more work on Nitpick's support for nonstandard models + fix in model reconstruction
1 (* Title: HOL/Nitpick.thy
2 Author: Jasmin Blanchette, TU Muenchen
5 Nitpick: Yet another counterexample generator for Isabelle/HOL.
8 header {* Nitpick: Yet Another Counterexample Generator for Isabelle/HOL *}
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 quot_normal :: "'a \<Rightarrow> 'a"
40 and Tha :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
42 datatype ('a, 'b) pair_box = PairBox 'a 'b
43 datatype ('a, 'b) fun_box = FunBox "('a \<Rightarrow> 'b)"
48 datatype 'a word = Word "('a set)"
51 Alternative definitions.
54 lemma If_def [nitpick_def]:
55 "(if P then Q else R) \<equiv> (P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> R)"
56 by (rule eq_reflection) (rule if_bool_eq_conj)
58 lemma Ex1_def [nitpick_def]:
59 "Ex1 P \<equiv> \<exists>x. P = {x}"
60 apply (rule eq_reflection)
61 apply (simp add: Ex1_def expand_set_eq)
65 apply (rule_tac x = x in exI)
68 apply (erule_tac x = y in allE)
69 by (auto simp: mem_def)
71 lemma rtrancl_def [nitpick_def]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
74 lemma rtranclp_def [nitpick_def]:
75 "rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"
76 by (rule eq_reflection) (auto dest: rtranclpD)
78 lemma tranclp_def [nitpick_def]:
79 "tranclp r a b \<equiv> trancl (split r) (a, b)"
80 by (simp add: trancl_def Collect_def mem_def)
82 definition refl' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
83 "refl' r \<equiv> \<forall>x. (x, x) \<in> r"
85 definition wf' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
86 "wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
88 axiomatization wf_wfrec :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
90 definition wf_wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
91 [nitpick_simp]: "wf_wfrec' R F x = F (Recdef.cut (wf_wfrec R F) R x) x"
93 definition wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
94 "wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x
95 else THE y. wfrec_rel R (%f x. F (Recdef.cut f R x) x) x y"
97 definition card' :: "('a \<Rightarrow> bool) \<Rightarrow> nat" where
98 "card' X \<equiv> length (SOME xs. set xs = X \<and> distinct xs)"
100 definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b" where
101 "setsum' f A \<equiv> if finite A then listsum (map f (SOME xs. set xs = A \<and> distinct xs)) else 0"
103 inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool" where
104 "fold_graph' f z {} z" |
105 "\<lbrakk>x \<in> A; fold_graph' f z (A - {x}) y\<rbrakk> \<Longrightarrow> fold_graph' f z A (f x y)"
108 The following lemmas are not strictly necessary but they help the
109 \textit{special\_level} optimization.
112 lemma The_psimp [nitpick_psimp]:
113 "P = {x} \<Longrightarrow> The P = x"
114 by (subgoal_tac "{x} = (\<lambda>y. y = x)") (auto simp: mem_def)
116 lemma Eps_psimp [nitpick_psimp]:
117 "\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> Eps P = x"
118 apply (case_tac "P (Eps P)")
120 apply (erule contrapos_np)
123 lemma unit_case_def [nitpick_def]:
124 "unit_case x u \<equiv> x"
125 apply (subgoal_tac "u = ()")
126 apply (simp only: unit.cases)
129 declare unit.cases [nitpick_simp del]
131 lemma nat_case_def [nitpick_def]:
132 "nat_case x f n \<equiv> if n = 0 then x else f (n - 1)"
133 apply (rule eq_reflection)
136 declare nat.cases [nitpick_simp del]
138 lemma list_size_simp [nitpick_simp]:
139 "list_size f xs = (if xs = [] then 0
140 else Suc (f (hd xs) + list_size f (tl xs)))"
141 "size xs = (if xs = [] then 0 else Suc (size (tl xs)))"
142 by (case_tac xs) auto
145 Auxiliary definitions used to provide an alternative representation for
146 @{text rat} and @{text real}.
