src/HOL/Bali/Basis.thy
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     1 (*  Title:      isabelle/Bali/Basis.thy
       
     2     ID:         $Id$
       
     3     Author:     David von Oheimb
       
     4     Copyright   1997 Technische Universitaet Muenchen
       
     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 quick_and_dirty:=true;
       
    16 
       
    17 Pretty.setmargin 77;
       
    18 goals_limit:=2;
       
    19 *}
       
    20 (*print_depth 100;*)
       
    21 (*Syntax.ambiguity_level := 1;*)
       
    22 
       
    23 section "misc"
       
    24 
       
    25 declare same_fstI [intro!] (*### TO HOL/Wellfounded_Relations *)
       
    26 
       
    27 (* ###TO HOL/???.ML?? *)
       
    28 ML {*
       
    29 fun make_simproc name pat pred thm = Simplifier.mk_simproc name
       
    30    [Thm.read_cterm (Thm.sign_of_thm thm) (pat, HOLogic.typeT)] 
       
    31    (K (K (fn s => if pred s then None else Some (standard (mk_meta_eq thm)))))
       
    32 *}
       
    33 
       
    34 declare split_if_asm  [split] option.split [split] option.split_asm [split]
       
    35 ML {*
       
    36 simpset_ref() := simpset() addloop ("split_all_tac", split_all_tac)
       
    37 *}
       
    38 declare if_weak_cong [cong del] option.weak_case_cong [cong del]
       
    39 declare length_Suc_conv [iff];
       
    40 
       
    41 (*###to be phased out *)
       
    42 ML {*
       
    43 bind_thm ("make_imp", rearrange_prems [1,0] mp)
       
    44 *}
       
    45 
       
    46 lemma Collect_split_eq: "{p. P (split f p)} = {(a,b). P (f a b)}"
       
    47 apply auto
       
    48 done
       
    49 
       
    50 lemma subset_insertD: 
       
    51   "A <= insert x B ==> A <= B & x ~: A | (EX B'. A = insert x B' & B' <= B)"
       
    52 apply (case_tac "x:A")
       
    53 apply (rule disjI2)
       
    54 apply (rule_tac x = "A-{x}" in exI)
       
    55 apply fast+
       
    56 done
       
    57 
       
    58 syntax
       
    59   "3" :: nat   ("3")
       
    60   "4" :: nat   ("4")
       
    61 translations
       
    62  "3" == "Suc 2"
       
    63  "4" == "Suc 3"
       
    64 
       
    65 (*unused*)
       
    66 lemma range_bool_domain: "range f = {f True, f False}"
       
    67 apply auto
       
    68 apply (case_tac "xa")
       
    69 apply auto
       
    70 done
       
    71 
       
    72 (* context (theory "Transitive_Closure") *)
       
    73 lemma irrefl_tranclI': "r^-1 Int r^+ = {} ==> !x. (x, x) ~: r^+"
       
    74 apply (rule allI)
       
    75 apply (erule irrefl_tranclI)
       
    76 done
       
    77 
       
    78 lemma trancl_rtrancl_trancl:
       
    79 "\<lbrakk>(x,y)\<in>r^+; (y,z)\<in>r^*\<rbrakk> \<Longrightarrow> (x,z)\<in>r^+"
       
    80 by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)
       
    81 
       
    82 lemma rtrancl_into_trancl3:
       
    83 "\<lbrakk>(a,b)\<in>r^*; a\<noteq>b\<rbrakk> \<Longrightarrow> (a,b)\<in>r^+"
       
    84 apply (drule rtranclD)
       
    85 apply auto
       
    86 done
       
    87 
       
    88 lemma rtrancl_into_rtrancl2: 
       
    89   "\<lbrakk> (a, b) \<in>  r; (b, c) \<in> r^* \<rbrakk> \<Longrightarrow> (a, c) \<in>  r^*"
       
    90 by (auto intro: r_into_rtrancl rtrancl_trans)
       
    91 
       
    92 lemma triangle_lemma:
       
    93  "\<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> 
       
    94  \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
       
    95 proof -
       
    96   note converse_rtrancl_induct = converse_rtrancl_induct [consumes 1]
       
