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.
On Tue, Feb 24, 2009 at 10:25:56AM -0500, Peter Hilal wrote:
Where do I go from here? Thank you so much.
A hint: I don't think you can do it by recursion on (/). You'll need an auxiliary function. Then prove that your function satisfies the constraint. Phil (Unless there's some clever way to repeatedly divide which comes out with the right answer that I'm not seeing of course...) -- http://www.kantaka.co.uk/ .oOo. public key: http://www.kantaka.co.uk/gpg.txt
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
Hi, I'm just re-reading the book again, this time doing the exercises though :) Is there a site with solutions for the exercises? Günther
Günther Schmidt wrote:
I'm just re-reading the book again, this time doing the exercises though :)
Is there a site with solutions for the exercises?
Unless you count the haskell-cafe and beginners mailing lists as sites, I don't know any sites which have the solutions. ;) Problem 3.3.3: Construct a program for division from the specification (m * n) / n = m and prove that it's correct. Sketch of a solution: To define a / n , the usual approach of subtracting n from the first argument until something less than n remains will work. Correctness can then be proven by induction on m . Regards, Heinrich Apfelmus -- http://apfelmus.nfshost.com
participants (5)
-
Günther Schmidt -
Heinrich Apfelmus -
Peter Hilal -
Philip Armstrong -
Ryan Ingram