src/HOL/Nitpick.thy
author blanchet
Fri Feb 12 19:44:37 2010 +0100 (2010-02-12)
changeset 35177 168041f24f80
parent 35079 592edca1dfb3
child 35180 c57dba973391
permissions -rw-r--r--
various cosmetic changes to Nitpick
     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_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 quot_normal :: "'a \<Rightarrow> 'a"
    40            and Silly :: "'a \<Rightarrow> 'b"
    41            and Tha :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
    42 
    43 datatype ('a, 'b) pair_box = PairBox 'a 'b
    44 datatype ('a, 'b) fun_box = FunBox "('a \<Rightarrow> 'b)"
    45 
    46 typedecl unsigned_bit
    47 typedecl signed_bit
    48 typedecl \<xi>
    49 
    50 datatype 'a word = Word "('a set)"
    51 
    52 text {*
    53 Alternative definitions.
    54 *}
    55 
    56 lemma If_def [nitpick_def]:
    57 "(if P then Q else R) \<equiv> (P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> R)"
    58 by (rule eq_reflection) (rule if_bool_eq_conj)
    59 
    60 lemma Ex1_def [nitpick_def]:
    61 "Ex1 P \<equiv> \<exists>x. P = {x}"
    62 apply (rule eq_reflection)
    63 apply (simp add: Ex1_def expand_set_eq)
    64 apply (rule iffI)
    65  apply (erule exE)
    66  apply (erule conjE)
    67  apply (rule_tac x = x in exI)
    68  apply (rule allI)
    69  apply (rename_tac y)
    70  apply (erule_tac x = y in allE)
    71 by (auto simp: mem_def)
    72 
    73 lemma rtrancl_def [nitpick_def]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
    74 by simp
    75 
    76 lemma rtranclp_def [nitpick_def]:
    77 "rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"
    78 by (rule eq_reflection) (auto dest: rtranclpD)
    79 
    80 lemma tranclp_def [nitpick_def]:
    81 "tranclp r a b \<equiv> trancl (split r) (a, b)"
    82 by (simp add: trancl_def Collect_def mem_def)
    83 
    84 definition refl' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
    85 "refl' r \<equiv> \<forall>x. (x, x) \<in> r"
    86 
    87 definition wf' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
    88 "wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
    89 
    90 axiomatization wf_wfrec :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
    91 
    92 definition wf_wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
    93 [nitpick_simp]: "wf_wfrec' R F x = F (Recdef.cut (wf_wfrec R F) R x) x"
    94 
    95 definition wfrec' ::  "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
    96 "wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x
    97                 else THE y. wfrec_rel R (%f x. F (Recdef.cut f R x) x) x y"
    98 
    99 definition card' :: "('a \<Rightarrow> bool) \<Rightarrow> nat" where
   100 "card' X \<equiv> length (SOME xs. set xs = X \<and> distinct xs)"
   101 
   102 definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b" where
   103 "setsum' f A \<equiv> if finite A then listsum (map f (SOME xs. set xs = A \<and> distinct xs)) else 0"
   104 
   105 inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool" where
   106 "fold_graph' f z {} z" |
   107 "\<lbrakk>x \<in> A; fold_graph' f z (A - {x}) y\<rbrakk> \<Longrightarrow> fold_graph' f z A (f x y)"
   108 
   109 text {*
   110 The following lemmas are not strictly necessary but they help the
   111 \textit{special\_level} optimization.
