src/HOL/Bali/Basis.thy
author schirmer
Fri Feb 22 11:26:44 2002 +0100 (2002-02-22)
changeset 12925 99131847fb93
parent 12919 d6a0d168291e
child 13462 56610e2ba220
permissions -rw-r--r--
Added check for field/method access to operational semantics and proved the acesses valid.
     1 (*  Title:      HOL/Bali/Basis.thy
     2     ID:         $Id$
     3     Author:     David von Oheimb
     4     License:    GPL (GNU GENERAL PUBLIC LICENSE)
     5 
     6 *)
     7 header {* Definitions extending HOL as logical basis of Bali *}
     8 
     9 theory Basis = Main:
    10 
    11 ML_setup {*
    12 Unify.search_bound := 40;
    13 Unify.trace_bound  := 40;
    14 *}
    15 (*print_depth 100;*)
    16 (*Syntax.ambiguity_level := 1;*)
    17 
    18 section "misc"
    19 
    20 declare same_fstI [intro!] (*### TO HOL/Wellfounded_Relations *)
    21 
    22 (* ###TO HOL/???.ML?? *)
    23 ML {*
    24 fun make_simproc name pat pred thm = Simplifier.mk_simproc name
    25    [Thm.read_cterm (Thm.sign_of_thm thm) (pat, HOLogic.typeT)] 
    26    (K (K (fn s => if pred s then None else Some (standard (mk_meta_eq thm)))))
    27 *}
    28 
    29 declare split_if_asm  [split] option.split [split] option.split_asm [split]
    30 ML {*
    31 simpset_ref() := simpset() addloop ("split_all_tac", split_all_tac)
    32 *}
    33 declare if_weak_cong [cong del] option.weak_case_cong [cong del]
    34 declare length_Suc_conv [iff];
    35 
    36 (*###to be phased out *)
    37 ML {*
    38 bind_thm ("make_imp", rearrange_prems [1,0] mp)
    39 *}
    40 
    41 lemma Collect_split_eq: "{p. P (split f p)} = {(a,b). P (f a b)}"
    42 apply auto
    43 done
    44 
    45 lemma subset_insertD: 
    46   "A <= insert x B ==> A <= B & x ~: A | (EX B'. A = insert x B' & B' <= B)"
    47 apply (case_tac "x:A")
    48 apply (rule disjI2)
    49 apply (rule_tac x = "A-{x}" in exI)
    50 apply fast+
    51 done
    52 
    53 syntax
    54   "3" :: nat   ("3") 
    55   "4" :: nat   ("4")
    56 translations
    57  "3" == "Suc 2"
    58  "4" == "Suc 3"
    59 
    60 (*unused*)
    61 lemma range_bool_domain: "range f = {f True, f False}"
    62 apply auto
    63 apply (case_tac "xa")
    64 apply auto
    65 done
    66 
    67 (* context (theory "Transitive_Closure") *)
    68 lemma irrefl_tranclI': "r^-1 Int r^+ = {} ==> !x. (x, x) ~: r^+"
    69 apply (rule allI)
    70 apply (erule irrefl_tranclI)
    71 done
    72 
    73 lemma trancl_rtrancl_trancl:
    74 "\<lbrakk>(x,y)\<in>r^+; (y,z)\<in>r^*\<rbrakk> \<Longrightarrow> (x,z)\<in>r^+"
    75 by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)
    76 
    77 lemma rtrancl_into_trancl3:
    78 "\<lbrakk>(a,b)\<in>r^*; a\<noteq>b\<rbrakk> \<Longrightarrow> (a,b)\<in>r^+" 
    79 apply (drule rtranclD)
    80 apply auto
    81 done
    82 
    83 lemma rtrancl_into_rtrancl2: 
    84   "\<lbrakk> (a, b) \<in>  r; (b, c) \<in> r^* \<rbrakk> \<Longrightarrow> (a, c) \<in>  r^*"
    85 by (auto intro: r_into_rtrancl rtrancl_trans)
    86 
    87 lemma triangle_lemma:
    88  "\<lbrakk> \<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c; (a,x)\<in>r\<^sup>*; (a,y)\<in>r\<^sup>*\<rbrakk> 
