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(* Title: HOL/ex/Quickcheck_Examples.thy |
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Author: Stefan Berghofer, Lukas Bulwahn |
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Copyright 2004 - 2010 TU Muenchen |
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*) |
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header {* Examples for the 'quickcheck' command *} |
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theory Quickcheck_Examples |
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imports Complex_Main |
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begin |
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text {* |
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The 'quickcheck' command allows to find counterexamples by evaluating |
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formulae. |
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Currently, there are two different exploration schemes: |
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- random testing: this is incomplete, but explores the search space faster. |
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- exhaustive testing: this is complete, but increasing the depth leads to |
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exponentially many assignments. |
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quickcheck can handle quantifiers on finite universes. |
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*} |
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subsection {* Lists *} |
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theorem "map g (map f xs) = map (g o f) xs" |
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quickcheck[random, expect = no_counterexample] |
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quickcheck[exhaustive, size = 3, expect = no_counterexample] |
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theorem "map g (map f xs) = map (f o g) xs" |
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quickcheck[random, expect = counterexample] |
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quickcheck[exhaustive, expect = counterexample] |
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theorem "rev (xs @ ys) = rev ys @ rev xs" |
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quickcheck[random, expect = no_counterexample] |
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quickcheck[exhaustive, expect = no_counterexample] |
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quickcheck[exhaustive, size = 1000, timeout = 0.1] |
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theorem "rev (xs @ ys) = rev xs @ rev ys" |
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quickcheck[random, expect = counterexample] |
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quickcheck[exhaustive, expect = counterexample] |
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theorem "rev (rev xs) = xs" |
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quickcheck[random, expect = no_counterexample] |
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quickcheck[exhaustive, expect = no_counterexample] |
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theorem "rev xs = xs" |
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quickcheck[tester = random, finite_types = true, report = false, expect = counterexample] |
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quickcheck[tester = random, finite_types = false, report = false, expect = counterexample] |
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quickcheck[tester = random, finite_types = true, report = true, expect = counterexample] |
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quickcheck[tester = random, finite_types = false, report = true, expect = counterexample] |
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quickcheck[tester = exhaustive, finite_types = true, expect = counterexample] |
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quickcheck[tester = exhaustive, finite_types = false, expect = counterexample] |
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oops |
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text {* An example involving functions inside other data structures *} |
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primrec app :: "('a \<Rightarrow> 'a) list \<Rightarrow> 'a \<Rightarrow> 'a" where |
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"app [] x = x" |
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| "app (f # fs) x = app fs (f x)" |
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lemma "app (fs @ gs) x = app gs (app fs x)" |
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quickcheck[random, expect = no_counterexample] |
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quickcheck[exhaustive, size = 4, expect = no_counterexample] |
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by (induct fs arbitrary: x) simp_all |
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lemma "app (fs @ gs) x = app fs (app gs x)" |
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quickcheck[random, expect = counterexample] |
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quickcheck[exhaustive, expect = counterexample] |
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primrec occurs :: "'a \<Rightarrow> 'a list \<Rightarrow> nat" where |
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"occurs a [] = 0" |
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| "occurs a (x#xs) = (if (x=a) then Suc(occurs a xs) else occurs a xs)" |
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primrec del1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
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"del1 a [] = []" |
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| "del1 a (x#xs) = (if (x=a) then xs else (x#del1 a xs))" |
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text {* A lemma, you'd think to be true from our experience with delAll *} |
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lemma "Suc (occurs a (del1 a xs)) = occurs a xs" |
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-- {* Wrong. Precondition needed.*} |
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quickcheck[random, expect = counterexample] |
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quickcheck[exhaustive, expect = counterexample] |
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lemma "xs ~= [] \<longrightarrow> Suc (occurs a (del1 a xs)) = occurs a xs" |
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quickcheck[random, expect = counterexample] |
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quickcheck[exhaustive, expect = counterexample] |
