src/HOL/Nitpick.thy
author blanchet
Wed Apr 23 10:23:27 2014 +0200 (2014-04-23)
changeset 56643 41d3596d8a64
parent 55642 63beb38e9258
child 57231 dca8d06ecbba
permissions -rw-r--r--
move size hooks together, with new one preceding old one and sharing same theory data
     1 (*  Title:      HOL/Nitpick.thy
     2     Author:     Jasmin Blanchette, TU Muenchen
     3     Copyright   2008, 2009, 2010
     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 BNF_FP_Base Map Record Sledgehammer
    12 keywords
    13   "nitpick" :: diag and
    14   "nitpick_params" :: thy_decl
    15 begin
    16 
    17 typedecl bisim_iterator
    18 
    19 axiomatization unknown :: 'a
    20            and is_unknown :: "'a \<Rightarrow> bool"
    21            and bisim :: "bisim_iterator \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
    22            and bisim_iterator_max :: bisim_iterator
    23            and Quot :: "'a \<Rightarrow> 'b"
    24            and safe_The :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
    25 
    26 datatype ('a, 'b) fun_box = FunBox "('a \<Rightarrow> 'b)"
    27 datatype ('a, 'b) pair_box = PairBox 'a 'b
    28 
    29 typedecl unsigned_bit
    30 typedecl signed_bit
    31 
    32 datatype 'a word = Word "('a set)"
    33 
    34 text {*
    35 Alternative definitions.
    36 *}
    37 
    38 lemma Ex1_unfold [nitpick_unfold]:
    39 "Ex1 P \<equiv> \<exists>x. {x. P x} = {x}"
    40 apply (rule eq_reflection)
    41 apply (simp add: Ex1_def set_eq_iff)
    42 apply (rule iffI)
    43  apply (erule exE)
    44  apply (erule conjE)
    45  apply (rule_tac x = x in exI)
    46  apply (rule allI)
    47  apply (rename_tac y)
    48  apply (erule_tac x = y in allE)
    49 by auto
    50 
    51 lemma rtrancl_unfold [nitpick_unfold]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
    52   by (simp only: rtrancl_trancl_reflcl)
    53 
    54 lemma rtranclp_unfold [nitpick_unfold]:
    55 "rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"
    56 by (rule eq_reflection) (auto dest: rtranclpD)
    57 
    58 lemma tranclp_unfold [nitpick_unfold]:
    59 "tranclp r a b \<equiv> (a, b) \<in> trancl {(x, y). r x y}"
    60 by (simp add: trancl_def)
    61 
    62 lemma [nitpick_simp]:
    63 "of_nat n = (if n = 0 then 0 else 1 + of_nat (n - 1))"
    64 by (cases n) auto
    65 
    66 definition prod :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
    67 "prod A B = {(a, b). a \<in> A \<and> b \<in> B}"
    68 
    69 definition refl' :: "('a \<times> 'a) set \<Rightarrow> bool" where
    70 "refl' r \<equiv> \<forall>x. (x, x) \<in> r"
    71 
    72 definition wf' :: "('a \<times> 'a) set \<Rightarrow> bool" where
    73 "wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
    74 
    75 definition card' :: "'a set \<Rightarrow> nat" where
    76 "card' A \<equiv> if finite A then length (SOME xs. set xs = A \<and> distinct xs) else 0"
    77 
    78 definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> 'a set \<Rightarrow> 'b" where
    79 "setsum' f A \<equiv> if finite A then listsum (map f (SOME xs. set xs = A \<and> distinct xs)) else 0"
    80 
    81 inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool" where
    82 "fold_graph' f z {} z" |
    83 "\<lbrakk>x \<in> A; fold_graph' f z (A - {x}) y\<rbrakk> \<Longrightarrow> fold_graph' f z A (f x y)"
    84 
    85 text {*
    86 The following lemmas are not strictly necessary but they help the
    87 \textit{specialize} optimization.
