# Theory Quickcheck_Examples

theory Quickcheck_Examples
imports Complex_Main Dlist DAList_Multiset
```(*  Title:      HOL/Quickcheck_Examples/Quickcheck_Examples.thy
Author:     Stefan Berghofer, Lukas Bulwahn
Copyright   2004 - 2010 TU Muenchen
*)

section ‹Examples for the 'quickcheck' command›

theory Quickcheck_Examples
imports Complex_Main "HOL-Library.Dlist" "HOL-Library.DAList_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 ⇒ 'a) list ⇒ 'a ⇒ '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 = 2, 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 ⇒ 'a list ⇒ nat" where
"occurs a [] = 0"
| "occurs a (x#xs) = (if (x=a) then Suc(occurs a xs) else occurs a xs)"

primrec del1 :: "'a ⇒ 'a list ⇒ '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 ~= [] ⟶ 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 ⟶ 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 ⇒ 'a ⇒ 'a list ⇒ '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 ∨ a=b ⟶ 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 ⇒ 'a list" where
"leaves Twig = []"
| "leaves (Leaf a) = [a]"
| "leaves (Branch l r) = (leaves l) @ (leaves r)"

primrec plant :: "'a list ⇒ 'a tree" where
"plant [] = Twig "
| "plant (x#xs) = Branch (Leaf x) (plant xs)"

primrec mirror :: "'a tree ⇒ '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 ⇒ 'a list" where
"inOrder (Tip a)= [a]"
| "inOrder (Node f x y) = (inOrder x)@[f]@(inOrder y)"

primrec root :: "'a ntree ⇒ '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 <= sum_list 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 ∈ {..<i}]"
quickcheck[random]
quickcheck[exhaustive, expect = counterexample]
oops

lemma
"(ys = zs) = (xs @ ys = splice xs zs)"
quickcheck[random]
quickcheck[exhaustive, expect = counterexample]
oops

subsection ‹Random Testing beats Exhaustive Testing›

lemma mult_inj_if_coprime_nat:
"inj_on f A ⟹ inj_on g B
⟹ inj_on (%(a,b). f a * g b::nat) (A × B)"
quickcheck[exhaustive]
quickcheck[random]
oops

subsection ‹Examples with quantifiers›

text ‹
These examples show that we can handle quantifiers.
›

lemma "(∃x. P x) ⟶ (∀x. P x)"
quickcheck[random, expect = counterexample]
quickcheck[exhaustive, expect = counterexample]
oops

lemma "(∀x. ∃y. P x y) ⟶ (∃y. ∀x. P x y)"
quickcheck[random, expect = counterexample]
quickcheck[expect = counterexample]
oops

lemma "(∃x. P x) ⟶ (∃!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) ==> ∃ 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 ≠ 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 ≠ 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 ≠ 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 ≠ 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 "⌊x⌋ + ⌊y⌋ = ⌊x + y :: rat⌋"
quickcheck[expect = counterexample]
oops

lemma "⌊x⌋ + ⌊y⌋ = ⌊x + y :: real⌋"
quickcheck[expect = counterexample]
oops

lemma "⌈x⌉ + ⌈y⌉ = ⌈x + y :: rat⌉"
quickcheck[expect = counterexample]
oops

lemma "⌈x⌉ + ⌈y⌉ = ⌈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"

interpretation equal : antisym "op =" by standard simp
interpretation order_nat : antisym "op <= :: nat => _ => _" by standard simp

lemma (in antisym)
"R x y --> R y z --> R x z"
quickcheck[exhaustive, finite_type_size = 2, expect = no_counterexample]
quickcheck[exhaustive, expect = counterexample]
oops

declare [[quickcheck_locale = "interpret"]]

lemma (in antisym)
"R x y --> R y z --> R x z"
quickcheck[exhaustive, expect = no_counterexample]
oops

declare [[quickcheck_locale = "expand"]]

lemma (in antisym)
"R x y --> R y z --> R x z"
quickcheck[exhaustive, finite_type_size = 2, expect = no_counterexample]
quickcheck[exhaustive, expect = counterexample]
oops

subsection ‹Examples with HOL quantifiers›

lemma
"∀ xs ys. xs = [] --> xs = ys"
quickcheck[exhaustive, expect = counterexample]
oops

lemma
"ys = [] --> (∀xs. xs = [] --> xs = y # ys)"
quickcheck[exhaustive, expect = counterexample]
oops

lemma
"∀xs. (∃ ys. ys = []) --> xs = ys"
quickcheck[exhaustive, expect = counterexample]
oops

subsection ‹Examples with underspecified/partial functions›

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"
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 ≠ 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]]

subsection ‹Equality Optimisation›

lemma
"f x = y ==> y = (0 :: nat)"
quickcheck
oops

lemma
"y = f x ==> y = (0 :: nat)"
quickcheck
oops

lemma
"f y = zz # zzs ==> zz = (0 :: nat) ∧ zzs = []"
quickcheck
oops

lemma
"f y = x # x' # xs ==> x = (0 :: nat) ∧ x' = 0 ∧ xs = []"
quickcheck
oops

lemma
"x = f x ⟹ x = (0 :: nat)"
quickcheck
oops

lemma
"f y = x # x # xs ==> x = (0 :: nat) ∧ xs = []"
quickcheck
oops

lemma
"m1 k = Some v ⟹ (m1 ++ m2) k = Some v"
quickcheck
oops

end
```