src/HOL/Nitpick.thy
author hoelzl
Thu Jan 31 11:31:27 2013 +0100 (2013-01-31)
changeset 50999 3de230ed0547
parent 49989 34d0ac1bdac6
child 52641 c56b6fa636e8
permissions -rw-r--r--
introduce order topology
     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 Hilbert_Choice List Map Quotient Record Sledgehammer
    12 keywords "nitpick" :: diag and "nitpick_params" :: thy_decl
    13 begin
    14 
    15 typedecl bisim_iterator
    16 
    17 axiomatization unknown :: 'a
    18            and is_unknown :: "'a \<Rightarrow> bool"
    19            and bisim :: "bisim_iterator \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
    20            and bisim_iterator_max :: bisim_iterator
    21            and Quot :: "'a \<Rightarrow> 'b"
    22            and safe_The :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
    23 
    24 datatype ('a, 'b) fun_box = FunBox "('a \<Rightarrow> 'b)"
    25 datatype ('a, 'b) pair_box = PairBox 'a 'b
    26 
    27 typedecl unsigned_bit
    28 typedecl signed_bit
    29 
    30 datatype 'a word = Word "('a set)"
    31 
    32 text {*
    33 Alternative definitions.
    34 *}
    35 
    36 lemma Ex1_unfold [nitpick_unfold, no_atp]:
    37 "Ex1 P \<equiv> \<exists>x. {x. P x} = {x}"
    38 apply (rule eq_reflection)
    39 apply (simp add: Ex1_def set_eq_iff)
    40 apply (rule iffI)
    41  apply (erule exE)
    42  apply (erule conjE)
    43  apply (rule_tac x = x in exI)
    44  apply (rule allI)
    45  apply (rename_tac y)
    46  apply (erule_tac x = y in allE)
    47 by auto
    48 
    49 lemma rtrancl_unfold [nitpick_unfold, no_atp]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
    50   by (simp only: rtrancl_trancl_reflcl)
    51 
    52 lemma rtranclp_unfold [nitpick_unfold, no_atp]:
    53 "rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"
    54 by (rule eq_reflection) (auto dest: rtranclpD)
    55 
    56 lemma tranclp_unfold [nitpick_unfold, no_atp]:
    57 "tranclp r a b \<equiv> (a, b) \<in> trancl {(x, y). r x y}"
    58 by (simp add: trancl_def)
    59 
    60 lemma [nitpick_simp, no_atp]:
    61 "of_nat n = (if n = 0 then 0 else 1 + of_nat (n - 1))"
    62 by (cases n) auto
    63 
    64 definition prod :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
    65 "prod A B = {(a, b). a \<in> A \<and> b \<in> B}"
    66 
    67 definition refl' :: "('a \<times> 'a) set \<Rightarrow> bool" where
    68 "refl' r \<equiv> \<forall>x. (x, x) \<in> r"
    69 
    70 definition wf' :: "('a \<times> 'a) set \<Rightarrow> bool" where
    71 "wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
    72 
    73 definition card' :: "'a set \<Rightarrow> nat" where
    74 "card' A \<equiv> if finite A then length (SOME xs. set xs = A \<and> distinct xs) else 0"
    75 
    76 definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> 'a set \<Rightarrow> 'b" where
    77 "setsum' f A \<equiv> if finite A then listsum (map f (SOME xs. set xs = A \<and> distinct xs)) else 0"
    78 
    79 inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool" where
    80 "fold_graph' f z {} z" |
    81 "\<lbrakk>x \<in> A; fold_graph' f z (A - {x}) y\<rbrakk> \<Longrightarrow> fold_graph' f z A (f x y)"
    82 
    83 text {*
    84 The following lemmas are not strictly necessary but they help the
    85 \textit{specialize} optimization.
