul_div (w_sub w_zdigits (w_head0 b)) w_0 r|] = [|r|]/2^[|w_head0 b|]). rewrite spec_add_mul_div. - rewrite spec_0;rewrite Zmult_0_l;rewrite Zplus_0_l. + rewrite spec_0;rewrite Z.mul_0_l;rewrite Z.add_0_l. replace (Zpos w_digits - [|w_sub w_zdigits (w_head0 b)|]) with ([|w_head0 b|]). rewrite Zmod_small;auto with zarith. assert (H9 := spec_to_Z r). split;auto with zarith. - apply Zle_lt_trans with ([|r|]);auto with zarith. + apply Z.le_lt_trans with ([|r|]);auto with zarith. apply Zdiv_le_upper_bound;auto with zarith. - pattern ([|r|]) at 1;rewrite <- Zmult_1_r. - apply Zmult_le_compat;auto with zarith. - assert (H10 := Zpower_gt_0 2 ([|w_head0 b|]));auto with zarith. + pattern ([|r|]) at 1;rewrite <- Z.mul_1_r. + apply Z.mul_le_mono_nonneg;auto with zarith. + assert (H10 := Z.pow_pos_nonneg 2 ([|w_head0 b|]));auto with zarith. rewrite spec_sub. rewrite Zmod_small; auto with zarith. split; auto with zarith. @@ -475,7 +475,7 @@ Section GENDIVN1. auto with zarith. rewrite H9. apply Zdiv_lt_upper_bound;auto with zarith. - rewrite Zmult_comm;auto with zarith. + rewrite Z.mul_comm;auto with zarith. exact (spec_double_to_Z w_digits w_to_Z spec_to_Z n a). Qed. @@ -498,7 +498,7 @@ Section GENDIVN1. double_modn1 n a b = snd (double_divn1 n a b). Proof. intros n a b;unfold double_divn1,double_modn1. - rewrite spec_compare; case Zcompare_spec; + rewrite spec_compare; case Z.compare_spec; rewrite spec_0; intros H2; auto with zarith. apply spec_double_modn1_0. apply spec_double_modn1_0. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v index a6a0fc8e..0a70dbf4 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -104,9 +104,9 @@ Section DoubleLift. Variable spec_w_W0 : forall h, [[w_W0 h]] = [|h|] * wB. Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. Variable spec_compare : forall x y, - w_compare x y = Zcompare [|x|] [|y|]. + w_compare x y = Z.compare [|x|] [|y|]. Variable spec_ww_compare : forall x y, - ww_compare x y = Zcompare [[x]] [[y]]. + ww_compare x y = Z.compare [[x]] [[y]]. Variable spec_ww_digits : ww_Digits = xO w_digits. Variable spec_w_head00 : forall x, [|x|] = 0 -> [|w_head0 x|] = Zpos w_digits. Variable spec_w_head0 : forall x, 0 < [|x|] -> @@ -140,20 +140,20 @@ Section DoubleLift. case (spec_to_Z xh); intros Hx1 Hx2. case (spec_to_Z xl); intros Hy1 Hy2. assert (F1: [|xh|] = 0). - case (Zle_lt_or_eq _ _ Hy1); auto; intros Hy3. - absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. - apply Zlt_le_trans with (1 := Hy3); auto with zarith. - pattern [|xl|] at 1; rewrite <- (Zplus_0_l [|xl|]). - apply Zplus_le_compat_r; auto with zarith. - case (Zle_lt_or_eq _ _ Hx1); auto; intros Hx3. - absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. - rewrite <- Hy3; rewrite Zplus_0_r; auto with zarith. - apply Zmult_lt_0_compat; auto with zarith. - rewrite spec_compare. case Zcompare_spec. + { Z.le_elim Hy1; auto. + - absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. + apply Z.lt_le_trans with (1 := Hy1); auto with zarith. + pattern [|xl|] at 1; rewrite <- (Z.add_0_l [|xl|]). + apply Z.add_le_mono_r; auto with zarith. + - Z.le_elim Hx1; auto. + absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. + rewrite <- Hy1; rewrite Z.add_0_r; auto with zarith. + apply Z.mul_pos_pos; auto with zarith. } + rewrite spec_compare. case Z.compare_spec. intros H; simpl. rewrite spec_w_add; rewrite spec_w_head00. rewrite spec_zdigits; rewrite spec_ww_digits. - rewrite Zpos_xO; auto with zarith. + rewrite Pos2Z.inj_xO; auto with zarith. rewrite F1 in Hx; auto with zarith. rewrite spec_w_0; auto with zarith. rewrite spec_w_0; auto with zarith. @@ -163,43 +163,43 @@ Section DoubleLift. wwB/ 2 <= 2 ^ [[ww_head0 x]] * [[x]] < wwB. Proof. clear spec_ww_zdigits. - rewrite wwB_div_2;rewrite Zmult_comm;rewrite wwB_wBwB. + rewrite wwB_div_2;rewrite Z.mul_comm;rewrite wwB_wBwB. assert (U:= lt_0_wB w_digits); destruct x as [ |xh xl];simpl ww_to_Z;intros H. - unfold Zlt in H;discriminate H. - rewrite spec_compare, spec_w_0. case Zcompare_spec; intros H0. - rewrite <- H0 in *. simpl Zplus. simpl in H. + unfold Z.lt in H;discriminate H. + rewrite spec_compare, spec_w_0. case Z.compare_spec; intros H0. + rewrite <- H0 in *. simpl Z.add. simpl in H. case (spec_to_Z w_zdigits); case (spec_to_Z (w_head0 xl)); intros HH1 HH2 HH3 HH4. rewrite spec_w_add. rewrite spec_zdigits; rewrite Zpower_exp; auto with zarith. case (spec_w_head0 H); intros H1 H2. - rewrite Zpower_2; fold wB; rewrite <- Zmult_assoc; split. - apply Zmult_le_compat_l; auto with zarith. - apply Zmult_lt_compat_l; auto with zarith. + rewrite Z.pow_2_r; fold wB; rewrite <- Z.mul_assoc; split. + apply Z.mul_le_mono_nonneg_l; auto with zarith. + apply Z.mul_lt_mono_pos_l; auto with zarith. assert (H1 := spec_w_head0 H0). rewrite spec_w_0W. split. - rewrite Zmult_plus_distr_r;rewrite Zmult_assoc. - apply Zle_trans with (2 ^ [|w_head0 xh|] * [|xh|] * wB). - rewrite Zmult_comm; zarith. + rewrite Z.mul_add_distr_l;rewrite Z.mul_assoc. + apply Z.le_trans with (2 ^ [|w_head0 xh|] * [|xh|] * wB). + rewrite Z.mul_comm; zarith. assert (0 <= 2 ^ [|w_head0 xh|] * [|xl|]);zarith. - assert (H2:=spec_to_Z xl);apply Zmult_le_0_compat;zarith. + assert (H2:=spec_to_Z xl);apply Z.mul_nonneg_nonneg;zarith. case (spec_to_Z (w_head0 xh)); intros H2 _. generalize ([|w_head0 xh|]) H1 H2;clear H1 H2; intros p H1 H2. assert (Eq1 : 2^p < wB). - rewrite <- (Zmult_1_r (2^p));apply Zle_lt_trans with (2^p*[|xh|]);zarith. + rewrite <- (Z.mul_1_r (2^p));apply Z.le_lt_trans with (2^p*[|xh|]);zarith. assert (Eq2: p < Zpos w_digits). - destruct (Zle_or_lt (Zpos w_digits) p);trivial;contradict Eq1. - apply Zle_not_lt;unfold base;apply Zpower_le_monotone;zarith. + destruct (Z.le_gt_cases (Zpos w_digits) p);trivial;contradict Eq1. + apply Z.le_ngt;unfold base;apply Zpower_le_monotone;zarith. assert (Zpos w_digits = p + (Zpos w_digits - p)). ring. - rewrite Zpower_2. + rewrite Z.pow_2_r. unfold base at 2;rewrite H3;rewrite Zpower_exp;zarith. - rewrite <- Zmult_assoc; apply Zmult_lt_compat_l; zarith. - rewrite <- (Zplus_0_r (2^(Zpos w_digits - p)*wB));apply beta_lex_inv;zarith. - apply Zmult_lt_reg_r with (2 ^ p); zarith. + rewrite <- Z.mul_assoc; apply Z.mul_lt_mono_pos_l; zarith. + rewrite <- (Z.add_0_r (2^(Zpos w_digits - p)*wB));apply beta_lex_inv;zarith. + apply Z.mul_lt_mono_pos_r with (2 ^ p); zarith. rewrite <- Zpower_exp;zarith. - rewrite Zmult_comm;ring_simplify (Zpos w_digits - p + p);fold wB;zarith. + rewrite Z.mul_comm;ring_simplify (Zpos w_digits - p + p);fold wB;zarith. assert (H1 := spec_to_Z xh);zarith. Qed. @@ -211,22 +211,22 @@ Section DoubleLift. case (spec_to_Z xh); intros Hx1 Hx2. case (spec_to_Z xl); intros Hy1 Hy2. assert (F1: [|xh|] = 0). - case (Zle_lt_or_eq _ _ Hy1); auto; intros Hy3. - absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. - apply Zlt_le_trans with (1 := Hy3); auto with zarith. - pattern [|xl|] at 1; rewrite <- (Zplus_0_l [|xl|]). - apply Zplus_le_compat_r; auto with zarith. - case (Zle_lt_or_eq _ _ Hx1); auto; intros Hx3. - absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. - rewrite <- Hy3; rewrite Zplus_0_r; auto with zarith. - apply Zmult_lt_0_compat; auto with zarith. + { Z.le_elim Hy1; auto. + - absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. + apply Z.lt_le_trans with (1 := Hy1); auto with zarith. + pattern [|xl|] at 1; rewrite <- (Z.add_0_l [|xl|]). + apply Z.add_le_mono_r; auto with zarith. + - Z.le_elim Hx1; auto. + absurd (0 < [|xh|] * wB + [|xl|]); auto with zarith. + rewrite <- Hy1; rewrite Z.add_0_r; auto with zarith. + apply Z.mul_pos_pos; auto with zarith. } assert (F2: [|xl|] = 0). rewrite F1 in Hx; auto with zarith. - rewrite spec_compare; case Zcompare_spec. + rewrite spec_compare; case Z.compare_spec. intros H; simpl. rewrite spec_w_add; rewrite spec_w_tail00; auto. rewrite spec_zdigits; rewrite spec_ww_digits. - rewrite Zpos_xO; auto with zarith. + rewrite Pos2Z.inj_xO; auto with zarith. rewrite spec_w_0; auto with zarith. rewrite spec_w_0; auto with zarith. Qed. @@ -236,51 +236,51 @@ Section DoubleLift. Proof. clear spec_ww_zdigits. destruct x as [ |xh xl];simpl ww_to_Z;intros H. - unfold Zlt in H;discriminate H. - rewrite spec_compare, spec_w_0. case Zcompare_spec; intros H0. - rewrite <- H0; rewrite Zplus_0_r. + unfold Z.lt in H;discriminate H. + rewrite spec_compare, spec_w_0. case Z.compare_spec; intros H0. + rewrite <- H0; rewrite Z.add_0_r. case (spec_to_Z (w_tail0 xh)); intros HH1 HH2. - generalize H; rewrite <- H0; rewrite Zplus_0_r; clear H; intros H. + generalize H; rewrite <- H0; rewrite Z.add_0_r; clear H; intros H. case (@spec_w_tail0 xh). - apply Zmult_lt_reg_r with wB; auto with zarith. + apply Z.mul_lt_mono_pos_r with wB; auto with zarith. unfold base; auto with zarith. intros z (Hz1, Hz2); exists z; split; auto. - rewrite spec_w_add; rewrite (fun x => Zplus_comm [|x|]). + rewrite spec_w_add; rewrite (fun x => Z.add_comm [|x|]). rewrite spec_zdigits; rewrite Zpower_exp; auto with zarith. - rewrite Zmult_assoc; rewrite <- Hz2; auto. + rewrite Z.mul_assoc; rewrite <- Hz2; auto. case (spec_to_Z (w_tail0 xh)); intros HH1 HH2. case (spec_w_tail0 H0); intros z (Hz1, Hz2). assert (Hp: [|w_tail0 xl|] < Zpos w_digits). - case (Zle_or_lt (Zpos w_digits) [|w_tail0 xl|]); auto; intros H1. + case (Z.le_gt_cases (Zpos w_digits) [|w_tail0 xl|]); auto; intros H1. absurd (2 ^ (Zpos w_digits) <= 2 ^ [|w_tail0 xl|]). - apply Zlt_not_le. + apply Z.lt_nge. case (spec_to_Z xl); intros HH3 HH4. - apply Zle_lt_trans with (2 := HH4). - apply Zle_trans with (1 * 2 ^ [|w_tail0 xl|]); auto with zarith. + apply Z.le_lt_trans with (2 := HH4). + apply Z.le_trans with (1 * 2 ^ [|w_tail0 xl|]); auto with zarith. rewrite Hz2. - apply Zmult_le_compat_r; auto with zarith. + apply Z.mul_le_mono_nonneg_r; auto with zarith. apply Zpower_le_monotone; auto with zarith. exists ([|xh|] * (2 ^ ((Zpos w_digits - [|w_tail0 xl|]) - 1)) + z); split. - apply Zplus_le_0_compat; auto. - apply Zmult_le_0_compat; auto with zarith. + apply Z.add_nonneg_nonneg; auto. + apply Z.mul_nonneg_nonneg; auto with zarith. case (spec_to_Z xh); auto. rewrite spec_w_0W. - rewrite (Zmult_plus_distr_r 2); rewrite <- Zplus_assoc. - rewrite Zmult_plus_distr_l; rewrite <- Hz2. - apply f_equal2 with (f := Zplus); auto. - rewrite (Zmult_comm 2). - repeat rewrite <- Zmult_assoc. - apply f_equal2 with (f := Zmult); auto. + rewrite (Z.mul_add_distr_l 2); rewrite <- Z.add_assoc. + rewrite Z.mul_add_distr_r; rewrite <- Hz2. + apply f_equal2 with (f := Z.add); auto. + rewrite (Z.mul_comm 2). + repeat rewrite <- Z.mul_assoc. + apply f_equal2 with (f := Z.mul); auto. case (spec_to_Z (w_tail0 xl)); intros HH3 HH4. - pattern 2 at 2; rewrite <- Zpower_1_r. + pattern 2 at 2; rewrite <- Z.pow_1_r. lazy beta; repeat rewrite <- Zpower_exp; auto with zarith. - unfold base; apply f_equal with (f := Zpower 2); auto with zarith. + unfold base; apply f_equal with (f := Z.pow 2); auto with zarith. contradict H0; case (spec_to_Z xl); auto with zarith. Qed. - Hint Rewrite Zdiv_0_l Zmult_0_l Zplus_0_l Zmult_0_r Zplus_0_r + Hint Rewrite Zdiv_0_l Z.mul_0_l Z.add_0_l Z.mul_0_r Z.add_0_r spec_w_W0 spec_w_0W spec_w_WW spec_w_0 (wB_div w_digits w_to_Z spec_to_Z) (wB_div_plus w_digits w_to_Z spec_to_Z) : w_rewrite. @@ -304,20 +304,20 @@ Section DoubleLift. intros xh xl yh yl p zdigits;assert (HwwB := wwB_pos w_digits). case (spec_to_w_Z p); intros Hv1 Hv2. replace (Zpos (xO w_digits)) with (Zpos w_digits + Zpos w_digits). - 2 : rewrite Zpos_xO;ring. + 2 : rewrite Pos2Z.inj_xO;ring. replace (Zpos w_digits + Zpos w_digits - [[p]]) with (Zpos w_digits + (Zpos w_digits - [[p]])). 2:ring. intros Hp; assert (Hxh := spec_to_Z xh);assert (Hxl:=spec_to_Z xl); assert (Hx := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh xl)); simpl in Hx;assert (Hyh := spec_to_Z yh);assert (Hyl:=spec_to_Z yl); assert (Hy:=spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW yh yl));simpl in Hy. - rewrite spec_ww_compare; case Zcompare_spec; intros H1. + rewrite spec_ww_compare; case Z.compare_spec; intros H1. rewrite H1; unfold zdigits; rewrite spec_w_0W. - rewrite spec_zdigits; rewrite Zminus_diag; rewrite Zplus_0_r. + rewrite spec_zdigits; rewrite Z.sub_diag; rewrite Z.add_0_r. simpl ww_to_Z; w_rewrite;zarith. fold wB. - rewrite Zmult_plus_distr_l;rewrite <- Zmult_assoc;rewrite <- Zplus_assoc. - rewrite <- Zpower_2. + rewrite Z.mul_add_distr_r;rewrite <- Z.mul_assoc;rewrite <- Z.add_assoc. + rewrite <- Z.pow_2_r. rewrite <- wwB_wBwB;apply Zmod_unique with [|xh|]. exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xl yh)). ring. simpl ww_to_Z; w_rewrite;zarith. @@ -327,7 +327,7 @@ Section DoubleLift. case (spec_to_w_Z p); intros HH1 HH2; split; auto. generalize H1; unfold zdigits; rewrite spec_w_0W; rewrite spec_zdigits; intros tmp. - apply Zlt_le_trans with (1 := tmp). + apply Z.lt_le_trans with (1 := tmp). unfold base. apply Zpower2_le_lin; auto with zarith. 2: generalize H1; unfold zdigits; rewrite spec_w_0W; @@ -338,16 +338,16 @@ Section DoubleLift. rewrite HH0; auto with zarith. repeat rewrite spec_w_add_mul_div with (1 := HH). rewrite HH0. - rewrite Zmult_plus_distr_l. + rewrite Z.mul_add_distr_r. pattern ([|xl|] * 2 ^ [[p]]) at 2; rewrite shift_unshift_mod with (n:= Zpos w_digits);fold wB;zarith. replace ([|xh|] * wB * 2^[[p]]) with ([|xh|] * 2^[[p]] * wB). 2:ring. - rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. rewrite <- Zplus_assoc. + rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. rewrite <- Z.add_assoc. unfold base at 5;rewrite <- Zmod_shift_r;zarith. unfold base;rewrite Zmod_shift_r with (b:= Zpos (ww_digits w_digits)); fold wB;fold wwB;zarith. - rewrite wwB_wBwB;rewrite Zpower_2; rewrite Zmult_mod_distr_r;zarith. - unfold ww_digits;rewrite Zpos_xO;zarith. apply Z_mod_lt;zarith. + rewrite wwB_wBwB;rewrite Z.pow_2_r; rewrite Zmult_mod_distr_r;zarith. + unfold ww_digits;rewrite Pos2Z.inj_xO;zarith. apply Z_mod_lt;zarith. split;zarith. apply Zdiv_lt_upper_bound;zarith. rewrite <- Zpower_exp;zarith. ring_simplify ([[p]] + (Zpos w_digits - [[p]]));fold wB;zarith. @@ -362,10 +362,10 @@ Section DoubleLift. rewrite <- Zmod_div_mod; auto with zarith. rewrite Zmod_small; auto with zarith. split; auto with zarith. - apply Zle_lt_trans with (Zpos w_digits); auto with zarith. + apply Z.le_lt_trans with (Zpos w_digits); auto with zarith. unfold base; apply Zpower2_lt_lin; auto with zarith. exists wB; unfold base. - unfold ww_digits; rewrite (Zpos_xO w_digits). + unfold ww_digits; rewrite (Pos2Z.inj_xO w_digits). rewrite <- Zpower_exp; auto with zarith. apply f_equal with (f := fun x => 2 ^ x); auto with zarith. assert (HH: [|low (ww_sub p zdigits)|] <= Zpos w_digits). @@ -378,25 +378,25 @@ Section DoubleLift. pattern wB at 5;replace wB with (2^(([[p]] - Zpos w_digits) + (Zpos w_digits - ([[p]] - Zpos w_digits)))). - rewrite Zpower_exp;zarith. rewrite Zmult_assoc. + rewrite Zpower_exp;zarith. rewrite Z.mul_assoc. rewrite Z_div_plus_l;zarith. rewrite shift_unshift_mod with (a:= [|yh|]) (p:= [[p]] - Zpos w_digits) (n := Zpos w_digits);zarith. fold wB. set (u := [[p]] - Zpos w_digits). replace [[p]] with (u + Zpos w_digits);zarith. - rewrite Zpower_exp;zarith. rewrite Zmult_assoc. fold wB. - repeat rewrite Zplus_assoc. rewrite <- Zmult_plus_distr_l. - repeat rewrite <- Zplus_assoc. + rewrite Zpower_exp;zarith. rewrite Z.mul_assoc. fold wB. + repeat rewrite Z.add_assoc. rewrite <- Z.mul_add_distr_r. + repeat rewrite <- Z.add_assoc. unfold base;rewrite Zmod_shift_r with (b:= Zpos (ww_digits w_digits)); fold wB;fold wwB;zarith. unfold base;rewrite Zmod_shift_r with (a:= Zpos w_digits) (b:= Zpos w_digits);fold wB;fold wwB;zarith. - rewrite wwB_wBwB; rewrite Zpower_2; rewrite Zmult_mod_distr_r;zarith. - rewrite Zmult_plus_distr_l. + rewrite wwB_wBwB; rewrite Z.pow_2_r; rewrite Zmult_mod_distr_r;zarith. + rewrite Z.mul_add_distr_r. replace ([|xh|] * wB * 2 ^ u) with ([|xh|]*2^u*wB). 2:ring. - repeat rewrite <- Zplus_assoc. - rewrite (Zplus_comm ([|xh|] * 2 ^ u * wB)). + repeat rewrite <- Z.add_assoc. + rewrite (Z.add_comm ([|xh|] * 2 ^ u * wB)). rewrite Z_mod_plus;zarith. rewrite Z_mod_mult;zarith. unfold base;rewrite <- Zmod_shift_r;zarith. fold base;apply Z_mod_lt;zarith. unfold u; split;zarith. @@ -404,7 +404,7 @@ Section DoubleLift. rewrite <- Zpower_exp;zarith. fold u. ring_simplify (u + (Zpos w_digits - u)); fold - wB;zarith. unfold ww_digits;rewrite Zpos_xO;zarith. + wB;zarith. unfold ww_digits;rewrite Pos2Z.inj_xO;zarith. unfold base;rewrite <- Zmod_shift_r;zarith. fold base;apply Z_mod_lt;zarith. unfold u; split;zarith. unfold u; split;zarith. @@ -434,14 +434,14 @@ Section DoubleLift. clear H1;w_rewrite);simpl ww_add_mul_div. replace [[WW w_0 w_0]] with 0;[w_rewrite|simpl;w_rewrite;trivial]. intros Heq;rewrite <- Heq;clear Heq; auto. - rewrite spec_ww_compare. case Zcompare_spec; intros H1; w_rewrite. + rewrite spec_ww_compare. case Z.compare_spec; intros H1; w_rewrite. rewrite (spec_w_add_mul_div w_0 w_0);w_rewrite;zarith. generalize H1; w_rewrite; rewrite spec_zdigits; clear H1; intros H1. assert (HH0: [|low p|] = [[p]]). rewrite spec_low. apply Zmod_small. case (spec_to_w_Z p); intros HH1 HH2; split; auto. - apply Zlt_le_trans with (1 := H1). + apply Z.lt_le_trans with (1 := H1). unfold base; apply Zpower2_le_lin; auto with zarith. rewrite HH0; auto with zarith. replace [[WW w_0 w_0]] with 0;[w_rewrite|simpl;w_rewrite;trivial]. @@ -449,7 +449,7 @@ Section DoubleLift. generalize (spec_ww_compare p (w_0W w_zdigits)); case ww_compare; intros H1; w_rewrite. rewrite (spec_w_add_mul_div w_0 w_0);w_rewrite;zarith. - rewrite Zpos_xO in H;zarith. + rewrite Pos2Z.inj_xO in H;zarith. assert (HH: [|low (ww_sub p (w_0W w_zdigits)) |] = [[p]] - Zpos w_digits). symmetry in H1; change ([[p]] > [[w_0W w_zdigits]]) in H1. revert H1. @@ -458,12 +458,12 @@ Section DoubleLift. rewrite <- Zmod_div_mod; auto with zarith. rewrite Zmod_small; auto with zarith. split; auto with zarith. - apply Zle_lt_trans with (Zpos w_digits); auto with zarith. + apply Z.le_lt_trans with (Zpos w_digits); auto with zarith. unfold base; apply Zpower2_lt_lin; auto with zarith. unfold base; auto with zarith. unfold base; auto with zarith. exists wB; unfold base. - unfold ww_digits; rewrite (Zpos_xO w_digits). + unfold ww_digits; rewrite (Pos2Z.inj_xO w_digits). rewrite <- Zpower_exp; auto with zarith. apply f_equal with (f := fun x => 2 ^ x); auto with zarith. case (spec_to_Z xh); auto with zarith. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v index 0032d2c3..7a92ff0c 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -246,7 +246,7 @@ Section DoubleMul. Variable spec_w_W0 : forall h, [[w_W0 h]] = [|h|] * wB. Variable spec_w_0W : forall l, [[w_0W l]] = [|l|]. Variable spec_w_compare : - forall x y, w_compare x y = Zcompare [|x|] [|y|]. + forall x y, w_compare x y = Z.compare [|x|] [|y|]. Variable spec_w_succ : forall x, [|w_succ x|] = ([|x|] + 1) mod wB. Variable spec_w_add_c : forall x y, [+|w_add_c x y|] = [|x|] + [|y|]. Variable spec_w_add : forall x y, [|w_add x y|] = ([|x|] + [|y|]) mod wB. @@ -325,7 +325,7 @@ Section DoubleMul. destruct cc as [ | cch ccl]; simpl zn2z_to_Z; simpl ww_to_Z. rewrite spec_ww_add;rewrite spec_w_W0;rewrite Zmod_small; rewrite wwB_wBwB. ring. - rewrite <- (Zplus_0_r ([|wc|]*wB));rewrite H;apply mult_add_ineq3;zarith. + rewrite <- (Z.add_0_r ([|wc|]*wB));rewrite H;apply mult_add_ineq3;zarith. simpl ww_to_Z in H1. assert (U:=spec_to_Z cch). assert ([|wc|]*wB + [|cch|] <= 2*wB - 3). destruct (Z_le_gt_dec ([|wc|]*wB + [|cch|]) (2*wB - 3));trivial. @@ -335,21 +335,21 @@ Section DoubleMul. assert (H5 := Zmult_lt_b _ _ _ (spec_to_Z xl) (spec_to_Z yh)). omega. generalize H3;clear H3;rewrite <- H1. - rewrite Zplus_assoc; rewrite Zpower_2; rewrite Zmult_assoc; - rewrite <- Zmult_plus_distr_l. + rewrite Z.add_assoc; rewrite Z.pow_2_r; rewrite Z.mul_assoc; + rewrite <- Z.mul_add_distr_r. assert (((2 * wB - 4) + 2)*wB <= ([|wc|] * wB + [|cch|])*wB). - apply Zmult_le_compat;zarith. - rewrite Zmult_plus_distr_l in H3. + apply Z.mul_le_mono_nonneg;zarith. + rewrite Z.mul_add_distr_r in H3. intros. assert (U2 := spec_to_Z ccl);omega. generalize (spec_ww_add_c (w_W0 ccl) ll);destruct (ww_add_c (w_W0 ccl) ll) - as [l|l];unfold interp_carry;rewrite spec_w_W0;try rewrite Zmult_1_l; + as [l|l];unfold interp_carry;rewrite spec_w_W0;try rewrite Z.mul_1_l; simpl zn2z_to_Z; try rewrite spec_ww_add;try rewrite spec_ww_add_carry;rewrite spec_w_WW; rewrite Zmod_small;rewrite wwB_wBwB;intros. rewrite H4;ring. rewrite H;apply mult_add_ineq2;zarith. - rewrite Zplus_assoc;rewrite Zmult_plus_distr_l. - rewrite Zmult_1_l;rewrite <- Zplus_assoc;rewrite H4;ring. - repeat rewrite <- Zplus_assoc;rewrite H;apply mult_add_ineq2;zarith. + rewrite Z.add_assoc;rewrite Z.mul_add_distr_r. + rewrite Z.mul_1_l;rewrite <- Z.add_assoc;rewrite H4;ring. + repeat rewrite <- Z.add_assoc;rewrite H;apply mult_add_ineq2;zarith. Qed. Lemma spec_double_mul_c : forall cross:w->w->w->w->zn2z w -> zn2z w -> w*zn2z w, @@ -361,7 +361,7 @@ Section DoubleMul. forall x y, [||double_mul_c cross x y||] = [[x]] * [[y]]. Proof. intros cross Hcross x y;destruct x as [ |xh xl];simpl;trivial. - destruct y as [ |yh yl];simpl. rewrite Zmult_0_r;trivial. + destruct y as [ |yh yl];simpl. rewrite Z.mul_0_r;trivial. assert (H1:= spec_w_mul_c xh yh);assert (H2:= spec_w_mul_c xl yl). generalize (Hcross _ _ _ _ _ _ H1 H2). destruct (cross xh xl yh yl (w_mul_c xh yh) (w_mul_c xl yl)) as (wc,cc). @@ -382,7 +382,7 @@ Section DoubleMul. Lemma spec_w_2: [|w_2|] = 2. unfold w_2; rewrite spec_w_add; rewrite spec_w_1; simpl. apply Zmod_small; split; auto with zarith. - rewrite <- (Zpower_1_r 2); unfold base; apply Zpower_lt_monotone; auto with zarith. + rewrite <- (Z.pow_1_r 2); unfold base; apply Zpower_lt_monotone; auto with zarith. Qed. Lemma kara_prod_aux : forall xh xl yh yl, @@ -401,19 +401,19 @@ Section DoubleMul. assert (Hyh := (spec_to_Z yh)); assert (Hyl := (spec_to_Z yl)). generalize (spec_ww_add_c hh ll); case (ww_add_c hh ll); intros z Hz; rewrite <- Hz; unfold interp_carry; assert (Hz1 := (spec_ww_to_Z z)). - rewrite spec_w_compare; case Zcompare_spec; intros Hxlh; + rewrite spec_w_compare; case Z.compare_spec; intros Hxlh; try rewrite Hxlh; try rewrite spec_w_0; try (ring; fail). - rewrite spec_w_compare; case Zcompare_spec; intros Hylh. + rewrite spec_w_compare; case Z.compare_spec; intros Hylh. rewrite Hylh; rewrite spec_w_0; try (ring; fail). rewrite spec_w_0; try (ring; fail). repeat (rewrite spec_ww_sub || rewrite spec_w_sub || rewrite spec_w_mul_c). repeat rewrite Zmod_small; auto with zarith; try (ring; fail). split; auto with zarith. simpl in Hz; rewrite Hz; rewrite H; rewrite H0. - rewrite kara_prod_aux; apply Zplus_le_0_compat; apply Zmult_le_0_compat; auto with zarith. - apply Zle_lt_trans with ([[z]]-0); auto with zarith. - unfold Zminus; apply Zplus_le_compat_l; apply Zle_left_rev; simpl; rewrite Zopp_involutive. - apply Zmult_le_0_compat; auto with zarith. + rewrite kara_prod_aux; apply Z.add_nonneg_nonneg; apply Z.mul_nonneg_nonneg; auto with zarith. + apply Z.le_lt_trans with ([[z]]-0); auto with zarith. + unfold Z.sub; apply Z.add_le_mono_l; apply Z.le_0_sub; simpl; rewrite Z.opp_involutive. + apply Z.mul_nonneg_nonneg; auto with zarith. match goal with |- context[ww_add_c ?x ?y] => generalize (spec_ww_add_c x y); case (ww_add_c x y); try rewrite spec_w_0; intros z1 Hz2 @@ -423,7 +423,7 @@ Section DoubleMul. rewrite spec_w_1; unfold interp_carry in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_compare; case Zcompare_spec; intros Hylh. + rewrite spec_w_compare; case Z.compare_spec; intros Hylh. rewrite Hylh; rewrite spec_w_0; try (ring; fail). match goal with |- context[ww_add_c ?x ?y] => generalize (spec_ww_add_c x y); case (ww_add_c x y); try rewrite spec_w_0; @@ -442,15 +442,15 @@ Section DoubleMul. replace ((x - y) * (z - t)) with ((y - x) * (t - z)); [idtac | ring] end. simpl in Hz; rewrite Hz; rewrite H; rewrite H0. - rewrite kara_prod_aux; apply Zplus_le_0_compat; apply Zmult_le_0_compat; auto with zarith. - apply Zle_lt_trans with ([[z]]-0); auto with zarith. - unfold Zminus; apply Zplus_le_compat_l; apply Zle_left_rev; simpl; rewrite Zopp_involutive. - apply Zmult_le_0_compat; auto with zarith. + rewrite kara_prod_aux; apply Z.add_nonneg_nonneg; apply Z.mul_nonneg_nonneg; auto with zarith. + apply Z.le_lt_trans with ([[z]]-0); auto with zarith. + unfold Z.sub; apply Z.add_le_mono_l; apply Z.le_0_sub; simpl; rewrite Z.opp_involutive. + apply Z.mul_nonneg_nonneg; auto with zarith. (** there is a carry in hh + ll **) - rewrite Zmult_1_l. - rewrite spec_w_compare; case Zcompare_spec; intros Hxlh; + rewrite Z.mul_1_l. + rewrite spec_w_compare; case Z.compare_spec; intros Hxlh; try rewrite Hxlh; try rewrite spec_w_1; try (ring; fail). - rewrite spec_w_compare; case Zcompare_spec; intros Hylh; + rewrite spec_w_compare; case Z.compare_spec; intros Hylh; try rewrite Hylh; try rewrite spec_w_1; try (ring; fail). match goal with |- context[ww_sub_c ?x ?y] => generalize (spec_ww_sub_c x y); case (ww_sub_c x y); try rewrite spec_w_1; @@ -458,7 +458,7 @@ Section DoubleMul. end. simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_0; rewrite Zmult_0_l; rewrite Zplus_0_l. + rewrite spec_w_0; rewrite Z.mul_0_l; rewrite Z.add_0_l. generalize Hz2; clear Hz2; unfold interp_carry. repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). repeat rewrite Zmod_small; auto with zarith; try (ring; fail). @@ -469,11 +469,11 @@ Section DoubleMul. simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). repeat rewrite Zmod_small; auto with zarith; try (ring; fail). rewrite spec_w_2; unfold interp_carry in Hz2. - apply trans_equal with (wwB + (1 * wwB + [[z1]])). + transitivity (wwB + (1 * wwB + [[z1]])). ring. rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_compare; case Zcompare_spec; intros Hylh; + rewrite spec_w_compare; case Z.compare_spec; intros Hylh; try rewrite Hylh; try rewrite spec_w_1; try (ring; fail). match goal with |- context[ww_add_c ?x ?y] => generalize (spec_ww_add_c x y); case (ww_add_c x y); try rewrite spec_w_1; @@ -482,7 +482,7 @@ Section DoubleMul. simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). repeat rewrite Zmod_small; auto with zarith; try (ring; fail). rewrite spec_w_2; unfold interp_carry in Hz2. - apply trans_equal with (wwB + (1 * wwB + [[z1]])). + transitivity (wwB + (1 * wwB + [[z1]])). ring. rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). repeat rewrite Zmod_small; auto with zarith; try (ring; fail). @@ -492,7 +492,7 @@ Section DoubleMul. end. simpl in Hz2; rewrite Hz2; repeat (rewrite spec_w_sub || rewrite spec_w_mul_c). repeat rewrite Zmod_small; auto with zarith; try (ring; fail). - rewrite spec_w_0; rewrite Zmult_0_l; rewrite Zplus_0_l. + rewrite spec_w_0; rewrite Z.mul_0_l; rewrite Z.add_0_l. match goal with |- context[(?x - ?y) * (?z - ?t)] => replace ((x - y) * (z - t)) with ((y - x) * (t - z)); [idtac | ring] end. @@ -513,7 +513,7 @@ Section DoubleMul. rewrite <- wwB_wBwB;intros H1 H2. assert (H3 := wB_pos w_digits). assert (2*wB <= wwB). - rewrite wwB_wBwB; rewrite Zpower_2; apply Zmult_le_compat;zarith. + rewrite wwB_wBwB; rewrite Z.pow_2_r; apply Z.mul_le_mono_nonneg;zarith. omega. Qed. @@ -537,14 +537,14 @@ Section DoubleMul. assert (U1:= lt_0_wwB w_digits). intros x y; case x; auto; intros xh xl. case y; auto. - simpl; rewrite Zmult_0_r; rewrite Zmod_small; auto with zarith. + simpl; rewrite Z.mul_0_r; rewrite Zmod_small; auto with zarith. intros yh yl;simpl. repeat (rewrite spec_ww_add || rewrite spec_w_W0 || rewrite spec_w_mul_c || rewrite spec_w_add || rewrite spec_w_mul). rewrite <- Zplus_mod; auto with zarith. - repeat (rewrite Zmult_plus_distr_l || rewrite Zmult_plus_distr_r). + repeat (rewrite Z.mul_add_distr_r || rewrite Z.mul_add_distr_l). rewrite <- Zmult_mod_distr_r; auto with zarith. - rewrite <- Zpower_2; rewrite <- wwB_wBwB; auto with zarith. + rewrite <- Z.pow_2_r; rewrite <- wwB_wBwB; auto with zarith. rewrite Zplus_mod; auto with zarith. rewrite Zmod_mod; auto with zarith. rewrite <- Zplus_mod; auto with zarith. @@ -564,10 +564,10 @@ Section DoubleMul. apply (spec_mul_aux xh xl xh xl wc cc);trivial. generalize Heq (spec_ww_add_c (w_mul_c xh xl) (w_mul_c xh xl));clear Heq. rewrite spec_w_mul_c;destruct (ww_add_c (w_mul_c xh xl) (w_mul_c xh xl)); - unfold interp_carry;try rewrite Zmult_1_l;intros Heq Heq';inversion Heq; - rewrite (Zmult_comm [|xl|]);subst. - rewrite spec_w_0;rewrite Zmult_0_l;rewrite Zplus_0_l;trivial. - rewrite spec_w_1;rewrite Zmult_1_l;rewrite <- wwB_wBwB;trivial. + unfold interp_carry;try rewrite Z.mul_1_l;intros Heq Heq';inversion Heq; + rewrite (Z.mul_comm [|xl|]);subst. + rewrite spec_w_0;rewrite Z.mul_0_l;rewrite Z.add_0_l;trivial. + rewrite spec_w_1;rewrite Z.mul_1_l;rewrite <- wwB_wBwB;trivial. Qed. Section DoubleMulAddn1Proof. @@ -589,8 +589,8 @@ Section DoubleMul. assert(H:=IHn xl y r);destruct (double_mul_add_n1 w_mul_add n xl y r)as(rl,l). assert(U:=IHn xh y rl);destruct(double_mul_add_n1 w_mul_add n xh y rl)as(rh,h). rewrite <- double_wB_wwB. rewrite spec_double_WW;simpl;trivial. - rewrite Zmult_plus_distr_l;rewrite <- Zplus_assoc;rewrite <- H. - rewrite Zmult_assoc;rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + rewrite Z.mul_add_distr_r;rewrite <- Z.add_assoc;rewrite <- H. + rewrite Z.mul_assoc;rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. rewrite U;ring. Qed. @@ -604,9 +604,9 @@ Section DoubleMul. destruct (w_mul_c x y) as [ |h l];simpl;rewrite <- H. rewrite spec_w_0;trivial. assert (U:=spec_w_add_c l r);destruct (w_add_c l r) as [lr|lr];unfold - interp_carry in U;try rewrite Zmult_1_l in H;simpl. + interp_carry in U;try rewrite Z.mul_1_l in H;simpl. rewrite U;ring. rewrite spec_w_succ. rewrite Zmod_small. - rewrite <- Zplus_assoc;rewrite <- U;ring. + rewrite <- Z.add_assoc;rewrite <- U;ring. simpl in H;assert (H1:= Zmult_lt_b _ _ _ (spec_to_Z x) (spec_to_Z y)). rewrite <- H in H1. assert (H2:=spec_to_Z h);split;zarith. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v index b073d6be..40556c4a 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -219,7 +219,7 @@ Section DoubleSqrt. Variable spec_w_is_even : forall x, if w_is_even x then [|x|] mod 2 = 0 else [|x|] mod 2 = 1. Variable spec_w_compare : forall x y, - w_compare x y = Zcompare [|x|] [|y|]. + w_compare x y = Z.compare [|x|] [|y|]. Variable spec_w_sub : forall x y, [|w_sub x y|] = ([|x|] - [|y|]) mod wB. Variable spec_w_square_c : forall x, [[ w_square_c x]] = [|x|] * [|x|]. Variable spec_w_div21 : forall a1 a2 b, @@ -232,7 +232,7 @@ Section DoubleSqrt. [|p|] <= Zpos w_digits -> [| w_add_mul_div p x y |] = ([|x|] * (2 ^ [|p|]) + - [|y|] / (Zpower 2 ((Zpos w_digits) - [|p|]))) mod wB. + [|y|] / (Z.pow 2 ((Zpos w_digits) - [|p|]))) mod wB. Variable spec_ww_add_mul_div : forall x y p, [[p]] <= Zpos (xO w_digits) -> [[ ww_add_mul_div p x y ]] = @@ -251,7 +251,7 @@ Section DoubleSqrt. Variable spec_ww_pred : forall x, [[ww_pred x]] = ([[x]] - 1) mod wwB. Variable spec_ww_add_c : forall x y, [+[ww_add_c x y]] = [[x]] + [[y]]. Variable spec_ww_compare : forall x y, - ww_compare x y = Zcompare [[x]] [[y]]. + ww_compare x y = Z.compare [[x]] [[y]]. Variable spec_ww_head0 : forall x, 0 < [[x]] -> wwB/ 2 <= 2 ^ [[ww_head0 x]] * [[x]] < wwB. Variable spec_low: forall x, [|low x|] = [[x]] mod wB. @@ -272,10 +272,9 @@ intros x; case x; simpl ww_is_even. unfold base. rewrite Zplus_mod; auto with zarith. rewrite (fun x y => (Zdivide_mod (x * y))); auto with zarith. - rewrite Zplus_0_l; rewrite Zmod_mod; auto with zarith. + rewrite Z.add_0_l; rewrite Zmod_mod; auto with zarith. apply spec_w_is_even; auto with zarith. - apply Zdivide_mult_r; apply Zpower_divide; auto with zarith. - red; simpl; auto. + apply Z.divide_mul_r; apply Zpower_divide; auto with zarith. Qed. @@ -286,10 +285,10 @@ intros x; case x; simpl ww_is_even. intros a1 a2 b Hb; unfold w_div21c. assert (H: 0 < [|b|]); auto with zarith. assert (U := wB_pos w_digits). - apply Zlt_le_trans with (2 := Hb); auto with zarith. - apply Zlt_le_trans with 1; auto with zarith. + apply Z.lt_le_trans with (2 := Hb); auto with zarith. + apply Z.lt_le_trans with 1; auto with zarith. apply Zdiv_le_lower_bound; auto with zarith. - rewrite !spec_w_compare. repeat case Zcompare_spec. + rewrite !spec_w_compare. repeat case Z.compare_spec. intros H1 H2; split. unfold interp_carry; autorewrite with w_rewrite rm10; auto with zarith. rewrite H1; rewrite H2; ring. @@ -308,7 +307,7 @@ intros x; case x; simpl ww_is_even. rewrite Zmod_small; auto with zarith. split; auto with zarith. assert ([|a2|] < 2 * [|b|]); auto with zarith. - apply Zlt_le_trans with (2 * (wB / 2)); auto with zarith. + apply Z.lt_le_trans with (2 * (wB / 2)); auto with zarith. rewrite wB_div_2; auto. intros H1. match goal with |- context[w_div21 ?y ?z ?t] => @@ -321,7 +320,7 @@ intros x; case x; simpl ww_is_even. rewrite spec_w_sub; auto with zarith. rewrite Zmod_small; auto with zarith. assert ([|a1|] < 2 * [|b|]); auto with zarith. - apply Zlt_le_trans with (2 * (wB / 2)); auto with zarith. + apply Z.lt_le_trans with (2 * (wB / 2)); auto with zarith. rewrite wB_div_2; auto. destruct (spec_to_Z a1);auto with zarith. destruct (spec_to_Z a1);auto with zarith. @@ -333,11 +332,11 @@ intros x; case x; simpl ww_is_even. intros w0 w1; replace [+|C1 w0|] with (wB + [|w0|]). rewrite Zmod_small; auto with zarith. intros (H3, H4); split; auto. - rewrite Zmult_plus_distr_l. - rewrite <- Zplus_assoc; rewrite <- H3; ring. + rewrite Z.mul_add_distr_r. + rewrite <- Z.add_assoc; rewrite <- H3; ring. split; auto with zarith. assert ([|a1|] < 2 * [|b|]); auto with zarith. - apply Zlt_le_trans with (2 * (wB / 2)); auto with zarith. + apply Z.lt_le_trans with (2 * (wB / 2)); auto with zarith. rewrite wB_div_2; auto. destruct (spec_to_Z a1);auto with zarith. destruct (spec_to_Z a1);auto with zarith. @@ -355,14 +354,14 @@ intros x; case x; simpl ww_is_even. rewrite spec_pred; rewrite spec_w_zdigits. rewrite Zmod_small; auto with zarith. split; auto with zarith. - apply Zlt_le_trans with (Zpos w_digits); auto with zarith. + apply Z.lt_le_trans with (Zpos w_digits); auto with zarith. unfold base; apply Zpower2_le_lin; auto with zarith. rewrite spec_w_add_mul_div; auto with zarith. autorewrite with w_rewrite rm10. match goal with |- context[?X - ?Y] => replace (X - Y) with 1 end. - rewrite Zpower_1_r; rewrite Zmod_small; auto with zarith. + rewrite Z.pow_1_r; rewrite Zmod_small; auto with zarith. destruct (spec_to_Z w1) as [H1 H2];auto with zarith. split; auto with zarith. apply Zdiv_lt_upper_bound; auto with zarith. @@ -377,15 +376,15 @@ intros x; case x; simpl ww_is_even. rewrite spec_pred; rewrite spec_w_zdigits. rewrite Zmod_small; auto with zarith. split; auto with zarith. - apply Zlt_le_trans with (Zpos w_digits); auto with zarith. + apply Z.lt_le_trans with (Zpos w_digits); auto with zarith. unfold base; apply Zpower2_le_lin; auto with zarith. autorewrite with w_rewrite rm10; auto with zarith. match goal with |- context[?X - ?Y] => replace (X - Y) with 1 end; rewrite Hp; try ring. - rewrite Zpos_minus; auto with zarith. - rewrite Zmax_right; auto with zarith. - rewrite Zpower_1_r; rewrite Zmod_small; auto with zarith. + rewrite Pos2Z.inj_sub_max; auto with zarith. + rewrite Z.max_r; auto with zarith. + rewrite Z.pow_1_r; rewrite Zmod_small; auto with zarith. destruct (spec_to_Z w1) as [H1 H2];auto with zarith. split; auto with zarith. unfold base. @@ -393,14 +392,14 @@ intros x; case x; simpl ww_is_even. assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; rewrite <- (tmp X); clear tmp end. - rewrite Zpower_exp; try rewrite Zpower_1_r; auto with zarith. + rewrite Zpower_exp; try rewrite Z.pow_1_r; auto with zarith. assert (tmp: forall p, 1 + (p -1) - 1 = p - 1); auto with zarith; rewrite tmp; clear tmp; auto with zarith. match goal with |- ?X + ?Y < _ => assert (Y < X); auto with zarith end. apply Zdiv_lt_upper_bound; auto with zarith. - pattern 2 at 2; rewrite <- Zpower_1_r; rewrite <- Zpower_exp; + pattern 2 at 2; rewrite <- Z.pow_1_r; rewrite <- Zpower_exp; auto with zarith. assert (tmp: forall p, (p - 1) + 1 = p); auto with zarith; rewrite tmp; clear tmp; auto with zarith. @@ -410,8 +409,8 @@ intros x; case x; simpl ww_is_even. [|w_add_mul_div w_1 w w_0|] = 2 * [|w|] mod wB. intros w1. autorewrite with w_rewrite rm10; auto with zarith. - rewrite Zpower_1_r; auto with zarith. - rewrite Zmult_comm; auto. + rewrite Z.pow_1_r; auto with zarith. + rewrite Z.mul_comm; auto. Qed. Theorem ww_add_mult_mult_2: forall w, @@ -420,8 +419,8 @@ intros x; case x; simpl ww_is_even. rewrite spec_ww_add_mul_div; auto with zarith. autorewrite with w_rewrite rm10. rewrite spec_w_0W; rewrite spec_w_1. - rewrite Zpower_1_r; auto with zarith. - rewrite Zmult_comm; auto. + rewrite Z.pow_1_r; auto with zarith. + rewrite Z.mul_comm; auto. rewrite spec_w_0W; rewrite spec_w_1; auto with zarith. red; simpl; intros; discriminate. Qed. @@ -432,18 +431,18 @@ intros x; case x; simpl ww_is_even. intros w1. rewrite spec_ww_add_mul_div; auto with zarith. rewrite spec_w_0W; rewrite spec_w_1; auto with zarith. - rewrite Zpower_1_r; auto with zarith. + rewrite Z.pow_1_r; auto with zarith. f_equal; auto. - rewrite Zmult_comm; f_equal; auto. + rewrite Z.mul_comm; f_equal; auto. autorewrite with w_rewrite