src/HOL/Nitpick.thy
author blanchet
Tue Feb 23 11:05:32 2010 +0100 (2010-02-23)
changeset 35311 8f9a66fc9f80
parent 35284 9edc2bd6d2bd
child 35665 ff2bf50505ab
permissions -rw-r--r--
improved Nitpick's support for quotient types
     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 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")
    28 begin
    29 
    30 typedecl bisim_iterator
    31 
    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 Tha :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
    40 
    41 datatype ('a, 'b) pair_box = PairBox 'a 'b
    42 datatype ('a, 'b) fun_box = FunBox "('a \<Rightarrow> 'b)"
    43 
    44 typedecl unsigned_bit
    45 typedecl signed_bit
    46 
    47 datatype 'a word = Word "('a set)"
    48 
    49 text {*
    50 Alternative definitions.
    51 *}
    52 
    53 lemma If_def [nitpick_def]:
    54 "(if P then Q else R) \<equiv> (P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> R)"
    55 by (rule eq_reflection) (rule if_bool_eq_conj)
    56 
    57 lemma Ex1_def [nitpick_def]:
    58 "Ex1 P \<equiv> \<exists>x. P = {x}"
    59 apply (rule eq_reflection)
    60 apply (simp add: Ex1_def expand_set_eq)
    61 apply (rule iffI)
    62  apply (erule exE)
    63  apply (erule conjE)
    64  apply (rule_tac x = x in exI)
    65  apply (rule allI)
    66  apply (rename_tac y)
    67  apply (erule_tac x = y in allE)
    68 by (auto simp: mem_def)
    69 
    70 lemma rtrancl_def [nitpick_def]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
    71 by simp
    72 
    73 lemma rtranclp_def [nitpick_def]:
    74 "rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"
    75 by (rule eq_reflection) (auto dest: rtranclpD)
    76 
    77 lemma tranclp_def [nitpick_def]:
    78 "tranclp r a b \<equiv> trancl (split r) (a, b)"
    79 by (simp add: trancl_def Collect_def mem_def)
    80 
    81 definition refl' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
    82 "refl' r \<equiv> \<forall>x. (x, x) \<in> r"
    83 
    84 definition wf' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
    85 "wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
    86 
    87 axiomatization wf_wfrec :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
    88 
    89 definition wf_wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
    90 [nitpick_simp]: "wf_wfrec' R F x = F (Recdef.cut (wf_wfrec R F) R x) x"
    91 
    92 definition wfrec' ::  "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
    93 "wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x
    94                 else THE y. wfrec_rel R (%f x. F (Recdef.cut f R x) x) x y"
    95 
    96 definition card' :: "('a \<Rightarrow> bool) \<Rightarrow> nat" where
    97 "card' X \<equiv> length (SOME xs. set xs = X \<and> distinct xs)"
    98 
    99 definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b" where
   100 "setsum' f A \<equiv> if finite A then listsum (map f (SOME xs. set xs = A \<and> distinct xs)) else 0"
   101 
   102 inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool" where
   103 "fold_graph' f z {} z" |
   104 "\<lbrakk>x \<in> A; fold_graph' f z (A - {x}) y\<rbrakk> \<Longrightarrow> fold_graph' f z A (f x y)"
   105 
   106 text {*
   107 The following lemmas are not strictly necessary but they help the
   108 \textit{special\_level} optimization.
   109 *}
   110 
   111 lemma The_psimp [nitpick_psimp]:
   112 "P = {x} \<Longrightarrow> The P = x"
   113 by (subgoal_tac "{x} = (\<lambda>y. y = x)") (auto simp: mem_def)
   114 
   115 lemma Eps_psimp [nitpick_psimp]:
   116 "\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> Eps P = x"
   117 apply (case_tac "P (Eps P)")
   118  apply auto
   119 apply (erule contrapos_np)
   120 by (rule someI)
   121 
   122 lemma unit_case_def [nitpick_def]:
   123 "unit_case x u \<equiv> x"
   124 apply (subgoal_tac "u = ()")
   125  apply (simp only: unit.cases)
   126 by simp
   127 
   128 declare unit.cases [nitpick_simp del]
   129 
   130 lemma nat_case_def [nitpick_def]:
   131 "nat_case x f n \<equiv> if n = 0 then x else f (n - 1)"
   132 apply (rule eq_reflection)
   133 by (case_tac n) auto
   134 
   135 declare nat.cases [nitpick_simp del]
   136 
   137 lemma list_size_simp [nitpick_simp]:
   138 "list_size f xs = (if xs = [] then 0
   139                    else Suc (f (hd xs) + list_size f (tl xs)))"
   140 "size xs = (if xs = [] then 0 else Suc (size (tl xs)))"
   141 by (case_tac xs) auto
   142 
   143 text {*
   144 Auxiliary definitions used to provide an alternative representation for
   145 @{text rat} and @{text real}.
