src/HOL/Quickcheck_Examples/Quickcheck_Narrowing_Examples.thy
 author blanchet Thu Sep 11 18:54:36 2014 +0200 (2014-09-11) changeset 58306 117ba6cbe414 parent 58148 9764b994a421 child 58310 91ea607a34d8 permissions -rw-r--r--
renamed 'rep_datatype' to 'old_rep_datatype' (HOL)
1 (*  Title:      HOL/Quickcheck_Examples/Quickcheck_Narrowing_Examples.thy
2     Author:     Lukas Bulwahn
3     Copyright   2011 TU Muenchen
4 *)
6 header {* Examples for narrowing-based testing  *}
8 theory Quickcheck_Narrowing_Examples
9 imports Main
10 begin
12 declare [[quickcheck_timeout = 3600]]
14 subsection {* Minimalistic examples *}
16 lemma
17   "x = y"
18 quickcheck[tester = narrowing, finite_types = false, default_type = int, expect = counterexample]
19 oops
21 lemma
22   "x = y"
23 quickcheck[tester = narrowing, finite_types = false, default_type = nat, expect = counterexample]
24 oops
26 subsection {* Simple examples with existentials *}
28 lemma
29   "\<exists> y :: nat. \<forall> x. x = y"
30 quickcheck[tester = narrowing, finite_types = false, expect = counterexample]
31 oops
32 (*
33 lemma
34   "\<exists> y :: int. \<forall> x. x = y"
35 quickcheck[tester = narrowing, size = 2]
36 oops
37 *)
38 lemma
39   "x > 1 --> (\<exists>y :: nat. x < y & y <= 1)"
40 quickcheck[tester = narrowing, finite_types = false, expect = counterexample]
41 oops
43 lemma
44   "x > 2 --> (\<exists>y :: nat. x < y & y <= 2)"
45 quickcheck[tester = narrowing, finite_types = false, size = 10]
46 oops
48 lemma
49   "\<forall> x. \<exists> y :: nat. x > 3 --> (y < x & y > 3)"
50 quickcheck[tester = narrowing, finite_types = false, size = 7]
51 oops
53 text {* A false conjecture derived from an partial copy-n-paste of @{thm not_distinct_decomp} *}
54 lemma
55   "~ distinct ws ==> EX xs ys zs y. ws = xs @ [y] @ ys @ [y]"
56 quickcheck[tester = narrowing, finite_types = false, default_type = nat, expect = counterexample]
57 oops
59 text {* A false conjecture derived from theorems @{thm split_list_first} and @{thm split_list_last} *}
60 lemma
61   "x : set xs ==> EX ys zs. xs = ys @ x # zs & x ~: set zs & x ~: set ys"
62 quickcheck[tester = narrowing, finite_types = false, default_type = nat, expect = counterexample]
63 oops
65 text {* A false conjecture derived from @{thm map_eq_Cons_conv} with a typo *}
66 lemma
67   "(map f xs = y # ys) = (EX z zs. xs = z' # zs & f z = y & map f zs = ys)"
68 quickcheck[tester = narrowing, finite_types = false, default_type = nat, expect = counterexample]
69 oops
71 lemma "a # xs = ys @ [a] ==> EX zs. xs = zs @ [a] & ys = a#zs"
72 quickcheck[narrowing, expect = counterexample]
73 quickcheck[exhaustive, random, expect = no_counterexample]
74 oops
76 subsection {* Simple list examples *}
78 lemma "rev xs = xs"
79 quickcheck[tester = narrowing, finite_types = false, default_type = nat, expect = counterexample]
80 oops
82 lemma "rev xs = xs"
83 quickcheck[tester = narrowing, finite_types = false, default_type = int, expect = counterexample]
84 oops
86 lemma "rev xs = xs"
87   quickcheck[tester = narrowing, finite_types = true, expect = counterexample]
88 oops
90 subsection {* Simple examples with functions *}
92 lemma "map f xs = map g xs"
93   quickcheck[tester = narrowing, finite_types = false, expect = counterexample]
94 oops
96 lemma "map f xs = map f ys ==> xs = ys"
97   quickcheck[tester = narrowing, finite_types = false, expect = counterexample]
98 oops
100 lemma
101   "list_all2 P (rev xs) (rev ys) = list_all2 P xs (rev ys)"
102   quickcheck[tester = narrowing, finite_types = false, expect = counterexample]
103 oops
105 lemma "map f xs = F f xs"
106   quickcheck[tester = narrowing, finite_types = false, expect = counterexample]
107 oops
109 lemma "map f xs = F f xs"
110   quickcheck[tester = narrowing, finite_types = false, expect = counterexample]
111 oops
113 (* requires some work...*)
114 (*
115 lemma "R O S = S O R"
116   quickcheck[tester = narrowing, size = 2]
117 oops
118 *)
120 subsection {* Simple examples with inductive predicates *}
122 inductive even where
123   "even 0" |
124   "even n ==> even (Suc (Suc n))"
126 code_pred even .
