src/HOL/Bali/Basis.thy
author berghofe
Mon Sep 30 16:14:02 2002 +0200 (2002-09-30)
changeset 13601 fd3e3d6b37b2
parent 13462 56610e2ba220
child 13688 a0b16d42d489
permissions -rw-r--r--
Adapted to new simplifier.
     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 ML {*
    23 fun cond_simproc name pat pred thm = Simplifier.simproc (Thm.sign_of_thm thm) name [pat]
    24   (fn _ => fn _ => fn t => if pred t then None else Some (mk_meta_eq thm));
    25 *}
    26 
    27 declare split_if_asm  [split] option.split [split] option.split_asm [split]
    28 ML {*
    29 simpset_ref() := simpset() addloop ("split_all_tac", split_all_tac)
    30 *}
    31 declare if_weak_cong [cong del] option.weak_case_cong [cong del]
    32 declare length_Suc_conv [iff];
    33 
    34 (*###to be phased out *)
    35 ML {*
    36 bind_thm ("make_imp", rearrange_prems [1,0] mp)
    37 *}
    38 
    39 lemma Collect_split_eq: "{p. P (split f p)} = {(a,b). P (f a b)}"
    40 apply auto
    41 done
    42 
    43 lemma subset_insertD: 
    44   "A <= insert x B ==> A <= B & x ~: A | (EX B'. A = insert x B' & B' <= B)"
    45 apply (case_tac "x:A")
    46 apply (rule disjI2)
    47 apply (rule_tac x = "A-{x}" in exI)
    48 apply fast+
    49 done
    50 
    51 syntax
    52   "3" :: nat   ("3") 
    53   "4" :: nat   ("4")
    54 translations
    55  "3" == "Suc 2"
    56  "4" == "Suc 3"
    57 
    58 (*unused*)
    59 lemma range_bool_domain: "range f = {f True, f False}"
    60 apply auto
    61 apply (case_tac "xa")
    62 apply auto
    63 done
    64 
    65 (* context (theory "Transitive_Closure") *)
    66 lemma irrefl_tranclI': "r^-1 Int r^+ = {} ==> !x. (x, x) ~: r^+"
    67 apply (rule allI)
    68 apply (erule irrefl_tranclI)
    69 done
    70 
    71 lemma trancl_rtrancl_trancl:
    72 "\<lbrakk>(x,y)\<in>r^+; (y,z)\<in>r^*\<rbrakk> \<Longrightarrow> (x,z)\<in>r^+"
    73 by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)
    74 
    75 lemma rtrancl_into_trancl3:
    76 "\<lbrakk>(a,b)\<in>r^*; a\<noteq>b\<rbrakk> \<Longrightarrow> (a,b)\<in>r^+" 
    77 apply (drule rtranclD)
    78 apply auto
    79 done
    80 
    81 lemma rtrancl_into_rtrancl2: 
    82   "\<lbrakk> (a, b) \<in>  r; (b, c) \<in> r^* \<rbrakk> \<Longrightarrow> (a, c) \<in>  r^*"
    83 by (auto intro: r_into_rtrancl rtrancl_trans)
    84 
    85 lemma triangle_lemma:
    86  "\<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> 
    87  \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
    88 proof -
    89   note converse_rtrancl_induct = converse_rtrancl_induct [consumes 1]
    90   note converse_rtranclE = converse_rtranclE [consumes 1] 
    91   assume unique: "\<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c"
    92   assume "(a,x)\<in>r\<^sup>*" 
    93   then show "(a,y)\<in>r\<^sup>* \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
    94   proof (induct rule: converse_rtrancl_induct)
    95     assume "(x,y)\<in>r\<^sup>*"
    96     then show ?thesis 
    97       by blast
    98   next
    99     fix a v
   100     assume a_v_r: "(a, v) \<in> r" and
   101           v_x_rt: "(v, x) \<in> r\<^sup>*" and
   102           a_y_rt: "(a, y) \<in> r\<^sup>*"  and
   103              hyp: "(v, y) \<in> r\<^sup>* \<Longrightarrow> (x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
   104     from a_y_rt 
   105     show "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
   106     proof (cases rule: converse_rtranclE)
   107       assume "a=y"
   108       with a_v_r v_x_rt have "(y,x) \<in> r\<^sup>*"
   109 	by (auto intro: r_into_rtrancl rtrancl_trans)
   110       then show ?thesis 
   111 	by blast
   112     next
   113       fix w 
   114       assume a_w_r: "(a, w) \<in> r" and
   115             w_y_rt: "(w, y) \<in> r\<^sup>*"
   116       from a_v_r a_w_r unique 
   117       have "v=w" 
   118 	by auto
   119       with w_y_rt hyp 
   120       show ?thesis
   121 	by blast
   122     qed
   123   qed
