src/HOL/Nitpick.thy
changeset 33192 08a39a957ed7
child 33235 cbe96b3cb3d0
child 33556 cba22e2999d5
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/HOL/Nitpick.thy	Thu Oct 22 14:51:47 2009 +0200
     1.3 @@ -0,0 +1,240 @@
     1.4 +(*  Title:      HOL/Nitpick.thy
     1.5 +    Author:     Jasmin Blanchette, TU Muenchen
     1.6 +    Copyright   2008, 2009
     1.7 +
     1.8 +Nitpick: Yet another counterexample generator for Isabelle/HOL.
     1.9 +*)
    1.10 +
    1.11 +header {* Nitpick: Yet Another Counterexample Generator for Isabelle/HOL *}
    1.12 +
    1.13 +theory Nitpick
    1.14 +imports Map SAT
    1.15 +uses ("Tools/Nitpick/kodkod.ML")
    1.16 +     ("Tools/Nitpick/kodkod_sat.ML")
    1.17 +     ("Tools/Nitpick/nitpick_util.ML")
    1.18 +     ("Tools/Nitpick/nitpick_hol.ML")
    1.19 +     ("Tools/Nitpick/nitpick_mono.ML")
    1.20 +     ("Tools/Nitpick/nitpick_scope.ML")
    1.21 +     ("Tools/Nitpick/nitpick_peephole.ML")
    1.22 +     ("Tools/Nitpick/nitpick_rep.ML")
    1.23 +     ("Tools/Nitpick/nitpick_nut.ML")
    1.24 +     ("Tools/Nitpick/nitpick_kodkod.ML")
    1.25 +     ("Tools/Nitpick/nitpick_model.ML")
    1.26 +     ("Tools/Nitpick/nitpick.ML")
    1.27 +     ("Tools/Nitpick/nitpick_isar.ML")
    1.28 +     ("Tools/Nitpick/nitpick_tests.ML")
    1.29 +     ("Tools/Nitpick/minipick.ML")
    1.30 +begin
    1.31 +
    1.32 +typedecl bisim_iterator
    1.33 +
    1.34 +(* FIXME: use axiomatization (here and elsewhere) *)
    1.35 +axiomatization unknown :: 'a
    1.36 +           and undefined_fast_The :: 'a
    1.37 +           and undefined_fast_Eps :: 'a
    1.38 +           and bisim :: "bisim_iterator \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
    1.39 +           and bisim_iterator_max :: bisim_iterator
    1.40 +           and Tha :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
    1.41 +
    1.42 +datatype ('a, 'b) pair_box = PairBox 'a 'b
    1.43 +datatype ('a, 'b) fun_box = FunBox "'a \<Rightarrow> 'b"
    1.44 +
    1.45 +text {*
    1.46 +Alternative definitions.
    1.47 +*}
    1.48 +
    1.49 +lemma If_def [nitpick_def]:
    1.50 +"(if P then Q else R) \<equiv> (P \<longrightarrow> Q) \<and> (\<not> P \<longrightarrow> R)"
    1.51 +by (rule eq_reflection) (rule if_bool_eq_conj)
    1.52 +
    1.53 +lemma Ex1_def [nitpick_def]:
    1.54 +"Ex1 P \<equiv> \<exists>x. P = {x}"
    1.55 +apply (rule eq_reflection)
    1.56 +apply (simp add: Ex1_def expand_set_eq)
    1.57 +apply (rule iffI)
    1.58 + apply (erule exE)
    1.59 + apply (erule conjE)
    1.60 + apply (rule_tac x = x in exI)
    1.61 + apply (rule allI)
    1.62 + apply (rename_tac y)
    1.63 + apply (erule_tac x = y in allE)
    1.64 +by (auto simp: mem_def)
    1.65 +
    1.66 +lemma rtrancl_def [nitpick_def]: "r\<^sup>* \<equiv> (r\<^sup>+)\<^sup>="
    1.67 +by simp
    1.68 +
    1.69 +lemma rtranclp_def [nitpick_def]:
    1.70 +"rtranclp r a b \<equiv> (a = b \<or> tranclp r a b)"
    1.71 +by (rule eq_reflection) (auto dest: rtranclpD)
    1.72 +
    1.73 +lemma tranclp_def [nitpick_def]:
    1.74 +"tranclp r a b \<equiv> trancl (split r) (a, b)"
    1.75 +by (simp add: trancl_def Collect_def mem_def)
    1.76 +
    1.77 +definition refl' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
    1.78 +"refl' r \<equiv> \<forall>x. (x, x) \<in> r"
    1.79 +
    1.80 +definition wf' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> bool" where
    1.81 +"wf' r \<equiv> acyclic r \<and> (finite r \<or> unknown)"
    1.82 +
    1.83 +axiomatization wf_wfrec :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
    1.84 +
    1.85 +definition wf_wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
    1.86 +[nitpick_simp]: "wf_wfrec' R F x = F (Recdef.cut (wf_wfrec R F) R x) x"
    1.87 +
    1.88 +definition wfrec' ::  "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where
    1.89 +"wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x
    1.90 +                else THE y. wfrec_rel R (%f x. F (Recdef.cut f R x) x) x y"
    1.91 +
    1.92 +definition card' :: "('a \<Rightarrow> bool) \<Rightarrow> nat" where
    1.93 +"card' X \<equiv> length (SOME xs. set xs = X \<and> distinct xs)"
    1.94 +
    1.95 +definition setsum' :: "('a \<Rightarrow> 'b\<Colon>comm_monoid_add) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b" where
    1.96 +"setsum' f A \<equiv> if finite A then listsum (map f (SOME xs. set xs = A \<and> distinct xs)) else 0"
    1.97 +
    1.98 +inductive fold_graph' :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool" where
    1.99 +"fold_graph' f z {} z" |
   1.100 +"\<lbrakk>x \<in> A; fold_graph' f z (A - {x}) y\<rbrakk> \<Longrightarrow> fold_graph' f z A (f x y)"
   1.101 +
   1.102 +text {*
   1.103 +The following lemmas are not strictly necessary but they help the
   1.104 +\textit{special\_level} optimization.
