src/HOL/Tools/Quotient/quotient_term.ML

do not open ML structures;

(*  Title:      HOL/Tools/Quotient/quotient_term.ML
    Author:     Cezary Kaliszyk and Christian Urban

Constructs terms corresponding to goals from lifting theorems to
quotient types.
*)

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

  (* Allows Nitpick to represent quotient types as single elements from raw type *)
  val absrep_const_chk: flag -> Proof.context -> string -> 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

  val derive_qtyp: Proof.context -> typ list -> typ -> typ
  val derive_qtrm: Proof.context -> typ list -> term -> term
  val derive_rtyp: Proof.context -> typ list -> typ -> typ
  val derive_rtrm: Proof.context -> typ list -> term -> term
end;

structure Quotient_Term: QUOTIENT_TERM =
struct

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 is_identity (Const (@{const_name id}, _)) = true
  | is_identity _ = false

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 mapfun = #mapfun (Quotient_Info.maps_lookup thy s) handle Quotient_Info.NotFound =>
      raise LIFT_MATCH ("No map function for type " ^ quote s ^ " found.")
  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 qdata = Quotient_Info.quotdata_lookup thy s handle Quotient_Info.NotFound =>
      raise LIFT_MATCH ("No quotient type " ^ quote s ^ " found.")
  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 Type.TYPE_MATCH => err ctxt ty_pat ty
  end

(* produces the rep or abs constant for a qty *)
fun absrep_const flag ctxt qty_str =
  let
    val qty_name = Long_Name.base_name qty_str
    val qualifier = Long_Name.qualifier qty_str
  in
    case flag of
      AbsF => Const (Long_Name.qualify qualifier ("abs_" ^ qty_name), dummyT)
    | RepF => Const (Long_Name.qualify qualifier ("rep_" ^ qty_name), dummyT)
  end

(* Lets Nitpick represent elements of quotient types as elements of the raw type *)
fun absrep_const_chk flag ctxt qty_str =
  Syntax.check_term ctxt (absrep_const flag ctxt qty_str)

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


(** generation of an aggregate absrep function **)

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

   - In case of TFrees we also return the identity.

   - In case of function types we recurse taking
     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 correspond -> error
       - 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 in case at least one argument produced
         a non-identity function /
         otherwise we just return 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

     The test is necessary in order to eliminate superfluous
     identity maps.
*)

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
          in
            if forall is_identity args
            then absrep_const flag ctxt s'
            else
              let
                val map_fun = mk_mapfun ctxt vs rty_pat
                val result = list_comb (map_fun, args)
              in
                mk_fun_compose flag (absrep_const flag ctxt s', result)
              end
          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 ***)


(* works very similar to the absrep generation,
   except there is no need for polarities
*)

(* 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 is_eq (Const (@{const_name HOL.eq}, _)) = true
  | is_eq _ = false

fun mk_rel_compose (trm1, trm2) =
  Const (@{const_abbrev "rel_conj"}, dummyT) $ trm1 $ trm2

fun get_relmap ctxt s =
  let
    val thy = ProofContext.theory_of ctxt
    val relmap = #relmap (Quotient_Info.maps_lookup thy s) handle Quotient_Info.NotFound =>
      raise LIFT_MATCH ("get_relmap (no relation map function found for type " ^ s ^ ")")
  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
  in
    #equiv_rel (Quotient_Info.quotdata_lookup thy s) handle Quotient_Info.NotFound =>
      raise LIFT_MATCH ("get_quotdata (no quotient found for type " ^ s ^ ")")
  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 eqv_rel = get_equiv_rel ctxt s'
            val eqv_rel' = force_typ ctxt eqv_rel ([rty, rty] ---> @{typ bool})
          in
            if forall is_eq args
            then eqv_rel'
            else
              let
                val rel_map = mk_relmap ctxt vs rty_pat
                val result = list_comb (rel_map, args)
              in
                mk_rel_compose (result, eqv_rel')
              end
          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


 The regularize_trm accepts raw theorems in which equalities
 and quantifiers match exactly the ones in the lifted theorem
 but also accepts partially regularized terms.

 This means that the raw theorems can have:
   Ball (Respects R),  Bex (Respects R), Bex1_rel (Respects R), Babs, R
 in the places where:
   All, Ex, Ex1, %, (op =)
 is required the lifted theorem.

