renamed "set_natural" to "set_map", reflecting {Bl,Po,Tr} concensus
authorblanchet
Wed, 24 Apr 2013 17:47:22 +0200
changeset 51766 f19a4d0ab1bf
parent 51765 224b6eb2313a
child 51767 bbcdd8519253
renamed "set_natural" to "set_map", reflecting {Bl,Po,Tr} concensus
src/HOL/BNF/Examples/Stream.thy
src/HOL/BNF/Examples/TreeFI.thy
src/HOL/BNF/More_BNFs.thy
src/HOL/BNF/Tools/bnf_comp.ML
src/HOL/BNF/Tools/bnf_comp_tactics.ML
src/HOL/BNF/Tools/bnf_def.ML
src/HOL/BNF/Tools/bnf_def_tactics.ML
src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
src/HOL/BNF/Tools/bnf_fp_def_sugar_tactics.ML
src/HOL/BNF/Tools/bnf_gfp.ML
src/HOL/BNF/Tools/bnf_gfp_tactics.ML
src/HOL/BNF/Tools/bnf_lfp.ML
src/HOL/BNF/Tools/bnf_lfp_tactics.ML
--- a/src/HOL/BNF/Examples/Stream.thy	Wed Apr 24 17:03:43 2013 +0200
+++ b/src/HOL/BNF/Examples/Stream.thy	Wed Apr 24 17:47:22 2013 +0200
@@ -442,12 +442,12 @@
 proof (cases "n \<le> m")
   case False thus ?thesis unfolding smerge_def
     by (subst stream_set_flat)
-      (auto simp: stream.set_natural' in_set_conv_nth simp del: stake.simps
+      (auto simp: stream.set_map' in_set_conv_nth simp del: stake.simps
         intro!: exI[of _ n, OF disjI2] exI[of _ m, OF mp])
 next
   case True thus ?thesis unfolding smerge_def
     by (subst stream_set_flat)
-      (auto simp: stream.set_natural' in_set_conv_nth image_iff simp del: stake.simps snth.simps
+      (auto simp: stream.set_map' in_set_conv_nth image_iff simp del: stake.simps snth.simps
         intro!: exI[of _ m, OF disjI1] bexI[of _ "ss !! n"] exI[of _ n, OF mp])
 qed
 
@@ -456,7 +456,7 @@
   fix x assume "x \<in> stream_set (smerge ss)"
   thus "x \<in> UNION (stream_set ss) stream_set"
     unfolding smerge_def by (subst (asm) stream_set_flat)
-      (auto simp: stream.set_natural' in_set_conv_nth stream_set_range simp del: stake.simps, fast+)
+      (auto simp: stream.set_map' in_set_conv_nth stream_set_range simp del: stake.simps, fast+)
 next
   fix s x assume "s \<in> stream_set ss" "x \<in> stream_set s"
   thus "x \<in> stream_set (smerge ss)" using snth_stream_set_smerge by (auto simp: stream_set_range)
@@ -469,7 +469,7 @@
   "sproduct s1 s2 = smerge (stream_map (\<lambda>x. stream_map (Pair x) s2) s1)"
 
 lemma stream_set_sproduct: "stream_set (sproduct s1 s2) = stream_set s1 \<times> stream_set s2"
-  unfolding sproduct_def stream_set_smerge by (auto simp: stream.set_natural')
+  unfolding sproduct_def stream_set_smerge by (auto simp: stream.set_map')
 
 
 subsection {* interleave two streams *}
--- a/src/HOL/BNF/Examples/TreeFI.thy	Wed Apr 24 17:03:43 2013 +0200
+++ b/src/HOL/BNF/Examples/TreeFI.thy	Wed Apr 24 17:47:22 2013 +0200
@@ -16,7 +16,7 @@
 
 lemma pre_treeFI_listF_set[simp]: "pre_treeFI_set2 (i, xs) = listF_set xs"
 unfolding pre_treeFI_set2_def collect_def[abs_def] prod_set_defs
-by (auto simp add: listF.set_natural')
+by (auto simp add: listF.set_map')
 
 lemma dtor[simp]: "treeFI_dtor tr = (lab tr, sub tr)"
 unfolding lab_def sub_def treeFI_case_def
--- a/src/HOL/BNF/More_BNFs.thy	Wed Apr 24 17:03:43 2013 +0200
+++ b/src/HOL/BNF/More_BNFs.thy	Wed Apr 24 17:47:22 2013 +0200
@@ -353,7 +353,7 @@
 apply transfer apply simp
 done
 
-lemmas [simp] = fset.map_comp' fset.map_id' fset.set_natural'
+lemmas [simp] = fset.map_comp' fset.map_id' fset.set_map'
 
 lemma fset_rel_fset: "set_rel \<chi> (fset A1) (fset A2) = fset_rel \<chi> A1 A2"
   unfolding fset_rel_def set_rel_def by auto 
@@ -422,7 +422,7 @@
 "{(x, y). R x y} \<inter> A \<times> B = {(x, y). R x y \<and> x \<in> A \<and> y \<in> B}"
 by auto
 
-lemma rcset_natural': "rcset (cIm f x) = f ` rcset x"
+lemma rcset_map': "rcset (cIm f x) = f ` rcset x"
 unfolding cIm_def[abs_def] by simp
 
 definition cset_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a cset \<Rightarrow> 'b cset \<Rightarrow> bool" where
@@ -455,9 +455,9 @@
   assume ?R thus ?L unfolding Gr_def relcomp_unfold converse_unfold
   apply (simp add: subset_eq Ball_def)
   apply (rule conjI)
-  apply (clarsimp, metis (lifting, no_types) rcset_natural' image_iff surjective_pairing)
+  apply (clarsimp, metis (lifting, no_types) rcset_map' image_iff surjective_pairing)
   apply (clarsimp)
-  by (metis Domain.intros Range.simps rcset_natural' fst_eq_Domain snd_eq_Range)
+  by (metis Domain.intros Range.simps rcset_map' fst_eq_Domain snd_eq_Range)
 qed
 
 bnf_def cIm [rcset] "\<lambda>_::'a cset. natLeq" ["cEmp"] cset_rel
@@ -1151,7 +1151,7 @@
 Plus: "\<lbrakk>R a b; multiset_rel' R M N\<rbrakk> \<Longrightarrow> multiset_rel' R (M + {#a#}) (N + {#b#})"
 
 lemma multiset_map_Zero_iff[simp]: "multiset_map f M = {#} \<longleftrightarrow> M = {#}"
-by (metis image_is_empty multiset.set_natural' set_of_eq_empty_iff)
+by (metis image_is_empty multiset.set_map' set_of_eq_empty_iff)
 
 lemma multiset_map_Zero[simp]: "multiset_map f {#} = {#}" by simp
 
@@ -1287,7 +1287,7 @@
 shows "\<exists> N1. N = N1 + {#f a#} \<and> multiset_map f M = N1"
 proof-
   have "f a \<in># N"
-  using assms multiset.set_natural'[of f "M + {#a#}"] by auto
+  using assms multiset.set_map'[of f "M + {#a#}"] by auto
   then obtain N1 where N: "N = N1 + {#f a#}" using multi_member_split by metis
   have "multiset_map f M = N1" using assms unfolding N by simp
   thus ?thesis using N by blast
@@ -1298,7 +1298,7 @@
 shows "\<exists> M1 a. M = M1 + {#a#} \<and> f a = b \<and> multiset_map f M1 = N"
 proof-
   obtain a where a: "a \<in># M" and fa: "f a = b"
-  using multiset.set_natural'[of f M] unfolding assms
+  using multiset.set_map'[of f M] unfolding assms
   by (metis image_iff mem_set_of_iff union_single_eq_member)
   then obtain M1 where M: "M = M1 + {#a#}" using multi_member_split by metis
   have "multiset_map f M1 = N" using assms unfolding M fa[symmetric] by simp
--- a/src/HOL/BNF/Tools/bnf_comp.ML	Wed Apr 24 17:03:43 2013 +0200
+++ b/src/HOL/BNF/Tools/bnf_comp.ML	Wed Apr 24 17:47:22 2013 +0200
@@ -157,12 +157,12 @@
       mk_comp_map_comp_tac (map_comp_of_bnf outer) (map_cong0_of_bnf outer)
         (map map_comp_of_bnf inners);
 
-    fun mk_single_set_natural_tac i _ =
-      mk_comp_set_natural_tac (map_comp_of_bnf outer) (map_cong0_of_bnf outer)
-        (collect_set_natural_of_bnf outer)
-        (map ((fn thms => nth thms i) o set_natural_of_bnf) inners);
+    fun mk_single_set_map_tac i _ =
+      mk_comp_set_map_tac (map_comp_of_bnf outer) (map_cong0_of_bnf outer)
+        (collect_set_map_of_bnf outer)
+        (map ((fn thms => nth thms i) o set_map_of_bnf) inners);
 
-    val set_natural_tacs = map mk_single_set_natural_tac (0 upto ilive - 1);
+    val set_map_tacs = map mk_single_set_map_tac (0 upto ilive - 1);
 
     fun bd_card_order_tac _ =
       mk_comp_bd_card_order_tac (map bd_card_order_of_bnf inners) (bd_card_order_of_bnf outer);
@@ -177,7 +177,7 @@
         map (fn goal =>
           Goal.prove_sorry lthy [] [] goal
             (fn {context = ctxt, prems = _} =>
-              mk_comp_set_alt_tac ctxt (collect_set_natural_of_bnf outer))
+              mk_comp_set_alt_tac ctxt (collect_set_map_of_bnf outer))
           |> Thm.close_derivation)
         (map2 (curry (HOLogic.mk_Trueprop o HOLogic.mk_eq)) sets sets_alt);
 
@@ -238,7 +238,7 @@
         unfold_thms_tac lthy basic_thms THEN rtac thm 1
       end;
 
-    val tacs = zip_axioms map_id_tac map_comp_tac map_cong0_tac set_natural_tacs bd_card_order_tac
+    val tacs = zip_axioms map_id_tac map_comp_tac map_cong0_tac set_map_tacs bd_card_order_tac
       bd_cinfinite_tac set_bd_tacs in_bd_tac map_wpull_tac srel_O_Gr_tac;
 
     val outer_wits = mk_wits_of_bnf (replicate onwits oDs) (replicate onwits CAs) outer;
@@ -257,7 +257,7 @@
       |> map (fn (frees, t) => fold absfree frees t);
 
     fun wit_tac {context = ctxt, prems = _} =
-      mk_comp_wit_tac ctxt (wit_thms_of_bnf outer) (collect_set_natural_of_bnf outer)
+      mk_comp_wit_tac ctxt (wit_thms_of_bnf outer) (collect_set_map_of_bnf outer)
         (maps wit_thms_of_bnf inners);
 
     val (bnf', lthy') =
@@ -311,7 +311,7 @@
       rtac refl 1;
     fun map_cong0_tac {context = ctxt, prems = _} =
       mk_kill_map_cong0_tac ctxt n (live - n) (map_cong0_of_bnf bnf);
-    val set_natural_tacs = map (fn thm => fn _ => rtac thm 1) (drop n (set_natural_of_bnf bnf));
+    val set_map_tacs = map (fn thm => fn _ => rtac thm 1) (drop n (set_map_of_bnf bnf));
     fun bd_card_order_tac _ = mk_kill_bd_card_order_tac n (bd_card_order_of_bnf bnf);
     fun bd_cinfinite_tac _ = mk_kill_bd_cinfinite_tac (bd_Cinfinite_of_bnf bnf);
     val set_bd_tacs =
@@ -348,7 +348,7 @@
         rtac thm 1
       end;
 
-    val tacs = zip_axioms map_id_tac map_comp_tac map_cong0_tac set_natural_tacs bd_card_order_tac
+    val tacs = zip_axioms map_id_tac map_comp_tac map_cong0_tac set_map_tacs bd_card_order_tac
       bd_cinfinite_tac set_bd_tacs in_bd_tac map_wpull_tac srel_O_Gr_tac;
 
     val bnf_wits = mk_wits_of_bnf (replicate nwits Ds) (replicate nwits As) bnf;
@@ -405,12 +405,12 @@
       rtac refl 1;
     fun map_cong0_tac {context = ctxt, prems = _} =
       rtac (map_cong0_of_bnf bnf) 1 THEN REPEAT_DETERM_N live (Goal.assume_rule_tac ctxt 1);
-    val set_natural_tacs =
+    val set_map_tacs =
       if ! quick_and_dirty then
         replicate (n + live) (K all_tac)
       else
         replicate n (K empty_natural_tac) @
-        map (fn thm => fn _ => rtac thm 1) (set_natural_of_bnf bnf);
+        map (fn thm => fn _ => rtac thm 1) (set_map_of_bnf bnf);
     fun bd_card_order_tac _ = rtac (bd_card_order_of_bnf bnf) 1;
     fun bd_cinfinite_tac _ = rtac (bd_cinfinite_of_bnf bnf) 1;
     val set_bd_tacs =
@@ -435,7 +435,7 @@
     fun srel_O_Gr_tac _ =
       mk_simple_srel_O_Gr_tac lthy (srel_def_of_bnf bnf) (srel_O_Gr_of_bnf bnf) in_alt_thm;
 
