(*  Title:      HOL/Tools/Lifting/lifting_def.ML

    Author:     Ondrej Kuncar

Definitions for constants on quotient types.

*)

signature LIFTING_DEF =

sig

  val generate_parametric_transfer_rule:

    Proof.context -> thm -> thm -> thm

  val add_lift_def:

    (binding * mixfix) -> typ -> term -> thm -> thm list -> local_theory -> local_theory

  val lift_def_cmd:

    (binding * string option * mixfix) * string * (Facts.ref * Args.src list) list -> local_theory -> Proof.state

  val can_generate_code_cert: thm -> bool

end

structure Lifting_Def: LIFTING_DEF =

struct

open Lifting_Util

infix 0 MRSL

(* Reflexivity prover *)

fun mono_eq_prover ctxt prop =

  let

    val refl_rules = Lifting_Info.get_reflexivity_rules ctxt

    val transfer_rules = Transfer.get_transfer_raw ctxt

    fun main_tac i = (REPEAT_ALL_NEW (DETERM o resolve_tac refl_rules) THEN_ALL_NEW 

      (REPEAT_ALL_NEW (DETERM o resolve_tac transfer_rules))) i

  in

    SOME (Goal.prove ctxt [] [] prop (K (main_tac 1)))

      handle ERROR _ => NONE

  end

fun try_prove_refl_rel ctxt rel =

  let

    fun mk_ge_eq x =

      let

        val T = fastype_of x

      in

        Const (@{const_name "less_eq"}, T --> T --> HOLogic.boolT) $ 

          (Const (@{const_name HOL.eq}, T)) $ x

      end;

    val goal = HOLogic.mk_Trueprop (mk_ge_eq rel);

  in

     mono_eq_prover ctxt goal

  end;

fun try_prove_reflexivity ctxt prop =

  let

    val thy = Proof_Context.theory_of ctxt

    val cprop = cterm_of thy prop

    val rule = @{thm ge_eq_refl}

    val concl_pat = Drule.strip_imp_concl (cprop_of rule)

    val insts = Thm.first_order_match (concl_pat, cprop)

    val rule = Drule.instantiate_normalize insts rule

    val prop = hd (prems_of rule)

  in

    case mono_eq_prover ctxt prop of

      SOME thm => SOME (thm RS rule)

      | NONE => NONE

  end

(* 

  Generates a parametrized transfer rule.

  transfer_rule - of the form T t f

  parametric_transfer_rule - of the form par_R t' t

  Result: par_T t' f, after substituing op= for relations in par_R that relate

    a type constructor to the same type constructor, it is a merge of (par_R' OO T) t' f

    using Lifting_Term.merge_transfer_relations

*)

fun generate_parametric_transfer_rule ctxt transfer_rule parametric_transfer_rule =

  let

    fun preprocess ctxt thm =

      let

        val tm = (strip_args 2 o HOLogic.dest_Trueprop o concl_of) thm;

        val param_rel = (snd o dest_comb o fst o dest_comb) tm;

        val thy = Proof_Context.theory_of ctxt;

        val free_vars = Term.add_vars param_rel [];

        fun make_subst (var as (_, typ)) subst = 

          let

            val [rty, rty'] = binder_types typ

          in

            if (Term.is_TVar rty andalso is_Type rty') then

              (Var var, HOLogic.eq_const rty')::subst

            else

              subst

          end;

        val subst = fold make_subst free_vars [];

        val csubst = map (pairself (cterm_of thy)) subst;

        val inst_thm = Drule.cterm_instantiate csubst thm;

      in

        Conv.fconv_rule 

          ((Conv.concl_conv (nprems_of inst_thm) o HOLogic.Trueprop_conv o Conv.fun2_conv o Conv.arg1_conv)

