signature QUOTIENT_TERM =

sig

   exception LIFT_MATCH of string

   datatype flag = absF | repF

   val absrep_fun: flag -> Proof.context -> (typ * typ) -> term

   val absrep_fun_chk: flag -> Proof.context -> (typ * typ) -> term

   val equiv_relation: Proof.context -> (typ * typ) -> term

   val equiv_relation_chk: Proof.context -> (typ * typ) -> term

   val regularize_trm: Proof.context -> (term * term) -> term

   val regularize_trm_chk: Proof.context -> (term * term) -> term

   val inj_repabs_trm: Proof.context -> (term * term) -> term

   val inj_repabs_trm_chk: Proof.context -> (term * term) -> term

end;

structure Quotient_Term: QUOTIENT_TERM =

struct

open Quotient_Info;

exception LIFT_MATCH of string

(******************************)

(* Aggregate Rep/Abs Function *)

(******************************)

(* The flag repF is for types in negative position; absF is for types *)

(* in positive position. Because of this, function types need to be   *)

(* treated specially, since there the polarity changes.               *)

datatype flag = absF | repF

fun negF absF = repF

  | negF repF = absF

fun mk_identity ty = Const (@{const_name "id"}, ty --> ty)

fun mk_fun_compose flag (trm1, trm2) = 

  case flag of

    absF => Const (@{const_name "comp"}, dummyT) $ trm1 $ trm2

  | repF => Const (@{const_name "comp"}, dummyT) $ trm2 $ trm1

fun get_mapfun ctxt s =

let

  val thy = ProofContext.theory_of ctxt

  val exc = LIFT_MATCH ("No map function for type " ^ (quote s) ^ " found.")

  val mapfun = #mapfun (maps_lookup thy s) handle NotFound => raise exc

in

  Const (mapfun, dummyT)

end

(* makes a Free out of a TVar *)

fun mk_Free (TVar ((x, i), _)) = Free (unprefix "'" x ^ string_of_int i, dummyT)

(* produces an aggregate map function for the       *)

(* rty-part of a quotient definition; abstracts     *)

(* over all variables listed in vs (these variables *)

(* correspond to the type variables in rty)         *)        

(*                                                  *)

(* for example for: (?'a list * ?'b)                *)

(* it produces:     %a b. prod_map (map a) b        *)

fun mk_mapfun ctxt vs rty =

let

  val vs' = map (mk_Free) vs

  fun mk_mapfun_aux rty =

    case rty of

      TVar _ => mk_Free rty

    | Type (_, []) => mk_identity rty

    | Type (s, tys) => list_comb (get_mapfun ctxt s, map mk_mapfun_aux tys)

    | _ => raise LIFT_MATCH ("mk_mapfun (default)")

in

  fold_rev Term.lambda vs' (mk_mapfun_aux rty)

end

(* looks up the (varified) rty and qty for *)

(* a quotient definition                   *)

fun get_rty_qty ctxt s =

let

  val thy = ProofContext.theory_of ctxt

  val exc = LIFT_MATCH ("No quotient type " ^ (quote s) ^ " found.")

  val qdata = (quotdata_lookup thy s) handle NotFound => raise exc

in

  (#rtyp qdata, #qtyp qdata)

end

(* takes two type-environments and looks    *)

(* up in both of them the variable v, which *)

(* must be listed in the environment        *)

fun double_lookup rtyenv qtyenv v =

let

  val v' = fst (dest_TVar v)

in

  (snd (the (Vartab.lookup rtyenv v')), snd (the (Vartab.lookup qtyenv v')))

end

(* matches a type pattern with a type *)

fun match ctxt err ty_pat ty =

let

  val thy = ProofContext.theory_of ctxt

in

  Sign.typ_match thy (ty_pat, ty) Vartab.empty

  handle MATCH_TYPE => err ctxt ty_pat ty

end

(* produces the rep or abs constant for a qty *)

fun absrep_const flag ctxt qty_str =

let

  val thy = ProofContext.theory_of ctxt

  val qty_name = Long_Name.base_name qty_str

in

  case flag of

    absF => Const (Sign.full_bname thy ("abs_" ^ qty_name), dummyT)

  | repF => Const (Sign.full_bname thy ("rep_" ^ qty_name), dummyT)

end

fun absrep_match_err ctxt ty_pat ty =

let

  val ty_pat_str = Syntax.string_of_typ ctxt ty_pat

  val ty_str = Syntax.string_of_typ ctxt ty 

in

  raise LIFT_MATCH (space_implode " " 

