src/HOL/Bali/Basis.thy
 author wenzelm Mon Jun 16 22:13:39 2008 +0200 (2008-06-16) changeset 27239 f2f42f9fa09d parent 27226 5a3e5e46d977 child 30235 58d147683393 permissions -rw-r--r--
pervasive RuleInsts;
```     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 declare [[unify_search_bound = 40, unify_trace_bound = 40]]
```
```    11
```
```    12
```
```    13 section "misc"
```
```    14
```
```    15 declare same_fstI [intro!] (*### TO HOL/Wellfounded_Relations *)
```
```    16
```
```    17 declare split_if_asm  [split] option.split [split] option.split_asm [split]
```
```    18 declaration {* K (Simplifier.map_ss (fn ss => ss addloop ("split_all_tac", split_all_tac))) *}
```
```    19 declare if_weak_cong [cong del] option.weak_case_cong [cong del]
```
```    20 declare length_Suc_conv [iff]
```
```    21
```
```    22 lemma Collect_split_eq: "{p. P (split f p)} = {(a,b). P (f a b)}"
```
```    23 apply auto
```
```    24 done
```
```    25
```
```    26 lemma subset_insertD:
```
```    27   "A <= insert x B ==> A <= B & x ~: A | (EX B'. A = insert x B' & B' <= B)"
```
```    28 apply (case_tac "x:A")
```
```    29 apply (rule disjI2)
```
```    30 apply (rule_tac x = "A-{x}" in exI)
```
```    31 apply fast+
```
```    32 done
```
```    33
```
```    34 syntax
```
```    35   "3" :: nat   ("3")
```
```    36   "4" :: nat   ("4")
```
```    37 translations
```
```    38  "3" == "Suc 2"
```
```    39  "4" == "Suc 3"
```
```    40
```
```    41 (*unused*)
```
```    42 lemma range_bool_domain: "range f = {f True, f False}"
```
```    43 apply auto
```
```    44 apply (case_tac "xa")
```
```    45 apply auto
```
```    46 done
```
```    47
```
```    48 (* irrefl_tranclI in Transitive_Closure.thy is more general *)
```
```    49 lemma irrefl_tranclI': "r^-1 Int r^+ = {} ==> !x. (x, x) ~: r^+"
```
```    50 by(blast elim: tranclE dest: trancl_into_rtrancl)
```
```    51
```
```    52
```
```    53 lemma trancl_rtrancl_trancl:
```
```    54 "\<lbrakk>(x,y)\<in>r^+; (y,z)\<in>r^*\<rbrakk> \<Longrightarrow> (x,z)\<in>r^+"
```
```    55 by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)
```
```    56
```
```    57 lemma rtrancl_into_trancl3:
```
```    58 "\<lbrakk>(a,b)\<in>r^*; a\<noteq>b\<rbrakk> \<Longrightarrow> (a,b)\<in>r^+"
```
```    59 apply (drule rtranclD)
```
```    60 apply auto
```
```    61 done
```
```    62
```
```    63 lemma rtrancl_into_rtrancl2:
```
```    64   "\<lbrakk> (a, b) \<in>  r; (b, c) \<in> r^* \<rbrakk> \<Longrightarrow> (a, c) \<in>  r^*"
```
```    65 by (auto intro: r_into_rtrancl rtrancl_trans)
```
```    66
```
```    67 lemma triangle_lemma:
```
```    68  "\<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>
```
```    69  \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
```
```    70 proof -
```
```    71   note converse_rtrancl_induct = converse_rtrancl_induct [consumes 1]
```
```    72   note converse_rtranclE = converse_rtranclE [consumes 1]
```
```    73   assume unique: "\<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c"
```
```    74   assume "(a,x)\<in>r\<^sup>*"
```
```    75   then show "(a,y)\<in>r\<^sup>* \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
```
```    76   proof (induct rule: converse_rtrancl_induct)
```
```    77     assume "(x,y)\<in>r\<^sup>*"
```
```    78     then show ?thesis
```
```    79       by blast
```
```    80   next
```
```    81     fix a v
```
```    82     assume a_v_r: "(a, v) \<in> r" and
```
```    83           v_x_rt: "(v, x) \<in> r\<^sup>*" and
```
```    84           a_y_rt: "(a, y) \<in> r\<^sup>*"  and
```
