theory FSet3

begin

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)

(* PROBLEM: these lemmas needs to be restated, since  *)

(* concat.simps(1) and concat.simps(2) contain the    *)

(* type variables ?'a1.0 (which are turned into frees *)

(* 'a_1                                               *)

lemma concat1:

  shows "concat [] \<approx> []"

lemma concat2:

  shows "concat (x # xs) \<approx> x @ concat xs"

by (simp)

lemma concat_rsp:

  "\<lbrakk>list_rel op \<approx> x x'; x' \<approx> y'; list_rel op \<approx> y' y\<rbrakk> \<Longrightarrow> concat x \<approx> concat y"

  apply (induct x y arbitrary: x' y' rule: list_induct2')

  apply simp

  defer defer

  apply (simp only: concat.simps)

  sorry

lemma [quot_respect]:

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

  apply (simp only: fun_rel_def)

  apply clarify

  apply (rule concat_rsp)

  apply assumption+

  done

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

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

  apply (rule eq_reflection)

  apply auto

  done

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

  unfolding list_eq.simps

  apply(simp only: set_map set_in_eq)

  done

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

  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")

  prefer 2

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

  apply (subgoal_tac "map abs_fset s = map abs_fset ba")

  prefer 2

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

  apply (simp only: map_rel_cong)

  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 (erule conjE)+

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

  prefer 2

  apply (metis Quotient_def Quotient_fset equivp_reflp fset_equivp)

  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] id_simps Quotient_abs_rep[OF Quotient_fset])

  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 {||} = {||}"

  apply(lifting concat1)

  apply(cleaning)

  done

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)

  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"

  apply(lifting concat2)

  apply(cleaning)

  apply (simp add: finsert_def fconcat_def comp_def)

  apply cleaning

  done

text {* raw section *}

lemma map_rsp_aux:

  assumes a: "a \<approx> b"

  shows "map f a \<approx> map f b"

  using a

apply(induct a arbitrary: b)

apply(auto)

apply(metis rev_image_eqI)

done

lemma map_rsp[quot_respect]:

  shows "(op = ===> op \<approx> ===> op \<approx>) map map"

by (auto simp add: map_rsp_aux)

text {* lifted section *}

(* 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 *)

lemma no_mem_nil: 

  "(\<forall>a. a \<notin> set A) = (A = [])"

by (induct A) (auto)

lemma none_mem_nil: 

  "(\<forall>a. a \<notin> set A) = (A \<approx> [])"

by simp

lemma mem_cons: 

  "a \<in> set A \<Longrightarrow> a # A \<approx> A"

by auto

lemma cons_left_comm:

  "x # y # A \<approx> y # x # A"

by (auto simp add: id_simps)

lemma cons_left_idem:

  "x # x # A \<approx> x # A"

by (auto simp add: id_simps)

lemma finite_set_raw_strong_cases:

  "(X = []) \<or> (\<exists>a Y. ((a \<notin> set Y) \<and> (X \<approx> a # Y)))"

  apply (induct X)

  apply (simp)

  apply (rule disjI2)

  apply (erule disjE)

  apply (rule_tac x="a" in exI)

  apply (rule_tac x="[]" in exI)

  apply (simp)

  apply (erule exE)+

  apply (case_tac "a = aa")

  apply (rule_tac x="a" in exI)

  apply (rule_tac x="Y" in exI)

  apply (simp)

  apply (rule_tac x="aa" in exI)

  apply (rule_tac x="a # Y" in exI)

  apply (auto)

  done

fun

  delete_raw :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list"

where

  "delete_raw [] x = []"

| "delete_raw (a # A) x = (if (a = x) then delete_raw A x else a # (delete_raw A x))"

lemma mem_delete_raw:

  "x \<in> set (delete_raw A a) = (x \<in> set A \<and> \<not>(x = a))"

  by (induct A arbitrary: x a) (auto)

lemma mem_delete_raw_ident:

  "\<not>(a \<in> set (delete_raw A a))"

by (induct A) (auto)

lemma not_mem_delete_raw_ident:

  "b \<notin> set A \<Longrightarrow> (delete_raw A b = A)"

by (induct A) (auto)

lemma delete_raw_RSP:

  "A \<approx> B \<Longrightarrow> delete_raw A a \<approx> delete_raw B a"

apply(induct A arbitrary: B a)

apply(auto)

sorry

lemma cons_delete_raw:

  "a # (delete_raw A a) \<approx> (if a \<in> set A then A else (a # A))"

sorry

lemma mem_cons_delete_raw:

    "a \<in> set A \<Longrightarrow> a # (delete_raw A a) \<approx> A"

sorry

lemma finite_set_raw_delete_raw_cases:

    "X = [] \<or> (\<exists>a. a mem X \<and> X \<approx> a # delete_raw X a)"

  by (induct X) (auto)

lemma list2set_thm:

  shows "set [] = {}"

  and "set (h # t) = insert h (set t)"

  by (auto)

lemma list2set_RSP:

  "A \<approx> B \<Longrightarrow> set A = set B"

  by auto

definition

  rsp_fold

where

  "rsp_fold f = (\<forall>u v w. (f u (f v w) = f v (f u w)))"

primrec

  fold_raw :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b"

where

  "fold_raw f z [] = z"

| "fold_raw f z (a # A) =

     (if (rsp_fold f) then

       if a mem A then fold_raw f z A

       else f a (fold_raw f z A)

     else z)"

lemma mem_lcommuting_fold_raw:

  "rsp_fold f \<Longrightarrow> h mem B \<Longrightarrow> fold_raw f z B = f h (fold_raw f z (delete_raw B h))"

sorry

lemma fold_rsp[quot_respect]:

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

apply(auto)

sorry

primrec

  inter_raw

where

  "inter_raw [] B = []"

| "inter_raw (a # A) B = (if a mem B then a # inter_raw A B else inter_raw A B)"

lemma mem_inter_raw:

  "x mem (inter_raw A B) = x mem A \<and> x mem B"

sorry

lemma inter_raw_RSP:

  "A1 \<approx> A2 \<and> B1 \<approx> B2 \<Longrightarrow> (inter_raw A1 B1) \<approx> (inter_raw A2 B2)"

sorry

(* LIFTING DEFS *)

section {* Constants on the Quotient Type *} 

quotient_definition

  "fdelete :: 'a fset \<Rightarrow> 'a \<Rightarrow> 'a fset" 

  is "delete_raw"

quotient_definition

  finter ("_ \<inter>f _" [70, 71] 70)

where

  "finter :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"

  is "inter_raw"

quotient_definition

  "ffold :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a fset \<Rightarrow> 'b" 

  is "fold_raw"

quotient_definition

  "fset_to_set :: 'a fset \<Rightarrow> 'a set" 

  is "set"

section {* Lifted Theorems *}

thm list.cases (* ??? *)

thm cons_left_comm

lemma "finsert a (finsert b S) = finsert b (finsert a S)"

by (lifting cons_left_comm)

thm cons_left_idem

lemma "finsert a (finsert a S) = finsert a S"

by (lifting cons_left_idem)

(* thm MEM:

  MEM x [] = F

  MEM x (h::t) = (x=h) \/ MEM x t *)

thm none_mem_nil

(*lemma "(\<forall>a. a \<notin>f A) = (A = fempty)"*)

thm mem_cons

thm finite_set_raw_strong_cases

(*thm card_raw.simps*)

(*thm not_mem_card_raw*)

(*thm card_raw_suc*)

lemma "fcard X = Suc n \<Longrightarrow> (\<exists>a S. a \<notin>f S & X = finsert a S)"

(*by (lifting card_raw_suc)*)

sorry

(*thm card_raw_cons_gt_0

thm mem_card_raw_gt_0

thm not_nil_equiv_cons

*)

thm delete_raw.simps

(*thm mem_delete_raw*)

(*thm card_raw_delete_raw*)

thm cons_delete_raw

thm mem_cons_delete_raw

thm finite_set_raw_delete_raw_cases

thm append.simps

(* MEM_APPEND: MEM e (APPEND l1 l2) = MEM e l1 \/ MEM e l2 *)

thm inter_raw.simps

thm mem_inter_raw

thm fold_raw.simps

thm list2set_thm

thm list_eq_def

thm list.induct

lemma "\<lbrakk>P fempty; \<And>a x. P x \<Longrightarrow> P (finsert a x)\<rbrakk> \<Longrightarrow> P l"

by (lifting list.induct)

(* We also have map and some properties of it in FSet *)

(* and the following which still lifts ok *)

lemma "funion (funion x xa) xb = funion x (funion xa xb)"

by (lifting append_assoc)

end