33192
|
1 |
(* Title: HOL/Nitpick.thy
|
|
2 |
Author: Jasmin Blanchette, TU Muenchen
|
|
3 |
Copyright 2008, 2009
|
|
4 |
|
|
5 |
Nitpick: Yet another counterexample generator for Isabelle/HOL.
|
|
6 |
*)
|
|
7 |
|
|
8 |
header {* Nitpick: Yet Another Counterexample Generator for Isabelle/HOL *}
|
|
9 |
|
|
10 |
theory Nitpick
|
|
11 |
imports Map 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_mono.ML")
|
|
17 |
("Tools/Nitpick/nitpick_scope.ML")
|
|
18 |
("Tools/Nitpick/nitpick_peephole.ML")
|
|
19 |
("Tools/Nitpick/nitpick_rep.ML")
|
|
20 |
("Tools/Nitpick/nitpick_nut.ML")
|
|
21 |
("Tools/Nitpick/nitpick_kodkod.ML")
|
|
22 |
("Tools/Nitpick/nitpick_model.ML")
|
|
23 |
("Tools/Nitpick/nitpick.ML")
|
|
24 |
("Tools/Nitpick/nitpick_isar.ML")
|
|
25 |
("Tools/Nitpick/nitpick_tests.ML")
|
|
26 |
("Tools/Nitpick/minipick.ML")
|
|
27 |
begin
|
|
28 |
|
|
29 |
typedecl bisim_iterator
|
|
30 |
|
|
31 |
(* FIXME: use axiomatization (here and elsewhere) *)
|
|
32 |
axiomatization unknown :: 'a
|
|
33 |
and undefined_fast_The :: 'a
|
|
34 |
and undefined_fast_Eps :: 'a
|
|
35 |
and bisim :: "bisim_iterator \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
|
|
36 |
and bisim_iterator_max :: bisim_iterator
|
|
37 |
and Tha :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
|
|
38 |
|
|
39 |
datatype ('a, 'b) pair_box = PairBox 'a 'b
|
|
40 |
datatype ('a, 'b) fun_box = FunBox "'a \<Rightarrow> 'b"
|
|
41 |
|
|
42 |
text {*
|
|
43 |
Alternative definitions.
|
|
44 |
*}
|
|
45 |
|
|
46 |
lemma If_def [nitpick_def]:
|
|
47 |
"(if P then Q else R) \<equiv> (P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> R)"
|
|
48 |
by (rule eq_reflection) (rule if_bool_eq_conj)
|
|
49 |
|
|
50 |
lemma Ex1_def [nitpick_def]:
|
|
51 |
"Ex1 P \<equiv> \<exists>x. P = {x}"
|
|
52 |
apply (rule eq_reflection)
|
|
53 |
apply (simp add: Ex1_def expand_set_eq)
|
|
54 |
apply (rule iffI)
|
|
55 |
apply (erule exE)
|
|
56 |
apply (erule conjE)
|
|
57 |
apply (rule_tac x = x in exI)
|
|
58 |
apply (rule allI)
|
|
59 |
apply (rename_tac y)
|
|
60 |
apply (erule_tac x = y in allE)
|
|
61 |
by (auto simp: mem_def)
|
|
62 |
|
|
63 |
lemma rtrancl_def [nitpick_def]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
|
|
64 |
by simp
|
|
65 |
|
|
66 |
lemma rtranclp_def [nitpick_def]:
|
|
67 |
"rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"
|
|
68 |
by (rule eq_reflection) (auto dest: rtranclpD)
|
|
69 |
|
|
70 |
lemma tranclp_def [nitpick_def]:
|
|
71 |
"tranclp r a b \<equiv> trancl (split r) (a, b)"
|
|
72 |
by (simp add: trancl_def Collect_def mem_def)
|
|
73 |
|
|
74 |
definition refl' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
|
|
75 |
"refl' r \<equiv> \<forall>x. (x, x) \<in> r"
|
|
76 |
|
|
77 |
definition wf' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
|
|
78 |
"wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
|
|
79 |
|
|
80 |
axiomatization wf_wfrec :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
|
|
81 |
|
|
82 |
definition wf_wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
|
|
83 |
[nitpick_simp]: "wf_wfrec' R F x = F (Recdef.cut (wf_wfrec R F) R x) x"
|
|
84 |
|
|
85 |
definition wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
|
|
86 |
"wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x
|
|
87 |
else THE y. wfrec_rel R (%f x. F (Recdef.cut f R x) x) x y"
|
|
88 |
|
|
89 |
definition card' :: "('a \<Rightarrow> bool) \<Rightarrow> nat" where
|
|
90 |
"card' X \<equiv> length (SOME xs. set xs = X \<and> distinct xs)"
|
|
91 |
|
|
92 |
definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b" where
|
|
93 |
"setsum' f A \<equiv> if finite A then listsum (map f (SOME xs. set xs = A \<and> distinct xs)) else 0"
|
|
94 |
|
|
95 |
inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool" where
|
|
96 |
"fold_graph' f z {} z" |
|
|
97 |
"\<lbrakk>x \<in> A; fold_graph' f z (A - {x}) y\<rbrakk> \<Longrightarrow> fold_graph' f z A (f x y)"
|
|
98 |
|
|
99 |
text {*
|
|
100 |
The following lemmas are not strictly necessary but they help the
|
|
101 |
\textit{special\_level} optimization.
