Quelle Basis.thy Sprache: Isabelle

(*  Title:      HOL/Bali/Basis.thy
    Author:     David von Oheimb
*)
subsection \<open>Definitions extending HOL as logical basis of Bali\<close>

theory Basis
imports Main
begin

subsubsection "misc"

ML \<open>fun strip_tac ctxt i = REPEAT (resolve_tac ctxt [impI, allI] i)\<close>

declare if_split_asm  [split] option.split [split] option.split_asm [split]
setup \<open>map_theory_simpset (fn ctxt => ctxt |> Simplifier.add_loop ("split_all_tac", split_all_tac))\<close>
declare if_weak_cong [cong del] option.case_cong_weak [cong del]
declare length_Suc_conv [iff]

lemma Collect_split_eq: "{p. P (case_prod f p)} = {(a,b). P (f a b)}"
  by auto

lemma subset_insertD: "A \ insert x B \ A \ B \ x \ A \ (\B'. A = insert x B' \ B' \ B)"
  apply (case_tac "x \ A")
   apply (rule disjI2)
   apply (rule_tac x = "A - {x}" in exI)
   apply fast+
  done

abbreviation nat3 :: nat  (\<open>3\<close>) where "3 \<equiv> Suc 2"
abbreviation nat4 :: nat  (\<open>4\<close>) where "4 \<equiv> Suc 3"

(* irrefl_tranclI in Transitive_Closure.thy is more general *)
lemma irrefl_tranclI': "r\ \ r\<^sup>+ = {} \ \x. (x, x) \ r\<^sup>+"
  by (blast elim: tranclE dest: trancl_into_rtrancl)

lemma trancl_rtrancl_trancl: "\(x, y) \ r\<^sup>+; (y, z) \ r\<^sup>*\ \ (x, z) \ r\<^sup>+"
  by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)

lemma rtrancl_into_trancl3: "\(a, b) \ r\<^sup>*; a \ b\ \ (a, b) \ r\<^sup>+"
  apply (drule rtranclD)
  apply auto
  done

lemma rtrancl_into_rtrancl2: "\(a, b) \ r; (b, c) \ r\<^sup>*\ \ (a, c) \ r\<^sup>*"
  by (auto intro: rtrancl_trans)

lemma triangle_lemma:
  assumes unique: "\a b c. \(a,b)\r; (a,c)\r\ \ b = c"
    and ax: "(a,x)\r\<^sup>*" and ay: "(a,y)\r\<^sup>*"
  shows "(x,y)\r\<^sup>* \ (y,x)\r\<^sup>*"
  using ax ay
proof (induct rule: converse_rtrancl_induct)
  assume "(x,y)\r\<^sup>*"
  then show ?thesis by blast
next
  fix a v
  assume a_v_r: "(a, v) \ r"
    and v_x_rt: "(v, x) \ r\<^sup>*"
    and a_y_rt: "(a, y) \ r\<^sup>*"
    and hyp: "(v, y) \ r\<^sup>* \ (x, y) \ r\<^sup>* \ (y, x) \ r\<^sup>*"
  from a_y_rt show "(x, y) \ r\<^sup>* \ (y, x) \ r\<^sup>*"
  proof (cases rule: converse_rtranclE)
    assume "a = y"
    with a_v_r v_x_rt have "(y,x) \ r\<^sup>*"
      by (auto intro: rtrancl_trans)
    then show ?thesis by blast
  next
    fix w
    assume a_w_r: "(a, w) \ r"
      and w_y_rt: "(w, y) \ r\<^sup>*"
    from a_v_r a_w_r unique have "v=w" by auto
    with w_y_rt hyp show ?thesis by blast
  qed
qed

lemma rtrancl_cases:
  assumes "(a,b)\r\<^sup>*"
  obtains (Refl) "a = b"
    | (Trancl) "(a,b)\r\<^sup>+"
  apply (rule rtranclE [OF assms])
   apply (auto dest: rtrancl_into_trancl1)
  done