```    85              hyp: "(v, y) \<in> r\<^sup>* \<Longrightarrow> (x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
```
```    86     from a_y_rt
```
```    87     show "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
```
```    88     proof (cases rule: converse_rtranclE)
```
```    89       assume "a=y"
```
```    90       with a_v_r v_x_rt have "(y,x) \<in> r\<^sup>*"
```
```    91 	by (auto intro: r_into_rtrancl rtrancl_trans)
```
```    92       then show ?thesis
```
```    93 	by blast
```
```    94     next
```
```    95       fix w
```
```    96       assume a_w_r: "(a, w) \<in> r" and
```
```    97             w_y_rt: "(w, y) \<in> r\<^sup>*"
```
```    98       from a_v_r a_w_r unique
```
```    99       have "v=w"
```
```   100 	by auto
```
```   101       with w_y_rt hyp
```
```   102       show ?thesis
```
```   103 	by blast
```
```   104     qed
```
```   105   qed
```
```   106 qed
```
```   107
```
```   108
```
```   109 lemma rtrancl_cases [consumes 1, case_names Refl Trancl]:
```
```   110  "\<lbrakk>(a,b)\<in>r\<^sup>*;  a = b \<Longrightarrow> P; (a,b)\<in>r\<^sup>+ \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
```
```   111 apply (erule rtranclE)
```
```   112 apply (auto dest: rtrancl_into_trancl1)
```
```   113 done
```
```   114
```
```   115 (* ### To Transitive_Closure *)
```
```   116 theorems converse_rtrancl_induct
```
```   117  = converse_rtrancl_induct [consumes 1,case_names Id Step]
```
```   118
```
```   119 theorems converse_trancl_induct
```
```   120          = converse_trancl_induct [consumes 1,case_names Single Step]
```
```   121
```
```   122 (* context (theory "Set") *)
```
```   123 lemma Ball_weaken:"\<lbrakk>Ball s P;\<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"
```
```   124 by auto
```
```   125
```
```   126 (* context (theory "Finite") *)
```
```   127 lemma finite_SetCompr2: "[| finite (Collect P); !y. P y --> finite (range (f y)) |] ==>
```
```   128   finite {f y x |x y. P y}"
```
```   129 apply (subgoal_tac "{f y x |x y. P y} = UNION (Collect P) (%y. range (f y))")
```
```   130 prefer 2 apply  fast
```
```   131 apply (erule ssubst)
```
```   132 apply (erule finite_UN_I)
```
```   133 apply fast
```
```   134 done
```
```   135
```
```   136
```
```   137 (* ### TO theory "List" *)
```
```   138 lemma list_all2_trans: "\<forall> a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>
```
```   139  \<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> list_all2 P3 xs1 xs3"
```
```   140 apply (induct_tac "xs1")
```
```   141 apply simp
```
```   142 apply (rule allI)
```
```   143 apply (induct_tac "xs2")
```
```   144 apply simp
```
```   145 apply (rule allI)
```
```   146 apply (induct_tac "xs3")
```
```   147 apply auto
```
```   148 done
```
```   149
```
```   150
```
```   151 section "pairs"
```
```   152
```
```   153 lemma surjective_pairing5: "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))),
```
```   154   snd (snd (snd (snd p))))"
```
```   155 apply auto
```
```   156 done
```
```   157
```
```   158 lemma fst_splitE [elim!]:
```
```   159 "[| fst s' = x';  !!x s. [| s' = (x,s);  x = x' |] ==> Q |] ==> Q"
```
```   160 by (cases s') auto
```
```   161
```
```   162 lemma fst_in_set_lemma [rule_format (no_asm)]: "(x, y) : set l --> x : fst ` set l"
```
```   163 apply (induct_tac "l")
```
```   164 apply  auto
```
```   165 done
```
```   166
```
```   167
```
```   168 section "quantifiers"
```
```   169
```
```   170 lemma All_Ex_refl_eq2 [simp]:
```
```   171  "(!x. (? b. x = f b & Q b) \<longrightarrow> P x) = (!b. Q b --> P (f b))"
```
```   172 apply auto
```
```   173 done
```
```   174
```