   124 qed
   125 
   126 
   127 lemma rtrancl_cases [consumes 1, case_names Refl Trancl]:
   128  "\<lbrakk>(a,b)\<in>r\<^sup>*;  a = b \<Longrightarrow> P; (a,b)\<in>r\<^sup>+ \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
   129 apply (erule rtranclE)
   130 apply (auto dest: rtrancl_into_trancl1)
   131 done
   132 
   133 (* ### To Transitive_Closure *)
   134 theorems converse_rtrancl_induct 
   135  = converse_rtrancl_induct [consumes 1,case_names Id Step]
   136 
   137 theorems converse_trancl_induct 
   138          = converse_trancl_induct [consumes 1,case_names Single Step]
   139 
   140 (* context (theory "Set") *)
   141 lemma Ball_weaken:"\<lbrakk>Ball s P;\<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"
   142 by auto
   143 
   144 (* context (theory "Finite") *)
   145 lemma finite_SetCompr2: "[| finite (Collect P); !y. P y --> finite (range (f y)) |] ==>  
   146   finite {f y x |x y. P y}"
   147 apply (subgoal_tac "{f y x |x y. P y} = UNION (Collect P) (%y. range (f y))")
   148 prefer 2 apply  fast
   149 apply (erule ssubst)
   150 apply (erule finite_UN_I)
   151 apply fast
   152 done
   153 
   154 
   155 (* ### TO theory "List" *)
   156 lemma list_all2_trans: "\<forall> a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>
   157  \<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> list_all2 P3 xs1 xs3"
   158 apply (induct_tac "xs1")
   159 apply simp
   160 apply (rule allI)
   161 apply (induct_tac "xs2")
   162 apply simp
   163 apply (rule allI)
   164 apply (induct_tac "xs3")
   165 apply auto
   166 done
   167 
   168 
   169 section "pairs"
   170 
   171 lemma surjective_pairing5: "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))), 
   172   snd (snd (snd (snd p))))"
   173 apply auto
   174 done
   175 
   176 lemma fst_splitE [elim!]: 
   177 "[| fst s' = x';  !!x s. [| s' = (x,s);  x = x' |] ==> Q |] ==> Q"
   178 apply (cut_tac p = "s'" in surjective_pairing)
   179 apply auto
   180 done
   181 
   182 lemma fst_in_set_lemma [rule_format (no_asm)]: "(x, y) : set l --> x : fst ` set l"
   183 apply (induct_tac "l")
   184 apply  auto
   185 done
   186 
   187 
   188 section "quantifiers"
   189 
   190 (*###to be phased out *)
   191 ML {* 
   192 fun noAll_simpset () = simpset() setmksimps 
   193 	mksimps (filter (fn (x,_) => x<>"All") mksimps_pairs)
   194 *}
   195 
   196 lemma All_Ex_refl_eq2 [simp]: 
   197  "(!x. (? b. x = f b & Q b) \<longrightarrow> P x) = (!b. Q b --> P (f b))"
   198 apply auto
   199 done
   200 
   201 lemma ex_ex_miniscope1 [simp]:
   202   "(EX w v. P w v & Q v) = (EX v. (EX w. P w v) & Q v)"
   203 apply auto
   204 done
   205 
   206 lemma ex_miniscope2 [simp]:
   207   "(EX v. P v & Q & R v) = (Q & (EX v. P v & R v))" 
   208 apply auto
   209 done
   210 
   211 lemma ex_reorder31: "(\<exists>z x y. P x y z) = (\<exists>x y z. P x y z)"
   212 apply auto
   213 done
   214 
   215 lemma All_Ex_refl_eq1 [simp]: "(!x. (? b. x = f b) --> P x) = (!b. P (f b))"
   216 apply auto
   217 done
   218 
   219 
   220 section "sums"
   221 
   222 hide const In0 In1
   223 
   224 syntax
   225   fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
   226 translations
   227  "fun_sum" == "sum_case"
   228 
   229 consts    the_Inl  :: "'a + 'b \<Rightarrow> 'a"
   230           the_Inr  :: "'a + 'b \<Rightarrow> 'b"
   231 primrec  "the_Inl (Inl a) = a"
   232 primrec  "the_Inr (Inr b) = b"
   233 
   234 datatype ('a, 'b, 'c) sum3 = In1 'a | In2 'b | In3 'c
   235 
   236 consts    the_In1  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"
   237           the_In2  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"
   238           the_In3  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"
   239 primrec  "the_In1 (In1 a) = a"
   240 primrec  "the_In2 (In2 b) = b"
   241 primrec  "the_In3 (In3 c) = c"
   242 
   243 syntax
   244 	 In1l	:: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
