(*  Title:      HOL/Bali/Basis.thy

    ID:         $Id$

    Author:     David von Oheimb

    License:    GPL (GNU GENERAL PUBLIC LICENSE)

*)

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

theory Basis = Main:

ML_setup {*

Unify.search_bound := 40;

Unify.trace_bound  := 40;

*}

(*print_depth 100;*)

(*Syntax.ambiguity_level := 1;*)

section "misc"

declare same_fstI [intro!] (*### TO HOL/Wellfounded_Relations *)

(* ###TO HOL/???.ML?? *)

ML {*

fun make_simproc name pat pred thm = Simplifier.mk_simproc name

   [Thm.read_cterm (Thm.sign_of_thm thm) (pat, HOLogic.typeT)] 

   (K (K (fn s => if pred s then None else Some (standard (mk_meta_eq thm)))))

*}

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

ML {*

simpset_ref() := simpset() addloop ("split_all_tac", split_all_tac)

*}

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

declare length_Suc_conv [iff];

(*###to be phased out *)

ML {*

bind_thm ("make_imp", rearrange_prems [1,0] mp)

*}

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

syntax

  "3" :: nat   ("3")

  "4" :: nat   ("4")

translations

 "3" == "Suc 2"

 "4" == "Suc 3"

(*unused*)

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

apply auto

apply (case_tac "xa")

apply auto

done

(* context (theory "Transitive_Closure") *)

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

apply (rule allI)

apply (erule irrefl_tranclI)

done

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 -

  note converse_rtrancl_induct = converse_rtrancl_induct [consumes 1]

  note converse_rtranclE = converse_rtranclE [consumes 1] 

  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

(* ### To Transitive_Closure *)

theorems converse_rtrancl_induct 

 = converse_rtrancl_induct [consumes 1,case_names Id Step]

theorems converse_trancl_induct 

         = converse_trancl_induct [consumes 1,case_names Single Step]

(* 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"

apply (cut_tac p = "s'" in surjective_pairing)

apply auto

done

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"

(*###to be phased out *)

ML {* 

fun noAll_simpset () = simpset() setmksimps 

	mksimps (filter (fn (x,_) => x<>"All") mksimps_pairs)

*}

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

syntax

  fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)

translations

 "fun_sum" == "sum_case"

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

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

primrec  "the_Inl (Inl a) = a"

primrec  "the_Inr (Inr b) = b"

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

consts    the_In1  :: "('a, 'b, 'c) sum3 \<Rightarrow> 'a"

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

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

primrec  "the_In1 (In1 a) = a"

primrec  "the_In2 (In2 b) = b"

primrec  "the_In3 (In3 c) = c"

syntax

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

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

translations

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

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

ML {*

fun sum3_instantiate thm = map (fn s => simplify(simpset()delsimps[not_None_eq])

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

*}

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

translations

  "option"<= (type) "Option.option"

  "list"  <= (type) "List.list"

  "sum3"  <= (type) "Basis.sum3"

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:o2s A. P"

  "? x:A: P"    == "? x:o2s A. P"

section "unique association lists"

constdefs

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

 "unique \<equiv> nodups \<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"

consts

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

defs

  lsplit_def:    "lsplit == %f l. f (hd l) (tl l)"

(*  list patterns -- extends pre-defined type "pttrn" used in abstractions *)

syntax

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

translations

  "%y#x#xs. b"  == "lsplit (%y x#xs. b)"

  "%x#xs  . b"  == "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 

section "dummy pattern for quantifiers, let, etc."

syntax

  "@dummy_pat"   :: pttrn    ("'_")

parse_translation {*

let fun dummy_pat_tr [] = Free ("_",dummyT)

  | dummy_pat_tr ts = raise TERM ("dummy_pat_tr", ts);

in [("@dummy_pat", dummy_pat_tr)] 

end

*}

end