10.11.3 Conditional Rewrite Rules

A rewrite rule can also be conditional, written in the form ‘old := new :: cond’. (There is also the obsolete form ‘[old, new, cond]’.) If a cond part is present in the rule, this is an additional condition that must be satisfied before the rule is accepted. Once old has been successfully matched to the target expression, cond is evaluated (with all the meta-variables substituted for the values they matched) and simplified with Calc’s algebraic simplifications. If the result is a nonzero number or any other object known to be nonzero (see Declarations), the rule is accepted. If the result is zero or if it is a symbolic formula that is not known to be nonzero, the rule is rejected. See Logical Operations, for a number of functions that return 1 or 0 according to the results of various tests.

For example, the formula ‘n > 0’ simplifies to 1 or 0 if ‘n’ is replaced by a positive or nonpositive number, respectively (or if ‘n’ has been declared to be positive or nonpositive). Thus, the rule ‘f(x,y) := g(y+x,x) :: x+y > 0’ would apply to ‘f(0, 4)’ but not to ‘f(-3, 2)’ or ‘f(12, a+1)’ (assuming no outstanding declarations for ‘a’). In the case of ‘f(-3, 2)’, the condition can be shown not to be satisfied; in the case of ‘f(12, a+1)’, the condition merely cannot be shown to be satisfied, but that is enough to reject the rule.

While Calc will use declarations to reason about variables in the formula being rewritten, declarations do not apply to meta-variables. For example, the rule ‘f(a) := g(a+1)’ will match for any values of ‘a’, such as complex numbers, vectors, or formulas, even if ‘a’ has been declared to be real or scalar. If you want the meta-variable ‘a’ to match only literal real numbers, use ‘f(a) := g(a+1) :: real(a)’. If you want ‘a’ to match only reals and formulas which are provably real, use ‘dreal(a)’ as the condition.

The ‘::’ operator is a shorthand for the condition function; ‘old := new :: cond’ is equivalent to the formula ‘condition(assign(old, new), cond)’.

If you have several conditions, you can use ‘... :: c1 :: c2 :: c3’ or ‘... :: c1 && c2 && c3’. The two are entirely equivalent.

It is also possible to embed conditions inside the pattern: ‘f(x :: x>0, y) := g(y+x, x)’. This is purely a notational convenience, though; where a condition appears in a rule has no effect on when it is tested. The rewrite-rule compiler automatically decides when it is best to test each condition while a rule is being matched.

Certain conditions are handled as special cases by the rewrite rule system and are tested very efficiently: Where ‘x’ is any meta-variable, these conditions are ‘integer(x)’, ‘real(x)’, ‘constant(x)’, ‘negative(x)’, ‘x >= y’ where ‘y’ is either a constant or another meta-variable and ‘>=’ may be replaced by any of the six relational operators, and ‘x % a = b’ where ‘a’ and ‘b’ are constants. Other conditions, like ‘x >= y+1’ or ‘dreal(x)’, will be less efficient to check since Calc must bring the whole evaluator and simplifier into play.

An interesting property of ‘::’ is that neither of its arguments will be touched by Calc’s default simplifications. This is important because conditions often are expressions that cannot safely be evaluated early. For example, the typeof function never remains in symbolic form; entering ‘typeof(a)’ will put the number 100 (the type code for variables like ‘a’) on the stack. But putting the condition ‘... :: typeof(a) = 6’ on the stack is safe since ‘::’ prevents the typeof from being evaluated until the condition is actually used by the rewrite system.

Since ‘::’ protects its lefthand side, too, you can use a dummy condition to protect a rule that must itself not evaluate early. For example, it’s not safe to put ‘a(f,x) := apply(f, [x])’ on the stack because it will immediately evaluate to ‘a(f,x) := f(x)’, where the meta-variable-ness of f on the righthand side has been lost. But ‘a(f,x) := apply(f, [x]) :: 1’ is safe, and of course the condition ‘1’ is always true (nonzero) so it has no effect on the functioning of the rule. (The rewrite compiler will ensure that it doesn’t even impact the speed of matching the rule.)