diff options
| author |  letouzey <letouzey@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2012-07-05 16:56:16 +0000 |
|---|---|---|
| committer | letouzey <letouzey@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2012-07-05 16:56:16 +0000 |
| commit | fc2613e871dffffa788d90044a81598f671d0a3b (patch) | |
| tree | f6f308b3d6b02e1235446b2eb4a2d04b135a0462 /theories/Numbers | |
| parent | f93f073df630bb46ddd07802026c0326dc72dafd (diff) | |
ZArith + other : favor the use of modern names instead of compat notations
- For instance, refl_equal --> eq_refl
- Npos, Zpos, Zneg now admit more uniform qualified aliases
N.pos, Z.pos, Z.neg.
- A new module BinInt.Pos2Z with results about injections from
positive to Z
- A result about Z.pow pushed in the generic layer
- Zmult_le_compat_{r,l} --> Z.mul_le_mono_nonneg_{r,l}
- Using tactic Z.le_elim instead of Zle_lt_or_eq
- Some cleanup in ring, field, micromega
(use of "Equivalence", "Proper" ...)
- Some adaptions in QArith (for instance changed Qpower.Qpower_decomp)
- In ZMake and ZMake, functor parameters are now named NN and ZZ
instead of N and Z for avoiding confusions
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@15515 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Numbers')
35 files changed, 1846 insertions, 1885 deletions
diff --git a/theories/Numbers/BigNumPrelude.v b/theories/Numbers/BigNumPrelude.v index 26850688e..bf9238010 100644 --- a/theories/Numbers/BigNumPrelude.v +++ b/theories/Numbers/BigNumPrelude.v @@ -30,7 +30,7 @@ Declare ML Module "numbers_syntax_plugin". Local Open Scope Z_scope. -(* For compatibility of scripts, weaker version of some lemmas of Zdiv *) +(* For compatibility of scripts, weaker version of some lemmas of Z.div *) Lemma Zlt0_not_eq : forall n, 0<n -> n<>0. Proof. @@ -43,22 +43,22 @@ Definition Z_div_plus_l a b c H := Zdiv.Z_div_plus_full_l a b c (Zlt0_not_eq _ H (* Automation *) -Hint Extern 2 (Zle _ _) => +Hint Extern 2 (Z.le _ _) => (match goal with - |- Zpos _ <= Zpos _ => exact (refl_equal _) -| H: _ <= ?p |- _ <= ?p => apply Zle_trans with (2 := H) -| H: _ < ?p |- _ <= ?p => apply Zlt_le_weak; apply Zle_lt_trans with (2 := H) + |- Zpos _ <= Zpos _ => exact (eq_refl _) +| H: _ <= ?p |- _ <= ?p => apply Z.le_trans with (2 := H) +| H: _ < ?p |- _ <= ?p => apply Z.lt_le_incl; apply Z.le_lt_trans with (2 := H) end). -Hint Extern 2 (Zlt _ _) => +Hint Extern 2 (Z.lt _ _) => (match goal with - |- Zpos _ < Zpos _ => exact (refl_equal _) -| H: _ <= ?p |- _ <= ?p => apply Zlt_le_trans with (2 := H) -| H: _ < ?p |- _ <= ?p => apply Zle_lt_trans with (2 := H) + |- Zpos _ < Zpos _ => exact (eq_refl _) +| H: _ <= ?p |- _ <= ?p => apply Z.lt_le_trans with (2 := H) +| H: _ < ?p |- _ <= ?p => apply Z.le_lt_trans with (2 := H) end). -Hint Resolve Zlt_gt Zle_ge Z_div_pos: zarith. +Hint Resolve Z.lt_gt Z.le_ge Z_div_pos: zarith. (************************************** Properties of order and product @@ -71,9 +71,9 @@ Hint Resolve Zlt_gt Zle_ge Z_div_pos: zarith. Proof. intros a b c d beta H1 (H3, H4) (H5, H6). assert (a - c < 1); auto with zarith. - apply Zmult_lt_reg_r with beta; auto with zarith. - apply Zle_lt_trans with (d - b); auto with zarith. - rewrite Zmult_minus_distr_r; auto with zarith. + apply Z.mul_lt_mono_pos_r with beta; auto with zarith. + apply Z.le_lt_trans with (d - b); auto with zarith. + rewrite Z.mul_sub_distr_r; auto with zarith. Qed. Theorem beta_lex_inv: forall a b c d beta, @@ -82,15 +82,15 @@ Hint Resolve Zlt_gt Zle_ge Z_div_pos: zarith. a * beta + b < c * beta + d. Proof. intros a b c d beta H1 (H3, H4) (H5, H6). - case (Zle_or_lt (c * beta + d) (a * beta + b)); auto with zarith. - intros H7; contradict H1;apply Zle_not_lt;apply beta_lex with (1 := H7);auto. + case (Z.le_gt_cases (c * beta + d) (a * beta + b)); auto with zarith. + intros H7. contradict H1. apply Z.le_ngt. apply beta_lex with (1 := H7); auto. Qed. Lemma beta_mult : forall h l beta, 0 <= h < beta -> 0 <= l < beta -> 0 <= h*beta+l < beta^2. Proof. intros h l beta H1 H2;split. auto with zarith. - rewrite <- (Zplus_0_r (beta^2)); rewrite Zpower_2; + rewrite <- (Z.add_0_r (beta^2)); rewrite Z.pow_2_r; apply beta_lex_inv;auto with zarith. Qed. @@ -98,9 +98,9 @@ Hint Resolve Zlt_gt Zle_ge Z_div_pos: zarith. forall b x y, 0 <= x < b -> 0 <= y < b -> 0 <= x * y <= b^2 - 2*b + 1. Proof. intros b x y (Hx1,Hx2) (Hy1,Hy2);split;auto with zarith. - apply Zle_trans with ((b-1)*(b-1)). - apply Zmult_le_compat;auto with zarith. - apply Zeq_le; ring. + apply Z.le_trans with ((b-1)*(b-1)). + apply Z.mul_le_mono_nonneg;auto with zarith. + apply Z.eq_le_incl; ring. Qed. Lemma sum_mul_carry : forall xh xl yh yl wc cc beta, @@ -129,11 +129,10 @@ Hint Resolve Zlt_gt Zle_ge Z_div_pos: zarith. Proof. intros x y cross beta HH HH1 HH2. split; auto with zarith. - apply Zle_lt_trans with ((beta-1)*(beta-1)+(beta-1)); auto with zarith. - apply Zplus_le_compat; auto with zarith. - apply Zmult_le_compat; auto with zarith. - repeat (rewrite Zmult_minus_distr_l || rewrite Zmult_minus_distr_r); - rewrite Zpower_2; auto with zarith. + apply Z.le_lt_trans with ((beta-1)*(beta-1)+(beta-1)); auto with zarith. + apply Z.add_le_mono; auto with zarith. + apply Z.mul_le_mono_nonneg; auto with zarith. + rewrite ?Z.mul_sub_distr_l, ?Z.mul_sub_distr_r, Z.pow_2_r; auto with zarith. Qed. Theorem mult_add_ineq2: forall x y c cross beta, @@ -144,11 +143,10 @@ Hint Resolve Zlt_gt Zle_ge Z_div_pos: zarith. Proof. intros x y c cross beta HH HH1 HH2. split; auto with zarith. - apply Zle_lt_trans with ((beta-1)*(beta-1)+(2*beta-2));auto with zarith. - apply Zplus_le_compat; auto with zarith. - apply Zmult_le_compat; auto with zarith. - repeat (rewrite Zmult_minus_distr_l || rewrite Zmult_minus_distr_r); - rewrite Zpower_2; auto with zarith. + apply Z.le_lt_trans with ((beta-1)*(beta-1)+(2*beta-2));auto with zarith. + apply Z.add_le_mono; auto with zarith. + apply Z.mul_le_mono_nonneg; auto with zarith. + rewrite ?Z.mul_sub_distr_l, ?Z.mul_sub_distr_r, Z.pow_2_r; auto with zarith. Qed. Theorem mult_add_ineq3: forall x y c cross beta, @@ -161,20 +159,20 @@ Theorem mult_add_ineq3: forall x y c cross beta, intros x y c cross beta HH HH1 HH2 HH3. apply mult_add_ineq2;auto with zarith. split;auto with zarith. - apply Zle_trans with (1*beta+cross);auto with zarith. + apply Z.le_trans with (1*beta+cross);auto with zarith. Qed. -Hint Rewrite Zmult_1_r Zmult_0_r Zmult_1_l Zmult_0_l Zplus_0_l Zplus_0_r Zminus_0_r: rm10. +Hint Rewrite Z.mul_1_r Z.mul_0_r Z.mul_1_l Z.mul_0_l Z.add_0_l Z.add_0_r Z.sub_0_r: rm10. (************************************** - Properties of Zdiv and Zmod + Properties of Z.div and Z.modulo **************************************) Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. Proof. intros a b H H1;case (Z_mod_lt a b);auto with zarith;intros H2 H3;split;auto. - case (Zle_or_lt b a); intros H4; auto with zarith. + case (Z.le_gt_cases b a); intros H4; auto with zarith. rewrite Zmod_small; auto with zarith. Qed. @@ -184,26 +182,26 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. Proof. intros a b r t (H1, H2) H3 (H4, H5). assert (t < 2 ^ b). - apply Zlt_le_trans with (1:= H5); auto with zarith. + apply Z.lt_le_trans with (1:= H5); auto with zarith. apply Zpower_le_monotone; auto with zarith. rewrite Zplus_mod; auto with zarith. rewrite Zmod_small with (a := t); auto with zarith. apply Zmod_small; auto with zarith. split; auto with zarith. assert (0 <= 2 ^a * r); auto with zarith. - apply Zplus_le_0_compat; auto with zarith. + apply Z.add_nonneg_nonneg; auto with zarith. match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; auto with zarith. pattern (2 ^ b) at 2; replace (2 ^ b) with ((2 ^ b - 2 ^a) + 2 ^ a); try ring. - apply Zplus_le_lt_compat; auto with zarith. + apply Z.add_le_lt_mono; auto with zarith. replace b with ((b - a) + a); try ring. rewrite Zpower_exp; auto with zarith. - pattern (2 ^a) at 4; rewrite <- (Zmult_1_l (2 ^a)); - try rewrite <- Zmult_minus_distr_r. - rewrite (Zmult_comm (2 ^(b - a))); rewrite Zmult_mod_distr_l; + pattern (2 ^a) at 4; rewrite <- (Z.mul_1_l (2 ^a)); + try rewrite <- Z.mul_sub_distr_r. + rewrite (Z.mul_comm (2 ^(b - a))); rewrite Zmult_mod_distr_l; auto with zarith. - rewrite (Zmult_comm (2 ^a)); apply Zmult_le_compat_r; auto with zarith. + rewrite (Z.mul_comm (2 ^a)); apply Z.mul_le_mono_nonneg_r; auto with zarith. match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; auto with zarith. Qed. @@ -214,25 +212,25 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. Proof. intros a b r t (H1, H2) H3 (H4, H5). assert (t < 2 ^ b). - apply Zlt_le_trans with (1:= H5); auto with zarith. + apply Z.lt_le_trans with (1:= H5); auto with zarith. apply Zpower_le_monotone; auto with zarith. rewrite Zplus_mod; auto with zarith. rewrite Zmod_small with (a := t); auto with zarith. apply Zmod_small; auto with zarith. split; auto with zarith. assert (0 <= 2 ^a * r); auto with zarith. - apply Zplus_le_0_compat; auto with zarith. + apply Z.add_nonneg_nonneg; auto with zarith. match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; auto with zarith. pattern (2 ^ b) at 2;replace (2 ^ b) with ((2 ^ b - 2 ^a) + 2 ^ a); try ring. - apply Zplus_le_lt_compat; auto with zarith. + apply Z.add_le_lt_mono; auto with zarith. replace b with ((b - a) + a); try ring. rewrite Zpower_exp; auto with zarith. - pattern (2 ^a) at 4; rewrite <- (Zmult_1_l (2 ^a)); - try rewrite <- Zmult_minus_distr_r. - repeat rewrite (fun x => Zmult_comm x (2 ^ a)); rewrite Zmult_mod_distr_l; + pattern (2 ^a) at 4; rewrite <- (Z.mul_1_l (2 ^a)); + try rewrite <- Z.mul_sub_distr_r. + repeat rewrite (fun x => Z.mul_comm x (2 ^ a)); rewrite Zmult_mod_distr_l; auto with zarith. - apply Zmult_le_compat_l; auto with zarith. + apply Z.mul_le_mono_nonneg_l; auto with zarith. match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; auto with zarith. Qed. @@ -243,13 +241,13 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. Proof. intros a b r t (H1, H2) H3 (H4, H5). assert (Eq: t < 2 ^ b); auto with zarith. - apply Zlt_le_trans with (1 := H5); auto with zarith. + apply Z.lt_le_trans with (1 := H5); auto with zarith. apply Zpower_le_monotone; auto with zarith. pattern (r * 2 ^ a) at 1; rewrite Z_div_mod_eq with (b := 2 ^ b); auto with zarith. - rewrite <- Zplus_assoc. + rewrite <- Z.add_assoc. rewrite <- Zmod_shift_r; auto with zarith. - rewrite (Zmult_comm (2 ^ b)); rewrite Z_div_plus_full_l; auto with zarith. + rewrite (Z.mul_comm (2 ^ b)); rewrite Z_div_plus_full_l; auto with zarith. rewrite (fun x y => @Zdiv_small (x mod y)); auto with zarith. match goal with |- context [?X mod ?Y] => case (Z_mod_lt X Y) end; auto with zarith. @@ -264,7 +262,7 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. intros n p a H1 H2. pattern (a*2^p) at 1;replace (a*2^p) with (a*2^p/2^n * 2^n + a*2^p mod 2^n). - 2:symmetry;rewrite (Zmult_comm (a*2^p/2^n));apply Z_div_mod_eq. + 2:symmetry;rewrite (Z.mul_comm (a*2^p/2^n));apply Z_div_mod_eq. replace (a * 2 ^ p / 2 ^ n) with (a / 2 ^ (n - p));trivial. replace (2^n) with (2^(n-p)*2^p). symmetry;apply Zdiv_mult_cancel_r. @@ -273,7 +271,7 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. rewrite <- Zpower_exp. replace (n-p+p) with n;trivial. ring. omega. omega. - apply Zlt_gt. apply Zpower_gt_0;auto with zarith. + apply Z.lt_gt. apply Z.pow_pos_nonneg;auto with zarith. Qed. @@ -284,15 +282,15 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. intros. rewrite Zmod_small. rewrite Zmod_eq by (auto with zarith). - unfold Zminus at 1. + unfold Z.sub at 1. rewrite Z_div_plus_l by (auto with zarith). assert (2^n = 2^(n-p)*2^p). rewrite <- Zpower_exp by (auto with zarith). replace (n-p+p) with n; auto with zarith. rewrite H0. rewrite <- Zdiv_Zdiv, Z_div_mult by (auto with zarith). - rewrite (Zmult_comm (2^(n-p))), Zmult_assoc. - rewrite Zopp_mult_distr_l. + rewrite (Z.mul_comm (2^(n-p))), Z.mul_assoc. + rewrite <- Z.mul_opp_l. rewrite Z_div_mult by (auto with zarith). symmetry; apply Zmod_eq; auto with zarith. @@ -301,9 +299,9 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. split. apply Z_div_pos; auto with zarith. apply Zdiv_lt_upper_bound; auto with zarith. - apply Zlt_le_trans with (2^n); auto with zarith. - rewrite <- (Zmult_1_r (2^n)) at 1. - apply Zmult_le_compat; auto with zarith. + apply Z.lt_le_trans with (2^n); auto with zarith. + rewrite <- (Z.mul_1_r (2^n)) at 1. + apply Z.mul_le_mono_nonneg; auto with zarith. cut (0 < 2 ^ (n-p)); auto with zarith. Qed. @@ -320,8 +318,8 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. Proof. intros p x y H;destruct (Z_le_gt_dec 0 p). apply Zdiv_lt_upper_bound;auto with zarith. - apply Zlt_le_trans with y;auto with zarith. - rewrite <- (Zmult_1_r y);apply Zmult_le_compat;auto with zarith. + apply Z.lt_le_trans with y;auto with zarith. + rewrite <- (Z.mul_1_r y);apply Z.mul_le_mono_nonneg;auto with zarith. assert (0 < 2^p);auto with zarith. replace (2^p) with 0. destruct x;change (0<y);auto with zarith. @@ -329,15 +327,13 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. Qed. Theorem Zgcd_div_pos a b: - 0 < b -> 0 < Zgcd a b -> 0 < b / Zgcd a b. + 0 < b -> 0 < Z.gcd a b -> 0 < b / Z.gcd a b. Proof. - intros Ha Hg. - case (Zle_lt_or_eq 0 (b/Zgcd a b)); auto. - apply Z_div_pos; auto with zarith. - intros H; generalize Ha. - pattern b at 1; rewrite (Zdivide_Zdiv_eq (Zgcd a b) b); auto. - rewrite <- H; auto with zarith. - assert (F := (Zgcd_is_gcd a b)); inversion F; auto. + intros Hb Hg. + assert (H : 0 <= b / Z.gcd a b) by (apply Z.div_pos; auto with zarith). + Z.le_elim H; trivial. + rewrite (Zdivide_Zdiv_eq (Z.gcd a b) b), <- H, Z.mul_0_r in Hb; + auto using Z.gcd_divide_r with zarith. Qed. Theorem Zdiv_neg a b: @@ -347,7 +343,7 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. assert (b > 0) by omega. generalize (Z_mult_div_ge a _ H); intros. assert (b * (a / b) < 0)%Z. - apply Zle_lt_trans with a; auto with zarith. + apply Z.le_lt_trans with a; auto with zarith. destruct b; try (compute in Hb; discriminate). destruct (a/Zpos p)%Z. compute in H1; discriminate. @@ -355,20 +351,20 @@ Theorem Zmod_le_first: forall a b, 0 <= a -> 0 < b -> 0 <= a mod b <= a. compute; auto. Qed. - Lemma Zdiv_gcd_zero : forall a b, b / Zgcd a b = 0 -> b <> 0 -> - Zgcd a b = 0. + Lemma Zdiv_gcd_zero : forall a b, b / Z.gcd a b = 0 -> b <> 0 -> + Z.gcd a b = 0. Proof. intros. generalize (Zgcd_is_gcd a b); destruct 1. destruct H2 as (k,Hk). generalize H; rewrite Hk at 1. - destruct (Z_eq_dec (Zgcd a b) 0) as [H'|H']; auto. + destruct (Z.eq_dec (Z.gcd a b) 0) as [H'|H']; auto. rewrite Z_div_mult_full; auto. intros; subst k; simpl in *; subst b; elim H0; auto. Qed. Lemma Zgcd_mult_rel_prime : forall a b c, - Zgcd a c = 1 -> Zgcd b c = 1 -> Zgcd (a*b) c = 1. + Z.gcd a c = 1 -> Z.gcd b c = 1 -> Z.gcd (a*b) c = 1. Proof. intros. rewrite Zgcd_1_rel_prime in *. @@ -396,23 +392,20 @@ intros Q b Q0 QS. set (Q' := fun n => (n < b /\ Q n) \/ (b <= n)). assert (H : forall n, 0 <= n -> Q' n). apply natlike_rec2; unfold Q'. -destruct (Zle_or_lt b 0) as [H | H]. now right. left; now split. +destruct (Z.le_gt_cases b 0) as [H | H]. now right. left; now split. intros n H IH. destruct IH as [[IH1 IH2] | IH]. -destruct (Zle_or_lt (b - 1) n) as [H1 | H1]. +destruct (Z.le_gt_cases (b - 1) n) as [H1 | H1]. right; auto with zarith. left. split; [auto with zarith | now apply (QS n)]. right; auto with zarith. unfold Q' in *; intros n H1 H2. destruct (H n H1) as [[H3 H4] | H3]. -assumption. apply Zle_not_lt in H3. false_hyp H2 H3. +assumption. now apply Z.le_ngt in H3. Qed. -Lemma Zsquare_le : forall x, x <= x*x. +Lemma Zsquare_le x : x <= x*x. Proof. -intros. -destruct (Z_lt_le_dec 0 x). -pattern x at 1; rewrite <- (Zmult_1_l x). -apply Zmult_le_compat; auto with zarith. -apply Zle_trans with 0; auto with zarith. -rewrite <- Zmult_opp_opp. -apply Zmult_le_0_compat; auto with zarith. +destruct (Z.lt_ge_cases 0 x). +- rewrite <- Z.mul_1_l at 1. + rewrite <- Z.mul_le_mono_pos_r; auto with zarith. +- pose proof (Z.square_nonneg x); auto with zarith. Qed. diff --git a/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v b/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v index 59656eedb..1cfefd3b9 100644 --- a/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v +++ b/theories/Numbers/Cyclic/Abstract/CyclicAxioms.v @@ -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 c52cbe105..82e4ad13f 100644 --- a/theories/Numbers/Cyclic/Abstract/NZCyclic.v +++ b/theories/Numbers/Cyclic/Abstract/NZCyclic.v @@ -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 deb216ddc..a9b976adb 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleAdd.v @@ -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 e6c5a0e04..3eda130fb 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleBase.v @@ -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 00a840520..55ecefa15 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleCyclic.v @@ -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 0cb6848e3..40bce95b4 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDiv.v @@ -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 062282f2e..8a8e90c3c 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleDivn1.v @@ -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 a6a0fc8e3..7086d0fd7 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleLift.v @@ -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 0032d2c3f..ceee1869d 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleMul.v @@ -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 b073d6bed..1318c6123 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSqrt.v @@ -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 e63e20c69..46a163cde 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleSub.v @@ -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 a274b8395..948203822 100644 --- a/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v +++ b/theories/Numbers/Cyclic/DoubleCyclic/DoubleType.v @@ -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 2dd1c3eec..8ed59632e 100644 --- a/theories/Numbers/Cyclic/Int31/Cyclic31.v +++ b/theories/Numbers/Cyclic/Int31/Cyclic31.v @@ -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 20f750f64..5415b379b 100644 --- a/theories/Numbers/Cyclic/Int31/Int31.v +++ b/theories/Numbers/Cyclic/Int31/Int31.v @@ -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 23e8bd338..ef3400223 100644 --- a/theories/Numbers/Cyclic/Int31/Ring31.v +++ b/theories/Numbers/Cyclic/Int31/Ring31.v @@ -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 d039fdcbf..6945d0757 100644 --- a/theories/Numbers/Cyclic/ZModulo/ZModulo.v +++ b/theories/Numbers/Cyclic/ZModulo/ZModulo.v @@ -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. diff --git a/theories/Numbers/Integer/Abstract/ZBits.v b/theories/Numbers/Integer/Abstract/ZBits.v index 92afbcb53..7c3a99215 100644 --- a/theories/Numbers/Integer/Abstract/ZBits.v +++ b/theories/Numbers/Integer/Abstract/ZBits.v @@ -292,7 +292,7 @@ Proof. Qed. (** Hence the number of bits of [a] is [1+log2 a] - (see [Psize] and [Psize_pos]). + (see [Pos.size_nat] and [Pos.size]). *) (** For negative numbers, things are the other ways around: diff --git a/theories/Numbers/Integer/Abstract/ZDivFloor.v b/theories/Numbers/Integer/Abstract/ZDivFloor.v index 14003d892..24c8dc8b0 100644 --- a/theories/Numbers/Integer/Abstract/ZDivFloor.v +++ b/theories/Numbers/Integer/Abstract/ZDivFloor.v @@ -16,7 +16,7 @@ Require Import ZAxioms ZMulOrder ZSgnAbs NZDiv. [a = bq+r /\ 0 <= |r| < |b| /\ Sign(r) = Sign(b)] - This is the convention followed historically by [Zdiv] in Coq, and + This is the convention followed historically by [Z.div] in Coq, and corresponds to convention "F" in the following paper: R. Boute, "The Euclidean definition of the functions div and mod", diff --git a/theories/Numbers/Integer/Abstract/ZPow.v b/theories/Numbers/Integer/Abstract/ZPow.v index 53d84dce1..c9c98b1e5 100644 --- a/theories/Numbers/Integer/Abstract/ZPow.v +++ b/theories/Numbers/Integer/Abstract/ZPow.v @@ -18,6 +18,17 @@ Module Type ZPowProp Include NZPowProp A A B. +(** A particular case of [pow_add_r], with no precondition *) + +Lemma pow_twice_r a b : a^(2*b) == a^b * a^b. +Proof. + rewrite two_succ. nzsimpl. + destruct (le_gt_cases 0 b). + - now rewrite pow_add_r. + - rewrite !pow_neg_r. now nzsimpl. trivial. + now apply add_neg_neg. +Qed. + (** Parity of power *) Lemma even_pow : forall a b, 0<b -> even (a^b) = even a. diff --git a/theories/Numbers/Integer/BigZ/BigZ.v b/theories/Numbers/Integer/BigZ/BigZ.v index 443777f52..ae1701be7 100644 --- a/theories/Numbers/Integer/BigZ/BigZ.v +++ b/theories/Numbers/Integer/BigZ/BigZ.v @@ -76,7 +76,7 @@ Infix "÷" := BigZ.quot (at level 40, left associativity) : bigZ_scope. (** Some additional results about [BigZ] *) Theorem spec_to_Z: forall n : bigZ, - BigN.to_Z (BigZ.to_N n) = ((Zsgn [n]) * [n])%Z. + BigN.to_Z (BigZ.to_N n) = ((Z.sgn [n]) * [n])%Z. Proof. intros n; case n; simpl; intros p; generalize (BigN.spec_pos p); case (BigN.to_Z p); auto. @@ -85,7 +85,7 @@ intros p1 H1; case H1; auto. Qed. Theorem spec_to_N n: - ([n] = Zsgn [n] * (BigN.to_Z (BigZ.to_N n)))%Z. + ([n] = Z.sgn [n] * (BigN.to_Z (BigZ.to_N n)))%Z. Proof. case n; simpl; intros p; generalize (BigN.spec_pos p); case (BigN.to_Z p); auto. @@ -118,7 +118,7 @@ Qed. Lemma BigZeqb_correct : forall x y, (x =? y) = true -> x==y. Proof. now apply BigZ.eqb_eq. Qed. -Definition BigZ_of_N n := BigZ.of_Z (Z_of_N n). +Definition BigZ_of_N n := BigZ.of_Z (Z.of_N n). Lemma BigZpower : power_theory 1 BigZ.mul BigZ.eq BigZ_of_N BigZ.pow. Proof. @@ -139,7 +139,7 @@ BigZ.zify. auto with zarith. intros NEQ. generalize (BigZ.spec_div_eucl a b). generalize (Z_div_mod_full [a] [b] NEQ). -destruct BigZ.div_eucl as (q,r), Zdiv_eucl as (q',r'). +destruct BigZ.div_eucl as (q,r), Z.div_eucl as (q',r'). intros (EQ,_). injection 1. intros EQr EQq. BigZ.zify. rewrite EQr, EQq; auto. Qed. diff --git a/theories/Numbers/Integer/BigZ/ZMake.v b/theories/Numbers/Integer/BigZ/ZMake.v index 0142b36be..d1c817f54 100644 --- a/theories/Numbers/Integer/BigZ/ZMake.v +++ b/theories/Numbers/Integer/BigZ/ZMake.v @@ -21,92 +21,92 @@ Open Scope Z_scope. [NSig.NType] to a structure of integers [ZSig.ZType]. *) -Module Make (N:NType) <: ZType. +Module Make (NN:NType) <: ZType. Inductive t_ := - | Pos : N.t -> t_ - | Neg : N.t -> t_. + | Pos : NN.t -> t_ + | Neg : NN.t -> t_. Definition t := t_. Bind Scope abstract_scope with t t_. - Definition zero := Pos N.zero. - Definition one := Pos N.one. - Definition two := Pos N.two. - Definition minus_one := Neg N.one. + Definition zero := Pos NN.zero. + Definition one := Pos NN.one. + Definition two := Pos NN.two. + Definition minus_one := Neg NN.one. Definition of_Z x := match x with - | Zpos x => Pos (N.of_N (Npos x)) + | Zpos x => Pos (NN.of_N (Npos x)) | Z0 => zero - | Zneg x => Neg (N.of_N (Npos x)) + | Zneg x => Neg (NN.of_N (Npos x)) end. Definition to_Z x := match x with - | Pos nx => N.to_Z nx - | Neg nx => Zopp (N.to_Z nx) + | Pos nx => NN.to_Z nx + | Neg nx => Z.opp (NN.to_Z nx) end. Theorem spec_of_Z: forall x, to_Z (of_Z x) = x. Proof. intros x; case x; unfold to_Z, of_Z, zero. - exact N.spec_0. - intros; rewrite N.spec_of_N; auto. - intros; rewrite N.spec_of_N; auto. + exact NN.spec_0. + intros; rewrite NN.spec_of_N; auto. + intros; rewrite NN.spec_of_N; auto. Qed. Definition eq x y := (to_Z x = to_Z y). Theorem spec_0: to_Z zero = 0. - exact N.spec_0. + exact NN.spec_0. Qed. Theorem spec_1: to_Z one = 1. - exact N.spec_1. + exact NN.spec_1. Qed. Theorem spec_2: to_Z two = 2. - exact N.spec_2. + exact NN.spec_2. Qed. Theorem spec_m1: to_Z minus_one = -1. - simpl; rewrite N.spec_1; auto. + simpl; rewrite NN.spec_1; auto. Qed. Definition compare x y := match x, y with - | Pos nx, Pos ny => N.compare nx ny + | Pos nx, Pos ny => NN.compare nx ny | Pos nx, Neg ny => - match N.compare nx N.zero with + match NN.compare nx NN.zero with | Gt => Gt - | _ => N.compare ny N.zero + | _ => NN.compare ny NN.zero end | Neg nx, Pos ny => - match N.compare N.zero nx with + match NN.compare NN.zero nx with | Lt => Lt - | _ => N.compare N.zero ny + | _ => NN.compare NN.zero ny end - | Neg nx, Neg ny => N.compare ny nx + | Neg nx, Neg ny => NN.compare ny nx end. Theorem spec_compare : - forall x y, compare x y = Zcompare (to_Z x) (to_Z y). + forall x y, compare x y = Z.compare (to_Z x) (to_Z y). Proof. unfold compare, to_Z. destruct x as [x|x], y as [y|y]; - rewrite ?N.spec_compare, ?N.spec_0, <-?Zcompare_opp; auto; - assert (Hx:=N.spec_pos x); assert (Hy:=N.spec_pos y); - set (X:=N.to_Z x) in *; set (Y:=N.to_Z y) in *; clearbody X Y. - destruct (Zcompare_spec X 0) as [EQ|LT|GT]. - rewrite EQ. rewrite <- Zopp_0 at 2. apply Zcompare_opp. - exfalso. omega. - symmetry. change (X > -Y). omega. - destruct (Zcompare_spec 0 X) as [EQ|LT|GT]. - rewrite <- EQ. rewrite Zopp_0; auto. - symmetry. change (-X < Y). omega. - exfalso. omega. + rewrite ?NN.spec_compare, ?NN.spec_0, ?Z.compare_opp; auto; + assert (Hx:=NN.spec_pos x); assert (Hy:=NN.spec_pos y); + set (X:=NN.to_Z x) in *; set (Y:=NN.to_Z y) in *; clearbody X Y. + - destruct (Z.compare_spec X 0) as [EQ|LT|GT]. + + rewrite <- Z.opp_0 in EQ. now rewrite EQ, Z.compare_opp. + + exfalso. omega. + + symmetry. change (X > -Y). omega. + - destruct (Z.compare_spec 0 X) as [EQ|LT|GT]. + + rewrite <- EQ, Z.opp_0; auto. + + symmetry. change (-X < Y). omega. + + exfalso. omega. Qed. Definition eqb x y := @@ -155,14 +155,14 @@ Module Make (N:NType) <: ZType. Definition min n m := match compare n m with Gt => m | _ => n end. Definition max n m := match compare n m with Lt => m | _ => n end. - Theorem spec_min : forall n m, to_Z (min n m) = Zmin (to_Z n) (to_Z m). + Theorem spec_min : forall n m, to_Z (min n m) = Z.min (to_Z n) (to_Z m). Proof. - unfold min, Zmin. intros. rewrite spec_compare. destruct Zcompare; auto. + unfold min, Z.min. intros. rewrite spec_compare. destruct Z.compare; auto. Qed. - Theorem spec_max : forall n m, to_Z (max n m) = Zmax (to_Z n) (to_Z m). + Theorem spec_max : forall n m, to_Z (max n m) = Z.max (to_Z n) (to_Z m). Proof. - unfold max, Zmax. intros. rewrite spec_compare. destruct Zcompare; auto. + unfold max, Z.max. intros. rewrite spec_compare. destruct Z.compare; auto. Qed. Definition to_N x := @@ -173,11 +173,11 @@ Module Make (N:NType) <: ZType. Definition abs x := Pos (to_N x). - Theorem spec_abs: forall x, to_Z (abs x) = Zabs (to_Z x). + Theorem spec_abs: forall x, to_Z (abs x) = Z.abs (to_Z x). Proof. - intros x; case x; clear x; intros x; assert (F:=N.spec_pos x). - simpl; rewrite Zabs_eq; auto. - simpl; rewrite Zabs_non_eq; simpl; auto with zarith. + intros x; case x; clear x; intros x; assert (F:=NN.spec_pos x). + simpl; rewrite Z.abs_eq; auto. + simpl; rewrite Z.abs_neq; simpl; auto with zarith. Qed. Definition opp x := @@ -193,10 +193,10 @@ Module Make (N:NType) <: ZType. Definition succ x := match x with - | Pos n => Pos (N.succ n) + | Pos n => Pos (NN.succ n) | Neg n => - match N.compare N.zero n with - | Lt => Neg (N.pred n) + match NN.compare NN.zero n with + | Lt => Neg (NN.pred n) | _ => one end end. @@ -204,134 +204,134 @@ Module Make (N:NType) <: ZType. Theorem spec_succ: forall n, to_Z (succ n) = to_Z n + 1. Proof. intros x; case x; clear x; intros x. - exact (N.spec_succ x). - simpl. rewrite N.spec_compare. case Zcompare_spec; rewrite ?N.spec_0; simpl. - intros HH; rewrite <- HH; rewrite N.spec_1; ring. - intros HH; rewrite N.spec_pred, Zmax_r; auto with zarith. - generalize (N.spec_pos x); auto with zarith. + exact (NN.spec_succ x). + simpl. rewrite NN.spec_compare. case Z.compare_spec; rewrite ?NN.spec_0; simpl. + intros HH; rewrite <- HH; rewrite NN.spec_1; ring. + intros HH; rewrite NN.spec_pred, Z.max_r; auto with zarith. + generalize (NN.spec_pos x); auto with zarith. Qed. Definition add x y := match x, y with - | Pos nx, Pos ny => Pos (N.add nx ny) + | Pos nx, Pos ny => Pos (NN.add nx ny) | Pos nx, Neg ny => - match N.compare nx ny with - | Gt => Pos (N.sub nx ny) + match NN.compare nx ny with + | Gt => Pos (NN.sub nx ny) | Eq => zero - | Lt => Neg (N.sub ny nx) + | Lt => Neg (NN.sub ny nx) end | Neg nx, Pos ny => - match N.compare nx ny with - | Gt => Neg (N.sub nx ny) + match NN.compare nx ny with + | Gt => Neg (NN.sub nx ny) | Eq => zero - | Lt => Pos (N.sub ny nx) + | Lt => Pos (NN.sub ny nx) end - | Neg nx, Neg ny => Neg (N.add nx ny) + | Neg nx, Neg ny => Neg (NN.add nx ny) end. Theorem spec_add: forall x y, to_Z (add x y) = to_Z x + to_Z y. Proof. unfold add, to_Z; intros [x | x] [y | y]; - try (rewrite N.spec_add; auto with zarith); - rewrite N.spec_compare; case Zcompare_spec; - unfold zero; rewrite ?N.spec_0, ?N.spec_sub; omega with *. + try (rewrite NN.spec_add; auto with zarith); + rewrite NN.spec_compare; case Z.compare_spec; + unfold zero; rewrite ?NN.spec_0, ?NN.spec_sub; omega with *. Qed. Definition pred x := match x with | Pos nx => - match N.compare N.zero nx with - | Lt => Pos (N.pred nx) + match NN.compare NN.zero nx with + | Lt => Pos (NN.pred nx) | _ => minus_one end - | Neg nx => Neg (N.succ nx) + | Neg nx => Neg (NN.succ nx) end. Theorem spec_pred: forall x, to_Z (pred x) = to_Z x - 1. Proof. unfold pred, to_Z, minus_one; intros [x | x]; - try (rewrite N.spec_succ; ring). - rewrite N.spec_compare; case Zcompare_spec; - rewrite ?N.spec_0, ?N.spec_1, ?N.spec_pred; - generalize (N.spec_pos x); omega with *. + try (rewrite NN.spec_succ; ring). + rewrite NN.spec_compare; case Z.compare_spec; + rewrite ?NN.spec_0, ?NN.spec_1, ?NN.spec_pred; + generalize (NN.spec_pos x); omega with *. Qed. Definition sub x y := match x, y with | Pos nx, Pos ny => - match N.compare nx ny with - | Gt => Pos (N.sub nx ny) + match NN.compare nx ny with + | Gt => Pos (NN.sub nx ny) | Eq => zero - | Lt => Neg (N.sub ny nx) + | Lt => Neg (NN.sub ny nx) end - | Pos nx, Neg ny => Pos (N.add nx ny) - | Neg nx, Pos ny => Neg (N.add nx ny) + | Pos nx, Neg ny => Pos (NN.add nx ny) + | Neg nx, Pos ny => Neg (NN.add nx ny) | Neg nx, Neg ny => - match N.compare nx ny with - | Gt => Neg (N.sub nx ny) + match NN.compare nx ny with + | Gt => Neg (NN.sub nx ny) | Eq => zero - | Lt => Pos (N.sub ny nx) + | Lt => Pos (NN.sub ny nx) end end. Theorem spec_sub: forall x y, to_Z (sub x y) = to_Z x - to_Z y. Proof. unfold sub, to_Z; intros [x | x] [y | y]; - try (rewrite N.spec_add; auto with zarith); - rewrite N.spec_compare; case Zcompare_spec; - unfold zero; rewrite ?N.spec_0, ?N.spec_sub; omega with *. + try (rewrite NN.spec_add; auto with zarith); + rewrite NN.spec_compare; case Z.compare_spec; + unfold zero; rewrite ?NN.spec_0, ?NN.spec_sub; omega with *. Qed. Definition mul x y := match x, y with - | Pos nx, Pos ny => Pos (N.mul nx ny) - | Pos nx, Neg ny => Neg (N.mul nx ny) - | Neg nx, Pos ny => Neg (N.mul nx ny) - | Neg nx, Neg ny => Pos (N.mul nx ny) + | Pos nx, Pos ny => Pos (NN.mul nx ny) + | Pos nx, Neg ny => Neg (NN.mul nx ny) + | Neg nx, Pos ny => Neg (NN.mul nx ny) + | Neg nx, Neg ny => Pos (NN.mul nx ny) end. Theorem spec_mul: forall x y, to_Z (mul x y) = to_Z x * to_Z y. Proof. - unfold mul, to_Z; intros [x | x] [y | y]; rewrite N.spec_mul; ring. + unfold mul, to_Z; intros [x | x] [y | y]; rewrite NN.spec_mul; ring. Qed. Definition square x := match x with - | Pos nx => Pos (N.square nx) - | Neg nx => Pos (N.square nx) + | Pos nx => Pos (NN.square nx) + | Neg nx => Pos (NN.square nx) end. Theorem spec_square: forall x, to_Z (square x) = to_Z x * to_Z x. Proof. - unfold square, to_Z; intros [x | x]; rewrite N.spec_square; ring. + unfold square, to_Z; intros [x | x]; rewrite NN.spec_square; ring. Qed. Definition pow_pos x p := match x with - | Pos nx => Pos (N.pow_pos nx p) + | Pos nx => Pos (NN.pow_pos nx p) | Neg nx => match p with | xH => x - | xO _ => Pos (N.pow_pos nx p) - | xI _ => Neg (N.pow_pos nx p) + | xO _ => Pos (NN.pow_pos nx p) + | xI _ => Neg (NN.pow_pos nx p) end end. Theorem spec_pow_pos: forall x n, to_Z (pow_pos x n) = to_Z x ^ Zpos n. Proof. assert (F0: forall x, (-x)^2 = x^2). - intros x; rewrite Zpower_2; ring. + intros x; rewrite Z.pow_2_r; ring. unfold pow_pos, to_Z; intros [x | x] [p | p |]; - try rewrite N.spec_pow_pos; try ring. + try rewrite NN.spec_pow_pos; try ring. assert (F: 0 <= 2 * Zpos p). assert (0 <= Zpos p); auto with zarith. - rewrite Zpos_xI; repeat rewrite Zpower_exp; auto with zarith. - repeat rewrite Zpower_mult; auto with zarith. + rewrite Pos2Z.inj_xI; repeat rewrite Zpower_exp; auto with zarith. + repeat rewrite Z.pow_mul_r; auto with zarith. rewrite F0; ring. assert (F: 0 <= 2 * Zpos p). assert (0 <= Zpos p); auto with zarith. - rewrite Zpos_xO; repeat rewrite Zpower_exp; auto with zarith. - repeat rewrite Zpower_mult; auto with zarith. + rewrite Pos2Z.inj_xO; repeat rewrite Zpower_exp; auto with zarith. + repeat rewrite Z.pow_mul_r; auto with zarith. rewrite F0; ring. Qed. @@ -341,9 +341,9 @@ Module Make (N:NType) <: ZType. | Npos p => pow_pos x p end. - Theorem spec_pow_N: forall x n, to_Z (pow_N x n) = to_Z