theory FSet3

imports "../../../Nominal/FSet"

begin

notation

  list_eq (infix "\<approx>" 50)

lemma fset_exhaust[case_names fempty finsert, cases type: fset]:

  shows "\<lbrakk>S = {||} \<Longrightarrow> P; \<And>x S'. S = finsert x S' \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"

by (lifting list.exhaust)

lemma list_rel_find_element:

  assumes a: "x \<in> set a"

  and b: "list_rel R a b"

  shows "\<exists>y. (y \<in> set b \<and> R x y)"

proof -

  have "length a = length b" using b by (rule list_rel_len)

  then show ?thesis using a b proof (induct a b rule: list_induct2)

    case Nil

    show ?case using Nil.prems by simp

  next

    case (Cons x xs y ys)

    show ?case using Cons by auto

  qed

qed

lemma concat_rsp_pre:

lemma [quot_respect]:

  shows "(list_rel op \<approx> OOO op \<approx> ===> op \<approx>) concat concat"

lemma nil_rsp2[quot_respect]: "(list_rel op \<approx> OOO op \<approx>) [] []"

  by (metis nil_rsp list_rel.simps(1) pred_compI)

lemma set_in_eq: "(\<forall>e. ((e \<in> A) \<longleftrightarrow> (e \<in> B))) \<equiv> A = B"

  by (rule eq_reflection) auto

lemma map_rel_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> map f ba"

  unfolding list_eq.simps

  by (simp only: set_map set_in_eq)

lemma quotient_compose_list_pre:

  "(list_rel op \<approx> OOO op \<approx>) r s =

  ((list_rel op \<approx> OOO op \<approx>) r r \<and> (list_rel op \<approx> OOO op \<approx>) s s \<and>

  abs_fset (map abs_fset r) = abs_fset (map abs_fset s))"

  apply rule

  apply rule

  apply rule

  apply (rule list_rel_refl)

  apply (metis equivp_def fset_equivp)

  apply rule

  apply (rule equivp_reflp[OF fset_equivp])

  apply (rule list_rel_refl)

  apply (metis equivp_def fset_equivp)

  apply rule

  apply (rule list_rel_refl)

  apply (metis equivp_def fset_equivp)

  apply rule

  apply (rule equivp_reflp[OF fset_equivp])

  apply (rule list_rel_refl)

  apply (metis equivp_def fset_equivp)

  apply (subgoal_tac "map abs_fset r \<approx> map abs_fset s")

  apply (metis Quotient_rel[OF Quotient_fset])

  apply (auto simp only:)[1]

  apply (subgoal_tac "map abs_fset r = map abs_fset b")

  apply (metis Quotient_rel[OF list_quotient[OF Quotient_fset]])

  apply (metis Quotient_rel[OF list_quotient[OF Quotient_fset]])

  apply rule

  apply (rule rep_abs_rsp[of "list_rel op \<approx>" "map abs_fset"])

  apply (tactic {* Quotient_Tacs.quotient_tac @{context} 1 *})

  apply (rule list_rel_refl)

  apply (metis equivp_def fset_equivp)

  apply rule

  prefer 2

  apply (rule rep_abs_rsp_left[of "list_rel op \<approx>" "map abs_fset"])

  apply (tactic {* Quotient_Tacs.quotient_tac @{context} 1 *})

  apply (rule list_rel_refl)

  apply (metis equivp_def fset_equivp)

  apply (subgoal_tac "map abs_fset r \<approx> map abs_fset s")

  apply (rule map_rel_cong)

  apply (assumption)

  done

lemma quotient_compose_list[quot_thm]:

  shows  "Quotient ((list_rel op \<approx>) OOO (op \<approx>))

    (abs_fset \<circ> (map abs_fset)) ((map rep_fset) \<circ> rep_fset)"

  unfolding Quotient_def comp_def

  apply (rule)+

  apply (simp add: abs_o_rep[OF Quotient_fset] Quotient_abs_rep[OF Quotient_fset] map_id)

  apply (rule)

  apply (rule)

  apply (rule)

  apply (rule list_rel_refl)

  apply (metis equivp_def fset_equivp)

  apply (rule)

  apply (rule equivp_reflp[OF fset_equivp])

  apply (rule list_rel_refl)

  apply (metis equivp_def fset_equivp)

  apply rule

  apply rule

  apply (rule quotient_compose_list_pre)

  done

lemma fconcat_empty:

  shows "fconcat {||} = {||}"

lemma insert_rsp2[quot_respect]:

  "(op \<approx> ===> list_rel op \<approx> OOO op \<approx> ===> list_rel op \<approx> OOO op \<approx>) op # op #"

  apply auto

  apply (simp add: set_in_eq)

  apply (rule_tac b="x # b" in pred_compI)

  apply auto

  apply (rule_tac b="x # ba" in pred_compI)

  apply auto

  done

lemma append_rsp[quot_respect]:

  "(op \<approx> ===> op \<approx> ===> op \<approx>) op @ op @"

  by (auto)

lemma fconcat_insert:

  shows "fconcat (finsert x S) = x |\<union>| fconcat S"

(* TBD *)

text {* syntax for fset comprehensions (adapted from lists) *}

nonterminals fsc_qual fsc_quals

syntax

"_fsetcompr" :: "'a \<Rightarrow> fsc_qual \<Rightarrow> fsc_quals \<Rightarrow> 'a fset"  ("{|_ . __")

"_fsc_gen" :: "'a \<Rightarrow> 'a fset \<Rightarrow> fsc_qual" ("_ <- _")

"_fsc_test" :: "bool \<Rightarrow> fsc_qual" ("_")

"_fsc_end" :: "fsc_quals" ("|}")

"_fsc_quals" :: "fsc_qual \<Rightarrow> fsc_quals \<Rightarrow> fsc_quals" (", __")

"_fsc_abs" :: "'a => 'b fset => 'b fset"

syntax (xsymbols)

"_fsc_gen" :: "'a \<Rightarrow> 'a fset \<Rightarrow> fsc_qual" ("_ \<leftarrow> _")

syntax (HTML output)

"_fsc_gen" :: "'a \<Rightarrow> 'a fset \<Rightarrow> fsc_qual" ("_ \<leftarrow> _")

parse_translation (advanced) {*

let

  val femptyC = Syntax.const @{const_name fempty};

  val finsertC = Syntax.const @{const_name finsert};

  val fmapC = Syntax.const @{const_name fmap};

  val fconcatC = Syntax.const @{const_name fconcat};

  val IfC = Syntax.const @{const_name If};

  fun fsingl x = finsertC $ x $ femptyC;

  fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)

    let

      val x = Free (Name.variant (fold Term.add_free_names [p, e] []) "x", dummyT);

      val e = if opti then fsingl e else e;

      val case1 = Syntax.const "_case1" $ p $ e;

      val case2 = Syntax.const "_case1" $ Syntax.const Term.dummy_patternN

                                        $ femptyC;

      val cs = Syntax.const "_case2" $ case1 $ case2

      val ft = Datatype_Case.case_tr false Datatype.info_of_constr

                 ctxt [x, cs]

    in lambda x ft end;

  fun abs_tr ctxt (p as Free(s,T)) e opti =

        let val thy = ProofContext.theory_of ctxt;

            val s' = Sign.intern_const thy s

        in if Sign.declared_const thy s'

           then (pat_tr ctxt p e opti, false)

           else (lambda p e, true)

        end

    | abs_tr ctxt p e opti = (pat_tr ctxt p e opti, false);

  fun fsc_tr ctxt [e, Const("_fsc_test",_) $ b, qs] =

        let 

          val res = case qs of 

                      Const("_fsc_end",_) => fsingl e

                    | Const("_fsc_quals",_)$ q $ qs => fsc_tr ctxt [e, q, qs];

        in 

          IfC $ b $ res $ femptyC 

        end

    | fsc_tr ctxt [e, Const("_fsc_gen",_) $ p $ es, Const("_fsc_end",_)] =

         (case abs_tr ctxt p e true of

            (f,true) => fmapC $ f $ es

          | (f, false) => fconcatC $ (fmapC $ f $ es))

    | fsc_tr ctxt [e, Const("_fsc_gen",_) $ p $ es, Const("_fsc_quals",_) $ q $ qs] =

        let

          val e' = fsc_tr ctxt [e, q, qs];

        in 

          fconcatC $ (fmapC $ (fst (abs_tr ctxt p e' false)) $ es) 

        end

in [("_fsetcompr", fsc_tr)] end

*}

(* NEEDS FIXING *)

(* examles *)

(*

term "{|(x,y,z). b|}"

term "{|x. x \<leftarrow> xs|}"

term "{|(x,y,z). x\<leftarrow>xs|}"

term "{|e x y. x\<leftarrow>xs, y\<leftarrow>ys|}"

term "{|(x,y,z). x<a, x>b|}"

term "{|(x,y,z). x\<leftarrow>xs, x>b|}"

term "{|(x,y,z). x<a, x\<leftarrow>xs|}"

term "{|(x,y). Cons True x \<leftarrow> xs|}"

term "{|(x,y,z). Cons x [] \<leftarrow> xs|}"

term "{|(x,y,z). x<a, x>b, x=d|}"

term "{|(x,y,z). x<a, x>b, y\<leftarrow>ys|}"

term "{|(x,y,z). x<a, x\<leftarrow>xs,y>b|}"

term "{|(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys|}"

term "{|(x,y,z). x\<leftarrow>xs, x>b, y<a|}"

term "{|(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys|}"

term "{|(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x|}"

term "{|(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs|}"

*)

(* BELOW CONSTRUCTION SITE *)

end