src/HOL/Bali/Basis.thy
 author wenzelm Mon Jul 26 17:41:26 2010 +0200 (2010-07-26) changeset 37956 ee939247b2fb parent 36176 3fe7e97ccca8 child 44011 f67c93f52d13 permissions -rw-r--r--
modernized/unified some specifications;
```     1 (*  Title:      HOL/Bali/Basis.thy
```
```     2     Author:     David von Oheimb
```
```     3 *)
```
```     4 header {* Definitions extending HOL as logical basis of Bali *}
```
```     5
```
```     6 theory Basis imports Main begin
```
```     7
```
```     8
```
```     9 section "misc"
```
```    10
```
```    11 declare same_fstI [intro!] (*### TO HOL/Wellfounded_Relations *)
```
```    12
```
```    13 declare split_if_asm  [split] option.split [split] option.split_asm [split]
```
```    14 declaration {* K (Simplifier.map_ss (fn ss => ss addloop ("split_all_tac", split_all_tac))) *}
```
```    15 declare if_weak_cong [cong del] option.weak_case_cong [cong del]
```
```    16 declare length_Suc_conv [iff]
```
```    17
```
```    18 lemma Collect_split_eq: "{p. P (split f p)} = {(a,b). P (f a b)}"
```
```    19 apply auto
```
```    20 done
```
```    21
```
```    22 lemma subset_insertD:
```
```    23   "A <= insert x B ==> A <= B & x ~: A | (EX B'. A = insert x B' & B' <= B)"
```
```    24 apply (case_tac "x:A")
```
```    25 apply (rule disjI2)
```
```    26 apply (rule_tac x = "A-{x}" in exI)
```
```    27 apply fast+
```
```    28 done
```
```    29
```
```    30 abbreviation nat3 :: nat  ("3") where "3 == Suc 2"
```
```    31 abbreviation nat4 :: nat  ("4") where "4 == Suc 3"
```
```    32
```
```    33 (*unused*)
```
```    34 lemma range_bool_domain: "range f = {f True, f False}"
```
```    35 apply auto
```
```    36 apply (case_tac "xa")
```
```    37 apply auto
```
```    38 done
```
```    39
```
```    40 (* irrefl_tranclI in Transitive_Closure.thy is more general *)
```
```    41 lemma irrefl_tranclI': "r^-1 Int r^+ = {} ==> !x. (x, x) ~: r^+"
```
```    42 by(blast elim: tranclE dest: trancl_into_rtrancl)
```
```    43
```
```    44
```
```    45 lemma trancl_rtrancl_trancl:
```
```    46 "\<lbrakk>(x,y)\<in>r^+; (y,z)\<in>r^*\<rbrakk> \<Longrightarrow> (x,z)\<in>r^+"
```
```    47 by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)
```
```    48
```
```    49 lemma rtrancl_into_trancl3:
```
```    50 "\<lbrakk>(a,b)\<in>r^*; a\<noteq>b\<rbrakk> \<Longrightarrow> (a,b)\<in>r^+"
```
```    51 apply (drule rtranclD)
```
```    52 apply auto
```
```    53 done
```
```    54
```
```    55 lemma rtrancl_into_rtrancl2:
```
```    56   "\<lbrakk> (a, b) \<in>  r; (b, c) \<in> r^* \<rbrakk> \<Longrightarrow> (a, c) \<in>  r^*"
```
```    57 by (auto intro: r_into_rtrancl rtrancl_trans)
```
```    58
```
```    59 lemma triangle_lemma:
```
```    60  "\<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>
```
```    61  \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
```
```    62 proof -
```
```    63   assume unique: "\<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c"
```
```    64   assume "(a,x)\<in>r\<^sup>*"
```
```    65   then show "(a,y)\<in>r\<^sup>* \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
```
```    66   proof (induct rule: converse_rtrancl_induct)
```
```    67     assume "(x,y)\<in>r\<^sup>*"
```
```    68     then show ?thesis
```
```    69       by blast
```
```    70   next
```
```    71     fix a v
```
```    72     assume a_v_r: "(a, v) \<in> r" and
```
```    73           v_x_rt: "(v, x) \<in> r\<^sup>*" and
```