   1.105 +*}
   1.106 +
   1.107 +lemma The_psimp [nitpick_psimp]:
   1.108 +"P = {x} \<Longrightarrow> The P = x"
   1.109 +by (subgoal_tac "{x} = (\<lambda>y. y = x)") (auto simp: mem_def)
   1.110 +
   1.111 +lemma Eps_psimp [nitpick_psimp]:
   1.112 +"\<lbrakk>P x; \<not> P y; Eps P = y\<rbrakk> \<Longrightarrow> Eps P = x"
   1.113 +apply (case_tac "P (Eps P)")
   1.114 + apply auto
   1.115 +apply (erule contrapos_np)
   1.116 +by (rule someI)
   1.117 +
   1.118 +lemma unit_case_def [nitpick_def]:
   1.119 +"unit_case x u \<equiv> x"
   1.120 +apply (subgoal_tac "u = ()")
   1.121 + apply (simp only: unit.cases)
   1.122 +by simp
   1.123 +
   1.124 +lemma nat_case_def [nitpick_def]:
   1.125 +"nat_case x f n \<equiv> if n = 0 then x else f (n - 1)"
   1.126 +apply (rule eq_reflection)
   1.127 +by (case_tac n) auto
   1.128 +
   1.129 +lemmas dvd_def = dvd_eq_mod_eq_0 [THEN eq_reflection, nitpick_def]
   1.130 +
   1.131 +lemma list_size_simp [nitpick_simp]:
   1.132 +"list_size f xs = (if xs = [] then 0
   1.133 +                   else Suc (f (hd xs) + list_size f (tl xs)))"
   1.134 +"size xs = (if xs = [] then 0 else Suc (size (tl xs)))"
   1.135 +by (case_tac xs) auto
   1.136 +
   1.137 +text {*
   1.138 +Auxiliary definitions used to provide an alternative representation for
   1.139 +@{text rat} and @{text real}.
   1.140 +*}
   1.141 +
   1.142 +function nat_gcd :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
   1.143 +[simp del]: "nat_gcd x y = (if y = 0 then x else nat_gcd y (x mod y))"
   1.144 +by auto
   1.145 +termination
   1.146 +apply (relation "measure (\<lambda>(x, y). x + y + (if y > x then 1 else 0))")
   1.147 + apply auto
   1.148 + apply (metis mod_less_divisor xt1(9))
   1.149 +by (metis mod_mod_trivial mod_self nat_neq_iff xt1(10))
   1.150 +
   1.151 +definition nat_lcm :: "nat \<Rightarrow> nat \<Rightarrow> nat" where
   1.152 +"nat_lcm x y = x * y div (nat_gcd x y)"
   1.153 +
   1.154 +definition int_gcd :: "int \<Rightarrow> int \<Rightarrow> int" where
   1.155 +"int_gcd x y = int (nat_gcd (nat (abs x)) (nat (abs y)))"
   1.156 +
   1.157 +definition int_lcm :: "int \<Rightarrow> int \<Rightarrow> int" where
   1.158 +"int_lcm x y = int (nat_lcm (nat (abs x)) (nat (abs y)))"
   1.159 +
   1.160 +definition Frac :: "int \<times> int \<Rightarrow> bool" where
   1.161 +"Frac \<equiv> \<lambda>(a, b). b > 0 \<and> int_gcd a b = 1"
   1.162 +
   1.163 +axiomatization Abs_Frac :: "int \<times> int \<Rightarrow> 'a"
   1.164 +           and Rep_Frac :: "'a \<Rightarrow> int \<times> int"
   1.165 +
   1.166 +definition zero_frac :: 'a where
   1.167 +"zero_frac \<equiv> Abs_Frac (0, 1)"
   1.168 +
   1.169 +definition one_frac :: 'a where
   1.170 +"one_frac \<equiv> Abs_Frac (1, 1)"
   1.171 +
   1.172 +definition num :: "'a \<Rightarrow> int" where
   1.173 +"num \<equiv> fst o Rep_Frac"
   1.174 +
   1.175 +definition denom :: "'a \<Rightarrow> int" where
   1.176 +"denom \<equiv> snd o Rep_Frac"
   1.177 +
   1.178 +function norm_frac :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
   1.179 +[simp del]: "norm_frac a b = (if b < 0 then norm_frac (- a) (- b)
   1.180 +                              else if a = 0 \<or> b = 0 then (0, 1)
   1.181 +                              else let c = int_gcd a b in (a div c, b div c))"
