src/HOL/Nitpick.thy
author blanchet
Thu Feb 04 16:03:15 2010 +0100 (2010-02-04)
changeset 35070 96136eb6218f
parent 34982 7b8c366e34a2
child 35079 592edca1dfb3
permissions -rw-r--r--
split "nitpick_hol.ML" into two files to make it more manageable;
more refactoring to come
blanchet@33192
     1
(*  Title:      HOL/Nitpick.thy
blanchet@33192
     2
    Author:     Jasmin Blanchette, TU Muenchen
blanchet@33192
     3
    Copyright   2008, 2009
blanchet@33192
     4
blanchet@33192
     5
Nitpick: Yet another counterexample generator for Isabelle/HOL.
blanchet@33192
     6
*)
blanchet@33192
     7
blanchet@33192
     8
header {* Nitpick: Yet Another Counterexample Generator for Isabelle/HOL *}
blanchet@33192
     9
blanchet@33192
    10
theory Nitpick
haftmann@33608
    11
imports Map SAT
blanchet@33192
    12
uses ("Tools/Nitpick/kodkod.ML")
blanchet@33192
    13
     ("Tools/Nitpick/kodkod_sat.ML")
blanchet@33192
    14
     ("Tools/Nitpick/nitpick_util.ML")
blanchet@33192
    15
     ("Tools/Nitpick/nitpick_hol.ML")
blanchet@35070
    16
     ("Tools/Nitpick/nitpick_preproc.ML")
blanchet@33192
    17
     ("Tools/Nitpick/nitpick_mono.ML")
blanchet@33192
    18
     ("Tools/Nitpick/nitpick_scope.ML")
blanchet@33192
    19
     ("Tools/Nitpick/nitpick_peephole.ML")
blanchet@33192
    20
     ("Tools/Nitpick/nitpick_rep.ML")
blanchet@33192
    21
     ("Tools/Nitpick/nitpick_nut.ML")
blanchet@33192
    22
     ("Tools/Nitpick/nitpick_kodkod.ML")
blanchet@33192
    23
     ("Tools/Nitpick/nitpick_model.ML")
blanchet@33192
    24
     ("Tools/Nitpick/nitpick.ML")
blanchet@33192
    25
     ("Tools/Nitpick/nitpick_isar.ML")
blanchet@33192
    26
     ("Tools/Nitpick/nitpick_tests.ML")
blanchet@33192
    27
     ("Tools/Nitpick/minipick.ML")
blanchet@33192
    28
begin
blanchet@33192
    29
blanchet@33192
    30
typedecl bisim_iterator
blanchet@33192
    31
blanchet@33192
    32
axiomatization unknown :: 'a
blanchet@34938
    33
           and is_unknown :: "'a \<Rightarrow> bool"
blanchet@33192
    34
           and undefined_fast_The :: 'a
blanchet@33192
    35
           and undefined_fast_Eps :: 'a
blanchet@33192
    36
           and bisim :: "bisim_iterator \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
blanchet@33192
    37
           and bisim_iterator_max :: bisim_iterator
blanchet@34938
    38
           and Quot :: "'a \<Rightarrow> 'b"
blanchet@34938
    39
           and quot_normal :: "'a \<Rightarrow> 'a"
blanchet@34982
    40
           and NonStd :: "'a \<Rightarrow> 'b"
blanchet@33192
    41
           and Tha :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
blanchet@33192
    42
blanchet@33192
    43
datatype ('a, 'b) pair_box = PairBox 'a 'b
blanchet@34124
    44
datatype ('a, 'b) fun_box = FunBox "('a \<Rightarrow> 'b)"
blanchet@34124
    45
blanchet@34124
    46
typedecl unsigned_bit
blanchet@34124
    47
typedecl signed_bit
blanchet@34982
    48
typedecl \<xi>
blanchet@34124
    49
blanchet@34124
    50
datatype 'a word = Word "('a set)"
blanchet@33192
    51
blanchet@33192
    52
text {*
blanchet@33192
    53
Alternative definitions.