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-- {* Also wrong.*} |
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lemma "0 < occurs a xs \<longrightarrow> Suc (occurs a (del1 a xs)) = occurs a xs" |
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quickcheck[random, expect = no_counterexample] |
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quickcheck[exhaustive, expect = no_counterexample] |
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by (induct xs) auto |
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primrec replace :: "'a \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
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"replace a b [] = []" |
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| "replace a b (x#xs) = (if (x=a) then (b#(replace a b xs)) |
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else (x#(replace a b xs)))" |
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lemma "occurs a xs = occurs b (replace a b xs)" |
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quickcheck[random, expect = counterexample] |
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quickcheck[exhaustive, expect = counterexample] |
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-- {* Wrong. Precondition needed.*} |
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lemma "occurs b xs = 0 \<or> a=b \<longrightarrow> occurs a xs = occurs b (replace a b xs)" |
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quickcheck[random, expect = no_counterexample] |
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quickcheck[exhaustive, expect = no_counterexample] |
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by (induct xs) simp_all |
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subsection {* Trees *} |
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datatype 'a tree = Twig | Leaf 'a | Branch "'a tree" "'a tree" |
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primrec leaves :: "'a tree \<Rightarrow> 'a list" where |
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"leaves Twig = []" |
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| "leaves (Leaf a) = [a]" |
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| "leaves (Branch l r) = (leaves l) @ (leaves r)" |
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primrec plant :: "'a list \<Rightarrow> 'a tree" where |
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"plant [] = Twig " |
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| "plant (x#xs) = Branch (Leaf x) (plant xs)" |
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primrec mirror :: "'a tree \<Rightarrow> 'a tree" where |
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"mirror (Twig) = Twig " |
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| "mirror (Leaf a) = Leaf a " |
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| "mirror (Branch l r) = Branch (mirror r) (mirror l)" |
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theorem "plant (rev (leaves xt)) = mirror xt" |
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quickcheck[random, expect = counterexample] |
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quickcheck[exhaustive, expect = counterexample] |
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--{* Wrong! *} |
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theorem "plant((leaves xt) @ (leaves yt)) = Branch xt yt" |
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quickcheck[random, expect = counterexample] |
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quickcheck[exhaustive, expect = counterexample] |
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--{* Wrong! *} |
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datatype 'a ntree = Tip "'a" | Node "'a" "'a ntree" "'a ntree" |
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primrec inOrder :: "'a ntree \<Rightarrow> 'a list" where |
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"inOrder (Tip a)= [a]" |
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| "inOrder (Node f x y) = (inOrder x)@[f]@(inOrder y)" |
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primrec root :: "'a ntree \<Rightarrow> 'a" where |
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"root (Tip a) = a" |
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| "root (Node f x y) = f" |
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theorem "hd (inOrder xt) = root xt" |
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quickcheck[random, expect = counterexample] |
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quickcheck[exhaustive, expect = counterexample] |
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--{* Wrong! *} |
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subsection {* Exhaustive Testing beats Random Testing *} |
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text {* Here are some examples from mutants from the List theory |
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where exhaustive testing beats random testing *} |
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lemma |
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"[] ~= xs ==> hd xs = last (x # xs)" |
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quickcheck[random] |
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quickcheck[exhaustive, expect = counterexample] |
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oops |
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lemma |
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assumes "!!i. [| i < n; i < length xs |] ==> P (xs ! i)" "n < length xs ==> ~ P (xs ! n)" |
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shows "drop n xs = takeWhile P xs" |
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quickcheck[random, iterations = 10000, quiet] |
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quickcheck[exhaustive, expect = counterexample] |
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oops |
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lemma |
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"i < length (List.transpose (List.transpose xs)) ==> xs ! i = map (%xs. xs ! i) [ys<-xs. i < length ys]" |
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quickcheck[random, iterations = 10000] |
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quickcheck[exhaustive, expect = counterexample] |
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oops |
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lemma |
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"i < n - m ==> f (lcm m i) = map f [m..<n] ! i" |
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quickcheck[random, iterations = 10000, finite_types = false] |
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quickcheck[exhaustive, finite_types = false, expect = counterexample] |
