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#### 10.11.2 Basic Rewrite Rules

To match a particular formula ‘x’ with a particular rewrite rule ‘old := new’, Calc compares the structure of ‘x’ with the structure of old. Variables that appear in old are treated as meta-variables; the corresponding positions in ‘x’ may contain any sub-formulas. For example, the pattern ‘f(x,y)’ would match the expression ‘f(12, a+1)’ with the meta-variable ‘x’ corresponding to 12 and with ‘y’ corresponding to ‘a+1’. However, this pattern would not match ‘f(12)’ or ‘g(12, a+1)’, since there is no assignment of the meta-variables that will make the pattern match these expressions. Notice that if the pattern is a single meta-variable, it will match any expression.

If a given meta-variable appears more than once in old, the corresponding sub-formulas of ‘x’ must be identical. Thus the pattern ‘f(x,x)’ would match ‘f(12, 12)’ and ‘f(a+1, a+1)’ but not ‘f(12, a+1)’ or ‘f(a+b, b+a)’. (See Conditional Rewrite Rules, for a way to match the latter.)

Things other than variables must match exactly between the pattern and the target formula. To match a particular variable exactly, use the pseudo-function ‘quote(v)’ in the pattern. For example, the pattern ‘x+quote(y)’ matches ‘x+y’, ‘2+y’, or ‘sin(a)+y’.

The special variable names ‘e’, ‘pi’, ‘i’, ‘phi’, ‘gamma’, ‘inf’, ‘uinf’, and ‘nan’ always match literally. Thus the pattern ‘sin(d + e + f)’ acts exactly like ‘sin(d + quote(e) + f)’.

If the old pattern is found to match a given formula, that formula is replaced by new, where any occurrences in new of meta-variables from the pattern are replaced with the sub-formulas that they matched. Thus, applying the rule ‘f(x,y) := g(y+x,x)’ to ‘f(12, a+1)’ would produce ‘g(a+13, 12)’.

The normal a r command applies rewrite rules over and over throughout the target formula until no further changes are possible (up to a limit of 100 times). Use C-u 1 a r to make only one change at a time.

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