    89  \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
    90 proof -
    91   note converse_rtrancl_induct = converse_rtrancl_induct [consumes 1]
    92   note converse_rtranclE = converse_rtranclE [consumes 1] 
    93   assume unique: "\<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c"
    94   assume "(a,x)\<in>r\<^sup>*" 
    95   then show "(a,y)\<in>r\<^sup>* \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
    96   proof (induct rule: converse_rtrancl_induct)
    97     assume "(x,y)\<in>r\<^sup>*"
    98     then show ?thesis 
    99       by blast
   100   next
   101     fix a v
   102     assume a_v_r: "(a, v) \<in> r" and
   103           v_x_rt: "(v, x) \<in> r\<^sup>*" and
   104           a_y_rt: "(a, y) \<in> r\<^sup>*"  and
   105              hyp: "(v, y) \<in> r\<^sup>* \<Longrightarrow> (x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
   106     from a_y_rt 
   107     show "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
   108     proof (cases rule: converse_rtranclE)
   109       assume "a=y"
   110       with a_v_r v_x_rt have "(y,x) \<in> r\<^sup>*"
   111 	by (auto intro: r_into_rtrancl rtrancl_trans)
   112       then show ?thesis 
   113 	by blast
   114     next
   115       fix w 
   116       assume a_w_r: "(a, w) \<in> r" and
   117             w_y_rt: "(w, y) \<in> r\<^sup>*"
   118       from a_v_r a_w_r unique 
   119       have "v=w" 
   120 	by auto
   121       with w_y_rt hyp 
   122       show ?thesis
   123 	by blast
   124     qed
   125   qed
   126 qed
   127 
   128 
   129 lemma rtrancl_cases [consumes 1, case_names Refl Trancl]:
   130  "\<lbrakk>(a,b)\<in>r\<^sup>*;  a = b \<Longrightarrow> P; (a,b)\<in>r\<^sup>+ \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
   131 apply (erule rtranclE)
   132 apply (auto dest: rtrancl_into_trancl1)
   133 done
   134 
   135 (* ### To Transitive_Closure *)
   136 theorems converse_rtrancl_induct 
   137  = converse_rtrancl_induct [consumes 1,case_names Id Step]
   138 
   139 theorems converse_trancl_induct 
   140          = converse_trancl_induct [consumes 1,case_names Single Step]
   141 
   142 (* context (theory "Set") *)
   143 lemma Ball_weaken:"\<lbrakk>Ball s P;\<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"
   144 by auto
   145 
   146 (* context (theory "Finite") *)
   147 lemma finite_SetCompr2: "[| finite (Collect P); !y. P y --> finite (range (f y)) |] ==>  
   148   finite {f y x |x y. P y}"
   149 apply (subgoal_tac "{f y x |x y. P y} = UNION (Collect P) (%y. range (f y))")
   150 prefer 2 apply  fast
   151 apply (erule ssubst)
   152 apply (erule finite_UN_I)
   153 apply fast
   154 done
   155 
   156 
   157 (* ### TO theory "List" *)
   158 lemma list_all2_trans: "\<forall> a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>
   159  \<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> list_all2 P3 xs1 xs3"
   160 apply (induct_tac "xs1")
   161 apply simp
   162 apply (rule allI)
   163 apply (induct_tac "xs2")
   164 apply simp
   165 apply (rule allI)
   166 apply (induct_tac "xs3")
   167 apply auto
   168 done
   169 
   170 
   171 section "pairs"