    97   note converse_rtranclE = converse_rtranclE [consumes 1] 
       
    98   assume unique: "\<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c"
       
    99   assume "(a,x)\<in>r\<^sup>*" 
       
   100   then show "(a,y)\<in>r\<^sup>* \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
       
   101   proof (induct rule: converse_rtrancl_induct)
       
   102     assume "(x,y)\<in>r\<^sup>*"
       
   103     then show ?thesis 
       
   104       by blast
       
   105   next
       
   106     fix a v
       
   107     assume a_v_r: "(a, v) \<in> r" and
       
   108           v_x_rt: "(v, x) \<in> r\<^sup>*" and
       
   109           a_y_rt: "(a, y) \<in> r\<^sup>*"  and
       
   110              hyp: "(v, y) \<in> r\<^sup>* \<Longrightarrow> (x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
       
   111     from a_y_rt 
       
   112     show "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
       
   113     proof (cases rule: converse_rtranclE)
       
   114       assume "a=y"
       
   115       with a_v_r v_x_rt have "(y,x) \<in> r\<^sup>*"
       
   116 	by (auto intro: r_into_rtrancl rtrancl_trans)
       
   117       then show ?thesis 
       
   118 	by blast
       
   119     next
       
   120       fix w 
       
   121       assume a_w_r: "(a, w) \<in> r" and
       
   122             w_y_rt: "(w, y) \<in> r\<^sup>*"
       
   123       from a_v_r a_w_r unique 
       
   124       have "v=w" 
       
   125 	by auto
       
   126       with w_y_rt hyp 
       
   127       show ?thesis
       
   128 	by blast
       
   129     qed
       
   130   qed
       
   131 qed
       
   132 
       
   133 
       
   134 lemma rtrancl_cases [consumes 1, case_names Refl Trancl]:
       
   135  "\<lbrakk>(a,b)\<in>r\<^sup>*;  a = b \<Longrightarrow> P; (a,b)\<in>r\<^sup>+ \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
       
   136 apply (erule rtranclE)
       
   137 apply (auto dest: rtrancl_into_trancl1)
       
   138 done
       
   139 
       
   140 (* ### To Transitive_Closure *)
       
   141 theorems converse_rtrancl_induct 
       
   142  = converse_rtrancl_induct [consumes 1,case_names Id Step]
       
   143 
       
   144 theorems converse_trancl_induct 
       
   145          = converse_trancl_induct [consumes 1,case_names Single Step]
       
   146 
       
   147 (* context (theory "Set") *)
       
   148 lemma Ball_weaken:"\<lbrakk>Ball s P;\<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"
       
   149 by auto
       
   150 
       
   151 (* context (theory "Finite") *)
       
   152 lemma finite_SetCompr2: "[| finite (Collect P); !y. P y --> finite (range (f y)) |] ==>  
       
   153   finite {f y x |x y. P y}"
       
   154 apply (subgoal_tac "{f y x |x y. P y} = UNION (Collect P) (%y. range (f y))")
       
   155 prefer 2 apply  fast
       
   156 apply (erule ssubst)
       
   157 apply (erule finite_UN_I)
       
   158 apply fast
       
   159 done
       
   160 
       
   161 
       
   162 (* ### TO theory "List" *)
       
   163 lemma list_all2_trans: "\<forall> a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>
       
   164  \<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> list_all2 P3 xs1 xs3"
       
   165 apply (induct_tac "xs1")
       
   166 apply simp
       
   167 apply (rule allI)
       
   168 apply (induct_tac "xs2")
       
   169 apply simp
       
   170 apply (rule allI)
       
   171 apply (induct_tac "xs3")
       
   172 apply auto
       
   173 done
       
   174 
       
   175 
       
   176 section "pairs"
       
   177 
       
   178 lemma surjective_pairing5: "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))), 
       
   179   snd (snd (snd (snd p))))"
       
   180 apply auto
       
   181 done
       
   182 
       
   183 lemma fst_splitE [elim!]: 
       
   184 "[| fst s' = x';  !!x s. [| s' = (x,s);  x = x' |] ==> Q |] ==> Q"
       
   185 apply (cut_tac p = "s'" in surjective_pairing)
       