            (Raw_Simplifier.rewrite ctxt false (Transfer.get_sym_relator_eq ctxt))) inst_thm

      end

    fun inst_relcomppI thy ant1 ant2 =

      let

        val t1 = (HOLogic.dest_Trueprop o concl_of) ant1

        val t2 = (HOLogic.dest_Trueprop o prop_of) ant2

        val fun1 = cterm_of thy (strip_args 2 t1)

        val args1 = map (cterm_of thy) (get_args 2 t1)

        val fun2 = cterm_of thy (strip_args 2 t2)

        val args2 = map (cterm_of thy) (get_args 1 t2)

        val relcomppI = Drule.incr_indexes2 ant1 ant2 @{thm relcomppI}

        val vars = (rev (Term.add_vars (prop_of relcomppI) []))

        val subst = map (apfst ((cterm_of thy) o Var)) (vars ~~ ([fun1] @ args1 @ [fun2] @ args2))

      in

        Drule.cterm_instantiate subst relcomppI

      end

    fun zip_transfer_rules ctxt thm =

      let

        val thy = Proof_Context.theory_of ctxt

        fun mk_POS ty = Const (@{const_name POS}, ty --> ty --> HOLogic.boolT)

        val rel = (Thm.dest_fun2 o Thm.dest_arg o cprop_of) thm

        val typ = (typ_of o ctyp_of_term) rel

        val POS_const = cterm_of thy (mk_POS typ)

        val var = cterm_of thy (Var (("X", #maxidx (rep_cterm (rel)) + 1), typ))

        val goal = Thm.apply (cterm_of thy HOLogic.Trueprop) (Thm.apply (Thm.apply POS_const rel) var)

      in

        [Lifting_Term.merge_transfer_relations ctxt goal, thm] MRSL @{thm POS_apply}

      end

    val thm = (inst_relcomppI (Proof_Context.theory_of ctxt) parametric_transfer_rule transfer_rule) 

                OF [parametric_transfer_rule, transfer_rule]

    val preprocessed_thm = preprocess ctxt thm

    val orig_ctxt = ctxt

    val (fixed_thm, ctxt) = yield_singleton (apfst snd oo Variable.import true) preprocessed_thm ctxt

    val assms = cprems_of fixed_thm

    val add_transfer_rule = Thm.attribute_declaration Transfer.transfer_add

    val (prems, ctxt) = fold_map Thm.assume_hyps assms ctxt

    val ctxt = Context.proof_map (fold add_transfer_rule prems) ctxt

    val zipped_thm =

      fixed_thm

      |> undisch_all

      |> zip_transfer_rules ctxt

      |> implies_intr_list assms

      |> singleton (Variable.export ctxt orig_ctxt)

  in

    zipped_thm

  end

fun print_generate_transfer_info msg = 

  let

    val error_msg = cat_lines 

      ["Generation of a parametric transfer rule failed.",

      (Pretty.string_of (Pretty.block

         [Pretty.str "Reason:", Pretty.brk 2, msg]))]

  in

    error error_msg

  end

fun map_ter _ x [] = x

    | map_ter f _ xs = map f xs

fun generate_transfer_rules lthy quot_thm rsp_thm def_thm par_thms =

  let

    val transfer_rule =

      ([quot_thm, rsp_thm, def_thm] MRSL @{thm Quotient_to_transfer})

      |> Lifting_Term.parametrize_transfer_rule lthy

  in

    (map_ter (generate_parametric_transfer_rule lthy transfer_rule) [transfer_rule] par_thms

    handle Lifting_Term.MERGE_TRANSFER_REL msg => (print_generate_transfer_info msg; [transfer_rule]))

  end

(* Generation of the code certificate from the rsp theorem *)

fun get_body_types (Type ("fun", [_, U]), Type ("fun", [_, V])) = get_body_types (U, V)

  | get_body_types (U, V)  = (U, V)

fun get_binder_types (Type ("fun", [T, U]), Type ("fun", [V, W])) = (T, V) :: get_binder_types (U, W)

  | get_binder_types _ = []

fun get_binder_types_by_rel (Const (@{const_name "rel_fun"}, _) $ _ $ S) (Type ("fun", [T, U]), Type ("fun", [V, W])) = 