-    val tacs = zip_axioms map_id_tac map_comp_tac map_cong0_tac set_natural_tacs bd_card_order_tac
+    val tacs = zip_axioms map_id_tac map_comp_tac map_cong0_tac set_map_tacs bd_card_order_tac
       bd_cinfinite_tac set_bd_tacs in_bd_tac map_wpull_tac srel_O_Gr_tac;
 
     val wits = map snd (mk_wits_of_bnf (replicate nwits Ds) (replicate nwits As) bnf);
@@ -490,7 +490,7 @@
     fun map_comp_tac _ = rtac (map_comp_of_bnf bnf) 1;
     fun map_cong0_tac {context = ctxt, prems = _} =
       rtac (map_cong0_of_bnf bnf) 1 THEN REPEAT_DETERM_N live (Goal.assume_rule_tac ctxt 1);
-    val set_natural_tacs = permute (map (fn thm => fn _ => rtac thm 1) (set_natural_of_bnf bnf));
+    val set_map_tacs = permute (map (fn thm => fn _ => rtac thm 1) (set_map_of_bnf bnf));
     fun bd_card_order_tac _ = rtac (bd_card_order_of_bnf bnf) 1;
     fun bd_cinfinite_tac _ = rtac (bd_cinfinite_of_bnf bnf) 1;
     val set_bd_tacs = permute (map (fn thm => fn _ => rtac thm 1) (set_bd_of_bnf bnf));
@@ -512,7 +512,7 @@
     fun srel_O_Gr_tac _ =
       mk_simple_srel_O_Gr_tac lthy (srel_def_of_bnf bnf) (srel_O_Gr_of_bnf bnf) in_alt_thm;
 
-    val tacs = zip_axioms map_id_tac map_comp_tac map_cong0_tac set_natural_tacs bd_card_order_tac
+    val tacs = zip_axioms map_id_tac map_comp_tac map_cong0_tac set_map_tacs bd_card_order_tac
       bd_cinfinite_tac set_bd_tacs in_bd_tac map_wpull_tac srel_O_Gr_tac;
 
     val wits = map snd (mk_wits_of_bnf (replicate nwits Ds) (replicate nwits As) bnf);
@@ -653,7 +653,7 @@
       SOLVE o REPEAT_DETERM o (atac ORELSE' Goal.assume_rule_tac ctxt)) 1;
 
     val tacs = zip_axioms (mk_tac (map_id_of_bnf bnf)) (mk_tac (map_comp_of_bnf bnf))
-      (mk_tac (map_cong0_of_bnf bnf)) (map mk_tac (set_natural_of_bnf bnf))
+      (mk_tac (map_cong0_of_bnf bnf)) (map mk_tac (set_map_of_bnf bnf))
       (K (rtac bd_card_order 1)) (K (rtac bd_cinfinite 1)) (map mk_tac set_bds) (mk_tac in_bd)
       (mk_tac (map_wpull_of_bnf bnf))
       (mk_tac (unfold_thms lthy [srel_def_of_bnf bnf] (srel_O_Gr_of_bnf bnf)));
--- a/src/HOL/BNF/Tools/bnf_comp_tactics.ML	Wed Apr 24 17:03:43 2013 +0200
+++ b/src/HOL/BNF/Tools/bnf_comp_tactics.ML	Wed Apr 24 17:47:22 2013 +0200
@@ -17,7 +17,7 @@
   val mk_comp_map_id_tac: thm -> thm -> thm list -> tactic
   val mk_comp_set_alt_tac: Proof.context -> thm -> tactic
   val mk_comp_set_bd_tac: Proof.context -> thm -> thm list -> tactic
-  val mk_comp_set_natural_tac: thm -> thm -> thm -> thm list -> tactic
+  val mk_comp_set_map_tac: thm -> thm -> thm -> thm list -> tactic
   val mk_comp_wit_tac: Proof.context -> thm list -> thm -> thm list -> tactic
 
   val mk_kill_bd_card_order_tac: int -> thm -> tactic
@@ -64,9 +64,9 @@
 
 (* Composition *)
 
-fun mk_comp_set_alt_tac ctxt collect_set_natural =
+fun mk_comp_set_alt_tac ctxt collect_set_map =
   unfold_thms_tac ctxt @{thms sym[OF o_assoc]} THEN
-  unfold_thms_tac ctxt [collect_set_natural RS sym] THEN
+  unfold_thms_tac ctxt [collect_set_map RS sym] THEN
   rtac refl 1;
 
 fun mk_comp_map_id_tac Gmap_id Gmap_cong0 map_ids =
@@ -78,21 +78,21 @@
     rtac (Gmap_comp RS sym RS o_eq_dest_lhs RS trans), rtac Gmap_cong0] @
     map (fn thm => rtac (thm RS sym RS fun_cong)) map_comps) 1;
 
-fun mk_comp_set_natural_tac Gmap_comp Gmap_cong0 Gset_natural set_naturals =
+fun mk_comp_set_map_tac Gmap_comp Gmap_cong0 Gset_map set_maps =
   EVERY' ([rtac ext] @
     replicate 3 (rtac trans_o_apply) @
     [rtac (arg_cong_Union RS trans),
      rtac (@{thm arg_cong2[of _ _ _ _ collect, OF refl]} RS trans),
      rtac (Gmap_comp RS sym RS o_eq_dest_lhs RS trans),
      rtac Gmap_cong0] @
-     map (fn thm => rtac (thm RS fun_cong)) set_naturals @
-     [rtac (Gset_natural RS o_eq_dest_lhs), rtac sym, rtac trans_o_apply,
+     map (fn thm => rtac (thm RS fun_cong)) set_maps @
+     [rtac (Gset_map RS o_eq_dest_lhs), rtac sym, rtac trans_o_apply,
      rtac trans_image_cong_o_apply, rtac trans_image_cong_o_apply,
-     rtac (@{thm image_cong} OF [Gset_natural RS o_eq_dest_lhs RS arg_cong_Union, refl] RS trans),
+     rtac (@{thm image_cong} OF [Gset_map RS o_eq_dest_lhs RS arg_cong_Union, refl] RS trans),
      rtac @{thm trans[OF pointfreeE[OF Union_natural[symmetric]]]}, rtac arg_cong_Union,
      rtac @{thm trans[OF o_eq_dest_lhs[OF image_o_collect[symmetric]]]},
      rtac @{thm fun_cong[OF arg_cong[of _ _ collect]]}] @
-     [REPEAT_DETERM_N (length set_naturals) o EVERY' [rtac @{thm trans[OF image_insert]},
+     [REPEAT_DETERM_N (length set_maps) o EVERY' [rtac @{thm trans[OF image_insert]},
         rtac @{thm arg_cong2[of _ _ _ _ insert]}, rtac ext, rtac trans_o_apply,
         rtac trans_image_cong_o_apply, rtac @{thm trans[OF image_image]},
         rtac @{thm sym[OF trans[OF o_apply]]}, rtac @{thm image_cong[OF refl o_apply]}],
@@ -217,9 +217,9 @@
 val comp_wit_thms = @{thms Union_empty_conv o_apply collect_def SUP_def
   Union_image_insert Union_image_empty};
 
-fun mk_comp_wit_tac ctxt Gwit_thms collect_set_natural Fwit_thms =
+fun mk_comp_wit_tac ctxt Gwit_thms collect_set_map Fwit_thms =
   ALLGOALS (dtac @{thm in_Union_o_assoc}) THEN
-  unfold_thms_tac ctxt (collect_set_natural :: comp_wit_thms) THEN
+  unfold_thms_tac ctxt (collect_set_map :: comp_wit_thms) THEN
   REPEAT_DETERM (
     atac 1 ORELSE
     REPEAT_DETERM (eresolve_tac @{thms UnionE UnE imageE} 1) THEN
--- a/src/HOL/BNF/Tools/bnf_def.ML	Wed Apr 24 17:03:43 2013 +0200
+++ b/src/HOL/BNF/Tools/bnf_def.ML	Wed Apr 24 17:47:22 2013 +0200
@@ -45,7 +45,7 @@
   val bd_Cnotzero_of_bnf: BNF -> thm
   val bd_card_order_of_bnf: BNF -> thm
   val bd_cinfinite_of_bnf: BNF -> thm
-  val collect_set_natural_of_bnf: BNF -> thm
+  val collect_set_map_of_bnf: BNF -> thm
   val in_bd_of_bnf: BNF -> thm
   val in_cong_of_bnf: BNF -> thm
   val in_mono_of_bnf: BNF -> thm
@@ -65,8 +65,8 @@
   val rel_srel_of_bnf: BNF -> thm
   val set_bd_of_bnf: BNF -> thm list
   val set_defs_of_bnf: BNF -> thm list
-  val set_natural'_of_bnf: BNF -> thm list
-  val set_natural_of_bnf: BNF -> thm list
+  val set_map'_of_bnf: BNF -> thm list
+  val set_map_of_bnf: BNF -> thm list
   val srel_def_of_bnf: BNF -> thm
   val srel_Gr_of_bnf: BNF -> thm
   val srel_Id_of_bnf: BNF -> thm
@@ -111,7 +111,7 @@
   map_id: thm,
   map_comp: thm,
   map_cong0: thm,
-  set_natural: thm list,
+  set_map: thm list,
   bd_card_order: thm,
   bd_cinfinite: thm,
   set_bd: thm list,
@@ -121,7 +121,7 @@
 };
 
 fun mk_axioms' (((((((((id, comp), cong), nat), c_o), cinf), set_bd), in_bd), wpull), srel) =
-  {map_id = id, map_comp = comp, map_cong0 = cong, set_natural = nat, bd_card_order = c_o,
+  {map_id = id, map_comp = comp, map_cong0 = cong, set_map = nat, bd_card_order = c_o,
    bd_cinfinite = cinf, set_bd = set_bd, in_bd = in_bd, map_wpull = wpull, srel_O_Gr = srel};
 
 fun dest_cons [] = raise Empty
@@ -144,17 +144,17 @@
 fun zip_axioms mid mcomp mcong snat bdco bdinf sbd inbd wpull srel =
   [mid, mcomp, mcong] @ snat @ [bdco, bdinf] @ sbd @ [inbd, wpull, srel];
 
-fun dest_axioms {map_id, map_comp, map_cong0, set_natural, bd_card_order, bd_cinfinite, set_bd,
-  in_bd, map_wpull, srel_O_Gr} =
-  zip_axioms map_id map_comp map_cong0 set_natural bd_card_order bd_cinfinite set_bd in_bd map_wpull
+fun dest_axioms {map_id, map_comp, map_cong0, set_map, bd_card_order, bd_cinfinite, set_bd, in_bd,
+  map_wpull, srel_O_Gr} =
+  zip_axioms map_id map_comp map_cong0 set_map bd_card_order bd_cinfinite set_bd in_bd map_wpull
     srel_O_Gr;
 
-fun map_axioms f {map_id, map_comp, map_cong0, set_natural, bd_card_order, bd_cinfinite, set_bd,
+fun map_axioms f {map_id, map_comp, map_cong0, set_map, bd_card_order, bd_cinfinite, set_bd,
   in_bd, map_wpull, srel_O_Gr} =
   {map_id = f map_id,
     map_comp = f map_comp,
     map_cong0 = f map_cong0,
-    set_natural = map f set_natural,
+    set_map = map f set_map,
     bd_card_order = f bd_card_order,
     bd_cinfinite = f bd_cinfinite,
     set_bd = map f set_bd,
@@ -182,7 +182,7 @@
   bd_Card_order: thm,
   bd_Cinfinite: thm,
   bd_Cnotzero: thm,
-  collect_set_natural: thm lazy,
+  collect_set_map: thm lazy,
   in_cong: thm lazy,
   in_mono: thm lazy,
   in_srel: thm lazy,
@@ -193,7 +193,7 @@
   rel_eq: thm lazy,
   rel_flip: thm lazy,
   rel_srel: thm lazy,
-  set_natural': thm lazy list,
+  set_map': thm lazy list,
   srel_cong: thm lazy,
   srel_mono: thm lazy,
   srel_Id: thm lazy,
@@ -202,13 +202,13 @@
   srel_O: thm lazy
 };
 
-fun mk_facts bd_Card_order bd_Cinfinite bd_Cnotzero collect_set_natural in_cong in_mono in_srel
-    map_comp' map_cong map_id' map_wppull rel_eq rel_flip rel_srel set_natural' srel_cong srel_mono
+fun mk_facts bd_Card_order bd_Cinfinite bd_Cnotzero collect_set_map in_cong in_mono in_srel
+    map_comp' map_cong map_id' map_wppull rel_eq rel_flip rel_srel set_map' srel_cong srel_mono
     srel_Id srel_Gr srel_converse srel_O = {
   bd_Card_order = bd_Card_order,
   bd_Cinfinite = bd_Cinfinite,
   bd_Cnotzero = bd_Cnotzero,
-  collect_set_natural = collect_set_natural,
+  collect_set_map = collect_set_map,
   in_cong = in_cong,
   in_mono = in_mono,
   in_srel = in_srel,
@@ -219,7 +219,7 @@
   rel_eq = rel_eq,
   rel_flip = rel_flip,
   rel_srel = rel_srel,
-  set_natural' = set_natural',
+  set_map' = set_map',
   srel_cong = srel_cong,
   srel_mono = srel_mono,
   srel_Id = srel_Id,
@@ -231,7 +231,7 @@
   bd_Card_order,
   bd_Cinfinite,
   bd_Cnotzero,
-  collect_set_natural,
+  collect_set_map,
   in_cong,
   in_mono,
   in_srel,
@@ -242,7 +242,7 @@
   rel_eq,
   rel_flip,
   rel_srel,
-  set_natural',
+  set_map',
   srel_cong,
   srel_mono,
   srel_Id,
@@ -252,7 +252,7 @@
   {bd_Card_order = f bd_Card_order,
     bd_Cinfinite = f bd_Cinfinite,
     bd_Cnotzero = f bd_Cnotzero,
-    collect_set_natural = Lazy.map f collect_set_natural,
+    collect_set_map = Lazy.map f collect_set_map,
     in_cong = Lazy.map f in_cong,
     in_mono = Lazy.map f in_mono,
     in_srel = Lazy.map f in_srel,
@@ -263,7 +263,7 @@
     rel_eq = Lazy.map f rel_eq,
     rel_flip = Lazy.map f rel_flip,
     rel_srel = Lazy.map f rel_srel,
-    set_natural' = map (Lazy.map f) set_natural',
+    set_map' = map (Lazy.map f) set_map',
     srel_cong = Lazy.map f srel_cong,
     srel_mono = Lazy.map f srel_mono,
     srel_Id = Lazy.map f srel_Id,
@@ -368,7 +368,7 @@
 val bd_Card_order_of_bnf = #bd_Card_order o #facts o rep_bnf;
 val bd_Cinfinite_of_bnf = #bd_Cinfinite o #facts o rep_bnf;
 val bd_Cnotzero_of_bnf = #bd_Cnotzero o #facts o rep_bnf;
-val collect_set_natural_of_bnf = Lazy.force o #collect_set_natural o #facts o rep_bnf;
+val collect_set_map_of_bnf = Lazy.force o #collect_set_map o #facts o rep_bnf;
 val in_bd_of_bnf = #in_bd o #axioms o rep_bnf;
 val in_cong_of_bnf = Lazy.force o #in_cong o #facts o rep_bnf;
 val in_mono_of_bnf = Lazy.force o #in_mono o #facts o rep_bnf;
@@ -388,8 +388,8 @@
 val rel_srel_of_bnf = Lazy.force o #rel_srel o #facts o rep_bnf;
 val set_bd_of_bnf = #set_bd o #axioms o rep_bnf;
 val set_defs_of_bnf = #set_defs o #defs o rep_bnf;
-val set_natural_of_bnf = #set_natural o #axioms o rep_bnf;
-val set_natural'_of_bnf = map Lazy.force o #set_natural' o #facts o rep_bnf;
+val set_map_of_bnf = #set_map o #axioms o rep_bnf;
+val set_map'_of_bnf = map Lazy.force o #set_map' o #facts o rep_bnf;
 val srel_cong_of_bnf = Lazy.force o #srel_cong o #facts o rep_bnf;
 val srel_mono_of_bnf = Lazy.force o #srel_mono o #facts o rep_bnf;
 val srel_def_of_bnf = #srel_def o #defs o rep_bnf;
@@ -507,7 +507,7 @@
 val bd_Card_orderN = "bd_Card_order";
 val bd_CinfiniteN = "bd_Cinfinite";
 val bd_CnotzeroN = "bd_Cnotzero";
-val collect_set_naturalN = "collect_set_natural";
+val collect_set_mapN = "collect_set_map";
 val in_bdN = "in_bd";
 val in_monoN = "in_mono";
 val in_srelN = "in_srel";
@@ -521,8 +521,8 @@
 val rel_eqN = "rel_eq";
 val rel_flipN = "rel_flip";
 val rel_srelN = "rel_srel";
-val set_naturalN = "set_natural";
-val set_natural'N = "set_natural'";
+val set_mapN = "set_map";
+val set_map'N = "set_map'";
 val set_bdN = "set_bd";
 val srel_IdN = "srel_Id";
 val srel_GrN = "srel_Gr";
@@ -835,7 +835,7 @@
         fold_rev Logic.all (x :: fs @ fs_copy) (Logic.list_implies (prems, eq))
       end;
 
-    val set_naturals_goal =
+    val set_maps_goal =
       let
         fun mk_goal setA setB f =
           let
@@ -897,8 +897,8 @@
 
     val srel_O_Gr_goal = fold_rev Logic.all Rs (mk_Trueprop_eq (Term.list_comb (srel, Rs), O_Gr));
 
-    val goals = zip_axioms map_id_goal map_comp_goal map_cong0_goal set_naturals_goal
-      card_order_bd_goal cinfinite_bd_goal set_bds_goal in_bd_goal map_wpull_goal srel_O_Gr_goal;
+    val goals = zip_axioms map_id_goal map_comp_goal map_cong0_goal set_maps_goal card_order_bd_goal
+      cinfinite_bd_goal set_bds_goal in_bd_goal map_wpull_goal srel_O_Gr_goal;
 
     fun mk_wit_goals (I, wit) =
       let
@@ -928,7 +928,7 @@
         val bd_Cinfinite = @{thm conjI} OF [#bd_cinfinite axioms, bd_Card_order];
         val bd_Cnotzero = bd_Cinfinite RS @{thm Cinfinite_Cnotzero};
 
-        fun mk_collect_set_natural () =
+        fun mk_collect_set_map () =
           let
             val defT = mk_bnf_T Ts CA --> HOLogic.mk_setT T;
             val collect_map = HOLogic.mk_comp
@@ -940,11 +940,11 @@
             (*collect {set1 ... setm} o map f1 ... fm = collect {f1` o set1 ... fm` o setm}*)
             val goal = fold_rev Logic.all hs (mk_Trueprop_eq (collect_map, image_collect));
           in
-            Goal.prove_sorry lthy [] [] goal (K (mk_collect_set_natural_tac (#set_natural axioms)))
+            Goal.prove_sorry lthy [] [] goal (K (mk_collect_set_map_tac (#set_map axioms)))
             |> Thm.close_derivation
           end;
 
-        val collect_set_natural = Lazy.lazy mk_collect_set_natural;
+        val collect_set_map = Lazy.lazy mk_collect_set_map;
 
         fun mk_in_mono () =
           let
@@ -992,8 +992,7 @@
 
         val map_cong = Lazy.lazy mk_map_cong;
 
-        val set_natural' =
-          map (fn thm => Lazy.lazy (fn () => mk_set_natural' thm)) (#set_natural axioms);
+        val set_map' = map (fn thm => Lazy.lazy (fn () => mk_set_map' thm)) (#set_map axioms);
 
         fun mk_map_wppull () =
           let
@@ -1027,7 +1026,7 @@
           in
             Goal.prove_sorry lthy [] [] goal
               (fn _ => mk_map_wppull_tac (#map_id axioms) (#map_cong0 axioms)
-                (#map_wpull axioms) (Lazy.force map_comp') (map Lazy.force set_natural'))
+                (#map_wpull axioms) (Lazy.force map_comp') (map Lazy.force set_map'))
             |> Thm.close_derivation
           end;
 
@@ -1043,7 +1042,7 @@
           in
             Goal.prove_sorry lthy [] [] goal
               (mk_srel_Gr_tac srel_O_Grs (#map_id axioms) (#map_cong0 axioms) (Lazy.force map_id')
-                (Lazy.force map_comp') (map Lazy.force set_natural'))
+                (Lazy.force map_comp') (map Lazy.force set_map'))
             |> Thm.close_derivation
           end;
 
@@ -1096,7 +1095,7 @@
             val le_goal = fold_rev Logic.all Rs (HOLogic.mk_Trueprop (mk_subset lhs rhs));
             val le_thm = Goal.prove_sorry lthy [] [] le_goal
               (mk_srel_converse_le_tac srel_O_Grs (Lazy.force srel_Id) (#map_cong0 axioms)
-                (Lazy.force map_comp') (map Lazy.force set_natural'))
+                (Lazy.force map_comp') (map Lazy.force set_map'))
               |> Thm.close_derivation
             val goal = fold_rev Logic.all Rs (mk_Trueprop_eq (lhs, rhs));
           in
@@ -1116,7 +1115,7 @@
           in
             Goal.prove_sorry lthy [] [] goal
               (mk_srel_O_tac srel_O_Grs (Lazy.force srel_Id) (#map_cong0 axioms)
-                (Lazy.force map_wppull) (Lazy.force map_comp') (map Lazy.force set_natural'))
+                (Lazy.force map_wppull) (Lazy.force map_comp') (map Lazy.force set_map'))
             |> Thm.close_derivation
           end;
 
@@ -1176,9 +1175,9 @@
 
         val defs = mk_defs bnf_map_def bnf_set_defs bnf_rel_def bnf_srel_def;
 
-        val facts = mk_facts bd_Card_order bd_Cinfinite bd_Cnotzero collect_set_natural in_cong
-          in_mono in_srel map_comp' map_cong map_id' map_wppull rel_eq rel_flip rel_srel
-          set_natural' srel_cong srel_mono srel_Id srel_Gr srel_converse srel_O;
+        val facts = mk_facts bd_Card_order bd_Cinfinite bd_Cnotzero collect_set_map in_cong in_mono
+          in_srel map_comp' map_cong map_id' map_wppull rel_eq rel_flip rel_srel set_map'
+          srel_cong srel_mono srel_Id srel_Gr srel_converse srel_O;
 
         val wits = map2 mk_witness bnf_wits wit_thms;
 
@@ -1200,14 +1199,14 @@
                     (bd_Card_orderN, [#bd_Card_order facts]),
                     (bd_CinfiniteN, [#bd_Cinfinite facts]),
                     (bd_CnotzeroN, [#bd_Cnotzero facts]),
-                    (collect_set_naturalN, [Lazy.force (#collect_set_natural facts)]),
+                    (collect_set_mapN, [Lazy.force (#collect_set_map facts)]),
                     (in_bdN, [#in_bd axioms]),
                     (in_monoN, [Lazy.force (#in_mono facts)]),
                     (in_srelN, [Lazy.force (#in_srel facts)]),
                     (map_compN, [#map_comp axioms]),
                     (map_idN, [#map_id axioms]),
                     (map_wpullN, [#map_wpull axioms]),
-                    (set_naturalN, #set_natural axioms),
+                    (set_mapN, #set_map axioms),
                     (set_bdN, #set_bd axioms)] @
                     (witNs ~~ wit_thms)
                     |> map (fn (thmN, thms) =>
@@ -1228,7 +1227,7 @@
                     (rel_eqN, [Lazy.force (#rel_eq facts)], []),
                     (rel_flipN, [Lazy.force (#rel_flip facts)], []),
                     (rel_srelN, [Lazy.force (#rel_srel facts)], []),
-                    (set_natural'N, map Lazy.force (#set_natural' facts), []),
+                    (set_map'N, map Lazy.force (#set_map' facts), []),
                     (srel_O_GrN, srel_O_Grs, []),
                     (srel_IdN, [Lazy.force (#srel_Id facts)], []),
                     (srel_GrN, [Lazy.force (#srel_Gr facts)], []),
--- a/src/HOL/BNF/Tools/bnf_def_tactics.ML	Wed Apr 24 17:03:43 2013 +0200
+++ b/src/HOL/BNF/Tools/bnf_def_tactics.ML	Wed Apr 24 17:47:22 2013 +0200
@@ -8,13 +8,13 @@
 
 signature BNF_DEF_TACTICS =
 sig
-  val mk_collect_set_natural_tac: thm list -> tactic
+  val mk_collect_set_map_tac: thm list -> tactic
   val mk_map_id': thm -> thm
   val mk_map_comp': thm -> thm
   val mk_map_cong_tac: thm -> tactic
   val mk_in_mono_tac: int -> tactic
   val mk_map_wppull_tac: thm -> thm -> thm -> thm -> thm list -> tactic
-  val mk_set_natural': thm -> thm
+  val mk_set_map': thm -> thm
 
   val mk_srel_Gr_tac: thm list -> thm -> thm -> thm -> thm -> thm list ->
     {prems: thm list, context: Proof.context} -> tactic
@@ -39,7 +39,7 @@
 fun mk_map_cong_tac cong0 =
   (hyp_subst_tac THEN' rtac cong0 THEN'
    REPEAT_DETERM o (dtac meta_spec THEN' etac meta_mp THEN' atac)) 1;
-fun mk_set_natural' set_natural = set_natural RS @{thm pointfreeE};
+fun mk_set_map' set_map = set_map RS @{thm pointfreeE};
 fun mk_in_mono_tac n = if n = 0 then rtac subset_UNIV 1
   else (rtac subsetI THEN'
   rtac CollectI) 1 THEN
@@ -48,15 +48,15 @@
     ((rtac conjI THEN' etac subset_trans THEN' atac) 1) THEN
   (etac subset_trans THEN' atac) 1;
 
