theory AbsRepTest

imports "../Quotient" "../Quotient_List" "../Quotient_Option" "../Quotient_Sum" "../Quotient_Product" List

begin

(*

ML_command "ProofContext.debug := false"

ML_command "ProofContext.verbose := false"

*)

ML {* open Quotient_Term *}

ML {*

fun test_funs flag ctxt (rty, qty) =

  (absrep_fun_chk flag ctxt (rty, qty)

   |> Syntax.string_of_term ctxt

   |> writeln;

   equiv_relation_chk ctxt (rty, qty) 

   |> Syntax.string_of_term ctxt

   |> writeln)

*}

definition

  erel1 (infixl "\<approx>1" 50)

where

  "erel1 \<equiv> \<lambda>xs ys. \<forall>e. e \<in> set xs \<longleftrightarrow> e \<in> set ys"

quotient_type 

  'a fset = "'a list" / erel1

  apply(rule equivpI)

  unfolding erel1_def reflp_def symp_def transp_def

  by auto

definition

  erel2 (infixl "\<approx>2" 50)

where

  "erel2 \<equiv> \<lambda>(xs::('a * 'a) list) ys. \<forall>e. e \<in> set xs \<longleftrightarrow> e \<in> set ys"

quotient_type 

  'a foo = "('a * 'a) list" / erel2

  apply(rule equivpI)

  unfolding erel2_def reflp_def symp_def transp_def

  by auto

definition

  erel3 (infixl "\<approx>3" 50)

where

  "erel3 \<equiv> \<lambda>(xs::('a * int) list) ys. \<forall>e. e \<in> set xs \<longleftrightarrow> e \<in> set ys"

quotient_type 

  'a bar = "('a * int) list" / "erel3"

  apply(rule equivpI)

  unfolding erel3_def reflp_def symp_def transp_def

  by auto

fun

  intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool" (infixl "\<approx>4" 50)

where

  "intrel (x, y) (u, v) = (x + v = u + y)"

quotient_type myint = "nat \<times> nat" / intrel

  by (auto simp add: equivp_def expand_fun_eq)

ML {*

test_funs AbsF @{context} 

     (@{typ "nat \<times> nat"}, 

      @{typ "myint"})

*}

ML {*

test_funs AbsF @{context} 

     (@{typ "('a * 'a) list"}, 

      @{typ "'a foo"})

*}

ML {*

test_funs RepF @{context} 

     (@{typ "(('a * 'a) list * 'b)"}, 

      @{typ "('a foo * 'b)"})

*}

ML {*

test_funs AbsF @{context} 

     (@{typ "(('a list) * int) list"}, 

      @{typ "('a fset) bar"})

*}

ML {*

test_funs AbsF @{context} 

     (@{typ "('a list)"}, 

      @{typ "('a fset)"})

*}

ML {*

test_funs AbsF @{context} 

     (@{typ "('a list) list"}, 

      @{typ "('a fset) fset"})

*}

ML {*

test_funs AbsF @{context} 

     (@{typ "((nat * nat) list) list"}, 

      @{typ "((myint) fset) fset"})

*}

ML {*

test_funs AbsF @{context} 

     (@{typ "(('a * 'a) list) list"}, 

      @{typ "(('a * 'a) fset) fset"})

*}

ML {*

test_funs AbsF @{context} 

      (@{typ "(nat * nat) list"}, 

       @{typ "myint fset"})

*}

ML {*

test_funs AbsF @{context} 

     (@{typ "('a list) list \<Rightarrow> 'a list"}, 

      @{typ "('a fset) fset \<Rightarrow> 'a fset"})

