src/HOL/ex/Quickcheck_Examples.thy
author bulwahn
Fri Oct 21 11:17:14 2011 +0200 (2011-10-21)
changeset 45231 d85a2fdc586c
parent 45118 7462f287189a
child 45441 fb4ac1dd4fde
permissions -rw-r--r--
replacing code_inline by code_unfold, removing obsolete code_unfold, code_inline del now that the ancient code generator is removed
     1 (*  Title:      HOL/ex/Quickcheck_Examples.thy
     2     Author:     Stefan Berghofer, Lukas Bulwahn
     3     Copyright   2004 - 2010 TU Muenchen
     4 *)
     5 
     6 header {* Examples for the 'quickcheck' command *}
     7 
     8 theory Quickcheck_Examples
     9 imports Complex_Main
    10 begin
    11 
    12 text {*
    13 The 'quickcheck' command allows to find counterexamples by evaluating
    14 formulae.
    15 Currently, there are two different exploration schemes:
    16 - random testing: this is incomplete, but explores the search space faster.
    17 - exhaustive testing: this is complete, but increasing the depth leads to
    18   exponentially many assignments.
    19 
    20 quickcheck can handle quantifiers on finite universes.
    21 
    22 *}
    23 
    24 declare [[quickcheck_timeout = 3600]]
    25 
    26 subsection {* Lists *}
    27 
    28 theorem "map g (map f xs) = map (g o f) xs"
    29   quickcheck[random, expect = no_counterexample]
    30   quickcheck[exhaustive, size = 3, expect = no_counterexample]
    31   oops
    32 
    33 theorem "map g (map f xs) = map (f o g) xs"
    34   quickcheck[random, expect = counterexample]
    35   quickcheck[exhaustive, expect = counterexample]
    36   oops
    37 
    38 theorem "rev (xs @ ys) = rev ys @ rev xs"
    39   quickcheck[random, expect = no_counterexample]
    40   quickcheck[exhaustive, expect = no_counterexample]
    41   quickcheck[exhaustive, size = 1000, timeout = 0.1]
    42   oops
    43 
    44 theorem "rev (xs @ ys) = rev xs @ rev ys"
    45   quickcheck[random, expect = counterexample]
    46   quickcheck[exhaustive, expect = counterexample]
    47   oops
    48 
    49 theorem "rev (rev xs) = xs"
    50   quickcheck[random, expect = no_counterexample]
    51   quickcheck[exhaustive, expect = no_counterexample]
    52   oops
    53 
    54 theorem "rev xs = xs"
    55   quickcheck[tester = random, finite_types = true, report = false, expect = counterexample]
    56   quickcheck[tester = random, finite_types = false, report = false, expect = counterexample]
    57   quickcheck[tester = random, finite_types = true, report = true, expect = counterexample]
    58   quickcheck[tester = random, finite_types = false, report = true, expect = counterexample]
    59   quickcheck[tester = exhaustive, finite_types = true, expect = counterexample]
    60   quickcheck[tester = exhaustive, finite_types = false, expect = counterexample]
    61 oops
    62 
    63 
    64 text {* An example involving functions inside other data structures *}
    65 
    66 primrec app :: "('a \<Rightarrow> 'a) list \<Rightarrow> 'a \<Rightarrow> 'a" where
    67   "app [] x = x"
    68   | "app (f # fs) x = app fs (f x)"
    69 
    70 lemma "app (fs @ gs) x = app gs (app fs x)"
    71   quickcheck[random, expect = no_counterexample]
    72   quickcheck[exhaustive, size = 4, expect = no_counterexample]
    73   by (induct fs arbitrary: x) simp_all
    74 
    75 lemma "app (fs @ gs) x = app fs (app gs x)"
    76   quickcheck[random, expect = counterexample]
    77   quickcheck[exhaustive, expect = counterexample]
    78   oops
    79 
    80 primrec occurs :: "'a \<Rightarrow> 'a list \<Rightarrow> nat" where
    81   "occurs a [] = 0"
    82   | "occurs a (x#xs) = (if (x=a) then Suc(occurs a xs) else occurs a xs)"
    83 
    84 primrec del1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
    85   "del1 a [] = []"
    86   | "del1 a (x#xs) = (if (x=a) then xs else (x#del1 a xs))"
    87 
    88 text {* A lemma, you'd think to be true from our experience with delAll *}
    89 lemma "Suc (occurs a (del1 a xs)) = occurs a xs"
    90   -- {* Wrong. Precondition needed.*}
    91   quickcheck[random, expect = counterexample]
    92   quickcheck[exhaustive, expect = counterexample]
    93   oops
    94 
    95 lemma "xs ~= [] \<longrightarrow> Suc (occurs a (del1 a xs)) = occurs a xs"
    96   quickcheck[random, expect = counterexample]
    97   quickcheck[exhaustive, expect = counterexample]
    98     -- {* Also wrong.*}
    99   oops
   100 
   101 lemma "0 < occurs a xs \<longrightarrow> Suc (occurs a (del1 a xs)) = occurs a xs"
   102   quickcheck[random, expect = no_counterexample]
   103   quickcheck[exhaustive, expect = no_counterexample]
   104   by (induct xs) auto
   105 
   106 primrec replace :: "'a \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   107   "replace a b [] = []"
   108   | "replace a b (x#xs) = (if (x=a) then (b#(replace a b xs)) 
   109                             else (x#(replace a b xs)))"
   110 
   111 lemma "occurs a xs = occurs b (replace a b xs)"
   112   quickcheck[random, expect = counterexample]
   113   quickcheck[exhaustive, expect = counterexample]
   114   -- {* Wrong. Precondition needed.*}
   115   oops
   116 
   117 lemma "occurs b xs = 0 \<or> a=b \<longrightarrow> occurs a xs = occurs b (replace a b xs)"
   118   quickcheck[random, expect = no_counterexample]
   119   quickcheck[exhaustive, expect = no_counterexample]
   120   by (induct xs) simp_all
   121 
   122 
   123 subsection {* Trees *}
   124 
   125 datatype 'a tree = Twig |  Leaf 'a | Branch "'a tree" "'a tree"
   126 
   127 primrec leaves :: "'a tree \<Rightarrow> 'a list" where
   128   "leaves Twig = []"
   129   | "leaves (Leaf a) = [a]"
   130   | "leaves (Branch l r) = (leaves l) @ (leaves r)"
   131 
   132 primrec plant :: "'a list \<Rightarrow> 'a tree" where
   133   "plant [] = Twig "
   134   | "plant (x#xs) = Branch (Leaf x) (plant xs)"
   135 
   136 primrec mirror :: "'a tree \<Rightarrow> 'a tree" where
   137   "mirror (Twig) = Twig "
   138   | "mirror (Leaf a) = Leaf a "
   139   | "mirror (Branch l r) = Branch (mirror r) (mirror l)"
   140 
   141 theorem "plant (rev (leaves xt)) = mirror xt"
   142   quickcheck[random, expect = counterexample]
   143   quickcheck[exhaustive, expect = counterexample]
   144     --{* Wrong! *} 
   145   oops
   146 
   147 theorem "plant((leaves xt) @ (leaves yt)) = Branch xt yt"
   148   quickcheck[random, expect = counterexample]
   149   quickcheck[exhaustive, expect = counterexample]
   150     --{* Wrong! *} 
   151   oops
   152 
   153 datatype 'a ntree = Tip "'a" | Node "'a" "'a ntree" "'a ntree"
   154 
   155 primrec inOrder :: "'a ntree \<Rightarrow> 'a list" where
   156   "inOrder (Tip a)= [a]"
   157   | "inOrder (Node f x y) = (inOrder x)@[f]@(inOrder y)"
   158 
   159 primrec root :: "'a ntree \<Rightarrow> 'a" where
   160   "root (Tip a) = a"
   161   | "root (Node f x y) = f"
   162 
   163 theorem "hd (inOrder xt) = root xt"
   164   quickcheck[random, expect = counterexample]
   165   quickcheck[exhaustive, expect = counterexample]
   166   --{* Wrong! *} 
   167   oops
   168 
   169 
   170 subsection {* Exhaustive Testing beats Random Testing *}
   171 
   172 text {* Here are some examples from mutants from the List theory
   173 where exhaustive testing beats random testing *}
   174 
   175 lemma
   176   "[] ~= xs ==> hd xs = last (x # xs)"
   177 quickcheck[random]
   178 quickcheck[exhaustive, expect = counterexample]
   179 oops
   180 
   181 lemma
   182   assumes "!!i. [| i < n; i < length xs |] ==> P (xs ! i)" "n < length xs ==> ~ P (xs ! n)"
   183   shows "drop n xs = takeWhile P xs"
   184 quickcheck[random, iterations = 10000, quiet]
   185 quickcheck[exhaustive, expect = counterexample]
   186 oops
   187 
   188 lemma
   189   "i < length (List.transpose (List.transpose xs)) ==> xs ! i = map (%xs. xs ! i) [ys<-xs. i < length ys]"
   190 quickcheck[random, iterations = 10000]
   191 quickcheck[exhaustive, expect = counterexample]
   192 oops
   193 
   194 lemma
   195   "i < n - m ==> f (lcm m i) = map f [m..<n] ! i"
   196 quickcheck[random, iterations = 10000, finite_types = false]
   197 quickcheck[exhaustive, finite_types = false, expect = counterexample]
   198 oops
   199 
   200 lemma
   201   "i < n - m ==> f (lcm m i) = map f [m..<n] ! i"
   202 quickcheck[random, iterations = 10000, finite_types = false]
   203 quickcheck[exhaustive, finite_types = false, expect = counterexample]
   204 oops
   205 
   206 lemma
   207   "ns ! k < length ns ==> k <= listsum ns"
   208 quickcheck[random, iterations = 10000, finite_types = false, quiet]
   209 quickcheck[exhaustive, finite_types = false, expect = counterexample]
   210 oops
   211 
   212 lemma
   213   "[| ys = x # xs1; zs = xs1 @ xs |] ==> ys @ zs = x # xs"
   214 quickcheck[random, iterations = 10000]
   215 quickcheck[exhaustive, expect = counterexample]
   216 oops
   217 
   218 lemma
   219 "i < length xs ==> take (Suc i) xs = [] @ xs ! i # take i xs"
   220 quickcheck[random, iterations = 10000]
   221 quickcheck[exhaustive, expect = counterexample]
   222 oops
   223 
   224 lemma
   225   "i < length xs ==> take (Suc i) xs = (xs ! i # xs) @ take i []"
   226 quickcheck[random, iterations = 10000]
   227 quickcheck[exhaustive, expect = counterexample]
   228 oops
   229 
   230 lemma
   231   "[| sorted (rev (map length xs)); i < length xs |] ==> xs ! i = map (%ys. ys ! i) [ys<-remdups xs. i < length ys]"
   232 quickcheck[random]
   233 quickcheck[exhaustive, expect = counterexample]
   234 oops
   235 
   236 lemma
   237   "[| sorted (rev (map length xs)); i < length xs |] ==> xs ! i = map (%ys. ys ! i) [ys<-List.transpose xs. length ys \<in> {..<i}]"
   238 quickcheck[random]
   239 quickcheck[exhaustive, expect = counterexample]
   240 oops
   241 
   242 lemma
   243   "(ys = zs) = (xs @ ys = splice xs zs)"
   244 quickcheck[random]
   245 quickcheck[exhaustive, expect = counterexample]
   246 oops
   247 
   248 subsection {* Examples with quantifiers *}
   249 
   250 text {*
   251   These examples show that we can handle quantifiers.
   252 *}
   253 
   254 lemma "(\<exists>x. P x) \<longrightarrow> (\<forall>x. P x)"
   255   quickcheck[random, expect = counterexample]
   256   quickcheck[exhaustive, expect = counterexample]
   257 oops
   258 
   259 lemma "(\<forall>x. \<exists>y. P x y) \<longrightarrow> (\<exists>y. \<forall>x. P x y)"
   260   quickcheck[random, expect = counterexample]
   261   quickcheck[expect = counterexample]
   262 oops
   263 
   264 lemma "(\<exists>x. P x) \<longrightarrow> (EX! x. P x)"
   265   quickcheck[random, expect = counterexample]
   266   quickcheck[expect = counterexample]
   267 oops
   268 
   269 
   270 subsection {* Examples with relations *}
   271 
   272 lemma
   273   "acyclic R ==> acyclic S ==> acyclic (R Un S)"
   274 quickcheck[expect = counterexample]
   275 oops
   276 
   277 lemma
   278   "(x, z) : rtrancl (R Un S) ==> \<exists> y. (x, y) : rtrancl R & (y, z) : rtrancl S"
   279 quickcheck[expect = counterexample]
   280 oops
   281 
   282 
   283 subsection {* Examples with numerical types *}
   284 
   285 text {*
   286 Quickcheck supports the common types nat, int, rat and real.
   287 *}
   288 
   289 lemma
   290   "(x :: nat) > 0 ==> y > 0 ==> z > 0 ==> x * x + y * y \<noteq> z * z"
   291 quickcheck[exhaustive, size = 10, expect = counterexample]
   292 quickcheck[random, size = 10]
   293 oops
   294 
   295 lemma
   296   "(x :: int) > 0 ==> y > 0 ==> z > 0 ==> x * x + y * y \<noteq> z * z"
   297 quickcheck[exhaustive, size = 10, expect = counterexample]
   298 quickcheck[random, size = 10]
   299 oops
   300 
   301 lemma
   302   "(x :: rat) > 0 ==> y > 0 ==> z > 0 ==> x * x + y * y \<noteq> z * z"
   303 quickcheck[exhaustive, size = 10, expect = counterexample]
   304 quickcheck[random, size = 10]
   305 oops
   306 
   307 lemma
   308   "(x :: real) > 0 ==> y > 0 ==> z > 0 ==> x * x + y * y \<noteq> z * z"
   309 quickcheck[exhaustive, size = 10, expect = counterexample]
   310 quickcheck[random, size = 10]
   311 oops
   312 
   313 subsubsection {* floor and ceiling functions *}
   314 
   315 lemma
   316   "floor x + floor y = floor (x + y :: rat)"
   317 quickcheck[expect = counterexample]
   318 oops
   319 
   320 lemma
   321   "floor x + floor y = floor (x + y :: real)"
   322 quickcheck[expect = counterexample]
   323 oops
   324 
   325 lemma
   326   "ceiling x + ceiling y = ceiling (x + y :: rat)"
   327 quickcheck[expect = counterexample]
   328 oops
   329 
   330 lemma
   331   "ceiling x + ceiling y = ceiling (x + y :: real)"
   332 quickcheck[expect = counterexample]
   333 oops
   334 
   335 
   336 subsection {* Examples with Records *}
   337 
   338 record point =
   339   xpos :: nat
   340   ypos :: nat
   341 
   342 lemma
   343   "xpos r = xpos r' ==> r = r'"
   344 quickcheck[exhaustive, expect = counterexample]
   345 quickcheck[random, expect = counterexample]
   346 oops
   347 
   348 datatype colour = Red | Green | Blue
   349 
   350 record cpoint = point +
   351   colour :: colour
   352 
   353 lemma
   354   "xpos r = xpos r' ==> ypos r = ypos r' ==> (r :: cpoint) = r'"
   355 quickcheck[exhaustive, expect = counterexample]
   356 quickcheck[random, expect = counterexample]
   357 oops
   358 
   359 subsection {* Examples with locales *}
   360 
   361 locale Truth
   362 
   363 context Truth
   364 begin
   365 
   366 lemma "False"
   367 quickcheck[exhaustive, expect = no_counterexample]
   368 oops
   369 
   370 end
   371 
   372 interpretation Truth .
   373 
   374 context Truth
   375 begin
   376 
   377 lemma "False"
   378 quickcheck[exhaustive, expect = counterexample]
   379 oops
   380 
   381 end
   382 
   383 end