128 lemma "even (n - 2) ==> even n"
129 quickcheck[narrowing, expect = counterexample]
130 oops
133 subsection {* AVL Trees *}
135 datatype_new 'a tree = ET |  MKT 'a "'a tree" "'a tree" nat
137 primrec set_of :: "'a tree \<Rightarrow> 'a set"
138 where
139 "set_of ET = {}" |
140 "set_of (MKT n l r h) = insert n (set_of l \<union> set_of r)"
142 primrec height :: "'a tree \<Rightarrow> nat"
143 where
144 "height ET = 0" |
145 "height (MKT x l r h) = max (height l) (height r) + 1"
147 primrec avl :: "'a tree \<Rightarrow> bool"
148 where
149 "avl ET = True" |
150 "avl (MKT x l r h) =
151  ((height l = height r \<or> height l = 1+height r \<or> height r = 1+height l) \<and>
152   h = max (height l) (height r) + 1 \<and> avl l \<and> avl r)"
154 primrec is_ord :: "('a::order) tree \<Rightarrow> bool"
155 where
156 "is_ord ET = True" |
157 "is_ord (MKT n l r h) =
158  ((\<forall>n' \<in> set_of l. n' < n) \<and> (\<forall>n' \<in> set_of r. n < n') \<and> is_ord l \<and> is_ord r)"
160 primrec is_in :: "('a::order) \<Rightarrow> 'a tree \<Rightarrow> bool"
161 where
162  "is_in k ET = False" |
163  "is_in k (MKT n l r h) = (if k = n then True else
164                            if k < n then (is_in k l)
165                            else (is_in k r))"
167 primrec ht :: "'a tree \<Rightarrow> nat"
168 where
169 "ht ET = 0" |
170 "ht (MKT x l r h) = h"
172 definition
173  mkt :: "'a \<Rightarrow> 'a tree \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
174 "mkt x l r = MKT x l r (max (ht l) (ht r) + 1)"
176 (* replaced MKT lrn lrl lrr by MKT lrr lrl *)
177 fun l_bal where
178 "l_bal(n, MKT ln ll lr h, r) =
179  (if ht ll < ht lr
180   then case lr of ET \<Rightarrow> ET (* impossible *)
181                 | MKT lrn lrr lrl lrh \<Rightarrow>
182                   mkt lrn (mkt ln ll lrl) (mkt n lrr r)
183   else mkt ln ll (mkt n lr r))"
185 fun r_bal where
186 "r_bal(n, l, MKT rn rl rr h) =
187  (if ht rl > ht rr
188   then case rl of ET \<Rightarrow> ET (* impossible *)
189                 | MKT rln rll rlr h \<Rightarrow> mkt rln (mkt n l rll) (mkt rn rlr rr)
190   else mkt rn (mkt n l rl) rr)"
192 primrec insrt :: "'a::order \<Rightarrow> 'a tree \<Rightarrow> 'a tree"
193 where
194 "insrt x ET = MKT x ET ET 1" |
195 "insrt x (MKT n l r h) =
196    (if x=n
197     then MKT n l r h
198     else if x<n
199          then let l' = insrt x l; hl' = ht l'; hr = ht r
200               in if hl' = 2+hr then l_bal(n,l',r)
201                  else MKT n l' r (1 + max hl' hr)
202          else let r' = insrt x r; hl = ht l; hr' = ht r'
203               in if hr' = 2+hl then r_bal(n,l,r')
204                  else MKT n l r' (1 + max hl hr'))"
207 subsubsection {* Necessary setup for code generation *}
209 primrec set_of'
210 where
211   "set_of' ET = []"
212 | "set_of' (MKT n l r h) = n # (set_of' l @ set_of' r)"
214 lemma set_of':
215   "set (set_of' t) = set_of t"
216 by (induct t) auto
218 lemma is_ord_mkt:
219   "is_ord (MKT n l r h) = ((ALL n': set (set_of' l). n' < n) & (ALL n': set (set_of' r). n < n') & is_ord l & is_ord r)"
220 by (simp add: set_of')
222 declare is_ord.simps(1)[code] is_ord_mkt[code]
224 subsubsection {* Invalid Lemma due to typo in @{const l_bal} *}
226 lemma is_ord_l_bal:
227  "\<lbrakk> is_ord(MKT (x :: nat) l r h); height l = height r + 2 \<rbrakk> \<Longrightarrow> is_ord(l_bal(x,l,r))"
228 quickcheck[tester = narrowing, finite_types = false, default_type = nat, size = 6, expect = counterexample]
229 oops
231 subsection {* Examples with hd and last *}
233 lemma
234   "hd (xs @ ys) = hd ys"
235 quickcheck[narrowing, expect = counterexample]
236 oops
238 lemma
239   "last(xs @ ys) = last xs"
240 quickcheck[narrowing, expect = counterexample]
241 oops
243 subsection {* Examples with underspecified/partial functions *}
245 lemma
246   "xs = [] ==> hd xs \<noteq> x"
247 quickcheck[narrowing, expect = no_counterexample]
248 oops
250 lemma
251   "xs = [] ==> hd xs = x"
252 quickcheck[narrowing, expect = no_counterexample]
253 oops
255 lemma "xs = [] ==> hd xs = x ==> x = y"
256 quickcheck[narrowing, expect = no_counterexample]
257 oops
259 lemma
260   "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
261 quickcheck[narrowing, expect = no_counterexample]
262 oops
264 lemma
265   "hd (map f xs) = f (hd xs)"
266 quickcheck[narrowing, expect = no_counterexample]
267 oops
269 end