new command for lifting BNF structure over typedefs

--- a/src/HOL/BNF_Composition.thy	Wed Aug 12 13:56:46 2015 +0200
+++ b/src/HOL/BNF_Composition.thy	Wed Aug 12 20:46:33 2015 +0200
@@ -10,8 +10,14 @@
 
 theory BNF_Composition
 imports BNF_Def
+keywords
+  "copy_bnf" :: thy_decl and
+  "lift_bnf" :: thy_goal
 begin
 
+lemma ssubst_mem: "\<lbrakk>t = s; s \<in> X\<rbrakk> \<Longrightarrow> t \<in> X"
+  by simp
+
 lemma empty_natural: "(\<lambda>_. {}) o f = image g o (\<lambda>_. {})"
   by (rule ext) simp
 
@@ -168,6 +174,7 @@
 
 ML_file "Tools/BNF/bnf_comp_tactics.ML"
 ML_file "Tools/BNF/bnf_comp.ML"
+ML_file "Tools/BNF/bnf_lift.ML"
 
 hide_fact
   DEADID.inj_map DEADID.inj_map_strong DEADID.map_comp DEADID.map_cong DEADID.map_cong0

--- a/src/HOL/BNF_Fixpoint_Base.thy	Wed Aug 12 13:56:46 2015 +0200
+++ b/src/HOL/BNF_Fixpoint_Base.thy	Wed Aug 12 20:46:33 2015 +0200
@@ -55,9 +55,6 @@
   thus "EX a. b = f a" by blast
 qed
 
-lemma ssubst_mem: "\<lbrakk>t = s; s \<in> X\<rbrakk> \<Longrightarrow> t \<in> X"
-  by simp
-
 lemma case_sum_step:
   "case_sum (case_sum f' g') g (Inl p) = case_sum f' g' p"
   "case_sum f (case_sum f' g') (Inr p) = case_sum f' g' p"

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Datatype_Examples/Lift_BNF.thy	Wed Aug 12 20:46:33 2015 +0200
@@ -0,0 +1,75 @@
+theory Lift_BNF
+imports Main
+begin
+
+typedef 'a nonempty_list = "{xs :: 'a list. xs \<noteq> []}"
+  by blast
+
+lift_bnf (no_warn_wits) (neset: 'a) nonempty_list
+  for map: nemap rel: nerel
+  by simp_all
+
+typedef ('a :: finite, 'b) fin_nonempty_list = "{(xs :: 'a set, ys :: 'b list). ys \<noteq> []}"
+  by blast
+
+lift_bnf (dead 'a :: finite, 'b) fin_nonempty_list
+  by auto
+
+datatype 'a tree = Leaf | Node 'a "'a tree nonempty_list"
+
+record 'a point =
+  xCoord :: 'a
+  yCoord :: 'a
+
+copy_bnf ('a, 's) point_ext
+
+typedef 'a it = "UNIV :: 'a set"
+  by blast
+
+copy_bnf (plugins del: size) 'a it
+
+typedef ('a, 'b) T_prod = "UNIV :: ('a \<times> 'b) set"
+  by blast
+
+copy_bnf ('a, 'b) T_prod
+
+typedef ('a, 'b, 'c) T_func = "UNIV :: ('a \<Rightarrow> 'b * 'c) set"
+  by blast
+
+copy_bnf ('a, 'b, 'c) T_func
+
+typedef ('a, 'b) sum_copy = "UNIV :: ('a + 'b) set"
+  by blast
+
+copy_bnf ('a, 'b) sum_copy
+
+typedef ('a, 'b) T_sum = "{Inl x | x. True} :: ('a + 'b) set"
+  by blast
+
+lift_bnf (no_warn_wits) ('a, 'b) T_sum [wits: "Inl :: 'a \<Rightarrow> 'a + 'b"]
+  by (auto simp: map_sum_def sum_set_defs split: sum.splits)
+
+typedef ('key, 'value) alist = "{xs :: ('key \<times> 'value) list. (distinct \<circ> map fst) xs}"
+  morphisms impl_of Alist
+proof
+  show "[] \<in> {xs. (distinct o map fst) xs}"
+    by simp
+qed
+
+lift_bnf (dead 'k, 'v) alist [wits: "Nil :: ('k \<times> 'v) list"]
+  by simp_all
+
+typedef 'a myopt = "{X :: 'a set. finite X \<and> card X \<le> 1}" by (rule exI[of _ "{}"]) auto
+lemma myopt_type_def: "type_definition
+  (\<lambda>X. if card (Rep_myopt X) = 1 then Some (the_elem (Rep_myopt X)) else None)
+  (\<lambda>x. Abs_myopt (case x of Some x \<Rightarrow> {x} | _ \<Rightarrow> {}))
+  (UNIV :: 'a option set)"
+  apply unfold_locales
+    apply (auto simp: Abs_myopt_inverse dest!: card_eq_SucD split: option.splits)
+   apply (metis Rep_myopt_inverse)
+  apply (metis One_nat_def Rep_myopt Rep_myopt_inverse Suc_le_mono card_0_eq le0 le_antisym mem_Collect_eq nat.exhaust)
+  done
+
+copy_bnf 'a myopt via myopt_type_def
+
+end

--- a/src/HOL/Tools/BNF/bnf_def.ML	Wed Aug 12 13:56:46 2015 +0200
+++ b/src/HOL/Tools/BNF/bnf_def.ML	Wed Aug 12 20:46:33 2015 +0200
@@ -104,6 +104,7 @@
     'a list
 
   val mk_witness: int list * term -> thm list -> nonemptiness_witness
+  val mk_wit_goals: term list -> term list -> term list -> int list * term -> term list
   val minimize_wits: (''a list * 'b) list -> (''a list * 'b) list
   val wits_of_bnf: bnf -> nonemptiness_witness list
 
@@ -759,6 +760,22 @@
     |>> (fn [] => bnf | noted => morph_bnf (substitute_noted_thm noted) bnf)
   end;
 
+fun mk_wit_goals zs bs sets (I, wit) =
+  let
+    val xs = map (nth bs) I;
+    fun wit_goal i =
+      let
+        val z = nth zs i;
+        val set_wit = nth sets i $ Term.list_comb (wit, xs);
+        val concl = HOLogic.mk_Trueprop
+          (if member (op =) I i then HOLogic.mk_eq (z, nth bs i) else @{term False});
+      in
+        fold_rev Logic.all (z :: xs) (Logic.mk_implies (mk_Trueprop_mem (z, set_wit), concl))
+      end;
+  in
+    map wit_goal (0 upto length sets - 1)
+  end;
+
 
 (* Define new BNFs *)
 
@@ -1117,22 +1134,7 @@
     val goals = zip_axioms map_id0_goal map_comp0_goal map_cong0_goal set_map0s_goal
       card_order_bd_goal cinfinite_bd_goal set_bds_goal le_rel_OO_goal rel_OO_Grp_goal;
 
-    fun mk_wit_goals (I, wit) =
-      let
-        val xs = map (nth bs) I;
-        fun wit_goal i =
-          let
-            val z = nth zs i;
-            val set_wit = nth bnf_sets_As i $ Term.list_comb (wit, xs);
-            val concl = HOLogic.mk_Trueprop
-              (if member (op =) I i then HOLogic.mk_eq (z, nth bs i) else @{term False});
-          in
-            fold_rev Logic.all (z :: xs) (Logic.mk_implies (mk_Trueprop_mem (z, set_wit), concl))
-          end;
-      in
-        map wit_goal (0 upto live - 1)
-      end;
-
+    val mk_wit_goals = mk_wit_goals bs zs bnf_sets_As;
     fun triv_wit_tac ctxt = mk_trivial_wit_tac ctxt bnf_wit_defs;
 
