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.
wenzelm@12857
     1
(*  Title:      HOL/Bali/Basis.thy
schirmer@12854
     2
    ID:         $Id$
schirmer@12854
     3
    Author:     David von Oheimb
wenzelm@12858
     4
    License:    GPL (GNU GENERAL PUBLIC LICENSE)
schirmer@12854
     5
schirmer@12854
     6
*)
schirmer@12854
     7
header {* Definitions extending HOL as logical basis of Bali *}
schirmer@12854
     8
schirmer@12854
     9
theory Basis = Main:
schirmer@12854
    10
schirmer@12854
    11
ML_setup {*
schirmer@12854
    12
Unify.search_bound := 40;
schirmer@12854
    13
Unify.trace_bound  := 40;
schirmer@12854
    14
*}
schirmer@12854
    15
(*print_depth 100;*)
schirmer@12854
    16
(*Syntax.ambiguity_level := 1;*)
schirmer@12854
    17
schirmer@12854
    18
section "misc"
schirmer@12854
    19
schirmer@12854
    20
declare same_fstI [intro!] (*### TO HOL/Wellfounded_Relations *)
schirmer@12854
    21
schirmer@12854
    22
ML {*
wenzelm@13462
    23
fun cond_simproc name pat pred thm = Simplifier.simproc (Thm.sign_of_thm thm) name [pat]
wenzelm@13462
    24
  (fn _ => fn _ => fn t => if pred t then None else Some (mk_meta_eq thm));
schirmer@12854
    25
*}
schirmer@12854
    26
schirmer@12854
    27
declare split_if_asm  [split] option.split [split] option.split_asm [split]
schirmer@12854
    28
ML {*
schirmer@12854
    29
simpset_ref() := simpset() addloop ("split_all_tac", split_all_tac)
schirmer@12854
    30
*}
schirmer@12854
    31
declare if_weak_cong [cong del] option.weak_case_cong [cong del]
schirmer@12854
    32
declare length_Suc_conv [iff];
schirmer@12854
    33
schirmer@12854
    34
(*###to be phased out *)
schirmer@12854
    35
ML {*
schirmer@12854
    36
bind_thm ("make_imp", rearrange_prems [1,0] mp)
schirmer@12854
    37
*}
schirmer@12854
    38
schirmer@12854
    39
lemma Collect_split_eq: "{p. P (split f p)} = {(a,b). P (f a b)}"
schirmer@12854
    40
apply auto
schirmer@12854
    41
done
schirmer@12854
    42
schirmer@12854
    43
lemma subset_insertD: 
schirmer@12854
    44
  "A <= insert x B ==> A <= B & x ~: A | (EX B'. A = insert x B' & B' <= B)"
schirmer@12854
    45
apply (case_tac "x:A")
schirmer@12854
    46
apply (rule disjI2)
schirmer@12854
    47
apply (rule_tac x = "A-{x}" in exI)
schirmer@12854
    48
apply fast+
schirmer@12854
    49
done
schirmer@12854
    50
schirmer@12854
    51
syntax
schirmer@12925
    52
  "3" :: nat   ("3") 
schirmer@12854
    53
  "4" :: nat   ("4")
schirmer@12854
    54
translations
schirmer@12854
    55
 "3" == "Suc 2"
schirmer@12854
    56
 "4" == "Suc 3"
schirmer@12854
    57
schirmer@12854
    58
(*unused*)
schirmer@12854
    59
lemma range_bool_domain: "range f = {f True, f False}"
schirmer@12854
    60
apply auto
schirmer@12854
    61
apply (case_tac "xa")
schirmer@12854
    62
apply auto
schirmer@12854
    63
done
schirmer@12854
    64
schirmer@12854
    65
(* context (theory "Transitive_Closure") *)
schirmer@12854
    66
lemma irrefl_tranclI': "r^-1 Int r^+ = {} ==> !x. (x, x) ~: r^+"
schirmer@12854
    67
apply (rule allI)
schirmer@12854
    68
apply (erule irrefl_tranclI)
schirmer@12854
    69
done
schirmer@12854
    70
schirmer@12854
    71
lemma trancl_rtrancl_trancl:
schirmer@12854
    72
"\<lbrakk>(x,y)\<in>r^+; (y,z)\<in>r^*\<rbrakk> \<Longrightarrow> (x,z)\<in>r^+"
schirmer@12854
    73
by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)
schirmer@12854
    74
schirmer@12854
    75
lemma rtrancl_into_trancl3:
schirmer@12925
    76
"\<lbrakk>(a,b)\<in>r^*; a\<noteq>b\<rbrakk> \<Longrightarrow> (a,b)\<in>r^+" 