blanchet@33192
    54
*}
blanchet@33192
    55
blanchet@33192
    56
lemma If_def [nitpick_def]:
blanchet@33192
    57
"(if P then Q else R) \<equiv> (P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> R)"
blanchet@33192
    58
by (rule eq_reflection) (rule if_bool_eq_conj)
blanchet@33192
    59
blanchet@33192
    60
lemma Ex1_def [nitpick_def]:
blanchet@33192
    61
"Ex1 P \<equiv> \<exists>x. P = {x}"
blanchet@33192
    62
apply (rule eq_reflection)
blanchet@33192
    63
apply (simp add: Ex1_def expand_set_eq)
blanchet@33192
    64
apply (rule iffI)
blanchet@33192
    65
 apply (erule exE)
blanchet@33192
    66
 apply (erule conjE)
blanchet@33192
    67
 apply (rule_tac x = x in exI)
blanchet@33192
    68
 apply (rule allI)
blanchet@33192
    69
 apply (rename_tac y)
blanchet@33192
    70
 apply (erule_tac x = y in allE)
blanchet@33192
    71
by (auto simp: mem_def)
blanchet@33192
    72
blanchet@33192
    73
lemma rtrancl_def [nitpick_def]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
blanchet@33192
    74
by simp
blanchet@33192
    75
blanchet@33192
    76
lemma rtranclp_def [nitpick_def]:
blanchet@33192
    77
"rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"
blanchet@33192
    78
by (rule eq_reflection) (auto dest: rtranclpD)
blanchet@33192
    79
blanchet@33192
    80
lemma tranclp_def [nitpick_def]:
blanchet@33192
    81
"tranclp r a b \<equiv> trancl (split r) (a, b)"
blanchet@33192
    82
by (simp add: trancl_def Collect_def mem_def)
blanchet@33192
    83
blanchet@33192
    84
definition refl' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
blanchet@33192
    85
"refl' r \<equiv> \<forall>x. (x, x) \<in> r"
blanchet@33192
    86
blanchet@33192
    87
definition wf' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
blanchet@33192
    88
"wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
blanchet@33192
    89
blanchet@33192
    90
axiomatization wf_wfrec :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
blanchet@33192
    91
blanchet@33192
    92
definition wf_wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
blanchet@33192
    93
[nitpick_simp]: "wf_wfrec' R F x = F (Recdef.cut (wf_wfrec R F) R x) x"
blanchet@33192
    94
blanchet@33192
    95
definition wfrec' ::  "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
blanchet@33192
    96
"wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x
blanchet@33192
    97
                else THE y. wfrec_rel R (%f x. F (Recdef.cut f R x) x) x y"
blanchet@33192
    98
blanchet@33192
    99
definition card' :: "('a \<Rightarrow> bool) \<Rightarrow> nat" where
blanchet@33192
   100
"card' X \<equiv> length (SOME xs. set xs = X \<and> distinct xs)"
blanchet@33192
   101
blanchet@33192
   102
definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b" where
blanchet@33192
   103
"setsum' f A \<equiv> if finite A then listsum (map f (SOME xs. set xs = A \<and> distinct xs)) else 0"
blanchet@33192
   104
blanchet@33192
   105
inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool" where
blanchet@33192
   106
"fold_graph' f z {} z" |
blanchet@33192
   107
"\<lbrakk>x \<in> A; fold_graph' f z (A - {x}) y\<rbrakk> \<Longrightarrow> fold_graph' f z A (f x y)"
blanchet@33192
   108
blanchet@33192
   109
text {*
blanchet@33192
   110
The following lemmas are not strictly necessary but they help the
blanchet@33192
   111
\textit{special\_level} optimization.