   245 	 In1r	:: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
   246 translations
   247 	"In1l e" == "In1 (Inl e)"
   248 	"In1r c" == "In1 (Inr c)"
   249 
   250 ML {*
   251 fun sum3_instantiate thm = map (fn s => simplify(simpset()delsimps[not_None_eq])
   252  (read_instantiate [("t","In"^s^" ?x")] thm)) ["1l","2","3","1r"]
   253 *}
   254 (* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)
   255 
   256 translations
   257   "option"<= (type) "Datatype.option"
   258   "list"  <= (type) "List.list"
   259   "sum3"  <= (type) "Basis.sum3"
   260 
   261 
   262 section "quantifiers for option type"
   263 
   264 syntax
   265   Oall :: "[pttrn, 'a option, bool] => bool"   ("(3! _:_:/ _)" [0,0,10] 10)
   266   Oex  :: "[pttrn, 'a option, bool] => bool"   ("(3? _:_:/ _)" [0,0,10] 10)
   267 
   268 syntax (symbols)
   269   Oall :: "[pttrn, 'a option, bool] => bool"   ("(3\<forall>_\<in>_:/ _)"  [0,0,10] 10)
   270   Oex  :: "[pttrn, 'a option, bool] => bool"   ("(3\<exists>_\<in>_:/ _)"  [0,0,10] 10)
   271 
   272 translations
   273   "! x:A: P"    == "! x:o2s A. P"
   274   "? x:A: P"    == "? x:o2s A. P"
   275 
   276 
   277 section "unique association lists"
   278 
   279 constdefs
   280   unique   :: "('a \<times> 'b) list \<Rightarrow> bool"
   281  "unique \<equiv> distinct \<circ> map fst"
   282 
   283 lemma uniqueD [rule_format (no_asm)]: 
   284 "unique l--> (!x y. (x,y):set l --> (!x' y'. (x',y'):set l --> x=x'-->  y=y'))"
   285 apply (unfold unique_def o_def)
   286 apply (induct_tac "l")
   287 apply  (auto dest: fst_in_set_lemma)
   288 done
   289 
   290 lemma unique_Nil [simp]: "unique []"
   291 apply (unfold unique_def)
   292 apply (simp (no_asm))
   293 done
   294 
   295 lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l & (!y. (x,y) ~: set l))"
   296 apply (unfold unique_def)
   297 apply  (auto dest: fst_in_set_lemma)
   298 done
   299 
   300 lemmas unique_ConsI = conjI [THEN unique_Cons [THEN iffD2], standard]
   301 
   302 lemma unique_single [simp]: "!!p. unique [p]"
   303 apply auto
   304 done
   305 
   306 lemma unique_ConsD: "unique (x#xs) ==> unique xs"
   307 apply (simp add: unique_def)
   308 done
   309 
   310 lemma unique_append [rule_format (no_asm)]: "unique l' ==> unique l -->  
   311   (!(x,y):set l. !(x',y'):set l'. x' ~= x) --> unique (l @ l')"
   312 apply (induct_tac "l")
   313 apply  (auto dest: fst_in_set_lemma)
   314 done
   315 
   316 lemma unique_map_inj [rule_format (no_asm)]: "unique l --> inj f --> unique (map (%(k,x). (f k, g k x)) l)"
   317 apply (induct_tac "l")
   318 apply  (auto dest: fst_in_set_lemma simp add: inj_eq)
   319 done
   320 
   321 lemma map_of_SomeI [rule_format (no_asm)]: "unique l --> (k, x) : set l --> map_of l k = Some x"
   322 apply (induct_tac "l")
   323 apply auto
   324 done
   325 
   326 
   327 section "list patterns"
   328 
   329 consts
   330   lsplit         :: "[['a, 'a list] => 'b, 'a list] => 'b"
   331 defs
   332   lsplit_def:    "lsplit == %f l. f (hd l) (tl l)"
   333 (*  list patterns -- extends pre-defined type "pttrn" used in abstractions *)
   334 syntax
   335   "_lpttrn"    :: "[pttrn,pttrn] => pttrn"     ("_#/_" [901,900] 900)
   336 translations
   337   "%y#x#xs. b"  == "lsplit (%y x#xs. b)"
   338   "%x#xs  . b"  == "lsplit (%x xs  . b)"
   339 
   340 lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"
   341 apply (unfold lsplit_def)
   342 apply (simp (no_asm))
   343 done
   344 
   345 lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"
   346 apply (unfold lsplit_def)
   347 apply simp
   348 done 
   349 
   350 
   351 section "dummy pattern for quantifiers, let, etc."
   352 
   353 syntax
   354   "@dummy_pat"   :: pttrn    ("'_")
   355 
   356 parse_translation {*
   357 let fun dummy_pat_tr [] = Free ("_",dummyT)
   358   | dummy_pat_tr ts = raise TERM ("dummy_pat_tr", ts);
   359 in [("@dummy_pat", dummy_pat_tr)] 
   360 end
   361 *}
   362 
   363 end