    (T, V) :: get_binder_types_by_rel S (U, W)

  | get_binder_types_by_rel _ _ = []

fun get_body_type_by_rel (Const (@{const_name "rel_fun"}, _) $ _ $ S) (Type ("fun", [_, U]), Type ("fun", [_, V])) = 

    get_body_type_by_rel S (U, V)

  | get_body_type_by_rel _ (U, V)  = (U, V)

fun get_binder_rels (Const (@{const_name "rel_fun"}, _) $ R $ S) = R :: get_binder_rels S

  | get_binder_rels _ = []

fun force_rty_type ctxt rty rhs = 

  let

    val thy = Proof_Context.theory_of ctxt

    val rhs_schematic = singleton (Variable.polymorphic ctxt) rhs

    val rty_schematic = fastype_of rhs_schematic

    val match = Sign.typ_match thy (rty_schematic, rty) Vartab.empty

  in

    Envir.subst_term_types match rhs_schematic

  end

fun unabs_def ctxt def = 

  let

    val (_, rhs) = Thm.dest_equals (cprop_of def)

    fun dest_abs (Abs (var_name, T, _)) = (var_name, T)

      | dest_abs tm = raise TERM("get_abs_var",[tm])

    val (var_name, T) = dest_abs (term_of rhs)

    val (new_var_names, ctxt') = Variable.variant_fixes [var_name] ctxt

    val thy = Proof_Context.theory_of ctxt'

    val refl_thm = Thm.reflexive (cterm_of thy (Free (hd new_var_names, T)))

  in

    Thm.combination def refl_thm |>

    singleton (Proof_Context.export ctxt' ctxt)

  end

fun unabs_all_def ctxt def = 

  let

    val (_, rhs) = Thm.dest_equals (cprop_of def)

    val xs = strip_abs_vars (term_of rhs)

  in  

    fold (K (unabs_def ctxt)) xs def

  end

val map_fun_unfolded = 

  @{thm map_fun_def[abs_def]} |>

  unabs_def @{context} |>

  unabs_def @{context} |>

  Local_Defs.unfold @{context} [@{thm comp_def}]

fun unfold_fun_maps ctm =

  let

    fun unfold_conv ctm =

      case (Thm.term_of ctm) of

        Const (@{const_name "map_fun"}, _) $ _ $ _ => 

          (Conv.arg_conv unfold_conv then_conv Conv.rewr_conv map_fun_unfolded) ctm

        | _ => Conv.all_conv ctm

  in

    (Conv.fun_conv unfold_conv) ctm

  end

fun unfold_fun_maps_beta ctm =

  let val try_beta_conv = Conv.try_conv (Thm.beta_conversion false)

  in 

    (unfold_fun_maps then_conv try_beta_conv) ctm 

  end

fun prove_rel ctxt rsp_thm (rty, qty) =

  let

    val ty_args = get_binder_types (rty, qty)

    fun disch_arg args_ty thm = 

      let

        val quot_thm = Lifting_Term.prove_quot_thm ctxt args_ty

      in

        [quot_thm, thm] MRSL @{thm apply_rsp''}

      end

  in

    fold disch_arg ty_args rsp_thm

  end

exception CODE_CERT_GEN of string

fun simplify_code_eq ctxt def_thm = 

  Local_Defs.unfold ctxt [@{thm o_apply}, @{thm map_fun_def}, @{thm id_apply}] def_thm

(*

  quot_thm - quotient theorem (Quotient R Abs Rep T).

  returns: whether the Lifting package is capable to generate code for the abstract type

    represented by quot_thm

*)

fun can_generate_code_cert quot_thm  =

  case quot_thm_rel quot_thm of

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

    | Const (@{const_name eq_onp}, _) $ _  => true

    | _ => false

fun generate_rep_eq ctxt def_thm rsp_thm (rty, qty) =

  let

    val unfolded_def = Conv.fconv_rule (Conv.arg_conv unfold_fun_maps_beta) def_thm

    val unabs_def = unabs_all_def ctxt unfolded_def

  in  

    if body_type rty = body_type qty then 

      SOME (simplify_code_eq ctxt (unabs_def RS @{thm meta_eq_to_obj_eq}))

    else 

      let

        val thy = Proof_Context.theory_of ctxt

        val quot_thm = Lifting_Term.prove_quot_thm ctxt (get_body_types (rty, qty))

        val rel_fun = prove_rel ctxt rsp_thm (rty, qty)