   1.182 +by pat_completeness auto
   1.183 +termination by (relation "measure (\<lambda>(_, b). if b < 0 then 1 else 0)") auto
   1.184 +
   1.185 +definition frac :: "int \<Rightarrow> int \<Rightarrow> 'a" where
   1.186 +"frac a b \<equiv> Abs_Frac (norm_frac a b)"
   1.187 +
   1.188 +definition plus_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   1.189 +[nitpick_simp]:
   1.190 +"plus_frac q r = (let d = int_lcm (denom q) (denom r) in
   1.191 +                    frac (num q * (d div denom q) + num r * (d div denom r)) d)"
   1.192 +
   1.193 +definition times_frac :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
   1.194 +[nitpick_simp]:
   1.195 +"times_frac q r = frac (num q * num r) (denom q * denom r)"
   1.196 +
   1.197 +definition uminus_frac :: "'a \<Rightarrow> 'a" where
   1.198 +"uminus_frac q \<equiv> Abs_Frac (- num q, denom q)"
   1.199 +
   1.200 +definition number_of_frac :: "int \<Rightarrow> 'a" where
   1.201 +"number_of_frac n \<equiv> Abs_Frac (n, 1)"
   1.202 +
   1.203 +definition inverse_frac :: "'a \<Rightarrow> 'a" where
   1.204 +"inverse_frac q \<equiv> frac (denom q) (num q)"
   1.205 +
   1.206 +definition less_eq_frac :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
   1.207 +[nitpick_simp]:
   1.208 +"less_eq_frac q r \<longleftrightarrow> num (plus_frac q (uminus_frac r)) \<le> 0"
   1.209 +
   1.210 +definition of_frac :: "'a \<Rightarrow> 'b\<Colon>{inverse,ring_1}" where
   1.211 +"of_frac q \<equiv> of_int (num q) / of_int (denom q)"
   1.212 +
   1.213 +use "Tools/Nitpick/kodkod.ML"
   1.214 +use "Tools/Nitpick/kodkod_sat.ML"
   1.215 +use "Tools/Nitpick/nitpick_util.ML"
   1.216 +use "Tools/Nitpick/nitpick_hol.ML"
   1.217 +use "Tools/Nitpick/nitpick_mono.ML"
   1.218 +use "Tools/Nitpick/nitpick_scope.ML"
   1.219 +use "Tools/Nitpick/nitpick_peephole.ML"
   1.220 +use "Tools/Nitpick/nitpick_rep.ML"
   1.221 +use "Tools/Nitpick/nitpick_nut.ML"
   1.222 +use "Tools/Nitpick/nitpick_kodkod.ML"
   1.223 +use "Tools/Nitpick/nitpick_model.ML"
   1.224 +use "Tools/Nitpick/nitpick.ML"
   1.225 +use "Tools/Nitpick/nitpick_isar.ML"
   1.226 +use "Tools/Nitpick/nitpick_tests.ML"
   1.227 +use "Tools/Nitpick/minipick.ML"
   1.228 +
   1.229 +hide (open) const unknown undefined_fast_The undefined_fast_Eps bisim 
   1.230 +    bisim_iterator_max Tha refl' wf' wf_wfrec wf_wfrec' wfrec' card' setsum'
   1.231 +    fold_graph' nat_gcd nat_lcm int_gcd int_lcm Frac Abs_Frac Rep_Frac zero_frac
   1.232 +    one_frac num denom norm_frac frac plus_frac times_frac uminus_frac
   1.233 +    number_of_frac inverse_frac less_eq_frac of_frac
   1.234 +hide (open) type bisim_iterator pair_box fun_box
   1.235 +hide (open) fact If_def Ex1_def rtrancl_def rtranclp_def tranclp_def refl'_def
   1.236 +    wf'_def wf_wfrec'_def wfrec'_def card'_def setsum'_def fold_graph'_def
   1.237 +    The_psimp Eps_psimp unit_case_def nat_case_def dvd_def list_size_simp
   1.238 +    nat_gcd_def nat_lcm_def int_gcd_def int_lcm_def Frac_def zero_frac_def
   1.239 +    one_frac_def num_def denom_def norm_frac_def frac_def plus_frac_def
   1.240 +    times_frac_def uminus_frac_def number_of_frac_def inverse_frac_def
   1.241 +    less_eq_frac_def of_frac_def
   1.242 +
   1.243 +end