-fun mk_collect_set_natural_tac set_naturals =
+fun mk_collect_set_map_tac set_maps =
   (rtac (@{thm collect_o} RS trans) THEN' rtac @{thm arg_cong[of _ _ collect]} THEN'
-  EVERY' (map (fn set_natural =>
+  EVERY' (map (fn set_map =>
     rtac (mk_trans @{thm image_insert} @{thm arg_cong2[of _ _ _ _ insert]}) THEN'
-    rtac set_natural) set_naturals) THEN'
+    rtac set_map) set_maps) THEN'
   rtac @{thm image_empty}) 1;
 
-fun mk_map_wppull_tac map_id map_cong0 map_wpull map_comp set_naturals =
-  if null set_naturals then
+fun mk_map_wppull_tac map_id map_cong0 map_wpull map_comp set_maps =
+  if null set_maps then
     EVERY' [rtac @{thm wppull_id}, rtac map_wpull, rtac map_id, rtac map_id] 1
   else EVERY' [REPEAT_DETERM o etac conjE, REPEAT_DETERM o dtac @{thm wppull_thePull},
     REPEAT_DETERM o etac exE, rtac @{thm wpull_wppull}, rtac map_wpull,
@@ -64,19 +64,19 @@
     REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac conjI, rtac CollectI,
     CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS @{thm ord_eq_le_trans}),
       rtac @{thm image_subsetI}, rtac conjunct1, etac bspec, etac set_mp, atac])
-      set_naturals,
+      set_maps,
     CONJ_WRAP' (fn thm => EVERY' [rtac (map_comp RS trans), rtac (map_comp RS trans),
       rtac (map_comp RS trans RS sym), rtac map_cong0,
-      REPEAT_DETERM_N (length set_naturals) o EVERY' [rtac (o_apply RS trans),
+      REPEAT_DETERM_N (length set_maps) o EVERY' [rtac (o_apply RS trans),
         rtac (o_apply RS trans RS sym), rtac (o_apply RS trans), rtac thm,
         rtac conjunct2, etac bspec, etac set_mp, atac]]) [conjunct1, conjunct2]] 1;
 
-fun mk_srel_Gr_tac srel_O_Grs map_id map_cong0 map_id' map_comp set_naturals
+fun mk_srel_Gr_tac srel_O_Grs map_id map_cong0 map_id' map_comp set_maps
   {context = ctxt, prems = _} =
   let
-    val n = length set_naturals;
+    val n = length set_maps;
   in
-    if null set_naturals then
+    if null set_maps then
       unfold_thms_tac ctxt srel_O_Grs THEN EVERY' [rtac @{thm Gr_UNIV_id}, rtac map_id] 1
     else unfold_thms_tac ctxt (@{thm Gr_def} :: srel_O_Grs) THEN
       EVERY' [rtac equalityI, rtac subsetI,
@@ -93,7 +93,7 @@
         CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS @{thm ord_eq_le_trans}),
           rtac @{thm image_subsetI}, dtac @{thm set_rev_mp}, atac,
           REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
-          stac @{thm fst_conv}, atac]) set_naturals,
+          stac @{thm fst_conv}, atac]) set_maps,
         rtac @{thm subrelI}, etac CollectE, REPEAT_DETERM o eresolve_tac [exE, conjE],
         REPEAT_DETERM o dtac Pair_eqD,
         REPEAT_DETERM o etac conjE, hyp_subst_tac,
@@ -108,7 +108,7 @@
             CONJ_WRAP' (fn thm =>
               EVERY' [rtac @{thm ord_eq_le_trans}, rtac thm, rtac @{thm image_subsetI},
                 rtac @{thm convol_memI[of id _ "%x. x", OF id_apply refl]}, etac set_mp, atac])
-            set_naturals])
+            set_maps])
           @{thms fst_convol snd_convol} [map_id', refl])] 1
   end;
 
@@ -125,12 +125,12 @@
       rtac @{thm Gr_mono}, rtac in_mono, REPEAT_DETERM o atac,
       rtac @{thm Gr_mono}, rtac in_mono, REPEAT_DETERM o atac] 1;
 
-fun mk_srel_converse_le_tac srel_O_Grs srel_Id map_cong0 map_comp set_naturals
+fun mk_srel_converse_le_tac srel_O_Grs srel_Id map_cong0 map_comp set_maps
   {context = ctxt, prems = _} =
   let
-    val n = length set_naturals;
+    val n = length set_maps;
   in
-    if null set_naturals then
+    if null set_maps then
       unfold_thms_tac ctxt [srel_Id] THEN rtac equalityD2 1 THEN rtac @{thm converse_Id} 1
     else unfold_thms_tac ctxt (@{thm Gr_def} :: srel_O_Grs) THEN
       EVERY' [rtac @{thm subrelI},
@@ -143,22 +143,22 @@
           rtac map_cong0, REPEAT_DETERM_N n o rtac thm,
           rtac (map_comp RS sym), rtac CollectI,
           CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS @{thm ord_eq_le_trans}),
-            etac @{thm flip_rel}]) set_naturals]) [@{thm snd_fst_flip}, @{thm fst_snd_flip}])] 1
+            etac @{thm flip_rel}]) set_maps]) [@{thm snd_fst_flip}, @{thm fst_snd_flip}])] 1
   end;
 
 fun mk_srel_converse_tac le_converse =
   EVERY' [rtac equalityI, rtac le_converse, rtac @{thm xt1(6)}, rtac @{thm converse_shift},
     rtac le_converse, REPEAT_DETERM o stac @{thm converse_converse}, rtac subset_refl] 1;
 
-fun mk_srel_O_tac srel_O_Grs srel_Id map_cong0 map_wppull map_comp set_naturals
+fun mk_srel_O_tac srel_O_Grs srel_Id map_cong0 map_wppull map_comp set_maps
   {context = ctxt, prems = _} =
   let
-    val n = length set_naturals;
+    val n = length set_maps;
     fun in_tac nthO_in = rtac CollectI THEN'
         CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS @{thm ord_eq_le_trans}),
-          rtac @{thm image_subsetI}, rtac nthO_in, etac set_mp, atac]) set_naturals;
+          rtac @{thm image_subsetI}, rtac nthO_in, etac set_mp, atac]) set_maps;
   in
-    if null set_naturals then unfold_thms_tac ctxt [srel_Id] THEN rtac (@{thm Id_O_R} RS sym) 1
+    if null set_maps then unfold_thms_tac ctxt [srel_Id] THEN rtac (@{thm Id_O_R} RS sym) 1
     else unfold_thms_tac ctxt (@{thm Gr_def} :: srel_O_Grs) THEN
       EVERY' [rtac equalityI, rtac @{thm subrelI},
         REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE, @{thm relcompE}, @{thm converseE}],
@@ -190,7 +190,7 @@
         REPEAT_DETERM o dtac Pair_eqD,
         REPEAT_DETERM o etac conjE, hyp_subst_tac,
         rtac allE, rtac subst, rtac @{thm wppull_def}, rtac map_wppull,
-        CONJ_WRAP' (K (rtac @{thm wppull_fstO_sndO})) set_naturals,
+        CONJ_WRAP' (K (rtac @{thm wppull_fstO_sndO})) set_maps,
         etac allE, etac impE, etac conjI, etac conjI, atac,
         REPEAT_DETERM o eresolve_tac [bexE, conjE],
         rtac @{thm relcompI}, rtac @{thm converseI},
--- a/src/HOL/BNF/Tools/bnf_fp_def_sugar.ML	Wed Apr 24 17:03:43 2013 +0200
+++ b/src/HOL/BNF/Tools/bnf_fp_def_sugar.ML	Wed Apr 24 17:47:22 2013 +0200
@@ -287,8 +287,8 @@
     val nested_map_comp's = map map_comp'_of_bnf nested_bnfs;
     val nested_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nested_bnfs;
     val nesting_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nesting_bnfs;
-    val nested_set_natural's = maps set_natural'_of_bnf nested_bnfs;
-    val nesting_set_natural's = maps set_natural'_of_bnf nesting_bnfs;
+    val nested_set_map's = maps set_map'_of_bnf nested_bnfs;
+    val nesting_set_map's = maps set_map'_of_bnf nesting_bnfs;
 
     val live = live_of_bnf any_fp_bnf;
 
@@ -548,7 +548,7 @@
 
             fun mk_set_thm fp_set_thm ctr_def' cxIn =
               fold_thms lthy [ctr_def']
-                (unfold_thms lthy (pre_set_defs @ nested_set_natural's @ nesting_set_natural's @
+                (unfold_thms lthy (pre_set_defs @ nested_set_map's @ nesting_set_map's @
                      (if lfp then [] else [dtor_ctor]) @ sum_prod_thms_set)
                    (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
@@ -814,8 +814,8 @@
 
             val thm =
               Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
-                mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct'
-                  nested_set_natural's pre_set_defss)
+                mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct' nested_set_map's
+                  pre_set_defss)
               |> singleton (Proof_Context.export names_lthy lthy)
               |> Thm.close_derivation;
           in
--- a/src/HOL/BNF/Tools/bnf_fp_def_sugar_tactics.ML	Wed Apr 24 17:03:43 2013 +0200
+++ b/src/HOL/BNF/Tools/bnf_fp_def_sugar_tactics.ML	Wed Apr 24 17:47:22 2013 +0200
@@ -133,26 +133,26 @@
     hyp_subst_tac ORELSE' resolve_tac @{thms disjI1 disjI2}) THEN'
   (rtac refl ORELSE' atac ORELSE' rtac @{thm singletonI});
 
-fun mk_induct_leverage_prem_prems_tac ctxt nn kks set_natural's pre_set_defs =
+fun mk_induct_leverage_prem_prems_tac ctxt nn kks set_map's pre_set_defs =
   EVERY' (maps (fn kk => [select_prem_tac nn (dtac meta_spec) kk, etac meta_mp,
-     SELECT_GOAL (unfold_thms_tac ctxt (pre_set_defs @ set_natural's @ sum_prod_thms_set0)),
+     SELECT_GOAL (unfold_thms_tac ctxt (pre_set_defs @ set_map's @ sum_prod_thms_set0)),
      solve_prem_prem_tac]) (rev kks)) 1;
 
-fun mk_induct_discharge_prem_tac ctxt nn n set_natural's pre_set_defs m k kks =
+fun mk_induct_discharge_prem_tac ctxt nn n set_map's pre_set_defs m k kks =
   let val r = length kks in
     EVERY' [select_prem_tac n (rotate_tac 1) k, rotate_tac ~1, hyp_subst_tac,
       REPEAT_DETERM_N m o (dtac meta_spec THEN' rotate_tac ~1)] 1 THEN
     EVERY [REPEAT_DETERM_N r
         (rotate_tac ~1 1 THEN dtac meta_mp 1 THEN rotate_tac 1 1 THEN prefer_tac 2),
       if r > 0 then PRIMITIVE Raw_Simplifier.norm_hhf else all_tac, atac 1,
-      mk_induct_leverage_prem_prems_tac ctxt nn kks set_natural's pre_set_defs]
+      mk_induct_leverage_prem_prems_tac ctxt nn kks set_map's pre_set_defs]
   end;
 