        val rep_abs_thm = [quot_thm, rel_fun] MRSL @{thm Quotient_rep_abs_eq}

      in

        case mono_eq_prover ctxt (hd(prems_of rep_abs_thm)) of

          SOME mono_eq_thm =>

            let

              val rep_abs_eq = mono_eq_thm RS rep_abs_thm

              val rep = (cterm_of thy o quot_thm_rep) quot_thm

              val rep_refl = Thm.reflexive rep RS @{thm meta_eq_to_obj_eq}

              val repped_eq = [rep_refl, unabs_def RS @{thm meta_eq_to_obj_eq}] MRSL @{thm cong}

              val code_cert = [repped_eq, rep_abs_eq] MRSL trans

            in

              SOME (simplify_code_eq ctxt code_cert)

            end

          | NONE => NONE

      end

  end

fun generate_abs_eq ctxt def_thm rsp_thm quot_thm =

  let

    val abs_eq_with_assms =

      let

        val (rty, qty) = quot_thm_rty_qty quot_thm

        val rel = quot_thm_rel quot_thm

        val ty_args = get_binder_types_by_rel rel (rty, qty)

        val body_type = get_body_type_by_rel rel (rty, qty)

        val quot_ret_thm = Lifting_Term.prove_quot_thm ctxt body_type

        val rep_abs_folded_unmapped_thm = 

          let

            val rep_id = [quot_thm, def_thm] MRSL @{thm Quotient_Rep_eq}

            val ctm = Thm.dest_equals_lhs (cprop_of rep_id)

            val unfolded_maps_eq = unfold_fun_maps ctm

            val t1 = [quot_thm, def_thm, rsp_thm] MRSL @{thm Quotient_rep_abs_fold_unmap}

            val prems_pat = (hd o Drule.cprems_of) t1

            val insts = Thm.first_order_match (prems_pat, cprop_of unfolded_maps_eq)

          in

            unfolded_maps_eq RS (Drule.instantiate_normalize insts t1)

          end

      in

        rep_abs_folded_unmapped_thm

        |> fold (fn _ => fn thm => thm RS @{thm rel_funD2}) ty_args

        |> (fn x => x RS (@{thm Quotient_rel_abs2} OF [quot_ret_thm]))

      end

    val prem_rels = get_binder_rels (quot_thm_rel quot_thm);

    val proved_assms = prem_rels |> map (try_prove_refl_rel ctxt) 

      |> map_index (apfst (fn x => x + 1)) |> filter (is_some o snd) |> map (apsnd the) 

      |> map (apsnd (fn assm => assm RS @{thm ge_eq_refl}))

    val abs_eq = fold_rev (fn (i, assm) => fn thm => assm RSN (i, thm)) proved_assms abs_eq_with_assms

  in

    simplify_code_eq ctxt abs_eq

  end

fun register_code_eq_thy abs_eq_thm opt_rep_eq_thm (rty, qty) thy =

  let

    fun no_abstr (t $ u) = no_abstr t andalso no_abstr u

      | no_abstr (Abs (_, _, t)) = no_abstr t

      | no_abstr (Const (name, _)) = not (Code.is_abstr thy name)

      | no_abstr _ = true

    fun is_valid_eq eqn = can (Code.assert_eqn thy) (mk_meta_eq eqn, true) 

      andalso no_abstr (prop_of eqn)

    fun is_valid_abs_eq abs_eq = can (Code.assert_abs_eqn thy NONE) (mk_meta_eq abs_eq)

  in

    if is_valid_eq abs_eq_thm then

      Code.add_default_eqn abs_eq_thm thy

    else

      let

        val (rty_body, qty_body) = get_body_types (rty, qty)

      in

        if rty_body = qty_body then

         Code.add_default_eqn (the opt_rep_eq_thm) thy

        else

          if is_some opt_rep_eq_thm andalso is_valid_abs_eq (the opt_rep_eq_thm)

          then

            Code.add_abs_eqn (the opt_rep_eq_thm) thy

          else

            thy

      end

  end

local

  fun encode_code_eq thy abs_eq opt_rep_eq (rty, qty) = 

    let

      fun mk_type typ = typ |> Logic.mk_type |> cterm_of thy |> Drule.mk_term

    in

      Conjunction.intr_balanced [abs_eq, (the_default TrueI opt_rep_eq), mk_type rty, mk_type qty]