     val wit_goalss =

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/BNF/bnf_lift.ML	Wed Aug 12 20:46:33 2015 +0200
@@ -0,0 +1,391 @@
+(*  Title:      HOL/Tools/BNF/bnf_lift.ML
+    Author:     Julian Biendarra, TU Muenchen
+    Author:     Dmitriy Traytel, ETH Zurich
+    Copyright   2015
+
+Lifting of BNFs through typedefs.
+*)
+
+signature BNF_LIFT = sig
+  datatype lift_bnf_option = Plugins_Option of Proof.context -> Plugin_Name.filter | No_Warn_Wits
+
+  val copy_bnf:
+    (((lift_bnf_option list * (binding option * (string * sort option)) list) *
+      string) * thm option) * (binding * binding) ->
+      local_theory -> local_theory
+  val copy_bnf_cmd:
+    (((lift_bnf_option list * (binding option * (string * string option)) list) *
+      string) * (Facts.ref * Token.src list) option) * (binding * binding) ->
+      local_theory -> local_theory
+  val lift_bnf:
+    (((lift_bnf_option list * (binding option * (string * sort option)) list) *
+      string) * thm option) * (binding * binding) ->
+      ({context: Proof.context, prems: thm list} -> tactic) list ->
+      local_theory -> local_theory
+  val lift_bnf_cmd:
+     ((((lift_bnf_option list * (binding option * (string * string option)) list) *
+       string) * string list) * (Facts.ref * Token.src list) option) * (binding * binding) ->
+       local_theory -> Proof.state
+  end
+
+structure BNF_Lift : BNF_LIFT = struct
+
+open Ctr_Sugar_Tactics
+open BNF_Util
+open BNF_Comp
+open BNF_Def
+
+datatype lift_bnf_option = Plugins_Option of Proof.context -> Plugin_Name.filter | No_Warn_Wits
+
+fun typedef_bnf thm wits specs map_b rel_b opts lthy =
+  let
+    val plugins = get_first (fn Plugins_Option f => SOME (f lthy) | _ => NONE) (rev opts)
+      |> the_default Plugin_Name.default_filter;
+    val no_warn_wits = exists (can (fn Sequential_Option => ())) opts;
+
+    (* extract Rep Abs F RepT AbsT *)
+    val (_, [Rep_G, Abs_G, F]) = Thm.prop_of thm
+      |> HOLogic.dest_Trueprop
+      |> Term.strip_comb;
+    val typ_Abs_G = fastype_of Abs_G |> dest_funT;
+    val RepT = fst typ_Abs_G; (* F *)
+    val AbsT = snd typ_Abs_G; (* G *)
+    val AbsT_name = fst (dest_Type AbsT);
+    val tvs = AbsT |> dest_Type |> snd |> map (fst o dest_TVar);
+    val alpha0s = map (TFree o snd) specs;
+    
+    (* instantiate the new type variables newtvs to oldtvs *)
+    val subst = subst_TVars (tvs ~~ alpha0s);
+    val typ_subst = typ_subst_TVars (tvs ~~ alpha0s);
+
+    val Rep_G = subst Rep_G;
+    val Abs_G = subst Abs_G;
+    val F = subst F;
+    val RepT = typ_subst RepT;
+    val AbsT = typ_subst AbsT;
+
+    fun flatten_tyargs Ass = map dest_TFree alpha0s |>
+      filter (fn T => exists (fn Ts => member (op =) Ts T) Ass);
+
+    val Ds0 = filter (is_none o fst) specs |> map snd;
+
+    (* get the bnf for RepT *)
+    val ((bnf, (deads, alphas)),((_, unfolds), lthy)) =
+      bnf_of_typ Dont_Inline (Binding.qualify true AbsT_name) flatten_tyargs []
+        Ds0 RepT ((empty_comp_cache, empty_unfolds), lthy);
