Library Coq.Bool.Bvector

Bit vectors. Contribution by Jean Duprat (ENS Lyon).

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.

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).