```    74           a_y_rt: "(a, y) \<in> r\<^sup>*"  and
```
```    75              hyp: "(v, y) \<in> r\<^sup>* \<Longrightarrow> (x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
```
```    76     from a_y_rt
```
```    77     show "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
```
```    78     proof (cases rule: converse_rtranclE)
```
```    79       assume "a=y"
```
```    80       with a_v_r v_x_rt have "(y,x) \<in> r\<^sup>*"
```
```    81         by (auto intro: r_into_rtrancl rtrancl_trans)
```
```    82       then show ?thesis
```
```    83         by blast
```
```    84     next
```
```    85       fix w
```
```    86       assume a_w_r: "(a, w) \<in> r" and
```
```    87             w_y_rt: "(w, y) \<in> r\<^sup>*"
```
```    88       from a_v_r a_w_r unique
```
```    89       have "v=w"
```
```    90         by auto
```
```    91       with w_y_rt hyp
```
```    92       show ?thesis
```
```    93         by blast
```
```    94     qed
```
```    95   qed
```
```    96 qed
```
```    97
```
```    98
```
```    99 lemma rtrancl_cases [consumes 1, case_names Refl Trancl]:
```
```   100  "\<lbrakk>(a,b)\<in>r\<^sup>*;  a = b \<Longrightarrow> P; (a,b)\<in>r\<^sup>+ \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
```
```   101 apply (erule rtranclE)
```
```   102 apply (auto dest: rtrancl_into_trancl1)
```
```   103 done
```
```   104
```
```   105 (* context (theory "Set") *)
```
```   106 lemma Ball_weaken:"\<lbrakk>Ball s P;\<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"
```
```   107 by auto
```
```   108
```
```   109 (* context (theory "Finite") *)
```
```   110 lemma finite_SetCompr2: "[| finite (Collect P); !y. P y --> finite (range (f y)) |] ==>
```
```   111   finite {f y x |x y. P y}"
```
```   112 apply (subgoal_tac "{f y x |x y. P y} = UNION (Collect P) (%y. range (f y))")
```
```   113 prefer 2 apply  fast
```
```   114 apply (erule ssubst)
```
```   115 apply (erule finite_UN_I)
```
```   116 apply fast
```
```   117 done
```
```   118
```
```   119
```
```   120 (* ### TO theory "List" *)
```
```   121 lemma list_all2_trans: "\<forall> a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>
```
```   122  \<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> list_all2 P3 xs1 xs3"
```
```   123 apply (induct_tac "xs1")
```
```   124 apply simp
```
```   125 apply (rule allI)
```
```   126 apply (induct_tac "xs2")
```
```   127 apply simp
```
```   128 apply (rule allI)
```
```   129 apply (induct_tac "xs3")
```
```   130 apply auto
```
```   131 done
```
```   132
```
```   133
```
```   134 section "pairs"
```
```   135
```
```   136 lemma surjective_pairing5: "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))),
```
```   137   snd (snd (snd (snd p))))"
```
```   138 apply auto
```
```   139 done
```
```   140
```
```   141 lemma fst_splitE [elim!]:
```
```   142 "[| fst s' = x';  !!x s. [| s' = (x,s);  x = x' |] ==> Q |] ==> Q"
```
```   143 by (cases s') auto
```
```   144
```
```   145 lemma fst_in_set_lemma [rule_format (no_asm)]: "(x, y) : set l --> x : fst ` set l"
```
```   146 apply (induct_tac "l")
```
```   147 apply  auto
```
```   148 done
```
```   149
```
```   150
```
```   151 section "quantifiers"
```
```   152
```
```   153 lemma All_Ex_refl_eq2 [simp]:
```
```   154  "(!x. (? b. x = f b & Q b) \<longrightarrow> P x) = (!b. Q b --> P (f b))"
```
```   155 apply auto
```
```   156 done
```
```   157
```
```   158 lemma ex_ex_miniscope1 [simp]:
```