    end

  exception DECODE

  fun decode_code_eq thm =

    if nprems_of thm > 0 then raise DECODE 

    else

      let

        val [abs_eq, rep_eq, rty, qty] = Conjunction.elim_balanced 4 thm

        val opt_rep_eq = if Thm.eq_thm_prop (rep_eq, TrueI) then NONE else SOME rep_eq

        fun dest_type typ = typ |> Drule.dest_term |> term_of |> Logic.dest_type

      in

        (abs_eq, opt_rep_eq, (dest_type rty, dest_type qty)) 

      end

  fun register_encoded_code_eq thm thy =

    let

      val (abs_eq_thm, opt_rep_eq_thm, (rty, qty)) = decode_code_eq thm

    in

      register_code_eq_thy abs_eq_thm opt_rep_eq_thm (rty, qty) thy

    end

    handle DECODE => thy

  val register_code_eq_attribute = Thm.declaration_attribute

    (fn thm => Context.mapping (register_encoded_code_eq thm) I)

  val register_code_eq_attrib = Attrib.internal (K register_code_eq_attribute)

  fun no_no_code ctxt (rty, qty) =

    if same_type_constrs (rty, qty) then

      forall (no_no_code ctxt) (Targs rty ~~ Targs qty)

    else

      if is_Type qty then

        if Lifting_Info.is_no_code_type ctxt (Tname qty) then false

        else 

          let 

            val (rty', rtyq) = Lifting_Term.instantiate_rtys ctxt (rty, qty)

            val (rty's, rtyqs) = (Targs rty', Targs rtyq)

          in

            forall (no_no_code ctxt) (rty's ~~ rtyqs)

          end

      else

        true

in

fun register_code_eq abs_eq_thm opt_rep_eq_thm (rty, qty) lthy =

  let

    val thy = Proof_Context.theory_of lthy

    val encoded_code_eq = encode_code_eq thy abs_eq_thm opt_rep_eq_thm (rty, qty)

  in

    if no_no_code lthy (rty, qty) then 

      (snd oo Local_Theory.note) ((Binding.empty, [register_code_eq_attrib]), [encoded_code_eq]) lthy

    else

      lthy

  end

end

(*

  Defines an operation on an abstract type in terms of a corresponding operation 

    on a representation type.

  var - a binding and a mixfix of the new constant being defined

  qty - an abstract type of the new constant

  rhs - a term representing the new constant on the raw level

  rsp_thm - a respectfulness theorem in the internal tagged form (like '(R ===> R ===> R) f f'),

    i.e. "(Lifting_Term.equiv_relation (fastype_of rhs, qty)) $ rhs $ rhs"

  par_thms - a parametricity theorem for rhs

*)

fun add_lift_def var qty rhs rsp_thm par_thms lthy =

  let

    val rty = fastype_of rhs

    val quot_thm = Lifting_Term.prove_quot_thm lthy (rty, qty)

    val absrep_trm =  quot_thm_abs quot_thm

    val rty_forced = (domain_type o fastype_of) absrep_trm

    val forced_rhs = force_rty_type lthy rty_forced rhs

    val lhs = Free (Binding.name_of (#1 var), qty)

    val prop = Logic.mk_equals (lhs, absrep_trm $ forced_rhs)

    val (_, prop') = Local_Defs.cert_def lthy prop

    val (_, newrhs) = Local_Defs.abs_def prop'

    val ((_, (_ , def_thm)), lthy') = 

      Local_Theory.define (var, ((Thm.def_binding (#1 var), []), newrhs)) lthy

    val transfer_rules = generate_transfer_rules lthy' quot_thm rsp_thm def_thm par_thms

    val abs_eq_thm = generate_abs_eq lthy' def_thm rsp_thm quot_thm

    val opt_rep_eq_thm = generate_rep_eq lthy' def_thm rsp_thm (rty_forced, qty)

    fun qualify defname suffix = Binding.qualified true suffix defname

    val lhs_name = (#1 var)

    val rsp_thm_name = qualify lhs_name "rsp"

    val abs_eq_thm_name = qualify lhs_name "abs_eq"

    val rep_eq_thm_name = qualify lhs_name "rep_eq"

