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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id$ i*)
(** Bit vectors. Contribution by Jean Duprat (ENS Lyon). *)
Require Export Bool.
Require Export Sumbool.
Require Import Arith.
Open Local Scope nat_scope.
(**
We build bit vectors in the spirit of List.v.
The size of the vector is a parameter which is too important
to be accessible only via function "length".
The first idea is to build a record with both the list and the length.
Unfortunately, this a posteriori verification leads to
numerous lemmas for handling lengths.
The second idea is to use a dependent type in which the length
is a building parameter. This leads to structural induction that
are slightly more complex and in some cases we will use a proof-term
as definition, since the type inference mechanism for pattern-matching
is sometimes weaker that the one implemented for elimination tactiques.
*)
Section VECTORS.
(**
A vector is a list of size n whose elements belongs to a set A.
If the size is non-zero, we can extract the first component and the
rest of the vector, as well as the last component, or adding or
removing a component (carry) or repeating the last component at the
end of the vector.
We can also truncate the vector and remove its p last components or
reciprocally extend the vector by concatenation.
A unary function over A generates a function on vectors of size n by
applying f pointwise.
A binary function over A generates a function on pairs of vectors of
size n by applying f pointwise.
*)
Variable A : Type.
Inductive vector : nat -> Type :=
| Vnil : vector 0
| Vcons : forall (a:A) (n:nat), vector n -> vector (S n).
Definition Vhead (n:nat) (v:vector (S n)) :=
match v with
| Vcons a _ _ => a
end.
Definition Vtail (n:nat) (v:vector (S n)) :=
match v with
| Vcons _ _ v => v
end.
Definition Vlast : forall n:nat, vector (S n) -> A.
Proof.
induction n as [| n f]; intro v.
inversion v.
exact a.
inversion v as [| n0 a H0 H1].
exact (f H0).
Defined.
Fixpoint Vconst (a:A) (n:nat) :=
match n return vector n with
| O => Vnil
| S n => Vcons a _ (Vconst a n)
end.
(** Shifting and truncating *)
Lemma Vshiftout : forall n:nat, vector (S n) -> vector n.
Proof.
induction n as [| n f]; intro v.
exact Vnil.
inversion v as [| a n0 H0 H1].
exact (Vcons a n (f H0)).
Defined.
Lemma Vshiftin : forall n:nat, A -> vector n -> vector (S n).
Proof.
induction n as [| n f]; intros a v.
exact (Vcons a 0 v).
inversion v as [| a0 n0 H0 H1 ].
exact (Vcons a (S n) (f a H0)).
Defined.
Lemma Vshiftrepeat : forall n:nat, vector (S n) -> vector (S (S n)).
Proof.
induction n as [| n f]; intro v.
inversion v.
exact (Vcons a 1 v).
inversion v as [| a n0 H0 H1 ].
exact (Vcons a (S (S n)) (f H0)).
Defined.
Lemma Vtrunc : forall n p:nat, n > p -> vector n -> vector (n - p).
Proof.
induction p as [| p f]; intros H v.
rewrite <- minus_n_O.
exact v.
apply (Vshiftout (n - S p)).
rewrite minus_Sn_m.
apply f.
auto with *.
exact v.
auto with *.
Defined.
(** Concatenation of two vectors *)
Fixpoint Vextend n p (v:vector n) (w:vector p) : vector (n+p) :=
match v with
| Vnil => w
| Vcons a n' v' => Vcons a (n'+p) (Vextend n' p v' w)
end.
(** Uniform application on the arguments of the vector *)
Variable f : A -> A.
Fixpoint Vunary n (v:vector n) : vector n :=
match v with
| Vnil => Vnil
| Vcons a n' v' => Vcons (f a) n' (Vunary n' v')
end.
Variable g : A -> A -> A.
Lemma Vbinary : forall n:nat, vector n -> vector n -> vector n.
Proof.
induction n as [| n h]; intros v v0.
exact Vnil.
inversion v as [| a n0 H0 H1]; inversion v0 as [| a0 n1 H2 H3].
exact (Vcons (g a a0) n (h H0 H2)).
Defined.
(** Eta-expansion of a vector *)
Definition Vid n : vector n -> vector n :=
match n with
| O => fun _ => Vnil
| _ => fun v => Vcons (Vhead _ v) _ (Vtail _ v)
end.
Lemma Vid_eq : forall (n:nat) (v:vector n), v = Vid n v.
Proof.
destruct v; auto.
Qed.
Lemma VSn_eq :
forall (n : nat) (v : vector (S n)), v = Vcons (Vhead _ v) _ (Vtail _ v).
Proof.
intros.
exact (Vid_eq _ v).
Qed.
Lemma V0_eq : forall (v : vector 0), v = Vnil.
Proof.
intros.
exact (Vid_eq _ v).
Qed.
End VECTORS.
(* suppressed: incompatible with Coq-Art book
Implicit Arguments Vnil [A].
Implicit Arguments Vcons [A n].
*)
Section BOOLEAN_VECTORS.
(**
A bit vector is a vector over booleans.
Notice that the LEAST significant bit comes first (little-endian representation).
We extract the least significant bit (head) and the rest of the vector (tail).
We compute bitwise operation on vector: negation, and, or, xor.
We compute size-preserving shifts: to the left (towards most significant bits,
we hence use Vshiftout) and to the right (towards least significant bits,
we use Vshiftin) by inserting a 'carry' bit (logical shift) or by repeating
the most significant bit (arithmetical shift).
NOTA BENE: all shift operations expect predecessor of size as parameter
(they only work on non-empty vectors).
*)
Definition Bvector := vector bool.
Definition Bnil := @Vnil bool.
Definition Bcons := @Vcons bool.
Definition Bvect_true := Vconst bool true.
Definition Bvect_false := Vconst bool false.
Definition Blow := Vhead bool.
Definition Bhigh := Vtail bool.
Definition Bsign := Vlast bool.
Definition Bneg := Vunary bool negb.
Definition BVand := Vbinary bool andb.
Definition BVor := Vbinary bool orb.
Definition BVxor := Vbinary bool xorb.
Definition BshiftL (n:nat) (bv:Bvector (S n)) (carry:bool) :=
Bcons carry n (Vshiftout bool n bv).
Definition BshiftRl (n:nat) (bv:Bvector (S n)) (carry:bool) :=
Bhigh (S n) (Vshiftin bool (S n) carry bv).
Definition BshiftRa (n:nat) (bv:Bvector (S n)) :=
Bhigh (S n) (Vshiftrepeat bool n bv).
Fixpoint BshiftL_iter (n:nat) (bv:Bvector (S n)) (p:nat) : Bvector (S n) :=
match p with
| O => bv
| S p' => BshiftL n (BshiftL_iter n bv p') false
end.
Fixpoint BshiftRl_iter (n:nat) (bv:Bvector (S n)) (p:nat) : Bvector (S n) :=
match p with
| O => bv
| S p' => BshiftRl n (BshiftRl_iter n bv p') false
end.
Fixpoint BshiftRa_iter (n:nat) (bv:Bvector (S n)) (p:nat) : Bvector (S n) :=
match p with
| O => bv
| S p' => BshiftRa n (BshiftRa_iter n bv p')
end.
End BOOLEAN_VECTORS.
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