src/HOL/Nitpick.thy
author haftmann
Tue Oct 27 15:32:20 2009 +0100 (2009-10-27)
changeset 33235 cbe96b3cb3d0
parent 33192 08a39a957ed7
child 33562 b1e2830ee31a
permissions -rw-r--r--
dropped obsolete comment
     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_mono.ML")
    17      ("Tools/Nitpick/nitpick_scope.ML")
    18      ("Tools/Nitpick/nitpick_peephole.ML")
    19      ("Tools/Nitpick/nitpick_rep.ML")
    20      ("Tools/Nitpick/nitpick_nut.ML")
    21      ("Tools/Nitpick/nitpick_kodkod.ML")
    22      ("Tools/Nitpick/nitpick_model.ML")
    23      ("Tools/Nitpick/nitpick.ML")
    24      ("Tools/Nitpick/nitpick_isar.ML")
    25      ("Tools/Nitpick/nitpick_tests.ML")
    26      ("Tools/Nitpick/minipick.ML")
    27 begin
    28 
    29 typedecl bisim_iterator
    30 
    31 axiomatization unknown :: 'a
    32            and undefined_fast_The :: 'a
    33            and undefined_fast_Eps :: 'a
    34            and bisim :: "bisim_iterator \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
    35            and bisim_iterator_max :: bisim_iterator
    36            and Tha :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
    37 
    38 datatype ('a, 'b) pair_box = PairBox 'a 'b
    39 datatype ('a, 'b) fun_box = FunBox "'a \<Rightarrow> 'b"
    40 
    41 text {*
    42 Alternative definitions.
    43 *}
    44 
    45 lemma If_def [nitpick_def]:
    46 "(if P then Q else R) \<equiv> (P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> R)"
    47 by (rule eq_reflection) (rule if_bool_eq_conj)
    48 
    49 lemma Ex1_def [nitpick_def]:
    50 "Ex1 P \<equiv> \<exists>x. P = {x}"
    51 apply (rule eq_reflection)
    52 apply (simp add: Ex1_def expand_set_eq)
    53 apply (rule iffI)
    54  apply (erule exE)
    55  apply (erule conjE)
    56  apply (rule_tac x = x in exI)
    57  apply (rule allI)
    58  apply (rename_tac y)
    59  apply (erule_tac x = y in allE)
    60 by (auto simp: mem_def)
    61 
    62 lemma rtrancl_def [nitpick_def]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
    63 by simp
    64 
    65 lemma rtranclp_def [nitpick_def]:
    66 "rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"
    67 by (rule eq_reflection) (auto dest: rtranclpD)
    68 
    69 lemma tranclp_def [nitpick_def]:
    70 "tranclp r a b \<equiv> trancl (split r) (a, b)"
    71 by (simp add: trancl_def Collect_def mem_def)
    72 
    73 definition refl' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
    74 "refl' r \<equiv> \<forall>x. (x, x) \<in> r"
    75 
    76 definition wf' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
    77 "wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
    78 
    79 axiomatization wf_wfrec :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
    80 
    81 definition wf_wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
    82 [nitpick_simp]: "wf_wfrec' R F x = F (Recdef.cut (wf_wfrec R F) R x) x"
    83 
    84 definition wfrec' ::  "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
    85 "wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x
    86                 else THE y. wfrec_rel R (%f x. F (Recdef.cut f R x) x) x y"
    87 
    88 definition card' :: "('a \<Rightarrow> bool) \<Rightarrow> nat" where
    89 "card' X \<equiv> length (SOME xs. set xs = X \<and> distinct xs)"
    90 
    91 definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b" where
    92 "setsum' f A \<equiv> if finite A then listsum (map f (SOME xs. set xs = A \<and> distinct xs)) else 0"
    93 
    94 inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool" where
    95 "fold_graph' f z {} z" |
    96 "\<lbrakk>x \<in> A; fold_graph' f z (A - {x}) y\<rbrakk> \<Longrightarrow> fold_graph' f z A (f x y)"
    97 
    98 text {*
    99 The following lemmas are not strictly necessary but they help the
   100 \textit{special\_level} optimization.
   101 *}
   102 
   103 lemma The_psimp [nitpick_psimp]:
   104 "P = {x} \<Longrightarrow> The P = x"
   105 by (subgoal_tac "{x} = (\<lambda>y. y = x)") (auto simp: mem_def)
   106 
   107 lemma Eps_psimp [nitpick_psimp]:
   108 "\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> Eps P = x"
   109 apply (case_tac "P (Eps P)")
   110  apply auto
   111 apply (erule contrapos_np)
   112 by (rule someI)
   113 
   114 lemma unit_case_def [nitpick_def]:
   115 "unit_case x u \<equiv> x"
   116 apply (subgoal_tac "u = ()")
   117  apply (simp only: unit.cases)
   118 by simp
   119 
   120 lemma nat_case_def [nitpick_def]:
   121 "nat_case x f n \<equiv> if n = 0 then x else f (n - 1)"
   122 apply (rule eq_reflection)
   123 by (case_tac n) auto
   124 
   125 lemmas dvd_def = dvd_eq_mod_eq_0 [THEN eq_reflection, nitpick_def]
   126 
   127 lemma list_size_simp [nitpick_simp]:
   128 "list_size f xs = (if xs = [] then 0
   129                    else Suc (f (hd xs) + list_size f (tl xs)))"
   130 "size xs = (if xs = [] then 0 else Suc (size (tl xs)))"
   131 by (case_tac xs) auto
   132 
   133 text {*
   134 Auxiliary definitions used to provide an alternative representation for
   135 @{text rat} and @{text real}.