```   159   "(EX w v. P w v & Q v) = (EX v. (EX w. P w v) & Q v)"
```
```   160 apply auto
```
```   161 done
```
```   162
```
```   163 lemma ex_miniscope2 [simp]:
```
```   164   "(EX v. P v & Q & R v) = (Q & (EX v. P v & R v))"
```
```   165 apply auto
```
```   166 done
```
```   167
```
```   168 lemma ex_reorder31: "(\<exists>z x y. P x y z) = (\<exists>x y z. P x y z)"
```
```   169 apply auto
```
```   170 done
```
```   171
```
```   172 lemma All_Ex_refl_eq1 [simp]: "(!x. (? b. x = f b) --> P x) = (!b. P (f b))"
```
```   173 apply auto
```
```   174 done
```
```   175
```
```   176
```
```   177 section "sums"
```
```   178
```
```   179 hide_const In0 In1
```
```   180
```
```   181 notation sum_case  (infixr "'(+')"80)
```
```   182
```
```   183 primrec the_Inl  :: "'a + 'b \<Rightarrow> 'a"
```
```   184   where "the_Inl (Inl a) = a"
```
```   185
```
```   186 primrec the_Inr  :: "'a + 'b \<Rightarrow> 'b"
```
```   187   where "the_Inr (Inr b) = b"
```
```   188
```
```   189 datatype ('a, 'b, 'c) sum3 = In1 'a | In2 'b | In3 'c
```
```   190
```
```   191 primrec the_In1  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"
```
```   192   where "the_In1 (In1 a) = a"
```
```   193
```
```   194 primrec the_In2  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"
```
```   195   where "the_In2 (In2 b) = b"
```
```   196
```
```   197 primrec the_In3  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"
```
```   198   where "the_In3 (In3 c) = c"
```
```   199
```
```   200 abbreviation In1l   :: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
```
```   201   where "In1l e == In1 (Inl e)"
```
```   202
```
```   203 abbreviation In1r   :: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
```
```   204   where "In1r c == In1 (Inr c)"
```
```   205
```
```   206 abbreviation the_In1l :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'al"
```
```   207   where "the_In1l == the_Inl \<circ> the_In1"
```
```   208
```
```   209 abbreviation the_In1r :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'ar"
```
```   210   where "the_In1r == the_Inr \<circ> the_In1"
```
```   211
```
```   212 ML {*
```
```   213 fun sum3_instantiate ctxt thm = map (fn s =>
```
```   214   simplify (simpset_of ctxt delsimps[@{thm not_None_eq}])
```
```   215     (read_instantiate ctxt [(("t", 0), "In" ^ s ^ " ?x")] thm)) ["1l","2","3","1r"]
```
```   216 *}
```
```   217 (* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)
```
```   218
```
```   219
```
```   220 section "quantifiers for option type"
```
```   221
```
```   222 syntax
```
```   223   "_Oall" :: "[pttrn, 'a option, bool] => bool"   ("(3! _:_:/ _)" [0,0,10] 10)
```
```   224   "_Oex"  :: "[pttrn, 'a option, bool] => bool"   ("(3? _:_:/ _)" [0,0,10] 10)
```
```   225
```
```   226 syntax (symbols)
```
```   227   "_Oall" :: "[pttrn, 'a option, bool] => bool"   ("(3\<forall>_\<in>_:/ _)"  [0,0,10] 10)
```
```   228   "_Oex"  :: "[pttrn, 'a option, bool] => bool"   ("(3\<exists>_\<in>_:/ _)"  [0,0,10] 10)
```
```   229
```
```   230 translations
```
```   231   "! x:A: P"    == "! x:CONST Option.set A. P"
```
```   232   "? x:A: P"    == "? x:CONST Option.set A. P"
```
```   233
```
```   234 section "Special map update"
```
```   235
```
```   236 text{* Deemed too special for theory Map. *}
```
```   237
```
```   238 definition
```
```   239   chg_map :: "('b => 'b) => 'a => ('a ~=> 'b) => ('a ~=> 'b)"
```
```   240   where "chg_map f a m = (case m a of None => m | Some b => m(a|->f b))"
