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 bla:

  assumes a1: "Quotient (op \<approx>1) abs_fset rep_fset"

  and     a2:  "Quotient r2 abs2 rep2"

  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] id_simps Quotient_abs_rep[OF Quotient_fset])

apply (rule)

apply (rule)

sledgehammer

apply (metis Quotient_def a2 equivp_reflp fset_equivp list_quotient list_rel_rel pred_comp.cases pred_comp.intros rep_fset_def)

using a1

apply -

sorry

lemma bla2:

  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 bla [OF Quotient_fset]

thm bla2[OF Quotient_fset]

thm bla [OF Quotient_fset Quotient_fset]

thm bla2[OF Quotient_fset Quotient_fset]

lemma bla:

  assumes a1: "Quotient r1 abs1 rep1"

  and     a2: "Quotient r2 abs2 rep2"

  shows  "Quotient r2 (abs1 \<circ> abs2) (rep2 \<circ> rep1)"

sorry

  unfolding Quotient_def

apply auto

term rep_fset

lemma

  assumes sr: "equivp r"

  and     ss: "equivp s"

  shows "r OO s = s OO r"

apply(rule ext)

apply(rule ext)

using sr ss

nitpick

apply(auto)

apply(rule pred_compI)

definition

  relation_compose

where

  "relation_compose R1 R2 = \<lambda>x y. \<exists> z. (R1 x z \<and> R2 z y)"

end