+
+    val set_bs = map (fn T => find_index (fn U => T = U) alpha0s) alphas
+      |> map (the_default Binding.empty o fst o nth specs);
+
+    val _ = case alphas of [] => error "No live variables." | alphas => alphas;
+
+    val defs = #map_unfolds unfolds @ flat (#set_unfoldss unfolds) @ #rel_unfolds unfolds;
+
+    (* number of live variables *)
+    val lives = length alphas;
+
+    (* state the three required properties *)
+    val sorts = map Type.sort_of_atyp alphas;
+    val names_lthy = fold Variable.declare_typ (alphas @ deads) lthy;
+    val (alphas', names_lthy) = mk_TFrees' sorts names_lthy;
+    val (betas, names_lthy) = mk_TFrees' sorts names_lthy;
+
+    val map_F = mk_map_of_bnf deads alphas betas bnf;
+
+    val (typ_fs, typ_aF) = fastype_of map_F |> strip_typeN lives ||> domain_type;
+    val typ_pairs = map HOLogic.mk_prodT (alphas ~~ alphas');
+    val typ_subst_pair = typ_subst_atomic (alphas ~~ typ_pairs);
+    val typ_pair = typ_subst_pair RepT;
+
+    val subst_b = subst_atomic_types (alphas ~~ betas);
+    val subst_a' = subst_atomic_types (alphas ~~ alphas');
+    val subst_pair = subst_atomic_types (alphas ~~ typ_pairs);
+    val aF_set = F;
+    val bF_set = subst_b F;
+    val aF_set' = subst_a' F;
+    val pairF_set = subst_pair F;
+    val map_F_fst = mk_map_of_bnf deads typ_pairs alphas bnf;
+    val map_F_snd = mk_map_of_bnf deads typ_pairs alphas' bnf;
+    val wits_F = mk_wits_of_bnf
+      (replicate (nwits_of_bnf bnf) deads)
+      (replicate (nwits_of_bnf bnf) alphas) bnf;
+
+    (* val map_closed_F = @{term "\<And>f x. x \<in> F \<Longrightarrow> map_F f x \<in> F"}; *)
+    val (var_fs, names_lthy) = mk_Frees "f" typ_fs names_lthy;
+    val (var_x, names_lthy) = mk_Frees "x" [typ_aF] names_lthy |>> the_single;
+    val mem_x = HOLogic.mk_mem (var_x, aF_set) |> HOLogic.mk_Trueprop;
+    val map_f = list_comb (map_F, var_fs);
+    val mem_map = HOLogic.mk_mem (map_f $ var_x, bF_set) |> HOLogic.mk_Trueprop;
+    val imp_map = Logic.mk_implies (mem_x, mem_map);
+    val map_closed_F = Library.foldr (Library.uncurry Logic.all) (var_fs, Logic.all var_x imp_map);
+
+    (* val zip_closed_F = @{term "\<And>z. map_F fst z \<in> F \<Longrightarrow> map_F snd z \<in> F \<Longrightarrow> z \<in> F"}; *)
+    val (var_zs, names_lthy) = mk_Frees "z" [typ_pair] names_lthy;
+    val (pairs, names_lthy) = mk_Frees "tmp" typ_pairs names_lthy;
+    val var_z = hd var_zs;
+    val fsts = map (fst o Term.strip_comb o HOLogic.mk_fst) pairs;
+    val snds = map (fst o Term.strip_comb o HOLogic.mk_snd) pairs;
+    val map_fst = list_comb (list_comb (map_F_fst, fsts), var_zs);
+    val mem_map_fst = HOLogic.mk_mem (map_fst, aF_set) |> HOLogic.mk_Trueprop;
+    val map_snd = list_comb (list_comb (map_F_snd, snds), var_zs);
+    val mem_map_snd = HOLogic.mk_mem (map_snd, aF_set') |> HOLogic.mk_Trueprop;
+    val mem_z = HOLogic.mk_mem (var_z, pairF_set) |> HOLogic.mk_Trueprop;
+    val imp_zip = Logic.mk_implies (mem_map_fst, Logic.mk_implies (mem_map_snd, mem_z));
