src/HOL/Bali/Basis.thy
author kuncar
Fri Dec 09 18:07:04 2011 +0100 (2011-12-09)
changeset 45802 b16f976db515
parent 45151 2dd44cd8f963
child 51304 0e71a248cacb
permissions -rw-r--r--
Quotient_Info stores only relation maps
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(*  Title:      HOL/Bali/Basis.thy
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    Author:     David von Oheimb
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*)
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header {* Definitions extending HOL as logical basis of Bali *}
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theory Basis
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imports Main "~~/src/HOL/Library/Old_Recdef"
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begin
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section "misc"
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declare split_if_asm  [split] option.split [split] option.split_asm [split]
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declaration {* K (Simplifier.map_ss (fn ss => ss addloop ("split_all_tac", split_all_tac))) *}
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declare if_weak_cong [cong del] option.weak_case_cong [cong del]
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declare length_Suc_conv [iff]
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lemma Collect_split_eq: "{p. P (split f p)} = {(a,b). P (f a b)}"
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  by auto
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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)"
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  apply (case_tac "x \<in> A")
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   apply (rule disjI2)
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   apply (rule_tac x = "A - {x}" in exI)
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   apply fast+
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  done
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abbreviation nat3 :: nat  ("3") where "3 \<equiv> Suc 2"
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abbreviation nat4 :: nat  ("4") where "4 \<equiv> Suc 3"
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(* irrefl_tranclI in Transitive_Closure.thy is more general *)
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lemma irrefl_tranclI': "r\<inverse> \<inter> r\<^sup>+ = {} \<Longrightarrow> \<forall>x. (x, x) \<notin> r\<^sup>+"
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  by (blast elim: tranclE dest: trancl_into_rtrancl)
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lemma trancl_rtrancl_trancl: "\<lbrakk>(x, y) \<in> r\<^sup>+; (y, z) \<in> r\<^sup>*\<rbrakk> \<Longrightarrow> (x, z) \<in> r\<^sup>+"
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  by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)
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lemma rtrancl_into_trancl3: "\<lbrakk>(a, b) \<in> r\<^sup>*; a \<noteq> b\<rbrakk> \<Longrightarrow> (a, b) \<in> r\<^sup>+"
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  apply (drule rtranclD)
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  apply auto
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  done
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lemma rtrancl_into_rtrancl2: "\<lbrakk>(a, b) \<in>  r; (b, c) \<in> r\<^sup>*\<rbrakk> \<Longrightarrow> (a, c) \<in> r\<^sup>*"
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  by (auto intro: rtrancl_trans)
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lemma triangle_lemma:
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  assumes unique: "\<And>a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b = c"
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    and ax: "(a,x)\<in>r\<^sup>*" and ay: "(a,y)\<in>r\<^sup>*"
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  shows "(x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
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  using ax ay
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proof (induct rule: converse_rtrancl_induct)
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  assume "(x,y)\<in>r\<^sup>*"
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  then show ?thesis by blast
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next
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  fix a v
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  assume a_v_r: "(a, v) \<in> r"
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    and v_x_rt: "(v, x) \<in> r\<^sup>*"
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    and a_y_rt: "(a, y) \<in> r\<^sup>*"
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    and hyp: "(v, y) \<in> r\<^sup>* \<Longrightarrow> (x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
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  from a_y_rt show "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
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  proof (cases rule: converse_rtranclE)
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    assume "a = y"
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    with a_v_r v_x_rt have "(y,x) \<in> r\<^sup>*"
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      by (auto intro: rtrancl_trans)
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    then show ?thesis by blast
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  next
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    fix w
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    assume a_w_r: "(a, w) \<in> r"
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      and w_y_rt: "(w, y) \<in> r\<^sup>*"
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    from a_v_r a_w_r unique have "v=w" by auto
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    with w_y_rt hyp show ?thesis by blast
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  qed
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qed
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lemma rtrancl_cases:
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  assumes "(a,b)\<in>r\<^sup>*"
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  obtains (Refl) "a = b"
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    | (Trancl) "(a,b)\<in>r\<^sup>+"
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  apply (rule rtranclE [OF assms])