   136 *}
   137 
   138 function nat_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
   139 [simp del]: "nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"
   140 by auto
   141 termination
   142 apply (relation "measure (\<lambda>(x, y). x + y + (if y > x then 1 else 0))")
   143  apply auto
   144  apply (metis mod_less_divisor xt1(9))
   145 by (metis mod_mod_trivial mod_self nat_neq_iff xt1(10))
   146 
   147 definition nat_lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
   148 "nat_lcm x y = x * y div (nat_gcd x y)"
   149 
   150 definition int_gcd :: "int \<Rightarrow> int \<Rightarrow> int" where
   151 "int_gcd x y = int (nat_gcd (nat (abs x)) (nat (abs y)))"
   152 
   153 definition int_lcm :: "int \<Rightarrow> int \<Rightarrow> int" where
   154 "int_lcm x y = int (nat_lcm (nat (abs x)) (nat (abs y)))"
   155 
   156 definition Frac :: "int \<times> int \<Rightarrow> bool" where
   157 "Frac \<equiv> \<lambda>(a, b). b > 0 \<and> int_gcd a b = 1"
   158 
   159 axiomatization Abs_Frac :: "int \<times> int \<Rightarrow> 'a"
   160            and Rep_Frac :: "'a \<Rightarrow> int \<times> int"
   161 
   162 definition zero_frac :: 'a where
   163 "zero_frac \<equiv> Abs_Frac (0, 1)"
   164 
   165 definition one_frac :: 'a where
   166 "one_frac \<equiv> Abs_Frac (1, 1)"
   167 
   168 definition num :: "'a \<Rightarrow> int" where
   169 "num \<equiv> fst o Rep_Frac"
   170 
   171 definition denom :: "'a \<Rightarrow> int" where
   172 "denom \<equiv> snd o Rep_Frac"
   173 
   174 function norm_frac :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
   175 [simp del]: "norm_frac a b = (if b < 0 then norm_frac (- a) (- b)
   176                               else if a = 0 \<or> b = 0 then (0, 1)
   177                               else let c = int_gcd a b in (a div c, b div c))"
   178 by pat_completeness auto
   179 termination by (relation "measure (\<lambda>(_, b). if b < 0 then 1 else 0)") auto
   180 
   181 definition frac :: "int \<Rightarrow> int \<Rightarrow> 'a" where
   182 "frac a b \<equiv> Abs_Frac (norm_frac a b)"
   183 
   184 definition plus_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   185 [nitpick_simp]:
   186 "plus_frac q r = (let d = int_lcm (denom q) (denom r) in
   187                     frac (num q * (d div denom q) + num r * (d div denom r)) d)"
   188 
   189 definition times_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   190 [nitpick_simp]:
   191 "times_frac q r = frac (num q * num r) (denom q * denom r)"
   192 
   193 definition uminus_frac :: "'a \<Rightarrow> 'a" where
   194 "uminus_frac q \<equiv> Abs_Frac (- num q, denom q)"
   195 
   196 definition number_of_frac :: "int \<Rightarrow> 'a" where
   197 "number_of_frac n \<equiv> Abs_Frac (n, 1)"
   198 
   199 definition inverse_frac :: "'a \<Rightarrow> 'a" where
   200 "inverse_frac q \<equiv> frac (denom q) (num q)"
   201 
   202 definition less_eq_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
   203 [nitpick_simp]:
   204 "less_eq_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) \<le> 0"
   205 
   206 definition of_frac :: "'a \<Rightarrow> 'b\<Colon>{inverse,ring_1}" where
   207 "of_frac q \<equiv> of_int (num q) / of_int (denom q)"
   208 
   209 use "Tools/Nitpick/kodkod.ML"
   210 use "Tools/Nitpick/kodkod_sat.ML"
   211 use "Tools/Nitpick/nitpick_util.ML"
   212 use "Tools/Nitpick/nitpick_hol.ML"
   213 use "Tools/Nitpick/nitpick_mono.ML"
   214 use "Tools/Nitpick/nitpick_scope.ML"
   215 use "Tools/Nitpick/nitpick_peephole.ML"
   216 use "Tools/Nitpick/nitpick_rep.ML"
   217 use "Tools/Nitpick/nitpick_nut.ML"
   218 use "Tools/Nitpick/nitpick_kodkod.ML"
   219 use "Tools/Nitpick/nitpick_model.ML"
   220 use "Tools/Nitpick/nitpick.ML"
   221 use "Tools/Nitpick/nitpick_isar.ML"
   222 use "Tools/Nitpick/nitpick_tests.ML"
   223 use "Tools/Nitpick/minipick.ML"
   224 
   225 hide (open) const unknown undefined_fast_The undefined_fast_Eps bisim 
   226     bisim_iterator_max Tha refl' wf' wf_wfrec wf_wfrec' wfrec' 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_eq_frac of_frac
   230 hide (open) type bisim_iterator pair_box fun_box
   231 hide (open) fact If_def Ex1_def rtrancl_def rtranclp_def tranclp_def refl'_def
   232     wf'_def wf_wfrec'_def wfrec'_def card'_def setsum'_def fold_graph'_def
   233     The_psimp Eps_psimp unit_case_def nat_case_def dvd_def list_size_simp
   234     nat_gcd_def nat_lcm_def int_gcd_def int_lcm_def Frac_def zero_frac_def
   235     one_frac_def num_def denom_def norm_frac_def frac_def plus_frac_def
   236     times_frac_def uminus_frac_def number_of_frac_def inverse_frac_def
   237     less_eq_frac_def of_frac_def
   238 
   239 end