    ["absrep_fun (Types ", quote ty_pat_str, "and", quote ty_str, " do not match.)"])

end

(* produces an aggregate absrep function                           *)

(*                                                                 *)

(* - In case of equal types we just return the identity.           *)

(*                                                                 *)

(* - In case of function types and TFrees, we recurse, taking in   *) 

(*   the first case the polarity change into account.              *)

(*                                                                 *)

(* - If the type constructors are equal, we recurse for the        *)

(*   arguments and build the appropriate map function.             *)

(*                                                                 *)

(* - If the type constructors are unequal, there must be an        *)

(*   instance of quotient types:                                   *)

(*     - we first look up the corresponding rty_pat and qty_pat    *)

(*       from the quotient definition; the arguments of qty_pat    *)

(*       must be some distinct TVars                               *)  

(*     - we then match the rty_pat with rty and qty_pat with qty;  *)

(*       if matching fails the types do not match                  *)

(*     - the matching produces two environments; we look up the    *)

(*       assignments for the qty_pat variables and recurse on the  *)

(*       assignments                                               *)

(*     - we prefix the aggregate map function for the rty_pat,     *)

(*       which is an abstraction over all type variables           *)

(*     - finally we compose the result with the appropriate        *)

(*       absrep function                                           *)    

(*                                                                 *)

(*   The composition is necessary for types like                   *)

(*                                                                 *)

(*      ('a list) list / ('a foo) foo                              *)

(*                                                                 *)

(*   The matching is necessary for types like                      *)

(*                                                                 *)

(*      ('a * 'a) list / 'a bar                                    *)  

fun absrep_fun flag ctxt (rty, qty) =

  if rty = qty  

  then mk_identity rty

  else

    case (rty, qty) of

      (Type ("fun", [ty1, ty2]), Type ("fun", [ty1', ty2'])) =>

        let

          val arg1 = absrep_fun (negF flag) ctxt (ty1, ty1')

          val arg2 = absrep_fun flag ctxt (ty2, ty2')

        in

          list_comb (get_mapfun ctxt "fun", [arg1, arg2])

        end

    | (Type (s, tys), Type (s', tys')) =>

        if s = s'

        then 

           let

             val args = map (absrep_fun flag ctxt) (tys ~~ tys')

           in

             list_comb (get_mapfun ctxt s, args)

           end 

        else

           let

             val (rty_pat, qty_pat as Type (_, vs)) = get_rty_qty ctxt s'

             val rtyenv = match ctxt absrep_match_err rty_pat rty

             val qtyenv = match ctxt absrep_match_err qty_pat qty

             val args_aux = map (double_lookup rtyenv qtyenv) vs 

             val args = map (absrep_fun flag ctxt) args_aux

             val map_fun = mk_mapfun ctxt vs rty_pat       

             val result = list_comb (map_fun, args) 

           in

             then absrep_const flag ctxt s'

           end

    | (TFree x, TFree x') =>

        if x = x'

        then mk_identity rty

        else raise (LIFT_MATCH "absrep_fun (frees)")

    | (TVar _, TVar _) => raise (LIFT_MATCH "absrep_fun (vars)")

    | _ => raise (LIFT_MATCH "absrep_fun (default)")

fun absrep_fun_chk flag ctxt (rty, qty) =

  absrep_fun flag ctxt (rty, qty)

  |> Syntax.check_term ctxt

(**********************************)

(* Aggregate Equivalence Relation *)

(**********************************)

(* instantiates TVars so that the term is of type ty *)

fun force_typ ctxt trm ty =

let

  val thy = ProofContext.theory_of ctxt 

  val trm_ty = fastype_of trm

  val ty_inst = Sign.typ_match thy (trm_ty, ty) Vartab.empty

in

  map_types (Envir.subst_type ty_inst) trm

end

fun mk_rel_compose (trm1, trm2) =

  Const (@{const_name "pred_comp"}, dummyT) $ trm1 $

   (Const (@{const_name "pred_comp"}, dummyT) $ trm2 $ trm1)

fun get_relmap ctxt s =

let

  val thy = ProofContext.theory_of ctxt

  val exc =  LIFT_MATCH ("get_relmap (no relation map function found for type " ^ s ^ ")") 

  val relmap =  #relmap (maps_lookup thy s) handle NotFound => raise exc

in

  Const (relmap, dummyT)

end

fun mk_relmap ctxt vs rty =

let

  val vs' = map (mk_Free) vs

  fun mk_relmap_aux rty =

    case rty of

      TVar _ => mk_Free rty

    | Type (_, []) => HOLogic.eq_const rty

    | Type (s, tys) => list_comb (get_relmap ctxt s, map mk_relmap_aux tys)