x ^ Z_of_N n. + Theorem spec_pow_N: forall x n, to_Z (pow_N x n) = to_Z x ^ Z.of_N n. Proof. - destruct n; simpl. apply N.spec_1. + destruct n; simpl. apply NN.spec_1. apply spec_pow_pos. Qed. @@ -357,38 +357,38 @@ Module Make (N:NType) <: ZType. Theorem spec_pow: forall x y, to_Z (pow x y) = to_Z x ^ to_Z y. Proof. intros. unfold pow. destruct (to_Z y); simpl. - apply N.spec_1. + apply NN.spec_1. apply spec_pow_pos. - apply N.spec_0. + apply NN.spec_0. Qed. Definition log2 x := match x with - | Pos nx => Pos (N.log2 nx) + | Pos nx => Pos (NN.log2 nx) | Neg nx => zero end. Theorem spec_log2: forall x, to_Z (log2 x) = Z.log2 (to_Z x). Proof. - intros. destruct x as [p|p]; simpl. apply N.spec_log2. - rewrite N.spec_0. - destruct (Z_le_lt_eq_dec _ _ (N.spec_pos p)) as [LT|EQ]. + intros. destruct x as [p|p]; simpl. apply NN.spec_log2. + rewrite NN.spec_0. + destruct (Z_le_lt_eq_dec _ _ (NN.spec_pos p)) as [LT|EQ]. rewrite Z.log2_nonpos; auto with zarith. now rewrite <- EQ. Qed. Definition sqrt x := match x with - | Pos nx => Pos (N.sqrt nx) - | Neg nx => Neg N.zero + | Pos nx => Pos (NN.sqrt nx) + | Neg nx => Neg NN.zero end. Theorem spec_sqrt: forall x, to_Z (sqrt x) = Z.sqrt (to_Z x). Proof. destruct x as [p|p]; simpl. - apply N.spec_sqrt. - rewrite N.spec_0. - destruct (Z_le_lt_eq_dec _ _ (N.spec_pos p)) as [LT|EQ]. + apply NN.spec_sqrt. + rewrite NN.spec_0. + destruct (Z_le_lt_eq_dec _ _ (NN.spec_pos p)) as [LT|EQ]. rewrite Z.sqrt_neg; auto with zarith. now rewrite <- EQ. Qed. @@ -396,68 +396,68 @@ Module Make (N:NType) <: ZType. Definition div_eucl x y := match x, y with | Pos nx, Pos ny => - let (q, r) := N.div_eucl nx ny in + let (q, r) := NN.div_eucl nx ny in (Pos q, Pos r) | Pos nx, Neg ny => - let (q, r) := N.div_eucl nx ny in - if N.eqb N.zero r + let (q, r) := NN.div_eucl nx ny in + if NN.eqb NN.zero r then (Neg q, zero) - else (Neg (N.succ q), Neg (N.sub ny r)) + else (Neg (NN.succ q), Neg (NN.sub ny r)) | Neg nx, Pos ny => - let (q, r) := N.div_eucl nx ny in - if N.eqb N.zero r + let (q, r) := NN.div_eucl nx ny in + if NN.eqb NN.zero r then (Neg q, zero) - else (Neg (N.succ q), Pos (N.sub ny r)) + else (Neg (NN.succ q), Pos (NN.sub ny r)) | Neg nx, Neg ny => - let (q, r) := N.div_eucl nx ny in + let (q, r) := NN.div_eucl nx ny in (Pos q, Neg r) end. Ltac break_nonneg x px EQx := let H := fresh "H" in - assert (H:=N.spec_pos x); - destruct (N.to_Z x) as [|px|px]_eqn:EQx; + assert (H:=NN.spec_pos x); + destruct (NN.to_Z x) as [|px|px]_eqn:EQx; [clear H|clear H|elim H; reflexivity]. Theorem spec_div_eucl: forall x y, let (q,r) := div_eucl x y in - (to_Z q, to_Z r) = Zdiv_eucl (to_Z x) (to_Z y). + (to_Z q, to_Z r) = Z.div_eucl (to_Z x) (to_Z y). Proof. unfold div_eucl, to_Z. intros [x | x] [y | y]. (* Pos Pos *) - generalize (N.spec_div_eucl x y); destruct (N.div_eucl x y); auto. + generalize (NN.spec_div_eucl x y); destruct (NN.div_eucl x y); auto. (* Pos Neg *) - generalize (N.spec_div_eucl x y); destruct (N.div_eucl x y) as (q,r). + generalize (NN.spec_div_eucl x y); destruct (NN.div_eucl x y) as (q,r). break_nonneg x px EQx; break_nonneg y py EQy; - try (injection 1; intros Hr Hq; rewrite N.spec_eqb, N.spec_0, Hr; - simpl; rewrite Hq, N.spec_0; auto). + try (injection 1; intros Hr Hq; rewrite NN.spec_eqb, NN.spec_0, Hr; + simpl; rewrite Hq, NN.spec_0; auto). change (- Zpos py) with (Zneg py). assert (GT : Zpos py > 0) by (compute; auto). generalize (Z_div_mod (Zpos px) (Zpos py) GT). - unfold Zdiv_eucl. destruct (Zdiv_eucl_POS px (Zpos py)) as (q',r'). + unfold Z.div_eucl. destruct (Z.pos_div_eucl px (Zpos py)) as (q',r'). intros (EQ,MOD). injection 1. intros Hr' Hq'. - rewrite N.spec_eqb, N.spec_0, Hr'. + rewrite NN.spec_eqb, NN.spec_0, Hr'. break_nonneg r pr EQr. - subst; simpl. rewrite N.spec_0; auto. + subst; simpl. rewrite NN.spec_0; auto. subst. lazy iota beta delta [Z.eqb]. - rewrite N.spec_sub, N.spec_succ, EQy, EQr. f_equal. omega with *. + rewrite NN.spec_sub, NN.spec_succ, EQy, EQr. f_equal. omega with *. (* Neg Pos *) - generalize (N.spec_div_eucl x y); destruct (N.div_eucl x y) as (q,r). + generalize (NN.spec_div_eucl x y); destruct (NN.div_eucl x y) as (q,r). break_nonneg x px EQx; break_nonneg y py EQy; - try (injection 1; intros Hr Hq; rewrite N.spec_eqb, N.spec_0, Hr; - simpl; rewrite Hq, N.spec_0; auto). + try (injection 1; intros Hr Hq; rewrite NN.spec_eqb, NN.spec_0, Hr; + simpl; rewrite Hq, NN.spec_0; auto). change (- Zpos px) with (Zneg px). assert (GT : Zpos py > 0) by (compute; auto). generalize (Z_div_mod (Zpos px) (Zpos py) GT). - unfold Zdiv_eucl. destruct (Zdiv_eucl_POS px (Zpos py)) as (q',r'). + unfold Z.div_eucl. destruct (Z.pos_div_eucl px (Zpos py)) as (q',r'). intros (EQ,MOD). injection 1. intros Hr' Hq'. - rewrite N.spec_eqb, N.spec_0, Hr'. + rewrite NN.spec_eqb, NN.spec_0, Hr'. break_nonneg r pr EQr. - subst; simpl. rewrite N.spec_0; auto. + subst; simpl. rewrite NN.spec_0; auto. subst. lazy iota beta delta [Z.eqb]. - rewrite N.spec_sub, N.spec_succ, EQy, EQr. f_equal. omega with *. + rewrite NN.spec_sub, NN.spec_succ, EQy, EQr. f_equal. omega with *. (* Neg Neg *) - generalize (N.spec_div_eucl x y); destruct (N.div_eucl x y) as (q,r). + generalize (NN.spec_div_eucl x y); destruct (NN.div_eucl x y) as (q,r). break_nonneg x px EQx; break_nonneg y py EQy; try (injection 1; intros Hr Hq; rewrite Hr, Hq; auto). simpl. intros <-; auto. @@ -468,8 +468,8 @@ Module Make (N:NType) <: ZType. Definition spec_div: forall x y, to_Z (div x y) = to_Z x / to_Z y. Proof. - intros x y; generalize (spec_div_eucl x y); unfold div, Zdiv. - case div_eucl; case Zdiv_eucl; simpl; auto. + intros x y; generalize (spec_div_eucl x y); unfold div, Z.div. + case div_eucl; case Z.div_eucl; simpl; auto. intros q r q11 r1 H; injection H; auto. Qed. @@ -478,38 +478,38 @@ Module Make (N:NType) <: ZType. Theorem spec_modulo: forall x y, to_Z (modulo x y) = to_Z x mod to_Z y. Proof. - intros x y; generalize (spec_div_eucl x y); unfold modulo, Zmod. - case div_eucl; case Zdiv_eucl; simpl; auto. + intros x y; generalize (spec_div_eucl x y); unfold modulo, Z.modulo. + case div_eucl; case Z.div_eucl; simpl; auto. intros q r q11 r1 H; injection H; auto. Qed. Definition quot x y := match x, y with - | Pos nx, Pos ny => Pos (N.div nx ny) - | Pos nx, Neg ny => Neg (N.div nx ny) - | Neg nx, Pos ny => Neg (N.div nx ny) - | Neg nx, Neg ny => Pos (N.div nx ny) + | Pos nx, Pos ny => Pos (NN.div nx ny) + | Pos nx, Neg ny => Neg (NN.div nx ny) + | Neg nx, Pos ny => Neg (NN.div nx ny) + | Neg nx, Neg ny => Pos (NN.div nx ny) end. Definition rem x y := if eqb y zero then x else match x, y with - | Pos nx, Pos ny => Pos (N.modulo nx ny) - | Pos nx, Neg ny => Pos (N.modulo nx ny) - | Neg nx, Pos ny => Neg (N.modulo nx ny) - | Neg nx, Neg ny => Neg (N.modulo nx ny) + | Pos nx, Pos ny => Pos (NN.modulo nx ny) + | Pos nx, Neg ny => Pos (NN.modulo nx ny) + | Neg nx, Pos ny => Neg (NN.modulo nx ny) + | Neg nx, Neg ny => Neg (NN.modulo nx ny) end. Lemma spec_quot : forall x y, to_Z (quot x y) = (to_Z x) ÷ (to_Z y). Proof. - intros [x|x] [y|y]; simpl; symmetry; rewrite N.spec_div; + intros [x|x] [y|y]; simpl; symmetry; rewrite NN.spec_div; (* Nota: we rely here on [forall a b, a ÷ 0 = b / 0] *) - destruct (Z.eq_dec (N.to_Z y) 0) as [EQ|NEQ]; - try (rewrite EQ; now destruct (N.to_Z x)); + destruct (Z.eq_dec (NN.to_Z y) 0) as [EQ|NEQ]; + try (rewrite EQ; now destruct (NN.to_Z x)); rewrite ?Z.quot_opp_r, ?Z.quot_opp_l, ?Z.opp_involutive, ?Z.opp_inj_wd; trivial; apply Z.quot_div_nonneg; - generalize (N.spec_pos x) (N.spec_pos y); Z.order. + generalize (NN.spec_pos x) (NN.spec_pos y); Z.order. Qed. Lemma spec_rem : forall x y, @@ -521,26 +521,26 @@ Module Make (N:NType) <: ZType. rewrite Hy. now destruct (to_Z x). destruct x as [x|x], y as [y|y]; simpl in *; symmetry; rewrite ?Z.eq_opp_l, ?Z.opp_0 in Hy; - rewrite N.spec_modulo, ?Z.rem_opp_r, ?Z.rem_opp_l, ?Z.opp_involutive, + rewrite NN.spec_modulo, ?Z.rem_opp_r, ?Z.rem_opp_l, ?Z.opp_involutive, ?Z.opp_inj_wd; trivial; apply Z.rem_mod_nonneg; - generalize (N.spec_pos x) (N.spec_pos y); Z.order. + generalize (NN.spec_pos x) (NN.spec_pos y); Z.order. Qed. Definition gcd x y := match x, y with - | Pos nx, Pos ny => Pos (N.gcd nx ny) - | Pos nx, Neg ny => Pos (N.gcd nx ny) - | Neg nx, Pos ny => Pos (N.gcd nx ny) - | Neg nx, Neg ny => Pos (N.gcd nx ny) + | Pos nx, Pos ny => Pos (NN.gcd nx ny) + | Pos nx, Neg ny => Pos (NN.gcd nx ny) + | Neg nx, Pos ny => Pos (NN.gcd nx ny) + | Neg nx, Neg ny => Pos (NN.gcd nx ny) end. - Theorem spec_gcd: forall a b, to_Z (gcd a b) = Zgcd (to_Z a) (to_Z b). + Theorem spec_gcd: forall a b, to_Z (gcd a b) = Z.gcd (to_Z a) (to_Z b). Proof. - unfold gcd, Zgcd, to_Z; intros [x | x] [y | y]; rewrite N.spec_gcd; unfold Zgcd; - auto; case N.to_Z; simpl; auto with zarith; - try rewrite Zabs_Zopp; auto; - case N.to_Z; simpl; auto with zarith. + unfold gcd, Z.gcd, to_Z; intros [x | x] [y | y]; rewrite NN.spec_gcd; unfold Z.gcd; + auto; case NN.to_Z; simpl; auto with zarith; + try rewrite Z.abs_opp; auto; + case NN.to_Z; simpl; auto with zarith. Qed. Definition sgn x := @@ -550,124 +550,124 @@ Module Make (N:NType) <: ZType. | Gt => minus_one end. - Lemma spec_sgn : forall x, to_Z (sgn x) = Zsgn (to_Z x). + Lemma spec_sgn : forall x, to_Z (sgn x) = Z.sgn (to_Z x). Proof. - intros. unfold sgn. rewrite spec_compare. case Zcompare_spec. + intros. unfold sgn. rewrite spec_compare. case Z.compare_spec. rewrite spec_0. intros <-; auto. - rewrite spec_0, spec_1. symmetry. rewrite Zsgn_pos; auto. - rewrite spec_0, spec_m1. symmetry. rewrite Zsgn_neg; auto with zarith. + rewrite spec_0, spec_1. symmetry. rewrite Z.sgn_pos_iff; auto. + rewrite spec_0, spec_m1. symmetry. rewrite Z.sgn_neg_iff; auto with zarith. Qed. Definition even z := match z with - | Pos n => N.even n - | Neg n => N.even n + | Pos n => NN.even n + | Neg n => NN.even n end. Definition odd z := match z with - | Pos n => N.odd n - | Neg n => N.odd n + | Pos n => NN.odd n + | Neg n => NN.odd n end. - Lemma spec_even : forall z, even z = Zeven_bool (to_Z z). + Lemma spec_even : forall z, even z = Z.even (to_Z z). Proof. - intros [n|n]; simpl; rewrite N.spec_even; trivial. - destruct (N.to_Z n) as [|p|p]; now try destruct p. + intros [n|n]; simpl; rewrite NN.spec_even; trivial. + destruct (NN.to_Z n) as [|p|p]; now try destruct p. Qed. - Lemma spec_odd : forall z, odd z = Zodd_bool (to_Z z). + Lemma spec_odd : forall z, odd z = Z.odd (to_Z z). Proof. - intros [n|n]; simpl; rewrite N.spec_odd; trivial. - destruct (N.to_Z n) as [|p|p]; now try destruct p. + intros [n|n]; simpl; rewrite NN.spec_odd; trivial. + destruct (NN.to_Z n) as [|p|p]; now try destruct p. Qed. Definition norm_pos z := match z with | Pos _ => z - | Neg n => if N.eqb n N.zero then Pos n else z + | Neg n => if NN.eqb n NN.zero then Pos n else z end. Definition testbit a n := match norm_pos n, norm_pos a with - | Pos p, Pos a => N.testbit a p - | Pos p, Neg a => negb (N.testbit (N.pred a) p) + | Pos p, Pos a => NN.testbit a p + | Pos p, Neg a => negb (NN.testbit (NN.pred a) p) | Neg p, _ => false end. Definition shiftl a n := match norm_pos a, n with - | Pos a, Pos n => Pos (N.shiftl a n) - | Pos a, Neg n => Pos (N.shiftr a n) - | Neg a, Pos n => Neg (N.shiftl a n) - | Neg a, Neg n => Neg (N.succ (N.shiftr (N.pred a) n)) + | Pos a, Pos n => Pos (NN.shiftl a n) + | Pos a, Neg n => Pos (NN.shiftr a n) + | Neg a, Pos n => Neg (NN.shiftl a n) + | Neg a, Neg n => Neg (NN.succ (NN.shiftr (NN.pred a) n)) end. Definition shiftr a n := shiftl a (opp n). Definition lor a b := match norm_pos a, norm_pos b with - | Pos a, Pos b => Pos (N.lor a b) - | Neg a, Pos b => Neg (N.succ (N.ldiff (N.pred a) b)) - | Pos a, Neg b => Neg (N.succ (N.ldiff (N.pred b) a)) - | Neg a, Neg b => Neg (N.succ (N.land (N.pred a) (N.pred b))) + | Pos a, Pos b => Pos (NN.lor a b) + | Neg a, Pos b => Neg (NN.succ (NN.ldiff (NN.pred a) b)) + | Pos a, Neg b => Neg (NN.succ (NN.ldiff (NN.pred b) a)) + | Neg a, Neg b => Neg (NN.succ (NN.land (NN.pred a) (NN.pred b))) end. Definition land a b := match norm_pos a, norm_pos b with - | Pos a, Pos b => Pos (N.land a b) - | Neg a, Pos b => Pos (N.ldiff b (N.pred a)) - | Pos a, Neg b => Pos (N.ldiff a (N.pred b)) - | Neg a, Neg b => Neg (N.succ (N.lor (N.pred a) (N.pred b))) + | Pos a, Pos b => Pos (NN.land a b) + | Neg a, Pos b => Pos (NN.ldiff b (NN.pred a)) + | Pos a, Neg b => Pos (NN.ldiff a (NN.pred b)) + | Neg a, Neg b => Neg (NN.succ (NN.lor (NN.pred a) (NN.pred b))) end. Definition ldiff a b := match norm_pos a, norm_pos b with - | Pos a, Pos b => Pos (N.ldiff a b) - | Neg a, Pos b => Neg (N.succ (N.lor (N.pred a) b)) - | Pos a, Neg b => Pos (N.land a (N.pred b)) - | Neg a, Neg b => Pos (N.ldiff (N.pred b) (N.pred a)) + | Pos a, Pos b => Pos (NN.ldiff a b) + | Neg a, Pos b => Neg (NN.succ (NN.lor (NN.pred a) b)) + | Pos a, Neg b => Pos (NN.land a (NN.pred b)) + | Neg a, Neg b => Pos (NN.ldiff (NN.pred b) (NN.pred a)) end. Definition lxor a b := match norm_pos a, norm_pos b with - | Pos a, Pos b => Pos (N.lxor a b) - | Neg a, Pos b => Neg (N.succ (N.lxor (N.pred a) b)) - | Pos a, Neg b => Neg (N.succ (N.lxor a (N.pred b))) - | Neg a, Neg b => Pos (N.lxor (N.pred a) (N.pred b)) + | Pos a, Pos b => Pos (NN.lxor a b) + | Neg a, Pos b => Neg (NN.succ (NN.lxor (NN.pred a) b)) + | Pos a, Neg b => Neg (NN.succ (NN.lxor a (NN.pred b))) + | Neg a, Neg b => Pos (NN.lxor (NN.pred a) (NN.pred b)) end. Definition div2 x := shiftr x one. Lemma Zlnot_alt1 : forall x, -(x+1) = Z.lnot x. Proof. - unfold Z.lnot, Zpred; auto with zarith. + unfold Z.lnot, Z.pred; auto with zarith. Qed. Lemma Zlnot_alt2 : forall x, Z.lnot (x-1) = -x. Proof. - unfold Z.lnot, Zpred; auto with zarith. + unfold Z.lnot, Z.pred; auto with zarith. Qed. Lemma Zlnot_alt3 : forall x, Z.lnot (-x) = x-1. Proof. - unfold Z.lnot, Zpred; auto with zarith. + unfold Z.lnot, Z.pred; auto with zarith. Qed. Lemma spec_norm_pos : forall x, to_Z (norm_pos x) = to_Z x. Proof. intros [x|x]; simpl; trivial. - rewrite N.spec_eqb, N.spec_0. + rewrite NN.spec_eqb, NN.spec_0. case Z.eqb_spec; simpl; auto with zarith. Qed. Lemma spec_norm_pos_pos : forall x y, norm_pos x = Neg y -> - 0 < N.to_Z y. + 0 < NN.to_Z y. Proof. intros [x|x] y; simpl; try easy. - rewrite N.spec_eqb, N.spec_0. + rewrite NN.spec_eqb, NN.spec_0. case Z.eqb_spec; simpl; try easy. - inversion 2. subst. generalize (N.spec_pos y); auto with zarith. + inversion 2. subst. generalize (NN.spec_pos y); auto with zarith. Qed. Ltac destr_norm_pos x := @@ -682,9 +682,9 @@ Module Make (N:NType) <: ZType. Proof. intros x p. unfold testbit. destr_norm_pos p; simpl. destr_norm_pos x; simpl. - apply N.spec_testbit. - rewrite N.spec_testbit, N.spec_pred, Zmax_r by auto with zarith. - symmetry. apply Z.bits_opp. apply N.spec_pos. + apply NN.spec_testbit. + rewrite NN.spec_testbit, NN.spec_pred, Z.max_r by auto with zarith. + symmetry. apply Z.bits_opp. apply NN.spec_pos. symmetry. apply Z.testbit_neg_r; auto with zarith. Qed. @@ -692,13 +692,13 @@ Module Make (N:NType) <: ZType. Proof. intros x p. unfold shiftl. destr_norm_pos x; destruct p as [p|p]; simpl; - assert (Hp := N.spec_pos p). - apply N.spec_shiftl. - rewrite Z.shiftl_opp_r. apply N.spec_shiftr. - rewrite !N.spec_shiftl. - rewrite !Z.shiftl_mul_pow2 by apply N.spec_pos. - apply Zopp_mult_distr_l. - rewrite Z.shiftl_opp_r, N.spec_succ, N.spec_shiftr, N.spec_pred, Zmax_r + assert (Hp := NN.spec_pos p). + apply NN.spec_shiftl. + rewrite Z.shiftl_opp_r. apply NN.spec_shiftr. + rewrite !NN.spec_shiftl. + rewrite !Z.shiftl_mul_pow2 by apply NN.spec_pos. + symmetry. apply Z.mul_opp_l. + rewrite Z.shiftl_opp_r, NN.spec_succ, NN.spec_shiftr, NN.spec_pred, Z.max_r by auto with zarith. now rewrite Zlnot_alt1, Z.lnot_shiftr, Zlnot_alt2. Qed. @@ -713,8 +713,8 @@ Module Make (N:NType) <: ZType. Proof. intros x y. unfold land. destr_norm_pos x; destr_norm_pos y; simpl; - rewrite ?N.spec_succ, ?N.spec_land, ?N.spec_ldiff, ?N.spec_lor, - ?N.spec_pred, ?Zmax_r, ?Zlnot_alt1; auto with zarith. + rewrite ?NN.spec_succ, ?NN.spec_land, ?NN.spec_ldiff, ?NN.spec_lor, + ?NN.spec_pred, ?Z.max_r, ?Zlnot_alt1; auto with zarith. now rewrite Z.ldiff_land, Zlnot_alt2. now rewrite Z.ldiff_land, Z.land_comm, Zlnot_alt2. now rewrite Z.lnot_lor, !Zlnot_alt2. @@ -724,8 +724,8 @@ Module Make (N:NType) <: ZType. Proof. intros x y. unfold lor. destr_norm_pos x; destr_norm_pos y; simpl; - rewrite ?N.spec_succ, ?N.spec_land, ?N.spec_ldiff, ?N.spec_lor, - ?N.spec_pred, ?Zmax_r, ?Zlnot_alt1; auto with zarith. + rewrite ?NN.spec_succ, ?NN.spec_land, ?NN.spec_ldiff, ?NN.spec_lor, + ?NN.spec_pred, ?Z.max_r, ?Zlnot_alt1; auto with zarith. now rewrite Z.lnot_ldiff, Z.lor_comm, Zlnot_alt2. now rewrite Z.lnot_ldiff, Zlnot_alt2. now rewrite Z.lnot_land, !Zlnot_alt2. @@ -735,8 +735,8 @@ Module Make (N:NType) <: ZType. Proof. intros x y. unfold ldiff. destr_norm_pos x; destr_norm_pos y; simpl; - rewrite ?N.spec_succ, ?N.spec_land, ?N.spec_ldiff, ?N.spec_lor, - ?N.spec_pred, ?Zmax_r, ?Zlnot_alt1; auto with zarith. + rewrite ?NN.spec_succ, ?NN.spec_land, ?NN.spec_ldiff, ?NN.spec_lor, + ?NN.spec_pred, ?Z.max_r, ?Zlnot_alt1; auto with zarith. now rewrite Z.ldiff_land, Zlnot_alt3. now rewrite Z.lnot_lor, Z.ldiff_land, <- Zlnot_alt2. now rewrite 2 Z.ldiff_land, Zlnot_alt2, Z.land_comm, Zlnot_alt3. @@ -746,7 +746,7 @@ Module Make (N:NType) <: ZType. Proof. intros x y. unfold lxor. destr_norm_pos x; destr_norm_pos y; simpl; - rewrite ?N.spec_succ, ?N.spec_lxor, ?N.spec_pred, ?Zmax_r, ?Zlnot_alt1; + rewrite ?NN.spec_succ, ?NN.spec_lxor, ?NN.spec_pred, ?Z.max_r, ?Zlnot_alt1; auto with zarith. now rewrite !Z.lnot_lxor_r, Zlnot_alt2. now rewrite !Z.lnot_lxor_l, Zlnot_alt2. diff --git a/theories/Numbers/Integer/Binary/ZBinary.v b/theories/Numbers/Integer/Binary/ZBinary.v index d7c0abd8a..a756eb256 100644 --- a/theories/Numbers/Integer/Binary/ZBinary.v +++ b/theories/Numbers/Integer/Binary/ZBinary.v @@ -36,7 +36,7 @@ End TestOrder. (** Z forms a ring *) -(*Lemma Zring : ring_theory 0 1 NZadd NZmul NZsub Zopp NZeq. +(*Lemma Zring : ring_theory 0 1 NZadd NZmul NZsub Z.opp NZeq. Proof. constructor. exact Zadd_0_l. diff --git a/theories/Numbers/Integer/NatPairs/ZNatPairs.v b/theories/Numbers/Integer/NatPairs/ZNatPairs.v index dbcc1961e..6abc78164 100644 --- a/theories/Numbers/Integer/NatPairs/ZNatPairs.v +++ b/theories/Numbers/Integer/NatPairs/ZNatPairs.v @@ -59,7 +59,7 @@ Definition le (n m : t) := n#1 + m#2 <= m#1 + n#2. Definition min (n m : t) : t := (min (n#1 + m#2) (m#1 + n#2), n#2 + m#2). Definition max (n m : t) : t := (max (n#1 + m#2) (m#1 + n#2), n#2 + m#2). -(** NB : We do not have [Zpred (Zsucc n) = n] but only [Zpred (Zsucc n) == n]. +(** NB : We do not have [Z.pred (Z.succ n) = n] but only [Z.pred (Z.succ n) == n]. It could be possible to consider as canonical only pairs where one of the elements is 0, and make all operations convert canonical values into other canonical values. In that case, we diff --git a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v index bfbc063ce..5d3836eff 100644 --- a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v +++ b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v @@ -93,7 +93,7 @@ replace z with (-(-z))%Z in * by (auto with zarith). remember (-z)%Z as z'. pattern z'; apply natlike_ind. apply B0. -intros; rewrite Zopp_succ; unfold Zpred; apply BP; auto. +intros; rewrite Z.opp_succ; unfold Z.pred; apply BP; auto. subst z'; auto with zarith. Qed. @@ -364,7 +364,7 @@ Program Instance mod_wd : Proper (eq==>eq==>eq) modulo. Theorem div_mod : forall a b, ~b==0 -> a == b*(div a b) + (modulo a b). Proof. -intros a b. zify. intros. apply Z_div_mod_eq_full; auto. +intros a b. zify. intros. apply Z.div_mod; auto. Qed. Theorem mod_pos_bound : diff --git a/theories/Numbers/NatInt/NZAxioms.v b/theories/Numbers/NatInt/NZAxioms.v index fcd98787f..ef68c3bca 100644 --- a/theories/Numbers/NatInt/NZAxioms.v +++ b/theories/Numbers/NatInt/NZAxioms.v @@ -43,7 +43,7 @@ End IsNZDomain. (** Axiomatization of some more constants Simply denoting "1" for (S 0) and so on works ok when implementing - by nat, but leaves some (Nsucc N0) when implementing by N. + by nat, but leaves some (N.succ N0) when implementing by N. *) Module Type OneTwo (Import T:Typ). diff --git a/theories/Numbers/NatInt/NZMulOrder.v b/theories/Numbers/NatInt/NZMulOrder.v index 97306f934..f6de98560 100644 --- a/theories/Numbers/NatInt/NZMulOrder.v +++ b/theories/Numbers/NatInt/NZMulOrder.v @@ -271,9 +271,9 @@ Definition mul_eq_0 := eq_mul_0. Definition mul_eq_0_l := eq_mul_0_l. Definition mul_eq_0_r := eq_mul_0_r. -Theorem lt_0_mul : forall n m, 0 < n * m <-> (0 < n /\ 0 < m) \/ (m < 0 /\ n < 0). +Theorem lt_0_mul n m : 0 < n * m <-> (0 < n /\ 0 < m) \/ (m < 0 /\ n < 0). Proof. -intros n m; split; [intro H | intros [[H1 H2] | [H1 H2]]]. +split; [intro H | intros [[H1 H2] | [H1 H2]]]. destruct (lt_trichotomy n 0) as [H1 | [H1 | H1]]; [| rewrite H1 in H; rewrite mul_0_l in H; false_hyp H lt_irrefl |]; (destruct (lt_trichotomy m 0) as [H2 | [H2 | H2]]; diff --git a/theories/Numbers/Natural/Abstract/NBits.v b/theories/Numbers/Natural/Abstract/NBits.v index c66f003ec..777e52eb2 100644 --- a/theories/Numbers/Natural/Abstract/NBits.v +++ b/theories/Numbers/Natural/Abstract/NBits.v @@ -236,7 +236,7 @@ Proof. Qed. (** Hence the number of bits of [a] is [1+log2 a] - (see [Psize] and [Psize_pos]). + (see [Pos.size_nat] and [Pos.size]). *) (** Testing bits after division or multiplication by a power of two *) diff --git a/theories/Numbers/Natural/BigN/BigN.v b/theories/Numbers/Natural/BigN/BigN.v index 7f205b388..10eb3c208 100644 --- a/theories/Numbers/Natural/BigN/BigN.v +++ b/theories/Numbers/Natural/BigN/BigN.v @@ -119,7 +119,7 @@ BigN.zify. auto with zarith. intros NEQ. generalize (BigN.spec_div_eucl a b). generalize (Z_div_mod_full [a] [b] NEQ). -destruct BigN.div_eucl as (q,r), Zdiv_eucl as (q',r'). +destruct BigN.div_eucl as (q,r), Z.div_eucl as (q',r'). intros (EQ,_). injection 1. intros EQr EQq. BigN.zify. rewrite EQr, EQq; auto. Qed. diff --git a/theories/Numbers/Natural/BigN/NMake.v b/theories/Numbers/Natural/BigN/NMake.v index 952f61833..020706290 100644 --- a/theories/Numbers/Natural/BigN/NMake.v +++ b/theories/Numbers/Natural/BigN/NMake.v @@ -65,8 +65,8 @@ Module Make (W0:CyclicType) <: NType. intros. change (Zpos (ZnZ.digits (dom_op n)) <= Zpos (ZnZ.digits (dom_op m))). rewrite !digits_dom_op, !Pshiftl_nat_Zpower. - apply Zmult_le_compat_l; auto with zarith. - apply Zpower_le_monotone2; auto with zarith. + apply Z.mul_le_mono_nonneg_l; auto with zarith. + apply Z.pow_le_mono_r; auto with zarith. Qed. Definition to_N (x : t) := Z.to_N (to_Z x). @@ -186,12 +186,12 @@ Module Make (W0:CyclicType) <: NType. exact spec_0. Qed. - Lemma spec_pred : forall x, [pred x] = Zmax 0 ([x]-1). + Lemma spec_pred x : [pred x] = Z.max 0 ([x]-1). Proof. - intros. destruct (Zle_lt_or_eq _ _ (spec_pos x)). - rewrite Zmax_r; auto with zarith. - apply spec_pred_pos; auto. - rewrite <- H; apply spec_pred0; auto. + rewrite Z.max_comm. + destruct (Z.max_spec ([x]-1) 0) as [(H,->)|(H,->)]. + - apply spec_pred0; generalize (spec_pos x); auto with zarith. + - apply spec_pred_pos; auto with zarith. Qed. (** * Subtraction *) @@ -230,11 +230,11 @@ Module Make (W0:CyclicType) <: NType. exact spec_0. Qed. - Lemma spec_sub : forall x y, [sub x y] = Zmax 0 ([x]-[y]). + Lemma spec_sub : forall x y, [sub x y] = Z.max 0 ([x]-[y]). Proof. - intros. destruct (Zle_or_lt [y] [x]). - rewrite Zmax_r; auto with zarith. apply spec_sub_pos; auto. - rewrite Zmax_l; auto with zarith. apply spec_sub0; auto. + intros. destruct (Z.le_gt_cases [y] [x]). + rewrite Z.max_r; auto with zarith. apply spec_sub_pos; auto. + rewrite Z.max_l; auto with zarith. apply spec_sub0; auto. Qed. (** * Comparison *) @@ -249,7 +249,7 @@ Module Make (W0:CyclicType) <: NType. Let spec_comparen_m: forall n m (x : word (dom_t n) (S m)) (y : dom_t n), - comparen_m n m x y = Zcompare (eval n (S m) x) (ZnZ.to_Z y). + comparen_m n m x y = Z.compare (eval n (S m) x) (ZnZ.to_Z y). Proof. intros n m x y. unfold comparen_m, eval. @@ -287,10 +287,8 @@ Module Make (W0:CyclicType) <: NType. lazy beta iota delta [iter_sym dom_op dom_t comparen_m]. reflexivity. Qed. -(** TODO: no need for ZnZ.Spec_rect , Spec_ind, and so on... *) - Theorem spec_compare : forall x y, - compare x y = Zcompare [x] [y]. + compare x y = Z.compare [x] [y]. Proof. intros x y. rewrite compare_fold. apply spec_iter_sym; clear x y. intros. apply ZnZ.spec_compare. @@ -298,7 +296,7 @@ Module Make (W0:CyclicType) <: NType. intros n m x y; unfold comparenm. rewrite (spec_cast_l n m x), (spec_cast_r n m y). unfold to_Z; apply ZnZ.spec_compare. - intros. subst. apply Zcompare_antisym. + intros. subst. now rewrite <- Z.compare_antisym. Qed. Definition eqb (x y : t) : bool := @@ -346,14 +344,14 @@ Module Make (W0:CyclicType) <: NType. Definition min (n m : t) : t := match compare n m with Gt => m | _ => n end. Definition max (n m : t) : t := match compare n m with Lt => m | _ => n end. - Theorem spec_max : forall n m, [max n m] = Zmax [n] [m]. + Theorem spec_max : forall n m, [max n m] = Z.max [n] [m]. Proof. - intros. unfold max, Zmax. rewrite spec_compare; destruct Zcompare; reflexivity. + intros. unfold max, Z.max. rewrite spec_compare; destruct Z.compare; reflexivity. Qed. - Theorem spec_min : forall n m, [min n m] = Zmin [n] [m]. + Theorem spec_min : forall n m, [min n m] = Z.min [n] [m]. Proof. - intros. unfold min, Zmin. rewrite spec_compare; destruct Zcompare; reflexivity. + intros. unfold min, Z.min. rewrite spec_compare; destruct Z.compare; reflexivity. Qed. (** * Multiplication *) @@ -437,7 +435,7 @@ Module Make (W0:CyclicType) <: NType. intros; unfold wn_mul. generalize (spec_mul_add_n1 n m x y ZnZ.zero). case DoubleMul.double_mul_add_n1; intros q r Hqr. - rewrite ZnZ.spec_0, Zplus_0_r in Hqr. rewrite <- Hqr. + rewrite ZnZ.spec_0, Z.add_0_r in Hqr. rewrite <- Hqr. generalize (ZnZ.spec_eq0 q); case ZnZ.eq0; intros HH. rewrite HH; auto. simpl. apply spec_mk_t_w'. clear. @@ -458,7 +456,7 @@ Module Make (W0:CyclicType) <: NType. intros n m x y; unfold mulnm. rewrite spec_reduce_n. rewrite (spec_cast_l n m x), (spec_cast_r n m y). apply spec_muln. - intros. rewrite Zmult_comm; auto. + intros. rewrite Z.mul_comm; auto. Qed. (** * Division by a smaller number *) @@ -519,7 +517,7 @@ Module Make (W0:CyclicType) <: NType. apply DoubleBase.spec_get_low. apply spec_zeron. exact ZnZ.spec_to_Z. - apply Zle_lt_trans with (ZnZ.to_Z y); auto. + apply Z.le_lt_trans with (ZnZ.to_Z y); auto. rewrite <- nmake_double; auto. case (ZnZ.spec_to_Z y); auto. Qed. @@ -580,9 +578,9 @@ Module Make (W0:CyclicType) <: NType. intros x y H1 H2; generalize (spec_div_gt_aux x y H1 H2); case div_gt. intros q r (H3, H4); split. apply (Zdiv_unique [x] [y] [q] [r]); auto. - rewrite Zmult_comm; auto. + rewrite Z.mul_comm; auto. apply (Zmod_unique [x] [y] [q] [r]); auto. - rewrite Zmult_comm; auto. + rewrite Z.mul_comm; auto. Qed. (** * General Division *) @@ -597,7 +595,7 @@ Module Make (W0:CyclicType) <: NType. Theorem spec_div_eucl: forall x y, let (q,r) := div_eucl x y in - ([q], [r]) = Zdiv_eucl [x] [y]. + ([q], [r]) = Z.div_eucl [x] [y]. Proof. intros x y. unfold div_eucl. rewrite spec_eqb, spec_compare, spec_0. @@ -606,16 +604,16 @@ Module Make (W0:CyclicType) <: NType. intros H'. assert (H : 0 < [y]) by (generalize (spec_pos y); auto with zarith). clear H'. - case Zcompare_spec; intros Cmp; + case Z.compare_spec; intros Cmp; rewrite ?spec_0, ?spec_1; intros; auto with zarith. - rewrite Cmp; generalize (Z_div_same [y] (Zlt_gt _ _ H)) - (Z_mod_same [y] (Zlt_gt _ _ H)); - unfold Zdiv, Zmod; case Zdiv_eucl; intros; subst; auto. + rewrite Cmp; generalize (Z_div_same [y] (Z.lt_gt _ _ H)) + (Z_mod_same [y] (Z.lt_gt _ _ H)); + unfold Z.div, Z.modulo; case Z.div_eucl; intros; subst; auto. assert (LeLt: 0 <= [x] < [y]) by (generalize (spec_pos x); auto). generalize (Zdiv_small _ _ LeLt) (Zmod_small _ _ LeLt); - unfold Zdiv, Zmod; case Zdiv_eucl; intros; subst; auto. - generalize (spec_div_gt _ _ (Zlt_gt _ _ Cmp) H); auto. - unfold Zdiv, Zmod; case Zdiv_eucl; case div_gt. + unfold Z.div, Z.modulo; case Z.div_eucl; intros; subst; auto. + generalize (spec_div_gt _ _ (Z.lt_gt _ _ Cmp) H); auto. + unfold Z.div, Z.modulo; case Z.div_eucl; case div_gt. intros a b c d (H1, H2); subst; auto. Qed. @@ -626,7 +624,7 @@ Module Make (W0:CyclicType) <: NType. Proof. intros x y; unfold div; generalize (spec_div_eucl x y); case div_eucl; simpl fst. - intros xx yy; unfold Zdiv; case Zdiv_eucl; intros qq rr H; + intros xx yy; unfold Z.div; case Z.div_eucl; intros qq rr H; injection H; auto. Qed. @@ -730,10 +728,10 @@ Module Make (W0:CyclicType) <: NType. intro H'. assert (H : 0 < [y]) by (generalize (spec_pos y); auto with zarith). clear H'. - case Zcompare_spec; + case Z.compare_spec; rewrite ?spec_0, ?spec_1; intros; try split; auto with zarith. - rewrite H0; apply sym_equal; apply Z_mod_same; auto with zarith. - apply sym_equal; apply Zmod_small; auto with zarith. + rewrite H0; symmetry; apply Z_mod_same; auto with zarith. + symmetry; apply Zmod_small; auto with zarith. generalize (spec_pos x); auto with zarith. apply spec_mod_gt; auto with zarith. Qed. @@ -775,7 +773,7 @@ Module Make (W0:CyclicType) <: NType. Proof. intros x. symmetry. apply Z.sqrt_unique. - rewrite <- ! Zpower_2. apply spec_sqrt_aux. + rewrite <- ! Z.pow_2_r. apply spec_sqrt_aux. Qed. (** * Power *) @@ -791,14 +789,14 @@ Module Make (W0:CyclicType) <: NType. Proof. intros x n; generalize x; elim n; clear n x; simpl pow_pos. intros; rewrite spec_mul; rewrite spec_square; rewrite H. - rewrite Zpos_xI; rewrite Zpower_exp; auto with zarith. - rewrite (Zmult_comm 2); rewrite Zpower_mult; auto with zarith. - rewrite Zpower_2; rewrite Zpower_1_r; auto. + rewrite Pos2Z.inj_xI; rewrite Zpower_exp; auto with zarith. + rewrite (Z.mul_comm 2); rewrite Z.pow_mul_r; auto with zarith. + rewrite Z.pow_2_r; rewrite Z.pow_1_r; auto. intros; rewrite spec_square; rewrite H. - rewrite Zpos_xO; auto with zarith. - rewrite (Zmult_comm 2); rewrite Zpower_mult; auto with zarith. - rewrite Zpower_2; auto. - intros; rewrite Zpower_1_r; auto. + rewrite Pos2Z.inj_xO; auto with zarith. + rewrite (Z.mul_comm 2); rewrite Z.pow_mul_r; auto with zarith. + rewrite Z.pow_2_r; auto. + intros; rewrite Z.pow_1_r; auto. Qed. Definition pow_N (x:t)(n:N) : t := match n with @@ -806,7 +804,7 @@ Module Make (W0:CyclicType) <: NType. | BinNat.Npos p => pow_pos x p end. - Theorem spec_pow_N: forall x n, [pow_N x n] = [x] ^ Z_of_N n. + Theorem spec_pow_N: forall x n, [pow_N x n] = [x] ^ Z.of_N n. Proof. destruct n; simpl. apply spec_1. apply spec_pow_pos. @@ -867,15 +865,15 @@ Module Make (W0:CyclicType) <: NType. Zis_gcd [a] [b] [gcd_gt_body a b cont]. Proof. intros a b cont p H2 H3 H4; unfold gcd_gt_body. - rewrite ! spec_compare, spec_0. case Zcompare_spec. + rewrite ! spec_compare, spec_0. case Z.compare_spec. intros ->; apply Zis_gcd_0. intros HH; absurd (0 <= [b]); auto with zarith. case (spec_digits b); auto with zarith. - intros H5; case Zcompare_spec. - intros H6; rewrite <- (Zmult_1_r [b]). + intros H5; case Z.compare_spec. + intros H6; rewrite <- (Z.mul_1_r [b]). rewrite (Z_div_mod_eq [a] [b]); auto with zarith. rewrite <- spec_mod_gt; auto with zarith. - rewrite H6; rewrite Zplus_0_r. + rewrite H6; rewrite Z.add_0_r. apply Zis_gcd_mult; apply Zis_gcd_1. intros; apply False_ind. case (spec_digits (mod_gt a b)); auto with zarith. @@ -890,24 +888,19 @@ Module Make (W0:CyclicType) <: NType. rewrite <- spec_mod_gt; auto with zarith. repeat rewrite <- spec_mod_gt; auto with zarith. apply H4; auto with zarith. - apply Zmult_lt_reg_r with 2; auto with zarith. - apply Zle_lt_trans with ([b] + [mod_gt a b]); auto with zarith. - apply Zle_lt_trans with (([a]/[b]) * [b] + [mod_gt a b]); auto with zarith. - apply Zplus_le_compat_r. - pattern [b] at 1; rewrite <- (Zmult_1_l [b]). - apply Zmult_le_compat_r; auto with zarith. - case (Zle_lt_or_eq 0 ([a]/[b])); auto with zarith. - intros HH; rewrite (Z_div_mod_eq [a] [b]) in H2; - try rewrite <- HH in H2; auto with zarith. - case (Z_mod_lt [a] [b]); auto with zarith. - rewrite Zmult_comm; rewrite spec_mod_gt; auto with zarith. - rewrite <- Z_div_mod_eq; auto with zarith. - pattern 2 at 2; rewrite <- (Zpower_1_r 2). - rewrite <- Zpower_exp; auto with zarith. - ring_simplify (p - 1 + 1); auto. - case (Zle_lt_or_eq 0 p); auto with zarith. - generalize H3; case p; simpl Zpower; auto with zarith. - intros HH; generalize H3; rewrite <- HH; simpl Zpower; auto with zarith. + apply Z.mul_lt_mono_pos_r with 2; auto with zarith. + apply Z.le_lt_trans with ([b] + [mod_gt a b]); auto with zarith. + apply Z.le_lt_trans with (([a]/[b]) * [b] + [mod_gt a b]); auto with zarith. + - apply Z.add_le_mono_r. + rewrite <- (Z.mul_1_l [b]) at 1. + apply Z.mul_le_mono_nonneg_r; auto with zarith. + change 1 with (Z.succ 0). apply Z.le_succ_l. + apply Z.div_str_pos; auto with zarith. + - rewrite Z.mul_comm; rewrite spec_mod_gt; auto with