*)

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_bex1_rel = Const (@{const_name Bex1_rel}, 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)

fun term_mismatch str ctxt t1 t2 =
  let
    val t1_str = Syntax.string_of_term ctxt t1
    val t2_str = Syntax.string_of_term ctxt t2
    val t1_ty_str = Syntax.string_of_typ ctxt (fastype_of t1)
    val t2_ty_str = Syntax.string_of_typ ctxt (fastype_of t2)
  in
    raise LIFT_MATCH (cat_lines [str, t1_str ^ "::" ^ t1_ty_str, t2_str ^ "::" ^ t2_ty_str])
  end

(* the major type of All and Ex quantifiers *)
fun qnt_typ ty = domain_type (domain_type ty)

(* Checks that two types match, for example:
     rty -> rty   matches   qty -> qty *)
fun matches_typ thy rT qT =
  if rT = qT then true
  else
    (case (rT, qT) of
      (Type (rs, rtys), Type (qs, qtys)) =>
        if rs = qs then
          if length rtys <> length qtys then false
          else forall (fn x => x = true) (map2 (matches_typ thy) rtys qtys)
        else
          (case Quotient_Info.quotdata_lookup_raw thy qs of
            SOME quotinfo => Sign.typ_instance thy (rT, #rtyp quotinfo)
          | NONE => false)
    | _ => false)


(* produces a regularized version of rtrm

   - the result might contain dummyTs

   - for regularization 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 Babs}, T) $ resrel $ (t as (Abs (_, ty, _))), t' as (Abs (_, ty', _))) =>
      let
        val subtrm = regularize_trm ctxt (t, t')
        val needres = mk_resp $ equiv_relation_chk ctxt (ty, ty')
      in
        if resrel <> needres
        then term_mismatch "regularize (Babs)" ctxt resrel needres
        else mk_babs $ resrel $ 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

  | (Const (@{const_name Ex1}, ty) $ (Abs (_, _,
      (Const (@{const_name HOL.conj}, _) $ (Const (@{const_name Set.member}, _) $ _ $
        (Const (@{const_name Respects}, _) $ resrel)) $ (t $ _)))),
     Const (@{const_name Ex1}, ty') $ t') =>
      let
        val t_ = incr_boundvars (~1) t
        val subtrm = apply_subt (regularize_trm ctxt) (t_, t')
        val needrel = equiv_relation_chk ctxt (qnt_typ ty, qnt_typ ty')
      in
        if resrel <> needrel
        then term_mismatch "regularize (Bex1)" ctxt resrel needrel
        else mk_bex1_rel $ resrel $ subtrm
      end

  | (Const (@{const_name Ex1}, ty) $ t, Const (@{const_name Ex1}, ty') $ t') =>
      let
        val subtrm = apply_subt (regularize_trm ctxt) (t, t')
      in
        if ty = ty' then Const (@{const_name Ex1}, ty) $ subtrm
        else mk_bex1_rel $ (equiv_relation ctxt (qnt_typ ty, qnt_typ ty')) $ subtrm
      end

  | (Const (@{const_name Ball}, ty) $ (Const (@{const_name Respects}, _) $ resrel) $ t,
     Const (@{const_name All}, ty') $ t') =>
      let
        val subtrm = apply_subt (regularize_trm ctxt) (t, t')
        val needrel = equiv_relation_chk ctxt (qnt_typ ty, qnt_typ ty')
      in
        if resrel <> needrel
        then term_mismatch "regularize (Ball)" ctxt resrel needrel
        else mk_ball $ (mk_resp $ resrel) $ subtrm
      end

  | (Const (@{const_name Bex}, ty) $ (Const (@{const_name Respects}, _) $ resrel) $ t,
     Const (@{const_name Ex}, ty') $ t') =>
      let
        val subtrm = apply_subt (regularize_trm ctxt) (t, t')
        val needrel = equiv_relation_chk ctxt (qnt_typ ty, qnt_typ ty')
      in
        if resrel <> needrel
        then term_mismatch "regularize (Bex)" ctxt resrel needrel
        else mk_bex $ (mk_resp $ resrel) $ subtrm
      end

  | (Const (@{const_name Bex1_rel}, ty) $ resrel $ t, Const (@{const_name Ex1}, ty') $ t') =>
      let
        val subtrm = apply_subt (regularize_trm ctxt) (t, t')
        val needrel = equiv_relation_chk ctxt (qnt_typ ty, qnt_typ ty')
      in
        if resrel <> needrel
        then term_mismatch "regularize (Bex1_res)" ctxt resrel needrel
        else mk_bex1_rel $ resrel $ subtrm
      end