    val transfer_rule_name = qualify lhs_name "transfer"

    val transfer_attr = Attrib.internal (K Transfer.transfer_add)

  in

    lthy'

      |> (snd oo Local_Theory.note) ((rsp_thm_name, []), [rsp_thm])

      |> (snd oo Local_Theory.note) ((transfer_rule_name, [transfer_attr]), transfer_rules)

      |> (snd oo Local_Theory.note) ((abs_eq_thm_name, []), [abs_eq_thm])

      |> (case opt_rep_eq_thm of 

            SOME rep_eq_thm => (snd oo Local_Theory.note) ((rep_eq_thm_name, []), [rep_eq_thm])

            | NONE => I)

      |> register_code_eq abs_eq_thm opt_rep_eq_thm (rty_forced, qty)

  end

local

  val eq_onp_assms_tac_fixed_rules = map (Transfer.prep_transfer_domain_thm @{context})

    [@{thm pcr_Domainp_total}, @{thm pcr_Domainp_par_left_total}, @{thm pcr_Domainp_par}, 

      @{thm pcr_Domainp}]

in

fun mk_readable_rsp_thm_eq tm lthy =

  let

    val ctm = cterm_of (Proof_Context.theory_of lthy) tm

    fun assms_rewr_conv tactic rule ct =

      let

        fun prove_extra_assms thm =

          let

            val assms = cprems_of thm

            fun finish thm = if Thm.no_prems thm then SOME (Goal.conclude thm) else NONE

            fun prove ctm = Option.mapPartial finish (SINGLE tactic (Goal.init ctm))

          in

            map_interrupt prove assms

          end

        fun cconl_of thm = Drule.strip_imp_concl (cprop_of thm)

        fun lhs_of thm = fst (Thm.dest_equals (cconl_of thm))

        fun rhs_of thm = snd (Thm.dest_equals (cconl_of thm))

        val rule1 = Thm.incr_indexes (#maxidx (Thm.rep_cterm ct) + 1) rule;

        val lhs = lhs_of rule1;

        val rule2 = Thm.rename_boundvars (Thm.term_of lhs) (Thm.term_of ct) rule1;

        val rule3 =

          Thm.instantiate (Thm.match (lhs, ct)) rule2

            handle Pattern.MATCH => raise CTERM ("assms_rewr_conv", [lhs, ct]);

        val proved_assms = prove_extra_assms rule3

      in

        case proved_assms of

          SOME proved_assms =>

            let

              val rule3 = proved_assms MRSL rule3

              val rule4 =

                if lhs_of rule3 aconvc ct then rule3

                else

                  let val ceq = Thm.dest_fun2 (Thm.cprop_of rule3)

                  in rule3 COMP Thm.trivial (Thm.mk_binop ceq ct (rhs_of rule3)) end

            in Thm.transitive rule4 (Thm.beta_conversion true (rhs_of rule4)) end

          | NONE => Conv.no_conv ct

      end

    fun assms_rewrs_conv tactic rules = Conv.first_conv (map (assms_rewr_conv tactic) rules)

    fun simp_arrows_conv ctm =

      let

        val unfold_conv = Conv.rewrs_conv 

          [@{thm rel_fun_eq_eq_onp[THEN eq_reflection]}, 

            @{thm rel_fun_eq_onp_rel[THEN eq_reflection]},

            @{thm rel_fun_eq[THEN eq_reflection]},

            @{thm rel_fun_eq_rel[THEN eq_reflection]}, 

            @{thm rel_fun_def[THEN eq_reflection]}]

        fun binop_conv2 cv1 cv2 = Conv.combination_conv (Conv.arg_conv cv1) cv2

        val eq_onp_assms_tac_rules = @{thm left_unique_OO} :: 

            eq_onp_assms_tac_fixed_rules @ (Transfer.get_transfer_raw lthy)

        val eq_onp_assms_tac = (TRY o REPEAT_ALL_NEW (resolve_tac eq_onp_assms_tac_rules) 

