src/HOL/Bali/Basis.thy
 author wenzelm Wed Nov 01 20:46:23 2017 +0100 (22 months ago) changeset 66983 df83b66f1d94 parent 66809 f6a30d48aab0 child 67613 ce654b0e6d69 permissions -rw-r--r--
proper merge (amending fb46c031c841);
```     1 (*  Title:      HOL/Bali/Basis.thy
```
```     2     Author:     David von Oheimb
```
```     3 *)
```
```     4 subsection \<open>Definitions extending HOL as logical basis of Bali\<close>
```
```     5
```
```     6 theory Basis
```
```     7 imports Main
```
```     8 begin
```
```     9
```
```    10 subsubsection "misc"
```
```    11
```
```    12 ML \<open>fun strip_tac ctxt i = REPEAT (resolve_tac ctxt [impI, allI] i)\<close>
```
```    13
```
```    14 declare if_split_asm  [split] option.split [split] option.split_asm [split]
```
```    15 setup \<open>map_theory_simpset (fn ctxt => ctxt addloop ("split_all_tac", split_all_tac))\<close>
```
```    16 declare if_weak_cong [cong del] option.case_cong_weak [cong del]
```
```    17 declare length_Suc_conv [iff]
```
```    18
```
```    19 lemma Collect_split_eq: "{p. P (case_prod f p)} = {(a,b). P (f a b)}"
```
```    20   by auto
```
```    21
```
```    22 lemma subset_insertD: "A \<subseteq> insert x B \<Longrightarrow> A \<subseteq> B \<and> x \<notin> A \<or> (\<exists>B'. A = insert x B' \<and> B' \<subseteq> B)"
```
```    23   apply (case_tac "x \<in> A")
```
```    24    apply (rule disjI2)
```
```    25    apply (rule_tac x = "A - {x}" in exI)
```
```    26    apply fast+
```
```    27   done
```
```    28
```
```    29 abbreviation nat3 :: nat  ("3") where "3 \<equiv> Suc 2"
```
```    30 abbreviation nat4 :: nat  ("4") where "4 \<equiv> Suc 3"
```
```    31
```
```    32 (* irrefl_tranclI in Transitive_Closure.thy is more general *)
```
```    33 lemma irrefl_tranclI': "r\<inverse> \<inter> r\<^sup>+ = {} \<Longrightarrow> \<forall>x. (x, x) \<notin> r\<^sup>+"
```
```    34   by (blast elim: tranclE dest: trancl_into_rtrancl)
```
```    35
```
```    36
```
```    37 lemma trancl_rtrancl_trancl: "\<lbrakk>(x, y) \<in> r\<^sup>+; (y, z) \<in> r\<^sup>*\<rbrakk> \<Longrightarrow> (x, z) \<in> r\<^sup>+"
```
```    38   by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)
```
```    39
```
```    40 lemma rtrancl_into_trancl3: "\<lbrakk>(a, b) \<in> r\<^sup>*; a \<noteq> b\<rbrakk> \<Longrightarrow> (a, b) \<in> r\<^sup>+"
```
```    41   apply (drule rtranclD)
```
```    42   apply auto
```
```    43   done
```
```    44
```
```    45 lemma rtrancl_into_rtrancl2: "\<lbrakk>(a, b) \<in>  r; (b, c) \<in> r\<^sup>*\<rbrakk> \<Longrightarrow> (a, c) \<in> r\<^sup>*"
```
```    46   by (auto intro: rtrancl_trans)
```
```    47
```
```    48 lemma triangle_lemma:
```
```    49   assumes unique: "\<And>a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b = c"
```
```    50     and ax: "(a,x)\<in>r\<^sup>*" and ay: "(a,y)\<in>r\<^sup>*"
```
```    51   shows "(x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
```
```    52   using ax ay
```
```    53 proof (induct rule: converse_rtrancl_induct)
```
```    54   assume "(x,y)\<in>r\<^sup>*"
```
```    55   then show ?thesis by blast
```
```    56 next
```
```    57   fix a v
```
```    58   assume a_v_r: "(a, v) \<in> r"
```
```    59     and v_x_rt: "(v, x) \<in> r\<^sup>*"
```
```    60     and a_y_rt: "(a, y) \<in> r\<^sup>*"
```
