diff options
Diffstat (limited to 'theories/ZArith/Zdigits.v')
| -rw-r--r-- | theories/ZArith/Zdigits.v | 58 |
1 files changed, 29 insertions, 29 deletions
diff --git a/theories/ZArith/Zdigits.v b/theories/ZArith/Zdigits.v index ff1d96df..fa8f5c27 100644 --- a/theories/ZArith/Zdigits.v +++ b/theories/ZArith/Zdigits.v @@ -1,7 +1,7 @@ (* -*- coding: utf-8 -*- *) (************************************************************************) (* 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 *) @@ -64,7 +64,7 @@ Section ENCODING_VALUE. (** We compute the binary value via a Horner scheme. Computation stops at the vector length without checks. - We define a function Zmod2 similar to Zdiv2 returning the + We define a function Zmod2 similar to Z.div2 returning the quotient of division z=2q+r with 0<=r<=1. The two's complement value is also computed via a Horner scheme with Zmod2, the parameter is the size minus one. @@ -88,16 +88,16 @@ Section ENCODING_VALUE. Lemma Zmod2_twice : - forall z:Z, z = (2 * Zmod2 z + bit_value (Zeven.Zodd_bool z))%Z. + forall z:Z, z = (2 * Zmod2 z + bit_value (Z.odd z))%Z. Proof. - destruct z; simpl in |- *. + destruct z; simpl. trivial. - destruct p; simpl in |- *; trivial. + destruct p; simpl; trivial. - destruct p; simpl in |- *. - destruct p as [p| p| ]; simpl in |- *. - rewrite <- (Pdouble_minus_one_o_succ_eq_xI p); trivial. + destruct p; simpl. + destruct p as [p| p| ]; simpl. + rewrite <- (Pos.pred_double_succ p); trivial. trivial. @@ -113,15 +113,15 @@ Section ENCODING_VALUE. simple induction n; intros. exact Bnil. - exact (Bcons (Zeven.Zodd_bool H0) n0 (H (Zeven.Zdiv2 H0))). + exact (Bcons (Z.odd H0) n0 (H (Z.div2 H0))). Defined. Lemma Z_to_two_compl : forall n:nat, Z -> Bvector (S n). Proof. simple induction n; intros. - exact (Bcons (Zeven.Zodd_bool H) 0 Bnil). + exact (Bcons (Z.odd H) 0 Bnil). - exact (Bcons (Zeven.Zodd_bool H0) (S n0) (H (Zmod2 H0))). + exact (Bcons (Z.odd H0) (S n0) (H (Zmod2 H0))). Defined. End ENCODING_VALUE. @@ -145,17 +145,17 @@ Section Z_BRIC_A_BRAC. (z >= 0)%Z -> Z_to_binary (S n) (bit_value b + 2 * z) = Bcons b n (Z_to_binary n z). Proof. - destruct b; destruct z; simpl in |- *; auto. + destruct b; destruct z; simpl; auto. intro H; elim H; trivial. Qed. Lemma binary_value_pos : forall (n:nat) (bv:Bvector n), (binary_value n bv >= 0)%Z. Proof. - induction bv as [| a n v IHbv]; simpl in |- *. + induction bv as [| a n v IHbv]; simpl. omega. - destruct a; destruct (binary_value n v); simpl in |- *; auto. + destruct a; destruct (binary_value n v); simpl; auto. auto with zarith. Qed. @@ -174,34 +174,34 @@ Section Z_BRIC_A_BRAC. Proof. destruct b; destruct z as [| p| p]; auto. destruct p as [p| p| ]; auto. - destruct p as [p| p| ]; simpl in |- *; auto. - intros; rewrite (Psucc_o_double_minus_one_eq_xO p); trivial. + destruct