   186 apply auto
       
   187 done
       
   188 
       
   189 lemma fst_in_set_lemma [rule_format (no_asm)]: "(x, y) : set l --> x : fst ` set l"
       
   190 apply (induct_tac "l")
       
   191 apply  auto
       
   192 done
       
   193 
       
   194 
       
   195 section "quantifiers"
       
   196 
       
   197 (*###to be phased out *)
       
   198 ML {* 
       
   199 fun noAll_simpset () = simpset() setmksimps 
       
   200 	mksimps (filter (fn (x,_) => x<>"All") mksimps_pairs)
       
   201 *}
       
   202 
       
   203 lemma All_Ex_refl_eq2 [simp]: 
       
   204  "(!x. (? b. x = f b & Q b) \<longrightarrow> P x) = (!b. Q b --> P (f b))"
       
   205 apply auto
       
   206 done
       
   207 
       
   208 lemma ex_ex_miniscope1 [simp]:
       
   209   "(EX w v. P w v & Q v) = (EX v. (EX w. P w v) & Q v)"
       
   210 apply auto
       
   211 done
       
   212 
       
   213 lemma ex_miniscope2 [simp]:
       
   214   "(EX v. P v & Q & R v) = (Q & (EX v. P v & R v))" 
       
   215 apply auto
       
   216 done
       
   217 
       
   218 lemma ex_reorder31: "(\<exists>z x y. P x y z) = (\<exists>x y z. P x y z)"
       
   219 apply auto
       
   220 done
       
   221 
       
   222 lemma All_Ex_refl_eq1 [simp]: "(!x. (? b. x = f b) --> P x) = (!b. P (f b))"
       
   223 apply auto
       
   224 done
       
   225 
       
   226 
       
   227 section "sums"
       
   228 
       
   229 hide const In0 In1
       
   230 
       
   231 syntax
       
   232   fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
       
   233 translations
       
   234  "fun_sum" == "sum_case"
       
   235 
       
   236 consts    the_Inl  :: "'a + 'b \<Rightarrow> 'a"
       
   237           the_Inr  :: "'a + 'b \<Rightarrow> 'b"
       
   238 primrec  "the_Inl (Inl a) = a"
       
   239 primrec  "the_Inr (Inr b) = b"
       
   240 
       
   241 datatype ('a, 'b, 'c) sum3 = In1 'a | In2 'b | In3 'c
       
   242 
       
   243 consts    the_In1  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"
       
   244           the_In2  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"
       
   245           the_In3  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"
       
   246 primrec  "the_In1 (In1 a) = a"
       
   247 primrec  "the_In2 (In2 b) = b"
       
   248 primrec  "the_In3 (In3 c) = c"
       
   249 
       
   250 syntax
       
   251 	 In1l	:: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
       
   252 	 In1r	:: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
       
   253 translations
       
   254 	"In1l e" == "In1 (Inl e)"
       
   255 	"In1r c" == "In1 (Inr c)"
       
   256 
       
   257 ML {*
       
   258 fun sum3_instantiate thm = map (fn s => simplify(simpset()delsimps[not_None_eq])
       
   259  (read_instantiate [("t","In"^s^" ?x")] thm)) ["1l","2","3","1r"]
       
   260 *}
       
   261 (* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)
       
   262 
       
   263 translations
       
   264   "option"<= (type) "Option.option"
       
   265   "list"  <= (type) "List.list"
       
   266   "sum3"  <= (type) "Basis.sum3"
       
   267 
       
   268 
       
   269 section "quantifiers for option type"
       
   270 
       
   271 syntax
       
   272   Oall :: "[pttrn, 'a option, bool] => bool"   ("(3! _:_:/ _)" [0,0,10] 10)
       
   273   Oex  :: "[pttrn, 'a option, bool] => bool"   ("(3? _:_:/ _)" [0,0,10] 10)
       
   274 
       
   275 syntax (symbols)
       
   276   Oall :: "[pttrn, 'a option, bool] => bool"   ("(3\<forall>_\<in>_:/ _)"  [0,0,10] 10)
       
   277   Oex  :: "[pttrn, 'a option, bool] => bool"   ("(3\<exists>_\<in>_:/ _)"  [0,0,10] 10)
       
   278 
       
   279 translations
       
   280   "! x:A: P"    == "! x:o2s A. P"
       