rm10. unfold ww_digits, base. - apply sym_equal; apply Zdiv_unique with (r := 2 ^ (Zpos (ww_digits w_digits) - 1) -1); + symmetry; apply Zdiv_unique with (r := 2 ^ (Zpos (ww_digits w_digits) - 1) -1); auto with zarith. unfold ww_digits; split; auto with zarith. match goal with |- 0 <= ?X - 1 => assert (0 < X); auto with zarith end. - apply Zpower_gt_0; auto with zarith. + apply Z.pow_pos_nonneg; auto with zarith. match goal with |- 0 <= ?X - 1 => assert (0 < X); auto with zarith; red; reflexivity end. @@ -453,7 +452,7 @@ intros x; case x; simpl ww_is_even. assert (tmp: forall p, p + p = 2 * p); auto with zarith; rewrite tmp; clear tmp. f_equal; auto. - pattern 2 at 2; rewrite <- Zpower_1_r; rewrite <- Zpower_exp; + pattern 2 at 2; rewrite <- Z.pow_1_r; rewrite <- Zpower_exp; auto with zarith. assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; rewrite tmp; clear tmp; auto. @@ -466,7 +465,7 @@ intros x; case x; simpl ww_is_even. Theorem Zplus_mod_one: forall a1 b1, 0 < b1 -> (a1 + b1) mod b1 = a1 mod b1. intros a1 b1 H; rewrite Zplus_mod; auto with zarith. - rewrite Z_mod_same; try rewrite Zplus_0_r; auto with zarith. + rewrite Z_mod_same; try rewrite Z.add_0_r; auto with zarith. apply Zmod_mod; auto. Qed. @@ -481,8 +480,8 @@ intros x; case x; simpl ww_is_even. intros a1 a2 b H. assert (HH: 0 < [|b|]); auto with zarith. assert (U := wB_pos w_digits). - apply Zlt_le_trans with (2 := H); auto with zarith. - apply Zlt_le_trans with 1; auto with zarith. + apply Z.lt_le_trans with (2 := H); auto with zarith. + apply Z.lt_le_trans with 1; auto with zarith. apply Zdiv_le_lower_bound; auto with zarith. unfold w_div2s; case a1; intros w0 H0. match goal with |- context[w_div21c ?y ?z ?t] => @@ -528,10 +527,10 @@ intros x; case x; simpl ww_is_even. match goal with |- context[_ ^ ?X] => assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; - try rewrite Zpower_1_r; auto with zarith + try rewrite Z.pow_1_r; auto with zarith end. - rewrite Zpos_minus; auto with zarith. - rewrite Zmax_right; auto with zarith. + rewrite Pos2Z.inj_sub_max; auto with zarith. + rewrite Z.max_r; auto with zarith. ring. repeat rewrite C0_id. rewrite spec_w_add_c; auto with zarith. @@ -545,10 +544,10 @@ intros x; case x; simpl ww_is_even. match goal with |- context[_ ^ ?X] => assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; - try rewrite Zpower_1_r; auto with zarith + try rewrite Z.pow_1_r; auto with zarith end. - rewrite Zpos_minus; auto with zarith. - rewrite Zmax_right; auto with zarith. + rewrite Pos2Z.inj_sub_max; auto with zarith. + rewrite Z.max_r; auto with zarith. ring. repeat rewrite C1_plus_wB in H0. rewrite C1_plus_wB. @@ -570,7 +569,7 @@ intros x; case x; simpl ww_is_even. rewrite add_mult_div_2_plus_1. replace (wB + [|w0|]) with ([|b|] + ([|w0|] - [|b|] + wB)); auto with zarith. - rewrite Zmult_plus_distr_l; rewrite <- Zplus_assoc. + rewrite Z.mul_add_distr_r; rewrite <- Z.add_assoc. rewrite Hw1. pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); auto with zarith. @@ -578,10 +577,10 @@ intros x; case x; simpl ww_is_even. match goal with |- context[_ ^ ?X] => assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; - try rewrite Zpower_1_r; auto with zarith + try rewrite Z.pow_1_r; auto with zarith end. - rewrite Zpos_minus; auto with zarith. - rewrite Zmax_right; auto with zarith. + rewrite Pos2Z.inj_sub_max; auto with zarith. + rewrite Z.max_r; auto with zarith. ring. repeat rewrite C0_id. rewrite add_mult_div_2_plus_1. @@ -589,7 +588,7 @@ intros x; case x; simpl ww_is_even. intros H1; split; auto with zarith. replace (wB + [|w0|]) with ([|b|] + ([|w0|] - [|b|] + wB)); auto with zarith. - rewrite Zmult_plus_distr_l; rewrite <- Zplus_assoc. + rewrite Z.mul_add_distr_r; rewrite <- Z.add_assoc. rewrite Hw1. pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); auto with zarith. @@ -597,10 +596,10 @@ intros x; case x; simpl ww_is_even. match goal with |- context[_ ^ ?X] => assert (tmp: forall p, 1 + (p - 1) = p); auto with zarith; rewrite <- (tmp X); clear tmp; rewrite Zpower_exp; - try rewrite Zpower_1_r; auto with zarith + try rewrite Z.pow_1_r; auto with zarith end. - rewrite Zpos_minus; auto with zarith. - rewrite Zmax_right; auto with zarith. + rewrite Pos2Z.inj_sub_max; auto with zarith. + rewrite Z.max_r; auto with zarith. ring. split; auto with zarith. destruct (spec_to_Z b);auto with zarith. @@ -620,7 +619,7 @@ intros x; case x; simpl ww_is_even. rewrite add_mult_div_2. replace (wB + [|w0|]) with ([|b|] + ([|w0|] - [|b|] + wB)); auto with zarith. - rewrite Zmult_plus_distr_l; rewrite <- Zplus_assoc. + rewrite Z.mul_add_distr_r; rewrite <- Z.add_assoc. rewrite Hw1. pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); auto with zarith. @@ -631,7 +630,7 @@ intros x; case x; simpl ww_is_even. rewrite add_mult_div_2. replace (wB + [|w0|]) with ([|b|] + ([|w0|] - [|b|] + wB)); auto with zarith. - rewrite Zmult_plus_distr_l; rewrite <- Zplus_assoc. + rewrite Z.mul_add_distr_r; rewrite <- Z.add_assoc. rewrite Hw1. pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] 2); auto with zarith. @@ -652,20 +651,20 @@ intros x; case x; simpl ww_is_even. rewrite <- Zpower_exp; auto with zarith. f_equal; auto with zarith. rewrite H. - rewrite (fun x => (Zmult_comm 4 (2 ^x))). + rewrite (fun x => (Z.mul_comm 4 (2 ^x))). rewrite Z_div_mult; auto with zarith. Qed. Theorem Zsquare_mult: forall p, p ^ 2 = p * p. intros p; change 2 with (1 + 1); rewrite Zpower_exp; - try rewrite Zpower_1_r; auto with zarith. + try rewrite Z.pow_1_r; auto with zarith. Qed. Theorem Zsquare_pos: forall p, 0 <= p ^ 2. - intros p; case (Zle_or_lt 0 p); intros H1. - rewrite Zsquare_mult; apply Zmult_le_0_compat; auto with zarith. + intros p; case (Z.le_gt_cases 0 p); intros H1. + rewrite Zsquare_mult; apply Z.mul_nonneg_nonneg; auto with zarith. rewrite Zsquare_mult; replace (p * p) with ((- p) * (- p)); try ring. - apply Zmult_le_0_compat; auto with zarith. + apply Z.mul_nonneg_nonneg; auto with zarith. Qed. Lemma spec_split: forall x, @@ -676,13 +675,12 @@ intros x; case x; simpl ww_is_even. Theorem mult_wwB: forall x y, [|x|] * [|y|] < wwB. Proof. - intros x y; rewrite wwB_wBwB; rewrite Zpower_2. + intros x y; rewrite wwB_wBwB; rewrite Z.pow_2_r. generalize (spec_to_Z x); intros U. generalize (spec_to_Z y); intros U1. - apply Zle_lt_trans with ((wB -1 ) * (wB - 1)); auto with zarith. - apply Zmult_le_compat; auto with zarith. - repeat (rewrite Zmult_minus_distr_r || rewrite Zmult_minus_distr_l); - auto with zarith. + apply Z.le_lt_trans with ((wB -1 ) * (wB - 1)); auto with zarith. + apply Z.mul_le_mono_nonneg; auto with zarith. + rewrite ?Z.mul_sub_distr_l, ?Z.mul_sub_distr_r; auto with zarith. Qed. Hint Resolve mult_wwB. @@ -697,22 +695,22 @@ intros x; case x; simpl ww_is_even. end; simpl fst; simpl snd. intros w0 w1 Hw0 w2 w3 Hw1. assert (U: wB/4 <= [|w2|]). - case (Zle_or_lt (wB / 4) [|w2|]); auto; intros H1. - contradict H; apply Zlt_not_le. - rewrite wwB_wBwB; rewrite Zpower_2. - pattern wB at 1; rewrite <- wB_div_4; rewrite <- Zmult_assoc; - rewrite Zmult_comm. + case (Z.le_gt_cases (wB / 4) [|w2|]); auto; intros H1. + contradict H; apply Z.lt_nge. + rewrite wwB_wBwB; rewrite Z.pow_2_r. + pattern wB at 1; rewrite <- wB_div_4; rewrite <- Z.mul_assoc; + rewrite Z.mul_comm. rewrite Z_div_mult; auto with zarith. rewrite <- Hw1. match goal with |- _ < ?X => - pattern X; rewrite <- Zplus_0_r; apply beta_lex_inv; + pattern X; rewrite <- Z.add_0_r; apply beta_lex_inv; auto with zarith end. destruct (spec_to_Z w3);auto with zarith. generalize (@spec_w_sqrt2 w2 w3 U); case (w_sqrt2 w2 w3). intros w4 c (H1, H2). assert (U1: wB/2 <= [|w4|]). - case (Zle_or_lt (wB/2) [|w4|]); auto with zarith. + case (Z.le_gt_cases (wB/2) [|w4|]); auto with zarith. intros U1. assert (U2 : [|w4|] <= wB/2 -1); auto with zarith. assert (U3 : [|w4|] ^ 2 <= wB/4 * wB - wB + 1); auto with zarith. @@ -720,19 +718,19 @@ intros x; case x; simpl ww_is_even. rewrite Zsquare_mult; replace Y with ((wB/2 - 1) * (wB/2 -1)) end. - apply Zmult_le_compat; auto with zarith. + apply Z.mul_le_mono_nonneg; auto with zarith. destruct (spec_to_Z w4);auto with zarith. destruct (spec_to_Z w4);auto with zarith. pattern wB at 4 5; rewrite <- wB_div_2. - rewrite Zmult_assoc. + rewrite Z.mul_assoc. replace ((wB / 4) * 2) with (wB / 2). ring. pattern wB at 1; rewrite <- wB_div_4. change 4 with (2 * 2). - rewrite <- Zmult_assoc; rewrite (Zmult_comm 2). + rewrite <- Z.mul_assoc; rewrite (Z.mul_comm 2). rewrite Z_div_mult; try ring; auto with zarith. assert (U4 : [+|c|] <= wB -2); auto with zarith. - apply Zle_trans with (1 := H2). + apply Z.le_trans with (1 := H2). match goal with |- ?X <= ?Y => replace Y with (2 * (wB/ 2 - 1)); auto with zarith end. @@ -741,10 +739,10 @@ intros x; case x; simpl ww_is_even. assert (U5: X < wB / 4 * wB) end. rewrite H1; auto with zarith. - contradict U; apply Zlt_not_le. - apply Zmult_lt_reg_r with wB; auto with zarith. + contradict U; apply Z.lt_nge. + apply Z.mul_lt_mono_pos_r with wB; auto with zarith. destruct (spec_to_Z w4);auto with zarith. - apply Zle_lt_trans with (2 := U5). + apply Z.le_lt_trans with (2 := U5). unfold ww_to_Z, zn2z_to_Z. destruct (spec_to_Z w3);auto with zarith. generalize (@spec_w_div2s c w0 w4 U1 H2). @@ -766,7 +764,7 @@ intros x; case x; simpl ww_is_even. unfold ww_to_Z, zn2z_to_Z in H1; rewrite H1. rewrite <- Hw0. match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - apply trans_equal with ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) + transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) end. repeat rewrite Zsquare_mult. rewrite wwB_wBwB; ring. @@ -779,17 +777,17 @@ intros x; case x; simpl ww_is_even. match goal with |- ?X - ?Y * ?Y <= _ => assert (V := Zsquare_pos Y); rewrite Zsquare_mult in V; - apply Zle_trans with X; auto with zarith; + apply Z.le_trans with X; auto with zarith; clear V end. match goal with |- ?X * wB + ?Y <= 2 * (?Z * wB + ?T) => - apply Zle_trans with ((2 * Z - 1) * wB + wB); auto with zarith + apply Z.le_trans with ((2 * Z - 1) * wB + wB); auto with zarith end. destruct (spec_to_Z w1);auto with zarith. match goal with |- ?X <= _ => replace X with (2 * [|w4|] * wB); auto with zarith end. - rewrite Zmult_plus_distr_r; rewrite Zmult_assoc. + rewrite Z.mul_add_distr_l; rewrite Z.mul_assoc. destruct (spec_to_Z w5); auto with zarith. ring. intros z; replace [-[C1 z]] with (- wwB + [[z]]). @@ -815,7 +813,7 @@ intros x; case x; simpl ww_is_even. unfold ww_to_Z, zn2z_to_Z in H1; rewrite H1. rewrite <- Hw0. match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - apply trans_equal with ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) + transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) end. repeat rewrite Zsquare_mult. rewrite wwB_wBwB; ring. @@ -828,11 +826,11 @@ intros x; case x; simpl ww_is_even. destruct (spec_ww_to_Z w_digits w_to_Z spec_to_Z z);auto with zarith. assert (V1 := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW w4 w5)). assert (0 < [[WW w4 w5]]); auto with zarith. - apply Zlt_le_trans with (wB/ 2 * wB + 0); auto with zarith. - autorewrite with rm10; apply Zmult_lt_0_compat; auto with zarith. - apply Zmult_lt_reg_r with 2; auto with zarith. + apply Z.lt_le_trans with (wB/ 2 * wB + 0); auto with zarith. + autorewrite with rm10; apply Z.mul_pos_pos; auto with zarith. + apply Z.mul_lt_mono_pos_r with 2; auto with zarith. autorewrite with rm10. - rewrite Zmult_comm; rewrite wB_div_2; auto with zarith. + rewrite Z.mul_comm; rewrite wB_div_2; auto with zarith. case (spec_to_Z w5);auto with zarith. case (spec_to_Z w5);auto with zarith. simpl. @@ -840,11 +838,11 @@ intros x; case x; simpl ww_is_even. assert (V1 := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW w4 w5)); auto with zarith. split; auto with zarith. assert (wwB <= 2 * [[WW w4 w5]]); auto with zarith. - apply Zle_trans with (2 * ([|w4|] * wB)). - rewrite wwB_wBwB; rewrite Zpower_2. - rewrite Zmult_assoc; apply Zmult_le_compat_r; auto with zarith. - rewrite <- wB_div_2; auto with zarith. + apply Z.le_trans with (2 * ([|w4|] * wB)). + rewrite wwB_wBwB; rewrite Z.pow_2_r. + rewrite Z.mul_assoc; apply Z.mul_le_mono_nonneg_r; auto with zarith. assert (V2 := spec_to_Z w5);auto with zarith. + rewrite <- wB_div_2; auto with zarith. simpl ww_to_Z; assert (V2 := spec_to_Z w5);auto with zarith. assert (V1 := spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW w4 w5)); auto with zarith. intros z1; change [-[C1 z1]] with (-wwB + [[z1]]). @@ -856,21 +854,21 @@ intros x; case x; simpl ww_is_even. rewrite ww_add_mult_mult_2. rename V1 into VV1. assert (VV2: 0 < [[WW w4 w5]]); auto with zarith. - apply Zlt_le_trans with (wB/ 2 * wB + 0); auto with zarith. - autorewrite with rm10; apply Zmult_lt_0_compat; auto with zarith. - apply Zmult_lt_reg_r with 2; auto with zarith. + apply Z.lt_le_trans with (wB/ 2 * wB + 0); auto with zarith. + autorewrite with rm10; apply Z.mul_pos_pos; auto with zarith. + apply Z.mul_lt_mono_pos_r with 2; auto with zarith. autorewrite with rm10. - rewrite Zmult_comm; rewrite wB_div_2; auto with zarith. + rewrite Z.mul_comm; rewrite wB_div_2; auto with zarith. assert (VV3 := spec_to_Z w5);auto with zarith. assert (VV3 := spec_to_Z w5);auto with zarith. simpl. assert (VV3 := spec_to_Z w5);auto with zarith. assert (VV3: wwB <= 2 * [[WW w4 w5]]); auto with zarith. - apply Zle_trans with (2 * ([|w4|] * wB)). - rewrite wwB_wBwB; rewrite Zpower_2. - rewrite Zmult_assoc; apply Zmult_le_compat_r; auto with zarith. - rewrite <- wB_div_2; auto with zarith. + apply Z.le_trans with (2 * ([|w4|] * wB)). + rewrite wwB_wBwB; rewrite Z.pow_2_r. + rewrite Z.mul_assoc; apply Z.mul_le_mono_nonneg_r; auto with zarith. case (spec_to_Z w5);auto with zarith. + rewrite <- wB_div_2; auto with zarith. simpl ww_to_Z; assert (V4 := spec_to_Z w5);auto with zarith. rewrite <- Zmod_unique with (q := 1) (r := -wwB + 2 * [[WW w4 w5]]); auto with zarith. @@ -892,7 +890,7 @@ intros x; case x; simpl ww_is_even. rewrite <- Hw0. split. match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - apply trans_equal with ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) + transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) end. repeat rewrite Zsquare_mult. rewrite wwB_wBwB; ring. @@ -905,17 +903,17 @@ intros x; case x; simpl ww_is_even. assert (V2 := spec_ww_to_Z w_digits w_to_Z spec_to_Z z);auto with zarith. assert (V3 := spec_ww_to_Z w_digits w_to_Z spec_to_Z z1);auto with zarith. split; auto with zarith. - rewrite (Zplus_comm (-wwB)); rewrite <- Zplus_assoc. + rewrite (Z.add_comm (-wwB)); rewrite <- Z.add_assoc. rewrite H5. match goal with |- 0 <= ?X + (?Y - ?Z) => - apply Zle_trans with (X - Z); auto with zarith + apply Z.le_trans with (X - Z); auto with zarith end. 2: generalize (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW w6 w1)); unfold ww_to_Z; auto with zarith. rewrite V1. match goal with |- 0 <= ?X - 1 - ?Y => assert (Y < X); auto with zarith end. - apply Zlt_le_trans with wwB; auto with zarith. + apply Z.lt_le_trans with wwB; auto with zarith. intros (H3, H4). match goal with |- context [ww_sub_c ?y ?z] => generalize (spec_ww_sub_c y z); case (ww_sub_c y z) @@ -933,7 +931,7 @@ intros x; case x; simpl ww_is_even. unfold ww_to_Z, zn2z_to_Z in H1; rewrite H1. rewrite <- Hw0. match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - apply trans_equal with ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) + transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) end. repeat rewrite Zsquare_mult. rewrite wwB_wBwB; ring. @@ -945,27 +943,27 @@ intros x; case x; simpl ww_is_even. simpl ww_to_Z. rewrite H5. simpl ww_to_Z. - rewrite wwB_wBwB; rewrite Zpower_2. + rewrite wwB_wBwB; rewrite Z.pow_2_r. match goal with |- ?X * ?Y + (?Z * ?Y + ?T - ?U) <= _ => - apply Zle_trans with (X * Y + (Z * Y + T - 0)); + apply Z.le_trans with (X * Y + (Z * Y + T - 0)); auto with zarith end. assert (V := Zsquare_pos [|w5|]); rewrite Zsquare_mult in V; auto with zarith. autorewrite with rm10. match goal with |- _ <= 2 * (?U * ?V + ?W) => - apply Zle_trans with (2 * U * V + 0); + apply Z.le_trans with (2 * U * V + 0); auto with zarith end. match goal with |- ?X * ?Y + (?Z * ?Y + ?T) <= _ => replace (X * Y + (Z * Y + T)) with ((X + Z) * Y + T); try ring end. - apply Zlt_le_weak; apply beta_lex_inv; auto with zarith. + apply Z.lt_le_incl; apply beta_lex_inv; auto with zarith. destruct (spec_to_Z w1);auto with zarith. destruct (spec_to_Z w5);auto with zarith. - rewrite Zmult_plus_distr_r; auto with zarith. - rewrite Zmult_assoc; auto with zarith. + rewrite Z.mul_add_distr_l; auto with zarith. + rewrite Z.mul_assoc; auto with zarith. intros z; replace [-[C1 z]] with (- wwB + [[z]]). 