-fun mk_induct_tac ctxt nn ns mss kkss ctr_defs ctor_induct' set_natural's pre_set_defss =
+fun mk_induct_tac ctxt nn ns mss kkss ctr_defs ctor_induct' set_map's pre_set_defss =
   let val n = Integer.sum ns in
     unfold_thms_tac ctxt ctr_defs THEN rtac ctor_induct' 1 THEN inst_as_projs_tac ctxt 1 THEN
-    EVERY (map4 (EVERY oooo map3 o mk_induct_discharge_prem_tac ctxt nn n set_natural's)
-      pre_set_defss mss (unflat mss (1 upto n)) kkss)
+    EVERY (map4 (EVERY oooo map3 o mk_induct_discharge_prem_tac ctxt nn n set_map's) pre_set_defss
+      mss (unflat mss (1 upto n)) kkss)
   end;
 
 fun mk_coinduct_same_ctr ctxt rel_eqs pre_rel_def dtor_ctor ctr_def discs sels =
--- a/src/HOL/BNF/Tools/bnf_gfp.ML	Wed Apr 24 17:03:43 2013 +0200
+++ b/src/HOL/BNF/Tools/bnf_gfp.ML	Wed Apr 24 17:47:22 2013 +0200
@@ -227,7 +227,7 @@
     val map_id's = map map_id'_of_bnf bnfs;
     val map_wpulls = map map_wpull_of_bnf bnfs;
     val set_bdss = map set_bd_of_bnf bnfs;
-    val set_natural'ss = map set_natural'_of_bnf bnfs;
+    val set_map'ss = map set_map'_of_bnf bnfs;
     val srel_congs = map srel_cong_of_bnf bnfs;
     val srel_converses = map srel_converse_of_bnf bnfs;
     val srel_defs = map srel_def_of_bnf bnfs;
@@ -762,7 +762,7 @@
               singleton (Proof_Context.export names_lthy lthy)
                 (Goal.prove_sorry lthy [] [] goal
                   (K (mk_mor_hset_rec_tac m n cts j hset_rec_0s hset_rec_Sucs
-                    morE_thms set_natural'ss coalg_set_thmss)))
+                    morE_thms set_map'ss coalg_set_thmss)))
               |> Thm.close_derivation)
             ls goals ctss hset_rec_0ss' hset_rec_Sucss'
           end;
@@ -866,7 +866,7 @@
         Goal.prove_sorry lthy [] []
           (fold_rev Logic.all (As @ Bs @ ss @ B's @ s's @ Rs)
             (mk_Trueprop_eq (mk_bis As Bs ss B's s's Rs, rhs)))
-          (K (mk_bis_srel_tac m bis_def srel_O_Grs map_comp's map_cong0s set_natural'ss))
+          (K (mk_bis_srel_tac m bis_def srel_O_Grs map_comp's map_cong0s set_map'ss))
         |> Thm.close_derivation
       end;
 
@@ -1284,7 +1284,7 @@
     val coalgT_thm =
       Goal.prove_sorry lthy [] []
         (fold_rev Logic.all As (HOLogic.mk_Trueprop (mk_coalg As carTAs strTAs)))
-        (mk_coalgT_tac m (coalg_def :: isNode_defs @ carT_defs) strT_defs set_natural'ss)
+        (mk_coalgT_tac m (coalg_def :: isNode_defs @ carT_defs) strT_defs set_map'ss)
       |> Thm.close_derivation;
 
     val card_of_carT_thms =
@@ -1323,14 +1323,13 @@
         val goalss = map3 (fn carT => fn strT => fn sets =>
           map3 (mk_goal carT strT) (drop m sets) kks ks) carTAs strTAs tree_setss;
       in
-        map6 (fn i => fn goals =>
-            fn carT_def => fn strT_def => fn isNode_def => fn set_naturals =>
-          map2 (fn goal => fn set_natural =>
+        map6 (fn i => fn goals => fn carT_def => fn strT_def => fn isNode_def => fn set_maps =>
+          map2 (fn goal => fn set_map =>
             Goal.prove_sorry lthy [] [] goal
-              (mk_carT_set_tac n i carT_def strT_def isNode_def set_natural)
+              (mk_carT_set_tac n i carT_def strT_def isNode_def set_map)
             |> Thm.close_derivation)
-          goals (drop m set_naturals))
-        ks goalss carT_defs strT_defs isNode_defs set_natural'ss
+          goals (drop m set_maps))
+        ks goalss carT_defs strT_defs isNode_defs set_map'ss
       end;
 
     val carT_set_thmss' = transpose carT_set_thmss;
@@ -1372,7 +1371,7 @@
                       (K (mk_strT_hset_tac n m j arg_cong_cTs cTs cts
                         carT_defs strT_defs isNode_defs
                         set_incl_hsets set_hset_incl_hsetss coalg_set_thmss' carT_set_thmss'
-                        coalgT_thm set_natural'ss)))
+                        coalgT_thm set_map'ss)))
                   |> Thm.close_derivation)
                 ls goals ctss set_incl_hset_thmss' set_hset_incl_hset_thmsss''
               end;
@@ -1770,7 +1769,7 @@
           to_sbd_inj_thmss from_to_sbd_thmss Lev_0s Lev_Sucs rv_Nils rv_Conss Lev_sbd_thms
           length_Lev_thms length_Lev'_thms prefCl_Lev_thms rv_last_thmss
           set_rv_Lev_thmsss set_Lev_thmsss set_image_Lev_thmsss
-          set_natural'ss coalg_set_thmss map_comp_id_thms map_cong0s map_arg_cong_thms)
+          set_map'ss coalg_set_thmss map_comp_id_thms map_cong0s map_arg_cong_thms)
       |> Thm.close_derivation;
 
     val timer = time (timer "Behavioral morphism");
@@ -1809,7 +1808,7 @@
     val coalg_final_thm = Goal.prove_sorry lthy [] [] (fold_rev Logic.all As
       (HOLogic.mk_Trueprop (mk_coalg As car_finalAs str_finalAs)))
       (K (mk_coalg_final_tac m coalg_def congruent_str_final_thms equiv_LSBIS_thms
-        set_natural'ss coalgT_set_thmss))
+        set_map'ss coalgT_set_thmss))
       |> Thm.close_derivation;
 
     val mor_T_final_thm = Goal.prove_sorry lthy [] [] (fold_rev Logic.all As
@@ -2517,7 +2516,7 @@
               map4 (fn goal => fn cts => fn rec_0s => fn rec_Sucs =>
                 singleton (Proof_Context.export names_lthy lthy)
                   (Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop goal)
-                    (mk_col_natural_tac cts rec_0s rec_Sucs dtor_map_thms set_natural'ss))
+                    (mk_col_natural_tac cts rec_0s rec_Sucs dtor_map_thms set_map'ss))
                 |> Thm.close_derivation)
               goals ctss hset_rec_0ss' hset_rec_Sucss';
           in
@@ -2585,7 +2584,7 @@
 
             val thm = singleton (Proof_Context.export names_lthy lthy)
               (Goal.prove_sorry lthy [] [] goal
-              (K (mk_mcong_tac m (rtac coinduct) map_comp's dtor_map_thms map_cong0s set_natural'ss
+              (K (mk_mcong_tac m (rtac coinduct) map_comp's dtor_map_thms map_cong0s set_map'ss
               set_hset_thmss set_hset_hset_thmsss)))
               |> Thm.close_derivation
           in
@@ -2621,7 +2620,7 @@
               (Logic.mk_implies (wpull_prem, coalg));
           in
             Goal.prove_sorry lthy [] [] goal (mk_coalg_thePull_tac m coalg_def map_wpull_thms
-              set_natural'ss pickWP_assms_tacs)
+              set_map'ss pickWP_assms_tacs)
             |> Thm.close_derivation
           end;
 
@@ -2672,7 +2671,7 @@
             val thms =
               map5 (fn j => fn goal => fn cts => fn rec_0s => fn rec_Sucs =>
                 singleton (Proof_Context.export names_lthy lthy) (Goal.prove_sorry lthy [] [] goal
-                  (mk_pick_col_tac m j cts rec_0s rec_Sucs dtor_unfold_thms set_natural'ss
+                  (mk_pick_col_tac m j cts rec_0s rec_Sucs dtor_unfold_thms set_map'ss
                     map_wpull_thms pickWP_assms_tacs))
                 |> Thm.close_derivation)
               ls goals ctss hset_rec_0ss' hset_rec_Sucss';
@@ -2687,7 +2686,7 @@
         val map_comp_tacs = map (fn thm => K (rtac (thm RS sym) 1)) map_comp_thms;
         val map_cong0_tacs = map (mk_map_cong0_tac m) map_cong0_thms;
         val set_nat_tacss =
-          map2 (map2 (K oo mk_set_natural_tac)) hset_defss (transpose col_natural_thmss);
+          map2 (map2 (K oo mk_set_map_tac)) hset_defss (transpose col_natural_thmss);
 
         val bd_co_tacs = replicate n (K (mk_bd_card_order_tac sbd_card_order));
         val bd_cinf_tacs = replicate n (K (mk_bd_cinfinite_tac sbd_Cinfinite));
@@ -2866,7 +2865,7 @@
           in
             map2 (fn goal => fn induct =>
               Goal.prove_sorry lthy [] [] goal
-                (mk_coind_wit_tac induct dtor_unfold_thms (flat set_natural'ss) wit_thms)
+                (mk_coind_wit_tac induct dtor_unfold_thms (flat set_map'ss) wit_thms)
               |> Thm.close_derivation)
             goals dtor_hset_induct_thms
             |> map split_conj_thm
@@ -2943,13 +2942,13 @@
           in
             map12 (fn i => fn goal => fn in_srel => fn map_comp => fn map_cong0 =>
               fn dtor_map => fn dtor_sets => fn dtor_inject => fn dtor_ctor =>
-              fn set_naturals => fn dtor_set_incls => fn dtor_set_set_inclss =>
+              fn set_maps => fn dtor_set_incls => fn dtor_set_set_inclss =>
               Goal.prove_sorry lthy [] [] goal
                 (K (mk_dtor_srel_tac in_Jsrels i in_srel map_comp map_cong0 dtor_map dtor_sets
-                  dtor_inject dtor_ctor set_naturals dtor_set_incls dtor_set_set_inclss))
+                  dtor_inject dtor_ctor set_maps dtor_set_incls dtor_set_set_inclss))
               |> Thm.close_derivation)
             ks goals in_srels map_comp's map_cong0s folded_dtor_map_thms folded_dtor_set_thmss'
-              dtor_inject_thms dtor_ctor_thms set_natural'ss dtor_set_incl_thmss
+              dtor_inject_thms dtor_ctor_thms set_map'ss dtor_set_incl_thmss
               dtor_set_set_incl_thmsss
           end;
 
--- a/src/HOL/BNF/Tools/bnf_gfp_tactics.ML	Wed Apr 24 17:03:43 2013 +0200
+++ b/src/HOL/BNF/Tools/bnf_gfp_tactics.ML	Wed Apr 24 17:47:22 2013 +0200
@@ -110,7 +110,7 @@
   val mk_set_incl_hin_tac: thm list -> tactic
   val mk_set_incl_hset_tac: thm -> thm -> tactic
   val mk_set_le_tac: int -> thm -> thm list -> thm list list -> tactic
-  val mk_set_natural_tac: thm -> thm -> tactic
+  val mk_set_map_tac: thm -> thm -> tactic
   val mk_set_rv_Lev_tac: int -> cterm option list -> thm list -> thm list -> thm list -> thm list ->
     thm list list -> thm list list -> tactic
   val mk_strT_hset_tac: int -> int -> int -> ctyp option list -> ctyp option list ->
@@ -230,38 +230,38 @@
     EVERY' (replicate ((n + 1) * n) (Goal.assume_rule_tac ctxt)), rtac impI,
     REPEAT_DETERM o eresolve_tac [allE, conjE], atac])) hset_defs) 1
 
-fun mk_mor_hset_rec_tac m n cts j rec_0s rec_Sucs morEs set_naturalss coalg_setss =
+fun mk_mor_hset_rec_tac m n cts j rec_0s rec_Sucs morEs set_mapss coalg_setss =
   EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
     REPEAT_DETERM o rtac allI,
     CONJ_WRAP' (fn thm => EVERY' (map rtac [impI, thm RS trans, thm RS sym])) rec_0s,
     REPEAT_DETERM o rtac allI,
     CONJ_WRAP'
-      (fn (rec_Suc, (morE, ((passive_set_naturals, active_set_naturals), coalg_sets))) =>
+      (fn (rec_Suc, (morE, ((passive_set_maps, active_set_maps), coalg_sets))) =>
         EVERY' [rtac impI, rtac (rec_Suc RS trans), rtac (rec_Suc RS trans RS sym),
           if m = 0 then K all_tac
           else EVERY' [rtac @{thm Un_cong}, rtac box_equals,
-            rtac (nth passive_set_naturals (j - 1) RS sym),
+            rtac (nth passive_set_maps (j - 1) RS sym),
             rtac trans_fun_cong_image_id_id_apply, etac (morE RS arg_cong), atac],
           CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_cong}))
-            (fn (i, (set_natural, coalg_set)) =>
+            (fn (i, (set_map, coalg_set)) =>
               EVERY' [rtac sym, rtac trans, rtac (refl RSN (2, @{thm UN_cong})),
-                etac (morE RS sym RS arg_cong RS trans), atac, rtac set_natural,
+                etac (morE RS sym RS arg_cong RS trans), atac, rtac set_map,
                 rtac (@{thm UN_simps(10)} RS trans), rtac (refl RS @{thm UN_cong}),
                 ftac coalg_set, atac, dtac set_mp, atac, rtac mp, rtac (mk_conjunctN n i),
                 REPEAT_DETERM o etac allE, atac, atac])
-            (rev ((1 upto n) ~~ (active_set_naturals ~~ coalg_sets)))])
-      (rec_Sucs ~~ (morEs ~~ (map (chop m) set_naturalss ~~ map (drop m) coalg_setss)))] 1;
+            (rev ((1 upto n) ~~ (active_set_maps ~~ coalg_sets)))])
+      (rec_Sucs ~~ (morEs ~~ (map (chop m) set_mapss ~~ map (drop m) coalg_setss)))] 1;
 
 fun mk_mor_hset_tac hset_def mor_hset_rec =
   EVERY' [rtac (hset_def RS trans), rtac (refl RS @{thm UN_cong} RS trans), etac mor_hset_rec,
     atac, atac, rtac (hset_def RS sym)] 1
 