```    61     and hyp: "(v, y) \<in> r\<^sup>* \<Longrightarrow> (x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
```
```    62   from a_y_rt show "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
```
```    63   proof (cases rule: converse_rtranclE)
```
```    64     assume "a = y"
```
```    65     with a_v_r v_x_rt have "(y,x) \<in> r\<^sup>*"
```
```    66       by (auto intro: rtrancl_trans)
```
```    67     then show ?thesis by blast
```
```    68   next
```
```    69     fix w
```
```    70     assume a_w_r: "(a, w) \<in> r"
```
```    71       and w_y_rt: "(w, y) \<in> r\<^sup>*"
```
```    72     from a_v_r a_w_r unique have "v=w" by auto
```
```    73     with w_y_rt hyp show ?thesis by blast
```
```    74   qed
```
```    75 qed
```
```    76
```
```    77
```
```    78 lemma rtrancl_cases:
```
```    79   assumes "(a,b)\<in>r\<^sup>*"
```
```    80   obtains (Refl) "a = b"
```
```    81     | (Trancl) "(a,b)\<in>r\<^sup>+"
```
```    82   apply (rule rtranclE [OF assms])
```
```    83    apply (auto dest: rtrancl_into_trancl1)
```
```    84   done
```
```    85
```
```    86 lemma Ball_weaken: "\<lbrakk>Ball s P; \<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"
```
```    87   by auto
```
```    88
```
```    89 lemma finite_SetCompr2:
```
```    90   "finite (Collect P) \<Longrightarrow> \<forall>y. P y \<longrightarrow> finite (range (f y)) \<Longrightarrow>
```
```    91     finite {f y x |x y. P y}"
```
```    92   apply (subgoal_tac "{f y x |x y. P y} = UNION (Collect P) (\<lambda>y. range (f y))")
```
```    93    prefer 2 apply fast
```
```    94   apply (erule ssubst)
```
```    95   apply (erule finite_UN_I)
```
```    96   apply fast
```
```    97   done
```
```    98
```
```    99 lemma list_all2_trans: "\<forall>a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>
```
```   100     \<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> list_all2 P3 xs1 xs3"
```
```   101   apply (induct_tac xs1)
```
```   102    apply simp
```
```   103   apply (rule allI)
```
```   104   apply (induct_tac xs2)
```
```   105    apply simp
```
```   106   apply (rule allI)
```
```   107   apply (induct_tac xs3)
```
```   108    apply auto
```
```   109   done
```
```   110
```
```   111
```
```   112 subsubsection "pairs"
```
```   113
```
```   114 lemma surjective_pairing5:
```
```   115   "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))),
```
```   116     snd (snd (snd (snd p))))"
```
```   117   by auto
```
```   118
```
```   119 lemma fst_splitE [elim!]:
```
```   120   assumes "fst s' = x'"
```
```   121   obtains x s where "s' = (x,s)" and "x = x'"
```
```   122   using assms by (cases s') auto
```
```   123
```
```   124 lemma fst_in_set_lemma: "(x, y) : set l \<Longrightarrow> x : fst ` set l"
```
```   125   by (induct l) auto
```
```   126
```
```   127
```
```   128 subsubsection "quantifiers"
```
```   129
```
```   130 lemma All_Ex_refl_eq2 [simp]: "(\<forall>x. (\<exists>b. x = f b \<and> Q b) \<longrightarrow> P x) = (\<forall>b. Q b \<longrightarrow> P (f b))"
```
```   131   by auto
```
```   132
```
```   133 lemma ex_ex_miniscope1 [simp]: "(\<exists>w v. P w v \<and> Q v) = (\<exists>v. (\<exists>w. P w v) \<and> Q v)"
```
```   134   by auto
```
```   135
```
```   136 lemma ex_miniscope2 [simp]: "(\<exists>v. P v \<and> Q \<and> R v) = (Q \<and> (\<exists>v. P v \<and> R v))"
