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CSC 236            Sample Solutions for Week  8 Tutorial          Fall 2007
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Prove that the following algorithm is correct, with respect to the
conditions given.

    Precondition:  x in R, y in N
    Postcondition:  pow(x,y) returns x^y (with convention 0^0 = 1)

    pow(x, y):
        if y == 0:
            return 1
        elif y % 2 == 0:
            t = pow(x, y/2)
            return t * t
        else:
            t = pow(x, (y-1)/2)
            return t * t * x

Q:  How do we measure input size?
A:  Input size = y (just like last week's recurrence).

Q:  What are we proving exactly, and how?
A:  For all y in N, for all x in R, pow(x,y) returns x^y,
    by induction on y.

Q:  What's first?
A:  Base Case:  y = 0
    Q:  Proof?
    A:  For all x in R, pow(x,y) = pow(x,0) returns 1 = x^0 = x^y.

Q:  What's next?
A:  Ind. Hyp.:
    Q:  Statement?
    A:  Let y > 0 and suppose pow(x,j) returns x^j for 0 <= j < y.

Q:  What's next?
A:  Ind. Step:  Let x in R and consider call pow(x,y).

    Q:  Approach?
    A:  Following algorithm, consider cases y even and y odd.

        Case 1:  y even (i.e., y % 2 = 0)

        Q:  What's next?
        A:  Ensure recursive call meets precondition and size is within
            range of I.H.:  y even and y > 0 implies 0 <= y/2 < y
        Q:  And now?
        A:  Apply IH:  so by IH, pow(x, y/2) returns x^{y/2}
            Conclude:  and current call returns
                x^{y/2} * x^{y/2} = x^{y/2+y/2} = x^y.

        Case 2:  y odd (i.e., y % 2 = 1)

        Q:  What's next?
        A:  Ensure recursive call meets precondition and size is within
            range of I.H.:  y odd and y > 0 implies 0 <= (y-1)/2 < y
        Q:  And now?
        A:  Apply IH:  so by IH, pow(x, (y-1)/2) returns x^{(y-1)/2}
            Conclude:  and current call returns
                x^{(y-1)/2} * x^{(y-1)/2} * x
              = x^{(y-1)/2+(y-1)/2+1} = x^{y-1+1} = x^y.

        In all cases, current call returns x^y, i.e., for all x in R,
        pow(x,y) returns x^y.

    By induction, for all y in N, for all x in R, pow(x,y) returns x^y.