Integer Module Interface Specification

(0) CHARACTERISTICS

-type specified: int

-features: deterministic, non-parameterized

-foreign types:

(1) SYNTAX

ACCESS-PROGRAMS

Program Name	Arg#1	Arg#2	Result Type
ASSIGN	int:O	int:V
DIV	int:V	int:V	int
EQUAL	int:V	int:V	bool
LESS	int:V	int:V	bool
MINUS	int:V	int:V	int
MOD	int:V	int:V	int
NEG	int:VO
PLUS	int:V	int:V	int
PRED	int:VO
SUCC	int:VO
TIMES	int:V	int:V	int

(2) CANONICAL TRACES

canonical(T)

T = _

T = SUCC(*).NEG(*)

T1:<<int>>T = SUCC(*).T1 [canonical(T1)]

AUXILIARY FUNCTIONS

pred: <int> <int>

pred(x) =

Condition	Value
! x1:<int> [x = x1.NEG(*) ]	SUCC(*).x
x = _	SUCC().NEG()
! x1:<int> [x = x1.SUCC(*) ]	x1

succ: <int> <int>

succ(x) =

Condition	Value
Ź ! x1:<int> [x = x1.NEG(*) ]	x.SUCC(*)
x = SUCC().NEG()	_
! x1:<int> [x = x1.SUCC().SUCC().NEG(*) ]	x1.SUCC().NEG()

plus: <int> × <int> <int>

plus(x, y) =

Condition	Value
x = _	y
x = SUCC().NEG()	pred(y)
! x1:<int> [x = x1.SUCC(*) ]	succ(plus(x1, y))
! x1:<int> [x = x1.SUCC().SUCC().NEG(*) ]	pred(plus(x1.SUCC().NEG(), y))

negate: <int> <int>

negate(x) =

Condition	Value
x = _	_
! x1:<int> [x = x1.SUCC(*) ]	x.NEG(*)
! x1:<int> [x = x1.NEG(*) ]	x1

minus: <int> × <int> <int>

minus(x, y) = plus(x, negate(y))

times: <int> × <int> <int>

times(x, y) =

Condition	Value
x = _	_
! x1:<int> [x = x1.SUCC(*) ]	plus(times(x1, y), y)
! x1:<int> [x = x1.NEG(*) ]	negate(times(x1, y))

div: <int> × <int> <int>

div(x, y) (y _) = z where z,r:<int> [plus(times(z, y), r) = x less(SUCC(*).NEG(*), r) less(r, y)]

modulo: <int> × <int> <int>

modulo(x, y) (y _) = minus(x, times(y, div(x, y)))

less: <int> × <int> <bool>

less(x, y) = z:<int> [minus(x, y) = z.NEG(*) ]

leq: <int> × <int> <bool>

leq(x, y) = x = y less(x, y)

geq: <int> × <int> <bool>

geq(x, y) = Źless(x, y)

greater: <int> × <int> <bool>

greater(x, y) = Źleq(x, y)

(3) SEMANTICS

ACCESS-PROGRAMS

Legality(ASSIGN(n, T)) = %legal%

ASSIGN(n , T) = T

Legality(DIV(T, U)) =

Condition	Value
U = _	%fatal%
U _	%legal%

DIV(T, U) = div(T, U)

Legality(EQUAL(T, U)) = %legal%

EQUAL(T, U) = T = U

Legality(LESS(T, U)) = %legal%

LESS(T, U) = less(T, U)

Legality(MINUS(T, U)) = %legal%

MINUS(T, U) = minus(T, U)

Legality(MOD(T, U)) =

Condition	Value
U = _	%fatal%
U _	%legal%

MOD(T, U) = modulo(T, U)

Legality(NEG((n, U))) = %legal%

NEG((n, U) ) = negate(U)

Legality(PLUS(T, U)) = %legal%

PLUS(T, U) = plus(T, U)

Legality(PRED((n, T))) = %legal%

PRED((n, T) ) = pred(T)

Legality(SUCC((n, T))) = %legal%

SUCC((n, T) ) = succ(T)

Legality(TIMES(T, U)) = %legal%

TIMES(T, U) = times(T, U)