-fun mk_bis_srel_tac m bis_def srel_O_Grs map_comps map_cong0s set_naturalss =
+fun mk_bis_srel_tac m bis_def srel_O_Grs map_comps map_cong0s set_mapss =
   let
     val n = length srel_O_Grs;
-    val thms = ((1 upto n) ~~ map_comps ~~ map_cong0s ~~ set_naturalss ~~ srel_O_Grs);
+    val thms = ((1 upto n) ~~ map_comps ~~ map_cong0s ~~ set_mapss ~~ srel_O_Grs);
 
-    fun mk_if_tac ((((i, map_comp), map_cong0), set_naturals), srel_O_Gr) =
+    fun mk_if_tac ((((i, map_comp), map_cong0), set_maps), srel_O_Gr) =
       EVERY' [rtac allI, rtac allI, rtac impI, dtac (mk_conjunctN n i),
         etac allE, etac allE, etac impE, atac, etac bexE, etac conjE,
         rtac (srel_O_Gr RS equalityD2 RS set_mp),
@@ -275,12 +275,12 @@
                 etac @{thm image_mono}, rtac @{thm image_subsetI}, etac @{thm Id_onI}]
               else EVERY' [rtac ord_eq_le_trans, rtac trans, rtac thm,
                 rtac trans_fun_cong_image_id_id_apply, atac])
-            (1 upto (m + n) ~~ set_naturals),
+            (1 upto (m + n) ~~ set_maps),
             rtac trans, rtac trans, rtac map_comp, rtac map_cong0, REPEAT_DETERM_N m o rtac thm,
             REPEAT_DETERM_N n o rtac (@{thm o_id} RS fun_cong), atac])
           @{thms fst_diag_id snd_diag_id})];
 
-    fun mk_only_if_tac ((((i, map_comp), map_cong0), set_naturals), srel_O_Gr) =
+    fun mk_only_if_tac ((((i, map_comp), map_cong0), set_maps), srel_O_Gr) =
       EVERY' [dtac (mk_conjunctN n i), rtac allI, rtac allI, rtac impI,
         etac allE, etac allE, etac impE, atac,
         dtac (srel_O_Gr RS equalityD1 RS set_mp),
@@ -306,7 +306,7 @@
             rtac @{thm Id_on_fst}, etac set_mp, atac]
           else EVERY' [rtac ord_eq_le_trans, rtac trans, rtac thm,
             rtac trans_fun_cong_image_id_id_apply, atac])
-        (1 upto (m + n) ~~ set_naturals)];
+        (1 upto (m + n) ~~ set_maps)];
   in
     EVERY' [rtac (bis_def RS trans),
       rtac iffI, etac conjE, etac conjI, CONJ_WRAP' mk_if_tac thms,
@@ -401,7 +401,7 @@
     rtac incl_lsbis, rtac bis_O, rtac sbis_lsbis, rtac sbis_lsbis,
     etac @{thm relcompI}, atac] 1;
 
-fun mk_coalgT_tac m defs strT_defs set_naturalss {context = ctxt, prems = _} =
+fun mk_coalgT_tac m defs strT_defs set_mapss {context = ctxt, prems = _} =
   let
     val n = length strT_defs;
     val ks = 1 upto n;
@@ -446,7 +446,7 @@
             rtac (@{thm append_Nil} RS sym RS arg_cong)])) ks]) (ks ~~ active_sets)];
   in
     unfold_thms_tac ctxt defs THEN
-    CONJ_WRAP' coalg_tac (ks ~~ (map (chop m) set_naturalss ~~ strT_defs)) 1
+    CONJ_WRAP' coalg_tac (ks ~~ (map (chop m) set_mapss ~~ strT_defs)) 1
   end;
 
 fun mk_card_of_carT_tac m isNode_defs sbd_sbd
@@ -539,7 +539,7 @@
       atac] 1
   end;
 
-fun mk_carT_set_tac n i carT_def strT_def isNode_def set_natural {context = ctxt, prems = _}=
+fun mk_carT_set_tac n i carT_def strT_def isNode_def set_map {context = ctxt, prems = _}=
   EVERY' [dtac (carT_def RS equalityD1 RS set_mp),
     REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE],
     dtac Pair_eqD,
@@ -550,20 +550,20 @@
     rtac (strT_def RS arg_cong RS trans),
     etac (arg_cong RS trans),
     fo_rtac (mk_sum_casesN n i RS arg_cong RS trans) ctxt,
-    rtac set_natural, rtac imageI, etac (equalityD2 RS set_mp), rtac CollectI,
+    rtac set_map, rtac imageI, etac (equalityD2 RS set_mp), rtac CollectI,
     etac @{thm prefCl_Succ}, atac] 1;
 
 fun mk_strT_hset_tac n m j arg_cong_cTs cTs cts carT_defs strT_defs isNode_defs
-  set_incl_hsets set_hset_incl_hsetss coalg_setss carT_setss coalgT set_naturalss =
+  set_incl_hsets set_hset_incl_hsetss coalg_setss carT_setss coalgT set_mapss =
   let
-    val set_naturals = map (fn xs => nth xs (j - 1)) set_naturalss;
+    val set_maps = map (fn xs => nth xs (j - 1)) set_mapss;
     val ks = 1 upto n;
-    fun base_tac (i, (cT, (strT_def, (set_incl_hset, set_natural)))) =
+    fun base_tac (i, (cT, (strT_def, (set_incl_hset, set_map)))) =
       CONJ_WRAP' (fn (i', (carT_def, isNode_def)) => rtac impI THEN' etac conjE THEN'
         (if i = i'
         then EVERY' [rtac @{thm xt1(4)}, rtac set_incl_hset,
           rtac (strT_def RS arg_cong RS trans), etac (arg_cong RS trans),
-          rtac (Thm.permute_prems 0 1 (set_natural RS box_equals)),
+          rtac (Thm.permute_prems 0 1 (set_map RS box_equals)),
           rtac (trans OF [@{thm image_id} RS fun_cong, id_apply]),
           rtac (mk_sum_casesN n i RS (Drule.instantiate' [cT] [] arg_cong) RS sym)]
         else EVERY' [dtac (carT_def RS equalityD1 RS set_mp),
@@ -585,7 +585,7 @@
     EVERY' [rtac (Drule.instantiate' cTs cts @{thm list.induct}),
       REPEAT_DETERM o rtac allI, rtac impI,
       CONJ_WRAP' base_tac
-        (ks ~~ (arg_cong_cTs ~~ (strT_defs ~~ (set_incl_hsets ~~ set_naturals)))),
+        (ks ~~ (arg_cong_cTs ~~ (strT_defs ~~ (set_incl_hsets ~~ set_maps)))),
       REPEAT_DETERM o rtac allI, rtac impI,
       REPEAT_DETERM o eresolve_tac [allE, impE], etac @{thm ShiftI},
       CONJ_WRAP' (fn i => dtac (mk_conjunctN n i) THEN' rtac (mk_sumEN n) THEN'
@@ -840,14 +840,14 @@
 
 fun mk_mor_beh_tac m mor_def mor_cong beh_defs carT_defs strT_defs isNode_defs
   to_sbd_injss from_to_sbdss Lev_0s Lev_Sucs rv_Nils rv_Conss Lev_sbds length_Levs length_Lev's
-  prefCl_Levs rv_lastss set_rv_Levsss set_Levsss set_image_Levsss set_naturalss coalg_setss
+  prefCl_Levs rv_lastss set_rv_Levsss set_Levsss set_image_Levsss set_mapss coalg_setss
   map_comp_ids map_cong0s map_arg_congs {context = ctxt, prems = _} =
   let
     val n = length beh_defs;
     val ks = 1 upto n;
 
     fun fbetw_tac (i, (carT_def, (isNode_def, (Lev_0, (rv_Nil, (Lev_sbd,
-      ((length_Lev, length_Lev'), (prefCl_Lev, (rv_lasts, (set_naturals,
+      ((length_Lev, length_Lev'), (prefCl_Lev, (rv_lasts, (set_maps,
         (coalg_sets, (set_rv_Levss, (set_Levss, set_image_Levss))))))))))))) =
       EVERY' [rtac ballI, rtac (carT_def RS equalityD2 RS set_mp),
         rtac CollectI, REPEAT_DETERM o rtac exI, rtac conjI, rtac refl, rtac conjI,
@@ -863,29 +863,29 @@
         rtac conjI,
           rtac ballI, etac @{thm UN_E}, rtac conjI,
           if n = 1 then K all_tac else rtac (mk_sumEN n),
-          EVERY' (map6 (fn i => fn isNode_def => fn set_naturals =>
+          EVERY' (map6 (fn i => fn isNode_def => fn set_maps =>
             fn set_rv_Levs => fn set_Levs => fn set_image_Levs =>
             EVERY' [rtac (mk_disjIN n i), rtac (isNode_def RS ssubst),
               rtac exI, rtac conjI,
               (if n = 1 then rtac @{thm if_P} THEN' etac length_Lev'
               else rtac (@{thm if_P} RS arg_cong RS trans) THEN' etac length_Lev' THEN'
                 etac (sum_case_weak_cong RS trans) THEN' rtac (mk_sum_casesN n i)),
-              EVERY' (map2 (fn set_natural => fn set_rv_Lev =>
-                EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac (set_natural RS trans),
+              EVERY' (map2 (fn set_map => fn set_rv_Lev =>
+                EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac (set_map RS trans),
                   rtac trans_fun_cong_image_id_id_apply,
                   etac set_rv_Lev, TRY o atac, etac conjI, atac])
-              (take m set_naturals) set_rv_Levs),
-              CONJ_WRAP' (fn (set_natural, (set_Lev, set_image_Lev)) =>
-                EVERY' [rtac (set_natural RS trans), rtac equalityI, rtac @{thm image_subsetI},
+              (take m set_maps) set_rv_Levs),
+              CONJ_WRAP' (fn (set_map, (set_Lev, set_image_Lev)) =>
+                EVERY' [rtac (set_map RS trans), rtac equalityI, rtac @{thm image_subsetI},
                   rtac CollectI, rtac @{thm SuccI}, rtac @{thm UN_I}, rtac UNIV_I, etac set_Lev,
                   if n = 1 then rtac refl else atac, atac, rtac subsetI,
                   REPEAT_DETERM o eresolve_tac [CollectE, @{thm SuccE}, @{thm UN_E}],
                   rtac set_image_Lev, atac, dtac length_Lev, hyp_subst_tac, dtac length_Lev',
                   etac @{thm set_mp[OF equalityD1[OF arg_cong[OF length_append_singleton]]]},
                   if n = 1 then rtac refl else atac])
-              (drop m set_naturals ~~ (set_Levs ~~ set_image_Levs))])
-          ks isNode_defs set_naturalss set_rv_Levss set_Levss set_image_Levss),
-          CONJ_WRAP' (fn (i, (rv_last, (isNode_def, (set_naturals,
+              (drop m set_maps ~~ (set_Levs ~~ set_image_Levs))])
+          ks isNode_defs set_mapss set_rv_Levss set_Levss set_image_Levss),
+          CONJ_WRAP' (fn (i, (rv_last, (isNode_def, (set_maps,
             (set_rv_Levs, (set_Levs, set_image_Levs)))))) =>
             EVERY' [rtac ballI,
               REPEAT_DETERM o eresolve_tac [CollectE, @{thm SuccE}, @{thm UN_E}],
@@ -894,13 +894,13 @@
               (if n = 1 then rtac @{thm if_P} THEN' etac length_Lev'
               else rtac (@{thm if_P} RS trans) THEN' etac length_Lev' THEN'
                 etac (sum_case_weak_cong RS trans) THEN' rtac (mk_sum_casesN n i)),
-              EVERY' (map2 (fn set_natural => fn set_rv_Lev =>
-                EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac (set_natural RS trans),
+              EVERY' (map2 (fn set_map => fn set_rv_Lev =>
+                EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac (set_map RS trans),
                   rtac trans_fun_cong_image_id_id_apply,
                   etac set_rv_Lev, TRY o atac, etac conjI, atac])
-              (take m set_naturals) set_rv_Levs),
-              CONJ_WRAP' (fn (set_natural, (set_Lev, set_image_Lev)) =>
-                EVERY' [rtac (set_natural RS trans), rtac equalityI, rtac @{thm image_subsetI},
+              (take m set_maps) set_rv_Levs),
+              CONJ_WRAP' (fn (set_map, (set_Lev, set_image_Lev)) =>
+                EVERY' [rtac (set_map RS trans), rtac equalityI, rtac @{thm image_subsetI},
                   rtac CollectI, rtac @{thm SuccI}, rtac @{thm UN_I}, rtac UNIV_I, etac set_Lev,
                   if n = 1 then rtac refl else atac, atac, rtac subsetI,
                   REPEAT_DETERM o eresolve_tac [CollectE, @{thm SuccE}, @{thm UN_E}],
@@ -909,8 +909,8 @@
                   atac, dtac length_Lev, hyp_subst_tac, dtac length_Lev',
                   etac @{thm set_mp[OF equalityD1[OF arg_cong[OF length_append_singleton]]]},
                   if n = 1 then rtac refl else atac])
-              (drop m set_naturals ~~ (set_Levs ~~ set_image_Levs))])
-          (ks ~~ (rv_lasts ~~ (isNode_defs ~~ (set_naturalss ~~
+              (drop m set_maps ~~ (set_Levs ~~ set_image_Levs))])
+          (ks ~~ (rv_lasts ~~ (isNode_defs ~~ (set_mapss ~~
             (set_rv_Levss ~~ (set_Levss ~~ set_image_Levss)))))),
         (**)
           rtac allI, rtac impI, rtac @{thm if_not_P}, rtac notI,
@@ -921,12 +921,12 @@
           CONVERSION (Conv.top_conv
             (K (Conv.try_conv (Conv.rewr_conv (rv_Nil RS eq_reflection)))) ctxt),
           if n = 1 then rtac refl else rtac (mk_sum_casesN n i),
-          EVERY' (map2 (fn set_natural => fn coalg_set =>
-            EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac (set_natural RS trans),
+          EVERY' (map2 (fn set_map => fn coalg_set =>
+            EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac (set_map RS trans),
               rtac trans_fun_cong_image_id_id_apply, etac coalg_set, atac])
-            (take m set_naturals) (take m coalg_sets)),
-          CONJ_WRAP' (fn (set_natural, (set_Lev, set_image_Lev)) =>
-            EVERY' [rtac (set_natural RS trans), rtac equalityI, rtac @{thm image_subsetI},
+            (take m set_maps) (take m coalg_sets)),
+          CONJ_WRAP' (fn (set_map, (set_Lev, set_image_Lev)) =>
+            EVERY' [rtac (set_map RS trans), rtac equalityI, rtac @{thm image_subsetI},
               rtac CollectI, rtac @{thm SuccI}, rtac @{thm UN_I}, rtac UNIV_I, rtac set_Lev,
               rtac (Lev_0 RS equalityD2 RS set_mp), rtac @{thm singletonI}, rtac rv_Nil,
               atac, rtac subsetI,
@@ -936,7 +936,7 @@
               etac @{thm set_mp[OF equalityD1[OF arg_cong[OF
                 trans[OF length_append_singleton arg_cong[of _ _ Suc, OF list.size(3)]]]]]},
               rtac rv_Nil])
-          (drop m set_naturals ~~ (nth set_Levss (i - 1) ~~ nth set_image_Levss (i - 1)))];
+          (drop m set_maps ~~ (nth set_Levss (i - 1) ~~ nth set_image_Levss (i - 1)))];
 
