Zdigits.v

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Bit vectors interpreted as integers. Contribution by Jean Duprat (ENS Lyon).

Require Import Bvector.
Require Import ZArith.
Require Export Zpower.
Require Import Omega.

The evaluation of boolean vector is done both in binary and two's complement. The computed number belongs to Z. We hence use Omega to perform computations in Z. Moreover, we use functions 2^n where n is a natural number (here the vector length).

Section VALUE_OF_BOOLEAN_VECTORS.

Computations are done in the usual convention. The values correspond either to the binary coding (nat) or to the two's complement coding (int). We perform the computation via Horner scheme. The two's complement coding only makes sense on vectors whose size is greater or equal to one (a sign bit should be present).

  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.forall n : nat, Bvector n -> Z
  Proof.forall n : nat, Bvector n -> Z
    refine (nat_rect _ _ _); intros.H:Bvector 0
Z
n:nat
H:Bvector n -> Z
H0:Bvector (S n)
Z
    exact 0%Z.n:nat
H:Bvector n -> Z
H0:Bvector (S n)
Z

    inversion H0.n:nat
H:Bvector n -> Z
H0:Bvector (S n)
n0:nat
h:bool
H2:Vector.t bool n
H1:n0 = n
Z
    exact (bit_value h + 2 * H H2)%Z.
  Defined.

  Lemma two_compl_value : forall n:nat, Bvector (S n) -> Z.forall n : nat, Bvector (S n) -> Z
  Proof.forall n : nat, Bvector (S n) -> Z
    simple induction n; intros.n:nat
H:Bvector 1
Z
n, n0:nat
H:Bvector (S n0) -> Z
H0:Bvector (S (S n0))
Z
    inversion H.n:nat
H:Bvector 1
n0:nat
h:bool
H1:Vector.t bool 0
H0:n0 = 0
Z
n, n0:nat
H:Bvector (S n0) -> Z
H0:Bvector (S (S n0))
Z
    exact (- bit_value h)%Z.n, n0:nat
H:Bvector (S n0) -> Z
H0:Bvector (S (S n0))
Z

    inversion H0.n, n0:nat
H:Bvector (S n0) -> Z
H0:Bvector (S (S n0))
n1:nat
h:bool
H2:Vector.t bool (S n0)
H1:n1 = S n0
Z
    exact (bit_value h + 2 * H H2)%Z.
  Defined.

End VALUE_OF_BOOLEAN_VECTORS.

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 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.

  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 (Z.odd z))%Z.forall z : Z,
z = (2 * Zmod2 z + bit_value (Z.odd z))%Z
  Proof.forall z : Z,
z = (2 * Zmod2 z + bit_value (Z.odd z))%Z
    destruct z; simpl.0%Z = 0%Z
p:positive
Z.pos p =
(match
   match p with
   | (q~1)%positive | (q~0)%positive => Z.pos q
   | 1%positive => 0
   end
 with
 | 0 => 0
 | Z.pos y' => Z.pos y'~0
 | Z.neg y' => Z.neg y'~0
 end +
 bit_value
   match p with
   | (_~0)%positive => false
   | _ => true
   end)%Z
p:positive
Z.neg p =
(match
   match p with
   | (q~1)%positive => Z.neg (q + 1)
   | (q~0)%positive => Z.neg q
   | 1%positive => -1
   end
 with
 | 0 => 0
 | Z.pos y' => Z.pos y'~0
 | Z.neg y' => Z.neg y'~0
 end +
 bit_value
   match p with
   | (_~0)%positive => false
   | _ => true
   end)%Z
    trivial.p:positive
Z.pos p =
(match
   match p with
   | (q~1)%positive | (q~0)%positive => Z.pos q
   | 1%positive => 0
   end
 with
 | 0 => 0
 | Z.pos y' => Z.pos y'~0
 | Z.neg y' => Z.neg y'~0
 end +
 bit_value
   match p with
   | (_~0)%positive => false
   | _ => true
   end)%Z
p:positive
Z.neg p =
(match
   match p with
   | (q~1)%positive => Z.neg (q + 1)
   | (q~0)%positive => Z.neg q
   | 1%positive => -1
   end
 with
 | 0 => 0
 | Z.pos y' => Z.pos y'~0
 | Z.neg y' => Z.neg y'~0
 end +
 bit_value
   match p with
   | (_~0)%positive => false
   | _ => true
   end)%Z