+    val zip_closed_F = Logic.all var_z imp_zip;
+
+    (* val wit_closed_F = @{term "wit_F a \<in> F"}; *)
+    val (var_as, names_lthy) = mk_Frees "a" alphas names_lthy;
+    val (var_bs, _) = mk_Frees "a" alphas names_lthy;
+    val Iwits = the_default wits_F (Option.map (map (`(map (fn T =>
+      find_index (fn U => T = U) alphas) o fst o strip_type o fastype_of))) wits);
+    val wit_closed_Fs =
+      map (fn (I, wit_F) =>
+        let
+          val vars = map (nth var_as) I;
+          val wit_a = list_comb (wit_F, vars);
+        in
+          Library.foldr (Library.uncurry Logic.all) (vars,
+            HOLogic.mk_mem (wit_a, aF_set) |> HOLogic.mk_Trueprop)
+        end)
+      Iwits;
+
+    val mk_wit_goals = mk_wit_goals var_as var_bs
+      (mk_sets_of_bnf (replicate lives deads)  (replicate lives alphas) bnf);
+
+    val goals = [map_closed_F, zip_closed_F] @ wit_closed_Fs @
+      (case wits of NONE => [] | _ => maps mk_wit_goals Iwits);
+
+    val lost_wits = filter_out (fn (J, _) => exists (fn (I, _) => I = J) Iwits) wits_F;
+    val _ = if null lost_wits orelse no_warn_wits then () else
+      lost_wits
+      |> map (Syntax.pretty_typ lthy o fastype_of o snd)
+      |> Pretty.big_list
+        "The following types of nonemptiness witnesses of the raw type's BNF were lost:"
+      |> (fn pt => Pretty.chunks [pt,
+        Pretty.para "You can specify a liftable witness (e.g., a term of one of the above types\
+          \ that satisfies the typedef's invariant)\
+          \ using the annotation [wits: <term>]."])
+      |> Pretty.string_of
+      |> warning;
+
+    fun after_qed ([map_closed_thm] :: [zip_closed_thm] :: wit_thmss) lthy =
+        let
+          val (wit_closed_thms, wit_thms) =
+            (case wits of
+              NONE => (map the_single wit_thmss, wit_thms_of_bnf bnf)
+            | _ => chop (length wit_closed_Fs) (map the_single wit_thmss))
+
+          (*  construct map set bd rel wit *)
+          (* val map_G = @{term "\<lambda>f. Abs_G o map_F f o Rep_G"}; *)
+          val Abs_Gb = subst_b Abs_G;
+          val map_G = Library.foldr (uncurry HOLogic.tupled_lambda)
+            (var_fs, HOLogic.mk_comp (HOLogic.mk_comp (Abs_Gb, map_f),
+            Rep_G));
+
+          (* val sets_G = [@{term "set_F o Rep_G"}]; *)
+          val sets_F = mk_sets_of_bnf (replicate lives deads) (replicate lives alphas) bnf;
+          val sets_G = map (fn set_F => HOLogic.mk_comp (set_F, Rep_G)) sets_F;
+
+          (* val bd_G = @{term "bd_F"}; *)
+          val bd_F = mk_bd_of_bnf deads alphas bnf;
+          val bd_G = bd_F;
+
+          (* val rel_G = @{term "\<lambda>R. BNF_Def.vimage2p Rep_G Rep_G (rel_F R)"}; *)
+          val rel_F = mk_rel_of_bnf deads alphas betas bnf;
+          val (typ_Rs, _) = fastype_of rel_F |> strip_typeN lives;
+
+          val (var_Rs, names_lthy) = mk_Frees "R" typ_Rs lthy;
+          val Rep_Gb = subst_b Rep_G;
+          val rel_G = fold_rev absfree (map dest_Free var_Rs)
