theory FSet3

imports "../QuotMain" "../QuotList" List

begin

fun

  list_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)

where

  "list_eq xs ys = (\<forall>x. x \<in> set xs \<longleftrightarrow> x \<in> set ys)"

lemma list_eq_equivp:

  shows "equivp list_eq"

unfolding equivp_reflp_symp_transp reflp_def symp_def transp_def

by auto

(* FIXME-TODO: because of beta-reduction, one cannot give the *)

(* FIXME-TODO: relation as a term or abbreviation             *) 

quotient_type 

  'a fset = "'a list" / "list_eq"

by (rule list_eq_equivp)

section {* empty fset, finsert and membership *} 

quotient_definition

  "fempty :: 'a fset" ("{||}")

as "[]::'a list"

quotient_definition

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

as "op #"

syntax

  "@Finset"     :: "args => 'a fset"  ("{|(_)|}")

translations

  "{|x, xs|}" == "CONST finsert x {|xs|}"

  "{|x|}"     == "CONST finsert x {||}"

definition 

  memb :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"

where

  "memb x xs \<equiv> x \<in> set xs"

quotient_definition

  "fin :: 'a \<Rightarrow> 'a fset \<Rightarrow> bool" ("_ |\<in>| _" [50, 51] 50)

as "memb"

abbreviation

  fnotin :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" ("_ |\<notin>| _" [50, 51] 50)

where

  "a |\<notin>| S \<equiv> \<not>(a |\<in>| S)"

lemma memb_rsp[quot_respect]: 

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

by (auto simp add: memb_def)

lemma nil_rsp[quot_respect]:

  shows "[] \<approx> []"

by simp

lemma cons_rsp[quot_respect]: 

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

by simp

section {* Augmenting a set -- @{const finsert} *}

text {* raw section *}

lemma nil_not_cons: 

  shows "\<not>[] \<approx> x # xs" 

by auto

lemma memb_cons_iff:

  shows "memb x (y # xs) = (x = y \<or> memb x xs)"

by (induct xs) (auto simp add: memb_def)

lemma memb_consI1:

  shows "memb x (x # xs)"

by (simp add: memb_def)

lemma memb_consI2: 

  shows "memb x xs \<Longrightarrow> memb x (y # xs)"

  by (simp add: memb_def)

lemma memb_absorb:

  shows "memb x xs \<Longrightarrow> (x # xs) \<approx> xs"

by (induct xs) (auto simp add: memb_def)

text {* lifted section *}

lemma fempty_not_finsert[simp]:

  shows "{||} \<noteq> finsert x S"

by (lifting nil_not_cons)

lemma fin_finsert_iff[simp]:

  "x |\<in>| finsert y S = (x = y \<or> x |\<in>| S)"

by (lifting memb_cons_iff) 

lemma

  shows finsertI1: "x |\<in>| finsert x S"

  and   finsertI2: "x |\<in>| S \<Longrightarrow> x |\<in>| finsert y S"

  by (lifting memb_consI1, lifting memb_consI2)

lemma finsert_absorb [simp]: 

  shows "x |\<in>| S \<Longrightarrow> finsert x S = S"

by (lifting memb_absorb)

section {* Singletons *}

text {* raw section *}

lemma singleton_list_eq:

  shows "[x] \<approx> [y] \<longleftrightarrow> x = y"

by auto

text {* lifted section *}

lemma fsingleton_eq[simp]:

  shows "{|x|} = {|y|} \<longleftrightarrow> x = y"

by (lifting singleton_list_eq)

section {* Union *}

quotient_definition

   "funion :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset" (infixl "|\<union>|" 65)

as

  "op @"

section {* Cardinality of finite sets *}

fun

  fcard_raw :: "'a list \<Rightarrow> nat"

where

  fcard_raw_nil:  "fcard_raw [] = 0"

| fcard_raw_cons: "fcard_raw (x # xs) = (if memb x xs then fcard_raw xs else Suc (fcard_raw xs))"

quotient_definition

  "fcard :: 'a fset \<Rightarrow> nat" 

as "fcard_raw"

text {* raw section *}

lemma fcard_raw_ge_0:

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

  shows "0 < fcard_raw xs"

using a

by (induct xs) (auto simp add: memb_def)

lemma fcard_raw_delete_one:

  "fcard_raw ([x \<leftarrow> xs. x \<noteq> y]) = (if memb y xs then fcard_raw xs - 1 else fcard_raw xs)"

by (induct xs) (auto dest: fcard_raw_ge_0 simp add: memb_def)

lemma fcard_raw_rsp_aux:

  assumes a: "a \<approx> b"

  shows "fcard_raw a = fcard_raw b"

using a

apply(induct a arbitrary: b)

apply(auto simp add: memb_def)

apply(metis)

apply(drule_tac x="[x \<leftarrow> b. x \<noteq> a1]" in meta_spec)

apply(simp add: fcard_raw_delete_one)

apply(metis Suc_pred' fcard_raw_ge_0 fcard_raw_delete_one memb_def)

done

lemma fcard_raw_rsp[quot_respect]:

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

  by (simp add: fcard_raw_rsp_aux)

text {* lifted section *}

lemma fcard_fempty [simp]:

  shows "fcard {||} = 0"

by (lifting fcard_raw_nil)

lemma fcard_finsert_if [simp]:

  shows "fcard (finsert x S) = (if x |\<in>| S then fcard S else Suc (fcard S))"

by (lifting fcard_raw_cons)

section {* Induction and Cases rules for finite sets *}

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 fset_induct[case_names fempty finsert]:

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

by (lifting list.induct)

lemma fset_induct2[case_names fempty finsert, induct type: fset]:

  assumes prem1: "P {||}" 

  and     prem2: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (finsert x S)"

  shows "P S"

proof(induct S rule: fset_induct)

  case fempty

  show "P {||}" by (rule prem1)

next

  case (finsert x S)

  have asm: "P S" by fact

  show "P (finsert x S)"

  proof(cases "x |\<in>| S")

    case True

    have "x |\<in>| S" by fact

    then show "P (finsert x S)" using asm by simp

  next

    case False

    have "x |\<notin>| S" by fact

    then show "P (finsert x S)" using prem2 asm by simp

  qed

qed

section {* fmap and fset comprehension *}

quotient_definition

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

as

 "map"

quotient_definition

  "fconcat :: ('a fset) fset => 'a fset"

as

 "concat"

lemma fconcat_rsp[quot_respect]:

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

apply(auto)

sorry

(* 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 [] = []"

by (simp)

lemma concat2: 

  shows "concat (x # xs) =  x @ concat xs"

by (simp)

lemma fconcat_empty:

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

apply(lifting concat1)

apply(injection)

defer

apply(cleaning)

apply(simp add: comp_def)

apply(cleaning)

apply(fold fempty_def[simplified id_simps])

apply(rule refl)

sorry

lemma fconcat_insert:

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

apply(lifting concat2)

apply(injection)

defer

apply(cleaning)

apply(simp add: comp_def)

apply(cleaning)

sorry

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

*}

(* 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

lemma cons_left_idem: 

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

by auto

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

lemma append_rsp[quot_respect]:

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

by auto

primrec