An Inductive Proof of the Correctness of power

(define power
  (lambda (b e)
    (if (= e 0)
        1
        (* b (power b (- e 1))))))

Claim: (power b e) returns b^e for non-negative integer e.

• Base Case: When e = 0, (power b e) returns 1, which is correct by definition.

• Induction Hypothesis (IH): For k = e-1, assume that (power b k) returns b^k.

• Inductive Step:
  1. (power b e) returns b times the result of calling (power b (- e 1))     [by inspection}

  2. (power b (- e 1)) returns b^e-1     [by the IH]

  3. (power b e) returns b × b^e-1     [by 1, 2]

  4. (power b e) returns b^e     [by 3 and elementary algebra]