    86 *}
    87 
    88 lemma The_psimp [nitpick_psimp, no_atp]:
    89   "P = (op =) x \<Longrightarrow> The P = x"
    90   by auto
    91 
    92 lemma Eps_psimp [nitpick_psimp, no_atp]:
    93 "\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> Eps P = x"
    94 apply (cases "P (Eps P)")
    95  apply auto
    96 apply (erule contrapos_np)
    97 by (rule someI)
    98 
    99 lemma unit_case_unfold [nitpick_unfold, no_atp]:
   100 "unit_case x u \<equiv> x"
   101 apply (subgoal_tac "u = ()")
   102  apply (simp only: unit.cases)
   103 by simp
   104 
   105 declare unit.cases [nitpick_simp del]
   106 
   107 lemma nat_case_unfold [nitpick_unfold, no_atp]:
   108 "nat_case x f n \<equiv> if n = 0 then x else f (n - 1)"
   109 apply (rule eq_reflection)
   110 by (cases n) auto
   111 
   112 declare nat.cases [nitpick_simp del]
   113 
   114 lemma list_size_simp [nitpick_simp, no_atp]:
   115 "list_size f xs = (if xs = [] then 0
   116                    else Suc (f (hd xs) + list_size f (tl xs)))"
   117 "size xs = (if xs = [] then 0 else Suc (size (tl xs)))"
   118 by (cases xs) auto
   119 
   120 text {*
   121 Auxiliary definitions used to provide an alternative representation for
   122 @{text rat} and @{text real}.
   123 *}
   124 
   125 function nat_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
   126 [simp del]: "nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"
   127 by auto
   128 termination
   129 apply (relation "measure (\<lambda>(x, y). x + y + (if y > x then 1 else 0))")
   130  apply auto
   131  apply (metis mod_less_divisor xt1(9))
   132 by (metis mod_mod_trivial mod_self nat_neq_iff xt1(10))
   133 
   134 definition nat_lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
   135 "nat_lcm x y = x * y div (nat_gcd x y)"
   136 
   137 definition int_gcd :: "int \<Rightarrow> int \<Rightarrow> int" where
   138 "int_gcd x y = int (nat_gcd (nat (abs x)) (nat (abs y)))"
   139 
   140 definition int_lcm :: "int \<Rightarrow> int \<Rightarrow> int" where
   141 "int_lcm x y = int (nat_lcm (nat (abs x)) (nat (abs y)))"
   142 
   143 definition Frac :: "int \<times> int \<Rightarrow> bool" where
   144 "Frac \<equiv> \<lambda>(a, b). b > 0 \<and> int_gcd a b = 1"
   145 
   146 axiomatization Abs_Frac :: "int \<times> int \<Rightarrow> 'a"
   147            and Rep_Frac :: "'a \<Rightarrow> int \<times> int"
   148 
   149 definition zero_frac :: 'a where
   150 "zero_frac \<equiv> Abs_Frac (0, 1)"
   151 
   152 definition one_frac :: 'a where
   153 "one_frac \<equiv> Abs_Frac (1, 1)"
   154 
   155 definition num :: "'a \<Rightarrow> int" where
   156 "num \<equiv> fst o Rep_Frac"
   157 
   158 definition denom :: "'a \<Rightarrow> int" where
   159 "denom \<equiv> snd o Rep_Frac"
   160 
   161 function norm_frac :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
   162 [simp del]: "norm_frac a b = (if b < 0 then norm_frac (- a) (- b)
   163                               else if a = 0 \<or> b = 0 then (0, 1)
   164                               else let c = int_gcd a b in (a div c, b div c))"
   165 by pat_completeness auto
   166 termination by (relation "measure (\<lambda>(_, b). if b < 0 then 1 else 0)") auto
   167 
   168 definition frac :: "int \<Rightarrow> int \<Rightarrow> 'a" where
   169 "frac a b \<equiv> Abs_Frac (norm_frac a b)"
   170 