2: simpl; case wwB; auto with zarith. intros H5; rewrite spec_w_square_c in H5; @@ -984,7 +982,7 @@ intros x; case x; simpl ww_is_even. rewrite <- Hw0. split. match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - apply trans_equal with ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) + transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) end. repeat rewrite Zsquare_mult. rewrite wwB_wBwB; ring. @@ -995,40 +993,38 @@ intros x; case x; simpl ww_is_even. repeat rewrite Zsquare_mult; ring. rewrite V. simpl ww_to_Z. - rewrite wwB_wBwB; rewrite Zpower_2. + rewrite wwB_wBwB; rewrite Z.pow_2_r. match goal with |- (?Z * ?Y + ?T - ?U) + ?X * ?Y <= _ => - apply Zle_trans with ((Z * Y + T - 0) + X * Y); + apply Z.le_trans with ((Z * Y + T - 0) + X * Y); auto with zarith end. assert (V1 := Zsquare_pos [|w5|]); rewrite Zsquare_mult in V1; auto with zarith. autorewrite with rm10. match goal with |- _ <= 2 * (?U * ?V + ?W) => - apply Zle_trans with (2 * U * V + 0); + apply Z.le_trans with (2 * U * V + 0); auto with zarith end. match goal with |- (?Z * ?Y + ?T) + ?X * ?Y <= _ => replace ((Z * Y + T) + X * Y) with ((X + Z) * Y + T); try ring end. - apply Zlt_le_weak; apply beta_lex_inv; auto with zarith. + apply Z.lt_le_incl; apply beta_lex_inv; auto with zarith. destruct (spec_to_Z w1);auto with zarith. destruct (spec_to_Z w5);auto with zarith. - rewrite Zmult_plus_distr_r; auto with zarith. - rewrite Zmult_assoc; auto with zarith. - case Zle_lt_or_eq with (1 := H2); clear H2; intros H2. + rewrite Z.mul_add_distr_l; auto with zarith. + rewrite Z.mul_assoc; auto with zarith. + Z.le_elim H2. intros c1 (H3, H4). - match type of H3 with ?X = ?Y => - absurd (X < Y) - end. - apply Zle_not_lt; rewrite <- H3; auto with zarith. - rewrite Zmult_plus_distr_l. - apply Zlt_le_trans with ((2 * [|w4|]) * wB + 0); + match type of H3 with ?X = ?Y => absurd (X < Y) end. + apply Z.le_ngt; rewrite <- H3; auto with zarith. + rewrite Z.mul_add_distr_r. + apply Z.lt_le_trans with ((2 * [|w4|]) * wB + 0); auto with zarith. apply beta_lex_inv; auto with zarith. destruct (spec_to_Z w0);auto with zarith. assert (V1 := spec_to_Z w5);auto with zarith. - rewrite (Zmult_comm wB); auto with zarith. + rewrite (Z.mul_comm wB); auto with zarith. assert (0 <= [|w5|] * (2 * [|w4|])); auto with zarith. intros c1 (H3, H4); rewrite H2 in H3. match type of H3 with ?X + ?Y = (?Z + ?T) * ?U + ?V => @@ -1038,20 +1034,19 @@ intros x; case x; simpl ww_is_even. end. assert (V1 := spec_to_Z w0);auto with zarith. assert (V2 := spec_to_Z w5);auto with zarith. - case (Zle_lt_or_eq 0 [|w5|]); auto with zarith; intros V3. - match type of VV with ?X = ?Y => - absurd (X < Y) - end. - apply Zle_not_lt; rewrite <- VV; auto with zarith. - apply Zlt_le_trans with wB; auto with zarith. + case V2; intros V3 _. + Z.le_elim V3; auto with zarith. + match type of VV with ?X = ?Y => absurd (X < Y) end. + apply Z.le_ngt; rewrite <- VV; auto with zarith. + apply Z.lt_le_trans with wB; auto with zarith. match goal with |- _ <= ?X + _ => - apply Zle_trans with X; auto with zarith + apply Z.le_trans with X; auto with zarith end. match goal with |- _ <= _ * ?X => - apply Zle_trans with (1 * X); auto with zarith + apply Z.le_trans with (1 * X); auto with zarith end. autorewrite with rm10. - rewrite <- wB_div_2; apply Zmult_le_compat_l; auto with zarith. + rewrite <- wB_div_2; apply Z.mul_le_mono_nonneg_l; auto with zarith. rewrite <- V3 in VV; generalize VV; autorewrite with rm10; clear VV; intros VV. rewrite spec_ww_add_c; auto with zarith. @@ -1067,7 +1062,7 @@ intros x; case x; simpl ww_is_even. simpl ww_to_Z in H1; rewrite H1. rewrite <- Hw0. match goal with |- (?X ^2 + ?Y) * wwB + (?Z * wB + ?T) = ?U => - apply trans_equal with ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) + transitivity ((X * wB) ^ 2 + (Y * wB + Z) * wB + T) end. repeat rewrite Zsquare_mult. rewrite wwB_wBwB; ring. @@ -1079,41 +1074,41 @@ intros x; case x; simpl ww_is_even. simpl ww_to_Z; unfold ww_to_Z. rewrite spec_w_Bm1; auto with zarith. split. - rewrite wwB_wBwB; rewrite Zpower_2. + rewrite wwB_wBwB; rewrite Z.pow_2_r. match goal with |- _ <= -?X + (2 * (?Z * ?T + ?U) + ?V) => assert (X <= 2 * Z * T); auto with zarith end. - apply Zmult_le_compat_r; auto with zarith. - rewrite <- wB_div_2; apply Zmult_le_compat_l; auto with zarith. - rewrite Zmult_plus_distr_r; auto with zarith. - rewrite Zmult_assoc; auto with zarith. + apply Z.mul_le_mono_nonneg_r; auto with zarith. + rewrite <- wB_div_2; apply Z.mul_le_mono_nonneg_l; auto with zarith. + rewrite Z.mul_add_distr_l; auto with zarith. + rewrite Z.mul_assoc; auto with zarith. match goal with |- _ + ?X < _ => replace X with ((2 * (([|w4|]) + 1) * wB) - 1); try ring end. assert (2 * ([|w4|] + 1) * wB <= 2 * wwB); auto with zarith. - rewrite <- Zmult_assoc; apply Zmult_le_compat_l; auto with zarith. - rewrite wwB_wBwB; rewrite Zpower_2. - apply Zmult_le_compat_r; auto with zarith. + rewrite <- Z.mul_assoc; apply Z.mul_le_mono_nonneg_l; auto with zarith. + rewrite wwB_wBwB; rewrite Z.pow_2_r. + apply Z.mul_le_mono_nonneg_r; auto with zarith. case (spec_to_Z w4);auto with zarith. Qed. Lemma spec_ww_is_zero: forall x, if ww_is_zero x then [[x]] = 0 else 0 < [[x]]. intro x; unfold ww_is_zero. - rewrite spec_ww_compare. case Zcompare_spec; + rewrite spec_ww_compare. case Z.compare_spec; auto with zarith. simpl ww_to_Z. assert (V4 := spec_ww_to_Z w_digits w_to_Z spec_to_Z x);auto with zarith. Qed. Lemma wwB_4_2: 2 * (wwB / 4) = wwB/ 2. - pattern wwB at 1; rewrite wwB_wBwB; rewrite Zpower_2. + pattern wwB at 1; rewrite wwB_wBwB; rewrite Z.pow_2_r. rewrite <- wB_div_2. match goal with |- context[(2 * ?X) * (2 * ?Z)] => replace ((2 * X) * (2 * Z)) with ((X * Z) * 4); try ring end. rewrite Z_div_mult; auto with zarith. - rewrite Zmult_assoc; rewrite wB_div_2. + rewrite Z.mul_assoc; rewrite wB_div_2. rewrite wwB_div_2; ring. Qed. @@ -1129,10 +1124,10 @@ Qed. intros H2. generalize (spec_ww_head0 x H2); case (ww_head0 x); autorewrite with rm10. intros (H3, H4); split; auto with zarith. - apply Zle_trans with (2 := H3). + apply Z.le_trans with (2 := H3). apply Zdiv_le_compat_l; auto with zarith. intros xh xl (H3, H4); split; auto with zarith. - apply Zle_trans with (2 := H3). + apply Z.le_trans with (2 := H3). apply Zdiv_le_compat_l; auto with zarith. intros H1. case (spec_to_w_Z (ww_head0 x)); intros Hv1 Hv2. @@ -1156,24 +1151,24 @@ Qed. case (spec_ww_head0 x); auto; intros Hv3 Hv4. assert (Hu: forall u, 0 < u -> 2 * 2 ^ (u - 1) = 2 ^u). intros u Hu. - pattern 2 at 1; rewrite <- Zpower_1_r. + pattern 2 at 1; rewrite <- Z.pow_1_r. rewrite <- Zpower_exp; auto with zarith. ring_simplify (1 + (u - 1)); auto with zarith. split; auto with zarith. - apply Zmult_le_reg_r with 2; auto with zarith. - repeat rewrite (fun x => Zmult_comm x 2). + apply Z.mul_le_mono_pos_r with 2; auto with zarith. + repeat rewrite (fun x => Z.mul_comm x 2). rewrite wwB_4_2. - rewrite Zmult_assoc; rewrite Hu; auto with zarith. - apply Zle_lt_trans with (2 * 2 ^ ([[ww_head0 x]] - 1) * [[x]]); auto with zarith; + rewrite Z.mul_assoc; rewrite Hu; auto with zarith. + apply Z.le_lt_trans with (2 * 2 ^ ([[ww_head0 x]] - 1) * [[x]]); auto with zarith; rewrite Hu; auto with zarith. - apply Zmult_le_compat_r; auto with zarith. + apply Z.mul_le_mono_nonneg_r; auto with zarith. apply Zpower_le_monotone; auto with zarith. Qed. Theorem wwB_4_wB_4: wwB / 4 = wB / 4 * wB. - apply sym_equal; apply Zdiv_unique with 0; - auto with zarith. - rewrite Zmult_assoc; rewrite wB_div_4; auto with zarith. + Proof. + symmetry; apply Zdiv_unique with 0; auto with zarith. + rewrite Z.mul_assoc; rewrite wB_div_4; auto with zarith. rewrite wwB_wBwB; ring. Qed. @@ -1182,10 +1177,10 @@ Qed. assert (U := wB_pos w_digits). intro x; unfold ww_sqrt. generalize (spec_ww_is_zero x); case (ww_is_zero x). - simpl ww_to_Z; simpl Zpower; unfold Zpower_pos; simpl; + simpl ww_to_Z; simpl Z.pow; unfold Z.pow_pos; simpl; auto with zarith. intros H1. - rewrite spec_ww_compare. case Zcompare_spec; + rewrite spec_ww_compare. case Z.compare_spec; simpl ww_to_Z; autorewrite with rm10. generalize H1; case x. intros HH; contradict HH; simpl ww_to_Z; auto with zarith. @@ -1203,7 +1198,7 @@ Qed. intros w3 (H6, H7); rewrite H6. assert (V1 := spec_to_Z w3);auto with zarith. split; auto with zarith. - apply Zle_lt_trans with ([|w2|] ^2 + 2 * [|w2|]); auto with zarith. + apply Z.le_lt_trans with ([|w2|] ^2 + 2 * [|w2|]); auto with zarith. match goal with |- ?X < ?Z => replace Z with (X + 1); auto with zarith end. @@ -1211,7 +1206,7 @@ Qed. intros w3 (H6, H7); rewrite H6. assert (V1 := spec_to_Z w3);auto with zarith. split; auto with zarith. - apply Zle_lt_trans with ([|w2|] ^2 + 2 * [|w2|]); auto with zarith. + apply Z.le_lt_trans with ([|w2|] ^2 + 2 * [|w2|]); auto with zarith. match goal with |- ?X < ?Z => replace Z with (X + 1); auto with zarith end. @@ -1221,42 +1216,42 @@ Qed. case (spec_ww_head1 x); intros Hp1 Hp2. generalize (Hp2 H1); clear Hp2; intros Hp2. assert (Hv2: [[ww_head1 x]] <= Zpos (xO w_digits)). - case (Zle_or_lt (Zpos (xO w_digits)) [[ww_head1 x]]); auto with zarith; intros HH1. + case (Z.le_gt_cases (Zpos (xO w_digits)) [[ww_head1 x]]); auto with zarith; intros HH1. case Hp2; intros _ HH2; contradict HH2. - apply Zle_not_lt; unfold base. - apply Zle_trans with (2 ^ [[ww_head1 x]]). + apply Z.le_ngt; unfold base. + apply Z.le_trans with (2 ^ [[ww_head1 x]]). apply Zpower_le_monotone; auto with zarith. pattern (2 ^ [[ww_head1 x]]) at 1; - rewrite <- (Zmult_1_r (2 ^ [[ww_head1 x]])). - apply Zmult_le_compat_l; auto with zarith. + rewrite <- (Z.mul_1_r (2 ^ [[ww_head1 x]])). + apply Z.mul_le_mono_nonneg_l; auto with zarith. generalize (spec_ww_add_mul_div x W0 (ww_head1 x) Hv2); case ww_add_mul_div. simpl ww_to_Z; autorewrite with w_rewrite rm10. rewrite Zmod_small; auto with zarith. - intros H2; case (Zmult_integral _ _ (sym_equal H2)); clear H2; intros H2. - rewrite H2; unfold Zpower, Zpower_pos; simpl; auto with zarith. + intros H2. symmetry in H2. rewrite Z.mul_eq_0 in H2. destruct H2 as [H2|H2]. + rewrite H2; unfold Z.pow, Z.pow_pos; simpl; auto with zarith. match type of H2 with ?X = ?Y => absurd (Y < X); try (rewrite H2; auto with zarith; fail) end. - apply Zpower_gt_0; auto with zarith. + apply Z.pow_pos_nonneg; auto with zarith. split; auto with zarith. - case Hp2; intros _ tmp; apply Zle_lt_trans with (2 := tmp); + case Hp2; intros _ tmp; apply Z.le_lt_trans with (2 := tmp); clear tmp. - rewrite Zmult_comm; apply Zmult_le_compat_r; auto with zarith. + rewrite Z.mul_comm; apply Z.mul_le_mono_nonneg_r; auto with zarith. assert (Hv0: [[ww_head1 x]] = 2 * ([[ww_head1 x]]/2)). pattern [[ww_head1 x]] at 1; rewrite (Z_div_mod_eq [[ww_head1 x]] 2); auto with zarith. generalize (spec_ww_is_even (ww_head1 x)); rewrite Hp1; - intros tmp; rewrite tmp; rewrite Zplus_0_r; auto. + intros tmp; rewrite tmp; rewrite Z.add_0_r; auto. intros w0 w1; autorewrite with w_rewrite rm10. rewrite Zmod_small; auto with zarith. - 2: rewrite Zmult_comm; auto with zarith. + 2: rewrite Z.mul_comm; auto with zarith. intros H2. assert (V: wB/4 <= [|w0|]). apply beta_lex with 0 [|w1|] wB; auto with zarith; autorewrite with rm10. simpl ww_to_Z in H2; rewrite H2. rewrite <- wwB_4_wB_4; auto with zarith. - rewrite Zmult_comm; auto with zarith. + rewrite Z.mul_comm; auto with zarith. assert (V1 := spec_to_Z w1);auto with zarith. generalize (@spec_w_sqrt2 w0 w1 V);auto with zarith. case (w_sqrt2 w0 w1); intros w2 c. @@ -1267,13 +1262,13 @@ Qed. rewrite spec_ww_pred; rewrite spec_ww_zdigits. rewrite Zmod_small; auto with zarith. split; auto with zarith. - apply Zlt_le_trans with (Zpos (xO w_digits)); auto with zarith. + apply Z.lt_le_trans with (Zpos (xO w_digits)); auto with zarith. unfold base; apply Zpower2_le_lin; auto with zarith. assert (Hv4: [[ww_head1 x]]/2 < wB). - apply Zle_lt_trans with (Zpos w_digits). - apply Zmult_le_reg_r with 2; auto with zarith. - repeat rewrite (fun x => Zmult_comm x 2). - rewrite <- Hv0; rewrite <- Zpos_xO; auto. + apply Z.le_lt_trans with (Zpos w_digits). + apply Z.mul_le_mono_pos_r with 2; auto with zarith. + repeat rewrite (fun x => Z.mul_comm x 2). + rewrite <- Hv0; rewrite <- Pos2Z.inj_xO; auto. unfold base; apply Zpower2_lt_lin; auto with zarith. assert (Hv5: [[(ww_add_mul_div (ww_pred ww_zdigits) W0 (ww_head1 x))]] = [[ww_head1 x]]/2). @@ -1281,12 +1276,12 @@ Qed. simpl ww_to_Z; autorewrite with rm10. rewrite Hv3. ring_simplify (Zpos (xO w_digits) - (Zpos (xO w_digits) - 1)). - rewrite Zpower_1_r. + rewrite Z.pow_1_r. rewrite Zmod_small; auto with zarith. split; auto with zarith. - apply Zlt_le_trans with (1 := Hv4); auto with zarith. + apply Z.lt_le_trans with (1 := Hv4); auto with zarith. unfold base; apply Zpower_le_monotone; auto with zarith. - split; unfold ww_digits; try rewrite Zpos_xO; auto with zarith. + split; unfold ww_digits; try rewrite Pos2Z.inj_xO; auto with zarith. rewrite Hv3; auto with zarith. assert (Hv6: [|low(ww_add_mul_div (ww_pred ww_zdigits) W0 (ww_head1 x))|] = [[ww_head1 x]]/2). @@ -1302,13 +1297,13 @@ Qed. rewrite Zmod_small. simpl ww_to_Z in H2; rewrite H2; auto with zarith. intros (H4, H5); split. - apply Zmult_le_reg_r with (2 ^ [[ww_head1 x]]); auto with zarith. + apply Z.mul_le_mono_pos_r with (2 ^ [[ww_head1 x]]); auto with zarith. rewrite H4. - apply Zle_trans with ([|w2|] ^ 2); auto with zarith. - rewrite Zmult_comm. + apply Z.le_trans with ([|w2|] ^ 2); auto with zarith. + rewrite Z.mul_comm. pattern [[ww_head1 x]] at 1; rewrite Hv0; auto with zarith. - rewrite (Zmult_comm 2); rewrite Zpower_mult; + rewrite (Z.mul_comm 2); rewrite Z.pow_mul_r; auto with zarith. assert (tmp: forall p q, p ^ 2 * q ^ 2 = (p * q) ^2); try (intros; repeat rewrite Zsquare_mult; ring); @@ -1324,17 +1319,17 @@ Qed. case (Z_mod_lt [|w2|] (2 ^ ([[ww_head1 x]] / 2))); auto with zarith. case c; unfold interp_carry; autorewrite with rm10; intros w3; assert (V3 := spec_to_Z w3);auto with zarith. - apply Zmult_lt_reg_r with (2 ^ [[ww_head1 x]]); auto with zarith. + apply Z.mul_lt_mono_pos_r with (2 ^ [[ww_head1 x]]); auto with zarith. rewrite H4. - apply Zle_lt_trans with ([|w2|] ^ 2 + 2 * [|w2|]); auto with zarith. - apply Zlt_le_trans with (([|w2|] + 1) ^ 2); auto with zarith. + apply Z.le_lt_trans with ([|w2|] ^ 2 + 2 * [|w2|]); auto with zarith. + apply Z.lt_le_trans with (([|w2|] + 1) ^ 2); auto with zarith. match goal with |- ?X < ?Y => replace Y with (X + 1); auto with zarith end. repeat rewrite (Zsquare_mult); ring. - rewrite Zmult_comm. + rewrite Z.mul_comm. pattern [[ww_head1 x]] at 1; rewrite Hv0. - rewrite (Zmult_comm 2); rewrite Zpower_mult; + rewrite (Z.mul_comm 2); rewrite Z.pow_mul_r; auto with zarith. assert (tmp: forall p q, p ^ 2 * q ^ 2 = (p * q) ^2); try (intros; repeat rewrite Zsquare_mult; ring); @@ -1343,20 +1338,20 @@ Qed. split; auto with zarith. pattern [|w2|] at 1; rewrite (Z_div_mod_eq [|w2|] (2 ^ ([[ww_head1 x]]/2))); auto with zarith. - rewrite <- Zplus_assoc; rewrite Zmult_plus_distr_r. - autorewrite with rm10; apply Zplus_le_compat_l; auto with zarith. + rewrite <- Z.add_assoc; rewrite Z.mul_add_distr_l. + autorewrite with rm10; apply Z.add_le_mono_l; auto with zarith. case (Z_mod_lt [|w2|] (2 ^ ([[ww_head1 x]]/2))); auto with zarith. split; auto with zarith. - apply Zle_lt_trans with ([|w2|]); auto with zarith. + apply Z.le_lt_trans with ([|w2|]); auto with zarith. apply Zdiv_le_upper_bound; auto with zarith. pattern [|w2|] at 1; replace [|w2|] with ([|w2|] * 2 ^0); auto with zarith. - apply Zmult_le_compat_l; auto with zarith. + apply Z.mul_le_mono_nonneg_l; auto with zarith. apply Zpower_le_monotone; auto with zarith. - rewrite Zpower_0_r; autorewrite with rm10; auto. + rewrite Z.pow_0_r; autorewrite with rm10; auto. split; auto with zarith. - rewrite Hv0 in Hv2; rewrite (Zpos_xO w_digits) in Hv2; auto with zarith. - apply Zle_lt_trans with (Zpos w_digits); auto with zarith. + rewrite Hv0 in Hv2; rewrite (Pos2Z.inj_xO w_digits) in Hv2; auto with zarith. + apply Z.le_lt_trans with (Zpos w_digits); auto with zarith. unfold base; apply Zpower2_lt_lin; auto with zarith. rewrite spec_w_sub; auto with zarith. rewrite Hv6; rewrite spec_w_zdigits; auto with zarith. @@ -1364,10 +1359,10 @@ Qed. rewrite Zmod_small; auto with zarith. split; auto with zarith. assert ([[ww_head1 x]]/2 <= Zpos w_digits); auto with zarith. - apply Zmult_le_reg_r with 2; auto with zarith. - repeat rewrite (fun x => Zmult_comm x 2). - rewrite <- Hv0; rewrite <- Zpos_xO; auto with zarith. - apply Zle_lt_trans with (Zpos w_digits); auto with zarith. + apply Z.mul_le_mono_pos_r with 2; auto with zarith. + repeat rewrite (fun x => Z.mul_comm x 2). + rewrite <- Hv0; rewrite <- Pos2Z.inj_xO; auto with zarith. + apply Z.le_lt_trans with (Zpos w_digits); auto with zarith. unfold base; apply Zpower2_lt_lin; auto with zarith. Qed. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v index e63e20c6..799c4e42 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -195,9 +195,9 @@ Section DoubleSub. Lemma spec_ww_opp_c : forall x, [-[ww_opp_c x]] = -[[x]]. Proof. destruct x as [ |xh xl];simpl. reflexivity. - rewrite Zopp_plus_distr;generalize (spec_opp_c xl);destruct (w_opp_c xl) + rewrite Z.opp_add_distr;generalize (spec_opp_c xl);destruct (w_opp_c xl) as [l|l];intros H;unfold interp_carry in H;rewrite <- H; - rewrite Zopp_mult_distr_l. + rewrite <- Z.mul_opp_l. assert ([|l|] = 0). assert (H1:= spec_to_Z l);assert (H2 := spec_to_Z xl);omega. rewrite H0;generalize (spec_opp_c xh);destruct (w_opp_c xh) @@ -213,13 +213,13 @@ Section DoubleSub. Lemma spec_ww_opp : forall x, [[ww_opp x]] = (-[[x]]) mod wwB. Proof. destruct x as [ |xh xl];simpl. reflexivity. - rewrite Zopp_plus_distr;rewrite Zopp_mult_distr_l. + rewrite Z.opp_add_distr, <- Z.mul_opp_l. generalize (spec_opp_c xl);destruct (w_opp_c xl) as [l|l];intros H;unfold interp_carry in H;rewrite <- H;simpl ww_to_Z. - rewrite spec_w_0;rewrite Zplus_0_r;rewrite wwB_wBwB. + rewrite spec_w_0;rewrite Z.add_0_r;rewrite wwB_wBwB. assert ([|l|] = 0). assert (H1:= spec_to_Z l);assert (H2 := spec_to_Z xl);omega. - rewrite H0;rewrite Zplus_0_r; rewrite Zpower_2; + rewrite H0;rewrite Z.add_0_r; rewrite Z.pow_2_r; rewrite Zmult_mod_distr_r;try apply lt_0_wB. rewrite spec_opp;trivial. apply Zmod_unique with (q:= -1). @@ -240,7 +240,7 @@ Section DoubleSub. simpl ww_to_Z;replace (([|xh|]*wB+[|xl|])-1) with ([|xh|]*wB+([|xl|]-1)). 2:ring. generalize (spec_pred_c xl);destruct (w_pred_c xl) as [l|l]; intros H;unfold interp_carry in H;rewrite <- H. simpl;apply spec_w_WW. - rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. assert ([|l|] = wB - 1). assert (H1:= spec_to_Z l);assert (H2 := spec_to_Z xl);omega. rewrite H0;change ([|xh|] + -1) with ([|xh|] - 1). @@ -263,7 +263,7 @@ Section DoubleSub. generalize (spec_sub_c xh yh);destruct (w_sub_c xh yh) as [h|h];intros H1; unfold interp_carry in H1;rewrite <- H1;unfold interp_carry; try rewrite spec_w_WW;simpl ww_to_Z;try rewrite wwB_wBwB;ring. - rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. change ([|xh|] - [|yh|] + -1) with ([|xh|] - [|yh|] - 1). generalize (spec_sub_carry_c xh yh);destruct (w_sub_carry_c xh yh) as [h|h]; intros H1;unfold interp_carry in *;rewrite <- H1;simpl ww_to_Z; @@ -274,7 +274,7 @@ Section DoubleSub. forall x y, [-[ww_sub_carry_c x y]] = [[x]] - [[y]] - 1. Proof. destruct y as [ |yh yl];simpl. - unfold Zminus;simpl;rewrite Zplus_0_r;exact (spec_ww_pred_c x). + unfold Z.sub;simpl;rewrite Z.add_0_r;exact (spec_ww_pred_c x). destruct x as [ |xh xl]. unfold interp_carry;rewrite spec_w_WW;simpl ww_to_Z;rewrite wwB_wBwB; repeat rewrite spec_opp_carry;ring. @@ -286,7 +286,7 @@ Section DoubleSub. generalize (spec_sub_c xh yh);destruct (w_sub_c xh yh) as [h|h];intros H1; unfold interp_carry in H1;rewrite <- H1;unfold interp_carry; try rewrite spec_w_WW;simpl ww_to_Z;try rewrite wwB_wBwB;ring. - rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. change ([|xh|] - [|yh|] + -1) with ([|xh|] - [|yh|] - 1). generalize (spec_sub_carry_c xh yh);destruct (w_sub_carry_c xh yh) as [h|h]; intros H1;unfold interp_carry in *;rewrite <- H1;try rewrite spec_w_WW; @@ -303,7 +303,7 @@ Section DoubleSub. unfold interp_carry in H;rewrite <- H;simpl ww_to_Z. rewrite Zmod_small. apply spec_w_WW. exact (spec_ww_to_Z w_digits w_to_Z spec_to_Z (WW xh l)). - rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. change ([|xh|] + -1) with ([|xh|] - 1). assert ([|l|] = wB - 1). assert (H1:= spec_to_Z l);assert (H2:= spec_to_Z xl);omega. @@ -322,7 +322,7 @@ Section DoubleSub. unfold interp_carry in H;rewrite <- H. rewrite spec_w_WW;rewrite (mod_wwB w_digits w_to_Z spec_to_Z). rewrite spec_sub;trivial. - simpl ww_to_Z;rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + simpl ww_to_Z;rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_sub_carry;trivial. Qed. @@ -341,7 +341,7 @@ Section DoubleSub. generalize (spec_sub_carry_c xl yl);destruct (w_sub_carry_c xl yl)as[l|l]; intros H;unfold interp_carry in H;rewrite <- H;rewrite spec_w_WW. rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_sub;trivial. - rewrite Zplus_assoc;rewrite <- Zmult_plus_distr_l. + rewrite Z.add_assoc;rewrite <- Z.mul_add_distr_r. rewrite (mod_wwB w_digits w_to_Z spec_to_Z);rewrite spec_sub_carry;trivial. Qed. diff --git a/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v b/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v index a274b839..ce1c0bef 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -13,7 +13,7 @@ Set Implicit Arguments. Require Import ZArith. Local Open Scope Z_scope. -Definition base digits := Zpower 2 (Zpos digits). +Definition base digits := Z.pow 2 (Zpos digits). Section Carry. diff --git a/theories/Numbers/Cyclic/Int31/Cyclic31.v b/theories/Numbers/Cyclic/Int31/Cyclic31.v index 2dd1c3ee..385217d0 100644 --- a/theories/Numbers/Cyclic/Int31/Cyclic31.v +++ b/theories/Numbers/Cyclic/Int31/Cyclic31.