*}

lemma OO_sym_inv:

  assumes sr: "symp r"

  and     ss: "symp s"

  shows "(r OO s) x y = (s OO r) y x"

  using sr ss

  unfolding symp_def

  apply (metis pred_comp.intros pred_compE ss symp_def)

  done

lemma abs_o_rep:

  assumes a: "Quotient r absf repf"

  shows "absf o repf = id"

  apply(rule ext)

  apply(simp add: Quotient_abs_rep[OF a])

  done

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>1 ba \<Longrightarrow> map f b \<approx>1 map f ba"

  unfolding erel1_def

  apply(simp only: set_map set_in_eq)

  done

lemma quotient_compose_list_gen_pre:

  assumes a: "equivp r2"

  and b: "Quotient r2 abs2 rep2"

  shows  "(list_rel r2 OOO op \<approx>1) r s =

          ((list_rel r2 OOO op \<approx>1) r r \<and> (list_rel r2 OOO op \<approx>1) s s \<and>

           abs_fset (map abs2 r) = abs_fset (map abs2 s))"

  apply rule

  apply rule

  apply rule

  apply (rule list_rel_refl)

  apply (metis equivp_def a)

  apply rule

  apply (rule equivp_reflp[OF fset_equivp])

  apply (rule list_rel_refl)

  apply (metis equivp_def a)

  apply(rule)

  apply rule

  apply (rule list_rel_refl)

  apply (metis equivp_def a)

  apply rule

  apply (rule equivp_reflp[OF fset_equivp])

  apply (rule list_rel_refl)

  apply (metis equivp_def a)

  apply (subgoal_tac "map abs2 r \<approx>1 map abs2 s")

  apply (metis Quotient_rel[OF Quotient_fset])

  apply (auto)[1]

  apply (subgoal_tac "map abs2 r = map abs2 b")

  prefer 2

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

  apply (subgoal_tac "map abs2 s = map abs2 ba")

  prefer 2

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

  apply (simp add: map_rel_cong)

  apply rule

  apply (rule rep_abs_rsp[of "list_rel r2" "map abs2"])

  apply (rule list_quotient)

  apply (rule b)

  apply (rule list_rel_refl)

  apply (metis equivp_def a)

  apply rule

  prefer 2

  apply (rule rep_abs_rsp_left[of "list_rel r2" "map abs2"])

  apply (rule list_quotient)

  apply (rule b)

  apply (rule list_rel_refl)

  apply (metis equivp_def a)

  apply (erule conjE)+

  apply (subgoal_tac "map abs2 r \<approx>1 map abs2 s")

  apply (rule map_rel_cong)

  apply (assumption)

  apply (metis Quotient_def Quotient_fset equivp_reflp fset_equivp a b)

  done

lemma quotient_compose_list_gen:

  assumes a: "Quotient r2 abs2 rep2"

  and     b: "equivp r2" (* reflp is not enough *)

  shows  "Quotient ((list_rel r2) OOO (op \<approx>1))

               (abs_fset \<circ> (map abs2)) ((map rep2) \<circ> rep_fset)"

  unfolding Quotient_def comp_def

  apply (rule)+

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

  apply (rule)

  apply (rule)

  apply (rule)

  apply (rule list_rel_refl)

  apply (metis b equivp_def)

  apply (rule)

  apply (rule equivp_reflp[OF fset_equivp])

  apply (rule list_rel_refl)

  apply (metis b equivp_def)

  apply rule

  apply rule

  apply(rule quotient_compose_list_gen_pre[OF b a])

  done

(* This is the general statement but the types of abs2 and rep2

   are wrong as can be seen in following exanples *)

lemma quotient_compose_general:

  assumes a2: "Quotient r1 abs1 rep1"

  and         "Quotient r2 abs2 rep2"

  shows  "Quotient ((list_rel r2) OOO r1)

               (abs1 \<circ> (map abs2)) ((map rep2) \<circ> rep1)"

sorry

thm quotient_compose_list_gen[OF Quotient_fset fset_equivp]

thm quotient_compose_general[OF Quotient_fset]

(* Doesn't work: *)

(* thm quotient_compose_general[OF Quotient_fset Quotient_fset] *)

end