blanchet@33192
   112
*}
blanchet@33192
   113
blanchet@33192
   114
lemma The_psimp [nitpick_psimp]:
blanchet@33192
   115
"P = {x} \<Longrightarrow> The P = x"
blanchet@33192
   116
by (subgoal_tac "{x} = (\<lambda>y. y = x)") (auto simp: mem_def)
blanchet@33192
   117
blanchet@33192
   118
lemma Eps_psimp [nitpick_psimp]:
blanchet@33192
   119
"\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> Eps P = x"
blanchet@33192
   120
apply (case_tac "P (Eps P)")
blanchet@33192
   121
 apply auto
blanchet@33192
   122
apply (erule contrapos_np)
blanchet@33192
   123
by (rule someI)
blanchet@33192
   124
blanchet@33192
   125
lemma unit_case_def [nitpick_def]:
blanchet@33192
   126
"unit_case x u \<equiv> x"
blanchet@33192
   127
apply (subgoal_tac "u = ()")
blanchet@33192
   128
 apply (simp only: unit.cases)
blanchet@33192
   129
by simp
blanchet@33192
   130
blanchet@33556
   131
declare unit.cases [nitpick_simp del]
blanchet@33556
   132
blanchet@33192
   133
lemma nat_case_def [nitpick_def]:
blanchet@33192
   134
"nat_case x f n \<equiv> if n = 0 then x else f (n - 1)"
blanchet@33192
   135
apply (rule eq_reflection)
blanchet@33192
   136
by (case_tac n) auto
blanchet@33192
   137
blanchet@33556
   138
declare nat.cases [nitpick_simp del]
blanchet@33556
   139
blanchet@33192
   140
lemma list_size_simp [nitpick_simp]:
blanchet@33192
   141
"list_size f xs = (if xs = [] then 0
blanchet@33192
   142
                   else Suc (f (hd xs) + list_size f (tl xs)))"
blanchet@33192
   143
"size xs = (if xs = [] then 0 else Suc (size (tl xs)))"
blanchet@33192
   144
by (case_tac xs) auto
blanchet@33192
   145
blanchet@33192
   146
text {*
blanchet@33192
   147
Auxiliary definitions used to provide an alternative representation for
blanchet@33192
   148
@{text rat} and @{text real}.
blanchet@33192
   149
*}
blanchet@33192
   150
blanchet@33192
   151
function nat_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
blanchet@33192
   152
[simp del]: "nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"
blanchet@33192
   153
by auto
blanchet@33192
   154
termination
blanchet@33192
   155
apply (relation "measure (\<lambda>(x, y). x + y + (if y > x then 1 else 0))")
blanchet@33192
   156
 apply auto
blanchet@33192
   157
 apply (metis mod_less_divisor xt1(9))
blanchet@33192
   158
by (metis mod_mod_trivial mod_self nat_neq_iff xt1(10))
blanchet@33192
   159
blanchet@33192
   160
definition nat_lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
blanchet@33192
   161
"nat_lcm x y = x * y div (nat_gcd x y)"
blanchet@33192
   162
blanchet@33192
   163
definition int_gcd :: "int \<Rightarrow> int \<Rightarrow> int" where
blanchet@33192
   164
"int_gcd x y = int (nat_gcd (nat (abs x)) (nat (abs y)))"
blanchet@33192
   165
blanchet@33192
   166
definition int_lcm :: "int \<Rightarrow> int \<Rightarrow> int" where
blanchet@33192
   167
"int_lcm x y = int (nat_lcm (nat (abs x)) (nat (abs y)))"
blanchet@33192
   168
blanchet@33192
   169
definition Frac :: "int \<times> int \<Rightarrow> bool" where
blanchet@33192
   170
"Frac \<equiv> \<lambda>(a, b). b > 0 \<and> int_gcd a b = 1"
blanchet@33192
   171
blanchet@33192
   172
axiomatization Abs_Frac :: "int \<times> int \<Rightarrow> 'a"
blanchet@33192
   173