     fun mor_tac (i, (strT_def, (((Lev_0, Lev_Suc), (rv_Nil, rv_Cons)),
       ((map_comp_id, (map_cong0, map_arg_cong)), (length_Lev', (from_to_sbds, to_sbd_injs)))))) =
@@ -997,7 +997,7 @@
     CONJ_WRAP' fbetw_tac
       (ks ~~ (carT_defs ~~ (isNode_defs ~~ (Lev_0s ~~ (rv_Nils ~~ (Lev_sbds ~~
         ((length_Levs ~~ length_Lev's) ~~ (prefCl_Levs ~~ (rv_lastss ~~
-          (set_naturalss ~~ (coalg_setss ~~
+          (set_mapss ~~ (coalg_setss ~~
             (set_rv_Levsss ~~ (set_Levsss ~~ set_image_Levsss))))))))))))) THEN'
     CONJ_WRAP' mor_tac
       (ks ~~ (strT_defs ~~ (((Lev_0s ~~ Lev_Sucs) ~~ (rv_Nils ~~ rv_Conss)) ~~
@@ -1015,22 +1015,22 @@
     equiv_LSBISs), rtac sym, rtac (o_apply RS trans),
     etac (sym RS arg_cong RS trans), rtac map_comp_id] 1;
 
-fun mk_coalg_final_tac m coalg_def congruent_str_finals equiv_LSBISs set_naturalss coalgT_setss =
+fun mk_coalg_final_tac m coalg_def congruent_str_finals equiv_LSBISs set_mapss coalgT_setss =
   EVERY' [stac coalg_def,
-    CONJ_WRAP' (fn ((set_naturals, coalgT_sets), (equiv_LSBIS, congruent_str_final)) =>
+    CONJ_WRAP' (fn ((set_maps, coalgT_sets), (equiv_LSBIS, congruent_str_final)) =>
       EVERY' [rtac @{thm univ_preserves}, rtac equiv_LSBIS, rtac congruent_str_final,
         rtac ballI, rtac @{thm ssubst_mem}, rtac o_apply, rtac CollectI,
-        EVERY' (map2 (fn set_natural => fn coalgT_set =>
-          EVERY' [rtac conjI, rtac (set_natural RS ord_eq_le_trans),
+        EVERY' (map2 (fn set_map => fn coalgT_set =>
+          EVERY' [rtac conjI, rtac (set_map RS ord_eq_le_trans),
             rtac ord_eq_le_trans_trans_fun_cong_image_id_id_apply,
             etac coalgT_set])
-        (take m set_naturals) (take m coalgT_sets)),
-        CONJ_WRAP' (fn (equiv_LSBIS, (set_natural, coalgT_set)) =>
-          EVERY' [rtac (set_natural RS ord_eq_le_trans),
+        (take m set_maps) (take m coalgT_sets)),
+        CONJ_WRAP' (fn (equiv_LSBIS, (set_map, coalgT_set)) =>
+          EVERY' [rtac (set_map RS ord_eq_le_trans),
             rtac @{thm image_subsetI}, rtac ssubst, rtac @{thm proj_in_iff},
             rtac equiv_LSBIS, etac set_rev_mp, etac coalgT_set])
-        (equiv_LSBISs ~~ drop m (set_naturals ~~ coalgT_sets))])
-    ((set_naturalss ~~ coalgT_setss) ~~ (equiv_LSBISs ~~ congruent_str_finals))] 1;
+        (equiv_LSBISs ~~ drop m (set_maps ~~ coalgT_sets))])
+    ((set_mapss ~~ coalgT_setss) ~~ (equiv_LSBISs ~~ congruent_str_finals))] 1;
 
 fun mk_mor_T_final_tac mor_def congruent_str_finals equiv_LSBISs =
   EVERY' [stac mor_def, rtac conjI,
@@ -1225,7 +1225,7 @@
         replicate n (@{thm o_id} RS fun_cong))))
     maps map_comps map_cong0s)] 1;
 
-fun mk_mcong_tac m coinduct_tac map_comp's dtor_maps map_cong0s set_naturalss set_hsetss
+fun mk_mcong_tac m coinduct_tac map_comp's dtor_maps map_cong0s set_mapss set_hsetss
   set_hset_hsetsss =
   let
     val n = length map_comp's;
@@ -1233,7 +1233,7 @@
   in
     EVERY' ([rtac rev_mp,
       coinduct_tac] @
-      maps (fn (((((map_comp'_trans, dtor_maps_trans), map_cong0), set_naturals), set_hsets),
+      maps (fn (((((map_comp'_trans, dtor_maps_trans), map_cong0), set_maps), set_hsets),
         set_hset_hsetss) =>
         [REPEAT_DETERM o eresolve_tac [exE, conjE], hyp_subst_tac, rtac exI, rtac conjI, rtac conjI,
          rtac map_comp'_trans, rtac sym, rtac dtor_maps_trans, rtac map_cong0,
@@ -1244,16 +1244,16 @@
            [rtac o_apply_trans_sym, rtac (id_apply RS trans), etac CollectE,
            REPEAT_DETERM o etac conjE, etac bspec, etac set_hset]) set_hsets),
          REPEAT_DETERM_N n o rtac snd_convol_fun_cong_sym,
-         CONJ_WRAP' (fn (set_natural, set_hset_hsets) =>
+         CONJ_WRAP' (fn (set_map, set_hset_hsets) =>
            EVERY' [REPEAT_DETERM o rtac allI, rtac impI, rtac @{thm image_convolD},
-             etac set_rev_mp, rtac ord_eq_le_trans, rtac set_natural,
+             etac set_rev_mp, rtac ord_eq_le_trans, rtac set_map,
              rtac @{thm image_mono}, rtac subsetI, rtac CollectI, etac CollectE,
              REPEAT_DETERM o etac conjE,
              CONJ_WRAP' (fn set_hset_hset =>
                EVERY' [rtac ballI, etac bspec, etac set_hset_hset, atac]) set_hset_hsets])
-           (drop m set_naturals ~~ set_hset_hsetss)])
+           (drop m set_maps ~~ set_hset_hsetss)])
         (map (fn th => th RS trans) map_comp's ~~ map (fn th => th RS trans) dtor_maps ~~
-          map_cong0s ~~ set_naturalss ~~ set_hsetss ~~ set_hset_hsetsss) @
+          map_cong0s ~~ set_mapss ~~ set_hsetss ~~ set_hset_hsetsss) @
       [rtac impI,
        CONJ_WRAP' (fn k =>
          EVERY' [rtac impI, dtac (mk_conjunctN n k), etac mp, rtac exI, rtac conjI, etac CollectI,
@@ -1265,7 +1265,7 @@
   unfold_thms_tac ctxt (map (fn thm => thm RS sym) map_comps @ @{thms o_assoc id_o o_id}) THEN
   ALLGOALS (etac sym);
 
-fun mk_col_natural_tac cts rec_0s rec_Sucs dtor_maps set_naturalss
+fun mk_col_natural_tac cts rec_0s rec_Sucs dtor_maps set_mapss
   {context = ctxt, prems = _} =
   let
     val n = length dtor_maps;
@@ -1281,10 +1281,10 @@
         CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_cong}))
           (fn i => EVERY' [rtac @{thm UN_cong[OF refl]},
             REPEAT_DETERM o etac allE, etac (mk_conjunctN n i)]) (n downto 1)])
-      (rec_Sucs ~~ (dtor_maps ~~ set_naturalss))] 1
+      (rec_Sucs ~~ (dtor_maps ~~ set_mapss))] 1
   end;
 
-fun mk_set_natural_tac hset_def col_natural =
+fun mk_set_map_tac hset_def col_natural =
   EVERY' (map rtac [ext, (o_apply RS trans), (hset_def RS trans), sym,
     (o_apply RS trans), (@{thm image_cong} OF [hset_def, refl] RS trans),
     (@{thm image_UN} RS trans), (refl RS @{thm UN_cong}), col_natural]) 1;
@@ -1370,10 +1370,10 @@
       REPEAT_DETERM o eresolve_tac [CollectE, conjE], etac map_eq]
   end;
 
-fun mk_coalg_thePull_tac m coalg_def map_wpulls set_naturalss pickWP_assms_tacs
+fun mk_coalg_thePull_tac m coalg_def map_wpulls set_mapss pickWP_assms_tacs
   {context = ctxt, prems = _} =
   unfold_thms_tac ctxt [coalg_def] THEN
-  CONJ_WRAP' (fn (map_wpull, (pickWP_assms_tac, set_naturals)) =>
+  CONJ_WRAP' (fn (map_wpull, (pickWP_assms_tac, set_maps)) =>
     EVERY' [rtac ballI, dtac @{thm set_mp[OF equalityD1[OF thePull_def]]},
       REPEAT_DETERM o eresolve_tac [CollectE, @{thm prod_caseE}],
       hyp_subst_tac, rtac rev_mp, rtac (map_wpull RS @{thm pickWP(1)}),
@@ -1383,11 +1383,11 @@
       REPEAT_DETERM o eresolve_tac [CollectE, conjE],
       rtac CollectI,
       REPEAT_DETERM_N m o (rtac conjI THEN' rtac subset_UNIV),
-      CONJ_WRAP' (fn set_natural =>
-        EVERY' [rtac ord_eq_le_trans, rtac trans, rtac set_natural,
+      CONJ_WRAP' (fn set_map =>
+        EVERY' [rtac ord_eq_le_trans, rtac trans, rtac set_map,
           rtac trans_fun_cong_image_id_id_apply, atac])
-      (drop m set_naturals)])
-  (map_wpulls ~~ (pickWP_assms_tacs ~~ set_naturalss)) 1;
+      (drop m set_maps)])
+  (map_wpulls ~~ (pickWP_assms_tacs ~~ set_mapss)) 1;
 
 fun mk_mor_thePull_nth_tac conv pick m mor_def map_wpulls map_comps pickWP_assms_tacs
   {context = ctxt, prems = _: thm list} =
@@ -1422,7 +1422,7 @@
       rtac refl])
   (unfolds ~~ map_comps) 1;
 
