src/HOL/Nitpick.thy
author haftmann
Sun Apr 23 14:53:33 2017 +0200 (24 months ago)
changeset 65555 85ed070017b7
parent 64267 b9a1486e79be
child 66011 f10bbfe07c41
permissions -rw-r--r--
include GCD as integral part of computational algebra in session HOL
     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 section \<open>Nitpick: Yet Another Counterexample Generator for Isabelle/HOL\<close>
     9 
    10 theory Nitpick
    11 imports Record GCD
    12 keywords
    13   "nitpick" :: diag and
    14   "nitpick_params" :: thy_decl
    15 begin
    16 
    17 datatype (plugins only: extraction) (dead 'a, dead 'b) fun_box = FunBox "'a \<Rightarrow> 'b"
    18 datatype (plugins only: extraction) (dead 'a, dead 'b) pair_box = PairBox 'a 'b
    19 datatype (plugins only: extraction) (dead 'a) word = Word "'a set"
    20 
    21 typedecl bisim_iterator
    22 typedecl unsigned_bit
    23 typedecl signed_bit
    24 
    25 consts
    26   unknown :: 'a
    27   is_unknown :: "'a \<Rightarrow> bool"
    28   bisim :: "bisim_iterator \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
    29   bisim_iterator_max :: bisim_iterator
    30   Quot :: "'a \<Rightarrow> 'b"
    31   safe_The :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
    32 
    33 text \<open>
    34 Alternative definitions.
    35 \<close>
    36 
    37 lemma Ex1_unfold[nitpick_unfold]: "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]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
    50   by (simp only: rtrancl_trancl_reflcl)
    51 
    52 lemma rtranclp_unfold[nitpick_unfold]: "rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"
    53   by (rule eq_reflection) (auto dest: rtranclpD)
    54 
    55 lemma tranclp_unfold[nitpick_unfold]:
    56   "tranclp r a b \<equiv> (a, b) \<in> trancl {(x, y). r x y}"
    57   by (simp add: trancl_def)
    58 
    59 lemma [nitpick_simp]:
    60   "of_nat n = (if n = 0 then 0 else 1 + of_nat (n - 1))"
    61   by (cases n) auto
    62 
    63 definition prod :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" where
    64   "prod A B = {(a, b). a \<in> A \<and> b \<in> B}"
    65 
    66 definition refl' :: "('a \<times> 'a) set \<Rightarrow> bool" where
    67   "refl' r \<equiv> \<forall>x. (x, x) \<in> r"
    68 
    69 definition wf' :: "('a \<times> 'a) set \<Rightarrow> bool" where
    70   "wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
    71 
    72 definition card' :: "'a set \<Rightarrow> nat" where
    73   "card' A \<equiv> if finite A then length (SOME xs. set xs = A \<and> distinct xs) else 0"
    74 
    75 definition sum' :: "('a \<Rightarrow> 'b::comm_monoid_add) \<Rightarrow> 'a set \<Rightarrow> 'b" where
    76   "sum' f A \<equiv> if finite A then sum_list (map f (SOME xs. set xs = A \<and> distinct xs)) else 0"
    77 
    78 inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool" where
    79   "fold_graph' f z {} z" |
    80   "\<lbrakk>x \<in> A; fold_graph' f z (A - {x}) y\<rbrakk> \<Longrightarrow> fold_graph' f z A (f x y)"
    81 
    82 text \<open>
    83 The following lemmas are not strictly necessary but they help the
    84 \textit{specialize} optimization.
