(* Title: HOL/ex/Quickcheck_Examples.thy
ID: $Id$
Author: Stefan Berghofer
Copyright 2004 TU Muenchen
*)
header {* Examples for the 'quickcheck' command *}
theory Quickcheck_Examples imports Main begin
text {*
The 'quickcheck' command allows to find counterexamples by evaluating
formulae under an assignment of free variables to random values.
In contrast to 'refute', it can deal with inductive datatypes,
but cannot handle quantifiers.
*}
subsection {* Lists *}
theorem "map g (map f xs) = map (g o f) xs"
quickcheck
oops
theorem "map g (map f xs) = map (f o g) xs"
quickcheck
oops
theorem "rev (xs @ ys) = rev ys @ rev xs"
quickcheck
oops
theorem "rev (xs @ ys) = rev xs @ rev ys"
quickcheck
oops
theorem "rev (rev xs) = xs"
quickcheck
oops
theorem "rev xs = xs"
quickcheck
oops
consts
occurs :: "'a \ 'a list \ nat"
primrec
"occurs a [] = 0"
"occurs a (x#xs) = (if (x=a) then Suc(occurs a xs) else occurs a xs)"
consts
del1 :: "'a \ 'a list \ 'a list"
primrec
"del1 a [] = []"
"del1 a (x#xs) = (if (x=a) then xs else (x#del1 a xs))"
(* A lemma, you'd think to be true from our experience with delAll*)
lemma "Suc (occurs a (del1 a xs)) = occurs a xs"
-- {* Wrong. Precondition needed.*}
quickcheck
oops
lemma "xs ~= [] \ Suc (occurs a (del1 a xs)) = occurs a xs"
quickcheck
-- {* Also wrong.*}
oops
lemma "0 < occurs a xs \ Suc (occurs a (del1 a xs)) = occurs a xs"
quickcheck
apply (induct_tac xs)
apply auto
-- {* Correct! *}
done
consts
replace :: "'a \ 'a \ 'a list \ 'a list"
primrec
"replace a b [] = []"
"replace a b (x#xs) = (if (x=a) then (b#(replace a b xs))
else (x#(replace a b xs)))"
lemma "occurs a xs = occurs b (replace a b xs)"
quickcheck
-- {* Wrong. Precondition needed.*}
oops
lemma "occurs b xs = 0 \ a=b \ occurs a xs = occurs b (replace a b xs)"
quickcheck
apply (induct_tac xs)
apply auto
done
subsection {* Trees *}
datatype 'a tree = Twig | Leaf 'a | Branch "'a tree" "'a tree"
consts
leaves :: "'a tree \ 'a list"
primrec
"leaves Twig = []"
"leaves (Leaf a) = [a]"
"leaves (Branch l r) = (leaves l) @ (leaves r)"
consts
plant :: "'a list \ 'a tree"
primrec
"plant [] = Twig "
"plant (x#xs) = Branch (Leaf x) (plant xs)"
consts
mirror :: "'a tree \ 'a tree"
primrec
"mirror (Twig) = Twig "
"mirror (Leaf a) = Leaf a "
"mirror (Branch l r) = Branch (mirror r) (mirror l)"
theorem "plant (rev (leaves xt)) = mirror xt"
quickcheck
--{* Wrong! *}
oops
theorem "plant((leaves xt) @ (leaves yt)) = Branch xt yt"
quickcheck
--{* Wrong! *}
oops
datatype 'a ntree = Tip "'a" | Node "'a" "'a ntree" "'a ntree"
consts
inOrder :: "'a ntree \ 'a list"
primrec
"inOrder (Tip a)= [a]"
"inOrder (Node f x y) = (inOrder x)@[f]@(inOrder y)"
consts
root :: "'a ntree \ 'a"
primrec
"root (Tip a) = a"
"root (Node f x y) = f"
theorem "hd(inOrder xt) = root xt"
quickcheck
--{* Wrong! *}
oops
end