signature QUOTIENT_DEF =
sig
  datatype flag = absF | repF
  val get_fun: flag -> (typ * typ) list -> Proof.context -> typ -> term * (typ * typ)
  val make_def: binding -> term -> typ -> mixfix -> Attrib.binding -> (typ * typ) list ->
    Proof.context -> (term * thm) * local_theory

  val quotdef: (binding * typ * mixfix) * (Attrib.binding * term) ->
    local_theory -> (term * thm) * local_theory
  val quotdef_cmd: (binding * string * mixfix) * (Attrib.binding * string) ->
    local_theory -> local_theory
end;

structure Quotient_Def: QUOTIENT_DEF =
struct

fun define name mx attr rhs lthy =
let
  val ((rhs, (_ , thm)), lthy') =
     LocalTheory.define Thm.internalK ((name, mx), (attr, rhs)) lthy
in
  ((rhs, thm), lthy')
end

fun lookup_qenv qenv qty =
  (case (AList.lookup (op=) qenv qty) of
    SOME rty => SOME (qty, rty)
  | NONE => NONE)


(* calculates the aggregate abs and rep functions for a given type; 
   repF is for constants' arguments; absF is for constants;
   function types need to be treated specially, since repF and absF
   change *)

datatype flag = absF | repF

fun negF absF = repF
  | negF repF = absF

fun get_fun flag qenv lthy ty =
let
  
  fun get_fun_aux s fs_tys =
  let
    val (fs, tys) = split_list fs_tys
    val (otys, ntys) = split_list tys
    val oty = Type (s, otys)
    val nty = Type (s, ntys)
    val ftys = map (op -->) tys
  in
   (case (maps_lookup (ProofContext.theory_of lthy) s) of
      SOME info => (list_comb (Const (#mapfun info, ftys ---> (oty --> nty)), fs), (oty, nty))
    | NONE      => error ("no map association for type " ^ s))
  end

  fun get_fun_fun fs_tys =
  let
    val (fs, tys) = split_list fs_tys
    val ([oty1, oty2], [nty1, nty2]) = split_list tys
    val oty = nty1 --> oty2
    val nty = oty1 --> nty2
    val ftys = map (op -->) tys
  in
    (list_comb (Const (@{const_name "fun_map"}, ftys ---> oty --> nty), fs), (oty, nty))
  end

  fun get_const flag (qty, rty) =
  let 
    val thy = ProofContext.theory_of lthy
    val qty_name = Long_Name.base_name (fst (dest_Type qty))
  in
    case flag of
      absF => (Const (Sign.full_bname thy ("ABS_" ^ qty_name), rty --> qty), (rty, qty))
    | repF => (Const (Sign.full_bname thy ("REP_" ^ qty_name), qty --> rty), (qty, rty))
  end

  fun mk_identity ty = Abs ("", ty, Bound 0)

in
  if (AList.defined (op=) qenv ty)
  then (get_const flag (the (lookup_qenv qenv ty)))
  else (case ty of
          TFree _ => (mk_identity ty, (ty, ty))
        | Type (_, []) => (mk_identity ty, (ty, ty)) 
        | Type ("fun" , [ty1, ty2]) => 
                 get_fun_fun [get_fun (negF flag) qenv lthy ty1, get_fun flag qenv lthy ty2]
        | Type (s, tys) => get_fun_aux s (map (get_fun flag qenv lthy) tys)
        | _ => raise ERROR ("no type variables"))
end

fun make_def nconst_bname rhs qty mx attr qenv lthy =
let
  val (arg_tys, res_ty) = strip_type qty

  val rep_fns = map (fst o get_fun repF qenv lthy) arg_tys
  val abs_fn  = (fst o get_fun absF qenv lthy) res_ty

  fun mk_fun_map t s = 
        Const (@{const_name "fun_map"}, dummyT) $ t $ s

  val absrep_fn = fold_rev mk_fun_map rep_fns abs_fn
                  |> Syntax.check_term lthy 
in
  define nconst_bname mx attr (absrep_fn $ rhs) lthy
end


(* returns all subterms where two types differ *)
fun diff (T, S) Ds =
  case (T, S) of
    (TVar v, TVar u) => if v = u then Ds else (T, S)::Ds 
  | (TFree x, TFree y) => if x = y then Ds else (T, S)::Ds
  | (Type (a, Ts), Type (b, Us)) => 
      if a = b then diffs (Ts, Us) Ds else (T, S)::Ds
  | _ => (T, S)::Ds
and diffs (T::Ts, U::Us) Ds = diffs (Ts, Us) (diff (T, U) Ds)
  | diffs ([], []) Ds = Ds
  | diffs _ _ = error "Unequal length of type arguments"


fun error_msg lthy (qty, rty) =
let 
  val qtystr = quote (Syntax.string_of_typ lthy qty)
  val rtystr = quote (Syntax.string_of_typ lthy rty)
in
  error (implode ["Quotient type ", qtystr, " does not match with ", rtystr])
end

fun sanity_chk lthy qenv =
let
   val qenv' = Quotient_Info.mk_qenv lthy
   val thy = ProofContext.theory_of lthy

   fun is_inst thy (qty, rty) (qty', rty') =
   if Sign.typ_instance thy (qty, qty')
   then let
          val inst = Sign.typ_match thy (qty', qty) Vartab.empty
        in
          rty = Envir.subst_type inst rty'
        end
   else false

   fun chk_inst (qty, rty) = 
     if exists (is_inst thy (qty, rty)) qenv' then true
     else error_msg lthy (qty, rty)
in
  forall chk_inst qenv
end


fun quotdef ((bind, qty, mx), (attr, prop)) lthy =
let   
  val (_, prop') = PrimitiveDefs.dest_def lthy (K true) (K false) (K false) prop
  val (_, rhs) = PrimitiveDefs.abs_def prop'

  val rty = fastype_of rhs
  val qenv = distinct (op=) (diff (qty, rty) []) 
in
  sanity_chk lthy qenv;
  make_def bind rhs qty mx attr qenv lthy 
end


val quotdef_parser =
  (OuterParse.binding --
    (OuterParse.$$$ "::" |-- OuterParse.!!! (OuterParse.typ -- 
      OuterParse.opt_mixfix' --| OuterParse.where_)) >> OuterParse.triple2) -- 
       (SpecParse.opt_thm_name ":" -- OuterParse.prop)

fun quotdef_cmd ((bind, qtystr, mx), (attr, propstr)) lthy = 
let
  val qty  = (Syntax.check_typ lthy o Syntax.parse_typ lthy) qtystr
  val prop = (Syntax.check_prop lthy o Syntax.parse_prop lthy) propstr
in
  quotdef ((bind, qty, mx), (attr, prop)) lthy |> snd
end

val _ = OuterSyntax.local_theory "quotient_def" "lifted definition of constants"
  OuterKeyword.thy_decl (quotdef_parser >> quotdef_cmd)

end; (* structure *)

open Quotient_Def;