    destruct p; simpl; trivial.p:positive
Z.neg p =
(match
   match p with
   | (q~1)%positive => Z.neg (q + 1)
   | (q~0)%positive => Z.neg q
   | 1%positive => -1
   end
 with
 | 0 => 0
 | Z.pos y' => Z.pos y'~0
 | Z.neg y' => Z.neg y'~0
 end +
 bit_value
   match p with
   | (_~0)%positive => false
   | _ => true
   end)%Z

    destruct p; simpl.p:positive
Z.neg p~1 = Z.neg (Pos.pred_double (p + 1))
p:positive
Z.neg p~0 = Z.neg p~0
(-1)%Z = (-1)%Z
    destruct p as [p| p| ]; simpl.p:positive
Z.neg p~1~1 = Z.neg (Pos.pred_double (Pos.succ p))~1
p:positive
Z.neg p~0~1 = Z.neg p~0~1
(-3)%Z = (-3)%Z
p:positive
Z.neg p~0 = Z.neg p~0
(-1)%Z = (-1)%Z
    rewrite <- (Pos.pred_double_succ p); trivial.p:positive
Z.neg p~0~1 = Z.neg p~0~1
(-3)%Z = (-3)%Z
p:positive
Z.neg p~0 = Z.neg p~0
(-1)%Z = (-1)%Z

    trivial.(-3)%Z = (-3)%Z
p:positive
Z.neg p~0 = Z.neg p~0
(-1)%Z = (-1)%Z

    trivial.p:positive
Z.neg p~0 = Z.neg p~0
(-1)%Z = (-1)%Z

    trivial.(-1)%Z = (-1)%Z

    trivial.
  Qed.

  Lemma Z_to_binary : forall n:nat, Z -> Bvector n.forall n : nat, Z -> Bvector n
  Proof.forall n : nat, Z -> Bvector n
    simple induction n; intros.n:nat
H:Z
Bvector 0
n, n0:nat
H:Z -> Bvector n0
H0:Z
Bvector (S n0)
    exact Bnil.n, n0:nat
H:Z -> Bvector n0
H0:Z
Bvector (S n0)

    exact (Bcons (Z.odd H0) n0 (H (Z.div2 H0))).
  Defined.

  Lemma Z_to_two_compl : forall n:nat, Z -> Bvector (S n).forall n : nat, Z -> Bvector (S n)
  Proof.forall n : nat, Z -> Bvector (S n)
    simple induction n; intros.n:nat
H:Z
Bvector 1
n, n0:nat
H:Z -> Bvector (S n0)
H0:Z
Bvector (S (S n0))
    exact (Bcons (Z.odd H) 0 Bnil).n, n0:nat
H:Z -> Bvector (S n0)
H0:Z
Bvector (S (S n0))

    exact (Bcons (Z.odd H0) (S n0) (H (Zmod2 H0))).
  Defined.

End ENCODING_VALUE.

Section Z_BRIC_A_BRAC.

Some auxiliary lemmas used in the next section. Large use of ZArith. Deserve to be properly rewritten.

  Lemma binary_value_Sn :
    forall (n:nat) (b:bool) (bv:Bvector n),
      binary_value (S n) ( b :: bv) =
      (bit_value b + 2 * binary_value n bv)%Z.forall (n : nat) (b : bool) (bv : Bvector n),
binary_value (S n) (b :: bv) =
(bit_value b + 2 * binary_value n bv)%Z
  Proof.forall (n : nat) (b : bool) (bv : Bvector n),
binary_value (S n) (b :: bv) =
(bit_value b + 2 * binary_value n bv)%Z
    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).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.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)
    destruct b; destruct z; simpl; auto.n:nat
p:positive
(Z.neg p >= 0)%Z ->
Bcons
  match Pos.pred_double p with
  | (_~0)%positive => false
  | _ => true
  end n
  (Z_to_binary n
     (Z.neg (Pos.div2_up (Pos.pred_double p)))) =
Bcons true n (Z_to_binary n (Z.neg p))
    intro H; elim H; trivial.
  Qed.