   172 
   173 lemma surjective_pairing5: "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))), 
   174   snd (snd (snd (snd p))))"
   175 apply auto
   176 done
   177 
   178 lemma fst_splitE [elim!]: 
   179 "[| fst s' = x';  !!x s. [| s' = (x,s);  x = x' |] ==> Q |] ==> Q"
   180 apply (cut_tac p = "s'" in surjective_pairing)
   181 apply auto
   182 done
   183 
   184 lemma fst_in_set_lemma [rule_format (no_asm)]: "(x, y) : set l --> x : fst ` set l"
   185 apply (induct_tac "l")
   186 apply  auto
   187 done
   188 
   189 
   190 section "quantifiers"
   191 
   192 (*###to be phased out *)
   193 ML {* 
   194 fun noAll_simpset () = simpset() setmksimps 
   195 	mksimps (filter (fn (x,_) => x<>"All") mksimps_pairs)
   196 *}
   197 
   198 lemma All_Ex_refl_eq2 [simp]: 
   199  "(!x. (? b. x = f b & Q b) \<longrightarrow> P x) = (!b. Q b --> P (f b))"
   200 apply auto
   201 done
   202 
   203 lemma ex_ex_miniscope1 [simp]:
   204   "(EX w v. P w v & Q v) = (EX v. (EX w. P w v) & Q v)"
   205 apply auto
   206 done
   207 
   208 lemma ex_miniscope2 [simp]:
   209   "(EX v. P v & Q & R v) = (Q & (EX v. P v & R v))" 
   210 apply auto
   211 done
   212 
   213 lemma ex_reorder31: "(\<exists>z x y. P x y z) = (\<exists>x y z. P x y z)"
   214 apply auto
   215 done
   216 
   217 lemma All_Ex_refl_eq1 [simp]: "(!x. (? b. x = f b) --> P x) = (!b. P (f b))"
   218 apply auto
   219 done
   220 
   221 
   222 section "sums"
   223 
   224 hide const In0 In1
   225 
   226 syntax
   227   fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
   228 translations
   229  "fun_sum" == "sum_case"
   230 
   231 consts    the_Inl  :: "'a + 'b \<Rightarrow> 'a"
   232           the_Inr  :: "'a + 'b \<Rightarrow> 'b"
   233 primrec  "the_Inl (Inl a) = a"
   234 primrec  "the_Inr (Inr b) = b"
   235 
   236 datatype ('a, 'b, 'c) sum3 = In1 'a | In2 'b | In3 'c
   237 
   238 consts    the_In1  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"
   239           the_In2  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"
   240           the_In3  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"
   241 primrec  "the_In1 (In1 a) = a"
   242 primrec  "the_In2 (In2 b) = b"
   243 primrec  "the_In3 (In3 c) = c"
   244 
   245 syntax
   246 	 In1l	:: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
   247 	 In1r	:: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
   248 translations
   249 	"In1l e" == "In1 (Inl e)"
   250 	"In1r c" == "In1 (Inr c)"
   251 
   252 ML {*
   253 fun sum3_instantiate thm = map (fn s => simplify(simpset()delsimps[not_None_eq])
   254  (read_instantiate [("t","In"^s^" ?x")] thm)) ["1l","2","3","1r"]
   255 *}
   256 (* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)
   257 
   258 translations
   259   "option"<= (type) "Datatype.option"
   260   "list"  <= (type) "List.list"
   261   "sum3"  <= (type) "Basis.sum3"
   262 
   263 
   264 section "quantifiers for option type"
   265 
   266 syntax
   267   Oall :: "[pttrn, 'a option, bool] => bool"   ("(3! _:_:/ _)" [0,0,10] 10)
   268   Oex  :: "[pttrn, 'a option, bool] => bool"   ("(3? _:_:/ _)" [0,0,10] 10)
   269 