+            (mk_vimage2p Rep_G Rep_Gb $ list_comb (rel_F, var_Rs));
+
+          (* val wits_G = [@{term "Abs_G o wit_F"}]; *)
+          val (var_as, _) = mk_Frees "a" alphas names_lthy;
+          val wits_G =
+            map (fn (I, wit_F) =>
+              let
+                val vs = map (nth var_as) I;
+              in fold_rev absfree (map dest_Free vs) (Abs_G $ (list_comb (wit_F, vs))) end)
+            Iwits;
+
+          (* tactics *)
+          val Rep_thm = thm RS @{thm type_definition.Rep};
+          val Abs_inverse_thm = thm RS @{thm type_definition.Abs_inverse};
+          val Abs_inject_thm = thm RS @{thm type_definition.Abs_inject};
+          val Rep_cases_thm = thm RS @{thm type_definition.Rep_cases};
+          val Rep_inverse_thm = thm RS @{thm type_definition.Rep_inverse};
+
+          fun map_id0_tac ctxt =
+            HEADGOAL (EVERY' [rtac ctxt ext,
+              SELECT_GOAL (unfold_thms_tac ctxt [map_id0_of_bnf bnf, id_apply, o_apply,
+                Rep_inverse_thm]),
+              rtac ctxt refl]);
+
+          fun map_comp0_tac ctxt =
+            HEADGOAL (EVERY' [rtac ctxt ext,
+              SELECT_GOAL (unfold_thms_tac ctxt [map_comp0_of_bnf bnf, o_apply,
+                Rep_thm RS (map_closed_thm RS Abs_inverse_thm)]),
+              rtac ctxt refl]);
+
+          fun map_cong0_tac ctxt =
+            HEADGOAL (EVERY' ([SELECT_GOAL (unfold_thms_tac ctxt [o_apply]),
+              rtac ctxt (([Rep_thm RS map_closed_thm, Rep_thm RS map_closed_thm] MRS
+                Abs_inject_thm) RS iffD2),
+              rtac ctxt (map_cong0_of_bnf bnf)] @ replicate lives (Goal.assume_rule_tac ctxt)));
+
+          val set_map0s_tac =
+            map (fn set_map => fn ctxt =>
+              HEADGOAL (EVERY' [rtac ctxt ext,
+                SELECT_GOAL (unfold_thms_tac ctxt [set_map, o_apply,
+                  Rep_thm RS (map_closed_thm RS Abs_inverse_thm)]),
+                rtac ctxt refl]))
+           (set_map_of_bnf bnf);
+
+          fun card_order_bd_tac ctxt = HEADGOAL (rtac ctxt (bd_card_order_of_bnf bnf));
+
+          fun cinfinite_bd_tac ctxt = HEADGOAL (rtac ctxt (bd_cinfinite_of_bnf bnf));
+
+          val set_bds_tac =
+            map (fn set_bd => fn ctxt =>
+              HEADGOAL (EVERY' [SELECT_GOAL (unfold_thms_tac ctxt [o_apply]), rtac ctxt set_bd]))
+            (set_bd_of_bnf bnf);
+
+          fun le_rel_OO_tac ctxt = 
+            HEADGOAL (EVERY' [rtac ctxt @{thm vimage2p_relcompp_mono},
+              rtac ctxt ((rel_OO_of_bnf bnf RS sym) RS @{thm ord_eq_le_trans}),
+              rtac ctxt @{thm order_refl}]);
+
+          fun rel_OO_Grp_tac ctxt =
+            HEADGOAL (EVERY' ([SELECT_GOAL (REPEAT_DETERM (HEADGOAL (rtac ctxt ext))),
+              SELECT_GOAL (unfold_thms_tac ctxt [@{thm OO_Grp_alt}, mem_Collect_eq,
+                o_apply, @{thm vimage2p_def}, in_rel_of_bnf bnf, Bex_def, mem_Collect_eq]),
+              rtac ctxt iffI,