schirmer@12854
    77
apply (drule rtranclD)
schirmer@12854
    78
apply auto
schirmer@12854
    79
done
schirmer@12854
    80
schirmer@12854
    81
lemma rtrancl_into_rtrancl2: 
schirmer@12854
    82
  "\<lbrakk> (a, b) \<in>  r; (b, c) \<in> r^* \<rbrakk> \<Longrightarrow> (a, c) \<in>  r^*"
schirmer@12854
    83
by (auto intro: r_into_rtrancl rtrancl_trans)
schirmer@12854
    84
schirmer@12854
    85
lemma triangle_lemma:
schirmer@12854
    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> 
schirmer@12854
    87
 \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
schirmer@12854
    88
proof -
schirmer@12854
    89
  note converse_rtrancl_induct = converse_rtrancl_induct [consumes 1]
schirmer@12854
    90
  note converse_rtranclE = converse_rtranclE [consumes 1] 
schirmer@12854
    91
  assume unique: "\<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c"
schirmer@12854
    92
  assume "(a,x)\<in>r\<^sup>*" 
schirmer@12854
    93
  then show "(a,y)\<in>r\<^sup>* \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
schirmer@12854
    94
  proof (induct rule: converse_rtrancl_induct)
schirmer@12854
    95
    assume "(x,y)\<in>r\<^sup>*"
schirmer@12854
    96
    then show ?thesis 
schirmer@12854
    97
      by blast
schirmer@12854
    98
  next
schirmer@12854
    99
    fix a v
schirmer@12854
   100
    assume a_v_r: "(a, v) \<in> r" and
schirmer@12854
   101
          v_x_rt: "(v, x) \<in> r\<^sup>*" and
schirmer@12854
   102
          a_y_rt: "(a, y) \<in> r\<^sup>*"  and
schirmer@12854
   103
             hyp: "(v, y) \<in> r\<^sup>* \<Longrightarrow> (x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
schirmer@12854
   104
    from a_y_rt 
schirmer@12854
   105
    show "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
schirmer@12854
   106
    proof (cases rule: converse_rtranclE)
schirmer@12854
   107
      assume "a=y"
schirmer@12854
   108
      with a_v_r v_x_rt have "(y,x) \<in> r\<^sup>*"
schirmer@12854
   109
	by (auto intro: r_into_rtrancl rtrancl_trans)
schirmer@12854
   110
      then show ?thesis 
schirmer@12854
   111
	by blast
schirmer@12854
   112
    next
schirmer@12854
   113
      fix w 
schirmer@12854
   114
      assume a_w_r: "(a, w) \<in> r" and
schirmer@12854
   115
            w_y_rt: "(w, y) \<in> r\<^sup>*"
schirmer@12854
   116
      from a_v_r a_w_r unique 
schirmer@12854
   117
      have "v=w" 
schirmer@12854
   118
	by auto
schirmer@12854
   119
      with w_y_rt hyp 
schirmer@12854
   120
      show ?thesis
schirmer@12854
   121
	by blast
schirmer@12854
   122
    qed
schirmer@12854
   123
  qed
schirmer@12854
   124
qed
schirmer@12854
   125
schirmer@12854
   126
schirmer@12854
   127
lemma rtrancl_cases [consumes 1, case_names Refl Trancl]:
schirmer@12854
   128
 "\<lbrakk>(a,b)\<in>r\<^sup>*;  a = b \<Longrightarrow> P; (a,b)\<in>r\<^sup>+ \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
schirmer@12854
   129
apply (erule rtranclE)
schirmer@12854
   130
apply (auto dest: rtrancl_into_trancl1)
schirmer@12854
   131
done
schirmer@12854
   132
schirmer@12854
   133
(* ### To Transitive_Closure *)
schirmer@12854
   134
theorems converse_rtrancl_induct 
schirmer@12854
   135
 = converse_rtrancl_induct [consumes 1,case_names Id Step]
schirmer@12854
   136
schirmer@12854
   137
theorems converse_trancl_induct 
schirmer@12854
   138
         = converse_trancl_induct [consumes 1,case_names Single Step]
schirmer@12854
   139
schirmer@12854
   140
(* context (theory "Set") *)
