(*  Title:      HOL/Bali/Basis.thy

     Author:     David von Oheimb

 *)

 header {* Definitions extending HOL as logical basis of Bali *}

 theory Basis imports Main begin

 section "misc"

 declare split_if_asm  [split] option.split [split] option.split_asm [split]

 declaration {* K (Simplifier.map_ss (fn ss => ss addloop ("split_all_tac", split_all_tac))) *}

 declare if_weak_cong [cong del] option.weak_case_cong [cong del]

 declare length_Suc_conv [iff]

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

 apply auto

 done

 lemma subset_insertD: 

   "A <= insert x B ==> A <= B & x ~: A | (EX 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  ("3") where "3 == Suc 2"

 abbreviation nat4 :: nat  ("4") where "4 == Suc 3"

 (*unused*)

 lemma range_bool_domain: "range f = {f True, f False}"

 apply auto

 apply (case_tac "xa")

 apply auto

 done

 (* irrefl_tranclI in Transitive_Closure.thy is more general *)

 lemma irrefl_tranclI': "r^-1 Int r^+ = {} ==> !x. (x, x) ~: r^+"

 by(blast elim: tranclE dest: trancl_into_rtrancl)

 lemma trancl_rtrancl_trancl:

 "\<lbrakk>(x,y)\<in>r^+; (y,z)\<in>r^*\<rbrakk> \<Longrightarrow> (x,z)\<in>r^+"

 by (auto dest: tranclD rtrancl_trans rtrancl_into_trancl2)

 lemma rtrancl_into_trancl3:

 "\<lbrakk>(a,b)\<in>r^*; a\<noteq>b\<rbrakk> \<Longrightarrow> (a,b)\<in>r^+" 

 apply (drule rtranclD)

 apply auto

 done

 lemma rtrancl_into_rtrancl2: 

   "\<lbrakk> (a, b) \<in>  r; (b, c) \<in> r^* \<rbrakk> \<Longrightarrow> (a, c) \<in>  r^*"

 by (auto intro: r_into_rtrancl rtrancl_trans)

 lemma triangle_lemma:

  "\<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> 

  \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"

 proof -

   assume unique: "\<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c"

   assume "(a,x)\<in>r\<^sup>*" 

   then show "(a,y)\<in>r\<^sup>* \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"

   proof (induct rule: converse_rtrancl_induct)

     assume "(x,y)\<in>r\<^sup>*"

     then show ?thesis 

       by blast

   next

     fix a v

     assume a_v_r: "(a, v) \<in> r" and

           v_x_rt: "(v, x) \<in> r\<^sup>*" and

           a_y_rt: "(a, y) \<in> r\<^sup>*"  and

              hyp: "(v, y) \<in> r\<^sup>* \<Longrightarrow> (x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"

     from a_y_rt 

     show "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"

     proof (cases rule: converse_rtranclE)

       assume "a=y"

       with a_v_r v_x_rt have "(y,x) \<in> r\<^sup>*"

         by (auto intro: r_into_rtrancl rtrancl_trans)

       then show ?thesis 

         by blast

     next

       fix w 

       assume a_w_r: "(a, w) \<in> r" and

             w_y_rt: "(w, y) \<in> 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

 qed

 lemma rtrancl_cases [consumes 1, case_names Refl Trancl]:

  "\<lbrakk>(a,b)\<in>r\<^sup>*;  a = b \<Longrightarrow> P; (a,b)\<in>r\<^sup>+ \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"

 apply (erule rtranclE)

 apply (auto dest: rtrancl_into_trancl1)

 done

 (* context (theory "Set") *)

 lemma Ball_weaken:"\<lbrakk>Ball s P;\<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"

 by auto

 (* context (theory "Finite") *)

 lemma finite_SetCompr2: "[| finite (Collect P); !y. P y --> finite (range (f y)) |] ==>  

   finite {f y x |x y. P y}"

 apply (subgoal_tac "{f y x |x y. P y} = UNION (Collect P) (%y. range (f y))")

 prefer 2 apply  fast

 apply (erule ssubst)

 apply (erule finite_UN_I)

 apply fast

 done

 (* ### TO theory "List" *)

 lemma list_all2_trans: "\<forall> a b c. P1 a b \<longrightarrow> P2 b c \<longrightarrow> P3 a c \<Longrightarrow>

  \<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

 section "pairs"

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

   snd (snd (snd (snd p))))"

 apply auto

 done

 lemma fst_splitE [elim!]: 

 "[| fst s' = x';  !!x s. [| s' = (x,s);  x = x' |] ==> Q |] ==> Q"

 by (cases s') auto

 lemma fst_in_set_lemma [rule_format (no_asm)]: "(x, y) : set l --> x : fst ` set l"

 apply (induct_tac "l")

 apply  auto

 done

 section "quantifiers"

 lemma All_Ex_refl_eq2 [simp]: 

  "(!x. (? b. x = f b & Q b) \<longrightarrow> P x) = (!b. Q b --> P (f b))"

 apply auto

 done

 lemma ex_ex_miniscope1 [simp]:

   "(EX w v. P w v & Q v) = (EX v. (EX w. P w v) & Q v)"

 apply auto

 done

 lemma ex_miniscope2 [simp]:

   "(EX v. P v & Q & R v) = (Q & (EX v. P v & R v))" 

 apply auto

 done

 lemma ex_reorder31: "(\<exists>z x y. P x y z) = (\<exists>x y z. P x y z)"

 apply auto

 done

 lemma All_Ex_refl_eq1 [simp]: "(!x. (? b. x = f b) --> P x) = (!b. P (f b))"

 apply auto

 done

 section "sums"

 hide_const In0 In1

 notation sum_case  (infixr "'(+')"80)

 primrec the_Inl  :: "'a + 'b \<Rightarrow> 'a"

   where "the_Inl (Inl a) = a"

 primrec the_Inr  :: "'a + 'b \<Rightarrow> '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 \<Rightarrow> 'a"

   where "the_In1 (In1 a) = a"

 primrec the_In2  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'b"

   where "the_In2 (In2 b) = b"

 primrec the_In3  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'c"

   where "the_In3 (In3 c) = c"

 abbreviation In1l   :: "'al \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"

   where "In1l e == In1 (Inl e)"

 abbreviation In1r   :: "'ar \<Rightarrow> ('al + 'ar, 'b, 'c) sum3"

   where "In1r c == In1 (Inr c)"

 abbreviation the_In1l :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'al"

   where "the_In1l == the_Inl \<circ> the_In1"

 abbreviation the_In1r :: "('al + 'ar, 'b, 'c) sum3 \<Rightarrow> 'ar"

   where "the_In1r == the_Inr \<circ> the_In1"

 ML {*

 fun sum3_instantiate ctxt thm = map (fn s =>

   simplify (simpset_of ctxt delsimps[@{thm not_None_eq}])

     (read_instantiate ctxt [(("t", 0), "In" ^ s ^ " ?x")] thm)) ["1l","2","3","1r"]

 *}

 (* e.g. lemmas is_stmt_rews = is_stmt_def [of "In1l x", simplified] *)

 section "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\<forall>_\<in>_:/ _)"  [0,0,10] 10)

   "_Oex"  :: "[pttrn, 'a option, bool] => bool"   ("(3\<exists>_\<in>_:/ _)"  [0,0,10] 10)

 translations

   "! x:A: P"    == "! x:CONST Option.set A. P"

   "? x:A: P"    == "? x:CONST Option.set A. P"

 section "Special map update"

 text{* Deemed too special for theory Map. *}

 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"

 by (unfold chg_map_def, auto)

 lemma chg_map_upd[simp]: "m a = Some b ==> chg_map f a m = m(a|->f b)"

 by (unfold chg_map_def, auto)

 lemma chg_map_other [simp]: "a \<noteq> b \<Longrightarrow> chg_map f a m b = m b"

 by (auto simp: chg_map_def split add: option.split)

 section "unique association lists"

 definition

   unique :: "('a \<times> 'b) list \<Rightarrow> bool"

   where "unique = distinct \<circ> map fst"

 lemma uniqueD [rule_format (no_asm)]: 

 "unique l--> (!x y. (x,y):set l --> (!x' y'. (x',y'):set l --> x=x'-->  y=y'))"

 apply (unfold unique_def o_def)

 apply (induct_tac "l")

 apply  (auto dest: fst_in_set_lemma)

 done

 lemma unique_Nil [simp]: "unique []"

 apply (unfold unique_def)

 apply (simp (no_asm))

 done

 lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l & (!y. (x,y) ~: set l))"

 apply (unfold unique_def)

 apply  (auto dest: fst_in_set_lemma)

 done

 lemmas unique_ConsI = conjI [THEN unique_Cons [THEN iffD2], standard]

 lemma unique_single [simp]: "!!p. unique [p]"

 apply auto

 done

 lemma unique_ConsD: "unique (x#xs) ==> unique xs"

 apply (simp add: unique_def)

 done

 lemma unique_append [rule_format (no_asm)]: "unique l' ==> unique l -->  

   (!(x,y):set l. !(x',y'):set l'. x' ~= x) --> unique (l @ l')"

 apply (induct_tac "l")

 apply  (auto dest: fst_in_set_lemma)

 done

 lemma unique_map_inj [rule_format (no_asm)]: "unique l --> inj f --> unique (map (%(k,x). (f k, g k x)) l)"

 apply (induct_tac "l")

 apply  (auto dest: fst_in_set_lemma simp add: inj_eq)

 done

 lemma map_of_SomeI [rule_format (no_asm)]: "unique l --> (k, x) : set l --> map_of l k = Some x"

 apply (induct_tac "l")

 apply auto

 done

 section "list patterns"

 definition

   lsplit :: "[['a, 'a list] => 'b, 'a list] => 'b" where

   "lsplit = (\<lambda>f l. f (hd l) (tl l))"

 text {* list patterns -- extends pre-defined type "pttrn" used in abstractions *}

 syntax

   "_lpttrn"    :: "[pttrn,pttrn] => pttrn"     ("_#/_" [901,900] 900)

 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"

 apply (unfold lsplit_def)

 apply (simp (no_asm))

 done

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

 apply (unfold lsplit_def)

 apply simp

 done 

 end