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 section 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.