1 (* Title: HOL/Nitpick.thy
2 Author: Jasmin Blanchette, TU Muenchen
3 Copyright 2008, 2009, 2010
5 Nitpick: Yet another counterexample generator for Isabelle/HOL.
8 header {* Nitpick: Yet Another Counterexample Generator for Isabelle/HOL *}
11 imports Map Quotient SAT
12 uses ("Tools/Nitpick/kodkod.ML")
13 ("Tools/Nitpick/kodkod_sat.ML")
14 ("Tools/Nitpick/nitpick_util.ML")
15 ("Tools/Nitpick/nitpick_hol.ML")
16 ("Tools/Nitpick/nitpick_preproc.ML")
17 ("Tools/Nitpick/nitpick_mono.ML")
18 ("Tools/Nitpick/nitpick_scope.ML")
19 ("Tools/Nitpick/nitpick_peephole.ML")
20 ("Tools/Nitpick/nitpick_rep.ML")
21 ("Tools/Nitpick/nitpick_nut.ML")
22 ("Tools/Nitpick/nitpick_kodkod.ML")
23 ("Tools/Nitpick/nitpick_model.ML")
24 ("Tools/Nitpick/nitpick.ML")
25 ("Tools/Nitpick/nitpick_isar.ML")
26 ("Tools/Nitpick/nitpick_tests.ML")
27 ("Tools/Nitpick/minipick.ML")
30 typedecl bisim_iterator
32 axiomatization unknown :: 'a
33 and is_unknown :: "'a \<Rightarrow> bool"
34 and undefined_fast_The :: 'a
35 and undefined_fast_Eps :: 'a
36 and bisim :: "bisim_iterator \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
37 and bisim_iterator_max :: bisim_iterator
38 and Quot :: "'a \<Rightarrow> 'b"
39 and safe_The :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
40 and safe_Eps :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
42 datatype ('a, 'b) fin_fun = FinFun "('a \<Rightarrow> 'b)"
43 datatype ('a, 'b) fun_box = FunBox "('a \<Rightarrow> 'b)"
44 datatype ('a, 'b) pair_box = PairBox 'a 'b
49 datatype 'a word = Word "('a set)"
52 Alternative definitions.
55 lemma If_def [nitpick_def, no_atp]:
56 "(if P then Q else R) \<equiv> (P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> R)"
57 by (rule eq_reflection) (rule if_bool_eq_conj)
59 lemma Ex1_def [nitpick_def, no_atp]:
60 "Ex1 P \<equiv> \<exists>x. P = {x}"
61 apply (rule eq_reflection)
62 apply (simp add: Ex1_def expand_set_eq)
66 apply (rule_tac x = x in exI)
69 apply (erule_tac x = y in allE)
70 by (auto simp: mem_def)
72 lemma rtrancl_def [nitpick_def, no_atp]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
75 lemma rtranclp_def [nitpick_def, no_atp]:
76 "rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"
77 by (rule eq_reflection) (auto dest: rtranclpD)
79 lemma tranclp_def [nitpick_def, no_atp]:
80 "tranclp r a b \<equiv> trancl (split r) (a, b)"
81 by (simp add: trancl_def Collect_def mem_def)
83 definition refl' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
84 "refl' r \<equiv> \<forall>x. (x, x) \<in> r"
86 definition wf' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
87 "wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
89 axiomatization wf_wfrec :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
91 definition wf_wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
92 [nitpick_simp]: "wf_wfrec' R F x = F (Recdef.cut (wf_wfrec R F) R x) x"
94 definition wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
95 "wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x
96 else THE y. wfrec_rel R (%f x. F (Recdef.cut f R x) x) x y"
98 definition card' :: "('a \<Rightarrow> bool) \<Rightarrow> nat" where
99 "card' A \<equiv> if finite A then length (safe_Eps (\<lambda>xs. set xs = A \<and> distinct xs)) else 0"
101 definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b" where
102 "setsum' f A \<equiv> if finite A then listsum (map f (safe_Eps (\<lambda>xs. set xs = A \<and> distinct xs))) else 0"
104 inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool" where
105 "fold_graph' f z {} z" |
106 "\<lbrakk>x \<in> A; fold_graph' f z (A - {x}) y\<rbrakk> \<Longrightarrow> fold_graph' f z A (f x y)"
109 The following lemmas are not strictly necessary but they help the
110 \textit{special\_level} optimization.
113 lemma The_psimp [nitpick_psimp, no_atp]:
114 "P = {x} \<Longrightarrow> The P = x"
115 by (subgoal_tac "{x} = (\<lambda>y. y = x)") (auto simp: mem_def)
117 lemma Eps_psimp [nitpick_psimp, no_atp]:
118 "\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> Eps P = x"
119 apply (case_tac "P (Eps P)")
121 apply (erule contrapos_np)
124 lemma unit_case_def [nitpick_def, no_atp]:
125 "unit_case x u \<equiv> x"
126 apply (subgoal_tac "u = ()")
127 apply (simp only: unit.cases)
130 declare unit.cases [nitpick_simp del]
132 lemma nat_case_def [nitpick_def, no_atp]:
133 "nat_case x f n \<equiv> if n = 0 then x else f (n - 1)"
134 apply (rule eq_reflection)
137 declare nat.cases [nitpick_simp del]
139 lemma list_size_simp [nitpick_simp, no_atp]:
140 "list_size f xs = (if xs = [] then 0
141 else Suc (f (hd xs) + list_size f (tl xs)))"
142 "size xs = (if xs = [] then 0 else Suc (size (tl xs)))"
143 by (case_tac xs) auto
146 Auxiliary definitions used to provide an alternative representation for
147 @{text rat} and @{text real}.