  Lemma binary_value_pos :
    forall (n:nat) (bv:Bvector n), (binary_value n bv >= 0)%Z.forall (n : nat) (bv : Bvector n),
(binary_value n bv >= 0)%Z
  Proof.forall (n : nat) (bv : Bvector n),
(binary_value n bv >= 0)%Z
    induction bv as [| a n v IHbv]; cbn.(0 >= 0)%Z
a:bool
n:nat
v:VectorDef.t bool n
IHbv:(binary_value n v >= 0)%Z
(bit_value a +
 match binary_value n v with
 | 0 => 0
 | Z.pos y' => Z.pos y'~0
 | Z.neg y' => Z.neg y'~0
 end >= 0)%Z
    omega.a:bool
n:nat
v:VectorDef.t bool n
IHbv:(binary_value n v >= 0)%Z
(bit_value a +
 match binary_value n v with
 | 0 => 0
 | Z.pos y' => Z.pos y'~0
 | Z.neg y' => Z.neg y'~0
 end >= 0)%Z

    destruct a; destruct (binary_value n v); simpl; auto.n:nat
v:VectorDef.t bool n
IHbv:(0 >= 0)%Z
(1 >= 0)%Z
    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.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.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
    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).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.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)
    destruct b; destruct z as [| p| p]; auto.n:nat
p:positive
Z_to_two_compl (S n) (bit_value true + 2 * Z.neg p) =
Bcons true (S n) (Z_to_two_compl n (Z.neg p))
    destruct p as [p| p| ]; auto.n:nat
p:positive
Z_to_two_compl (S n) (bit_value true + 2 * Z.neg p~0) =
Bcons true (S n) (Z_to_two_compl n (Z.neg p~0))
    destruct p as [p| p| ]; simpl; auto.n:nat
p:positive
Bcons true (S n)
  (Z_to_two_compl n
     (Z.neg (Pos.succ (Pos.pred_double p))~0)) =
Bcons true (S n) (Z_to_two_compl n (Z.neg p~0~0))
    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 (Z.odd z) n (Z_to_binary n (Z.div2 z)).forall (n : nat) (z : Z),
Z_to_binary (S n) z =
Bcons (Z.odd z) n (Z_to_binary n (Z.div2 z))
  Proof.forall (n : nat) (z : Z),
Z_to_binary (S n) z =
Bcons (Z.odd z) n (Z_to_binary n (Z.div2 z))
    intros; auto.
  Qed.

  Lemma Z_div2_value :
    forall z:Z,
      (z >= 0)%Z -> (bit_value (Z.odd z) + 2 * Z.div2 z)%Z = z.forall z : Z,
(z >= 0)%Z ->
(bit_value (Z.odd z) + 2 * Z.div2 z)%Z = z
  Proof.forall z : Z,
(z >= 0)%Z ->
(bit_value (Z.odd z) + 2 * Z.div2 z)%Z = z
    destruct z as [| p| p]; auto.p:positive
(Z.pos p >= 0)%Z ->
(bit_value (Z.odd (Z.pos p)) + 2 * Z.div2 (Z.pos p))%Z =
Z.pos p
p:positive
(Z.neg p >= 0)%Z ->
(bit_value (Z.odd (Z.neg p)) + 2 * Z.div2 (Z.neg p))%Z =
Z.neg p
    destruct p; auto.p:positive
(Z.neg p >= 0)%Z ->
(bit_value (Z.odd (Z.neg p)) + 2 * Z.div2 (Z.neg p))%Z =
Z.neg p
    intro H; elim H; trivial.
  Qed.

  Lemma Pdiv2 : forall z:Z, (z >= 0)%Z -> (Z.div2 z >= 0)%Z.forall z : Z, (z >= 0)%Z -> (Z.div2 z >= 0)%Z
  Proof.forall z : Z, (z >= 0)%Z -> (Z.div2 z >= 0)%Z
    destruct z as [| p| p].(0 >= 0)%Z -> (Z.div2 0 >= 0)%Z
p:positive
(Z.pos p >= 0)%Z -> (Z.div2 (Z.pos p) >= 0)%Z
p:positive
(Z.neg p >= 0)%Z -> (Z.div2 (Z.neg p) >= 0)%Z
    auto.p:positive
(Z.pos p >= 0)%Z -> (Z.div2 (Z.pos p) >= 0)%Z
p:positive
(Z.neg p >= 0)%Z -> (Z.div2 (Z.neg p) >= 0)%Z

    destruct p; auto.(1 >= 0)%Z -> (Z.div2 1 >= 0)%Z
p:positive
(Z.neg p >= 0)%Z -> (Z.div2 (Z.neg p) >= 0)%Z
    simpl; intros; omega.p:positive
(Z.neg p >= 0)%Z -> (Z.div2 (Z.neg p) >= 0)%Z

