src/HOL/Bali/Basis.thy
author berghofe
Tue Apr 24 15:17:22 2007 +0200 (2007-04-24)
changeset 22781 18fbba942a80
parent 22633 a47e4fd7ebc1
child 24019 67bde7cfcf10
permissions -rw-r--r--
sum_case is now authentic.
     1 (*  Title:      HOL/Bali/Basis.thy
     2     ID:         $Id$
     3     Author:     David von Oheimb
     4 
     5 *)
     6 header {* Definitions extending HOL as logical basis of Bali *}
     7 
     8 theory Basis imports Main begin
     9 
    10 ML {*
    11 Unify.search_bound := 40;
    12 Unify.trace_bound  := 40;
    13 *}
    14 (*print_depth 100;*)
    15 (*Syntax.ambiguity_level := 1;*)
    16 
    17 section "misc"
    18 
    19 declare same_fstI [intro!] (*### TO HOL/Wellfounded_Relations *)
    20 
    21 ML {*
    22 fun cond_simproc name pat pred thm = Simplifier.simproc (Thm.theory_of_thm thm) name [pat]
    23   (fn _ => fn _ => fn t => if pred t then NONE else SOME (mk_meta_eq thm));
    24 *}
    25 
    26 declare split_if_asm  [split] option.split [split] option.split_asm [split]
    27 ML {*
    28 change_simpset (fn ss => ss addloop ("split_all_tac", split_all_tac));
    29 *}
    30 declare if_weak_cong [cong del] option.weak_case_cong [cong del]
    31 declare length_Suc_conv [iff]
    32 
    33 (*###to be phased out *)
    34 ML {*
    35 bind_thm ("make_imp", rearrange_prems [1,0] mp)
    36 *}
    37 
    38 lemma Collect_split_eq: "{p. P (split f p)} = {(a,b). P (f a b)}"
    39 apply auto
    40 done
    41 
    42 lemma subset_insertD: 
    43   "A <= insert x B ==> A <= B & x ~: A | (EX B'. A = insert x B' & B' <= B)"
    44 apply (case_tac "x:A")
    45 apply (rule disjI2)
    46 apply (rule_tac x = "A-{x}" in exI)
    47 apply fast+
    48 done
    49 
    50 syntax
    51   "3" :: nat   ("3") 
    52   "4" :: nat   ("4")
    53 translations
    54  "3" == "Suc 2"
    55  "4" == "Suc 3"
    56 
    57 (*unused*)
    58 lemma range_bool_domain: "range f = {f True, f False}"
    59 apply auto
    60 apply (case_tac "xa")
    61 apply auto
    62 done
    63 
    64 (* irrefl_tranclI in Transitive_Closure.thy is more general *)
    65 lemma irrefl_tranclI': "r^-1 Int r^+ = {} ==> !x. (x, x) ~: r^+"
    66 by(blast elim: tranclE dest: trancl_into_rtrancl)
    67 
    68 
    69 lemma trancl_rtrancl_trancl:
    70 "\<lbrakk>(x,y)\<in>r^+; (y,z)\<in>r^*\<rbrakk> \<Longrightarrow> (x,z)\<in>r^+"
    71 by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)
    72 
    73 lemma rtrancl_into_trancl3:
    74 "\<lbrakk>(a,b)\<in>r^*; a\<noteq>b\<rbrakk> \<Longrightarrow> (a,b)\<in>r^+" 
    75 apply (drule rtranclD)
    76 apply auto
    77 done
    78 
    79 lemma rtrancl_into_rtrancl2: 
    80   "\<lbrakk> (a, b) \<in>  r; (b, c) \<in> r^* \<rbrakk> \<Longrightarrow> (a, c) \<in>  r^*"
    81 by (auto intro: r_into_rtrancl rtrancl_trans)
    82 
    83 lemma triangle_lemma:
    84  "\<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> 
    85  \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
    86 proof -
    87   note converse_rtrancl_induct = converse_rtrancl_induct [consumes 1]
    88   note converse_rtranclE = converse_rtranclE [consumes 1] 
    89   assume unique: "\<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c"