           and Rep_Frac :: "'a \<Rightarrow> int \<times> int"
blanchet@33192
   174
blanchet@33192
   175
definition zero_frac :: 'a where
blanchet@33192
   176
"zero_frac \<equiv> Abs_Frac (0, 1)"
blanchet@33192
   177
blanchet@33192
   178
definition one_frac :: 'a where
blanchet@33192
   179
"one_frac \<equiv> Abs_Frac (1, 1)"
blanchet@33192
   180
blanchet@33192
   181
definition num :: "'a \<Rightarrow> int" where
blanchet@33192
   182
"num \<equiv> fst o Rep_Frac"
blanchet@33192
   183
blanchet@33192
   184
definition denom :: "'a \<Rightarrow> int" where
blanchet@33192
   185
"denom \<equiv> snd o Rep_Frac"
blanchet@33192
   186
blanchet@33192
   187
function norm_frac :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
blanchet@33192
   188
[simp del]: "norm_frac a b = (if b < 0 then norm_frac (- a) (- b)
blanchet@33192
   189
                              else if a = 0 \<or> b = 0 then (0, 1)
blanchet@33192
   190
                              else let c = int_gcd a b in (a div c, b div c))"
blanchet@33192
   191
by pat_completeness auto
blanchet@33192
   192
termination by (relation "measure (\<lambda>(_, b). if b < 0 then 1 else 0)") auto
blanchet@33192
   193
blanchet@33192
   194
definition frac :: "int \<Rightarrow> int \<Rightarrow> 'a" where
blanchet@33192
   195
"frac a b \<equiv> Abs_Frac (norm_frac a b)"
blanchet@33192
   196
blanchet@33192
   197
definition plus_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
blanchet@33192
   198
[nitpick_simp]:
blanchet@33192
   199
"plus_frac q r = (let d = int_lcm (denom q) (denom r) in
blanchet@33192
   200
                    frac (num q * (d div denom q) + num r * (d div denom r)) d)"
blanchet@33192
   201
blanchet@33192
   202
definition times_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
blanchet@33192
   203
[nitpick_simp]:
blanchet@33192
   204
"times_frac q r = frac (num q * num r) (denom q * denom r)"
blanchet@33192
   205
blanchet@33192
   206
definition uminus_frac :: "'a \<Rightarrow> 'a" where
blanchet@33192
   207
"uminus_frac q \<equiv> Abs_Frac (- num q, denom q)"
blanchet@33192
   208
blanchet@33192
   209
definition number_of_frac :: "int \<Rightarrow> 'a" where
blanchet@33192
   210
"number_of_frac n \<equiv> Abs_Frac (n, 1)"
blanchet@33192
   211
blanchet@33192
   212
definition inverse_frac :: "'a \<Rightarrow> 'a" where
blanchet@33192
   213
"inverse_frac q \<equiv> frac (denom q) (num q)"
blanchet@33192
   214
blanchet@33192
   215
definition less_eq_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
blanchet@33192
   216
[nitpick_simp]:
blanchet@33192
   217
"less_eq_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) \<le> 0"
blanchet@33192
   218
blanchet@33192
   219
definition of_frac :: "'a \<Rightarrow> 'b\<Colon>{inverse,ring_1}" where
blanchet@33192
   220
"of_frac q \<equiv> of_int (num q) / of_int (denom q)"
blanchet@33192
   221
blanchet@33556
   222
(* While Nitpick normally avoids to unfold definitions for locales, it
blanchet@33556
   223
   unfortunately needs to unfold them when dealing with the following built-in
blanchet@33556
   224
   constants. A cleaner approach would be to change "Nitpick_HOL" and
blanchet@33747
   225
   "Nitpick_Nut" so that they handle the unexpanded overloaded constants