    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 -> (Z.div2 z < two_power_nat n)%Z.forall (z : Z) (n : nat),
(z >= 0)%Z ->
(z < two_power_nat (S n))%Z ->
(Z.div2 z < two_power_nat n)%Z
  Proof.forall (z : Z) (n : nat),
(z >= 0)%Z ->
(z < two_power_nat (S n))%Z ->
(Z.div2 z < two_power_nat n)%Z
    intros.z:Z
n:nat
H:(z >= 0)%Z
H0:(z < two_power_nat (S n))%Z
(Z.div2 z < two_power_nat n)%Z
    enough (2 * Z.div2 z < 2 * two_power_nat n)%Z by omega.z:Z
n:nat
H:(z >= 0)%Z
H0:(z < two_power_nat (S n))%Z
(2 * Z.div2 z < 2 * two_power_nat n)%Z
    rewrite <- two_power_nat_S.z:Z
n:nat
H:(z >= 0)%Z
H0:(z < two_power_nat (S n))%Z
(2 * Z.div2 z < two_power_nat (S n))%Z
    destruct (Zeven.Zeven_odd_dec z) as [Heven|Hodd]; intros.z:Z
n:nat
H:(z >= 0)%Z
H0:(z < two_power_nat (S n))%Z
Heven:Zeven z
(2 * Z.div2 z < two_power_nat (S n))%Z
z:Z
n:nat
H:(z >= 0)%Z
H0:(z < two_power_nat (S n))%Z
Hodd:Zodd z
(2 * Z.div2 z < two_power_nat (S n))%Z
    rewrite <- Zeven.Zeven_div2; auto.z:Z
n:nat
H:(z >= 0)%Z
H0:(z < two_power_nat (S n))%Z
Hodd:Zodd z
(2 * Z.div2 z < two_power_nat (S n))%Z
    generalize (Zeven.Zodd_div2 z Hodd); omega.
  Qed.

  Lemma Z_to_two_compl_Sn_z :
    forall (n:nat) (z:Z),
      Z_to_two_compl (S n) z =
      Bcons (Z.odd z) (S n) (Z_to_two_compl n (Zmod2 z)).forall (n : nat) (z : Z),
Z_to_two_compl (S n) z =
Bcons (Z.odd z) (S n) (Z_to_two_compl n (Zmod2 z))
  Proof.forall (n : nat) (z : Z),
Z_to_two_compl (S n) z =
Bcons (Z.odd z) (S n) (Z_to_two_compl n (Zmod2 z))
    intros; auto.
  Qed.

  Lemma Zeven_bit_value :
    forall z:Z, Zeven.Zeven z -> bit_value (Z.odd z) = 0%Z.forall z : Z, Zeven z -> bit_value (Z.odd z) = 0%Z
  Proof.forall z : Z, Zeven z -> bit_value (Z.odd z) = 0%Z
    destruct z; unfold bit_value; auto.p:positive
Zeven (Z.pos p) ->
(if Z.odd (Z.pos p) then 1%Z else 0%Z) = 0%Z
p:positive
Zeven (Z.neg p) ->
(if Z.odd (Z.neg p) then 1%Z else 0%Z) = 0%Z
    destruct p; tauto || (intro H; elim H).p:positive
Zeven (Z.neg p) ->
(if Z.odd (Z.neg p) then 1%Z else 0%Z) = 0%Z
    destruct p; tauto || (intro H; elim H).
  Qed.