    | _ => raise LIFT_MATCH ("mk_relmap (default)")

in

  fold_rev Term.lambda vs' (mk_relmap_aux rty)

end

fun get_equiv_rel ctxt s =

let

  val thy = ProofContext.theory_of ctxt

  val exc = LIFT_MATCH ("get_quotdata (no quotient found for type " ^ s ^ ")") 

in

  #equiv_rel (quotdata_lookup thy s) handle NotFound => raise exc

end

fun equiv_match_err ctxt ty_pat ty =

let

  val ty_pat_str = Syntax.string_of_typ ctxt ty_pat

  val ty_str = Syntax.string_of_typ ctxt ty 

in

  raise LIFT_MATCH (space_implode " " 

    ["equiv_relation (Types ", quote ty_pat_str, "and", quote ty_str, " do not match.)"])

end

(* builds the aggregate equivalence relation *)

(* that will be the argument of Respects     *)

fun equiv_relation ctxt (rty, qty) =

  if rty = qty

  then HOLogic.eq_const rty

  else

    case (rty, qty) of

      (Type (s, tys), Type (s', tys')) =>

       if s = s' 

       then 

         let

           val args = map (equiv_relation ctxt) (tys ~~ tys')

         in

           list_comb (get_relmap ctxt s, args) 

         end  

       else 

         let

           val (rty_pat, qty_pat as Type (_, vs)) = get_rty_qty ctxt s'

           val rtyenv = match ctxt equiv_match_err rty_pat rty

           val qtyenv = match ctxt equiv_match_err qty_pat qty

           val args_aux = map (double_lookup rtyenv qtyenv) vs 

           val args = map (equiv_relation ctxt) args_aux

           val rel_map = mk_relmap ctxt vs rty_pat       

           val result = list_comb (rel_map, args)

           val eqv_rel = get_equiv_rel ctxt s'

           val eqv_rel' = force_typ ctxt eqv_rel ([rty, rty] ---> @{typ bool})

         in

           then eqv_rel'

         end

      | _ => HOLogic.eq_const rty

fun equiv_relation_chk ctxt (rty, qty) =

  equiv_relation ctxt (rty, qty)

  |> Syntax.check_term ctxt

(******************)

(* Regularization *)

(******************)

(* Regularizing an rtrm means:

 - Quantifiers over types that need lifting are replaced 

   by bounded quantifiers, for example:

      All P  ----> All (Respects R) P

   where the aggregate relation R is given by the rty and qty;

 - Abstractions over types that need lifting are replaced

   by bounded abstractions, for example:

      %x. P  ----> Ball (Respects R) %x. P

 - Equalities over types that need lifting are replaced by

   corresponding equivalence relations, for example:

      A = B  ----> R A B

   or 

      A = B  ----> (R ===> R) A B

   for more complicated types of A and B

*)

val mk_babs = Const (@{const_name Babs}, dummyT)

val mk_ball = Const (@{const_name Ball}, dummyT)

val mk_bex  = Const (@{const_name Bex}, dummyT)

val mk_resp = Const (@{const_name Respects}, dummyT)

(* - applies f to the subterm of an abstraction,   *)

(*   otherwise to the given term,                  *)

(* - used by regularize, therefore abstracted      *)

(*   variables do not have to be treated specially *)

fun apply_subt f (trm1, trm2) =

  case (trm1, trm2) of

    (Abs (x, T, t), Abs (_ , _, t')) => Abs (x, T, f (t, t'))

  | _ => f (trm1, trm2)

(* the major type of All and Ex quantifiers *)

fun qnt_typ ty = domain_type (domain_type ty)  

(* produces a regularized version of rtrm       *)

(*                                              *)

(* - the result might contain dummyTs           *)

(*                                              *)

(* - for regularisation we do not need any      *)