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -368,7 +368,7 @@ Section Basics. (** Variant of [phi] via [recrbis] *) Let Phi := fun b (_:int31) => - match b with D0 => Zdouble | D1 => Zdouble_plus_one end. + match b with D0 => Z.double | D1 => Z.succ_double end. Definition phibis_aux n x := recrbis_aux n _ Z0 Phi x. @@ -381,7 +381,7 @@ Section Basics. (** Recursive equations satisfied by [phi] *) Lemma phi_eqn1 : forall x, firstr x = D0 -> - phi x = Zdouble (phi (shiftr x)). + phi x = Z.double (phi (shiftr x)). Proof. intros. case_eq (iszero x); intros. @@ -391,7 +391,7 @@ Section Basics. Qed. Lemma phi_eqn2 : forall x, firstr x = D1 -> - phi x = Zdouble_plus_one (phi (shiftr x)). + phi x = Z.succ_double (phi (shiftr x)). Proof. intros. case_eq (iszero x); intros. @@ -401,7 +401,7 @@ Section Basics. Qed. Lemma phi_twice_firstl : forall x, firstl x = D0 -> - phi (twice x) = Zdouble (phi x). + phi (twice x) = Z.double (phi x). Proof. intros. rewrite phi_eqn1; auto; [ | destruct x; auto ]. @@ -410,7 +410,7 @@ Section Basics. Qed. Lemma phi_twice_plus_one_firstl : forall x, firstl x = D0 -> - phi (twice_plus_one x) = Zdouble_plus_one (phi x). + phi (twice_plus_one x) = Z.succ_double (phi x). Proof. intros. rewrite phi_eqn2; auto; [ | destruct x; auto ]. @@ -430,13 +430,13 @@ Section Basics. unfold phibis_aux, recrbis_aux; fold recrbis_aux; fold (phibis_aux n (shiftr x)). destruct (firstr x). - specialize IHn with (shiftr x); rewrite Zdouble_mult; omega. - specialize IHn with (shiftr x); rewrite Zdouble_plus_one_mult; omega. + specialize IHn with (shiftr x); rewrite Z.double_spec; omega. + specialize IHn with (shiftr x); rewrite Z.succ_double_spec; omega. Qed. Lemma phibis_aux_bounded : forall n x, n <= size -> - (phibis_aux n (nshiftr (size-n) x) < 2 ^ (Z_of_nat n))%Z. + (phibis_aux n (nshiftr (size-n) x) < 2 ^ (Z.of_nat n))%Z. Proof. induction n. simpl; unfold phibis_aux; simpl; auto with zarith. @@ -450,13 +450,13 @@ Section Basics. assert (H1 : n <= size) by omega. specialize (IHn x H1). set (y:=phibis_aux n (nshiftr (size - n) x)) in *. - rewrite inj_S, Zpower_Zsucc; auto with zarith. + rewrite Nat2Z.inj_succ, Z.pow_succ_r; auto with zarith. case_eq (firstr (nshiftr (size - S n) x)); intros. - rewrite Zdouble_mult; auto with zarith. - rewrite Zdouble_plus_one_mult; auto with zarith. + rewrite Z.double_spec; auto with zarith. + rewrite Z.succ_double_spec; auto with zarith. Qed. - Lemma phi_bounded : forall x, (0 <= phi x < 2 ^ (Z_of_nat size))%Z. + Lemma phi_bounded : forall x, (0 <= phi x < 2 ^ (Z.of_nat size))%Z. Proof. intros. rewrite <- phibis_aux_equiv. @@ -468,32 +468,32 @@ Section Basics. Lemma phibis_aux_lowerbound : forall n x, firstr (nshiftr n x) = D1 -> - (2 ^ Z_of_nat n <= phibis_aux (S n) x)%Z. + (2 ^ Z.of_nat n <= phibis_aux (S n) x)%Z. Proof. induction n. intros. unfold nshiftr in H; simpl in *. unfold phibis_aux, recrbis_aux. - rewrite H, Zdouble_plus_one_mult; omega. + rewrite H, Z.succ_double_spec; omega. intros. remember (S n) as m. unfold phibis_aux, recrbis_aux; fold recrbis_aux; fold (phibis_aux m (shiftr x)). subst m. - rewrite inj_S, Zpower_Zsucc; auto with zarith. - assert (2^(Z_of_nat n) <= phibis_aux (S n) (shiftr x))%Z. + rewrite Nat2Z.inj_succ, Z.pow_succ_r; auto with zarith. + assert (2^(Z.of_nat n) <= phibis_aux (S n) (shiftr x))%Z. apply IHn. rewrite <- nshiftr_S_tail; auto. destruct (firstr x). - change (Zdouble (phibis_aux (S n) (shiftr x))) with + change (Z.double (phibis_aux (S n) (shiftr x))) with (2*(phibis_aux (S n) (shiftr x)))%Z. omega. - rewrite Zdouble_plus_one_mult; omega. + rewrite Z.succ_double_spec; omega. Qed. Lemma phi_lowerbound : - forall x, firstl x = D1 -> (2^(Z_of_nat (pred size)) <= phi x)%Z. + forall x, firstl x = D1 -> (2^(Z.of_nat (pred size)) <= phi x)%Z. Proof. intros. generalize (phibis_aux_lowerbound (pred size) x). @@ -776,7 +776,7 @@ Section Basics. (** First, recursive equations *) Lemma phi_inv_double_plus_one : forall z, - phi_inv (Zdouble_plus_one z) = twice_plus_one (phi_inv z). + phi_inv (Z.succ_double z) = twice_plus_one (phi_inv z). Proof. destruct z; simpl; auto. induction p; simpl. @@ -788,20 +788,20 @@ Section Basics. Qed. Lemma phi_inv_double : forall z, - phi_inv (Zdouble z) = twice (phi_inv z). + phi_inv (Z.double z) = twice (phi_inv z). Proof. destruct z; simpl; auto. rewrite incr_twice_plus_one; auto. Qed. Lemma phi_inv_incr : forall z, - phi_inv (Zsucc z) = incr (phi_inv z). + phi_inv (Z.succ z) = incr (phi_inv z). Proof. destruct z. simpl; auto. simpl; auto. induction p; simpl; auto. - rewrite Pplus_one_succ_r, IHp, incr_twice_plus_one; auto. + rewrite <- Pos.add_1_r, IHp, incr_twice_plus_one; auto. rewrite incr_twice; auto. simpl; auto. destruct p; simpl; auto. @@ -908,15 +908,15 @@ Section Basics. Local Open Scope Z_scope. Lemma p2ibis_spec : forall n p, (n<=size)%nat -> - Zpos p = (Z_of_N (fst (p2ibis n p)))*2^(Z_of_nat n) + + Zpos p = (Z.of_N (fst (p2ibis n p)))*2^(Z.of_nat n) + phi (snd (p2ibis n p)). Proof. induction n; intros. - simpl; rewrite Pmult_1_r; auto. - replace (2^(Z_of_nat (S n)))%Z with (2*2^(Z_of_nat n))%Z by - (rewrite <- Zpower_Zsucc, <- Zpos_P_of_succ_nat; + simpl; rewrite Pos.mul_1_r; auto. + replace (2^(Z.of_nat (S n)))%Z with (2*2^(Z.of_nat n))%Z by + (rewrite <- Z.pow_succ_r, <- Zpos_P_of_succ_nat; auto with zarith). - rewrite (Zmult_comm 2). + rewrite (Z.mul_comm 2). assert (n<=size)%nat by omega. destruct p; simpl; [ | | auto]; specialize (IHn p H0); @@ -924,13 +924,13 @@ Section Basics. destruct (p2ibis n p) as (r,i); simpl in *; intros. change (Zpos p~1) with (2*Zpos p + 1)%Z. - rewrite phi_twice_plus_one_firstl, Zdouble_plus_one_mult. + rewrite phi_twice_plus_one_firstl, Z.succ_double_spec. rewrite IHn; ring. apply (nshiftr_0_firstl n); auto; try omega. change (Zpos p~0) with (2*Zpos p)%Z. rewrite phi_twice_firstl. - change (Zdouble (phi i)) with (2*(phi i))%Z. + change (Z.double (phi i)) with (2*(phi i))%Z. rewrite IHn; ring. apply (nshiftr_0_firstl n); auto; try omega. Qed. @@ -956,12 +956,12 @@ Section Basics. for the positive case. *) Lemma phi_phi_inv_positive : forall p, - phi (phi_inv_positive p) = (Zpos p) mod (2^(Z_of_nat size)). + phi (phi_inv_positive p) = (Zpos p) mod (2^(Z.of_nat size)). Proof. intros. replace (phi_inv_positive p) with (snd (p2ibis size p)). rewrite (p2ibis_spec size p) by auto. - rewrite Zplus_comm, Z_mod_plus. + rewrite Z.add_comm, Z_mod_plus. symmetry; apply Zmod_small. apply phi_bounded. auto with zarith. @@ -978,7 +978,7 @@ Section Basics. Proof. intros. unfold mul31. - rewrite <- Zdouble_mult, <- phi_twice_firstl, phi_inv_phi; auto. + rewrite <- Z.double_spec, <- phi_twice_firstl, phi_inv_phi; auto. Qed. Lemma double_twice_plus_one_firstl : forall x, firstl x = D0 -> @@ -987,7 +987,7 @@ Section Basics. intros. rewrite double_twice_firstl; auto. unfold add31. - rewrite phi_twice_firstl, <- Zdouble_plus_one_mult, + rewrite phi_twice_firstl, <- Z.succ_double_spec, <- phi_twice_plus_one_firstl, phi_inv_phi; auto. Qed. @@ -1016,7 +1016,7 @@ Section Basics. Qed. Lemma positive_to_int31_spec : forall p, - Zpos p = (Z_of_N (fst (positive_to_int31 p)))*2^(Z_of_nat size) + + Zpos p = (Z.of_N (fst (positive_to_int31 p)))*2^(Z.of_nat size) + phi (snd (positive_to_int31 p)). Proof. unfold positive_to_int31. @@ -1029,43 +1029,43 @@ Section Basics. [phi o twice] and so one. *) Lemma phi_twice : forall x, - phi (twice x) = (Zdouble (phi x)) mod 2^(Z_of_nat size). + phi (twice x) = (Z.double (phi x)) mod 2^(Z.of_nat size). Proof. intros. pattern x at 1; rewrite <- (phi_inv_phi x). rewrite <- phi_inv_double. - assert (0 <= Zdouble (phi x)). - rewrite Zdouble_mult; generalize (phi_bounded x); omega. - destruct (Zdouble (phi x)). + assert (0 <= Z.double (phi x)). + rewrite Z.double_spec; generalize (phi_bounded x); omega. + destruct (Z.double (phi x)). simpl; auto. apply phi_phi_inv_positive. compute in H; elim H; auto. Qed. Lemma phi_twice_plus_one : forall x, - phi (twice_plus_one x) = (Zdouble_plus_one (phi x)) mod 2^(Z_of_nat size). + phi (twice_plus_one x) = (Z.succ_double (phi x)) mod 2^(Z.of_nat size). Proof. intros. pattern x at 1; rewrite <- (phi_inv_phi x). rewrite <- phi_inv_double_plus_one. - assert (0 <= Zdouble_plus_one (phi x)). - rewrite Zdouble_plus_one_mult; generalize (phi_bounded x); omega. - destruct (Zdouble_plus_one (phi x)). + assert (0 <= Z.succ_double (phi x)). + rewrite Z.succ_double_spec; generalize (phi_bounded x); omega. + destruct (Z.succ_double (phi x)). simpl; auto. apply phi_phi_inv_positive. compute in H; elim H; auto. Qed. Lemma phi_incr : forall x, - phi (incr x) = (Zsucc (phi x)) mod 2^(Z_of_nat size). + phi (incr x) = (Z.succ (phi x)) mod 2^(Z.of_nat size). Proof. intros. pattern x at 1; rewrite <- (phi_inv_phi x). rewrite <- phi_inv_incr. - assert (0 <= Zsucc (phi x)). - change (Zsucc (phi x)) with ((phi x)+1)%Z; + assert (0 <= Z.succ (phi x)). + change (Z.succ (phi x)) with ((phi x)+1)%Z; generalize (phi_bounded x); omega. - destruct (Zsucc (phi x)). + destruct (Z.succ (phi x)). simpl; auto. apply phi_phi_inv_positive. compute in H; elim H; auto. @@ -1075,7 +1075,7 @@ Section Basics. in the negative case *) Lemma phi_phi_inv_negative : - forall p, phi (incr (complement_negative p)) = (Zneg p) mod 2^(Z_of_nat size). + forall p, phi (incr (complement_negative p)) = (Zneg p) mod 2^(Z.of_nat size). Proof. induction p. @@ -1083,21 +1083,21 @@ Section Basics. rewrite phi_incr in IHp. rewrite incr_twice, phi_twice_plus_one. remember (phi (complement_negative p)) as q. - rewrite Zdouble_plus_one_mult. - replace (2*q+1) with (2*(Zsucc q)-1) by omega. + rewrite Z.succ_double_spec. + replace (2*q+1) with (2*(Z.succ q)-1) by omega. rewrite <- Zminus_mod_idemp_l, <- Zmult_mod_idemp_r, IHp. rewrite Zmult_mod_idemp_r, Zminus_mod_idemp_l; auto with zarith. simpl complement_negative. rewrite incr_twice_plus_one, phi_twice. remember (phi (incr (complement_negative p))) as q. - rewrite Zdouble_mult, IHp, Zmult_mod_idemp_r; auto with zarith. + rewrite Z.double_spec, IHp, Zmult_mod_idemp_r; auto with zarith. simpl; auto. Qed. Lemma phi_phi_inv : - forall z, phi (phi_inv z) = z mod 2 ^ (Z_of_nat size). + forall z, phi (phi_inv z) = z mod 2 ^ (Z.of_nat size). Proof. destruct z. simpl; auto. @@ -1167,7 +1167,7 @@ Section Int31_Specs. Notation "[| x |]" := (phi x) (at level 0, x at level 99). - Local Notation wB := (2 ^ (Z_of_nat size)). + Local Notation wB := (2 ^ (Z.of_nat size)). Lemma wB_pos : wB > 0. Proof. @@ -1221,14 +1221,14 @@ Section Int31_Specs. set (X:=[|x|]) in *; set (Y:=[|y|]) in *; clearbody X Y. assert ((X+Y) mod wB ?= X+Y <> Eq -> [+|C1 (phi_inv (X+Y))|] = X+Y). - unfold interp_carry; rewrite phi_phi_inv, Zcompare_Eq_iff_eq; intros. + unfold interp_carry; rewrite phi_phi_inv, Z.compare_eq_iff; intros. destruct (Z_lt_le_dec (X+Y) wB). contradict H1; auto using Zmod_small with zarith. rewrite <- (Z_mod_plus_full (X+Y) (-1) wB). rewrite Zmod_small; romega. - generalize (Zcompare_Eq_eq ((X+Y) mod wB) (X+Y)); intros Heq. - destruct Zcompare; intros; + generalize (Z.compare_eq ((X+Y) mod wB) (X+Y)); intros Heq. + destruct Z.compare; intros; [ rewrite phi_phi_inv; auto | now apply H1 | now apply H1]. Qed. @@ -1245,14 +1245,14 @@ Section Int31_Specs. set (X:=[|x|]) in *; set (Y:=[|y|]) in *; clearbody X Y. assert ((X+Y+1) mod wB ?= X+Y+1 <> Eq -> [+|C1 (phi_inv (X+Y+1))|] = X+Y+1). - unfold interp_carry; rewrite phi_phi_inv, Zcompare_Eq_iff_eq; intros. + unfold interp_carry; rewrite phi_phi_inv, Z.compare_eq_iff; intros. destruct (Z_lt_le_dec (X+Y+1) wB). contradict H1; auto using Zmod_small with zarith. rewrite <- (Z_mod_plus_full (X+Y+1) (-1) wB). rewrite Zmod_small; romega. - generalize (Zcompare_Eq_eq ((X+Y+1) mod wB) (X+Y+1)); intros Heq. - destruct Zcompare; intros; + generalize (Z.compare_eq ((X+Y+1) mod wB) (X+Y+1)); intros Heq. + destruct Z.compare; intros; [ rewrite phi_phi_inv; auto | now apply H1 | now apply H1]. Qed. @@ -1284,14 +1284,14 @@ Section Int31_Specs. set (X:=[|x|]) in *; set (Y:=[|y|]) in *; clearbody X Y. assert ((X-Y) mod wB ?= X-Y <> Eq -> [-|C1 (phi_inv (X-Y))|] = X-Y). - unfold interp_carry; rewrite phi_phi_inv, Zcompare_Eq_iff_eq; intros. + unfold interp_carry; rewrite phi_phi_inv, Z.compare_eq_iff; intros. destruct (Z_lt_le_dec (X-Y) 0). rewrite <- (Z_mod_plus_full (X-Y) 1 wB). rewrite Zmod_small; romega. contradict H1; apply Zmod_small; romega. - generalize (Zcompare_Eq_eq ((X-Y) mod wB) (X-Y)); intros Heq. - destruct Zcompare; intros; + generalize (Z.compare_eq ((X-Y) mod wB) (X-Y)); intros Heq. + destruct Z.compare; intros; [ rewrite phi_phi_inv; auto | now apply H1 | now apply H1]. Qed. @@ -1303,14 +1303,14 @@ Section Int31_Specs. set (X:=[|x|]) in *; set (Y:=[|y|]) in *; clearbody X Y. assert ((X-Y-1) mod wB ?= X-Y-1 <> Eq -> [-|C1 (phi_inv (X-Y-1))|] = X-Y-1). - unfold interp_carry; rewrite phi_phi_inv, Zcompare_Eq_iff_eq; intros. + unfold interp_carry; rewrite phi_phi_inv, Z.compare_eq_iff; intros. destruct (Z_lt_le_dec (X-Y-1) 0). rewrite <- (Z_mod_plus_full (X-Y-1) 1 wB). rewrite Zmod_small; romega. contradict H1; apply Zmod_small; romega. - generalize (Zcompare_Eq_eq ((X-Y-1) mod wB) (X-Y-1)); intros Heq. - destruct Zcompare; intros; + generalize (Z.compare_eq ((X-Y-1) mod wB) (X-Y-1)); intros Heq. + destruct Z.compare; intros; [ rewrite phi_phi_inv; auto | now apply H1 | now apply H1]. Qed. @@ -1386,7 +1386,7 @@ Section Int31_Specs. apply Zmod_small. generalize (phi_bounded x)(phi_bounded y); intros. change (wB^2) with (wB * wB). - auto using Zmult_lt_compat with zarith. + auto using Z.mul_lt_mono_nonneg with zarith. Qed. Lemma spec_mul : forall x y, [|x*y|] = ([|x|] * [|y|]) mod wB. @@ -1412,29 +1412,26 @@ Section Int31_Specs. generalize (phi_bounded a1)(phi_bounded a2)(phi_bounded b); intros. assert ([|b|]>0) by (auto with zarith). generalize (Z_div_mod (phi2 a1 a2) [|b|] H4) (Z_div_pos (phi2 a1 a2) [|b|] H4). - unfold Zdiv; destruct (Zdiv_eucl (phi2 a1 a2) [|b|]); simpl. + unfold Z.div; destruct (Z.div_eucl (phi2 a1 a2) [|b|]); simpl. rewrite ?phi_phi_inv. destruct 1; intros. unfold phi2 in *. change base with wB; change base with wB in H5. - change (Zpower_pos 2 31) with wB; change (Zpower_pos 2 31) with wB in H. - rewrite H5, Zmult_comm. + change (Z.pow_pos 2 31) with wB; change (Z.pow_pos 2 31) with wB in H. + rewrite H5, Z.mul_comm. replace (z0 mod wB) with z0 by (symmetry; apply Zmod_small; omega). replace (z mod wB) with z; auto with zarith. symmetry; apply Zmod_small. split. apply H7; change base with wB; auto with zarith. - apply Zmult_gt_0_lt_reg_r with [|b|]. - omega. - rewrite Zmult_comm. - apply Zle_lt_trans with ([|b|]*z+z0). - omega. + apply Z.mul_lt_mono_pos_r with [|b|]; [omega| ]. + rewrite Z.mul_comm. + apply Z.le_lt_trans with ([|b|]*z+z0); [omega| ]. rewrite <- H5. - apply Zle_lt_trans with ([|a1|]*wB+(wB-1)). - omega. + apply Z.le_lt_trans with ([|a1|]*wB+(wB-1)); [omega | ]. replace ([|a1|]*wB+(wB-1)) with (wB*([|a1|]+1)-1) by ring. assert (wB*([|a1|]+1) <= wB*[|b|]); try omega. - apply Zmult_le_compat; omega. + apply Z.mul_le_mono_nonneg; omega. Qed. Lemma spec_div : forall a b, 0 < [|b|] -> @@ -1445,20 +1442,20 @@ Section Int31_Specs. unfold div31; intros. assert ([|b|]>0) by (auto with zarith). generalize (Z_div_mod [|a|] [|b|] H0) (Z_div_pos [|a|] [|b|] H0). - unfold Zdiv; destruct (Zdiv_eucl [|a|] [|b|]); simpl. + unfold Z.div; destruct (Z.div_eucl [|a|] [|b|]); simpl. rewrite ?phi_phi_inv. destruct 1; intros. - rewrite H1, Zmult_comm. + rewrite H1, Z.mul_comm. generalize (phi_bounded a)(phi_bounded b); intros. replace (z0 mod wB) with z0 by (symmetry; apply Zmod_small; omega). replace (z mod wB) with z; auto with zarith. symmetry; apply Zmod_small. split; auto with zarith. - apply Zle_lt_trans with [|a|]; auto with zarith. + apply Z.le_lt_trans with [|a|]; auto with zarith. rewrite H1. - apply Zle_trans with ([|b|]*z); try omega. - rewrite <- (Zmult_1_l z) at 1. - apply Zmult_le_compat; auto with zarith. + apply Z.le_trans with ([|b|]*z); try omega. + rewrite <- (Z.mul_1_l z) at 1. + apply Z.mul_le_mono_nonneg; auto with zarith. Qed. Lemma spec_mod : forall a b, 0 < [|b|] -> @@ -1466,9 +1463,9 @@ Section Int31_Specs. Proof. unfold div31; intros. assert ([|b|]>0) by (auto with zarith). - unfold Zmod. + unfold Z.modulo. generalize (Z_div_mod [|a|] [|b|] H0). - destruct (Zdiv_eucl [|a|] [|b|]); simpl. + destruct (Z.div_eucl [|a|] [|b|]); simpl. rewrite ?phi_phi_inv. destruct 1; intros. generalize (phi_bounded b); intros. @@ -1506,12 +1503,12 @@ Section Int31_Specs. destruct [|b|]. unfold size; auto with zarith. intros (_,H). - cut (Psize p <= size)%nat; [ omega | rewrite <- Zpower2_Psize; auto]. + cut (Pos.size_nat p <= size)%nat; [ omega | rewrite <- Zpower2_Psize; auto]. intros (H,_); compute in H; elim H; auto. Qed. Lemma iter_int31_iter_nat : forall A f i a, - iter_int31 i A f a = iter_nat (Zabs_nat [|i|]) A f a. + iter_int31 i A f a = iter_nat (Z.abs_nat [|i|]) A f a. Proof. intros. unfold iter_int31. @@ -1528,15 +1525,15 @@ Section Int31_Specs. rewrite <- iter_nat_plus. f_equal. - rewrite Zdouble_mult, Zmult_comm, <- Zplus_diag_eq_mult_2. - symmetry; apply Zabs_nat_Zplus; auto with zarith. + rewrite Z.double_spec, <- Z.add_diag. + symmetry; apply Zabs2Nat.inj_add; auto with zarith. - change (iter_nat (S (Zabs_nat z + Zabs_nat z)) A f a = - iter_nat (Zabs_nat (Zdouble_plus_one z)) A f a); f_equal. - rewrite