Extracting Coq to Haskell

Question

I'm experimenting with Coq's extraction mechanism to Haskell. I wrote a naive predicate for prime numbers in Coq, here it is:

(***********)
(* IMPORTS *)
(***********)
Require Import Coq.Arith.PeanoNat.

(************)
(* helper'' *)
(************)
Fixpoint helper' (p m n : nat) : bool :=
  match m,n with
  | 0,_ => false
  | 1,_ => false
  | _,0 => false
  | _,1 => false
  | S m',S n' => (orb ((mult m n) =? p) (helper' p m' n))
  end.

(**********)
(* helper *)
(**********)
Fixpoint helper (p m : nat) : bool :=
  match m with
  | 0 => false
  | S m' => (orb ((mult m m) =? p) (orb (helper' p m' m) (helper p m')))
  end.

(***********)
(* isPrime *)
(***********)
Fixpoint isPrime (p : nat) : bool :=
  match p with
  | 0 => false
  | 1 => false
  | S p' => (negb (helper p p'))
  end.

Compute (isPrime 220).

(*****************)
(* isPrimeHelper *)
(*****************)
Extraction Language Haskell.

(*****************)
(* isPrimeHelper *)
(*****************)
Extraction "/home/oren/GIT/CoqIt/Primes.hs" isPrime helper helper'.

And after extracting the Haskell code, I wrote a simple driver to test it. I ran into two issues:

1. Coq exported its own Bool instead of using Haskell's built in boolean type.
2. Coq also used its own nat, so I can't ask isPrime 6 and I have to use S (S (...)).

module Main( main ) where    
import Primes    
main = do
    if ((isPrime (
        Primes.S (
        Primes.S (
        Primes.S (
        Primes.S (
        Primes.S (
        Primes.S ( O ))))))))
        ==
        Primes.True)
    then
        print "Prime"
    else
        print "Non Prime"

Answer 1

Regarding your first point - try to add

Require Import ExtrHaskellBasic.

to your Coq source. It specifies that the extraction should use Haskell's prelude definitions for some basic types. Documentation can be found here. There is also a similar module for strings.