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   apply (auto dest: rtrancl_into_trancl1)
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  done
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lemma Ball_weaken: "\<lbrakk>Ball s P; \<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"
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  by auto
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lemma finite_SetCompr2:
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  "finite (Collect P) \<Longrightarrow> \<forall>y. P y \<longrightarrow> finite (range (f y)) \<Longrightarrow>
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    finite {f y x |x y. P y}"
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  apply (subgoal_tac "{f y x |x y. P y} = UNION (Collect P) (\<lambda>y. range (f y))")
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   prefer 2 apply fast
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  apply (erule ssubst)
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  apply (erule finite_UN_I)
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  apply fast
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  done
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lemma list_all2_trans: "\<forall>a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>
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    \<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> list_all2 P3 xs1 xs3"
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  apply (induct_tac xs1)
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   apply simp
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  apply (rule allI)
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  apply (induct_tac xs2)
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   apply simp
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  apply (rule allI)
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  apply (induct_tac xs3)
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   apply auto
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  done
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section "pairs"
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lemma surjective_pairing5:
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  "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))),
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    snd (snd (snd (snd p))))"
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  by auto
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lemma fst_splitE [elim!]:
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  assumes "fst s' = x'"
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  obtains x s where "s' = (x,s)" and "x = x'"
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  using assms by (cases s') auto
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lemma fst_in_set_lemma: "(x, y) : set l \<Longrightarrow> x : fst ` set l"
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  by (induct l) auto
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section "quantifiers"
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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))"
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  by auto
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lemma ex_ex_miniscope1 [simp]: "(\<exists>w v. P w v \<and> Q v) = (\<exists>v. (\<exists>w. P w v) \<and> Q v)"
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  by auto
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lemma ex_miniscope2 [simp]: "(\<exists>v. P v \<and> Q \<and> R v) = (Q \<and> (\<exists>v. P v \<and> R v))"
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  by auto
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lemma ex_reorder31: "(\<exists>z x y. P x y z) = (\<exists>x y z. P x y z)"
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  by auto
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lemma All_Ex_refl_eq1 [simp]: "(\<forall>x. (\<exists>b. x = f b) \<longrightarrow> P x) = (\<forall>b. P (f b))"
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  by auto
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section "sums"
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hide_const In0 In1
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notation sum_case  (infixr "'(+')"80)
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primrec the_Inl :: "'a + 'b \<Rightarrow> 'a"
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  where "the_Inl (Inl a) = a"
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primrec the_Inr :: "'a + 'b \<Rightarrow> 'b"
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  where "the_Inr (Inr b) = b"
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datatype ('a, 'b, 'c) sum3 = In1 'a | In2 'b | In3 'c
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primrec the_In1 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"
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  where "the_In1 (In1 a) = a"
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primrec the_In2 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"
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  where "the_In2 (In2 b) = b"
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primrec the_In3 :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"
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  where "the_In3 (In3 c) = c"
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abbreviation In1l :: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
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  where "In1l e \<equiv> In1 (Inl e)"
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abbreviation In1r :: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"
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  where "In1r c \<equiv> In1 (Inr c)"
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abbreviation the_In1l :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'al"
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  where "the_In1l \<equiv> the_Inl \<circ> the_In1"
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abbreviation the_In1r :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'ar"
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  where "the_In1r \<equiv> the_Inr \<circ> the_In1"
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ML {*
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fun sum3_instantiate ctxt thm = map (fn s =>
