ARITH_FORM_NORM_CONV

Arith.ARITH_FORM_NORM_CONV : conv

Normalises an unquantified formula of linear natural number arithmetic.

ARITH_FORM_NORM_CONV converts a formula of natural number arithmetic into a disjunction of conjunctions of less-than-or-equal-to inequalities. The arithmetic expressions are only allowed to contain natural number constants, numeric variables, addition, the SUC function, and multiplication by a constant. The formula must not contain quantifiers, but may have disjunction, conjunction, negation, implication, equality on Booleans (if-and-only-if), and the natural number relations: <, <=, =, >=, >. The formula must not contain products of two expressions which both contain variables.

The inequalities in the result are normalised so that each variable appears on only one side of the inequality, and each side is a linear sum in which any constant appears first followed by products of a constant and a variable. The variables are ordered lexicographically, and if the coefficient of the variable is 1, the product of 1 and the variable appears in the term rather than the variable on its own.

Failure

The function fails if the argument term is not a formula in the specified subset.

Example

#ARITH_FORM_NORM_CONV "m < n";;
|- m < n = (1 + (1 * m)) <= (1 * n)

#ARITH_FORM_NORM_CONV
# "(n < 4) ==> ((n = 0) \/ (n = 1) \/ (n = 2) \/ (n = 3))";;
|- n < 4 ==> (n = 0) \/ (n = 1) \/ (n = 2) \/ (n = 3) =
   4 <= (1 * n) \/
   (1 * n) <= 0 /\ 0 <= (1 * n) \/
   (1 * n) <= 1 /\ 1 <= (1 * n) \/
   (1 * n) <= 2 /\ 2 <= (1 * n) \/
   (1 * n) <= 3 /\ 3 <= (1 * n)

Useful in constructing decision procedures for linear arithmetic.