```
```   137   by auto
```
```   138
```
```   139 lemma ex_reorder31: "(\<exists>z x y. P x y z) = (\<exists>x y z. P x y z)"
```
```   140   by auto
```
```   141
```
```   142 lemma All_Ex_refl_eq1 [simp]: "(\<forall>x. (\<exists>b. x = f b) \<longrightarrow> P x) = (\<forall>b. P (f b))"
```
```   143   by auto
```
```   144
```
```   145
```
```   146 subsubsection "sums"
```
```   147
```
```   148 notation case_sum  (infixr "'(+')" 80)
```
```   149
```
```   150 primrec the_Inl :: "'a + 'b \<Rightarrow> 'a"
```
```   151   where "the_Inl (Inl a) = a"
```
```   152
```
```   153 primrec the_Inr :: "'a + 'b \<Rightarrow> 'b"
```
```   154   where "the_Inr (Inr b) = b"
```
```   155
```
```   156 datatype ('a, 'b, 'c) sum3 = In1 'a | In2 'b | In3 'c
```
```   157
```
```   158 primrec the_In1 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"
```
```   159   where "the_In1 (In1 a) = a"
```
```   160
```
```   161 primrec the_In2 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"
```
```   162   where "the_In2 (In2 b) = b"
```
```   163
```
```   164 primrec the_In3 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"
```
```   165   where "the_In3 (In3 c) = c"
```
```   166
```
```   167 abbreviation In1l :: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
```
```   168   where "In1l e \<equiv> In1 (Inl e)"
```
```   169
```
```   170 abbreviation In1r :: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
```
```   171   where "In1r c \<equiv> In1 (Inr c)"
```
```   172
```
```   173 abbreviation the_In1l :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'al"
```
```   174   where "the_In1l \<equiv> the_Inl \<circ> the_In1"
```
```   175
```
```   176 abbreviation the_In1r :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'ar"
```
```   177   where "the_In1r \<equiv> the_Inr \<circ> the_In1"
```
```   178
```
```   179 ML \<open>
```
```   180 fun sum3_instantiate ctxt thm =
```
```   181   map (fn s =>
```
```   182     simplify (ctxt delsimps @{thms not_None_eq})
```
```   183       (Rule_Insts.read_instantiate ctxt [((("t", 0), Position.none), "In" ^ s ^ " x")] ["x"] thm))
```
```   184     ["1l","2","3","1r"]
```
```   185 \<close>
```
```   186 (* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)
```
```   187
```
```   188
```
```   189 subsubsection "quantifiers for option type"
```
```   190
```
```   191 syntax
```
```   192   "_Oall" :: "[pttrn, 'a option, bool] \<Rightarrow> bool"   ("(3! _:_:/ _)" [0,0,10] 10)
```
```   193   "_Oex"  :: "[pttrn, 'a option, bool] \<Rightarrow> bool"   ("(3? _:_:/ _)" [0,0,10] 10)
```
```   194
```
```   195 syntax (symbols)
```
```   196   "_Oall" :: "[pttrn, 'a option, bool] \<Rightarrow> bool"   ("(3\<forall>_\<in>_:/ _)"  [0,0,10] 10)
```
```   197   "_Oex"  :: "[pttrn, 'a option, bool] \<Rightarrow> bool"   ("(3\<exists>_\<in>_:/ _)"  [0,0,10] 10)
```
```   198
```
```   199 translations
```
```   200   "\<forall>x\<in>A: P" \<rightleftharpoons> "\<forall>x\<in>CONST set_option A. P"
```
```   201   "\<exists>x\<in>A: P" \<rightleftharpoons> "\<exists>x\<in>CONST set_option A. P"
```
```   202
```
```   203
```
```   204 subsubsection "Special map update"
```
```   205
```
```   206 text\<open>Deemed too special for theory Map.\<close>
```
```   207
```
```   208 definition chg_map :: "('b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b)"
