structure Quotient =
struct

(* constructs the term lambda (c::rty => bool). EX x. c= rel x *)
fun typedef_term rel rty lthy =
let
  val [x, c] = [("x", rty), ("c", rty --> @{typ bool})]
               |> Variable.variant_frees lthy [rel]
               |> map Free
in
  lambda c
    (HOLogic.exists_const rty $
       lambda x (HOLogic.mk_eq (c, (rel $ x))))
end

(* makes the new type definitions and proves non-emptyness*)
fun typedef_make (qty_name, mx, rel, rty) lthy =
let
  val typedef_tac =
     EVERY1 [rewrite_goal_tac @{thms mem_def},
             rtac @{thm exI},
             rtac @{thm exI},
             rtac @{thm refl}]
  val tfrees = map fst (Term.add_tfreesT rty [])
in
  LocalTheory.theory_result
    (Typedef.add_typedef false NONE
       (qty_name, tfrees, mx)
         (typedef_term rel rty lthy)
           NONE typedef_tac) lthy
end

(* tactic to prove the QUOT_TYPE theorem for the new type *)
fun typedef_quot_type_tac equiv_thm (typedef_info: Typedef.info) =
let
  val unfold_mem = MetaSimplifier.rewrite_rule @{thms mem_def}
  val rep_thm = #Rep typedef_info |> unfold_mem
  val rep_inv = #Rep_inverse typedef_info
  val abs_inv = #Abs_inverse typedef_info |> unfold_mem
  val rep_inj = #Rep_inject typedef_info
in
  EVERY1 [rtac @{thm QUOT_TYPE.intro},
          rtac equiv_thm,
          rtac rep_thm,
          rtac rep_inv,
          rtac abs_inv,
          rtac @{thm exI}, 
          rtac @{thm refl},
          rtac rep_inj]
end

(* proves the QUOT_TYPE theorem *)
fun typedef_quot_type_thm (rel, abs, rep, equiv_thm, typedef_info) lthy =
let
  val quot_type_const = Const (@{const_name "QUOT_TYPE"}, dummyT)
  val goal = HOLogic.mk_Trueprop (quot_type_const $ rel $ abs $ rep)
             |> Syntax.check_term lthy
in
  Goal.prove lthy [] [] goal
    (K (typedef_quot_type_tac equiv_thm typedef_info))
end

(* proves the quotient theorem *)
fun typedef_quotient_thm (rel, abs, rep, abs_def, rep_def, quot_type_thm) lthy =
let
  val quotient_const = Const (@{const_name "QUOTIENT"}, dummyT)
  val goal = HOLogic.mk_Trueprop (quotient_const $ rel $ abs $ rep)
             |> Syntax.check_term lthy

  val typedef_quotient_thm_tac =
    EVERY1 [K (rewrite_goals_tac [abs_def, rep_def]),
            rtac @{thm QUOT_TYPE.QUOTIENT},
            rtac quot_type_thm]
in
  Goal.prove lthy [] [] goal
    (K typedef_quotient_thm_tac)
end

(* two wrappers for define and note *)
fun make_def (name, mx, rhs) lthy =
let
  val ((rhs, (_ , thm)), lthy') =
     LocalTheory.define Thm.internalK ((name, mx), (Attrib.empty_binding, rhs)) lthy
in
  ((rhs, thm), lthy')
end

fun note_thm (name, thm) lthy =
let
  val ((_,[thm']), lthy') = LocalTheory.note Thm.theoremK ((name, []), [thm]) lthy
in
  (thm', lthy')
end

(* main function for constructing the quotient type *)
fun typedef_main (qty_name, mx, rel, rty, equiv_thm) lthy =
let
  (* generates typedef *)
  val ((_, typedef_info), lthy1) = typedef_make (qty_name, mx, rel, rty) lthy