    85 \<close>
    86 
    87 lemma The_psimp[nitpick_psimp]: "P = (op =) x \<Longrightarrow> The P = x"
    88   by auto
    89 
    90 lemma Eps_psimp[nitpick_psimp]:
    91   "\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> Eps P = x"
    92   apply (cases "P (Eps P)")
    93    apply auto
    94   apply (erule contrapos_np)
    95   by (rule someI)
    96 
    97 lemma case_unit_unfold[nitpick_unfold]:
    98   "case_unit x u \<equiv> x"
    99   apply (subgoal_tac "u = ()")
   100    apply (simp only: unit.case)
   101   by simp
   102 
   103 declare unit.case[nitpick_simp del]
   104 
   105 lemma case_nat_unfold[nitpick_unfold]:
   106   "case_nat x f n \<equiv> if n = 0 then x else f (n - 1)"
   107   apply (rule eq_reflection)
   108   by (cases n) auto
   109 
   110 declare nat.case[nitpick_simp del]
   111 
   112 lemma size_list_simp[nitpick_simp]:
   113   "size_list f xs = (if xs = [] then 0 else Suc (f (hd xs) + size_list f (tl xs)))"
   114   "size xs = (if xs = [] then 0 else Suc (size (tl xs)))"
   115   by (cases xs) auto
   116 
   117 text \<open>
   118 Auxiliary definitions used to provide an alternative representation for
   119 \<open>rat\<close> and \<open>real\<close>.
   120 \<close>
   121 
   122 function nat_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
   123   "nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"
   124   by auto
   125 termination
   126   apply (relation "measure (\<lambda>(x, y). x + y + (if y > x then 1 else 0))")
   127    apply auto
   128    apply (metis mod_less_divisor xt1(9))
   129   apply (metis mod_mod_trivial mod_self nat_neq_iff xt1(10))
   130   done
   131 
   132 declare nat_gcd.simps [simp del]
   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 \<bar>x\<bar>) (nat \<bar>y\<bar>))"
   139 
   140 definition int_lcm :: "int \<Rightarrow> int \<Rightarrow> int" where
   141   "int_lcm x y = int (nat_lcm (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
   142 
   143 lemma gcd_eq_nitpick_gcd [nitpick_unfold]: "gcd x y = Nitpick.nat_gcd x y"
   144   by (induct x y rule: nat_gcd.induct)
   145     (simp add: gcd_nat.simps Nitpick.nat_gcd.simps)
   146 
   147 lemma lcm_eq_nitpick_lcm [nitpick_unfold]: "lcm x y = Nitpick.nat_lcm x y"
   148   by (simp only: lcm_nat_def Nitpick.nat_lcm_def gcd_eq_nitpick_gcd)
   149 
   150 definition Frac :: "int \<times> int \<Rightarrow> bool" where
   151   "Frac \<equiv> \<lambda>(a, b). b > 0 \<and> int_gcd a b = 1"
   152 
   153 consts
   154   Abs_Frac :: "int \<times> int \<Rightarrow> 'a"
   155   Rep_Frac :: "'a \<Rightarrow> int \<times> int"
   156 
   157 definition zero_frac :: 'a where
   158   "zero_frac \<equiv> Abs_Frac (0, 1)"
   159 
   160 definition one_frac :: 'a where
   161   "one_frac \<equiv> Abs_Frac (1, 1)"
   162 
   163 definition num :: "'a \<Rightarrow> int" where
   164   "num \<equiv> fst o Rep_Frac"
   165 
   166 definition denom :: "'a \<Rightarrow> int" where
   167   "denom \<equiv> snd o Rep_Frac"
   168 
   169 function norm_frac :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
   170   "norm_frac a b =
   171     (if b < 0 then norm_frac (- a) (- b)
   172      else if a = 0 \<or> b = 0 then (0, 1)
   173      else let c = int_gcd a b in (a div c, b div c))"
   174   by pat_completeness auto
   175   termination by (relation "measure (\<lambda>(_, b). if b < 0 then 1 else 0)") auto
   176 
   177 declare norm_frac.simps[simp del]
   178 
   179 definition frac :: "int \<Rightarrow> int \<Rightarrow> 'a" where
   180   "frac a b \<equiv> Abs_Frac (norm_frac a b)"
   181 
   182 definition plus_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   183   [nitpick_simp]: "plus_frac q r = (let d = int_lcm (denom q) (denom r) in