zarith. + rewrite <- Z_div_mod_eq; auto with zarith. + rewrite Z.mul_comm, <- Z.pow_succ_r, Z.sub_1_r, Z.succ_pred; auto. + apply Z.le_0_sub. change 1 with (Z.succ 0). apply Z.le_succ_l. + destruct p; simpl in H3; auto with zarith. Qed. Fixpoint gcd_gt_aux (p:positive) (cont:t->t->t) (a b:t) : t := @@ -931,7 +924,7 @@ Module Make (W0:CyclicType) <: NType. apply Hrec with (Zpos p + n); auto. replace (Zpos p + (Zpos p + n)) with (Zpos (xI p) + n - 1); auto. - rewrite Zpos_xI; ring. + rewrite Pos2Z.inj_xI; ring. intros a2 b2 H9 H10. apply Hrec with n; auto. intros p Hrec n a b cont H2 H3 H4. @@ -940,18 +933,18 @@ Module Make (W0:CyclicType) <: NType. apply Hrec with (Zpos p + n - 1); auto. replace (Zpos p + (Zpos p + n - 1)) with (Zpos (xO p) + n - 1); auto. - rewrite Zpos_xO; ring. + rewrite Pos2Z.inj_xO; ring. intros a2 b2 H9 H10. apply Hrec with (n - 1); auto. replace (Zpos p + (n - 1)) with (Zpos p + n - 1); auto with zarith. intros a3 b3 H12 H13; apply H4; auto with zarith. - apply Zlt_le_trans with (1 := H12). - apply Zpower_le_monotone2; auto with zarith. + apply Z.lt_le_trans with (1 := H12). + apply Z.pow_le_mono_r; auto with zarith. intros n a b cont H H2 H3. simpl gcd_gt_aux. apply Zspec_gcd_gt_body with (n + 1); auto with zarith. - rewrite Zplus_comm; auto. + rewrite Z.add_comm; auto. intros a1 b1 H5 H6; apply H3; auto. replace n with (n + 1 - 1); auto; try ring. Qed. @@ -965,14 +958,14 @@ Module Make (W0:CyclicType) <: NType. Definition gcd_gt a b := gcd_gt_aux (digits a) gcd_cont a b. Theorem spec_gcd_gt: forall a b, - [a] > [b] -> [gcd_gt a b] = Zgcd [a] [b]. + [a] > [b] -> [gcd_gt a b] = Z.gcd [a] [b]. Proof. intros a b H2. case (spec_digits (gcd_gt a b)); intros H3 H4. case (spec_digits a); intros H5 H6. - apply sym_equal; apply Zis_gcd_gcd; auto with zarith. + symmetry; apply Zis_gcd_gcd; auto with zarith. unfold gcd_gt; apply Zspec_gcd_gt_aux with 0; auto with zarith. - intros a1 a2; rewrite Zpower_0_r. + intros a1 a2; rewrite Z.pow_0_r. case (spec_digits a2); intros H7 H8; intros; apply False_ind; auto with zarith. Qed. @@ -984,18 +977,18 @@ Module Make (W0:CyclicType) <: NType. | Gt => gcd_gt a b end. - Theorem spec_gcd: forall a b, [gcd a b] = Zgcd [a] [b]. + Theorem spec_gcd: forall a b, [gcd a b] = Z.gcd [a] [b]. Proof. intros a b. case (spec_digits a); intros H1 H2. case (spec_digits b); intros H3 H4. - unfold gcd. rewrite spec_compare. case Zcompare_spec. - intros HH; rewrite HH; apply sym_equal; apply Zis_gcd_gcd; auto. + unfold gcd. rewrite spec_compare. case Z.compare_spec. + intros HH; rewrite HH; symmetry; apply Zis_gcd_gcd; auto. apply Zis_gcd_refl. - intros; apply trans_equal with (Zgcd [b] [a]). + intros; transitivity (Z.gcd [b] [a]). apply spec_gcd_gt; auto with zarith. apply Zis_gcd_gcd; auto with zarith. - apply Zgcd_is_pos. + apply Z.gcd_nonneg. apply Zis_gcd_sym; apply Zgcd_is_gcd. intros; apply spec_gcd_gt; auto with zarith. Qed. @@ -1017,22 +1010,22 @@ Module Make (W0:CyclicType) <: NType. exact (ZnZ.spec_is_even x). Qed. - Theorem spec_even: forall x, even x = Zeven_bool [x]. + Theorem spec_even: forall x, even x = Z.even [x]. Proof. intros x. assert (H := spec_even_aux x). symmetry. - rewrite (Z_div_mod_eq_full [x] 2); auto with zarith. - destruct (even x); rewrite H, ?Zplus_0_r. + rewrite (Z.div_mod [x] 2); auto with zarith. + destruct (even x); rewrite H, ?Z.add_0_r. rewrite Zeven_bool_iff. apply Zeven_2p. apply not_true_is_false. rewrite Zeven_bool_iff. apply Zodd_not_Zeven. apply Zodd_2p_plus_1. Qed. - Theorem spec_odd: forall x, odd x = Zodd_bool [x]. + Theorem spec_odd: forall x, odd x = Z.odd [x]. Proof. intros x. unfold odd. assert (H := spec_even_aux x). symmetry. - rewrite (Z_div_mod_eq_full [x] 2); auto with zarith. - destruct (even x); rewrite H, ?Zplus_0_r; simpl negb. + rewrite (Z.div_mod [x] 2); auto with zarith. + destruct (even x); rewrite H, ?Z.add_0_r; simpl negb. apply not_true_is_false. rewrite Zodd_bool_iff. apply Zeven_not_Zodd. apply Zeven_2p. apply Zodd_bool_iff. apply Zodd_2p_plus_1. @@ -1041,27 +1034,21 @@ Module Make (W0:CyclicType) <: NType. (** * Conversion *) Definition pheight p := - Peano.pred (nat_of_P (get_height (ZnZ.digits (dom_op 0)) (plength p))). + Peano.pred (Pos.to_nat (get_height (ZnZ.digits (dom_op 0)) (plength p))). Theorem pheight_correct: forall p, - Zpos p < 2 ^ (Zpos (ZnZ.digits (dom_op 0)) * 2 ^ (Z_of_nat (pheight p))). + Zpos p < 2 ^ (Zpos (ZnZ.digits (dom_op 0)) * 2 ^ (Z.of_nat (pheight p))). Proof. intros p; unfold pheight. - assert (F1: forall x, Z_of_nat (Peano.pred (nat_of_P x)) = Zpos x - 1). - intros x. - assert (Zsucc (Z_of_nat (Peano.pred (nat_of_P x))) = Zpos x); auto with zarith. - rewrite <- inj_S. - rewrite <- (fun x => S_pred x 0); auto with zarith. - rewrite Zpos_eq_Z_of_nat_o_nat_of_P; auto. - apply lt_le_trans with 1%nat; auto with zarith. - exact (le_Pmult_nat x 1). - rewrite F1; clear F1. + rewrite Nat2Z.inj_pred by apply Pos2Nat.is_pos. + rewrite positive_nat_Z. + rewrite <- Z.sub_1_r. assert (F2:= (get_height_correct (ZnZ.digits (dom_op 0)) (plength p))). - apply Zlt_le_trans with (Zpos (Psucc p)). - rewrite Zpos_succ_morphism; auto with zarith. - apply Zle_trans with (1 := plength_pred_correct (Psucc p)). - rewrite Ppred_succ. - apply Zpower_le_monotone2; auto with zarith. + apply Z.lt_le_trans with (Zpos (Pos.succ p)). + rewrite Pos2Z.inj_succ; auto with zarith. + apply Z.le_trans with (1 := plength_pred_correct (Pos.succ p)). + rewrite Pos.pred_succ. + apply Z.pow_le_mono_r; auto with zarith. Qed. Definition of_pos (x:positive) : t := @@ -1076,8 +1063,8 @@ Module Make (W0:CyclicType) <: NType. simpl. apply ZnZ.of_pos_correct. unfold base. - apply Zlt_le_trans with (1 := pheight_correct x). - apply Zpower_le_monotone2; auto with zarith. + apply Z.lt_le_trans with (1 := pheight_correct x). + apply Z.pow_le_mono_r; auto with zarith. rewrite (digits_dom_op (_ _)), Pshiftl_nat_Zpower. auto with zarith. Qed. @@ -1088,7 +1075,7 @@ Module Make (W0:CyclicType) <: NType. end. Theorem spec_of_N: forall x, - [of_N x] = Z_of_N x. + [of_N x] = Z.of_N x. Proof. intros x; case x. simpl of_N. exact spec_0. @@ -1122,7 +1109,7 @@ Module Make (W0:CyclicType) <: NType. intros. apply Zdiv_unique with 0; auto with zarith. change 2 with (2^1) at 2. rewrite <- Zpower_exp; auto with zarith. - rewrite Zplus_0_r. f_equal. auto with zarith. + rewrite Z.add_0_r. f_equal. auto with zarith. Qed. Theorem spec_head0: forall x, 0 < [x] -> @@ -1212,9 +1199,9 @@ Module Make (W0:CyclicType) <: NType. set (d := ZnZ.digits (dom_op n)) in *; clearbody d. destruct (Z_lt_le_dec h (Zpos d)); auto. exfalso. assert (1 * 2^Zpos d <= ZnZ.to_Z x * 2^h). - apply Zmult_le_compat; auto with zarith. - apply Zpower_le_monotone2; auto with zarith. - rewrite Zmult_comm in H0. auto with zarith. + apply Z.mul_le_mono_nonneg; auto with zarith. + apply Z.pow_le_mono_r; auto with zarith. + rewrite Z.mul_comm in H0. auto with zarith. Qed. Lemma spec_log2_pos : forall x, [x]<>0 -> @@ -1232,13 +1219,13 @@ Module Make (W0:CyclicType) <: NType. assert (H2 := ZnZ.spec_to_Z (ZnZ.zdigits (dom_op n))). assert (H3 := head0_zdigits n x). rewrite Zmod_small by auto with zarith. + rewrite Z.sub_simpl_r. rewrite (Z.mul_lt_mono_pos_l (2^(ZnZ.to_Z (ZnZ.head0 x)))); auto with zarith. rewrite (Z.mul_le_mono_pos_l _ _ (2^(ZnZ.to_Z (ZnZ.head0 x)))); auto with zarith. rewrite <- 2 Zpower_exp; auto with zarith. - rewrite Z.add_sub_assoc, Zplus_minus. - rewrite Z.sub_simpl_r, Zplus_minus. + rewrite !Z.add_sub_assoc, !Z.add_simpl_l. rewrite ZnZ.spec_zdigits. rewrite pow2_pos_minus_1 by (red; auto). apply ZnZ.spec_head0; auto with zarith. @@ -1294,12 +1281,12 @@ Module Make (W0:CyclicType) <: NType. Proof. intros x y z HH HH1 HH2. split; auto with zarith. - apply Zle_lt_trans with (2 := HH2); auto with zarith. + apply Z.le_lt_trans with (2 := HH2); auto with zarith. apply Zdiv_le_upper_bound; auto with zarith. pattern x at 1; replace x with (x * 2 ^ 0); auto with zarith. - apply Zmult_le_compat_l; auto. - apply Zpower_le_monotone2; auto with zarith. - rewrite Zpower_0_r; ring. + apply Z.mul_le_mono_nonneg_l; auto. + apply Z.pow_le_mono_r; auto with zarith. + rewrite Z.pow_0_r; ring. Qed. Theorem spec_shiftr_pow2 : forall x n, @@ -1315,7 +1302,7 @@ Module Make (W0:CyclicType) <: NType. rewrite spec_reduce. rewrite ZnZ.spec_zdigits in H. rewrite ZnZ.spec_add_mul_div by auto with zarith. - rewrite ZnZ.spec_0, Zmult_0_l, Zplus_0_l. + rewrite ZnZ.spec_0, Z.mul_0_l, Z.add_0_l. rewrite Zmod_small. f_equal. f_equal. auto with zarith. split. auto with zarith. @@ -1324,8 +1311,8 @@ Module Make (W0:CyclicType) <: NType. rewrite ZnZ.spec_0. symmetry. apply Zdiv_small. split; auto with zarith. - apply Zlt_le_trans with (base (ZnZ.digits (dom_op n))); auto with zarith. - unfold base. apply Zpower_le_monotone2; auto with zarith. + apply Z.lt_le_trans with (base (ZnZ.digits (dom_op n))); auto with zarith. + unfold base. apply Z.pow_le_mono_r; auto with zarith. rewrite ZnZ.spec_zdigits in H. generalize (ZnZ.spec_to_Z d); auto with zarith. Qed. @@ -1370,21 +1357,21 @@ Module Make (W0:CyclicType) <: NType. destruct (ZnZ.spec_to_Z x). destruct (ZnZ.spec_to_Z p). rewrite ZnZ.spec_add_mul_div by (omega with *). - rewrite ZnZ.spec_0, Zdiv_0_l, Zplus_0_r. + rewrite ZnZ.spec_0, Zdiv_0_l, Z.add_0_r. apply Zmod_small. unfold base. split; auto with zarith. - rewrite Zmult_comm. - apply Zlt_le_trans with (2^(ZnZ.to_Z p + K)). + rewrite Z.mul_comm. + apply Z.lt_le_trans with (2^(ZnZ.to_Z p + K)). rewrite Zpower_exp; auto with zarith. - apply Zmult_lt_compat_l; auto with zarith. - apply Zpower_le_monotone2; auto with zarith. + apply Z.mul_lt_mono_pos_l; auto with zarith. + apply Z.pow_le_mono_r; auto with zarith. Qed. Theorem spec_unsafe_shiftl: forall x p, [p] <= [head0 x] -> [unsafe_shiftl x p] = [x] * 2 ^ [p]. Proof. intros. - destruct (Z_eq_dec [x] 0) as [EQ|NEQ]. + destruct (Z.eq_dec [x] 0) as [EQ|NEQ]. (* [x] = 0 *) apply spec_unsafe_shiftl_aux with 0; auto with zarith. now rewrite EQ. @@ -1421,7 +1408,7 @@ Module Make (W0:CyclicType) <: NType. Proof. intros x. rewrite ! digits_level, double_size_level. rewrite 2 digits_dom_op, 2 Pshiftl_nat_Zpower, - inj_S, Zpower_Zsucc; auto with zarith. + Nat2Z.inj_succ, Z.pow_succ_r; auto with zarith. ring. Qed. @@ -1438,46 +1425,47 @@ Module Make (W0:CyclicType) <: NType. assert (F1:= spec_pos (head0 x)). assert (F2: 0 < Zpos (digits x)). red; auto. - case (Zle_lt_or_eq _ _ (spec_pos x)); intros HH. + assert (HH := spec_pos x). Z.le_elim HH. generalize HH; rewrite <- (spec_double_size x); intros HH1. case (spec_head0 x HH); intros _ HH2. case (spec_head0 _ HH1). rewrite (spec_double_size x); rewrite (spec_double_size_digits x). intros HH3 _. - case (Zle_or_lt ([head0 (double_size x)]) (2 * [head0 x])); auto; intros HH4. + case (Z.le_gt_cases ([head0 (double_size x)]) (2 * [head0 x])); auto; intros HH4. absurd (2 ^ (2 * [head0 x] )* [x] < 2 ^ [head0 (double_size x)] * [x]); auto. - apply Zle_not_lt. - apply Zmult_le_compat_r; auto with zarith. - apply Zpower_le_monotone2; auto; auto with zarith. + apply Z.le_ngt. + apply Z.mul_le_mono_nonneg_r; auto with zarith. + apply Z.pow_le_mono_r; auto; auto with zarith. assert (HH5: 2 ^[head0 x] <= 2 ^(Zpos (digits x) - 1)). - case (Zle_lt_or_eq 1 [x]); auto with zarith; intros HH5. - apply Zmult_le_reg_r with (2 ^ 1); auto with zarith. - rewrite <- (fun x y z => Zpower_exp x (y - z)); auto with zarith. - assert (tmp: forall x, x - 1 + 1 = x); [intros; ring | rewrite tmp; clear tmp]. - apply Zle_trans with (2 := Zlt_le_weak _ _ HH2). - apply Zmult_le_compat_l; auto with zarith. - rewrite Zpower_1_r; auto with zarith. - apply Zpower_le_monotone2; auto with zarith. - case (Zle_or_lt (Zpos (digits x)) [head0 x]); auto with zarith; intros HH6. - absurd (2 ^ Zpos (digits x) <= 2 ^ [head0 x] * [x]); auto with zarith. - rewrite <- HH5; rewrite Zmult_1_r. - apply Zpower_le_monotone2; auto with zarith. - rewrite (Zmult_comm 2). - rewrite Zpower_mult; auto with zarith. - rewrite Zpower_2. - apply Zlt_le_trans with (2 := HH3). - rewrite <- Zmult_assoc. + { apply Z.le_succ_l in HH. change (1 <= [x]) in HH. + Z.le_elim HH. + - apply Z.mul_le_mono_pos_r with (2 ^ 1); auto with zarith. + rewrite <- (fun x y z => Z.pow_add_r x (y - z)); auto with zarith. + rewrite Z.sub_add. + apply Z.le_trans with (2 := Z.lt_le_incl _ _ HH2). + apply Z.mul_le_mono_nonneg_l; auto with zarith. + rewrite Z.pow_1_r; auto with zarith. + - apply Z.pow_le_mono_r; auto with zarith. + case (Z.le_gt_cases (Zpos (digits x)) [head0 x]); auto with zarith; intros HH6. + absurd (2 ^ Zpos (digits x) <= 2 ^ [head0 x] * [x]); auto with zarith. + rewrite <- HH; rewrite Z.mul_1_r. + apply Z.pow_le_mono_r; auto with zarith. } + rewrite (Z.mul_comm 2). + rewrite Z.pow_mul_r; auto with zarith. + rewrite Z.pow_2_r. + apply Z.lt_le_trans with (2 := HH3). + rewrite <- Z.mul_assoc. replace (2 * Zpos (digits x) - 1) with ((Zpos (digits x) - 1) + (Zpos (digits x))). rewrite Zpower_exp; auto with zarith. apply Zmult_lt_compat2; auto with zarith. split; auto with zarith. - apply Zmult_lt_0_compat; auto with zarith. - rewrite Zpos_xO; ring. - apply Zlt_le_weak; auto. + apply Z.mul_pos_pos; auto with zarith. + rewrite Pos2Z.inj_xO; ring. + apply Z.lt_le_incl; auto. repeat rewrite spec_head00; auto. rewrite spec_double_size_digits. - rewrite Zpos_xO; auto with zarith. + rewrite Pos2Z.inj_xO; auto with zarith. rewrite spec_double_size; auto. Qed. @@ -1485,24 +1473,26 @@ Module Make (W0:CyclicType) <: NType. forall x, 0 < [head0 (double_size x)]. Proof. intros x. - assert (F: 0 < Zpos (digits x)). - red; auto. - case (Zle_lt_or_eq _ _ (spec_pos (head0 (double_size x)))); auto; intros F0. - case (Zle_lt_or_eq _ _ (spec_pos (head0 x))); intros F1. - apply Zlt_le_trans with (2 := (spec_double_size_head0 x)); auto with zarith. - case (Zle_lt_or_eq _ _ (spec_pos x)); intros F3. + assert (F := Pos2Z.is_pos (digits x)). + assert (F0 := spec_pos (head0 (double_size x))). + Z.le_elim F0; auto. + assert (F1 := spec_pos (head0 x)). + Z.le_elim F1. + apply Z.lt_le_trans with (2 := (spec_double_size_head0 x)); auto with zarith. + assert (F3 := spec_pos x). + Z.le_elim F3. generalize F3; rewrite <- (spec_double_size x); intros F4. absurd (2 ^ (Zpos (xO (digits x)) - 1) < 2 ^ (Zpos (digits x))). - apply Zle_not_lt. - apply Zpower_le_monotone2; auto with zarith. - rewrite Zpos_xO; auto with zarith. + { apply Z.le_ngt. + apply Z.pow_le_mono_r; auto with zarith. + rewrite Pos2Z.inj_xO; auto with zarith. } case (spec_head0 x F3). - rewrite <- F1; rewrite Zpower_0_r; rewrite Zmult_1_l; intros _ HH. - apply Zle_lt_trans with (2 := HH). + rewrite <- F1; rewrite Z.pow_0_r; rewrite Z.mul_1_l; intros _ HH. + apply Z.le_lt_trans with (2 := HH). case (spec_head0 _ F4). rewrite (spec_double_size x); rewrite (spec_double_size_digits x). - rewrite <- F0; rewrite Zpower_0_r; rewrite Zmult_1_l; auto. - generalize F1; rewrite (spec_head00 _ (sym_equal F3)); auto with zarith. + rewrite <- F0; rewrite Z.pow_0_r; rewrite Z.mul_1_l; auto. + generalize F1; rewrite (spec_head00 _ (eq_sym F3)); auto with zarith. Qed. (** Finally we iterate [double_size] enough before [unsafe_shiftl] @@ -1521,14 +1511,14 @@ Module Make (W0:CyclicType) <: NType. [shiftl_aux_body cont x n] = [x] * 2 ^ [n]. Proof. intros n x p cont H1 H2; unfold shiftl_aux_body. - rewrite spec_compare; case Zcompare_spec; intros H. + rewrite spec_compare; case Z.compare_spec; intros H. apply spec_unsafe_shiftl; auto with zarith. apply spec_unsafe_shiftl; auto with zarith. rewrite H2. rewrite spec_double_size; auto. - rewrite Zplus_comm; rewrite Zpower_exp; auto with zarith. - apply Zle_trans with (2 := spec_double_size_head0 x). - rewrite Zpower_1_r; apply Zmult_le_compat_l; auto with zarith. + rewrite Z.add_comm; rewrite Zpower_exp; auto with zarith. + apply Z.le_trans with (2 := spec_double_size_head0 x). + rewrite Z.pow_1_r; apply Z.mul_le_mono_nonneg_l; auto with zarith. Qed. Fixpoint shiftl_aux p cont x n := @@ -1550,27 +1540,27 @@ Module Make (W0:CyclicType) <: NType. apply spec_shiftl_aux_body with (q); auto. intros x1 H3; apply Hrec with (q + 1)%positive; auto. intros x2 H4; apply Hrec with (p + q + 1)%positive; auto. - rewrite <- Pplus_assoc. - rewrite Zpos_plus_distr; auto. + rewrite <- Pos.add_assoc. + rewrite Pos2Z.inj_add; auto. intros x3 H5; apply H2. - rewrite Zpos_xI. + rewrite Pos2Z.inj_xI. replace (2 * Zpos p + 1 + Zpos q) with (Zpos p + Zpos (p + q + 1)); auto. - repeat rewrite Zpos_plus_distr; ring. + rewrite !Pos2Z.inj_add; ring. intros