```
```   209   where "chg_map f a m = (case m a of None \<Rightarrow> m | Some b \<Rightarrow> m(a\<mapsto>f b))"
```
```   210
```
```   211 lemma chg_map_new[simp]: "m a = None \<Longrightarrow> chg_map f a m = m"
```
```   212   unfolding chg_map_def by auto
```
```   213
```
```   214 lemma chg_map_upd[simp]: "m a = Some b \<Longrightarrow> chg_map f a m = m(a\<mapsto>f b)"
```
```   215   unfolding chg_map_def by auto
```
```   216
```
```   217 lemma chg_map_other [simp]: "a \<noteq> b \<Longrightarrow> chg_map f a m b = m b"
```
```   218   by (auto simp: chg_map_def)
```
```   219
```
```   220
```
```   221 subsubsection "unique association lists"
```
```   222
```
```   223 definition unique :: "('a \<times> 'b) list \<Rightarrow> bool"
```
```   224   where "unique = distinct \<circ> map fst"
```
```   225
```
```   226 lemma uniqueD: "unique l \<Longrightarrow> (x, y) \<in> set l \<Longrightarrow> (x', y') \<in> set l \<Longrightarrow> x = x' \<Longrightarrow> y = y'"
```
```   227   unfolding unique_def o_def
```
```   228   by (induct l) (auto dest: fst_in_set_lemma)
```
```   229
```
```   230 lemma unique_Nil [simp]: "unique []"
```
```   231   by (simp add: unique_def)
```
```   232
```
```   233 lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l \<and> (\<forall>y. (x,y) \<notin> set l))"
```
```   234   by (auto simp: unique_def dest: fst_in_set_lemma)
```
```   235
```
```   236 lemma unique_ConsD: "unique (x#xs) \<Longrightarrow> unique xs"
```
```   237   by (simp add: unique_def)
```
```   238
```
```   239 lemma unique_single [simp]: "\<And>p. unique [p]"
```
```   240   by simp
```
```   241
```
```   242 lemma unique_append [rule_format (no_asm)]: "unique l' \<Longrightarrow> unique l \<Longrightarrow>
```
```   243     (\<forall>(x,y)\<in>set l. \<forall>(x',y')\<in>set l'. x' \<noteq> x) \<longrightarrow> unique (l @ l')"
```
```   244   by (induct l) (auto dest: fst_in_set_lemma)
```
```   245
```
```   246 lemma unique_map_inj: "unique l \<Longrightarrow> inj f \<Longrightarrow> unique (map (\<lambda>(k,x). (f k, g k x)) l)"
```
```   247   by (induct l) (auto dest: fst_in_set_lemma simp add: inj_eq)
```
```   248
```
```   249 lemma map_of_SomeI: "unique l \<Longrightarrow> (k, x) : set l \<Longrightarrow> map_of l k = Some x"
```
```   250   by (induct l) auto
```
```   251
```
```   252
```
```   253 subsubsection "list patterns"
```
```   254
```
```   255 definition lsplit :: "[['a, 'a list] \<Rightarrow> 'b, 'a list] \<Rightarrow> 'b"
```
```   256   where "lsplit = (\<lambda>f l. f (hd l) (tl l))"
```
```   257
```
```   258 text \<open>list patterns -- extends pre-defined type "pttrn" used in abstractions\<close>
```
```   259 syntax
```
```   260   "_lpttrn" :: "[pttrn, pttrn] \<Rightarrow> pttrn"    ("_#/_" [901,900] 900)
```
```   261 translations
```
```   262   "\<lambda>y # x # xs. b" \<rightleftharpoons> "CONST lsplit (\<lambda>y x # xs. b)"
```
```   263   "\<lambda>x # xs. b" \<rightleftharpoons> "CONST lsplit (\<lambda>x xs. b)"
```
```   264
```
```   265 lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"
```
```   266   by (simp add: lsplit_def)
```
```   267
```
```   268 lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"
```
```   269   by (simp add: lsplit_def)
```
```   270
```
```   271 end
```