   184     frac (num q * (d div denom q) + num r * (d div denom r)) d)"
   185 
   186 definition times_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   187   [nitpick_simp]: "times_frac q r = frac (num q * num r) (denom q * denom r)"
   188 
   189 definition uminus_frac :: "'a \<Rightarrow> 'a" where
   190   "uminus_frac q \<equiv> Abs_Frac (- num q, denom q)"
   191 
   192 definition number_of_frac :: "int \<Rightarrow> 'a" where
   193   "number_of_frac n \<equiv> Abs_Frac (n, 1)"
   194 
   195 definition inverse_frac :: "'a \<Rightarrow> 'a" where
   196   "inverse_frac q \<equiv> frac (denom q) (num q)"
   197 
   198 definition less_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
   199   [nitpick_simp]: "less_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) < 0"
   200 
   201 definition less_eq_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
   202   [nitpick_simp]: "less_eq_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) \<le> 0"
   203 
   204 definition of_frac :: "'a \<Rightarrow> 'b::{inverse,ring_1}" where
   205   "of_frac q \<equiv> of_int (num q) / of_int (denom q)"
   206 
   207 axiomatization wf_wfrec :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
   208 
   209 definition wf_wfrec' :: "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
   210   [nitpick_simp]: "wf_wfrec' R F x = F (cut (wf_wfrec R F) R x) x"
   211 
   212 definition wfrec' ::  "('a \<times> 'a) set \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
   213   "wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x else THE y. wfrec_rel R (\<lambda>f x. F (cut f R x) x) x y"
   214 
   215 ML_file "Tools/Nitpick/kodkod.ML"
   216 ML_file "Tools/Nitpick/kodkod_sat.ML"
   217 ML_file "Tools/Nitpick/nitpick_util.ML"
   218 ML_file "Tools/Nitpick/nitpick_hol.ML"
   219 ML_file "Tools/Nitpick/nitpick_mono.ML"
   220 ML_file "Tools/Nitpick/nitpick_preproc.ML"
   221 ML_file "Tools/Nitpick/nitpick_scope.ML"
   222 ML_file "Tools/Nitpick/nitpick_peephole.ML"
   223 ML_file "Tools/Nitpick/nitpick_rep.ML"
   224 ML_file "Tools/Nitpick/nitpick_nut.ML"
   225 ML_file "Tools/Nitpick/nitpick_kodkod.ML"
   226 ML_file "Tools/Nitpick/nitpick_model.ML"
   227 ML_file "Tools/Nitpick/nitpick.ML"
   228 ML_file "Tools/Nitpick/nitpick_commands.ML"
   229 ML_file "Tools/Nitpick/nitpick_tests.ML"
   230 
   231 setup \<open>
   232   Nitpick_HOL.register_ersatz_global
   233     [(@{const_name card}, @{const_name card'}),
   234      (@{const_name sum}, @{const_name sum'}),
   235      (@{const_name fold_graph}, @{const_name fold_graph'}),
   236      (@{const_name wf}, @{const_name wf'}),
   237      (@{const_name wf_wfrec}, @{const_name wf_wfrec'}),
   238      (@{const_name wfrec}, @{const_name wfrec'})]
   239 \<close>
   240 
   241 hide_const (open) unknown is_unknown bisim bisim_iterator_max Quot safe_The FunBox PairBox Word prod
   242   refl' wf' card' sum' fold_graph' nat_gcd nat_lcm int_gcd int_lcm Frac Abs_Frac Rep_Frac
   243   zero_frac one_frac num denom norm_frac frac plus_frac times_frac uminus_frac number_of_frac
   244   inverse_frac less_frac less_eq_frac of_frac wf_wfrec wf_wfrec wfrec'
   245 
   246 hide_type (open) bisim_iterator fun_box pair_box unsigned_bit signed_bit word
   247 
   248 hide_fact (open) Ex1_unfold rtrancl_unfold rtranclp_unfold tranclp_unfold prod_def refl'_def wf'_def
   249   card'_def sum'_def The_psimp Eps_psimp case_unit_unfold case_nat_unfold
   250   size_list_simp nat_lcm_def int_gcd_def int_lcm_def Frac_def zero_frac_def one_frac_def
   251   num_def denom_def frac_def plus_frac_def times_frac_def uminus_frac_def
   252   number_of_frac_def inverse_frac_def less_frac_def less_eq_frac_def of_frac_def wf_wfrec'_def
   253   wfrec'_def
   254 
   255 end