p Hrec q n x cont H1 H2. apply spec_shiftl_aux_body with (q); auto. intros x1 H3; apply Hrec with (q); auto. - apply Zle_trans with (2 := H3); auto with zarith. - apply Zpower_le_monotone2; auto with zarith. + apply Z.le_trans with (2 := H3); auto with zarith. + apply Z.pow_le_mono_r; auto with zarith. intros x2 H4; apply Hrec with (p + q)%positive; auto. intros x3 H5; apply H2. - rewrite (Zpos_xO p). + rewrite (Pos2Z.inj_xO p). replace (2 * Zpos p + Zpos q) with (Zpos p + Zpos (p + q)); auto. - repeat rewrite Zpos_plus_distr; ring. + rewrite Pos2Z.inj_add; ring. intros q n x cont H1 H2. apply spec_shiftl_aux_body with (q); auto. - rewrite Zplus_comm; auto. + rewrite Z.add_comm; auto. Qed. Definition shiftl x n := @@ -1582,25 +1572,25 @@ Module Make (W0:CyclicType) <: NType. [shiftl x n] = [x] * 2 ^ [n]. Proof. intros x n; unfold shiftl, shiftl_aux_body. - rewrite spec_compare; case Zcompare_spec; intros H. + rewrite spec_compare; case Z.compare_spec; intros H. apply spec_unsafe_shiftl; auto with zarith. apply spec_unsafe_shiftl; auto with zarith. rewrite <- (spec_double_size x). - rewrite spec_compare; case Zcompare_spec; intros H1. + rewrite spec_compare; case Z.compare_spec; intros H1. apply spec_unsafe_shiftl; auto with zarith. apply spec_unsafe_shiftl; auto with zarith. rewrite <- (spec_double_size (double_size x)). apply spec_shiftl_aux with 1%positive. - apply Zle_trans with (2 := spec_double_size_head0 (double_size x)). + apply Z.le_trans with (2 := spec_double_size_head0 (double_size x)). replace (2 ^ 1) with (2 * 1). - apply Zmult_le_compat_l; auto with zarith. + apply Z.mul_le_mono_nonneg_l; auto with zarith. generalize (spec_double_size_head0_pos x); auto with zarith. - rewrite Zpower_1_r; ring. + rewrite Z.pow_1_r; ring. intros x1 H2; apply spec_unsafe_shiftl. - apply Zle_trans with (2 := H2). - apply Zle_trans with (2 ^ Zpos (digits n)); auto with zarith. + apply Z.le_trans with (2 := H2). + apply Z.le_trans with (2 ^ Zpos (digits n)); auto with zarith. case (spec_digits n); auto with zarith. - apply Zpower_le_monotone2; auto with zarith. + apply Z.pow_le_mono_r; auto with zarith. Qed. Lemma spec_shiftl: forall x p, [shiftl x p] = Z.shiftl [x] [p]. diff --git a/theories/Numbers/Natural/BigN/Nbasic.v b/theories/Numbers/Natural/BigN/Nbasic.v index 4717d0b23..756c7f9a2 100644 --- a/theories/Numbers/Natural/BigN/Nbasic.v +++ b/theories/Numbers/Natural/BigN/Nbasic.v @@ -32,7 +32,7 @@ Proof. transitivity (2 * (2 ^ Z.of_nat n * Zpos p)). rewrite <- IHn. auto. rewrite Z.mul_assoc. - rewrite inj_S. + rewrite Nat2Z.inj_succ. rewrite <- Z.pow_succ_r; auto with zarith. Qed. @@ -41,39 +41,39 @@ Qed. Fixpoint plength (p: positive) : positive := match p with xH => xH - | xO p1 => Psucc (plength p1) - | xI p1 => Psucc (plength p1) + | xO p1 => Pos.succ (plength p1) + | xI p1 => Pos.succ (plength p1) end. Theorem plength_correct: forall p, (Zpos p < 2 ^ Zpos (plength p))%Z. -assert (F: (forall p, 2 ^ (Zpos (Psucc p)) = 2 * 2 ^ Zpos p)%Z). -intros p; replace (Zpos (Psucc p)) with (1 + Zpos p)%Z. +assert (F: (forall p, 2 ^ (Zpos (Pos.succ p)) = 2 * 2 ^ Zpos p)%Z). +intros p; replace (Zpos (Pos.succ p)) with (1 + Zpos p)%Z. rewrite Zpower_exp; auto with zarith. -rewrite Zpos_succ_morphism; unfold Zsucc; auto with zarith. +rewrite Pos2Z.inj_succ; unfold Z.succ; auto with zarith. intros p; elim p; simpl plength; auto. -intros p1 Hp1; rewrite F; repeat rewrite Zpos_xI. +intros p1 Hp1; rewrite F; repeat rewrite Pos2Z.inj_xI. assert (tmp: (forall p, 2 * p = p + p)%Z); try repeat rewrite tmp; auto with zarith. -intros p1 Hp1; rewrite F; rewrite (Zpos_xO p1). +intros p1 Hp1; rewrite F; rewrite (Pos2Z.inj_xO p1). assert (tmp: (forall p, 2 * p = p + p)%Z); try repeat rewrite tmp; auto with zarith. -rewrite Zpower_1_r; auto with zarith. +rewrite Z.pow_1_r; auto with zarith. Qed. -Theorem plength_pred_correct: forall p, (Zpos p <= 2 ^ Zpos (plength (Ppred p)))%Z. -intros p; case (Psucc_pred p); intros H1. +Theorem plength_pred_correct: forall p, (Zpos p <= 2 ^ Zpos (plength (Pos.pred p)))%Z. +intros p; case (Pos.succ_pred_or p); intros H1. subst; simpl plength. -rewrite Zpower_1_r; auto with zarith. +rewrite Z.pow_1_r; auto with zarith. pattern p at 1; rewrite <- H1. -rewrite Zpos_succ_morphism; unfold Zsucc; auto with zarith. -generalize (plength_correct (Ppred p)); auto with zarith. +rewrite Pos2Z.inj_succ; unfold Z.succ; auto with zarith. +generalize (plength_correct (Pos.pred p)); auto with zarith. Qed. Definition Pdiv p q := - match Zdiv (Zpos p) (Zpos q) with + match Z.div (Zpos p) (Zpos q) with Zpos q1 => match (Zpos p) - (Zpos q) * (Zpos q1) with Z0 => q1 - | _ => (Psucc q1) + | _ => (Pos.succ q1) end | _ => xH end. @@ -85,20 +85,20 @@ unfold Pdiv. assert (H1: Zpos q > 0); auto with zarith. assert (H1b: Zpos p >= 0); auto with zarith. generalize (Z_div_ge0 (Zpos p) (Zpos q) H1 H1b). -generalize (Z_div_mod_eq (Zpos p) (Zpos q) H1); case Zdiv. - intros HH _; rewrite HH; rewrite Zmult_0_r; rewrite Zmult_1_r; simpl. +generalize (Z_div_mod_eq (Zpos p) (Zpos q) H1); case Z.div. + intros HH _; rewrite HH; rewrite Z.mul_0_r; rewrite Z.mul_1_r; simpl. case (Z_mod_lt (Zpos p) (Zpos q) H1); auto with zarith. intros q1 H2. replace (Zpos p - Zpos q * Zpos q1) with (Zpos p mod Zpos q). 2: pattern (Zpos p) at 2; rewrite H2; auto with zarith. generalize H2 (Z_mod_lt (Zpos p) (Zpos q) H1); clear H2; - case Zmod. + case Z.modulo. intros HH _; rewrite HH; auto with zarith. - intros r1 HH (_,HH1); rewrite HH; rewrite Zpos_succ_morphism. - unfold Zsucc; rewrite Zmult_plus_distr_r; auto with zarith. + intros r1 HH (_,HH1); rewrite HH; rewrite Pos2Z.inj_succ. + unfold Z.succ; rewrite Z.mul_add_distr_l; auto with zarith. intros r1 _ (HH,_); case HH; auto. intros q1 HH; rewrite HH. -unfold Zge; simpl Zcompare; intros HH1; case HH1; auto. +unfold Z.ge; simpl Z.compare; intros HH1; case HH1; auto. Qed. Definition is_one p := match p with xH => true | _ => false end. @@ -109,7 +109,7 @@ Qed. Definition get_height digits p := let r := Pdiv p digits in - if is_one r then xH else Psucc (plength (Ppred r)). + if is_one r then xH else Pos.succ (plength (Pos.pred r)). Theorem get_height_correct: forall digits N, @@ -119,13 +119,13 @@ unfold get_height. assert (H1 := Pdiv_le N digits). case_eq (is_one (Pdiv N digits)); intros H2. rewrite (is_one_one _ H2) in H1. -rewrite Zmult_1_r in H1. -change (2^(1-1))%Z with 1; rewrite Zmult_1_r; auto. +rewrite Z.mul_1_r in H1. +change (2^(1-1))%Z with 1; rewrite Z.mul_1_r; auto. clear H2. -apply Zle_trans with (1 := H1). -apply Zmult_le_compat_l; auto with zarith. -rewrite Zpos_succ_morphism; unfold Zsucc. -rewrite Zplus_comm; rewrite Zminus_plus. +apply Z.le_trans with (1 := H1). +apply Z.mul_le_mono_nonneg_l; auto with zarith. +rewrite Pos2Z.inj_succ; unfold Z.succ. +rewrite Z.add_comm; rewrite Z.add_simpl_l. apply plength_pred_correct. Qed. @@ -152,18 +152,18 @@ Open Scope nat_scope. Fixpoint plusnS (n m: nat) {struct n} : (n + S m = S (n + m))%nat := match n return (n + S m = S (n + m))%nat with - | 0 => refl_equal (S m) + | 0 => eq_refl (S m) | S n1 => let v := S (S n1 + m) in - eq_ind_r (fun n => S n = v) (refl_equal v) (plusnS n1 m) + eq_ind_r (fun n => S n = v) (eq_refl v) (plusnS n1 m) end. Fixpoint plusn0 n : n + 0 = n := match n return (n + 0 = n) with - | 0 => refl_equal 0 + | 0 => eq_refl 0 | S n1 => let v := S n1 in - eq_ind_r (fun n : nat => S n = v) (refl_equal v) (plusn0 n1) + eq_ind_r (fun n : nat => S n = v) (eq_refl v) (plusn0 n1) end. Fixpoint diff (m n: nat) {struct m}: nat * nat := @@ -177,8 +177,8 @@ Fixpoint diff_l (m n : nat) {struct m} : fst (diff m n) + n = max m n := match m return fst (diff m n) + n = max m n with | 0 => match n return (n = max 0 n) with - | 0 => refl_equal _ - | S n0 => refl_equal _ + | 0 => eq_refl _ + | S n0 => eq_refl _ end | S m1 => match n return (fst (diff (S m1) n) + n = max (S m1) n) @@ -188,7 +188,7 @@ Fixpoint diff_l (m n : nat) {struct m} : fst (diff m n) + n = max m n := let v := fst (diff m1 n1) + n1 in let v1 := fst (diff m1 n1) + S n1 in eq_ind v (fun n => v1 = S n) - (eq_ind v1 (fun n => v1 = n) (refl_equal v1) (S v) (plusnS _ _)) + (eq_ind v1 (fun n => v1 = n) (eq_refl v1) (S v) (plusnS _ _)) _ (diff_l _ _) end end. @@ -197,17 +197,17 @@ Fixpoint diff_r (m n: nat) {struct m}: snd (diff m n) + m = max m n := match m return (snd (diff m n) + m = max m n) with | 0 => match n return (snd (diff 0 n) + 0 = max 0 n) with - | 0 => refl_equal _ + | 0 => eq_refl _ | S _ => plusn0 _ end | S m => match n return (snd (diff (S m) n) + S m = max (S m) n) with - | 0 => refl_equal (snd (diff (S m) 0) + S m) + | 0 => eq_refl (snd (diff (S m) 0) + S m) | S n1 => let v := S (max m n1) in eq_ind_r (fun n => n = v) (eq_ind_r (fun n => S n = v) - (refl_equal v) (diff_r _ _)) (plusnS _ _) + (eq_refl v) (diff_r _ _)) (plusnS _ _) end end. @@ -216,7 +216,7 @@ Fixpoint diff_r (m n: nat) {struct m}: snd (diff m n) + m = max m n := Definition castm (m n: nat) (H: m = n) (x: word w (S m)): (word w (S n)) := match H in (_ = y) return (word w (S y)) with - | refl_equal => x + | eq_refl => x end. Variable m: nat. @@ -314,7 +314,7 @@ Section CompareRec. Lemma base_xO: forall n, base (xO n) = (base n)^2. Proof. intros n1; unfold base. - rewrite (Zpos_xO n1); rewrite Zmult_comm; rewrite Zpower_mult; auto with zarith. + rewrite (Pos2Z.inj_xO n1); rewrite Z.mul_comm; rewrite Z.pow_mul_r; auto with zarith. Qed. Let double_to_Z_pos: forall n x, 0 <= double_to_Z n x < double_wB n := @@ -332,13 +332,13 @@ Section CompareRec. rewrite 2 Hrec. simpl double_to_Z. set (wB := DoubleBase.double_wB wm_base n). - case Zcompare_spec; intros Cmp. + case Z.compare_spec; intros Cmp. rewrite <- Cmp. reflexivity. - symmetry. apply Zgt_lt, Zlt_gt. (* ;-) *) + symmetry. apply Z.gt_lt, Z.lt_gt. (* ;-) *) assert (0 < wB). unfold wB, DoubleBase.double_wB, base; auto with zarith. - change 0 with (0 + 0); apply Zplus_lt_le_compat; auto with zarith. - apply Zmult_lt_0_compat; auto with zarith. + change 0 with (0 + 0); apply Z.add_lt_le_mono; auto with zarith. + apply Z.mul_pos_pos; auto with zarith. case (double_to_Z_pos n xl); auto with zarith. case (double_to_Z_pos n xh); intros; exfalso; omega. Qed. @@ -358,9 +358,9 @@ Section CompareRec. end. Variable spec_compare: forall x y, - compare x y = Zcompare (w_to_Z x) (w_to_Z y). + compare x y = Z.compare (w_to_Z x) (w_to_Z y). Variable spec_compare_m: forall x y, - compare_m x y = Zcompare (wm_to_Z x) (w_to_Z y). + compare_m x y = Z.compare (wm_to_Z x) (w_to_Z y). Variable wm_base_lt: forall x, 0 <= w_to_Z x < base (wm_base). @@ -369,35 +369,35 @@ Section CompareRec. Proof. intros n x; elim n; simpl; auto; clear n. intros n (H0, H); split; auto. - apply Zlt_le_trans with (1:= H). + apply Z.lt_le_trans with (1:= H). unfold double_wB, DoubleBase.double_wB; simpl. rewrite Pshiftl_nat_S, base_xO. set (u := base (Pos.shiftl_nat wm_base n)). assert (0 < u). unfold u, base; auto with zarith. replace (u^2) with (u * u); simpl; auto with zarith. - apply Zle_trans with (1 * u); auto with zarith. - unfold Zpower_pos; simpl; ring. + apply Z.le_trans with (1 * u); auto with zarith. + unfold Z.pow_pos; simpl; ring. Qed. Lemma spec_compare_mn_1: forall n x y, - compare_mn_1 n x y = Zcompare (double_to_Z n x) (w_to_Z y). + compare_mn_1 n x y = Z.compare (double_to_Z n x) (w_to_Z y). Proof. intros n; elim n; simpl; auto; clear n. intros n Hrec x; case x; clear x; auto. intros y; rewrite spec_compare; rewrite w_to_Z_0. reflexivity. intros xh xl y; simpl; - rewrite spec_compare0_mn, Hrec. case Zcompare_spec. + rewrite spec_compare0_mn, Hrec. case Z.compare_spec. intros H1b. - rewrite <- H1b; rewrite Zmult_0_l; rewrite Zplus_0_l; auto. - symmetry. apply Zlt_gt. + rewrite <- H1b; rewrite Z.mul_0_l; rewrite Z.add_0_l; auto. + symmetry. apply Z.lt_gt. case (double_wB_lt n y); intros _ H0. - apply Zlt_le_trans with (1:= H0). + apply Z.lt_le_trans with (1:= H0). fold double_wB. case (double_to_Z_pos n xl); intros H1 H2. - apply Zle_trans with (double_to_Z n xh * double_wB n); auto with zarith. - apply Zle_trans with (1 * double_wB n); auto with zarith. + apply Z.le_trans with (double_to_Z n xh * double_wB n); auto with zarith. + apply Z.le_trans with (1 * double_wB n); auto with zarith. case (double_to_Z_pos n xh); intros; exfalso; omega. Qed. @@ -440,8 +440,8 @@ End AddS. Proof. intros x; elim x; clear x; [intros x1 Hrec | intros x1 Hrec | idtac]; intros y; case y; clear y; intros y1 H || intros H; simpl length_pos; - try (rewrite (Zpos_xI x1) || rewrite (Zpos_xO x1)); - try (rewrite (Zpos_xI y1) || rewrite (Zpos_xO y1)); + try (rewrite (Pos2Z.inj_xI x1) || rewrite (Pos2Z.inj_xO x1)); + try (rewrite (Pos2Z.inj_xI y1) || rewrite (Pos2Z.inj_xO y1)); try (inversion H; fail); try (assert (Zpos x1 < Zpos y1); [apply Hrec; apply lt_S_n | idtac]; auto with zarith); assert (0 < Zpos y1); auto with zarith; red; auto. diff --git a/theories/Numbers/Natural/Binary/NBinary.v b/theories/Numbers/Natural/Binary/NBinary.v index 43ca67dd3..3111d327c 100644 --- a/theories/Numbers/Natural/Binary/NBinary.v +++ b/theories/Numbers/Natural/Binary/NBinary.v @@ -31,8 +31,8 @@ Time Eval vm_compute in (log 500000). (* 11 sec *) Fixpoint binposlog (p : positive) : N := match p with | xH => 0 -| xO p' => Nsucc (binposlog p') -| xI p' => Nsucc (binposlog p') +| xO p' => N.succ (binposlog p') +| xI p' => N.succ (binposlog p') end. Definition binlog (n : N) : N := diff --git a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v index 2c7884ac4..ce92f44d9 100644 --- a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v +++ b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v @@ -318,7 +318,7 @@ Program Instance mod_wd : Proper (eq==>eq==>eq) modulo. Theorem div_mod : forall a b, ~b==0 -> a == b*(div a b) + (modulo a b). Proof. -intros a b. zify. intros. apply Z_div_mod_eq_full; auto. +intros a b. zify. intros. apply Z.div_mod; auto. Qed. Theorem mod_bound_pos : forall a b, 0<=a -> 0<b -> @@ -444,7 +444,7 @@ Qed. (** Recursion *) Definition recursion (A : Type) (a : A) (f : NN.t -> A -> A) (n : NN.t) := - Nrect (fun _ => A) a (fun n a => f (NN.of_N n) a) (NN.to_N n). + N.peano_rect (fun _ => A) a (fun n a => f (NN.of_N n) a) (NN.to_N n). Arguments recursion [A] a f n. Instance recursion_wd (A : Type) (Aeq : relation A) : @@ -457,7 +457,7 @@ unfold NN.to_N. rewrite <- Exx'; clear x' Exx'. induction (Z.to_N [x]) using N.peano_ind. simpl; auto. -rewrite 2 Nrect_step. now apply Eff'. +rewrite 2 N.peano_rect_succ. now apply Eff'. Qed. Theorem recursion_0 : @@ -474,7 +474,7 @@ Proof. unfold eq, recursion; intros A Aeq a f EAaa f_wd n. replace (to_N (succ n)) with (N.succ (to_N n)) by (zify; now rewrite <- Z2N.inj_succ by apply spec_pos). -rewrite Nrect_step. +rewrite N.peano_rect_succ. apply f_wd; auto. zify. now rewrite Z2N.id by apply spec_pos. fold (recursion a f n). apply recursion_wd; auto. red; auto. diff --git a/theories/Numbers/Rational/BigQ/BigQ.v b/theories/Numbers/Rational/BigQ/BigQ.v index 424db5b75..bc50c4148 100644 --- a/theories/Numbers/Rational/BigQ/BigQ.v +++ b/theories/Numbers/Rational/BigQ/BigQ.v @@ -26,10 +26,10 @@ Module BigN_BigZ <: NType_ZType BigN.BigN BigZ. reflexivity. Qed. Definition Zabs_N := BigZ.to_N. - Lemma spec_Zabs_N : forall z, BigN.to_Z (Zabs_N z) = Zabs (BigZ.to_Z z). + Lemma spec_Zabs_N : forall z, BigN.to_Z (Zabs_N z) = Z.abs (BigZ.to_Z z). Proof. unfold Zabs_N; intros. - rewrite BigZ.spec_to_Z, Zmult_comm; apply Zsgn_Zabs. + rewrite BigZ.spec_to_Z, Z.mul_comm; apply Z.sgn_abs. Qed. End BigN_BigZ. @@ -89,10 +89,10 @@ exact BigQ.div_mul_inv. exact BigQ.mul_inv_diag_l. Qed. Lemma BigQpowerth : - power_theory 1 BigQ.mul BigQ.eq Z_of_N BigQ.power. + power_theory 1 BigQ.mul BigQ.eq Z.of_N BigQ.power. Proof. constructor. intros. BigQ.qify. -replace ([r] ^ Z_of_N n)%Q with (pow_N 1 Qmult [r] n)%Q by (now destruct n). +replace ([r] ^ Z.of_N n)%Q with (pow_N 1 Qmult [r] n)%Q by (now destruct n). destruct n. reflexivity. induction p; simpl; auto; rewrite ?BigQ.spec_mul, ?IHp; reflexivity. Qed. diff --git a/theories/Numbers/Rational/BigQ/QMake.v b/theories/Numbers/Rational/BigQ/QMake.v index 995fbb9ee..78ad7c470 100644 --- a/theories/Numbers/Rational/BigQ/QMake.v +++ b/theories/Numbers/Rational/BigQ/QMake.v @@ -19,14 +19,14 @@ Require Import NSig ZSig QSig. denominators. But first we will need some glue between [NType] and [ZType]. *) -Module Type NType_ZType (N:NType)(Z:ZType). - Parameter Z_of_N : N.t -> Z.t. - Parameter spec_Z_of_N : forall n, Z.to_Z (Z_of_N n) = N.to_Z