|
|
102 |
*}
|
|
103 |
|
|
104 |
lemma The_psimp [nitpick_psimp]:
|
|
105 |
"P = {x} \<Longrightarrow> The P = x"
|
|
106 |
by (subgoal_tac "{x} = (\<lambda>y. y = x)") (auto simp: mem_def)
|
|
107 |
|
|
108 |
lemma Eps_psimp [nitpick_psimp]:
|
|
109 |
"\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> Eps P = x"
|
|
110 |
apply (case_tac "P (Eps P)")
|
|
111 |
apply auto
|
|
112 |
apply (erule contrapos_np)
|
|
113 |
by (rule someI)
|
|
114 |
|
|
115 |
lemma unit_case_def [nitpick_def]:
|
|
116 |
"unit_case x u \<equiv> x"
|
|
117 |
apply (subgoal_tac "u = ()")
|
|
118 |
apply (simp only: unit.cases)
|
|
119 |
by simp
|
|
120 |
|
|
121 |
lemma nat_case_def [nitpick_def]:
|
|
122 |
"nat_case x f n \<equiv> if n = 0 then x else f (n - 1)"
|
|
123 |
apply (rule eq_reflection)
|
|
124 |
by (case_tac n) auto
|
|
125 |
|
|
126 |
lemmas dvd_def = dvd_eq_mod_eq_0 [THEN eq_reflection, nitpick_def]
|
|
127 |
|
|
128 |
lemma list_size_simp [nitpick_simp]:
|
|
129 |
"list_size f xs = (if xs = [] then 0
|
|
130 |
else Suc (f (hd xs) + list_size f (tl xs)))"
|
|
131 |
"size xs = (if xs = [] then 0 else Suc (size (tl xs)))"
|
|
132 |
by (case_tac xs) auto
|
|
133 |
|
|
134 |
text {*
|
|
135 |
Auxiliary definitions used to provide an alternative representation for
|
|
136 |
@{text rat} and @{text real}.
|
|
137 |
*}
|
|
138 |
|
|
139 |
function nat_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
|
|
140 |
[simp del]: "nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"
|
|
141 |
by auto
|
|
142 |
termination
|
|
143 |
apply (relation "measure (\<lambda>(x, y). x + y + (if y > x then 1 else 0))")
|
|
144 |
apply auto
|
|
145 |
apply (metis mod_less_divisor xt1(9))
|
|
146 |
by (metis mod_mod_trivial mod_self nat_neq_iff xt1(10))
|
|
147 |
|
|
148 |
definition nat_lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
|
|
149 |
"nat_lcm x y = x * y div (nat_gcd x y)"
|
|
150 |
|
|
151 |
definition int_gcd :: "int \<Rightarrow> int \<Rightarrow> int" where
|
|
152 |
"int_gcd x y = int (nat_gcd (nat (abs x)) (nat (abs y)))"
|
|
153 |
|
|
154 |
definition int_lcm :: "int \<Rightarrow> int \<Rightarrow> int" where
|
|
155 |
"int_lcm x y = int (nat_lcm (nat (abs x)) (nat (abs y)))"
|
|
156 |
|
|
157 |
definition Frac :: "int \<times> int \<Rightarrow> bool" where
|
|
158 |
"Frac \<equiv> \<lambda>(a, b). b > 0 \<and> int_gcd a b = 1"
|
|
159 |
|
|
160 |
axiomatization Abs_Frac :: "int \<times> int \<Rightarrow> 'a"
|
|
161 |
and Rep_Frac :: "'a \<Rightarrow> int \<times> int"
|
|
162 |
|
|
163 |
definition zero_frac :: 'a where
|
|
164 |
"zero_frac \<equiv> Abs_Frac (0, 1)"
|
|
165 |
|
|
166 |
definition one_frac :: 'a where
|
|
167 |
"one_frac \<equiv> Abs_Frac (1, 1)"
|
|
168 |
|
|
169 |
definition num :: "'a \<Rightarrow> int" where
|
|
170 |
"num \<equiv> fst o Rep_Frac"
|
|
171 |
|
|
172 |
definition denom :: "'a \<Rightarrow> int" where
|
|
173 |
"denom \<equiv> snd o Rep_Frac"
|
|
174 |
|
|
175 |
function norm_frac :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
|
|
176 |
[simp del]: "norm_frac a b = (if b < 0 then norm_frac (- a) (- b)
|
|
177 |
else if a = 0 \<or> b = 0 then (0, 1)
|
|
178 |
else let c = int_gcd a b in (a div c, b div c))"