  Lemma Zodd_bit_value :
    forall z:Z, Zeven.Zodd z -> bit_value (Z.odd z) = 1%Z.forall z : Z, Zodd z -> bit_value (Z.odd z) = 1%Z
  Proof.forall z : Z, Zodd z -> bit_value (Z.odd z) = 1%Z
    destruct z; unfold bit_value; auto.Zodd 0 -> (if Z.odd 0 then 1%Z else 0%Z) = 1%Z
p:positive
Zodd (Z.pos p) ->
(if Z.odd (Z.pos p) then 1%Z else 0%Z) = 1%Z
p:positive
Zodd (Z.neg p) ->
(if Z.odd (Z.neg p) then 1%Z else 0%Z) = 1%Z
    intros; elim H.p:positive
Zodd (Z.pos p) ->
(if Z.odd (Z.pos p) then 1%Z else 0%Z) = 1%Z
p:positive
Zodd (Z.neg p) ->
(if Z.odd (Z.neg p) then 1%Z else 0%Z) = 1%Z
    destruct p; tauto || (intros; elim H).p:positive
Zodd (Z.neg p) ->
(if Z.odd (Z.neg p) then 1%Z else 0%Z) = 1%Z
    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.forall (n : nat) (z : Z),
(z >= - two_power_nat (S n))%Z ->
(Zmod2 z >= - two_power_nat n)%Z
  Proof.forall (n : nat) (z : Z),
(z >= - two_power_nat (S n))%Z ->
(Zmod2 z >= - two_power_nat n)%Z
    intros n z; rewrite (two_power_nat_S n).n:nat
z:Z
(z >= - (2 * two_power_nat n))%Z ->
(Zmod2 z >= - two_power_nat n)%Z
    generalize (Zmod2_twice z).n:nat
z:Z
z = (2 * Zmod2 z + bit_value (Z.odd z))%Z ->
(z >= - (2 * two_power_nat n))%Z ->
(Zmod2 z >= - two_power_nat n)%Z
    destruct (Zeven.Zeven_odd_dec z) as [H| H].n:nat
z:Z
H:Zeven z
z = (2 * Zmod2 z + bit_value (Z.odd z))%Z ->
(z >= - (2 * two_power_nat n))%Z ->
(Zmod2 z >= - two_power_nat n)%Z
n:nat
z:Z
H:Zodd z
z = (2 * Zmod2 z + bit_value (Z.odd z))%Z ->
(z >= - (2 * two_power_nat n))%Z ->
(Zmod2 z >= - two_power_nat n)%Z
    rewrite (Zeven_bit_value z H); intros; omega.n:nat
z:Z
H:Zodd z
z = (2 * Zmod2 z + bit_value (Z.odd z))%Z ->
(z >= - (2 * two_power_nat n))%Z ->
(Zmod2 z >= - two_power_nat n)%Z

    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.forall (n : nat) (z : Z),
(z < two_power_nat (S n))%Z ->
(Zmod2 z < two_power_nat n)%Z
  Proof.forall (n : nat) (z : Z),
(z < two_power_nat (S n))%Z ->
(Zmod2 z < two_power_nat n)%Z
    intros n z; rewrite (two_power_nat_S n).n:nat
z:Z
(z < 2 * two_power_nat n)%Z ->
(Zmod2 z < two_power_nat n)%Z
    generalize (Zmod2_twice z).n:nat
z:Z
z = (2 * Zmod2 z + bit_value (Z.odd z))%Z ->
(z < 2 * two_power_nat n)%Z ->
(Zmod2 z < two_power_nat n)%Z
    destruct (Zeven.Zeven_odd_dec z) as [H| H].n:nat
z:Z
H:Zeven z
z = (2 * Zmod2 z + bit_value (Z.odd z))%Z ->
(z < 2 * two_power_nat n)%Z ->
(Zmod2 z < two_power_nat n)%Z
n:nat
z:Z
H:Zodd z
z = (2 * Zmod2 z + bit_value (Z.odd z))%Z ->
(z < 2 * two_power_nat n)%Z ->
(Zmod2 z < two_power_nat n)%Z
    rewrite (Zeven_bit_value z H); intros; omega.n:nat
z:Z
H:Zodd z
z = (2 * Zmod2 z + bit_value (Z.odd z))%Z ->
(z < 2 * two_power_nat n)%Z ->
(Zmod2 z < two_power_nat n)%Z

    rewrite (Zodd_bit_value z H); intros; omega.
  Qed.

End Z_BRIC_A_BRAC.

Section COHERENT_VALUE.

We check that the functions are reciprocal on the definition interval. This uses earlier library lemmas.

  Lemma binary_to_Z_to_binary :
    forall (n:nat) (bv:Bvector n), Z_to_binary n (binary_value n bv) = bv.forall (n : nat) (bv : Bvector n),
Z_to_binary n (binary_value n bv) = bv
  Proof.forall (n : nat) (bv : Bvector n),
Z_to_binary n (binary_value n bv) = bv
    induction bv as [| a n bv IHbv].Z_to_binary 0 (binary_value 0 []) = []
a:bool
n:nat
bv:VectorDef.t bool n
IHbv:Z_to_binary n (binary_value n bv) = bv
Z_to_binary (S n) (binary_value (S n) (a :: bv)) =
a :: bv
    auto.a:bool
n:nat
bv:VectorDef.t bool n
IHbv:Z_to_binary n (binary_value n bv) = bv
Z_to_binary (S n) (binary_value (S n) (a :: bv)) =
a :: bv

