diff options
Diffstat (limited to 'theories/Numbers/Natural/BigN/Nbasic.v')
| -rw-r--r-- | theories/Numbers/Natural/BigN/Nbasic.v | 120 |
1 files changed, 60 insertions, 60 deletions
diff --git a/theories/Numbers/Natural/BigN/Nbasic.v b/theories/Numbers/Natural/BigN/Nbasic.v index 4717d0b2..5bde1008 100644 --- a/theories/Numbers/Natural/BigN/Nbasic.v +++ b/theories/Numbers/Natural/BigN/Nbasic.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -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. |