          THEN_ALL_NEW (DETERM o Transfer.eq_tac lthy)) 1

        val relator_eq_onp_conv = Conv.bottom_conv

          (K (Conv.try_conv (assms_rewrs_conv eq_onp_assms_tac

            (Lifting_Info.get_relator_eq_onp_rules lthy)))) lthy

        val relator_eq_conv = Conv.bottom_conv

          (K (Conv.try_conv (Conv.rewrs_conv (Transfer.get_relator_eq lthy)))) lthy

      in

        case (Thm.term_of ctm) of

          Const (@{const_name "rel_fun"}, _) $ _ $ _ => 

            (binop_conv2 simp_arrows_conv simp_arrows_conv then_conv unfold_conv) ctm

          | _ => (relator_eq_onp_conv then_conv relator_eq_conv) ctm

      end

    val unfold_ret_val_invs = Conv.bottom_conv 

      (K (Conv.try_conv (Conv.rewr_conv @{thm eq_onp_same_args[THEN eq_reflection]}))) lthy

    val unfold_inv_conv = 

      Conv.top_sweep_conv (K (Conv.rewr_conv @{thm eq_onp_def[THEN eq_reflection]})) lthy

    val simp_conv = HOLogic.Trueprop_conv (Conv.fun2_conv simp_arrows_conv)

    val univq_conv = Conv.rewr_conv @{thm HOL.all_simps(6)[symmetric, THEN eq_reflection]}

    val univq_prenex_conv = Conv.top_conv (K (Conv.try_conv univq_conv)) lthy

    val beta_conv = Thm.beta_conversion true

    val eq_thm = 

      (simp_conv then_conv univq_prenex_conv then_conv beta_conv then_conv unfold_ret_val_invs

         then_conv unfold_inv_conv) ctm

  in

    Object_Logic.rulify lthy (eq_thm RS Drule.equal_elim_rule2)

  end

end

fun rename_to_tnames ctxt term =

  let

    fun all_typs (Const (@{const_name Pure.all}, _) $ Abs (_, T, t)) = T :: all_typs t

      | all_typs _ = []

    fun rename (Const (@{const_name Pure.all}, T1) $ Abs (_, T2, t)) (new_name :: names) = 

        (Const (@{const_name Pure.all}, T1) $ Abs (new_name, T2, rename t names)) 

      | rename t _ = t

    val (fixed_def_t, _) = yield_singleton (Variable.importT_terms) term ctxt

    val new_names = Datatype_Prop.make_tnames (all_typs fixed_def_t)

  in

    rename term new_names

  end

(* This is not very cheap way of getting the rules but we have only few active

  liftings in the current setting *)

fun get_cr_pcr_eqs ctxt =

  let

    fun collect (data : Lifting_Info.quotient) l =

      if is_some (#pcr_info data) 

      then ((Thm.symmetric o safe_mk_meta_eq o #pcr_cr_eq o the o #pcr_info) data :: l) 

      else l

    val table = Lifting_Info.get_quotients ctxt

  in

    Symtab.fold (fn (_, data) => fn l => collect data l) table []

  end

(*

  lifting_definition command. It opens a proof of a corresponding respectfulness 

  theorem in a user-friendly, readable form. Then add_lift_def is called internally.

*)

fun lift_def_cmd (raw_var, rhs_raw, par_xthms) lthy =

  let

    val ((binding, SOME qty, mx), lthy) = yield_singleton Proof_Context.read_vars raw_var lthy 

    val rhs = (Syntax.check_term lthy o Syntax.parse_term lthy) rhs_raw

    val rsp_rel = Lifting_Term.equiv_relation lthy (fastype_of rhs, qty)

    val rty_forced = (domain_type o fastype_of) rsp_rel;

    val forced_rhs = force_rty_type lthy rty_forced rhs;

    val cr_to_pcr_conv = HOLogic.Trueprop_conv (Conv.fun2_conv 

      (Raw_Simplifier.rewrite lthy false (get_cr_pcr_eqs lthy)))

    val (prsp_tm, rsp_prsp_eq) = HOLogic.mk_Trueprop (rsp_rel $ forced_rhs $ forced_rhs)

      |> cterm_of (Proof_Context.theory_of lthy)