   112 *}
   113 
   114 lemma The_psimp [nitpick_psimp]:
   115 "P = {x} \<Longrightarrow> The P = x"
   116 by (subgoal_tac "{x} = (\<lambda>y. y = x)") (auto simp: mem_def)
   117 
   118 lemma Eps_psimp [nitpick_psimp]:
   119 "\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> Eps P = x"
   120 apply (case_tac "P (Eps P)")
   121  apply auto
   122 apply (erule contrapos_np)
   123 by (rule someI)
   124 
   125 lemma unit_case_def [nitpick_def]:
   126 "unit_case x u \<equiv> x"
   127 apply (subgoal_tac "u = ()")
   128  apply (simp only: unit.cases)
   129 by simp
   130 
   131 declare unit.cases [nitpick_simp del]
   132 
   133 lemma nat_case_def [nitpick_def]:
   134 "nat_case x f n \<equiv> if n = 0 then x else f (n - 1)"
   135 apply (rule eq_reflection)
   136 by (case_tac n) auto
   137 
   138 declare nat.cases [nitpick_simp del]
   139 
   140 lemma list_size_simp [nitpick_simp]:
   141 "list_size f xs = (if xs = [] then 0
   142                    else Suc (f (hd xs) + list_size f (tl xs)))"
   143 "size xs = (if xs = [] then 0 else Suc (size (tl xs)))"
   144 by (case_tac xs) auto
   145 
   146 text {*
   147 Auxiliary definitions used to provide an alternative representation for
   148 @{text rat} and @{text real}.
   149 *}
   150 
   151 function nat_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
   152 [simp del]: "nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"
   153 by auto
   154 termination
   155 apply (relation "measure (\<lambda>(x, y). x + y + (if y > x then 1 else 0))")
   156  apply auto
   157  apply (metis mod_less_divisor xt1(9))
   158 by (metis mod_mod_trivial mod_self nat_neq_iff xt1(10))
   159 
   160 definition nat_lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
   161 "nat_lcm x y = x * y div (nat_gcd x y)"
   162 
   163 definition int_gcd :: "int \<Rightarrow> int \<Rightarrow> int" where
   164 "int_gcd x y = int (nat_gcd (nat (abs x)) (nat (abs y)))"
   165 
   166 definition int_lcm :: "int \<Rightarrow> int \<Rightarrow> int" where
   167 "int_lcm x y = int (nat_lcm (nat (abs x)) (nat (abs y)))"
   168 
   169 definition Frac :: "int \<times> int \<Rightarrow> bool" where
   170 "Frac \<equiv> \<lambda>(a, b). b > 0 \<and> int_gcd a b = 1"
   171 
   172 axiomatization Abs_Frac :: "int \<times> int \<Rightarrow> 'a"
   173            and Rep_Frac :: "'a \<Rightarrow> int \<times> int"
   174 
   175 definition zero_frac :: 'a where
   176 "zero_frac \<equiv> Abs_Frac (0, 1)"
   177 
   178 definition one_frac :: 'a where
   179 "one_frac \<equiv> Abs_Frac (1, 1)"
   180 
   181 definition num :: "'a \<Rightarrow> int" where
   182 "num \<equiv> fst o Rep_Frac"
   183 
   184 definition denom :: "'a \<Rightarrow> int" where
   185 "denom \<equiv> snd o Rep_Frac"
   186 
   187 function norm_frac :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
   188 [simp del]: "norm_frac a b = (if b < 0 then norm_frac (- a) (- b)
   189                               else if a = 0 \<or> b = 0 then (0, 1)
   190                               else let c = int_gcd a b in (a div c, b div c))"
   191 by pat_completeness auto
   192 termination by (relation "measure (\<lambda>(_, b). if b < 0 then 1 else 0)") auto
   193 
   194 definition frac :: "int \<Rightarrow> int \<Rightarrow> 'a" where
   195 "frac a b \<equiv> Abs_Frac (norm_frac a b)"
   196 
   197 definition plus_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   198 [nitpick_simp]:
   199 "plus_frac q r = (let d = int_lcm (denom q) (denom r) in
   200                     frac (num q * (d div denom q) + num r * (d div denom r)) d)"
   201 
   202 definition times_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   203 [nitpick_simp]:
   204 "times_frac q r = frac (num q * num r) (denom q * denom r)"
   205 
   206 definition uminus_frac :: "'a \<Rightarrow> 'a" where
   207 "uminus_frac q \<equiv> Abs_Frac (- num q, denom q)"
   208 
   209 definition number_of_frac :: "int \<Rightarrow> 'a" where
   210 "number_of_frac n \<equiv> Abs_Frac (n, 1)"
   211 
   212 definition inverse_frac :: "'a \<Rightarrow> 'a" where
   213 "inverse_frac q \<equiv> frac (denom q) (num q)"
   214 
   215 definition less_eq_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
   216 [nitpick_simp]:
   217 "less_eq_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) \<le> 0"
   218 
   219 definition of_frac :: "'a \<Rightarrow> 'b\<Colon>{inverse,ring_1}" where
   220 "of_frac q \<equiv> of_int (num q) / of_int (denom q)"
   221 
   222 (* While Nitpick normally avoids to unfold definitions for locales, it
   223    unfortunately needs to unfold them when dealing with the following built-in
   224    constants. A cleaner approach would be to change "Nitpick_HOL" and
   225    "Nitpick_Nut" so that they handle the unexpanded overloaded constants
   226    directly, but this is slightly more tricky to implement. *)
   227 lemmas [nitpick_def] = div_int_inst.div_int div_int_inst.mod_int
   228     div_nat_inst.div_nat div_nat_inst.mod_nat semilattice_inf_fun_inst.inf_fun
   229     minus_fun_inst.minus_fun minus_int_inst.minus_int minus_nat_inst.minus_nat
   230     one_int_inst.one_int one_nat_inst.one_nat ord_fun_inst.less_eq_fun
   231     ord_int_inst.less_eq_int ord_int_inst.less_int ord_nat_inst.less_eq_nat
   232     ord_nat_inst.less_nat plus_int_inst.plus_int plus_nat_inst.plus_nat
   233     times_int_inst.times_int times_nat_inst.times_nat uminus_int_inst.uminus_int
   234     semilattice_sup_fun_inst.sup_fun zero_int_inst.zero_int
   235     zero_nat_inst.zero_nat
   236 
   237 use "Tools/Nitpick/kodkod.ML"
   238 use "Tools/Nitpick/kodkod_sat.ML"
   239 use "Tools/Nitpick/nitpick_util.ML"
   240 use "Tools/Nitpick/nitpick_hol.ML"
   241 use "Tools/Nitpick/nitpick_preproc.ML"
   242 use "Tools/Nitpick/nitpick_mono.ML"
   243 use "Tools/Nitpick/nitpick_scope.ML"
   244 use "Tools/Nitpick/nitpick_peephole.ML"
   245 use "Tools/Nitpick/nitpick_rep.ML"
   246 use "Tools/Nitpick/nitpick_nut.ML"
   247 use "Tools/Nitpick/nitpick_kodkod.ML"
   248 use "Tools/Nitpick/nitpick_model.ML"
   249 use "Tools/Nitpick/nitpick.ML"
   250 use "Tools/Nitpick/nitpick_isar.ML"
   251 use "Tools/Nitpick/nitpick_tests.ML"
   252 use "Tools/Nitpick/minipick.ML"
   253 
   254 setup {* Nitpick_Isar.setup *}
   255 
   256 hide (open) const unknown is_unknown undefined_fast_The undefined_fast_Eps bisim 
   257     bisim_iterator_max Quot quot_normal Silly Tha PairBox FunBox Word refl' wf'
   258     wf_wfrec wf_wfrec' wfrec' card' setsum' fold_graph' nat_gcd nat_lcm int_gcd
   259     int_lcm Frac Abs_Frac Rep_Frac zero_frac one_frac num denom norm_frac frac
   260     plus_frac times_frac uminus_frac number_of_frac inverse_frac less_eq_frac
   261     of_frac
   262 hide (open) type bisim_iterator pair_box fun_box unsigned_bit signed_bit \<xi> word
   263 hide (open) fact If_def Ex1_def rtrancl_def rtranclp_def tranclp_def refl'_def
   264     wf'_def wf_wfrec'_def wfrec'_def card'_def setsum'_def fold_graph'_def
   265     The_psimp Eps_psimp unit_case_def nat_case_def list_size_simp nat_gcd_def
   266     nat_lcm_def int_gcd_def int_lcm_def Frac_def zero_frac_def one_frac_def
   267     num_def denom_def norm_frac_def frac_def plus_frac_def times_frac_def
   268     uminus_frac_def number_of_frac_def inverse_frac_def less_eq_frac_def
   269     of_frac_def
   270 
   271 end