schirmer@12854
   141
lemma Ball_weaken:"\<lbrakk>Ball s P;\<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"
schirmer@12854
   142
by auto
schirmer@12854
   143
schirmer@12854
   144
(* context (theory "Finite") *)
schirmer@12854
   145
lemma finite_SetCompr2: "[| finite (Collect P); !y. P y --> finite (range (f y)) |] ==>  
schirmer@12854
   146
  finite {f y x |x y. P y}"
schirmer@12854
   147
apply (subgoal_tac "{f y x |x y. P y} = UNION (Collect P) (%y. range (f y))")
schirmer@12854
   148
prefer 2 apply  fast
schirmer@12854
   149
apply (erule ssubst)
schirmer@12854
   150
apply (erule finite_UN_I)
schirmer@12854
   151
apply fast
schirmer@12854
   152
done
schirmer@12854
   153
schirmer@12854
   154
schirmer@12854
   155
(* ### TO theory "List" *)
schirmer@12854
   156
lemma list_all2_trans: "\<forall> a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>
schirmer@12854
   157
 \<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> list_all2 P3 xs1 xs3"
schirmer@12854
   158
apply (induct_tac "xs1")
schirmer@12854
   159
apply simp
schirmer@12854
   160
apply (rule allI)
schirmer@12854
   161
apply (induct_tac "xs2")
schirmer@12854
   162
apply simp
schirmer@12854
   163
apply (rule allI)
schirmer@12854
   164
apply (induct_tac "xs3")
schirmer@12854
   165
apply auto
schirmer@12854
   166
done
schirmer@12854
   167
schirmer@12854
   168
schirmer@12854
   169
section "pairs"
schirmer@12854
   170
schirmer@12854
   171
lemma surjective_pairing5: "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))), 
schirmer@12854
   172
  snd (snd (snd (snd p))))"
schirmer@12854
   173
apply auto
schirmer@12854
   174
done
schirmer@12854
   175
schirmer@12854
   176
lemma fst_splitE [elim!]: 
schirmer@12854
   177
"[| fst s' = x';  !!x s. [| s' = (x,s);  x = x' |] ==> Q |] ==> Q"
schirmer@12854
   178
apply (cut_tac p = "s'" in surjective_pairing)
schirmer@12854
   179
apply auto
schirmer@12854
   180
done
schirmer@12854
   181
schirmer@12854
   182
lemma fst_in_set_lemma [rule_format (no_asm)]: "(x, y) : set l --> x : fst ` set l"
schirmer@12854
   183
apply (induct_tac "l")
schirmer@12854
   184
apply  auto
schirmer@12854
   185
done
schirmer@12854
   186
schirmer@12854
   187
schirmer@12854
   188
section "quantifiers"
schirmer@12854
   189
schirmer@12854
   190
(*###to be phased out *)
schirmer@12854
   191
ML {* 
schirmer@12854
   192
fun noAll_simpset () = simpset() setmksimps 
schirmer@12854
   193
	mksimps (filter (fn (x,_) => x<>"All") mksimps_pairs)
schirmer@12854
   194
*}
schirmer@12854
   195
schirmer@12854
   196
lemma All_Ex_refl_eq2 [simp]: 
schirmer@12854
   197
 "(!x. (? b. x = f b & Q b) \<longrightarrow> P x) = (!b. Q b --> P (f b))"
schirmer@12854
   198
apply auto
schirmer@12854
   199
done
schirmer@12854
   200
schirmer@12854
   201
lemma ex_ex_miniscope1 [simp]:
schirmer@12854
   202
  "(EX w v. P w v & Q v) = (EX v. (EX w. P w v) & Q v)"
schirmer@12854
   203
apply auto
schirmer@12854
   204
done
schirmer@12854
   205
schirmer@12854
   206
lemma ex_miniscope2 [simp]:
schirmer@12854
   207
  "(EX v. P v & Q & R v) = (Q & (EX v. P v & R v))" 
schirmer@12854
   208
apply auto
schirmer@12854
   209
done
schirmer@12854
   210
schirmer@12854
   211
lemma ex_reorder31: "(\<exists>z x y. P x y z) = (\<exists>x y z. P x y z)"
schirmer@12854
   212
apply auto
schirmer@12854
   213
done
schirmer@12854
   214
schirmer@12854
   215
lemma All_Ex_refl_eq1 [simp]: "(!x. (? b. x = f b) --> P x) = (!b. P (f b))"
schirmer@12854
   216
apply auto
schirmer@12854
   217
done