      |> cr_to_pcr_conv

      |> ` concl_of

      |>> Logic.dest_equals

      |>> snd

    val to_rsp = rsp_prsp_eq RS Drule.equal_elim_rule2

    val opt_proven_rsp_thm = try_prove_reflexivity lthy prsp_tm

    val par_thms = Attrib.eval_thms lthy par_xthms

    fun after_qed internal_rsp_thm lthy = 

      add_lift_def (binding, mx) qty rhs (internal_rsp_thm RS to_rsp) par_thms lthy

  in

    case opt_proven_rsp_thm of

      SOME thm => Proof.theorem NONE (K (after_qed thm)) [] lthy

      | NONE =>  

        let

          val readable_rsp_thm_eq = mk_readable_rsp_thm_eq prsp_tm lthy

          val (readable_rsp_tm, _) = Logic.dest_implies (prop_of readable_rsp_thm_eq)

          val readable_rsp_tm_tnames = rename_to_tnames lthy readable_rsp_tm

          fun after_qed' thm_list lthy = 

            let

              val internal_rsp_thm = Goal.prove lthy [] [] prsp_tm 

                  (fn {context = ctxt, ...} =>

                    rtac readable_rsp_thm_eq 1 THEN Proof_Context.fact_tac ctxt (hd thm_list) 1)

            in

              after_qed internal_rsp_thm lthy

            end

        in

          Proof.theorem NONE after_qed' [[(readable_rsp_tm_tnames,[])]] lthy

        end 

  end

fun quot_thm_err ctxt (rty, qty) pretty_msg =

  let

    val error_msg = cat_lines

       ["Lifting failed for the following types:",

        Pretty.string_of (Pretty.block

         [Pretty.str "Raw type:", Pretty.brk 2, Syntax.pretty_typ ctxt rty]),

        Pretty.string_of (Pretty.block

         [Pretty.str "Abstract type:", Pretty.brk 2, Syntax.pretty_typ ctxt qty]),

        "",

        (Pretty.string_of (Pretty.block

         [Pretty.str "Reason:", Pretty.brk 2, pretty_msg]))]

  in

    error error_msg

  end

fun check_rty_err ctxt (rty_schematic, rty_forced) (raw_var, rhs_raw) =

  let

    val (_, ctxt') = yield_singleton Proof_Context.read_vars raw_var ctxt 

    val rhs = (Syntax.check_term ctxt' o Syntax.parse_term ctxt') rhs_raw

    val error_msg = cat_lines

       ["Lifting failed for the following term:",

        Pretty.string_of (Pretty.block

         [Pretty.str "Term:", Pretty.brk 2, Syntax.pretty_term ctxt rhs]),

        Pretty.string_of (Pretty.block

         [Pretty.str "Type:", Pretty.brk 2, Syntax.pretty_typ ctxt rty_schematic]),

        "",

        (Pretty.string_of (Pretty.block

         [Pretty.str "Reason:", 

          Pretty.brk 2, 

          Pretty.str "The type of the term cannot be instantiated to",

          Pretty.brk 1,

          Pretty.quote (Syntax.pretty_typ ctxt rty_forced),

          Pretty.str "."]))]

    in

      error error_msg

    end

fun lift_def_cmd_with_err_handling (raw_var, rhs_raw, par_xthms) lthy =

  (lift_def_cmd (raw_var, rhs_raw, par_xthms) lthy

    handle Lifting_Term.QUOT_THM (rty, qty, msg) => quot_thm_err lthy (rty, qty) msg)

    handle Lifting_Term.CHECK_RTY (rty_schematic, rty_forced) => 

      check_rty_err lthy (rty_schematic, rty_forced) (raw_var, rhs_raw)

(* parser and command *)

val liftdef_parser =

  (((Parse.binding -- (@{keyword "::"} |-- (Parse.typ >> SOME) -- Parse.opt_mixfix')) >> Parse.triple2)

    --| @{keyword "is"} -- Parse.term -- 

      Scan.optional (@{keyword "parametric"} |-- Parse.!!! Parse_Spec.xthms1) []) >> Parse.triple1

val _ =

  Outer_Syntax.local_theory_to_proof @{command_spec "lift_definition"}

    "definition for constants over the quotient type"

      (liftdef_parser >> lift_def_cmd_with_err_handling)

end (* structure *)