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diff --git a/theories/ZArith/Zbinary.v b/theories/ZArith/Zbinary.v new file mode 100644 index 00000000..fa5f00dc --- /dev/null +++ b/theories/ZArith/Zbinary.v @@ -0,0 +1,426 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Zbinary.v,v 1.6.2.1 2004/07/16 19:31:21 herbelin Exp $ i*) + +(** Bit vectors interpreted as integers. + Contribution by Jean Duprat (ENS Lyon). *) + +Require Import Bvector. +Require Import ZArith. +Require Export Zpower. +Require Import Omega. + +(* +L'évaluation des vecteurs de booléens se font à la fois en binaire et +en complément à deux. Le nombre appartient à Z. +On utilise donc Omega pour faire les calculs dans Z. +De plus, on utilise les fonctions 2^n où n est un naturel, ici la longueur. + two_power_nat = [n:nat](POS (shift_nat n xH)) + : nat->Z + two_power_nat_S + : (n:nat)`(two_power_nat (S n)) = 2*(two_power_nat n)` + Z_lt_ge_dec + : (x,y:Z){`x < y`}+{`x >= y`} +*) + + +Section VALUE_OF_BOOLEAN_VECTORS. + +(* +Les calculs sont effectués dans la convention positive usuelle. +Les valeurs correspondent soit à l'écriture binaire (nat), +soit au complément à deux (int). +On effectue le calcul suivant le schéma de Horner. +Le complément à deux n'a de sens que sur les vecteurs de taille +supérieure ou égale à un, le bit de signe étant évalué négativement. +*) + +Definition bit_value (b:bool) : Z := + match b with + | true => 1%Z + | false => 0%Z + end. + +Lemma binary_value : forall n:nat, Bvector n -> Z. +Proof. + simple induction n; intros. + exact 0%Z. + + inversion H0. + exact (bit_value a + 2 * H H2)%Z. +Defined. + +Lemma two_compl_value : forall n:nat, Bvector (S n) -> Z. +Proof. + simple induction n; intros. + inversion H. + exact (- bit_value a)%Z. + + inversion H0. + exact (bit_value a + 2 * H H2)%Z. +Defined. + +(* +Coq < Eval Compute in (binary_value (3) (Bcons true (2) (Bcons false (1) (Bcons true (0) Bnil)))). + = `5` + : Z +*) + +(* +Coq < Eval Compute in (two_compl_value (3) (Bcons true (3) (Bcons false (2) (Bcons true (1) (Bcons true (0) Bnil))))). + = `-3` + : Z +*) + +End VALUE_OF_BOOLEAN_VECTORS. + +Section ENCODING_VALUE. + +(* +On calcule la valeur binaire selon un schema de Horner. +Le calcul s'arrete à la longueur du vecteur sans vérification. +On definit une fonction Zmod2 calquee sur Zdiv2 mais donnant le quotient +de la division z=2q+r avec 0<=r<=1. +La valeur en complément à deux est calculée selon un schema de Horner +avec Zmod2, le paramètre est la taille moins un. +*) + +Definition Zmod2 (z:Z) := + match z with + | Z0 => 0%Z + | Zpos p => match p with + | xI q => Zpos q + | xO q => Zpos q + | xH => 0%Z + end + | Zneg p => + match p with + | xI q => (Zneg q - 1)%Z + | xO q => Zneg q + | xH => (-1)%Z + end + end. + + +Lemma Zmod2_twice : + forall z:Z, z = (2 * Zmod2 z + bit_value (Zeven.Zodd_bool z))%Z. +Proof. + destruct z; simpl in |- *. + trivial. + + destruct p; simpl in |- *; trivial. + + destruct p; simpl in |- *. + destruct p as [p| p| ]; simpl in |- *. + rewrite <- (Pdouble_minus_one_o_succ_eq_xI p); trivial. + + trivial. + + trivial. + + trivial. + + trivial. +Qed. + +Lemma Z_to_binary : forall n:nat, Z -> Bvector n. +Proof. + simple induction n; intros. + exact Bnil. + + exact (Bcons (Zeven.Zodd_bool H0) n0 (H (Zeven.Zdiv2 H0))). +Defined. + +(* +Eval Compute in (Z_to_binary (5) `5`). + = (Vcons bool true (4) + (Vcons bool false (3) + (Vcons bool true (2) + (Vcons bool false (1) (Vcons bool false (0) (Vnil bool)))))) + : (Bvector (5)) +*) + +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 (Zeven.Zodd_bool H0) (S n0) (H (Zmod2 H0))). +Defined. + +(* +Eval Compute in (Z_to_two_compl (3) `0`). + = (Vcons bool false (3) + (Vcons bool false (2) + (Vcons bool false (1) (Vcons bool false (0) (Vnil bool))))) + : (vector bool (4)) + +Eval Compute in (Z_to_two_compl (3) `5`). + = (Vcons bool true (3) + (Vcons bool false (2) + (Vcons bool true (1) (Vcons bool false (0) (Vnil bool))))) + : (vector bool (4)) + +Eval Compute in (Z_to_two_compl (3) `-5`). + = (Vcons bool true (3) + (Vcons bool true (2) + (Vcons bool false (1) (Vcons bool true (0) (Vnil bool))))) + : (vector bool (4)) +*) + +End ENCODING_VALUE. + +Section Z_BRIC_A_BRAC. + +(* +Bibliotheque de lemmes utiles dans la section suivante. +Utilise largement ZArith. +Meriterait d'etre reecrite. +*) + +Lemma binary_value_Sn : + forall (n:nat) (b:bool) (bv:Bvector