```   175 lemma ex_ex_miniscope1 [simp]:
```
```   176   "(EX w v. P w v & Q v) = (EX v. (EX w. P w v) & Q v)"
```
```   177 apply auto
```
```   178 done
```
```   179
```
```   180 lemma ex_miniscope2 [simp]:
```
```   181   "(EX v. P v & Q & R v) = (Q & (EX v. P v & R v))"
```
```   182 apply auto
```
```   183 done
```
```   184
```
```   185 lemma ex_reorder31: "(\<exists>z x y. P x y z) = (\<exists>x y z. P x y z)"
```
```   186 apply auto
```
```   187 done
```
```   188
```
```   189 lemma All_Ex_refl_eq1 [simp]: "(!x. (? b. x = f b) --> P x) = (!b. P (f b))"
```
```   190 apply auto
```
```   191 done
```
```   192
```
```   193
```
```   194 section "sums"
```
```   195
```
```   196 hide const In0 In1
```
```   197
```
```   198 syntax
```
```   199   fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
```
```   200 translations
```
```   201  "fun_sum" == "CONST sum_case"
```
```   202
```
```   203 consts    the_Inl  :: "'a + 'b \<Rightarrow> 'a"
```
```   204           the_Inr  :: "'a + 'b \<Rightarrow> 'b"
```
```   205 primrec  "the_Inl (Inl a) = a"
```
```   206 primrec  "the_Inr (Inr b) = b"
```
```   207
```
```   208 datatype ('a, 'b, 'c) sum3 = In1 'a | In2 'b | In3 'c
```
```   209
```
```   210 consts    the_In1  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"
```
```   211           the_In2  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"
```
```   212           the_In3  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"
```
```   213 primrec  "the_In1 (In1 a) = a"
```
```   214 primrec  "the_In2 (In2 b) = b"
```
```   215 primrec  "the_In3 (In3 c) = c"
```
```   216
```
```   217 syntax
```
```   218 	 In1l	:: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
```
```   219 	 In1r	:: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
```
```   220 translations
```
```   221 	"In1l e" == "In1 (Inl e)"
```
```   222 	"In1r c" == "In1 (Inr c)"
```
```   223
```
```   224 syntax the_In1l :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'al"
```
```   225        the_In1r :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'ar"
```
```   226 translations
```
```   227    "the_In1l" == "the_Inl \<circ> the_In1"
```
```   228    "the_In1r" == "the_Inr \<circ> the_In1"
```
```   229
```
```   230 ML {*
```
```   231 fun sum3_instantiate ctxt thm = map (fn s =>
```
```   232   simplify (Simplifier.local_simpset_of ctxt delsimps[@{thm not_None_eq}])
```
```   233     (read_instantiate ctxt [(("t", 0), "In" ^ s ^ " ?x")] thm)) ["1l","2","3","1r"]
```
```   234 *}
```
```   235 (* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)
```
```   236
```
```   237 translations
```
```   238   "option"<= (type) "Datatype.option"
```
```   239   "list"  <= (type) "List.list"
```
```   240   "sum3"  <= (type) "Basis.sum3"
```
```   241
```
```   242
```
```   243 section "quantifiers for option type"
```
```   244
```
```   245 syntax
```
```   246   Oall :: "[pttrn, 'a option, bool] => bool"   ("(3! _:_:/ _)" [0,0,10] 10)
```
```   247   Oex  :: "[pttrn, 'a option, bool] => bool"   ("(3? _:_:/ _)" [0,0,10] 10)
```
```   248
```
```   249 syntax (symbols)
```
```   250   Oall :: "[pttrn, 'a option, bool] => bool"   ("(3\<forall>_\<in>_:/ _)"  [0,0,10] 10)
```
```   251   Oex  :: "[pttrn, 'a option, bool] => bool"   ("(3\<exists>_\<in>_:/ _)"  [0,0,10] 10)
```
```   252
```
```   253 translations
```
```   254   "! x:A: P"    == "! x:o2s A. P"
```
```   255   "? x:A: P"    == "? x:o2s A. P"
```
```   256
```
```   257 section "Special map update"
```
```   258
```
```   259 text{* Deemed too special for theory Map. *}
```
```   260
```
```   261 constdefs