blanchet@33556
   226
   directly, but this is slightly more tricky to implement. *)
blanchet@33556
   227
lemmas [nitpick_def] = div_int_inst.div_int div_int_inst.mod_int
blanchet@33556
   228
    div_nat_inst.div_nat div_nat_inst.mod_nat lower_semilattice_fun_inst.inf_fun
blanchet@33556
   229
    minus_fun_inst.minus_fun minus_int_inst.minus_int minus_nat_inst.minus_nat
blanchet@33556
   230
    one_int_inst.one_int one_nat_inst.one_nat ord_fun_inst.less_eq_fun
blanchet@33556
   231
    ord_int_inst.less_eq_int ord_int_inst.less_int ord_nat_inst.less_eq_nat
blanchet@33556
   232
    ord_nat_inst.less_nat plus_int_inst.plus_int plus_nat_inst.plus_nat
blanchet@33556
   233
    times_int_inst.times_int times_nat_inst.times_nat uminus_int_inst.uminus_int
blanchet@33556
   234
    upper_semilattice_fun_inst.sup_fun zero_int_inst.zero_int
blanchet@33556
   235
    zero_nat_inst.zero_nat
blanchet@33556
   236
blanchet@33192
   237
use "Tools/Nitpick/kodkod.ML"
blanchet@33192
   238
use "Tools/Nitpick/kodkod_sat.ML"
blanchet@33192
   239
use "Tools/Nitpick/nitpick_util.ML"
blanchet@33192
   240
use "Tools/Nitpick/nitpick_hol.ML"
blanchet@35070
   241
use "Tools/Nitpick/nitpick_preproc.ML"
blanchet@33192
   242
use "Tools/Nitpick/nitpick_mono.ML"
blanchet@33192
   243
use "Tools/Nitpick/nitpick_scope.ML"
blanchet@33192
   244
use "Tools/Nitpick/nitpick_peephole.ML"
blanchet@33192
   245
use "Tools/Nitpick/nitpick_rep.ML"
blanchet@33192
   246
use "Tools/Nitpick/nitpick_nut.ML"
blanchet@33192
   247
use "Tools/Nitpick/nitpick_kodkod.ML"
blanchet@33192
   248
use "Tools/Nitpick/nitpick_model.ML"
blanchet@33192
   249
use "Tools/Nitpick/nitpick.ML"
blanchet@33192
   250
use "Tools/Nitpick/nitpick_isar.ML"
blanchet@33192
   251
use "Tools/Nitpick/nitpick_tests.ML"
blanchet@33192
   252
use "Tools/Nitpick/minipick.ML"
blanchet@33192
   253
blanchet@33561
   254
setup {* Nitpick_Isar.setup *}
blanchet@33561
   255
blanchet@34938
   256
hide (open) const unknown is_unknown undefined_fast_The undefined_fast_Eps bisim 
blanchet@34982
   257
    bisim_iterator_max Quot quot_normal NonStd Tha PairBox FunBox Word refl' wf'
blanchet@34938
   258
    wf_wfrec wf_wfrec' wfrec' card' setsum' fold_graph' nat_gcd nat_lcm int_gcd
blanchet@34938
   259
    int_lcm Frac Abs_Frac Rep_Frac zero_frac one_frac num denom norm_frac frac
blanchet@34938
   260
    plus_frac times_frac uminus_frac number_of_frac inverse_frac less_eq_frac
blanchet@34938
   261
    of_frac
blanchet@34982
   262
hide (open) type bisim_iterator pair_box fun_box unsigned_bit signed_bit \<xi> word
blanchet@33192
   263
hide (open) fact If_def Ex1_def rtrancl_def rtranclp_def tranclp_def refl'_def
blanchet@33192
   264
    wf'_def wf_wfrec'_def wfrec'_def card'_def setsum'_def fold_graph'_def
blanchet@33556
   265
    The_psimp Eps_psimp unit_case_def nat_case_def list_size_simp nat_gcd_def
blanchet@33556
   266
    nat_lcm_def int_gcd_def int_lcm_def Frac_def zero_frac_def one_frac_def
blanchet@33556
   267
    num_def denom_def norm_frac_def frac_def plus_frac_def times_frac_def
blanchet@33556
   268
    uminus_frac_def number_of_frac_def inverse_frac_def less_eq_frac_def
blanchet@33556
   269
    of_frac_def
blanchet@33192
   270
blanchet@33192
   271
end