    90   assume "(a,x)\<in>r\<^sup>*" 
    91   then show "(a,y)\<in>r\<^sup>* \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
    92   proof (induct rule: converse_rtrancl_induct)
    93     assume "(x,y)\<in>r\<^sup>*"
    94     then show ?thesis 
    95       by blast
    96   next
    97     fix a v
    98     assume a_v_r: "(a, v) \<in> r" and
    99           v_x_rt: "(v, x) \<in> r\<^sup>*" and
   100           a_y_rt: "(a, y) \<in> r\<^sup>*"  and
   101              hyp: "(v, y) \<in> r\<^sup>* \<Longrightarrow> (x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
   102     from a_y_rt 
   103     show "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
   104     proof (cases rule: converse_rtranclE)
   105       assume "a=y"
   106       with a_v_r v_x_rt have "(y,x) \<in> r\<^sup>*"
   107 	by (auto intro: r_into_rtrancl rtrancl_trans)
   108       then show ?thesis 
   109 	by blast
   110     next
   111       fix w 
   112       assume a_w_r: "(a, w) \<in> r" and
   113             w_y_rt: "(w, y) \<in> r\<^sup>*"
   114       from a_v_r a_w_r unique 
   115       have "v=w" 
   116 	by auto
   117       with w_y_rt hyp 
   118       show ?thesis
   119 	by blast
   120     qed
   121   qed
   122 qed
   123 
   124 
   125 lemma rtrancl_cases [consumes 1, case_names Refl Trancl]:
   126  "\<lbrakk>(a,b)\<in>r\<^sup>*;  a = b \<Longrightarrow> P; (a,b)\<in>r\<^sup>+ \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
   127 apply (erule rtranclE)
   128 apply (auto dest: rtrancl_into_trancl1)
   129 done
   130 
   131 (* ### To Transitive_Closure *)
   132 theorems converse_rtrancl_induct 
   133  = converse_rtrancl_induct [consumes 1,case_names Id Step]
   134 
   135 theorems converse_trancl_induct 
   136          = converse_trancl_induct [consumes 1,case_names Single Step]
   137 
   138 (* context (theory "Set") *)
   139 lemma Ball_weaken:"\<lbrakk>Ball s P;\<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"
   140 by auto
   141 
   142 (* context (theory "Finite") *)
   143 lemma finite_SetCompr2: "[| finite (Collect P); !y. P y --> finite (range (f y)) |] ==>  
   144   finite {f y x |x y. P y}"
   145 apply (subgoal_tac "{f y x |x y. P y} = UNION (Collect P) (%y. range (f y))")
   146 prefer 2 apply  fast
   147 apply (erule ssubst)
   148 apply (erule finite_UN_I)
   149 apply fast
   150 done
   151 
   152 
   153 (* ### TO theory "List" *)
   154 lemma list_all2_trans: "\<forall> a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>
   155  \<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> list_all2 P3 xs1 xs3"
   156 apply (induct_tac "xs1")
   157 apply simp
   158 apply (rule allI)
   159 apply (induct_tac "xs2")
   160 apply simp
   161 apply (rule allI)
   162 apply (induct_tac "xs3")
   163 apply auto
   164 done
   165 
   166 
   167 section "pairs"
   168 
   169 lemma surjective_pairing5: "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))), 
   170   snd (snd (snd (snd p))))"
   171 apply auto
   172 done
   173 
   174 lemma fst_splitE [elim!]: 
   175 "[| fst s' = x';  !!x s. [| s' = (x,s);  x = x' |] ==> Q |] ==> Q"
   176 apply (cut_tac p = "s'" in surjective_pairing)