    rewrite binary_value_Sn.a:bool
n:nat
bv:VectorDef.t bool n
IHbv:Z_to_binary n (binary_value n bv) = bv
Z_to_binary (S n)
  (bit_value a + 2 * binary_value n bv) = a :: bv
    rewrite Z_to_binary_Sn.a:bool
n:nat
bv:VectorDef.t bool n
IHbv:Z_to_binary n (binary_value n bv) = bv
Bcons a n (Z_to_binary n (binary_value n bv)) =
a :: bv
a:bool
n:nat
bv:VectorDef.t bool n
IHbv:Z_to_binary n (binary_value n bv) = bv
(binary_value n bv >= 0)%Z
    rewrite IHbv; trivial.a:bool
n:nat
bv:VectorDef.t bool n
IHbv:Z_to_binary n (binary_value n bv) = bv
(binary_value n bv >= 0)%Z

    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.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.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
    induction bv as [| a n bv IHbv]; intro b.b:bool
Z_to_two_compl 0 (two_compl_value 0 (Bcons b 0 [])) =
Bcons b 0 []
a:bool
n:nat
bv:VectorDef.t bool n
IHbv:forall b0 : bool,
Z_to_two_compl n
  (two_compl_value n (Bcons b0 n bv)) =
Bcons b0 n bv
b:bool
Z_to_two_compl (S n)
  (two_compl_value (S n) (Bcons b (S n) (a :: bv))) =
Bcons b (S n) (a :: bv)
    destruct b; auto.a:bool
n:nat
bv:VectorDef.t bool n
IHbv:forall b0 : bool,
Z_to_two_compl n
  (two_compl_value n (Bcons b0 n bv)) =
Bcons b0 n bv
b:bool
Z_to_two_compl (S n)
  (two_compl_value (S n) (Bcons b (S n) (a :: bv))) =
Bcons b (S n) (a :: bv)

    rewrite two_compl_value_Sn.a:bool
n:nat
bv:VectorDef.t bool n
IHbv:forall b0 : bool,
Z_to_two_compl n
  (two_compl_value n (Bcons b0 n bv)) =
Bcons b0 n bv
b:bool
Z_to_two_compl (S n)
  (bit_value b + 2 * two_compl_value n (a :: bv)) =
Bcons b (S n) (a :: bv)
    rewrite Z_to_two_compl_Sn.a:bool
n:nat
bv:VectorDef.t bool n
IHbv:forall b0 : bool,
Z_to_two_compl n
  (two_compl_value n (Bcons b0 n bv)) =
Bcons b0 n bv
b:bool
Bcons b (S n)
  (Z_to_two_compl n (two_compl_value n (a :: bv))) =
Bcons b (S n) (a :: bv)
    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.forall (n : nat) (z : Z),
(z >= 0)%Z ->
(z < two_power_nat n)%Z ->
binary_value n (Z_to_binary n z) = z
  Proof.forall (n : nat) (z : Z),
(z >= 0)%Z ->
(z < two_power_nat n)%Z ->
binary_value n (Z_to_binary n z) = z
    induction n as [| n IHn].forall z : Z,
(z >= 0)%Z ->
(z < two_power_nat 0)%Z ->
binary_value 0 (Z_to_binary 0 z) = z
n:nat
IHn:forall z : Z,
(z >= 0)%Z -> (z < two_power_nat n)%Z -> binary_value n (Z_to_binary n z) = z
forall z : Z,
(z >= 0)%Z ->
(z < two_power_nat (S n))%Z ->
binary_value (S n) (Z_to_binary (S n) z) = z
    unfold two_power_nat, shift_nat; simpl; intros; omega.n:nat
IHn:forall z : Z,
(z >= 0)%Z -> (z < two_power_nat n)%Z -> binary_value n (Z_to_binary n z) = z
forall z : Z,
(z >= 0)%Z ->
(z < two_power_nat (S n))%Z ->
binary_value (S n) (Z_to_binary (S n) z) = z

