Try starting with
(m * n) / m = n -- given m /= 0
Then do case analysis on n.
I found this process quite enlightening, thanks for posting.
-- ryan
2009/2/24 Peter Hilal
I'm working my way through Bird's _Introduction to Functional Programming Using Haskell_. I'd appreciate any help with problem 3.3.3, which is: "Division of natural numbers can be specified by the condition (n * m) / n = m for all positive n and all m. Construct a program for division and prove that it meets the specification." The required construction relies on the following definitions:
data Nat = Zero| Succ Nat (+) :: Nat -> Nat m + Zero = m m + Succ n = Succ (m + n) (*) :: Nat -> Nat m * Zero = Zero m * Succ n = m * n + m Proceeding as Bird does in Sec. 3.2.2, where he derives the definition of "-" from the specification "(m + n) - n = m", I've so far gotten the first case, in which m matches the pattern "Zero", simply by (i) substituting Zero for m in the specification, (ii) substituting Succ n for n in the specification (solely because n is constrained to be positive), and (iii) reducing by applying the first equation of "*": Case Zero:
(Succ n * Zero) / Succ n = Zero ≡ {first equation of "*"} Zero / Succ n = Zero Easy enough, and completely intuitive, since we expect Zero divided by any non-Zero finite number to be Zero. The next case, in which m matches the pattern "Succ m", is where I get stuck, and I really have no intuition about what the definition is supposed to be. My first step is to substitute "Succ m" for "m" in the given specification, and to substitute Succ n for n in the specification (for the same reason, as above, that n is constrained to be positive), and then to use the definition of "*" to reduce the equation: Case Succ m: Succ n * Succ m / Succ n = Succ m ≡ {second equation of "*"} (Succ n * m + Succ n) / Succ n = Succ m Where do I go from here? Thank you so much. _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe