src/HOL/ex/Quickcheck_Examples.thy

--- a/src/HOL/ex/Quickcheck_Examples.thy	Mon Sep 22 13:56:01 2008 +0200
+++ b/src/HOL/ex/Quickcheck_Examples.thy	Mon Sep 22 13:56:03 2008 +0200
@@ -6,7 +6,9 @@
 
 header {* Examples for the 'quickcheck' command *}
 
-theory Quickcheck_Examples imports Main begin
+theory Quickcheck_Examples
+imports Main
+begin
 
 text {*
 The 'quickcheck' command allows to find counterexamples by evaluating
@@ -18,144 +20,121 @@
 subsection {* Lists *}
 
 theorem "map g (map f xs) = map (g o f) xs"
-  quickcheck
+  quickcheck []
   oops
 
 theorem "map g (map f xs) = map (f o g) xs"
-  quickcheck
+  quickcheck []
   oops
 
 theorem "rev (xs @ ys) = rev ys @ rev xs"
-  quickcheck
+  quickcheck []
   oops
 
 theorem "rev (xs @ ys) = rev xs @ rev ys"
-  quickcheck
+  quickcheck []
   oops
 
 theorem "rev (rev xs) = xs"
-  quickcheck
+  quickcheck []
   oops
 
 theorem "rev xs = xs"
-  quickcheck
+  quickcheck []
   oops
 
 text {* An example involving functions inside other data structures *}
 
-consts app :: "('a \<Rightarrow> 'a) list \<Rightarrow> 'a \<Rightarrow> 'a"
-
-primrec
+primrec app :: "('a \<Rightarrow> 'a) list \<Rightarrow> 'a \<Rightarrow> 'a" where
   "app [] x = x"
-  "app (f # fs) x = app fs (f x)"
+  | "app (f # fs) x = app fs (f x)"
 
 lemma "app (fs @ gs) x = app gs (app fs x)"
-  quickcheck
+  quickcheck []
   by (induct fs arbitrary: x) simp_all
 
 lemma "app (fs @ gs) x = app fs (app gs x)"
-  quickcheck
+  quickcheck []
   oops
 
-consts
-  occurs :: "'a \<Rightarrow> 'a list \<Rightarrow> nat"
-primrec
+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)"
+  | "occurs a (x#xs) = (if (x=a) then Suc(occurs a xs) else occurs a xs)"
 
-consts
-  del1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list"
-primrec
+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))"
+  | "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
+  quickcheck []
   oops
 
 lemma "xs ~= [] \<longrightarrow> Suc (occurs a (del1 a xs)) = occurs a xs"
-  quickcheck
+  quickcheck []
     -- {* Also wrong.*}
   oops
 
 lemma "0 < occurs a xs \<longrightarrow> Suc (occurs a (del1 a xs)) = occurs a xs"
-  quickcheck
-  apply (induct_tac xs)  
-  apply auto
-    -- {* Correct! *}
-  done
+  quickcheck []
+  by (induct xs) auto
 
-consts
-  replace :: "'a \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'a list"
-primrec
+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)) 
+  | "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
+  quickcheck []
   -- {* Wrong. Precondition needed.*}
   oops
 
 lemma "occurs b xs = 0 \<or> a=b \<longrightarrow> occurs a xs = occurs b (replace a b xs)"
-  quickcheck
-  apply (induct_tac xs)  
-  apply auto
-  done
+  quickcheck []
+  by (induct xs) simp_all
 
 
 subsection {* Trees *}
 
 datatype 'a tree = Twig |  Leaf 'a | Branch "'a tree" "'a tree"
 
-consts
-  leaves :: "'a tree \<Rightarrow> 'a list"
-primrec
+primrec leaves :: "'a tree \<Rightarrow> 'a list" where
   "leaves Twig = []"
-  "leaves (Leaf a) = [a]"
-  "leaves (Branch l r) = (leaves l) @ (leaves r)"
+  | "leaves (Leaf a) = [a]"
+  | "leaves (Branch l r) = (leaves l) @ (leaves r)"
 
-consts
-  plant :: "'a list \<Rightarrow> 'a tree"
-primrec
+primrec plant :: "'a list \<Rightarrow> 'a tree" where
   "plant [] = Twig "
-  "plant (x#xs) = Branch (Leaf x) (plant xs)"
+  | "plant (x#xs) = Branch (Leaf x) (plant xs)"
 
-consts
-  mirror :: "'a tree \<Rightarrow> 'a tree"
-primrec
+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)"
+  | "mirror (Leaf a) = Leaf a "
+  | "mirror (Branch l r) = Branch (mirror r) (mirror l)"
 
 theorem "plant (rev (leaves xt)) = mirror xt"
-  quickcheck
+  quickcheck []
     --{* Wrong! *} 
   oops
 
 theorem "plant((leaves xt) @ (leaves yt)) = Branch xt yt"
-  quickcheck
+  quickcheck []
     --{* Wrong! *} 
   oops
 
 datatype 'a ntree = Tip "'a" | Node "'a" "'a ntree" "'a ntree"
 
-consts
-  inOrder :: "'a ntree \<Rightarrow> 'a list"
-primrec
+primrec inOrder :: "'a ntree \<Rightarrow> 'a list" where
   "inOrder (Tip a)= [a]"
-  "inOrder (Node f x y) = (inOrder x)@[f]@(inOrder y)"
+  | "inOrder (Node f x y) = (inOrder x)@[f]@(inOrder y)"
 
-consts
-  root :: "'a ntree \<Rightarrow> 'a"
-primrec
+primrec root :: "'a ntree \<Rightarrow> 'a" where
   "root (Tip a) = a"
-  "root (Node f x y) = f"
+  | "root (Node f x y) = f"
 
-theorem "hd(inOrder xt) = root xt"
-  quickcheck
+theorem "hd (inOrder xt) = root xt"
+  quickcheck []
     --{* Wrong! *} 
   oops