   177 apply auto
   178 done
   179 
   180 lemma fst_in_set_lemma [rule_format (no_asm)]: "(x, y) : set l --> x : fst ` set l"
   181 apply (induct_tac "l")
   182 apply  auto
   183 done
   184 
   185 
   186 section "quantifiers"
   187 
   188 lemma All_Ex_refl_eq2 [simp]: 
   189  "(!x. (? b. x = f b & Q b) \<longrightarrow> P x) = (!b. Q b --> P (f b))"
   190 apply auto
   191 done
   192 
   193 lemma ex_ex_miniscope1 [simp]:
   194   "(EX w v. P w v & Q v) = (EX v. (EX w. P w v) & Q v)"
   195 apply auto
   196 done
   197 
   198 lemma ex_miniscope2 [simp]:
   199   "(EX v. P v & Q & R v) = (Q & (EX v. P v & R v))" 
   200 apply auto
   201 done
   202 
   203 lemma ex_reorder31: "(\<exists>z x y. P x y z) = (\<exists>x y z. P x y z)"
   204 apply auto
   205 done
   206 
   207 lemma All_Ex_refl_eq1 [simp]: "(!x. (? b. x = f b) --> P x) = (!b. P (f b))"
   208 apply auto
   209 done
   210 
   211 
   212 section "sums"
   213 
   214 hide const In0 In1
   215 
   216 syntax
   217   fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
   218 translations
   219  "fun_sum" == "CONST sum_case"
   220 
   221 consts    the_Inl  :: "'a + 'b \<Rightarrow> 'a"
   222           the_Inr  :: "'a + 'b \<Rightarrow> 'b"
   223 primrec  "the_Inl (Inl a) = a"
   224 primrec  "the_Inr (Inr b) = b"
   225 
   226 datatype ('a, 'b, 'c) sum3 = In1 'a | In2 'b | In3 'c
   227 
   228 consts    the_In1  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"
   229           the_In2  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"
   230           the_In3  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"
   231 primrec  "the_In1 (In1 a) = a"
   232 primrec  "the_In2 (In2 b) = b"
   233 primrec  "the_In3 (In3 c) = c"
   234 
   235 syntax
   236 	 In1l	:: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
   237 	 In1r	:: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
   238 translations
   239 	"In1l e" == "In1 (Inl e)"
   240 	"In1r c" == "In1 (Inr c)"
   241 
   242 syntax the_In1l :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'al"
   243        the_In1r :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'ar"
   244 translations
   245    "the_In1l" == "the_Inl \<circ> the_In1"
   246    "the_In1r" == "the_Inr \<circ> the_In1"
   247 
   248 ML {*
   249 fun sum3_instantiate thm = map (fn s => simplify(simpset()delsimps[@{thm not_None_eq}])
   250  (read_instantiate [("t","In"^s^" ?x")] thm)) ["1l","2","3","1r"]
   251 *}
   252 (* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)
   253 
   254 translations
   255   "option"<= (type) "Datatype.option"
   256   "list"  <= (type) "List.list"
   257   "sum3"  <= (type) "Basis.sum3"
   258 
   259 
   260 section "quantifiers for option type"
   261 
   262 syntax
   263   Oall :: "[pttrn, 'a option, bool] => bool"   ("(3! _:_:/ _)" [0,0,10] 10)
   264   Oex  :: "[pttrn, 'a option, bool] => bool"   ("(3? _:_:/ _)" [0,0,10] 10)
   265 
   266 syntax (symbols)
   267   Oall :: "[pttrn, 'a option, bool] => bool"   ("(3\<forall>_\<in>_:/ _)"  [0,0,10] 10)
   268   Oex  :: "[pttrn, 'a option, bool] => bool"   ("(3\<exists>_\<in>_:/ _)"  [0,0,10] 10)
   269 
   270 translations
   271   "! x:A: P"    == "! x:o2s A. P"