  | (* equalities need to be replaced by appropriate equivalence relations *)
    (Const (@{const_name HOL.eq}, ty), Const (@{const_name HOL.eq}, 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 HOL.eq}, ty')) =>
      let
        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 term_mismatch "regularize (relation mismatch)" ctxt rel rel'
      end

  | (_, Const _) =>
      let
        val thy = ProofContext.theory_of ctxt
        fun same_const (Const (s, T)) (Const (s', T')) = (s = s') andalso matches_typ thy T T'
          | same_const _ _ = false
      in
        if same_const rtrm qtrm then rtrm
        else
          let
            val rtrm' = #rconst (Quotient_Info.qconsts_lookup thy qtrm)
              handle Quotient_Info.NotFound =>
                term_mismatch "regularize (constant not found)" ctxt rtrm qtrm
          in
            if Pattern.matches thy (rtrm', rtrm)
            then rtrm else term_mismatch "regularize (constant mismatch)" ctxt rtrm qtrm
          end
      end

  | (((t1 as Const (@{const_name prod_case}, _)) $ Abs (v1, ty, Abs(v1', ty', s1))),
     ((t2 as Const (@{const_name prod_case}, _)) $ Abs (v2, _ , Abs(v2', _  , s2)))) =>
       regularize_trm ctxt (t1, t2) $ Abs (v1, ty, Abs (v1', ty', regularize_trm ctxt (s1, s2)))

  | (((t1 as Const (@{const_name prod_case}, _)) $ Abs (v1, ty, s1)),
     ((t2 as Const (@{const_name prod_case}, _)) $ Abs (v2, _ , s2))) =>
       regularize_trm ctxt (t1, t2) $ Abs (v1, ty, regularize_trm ctxt (s1, s2))

  | (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
    and Rep/Abs around it.

  For free variables:

  * We put a Rep/Abs around it if the type needs lifting.

  Vars case cannot occur.
*)

fun mk_repabs ctxt (T, T') trm =
  absrep_fun RepF ctxt (T, T') $ (absrep_fun AbsF ctxt (T, T') $ trm)

fun inj_repabs_err ctxt msg rtrm qtrm =
  let
    val rtrm_str = Syntax.string_of_term ctxt rtrm
    val qtrm_str = Syntax.string_of_term ctxt qtrm
  in
    raise LIFT_MATCH (space_implode " " [msg, quote rtrm_str, "and", quote qtrm_str])
  end


(* bound variables need to be treated properly,
   as the type of subterms needs to be calculated   *)
fun inj_repabs_trm ctxt (rtrm, qtrm) =
 case (rtrm, qtrm) of
    (Const (@{const_name Ball}, T) $ r $ t, Const (@{const_name All}, _) $ t') =>
       Const (@{const_name Ball}, T) $ r $ (inj_repabs_trm ctxt (t, t'))

  | (Const (@{const_name Bex}, T) $ r $ t, Const (@{const_name Ex}, _) $ t') =>
       Const (@{const_name Bex}, T) $ r $ (inj_repabs_trm ctxt (t, t'))

  | (Const (@{const_name Babs}, T) $ r $ t, t' as (Abs _)) =>
      let
        val rty = fastype_of rtrm
        val qty = fastype_of qtrm
      in
        mk_repabs ctxt (rty, qty) (Const (@{const_name Babs}, T) $ r $ (inj_repabs_trm ctxt (t, t')))
      end

  | (Abs (x, T, t), Abs (x', T', t')) =>
      let
        val rty = fastype_of rtrm
        val qty = fastype_of qtrm
        val (y, s) = Term.dest_abs (x, T, t)
        val (_, s') = Term.dest_abs (x', T', t')
        val yvar = Free (y, T)
        val result = Term.lambda_name (y, yvar) (inj_repabs_trm ctxt (s, s'))
      in
        if rty = qty then result
        else mk_repabs ctxt (rty, qty) result
      end

  | (t $ s, t' $ s') =>
       (inj_repabs_trm ctxt (t, t')) $ (inj_repabs_trm ctxt (s, s'))

  | (Free (_, T), Free (_, T')) =>
        if T = T' then rtrm
        else mk_repabs ctxt (T, T') rtrm

  | (_, Const (@{const_name HOL.eq}, _)) => rtrm

  | (_, Const (_, T')) =>
      let
        val rty = fastype_of rtrm
      in
        if rty = T' then rtrm
        else mk_repabs ctxt (rty, T') rtrm
      end

  | _ => inj_repabs_err ctxt "injection (default):" rtrm qtrm

fun inj_repabs_trm_chk ctxt (rtrm, qtrm) =
  inj_repabs_trm ctxt (rtrm, qtrm)
  |> Syntax.check_term ctxt



