Pattern matching natural powers as products

Question

I thought that Mathematica would be smart enough to know that f[t]^2 should match a_ * b_ with a and b both taking the value f[t], but it doesn't.

I'm writing some rules for identities involving products. It's mainly finding different ways of doing integration by parts. Am I forced to write separate rules for handling powers, or can Mathematica be persuaded to matchf[t]^2 as a_ * b_?

I would like to handle arbitrarily large natural powers, so f[t]^3 should match a_ * b_ with a as f[t] and b as f[t]^2, or vice versa. f[t]^4 should match as both f[t] * f[t]^3 and f[t]^2 * f[t]^2. I'm using ReplaceList, so the order is unimportant. I want the behaviour to be identical to if the powers were represented as f[t] * f[t] * f[t] ....

If not, is there a nice single pattern I can use to match all cases? I've been looking at Alternatives, but I don't know how to capture the a and b values properly when dealing with the powers forms.

Answer 1

By replacing multiplication with NonCommutativeMultiply, which is an associative but noncommutative symbol with no built-in meaning, you can preprocess your expressions in the desired explicit form:

expr = a^3 + b c (1 + c + c^2);

A = Expand[expr] /. {Times -> NonCommutativeMultiply,
                     x_^n_Integer :> NonCommutativeMultiply @@ ConstantArray[x, n]}

(*    b ** c + a ** a ** a + b ** c ** c + b ** c ** c ** c    *)

From here you can continue applying your rules by pattern-matching, for example

A /. {x_ ** y_ ** z_ ** w_ :> func4[x, y, z, w],
      x_ ** y_ ** z_ :> func3[x, y, z],
      x_ ** y_ :> func2[x, y]}

(*    func2[b, c] + func3[a, a, a] + func3[b, c, c] + func4[b, c, c, c]    *)