    88 *}
    89 
    90 lemma The_psimp [nitpick_psimp]:
    91   "P = (op =) x \<Longrightarrow> The P = x"
    92   by auto
    93 
    94 lemma Eps_psimp [nitpick_psimp]:
    95 "\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> Eps P = x"
    96 apply (cases "P (Eps P)")
    97  apply auto
    98 apply (erule contrapos_np)
    99 by (rule someI)
   100 
   101 lemma case_unit_unfold [nitpick_unfold]:
   102 "case_unit x u \<equiv> x"
   103 apply (subgoal_tac "u = ()")
   104  apply (simp only: unit.case)
   105 by simp
   106 
   107 declare unit.case [nitpick_simp del]
   108 
   109 lemma case_nat_unfold [nitpick_unfold]:
   110 "case_nat x f n \<equiv> if n = 0 then x else f (n - 1)"
   111 apply (rule eq_reflection)
   112 by (cases n) auto
   113 
   114 declare nat.case [nitpick_simp del]
   115 
   116 lemma size_list_simp [nitpick_simp]:
   117 "size_list f xs = (if xs = [] then 0 else Suc (f (hd xs) + size_list f (tl xs)))"
   118 "size xs = (if xs = [] then 0 else Suc (size (tl xs)))"
   119 by (cases xs) auto
   120 
   121 text {*
   122 Auxiliary definitions used to provide an alternative representation for
   123 @{text rat} and @{text real}.
   124 *}
   125 
   126 function nat_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
   127 [simp del]: "nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"
   128 by auto
   129 termination
   130 apply (relation "measure (\<lambda>(x, y). x + y + (if y > x then 1 else 0))")
   131  apply auto
   132  apply (metis mod_less_divisor xt1(9))
   133 by (metis mod_mod_trivial mod_self nat_neq_iff xt1(10))
   134 
   135 definition nat_lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
   136 "nat_lcm x y = x * y div (nat_gcd x y)"
   137 
   138 definition int_gcd :: "int \<Rightarrow> int \<Rightarrow> int" where
   139 "int_gcd x y = int (nat_gcd (nat (abs x)) (nat (abs y)))"
   140 
   141 definition int_lcm :: "int \<Rightarrow> int \<Rightarrow> int" where
   142 "int_lcm x y = int (nat_lcm (nat (abs x)) (nat (abs y)))"
   143 
   144 definition Frac :: "int \<times> int \<Rightarrow> bool" where
   145 "Frac \<equiv> \<lambda>(a, b). b > 0 \<and> int_gcd a b = 1"
   146 
   147 axiomatization
   148   Abs_Frac :: "int \<times> int \<Rightarrow> 'a" and
   149   Rep_Frac :: "'a \<Rightarrow> int \<times> int"
   150 
   151 definition zero_frac :: 'a where
   152 "zero_frac \<equiv> Abs_Frac (0, 1)"
   153 
   154 definition one_frac :: 'a where
   155 "one_frac \<equiv> Abs_Frac (1, 1)"
   156 
   157 definition num :: "'a \<Rightarrow> int" where
   158 "num \<equiv> fst o Rep_Frac"
   159 
   160 definition denom :: "'a \<Rightarrow> int" where
   161 "denom \<equiv> snd o Rep_Frac"
   162 
   163 function norm_frac :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
   164 [simp del]: "norm_frac a b = (if b < 0 then norm_frac (- a) (- b)
   165                               else if a = 0 \<or> b = 0 then (0, 1)
   166                               else let c = int_gcd a b in (a div c, b div c))"
   167 by pat_completeness auto
   168 termination by (relation "measure (\<lambda>(_, b). if b < 0 then 1 else 0)") auto
   169 
   170 definition frac :: "int \<Rightarrow> int \<Rightarrow> 'a" where
   171 "frac a b \<equiv> Abs_Frac (norm_frac a b)"
   172 
   173 definition plus_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   174 [nitpick_simp]:
   175 "plus_frac q r = (let d = int_lcm (denom q) (denom r) in
   176                     frac (num q * (d div denom q) + num r * (d div denom r)) d)"
   177 
   178 definition times_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   179 [nitpick_simp]:
   180 "times_frac q r = frac (num q * num r) (denom q * denom r)"
   181 
   182 definition uminus_frac :: "'a \<Rightarrow> 'a" where
   183 "uminus_frac q \<equiv> Abs_Frac (- num q, denom q)"
   184 
   185 definition number_of_frac :: "int \<Rightarrow> 'a" where
   186 "number_of_frac n \<equiv> Abs_Frac (n, 1)"
   187 
   188 definition inverse_frac :: "'a \<Rightarrow> 'a" where
   189 "inverse_frac q \<equiv> frac (denom q) (num q)"
   190 
   191 definition less_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
   192 [nitpick_simp]:
   193 "less_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) < 0"
   194 
   195 definition less_eq_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
   196 [nitpick_simp]:
   197 "less_eq_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) \<le> 0"
   198 
   199 definition of_frac :: "'a \<Rightarrow> 'b\<Colon>{inverse,ring_1}" where
   200 "of_frac q \<equiv> of_int (num q) / of_int (denom q)"
   201 
   202 axiomatization wf_wfrec :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
   203 
   204 definition wf_wfrec' :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
   205 [nitpick_simp]: "wf_wfrec' R F x = F (cut (wf_wfrec R F) R x) x"
   206 
   207 definition wfrec' ::  "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
   208 "wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x
   209                 else THE y. wfrec_rel R (%f x. F (cut f R x) x) x y"
   210 
   211 ML_file "Tools/Nitpick/kodkod.ML"
   212 ML_file "Tools/Nitpick/kodkod_sat.ML"
   213 ML_file "Tools/Nitpick/nitpick_util.ML"
   214 ML_file "Tools/Nitpick/nitpick_hol.ML"
   215 ML_file "Tools/Nitpick/nitpick_mono.ML"
   216 ML_file "Tools/Nitpick/nitpick_preproc.ML"
   217 ML_file "Tools/Nitpick/nitpick_scope.ML"
   218 ML_file "Tools/Nitpick/nitpick_peephole.ML"
   219 ML_file "Tools/Nitpick/nitpick_rep.ML"
   220 ML_file "Tools/Nitpick/nitpick_nut.ML"
   221 ML_file "Tools/Nitpick/nitpick_kodkod.ML"
   222 ML_file "Tools/Nitpick/nitpick_model.ML"
   223 ML_file "Tools/Nitpick/nitpick.ML"
   224 ML_file "Tools/Nitpick/nitpick_commands.ML"
   225 ML_file "Tools/Nitpick/nitpick_tests.ML"
   226 
   227 setup {*
   228   Nitpick_HOL.register_ersatz_global
   229     [(@{const_name card}, @{const_name card'}),
   230      (@{const_name setsum}, @{const_name setsum'}),
   231      (@{const_name fold_graph}, @{const_name fold_graph'}),
   232      (@{const_name wf}, @{const_name wf'}),
   233      (@{const_name wf_wfrec}, @{const_name wf_wfrec'}),
   234      (@{const_name wfrec}, @{const_name wfrec'})]
   235 *}
   236 
   237 hide_const (open) unknown is_unknown bisim bisim_iterator_max Quot safe_The
   238     FunBox PairBox Word prod refl' wf' card' setsum'
   239     fold_graph' nat_gcd nat_lcm int_gcd int_lcm Frac Abs_Frac Rep_Frac zero_frac
   240     one_frac num denom norm_frac frac plus_frac times_frac uminus_frac
   241     number_of_frac inverse_frac less_frac less_eq_frac of_frac wf_wfrec wf_wfrec
   242     wfrec'
   243 hide_type (open) bisim_iterator fun_box pair_box unsigned_bit signed_bit word
   244 hide_fact (open) Ex1_unfold rtrancl_unfold rtranclp_unfold tranclp_unfold
   245     prod_def refl'_def wf'_def card'_def setsum'_def
   246     fold_graph'_def The_psimp Eps_psimp case_unit_unfold case_nat_unfold
   247     size_list_simp nat_gcd_def nat_lcm_def int_gcd_def int_lcm_def Frac_def
   248     zero_frac_def one_frac_def num_def denom_def norm_frac_def frac_def
   249     plus_frac_def times_frac_def uminus_frac_def number_of_frac_def
   250     inverse_frac_def less_frac_def less_eq_frac_def of_frac_def wf_wfrec'_def
   251     wfrec'_def
   252 
   253 end