author | bulwahn |
Wed, 21 Jul 2010 19:21:07 +0200 | |
changeset 37929 | 22e0797857e6 |
parent 37914 | 49b908e43d61 |
child 40645 | 03ce94672ee6 |
permissions | -rw-r--r-- |
14592 | 1 |
(* Title: HOL/ex/Quickcheck_Examples.thy |
2 |
Author: Stefan Berghofer |
|
3 |
Copyright 2004 TU Muenchen |
|
4 |
*) |
|
5 |
||
6 |
header {* Examples for the 'quickcheck' command *} |
|
7 |
||
28314 | 8 |
theory Quickcheck_Examples |
9 |
imports Main |
|
10 |
begin |
|
14592 | 11 |
|
12 |
text {* |
|
13 |
The 'quickcheck' command allows to find counterexamples by evaluating |
|
14 |
formulae under an assignment of free variables to random values. |
|
15 |
In contrast to 'refute', it can deal with inductive datatypes, |
|
16 |
but cannot handle quantifiers. |
|
17 |
*} |
|
18 |
||
19 |
subsection {* Lists *} |
|
20 |
||
21 |
theorem "map g (map f xs) = map (g o f) xs" |
|
37929
22e0797857e6
adding checking of expected result for the tool quickcheck; annotated a few quickcheck examples
bulwahn
parents:
37914
diff
changeset
|
22 |
quickcheck[expect = no_counterexample] |
14592 | 23 |
oops |
24 |
||
25 |
theorem "map g (map f xs) = map (f o g) xs" |
|
37929
22e0797857e6
adding checking of expected result for the tool quickcheck; annotated a few quickcheck examples
bulwahn
parents:
37914
diff
changeset
|
26 |
quickcheck[expect = counterexample] |
14592 | 27 |
oops |
28 |
||
29 |
theorem "rev (xs @ ys) = rev ys @ rev xs" |
|
37929
22e0797857e6
adding checking of expected result for the tool quickcheck; annotated a few quickcheck examples
bulwahn
parents:
37914
diff
changeset
|
30 |
quickcheck[expect = no_counterexample] |
14592 | 31 |
oops |
32 |
||
33 |
theorem "rev (xs @ ys) = rev xs @ rev ys" |
|
37929
22e0797857e6
adding checking of expected result for the tool quickcheck; annotated a few quickcheck examples
bulwahn
parents:
37914
diff
changeset
|
34 |
quickcheck[expect = counterexample] |
14592 | 35 |
oops |
36 |
||
37 |
theorem "rev (rev xs) = xs" |
|
37929
22e0797857e6
adding checking of expected result for the tool quickcheck; annotated a few quickcheck examples
bulwahn
parents:
37914
diff
changeset
|
38 |
quickcheck[expect = no_counterexample] |
14592 | 39 |
oops |
40 |
||
41 |
theorem "rev xs = xs" |
|
37929
22e0797857e6
adding checking of expected result for the tool quickcheck; annotated a few quickcheck examples
bulwahn
parents:
37914
diff
changeset
|
42 |
quickcheck[expect = counterexample] |
14592 | 43 |
oops |
44 |
||
25891 | 45 |
text {* An example involving functions inside other data structures *} |
46 |
||
28314 | 47 |
primrec app :: "('a \<Rightarrow> 'a) list \<Rightarrow> 'a \<Rightarrow> 'a" where |
25891 | 48 |
"app [] x = x" |
28314 | 49 |
| "app (f # fs) x = app fs (f x)" |
25891 | 50 |
|
51 |
lemma "app (fs @ gs) x = app gs (app fs x)" |
|
37929
22e0797857e6
adding checking of expected result for the tool quickcheck; annotated a few quickcheck examples
bulwahn
parents:
37914
diff
changeset
|
52 |
quickcheck[expect = no_counterexample] |
25891 | 53 |
by (induct fs arbitrary: x) simp_all |
54 |
||
55 |
lemma "app (fs @ gs) x = app fs (app gs x)" |
|
37929
22e0797857e6
adding checking of expected result for the tool quickcheck; annotated a few quickcheck examples
bulwahn
parents:
37914
diff
changeset
|
56 |
quickcheck[expect = counterexample] |
25891 | 57 |
oops |
58 |
||
28314 | 59 |
primrec occurs :: "'a \<Rightarrow> 'a list \<Rightarrow> nat" where |
14592 | 60 |
"occurs a [] = 0" |
28314 | 61 |
| "occurs a (x#xs) = (if (x=a) then Suc(occurs a xs) else occurs a xs)" |
14592 | 62 |
|
28314 | 63 |
primrec del1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
14592 | 64 |
"del1 a [] = []" |
28314 | 65 |
| "del1 a (x#xs) = (if (x=a) then xs else (x#del1 a xs))" |
14592 | 66 |
|
25891 | 67 |
text {* A lemma, you'd think to be true from our experience with delAll *} |
14592 | 68 |
lemma "Suc (occurs a (del1 a xs)) = occurs a xs" |
69 |
-- {* Wrong. Precondition needed.*} |
|
37929
22e0797857e6