lemma Ball_weaken: "\Ball s P; \ x. P x\Q x\\Ball s Q"
  by auto

lemma finite_SetCompr2:
  "finite {f y x |x y. P y}" if "finite (Collect P)"
    "\y. P y \ finite (range (f y))"
proof -
  have "{f y x |x y. P y} = (\y\Collect P. range (f y))"
    by auto
  with that show ?thesis by simp
qed

lemma list_all2_trans: "\a b c. P1 a b \ P2 b c \ P3 a c \
    \<forall>xs2 xs3. list_all2 P1 xs1 xs2 \<longrightarrow> list_all2 P2 xs2 xs3 \<longrightarrow> list_all2 P3 xs1 xs3"
  apply (induct_tac xs1)
   apply simp
  apply (rule allI)
  apply (induct_tac xs2)
   apply simp
  apply (rule allI)
  apply (induct_tac xs3)
   apply auto
  done

subsubsection "pairs"

lemma surjective_pairing5:
  "p = (fst p, fst (snd p), fst (snd (snd p)), fst (snd (snd (snd p))),
    snd (snd (snd (snd p))))"
  by auto

lemma fst_splitE [elim!]:
  assumes "fst s' = x'"
  obtains x s where "s' = (x,s)" and "x = x'"
  using assms by (cases s') auto

lemma fst_in_set_lemma: "(x, y) \ set l \ x \ fst ` set l"
  by (induct l) auto

subsubsection "quantifiers"

lemma All_Ex_refl_eq2 [simp]: "(\x. (\b. x = f b \ Q b) \ P x) = (\b. Q b \ P (f b))"
  by auto

lemma ex_ex_miniscope1 [simp]: "(\w v. P w v \ Q v) = (\v. (\w. P w v) \ Q v)"
  by auto

lemma ex_miniscope2 [simp]: "(\v. P v \ Q \ R v) = (Q \ (\v. P v \ R v))"
  by auto

lemma ex_reorder31: "(\z x y. P x y z) = (\x y z. P x y z)"
  by auto

lemma All_Ex_refl_eq1 [simp]: "(\x. (\b. x = f b) \ P x) = (\b. P (f b))"
  by auto

subsubsection "sums"

notation case_sum  (infixr \<open>'(+')\<close> 80)

primrec the_Inl :: "'a + 'b \ 'a"
  where "the_Inl (Inl a) = a"

primrec the_Inr :: "'a + 'b \ 'b"
  where "the_Inr (Inr b) = b"

datatype ('a, 'b, 'c) sum3 = In1 'a | In2 'b | In3 'c

primrec the_In1 :: "('a, 'b, 'c) sum3 \ 'a"
  where "the_In1 (In1 a) = a"

primrec the_In2 :: "('a, 'b, 'c) sum3 \ 'b"
  where "the_In2 (In2 b) = b"

primrec the_In3 :: "('a, 'b, 'c) sum3 \ 'c"
  where "the_In3 (In3 c) = c"

abbreviation In1l :: "'al \ ('al + 'ar, 'b, 'c) sum3"
  where "In1l e \ In1 (Inl e)"

abbreviation In1r :: "'ar \ ('al + 'ar, 'b, 'c) sum3"
  where "In1r c \ In1 (Inr c)"

abbreviation the_In1l :: "('al + 'ar, 'b, 'c) sum3 \ 'al"
  where "the_In1l \ the_Inl \ the_In1"

abbreviation the_In1r :: "('al + 'ar, 'b, 'c) sum3 \ 'ar"
  where "the_In1r \ the_Inr \ the_In1"