p as [p| p| ]; simpl; auto. + intros; rewrite (Pos.succ_pred_double p); trivial. Qed. Lemma Z_to_binary_Sn_z : forall (n:nat) (z:Z), Z_to_binary (S n) z = - Bcons (Zeven.Zodd_bool z) n (Z_to_binary n (Zeven.Zdiv2 z)). + Bcons (Z.odd z) n (Z_to_binary n (Z.div2 z)). Proof. intros; auto. Qed. Lemma Z_div2_value : forall z:Z, - (z >= 0)%Z -> (bit_value (Zeven.Zodd_bool z) + 2 * Zeven.Zdiv2 z)%Z = z. + (z >= 0)%Z -> (bit_value (Z.odd z) + 2 * Z.div2 z)%Z = z. Proof. destruct z as [| p| p]; auto. destruct p; auto. intro H; elim H; trivial. Qed. - Lemma Pdiv2 : forall z:Z, (z >= 0)%Z -> (Zeven.Zdiv2 z >= 0)%Z. + Lemma Pdiv2 : forall z:Z, (z >= 0)%Z -> (Z.div2 z >= 0)%Z. Proof. destruct z as [| p| p]. auto. destruct p; auto. - simpl in |- *; intros; omega. + simpl; intros; omega. intro H; elim H; trivial. Qed. @@ -209,10 +209,10 @@ Section Z_BRIC_A_BRAC. Lemma Zdiv2_two_power_nat : forall (z:Z) (n:nat), (z >= 0)%Z -> - (z < two_power_nat (S n))%Z -> (Zeven.Zdiv2 z < two_power_nat n)%Z. + (z < two_power_nat (S n))%Z -> (Z.div2 z < two_power_nat n)%Z. Proof. intros. - cut (2 * Zeven.Zdiv2 z < 2 * two_power_nat n)%Z; intros. + cut (2 * Z.div2 z < 2 * two_power_nat n)%Z; intros. omega. rewrite <- two_power_nat_S. @@ -225,23 +225,23 @@ Section Z_BRIC_A_BRAC. Lemma Z_to_two_compl_Sn_z : forall (n:nat) (z:Z), Z_to_two_compl (S n) z = - Bcons (Zeven.Zodd_bool z) (S n) (Z_to_two_compl n (Zmod2 z)). + Bcons (Z.odd z) (S n) (Z_to_two_compl n (Zmod2 z)). Proof. intros; auto. Qed. Lemma Zeven_bit_value : - forall z:Z, Zeven.Zeven z -> bit_value (Zeven.Zodd_bool z) = 0%Z. + forall z:Z, Zeven.Zeven z -> bit_value (Z.odd z) = 0%Z. Proof. - destruct z; unfold bit_value in |- *; auto. + destruct z; unfold bit_value; auto. destruct p; tauto || (intro H; elim H). destruct p; tauto || (intro H; elim H). Qed. Lemma Zodd_bit_value : - forall z:Z, Zeven.Zodd z -> bit_value (Zeven.Zodd_bool z) = 1%Z. + forall z:Z, Zeven.Zodd z -> bit_value (Z.odd z) = 1%Z. Proof. - destruct z; unfold bit_value in |- *; auto. + destruct z; unfold bit_value; auto. intros; elim H. destruct p; tauto || (intros; elim H). destruct p; tauto || (intros; elim H). @@ -310,7 +310,7 @@ Section COHERENT_VALUE. (z < two_power_nat n)%Z -> binary_value n (Z_to_binary n z) = z. Proof. induction n as [| n IHn]. - unfold two_power_nat, shift_nat in |- *; simpl in |- *; intros; omega. + unfold two_power_nat, shift_nat; simpl; intros; omega. intros; rewrite Z_to_binary_Sn_z. rewrite binary_value_Sn. @@ -328,7 +328,7 @@ Section COHERENT_VALUE. (z < two_power_nat n)%Z -> two_compl_value n (Z_to_two_compl n z) = z. Proof. induction n as [| n IHn]. - unfold two_power_nat, shift_nat in |- *; simpl in |- *; intros. + unfold two_power_nat, shift_nat; simpl; intros. assert (z = (-1)%Z \/ z = 0%Z). omega. intuition; subst z; trivial. |