   272   "? x:A: P"    == "? x:o2s A. P"
   273 
   274 section "Special map update"
   275 
   276 text{* Deemed too special for theory Map. *}
   277 
   278 constdefs
   279   chg_map :: "('b => 'b) => 'a => ('a ~=> 'b) => ('a ~=> 'b)"
   280  "chg_map f a m == case m a of None => m | Some b => m(a|->f b)"
   281 
   282 lemma chg_map_new[simp]: "m a = None   ==> chg_map f a m = m"
   283 by (unfold chg_map_def, auto)
   284 
   285 lemma chg_map_upd[simp]: "m a = Some b ==> chg_map f a m = m(a|->f b)"
   286 by (unfold chg_map_def, auto)
   287 
   288 lemma chg_map_other [simp]: "a \<noteq> b \<Longrightarrow> chg_map f a m b = m b"
   289 by (auto simp: chg_map_def split add: option.split)
   290 
   291 
   292 section "unique association lists"
   293 
   294 constdefs
   295   unique   :: "('a \<times> 'b) list \<Rightarrow> bool"
   296  "unique \<equiv> distinct \<circ> map fst"
   297 
   298 lemma uniqueD [rule_format (no_asm)]: 
   299 "unique l--> (!x y. (x,y):set l --> (!x' y'. (x',y'):set l --> x=x'-->  y=y'))"
   300 apply (unfold unique_def o_def)
   301 apply (induct_tac "l")
   302 apply  (auto dest: fst_in_set_lemma)
   303 done
   304 
   305 lemma unique_Nil [simp]: "unique []"
   306 apply (unfold unique_def)
   307 apply (simp (no_asm))
   308 done
   309 
   310 lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l & (!y. (x,y) ~: set l))"
   311 apply (unfold unique_def)
   312 apply  (auto dest: fst_in_set_lemma)
   313 done
   314 
   315 lemmas unique_ConsI = conjI [THEN unique_Cons [THEN iffD2], standard]
   316 
   317 lemma unique_single [simp]: "!!p. unique [p]"
   318 apply auto
   319 done
   320 
   321 lemma unique_ConsD: "unique (x#xs) ==> unique xs"
   322 apply (simp add: unique_def)
   323 done
   324 
   325 lemma unique_append [rule_format (no_asm)]: "unique l' ==> unique l -->  
   326   (!(x,y):set l. !(x',y'):set l'. x' ~= x) --> unique (l @ l')"
   327 apply (induct_tac "l")
   328 apply  (auto dest: fst_in_set_lemma)
   329 done
   330 
   331 lemma unique_map_inj [rule_format (no_asm)]: "unique l --> inj f --> unique (map (%(k,x). (f k, g k x)) l)"
   332 apply (induct_tac "l")
   333 apply  (auto dest: fst_in_set_lemma simp add: inj_eq)
   334 done
   335 
   336 lemma map_of_SomeI [rule_format (no_asm)]: "unique l --> (k, x) : set l --> map_of l k = Some x"
   337 apply (induct_tac "l")
   338 apply auto
   339 done
   340 
   341 
   342 section "list patterns"
   343 
   344 consts
   345   lsplit         :: "[['a, 'a list] => 'b, 'a list] => 'b"
   346 defs
   347   lsplit_def:    "lsplit == %f l. f (hd l) (tl l)"
   348 (*  list patterns -- extends pre-defined type "pttrn" used in abstractions *)
   349 syntax
   350   "_lpttrn"    :: "[pttrn,pttrn] => pttrn"     ("_#/_" [901,900] 900)
   351 translations
   352   "%y#x#xs. b"  == "lsplit (%y x#xs. b)"
   353   "%x#xs  . b"  == "lsplit (%x xs  . b)"
   354 
   355 lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"
   356 apply (unfold lsplit_def)
   357 apply (simp (no_asm))
   358 done
   359 
   360 lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"
   361 apply (unfold lsplit_def)
   362 apply simp
   363 done 
   364 
   365 
   366 end