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oops |
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lemma |
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"i < n - m ==> f (lcm m i) = map f [m..<n] ! i" |
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quickcheck[random, iterations = 10000, finite_types = false] |
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quickcheck[exhaustive, finite_types = false, expect = counterexample] |
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lemma |
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"ns ! k < length ns ==> k <= listsum ns" |
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quickcheck[random, iterations = 10000, finite_types = false, quiet] |
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quickcheck[exhaustive, finite_types = false, expect = counterexample] |
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lemma |
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"[| ys = x # xs1; zs = xs1 @ xs |] ==> ys @ zs = x # xs" |
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quickcheck[random, iterations = 10000] |
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quickcheck[exhaustive, expect = counterexample] |
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oops |
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lemma |
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"i < length xs ==> take (Suc i) xs = [] @ xs ! i # take i xs" |
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quickcheck[random, iterations = 10000] |
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quickcheck[exhaustive, expect = counterexample] |
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oops |
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lemma |
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"i < length xs ==> take (Suc i) xs = (xs ! i # xs) @ take i []" |
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quickcheck[random, iterations = 10000] |
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quickcheck[exhaustive, expect = counterexample] |
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oops |
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lemma |
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"[| sorted (rev (map length xs)); i < length xs |] ==> xs ! i = map (%ys. ys ! i) [ys<-remdups xs. i < length ys]" |
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quickcheck[random] |
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quickcheck[exhaustive, expect = counterexample] |
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oops |
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lemma |
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"[| sorted (rev (map length xs)); i < length xs |] ==> xs ! i = map (%ys. ys ! i) [ys<-List.transpose xs. {..<i} (length ys)]" |
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quickcheck[random] |
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quickcheck[exhaustive, expect = counterexample] |
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oops |
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lemma |
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"(ys = zs) = (xs @ ys = splice xs zs)" |
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quickcheck[random] |
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quickcheck[exhaustive, expect = counterexample] |
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subsection {* Examples with quantifiers *} |
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text {* |
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These examples show that we can handle quantifiers. |
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lemma "(\<exists>x. P x) \<longrightarrow> (\<forall>x. P x)" |
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quickcheck[random, expect = counterexample] |
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quickcheck[exhaustive, expect = counterexample] |
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lemma "(\<forall>x. \<exists>y. P x y) \<longrightarrow> (\<exists>y. \<forall>x. P x y)" |
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quickcheck[random, expect = counterexample] |
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quickcheck[expect = counterexample] |
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lemma "(\<exists>x. P x) \<longrightarrow> (EX! x. P x)" |
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quickcheck[random, expect = counterexample] |
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quickcheck[expect = counterexample] |
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subsection {* Examples with numerical types *} |
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text {* |
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Quickcheck supports the common types nat, int, rat and real. |
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*} |
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lemma |
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"(x :: nat) > 0 ==> y > 0 ==> z > 0 ==> x * x + y * y \<noteq> z * z" |
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quickcheck[exhaustive, size = 10, expect = counterexample] |
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quickcheck[random, size = 10] |
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oops |
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lemma |
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"(x :: int) > 0 ==> y > 0 ==> z > 0 ==> x * x + y * y \<noteq> z * z" |
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quickcheck[exhaustive, size = 10, expect = counterexample] |
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quickcheck[random, size = 10] |
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oops |
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lemma |
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"(x :: rat) > 0 ==> y > 0 ==> z > 0 ==> x * x + y * y \<noteq> z * z" |
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quickcheck[exhaustive, size = 10, expect = counterexample] |
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quickcheck[random, size = 10] |
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oops |
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lemma |
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"(x :: real) > 0 ==> y > 0 ==> z > 0 ==> x * x + y * y \<noteq> z * z" |
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quickcheck[exhaustive, size = 10, expect = counterexample] |
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quickcheck[random, size = 10] |
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oops |
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end |