```
```   241
```
```   242 lemma chg_map_new[simp]: "m a = None   ==> chg_map f a m = m"
```
```   243 by (unfold chg_map_def, auto)
```
```   244
```
```   245 lemma chg_map_upd[simp]: "m a = Some b ==> chg_map f a m = m(a|->f b)"
```
```   246 by (unfold chg_map_def, auto)
```
```   247
```
```   248 lemma chg_map_other [simp]: "a \<noteq> b \<Longrightarrow> chg_map f a m b = m b"
```
```   249 by (auto simp: chg_map_def split add: option.split)
```
```   250
```
```   251
```
```   252 section "unique association lists"
```
```   253
```
```   254 definition
```
```   255   unique :: "('a \<times> 'b) list \<Rightarrow> bool"
```
```   256   where "unique = distinct \<circ> map fst"
```
```   257
```
```   258 lemma uniqueD [rule_format (no_asm)]:
```
```   259 "unique l--> (!x y. (x,y):set l --> (!x' y'. (x',y'):set l --> x=x'-->  y=y'))"
```
```   260 apply (unfold unique_def o_def)
```
```   261 apply (induct_tac "l")
```
```   262 apply  (auto dest: fst_in_set_lemma)
```
```   263 done
```
```   264
```
```   265 lemma unique_Nil [simp]: "unique []"
```
```   266 apply (unfold unique_def)
```
```   267 apply (simp (no_asm))
```
```   268 done
```
```   269
```
```   270 lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l & (!y. (x,y) ~: set l))"
```
```   271 apply (unfold unique_def)
```
```   272 apply  (auto dest: fst_in_set_lemma)
```
```   273 done
```
```   274
```
```   275 lemmas unique_ConsI = conjI [THEN unique_Cons [THEN iffD2], standard]
```
```   276
```
```   277 lemma unique_single [simp]: "!!p. unique [p]"
```
```   278 apply auto
```
```   279 done
```
```   280
```
```   281 lemma unique_ConsD: "unique (x#xs) ==> unique xs"
```
```   282 apply (simp add: unique_def)
```
```   283 done
```
```   284
```
```   285 lemma unique_append [rule_format (no_asm)]: "unique l' ==> unique l -->
```
```   286   (!(x,y):set l. !(x',y'):set l'. x' ~= x) --> unique (l @ l')"
```
```   287 apply (induct_tac "l")
```
```   288 apply  (auto dest: fst_in_set_lemma)
```
```   289 done
```
```   290
```
```   291 lemma unique_map_inj [rule_format (no_asm)]: "unique l --> inj f --> unique (map (%(k,x). (f k, g k x)) l)"
```
```   292 apply (induct_tac "l")
```
```   293 apply  (auto dest: fst_in_set_lemma simp add: inj_eq)
```
```   294 done
```
```   295
```
```   296 lemma map_of_SomeI [rule_format (no_asm)]: "unique l --> (k, x) : set l --> map_of l k = Some x"
```
```   297 apply (induct_tac "l")
```
```   298 apply auto
```
```   299 done
```
```   300
```
```   301
```
```   302 section "list patterns"
```
```   303
```
```   304 definition
```
```   305   lsplit :: "[['a, 'a list] => 'b, 'a list] => 'b" where
```
```   306   "lsplit = (\<lambda>f l. f (hd l) (tl l))"
```
```   307
```
```   308 text {* list patterns -- extends pre-defined type "pttrn" used in abstractions *}
```
```   309 syntax
```
```   310   "_lpttrn"    :: "[pttrn,pttrn] => pttrn"     ("_#/_" [901,900] 900)
```
```   311 translations
```
```   312   "%y#x#xs. b"  == "CONST lsplit (%y x#xs. b)"
```
```   313   "%x#xs  . b"  == "CONST lsplit (%x xs  . b)"
```
```   314
```
```   315 lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"
```
```   316 apply (unfold lsplit_def)
```
```   317 apply (simp (no_asm))
```
```   318 done
```
```   319
```
```   320 lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"
```
```   321 apply (unfold lsplit_def)
```
```   322 apply simp
```
```   323 done
```
```   324
```
```   325
```
```   326 end
```