schirmer@12854
   218
schirmer@12854
   219
schirmer@12854
   220
section "sums"
schirmer@12854
   221
schirmer@12854
   222
hide const In0 In1
schirmer@12854
   223
schirmer@12854
   224
syntax
schirmer@12854
   225
  fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
schirmer@12854
   226
translations
schirmer@12854
   227
 "fun_sum" == "sum_case"
schirmer@12854
   228
schirmer@12854
   229
consts    the_Inl  :: "'a + 'b \<Rightarrow> 'a"
schirmer@12854
   230
          the_Inr  :: "'a + 'b \<Rightarrow> 'b"
schirmer@12854
   231
primrec  "the_Inl (Inl a) = a"
schirmer@12854
   232
primrec  "the_Inr (Inr b) = b"
schirmer@12854
   233
schirmer@12854
   234
datatype ('a, 'b, 'c) sum3 = In1 'a | In2 'b | In3 'c
schirmer@12854
   235
schirmer@12854
   236
consts    the_In1  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"
schirmer@12854
   237
          the_In2  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"
schirmer@12854
   238
          the_In3  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"
schirmer@12854
   239
primrec  "the_In1 (In1 a) = a"
schirmer@12854
   240
primrec  "the_In2 (In2 b) = b"
schirmer@12854
   241
primrec  "the_In3 (In3 c) = c"
schirmer@12854
   242
schirmer@12854
   243
syntax
schirmer@12854
   244
	 In1l	:: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
schirmer@12854
   245
	 In1r	:: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
schirmer@12854
   246
translations
schirmer@12854
   247
	"In1l e" == "In1 (Inl e)"
schirmer@12854
   248
	"In1r c" == "In1 (Inr c)"
schirmer@12854
   249
schirmer@12854
   250
ML {*
schirmer@12854
   251
fun sum3_instantiate thm = map (fn s => simplify(simpset()delsimps[not_None_eq])
schirmer@12854
   252
 (read_instantiate [("t","In"^s^" ?x")] thm)) ["1l","2","3","1r"]
schirmer@12854
   253
*}
schirmer@12854
   254
(* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)
schirmer@12854
   255
schirmer@12854
   256
translations
wenzelm@12919
   257
  "option"<= (type) "Datatype.option"
schirmer@12854
   258
  "list"  <= (type) "List.list"
schirmer@12854
   259
  "sum3"  <= (type) "Basis.sum3"
schirmer@12854
   260
schirmer@12854
   261
schirmer@12854
   262
section "quantifiers for option type"
schirmer@12854
   263
schirmer@12854
   264
syntax
schirmer@12854
   265
  Oall :: "[pttrn, 'a option, bool] => bool"   ("(3! _:_:/ _)" [0,0,10] 10)
schirmer@12854
   266
  Oex  :: "[pttrn, 'a option, bool] => bool"   ("(3? _:_:/ _)" [0,0,10] 10)
schirmer@12854
   267
schirmer@12854
   268
syntax (symbols)
schirmer@12854
   269
  Oall :: "[pttrn, 'a option, bool] => bool"   ("(3\<forall>_\<in>_:/ _)"  [0,0,10] 10)
schirmer@12854
   270
  Oex  :: "[pttrn, 'a option, bool] => bool"   ("(3\<exists>_\<in>_:/ _)"  [0,0,10] 10)
schirmer@12854
   271
schirmer@12854
   272
translations
schirmer@12854
   273
  "! x:A: P"    == "! x:o2s A. P"
schirmer@12854
   274
  "? x:A: P"    == "? x:o2s A. P"
schirmer@12854
   275
schirmer@12854
   276
schirmer@12854
   277
section "unique association lists"
schirmer@12854
   278
schirmer@12854
   279
constdefs
schirmer@12854
   280
  unique   :: "('a \<times> 'b) list \<Rightarrow> bool"
wenzelm@12893
   281
 "unique \<equiv> distinct \<circ> map fst"
schirmer@12854
   282
schirmer@12854
   283
lemma uniqueD [rule_format (no_asm)]: 
schirmer@12854
   284
"unique l--> (!x y. (x,y):set l --> (!x' y'. (x',y'):set l --> x=x'-->  y=y'))"
schirmer@12854
   285
apply (unfold unique_def o_def)
schirmer@12854
   286
apply (induct_tac "l")
schirmer@12854
   287
apply  (auto dest: fst_in_set_lemma)
