(*  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 (t, i) =

       case HOLogic.dest_Trueprop t of 

         Const (@{const_name "less_eq"}, _) $ _ $ _ => REPEAT_ALL_NEW (resolve_tac refl_rules) i

         |  _ => REPEAT_ALL_NEW (resolve_tac transfer_rules) i

   in

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

       handle ERROR _ => NONE

   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 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 prems = prems_of abs_eq_with_assms

     val indexed_prems = map_index (apfst (fn x => x + 1)) prems

     val indexed_assms = map (apsnd (try_prove_reflexivity ctxt)) indexed_prems

     val proved_assms = map (apsnd the) (filter (is_some o snd) indexed_assms)

     val abs_eq = fold_rev (fn (i, assms) => fn thm => assms 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

   fun decode_code_eq thm =

     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

   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)

 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

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

       [encoded_code_eq]) 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

     (* 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

     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 cr_to_pcr_conv = Raw_Simplifier.rewrite lthy false (get_cr_pcr_eqs 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 

       (cr_to_pcr_conv then_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

 (*

   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 internal_rsp_tm = HOLogic.mk_Trueprop (rsp_rel $ forced_rhs $ forced_rhs)

     val opt_proven_rsp_thm = try_prove_reflexivity lthy internal_rsp_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 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 internal_rsp_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 [] [] internal_rsp_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 *)