src/HOL/Quickcheck_Examples/Quickcheck_Examples.thy

+(*  Title:      HOL/ex/Quickcheck_Examples.thy
+    Author:     Stefan Berghofer, Lukas Bulwahn
+    Copyright   2004 - 2010 TU Muenchen
+*)
+
+header {* Examples for the 'quickcheck' command *}
+
+theory Quickcheck_Examples
+imports Complex_Main "~~/src/HOL/Library/Dlist" "~~/src/HOL/Library/Multiset"
+begin
+
+text {*
+The 'quickcheck' command allows to find counterexamples by evaluating
+formulae.
+Currently, there are two different exploration schemes:
+- random testing: this is incomplete, but explores the search space faster.
+- exhaustive testing: this is complete, but increasing the depth leads to
+  exponentially many assignments.
+
+quickcheck can handle quantifiers on finite universes.
+
+*}
+
+declare [[quickcheck_timeout = 3600]]
+
+subsection {* Lists *}
+
+theorem "map g (map f xs) = map (g o f) xs"
+  quickcheck[random, expect = no_counterexample]
+  quickcheck[exhaustive, size = 3, expect = no_counterexample]
+  oops
+
+theorem "map g (map f xs) = map (f o g) xs"
+  quickcheck[random, expect = counterexample]
+  quickcheck[exhaustive, expect = counterexample]
+  oops
+
+theorem "rev (xs @ ys) = rev ys @ rev xs"
+  quickcheck[random, expect = no_counterexample]
+  quickcheck[exhaustive, expect = no_counterexample]
+  quickcheck[exhaustive, size = 1000, timeout = 0.1]
+  oops
+
+theorem "rev (xs @ ys) = rev xs @ rev ys"
+  quickcheck[random, expect = counterexample]
+  quickcheck[exhaustive, expect = counterexample]
+  oops
+
+theorem "rev (rev xs) = xs"
+  quickcheck[random, expect = no_counterexample]
+  quickcheck[exhaustive, expect = no_counterexample]
+  oops
+
+theorem "rev xs = xs"
+  quickcheck[tester = random, finite_types = true, report = false, expect = counterexample]
+  quickcheck[tester = random, finite_types = false, report = false, expect = counterexample]
+  quickcheck[tester = random, finite_types = true, report = true, expect = counterexample]
+  quickcheck[tester = random, finite_types = false, report = true, expect = counterexample]
+  quickcheck[tester = exhaustive, finite_types = true, expect = counterexample]
+  quickcheck[tester = exhaustive, finite_types = false, 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[random, expect = no_counterexample]
+  quickcheck[exhaustive, size = 4, expect = no_counterexample]
+  by (induct fs arbitrary: x) simp_all
+
+lemma "app (fs @ gs) x = app fs (app gs x)"
+  quickcheck[random, expect = counterexample]
+  quickcheck[exhaustive, 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[random, expect = counterexample]
+  quickcheck[exhaustive, expect = counterexample]
+  oops
+
+lemma "xs ~= [] \<longrightarrow> Suc (occurs a (del1 a xs)) = occurs a xs"
+  quickcheck[random, expect = counterexample]
+  quickcheck[exhaustive, expect = counterexample]
+    -- {* Also wrong.*}
+  oops
+
+lemma "0 < occurs a xs \<longrightarrow> Suc (occurs a (del1 a xs)) = occurs a xs"
+  quickcheck[random, expect = no_counterexample]
+  quickcheck[exhaustive, 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[random, expect = counterexample]
+  quickcheck[exhaustive, 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[random, expect = no_counterexample]
+  quickcheck[exhaustive, 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[random, expect = counterexample]
+  quickcheck[exhaustive, expect = counterexample]
+    --{* Wrong! *} 
+  oops
+
+theorem "plant((leaves xt) @ (leaves yt)) = Branch xt yt"
+  quickcheck[random, expect = counterexample]
+  quickcheck[exhaustive, 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[random, expect = counterexample]
+  quickcheck[exhaustive, expect = counterexample]
+  --{* Wrong! *} 
+  oops
+
+
+subsection {* Exhaustive Testing beats Random Testing *}
+
+text {* Here are some examples from mutants from the List theory
+where exhaustive testing beats random testing *}
+
+lemma
+  "[] ~= xs ==> hd xs = last (x # xs)"
+quickcheck[random]
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+  assumes "!!i. [| i < n; i < length xs |] ==> P (xs ! i)" "n < length xs ==> ~ P (xs ! n)"
+  shows "drop n xs = takeWhile P xs"
+quickcheck[random, iterations = 10000, quiet]
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+  "i < length (List.transpose (List.transpose xs)) ==> xs ! i = map (%xs. xs ! i) [ys<-xs. i < length ys]"
+quickcheck[random, iterations = 10000]
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+  "i < n - m ==> f (lcm m i) = map f [m..<n] ! i"
+quickcheck[random, iterations = 10000, finite_types = false]
+quickcheck[exhaustive, finite_types = false, expect = counterexample]
+oops
+
+lemma
+  "i < n - m ==> f (lcm m i) = map f [m..<n] ! i"
+quickcheck[random, iterations = 10000, finite_types = false]
+quickcheck[exhaustive, finite_types = false, expect = counterexample]
+oops
+
+lemma
+  "ns ! k < length ns ==> k <= listsum ns"
+quickcheck[random, iterations = 10000, finite_types = false, quiet]
+quickcheck[exhaustive, finite_types = false, expect = counterexample]
+oops
+
+lemma
+  "[| ys = x # xs1; zs = xs1 @ xs |] ==> ys @ zs = x # xs"
+quickcheck[random, iterations = 10000]
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+"i < length xs ==> take (Suc i) xs = [] @ xs ! i # take i xs"
+quickcheck[random, iterations = 10000]
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+  "i < length xs ==> take (Suc i) xs = (xs ! i # xs) @ take i []"
+quickcheck[random, iterations = 10000]
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+  "[| sorted (rev (map length xs)); i < length xs |] ==> xs ! i = map (%ys. ys ! i) [ys<-remdups xs. i < length ys]"
+quickcheck[random]
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+  "[| sorted (rev (map length xs)); i < length xs |] ==> xs ! i = map (%ys. ys ! i) [ys<-List.transpose xs. length ys \<in> {..<i}]"
+quickcheck[random]
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+  "(ys = zs) = (xs @ ys = splice xs zs)"
+quickcheck[random]
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+subsection {* Examples with quantifiers *}
+
+text {*
+  These examples show that we can handle quantifiers.
+*}
+
+lemma "(\<exists>x. P x) \<longrightarrow> (\<forall>x. P x)"
+  quickcheck[random, expect = counterexample]
+  quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma "(\<forall>x. \<exists>y. P x y) \<longrightarrow> (\<exists>y. \<forall>x. P x y)"
+  quickcheck[random, expect = counterexample]
+  quickcheck[expect = counterexample]
+oops
+
+lemma "(\<exists>x. P x) \<longrightarrow> (EX! x. P x)"
+  quickcheck[random, expect = counterexample]
+  quickcheck[expect = counterexample]
+oops
+
+
+subsection {* Examples with sets *}
+
+lemma
+  "{} = A Un - A"
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+  "[| bij_betw f A B; bij_betw f C D |] ==> bij_betw f (A Un C) (B Un D)"
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+
+subsection {* Examples with relations *}
+
+lemma
+  "acyclic (R :: ('a * 'a) set) ==> acyclic S ==> acyclic (R Un S)"
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+  "acyclic (R :: (nat * nat) set) ==> acyclic S ==> acyclic (R Un S)"
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+(* FIXME: some dramatic performance decrease after changing the code equation of the ntrancl *)
+lemma
+  "(x, z) : rtrancl (R Un S) ==> \<exists> y. (x, y) : rtrancl R & (y, z) : rtrancl S"
+(*quickcheck[exhaustive, expect = counterexample]*)
+oops
+
+lemma
+  "wf (R :: ('a * 'a) set) ==> wf S ==> wf (R Un S)"
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+  "wf (R :: (nat * nat) set) ==> wf S ==> wf (R Un S)"
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+  "wf (R :: (int * int) set) ==> wf S ==> wf (R Un S)"
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+
+subsection {* Examples with the descriptive operator *}
+
+lemma
+  "(THE x. x = a) = b"
+quickcheck[random, expect = counterexample]
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+subsection {* Examples with Multisets *}
+
+lemma
+  "X + Y = Y + (Z :: 'a multiset)"
+quickcheck[random, expect = counterexample]
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+  "X - Y = Y - (Z :: 'a multiset)"
+quickcheck[random, expect = counterexample]
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+  "N + M - N = (N::'a multiset)"
+quickcheck[random, expect = counterexample]
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+subsection {* Examples with numerical types *}
+
+text {*
+Quickcheck supports the common types nat, int, rat and real.
+*}
+
+lemma
+  "(x :: nat) > 0 ==> y > 0 ==> z > 0 ==> x * x + y * y \<noteq> z * z"
+quickcheck[exhaustive, size = 10, expect = counterexample]
+quickcheck[random, size = 10]
+oops
+
+lemma
+  "(x :: int) > 0 ==> y > 0 ==> z > 0 ==> x * x + y * y \<noteq> z * z"
+quickcheck[exhaustive, size = 10, expect = counterexample]
+quickcheck[random, size = 10]
+oops
+
+lemma
+  "(x :: rat) > 0 ==> y > 0 ==> z > 0 ==> x * x + y * y \<noteq> z * z"
+quickcheck[exhaustive, size = 10, expect = counterexample]
+quickcheck[random, size = 10]
+oops
+
+lemma "(x :: rat) >= 0"
+quickcheck[random, expect = counterexample]
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+  "(x :: real) > 0 ==> y > 0 ==> z > 0 ==> x * x + y * y \<noteq> z * z"
+quickcheck[exhaustive, size = 10, expect = counterexample]
+quickcheck[random, size = 10]
+oops
+
+lemma "(x :: real) >= 0"
+quickcheck[random, expect = counterexample]
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+subsubsection {* floor and ceiling functions *}
+
+lemma
+  "floor x + floor y = floor (x + y :: rat)"
+quickcheck[expect = counterexample]
+oops
+
+lemma
+  "floor x + floor y = floor (x + y :: real)"
+quickcheck[expect = counterexample]
+oops
+
+lemma
+  "ceiling x + ceiling y = ceiling (x + y :: rat)"
+quickcheck[expect = counterexample]
+oops
+
+lemma
+  "ceiling x + ceiling y = ceiling (x + y :: real)"
+quickcheck[expect = counterexample]
+oops
+
+subsection {* Examples with abstract types *}
+
+lemma
+  "Dlist.length (Dlist.remove x xs) = Dlist.length xs - 1"
+quickcheck[exhaustive]
+quickcheck[random]
+oops
+
+lemma
+  "Dlist.length (Dlist.insert x xs) = Dlist.length xs + 1"
+quickcheck[exhaustive]
+quickcheck[random]
+oops
+
+subsection {* Examples with Records *}
+
+record point =
+  xpos :: nat
+  ypos :: nat
+
+lemma
+  "xpos r = xpos r' ==> r = r'"
+quickcheck[exhaustive, expect = counterexample]
+quickcheck[random, expect = counterexample]
+oops
+
+datatype colour = Red | Green | Blue
+
+record cpoint = point +
+  colour :: colour
+
+lemma
+  "xpos r = xpos r' ==> ypos r = ypos r' ==> (r :: cpoint) = r'"
+quickcheck[exhaustive, expect = counterexample]
+quickcheck[random, expect = counterexample]
+oops
+
+subsection {* Examples with locales *}
+
+locale Truth
+
+context Truth
+begin
+
+lemma "False"
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+end
+
+interpretation Truth .
+
+context Truth
+begin
+
+lemma "False"
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+end
+
+locale antisym =
+  fixes R
+  assumes "R x y --> R y x --> x = y"
+begin
+
+lemma
+  "R x y --> R y z --> R x z"
+quickcheck[exhaustive, finite_type_size = 2, expect = no_counterexample]
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+end
+
+subsection {* Examples with HOL quantifiers *}
+
+lemma
+  "\<forall> xs ys. xs = [] --> xs = ys"
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+  "ys = [] --> (\<forall>xs. xs = [] --> xs = y # ys)"
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+lemma
+  "\<forall>xs. (\<exists> ys. ys = []) --> xs = ys"
+quickcheck[exhaustive, expect = counterexample]
+oops
+
+subsection {* Examples with underspecified/partial functions *}
+
+lemma
+  "xs = [] ==> hd xs \<noteq> x"
+quickcheck[exhaustive, expect = no_counterexample]
+quickcheck[random, report = false, expect = no_counterexample]
+quickcheck[random, report = true, expect = no_counterexample]
+oops
+
+lemma
+  "xs = [] ==> hd xs = x"
+quickcheck[exhaustive, expect = no_counterexample]
+quickcheck[random, report = false, expect = no_counterexample]
+quickcheck[random, report = true, expect = no_counterexample]
+oops
+
+lemma "xs = [] ==> hd xs = x ==> x = y"
+quickcheck[exhaustive, expect = no_counterexample]
+quickcheck[random, report = false, expect = no_counterexample]
+quickcheck[random, report = true, expect = no_counterexample]
+oops
+
+text {* with the simple testing scheme *}
+
+setup {* Exhaustive_Generators.setup_exhaustive_datatype_interpretation *}
+declare [[quickcheck_full_support = false]]
+
+lemma
+  "xs = [] ==> hd xs \<noteq> x"
+quickcheck[exhaustive, expect = no_counterexample]
+oops
+
+lemma
+  "xs = [] ==> hd xs = x"
+quickcheck[exhaustive, expect = no_counterexample]
+oops
+
+lemma "xs = [] ==> hd xs = x ==> x = y"
+quickcheck[exhaustive, expect = no_counterexample]
+oops
+
+declare [[quickcheck_full_support = true]]
+
+end