   270 syntax (symbols)
   271   Oall :: "[pttrn, 'a option, bool] => bool"   ("(3\<forall>_\<in>_:/ _)"  [0,0,10] 10)
   272   Oex  :: "[pttrn, 'a option, bool] => bool"   ("(3\<exists>_\<in>_:/ _)"  [0,0,10] 10)
   273 
   274 translations
   275   "! x:A: P"    == "! x:o2s A. P"
   276   "? x:A: P"    == "? x:o2s A. P"
   277 
   278 
   279 section "unique association lists"
   280 
   281 constdefs
   282   unique   :: "('a \<times> 'b) list \<Rightarrow> bool"
   283  "unique \<equiv> distinct \<circ> map fst"
   284 
   285 lemma uniqueD [rule_format (no_asm)]: 
   286 "unique l--> (!x y. (x,y):set l --> (!x' y'. (x',y'):set l --> x=x'-->  y=y'))"
   287 apply (unfold unique_def o_def)
   288 apply (induct_tac "l")
   289 apply  (auto dest: fst_in_set_lemma)
   290 done
   291 
   292 lemma unique_Nil [simp]: "unique []"
   293 apply (unfold unique_def)
   294 apply (simp (no_asm))
   295 done
   296 
   297 lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l & (!y. (x,y) ~: set l))"
   298 apply (unfold unique_def)
   299 apply  (auto dest: fst_in_set_lemma)
   300 done
   301 
   302 lemmas unique_ConsI = conjI [THEN unique_Cons [THEN iffD2], standard]
   303 
   304 lemma unique_single [simp]: "!!p. unique [p]"
   305 apply auto
   306 done
   307 
   308 lemma unique_ConsD: "unique (x#xs) ==> unique xs"
   309 apply (simp add: unique_def)
   310 done
   311 
   312 lemma unique_append [rule_format (no_asm)]: "unique l' ==> unique l -->  
   313   (!(x,y):set l. !(x',y'):set l'. x' ~= x) --> unique (l @ l')"
   314 apply (induct_tac "l")
   315 apply  (auto dest: fst_in_set_lemma)
   316 done
   317 
   318 lemma unique_map_inj [rule_format (no_asm)]: "unique l --> inj f --> unique (map (%(k,x). (f k, g k x)) l)"
   319 apply (induct_tac "l")
   320 apply  (auto dest: fst_in_set_lemma simp add: inj_eq)
   321 done
   322 
   323 lemma map_of_SomeI [rule_format (no_asm)]: "unique l --> (k, x) : set l --> map_of l k = Some x"
   324 apply (induct_tac "l")
   325 apply auto
   326 done
   327 
   328 
   329 section "list patterns"
   330 
   331 consts
   332   lsplit         :: "[['a, 'a list] => 'b, 'a list] => 'b"
   333 defs
   334   lsplit_def:    "lsplit == %f l. f (hd l) (tl l)"
   335 (*  list patterns -- extends pre-defined type "pttrn" used in abstractions *)
   336 syntax
   337   "_lpttrn"    :: "[pttrn,pttrn] => pttrn"     ("_#/_" [901,900] 900)
   338 translations
   339   "%y#x#xs. b"  == "lsplit (%y x#xs. b)"
   340   "%x#xs  . b"  == "lsplit (%x xs  . b)"
   341 
   342 lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"
   343 apply (unfold lsplit_def)
   344 apply (simp (no_asm))
   345 done
   346 
   347 lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"
   348 apply (unfold lsplit_def)
   349 apply simp
   350 done 
   351 
   352 
   353 section "dummy pattern for quantifiers, let, etc."
   354 
   355 syntax
   356   "@dummy_pat"   :: pttrn    ("'_")
   357 
   358 parse_translation {*
   359 let fun dummy_pat_tr [] = Free ("_",dummyT)
   360   | dummy_pat_tr ts = raise TERM ("dummy_pat_tr", ts);
   361 in [("@dummy_pat", dummy_pat_tr)] 
   362 end
   363 *}
   364 
   365 end