|
|
179 |
by pat_completeness auto
|
|
180 |
termination by (relation "measure (\<lambda>(_, b). if b < 0 then 1 else 0)") auto
|
|
181 |
|
|
182 |
definition frac :: "int \<Rightarrow> int \<Rightarrow> 'a" where
|
|
183 |
"frac a b \<equiv> Abs_Frac (norm_frac a b)"
|
|
184 |
|
|
185 |
definition plus_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
|
|
186 |
[nitpick_simp]:
|
|
187 |
"plus_frac q r = (let d = int_lcm (denom q) (denom r) in
|
|
188 |
frac (num q * (d div denom q) + num r * (d div denom r)) d)"
|
|
189 |
|
|
190 |
definition times_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
|
|
191 |
[nitpick_simp]:
|
|
192 |
"times_frac q r = frac (num q * num r) (denom q * denom r)"
|
|
193 |
|
|
194 |
definition uminus_frac :: "'a \<Rightarrow> 'a" where
|
|
195 |
"uminus_frac q \<equiv> Abs_Frac (- num q, denom q)"
|
|
196 |
|
|
197 |
definition number_of_frac :: "int \<Rightarrow> 'a" where
|
|
198 |
"number_of_frac n \<equiv> Abs_Frac (n, 1)"
|
|
199 |
|
|
200 |
definition inverse_frac :: "'a \<Rightarrow> 'a" where
|
|
201 |
"inverse_frac q \<equiv> frac (denom q) (num q)"
|
|
202 |
|
|
203 |
definition less_eq_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
|
|
204 |
[nitpick_simp]:
|
|
205 |
"less_eq_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) \<le> 0"
|
|
206 |
|
|
207 |
definition of_frac :: "'a \<Rightarrow> 'b\<Colon>{inverse,ring_1}" where
|
|
208 |
"of_frac q \<equiv> of_int (num q) / of_int (denom q)"
|
|
209 |
|
|
210 |
use "Tools/Nitpick/kodkod.ML"
|
|
211 |
use "Tools/Nitpick/kodkod_sat.ML"
|
|
212 |
use "Tools/Nitpick/nitpick_util.ML"
|
|
213 |
use "Tools/Nitpick/nitpick_hol.ML"
|
|
214 |
use "Tools/Nitpick/nitpick_mono.ML"
|
|
215 |
use "Tools/Nitpick/nitpick_scope.ML"
|
|
216 |
use "Tools/Nitpick/nitpick_peephole.ML"
|
|
217 |
use "Tools/Nitpick/nitpick_rep.ML"
|
|
218 |
use "Tools/Nitpick/nitpick_nut.ML"
|
|
219 |
use "Tools/Nitpick/nitpick_kodkod.ML"
|
|
220 |
use "Tools/Nitpick/nitpick_model.ML"
|
|
221 |
use "Tools/Nitpick/nitpick.ML"
|
|
222 |
use "Tools/Nitpick/nitpick_isar.ML"
|
|
223 |
use "Tools/Nitpick/nitpick_tests.ML"
|
|
224 |
use "Tools/Nitpick/minipick.ML"
|
|
225 |
|
|
226 |
hide (open) const unknown undefined_fast_The undefined_fast_Eps bisim
|
|
227 |
bisim_iterator_max Tha refl' wf' wf_wfrec wf_wfrec' wfrec' card' setsum'
|
|
228 |
fold_graph' nat_gcd nat_lcm int_gcd int_lcm Frac Abs_Frac Rep_Frac zero_frac
|
|
229 |
one_frac num denom norm_frac frac plus_frac times_frac uminus_frac
|
|
230 |
number_of_frac inverse_frac less_eq_frac of_frac
|
|
231 |
hide (open) type bisim_iterator pair_box fun_box
|
|
232 |
hide (open) fact If_def Ex1_def rtrancl_def rtranclp_def tranclp_def refl'_def
|
|
233 |
wf'_def wf_wfrec'_def wfrec'_def card'_def setsum'_def fold_graph'_def
|
|
234 |
The_psimp Eps_psimp unit_case_def nat_case_def dvd_def list_size_simp
|
|
235 |
nat_gcd_def nat_lcm_def int_gcd_def int_lcm_def Frac_def zero_frac_def
|
|
236 |
one_frac_def num_def denom_def norm_frac_def frac_def plus_frac_def
|
|
237 |
times_frac_def uminus_frac_def number_of_frac_def inverse_frac_def
|
|
238 |
less_eq_frac_def of_frac_def
|
|
239 |
|
|
240 |
end
|