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Funcons-beta : Bits.cbs | PRETTY | PDF

Outline

Bits and bit vectors 

[
  Type     bits
  Datatype bit-vectors
  Funcon   bit-vector
  Type     bytes             Alias octets
  Funcon   bit-vector-not
  Funcon   bit-vector-and
  Funcon   bit-vector-or
  Funcon   bit-vector-xor
  Funcon   bit-vector-shift-left
  Funcon   bit-vector-logical-shift-right
  Funcon   bit-vector-arithmetic-shift-right
  Funcon   integer-to-bit-vector
  Funcon   bit-vector-to-integer
  Funcon   bit-vector-to-natural
  Funcon   unsigned-bit-vector-maximum
  Funcon   signed-bit-vector-maximum
  Funcon   signed-bit-vector-minimum
  Funcon   is-in-signed-bit-vector
  Funcon   is-in-unsigned-bit-vector
]

Bits 

Type
  bits ~> booleans

false represents the absence of a bit, true its presence.

Bit vectors 

Datatype
  bit-vectors(N:natural-numbers) ::= bit-vector(_:bits^N)
Type
  bytes ~> bit-vectors(8)
Alias
  octets = bytes
Meta-variables
  BT <: bit-vectors(_)
Built-in Funcon
  bit-vector-not(_:BT) : =>BT
Built-in Funcon
  bit-vector-and(_:BT, _:BT) : =>BT
Built-in Funcon
  bit-vector-or(_:BT, _:BT) : =>BT
Built-in Funcon
  bit-vector-xor(_:BT, _:BT) : =>BT

The above four funcons are the natural extensions of funcons from booleans to bit-vectors(N) of the same length.

Built-in Funcon
  bit-vector-shift-left(_:BT, _:natural-numbers) : BT
Built-in Funcon
  bit-vector-logical-shift-right(_:BT, _:natural-numbers) : BT
Built-in Funcon
  bit-vector-arithmetic-shift-right(_:BT, _:natural-numbers) : BT
Built-in Funcon
  integer-to-bit-vector(_:integers, N:natural-numbers) : bit-vectors(N)

integer-to-bit-vector(M, N) converts an integer M to a bit-vector of length N, using Two’s Complement representation. If the integer is out of range of the representation, it will wrap around (modulo 2^N).

Built-in Funcon
  bit-vector-to-integer(_:BT) : =>integers

bit-vector-to-integer(B) interprets a bit-vector BV as an integer in Two’s Complement representation.

Built-in Funcon
  bit-vector-to-natural(_:BT) : =>natural-numbers

bit-vector-to-natural(BV) interprets a bit-vector BV as a natural number in unsigned representation.

Funcon
  unsigned-bit-vector-maximum(N:natural-numbers) : =>natural-numbers
    ~> integer-subtract(integer-power(2, N), 1)
Funcon
  signed-bit-vector-maximum(N:natural-numbers) : =>integers
    ~> integer-subtract(integer-power(2, integer-subtract(N, 1)), 1)
Funcon
  signed-bit-vector-minimum(N:natural-numbers) : =>integers
    ~> integer-negate(integer-power(2, integer-subtract(N, 1)))
Funcon
  is-in-signed-bit-vector(M:integers, N:natural-numbers) : =>booleans
    ~> and(integer-is-less-or-equal(M, signed-bit-vector-maximum(N)),
           integer-is-greater-or-equal(M, signed-bit-vector-minimum(N)))
Funcon
  is-in-unsigned-bit-vector(M:integers, N:natural-numbers) : =>booleans
    ~> and(integer-is-less-or-equal(M, unsigned-bit-vector-maximum(N)),
           integer-is-greater-or-equal(M, 0))
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