  (* abs and rep functions *)
  val abs_ty = #abs_type typedef_info
  val rep_ty = #rep_type typedef_info
  val abs_name = #Abs_name typedef_info
  val rep_name = #Rep_name typedef_info
  val abs = Const (abs_name, rep_ty --> abs_ty)
  val rep = Const (rep_name, abs_ty --> rep_ty)

  (* ABS and REP definitions *)
  val ABS_const = Const (@{const_name "QUOT_TYPE.ABS"}, dummyT )
  val REP_const = Const (@{const_name "QUOT_TYPE.REP"}, dummyT )
  val ABS_trm = Syntax.check_term lthy1 (ABS_const $ rel $ abs)
  val REP_trm = Syntax.check_term lthy1 (REP_const $ rep)
  val ABS_name = Binding.prefix_name "ABS_" qty_name
  val REP_name = Binding.prefix_name "REP_" qty_name
  val (((ABS, ABS_def), (REP, REP_def)), lthy2) =
         lthy1 |> make_def (ABS_name, NoSyn, ABS_trm)
               ||>> make_def (REP_name, NoSyn, REP_trm)

  (* quot_type theorem *)
  val quot_thm = typedef_quot_type_thm (rel, abs, rep, equiv_thm, typedef_info) lthy2
  val quot_thm_name = Binding.prefix_name "QUOT_TYPE_" qty_name

  (* quotient theorem *)
  val quotient_thm = typedef_quotient_thm (rel, ABS, REP, ABS_def, REP_def, quot_thm) lthy2
  val quotient_thm_name = Binding.prefix_name "QUOTIENT_" qty_name

  (* interpretation *)
  val bindd = ((Binding.make ("", Position.none)), ([]: Attrib.src list))
  val ((_, [eqn1pre]), lthy3) = Variable.import true [ABS_def] lthy2;
  val eqn1i = Thm.prop_of (symmetric eqn1pre)
  val ((_, [eqn2pre]), lthy4) = Variable.import true [REP_def] lthy3;
  val eqn2i = Thm.prop_of (symmetric eqn2pre)

  val exp_morphism = ProofContext.export_morphism lthy4 (ProofContext.init (ProofContext.theory_of lthy4));
  val exp_term = Morphism.term exp_morphism;
  val exp = Morphism.thm exp_morphism;

  val mthd = Method.SIMPLE_METHOD ((rtac quot_thm 1) THEN
    ALLGOALS (simp_tac (HOL_basic_ss addsimps [(symmetric (exp ABS_def)), (symmetric (exp REP_def))])))
  val mthdt = Method.Basic (fn _ => mthd)
  val bymt = Proof.global_terminal_proof (mthdt, NONE)
  val exp_i = [(@{const_name QUOT_TYPE}, ((("QUOT_TYPE_I_" ^ (Binding.name_of qty_name)), true),
    Expression.Named [
     ("R", rel),
     ("Abs", abs),
     ("Rep", rep)
    ]))]
in
  lthy4
  |> note_thm (quot_thm_name, quot_thm)
  ||>> note_thm (quotient_thm_name, quotient_thm)
  ||> LocalTheory.theory (fn thy =>
      let
        val global_eqns = map exp_term [eqn2i, eqn1i];
        (* Not sure if the following context should not be used *)
        val (global_eqns2, lthy5) = Variable.import_terms true global_eqns lthy4;
        val global_eqns3 = map (fn t => (bindd, t)) global_eqns2;
      in ProofContext.theory_of (bymt (Expression.interpretation (exp_i, []) global_eqns3 thy)) end)
end

(* syntax setup *)
local structure P = OuterParse in

val quottype_parser = 
    (P.type_args -- P.binding) -- 
      P.opt_infix -- 
       (P.$$$ "=" |-- P.term) -- 
         (P.$$$ "/" |-- P.term)
           
(*
val _ =
  OuterSyntax.command "quotient" "quotient type definition (requires equivalence proof)"
    OuterKeyword.thy_goal
    (typedef_decl >> (Toplevel.print oo (Toplevel.theory_to_proof o mk_typedef)));

end;
*)

end;

end;

open Quotient