Zdouble_plus_one_mult, Zmult_comm, <- Zplus_diag_eq_mult_2. - rewrite Zabs_nat_Zplus; auto with zarith. - rewrite Zabs_nat_Zplus; auto with zarith. - change (Zabs_nat 1) with 1%nat; omega. + change (iter_nat (S (Z.abs_nat z + Z.abs_nat z)) A f a = + iter_nat (Z.abs_nat (Z.succ_double z)) A f a); f_equal. + rewrite Z.succ_double_spec, <- Z.add_diag. + rewrite Zabs2Nat.inj_add; auto with zarith. + rewrite Zabs2Nat.inj_add; auto with zarith. + change (Z.abs_nat 1) with 1%nat; omega. Qed. Fixpoint addmuldiv31_alt n i j := @@ -1546,12 +1543,12 @@ Section Int31_Specs. end. Lemma addmuldiv31_equiv : forall p x y, - addmuldiv31 p x y = addmuldiv31_alt (Zabs_nat [|p|]) x y. + addmuldiv31 p x y = addmuldiv31_alt (Z.abs_nat [|p|]) x y. Proof. intros. unfold addmuldiv31. rewrite iter_int31_iter_nat. - set (n:=Zabs_nat [|p|]); clearbody n; clear p. + set (n:=Z.abs_nat [|p|]); clearbody n; clear p. revert x y; induction n. simpl; auto. intros. @@ -1566,21 +1563,21 @@ Section Int31_Specs. Proof. intros. rewrite addmuldiv31_equiv. - assert ([|p|] = Z_of_nat (Zabs_nat [|p|])). - rewrite inj_Zabs_nat; symmetry; apply Zabs_eq. + assert ([|p|] = Z.of_nat (Z.abs_nat [|p|])). + rewrite Zabs2Nat.id_abs; symmetry; apply Z.abs_eq. destruct (phi_bounded p); auto. - rewrite H0; rewrite H0 in H; clear H0; rewrite Zabs_nat_Z_of_nat. - set (n := Zabs_nat [|p|]) in *; clearbody n. + rewrite H0; rewrite H0 in H; clear H0; rewrite Zabs2Nat.id. + set (n := Z.abs_nat [|p|]) in *; clearbody n. assert (n <= 31)%nat. - rewrite inj_le_iff; auto with zarith. + rewrite Nat2Z.inj_le; auto with zarith. clear p H; revert x y. induction n. simpl; intros. - change (Zpower_pos 2 31) with (2^31). - rewrite Zmult_1_r. + change (Z.pow_pos 2 31) with (2^31). + rewrite Z.mul_1_r. replace ([|y|] / 2^31) with 0. - rewrite Zplus_0_r. + rewrite Z.add_0_r. symmetry; apply Zmod_small; apply phi_bounded. symmetry; apply Zdiv_small; apply phi_bounded. @@ -1588,43 +1585,43 @@ Section Int31_Specs. rewrite IHn; [ | omega ]. case_eq (firstl y); intros. - rewrite phi_twice, Zdouble_mult. + rewrite phi_twice, Z.double_spec. rewrite phi_twice_firstl; auto. - change (Zdouble [|y|]) with (2*[|y|]). - rewrite inj_S, Zpower_Zsucc; auto with zarith. + change (Z.double [|y|]) with (2*[|y|]). + rewrite Nat2Z.inj_succ, Z.pow_succ_r; auto with zarith. rewrite Zplus_mod; rewrite Zmult_mod_idemp_l; rewrite <- Zplus_mod. f_equal. - apply Zplus_eq_compat. + f_equal. ring. - replace (31-Z_of_nat n) with (Zsucc(31-Zsucc(Z_of_nat n))) by ring. - rewrite Zpower_Zsucc, <- Zdiv_Zdiv; auto with zarith. - rewrite Zmult_comm, Z_div_mult; auto with zarith. + replace (31-Z.of_nat n) with (Z.succ(31-Z.succ(Z.of_nat n))) by ring. + rewrite Z.pow_succ_r, <- Zdiv_Zdiv; auto with zarith. + rewrite Z.mul_comm, Z_div_mult; auto with zarith. - rewrite phi_twice_plus_one, Zdouble_plus_one_mult. + rewrite phi_twice_plus_one, Z.succ_double_spec. rewrite phi_twice; auto. - change (Zdouble [|y|]) with (2*[|y|]). - rewrite inj_S, Zpower_Zsucc; auto with zarith. + change (Z.double [|y|]) with (2*[|y|]). + rewrite Nat2Z.inj_succ, Z.pow_succ_r; auto with zarith. rewrite Zplus_mod; rewrite Zmult_mod_idemp_l; rewrite <- Zplus_mod. - rewrite Zmult_plus_distr_l, Zmult_1_l, <- Zplus_assoc. + rewrite Z.mul_add_distr_r, Z.mul_1_l, <- Z.add_assoc. + f_equal. f_equal. - apply Zplus_eq_compat. ring. assert ((2*[|y|]) mod wB = 2*[|y|] - wB). clear - H. symmetry. apply Zmod_unique with 1; [ | ring ]. generalize (phi_lowerbound _ H) (phi_bounded y). - set (wB' := 2^Z_of_nat (pred size)). + set (wB' := 2^Z.of_nat (pred size)). replace wB with (2*wB'); [ omega | ]. - unfold wB'. rewrite <- Zpower_Zsucc, <- inj_S by (auto with zarith). + unfold wB'. rewrite <- Z.pow_succ_r, <- Nat2Z.inj_succ by (auto with zarith). f_equal. rewrite H1. - replace wB with (2^(Z_of_nat n)*2^(31-Z_of_nat n)) by + replace wB with (2^(Z.of_nat n)*2^(31-Z.of_nat n)) by (rewrite <- Zpower_exp; auto with zarith; f_equal; unfold size; ring). - unfold Zminus; rewrite Zopp_mult_distr_l. + unfold Z.sub; rewrite <- Z.mul_opp_l. rewrite Z_div_plus; auto with zarith. ring_simplify. - replace (31+-Z_of_nat n) with (Zsucc(31-Zsucc(Z_of_nat n))) by ring. - rewrite Zpower_Zsucc, <- Zdiv_Zdiv; auto with zarith. - rewrite Zmult_comm, Z_div_mult; auto with zarith. + replace (31+-Z.of_nat n) with (Z.succ(31-Z.succ(Z.of_nat n))) by ring. + rewrite Z.pow_succ_r, <- Zdiv_Zdiv; auto with zarith. + rewrite Z.mul_comm, Z_div_mult; auto with zarith. Qed. Lemma spec_pos_mod : forall w p, @@ -1637,25 +1634,25 @@ Section Int31_Specs. generalize (phi_bounded w). symmetry; apply Zmod_small. split; auto with zarith. - apply Zlt_le_trans with wB; auto with zarith. + apply Z.lt_le_trans with wB; auto with zarith. apply Zpower_le_monotone; auto with zarith. intros. case_eq ([|p|] ?= 31); intros; - [ apply H; rewrite (Zcompare_Eq_eq _ _ H0); auto with zarith | | + [ apply H; rewrite (Z.compare_eq _ _ H0); auto with zarith | | apply H; change ([|p|]>31)%Z in H0; auto with zarith ]. change ([|p|]<31) in H0. rewrite spec_add_mul_div by auto with zarith. - change [|0|] with 0%Z; rewrite Zmult_0_l, Zplus_0_l. + change [|0|] with 0%Z; rewrite Z.mul_0_l, Z.add_0_l. generalize (phi_bounded p)(phi_bounded w); intros. assert (31-[|p|]<wB). - apply Zle_lt_trans with 31%Z; auto with zarith. + apply Z.le_lt_trans with 31%Z; auto with zarith. compute; auto. assert ([|31-p|]=31-[|p|]). unfold sub31; rewrite phi_phi_inv. change [|31|] with 31%Z. apply Zmod_small; auto with zarith. rewrite spec_add_mul_div by (rewrite H4; auto with zarith). - change [|0|] with 0%Z; rewrite Zdiv_0_l, Zplus_0_r. + change [|0|] with 0%Z; rewrite Zdiv_0_l, Z.add_0_r. rewrite H4. apply shift_unshift_mod_2; auto with zarith. Qed. @@ -1682,7 +1679,7 @@ Section Int31_Specs. end. Lemma head031_equiv : - forall x, [|head031 x|] = Z_of_nat (head031_alt size x). + forall x, [|head031 x|] = Z.of_nat (head031_alt size x). Proof. intros. case_eq (iszero x); intros. @@ -1690,7 +1687,7 @@ Section Int31_Specs. simpl; auto. unfold head031, recl. - change On with (phi_inv (Z_of_nat (31-size))). + change On with (phi_inv (Z.of_nat (31-size))). replace (head031_alt size x) with (head031_alt size x + (31 - size))%nat by auto. assert (size <= 31)%nat by auto with arith. @@ -1700,12 +1697,12 @@ Section Int31_Specs. unfold recl_aux; fold recl_aux. unfold head031_alt; fold head031_alt. rewrite H. - assert ([|phi_inv (Z_of_nat (31-S n))|] = Z_of_nat (31 - S n)). + assert ([|phi_inv (Z.of_nat (31-S n))|] = Z.of_nat (31 - S n)). rewrite phi_phi_inv. apply Zmod_small. split. - change 0 with (Z_of_nat O); apply inj_le; omega. - apply Zle_lt_trans with (Z_of_nat 31). + change 0 with (Z.of_nat O); apply inj_le; omega. + apply Z.le_lt_trans with (Z.of_nat 31). apply inj_le; omega. compute; auto. case_eq (firstl x); intros; auto. @@ -1718,7 +1715,7 @@ Section Int31_Specs. f_equal. change [|In|] with 1. replace (31-n)%nat with (S (31 - S n))%nat by omega. - rewrite inj_S; ring. + rewrite Nat2Z.inj_succ; ring. clear - H H2. rewrite (sneakr_shiftl x) in H. @@ -1747,16 +1744,16 @@ Section Int31_Specs. revert x H H0. unfold size at 2 5. induction size. - simpl Z_of_nat. + simpl Z.of_nat. intros. compute in H0; rewrite H0 in H; discriminate. intros. simpl head031_alt. case_eq (firstl x); intros. - rewrite (inj_S (head031_alt n (shiftl x))), Zpower_Zsucc; auto with zarith. - rewrite <- Zmult_assoc, Zmult_comm, <- Zmult_assoc, <-(Zmult_comm 2). - rewrite <- Zdouble_mult, <- (phi_twice_firstl _ H1). + rewrite (Nat2Z.inj_succ (head031_alt n (shiftl x))), Z.pow_succ_r; auto with zarith. + rewrite <- Z.mul_assoc, Z.mul_comm, <- Z.mul_assoc, <-(Z.mul_comm 2). + rewrite <- Z.double_spec, <- (phi_twice_firstl _ H1). apply IHn. rewrite phi_nz; rewrite phi_nz in H; contradict H. @@ -1765,9 +1762,9 @@ Section Int31_Specs. rewrite <- nshiftl_S_tail; auto. - change (2^(Z_of_nat 0)) with 1; rewrite Zmult_1_l. + change (2^(Z.of_nat 0)) with 1; rewrite Z.mul_1_l. generalize (phi_bounded x); unfold size; split; auto with zarith. - change (2^(Z_of_nat 31)/2) with (2^(Z_of_nat (pred size))). + change (2^(Z.of_nat 31)/2) with (2^(Z.of_nat (pred size))). apply phi_lowerbound; auto. Qed. @@ -1790,7 +1787,7 @@ Section Int31_Specs. end. Lemma tail031_equiv : - forall x, [|tail031 x|] = Z_of_nat (tail031_alt size x). + forall x, [|tail031 x|] = Z.of_nat (tail031_alt size x). Proof. intros. case_eq (iszero x); intros. @@ -1798,7 +1795,7 @@ Section Int31_Specs. simpl; auto. unfold tail031, recr. - change On with (phi_inv (Z_of_nat (31-size))). + change On with (phi_inv (Z.of_nat (31-size))). replace (tail031_alt size x) with (tail031_alt size x + (31 - size))%nat by auto. assert (size <= 31)%nat by auto with arith. @@ -1808,12 +1805,12 @@ Section Int31_Specs. unfold recr_aux; fold recr_aux. unfold tail031_alt; fold tail031_alt. rewrite H. - assert ([|phi_inv (Z_of_nat (31-S n))|] = Z_of_nat (31 - S n)). + assert ([|phi_inv (Z.of_nat (31-S n))|] = Z.of_nat (31 - S n)). rewrite phi_phi_inv. apply Zmod_small. split. - change 0 with (Z_of_nat O); apply inj_le; omega. - apply Zle_lt_trans with (Z_of_nat 31). + change 0 with (Z.of_nat O); apply inj_le; omega. + apply Z.le_lt_trans with (Z.of_nat 31). apply inj_le; omega. compute; auto. case_eq (firstr x); intros; auto. @@ -1826,7 +1823,7 @@ Section Int31_Specs. f_equal. change [|In|] with 1. replace (31-n)%nat with (S (31 - S n))%nat by omega. - rewrite inj_S; ring. + rewrite Nat2Z.inj_succ; ring. clear - H H2. rewrite (sneakl_shiftr x) in H. @@ -1844,14 +1841,14 @@ Section Int31_Specs. apply nshiftr_size. revert x H H0. induction size. - simpl Z_of_nat. + simpl Z.of_nat. intros. compute in H0; rewrite H0 in H; discriminate. intros. simpl tail031_alt. case_eq (firstr x); intros. - rewrite (inj_S (tail031_alt n (shiftr x))), Zpower_Zsucc; auto with zarith. + rewrite (Nat2Z.inj_succ (tail031_alt n (shiftr x))), Z.pow_succ_r; auto with zarith. destruct (IHn (shiftr x)) as (y & Hy1 & Hy2). rewrite phi_nz; rewrite phi_nz in H; contradict H. @@ -1861,13 +1858,13 @@ Section Int31_Specs. exists y; split; auto. rewrite phi_eqn1; auto. - rewrite Zdouble_mult, Hy2; ring. + rewrite Z.double_spec, Hy2; ring. exists [|shiftr x|]. split. generalize (phi_bounded (shiftr x)); auto with zarith. rewrite phi_eqn2; auto. - rewrite Zdouble_plus_one_mult; simpl; ring. + rewrite Z.succ_double_spec; simpl; ring. Qed. (* Sqrt *) @@ -1886,30 +1883,30 @@ Section Int31_Specs. Proof. intros Hj; generalize Hj k; pattern j; apply natlike_ind; auto; clear k j Hj. - intros _ k Hk; repeat rewrite Zplus_0_l. - apply Zmult_le_0_compat; generalize (Z_div_pos k 2); auto with zarith. + intros _ k Hk; repeat rewrite Z.add_0_l. + apply Z.mul_nonneg_nonneg; generalize (Z_div_pos k 2); auto with zarith. intros j Hj Hrec _ k Hk; pattern k; apply natlike_ind; auto; clear k Hk. - rewrite Zmult_0_r, Zplus_0_r, Zplus_0_l. - generalize (sqr_pos (Zsucc j / 2)) (quotient_by_2 (Zsucc j)); - unfold Zsucc. - rewrite Zpower_2, Zmult_plus_distr_l; repeat rewrite Zmult_plus_distr_r. + rewrite Z.mul_0_r, Z.add_0_r, Z.add_0_l. + generalize (sqr_pos (Z.succ j / 2)) (quotient_by_2 (Z.succ j)); + unfold Z.succ. + rewrite Z.pow_2_r, Z.mul_add_distr_r; repeat rewrite Z.mul_add_distr_l. auto with zarith. intros k Hk _. - replace ((Zsucc j + Zsucc k) / 2) with ((j + k)/2 + 1). + replace ((Z.succ j + Z.succ k) / 2) with ((j + k)/2 + 1). generalize (Hrec Hj k Hk) (quotient_by_2 (j + k)). - unfold Zsucc; repeat rewrite Zpower_2; - repeat rewrite Zmult_plus_distr_l; repeat rewrite Zmult_plus_distr_r. - repeat rewrite Zmult_1_l; repeat rewrite Zmult_1_r. + unfold Z.succ; repeat rewrite Z.pow_2_r; + repeat rewrite Z.mul_add_distr_r; repeat rewrite Z.mul_add_distr_l. + repeat rewrite Z.mul_1_l; repeat rewrite Z.mul_1_r. auto with zarith. - rewrite Zplus_comm, <- Z_div_plus_full_l; auto with zarith. - apply f_equal2 with (f := Zdiv); auto with zarith. + rewrite Z.add_comm, <- Z_div_plus_full_l; auto with zarith. + apply f_equal2 with (f := Z.div); auto with zarith. Qed. Lemma sqrt_main i j: 0 <= i -> 0 < j -> i < ((j + (i/j))/2 + 1) ^ 2. Proof. intros Hi Hj. assert (Hij: 0 <= i/j) by (apply Z_div_pos; auto with zarith). - apply Zlt_le_trans with (2 := sqrt_main_trick _ _ (Zlt_le_weak _ _ Hj) Hij). + apply Z.lt_le_trans with (2 := sqrt_main_trick _ _ (Z.lt_le_incl _ _ Hj) Hij). pattern i at 1; rewrite (Z_div_mod_eq i j); case (Z_mod_lt i j); auto with zarith. Qed. @@ -1919,48 +1916,34 @@ Section Int31_Specs. assert (H1: 0 <= i - 2) by auto with zarith. assert (H2: 1 <= (i / 2) ^ 2); auto with zarith. replace i with (1* 2 + (i - 2)); auto with zarith. - rewrite Zpower_2, Z_div_plus_full_l; auto with zarith. + rewrite Z.pow_2_r, Z_div_plus_full_l; auto with zarith. generalize (sqr_pos ((i - 2)/ 2)) (Z_div_pos (i - 2) 2). - rewrite Zmult_plus_distr_l; repeat rewrite Zmult_plus_distr_r. + rewrite Z.mul_add_distr_r; repeat rewrite Z.mul_add_distr_l. auto with zarith. generalize (quotient_by_2 i). - rewrite Zpower_2 in H2 |- *; - repeat (rewrite Zmult_plus_distr_l || - rewrite Zmult_plus_distr_r || - rewrite Zmult_1_l || rewrite Zmult_1_r). + rewrite Z.pow_2_r in H2 |- *; + repeat (rewrite Z.mul_add_distr_r || + rewrite Z.mul_add_distr_l || + rewrite Z.mul_1_l || rewrite Z.mul_1_r). auto with zarith. Qed. Lemma sqrt_test_true i j: 0 <= i -> 0 < j -> i/j >= j -> j ^ 2 <= i. Proof. - intros Hi Hj Hd; rewrite Zpower_2. - apply Zle_trans with (j * (i/j)); auto with zarith. + intros Hi Hj Hd; rewrite Z.pow_2_r. + apply Z.le_trans with (j * (i/j)); auto with zarith. apply Z_mult_div_ge; auto with zarith. Qed. Lemma sqrt_test_false i j: 0 <= i -> 0 < j -> i/j < j -> (j + (i/j))/2 < j. Proof. - intros Hi Hj H; case (Zle_or_lt j ((j + (i/j))/2)); auto. - intros H1; contradict H; apply Zle_not_lt. + intros Hi Hj H; case (Z.le_gt_cases j ((j + (i/j))/2)); auto. + intros H1; contradict H; apply Z.le_ngt. assert (2 * j <= j + (i/j)); auto with zarith. - apply Zle_trans with (2 * ((j + (i/j))/2)); auto with zarith. + apply Z.le_trans with (2 * ((j + (i/j))/2)); auto with zarith. apply Z_mult_div_ge; auto with zarith. Qed. - (* George's trick *) - Inductive ZcompareSpec (i j: Z): comparison -> Prop := - ZcompareSpecEq: i = j -> ZcompareSpec i j Eq - | ZcompareSpecLt: i < j -> ZcompareSpec i j Lt - | ZcompareSpecGt: j < i -> ZcompareSpec i j Gt. - - Lemma Zcompare_spec i j: ZcompareSpec i j (i ?= j). - Proof. - case_eq (Zcompare i j); intros H. - apply ZcompareSpecEq; apply Zcompare_Eq_eq; auto. - apply ZcompareSpecLt; auto. - apply ZcompareSpecGt; apply Zgt_lt; auto. - Qed. - Lemma sqrt31_step_def rec i j: sqrt31_step rec i j = match (fst (i/j) ?= j)%int31 with @@ -1987,65 +1970,66 @@ Section Int31_Specs. [|rec i j1|] ^ 2 <= [|i|] < ([|rec i j1|] + 1) ^ 2) -> [|sqrt31_step rec i j|] ^ 2 <= [|i|] < ([|sqrt31_step rec i j|] + 1) ^ 2. Proof. - assert (Hp2: 0 < [|2|]) by exact (refl_equal Lt). + assert (Hp2: 0 < [|2|]) by exact (eq_refl Lt). intros Hi Hj Hij H31 Hrec; rewrite sqrt31_step_def. rewrite spec_compare, div31_phi; auto. - case Zcompare_spec; auto; intros Hc; + case Z.compare_spec; auto; intros Hc; try (split; auto; apply sqrt_test_true; auto with zarith; fail). apply Hrec; repeat rewrite div31_phi; auto with zarith. replace [|(j + fst (i / j)%int31)|] with ([|j|] + [|i|] / [|j|]). split. - case (Zle_lt_or_eq 1 [|j|]); auto with zarith; intros Hj1. + apply Z.le_succ_l in Hj. change (1 <= [|j|]) in Hj. + Z.le_elim Hj. replace ([|j|] + [|i|]/[|j|]) with (1 * 2 + (([|j|] - 2) + [|i|] / [|j|])); try ring. rewrite Z_div_plus_full_l; auto with zarith. assert (0 <= [|i|]/ [|j|]) by (apply Z_div_pos; auto with zarith). assert (0 <= ([|j|] - 2 + [|i|] / [|j|]) / [|2|]) ; auto with zarith. - rewrite <- Hj1, Zdiv_1_r. + rewrite <- Hj, Zdiv_1_r. replace (1 + [|i|])%Z with (1 * 2 + ([|i|] - 1))%Z; try ring. rewrite Z_div_plus_full_l; auto with zarith. assert (0 <= ([|i|] - 1) /2)%Z by (apply Z_div_pos; auto with zarith). change ([|2|]) with 2%Z; auto with zarith. apply sqrt_test_false; auto with zarith. rewrite spec_add, div31_phi; auto. - apply sym_equal; apply Zmod_small. + symmetry; apply Zmod_small. split; auto with zarith. replace [|j + fst (i / j)%int31|] with ([|j|] + [|i|] / [|j|]). apply sqrt_main; auto with zarith. rewrite spec_add, div31_phi; auto. - apply sym_equal; apply Zmod_small. + symmetry; apply Zmod_small. split; auto with zarith. Qed. Lemma iter31_sqrt_correct n rec i j: 0 < [|i|] -> 0 < [|j|] -> - [|i|] < ([|j|] + 1) ^ 2 -> 2 * [|j|] < 2 ^ (Z_of_nat size) -> - (forall j1, 0 < [|j1|] -> 2^(Z_of_nat n) + [|j1|] <= [|j|] -> - [|i|] < ([|j1|] + 1) ^ 2 -> 2 * [|j1|] < 2 ^ (Z_of_nat size) -> + [|i|] < ([|j|] + 1) ^ 2 -> 2 * [|j|] < 2 ^ (Z.of_nat size) -> + (forall j1, 0 < [|j1|] -> 2^(Z.of_nat n) + [|j1|] <= [|j|] -> + [|i|] < ([|j1|] + 1) ^ 2 -> 2 * [|j1|] < 2 ^ (Z.of_nat size) -> [|rec i j1|] ^ 2 <= [|i|] < ([|rec i j1|] + 1) ^ 2) -> [|iter31_sqrt n rec i j|] ^ 2 <= [|i|] < ([|iter31_sqrt n rec i j|] + 1) ^ 2. Proof. revert rec i j; elim n; unfold iter31_sqrt; fold iter31_sqrt; clear n. intros rec i j Hi Hj Hij H31 Hrec; apply sqrt31_step_correct; auto with zarith. intros; apply Hrec; auto with zarith. - rewrite Zpower_0_r; auto with zarith. + rewrite Z.pow_0_r; auto with zarith. intros n Hrec rec i j Hi Hj Hij H31 HHrec. apply sqrt31_step_correct; auto. intros j1 Hj1 Hjp1; apply Hrec; auto with zarith. intros j2 Hj2 H2j2 Hjp2 Hj31; apply Hrec; auto with zarith. intros j3 Hj3 Hpj3. apply HHrec; auto. - rewrite inj_S, Zpower_Zsucc. - apply Zle_trans with (2 ^Z_of_nat n + [|j2|]); auto with zarith. - apply Zle_0_nat. + rewrite Nat2Z.inj_succ, Z.pow_succ_r. + apply Z.le_trans with (2 ^Z.of_nat n + [|j2|]); auto with zarith. + apply Nat2Z.is_nonneg. Qed. Lemma spec_sqrt : forall x, [|sqrt31 x|] ^ 2 <= [|x|] < ([|sqrt31 x|] + 1) ^ 2. Proof. intros i; unfold sqrt31. - rewrite spec_compare. case Zcompare_spec; change [|1|] with 1; + rewrite spec_compare. case Z.compare_spec; change [|1|] with 1; intros Hi; auto with zarith. - repeat rewrite Zpower_2; auto with zarith. + repeat rewrite Z.pow_2_r; auto with zarith. apply iter31_sqrt_correct; auto with zarith. rewrite div31_phi; change ([|2|]) with 2; auto with zarith. replace ([|i|]) with (1 * 2 + ([|i|] - 2))%Z; try ring. @@ -2054,18 +2038,18 @@ Section Int31_Specs. rewrite div31_phi; change ([|2|]) with 2; auto with zarith. apply sqrt_init; auto. rewrite div31_phi; change ([|2|]) with 2; auto with zarith. - apply Zle_lt_trans with ([|i|]). + apply Z.le_lt_trans with ([|i|]). apply Z_mult_div_ge; auto with zarith. case (phi_bounded i); auto. - intros j2 H1 H2; contradict H2; apply Zlt_not_le. + intros j2 H1 H2; contradict H2; apply Z.lt_nge. rewrite div31_phi; change ([|2|]) with 2; auto