+              SELECT_GOAL (REPEAT_DETERM (HEADGOAL (eresolve0_tac [exE,conjE]))),
+              rtac ctxt (zip_closed_thm OF (replicate 2 (Rep_thm RSN (2, @{thm ssubst_mem}))) RS
+                Rep_cases_thm),
+              assume_tac ctxt,
+              assume_tac ctxt,
+              hyp_subst_tac ctxt,
+              SELECT_GOAL (REPEAT_DETERM (HEADGOAL (rtac ctxt exI))),
+              rtac ctxt conjI] @ 
+              replicate (lives - 1) (rtac ctxt conjI THEN' assume_tac ctxt) @
+              [assume_tac ctxt,
+              SELECT_GOAL (REPEAT_DETERM (HEADGOAL (rtac ctxt conjI))),
+              REPEAT_DETERM_N 2 o
+                etac ctxt (trans OF [iffD2 OF [Abs_inject_thm OF
+                  [map_closed_thm OF [Rep_thm], Rep_thm]], Rep_inverse_thm]),
+              SELECT_GOAL (REPEAT_DETERM (HEADGOAL (eresolve0_tac [exE,conjE]))),
+              rtac ctxt exI,
+              rtac ctxt conjI] @ 
+              replicate (lives - 1) (rtac ctxt conjI THEN' assume_tac ctxt) @
+              [assume_tac ctxt,
+              rtac ctxt conjI,
+              REPEAT_DETERM_N 2 o EVERY'
+                [rtac ctxt (iffD1 OF [Abs_inject_thm OF [map_closed_thm OF [Rep_thm], Rep_thm]]),
+                etac ctxt (Rep_inverse_thm RS sym RSN (2, trans))]]));
+
+          fun wit_tac ctxt =
+            HEADGOAL (EVERY'
+              (map (fn thm => (EVERY'
+                [SELECT_GOAL (unfold_thms_tac ctxt (o_apply ::
+                  (wit_closed_thms RL [Abs_inverse_thm]))),
+                dtac ctxt thm, assume_tac ctxt]))
+              wit_thms));
+
+          val tactics = [map_id0_tac, map_comp0_tac, map_cong0_tac] @ set_map0s_tac @
+            [card_order_bd_tac, cinfinite_bd_tac] @ set_bds_tac @ [le_rel_OO_tac, rel_OO_Grp_tac];
+
+          val (bnf, lthy) = bnf_def Dont_Inline (user_policy Note_Some) false I
+            tactics wit_tac NONE map_b rel_b set_bs
+            ((((((Binding.empty, AbsT), map_G), sets_G), bd_G), wits_G), SOME rel_G)
+            lthy;
+        in
+          lthy |> BNF_Def.register_bnf plugins AbsT_name bnf
+        end
+      | after_qed _ _ = error "should not happen";
+  in
+    (goals, after_qed, defs, lthy)
+  end;
+
+fun prepare_common prepare_name prepare_sort prepare_term prepare_thm
+    (((((plugins, raw_specs), raw_Tname), raw_wits), xthm_opt), (map_b, rel_b)) lthy =
+  let
+    val Tname = prepare_name lthy raw_Tname;
+    val input_thm =
+      (case xthm_opt of
+        SOME xthm => prepare_thm lthy xthm
+      | NONE => Typedef.get_info lthy Tname |> hd |> snd |> #type_definition);
+    val wits = Option.map (map (prepare_term lthy)) raw_wits;
+    val specs = map (apsnd (apsnd
+      (the_default @{sort type} o Option.map (prepare_sort lthy)))) raw_specs;
+
+    (* analyze theorem here*)
+    fun is_typedef (t as (Const ("Typedef.type_definition", _) $ _ $ _ $ _)) = t
+      | is_typedef t = raise TERM("not a typedef",[t]);
+
+    val _ = (HOLogic.dest_Trueprop o Thm.prop_of) input_thm |> is_typedef