adding checking of expected result for the tool quickcheck; annotated a few quickcheck examples
bulwahn
parents:
37914
diff
changeset
|
70 |
quickcheck[expect = counterexample] |
14592 | 71 |
oops |
72 |
||
73 |
lemma "xs ~= [] \<longrightarrow> Suc (occurs a (del1 a xs)) = occurs a xs" |
|
37929
22e0797857e6
adding checking of expected result for the tool quickcheck; annotated a few quickcheck examples
bulwahn
parents:
37914
diff
changeset
|
74 |
quickcheck[expect = counterexample] |
14592 | 75 |
-- {* Also wrong.*} |
76 |
oops |
|
77 |
||
78 |
lemma "0 < occurs a xs \<longrightarrow> Suc (occurs a (del1 a xs)) = occurs a xs" |
|
37929
22e0797857e6
adding checking of expected result for the tool quickcheck; annotated a few quickcheck examples
bulwahn
parents:
37914
diff
changeset
|
79 |
quickcheck[expect = no_counterexample] |
28314 | 80 |
by (induct xs) auto |
14592 | 81 |
|
28314 | 82 |
primrec replace :: "'a \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
14592 | 83 |
"replace a b [] = []" |
28314 | 84 |
| "replace a b (x#xs) = (if (x=a) then (b#(replace a b xs)) |
14592 | 85 |
else (x#(replace a b xs)))" |
86 |
||
87 |
lemma "occurs a xs = occurs b (replace a b xs)" |
|
37929
22e0797857e6
adding checking of expected result for the tool quickcheck; annotated a few quickcheck examples
bulwahn
parents:
37914
diff
changeset
|
88 |
quickcheck[expect = counterexample] |
14592 | 89 |
-- {* Wrong. Precondition needed.*} |
90 |
oops |
|
91 |
||
92 |
lemma "occurs b xs = 0 \<or> a=b \<longrightarrow> occurs a xs = occurs b (replace a b xs)" |
|
37929
22e0797857e6
adding checking of expected result for the tool quickcheck; annotated a few quickcheck examples
bulwahn
parents:
37914
diff
changeset
|
93 |
quickcheck[expect = no_counterexample] |
28314 | 94 |
by (induct xs) simp_all |
14592 | 95 |
|
96 |
||
97 |
subsection {* Trees *} |
|
98 |
||
99 |
datatype 'a tree = Twig | Leaf 'a | Branch "'a tree" "'a tree" |
|
100 |
||
28314 | 101 |
primrec leaves :: "'a tree \<Rightarrow> 'a list" where |
14592 | 102 |
"leaves Twig = []" |
28314 | 103 |
| "leaves (Leaf a) = [a]" |
104 |
| "leaves (Branch l r) = (leaves l) @ (leaves r)" |
|
14592 | 105 |
|
28314 | 106 |
primrec plant :: "'a list \<Rightarrow> 'a tree" where |
14592 | 107 |
"plant [] = Twig " |
28314 | 108 |
| "plant (x#xs) = Branch (Leaf x) (plant xs)" |
14592 | 109 |
|
28314 | 110 |
primrec mirror :: "'a tree \<Rightarrow> 'a tree" where |
14592 | 111 |
"mirror (Twig) = Twig " |
28314 | 112 |
| "mirror (Leaf a) = Leaf a " |
113 |
| "mirror (Branch l r) = Branch (mirror r) (mirror l)" |
|
14592 | 114 |
|
115 |
theorem "plant (rev (leaves xt)) = mirror xt" |
|
37929
22e0797857e6
adding checking of expected result for the tool quickcheck; annotated a few quickcheck examples
bulwahn
parents:
37914
diff
changeset
|
116 |
quickcheck[expect = counterexample] |
14592 | 117 |
--{* Wrong! *} |
118 |
oops |
|
119 |
||
120 |
theorem "plant((leaves xt) @ (leaves yt)) = Branch xt yt" |
|
37929
22e0797857e6
adding checking of expected result for the tool quickcheck; annotated a few quickcheck examples
bulwahn
parents:
37914
diff
changeset
|
121 |
quickcheck[expect = counterexample] |
14592 | 122 |
--{* Wrong! *} |
123 |
oops |
|
124 |
||
125 |
datatype 'a ntree = Tip "'a" | Node "'a" "'a ntree" "'a ntree" |
|
126 |
||
28314 | 127 |
primrec inOrder :: "'a ntree \<Rightarrow> 'a list" where |
14592 | 128 |
"inOrder (Tip a)= [a]" |
28314 | 129 |
| "inOrder (Node f x y) = (inOrder x)@[f]@(inOrder y)" |
14592 | 130 |
|
28314 | 131 |
primrec root :: "'a ntree \<Rightarrow> 'a" where |
14592 | 132 |
"root (Tip a) = a" |
28314 | 133 |
| "root (Node f x y) = f" |
14592 | 134 |
|
28314 | 135 |
theorem "hd (inOrder xt) = root xt" |
37929
22e0797857e6
adding checking of expected result for the tool quickcheck; annotated a few quickcheck examples
bulwahn
parents:
37914
diff
changeset
|
136 |
quickcheck[expect = counterexample] |
14592 | 137 |
--{* Wrong! *} |
138 |
oops |
|
139 |
||
140 |
end |