src/HOL/ex/Quickcheck_Examples.thy

rename quantified variables earlier in the new skolemizer, to ensure consistent naming later (in the absence of the choice axiom)

(*  Title:      HOL/ex/Quickcheck_Examples.thy
    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[expect = no_counterexample]
  oops

theorem "map g (map f xs) = map (f o g) xs"
  quickcheck[expect = counterexample]
  oops

theorem "rev (xs @ ys) = rev ys @ rev xs"
  quickcheck[expect = no_counterexample]
  oops

theorem "rev (xs @ ys) = rev xs @ rev ys"
  quickcheck[expect = counterexample]
  oops

theorem "rev (rev xs) = xs"
  quickcheck[expect = no_counterexample]
  oops

theorem "rev xs = xs"
  quickcheck[expect = counterexample]
  oops

text {* An example involving functions inside other data structures *}

primrec app :: "('a \<Rightarrow> 'a) list \<Rightarrow> 'a \<Rightarrow> 'a" where
  "app [] x = x"
  | "app (f # fs) x = app fs (f x)"

lemma "app (fs @ gs) x = app gs (app fs x)"
  quickcheck[expect = no_counterexample]
  by (induct fs arbitrary: x) simp_all

lemma "app (fs @ gs) x = app fs (app gs x)"
  quickcheck[expect = counterexample]
  oops

primrec occurs :: "'a \<Rightarrow> 'a list \<Rightarrow> nat" where
  "occurs a [] = 0"
  | "occurs a (x#xs) = (if (x=a) then Suc(occurs a xs) else occurs a xs)"

primrec del1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
  "del1 a [] = []"
  | "del1 a (x#xs) = (if (x=a) then xs else (x#del1 a xs))"

text {* 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[expect = counterexample]
  oops

lemma "xs ~= [] \<longrightarrow> Suc (occurs a (del1 a xs)) = occurs a xs"
  quickcheck[expect = counterexample]
    -- {* Also wrong.*}
  oops

lemma "0 < occurs a xs \<longrightarrow> Suc (occurs a (del1 a xs)) = occurs a xs"
  quickcheck[expect = no_counterexample]
  by (induct xs) auto

primrec replace :: "'a \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
  "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[expect = counterexample]
  -- {* Wrong. Precondition needed.*}
  oops

lemma "occurs b xs = 0 \<or> a=b \<longrightarrow> occurs a xs = occurs b (replace a b xs)"
  quickcheck[expect = no_counterexample]
  by (induct xs) simp_all


subsection {* Trees *}

datatype 'a tree = Twig |  Leaf 'a | Branch "'a tree" "'a tree"

primrec leaves :: "'a tree \<Rightarrow> 'a list" where
  "leaves Twig = []"
  | "leaves (Leaf a) = [a]"
  | "leaves (Branch l r) = (leaves l) @ (leaves r)"

primrec plant :: "'a list \<Rightarrow> 'a tree" where
  "plant [] = Twig "
  | "plant (x#xs) = Branch (Leaf x) (plant xs)"

primrec mirror :: "'a tree \<Rightarrow> 'a tree" where
  "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[expect = counterexample]
    --{* Wrong! *} 
  oops

theorem "plant((leaves xt) @ (leaves yt)) = Branch xt yt"
  quickcheck[expect = counterexample]
    --{* Wrong! *} 
  oops

datatype 'a ntree = Tip "'a" | Node "'a" "'a ntree" "'a ntree"

primrec inOrder :: "'a ntree \<Rightarrow> 'a list" where
  "inOrder (Tip a)= [a]"
  | "inOrder (Node f x y) = (inOrder x)@[f]@(inOrder y)"

primrec root :: "'a ntree \<Rightarrow> 'a" where
  "root (Tip a) = a"
  | "root (Node f x y) = f"

theorem "hd (inOrder xt) = root xt"
  quickcheck[expect = counterexample]
    --{* Wrong! *} 
  oops

end