theory AbsRepTest
imports "../QuotMain" "../QuotList" "../QuotOption" "../QuotSum" "../QuotProd" List
begin

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;
   new_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
  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) = (e \<in> B))) \<equiv> A = B"
apply (rule eq_reflection)
apply auto
done

lemma map_rep_ok: "b \<approx>1 ba \<Longrightarrow> map rep_fset b \<approx>1 map rep_fset ba"
unfolding erel1_def
apply(simp only: set_map set_in_eq)
done

lemma map_abs_ok: "b \<approx>1 ba \<Longrightarrow> map abs_fset b \<approx>1 map abs_fset ba"
unfolding erel1_def
apply(simp only: set_map set_in_eq)
done

lemma quotient_compose_list_pre:
 "(list_rel op \<approx>1 OO op \<approx>1 OO list_rel op \<approx>1) r s =
       ((list_rel op \<approx>1 OO op \<approx>1 OO list_rel op \<approx>1) r r \<and>
        (list_rel op \<approx>1 OO op \<approx>1 OO list_rel op \<approx>1) 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>1 map abs_fset s")
apply (metis Quotient_rel[OF Quotient_fset])
apply (auto)[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 add: map_abs_ok)
apply rule
apply (rule rep_abs_rsp[of "list_rel op \<approx>1" "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>1" "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>1 map abs_fset s")
prefer 2
apply (metis Quotient_def Quotient_fset equivp_reflp fset_equivp)
apply (rule map_rep_ok)
apply (assumption)
done

lemma quotient_compose_list:
  shows  "Quotient ((list_rel op \<approx>1) OO (op \<approx>1) OO (list_rel op \<approx>1))
               (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 quotient_compose_list_gen:
  assumes a1: "Quotient (op \<approx>1) abs_fset rep_fset"
  and     a2:  "Quotient r2 abs2 rep2" "equivp r2"
  shows  "Quotient ((list_rel r2) OO (op \<approx>1) OO (list_rel r2))
               (abs_fset \<circ> (map abs2)) ((map rep2) \<circ> rep_fset)"
  unfolding Quotient_def comp_def
apply (rule)+
apply (simp add: abs_o_rep[OF a2(1)] id_simps Quotient_abs_rep[OF Quotient_fset])
apply (rule)
apply (rule)
apply (rule)
apply (rule list_rel_refl)
apply (metis a2(2) equivp_def)
apply (rule)
apply (rule equivp_reflp[OF fset_equivp])
apply (rule list_rel_refl)
apply (metis a2(2) equivp_def)
apply rule
apply rule
apply(rule quotient_compose_list_gen_pre)
done

(* This is the general statement but the types are wrong as in following exanples *)
lemma quotient_compose_general:
  assumes a2: "Quotient r1 abs1 rep_fset"
  and         "Quotient r2 abs2 rep2"
  shows  "Quotient ((list_rel r2) OO r1 OO (list_rel r2))
               (abs1 \<circ> (map abs2)) ((map rep2) \<circ> rep_fset)"
sorry

thm quotient_compose_ok     [OF Quotient_fset]
thm quotient_compose_general[OF Quotient_fset]

thm quotient_compose_ok     [OF Quotient_fset Quotient_fset]
(* Doesn't work: *)
(* thm quotient_compose_general[OF Quotient_fset Quotient_fset] *)

end