schirmer@12854
   288
done
schirmer@12854
   289
schirmer@12854
   290
lemma unique_Nil [simp]: "unique []"
schirmer@12854
   291
apply (unfold unique_def)
schirmer@12854
   292
apply (simp (no_asm))
schirmer@12854
   293
done
schirmer@12854
   294
schirmer@12854
   295
lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l & (!y. (x,y) ~: set l))"
schirmer@12854
   296
apply (unfold unique_def)
schirmer@12854
   297
apply  (auto dest: fst_in_set_lemma)
schirmer@12854
   298
done
schirmer@12854
   299
schirmer@12854
   300
lemmas unique_ConsI = conjI [THEN unique_Cons [THEN iffD2], standard]
schirmer@12854
   301
schirmer@12854
   302
lemma unique_single [simp]: "!!p. unique [p]"
schirmer@12854
   303
apply auto
schirmer@12854
   304
done
schirmer@12854
   305
schirmer@12854
   306
lemma unique_ConsD: "unique (x#xs) ==> unique xs"
schirmer@12854
   307
apply (simp add: unique_def)
schirmer@12854
   308
done
schirmer@12854
   309
schirmer@12854
   310
lemma unique_append [rule_format (no_asm)]: "unique l' ==> unique l -->  
schirmer@12854
   311
  (!(x,y):set l. !(x',y'):set l'. x' ~= x) --> unique (l @ l')"
schirmer@12854
   312
apply (induct_tac "l")
schirmer@12854
   313
apply  (auto dest: fst_in_set_lemma)
schirmer@12854
   314
done
schirmer@12854
   315
schirmer@12854
   316
lemma unique_map_inj [rule_format (no_asm)]: "unique l --> inj f --> unique (map (%(k,x). (f k, g k x)) l)"
schirmer@12854
   317
apply (induct_tac "l")
schirmer@12854
   318
apply  (auto dest: fst_in_set_lemma simp add: inj_eq)
schirmer@12854
   319
done
schirmer@12854
   320
schirmer@12854
   321
lemma map_of_SomeI [rule_format (no_asm)]: "unique l --> (k, x) : set l --> map_of l k = Some x"
schirmer@12854
   322
apply (induct_tac "l")
schirmer@12854
   323
apply auto
schirmer@12854
   324
done
schirmer@12854
   325
schirmer@12854
   326
schirmer@12854
   327
section "list patterns"
schirmer@12854
   328
schirmer@12854
   329
consts
schirmer@12854
   330
  lsplit         :: "[['a, 'a list] => 'b, 'a list] => 'b"
schirmer@12854
   331
defs
schirmer@12854
   332
  lsplit_def:    "lsplit == %f l. f (hd l) (tl l)"
schirmer@12854
   333
(*  list patterns -- extends pre-defined type "pttrn" used in abstractions *)
schirmer@12854
   334
syntax
schirmer@12854
   335
  "_lpttrn"    :: "[pttrn,pttrn] => pttrn"     ("_#/_" [901,900] 900)
schirmer@12854
   336
translations
schirmer@12854
   337
  "%y#x#xs. b"  == "lsplit (%y x#xs. b)"
schirmer@12854
   338
  "%x#xs  . b"  == "lsplit (%x xs  . b)"
schirmer@12854
   339
schirmer@12854
   340
lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"
schirmer@12854
   341
apply (unfold lsplit_def)
schirmer@12854
   342
apply (simp (no_asm))
schirmer@12854
   343
done
schirmer@12854
   344
schirmer@12854
   345
lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"
schirmer@12854
   346
apply (unfold lsplit_def)
schirmer@12854
   347
apply simp
schirmer@12854
   348
done 
schirmer@12854
   349
schirmer@12854
   350
schirmer@12854
   351
section "dummy pattern for quantifiers, let, etc."
schirmer@12854
   352
schirmer@12854
   353
syntax
schirmer@12854
   354
  "@dummy_pat"   :: pttrn    ("'_")
schirmer@12854
   355
schirmer@12854
   356
parse_translation {*
schirmer@12854
   357
let fun dummy_pat_tr [] = Free ("_",dummyT)
schirmer@12854
   358
  | dummy_pat_tr ts = raise TERM ("dummy_pat_tr", ts);
schirmer@12854
   359
in [("@dummy_pat", dummy_pat_tr)] 
schirmer@12854
   360
end
schirmer@12854
   361
*}
schirmer@12854
   362
schirmer@12854
   363
end