n. - Parameter Zabs_N : Z.t -> N.t. - Parameter spec_Zabs_N : forall z, N.to_Z (Zabs_N z) = Zabs (Z.to_Z z). +Module Type NType_ZType (NN:NType)(ZZ:ZType). + Parameter Z_of_N : NN.t -> ZZ.t. + Parameter spec_Z_of_N : forall n, ZZ.to_Z (Z_of_N n) = NN.to_Z n. + Parameter Zabs_N : ZZ.t -> NN.t. + Parameter spec_Zabs_N : forall z, NN.to_Z (Zabs_N z) = Z.abs (ZZ.to_Z z). End NType_ZType. -Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. +Module Make (NN:NType)(ZZ:ZType)(Import NZ:NType_ZType NN ZZ) <: QType. (** The notation of a rational number is either an integer x, interpreted as itself or a pair (x,y) of an integer x and a natural @@ -34,8 +34,8 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. interpreted as 0. *) Inductive t_ := - | Qz : Z.t -> t_ - | Qq : Z.t -> N.t -> t_. + | Qz : ZZ.t -> t_ + | Qq : ZZ.t -> NN.t -> t_. Definition t := t_. @@ -45,41 +45,41 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Local Open Scope Q_scope. - Definition of_Z x: t := Qz (Z.of_Z x). + Definition of_Z x: t := Qz (ZZ.of_Z x). Definition of_Q (q:Q) : t := let (x,y) := q in match y with - | 1%positive => Qz (Z.of_Z x) - | _ => Qq (Z.of_Z x) (N.of_N (Npos y)) + | 1%positive => Qz (ZZ.of_Z x) + | _ => Qq (ZZ.of_Z x) (NN.of_N (Npos y)) end. Definition to_Q (q: t) := match q with - | Qz x => Z.to_Z x # 1 - | Qq x y => if N.eqb y N.zero then 0 - else Z.to_Z x # Z2P (N.to_Z y) + | Qz x => ZZ.to_Z x # 1 + | Qq x y => if NN.eqb y NN.zero then 0 + else ZZ.to_Z x # Z.to_pos (NN.to_Z y) end. Notation "[ x ]" := (to_Q x). Lemma N_to_Z_pos : - forall x, (N.to_Z x <> N.to_Z N.zero)%Z -> (0 < N.to_Z x)%Z. + forall x, (NN.to_Z x <> NN.to_Z NN.zero)%Z -> (0 < NN.to_Z x)%Z. Proof. - intros x; rewrite N.spec_0; generalize (N.spec_pos x). romega. + intros x; rewrite NN.spec_0; generalize (NN.spec_pos x). romega. Qed. Ltac destr_zcompare := case Z.compare_spec; intros ?H. Ltac destr_eqb := match goal with - | |- context [Z.eqb ?x ?y] => - rewrite (Z.spec_eqb x y); - case (Z.eqb_spec (Z.to_Z x) (Z.to_Z y)); + | |- context [ZZ.eqb ?x ?y] => + rewrite (ZZ.spec_eqb x y); + case (Z.eqb_spec (ZZ.to_Z x) (ZZ.to_Z y)); destr_eqb - | |- context [N.eqb ?x ?y] => - rewrite (N.spec_eqb x y); - case (Z.eqb_spec (N.to_Z x) (N.to_Z y)); + | |- context [NN.eqb ?x ?y] => + rewrite (NN.spec_eqb x y); + case (Z.eqb_spec (NN.to_Z x) (NN.to_Z y)); [ | let H:=fresh "H" in try (intro H;generalize (N_to_Z_pos _ H); clear H)]; destr_eqb @@ -87,11 +87,11 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. end. Hint Rewrite - Zplus_0_r Zplus_0_l Zmult_0_r Zmult_0_l Zmult_1_r Zmult_1_l - Z.spec_0 N.spec_0 Z.spec_1 N.spec_1 Z.spec_m1 Z.spec_opp - Z.spec_compare N.spec_compare - Z.spec_add N.spec_add Z.spec_mul N.spec_mul Z.spec_div N.spec_div - Z.spec_gcd N.spec_gcd Zgcd_Zabs Zgcd_1 + Z.add_0_r Z.add_0_l Z.mul_0_r Z.mul_0_l Z.mul_1_r Z.mul_1_l + ZZ.spec_0 NN.spec_0 ZZ.spec_1 NN.spec_1 ZZ.spec_m1 ZZ.spec_opp + ZZ.spec_compare NN.spec_compare + ZZ.spec_add NN.spec_add ZZ.spec_mul NN.spec_mul ZZ.spec_div NN.spec_div + ZZ.spec_gcd NN.spec_gcd Z.gcd_abs_l Z.gcd_1_r spec_Z_of_N spec_Zabs_N : nz. @@ -99,13 +99,13 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Ltac qsimpl := try red; unfold to_Q; simpl; intros; destr_eqb; simpl; nzsimpl; intros; - rewrite ?Z2P_correct by auto; + rewrite ?Z2Pos.id by auto; auto. Theorem strong_spec_of_Q: forall q: Q, [of_Q q] = q. Proof. - intros(x,y); destruct y; simpl; rewrite ?Z.spec_of_Z; auto; - destr_eqb; now rewrite ?N.spec_0, ?N.spec_of_N. + intros(x,y); destruct y; simpl; rewrite ?ZZ.spec_of_Z; auto; + destr_eqb; now rewrite ?NN.spec_0, ?NN.spec_of_N. Qed. Theorem spec_of_Q: forall q: Q, [of_Q q] == q. @@ -115,9 +115,9 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Definition eq x y := [x] == [y]. - Definition zero: t := Qz Z.zero. - Definition one: t := Qz Z.one. - Definition minus_one: t := Qz Z.minus_one. + Definition zero: t := Qz ZZ.zero. + Definition one: t := Qz ZZ.one. + Definition minus_one: t := Qz ZZ.minus_one. Lemma spec_0: [zero] == 0. Proof. @@ -136,20 +136,20 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Definition compare (x y: t) := match x, y with - | Qz zx, Qz zy => Z.compare zx zy + | Qz zx, Qz zy => ZZ.compare zx zy | Qz zx, Qq ny dy => - if N.eqb dy N.zero then Z.compare zx Z.zero - else Z.compare (Z.mul zx (Z_of_N dy)) ny + if NN.eqb dy NN.zero then ZZ.compare zx ZZ.zero + else ZZ.compare (ZZ.mul zx (Z_of_N dy)) ny | Qq nx dx, Qz zy => - if N.eqb dx N.zero then Z.compare Z.zero zy - else Z.compare nx (Z.mul zy (Z_of_N dx)) + if NN.eqb dx NN.zero then ZZ.compare ZZ.zero zy + else ZZ.compare nx (ZZ.mul zy (Z_of_N dx)) | Qq nx dx, Qq ny dy => - match N.eqb dx N.zero, N.eqb dy N.zero with + match NN.eqb dx NN.zero, NN.eqb dy NN.zero with | true, true => Eq - | true, false => Z.compare Z.zero ny - | false, true => Z.compare nx Z.zero - | false, false => Z.compare (Z.mul nx (Z_of_N dy)) - (Z.mul ny (Z_of_N dx)) + | true, false => ZZ.compare ZZ.zero ny + | false, true => ZZ.compare nx ZZ.zero + | false, false => ZZ.compare (ZZ.mul nx (Z_of_N dy)) + (ZZ.mul ny (Z_of_N dx)) end end. @@ -188,7 +188,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. (** [check_int] : is a reduced fraction [n/d] in fact a integer ? *) Definition check_int n d := - match N.compare N.one d with + match NN.compare NN.one d with | Lt => Qq n d | Eq => Qz n | Gt => zero (* n/0 encodes 0 *) @@ -207,9 +207,9 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. (** Normalisation function *) Definition norm n d : t := - let gcd := N.gcd (Zabs_N n) d in - match N.compare N.one gcd with - | Lt => check_int (Z.div n (Z_of_N gcd)) (N.div d gcd) + let gcd := NN.gcd (Zabs_N n) d in + match NN.compare NN.one gcd with + | Lt => check_int (ZZ.div n (Z_of_N gcd)) (NN.div d gcd) | Eq => check_int n d | Gt => zero (* gcd = 0 => both numbers are 0 *) end. @@ -217,8 +217,8 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Theorem spec_norm: forall n q, [norm n q] == [Qq n q]. Proof. intros p q; unfold norm. - assert (Hp := N.spec_pos (Zabs_N p)). - assert (Hq := N.spec_pos q). + assert (Hp := NN.spec_pos (Zabs_N p)). + assert (Hq := NN.spec_pos q). nzsimpl. destr_zcompare. (* Eq *) @@ -226,15 +226,15 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. (* Lt *) rewrite strong_spec_check_int. qsimpl. - generalize (Zgcd_div_pos (Z.to_Z p) (N.to_Z q)). romega. - replace (N.to_Z q) with 0%Z in * by assumption. + generalize (Zgcd_div_pos (ZZ.to_Z p) (NN.to_Z q)). romega. + replace (NN.to_Z q) with 0%Z in * by assumption. rewrite Zdiv_0_l in *; auto with zarith. apply Zgcd_div_swap0; romega. (* Gt *) qsimpl. - assert (H' : Zgcd (Z.to_Z p) (N.to_Z q) = 0%Z). - generalize (Zgcd_is_pos (Z.to_Z p) (N.to_Z q)); romega. - symmetry; apply (Zgcd_inv_0_l _ _ H'); auto. + assert (H' : Z.gcd (ZZ.to_Z p) (NN.to_Z q) = 0%Z). + generalize (Z.gcd_nonneg (ZZ.to_Z p) (NN.to_Z q)); romega. + symmetry; apply (Z.gcd_eq_0_l _ _ H'); auto. Qed. Theorem strong_spec_norm : forall p q, [norm p q] = Qred [Qq p q]. @@ -244,8 +244,8 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. (apply Qred_complete; apply spec_norm). symmetry; apply Qred_identity. unfold norm. - assert (Hp := N.spec_pos (Zabs_N p)). - assert (Hq := N.spec_pos q). + assert (Hp := NN.spec_pos (Zabs_N p)). + assert (Hq := NN.spec_pos q). nzsimpl. destr_zcompare; rewrite ?strong_spec_check_int. (* Eq *) @@ -253,10 +253,10 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. (* Lt *) qsimpl. rewrite Zgcd_1_rel_prime. - destruct (Z_lt_le_dec 0 (N.to_Z q)). + destruct (Z_lt_le_dec 0 (NN.to_Z q)). apply Zis_gcd_rel_prime; auto with zarith. apply Zgcd_is_gcd. - replace (N.to_Z q) with 0%Z in * by romega. + replace (NN.to_Z q) with 0%Z in * by romega. rewrite Zdiv_0_l in *; romega. (* Gt *) simpl; auto with zarith. @@ -292,20 +292,20 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. match x with | Qz zx => match y with - | Qz zy => Qz (Z.add zx zy) + | Qz zy => Qz (ZZ.add zx zy) | Qq ny dy => - if N.eqb dy N.zero then x - else Qq (Z.add (Z.mul zx (Z_of_N dy)) ny) dy + if NN.eqb dy NN.zero then x + else Qq (ZZ.add (ZZ.mul zx (Z_of_N dy)) ny) dy end | Qq nx dx => - if N.eqb dx N.zero then y + if NN.eqb dx NN.zero then y else match y with - | Qz zy => Qq (Z.add nx (Z.mul zy (Z_of_N dx))) dx + | Qz zy => Qq (ZZ.add nx (ZZ.mul zy (Z_of_N dx))) dx | Qq ny dy => - if N.eqb dy N.zero then x + if NN.eqb dy NN.zero then x else - let n := Z.add (Z.mul nx (Z_of_N dy)) (Z.mul ny (Z_of_N dx)) in - let d := N.mul dx dy in + let n := ZZ.add (ZZ.mul nx (Z_of_N dy)) (ZZ.mul ny (Z_of_N dx)) in + let d := NN.mul dx dy in Qq n d end end. @@ -314,30 +314,30 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Proof. intros [x | nx dx] [y | ny dy]; unfold Qplus; qsimpl; auto with zarith. - rewrite Pmult_1_r, Z2P_correct; auto. - rewrite Pmult_1_r, Z2P_correct; auto. - destruct (Zmult_integral (N.to_Z dx) (N.to_Z dy)); intuition. - rewrite Zpos_mult_morphism, 2 Z2P_correct; auto. + rewrite Pos.mul_1_r, Z2Pos.id; auto. + rewrite Pos.mul_1_r, Z2Pos.id; auto. + rewrite Z.mul_eq_0 in *; intuition. + rewrite Pos2Z.inj_mul, 2 Z2Pos.id; auto. Qed. Definition add_norm (x y: t): t := match x with | Qz zx => match y with - | Qz zy => Qz (Z.add zx zy) + | Qz zy => Qz (ZZ.add zx zy) | Qq ny dy => - if N.eqb dy N.zero then x - else norm (Z.add (Z.mul zx (Z_of_N dy)) ny) dy + if NN.eqb dy NN.zero then x + else norm (ZZ.add (ZZ.mul zx (Z_of_N dy)) ny) dy end | Qq nx dx => - if N.eqb dx N.zero then y + if NN.eqb dx NN.zero then y else match y with - | Qz zy => norm (Z.add nx (Z.mul zy (Z_of_N dx))) dx + | Qz zy => norm (ZZ.add nx (ZZ.mul zy (Z_of_N dx))) dx | Qq ny dy => - if N.eqb dy N.zero then x + if NN.eqb dy NN.zero then x else - let n := Z.add (Z.mul nx (Z_of_N dy)) (Z.mul ny (Z_of_N dx)) in - let d := N.mul dx dy in + let n := ZZ.add (ZZ.mul nx (Z_of_N dy)) (ZZ.mul ny (Z_of_N dx)) in + let d := NN.mul dx dy in norm n d end end. @@ -363,18 +363,18 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Definition opp (x: t): t := match x with - | Qz zx => Qz (Z.opp zx) - | Qq nx dx => Qq (Z.opp nx) dx + | Qz zx => Qz (ZZ.opp zx) + | Qq nx dx => Qq (ZZ.opp nx) dx end. Theorem strong_spec_opp: forall q, [opp q] = -[q]. Proof. intros [z | x y]; simpl. - rewrite Z.spec_opp; auto. - match goal with |- context[N.eqb ?X ?Y] => - generalize (N.spec_eqb X Y); case N.eqb - end; auto; rewrite N.spec_0. - rewrite Z.spec_opp; auto. + rewrite ZZ.spec_opp; auto. + match goal with |- context[NN.eqb ?X ?Y] => + generalize (NN.spec_eqb X Y); case NN.eqb + end; auto; rewrite NN.spec_0. + rewrite ZZ.spec_opp; auto. Qed. Theorem spec_opp : forall q, [opp q] == -[q]. @@ -416,28 +416,28 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Definition mul (x y: t): t := match x, y with - | Qz zx, Qz zy => Qz (Z.mul zx zy) - | Qz zx, Qq ny dy => Qq (Z.mul zx ny) dy - | Qq nx dx, Qz zy => Qq (Z.mul nx zy) dx - | Qq nx dx, Qq ny dy => Qq (Z.mul nx ny) (N.mul dx dy) + | Qz zx, Qz zy => Qz (ZZ.mul zx zy) + | Qz zx, Qq ny dy => Qq (ZZ.mul zx ny) dy + | Qq nx dx, Qz zy => Qq (ZZ.mul nx zy) dx + | Qq nx dx, Qq ny dy => Qq (ZZ.mul nx ny) (NN.mul dx dy) end. Ltac nsubst := - match goal with E : N.to_Z _ = _ |- _ => rewrite E in * end. + match goal with E : NN.to_Z _ = _ |- _ => rewrite E in * end. Theorem spec_mul : forall x y, [mul x y] == [x] * [y]. Proof. intros [x | nx dx] [y | ny dy]; unfold Qmult; simpl; qsimpl. - rewrite Pmult_1_r, Z2P_correct; auto. - destruct (Zmult_integral (N.to_Z dx) (N.to_Z dy)); intuition. + rewrite Pos.mul_1_r, Z2Pos.id; auto. + rewrite Z.mul_eq_0 in *; intuition. nsubst; auto with zarith. nsubst; auto with zarith. nsubst; nzsimpl; auto with zarith. - rewrite Zpos_mult_morphism, 2 Z2P_correct; auto. + rewrite Pos2Z.inj_mul, 2 Z2Pos.id; auto. Qed. Definition norm_denum n d := - if N.eqb d N.one then Qz n else Qq n d. + if NN.eqb d NN.one then Qz n else Qq n d. Lemma spec_norm_denum : forall n d, [norm_denum n d] == [Qq n d]. @@ -448,40 +448,40 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Qed. Definition irred n d := - let gcd := N.gcd (Zabs_N n) d in - match N.compare gcd N.one with - | Gt => (Z.div n (Z_of_N gcd), N.div d gcd) + let gcd := NN.gcd (Zabs_N n) d in + match NN.compare gcd NN.one with + | Gt => (ZZ.div n (Z_of_N gcd), NN.div d gcd) | _ => (n, d) end. Lemma spec_irred : forall n d, exists g, let (n',d') := irred n d in - (Z.to_Z n' * g = Z.to_Z n)%Z /\ (N.to_Z d' * g = N.to_Z d)%Z. + (ZZ.to_Z n' * g = ZZ.to_Z n)%Z /\ (NN.to_Z d' * g = NN.to_Z d)%Z. Proof. intros. unfold irred; nzsimpl; simpl. destr_zcompare. exists 1%Z; nzsimpl; auto. exists 0%Z; nzsimpl. - assert (Zgcd (Z.to_Z n) (N.to_Z d) = 0%Z). - generalize (Zgcd_is_pos (Z.to_Z n) (N.to_Z d)); romega. + assert (Z.gcd (ZZ.to_Z n) (NN.to_Z d) = 0%Z). + generalize (Z.gcd_nonneg (ZZ.to_Z n) (NN.to_Z d)); romega. clear H. split. - symmetry; apply (Zgcd_inv_0_l _ _ H0). - symmetry; apply (Zgcd_inv_0_r _ _ H0). - exists (Zgcd (Z.to_Z n) (N.to_Z d)). + symmetry; apply (Z.gcd_eq_0_l _ _ H0). + symmetry; apply (Z.gcd_eq_0_r _ _ H0). + exists (Z.gcd (ZZ.to_Z n) (NN.to_Z d)). simpl. split. nzsimpl. - destruct (Zgcd_is_gcd (Z.to_Z n) (N.to_Z d)). - rewrite Zmult_comm; symmetry; apply Zdivide_Zdiv_eq; auto with zarith. + destruct (Zgcd_is_gcd (ZZ.to_Z n) (NN.to_Z d)). + rewrite Z.mul_comm; symmetry; apply Zdivide_Zdiv_eq; auto with zarith. nzsimpl. - destruct (Zgcd_is_gcd (Z.to_Z n) (N.to_Z d)). - rewrite Zmult_comm; symmetry; apply Zdivide_Zdiv_eq; auto with zarith. + destruct (Zgcd_is_gcd (ZZ.to_Z n) (NN.to_Z d)). + rewrite Z.mul_comm; symmetry; apply Zdivide_Zdiv_eq; auto with zarith. Qed. Lemma spec_irred_zero : forall n d, - (N.to_Z d = 0)%Z <-> (N.to_Z (snd (irred n d)) = 0)%Z. + (NN.to_Z d = 0)%Z <-> (NN.to_Z (snd (irred n d)) = 0)%Z. Proof. intros. unfold irred. @@ -494,8 +494,8 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. nzsimpl; destr_zcompare; simpl; auto. nzsimpl. intros. - generalize (N.spec_pos d); intros. - destruct (N.to_Z d); auto. + generalize (NN.spec_pos d); intros. + destruct (NN.to_Z d); auto. assert (0 < 0)%Z. rewrite <- H0 at 2. apply Zgcd_div_pos; auto with zarith. @@ -505,49 +505,49 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Qed. Lemma strong_spec_irred : forall n d, - (N.to_Z d <> 0%Z) -> - let (n',d') := irred n d in Zgcd (Z.to_Z n') (N.to_Z d') = 1%Z. + (NN.to_Z d <> 0%Z) -> + let (n',d') := irred n d in Z.gcd (ZZ.to_Z n') (NN.to_Z d') = 1%Z. Proof. unfold irred; intros. nzsimpl. destr_zcompare; simpl; auto. elim H. - apply (Zgcd_inv_0_r (Z.to_Z n)). - generalize (Zgcd_is_pos (Z.to_Z n) (N.to_Z d)); romega. + apply (Z.gcd_eq_0_r (ZZ.to_Z n)). + generalize (Z.gcd_nonneg (ZZ.to_Z n) (NN.to_Z d)); romega. nzsimpl. rewrite Zgcd_1_rel_prime. apply Zis_gcd_rel_prime. - generalize (N.spec_pos d); romega. - generalize (Zgcd_is_pos (Z.to_Z n) (N.to_Z d)); romega. + generalize (NN.spec_pos d); romega. + generalize (Z.gcd_nonneg (ZZ.to_Z n) (NN.to_Z d)); romega. apply Zgcd_is_gcd; auto. Qed. Definition mul_norm_Qz_Qq z n d := - if Z.eqb z Z.zero then zero + if ZZ.eqb z ZZ.zero then zero else - let gcd := N.gcd (Zabs_N z) d in - match N.compare gcd N.one with + let gcd := NN.gcd (Zabs_N z) d in + match NN.compare gcd NN.one with | Gt => - let z := Z.div z (Z_of_N gcd) in - let d := N.div d gcd in - norm_denum (Z.mul z n) d - | _ => Qq (Z.mul z n) d + let z := ZZ.div z (Z_of_N gcd) in + let d := NN.div d gcd in + norm_denum (ZZ.mul z n) d + | _ => Qq (ZZ.mul z n) d end. Definition mul_norm (x y: t): t := match x, y with - | Qz zx, Qz zy => Qz (Z.mul zx zy) + | Qz zx, Qz zy => Qz (ZZ.mul zx zy) | Qz zx, Qq ny dy => mul_norm_Qz_Qq zx ny dy | Qq nx dx, Qz zy => mul_norm_Qz_Qq zy nx dx | Qq nx dx, Qq ny dy => let (nx, dy) := irred nx dy in let (ny, dx) := irred ny dx in - norm_denum (Z.mul ny nx) (N.mul dx dy) + norm_denum (ZZ.mul ny nx) (NN.mul dx dy) end. Lemma spec_mul_norm_Qz_Qq : forall z n d, - [mul_norm_Qz_Qq z n d] == [Qq (Z.mul z n) d]. + [mul_norm_Qz_Qq z n d] == [Qq (ZZ.mul z n) d]. Proof. intros z n d; unfold mul_norm_Qz_Qq; nzsimpl; rewrite Zcompare_gt. destr_eqb; nzsimpl; intros Hz. @@ -558,7 +558,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. qsimpl. rewrite Zdiv_gcd_zero in GT; auto with zarith. nsubst. rewrite Zdiv_0_l in *; discriminate. - rewrite <- Zmult_assoc, (Zmult_comm (Z.to_Z n)), Zmult_assoc. + rewrite <- Z.mul_assoc, (Z.mul_comm (ZZ.to_Z n)), Z.mul_assoc. rewrite Zgcd_div_swap0; try romega. ring. Qed. @@ -582,34 +582,34 