src/HOL/Nitpick.thy
 author blanchet Mon Feb 17 22:54:38 2014 +0100 (2014-02-17) changeset 55539 0819931d652d parent 55415 05f5fdb8d093 child 55642 63beb38e9258 permissions -rw-r--r--
simplified data structure by reducing the incidence of clumsy indices
     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.cases)

   105 by simp

   106

   107 declare unit.cases [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.cases [nitpick_simp del]

   115

   116 lemma list_size_simp [nitpick_simp]:

   117 "list_size f xs = (if xs = [] then 0

   118                    else Suc (f (hd xs) + list_size f (tl xs)))"

   119 "size xs = (if xs = [] then 0 else Suc (size (tl xs)))"

   120 by (cases xs) auto

   121

   122 text {*

   123 Auxiliary definitions used to provide an alternative representation for

   124 @{text rat} and @{text real}.

   125 *}

   126

   127 function nat_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat" where

   128 [simp del]: "nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"

   129 by auto

   130 termination

   131 apply (relation "measure (\<lambda>(x, y). x + y + (if y > x then 1 else 0))")

   132  apply auto

   133  apply (metis mod_less_divisor xt1(9))

   134 by (metis mod_mod_trivial mod_self nat_neq_iff xt1(10))

   135

   136 definition nat_lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat" where

   137 "nat_lcm x y = x * y div (nat_gcd x y)"

   138

   139 definition int_gcd :: "int \<Rightarrow> int \<Rightarrow> int" where

   140 "int_gcd x y = int (nat_gcd (nat (abs x)) (nat (abs y)))"

   141

   142 definition int_lcm :: "int \<Rightarrow> int \<Rightarrow> int" where

   143 "int_lcm x y = int (nat_lcm (nat (abs x)) (nat (abs y)))"

   144

   145 definition Frac :: "int \<times> int \<Rightarrow> bool" where

   146 "Frac \<equiv> \<lambda>(a, b). b > 0 \<and> int_gcd a b = 1"

   147

   148 axiomatization Abs_Frac :: "int \<times> int \<Rightarrow> 'a"

   149            and 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     list_size_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