    intros; rewrite Z_to_binary_Sn_z.n:nat
IHn:forall z0 : Z,
(z0 >= 0)%Z ->
(z0 < two_power_nat n)%Z -> binary_value n (Z_to_binary n z0) = z0
z:Z
H:(z >= 0)%Z
H0:(z < two_power_nat (S n))%Z
binary_value (S n)
  (Bcons (Z.odd z) n (Z_to_binary n (Z.div2 z))) = z
    rewrite binary_value_Sn.n:nat
IHn:forall z0 : Z,
(z0 >= 0)%Z ->
(z0 < two_power_nat n)%Z -> binary_value n (Z_to_binary n z0) = z0
z:Z
H:(z >= 0)%Z
H0:(z < two_power_nat (S n))%Z
(bit_value (Z.odd z) +
 2 * binary_value n (Z_to_binary n (Z.div2 z)))%Z = z
    rewrite IHn.n:nat
IHn:forall z0 : Z,
(z0 >= 0)%Z ->
(z0 < two_power_nat n)%Z -> binary_value n (Z_to_binary n z0) = z0
z:Z
H:(z >= 0)%Z
H0:(z < two_power_nat (S n))%Z
(bit_value (Z.odd z) + 2 * Z.div2 z)%Z = z
n:nat
IHn:forall z0 : Z,
(z0 >= 0)%Z ->
(z0 < two_power_nat n)%Z -> binary_value n (Z_to_binary n z0) = z0
z:Z
H:(z >= 0)%Z
H0:(z < two_power_nat (S n))%Z
(Z.div2 z >= 0)%Z
n:nat
IHn:forall z0 : Z,
(z0 >= 0)%Z ->
(z0 < two_power_nat n)%Z -> binary_value n (Z_to_binary n z0) = z0
z:Z
H:(z >= 0)%Z
H0:(z < two_power_nat (S n))%Z
(Z.div2 z < two_power_nat n)%Z
    apply Z_div2_value; auto.n:nat
IHn:forall z0 : Z,
(z0 >= 0)%Z ->
(z0 < two_power_nat n)%Z -> binary_value n (Z_to_binary n z0) = z0
z:Z
H:(z >= 0)%Z
H0:(z < two_power_nat (S n))%Z
(Z.div2 z >= 0)%Z
n:nat
IHn:forall z0 : Z,
(z0 >= 0)%Z ->
(z0 < two_power_nat n)%Z -> binary_value n (Z_to_binary n z0) = z0
z:Z
H:(z >= 0)%Z
H0:(z < two_power_nat (S n))%Z
(Z.div2 z < two_power_nat n)%Z

    apply Pdiv2; trivial.n:nat
IHn:forall z0 : Z,
(z0 >= 0)%Z ->
(z0 < two_power_nat n)%Z -> binary_value n (Z_to_binary n z0) = z0
z:Z
H:(z >= 0)%Z
H0:(z < two_power_nat (S n))%Z
(Z.div2 z < two_power_nat n)%Z

    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.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.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
    induction n as [| n IHn].forall z : Z,
(z >= - two_power_nat 0)%Z ->
(z < two_power_nat 0)%Z ->
two_compl_value 0 (Z_to_two_compl 0 z) = z
n:nat
IHn:forall 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
forall z : Z,
(z >= - two_power_nat (S n))%Z ->
(z < two_power_nat (S n))%Z ->
two_compl_value (S n) (Z_to_two_compl (S n) z) = z
    unfold two_power_nat, shift_nat; simpl; intros.z:Z
H:(z >= -1)%Z
H0:(z < 1)%Z
eq_rec_r (fun n : nat => bool -> Vector.t bool n -> Z)
  (fun (h : bool) (_ : Vector.t bool 0) =>
   (- bit_value h)%Z) eq_refl (Z.odd z) Bnil = z
n:nat
IHn:forall 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
forall z : Z,
(z >= - two_power_nat (S n))%Z ->
(z < two_power_nat (S n))%Z ->
two_compl_value (S n) (Z_to_two_compl (S n) z) = z
    assert (z = (-1)%Z \/ z = 0%Z).z:Z
H:(z >= -1)%Z
H0:(z < 1)%Z
z = (-1)%Z \/ z = 0%Z
z:Z
H:(z >= -1)%Z
H0:(z < 1)%Z
H1:z = (-1)%Z \/ z = 0%Z
eq_rec_r (fun n : nat => bool -> Vector.t bool n -> Z)
  (fun (h : bool) (_ : Vector.t bool 0) =>
   (- bit_value h)%Z) eq_refl (Z.odd z) Bnil = z
n:nat
IHn:forall 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
forall z : Z,
(z >= - two_power_nat (S n))%Z ->
(z < two_power_nat (S n))%Z ->
two_compl_value (S n) (Z_to_two_compl (S n) z) = z omega.z:Z
H:(z >= -1)%Z
H0:(z < 1)%Z
H1:z = (-1)%Z \/ z = 0%Z
eq_rec_r (fun n : nat => bool -> Vector.t bool n -> Z)
  (fun (h : bool) (_ : Vector.t bool 0) =>
   (- bit_value h)%Z) eq_refl (Z.odd z) Bnil = z
n:nat
IHn:forall 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
forall z : Z,
(z >= - two_power_nat (S n))%Z ->
(z < two_power_nat (S n))%Z ->
two_compl_value (S n) (Z_to_two_compl (S n) z) = z
    intuition; subst z; trivial.n:nat
IHn:forall 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
forall z : Z,
(z >= - two_power_nat (S n))%Z ->
(z < two_power_nat (S n))%Z ->
two_compl_value (S n) (Z_to_two_compl (S n) z) = z