149 function nat_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
150 [simp del]: "nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"
153 apply (relation "measure (\<lambda>(x, y). x + y + (if y > x then 1 else 0))")
155 apply (metis mod_less_divisor xt1(9))
156 by (metis mod_mod_trivial mod_self nat_neq_iff xt1(10))
158 definition nat_lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
159 "nat_lcm x y = x * y div (nat_gcd x y)"
161 definition int_gcd :: "int \<Rightarrow> int \<Rightarrow> int" where
162 "int_gcd x y = int (nat_gcd (nat (abs x)) (nat (abs y)))"
164 definition int_lcm :: "int \<Rightarrow> int \<Rightarrow> int" where
165 "int_lcm x y = int (nat_lcm (nat (abs x)) (nat (abs y)))"
167 definition Frac :: "int \<times> int \<Rightarrow> bool" where
168 "Frac \<equiv> \<lambda>(a, b). b > 0 \<and> int_gcd a b = 1"
170 axiomatization Abs_Frac :: "int \<times> int \<Rightarrow> 'a"
171 and Rep_Frac :: "'a \<Rightarrow> int \<times> int"
173 definition zero_frac :: 'a where
174 "zero_frac \<equiv> Abs_Frac (0, 1)"
176 definition one_frac :: 'a where
177 "one_frac \<equiv> Abs_Frac (1, 1)"
179 definition num :: "'a \<Rightarrow> int" where
180 "num \<equiv> fst o Rep_Frac"
182 definition denom :: "'a \<Rightarrow> int" where
183 "denom \<equiv> snd o Rep_Frac"
185 function norm_frac :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
186 [simp del]: "norm_frac a b = (if b < 0 then norm_frac (- a) (- b)
187 else if a = 0 \<or> b = 0 then (0, 1)
188 else let c = int_gcd a b in (a div c, b div c))"
189 by pat_completeness auto
190 termination by (relation "measure (\<lambda>(_, b). if b < 0 then 1 else 0)") auto
192 definition frac :: "int \<Rightarrow> int \<Rightarrow> 'a" where
193 "frac a b \<equiv> Abs_Frac (norm_frac a b)"
195 definition plus_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
197 "plus_frac q r = (let d = int_lcm (denom q) (denom r) in
198 frac (num q * (d div denom q) + num r * (d div denom r)) d)"
200 definition times_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
202 "times_frac q r = frac (num q * num r) (denom q * denom r)"
204 definition uminus_frac :: "'a \<Rightarrow> 'a" where
205 "uminus_frac q \<equiv> Abs_Frac (- num q, denom q)"
207 definition number_of_frac :: "int \<Rightarrow> 'a" where
208 "number_of_frac n \<equiv> Abs_Frac (n, 1)"
210 definition inverse_frac :: "'a \<Rightarrow> 'a" where
211 "inverse_frac q \<equiv> frac (denom q) (num q)"
213 definition less_eq_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
215 "less_eq_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) \<le> 0"
217 definition of_frac :: "'a \<Rightarrow> 'b\<Colon>{inverse,ring_1}" where
218 "of_frac q \<equiv> of_int (num q) / of_int (denom q)"
220 (* While Nitpick normally avoids to unfold definitions for locales, it
221 unfortunately needs to unfold them when dealing with the following built-in
222 constants. A cleaner approach would be to change "Nitpick_HOL" and
223 "Nitpick_Nut" so that they handle the unexpanded overloaded constants
224 directly, but this is slightly more tricky to implement. *)
225 lemmas [nitpick_def] = div_int_inst.div_int div_int_inst.mod_int
226 div_nat_inst.div_nat div_nat_inst.mod_nat semilattice_inf_fun_inst.inf_fun
227 minus_fun_inst.minus_fun minus_int_inst.minus_int minus_nat_inst.minus_nat
228 one_int_inst.one_int one_nat_inst.one_nat ord_fun_inst.less_eq_fun
229 ord_int_inst.less_eq_int ord_int_inst.less_int ord_nat_inst.less_eq_nat
230 ord_nat_inst.less_nat plus_int_inst.plus_int plus_nat_inst.plus_nat
231 times_int_inst.times_int times_nat_inst.times_nat uminus_int_inst.uminus_int
232 semilattice_sup_fun_inst.sup_fun zero_int_inst.zero_int
233 zero_nat_inst.zero_nat
235 use "Tools/Nitpick/kodkod.ML"
236 use "Tools/Nitpick/kodkod_sat.ML"
237 use "Tools/Nitpick/nitpick_util.ML"
238 use "Tools/Nitpick/nitpick_hol.ML"
239 use "Tools/Nitpick/nitpick_preproc.ML"
240 use "Tools/Nitpick/nitpick_mono.ML"
241 use "Tools/Nitpick/nitpick_scope.ML"
242 use "Tools/Nitpick/nitpick_peephole.ML"
243 use "Tools/Nitpick/nitpick_rep.ML"
244 use "Tools/Nitpick/nitpick_nut.ML"
245 use "Tools/Nitpick/nitpick_kodkod.ML"
246 use "Tools/Nitpick/nitpick_model.ML"
247 use "Tools/Nitpick/nitpick.ML"
248 use "Tools/Nitpick/nitpick_isar.ML"
249 use "Tools/Nitpick/nitpick_tests.ML"
250 use "Tools/Nitpick/minipick.ML"
252 setup {* Nitpick_Isar.setup *}
254 hide (open) const unknown is_unknown undefined_fast_The undefined_fast_Eps bisim
255 bisim_iterator_max Quot quot_normal Tha PairBox FunBox Word refl' wf'
256 wf_wfrec wf_wfrec' wfrec' card' setsum' fold_graph' nat_gcd nat_lcm int_gcd
257 int_lcm Frac Abs_Frac Rep_Frac zero_frac one_frac num denom norm_frac frac
258 plus_frac times_frac uminus_frac number_of_frac inverse_frac less_eq_frac
260 hide (open) type bisim_iterator pair_box fun_box unsigned_bit signed_bit word
261 hide (open) fact If_def Ex1_def rtrancl_def rtranclp_def tranclp_def refl'_def
262 wf'_def wf_wfrec'_def wfrec'_def card'_def setsum'_def fold_graph'_def
263 The_psimp Eps_psimp unit_case_def nat_case_def list_size_simp nat_gcd_def
264 nat_lcm_def int_gcd_def int_lcm_def Frac_def zero_frac_def one_frac_def
265 num_def denom_def norm_frac_def frac_def plus_frac_def times_frac_def
266 uminus_frac_def number_of_frac_def inverse_frac_def less_eq_frac_def