+      handle TERM _ => error "Unsupported type of a theorem. Only type_definition is supported.";
+  in
+    typedef_bnf input_thm wits specs map_b rel_b plugins lthy
+  end;
+
+fun prepare_lift_bnf prepare_name prepare_sort prepare_term prepare_thm =
+  (fn (goals, after_qed, definitions, lthy) =>
+    lthy
+    |> Proof.theorem NONE after_qed (map (single o rpair []) goals)
+    |> Proof.refine (Method.Basic (fn ctxt => SIMPLE_METHOD (unfold_thms_tac ctxt definitions)))
+    |> Seq.hd
+    |> Proof.refine (Method.primitive_text (K I))
+    |> Seq.hd) oo
+  prepare_common prepare_name prepare_sort prepare_term prepare_thm o apfst (apfst (apsnd SOME));
+
+val lift_bnf_cmd = prepare_lift_bnf
+  (fst o dest_Type oo Proof_Context.read_type_name {proper = true, strict = false})
+  Syntax.read_sort Syntax.read_term (singleton o Attrib.eval_thms);
+
+fun prepare_solve prepare_name prepare_typ prepare_sort prepare_thm tacs =
+  (fn (goals, after_qed, _, lthy) =>
+    lthy
+    |> after_qed (map2 (single oo Goal.prove lthy [] []) goals (tacs (length goals)))) oo
+  prepare_common prepare_name prepare_typ prepare_sort prepare_thm o apfst (apfst (rpair NONE));
+
+fun lift_bnf args tacs = prepare_solve (K I) (K I) (K I) (K I) (K tacs) args;
+val copy_bnf = prepare_solve (K I) (K I) (K I) (K I)
+  (fn n => replicate n (fn {context = ctxt, prems = _} => rtac ctxt UNIV_I 1));
+val copy_bnf_cmd = prepare_solve
+  (fst o dest_Type oo Proof_Context.read_type_name {proper = true, strict = false})
+  Syntax.read_sort Syntax.read_term (singleton o Attrib.eval_thms)
+  (fn n => replicate n (fn {context = ctxt, prems = _} => rtac ctxt UNIV_I 1));
+
+val parse_wits =
+  @{keyword "["} |-- (Parse.name --| @{keyword ":"} -- Scan.repeat Parse.term >>
+    (fn ("wits", Ts) => Ts
+      | (s, _) => error ("Unknown label " ^ quote s ^ " (expected \"wits\")"))) --|
+  @{keyword "]"} || Scan.succeed [];
+
+val parse_options =
+  Scan.optional (@{keyword "("} |--
+    Parse.list1 (Parse.group (K "option")
+      (Plugin_Name.parse_filter >> Plugins_Option
+      || Parse.reserved "no_warn_wits" >> K No_Warn_Wits))
+    --| @{keyword ")"}) [];
+
+val parse_plugins =
+  Scan.optional (@{keyword "("} |-- Plugin_Name.parse_filter --| @{keyword ")"})
+    (K Plugin_Name.default_filter) >> Plugins_Option >> single;
+
+val parse_typedef_thm = Scan.option (Parse.reserved "via" |-- Parse.xthm);
+
+val _ =
+  Outer_Syntax.local_theory_to_proof @{command_keyword lift_bnf}
+    "register a subtype of a bounded natural functor (BNF) as a BNF"
+    ((parse_options -- parse_type_args_named_constrained -- Parse.type_const -- parse_wits --
+      parse_typedef_thm -- parse_map_rel_bindings) >> lift_bnf_cmd);
+
+val _ =
+  Outer_Syntax.local_theory @{command_keyword copy_bnf}
+    "register a type copy of a bounded natural functor (BNF) as a BNF"
+    ((parse_plugins -- parse_type_args_named_constrained -- Parse.type_const --
+      parse_typedef_thm -- parse_map_rel_bindings) >> copy_bnf_cmd);
+
+end
\ No newline at end of file