```
```   262   chg_map :: "('b => 'b) => 'a => ('a ~=> 'b) => ('a ~=> 'b)"
```
```   263  "chg_map f a m == case m a of None => m | Some b => m(a|->f b)"
```
```   264
```
```   265 lemma chg_map_new[simp]: "m a = None   ==> chg_map f a m = m"
```
```   266 by (unfold chg_map_def, auto)
```
```   267
```
```   268 lemma chg_map_upd[simp]: "m a = Some b ==> chg_map f a m = m(a|->f b)"
```
```   269 by (unfold chg_map_def, auto)
```
```   270
```
```   271 lemma chg_map_other [simp]: "a \<noteq> b \<Longrightarrow> chg_map f a m b = m b"
```
```   272 by (auto simp: chg_map_def split add: option.split)
```
```   273
```
```   274
```
```   275 section "unique association lists"
```
```   276
```
```   277 constdefs
```
```   278   unique   :: "('a \<times> 'b) list \<Rightarrow> bool"
```
```   279  "unique \<equiv> distinct \<circ> map fst"
```
```   280
```
```   281 lemma uniqueD [rule_format (no_asm)]:
```
```   282 "unique l--> (!x y. (x,y):set l --> (!x' y'. (x',y'):set l --> x=x'-->  y=y'))"
```
```   283 apply (unfold unique_def o_def)
```
```   284 apply (induct_tac "l")
```
```   285 apply  (auto dest: fst_in_set_lemma)
```
```   286 done
```
```   287
```
```   288 lemma unique_Nil [simp]: "unique []"
```
```   289 apply (unfold unique_def)
```
```   290 apply (simp (no_asm))
```
```   291 done
```
```   292
```
```   293 lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l & (!y. (x,y) ~: set l))"
```
```   294 apply (unfold unique_def)
```
```   295 apply  (auto dest: fst_in_set_lemma)
```
```   296 done
```
```   297
```
```   298 lemmas unique_ConsI = conjI [THEN unique_Cons [THEN iffD2], standard]
```
```   299
```
```   300 lemma unique_single [simp]: "!!p. unique [p]"
```
```   301 apply auto
```
```   302 done
```
```   303
```
```   304 lemma unique_ConsD: "unique (x#xs) ==> unique xs"
```
```   305 apply (simp add: unique_def)
```
```   306 done
```
```   307
```
```   308 lemma unique_append [rule_format (no_asm)]: "unique l' ==> unique l -->
```
```   309   (!(x,y):set l. !(x',y'):set l'. x' ~= x) --> unique (l @ l')"
```
```   310 apply (induct_tac "l")
```
```   311 apply  (auto dest: fst_in_set_lemma)
```
```   312 done
```
```   313
```
```   314 lemma unique_map_inj [rule_format (no_asm)]: "unique l --> inj f --> unique (map (%(k,x). (f k, g k x)) l)"
```
```   315 apply (induct_tac "l")
```
```   316 apply  (auto dest: fst_in_set_lemma simp add: inj_eq)
```
```   317 done
```
```   318
```
```   319 lemma map_of_SomeI [rule_format (no_asm)]: "unique l --> (k, x) : set l --> map_of l k = Some x"
```
```   320 apply (induct_tac "l")
```
```   321 apply auto
```
```   322 done
```
```   323
```
```   324
```
```   325 section "list patterns"
```
```   326
```
```   327 consts
```
```   328   lsplit         :: "[['a, 'a list] => 'b, 'a list] => 'b"
```
```   329 defs
```
```   330   lsplit_def:    "lsplit == %f l. f (hd l) (tl l)"
```
```   331 (*  list patterns -- extends pre-defined type "pttrn" used in abstractions *)
```
```   332 syntax
```
```   333   "_lpttrn"    :: "[pttrn,pttrn] => pttrn"     ("_#/_" [901,900] 900)
```
```   334 translations
```
```   335   "%y#x#xs. b"  == "lsplit (%y x#xs. b)"
```
```   336   "%x#xs  . b"  == "lsplit (%x xs  . b)"
```
```   337
```
```   338 lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"
```
```   339 apply (unfold lsplit_def)
```
```   340 apply (simp (no_asm))
```
```   341 done
```
```   342
```
```   343 lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"
```
```   344 apply (unfold lsplit_def)
```
```   345 apply simp
```
```   346 done
```
```   347
```
```   348
```
```   349 end
```