   171 definition plus_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   172 [nitpick_simp]:
   173 "plus_frac q r = (let d = int_lcm (denom q) (denom r) in
   174                     frac (num q * (d div denom q) + num r * (d div denom r)) d)"
   175 
   176 definition times_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   177 [nitpick_simp]:
   178 "times_frac q r = frac (num q * num r) (denom q * denom r)"
   179 
   180 definition uminus_frac :: "'a \<Rightarrow> 'a" where
   181 "uminus_frac q \<equiv> Abs_Frac (- num q, denom q)"
   182 
   183 definition number_of_frac :: "int \<Rightarrow> 'a" where
   184 "number_of_frac n \<equiv> Abs_Frac (n, 1)"
   185 
   186 definition inverse_frac :: "'a \<Rightarrow> 'a" where
   187 "inverse_frac q \<equiv> frac (denom q) (num q)"
   188 
   189 definition less_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
   190 [nitpick_simp]:
   191 "less_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) < 0"
   192 
   193 definition less_eq_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
   194 [nitpick_simp]:
   195 "less_eq_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) \<le> 0"
   196 
   197 definition of_frac :: "'a \<Rightarrow> 'b\<Colon>{inverse,ring_1}" where
   198 "of_frac q \<equiv> of_int (num q) / of_int (denom q)"
   199 
   200 ML_file "Tools/Nitpick/kodkod.ML"
   201 ML_file "Tools/Nitpick/kodkod_sat.ML"
   202 ML_file "Tools/Nitpick/nitpick_util.ML"
   203 ML_file "Tools/Nitpick/nitpick_hol.ML"
   204 ML_file "Tools/Nitpick/nitpick_mono.ML"
   205 ML_file "Tools/Nitpick/nitpick_preproc.ML"
   206 ML_file "Tools/Nitpick/nitpick_scope.ML"
   207 ML_file "Tools/Nitpick/nitpick_peephole.ML"
   208 ML_file "Tools/Nitpick/nitpick_rep.ML"
   209 ML_file "Tools/Nitpick/nitpick_nut.ML"
   210 ML_file "Tools/Nitpick/nitpick_kodkod.ML"
   211 ML_file "Tools/Nitpick/nitpick_model.ML"
   212 ML_file "Tools/Nitpick/nitpick.ML"
   213 ML_file "Tools/Nitpick/nitpick_isar.ML"
   214 ML_file "Tools/Nitpick/nitpick_tests.ML"
   215 
   216 setup {*
   217   Nitpick_Isar.setup #>
   218   Nitpick_HOL.register_ersatz_global
   219     [(@{const_name card}, @{const_name card'}),
   220      (@{const_name setsum}, @{const_name setsum'}),
   221      (@{const_name fold_graph}, @{const_name fold_graph'}),
   222      (@{const_name wf}, @{const_name wf'})]
   223 *}
   224 
   225 hide_const (open) unknown is_unknown bisim bisim_iterator_max Quot safe_The
   226     FunBox PairBox Word prod refl' wf' card' setsum'
   227     fold_graph' nat_gcd nat_lcm int_gcd int_lcm Frac Abs_Frac Rep_Frac zero_frac
   228     one_frac num denom norm_frac frac plus_frac times_frac uminus_frac
   229     number_of_frac inverse_frac less_frac less_eq_frac of_frac
   230 hide_type (open) bisim_iterator fun_box pair_box unsigned_bit signed_bit word
   231 hide_fact (open) Ex1_unfold rtrancl_unfold rtranclp_unfold tranclp_unfold
   232     prod_def refl'_def wf'_def card'_def setsum'_def
   233     fold_graph'_def The_psimp Eps_psimp unit_case_unfold nat_case_unfold
   234     list_size_simp nat_gcd_def nat_lcm_def int_gcd_def int_lcm_def Frac_def
   235     zero_frac_def one_frac_def num_def denom_def norm_frac_def frac_def
   236     plus_frac_def times_frac_def uminus_frac_def number_of_frac_def
   237     inverse_frac_def less_frac_def less_eq_frac_def of_frac_def
   238 
   239 end