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. destr_eqb; simpl; nzsimpl; auto. nzsimpl; rewrite Hd, Zdiv_0_l; auto with zarith. - rewrite Z2P_correct in H; auto. + rewrite Z2Pos.id in H; auto. unfold mul_norm_Qz_Qq; nzsimpl; rewrite Zcompare_gt. destr_eqb; intros Hz; simpl; nzsimpl; simpl; auto. destruct Z_le_gt_dec as [H'|H']. simpl; nzsimpl. destr_eqb; simpl; nzsimpl; auto. intros. - rewrite Z2P_correct; auto. + rewrite Z2Pos.id; auto. apply Zgcd_mult_rel_prime; auto. - generalize (Zgcd_inv_0_l (Z.to_Z z) (N.to_Z d)) - (Zgcd_is_pos (Z.to_Z z) (N.to_Z d)); romega. + generalize (Z.gcd_eq_0_l (ZZ.to_Z z) (NN.to_Z d)) + (Z.gcd_nonneg (ZZ.to_Z z) (NN.to_Z d)); romega. destr_eqb; simpl; nzsimpl; auto. unfold norm_denum. destr_eqb; nzsimpl; simpl; destr_eqb; simpl; auto. intros; nzsimpl. - rewrite Z2P_correct; auto. + rewrite Z2Pos.id; auto. apply Zgcd_mult_rel_prime. rewrite Zgcd_1_rel_prime. apply Zis_gcd_rel_prime. - generalize (N.spec_pos d); romega. - generalize (Zgcd_is_pos (Z.to_Z z) (N.to_Z d)); romega. + generalize (NN.spec_pos d); romega. + generalize (Z.gcd_nonneg (ZZ.to_Z z) (NN.to_Z d)); romega. apply Zgcd_is_gcd. - destruct (Zgcd_is_gcd (Z.to_Z z) (N.to_Z d)) as [ (z0,Hz0) (d0,Hd0) Hzd]. - replace (N.to_Z d / Zgcd (Z.to_Z z) (N.to_Z d))%Z with d0. + destruct (Zgcd_is_gcd (ZZ.to_Z z) (NN.to_Z d)) as [ (z0,Hz0) (d0,Hd0) Hzd]. + replace (NN.to_Z d / Z.gcd (ZZ.to_Z z) (NN.to_Z d))%Z with d0. rewrite Zgcd_1_rel_prime in *. apply bezout_rel_prime. destruct (rel_prime_bezout _ _ H) as [u v Huv]. - apply Bezout_intro with u (v*(Zgcd (Z.to_Z z) (N.to_Z d)))%Z. + apply Bezout_intro with u (v*(Z.gcd (ZZ.to_Z z) (NN.to_Z d)))%Z. rewrite <- Huv; rewrite Hd0 at 2; ring. rewrite Hd0 at 1. symmetry; apply Z_div_mult_full; auto with zarith. @@ -634,14 +634,14 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. qsimpl. match goal with E : (_ * _ = 0)%Z |- _ => - destruct (Zmult_integral _ _ E) as [Eq|Eq] end. + rewrite Z.mul_eq_0 in E; destruct E as [Eq|Eq] end. rewrite Eq in *; simpl in *. rewrite <- Hg2' in *; auto with zarith. rewrite Eq in *; simpl in *. rewrite <- Hg2 in *; auto with zarith. match goal with E : (_ * _ = 0)%Z |- _ => - destruct (Zmult_integral _ _ E) as [Eq|Eq] end. + rewrite Z.mul_eq_0 in E; destruct E as [Eq|Eq] end. rewrite Hz' in Eq; rewrite Eq in *; auto with zarith. rewrite Hz in Eq; rewrite Eq in *; auto with zarith. @@ -671,31 +671,31 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. unfold norm_denum; qsimpl. - assert (NEQ : N.to_Z dy <> 0%Z) by + assert (NEQ : NN.to_Z dy <> 0%Z) by (rewrite Hz; intros EQ; rewrite EQ in *; romega). specialize (Hgc NEQ). - assert (NEQ' : N.to_Z dx <> 0%Z) by + assert (NEQ' : NN.to_Z dx <> 0%Z) by (rewrite Hz'; intro EQ; rewrite EQ in *; romega). specialize (Hgc' NEQ'). revert H H0. rewrite 2 strong_spec_red, 2 Qred_iff; simpl. destr_eqb; simpl; nzsimpl; try romega; intros. - rewrite Z2P_correct in *; auto. + rewrite Z2Pos.id in *; auto. - apply Zgcd_mult_rel_prime; rewrite Zgcd_comm; - apply Zgcd_mult_rel_prime; rewrite Zgcd_comm; auto. + apply Zgcd_mult_rel_prime; rewrite Z.gcd_comm; + apply Zgcd_mult_rel_prime; rewrite Z.gcd_comm; auto. rewrite Zgcd_1_rel_prime in *. apply bezout_rel_prime. - destruct (rel_prime_bezout (Z.to_Z ny) (N.to_Z dy)) as [u v Huv]; trivial. + destruct (rel_prime_bezout (ZZ.to_Z ny) (NN.to_Z dy)) as [u v Huv]; trivial. apply Bezout_intro with (u*g')%Z (v*g)%Z. rewrite <- Huv, <- Hg1', <- Hg2. ring. rewrite Zgcd_1_rel_prime in *. apply bezout_rel_prime. - destruct (rel_prime_bezout (Z.to_Z nx) (N.to_Z dx)) as [u v Huv]; trivial. + destruct (rel_prime_bezout (ZZ.to_Z nx) (NN.to_Z dx)) as [u v Huv]; trivial. apply Bezout_intro with (u*g)%Z (v*g')%Z. rewrite <- Huv, <- Hg2', <- Hg1. ring. Qed. @@ -703,16 +703,16 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Definition inv (x: t): t := match x with | Qz z => - match Z.compare Z.zero z with + match ZZ.compare ZZ.zero z with | Eq => zero - | Lt => Qq Z.one (Zabs_N z) - | Gt => Qq Z.minus_one (Zabs_N z) + | Lt => Qq ZZ.one (Zabs_N z) + | Gt => Qq ZZ.minus_one (Zabs_N z) end | Qq n d => - match Z.compare Z.zero n with + match ZZ.compare ZZ.zero n with | Eq => zero | Lt => Qq (Z_of_N d) (Zabs_N n) - | Gt => Qq (Z.opp (Z_of_N d)) (Zabs_N n) + | Gt => Qq (ZZ.opp (Z_of_N d)) (Zabs_N n) end end. @@ -721,29 +721,29 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. destruct x as [ z | n d ]. (* Qz z *) simpl. - rewrite Z.spec_compare; destr_zcompare. + rewrite ZZ.spec_compare; destr_zcompare. (* 0 = z *) rewrite <- H. simpl; nzsimpl; compute; auto. (* 0 < z *) simpl. - destr_eqb; nzsimpl; [ intros; rewrite Zabs_eq in *; romega | intros _ ]. - set (z':=Z.to_Z z) in *; clearbody z'. + destr_eqb; nzsimpl; [ intros; rewrite Z.abs_eq in *; romega | intros _ ]. + set (z':=ZZ.to_Z z) in *; clearbody z'. red; simpl. - rewrite Zabs_eq by romega. - rewrite Z2P_correct by auto. + rewrite Z.abs_eq by romega. + rewrite Z2Pos.id by auto. unfold Qinv; simpl; destruct z'; simpl; auto; discriminate. (* 0 > z *) simpl. - destr_eqb; nzsimpl; [ intros; rewrite Zabs_non_eq in *; romega | intros _ ]. - set (z':=Z.to_Z z) in *; clearbody z'. + destr_eqb; nzsimpl; [ intros; rewrite Z.abs_neq in *; romega | intros _ ]. + set (z':=ZZ.to_Z z) in *; clearbody z'. red; simpl. - rewrite Zabs_non_eq by romega. - rewrite Z2P_correct by romega. + rewrite Z.abs_neq by romega. + rewrite Z2Pos.id by romega. unfold Qinv; simpl; destruct z'; simpl; auto; discriminate. (* Qq n d *) simpl. - rewrite Z.spec_compare; destr_zcompare. + rewrite ZZ.spec_compare; destr_zcompare. (* 0 = n *) rewrite <- H. simpl; nzsimpl. @@ -751,51 +751,51 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. (* 0 < n *) simpl. destr_eqb; nzsimpl; intros. - intros; rewrite Zabs_eq in *; romega. - intros; rewrite Zabs_eq in *; romega. + intros; rewrite Z.abs_eq in *; romega. + intros; rewrite Z.abs_eq in *; romega. nsubst; compute; auto. - set (n':=Z.to_Z n) in *; clearbody n'. - rewrite Zabs_eq by romega. + set (n':=ZZ.to_Z n) in *; clearbody n'. + rewrite Z.abs_eq by romega. red; simpl. - rewrite Z2P_correct by auto. + rewrite Z2Pos.id by auto. unfold Qinv; simpl; destruct n'; simpl; auto; try discriminate. - rewrite Zpos_mult_morphism, Z2P_correct; auto. + rewrite Pos2Z.inj_mul, Z2Pos.id; auto. (* 0 > n *) simpl. destr_eqb; nzsimpl; intros. - intros; rewrite Zabs_non_eq in *; romega. - intros; rewrite Zabs_non_eq in *; romega. + intros; rewrite Z.abs_neq in *; romega. + intros; rewrite Z.abs_neq in *; romega. nsubst; compute; auto. - set (n':=Z.to_Z n) in *; clearbody n'. + set (n':=ZZ.to_Z n) in *; clearbody n'. red; simpl; nzsimpl. - rewrite Zabs_non_eq by romega. - rewrite Z2P_correct by romega. + rewrite Z.abs_neq by romega. + rewrite Z2Pos.id by romega. unfold Qinv; simpl; destruct n'; simpl; auto; try discriminate. - assert (T : forall x, Zneg x = Zopp (Zpos x)) by auto. - rewrite T, Zpos_mult_morphism, Z2P_correct; auto; ring. + assert (T : forall x, Zneg x = Z.opp (Zpos x)) by auto. + rewrite T, Pos2Z.inj_mul, Z2Pos.id; auto; ring. Qed. Definition inv_norm (x: t): t := match x with | Qz z => - match Z.compare Z.zero z with + match ZZ.compare ZZ.zero z with | Eq => zero - | Lt => Qq Z.one (Zabs_N z) - | Gt => Qq Z.minus_one (Zabs_N z) + | Lt => Qq ZZ.one (Zabs_N z) + | Gt => Qq ZZ.minus_one (Zabs_N z) end | Qq n d => - if N.eqb d N.zero then zero else - match Z.compare Z.zero n with + if NN.eqb d NN.zero then zero else + match ZZ.compare ZZ.zero n with | Eq => zero | Lt => - match Z.compare n Z.one with + match ZZ.compare n ZZ.one with | Gt => Qq (Z_of_N d) (Zabs_N n) | _ => Qz (Z_of_N d) end | Gt => - match Z.compare n Z.minus_one with - | Lt => Qq (Z.opp (Z_of_N d)) (Zabs_N n) - | _ => Qz (Z.opp (Z_of_N d)) + match ZZ.compare n ZZ.minus_one with + | Lt => Qq (ZZ.opp (Z_of_N d)) (Zabs_N n) + | _ => Qz (ZZ.opp (Z_of_N d)) end end end. @@ -807,7 +807,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. destruct x as [ z | n d ]. (* Qz z *) simpl. - rewrite Z.spec_compare; destr_zcompare; auto with qarith. + rewrite ZZ.spec_compare; destr_zcompare; auto with qarith. (* Qq n d *) simpl; nzsimpl; destr_eqb. destr_zcompare; simpl; auto with qarith. @@ -818,12 +818,12 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. (* 0 < n *) destr_zcompare; auto with qarith. destr_zcompare; nzsimpl; simpl; auto with qarith; intros. - destr_eqb; nzsimpl; [ intros; rewrite Zabs_eq in *; romega | intros _ ]. + destr_eqb; nzsimpl; [ intros; rewrite Z.abs_eq in *; romega | intros _ ]. rewrite H0; auto with qarith. romega. (* 0 > n *) destr_zcompare; nzsimpl; simpl; auto with qarith. - destr_eqb; nzsimpl; [ intros; rewrite Zabs_non_eq in *; romega | intros _ ]. + destr_eqb; nzsimpl; [ intros; rewrite Z.abs_neq in *; romega | intros _ ]. rewrite H0; auto with qarith. romega. Qed. @@ -847,36 +847,36 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. (* 0 < n *) destr_zcompare; simpl; nzsimpl; auto. destr_eqb; nzsimpl; simpl; auto. - rewrite Zabs_eq; romega. + rewrite Z.abs_eq; romega. intros _. rewrite strong_spec_norm; simpl; nzsimpl. destr_eqb; nzsimpl. - rewrite Zabs_eq; romega. + rewrite Z.abs_eq; romega. intros _. rewrite Qred_iff. simpl. - rewrite Zabs_eq; auto with zarith. - rewrite Z2P_correct in *; auto. - rewrite Zgcd_comm; auto. + rewrite Z.abs_eq; auto with zarith. + rewrite Z2Pos.id in *; auto. + rewrite Z.gcd_comm; auto. (* 0 > n *) destr_eqb; nzsimpl; simpl; auto; intros. destr_zcompare; simpl; nzsimpl; auto. destr_eqb; nzsimpl. - rewrite Zabs_non_eq; romega. + rewrite Z.abs_neq; romega. intros _. rewrite strong_spec_norm; simpl; nzsimpl. destr_eqb; nzsimpl. - rewrite Zabs_non_eq; romega. + rewrite Z.abs_neq; romega. intros _. rewrite Qred_iff. simpl. - rewrite Z2P_correct in *; auto. + rewrite Z2Pos.id in *; auto. intros. - rewrite Zgcd_comm, Zgcd_Zabs, Zgcd_comm. + rewrite Z.gcd_comm, Z.gcd_abs_l, Z.gcd_comm. apply Zis_gcd_gcd; auto with zarith. apply Zis_gcd_minus. - rewrite Zopp_involutive, <- H1; apply Zgcd_is_gcd. - rewrite Zabs_non_eq; romega. + rewrite Z.opp_involutive, <- H1; apply Zgcd_is_gcd. + rewrite Z.abs_neq; romega. Qed. Definition div x y := mul x (inv y). @@ -909,31 +909,30 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Definition square (x: t): t := match x with - | Qz zx => Qz (Z.square zx) - | Qq nx dx => Qq (Z.square nx) (N.square dx) + | Qz zx => Qz (ZZ.square zx) + | Qq nx dx => Qq (ZZ.square nx) (NN.square dx) end. Theorem spec_square : forall x, [square x] == [x] ^ 2. Proof. destruct x as [ z | n d ]. - simpl; rewrite Z.spec_square; red; auto. + simpl; rewrite ZZ.spec_square; red; auto. simpl. destr_eqb; nzsimpl; intros. apply Qeq_refl. - rewrite N.spec_square in *; nzsimpl. - match goal with E : (_ * _ = 0)%Z |- _ => - elim (Zmult_integral _ _ E); romega end. - rewrite N.spec_square in *; nzsimpl; nsubst; romega. - rewrite Z.spec_square, N.spec_square. + rewrite NN.spec_square in *; nzsimpl. + rewrite Z.mul_eq_0 in *; romega. + rewrite NN.spec_square in *; nzsimpl; nsubst; romega. + rewrite ZZ.spec_square, NN.spec_square. red; simpl. - rewrite Zpos_mult_morphism; rewrite !Z2P_correct; auto. - apply Zmult_lt_0_compat; auto. + rewrite Pos2Z.inj_mul; rewrite !Z2Pos.id; auto. + apply Z.mul_pos_pos; auto. Qed. Definition power_pos (x : t) p : t := match x with - | Qz zx => Qz (Z.pow_pos zx p) - | Qq nx dx => Qq (Z.pow_pos nx p) (N.pow_pos dx p) + | Qz zx => Qz (ZZ.pow_pos zx p) + | Qq nx dx => Qq (ZZ.pow_pos nx p) (NN.pow_pos dx p) end. Theorem spec_power_pos : forall x p, [power_pos x p] == [x] ^ Zpos p. @@ -941,26 +940,26 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. intros [ z | n d ] p; unfold power_pos. (* Qz *) simpl. - rewrite Z.spec_pow_pos. - rewrite Qpower_decomp. + rewrite ZZ.spec_pow_pos, Qpower_decomp. red; simpl; f_equal. - rewrite Zpower_pos_1_l; auto. + now rewrite Pos2Z.inj_pow, Z.pow_1_l. (* Qq *) simpl. - rewrite Z.spec_pow_pos. + rewrite ZZ.spec_pow_pos. destr_eqb; nzsimpl; intros. - apply Qeq_sym; apply Qpower_positive_0. - rewrite N.spec_pow_pos in *. - assert (0 < N.to_Z d ^ ' p)%Z by - (apply Zpower_gt_0; auto with zarith). - romega. - exfalso. - rewrite N.spec_pow_pos in *. nsubst. - rewrite Zpower_0_l in *; [romega|discriminate]. - rewrite Qpower_decomp. - red; simpl; do 3 f_equal. - rewrite Z2P_correct by (generalize (N.spec_pos d); romega). - rewrite N.spec_pow_pos. auto. + - apply Qeq_sym; apply Qpower_positive_0. + - rewrite NN.spec_pow_pos in *. + assert (0 < NN.to_Z d ^ ' p)%Z by + (apply Z.pow_pos_nonneg; auto with zarith). + romega. + - exfalso. + rewrite NN.spec_pow_pos in *. nsubst. + rewrite Z.pow_0_l' in *; [romega|discriminate]. + - rewrite Qpower_decomp. + red; simpl; do 3 f_equal. + apply Pos2Z.inj. rewrite Pos2Z.inj_pow. + rewrite 2 Z2Pos.id by (generalize (NN.spec_pos d); romega). + now rewrite NN.spec_pow_pos. Qed. Instance strong_spec_power_pos x p `(Reduced x) : Reduced (power_pos x p). @@ -976,10 +975,10 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. unfold Reduced; rewrite strong_spec_red, Qred_iff; simpl. destr_eqb; nzsimpl; simpl; intros. exfalso. - rewrite N.spec_pow_pos in *. nsubst. - rewrite Zpower_0_l in *; [romega|discriminate]. - rewrite Z2P_correct in *; auto. - rewrite N.spec_pow_pos, Z.spec_pow_pos; auto. + rewrite NN.spec_pow_pos in *. nsubst. + rewrite Z.pow_0_l' in *; [romega|discriminate]. + rewrite Z2Pos.id in *; auto. + rewrite NN.spec_pow_pos, ZZ.spec_pow_pos; auto. rewrite Zgcd_1_rel_prime in *. apply rel_prime_Zpower; auto with zarith. Qed. @@ -1086,7 +1085,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. [[add x y]] = [[x]] + [[y]]. Proof. unfold to_Qc. - apply trans_equal with (!! ([x] + [y])). + transitivity (!! ([x] + [y])). unfold Q2Qc. apply Qc_decomp; intros _ _; unfold this. apply Qred_complete; apply spec_add; auto. @@ -1100,7 +1099,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. [[add_norm x y]] = [[x]] + [[y]]. Proof. unfold to_Qc. - apply trans_equal with (!! ([x] + [y])). + transitivity (!! ([x] + [y])). unfold Q2Qc. apply Qc_decomp; intros _ _; unfold this. apply Qred_complete; apply spec_add_norm; auto. @@ -1148,7 +1147,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. [[mul x y]] = [[x]] * [[y]]. Proof. unfold to_Qc. - apply trans_equal with (!! ([x] * [y])). + transitivity (!! ([x] * [y])). unfold Q2Qc. apply Qc_decomp; intros _ _; unfold this. apply Qred_complete; apply spec_mul; auto. @@ -1162,7 +1161,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. [[mul_norm x y]] = [[x]] * [[y]]. Proof. unfold to_Qc. - apply trans_equal with (!! ([x] * [y])). + transitivity (!! ([x] * [y])). unfold Q2Qc. apply Qc_decomp; intros _ _; unfold this. apply Qred_complete; apply spec_mul_norm; auto. @@ -1186,7 +1185,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. [[inv x]] = /[[x]]. Proof. unfold to_Qc. - apply trans_equal with (!! (/[x])). + transitivity (!! (/[x])). unfold Q2Qc. apply Qc_decomp; intros _ _; unfold this. apply Qred_complete; apply spec_inv; auto. @@ -1200,7 +1199,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. [[inv_norm x]] = /[[x]]. Proof. unfold to_Qc. - apply trans_equal with (!! (/[x])). + transitivity (!! (/[x])). unfold Q2Qc. apply Qc_decomp; intros _ _; unfold this. apply Qred_complete; apply spec_inv_norm; auto. @@ -1248,7 +1247,7 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Theorem spec_squarec x: [[square x]] = [[x]]^2. Proof. unfold to_Qc. - apply trans_equal with (!! ([x]^2)). + transitivity (!! ([x]^2)). unfold Q2Qc. apply Qc_decomp; intros _ _; unfold this. apply Qred_complete; apply spec_square; auto. @@ -1262,24 +1261,24 @@ Module Make (N:NType)(Z:ZType)(Import NZ:NType_ZType N Z) <: QType. Qed. Theorem spec_power_posc x p: - [[power_pos x p]] = [[x]] ^ nat_of_P p. + [[power_pos x p]] = [[x]] ^ Pos.to_nat p. Proof. unfold to_Qc. - apply trans_equal with (!! ([x]^Zpos p)). + transitivity (!! ([x]^Zpos p)). unfold Q2Qc. apply Qc_decomp; intros _ _; unfold this. apply Qred_complete; apply spec_power_pos; auto. - induction p using Pind. + induction p using Pos.peano_ind. simpl; ring. - rewrite Psucc_S; simpl Qcpower. + rewrite Pos2Nat.inj_succ; simpl Qcpower. rewrite <- IHp; clear IHp. unfold Qcmult, Q2Qc. apply Qc_decomp; intros _ _; unfold this. apply Qred_complete. - setoid_replace ([x] ^ ' Psucc p)%Q with ([x] * [x] ^ ' p)%Q. + setoid_replace ([x] ^ ' Pos.succ p)%Q with ([x] * [x] ^ ' p)%Q. apply Qmult_comp; apply Qeq_sym; apply Qred_correct. simpl. - rewrite Pplus_one_succ_l. + rewrite <- Pos.add_1_l. rewrite Qpower_plus_positive; simpl; apply Qeq_refl. Qed. |