   281   "? x:A: P"    == "? x:o2s A. P"
       
   282 
       
   283 
       
   284 section "unique association lists"
       
   285 
       
   286 constdefs
       
   287   unique   :: "('a \<times> 'b) list \<Rightarrow> bool"
       
   288  "unique \<equiv> nodups \<circ> map fst"
       
   289 
       
   290 lemma uniqueD [rule_format (no_asm)]: 
       
   291 "unique l--> (!x y. (x,y):set l --> (!x' y'. (x',y'):set l --> x=x'-->  y=y'))"
       
   292 apply (unfold unique_def o_def)
       
   293 apply (induct_tac "l")
       
   294 apply  (auto dest: fst_in_set_lemma)
       
   295 done
       
   296 
       
   297 lemma unique_Nil [simp]: "unique []"
       
   298 apply (unfold unique_def)
       
   299 apply (simp (no_asm))
       
   300 done
       
   301 
       
   302 lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l & (!y. (x,y) ~: set l))"
       
   303 apply (unfold unique_def)
       
   304 apply  (auto dest: fst_in_set_lemma)
       
   305 done
       
   306 
       
   307 lemmas unique_ConsI = conjI [THEN unique_Cons [THEN iffD2], standard]
       
   308 
       
   309 lemma unique_single [simp]: "!!p. unique [p]"
       
   310 apply auto
       
   311 done
       
   312 
       
   313 lemma unique_ConsD: "unique (x#xs) ==> unique xs"
       
   314 apply (simp add: unique_def)
       
   315 done
       
   316 
       
   317 lemma unique_append [rule_format (no_asm)]: "unique l' ==> unique l -->  
       
   318   (!(x,y):set l. !(x',y'):set l'. x' ~= x) --> unique (l @ l')"
       
   319 apply (induct_tac "l")
       
   320 apply  (auto dest: fst_in_set_lemma)
       
   321 done
       
   322 
       
   323 lemma unique_map_inj [rule_format (no_asm)]: "unique l --> inj f --> unique (map (%(k,x). (f k, g k x)) l)"
       
   324 apply (induct_tac "l")
       
   325 apply  (auto dest: fst_in_set_lemma simp add: inj_eq)
       
   326 done
       
   327 
       
   328 lemma map_of_SomeI [rule_format (no_asm)]: "unique l --> (k, x) : set l --> map_of l k = Some x"
       
   329 apply (induct_tac "l")
       
   330 apply auto
       
   331 done
       
   332 
       
   333 
       
   334 section "list patterns"
       
   335 
       
   336 consts
       
   337   lsplit         :: "[['a, 'a list] => 'b, 'a list] => 'b"
       
   338 defs
       
   339   lsplit_def:    "lsplit == %f l. f (hd l) (tl l)"
       
   340 (*  list patterns -- extends pre-defined type "pttrn" used in abstractions *)
       
   341 syntax
       
   342   "_lpttrn"    :: "[pttrn,pttrn] => pttrn"     ("_#/_" [901,900] 900)
       
   343 translations
       
   344   "%y#x#xs. b"  == "lsplit (%y x#xs. b)"
       
   345   "%x#xs  . b"  == "lsplit (%x xs  . b)"
       
   346 
       
   347 lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"
       
   348 apply (unfold lsplit_def)
       
   349 apply (simp (no_asm))
       
   350 done
       
   351 
       
   352 lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"
       
   353 apply (unfold lsplit_def)
       
   354 apply simp
       
   355 done 
       
   356 
       
   357 
       
   358 section "dummy pattern for quantifiers, let, etc."
       
   359 
       
   360 syntax
       
   361   "@dummy_pat"   :: pttrn    ("'_")
       
   362 
       
   363 parse_translation {*
       
   364 let fun dummy_pat_tr [] = Free ("_",dummyT)
       
   365   | dummy_pat_tr ts = raise TERM ("dummy_pat_tr", ts);
       
   366 in [("@dummy_pat", dummy_pat_tr)] 
       
   367 end
       
   368 *}
       
   369 
       
   370 end