signature QUOTIENT =
sig
  val quotient_type: ((binding * mixfix) * (typ * term)) list -> Proof.context -> Proof.state
  val quotient_type_cmd: (((bstring * mixfix) * string) * string) list -> Proof.context -> Proof.state

  val note: binding * thm -> local_theory -> thm * local_theory
end;

structure Quotient: QUOTIENT =
struct

(* wrappers for define, note and theorem_i *)
fun define (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 (name, thm) lthy =
let
  val ((_,[thm']), lthy') = LocalTheory.note Thm.theoremK ((name, []), [thm]) lthy
in
  (thm', lthy')
end

fun theorem after_qed goals ctxt =
let
  val goals' = map (rpair []) goals
  fun after_qed' thms = after_qed (the_single thms)
in 
  Proof.theorem_i NONE after_qed' [goals'] ctxt
end


(* definition of the quotient type *)
(***********************************)

(* constructs the term lambda (c::rty => bool). EX (x::rty). c = rel x *)
fun typedef_term rel rty lthy =
let
  val [x, c] = [("x", rty), ("c", HOLogic.mk_setT rty)]
               |> 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 [@{thm 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

(* main function for constructing the quotient type *)
fun mk_typedef_main (((qty_name, mx), (rty, rel)), 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 |> define (ABS_name, NoSyn, ABS_trm)
               ||>> define (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

  (* storing the quot-info *)
  val lthy3 = quotdata_update (Logic.varifyT abs_ty, Logic.varifyT rty, rel, equiv_thm) lthy2

  (* interpretation *)
  val bindd = ((Binding.make ("", Position.none)), ([]: Attrib.src list))
  val ((_, [eqn1pre]), lthy4) = Variable.import true [ABS_def] lthy3;
  val eqn1i = Thm.prop_of (symmetric eqn1pre)
  val ((_, [eqn2pre]), lthy5) = Variable.import true [REP_def] lthy4;
  val eqn2i = Thm.prop_of (symmetric eqn2pre)

  val exp_morphism = ProofContext.export_morphism lthy5 (ProofContext.init (ProofContext.theory_of lthy5));
  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
  lthy5
  |> note (quot_thm_name, quot_thm)
  ||>> note (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, lthy6) = Variable.import_terms true global_eqns lthy5;
        val global_eqns3 = map (fn t => (bindd, t)) global_eqns2;
      in ProofContext.theory_of (bymt (Expression.interpretation (exp_i, []) global_eqns3 thy)) end)
end




(* interface and syntax setup *)

(* the ML-interface takes a list of 4-tuples consisting of  *)
(*                                                          *)
(* - the name of the quotient type                          *)
(* - its mixfix annotation                                  *)
(* - the type to be quotient                                *)
(* - the relation according to which the type is quotient   *)

fun quotient_type quot_list lthy = 
let
  fun mk_goal (rty, rel) =
  let
    val EQUIV_ty = ([rty, rty] ---> @{typ bool}) --> @{typ bool}
  in 
    HOLogic.mk_Trueprop (Const (@{const_name EQUIV}, EQUIV_ty) $ rel)
  end

  val goals = map (mk_goal o snd) quot_list
              
  fun after_qed thms lthy =
    fold_map mk_typedef_main (quot_list ~~ thms) lthy |> snd
in
  theorem after_qed goals lthy
end
           
fun quotient_type_cmd spec lthy = 
let
  fun parse_spec (((qty_str, mx), rty_str), rel_str) =
  let
    val qty_name = Binding.name qty_str
    val rty = Syntax.read_typ lthy rty_str
    val rel = Syntax.read_term lthy rel_str 
  in
    ((qty_name, mx), (rty, rel))
  end
in
  quotient_type (map parse_spec spec) lthy
end

val quotspec_parser = 
    OuterParse.and_list1
     (OuterParse.short_ident -- OuterParse.opt_infix -- 
       (OuterParse.$$$ "=" |-- OuterParse.typ) -- 
         (OuterParse.$$$ "/" |-- OuterParse.term))

val _ = OuterKeyword.keyword "/"

val _ = 
    OuterSyntax.local_theory_to_proof "quotient" 
      "quotient type definitions (requires equivalence proofs)"
         OuterKeyword.thy_goal (quotspec_parser >> quotient_type_cmd)

end; (* structure *)

open Quotient