n), + binary_value (S n) (Vcons bool b n bv) = + (bit_value b + 2 * binary_value n bv)%Z. +Proof. + intros; auto. +Qed. + +Lemma Z_to_binary_Sn : + forall (n:nat) (b:bool) (z:Z), + (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. + 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 |- *. + omega. + + destruct a; destruct (binary_value n v); simpl in |- *; auto. + auto with zarith. +Qed. + + +Lemma two_compl_value_Sn : + forall (n:nat) (bv:Bvector (S n)) (b:bool), + two_compl_value (S n) (Bcons b (S n) bv) = + (bit_value b + 2 * two_compl_value n bv)%Z. +Proof. + intros; auto. +Qed. + +Lemma Z_to_two_compl_Sn : + forall (n:nat) (b:bool) (z:Z), + Z_to_two_compl (S n) (bit_value b + 2 * z) = + Bcons b (S n) (Z_to_two_compl n z). +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. +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)). +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. +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. +Proof. + destruct z as [| p| p]. + auto. + + destruct p; auto. + simpl in |- *; intros; omega. + + intro H; elim H; trivial. + +Qed. + +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. +Proof. + intros. + cut (2 * Zeven.Zdiv2 z < 2 * two_power_nat n)%Z; intros. + omega. + + rewrite <- two_power_nat_S. + destruct (Zeven.Zeven_odd_dec z); intros. + rewrite <- Zeven.Zeven_div2; auto. + + generalize (Zeven.Zodd_div2 z H z0); omega. +Qed. + +(* + +Lemma Z_minus_one_or_zero : (z:Z) + `z >= -1` -> + `z < 1` -> + {`z=-1`} + {`z=0`}. +Proof. + NewDestruct z; Auto. + NewDestruct p; Auto. + Tauto. + + Tauto. + + Intros. + Right; Omega. + + NewDestruct p. + Tauto. + + Tauto. + + Intros; Left; Omega. +Save. +*) + +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)). +Proof. + intros; auto. +Qed. + +Lemma Zeven_bit_value : + forall z:Z, Zeven.Zeven z -> bit_value (Zeven.Zodd_bool z) = 0%Z. +Proof. + destruct z; unfold bit_value in |- *; 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. +Proof. + destruct z; unfold bit_value in |- *; auto. + intros; elim H. + destruct p; tauto || (intros; elim H). + destruct p; tauto || (intros; elim H). +Qed. + +Lemma Zge_minus_two_power_nat_S : + forall (n:nat) (z:Z), + (z >= - two_power_nat (S n))%Z -> (Zmod2 z >= - two_power_nat n)%Z. +Proof. + intros n z; rewrite (two_power_nat_S n). + generalize (Zmod2_twice z). + destruct (Zeven.Zeven_odd_dec z) as [H| H]. + rewrite (Zeven_bit_value z H); intros; omega. + + rewrite (Zodd_bit_value z H); intros; omega. +Qed. + +Lemma Zlt_two_power_nat_S : + forall (n:nat) (z:Z), + (z < two_power_nat (S n))%Z -> (Zmod2 z < two_power_nat n)%Z. +Proof. + intros n z; rewrite (two_power_nat_S n). + generalize (Zmod2_twice z). + destruct (Zeven.Zeven_odd_dec z) as [H| H]. + rewrite (Zeven_bit_value z H); intros; omega. + + rewrite (Zodd_bit_value z H); intros; omega. +Qed. + +End Z_BRIC_A_BRAC. + +Section COHERENT_VALUE. + +(* +On vérifie que dans l'intervalle de définition les fonctions sont +réciproques l'une de l'autre. +Elles utilisent les lemmes du bric-a-brac. +*) + +Lemma binary_to_Z_to_binary : + forall (n:nat) (bv:Bvector n), Z_to_binary n (binary_value n bv) = bv. +Proof. + induction bv as [| a n bv IHbv]. + auto. + + rewrite binary_value_Sn. + rewrite Z_to_binary_Sn. + rewrite IHbv; trivial. + + apply binary_value_pos. +Qed. + +Lemma two_compl_to_Z_to_two_compl : + forall (n:nat) (bv:Bvector n) (b:bool), + Z_to_two_compl n (two_compl_value n (Bcons b n bv)) = Bcons b n bv. +Proof. + induction bv as [| a n bv IHbv]; intro b. + destruct b; auto. + + rewrite two_compl_value_Sn. + rewrite Z_to_two_compl_Sn. + rewrite IHbv; trivial. +Qed. + +Lemma Z_to_binary_to_Z : + forall (n:nat) (z:Z), + (z >= 0)%Z -> + (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. + + intros; rewrite Z_to_binary_Sn_z. + rewrite binary_value_Sn. + rewrite IHn. + apply Z_div2_value; auto. + + apply Pdiv2; trivial. + + apply Zdiv2_two_power_nat; trivial. +Qed. + +Lemma Z_to_two_compl_to_Z : + forall (n:nat) (z:Z), + (z >= - two_power_nat n)%Z -> + (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. + assert (z = (-1)%Z \/ z = 0%Z). omega. +intuition; subst z; trivial. + + intros; rewrite Z_to_two_compl_Sn_z. + rewrite two_compl_value_Sn. + rewrite IHn. + generalize (Zmod2_twice z); omega. + + apply Zge_minus_two_power_nat_S; auto. + + apply Zlt_two_power_nat_S; auto. +Qed. + +End COHERENT_VALUE. |