with zarith. - apply Zle_lt_trans with ([|i|]); auto with zarith. + apply Z.le_lt_trans with ([|i|]); auto with zarith. assert (0 <= [|i|]/2)%Z by (apply Z_div_pos; auto with zarith). - apply Zle_trans with (2 * ([|i|]/2)); auto with zarith. + apply Z.le_trans with (2 * ([|i|]/2)); auto with zarith. apply Z_mult_div_ge; auto with zarith. case (phi_bounded i); unfold size; auto with zarith. change [|0|] with 0; auto with zarith. - case (phi_bounded i); repeat rewrite Zpower_2; auto with zarith. + case (phi_bounded i); repeat rewrite Z.pow_2_r; auto with zarith. Qed. Lemma sqrt312_step_def rec ih il j: @@ -2095,10 +2079,10 @@ Section Int31_Specs. case (phi_bounded il); intros Hbil _. case (phi_bounded ih); intros Hbih Hbih1. assert (([|ih|] < [|j|] + 1)%Z); auto with zarith. - apply Zlt_square_simpl; auto with zarith. - repeat rewrite <-Zpower_2; apply Zle_lt_trans with (2 := H1). - apply Zle_trans with ([|ih|] * base)%Z; unfold phi2, base; - try rewrite Zpower_2; auto with zarith. + apply Z.square_lt_simpl_nonneg; auto with zarith. + repeat rewrite <-Z.pow_2_r; apply Z.le_lt_trans with (2 := H1). + apply Z.le_trans with ([|ih|] * base)%Z; unfold phi2, base; + try rewrite Z.pow_2_r; auto with zarith. Qed. Lemma div312_phi ih il j: (2^30 <= [|j|] -> [|ih|] < [|j|] -> @@ -2108,7 +2092,7 @@ Section Int31_Specs. generalize (spec_div21 ih il j Hj Hj1). case div3121; intros q r (Hq, Hr). apply Zdiv_unique with (phi r); auto with zarith. - simpl fst; apply trans_equal with (1 := Hq); ring. + simpl fst; apply eq_trans with (1 := Hq); ring. Qed. Lemma sqrt312_step_correct rec ih il j: @@ -2118,32 +2102,33 @@ Section Int31_Specs. [|sqrt312_step rec ih il j|] ^ 2 <= phi2 ih il < ([|sqrt312_step rec ih il j|] + 1) ^ 2. Proof. - assert (Hp2: (0 < [|2|])%Z) by exact (refl_equal Lt). + assert (Hp2: (0 < [|2|])%Z) by exact (eq_refl Lt). intros Hih Hj Hij Hrec; rewrite sqrt312_step_def. assert (H1: ([|ih|] <= [|j|])%Z) by (apply sqrt312_lower_bound with il; auto). case (phi_bounded ih); intros Hih1 _. case (phi_bounded il); intros Hil1 _. case (phi_bounded j); intros _ Hj1. assert (Hp3: (0 < phi2 ih il)). - unfold phi2; apply Zlt_le_trans with ([|ih|] * base)%Z; auto with zarith. - apply Zmult_lt_0_compat; auto with zarith. - apply Zlt_le_trans with (2:= Hih); auto with zarith. - rewrite spec_compare. case Zcompare_spec; intros Hc1. + unfold phi2; apply Z.lt_le_trans with ([|ih|] * base)%Z; auto with zarith. + apply Z.mul_pos_pos; auto with zarith. + apply Z.lt_le_trans with (2:= Hih); auto with zarith. + rewrite spec_compare. case Z.compare_spec; intros Hc1. split; auto. apply sqrt_test_true; auto. unfold phi2, base; auto with zarith. unfold phi2; rewrite Hc1. assert (0 <= [|il|]/[|j|]) by (apply Z_div_pos; auto with zarith). - rewrite Zmult_comm, Z_div_plus_full_l; unfold base; auto with zarith. - unfold Zpower, Zpower_pos in Hj1; simpl in Hj1; auto with zarith. - case (Zle_or_lt (2 ^ 30) [|j|]); intros Hjj. - rewrite spec_compare; case Zcompare_spec; + rewrite Z.mul_comm, Z_div_plus_full_l; unfold base; auto with zarith. + unfold Z.pow, Z.pow_pos in Hj1; simpl in Hj1; auto with zarith. + case (Z.le_gt_cases (2 ^ 30) [|j|]); intros Hjj. + rewrite spec_compare; case Z.compare_spec; rewrite div312_phi; auto; intros Hc; try (split; auto; apply sqrt_test_true; auto with zarith; fail). apply Hrec. assert (Hf1: 0 <= phi2 ih il/ [|j|]) by (apply Z_div_pos; auto with zarith). - case (Zle_lt_or_eq 1 ([|j|])); auto with zarith; intros Hf2. - 2: contradict Hc; apply Zle_not_lt; rewrite <- Hf2, Zdiv_1_r; auto with zarith. + apply Z.le_succ_l in Hj. change (1 <= [|j|]) in Hj. + Z.le_elim Hj. + 2: contradict Hc; apply Z.le_ngt; rewrite <- Hj, Zdiv_1_r; auto with zarith. assert (Hf3: 0 < ([|j|] + phi2 ih il / [|j|]) / 2). replace ([|j|] + phi2 ih il/ [|j|])%Z with (1 * 2 + (([|j|] - 2) + phi2 ih il / [|j|])); try ring. @@ -2157,9 +2142,9 @@ Section Int31_Specs. rewrite div31_phi; change [|2|] with 2%Z; auto with zarith. intros HH; rewrite HH; clear HH; auto with zarith. rewrite spec_add, div31_phi; change [|2|] with 2%Z; auto. - rewrite Zmult_1_l; intros HH. - rewrite Zplus_comm, <- Z_div_plus_full_l; auto with zarith. - change (phi v30 * 2) with (2 ^ Z_of_nat size). + rewrite Z.mul_1_l; intros HH. + rewrite Z.add_comm, <- Z_div_plus_full_l; auto with zarith. + change (phi v30 * 2) with (2 ^ Z.of_nat size). rewrite HH, Zmod_small; auto with zarith. replace (phi match j +c fst (div3121 ih il j) with @@ -2173,41 +2158,41 @@ Section Int31_Specs. rewrite div31_phi; auto with zarith. intros HH; rewrite HH; auto with zarith. intros HH; rewrite <- HH. - change (1 * 2 ^ Z_of_nat size) with (phi (v30) * 2). + change (1 * 2 ^ Z.of_nat size) with (phi (v30) * 2). rewrite Z_div_plus_full_l; auto with zarith. - rewrite Zplus_comm. + rewrite Z.add_comm. rewrite spec_add, Zmod_small. rewrite div31_phi; auto. split; auto with zarith. case (phi_bounded (fst (r/2)%int31)); case (phi_bounded v30); auto with zarith. rewrite div31_phi; change (phi 2) with 2%Z; auto. - change (2 ^Z_of_nat size) with (base/2 + phi v30). + change (2 ^Z.of_nat size) with (base/2 + phi v30). assert (phi r / 2 < base/2); auto with zarith. - apply Zmult_gt_0_lt_reg_r with 2; auto with zarith. + apply Z.mul_lt_mono_pos_r with 2; auto with zarith. change (base/2 * 2) with base. - apply Zle_lt_trans with (phi r). - rewrite Zmult_comm; apply Z_mult_div_ge; auto with zarith. + apply Z.le_lt_trans with (phi r). + rewrite Z.mul_comm; apply Z_mult_div_ge; auto with zarith. case (phi_bounded r); auto with zarith. - contradict Hij; apply Zle_not_lt. + contradict Hij; apply Z.le_ngt. assert ((1 + [|j|]) <= 2 ^ 30); auto with zarith. - apply Zle_trans with ((2 ^ 30) * (2 ^ 30)); auto with zarith. + apply Z.le_trans with ((2 ^ 30) * (2 ^ 30)); auto with zarith. assert (0 <= 1 + [|j|]); auto with zarith. - apply Zmult_le_compat; auto with zarith. + apply Z.mul_le_mono_nonneg; auto with zarith. change ((2 ^ 30) * (2 ^ 30)) with ((2 ^ 29) * base). - apply Zle_trans with ([|ih|] * base); auto with zarith. + apply Z.le_trans with ([|ih|] * base); auto with zarith. unfold phi2, base; auto with zarith. split; auto. apply sqrt_test_true; auto. unfold phi2, base; auto with zarith. - apply Zle_ge; apply Zle_trans with (([|j|] * base)/[|j|]). - rewrite Zmult_comm, Z_div_mult; auto with zarith. - apply Zge_le; apply Z_div_ge; auto with zarith. + apply Z.le_ge; apply Z.le_trans with (([|j|] * base)/[|j|]). + rewrite Z.mul_comm, Z_div_mult; auto with zarith. + apply Z.ge_le; apply Z_div_ge; auto with zarith. Qed. Lemma iter312_sqrt_correct n rec ih il j: 2^29 <= [|ih|] -> 0 < [|j|] -> phi2 ih il < ([|j|] + 1) ^ 2 -> - (forall j1, 0 < [|j1|] -> 2^(Z_of_nat n) + [|j1|] <= [|j|] -> + (forall j1, 0 < [|j1|] -> 2^(Z.of_nat n) + [|j1|] <= [|j|] -> phi2 ih il < ([|j1|] + 1) ^ 2 -> [|rec ih il j1|] ^ 2 <= phi2 ih il < ([|rec ih il j1|] + 1) ^ 2) -> [|iter312_sqrt n rec ih il j|] ^ 2 <= phi2 ih il @@ -2216,16 +2201,16 @@ Section Int31_Specs. revert rec ih il j; elim n; unfold iter312_sqrt; fold iter312_sqrt; clear n. intros rec ih il j Hi Hj Hij Hrec; apply sqrt312_step_correct; auto with zarith. intros; apply Hrec; auto with zarith. - rewrite Zpower_0_r; auto with zarith. + rewrite Z.pow_0_r; auto with zarith. intros n Hrec rec ih il j Hi Hj Hij HHrec. apply sqrt312_step_correct; auto. intros j1 Hj1 Hjp1; apply Hrec; auto with zarith. intros j2 Hj2 H2j2 Hjp2; apply Hrec; auto with zarith. intros j3 Hj3 Hpj3. apply HHrec; auto. - rewrite inj_S, Zpower_Zsucc. - apply Zle_trans with (2 ^Z_of_nat n + [|j2|])%Z; auto with zarith. - apply Zle_0_nat. + rewrite Nat2Z.inj_succ, Z.pow_succ_r. + apply Z.le_trans with (2 ^Z.of_nat n + [|j2|])%Z; auto with zarith. + apply Nat2Z.is_nonneg. Qed. Lemma spec_sqrt2 : forall x y, @@ -2240,30 +2225,30 @@ Section Int31_Specs. (intros s; ring). assert (Hb: 0 <= base) by (red; intros HH; discriminate). assert (Hi2: phi2 ih il < (phi Tn + 1) ^ 2). - change ((phi Tn + 1) ^ 2) with (2^62). - apply Zle_lt_trans with ((2^31 -1) * base + (2^31 - 1)); auto with zarith. - 2: simpl; unfold Zpower_pos; simpl; auto with zarith. - case (phi_bounded ih); case (phi_bounded il); intros H1 H2 H3 H4. - unfold base, Zpower, Zpower_pos in H2,H4; simpl in H2,H4. - unfold phi2,Zpower, Zpower_pos. simpl Pos.iter; auto with zarith. + { change ((phi Tn + 1) ^ 2) with (2^62). + apply Z.le_lt_trans with ((2^31 -1) * base + (2^31 - 1)); auto with zarith. + 2: simpl; unfold Z.pow_pos; simpl; auto with zarith. + case (phi_bounded ih); case (phi_bounded il); intros H1 H2 H3 H4. + unfold base, Z.pow, Z.pow_pos in H2,H4; simpl in H2,H4. + unfold phi2,Z.pow, Z.pow_pos. simpl Pos.iter; auto with zarith. } case (iter312_sqrt_correct 31 (fun _ _ j => j) ih il Tn); auto with zarith. change [|Tn|] with 2147483647; auto with zarith. intros j1 _ HH; contradict HH. - apply Zlt_not_le. + apply Z.lt_nge. change [|Tn|] with 2147483647; auto with zarith. - change (2 ^ Z_of_nat 31) with 2147483648; auto with zarith. + change (2 ^ Z.of_nat 31) with 2147483648; auto with zarith. case (phi_bounded j1); auto with zarith. set (s := iter312_sqrt 31 (fun _ _ j : int31 => j) ih il Tn). intros Hs1 Hs2. generalize (spec_mul_c s s); case mul31c. simpl zn2z_to_Z; intros HH. assert ([|s|] = 0). - case (Zmult_integral _ _ (sym_equal HH)); auto. - contradict Hs2; apply Zle_not_lt; rewrite H. + { symmetry in HH. rewrite Z.mul_eq_0 in HH. destruct HH; auto. } + contradict Hs2; apply Z.le_ngt; rewrite H. change ((0 + 1) ^ 2) with 1. - apply Zle_trans with (2 ^ Z_of_nat size / 4 * base). + apply Z.le_trans with (2 ^ Z.of_nat size / 4 * base). simpl; auto with zarith. - apply Zle_trans with ([|ih|] * base); auto with zarith. + apply Z.le_trans with ([|ih|] * base); auto with zarith. unfold phi2; case (phi_bounded il); auto with zarith. intros ih1 il1. change [||WW ih1 il1||] with (phi2 ih1 il1). @@ -2271,10 +2256,10 @@ Section Int31_Specs. generalize (spec_sub_c il il1). case sub31c; intros il2 Hil2. simpl interp_carry in Hil2. - rewrite spec_compare; case Zcompare_spec. + rewrite spec_compare; case Z.compare_spec. unfold interp_carry. intros H1; split. - rewrite Zpower_2, <- Hihl1. + rewrite Z.pow_2_r, <- Hihl1. unfold phi2; ring[Hil2 H1]. replace [|il2|] with (phi2 ih il - phi2 ih1 il1). rewrite Hihl1. @@ -2282,109 +2267,111 @@ Section Int31_Specs. unfold phi2; rewrite H1, Hil2; ring. unfold interp_carry. intros H1; contradict Hs1. - apply Zlt_not_le; rewrite Zpower_2, <-Hihl1. + apply Z.lt_nge; rewrite Z.pow_2_r, <-Hihl1. unfold phi2. case (phi_bounded il); intros _ H2. - apply Zlt_le_trans with (([|ih|] + 1) * base + 0). - rewrite Zmult_plus_distr_l, Zplus_0_r; auto with zarith. + apply Z.lt_le_trans with (([|ih|] + 1) * base + 0). + rewrite Z.mul_add_distr_r, Z.add_0_r; auto with zarith. case (phi_bounded il1); intros H3 _. - apply Zplus_le_compat; auto with zarith. - unfold interp_carry; change (1 * 2 ^ Z_of_nat size) with base. - rewrite Zpower_2, <- Hihl1, Hil2. + apply Z.add_le_mono; auto with zarith. + unfold interp_carry; change (1 * 2 ^ Z.of_nat size) with base. + rewrite Z.pow_2_r, <- Hihl1, Hil2. intros H1. - case (Zle_lt_or_eq ([|ih1|] + 1) ([|ih|])); auto with zarith. - intros H2; contradict Hs2; apply Zle_not_lt. + rewrite <- Z.le_succ_l, <- Z.add_1_r in H1. + Z.le_elim H1. + contradict Hs2; apply Z.le_ngt. replace (([|s|] + 1) ^ 2) with (phi2 ih1 il1 + 2 * [|s|] + 1). unfold phi2. case (phi_bounded il); intros Hpil _. assert (Hl1l: [|il1|] <= [|il|]). - case (phi_bounded il2); rewrite Hil2; auto with zarith. + { case (phi_bounded il2); rewrite Hil2; auto with zarith. } assert ([|ih1|] * base + 2 * [|s|] + 1 <= [|ih|] * base); auto with zarith. - case (phi_bounded s); change (2 ^ Z_of_nat size) with base; intros _ Hps. + case (phi_bounded s); change (2 ^ Z.of_nat size) with base; intros _ Hps. case (phi_bounded ih1); intros Hpih1 _; auto with zarith. - apply Zle_trans with (([|ih1|] + 2) * base); auto with zarith. - rewrite Zmult_plus_distr_l. + apply Z.le_trans with (([|ih1|] + 2) * base); auto with zarith. + rewrite Z.mul_add_distr_r. assert (2 * [|s|] + 1 <= 2 * base); auto with zarith. rewrite Hihl1, Hbin; auto. - intros H2; split. - unfold phi2; rewrite <- H2; ring. + split. + unfold phi2; rewrite <- H1; ring. replace (base + ([|il|] - [|il1|])) with (phi2 ih il - ([|s|] * [|s|])). rewrite <-Hbin in Hs2; auto with zarith. - rewrite <- Hihl1; unfold phi2; rewrite <- H2; ring. + rewrite <- Hihl1; unfold phi2; rewrite <- H1; ring. unfold interp_carry in Hil2 |- *. - unfold interp_carry; change (1 * 2 ^ Z_of_nat size) with base. + unfold interp_carry; change (1 * 2 ^ Z.of_nat size) with base. assert (Hsih: [|ih - 1|] = [|ih|] - 1). - rewrite spec_sub, Zmod_small; auto; change [|1|] with 1. - case (phi_bounded ih); intros H1 H2. - generalize Hih; change (2 ^ Z_of_nat size / 4) with 536870912. - split; auto with zarith. - rewrite spec_compare; case Zcompare_spec. + { rewrite spec_sub, Zmod_small; auto; change [|1|] with 1. + case (phi_bounded ih); intros H1 H2. + generalize Hih; change (2 ^ Z.of_nat size / 4) with 536870912. + split; auto with zarith. } + rewrite spec_compare; case Z.compare_spec. rewrite Hsih. intros H1; split. - rewrite Zpower_2, <- Hihl1. + rewrite Z.pow_2_r, <- Hihl1. unfold phi2; rewrite <-H1. - apply trans_equal with ([|ih|] * base + [|il1|] + ([|il|] - [|il1|])). + transitivity ([|ih|] * base + [|il1|] + ([|il|] - [|il1|])). ring. rewrite <-Hil2. - change (2 ^ Z_of_nat size) with base; ring. + change (2 ^ Z.of_nat size) with base; ring. replace [|il2|] with (phi2 ih il - phi2 ih1 il1). rewrite Hihl1. rewrite <-Hbin in Hs2; auto with zarith. unfold phi2. rewrite <-H1. ring_simplify. - apply trans_equal with (base + ([|il|] - [|il1|])). + transitivity (base + ([|il|] - [|il1|])). ring. rewrite <-Hil2. - change (2 ^ Z_of_nat size) with base; ring. + change (2 ^ Z.of_nat size) with base; ring. rewrite Hsih; intros H1. assert (He: [|ih|] = [|ih1|]). - apply Zle_antisym; auto with zarith. - case (Zle_or_lt [|ih1|] [|ih|]); auto; intros H2. - contradict Hs1; apply Zlt_not_le; rewrite Zpower_2, <-Hihl1. - unfold phi2. - case (phi_bounded il); change (2 ^ Z_of_nat size) with base; + { apply Z.le_antisymm; auto with zarith. + case (Z.le_gt_cases [|ih1|] [|ih|]); auto; intros H2. + contradict Hs1; apply Z.lt_nge; rewrite Z.pow_2_r, <-Hihl1. + unfold phi2. + case (phi_bounded il); change (2 ^ Z.of_nat size) with base; intros _ Hpil1. - apply Zlt_le_trans with (([|ih|] + 1) * base). - rewrite Zmult_plus_distr_l, Zmult_1_l; auto with zarith. - case (phi_bounded il1); intros Hpil2 _. - apply Zle_trans with (([|ih1|]) * base); auto with zarith. - rewrite Zpower_2, <-Hihl1; unfold phi2; rewrite <-He. - contradict Hs1; apply Zlt_not_le; rewrite Zpower_2, <-Hihl1. + apply Z.lt_le_trans with (([|ih|] + 1) * base). + rewrite Z.mul_add_distr_r, Z.mul_1_l; auto with zarith. + case (phi_bounded il1); intros Hpil2 _. + apply Z.le_trans with (([|ih1|]) * base); auto with zarith. } + rewrite Z.pow_2_r, <-Hihl1; unfold phi2; rewrite <-He. + contradict Hs1; apply Z.lt_nge; rewrite Z.pow_2_r, <-Hihl1. unfold phi2; rewrite He. assert (phi il - phi il1 < 0); auto with zarith. rewrite <-Hil2. case (phi_bounded il2); auto with zarith. intros H1. - rewrite Zpower_2, <-Hihl1. - case (Zle_lt_or_eq ([|ih1|] + 2) [|ih|]); auto with zarith. - intros H2; contradict Hs2; apply Zle_not_lt. + rewrite Z.pow_2_r, <-Hihl1. + assert (H2 : [|ih1|]+2 <= [|ih|]); auto with zarith. + Z.le_elim H2. + contradict Hs2; apply Z.le_ngt. replace (([|s|] + 1) ^ 2) with (phi2 ih1 il1 + 2 * [|s|] + 1). unfold phi2. assert ([|ih1|] * base + 2 * phi s + 1 <= [|ih|] * base + ([|il|] - [|il1|])); auto with zarith. rewrite <-Hil2. - change (-1 * 2 ^ Z_of_nat size) with (-base). + change (-1 * 2 ^ Z.of_nat size) with (-base). case (phi_bounded il2); intros Hpil2 _. - apply Zle_trans with ([|ih|] * base + - base); auto with zarith. - case (phi_bounded s); change (2 ^ Z_of_nat size) with base; intros _ Hps. + apply Z.le_trans with ([|ih|] * base + - base); auto with zarith. + case (phi_bounded s); change (2 ^ Z.of_nat size) with base; intros _ Hps. assert (2 * [|s|] + 1 <= 2 * base); auto with zarith. - apply Zle_trans with ([|ih1|] * base + 2 * base); auto with zarith. + apply Z.le_trans with ([|ih1|] * base + 2 * base); auto with zarith. assert (Hi: ([|ih1|] + 3) * base <= [|ih|] * base); auto with zarith. - rewrite Zmult_plus_distr_l in Hi; auto with zarith. + rewrite Z.mul_add_distr_r in Hi; auto with zarith. rewrite Hihl1, Hbin; auto. - intros H2; unfold phi2; rewrite <-H2. + unfold phi2; rewrite <-H2. split. replace [|il|] with (([|il|] - [|il1|]) + [|il1|]); try ring. rewrite <-Hil2. - change (-1 * 2 ^ Z_of_nat size) with (-base); ring. + change (-1 * 2 ^ Z.of_nat size) with (-base); ring. replace (base + [|il2|]) with (phi2 ih il - phi2 ih1 il1). rewrite Hihl1. rewrite <-Hbin in Hs2; auto with zarith. unfold phi2; rewrite <-H2. replace [|il|] with (([|il|] - [|il1|]) + [|il1|]); try ring. rewrite <-Hil2. - change (-1 * 2 ^ Z_of_nat size) with (-base); ring. + change (-1 * 2 ^ Z.of_nat size) with (-base); ring. Qed. (** [iszero] *) @@ -2394,7 +2381,7 @@ Qed. clear; unfold ZnZ.eq0; simpl. unfold compare31; simpl; intros. change [|0|] with 0 in H. - apply Zcompare_Eq_eq. + apply Z.compare_eq. now destruct ([|x|] ?= 0). Qed. @@ -2412,7 +2399,7 @@ Qed. destruct H; auto with zarith. replace ([|x|] mod 2) with [|r|]. destruct H; auto with zarith. - case Zcompare_spec; auto with zarith. + case Z.compare_spec; auto with zarith. apply Zmod_unique with [|q|]; auto with zarith. Qed. diff --git a/theories/Numbers/Cyclic/Int31/Int31.v b/theories/Numbers/Cyclic/Int31/Int31.v index 20f750f6..f414663a 100644 --- a/theories/Numbers/Cyclic/Int31/Int31.v +++ b/theories/Numbers/Cyclic/Int31/Int31.