diff options
Diffstat (limited to 'theories/ZArith/Zdigits.v')
| -rw-r--r-- | theories/ZArith/Zdigits.v | 30 |
1 files changed, 15 insertions, 15 deletions
diff --git a/theories/ZArith/Zdigits.v b/theories/ZArith/Zdigits.v index ff1d96df4..a9348785a 100644 --- a/theories/ZArith/Zdigits.v +++ b/theories/ZArith/Zdigits.v @@ -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. @@ -88,7 +88,7 @@ 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 |- *. trivial. @@ -97,7 +97,7 @@ Section ENCODING_VALUE. destruct p; simpl in |- *. destruct p as [p| p| ]; simpl in |- *. - rewrite <- (Pdouble_minus_one_o_succ_eq_xI p); trivial. + 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. @@ -175,27 +175,27 @@ Section Z_BRIC_A_BRAC. 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. + 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. @@ -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. @@ -225,13 +225,13 @@ 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 p; tauto || (intro H; elim H). @@ -239,7 +239,7 @@ Section Z_BRIC_A_BRAC. 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. intros; elim H. |