(*** Wrapper for automatically transforming an rthm into a qthm ***)

(* substitutions functions for r/q-types and
   r/q-constants, respectively
*)
fun subst_typ ctxt ty_subst rty =
  case rty of
    Type (s, rtys) =>
      let
        val thy = ProofContext.theory_of ctxt
        val rty' = Type (s, map (subst_typ ctxt ty_subst) rtys)

        fun matches [] = rty'
          | matches ((rty, qty)::tail) =
              case try (Sign.typ_match thy (rty, rty')) Vartab.empty of
                NONE => matches tail
              | SOME inst => Envir.subst_type inst qty
      in
        matches ty_subst
      end
  | _ => rty

fun subst_trm ctxt ty_subst trm_subst rtrm =
  case rtrm of
    t1 $ t2 => (subst_trm ctxt ty_subst trm_subst t1) $ (subst_trm ctxt ty_subst trm_subst t2)
  | Abs (x, ty, t) => Abs (x, subst_typ ctxt ty_subst ty, subst_trm ctxt ty_subst trm_subst t)
  | Free(n, ty) => Free(n, subst_typ ctxt ty_subst ty)
  | Var(n, ty) => Var(n, subst_typ ctxt ty_subst ty)
  | Bound i => Bound i
  | Const (a, ty) =>
      let
        val thy = ProofContext.theory_of ctxt

        fun matches [] = Const (a, subst_typ ctxt ty_subst ty)
          | matches ((rconst, qconst)::tail) =
              case try (Pattern.match thy (rconst, rtrm)) (Vartab.empty, Vartab.empty) of
                NONE => matches tail
              | SOME inst => Envir.subst_term inst qconst
      in
        matches trm_subst
      end

(* generate type and term substitutions out of the
   qtypes involved in a quotient; the direction flag
   indicates in which direction the substitutions work:

     true:  quotient -> raw
     false: raw -> quotient
*)
fun mk_ty_subst qtys direction ctxt =
  let
    val thy = ProofContext.theory_of ctxt
  in
    Quotient_Info.quotdata_dest ctxt
    |> map (fn x => (#rtyp x, #qtyp x))
    |> filter (fn (_, qty) => member (Sign.typ_instance thy o swap) qtys qty)
    |> map (if direction then swap else I)
  end

fun mk_trm_subst qtys direction ctxt =
  let
    val subst_typ' = subst_typ ctxt (mk_ty_subst qtys direction ctxt)
    fun proper (t1, t2) = subst_typ' (fastype_of t1) = fastype_of t2

    val const_substs =
      Quotient_Info.qconsts_dest ctxt
      |> map (fn x => (#rconst x, #qconst x))
      |> map (if direction then swap else I)

    val rel_substs =
      Quotient_Info.quotdata_dest ctxt
      |> map (fn x => (#equiv_rel x, HOLogic.eq_const (#qtyp x)))
      |> map (if direction then swap else I)
  in
    filter proper (const_substs @ rel_substs)
  end


(* derives a qtyp and qtrm out of a rtyp and rtrm,
   respectively
*)
fun derive_qtyp ctxt qtys rty =
  subst_typ ctxt (mk_ty_subst qtys false ctxt) rty

fun derive_qtrm ctxt qtys rtrm =
  subst_trm ctxt (mk_ty_subst qtys false ctxt) (mk_trm_subst qtys false ctxt) rtrm

(* derives a rtyp and rtrm out of a qtyp and qtrm,
   respectively
*)
fun derive_rtyp ctxt qtys qty =
  subst_typ ctxt (mk_ty_subst qtys true ctxt) qty

fun derive_rtrm ctxt qtys qtrm =
  subst_trm ctxt (mk_ty_subst qtys true ctxt) (mk_trm_subst qtys true ctxt) qtrm


end; (* structure *)