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -117,12 +117,12 @@ Definition iszero : int31 -> bool := Eval compute in It seems to work, but later "unfold iszero" takes forever. *) -(** [base] is [2^31], obtained via iterations of [Zdouble]. +(** [base] is [2^31], obtained via iterations of [Z.double]. It can also be seen as the smallest b > 0 s.t. phi_inv b = 0 (see below) *) Definition base := Eval compute in - iter_nat size Z Zdouble 1%Z. + iter_nat size Z Z.double 1%Z. (** * Recursors *) @@ -155,11 +155,11 @@ Definition recr := recr_aux size. (** * Conversions *) -(** From int31 to Z, we simply iterates [Zdouble] or [Zdouble_plus_one]. *) +(** From int31 to Z, we simply iterates [Z.double] or [Z.succ_double]. *) Definition phi : int31 -> Z := recr Z (0%Z) - (fun b _ => match b with D0 => Zdouble | D1 => Zdouble_plus_one end). + (fun b _ => match b with D0 => Z.double | D1 => Z.succ_double end). (** From positive to int31. An abstract definition could be : [ phi_inv (2n) = 2*(phi_inv n) /\ @@ -293,13 +293,13 @@ Notation "n '*c' m" := (mul31c n m) (at level 40, no associativity) : int31_scop (** Division of a double size word modulo [2^31] *) Definition div3121 (nh nl m : int31) := - let (q,r) := Zdiv_eucl (phi2 nh nl) (phi m) in + let (q,r) := Z.div_eucl (phi2 nh nl) (phi m) in (phi_inv q, phi_inv r). (** Division modulo [2^31] *) Definition div31 (n m : int31) := - let (q,r) := Zdiv_eucl (phi n) (phi m) in + let (q,r) := Z.div_eucl (phi n) (phi m) in (phi_inv q, phi_inv r). Notation "n / m" := (div31 n m) : int31_scope. @@ -391,7 +391,7 @@ Eval lazy delta [On In Twon] in | Lt => iter31_sqrt 31 (fun i j => j) i (fst (i/Twon)) end. -Definition v30 := Eval compute in (addmuldiv31 (phi_inv (Z_of_nat size - 1)) In On). +Definition v30 := Eval compute in (addmuldiv31 (phi_inv (Z.of_nat size - 1)) In On). Definition sqrt312_step (rec: int31 -> int31 -> int31 -> int31) (ih il j: int31) := @@ -452,7 +452,7 @@ Definition positive_to_int31 (p:positive) := p2i size p. It is used as default answer for numbers of zeros in [head0] and [tail0] *) -Definition T31 : int31 := Eval compute in phi_inv (Z_of_nat size). +Definition T31 : int31 := Eval compute in phi_inv (Z.of_nat size). Definition head031 (i:int31) := recl _ (fun _ => T31) diff --git a/theories/Numbers/Cyclic/Int31/Ring31.v b/theories/Numbers/Cyclic/Int31/Ring31.v index 23e8bd33..f5a08438 100644 --- a/theories/Numbers/Cyclic/Int31/Ring31.v +++ b/theories/Numbers/Cyclic/Int31/Ring31.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -81,7 +81,7 @@ Qed. Lemma eqb31_eq : forall x y, eqb31 x y = true <-> x=y. Proof. unfold eqb31. intros x y. -rewrite Cyclic31.spec_compare. case Zcompare_spec. +rewrite Cyclic31.spec_compare. case Z.compare_spec. intuition. apply Int31_canonic; auto. intuition; subst; auto with zarith; try discriminate. intuition; subst; auto with zarith; try discriminate. diff --git a/theories/Numbers/Cyclic/ZModulo/ZModulo.v b/theories/Numbers/Cyclic/ZModulo/ZModulo.v index d039fdcb..9e3f4ef4 100644 --- a/theories/Numbers/Cyclic/ZModulo/ZModulo.v +++ b/theories/Numbers/Cyclic/ZModulo/ZModulo.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -76,22 +76,22 @@ Section ZModulo. Qed. Definition of_pos x := - let (q,r) := Zdiv_eucl_POS x wB in (N_of_Z q, r). + let (q,r) := Z.pos_div_eucl x wB in (N_of_Z q, r). Lemma spec_of_pos : forall p, - Zpos p = (Z_of_N (fst (of_pos p)))*wB + [|(snd (of_pos p))|]. + Zpos p = (Z.of_N (fst (of_pos p)))*wB + [|(snd (of_pos p))|]. Proof. intros; unfold of_pos; simpl. generalize (Z_div_mod_POS wB wB_pos p). - destruct (Zdiv_eucl_POS p wB); simpl; destruct 1. + destruct (Z.pos_div_eucl p wB); simpl; destruct 1. unfold to_Z; rewrite Zmod_small; auto. assert (0 <= z). replace z with (Zpos p / wB) by (symmetry; apply Zdiv_unique with z0; auto). apply Z_div_pos; auto with zarith. - replace (Z_of_N (N_of_Z z)) with z by + replace (Z.of_N (N_of_Z z)) with z by (destruct z; simpl; auto; elim H1; auto). - rewrite Zmult_comm; auto. + rewrite Z.mul_comm; auto. Qed. Lemma spec_zdigits : [|zdigits|] = Zpos digits. @@ -118,7 +118,7 @@ Section ZModulo. unfold to_Z, one. apply Zmod_small; split; auto with zarith. unfold wB, base. - apply Zlt_trans with (Zpos digits); auto. + apply Z.lt_trans with (Zpos digits); auto. apply Zpower2_lt_lin; auto with zarith. Qed. @@ -128,14 +128,14 @@ Section ZModulo. apply Zmod_small; split; auto with zarith. unfold wB, base. cut (1 <= 2 ^ Zpos digits); auto with zarith. - apply Zle_trans with (Zpos digits); auto with zarith. + apply Z.le_trans with (Zpos digits); auto with zarith. apply Zpower2_le_lin; auto with zarith. Qed. - Definition compare x y := Zcompare [|x|] [|y|]. + Definition compare x y := Z.compare [|x|] [|y|]. Lemma spec_compare : forall x y, - compare x y = Zcompare [|x|] [|y|]. + compare x y = Z.compare [|x|] [|y|]. Proof. reflexivity. Qed. Definition eq0 x := @@ -183,7 +183,7 @@ Section ZModulo. Qed. Definition succ_c x := - let y := Zsucc x in + let y := Z.succ x in if eq0 y then C1 0 else C0 y. Definition add_c x y := @@ -194,29 +194,28 @@ Section ZModulo. let z := [|x|]+[|y|]+1 in if Z_lt_le_dec z wB then C0 z else C1 (z-wB). - Definition succ := Zsucc. - Definition add := Zplus. + Definition succ := Z.succ. + Definition add := Z.add. Definition add_carry x y := x + y + 1. Lemma Zmod_equal : forall x y z, z>0 -> (x-y) mod z = 0 -> x mod z = y mod z. Proof. intros. - generalize (Z_div_mod_eq (x-y) z H); rewrite H0, Zplus_0_r. + generalize (Z_div_mod_eq (x-y) z H); rewrite H0, Z.add_0_r. remember ((x-y)/z) as k. - intros H1; symmetry in H1; rewrite <- Zeq_plus_swap in H1. - subst x. - rewrite Zplus_comm, Zmult_comm, Z_mod_plus; auto. + rewrite Z.sub_move_r, Z.add_comm, Z.mul_comm. intros ->. + now apply Z_mod_plus. Qed. Lemma spec_succ_c : forall x, [+|succ_c x|] = [|x|] + 1. Proof. - intros; unfold succ_c, to_Z, Zsucc. + intros; unfold succ_c, to_Z, Z.succ. case_eq (eq0 (x+1)); intros; unfold interp_carry. - rewrite Zmult_1_l. + rewrite Z.mul_1_l. replace (wB + 0 mod wB) with wB by auto with zarith. - symmetry; rewrite Zeq_plus_swap. + symmetry. rewrite Z.add_move_r. assert ((x+1) mod wB = 0) by (apply spec_eq0; auto). replace (wB-1) with ((wB-1) mod wB) by (apply Zmod_small; generalize wB_pos; omega). @@ -227,7 +226,7 @@ Section ZModulo. unfold eq0, to_Z in *; now destruct ((x+1) mod wB). assert (x mod wB + 1 <> wB). contradict H0. - rewrite Zeq_plus_swap in H0; simpl in H0. + rewrite Z.add_move_r in H0; simpl in H0. rewrite <- Zplus_mod_idemp_l; rewrite H0. replace (wB-1+1) with wB; auto with zarith; apply Z_mod_same; auto. rewrite <- Zplus_mod_idemp_l. @@ -241,7 +240,7 @@ Section ZModulo. destruct Z_lt_le_dec. apply Zmod_small; generalize (Z_mod_lt x wB wB_pos) (Z_mod_lt y wB wB_pos); omega. - rewrite Zmult_1_l, Zplus_comm, Zeq_plus_swap. + rewrite Z.mul_1_l, Z.add_comm, Z.add_move_r. apply Zmod_small; generalize (Z_mod_lt x wB wB_pos) (Z_mod_lt y wB wB_pos); omega. Qed. @@ -252,14 +251,14 @@ Section ZModulo. destruct Z_lt_le_dec. apply Zmod_small; generalize (Z_mod_lt x wB wB_pos) (Z_mod_lt y wB wB_pos); omega. - rewrite Zmult_1_l, Zplus_comm, Zeq_plus_swap. + rewrite Z.mul_1_l, Z.add_comm, Z.add_move_r. apply Zmod_small; generalize (Z_mod_lt x wB wB_pos) (Z_mod_lt y wB wB_pos); omega. Qed. Lemma spec_succ : forall x, [|succ x|] = ([|x|] + 1) mod wB. Proof. - intros; unfold succ, to_Z, Zsucc. + intros; unfold succ, to_Z, Z.succ. symmetry; apply Zplus_mod_idemp_l. Qed. @@ -288,8 +287,8 @@ Section ZModulo. let z := [|x|]-[|y|]-1 in if Z_lt_le_dec z 0 then C1 (wB+z) else C0 z. - Definition pred := Zpred. - Definition sub := Zminus. + Definition pred := Z.pred. + Definition sub := Z.sub. Definition sub_carry x y := x - y - 1. Lemma spec_pred_c : forall x, [-|pred_c x|] = [|x|] - 1. @@ -337,7 +336,7 @@ Section ZModulo. Lemma spec_pred : forall x, [|pred x|] = ([|x|] - 1) mod wB. Proof. - intros; unfold pred, to_Z, Zpred. + intros; unfold pred, to_Z, Z.pred. rewrite <- Zplus_mod_idemp_l; auto. Qed. @@ -357,19 +356,19 @@ Section ZModulo. Qed. Definition mul_c x y := - let (h,l) := Zdiv_eucl ([|x|]*[|y|]) wB in + let (h,l) := Z.div_eucl ([|x|]*[|y|]) wB in if eq0 h then if eq0 l then W0 else WW h l else WW h l. - Definition mul := Zmult. + Definition mul := Z.mul. Definition square_c x := mul_c x x. Lemma spec_mul_c : forall x y, [|| mul_c x y ||] = [|x|] * [|y|]. Proof. intros; unfold mul_c, zn2z_to_Z. - assert (Zdiv_eucl ([|x|]*[|y|]) wB = (([|x|]*[|y|])/wB,([|x|]*[|y|]) mod wB)). - unfold Zmod, Zdiv; destruct Zdiv_eucl; auto. - generalize (Z_div_mod ([|x|]*[|y|]) wB wB_pos); destruct Zdiv_eucl as (h,l). + assert (Z.div_eucl ([|x|]*[|y|]) wB = (([|x|]*[|y|])/wB,([|x|]*[|y|]) mod wB)). + unfold Z.modulo, Z.div; destruct Z.div_eucl; auto. + generalize (Z_div_mod ([|x|]*[|y|]) wB wB_pos); destruct Z.div_eucl as (h,l). destruct 1; injection H; clear H; intros. rewrite H0. assert ([|l|] = l). @@ -380,7 +379,7 @@ Section ZModulo. split. apply Z_div_pos; auto with zarith. apply Zdiv_lt_upper_bound; auto with zarith. - apply Zmult_lt_compat; auto with zarith. + apply Z.mul_lt_mono_nonneg; auto with zarith. clear H H0 H1 H2. case_eq (eq0 h); simpl; intros. case_eq (eq0 l); simpl; intros. @@ -399,7 +398,7 @@ Section ZModulo. intros x; exact (spec_mul_c x x). Qed. - Definition div x y := Zdiv_eucl [|x|] [|y|]. + Definition div x y := Z.div_eucl [|x|] [|y|]. Lemma spec_div : forall a b, 0 < [|b|] -> let (q,r) := div a b in @@ -408,10 +407,10 @@ Section ZModulo. Proof. intros; unfold div. assert ([|b|]>0) by auto with zarith. - assert (Zdiv_eucl [|a|] [|b|] = ([|a|]/[|b|], [|a|] mod [|b|])). - unfold Zmod, Zdiv; destruct Zdiv_eucl; auto. + assert (Z.div_eucl [|a|] [|b|] = ([|a|]/[|b|], [|a|] mod [|b|])). + unfold Z.modulo, Z.div; destruct Z.div_eucl; auto. generalize (Z_div_mod [|a|] [|b|] H0). - destruct Zdiv_eucl as (q,r); destruct 1; intros. + destruct Z.div_eucl as (q,r); destruct 1; intros. injection H1; clear H1; intros. assert ([|r|]=r). apply Zmod_small; generalize (Z_mod_lt b wB wB_pos); fold [|b|]; @@ -422,10 +421,10 @@ Section ZModulo. split. apply Z_div_pos; auto with zarith. apply Zdiv_lt_upper_bound; auto with zarith. - apply Zlt_le_trans with (wB*1). - rewrite Zmult_1_r; auto with zarith. - apply Zmult_le_compat; generalize wB_pos; auto with zarith. - rewrite H5, H6; rewrite Zmult_comm; auto with zarith. + apply Z.lt_le_trans with (wB*1). + rewrite Z.mul_1_r; auto with zarith. + apply Z.mul_le_mono_nonneg; generalize wB_pos; auto with zarith. + rewrite H5, H6; rewrite Z.mul_comm; auto with zarith. Qed. Definition div_gt := div. @@ -458,28 +457,28 @@ Section ZModulo. intros; apply spec_modulo; auto. Qed. - Definition gcd x y := Zgcd [|x|] [|y|]. - Definition gcd_gt x y := Zgcd [|x|] [|y|]. + Definition gcd x y := Z.gcd [|x|] [|y|]. + Definition gcd_gt x y := Z.gcd [|x|] [|y|]. - Lemma Zgcd_bound : forall a b, 0<=a -> 0<=b -> Zgcd a b <= Zmax a b. + Lemma Zgcd_bound : forall a b, 0<=a -> 0<=b -> Z.gcd a b <= Z.max a b. Proof. intros. generalize (Zgcd_is_gcd a b); inversion_clear 1. destruct H2 as (q,H2); destruct H3 as (q',H3); clear H4. - assert (H4:=Zgcd_is_pos a b). - destruct (Z_eq_dec (Zgcd a b) 0). + assert (H4:=Z.gcd_nonneg a b). + destruct (Z.eq_dec (Z.gcd a b) 0). rewrite e; generalize (Zmax_spec a b); omega. assert (0 <= q). - apply Zmult_le_reg_r with (Zgcd a b); auto with zarith. - destruct (Z_eq_dec q 0). + apply Z.mul_le_mono_pos_r with (Z.gcd a b); auto with zarith. + destruct (Z.eq_dec q 0). subst q; simpl in *; subst a; simpl; auto. generalize (Zmax_spec 0 b) (Zabs_spec b); omega. - apply Zle_trans with a. + apply Z.le_trans with a. rewrite H2 at 2. - rewrite <- (Zmult_1_l (Zgcd a b)) at 1. - apply Zmult_le_compat; auto with zarith. + rewrite <- (Z.mul_1_l (Z.gcd a b)) at 1. + apply Z.mul_le_mono_nonneg; auto with zarith. generalize (Zmax_spec a b); omega. Qed. @@ -488,12 +487,12 @@ Section ZModulo. intros; unfold gcd. generalize (Z_mod_lt a wB wB_pos)(Z_mod_lt b wB wB_pos); intros. fold [|a|] in *; fold [|b|] in *. - replace ([|Zgcd [|a|] [|b|]|]) with (Zgcd [|a|] [|b|]). + replace ([|Z.gcd [|a|] [|b|]|]) with (Z.gcd [|a|] [|b|]). apply Zgcd_is_gcd. symmetry; apply Zmod_small. split. - apply Zgcd_is_pos. - apply Zle_lt_trans with (Zmax [|a|] [|b|]). + apply Z.gcd_nonneg. + apply Z.le_lt_trans with (Z.max [|a|] [|b|]). apply Zgcd_bound; auto with zarith. generalize (Zmax_spec [|a|] [|b|]); omega. Qed. @@ -505,7 +504,7 @@ Section ZModulo. Qed. Definition div21 a1 a2 b := - Zdiv_eucl ([|a1|]*wB+[|a2|]) [|b|]. + Z.div_eucl ([|a1|]*wB+[|a2|]) [|b|]. Lemma spec_div21 : forall a1 a2 b, wB/2 <= [|b|] -> @@ -519,10 +518,10 @@ Section ZModulo. generalize (Z_mod_lt a2 wB wB_pos); fold [|a2|]; intros. assert ([|b|]>0) by auto with zarith. remember ([|a1|]*wB+[|a2|]) as a. - assert (Zdiv_eucl a [|b|] = (a/[|b|], a mod [|b|])). - unfold Zmod, Zdiv; destruct Zdiv_eucl; auto. + assert (Z.div_eucl a [|b|] = (a/[|b|], a mod [|b|])). + unfold Z.modulo, Z.div; destruct Z.div_eucl; auto. generalize (Z_div_mod a [|b|] H3). - destruct Zdiv_eucl as (q,r); destruct 1; intros. + destruct Z.div_eucl as (q,r); destruct 1; intros. injection H4; clear H4; intros. assert ([|r|]=r). apply Zmod_small; generalize (Z_mod_lt b wB wB_pos); fold [|b|]; @@ -536,8 +535,8 @@ Section ZModulo. apply Zdiv_lt_upper_bound; auto with zarith. subst a. replace (wB*[|b|]) with (([|b|]-1)*wB + wB) by ring. - apply Zlt_le_trans with ([|a1|]*wB+wB); auto with zarith. - rewrite H8, H9; rewrite Zmult_comm; auto with zarith. + apply Z.lt_le_trans with ([|a1|]*wB+wB); auto with zarith. + rewrite H8, H9; rewrite Z.mul_comm; auto with zarith. Qed. Definition add_mul_div p x y := @@ -560,17 +559,17 @@ Section ZModulo. generalize (Z_mod_lt [|w|] (2 ^ [|p|])); intros. split. destruct H; auto with zarith. - apply Zle_lt_trans with [|w|]; auto with zarith. + apply Z.le_lt_trans with [|w|]; auto with zarith. apply Zmod_le; auto with zarith. Qed. Definition is_even x := - if Z_eq_dec ([|x|] mod 2) 0 then true else false. + if Z.eq_dec ([|x|] mod 2) 0 then true else false. Lemma spec_is_even : forall x, if is_even x then [|x|] mod 2 = 0 else [|x|] mod 2 = 1. Proof. - intros; unfold is_even; destruct Z_eq_dec; auto. + intros; unfold is_even; destruct Z.eq_dec; auto. generalize (Z_mod_lt [|x|] 2); omega. Qed. @@ -580,12 +579,12 @@ Section ZModulo. Proof. intros. unfold sqrt. - repeat rewrite Zpower_2. + repeat rewrite Z.pow_2_r. replace [|Z.sqrt [|x|]|] with (Z.sqrt [|x|]). apply Z.sqrt_spec; auto with zarith. symmetry; apply Zmod_small. split. apply Z.sqrt_nonneg; auto. - apply Zle_lt_trans with [|x|]; auto. + apply Z.le_lt_trans with [|x|]; auto. apply Z.sqrt_le_lin; auto. Qed. @@ -616,22 +615,22 @@ Section ZModulo. destruct (Z_lt_le_dec s wB); auto. assert (wB * wB <= Zpos p). rewrite U. - apply Zle_trans with (s*s); try omega. - apply Zmult_le_compat; generalize wB_pos; auto with zarith. + apply Z.le_trans with (s*s); try omega. + apply Z.mul_le_mono_nonneg; generalize wB_pos; auto with zarith. assert (Zpos p < wB*wB). rewrite Heqz. replace (wB*wB) with ((wB-1)*wB+wB) by ring. - apply Zplus_le_lt_compat; auto with zarith. - apply Zmult_le_compat; auto with zarith. + apply Z.add_le_lt_mono; auto with zarith. + apply Z.mul_le_mono_nonneg; auto with zarith. generalize (spec_to_Z x); auto with zarith. generalize wB_pos; auto with zarith. omega. replace [|s|] with s by (symmetry; apply Zmod_small; auto with zarith). destruct Z_lt_le_dec; unfold interp_carry. replace [|r|] with r by (symmetry; apply Zmod_small; auto with zarith). - rewrite Zpower_2; auto with zarith. + rewrite Z.pow_2_r; auto with zarith. replace [|r-wB|] with (r-wB) by (symmetry; apply Zmod_small; auto with zarith). - rewrite Zpower_2; omega. + rewrite Z.pow_2_r; omega. assert (0<=Zneg p). rewrite Heqz; generalize wB_pos; auto with zarith. @@ -667,15 +666,15 @@ Section ZModulo. cut (log_inf x < p - 1); [omega| ]. apply IHx. change (Zpos x~1) with (2*(Zpos x)+1) in H. - replace p with (Zsucc (p-1)) in H; auto with zarith. - rewrite Zpower_Zsucc in H; auto with zarith. + replace p with (Z.succ (p-1)) in H; auto with zarith. + rewrite Z.pow_succ_r in H; auto with zarith. assert (0 < p) by (destruct p; compute; auto with zarith; discriminate). cut (log_inf x < p - 1); [omega| ]. apply IHx. change (Zpos x~0) with (2*(Zpos x)) in H. - replace p with (Zsucc (p-1)) in H; auto with zarith. - rewrite Zpower_Zsucc in H; auto with zarith. + replace p with (Z.succ (p-1)) in H; auto with zarith. + rewrite Z.pow_succ_r in H; auto with zarith. simpl; intros; destruct p; compute; auto with zarith. Qed. @@ -696,27 +695,27 @@ Section ZModulo. unfold zdigits. unfold wB, base in *. apply log_inf_bounded; auto with zarith. - apply Zlt_trans with zdigits. + apply Z.lt_trans with zdigits. omega. unfold zdigits, wB, base; apply Zpower2_lt_lin; auto with zarith. unfold to_Z; rewrite (Zmod_small _ _ H3). destruct H2. split. - apply Zle_trans with (2^(zdigits - log_inf p - 1)*(2^log_inf p)). + apply Z.le_trans with (2^(zdigits - log_inf p - 1)*(2^log_inf p)). apply Zdiv_le_upper_bound; auto with zarith. rewrite <- Zpower_exp; auto with zarith. - rewrite Zmult_comm; rewrite <- Zpower_Zsucc; auto with zarith. - replace (Zsucc (zdigits - log_inf p -1 +log_inf p)) with zdigits + rewrite Z.mul_comm; rewrite <- Z.pow_succ_r; auto with zarith. + replace (Z.succ (zdigits - log_inf p -1 +log_inf p)) with zdigits by ring. unfold wB, base, zdigits; auto with zarith. - apply Zmult_le_compat; auto with zarith. + apply Z.mul_le_mono_nonneg; auto with zarith. - apply Zlt_le_trans - with (2^(zdigits - log_inf p - 1)*(2^(Zsucc (log_inf p)))). - apply Zmult_lt_compat_l; auto with zarith. + apply Z.lt_le_trans + with (2^(zdigits - log_inf p - 1)*(2^(Z.succ (log_inf p)))). + apply Z.mul_lt_mono_pos_l; auto with zarith. rewrite <- Zpower_exp; auto with zarith. - replace (zdigits - log_inf p -1 +Zsucc (log_inf p)) with zdigits + replace (zdigits - log_inf p -1 +Z.succ (log_inf p)) with zdigits by ring. unfold wB, base, zdigits; auto with zarith. Qed. @@ -739,18 +738,18 @@ Section ZModulo. assert (d <> xH). intro; subst. compute in H; destruct p; discriminate. - assert (Zsucc (Zpos (Ppred d)) = Zpos d). + assert (Z.succ (Zpos (Pos.pred d)) = Zpos d). simpl; f_equal. - rewrite <- Pplus_one_succ_r. - destruct (Psucc_pred d); auto. + rewrite Pos.add_1_r. + destruct (Pos.succ_pred_or d); auto. rewrite H1 in H0; elim H0; auto. - assert (Ptail p < Zpos (Ppred d)). + assert (Ptail p < Zpos (Pos.pred d)). apply IHp. - apply Zmult_lt_reg_r with 2; auto with zarith. - rewrite (Zmult_comm (Zpos p)). + apply Z.mul_lt_mono_pos_r with 2; auto with zarith. + rewrite (Z.mul_comm (Zpos p)). change (2 * Zpos p) with (Zpos p~0). - rewrite Zmult_comm. - rewrite <- Zpower_Zsucc; auto with zarith. + rewrite Z.mul_comm. + rewrite <- Z.pow_succ_r; auto with zarith. rewrite H1; auto. rewrite <- H1; omega. Qed. @@ -779,20 +778,20 @@ Section ZModulo. apply Zmod_small. split; auto. unfold wB, base in *. - apply Zlt_trans with (Zpos digits). + apply Z.lt_trans with (Zpos digits). apply Ptail_bounded; auto with zarith. apply Zpower2_lt_lin; auto with zarith. rewrite H1. clear; induction p. - exists (Zpos p); simpl; rewrite Pmult_1_r; auto with zarith. + exists (Zpos p); simpl; rewrite Pos.mul_1_r; auto with zarith. destruct IHp as (y & Yp & Ye). exists y. split; auto. change (Zpos p~0) with (2*Zpos p). rewrite Ye. - change (Ptail p~0) with (Zsucc (Ptail p)). - rewrite Zpower_Zsucc; auto; ring. + change (Ptail p~0) with (Z.succ (Ptail p)). + rewrite Z.pow_succ_r; auto; ring. exists 0; simpl; auto with zarith. Qed. |