(*   special treatment of bound variables       *)

fun regularize_trm ctxt (rtrm, qtrm) =

  case (rtrm, qtrm) of

    (Abs (x, ty, t), Abs (_, ty', t')) =>

       let

         val subtrm = Abs(x, ty, regularize_trm ctxt (t, t'))

       in

         if ty = ty' then subtrm

         else mk_babs $ (mk_resp $ equiv_relation ctxt (ty, ty')) $ subtrm

       end

  | (Const (@{const_name "All"}, ty) $ t, Const (@{const_name "All"}, ty') $ t') =>

       let

         val subtrm = apply_subt (regularize_trm ctxt) (t, t')

       in

         if ty = ty' then Const (@{const_name "All"}, ty) $ subtrm

         else mk_ball $ (mk_resp $ equiv_relation ctxt (qnt_typ ty, qnt_typ ty')) $ subtrm

       end

  | (Const (@{const_name "Ex"}, ty) $ t, Const (@{const_name "Ex"}, ty') $ t') =>

       let

         val subtrm = apply_subt (regularize_trm ctxt) (t, t')

       in

         if ty = ty' then Const (@{const_name "Ex"}, ty) $ subtrm

         else mk_bex $ (mk_resp $ equiv_relation ctxt (qnt_typ ty, qnt_typ ty')) $ subtrm

       end

  | (* equalities need to be replaced by appropriate equivalence relations *) 

    (Const (@{const_name "op ="}, ty), Const (@{const_name "op ="}, ty')) =>

         if ty = ty' then rtrm

         else equiv_relation ctxt (domain_type ty, domain_type ty') 

  | (* in this case we just check whether the given equivalence relation is correct *) 

    (rel, Const (@{const_name "op ="}, ty')) =>

       let 

         fun exc rel rel' = LIFT_MATCH ("regularise (relation mismatch)\n[" ^

           Syntax.string_of_term ctxt rel ^ " :: " ^

           Syntax.string_of_typ ctxt (fastype_of rel) ^ "]\n[" ^

           Syntax.string_of_term ctxt rel' ^ " :: " ^

           Syntax.string_of_typ ctxt (fastype_of rel') ^ "]")

         val rel_ty = fastype_of rel

         val rel' = equiv_relation_chk ctxt (domain_type rel_ty, domain_type ty') 

       in 

         if rel' aconv rel then rtrm else raise (exc rel rel')

       end  

  | (_, Const _) =>

       let 

         fun same_name (Const (s, T)) (Const (s', T')) = (s = s') (*andalso (T = T')*)

           | same_name _ _ = false

          (* TODO/FIXME: This test is not enough. *) 

          (*             Why?                     *)

          (* Because constants can have the same name but not be the same

             constant.  All overloaded constants have the same name but because

             of different types they do differ.

             This code will let one write a theorem where plus on nat is

             matched to plus on int, even if the latter is defined differently.

             This would result in hard to understand failures in injection and

             cleaning. *)

           (* cu: if I also test the type, then something else breaks *)

       in

         if same_name rtrm qtrm then rtrm

         else 

           let 

             val thy = ProofContext.theory_of ctxt

             val qtrm_str = Syntax.string_of_term ctxt qtrm

             val exc1 = LIFT_MATCH ("regularize (constant " ^ qtrm_str ^ " not found)")

             val exc2 = LIFT_MATCH ("regularize (constant " ^ qtrm_str ^ " mismatch)")

             val rtrm' = #rconst (qconsts_lookup thy qtrm) handle NotFound => raise exc1

           in 

             if Pattern.matches thy (rtrm', rtrm) 

             then rtrm else raise exc2

           end

       end 

  | (t1 $ t2, t1' $ t2') =>

       (regularize_trm ctxt (t1, t1')) $ (regularize_trm ctxt (t2, t2'))

  | (Bound i, Bound i') =>

       if i = i' then rtrm 

       else raise (LIFT_MATCH "regularize (bounds mismatch)")

  | _ =>

       let 

         val rtrm_str = Syntax.string_of_term ctxt rtrm

         val qtrm_str = Syntax.string_of_term ctxt qtrm

       in

         raise (LIFT_MATCH ("regularize failed (default: " ^ rtrm_str ^ "," ^ qtrm_str ^ ")"))

       end

fun regularize_trm_chk ctxt (rtrm, qtrm) =

  regularize_trm ctxt (rtrm, qtrm) 

  |> Syntax.check_term ctxt

(*********************)

(* Rep/Abs Injection *)

(*********************)

(*

Injection of Rep/Abs means:

  For abstractions

:

  * If the type of the abstraction needs lifting, then we add Rep/Abs 

    around the abstraction; otherwise we leave it unchanged.

  For applications:

  * If the application involves a bounded quantifier, we recurse on 

    the second argument. If the application is a bounded abstraction,

    we always put an Rep/Abs around it (since bounded abstractions

    are assumed to always need lifting). Otherwise we recurse on both 

    arguments.

  For constants:

  * If the constant is (op =), we leave it always unchanged. 

    Otherwise the type of the constant needs lifting, we put