   146 *}
   147 
   148 function nat_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
   149 [simp del]: "nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"
   150 by auto
   151 termination
   152 apply (relation "measure (\<lambda>(x, y). x + y + (if y > x then 1 else 0))")
   153  apply auto
   154  apply (metis mod_less_divisor xt1(9))
   155 by (metis mod_mod_trivial mod_self nat_neq_iff xt1(10))
   156 
   157 definition nat_lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
   158 "nat_lcm x y = x * y div (nat_gcd x y)"
   159 
   160 definition int_gcd :: "int \<Rightarrow> int \<Rightarrow> int" where
   161 "int_gcd x y = int (nat_gcd (nat (abs x)) (nat (abs y)))"
   162 
   163 definition int_lcm :: "int \<Rightarrow> int \<Rightarrow> int" where
   164 "int_lcm x y = int (nat_lcm (nat (abs x)) (nat (abs y)))"
   165 
   166 definition Frac :: "int \<times> int \<Rightarrow> bool" where
   167 "Frac \<equiv> \<lambda>(a, b). b > 0 \<and> int_gcd a b = 1"
   168 
   169 axiomatization Abs_Frac :: "int \<times> int \<Rightarrow> 'a"
   170            and Rep_Frac :: "'a \<Rightarrow> int \<times> int"
   171 
   172 definition zero_frac :: 'a where
   173 "zero_frac \<equiv> Abs_Frac (0, 1)"
   174 
   175 definition one_frac :: 'a where
   176 "one_frac \<equiv> Abs_Frac (1, 1)"
   177 
   178 definition num :: "'a \<Rightarrow> int" where
   179 "num \<equiv> fst o Rep_Frac"
   180 
   181 definition denom :: "'a \<Rightarrow> int" where
   182 "denom \<equiv> snd o Rep_Frac"
   183 
   184 function norm_frac :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
   185 [simp del]: "norm_frac a b = (if b < 0 then norm_frac (- a) (- b)
   186                               else if a = 0 \<or> b = 0 then (0, 1)
   187                               else let c = int_gcd a b in (a div c, b div c))"
   188 by pat_completeness auto
   189 termination by (relation "measure (\<lambda>(_, b). if b < 0 then 1 else 0)") auto
   190 
   191 definition frac :: "int \<Rightarrow> int \<Rightarrow> 'a" where
   192 "frac a b \<equiv> Abs_Frac (norm_frac a b)"
   193 
   194 definition plus_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   195 [nitpick_simp]:
   196 "plus_frac q r = (let d = int_lcm (denom q) (denom r) in
   197                     frac (num q * (d div denom q) + num r * (d div denom r)) d)"
   198 
   199 definition times_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   200 [nitpick_simp]:
   201 "times_frac q r = frac (num q * num r) (denom q * denom r)"
   202 
   203 definition uminus_frac :: "'a \<Rightarrow> 'a" where
   204 "uminus_frac q \<equiv> Abs_Frac (- num q, denom q)"
   205 
   206 definition number_of_frac :: "int \<Rightarrow> 'a" where
   207 "number_of_frac n \<equiv> Abs_Frac (n, 1)"
   208 
   209 definition inverse_frac :: "'a \<Rightarrow> 'a" where
   210 "inverse_frac q \<equiv> frac (denom q) (num q)"
   211 
   212 definition less_eq_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
   213 [nitpick_simp]:
   214 "less_eq_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) \<le> 0"
   215 
   216 definition of_frac :: "'a \<Rightarrow> 'b\<Colon>{inverse,ring_1}" where
   217 "of_frac q \<equiv> of_int (num q) / of_int (denom q)"
   218 
   219 use "Tools/Nitpick/kodkod.ML"
   220 use "Tools/Nitpick/kodkod_sat.ML"
   221 use "Tools/Nitpick/nitpick_util.ML"
   222 use "Tools/Nitpick/nitpick_hol.ML"
   223 use "Tools/Nitpick/nitpick_preproc.ML"
   224 use "Tools/Nitpick/nitpick_mono.ML"
   225 use "Tools/Nitpick/nitpick_scope.ML"
   226 use "Tools/Nitpick/nitpick_peephole.ML"
   227 use "Tools/Nitpick/nitpick_rep.ML"
   228 use "Tools/Nitpick/nitpick_nut.ML"
   229 use "Tools/Nitpick/nitpick_kodkod.ML"
   230 use "Tools/Nitpick/nitpick_model.ML"
   231 use "Tools/Nitpick/nitpick.ML"
   232 use "Tools/Nitpick/nitpick_isar.ML"
   233 use "Tools/Nitpick/nitpick_tests.ML"
   234 use "Tools/Nitpick/minipick.ML"
   235 
   236 setup {* Nitpick_Isar.setup *}
   237 
   238 hide (open) const unknown is_unknown undefined_fast_The undefined_fast_Eps bisim 
   239     bisim_iterator_max Quot Tha PairBox FunBox Word refl' wf' wf_wfrec wf_wfrec'
   240     wfrec' card' setsum' fold_graph' nat_gcd nat_lcm int_gcd int_lcm Frac
   241     Abs_Frac Rep_Frac zero_frac one_frac num denom norm_frac frac plus_frac
   242     times_frac uminus_frac number_of_frac inverse_frac less_eq_frac of_frac
   243 hide (open) type bisim_iterator pair_box fun_box unsigned_bit signed_bit word
   244 hide (open) fact If_def Ex1_def rtrancl_def rtranclp_def tranclp_def refl'_def
   245     wf'_def wf_wfrec'_def wfrec'_def card'_def setsum'_def fold_graph'_def
   246     The_psimp Eps_psimp unit_case_def nat_case_def list_size_simp nat_gcd_def
   247     nat_lcm_def int_gcd_def int_lcm_def Frac_def zero_frac_def one_frac_def
   248     num_def denom_def norm_frac_def frac_def plus_frac_def times_frac_def
   249     uminus_frac_def number_of_frac_def inverse_frac_def less_eq_frac_def
   250     of_frac_def
   251 
   252 end