    intros; rewrite Z_to_two_compl_Sn_z.n:nat
IHn:forall z0 : Z,
(z0 >= - two_power_nat n)%Z ->
(z0 < two_power_nat n)%Z -> two_compl_value n (Z_to_two_compl n z0) = z0
z:Z
H:(z >= - two_power_nat (S n))%Z
H0:(z < two_power_nat (S n))%Z
two_compl_value (S n)
  (Bcons (Z.odd z) (S n) (Z_to_two_compl n (Zmod2 z))) =
z
    rewrite two_compl_value_Sn.n:nat
IHn:forall z0 : Z,
(z0 >= - two_power_nat n)%Z ->
(z0 < two_power_nat n)%Z -> two_compl_value n (Z_to_two_compl n z0) = z0
z:Z
H:(z >= - two_power_nat (S n))%Z
H0:(z < two_power_nat (S n))%Z
(bit_value (Z.odd z) +
 2 * two_compl_value n (Z_to_two_compl n (Zmod2 z)))%Z =
z
    rewrite IHn.n:nat
IHn:forall z0 : Z,
(z0 >= - two_power_nat n)%Z ->
(z0 < two_power_nat n)%Z -> two_compl_value n (Z_to_two_compl n z0) = z0
z:Z
H:(z >= - two_power_nat (S n))%Z
H0:(z < two_power_nat (S n))%Z
(bit_value (Z.odd z) + 2 * Zmod2 z)%Z = z
n:nat
IHn:forall z0 : Z,
(z0 >= - two_power_nat n)%Z ->
(z0 < two_power_nat n)%Z -> two_compl_value n (Z_to_two_compl n z0) = z0
z:Z
H:(z >= - two_power_nat (S n))%Z
H0:(z < two_power_nat (S n))%Z
(Zmod2 z >= - two_power_nat n)%Z
n:nat
IHn:forall z0 : Z,
(z0 >= - two_power_nat n)%Z ->
(z0 < two_power_nat n)%Z -> two_compl_value n (Z_to_two_compl n z0) = z0
z:Z
H:(z >= - two_power_nat (S n))%Z
H0:(z < two_power_nat (S n))%Z
(Zmod2 z < two_power_nat n)%Z
    generalize (Zmod2_twice z); omega.n:nat
IHn:forall z0 : Z,
(z0 >= - two_power_nat n)%Z ->
(z0 < two_power_nat n)%Z -> two_compl_value n (Z_to_two_compl n z0) = z0
z:Z
H:(z >= - two_power_nat (S n))%Z
H0:(z < two_power_nat (S n))%Z
(Zmod2 z >= - two_power_nat n)%Z
n:nat
IHn:forall z0 : Z,
(z0 >= - two_power_nat n)%Z ->
(z0 < two_power_nat n)%Z -> two_compl_value n (Z_to_two_compl n z0) = z0
z:Z
H:(z >= - two_power_nat (S n))%Z
H0:(z < two_power_nat (S n))%Z
(Zmod2 z < two_power_nat n)%Z

    apply Zge_minus_two_power_nat_S; auto.n:nat
IHn:forall z0 : Z,
(z0 >= - two_power_nat n)%Z ->
(z0 < two_power_nat n)%Z -> two_compl_value n (Z_to_two_compl n z0) = z0
z:Z
H:(z >= - two_power_nat (S n))%Z
H0:(z < two_power_nat (S n))%Z
(Zmod2 z < two_power_nat n)%Z

    apply Zlt_two_power_nat_S; auto.
  Qed.

End COHERENT_VALUE.