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  simplify (simpset_of ctxt delsimps [@{thm not_None_eq}])
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    (read_instantiate ctxt [(("t", 0), "In" ^ s ^ " ?x")] thm)) ["1l","2","3","1r"]
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*}
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(* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)
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section "quantifiers for option type"
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syntax
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  "_Oall" :: "[pttrn, 'a option, bool] \<Rightarrow> bool"   ("(3! _:_:/ _)" [0,0,10] 10)
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  "_Oex"  :: "[pttrn, 'a option, bool] \<Rightarrow> bool"   ("(3? _:_:/ _)" [0,0,10] 10)
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syntax (symbols)
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  "_Oall" :: "[pttrn, 'a option, bool] \<Rightarrow> bool"   ("(3\<forall>_\<in>_:/ _)"  [0,0,10] 10)
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  "_Oex"  :: "[pttrn, 'a option, bool] \<Rightarrow> bool"   ("(3\<exists>_\<in>_:/ _)"  [0,0,10] 10)
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translations
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  "\<forall>x\<in>A: P" \<rightleftharpoons> "\<forall>x\<in>CONST Option.set A. P"
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  "\<exists>x\<in>A: P" \<rightleftharpoons> "\<exists>x\<in>CONST Option.set A. P"
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section "Special map update"
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text{* Deemed too special for theory Map. *}
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definition chg_map :: "('b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> ('a \<rightharpoonup> 'b) \<Rightarrow> ('a \<rightharpoonup> 'b)"
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  where "chg_map f a m = (case m a of None \<Rightarrow> m | Some b \<Rightarrow> m(a\<mapsto>f b))"
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lemma chg_map_new[simp]: "m a = None \<Longrightarrow> chg_map f a m = m"
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  unfolding chg_map_def by auto
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lemma chg_map_upd[simp]: "m a = Some b \<Longrightarrow> chg_map f a m = m(a\<mapsto>f b)"
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  unfolding chg_map_def by auto
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lemma chg_map_other [simp]: "a \<noteq> b \<Longrightarrow> chg_map f a m b = m b"
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  by (auto simp: chg_map_def)
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section "unique association lists"
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definition unique :: "('a \<times> 'b) list \<Rightarrow> bool"
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  where "unique = distinct \<circ> map fst"
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lemma uniqueD: "unique l \<Longrightarrow> (x, y) \<in> set l \<Longrightarrow> (x', y') \<in> set l \<Longrightarrow> x = x' \<Longrightarrow> y = y'"
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  unfolding unique_def o_def
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  by (induct l) (auto dest: fst_in_set_lemma)
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lemma unique_Nil [simp]: "unique []"
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  by (simp add: unique_def)
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lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l \<and> (\<forall>y. (x,y) \<notin> set l))"
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  by (auto simp: unique_def dest: fst_in_set_lemma)
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lemma unique_ConsD: "unique (x#xs) \<Longrightarrow> unique xs"
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  by (simp add: unique_def)
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lemma unique_single [simp]: "\<And>p. unique [p]"
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  by simp
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lemma unique_append [rule_format (no_asm)]: "unique l' \<Longrightarrow> unique l \<Longrightarrow>
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    (\<forall>(x,y)\<in>set l. \<forall>(x',y')\<in>set l'. x' \<noteq> x) \<longrightarrow> unique (l @ l')"
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  by (induct l) (auto dest: fst_in_set_lemma)
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lemma unique_map_inj: "unique l \<Longrightarrow> inj f \<Longrightarrow> unique (map (\<lambda>(k,x). (f k, g k x)) l)"
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  by (induct l) (auto dest: fst_in_set_lemma simp add: inj_eq)
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lemma map_of_SomeI: "unique l \<Longrightarrow> (k, x) : set l \<Longrightarrow> map_of l k = Some x"
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  by (induct l) auto
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section "list patterns"
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definition lsplit :: "[['a, 'a list] \<Rightarrow> 'b, 'a list] \<Rightarrow> 'b"
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  where "lsplit = (\<lambda>f l. f (hd l) (tl l))"
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text {* list patterns -- extends pre-defined type "pttrn" used in abstractions *}
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syntax
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  "_lpttrn" :: "[pttrn, pttrn] \<Rightarrow> pttrn"    ("_#/_" [901,900] 900)
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translations
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  "\<lambda>y # x # xs. b" \<rightleftharpoons> "CONST lsplit (\<lambda>y x # xs. b)"
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  "\<lambda>x # xs. b" \<rightleftharpoons> "CONST lsplit (\<lambda>x xs. b)"
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lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"
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  by (simp add: lsplit_def)
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lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"
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  by (simp add: lsplit_def)
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end