Arith.is_presburger

STRUCTURE

SYNOPSIS

Determines whether a formula is in the Presburger subset of arithmetic.

DESCRIPTION

This function returns true if the argument term is a formula in the Presburger subset of natural number arithmetic. Presburger natural arithmetic is the subset of arithmetic formulae made up from natural number constants, numeric variables, addition, multiplication by a constant, the natural number relations <, <=, =, >=, > and the logical connectives ~, /\, \/, ==>, = (if-and-only-if), ! (‘forall’) and ? (‘there exists’).

Products of two expressions which both contain variables are not included in the subset, but the function SUC which is not normally included in a specification of Presburger arithmetic is allowed in this HOL implementation. This function also considers subtraction and the predecessor function, PRE, to be part of the subset.

FAILURE

Never fails.

EXAMPLE

- is_presburger ``!m n p. m < (2 * n) /\ (n + n) <= p ==> m < SUC p``;
> val it = true : bool

- is_presburger ``!m n p q. m < (n * p) /\ (n * p) < q ==> m < q``;
> val it = false : bool

- is_presburger ``(m <= n) ==> !p. (m < SUC(n + p))``;
> val it = true : bool

- is_presburger ``(m + n) - m = n``;
> val it = true : bool

USES

Useful for determining whether a decision procedure for Presburger arithmetic is applicable to a term.

SEEALSO

non_presburger_subterms, FORALL_ARITH_CONV, EXISTS_ARITH_CONV, is_prenex