150 function nat_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
151 [simp del]: "nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"
154 apply (relation "measure (\<lambda>(x, y). x + y + (if y > x then 1 else 0))")
156 apply (metis mod_less_divisor xt1(9))
157 by (metis mod_mod_trivial mod_self nat_neq_iff xt1(10))
159 definition nat_lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
160 "nat_lcm x y = x * y div (nat_gcd x y)"
162 definition int_gcd :: "int \<Rightarrow> int \<Rightarrow> int" where
163 "int_gcd x y = int (nat_gcd (nat (abs x)) (nat (abs y)))"
165 definition int_lcm :: "int \<Rightarrow> int \<Rightarrow> int" where
166 "int_lcm x y = int (nat_lcm (nat (abs x)) (nat (abs y)))"
168 definition Frac :: "int \<times> int \<Rightarrow> bool" where
169 "Frac \<equiv> \<lambda>(a, b). b > 0 \<and> int_gcd a b = 1"
171 axiomatization Abs_Frac :: "int \<times> int \<Rightarrow> 'a"
172 and Rep_Frac :: "'a \<Rightarrow> int \<times> int"
174 definition zero_frac :: 'a where
175 "zero_frac \<equiv> Abs_Frac (0, 1)"
177 definition one_frac :: 'a where
178 "one_frac \<equiv> Abs_Frac (1, 1)"
180 definition num :: "'a \<Rightarrow> int" where
181 "num \<equiv> fst o Rep_Frac"
183 definition denom :: "'a \<Rightarrow> int" where
184 "denom \<equiv> snd o Rep_Frac"
186 function norm_frac :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
187 [simp del]: "norm_frac a b = (if b < 0 then norm_frac (- a) (- b)
188 else if a = 0 \<or> b = 0 then (0, 1)
189 else let c = int_gcd a b in (a div c, b div c))"
190 by pat_completeness auto
191 termination by (relation "measure (\<lambda>(_, b). if b < 0 then 1 else 0)") auto
193 definition frac :: "int \<Rightarrow> int \<Rightarrow> 'a" where
194 "frac a b \<equiv> Abs_Frac (norm_frac a b)"
196 definition plus_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
198 "plus_frac q r = (let d = int_lcm (denom q) (denom r) in
199 frac (num q * (d div denom q) + num r * (d div denom r)) d)"
201 definition times_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
203 "times_frac q r = frac (num q * num r) (denom q * denom r)"
205 definition uminus_frac :: "'a \<Rightarrow> 'a" where
206 "uminus_frac q \<equiv> Abs_Frac (- num q, denom q)"
208 definition number_of_frac :: "int \<Rightarrow> 'a" where
209 "number_of_frac n \<equiv> Abs_Frac (n, 1)"
211 definition inverse_frac :: "'a \<Rightarrow> 'a" where
212 "inverse_frac q \<equiv> frac (denom q) (num q)"
214 definition less_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
216 "less_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) < 0"
218 definition less_eq_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
220 "less_eq_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) \<le> 0"
222 definition of_frac :: "'a \<Rightarrow> 'b\<Colon>{inverse,ring_1}" where
223 "of_frac q \<equiv> of_int (num q) / of_int (denom q)"
225 use "Tools/Nitpick/kodkod.ML"
226 use "Tools/Nitpick/kodkod_sat.ML"
227 use "Tools/Nitpick/nitpick_util.ML"
228 use "Tools/Nitpick/nitpick_hol.ML"
229 use "Tools/Nitpick/nitpick_mono.ML"
230 use "Tools/Nitpick/nitpick_preproc.ML"
231 use "Tools/Nitpick/nitpick_scope.ML"
232 use "Tools/Nitpick/nitpick_peephole.ML"
233 use "Tools/Nitpick/nitpick_rep.ML"
234 use "Tools/Nitpick/nitpick_nut.ML"
235 use "Tools/Nitpick/nitpick_kodkod.ML"
236 use "Tools/Nitpick/nitpick_model.ML"
237 use "Tools/Nitpick/nitpick.ML"
238 use "Tools/Nitpick/nitpick_isar.ML"
239 use "Tools/Nitpick/nitpick_tests.ML"
240 use "Tools/Nitpick/minipick.ML"
242 setup {* Nitpick_Isar.setup *}
244 hide_const (open) unknown is_unknown undefined_fast_The undefined_fast_Eps bisim
245 bisim_iterator_max Quot safe_The safe_Eps FinFun FunBox PairBox Word refl'
246 wf' wf_wfrec wf_wfrec' wfrec' card' setsum' fold_graph' nat_gcd nat_lcm
247 int_gcd int_lcm Frac Abs_Frac Rep_Frac zero_frac one_frac num denom
248 norm_frac frac plus_frac times_frac uminus_frac number_of_frac inverse_frac
249 less_frac less_eq_frac of_frac
250 hide_type (open) bisim_iterator fin_fun fun_box pair_box unsigned_bit signed_bit
252 hide_fact (open) If_def Ex1_def rtrancl_def rtranclp_def tranclp_def refl'_def
253 wf'_def wf_wfrec'_def wfrec'_def card'_def setsum'_def fold_graph'_def
254 The_psimp Eps_psimp unit_case_def nat_case_def list_size_simp nat_gcd_def
255 nat_lcm_def int_gcd_def int_lcm_def Frac_def zero_frac_def one_frac_def
256 num_def denom_def norm_frac_def frac_def plus_frac_def times_frac_def
257 uminus_frac_def number_of_frac_def inverse_frac_def less_frac_def
258 less_eq_frac_def of_frac_def