ML \<open>
fun sum3_instantiate ctxt thm =
  map (fn s =>
    simplify (ctxt |> Simplifier.del_simps @{thms not_None_eq})
      (Rule_Insts.read_instantiate ctxt [((("t", 0), Position.none), "In" ^ s ^ " x")] ["x"] thm))
    ["1l","2","3","1r"]
\<close>
(* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)

subsubsection "quantifiers for option type"

syntax
  "_Oall" :: "[pttrn, 'a option, bool] \ bool" (\(3! _:_:/ _)\ [0,0,10] 10)
  "_Oex"  :: "[pttrn, 'a option, bool] \ bool" (\(3? _:_:/ _)\ [0,0,10] 10)

syntax (symbols)
  "_Oall" :: "[pttrn, 'a option, bool] \ bool" (\(3\_\_:/ _)\ [0,0,10] 10)
  "_Oex"  :: "[pttrn, 'a option, bool] \ bool" (\(3\_\_:/ _)\ [0,0,10] 10)

syntax_consts
  "_Oall" \<rightleftharpoons> Ball and
  "_Oex" \<rightleftharpoons> Bex

translations
  "\x\A: P" \ "\x\CONST set_option A. P"
  "\x\A: P" \ "\x\CONST set_option A. P"

subsubsection "Special map update"

text\<open>Deemed too special for theory Map.\<close>

definition chg_map :: "('b \ 'b) \ 'a \ ('a \ 'b) \ ('a \ 'b)"
  where "chg_map f a m = (case m a of None \ m | Some b \ m(a\f b))"

lemma chg_map_new[simp]: "m a = None \ chg_map f a m = m"
  unfolding chg_map_def by auto

lemma chg_map_upd[simp]: "m a = Some b \ chg_map f a m = m(a\f b)"
  unfolding chg_map_def by auto

lemma chg_map_other [simp]: "a \ b \ chg_map f a m b = m b"
  by (auto simp: chg_map_def)

subsubsection "unique association lists"

definition unique :: "('a \ 'b) list \ bool"
  where "unique = distinct \ map fst"

lemma uniqueD: "unique l \ (x, y) \ set l \ (x', y') \ set l \ x = x' \ y = y'"
  unfolding unique_def o_def
  by (induct l) (auto dest: fst_in_set_lemma)

lemma unique_Nil [simp]: "unique []"
  by (simp add: unique_def)

lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l \ (\y. (x,y) \ set l))"
  by (auto simp: unique_def dest: fst_in_set_lemma)

lemma unique_ConsD: "unique (x#xs) \ unique xs"
  by (simp add: unique_def)

lemma unique_single [simp]: "\p. unique [p]"
  by simp

lemma unique_append [rule_format (no_asm)]: "unique l' \ unique l \
    (\<forall>(x,y)\<in>set l. \<forall>(x',y')\<in>set l'. x' \<noteq> x) \<longrightarrow> unique (l @ l')"
  by (induct l) (auto dest: fst_in_set_lemma)

lemma unique_map_inj: "unique l \ inj f \ unique (map (\(k,x). (f k, g k x)) l)"
  by (induct l) (auto dest: fst_in_set_lemma simp add: inj_eq)

lemma map_of_SomeI: "unique l \ (k, x) \ set l \ map_of l k = Some x"
  by (induct l) auto

subsubsection "list patterns"

definition lsplit :: "[['a, 'a list] \ 'b, 'a list] \ 'b"
  where "lsplit = (\f l. f (hd l) (tl l))"

text \<open>list patterns -- extends pre-defined type "pttrn" used in abstractions\<close>
syntax
  "_lpttrn" :: "[pttrn, pttrn] \ pttrn" (\_#/_\ [901,900] 900)
syntax_consts
  "_lpttrn" \<rightleftharpoons> lsplit
translations
  "\y # x # xs. b" \ "CONST lsplit (\y x # xs. b)"
  "\x # xs. b" \ "CONST lsplit (\x xs. b)"

lemma lsplit [simp]: "lsplit c (x#xs) = c x xs"
  by (simp add: lsplit_def)

lemma lsplit2 [simp]: "lsplit P (x#xs) y z = P x xs y z"
  by (simp add: lsplit_def)

end

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