-fun mk_pick_col_tac m j cts rec_0s rec_Sucs unfolds set_naturalss map_wpulls pickWP_assms_tacs
+fun mk_pick_col_tac m j cts rec_0s rec_Sucs unfolds set_mapss map_wpulls pickWP_assms_tacs
   {context = ctxt, prems = _} =
   let
     val n = length rec_0s;
@@ -1433,7 +1433,7 @@
       CONJ_WRAP' (fn thm => EVERY'
         [rtac impI, rtac ord_eq_le_trans, rtac thm, rtac @{thm empty_subsetI}]) rec_0s,
       REPEAT_DETERM o rtac allI,
-      CONJ_WRAP' (fn (rec_Suc, ((unfold, set_naturals), (map_wpull, pickWP_assms_tac))) =>
+      CONJ_WRAP' (fn (rec_Suc, ((unfold, set_maps), (map_wpull, pickWP_assms_tac))) =>
         EVERY' [rtac impI, dtac @{thm set_mp[OF equalityD1[OF thePull_def]]},
           REPEAT_DETERM o eresolve_tac [CollectE, @{thm prod_caseE}],
           hyp_subst_tac, rtac rev_mp, rtac (map_wpull RS @{thm pickWP(1)}),
@@ -1442,16 +1442,16 @@
           rtac impI, REPEAT_DETERM o eresolve_tac [CollectE, conjE],
           rtac ord_eq_le_trans, rtac rec_Suc,
           rtac @{thm Un_least},
-          SELECT_GOAL (unfold_thms_tac ctxt [unfold, nth set_naturals (j - 1),
+          SELECT_GOAL (unfold_thms_tac ctxt [unfold, nth set_maps (j - 1),
             @{thm prod.cases}]),
           etac ord_eq_le_trans_trans_fun_cong_image_id_id_apply,
-          CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least})) (fn (i, set_natural) =>
+          CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least})) (fn (i, set_map) =>
             EVERY' [rtac @{thm UN_least},
-              SELECT_GOAL (unfold_thms_tac ctxt [unfold, set_natural, @{thm prod.cases}]),
+              SELECT_GOAL (unfold_thms_tac ctxt [unfold, set_map, @{thm prod.cases}]),
               etac imageE, hyp_subst_tac, REPEAT_DETERM o etac allE,
               dtac (mk_conjunctN n i), etac mp, etac set_mp, atac])
-          (ks ~~ rev (drop m set_naturals))])
-      (rec_Sucs ~~ ((unfolds ~~ set_naturalss) ~~ (map_wpulls ~~ pickWP_assms_tacs)))] 1
+          (ks ~~ rev (drop m set_maps))])
+      (rec_Sucs ~~ ((unfolds ~~ set_mapss) ~~ (map_wpulls ~~ pickWP_assms_tacs)))] 1
   end;
 
 fun mk_wpull_tac m coalg_thePull mor_thePull_fst mor_thePull_snd mor_thePull_pick
@@ -1500,26 +1500,26 @@
     FIRST' [rtac TrueI, rtac refl, etac (refl RSN (2, mp)), dresolve_tac wits THEN' etac FalseE])
 
 fun mk_dtor_srel_tac in_Jsrels i in_srel map_comp map_cong0 dtor_map dtor_sets dtor_inject dtor_ctor
-  set_naturals dtor_set_incls dtor_set_set_inclss =
+  set_maps dtor_set_incls dtor_set_set_inclss =
   let
     val m = length dtor_set_incls;
     val n = length dtor_set_set_inclss;
-    val (passive_set_naturals, active_set_naturals) = chop m set_naturals;
+    val (passive_set_maps, active_set_maps) = chop m set_maps;
     val in_Jsrel = nth in_Jsrels (i - 1);
     val if_tac =
       EVERY' [dtac (in_Jsrel RS iffD1), REPEAT_DETERM o eresolve_tac [exE, conjE, CollectE],
         rtac (in_srel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
-        EVERY' (map2 (fn set_natural => fn set_incl =>
-          EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac set_natural,
+        EVERY' (map2 (fn set_map => fn set_incl =>
+          EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac set_map,
             rtac ord_eq_le_trans, rtac trans_fun_cong_image_id_id_apply,
             etac (set_incl RS @{thm subset_trans})])
-        passive_set_naturals dtor_set_incls),
-        CONJ_WRAP' (fn (in_Jsrel, (set_natural, dtor_set_set_incls)) =>
-          EVERY' [rtac ord_eq_le_trans, rtac set_natural, rtac @{thm image_subsetI},
+        passive_set_maps dtor_set_incls),
+        CONJ_WRAP' (fn (in_Jsrel, (set_map, dtor_set_set_incls)) =>
+          EVERY' [rtac ord_eq_le_trans, rtac set_map, rtac @{thm image_subsetI},
             rtac (in_Jsrel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
             CONJ_WRAP' (fn thm => etac (thm RS @{thm subset_trans}) THEN' atac) dtor_set_set_incls,
             rtac conjI, rtac refl, rtac refl])
-        (in_Jsrels ~~ (active_set_naturals ~~ dtor_set_set_inclss)),
+        (in_Jsrels ~~ (active_set_maps ~~ dtor_set_set_inclss)),
         CONJ_WRAP' (fn conv =>
           EVERY' [rtac trans, rtac map_comp, rtac trans, rtac map_cong0,
           REPEAT_DETERM_N m o rtac @{thm fun_cong[OF o_id]},
@@ -1529,21 +1529,21 @@
     val only_if_tac =
       EVERY' [dtac (in_srel RS iffD1), REPEAT_DETERM o eresolve_tac [exE, conjE, CollectE],
         rtac (in_Jsrel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
-        CONJ_WRAP' (fn (dtor_set, passive_set_natural) =>
+        CONJ_WRAP' (fn (dtor_set, passive_set_map) =>
           EVERY' [rtac ord_eq_le_trans, rtac dtor_set, rtac @{thm Un_least},
-            rtac ord_eq_le_trans, rtac box_equals, rtac passive_set_natural,
+            rtac ord_eq_le_trans, rtac box_equals, rtac passive_set_map,
             rtac (dtor_ctor RS sym RS arg_cong), rtac trans_fun_cong_image_id_id_apply, atac,
             CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least}))
-              (fn (active_set_natural, in_Jsrel) => EVERY' [rtac ord_eq_le_trans,
+              (fn (active_set_map, in_Jsrel) => EVERY' [rtac ord_eq_le_trans,
                 rtac @{thm UN_cong[OF _ refl]}, rtac @{thm box_equals[OF _ _ refl]},
-                rtac active_set_natural, rtac (dtor_ctor RS sym RS arg_cong), rtac @{thm UN_least},
+                rtac active_set_map, rtac (dtor_ctor RS sym RS arg_cong), rtac @{thm UN_least},
                 dtac set_rev_mp, etac @{thm image_mono}, etac imageE,
                 dtac @{thm ssubst_mem[OF pair_collapse]}, dtac (in_Jsrel RS iffD1),
                 dtac @{thm someI_ex}, REPEAT_DETERM o etac conjE,
                 dtac (Thm.permute_prems 0 1 @{thm ssubst_mem}), atac,
                 hyp_subst_tac, REPEAT_DETERM o eresolve_tac [CollectE, conjE], atac])
-            (rev (active_set_naturals ~~ in_Jsrels))])
-        (dtor_sets ~~ passive_set_naturals),
+            (rev (active_set_maps ~~ in_Jsrels))])
+        (dtor_sets ~~ passive_set_maps),
         rtac conjI,
         REPEAT_DETERM_N 2 o EVERY'[rtac (dtor_inject RS iffD1), rtac trans, rtac dtor_map,
           rtac box_equals, rtac map_comp, rtac (dtor_ctor RS sym RS arg_cong), rtac trans,
--- a/src/HOL/BNF/Tools/bnf_lfp.ML	Wed Apr 24 17:03:43 2013 +0200
+++ b/src/HOL/BNF/Tools/bnf_lfp.ML	Wed Apr 24 17:47:22 2013 +0200
@@ -149,7 +149,7 @@
     val map_id's = map map_id'_of_bnf bnfs;
     val map_wpulls = map map_wpull_of_bnf bnfs;
     val set_bdss = map set_bd_of_bnf bnfs;
-    val set_natural'ss = map set_natural'_of_bnf bnfs;
+    val set_map'ss = map set_map'_of_bnf bnfs;
 
     val timer = time (timer "Extracted terms & thms");
 
@@ -388,7 +388,7 @@
         Goal.prove_sorry lthy [] []
           (fold_rev Logic.all (Bs @ ss @ B's @ s's @ B''s @ s''s @ fs @ gs)
              (Logic.list_implies (prems, concl)))
-          (K (mk_mor_comp_tac mor_def set_natural'ss map_comp_id_thms))
+          (K (mk_mor_comp_tac mor_def set_map'ss map_comp_id_thms))
         |> Thm.close_derivation
       end;
 
@@ -407,7 +407,7 @@
           (fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs @ inv_fs)
             (Logic.list_implies (prems, concl)))
           (K (mk_mor_inv_tac alg_def mor_def
-            set_natural'ss morE_thms map_comp_id_thms map_cong0L_thms))
+            set_map'ss morE_thms map_comp_id_thms map_cong0L_thms))
         |> Thm.close_derivation
       end;
 
@@ -512,7 +512,7 @@
         val copy_str_thm = Goal.prove_sorry lthy [] []
           (fold_rev Logic.all (Bs @ ss @ B's @ inv_fs @ fs)
             (Logic.list_implies (all_prems, alg)))
-          (K (mk_copy_str_tac set_natural'ss alg_def alg_set_thms))
+          (K (mk_copy_str_tac set_map'ss alg_def alg_set_thms))
           |> Thm.close_derivation;
 
         val iso = HOLogic.mk_Trueprop
@@ -520,7 +520,7 @@
         val copy_alg_thm = Goal.prove_sorry lthy [] []
           (fold_rev Logic.all (Bs @ ss @ B's @ inv_fs @ fs)
             (Logic.list_implies (all_prems, iso)))
-          (K (mk_copy_alg_tac set_natural'ss alg_set_thms mor_def iso_alt_thm copy_str_thm))
+          (K (mk_copy_alg_tac set_map'ss alg_set_thms mor_def iso_alt_thm copy_str_thm))
           |> Thm.close_derivation;
 
         val ex = HOLogic.mk_Trueprop
@@ -871,7 +871,7 @@
         Goal.prove_sorry lthy [] []
           (fold_rev Logic.all (iidx :: Bs @ ss @ Asuc_fs) (Logic.list_implies (prems, concl)))
           (K (mk_mor_select_tac mor_def mor_cong_thm mor_comp_thm mor_incl_min_alg_thm alg_def
-            alg_select_thm alg_set_thms set_natural'ss str_init_defs))
+            alg_select_thm alg_set_thms set_map'ss str_init_defs))
         |> Thm.close_derivation
       end;
 
@@ -1318,7 +1318,7 @@
       in
         (Goal.prove_sorry lthy [] []
           (fold_rev Logic.all (phis @ Izs) goal)
-          (K (mk_ctor_induct_tac m set_natural'ss init_induct_thm morE_thms mor_Abs_thm
+          (K (mk_ctor_induct_tac m set_map'ss init_induct_thm morE_thms mor_Abs_thm
             Rep_inverses Abs_inverses Reps))
         |> Thm.close_derivation,
         rev (Term.add_tfrees goal []))
@@ -1509,7 +1509,7 @@
                 Goal.prove_sorry lthy [] [] goal
                   (K (mk_ctor_set_tac set (nth set_nats (i - 1)) (drop m set_nats)))
                 |> Thm.close_derivation)
-              set_natural'ss) ls simp_goalss setss;
+              set_map'ss) ls simp_goalss setss;
           in
             ctor_setss
           end;
@@ -1528,13 +1528,13 @@
           map (map (Term.subst_atomic_types (passiveAs ~~ passiveBs))) setss_by_bnf;
         val setss_by_range' = transpose setss_by_bnf';
 
-        val set_natural_thmss =
+        val set_map_thmss =
           let
-            fun mk_set_natural f map z set set' =
+            fun mk_set_map f map z set set' =
               HOLogic.mk_eq (mk_image f $ (set $ z), set' $ (map $ z));
 
             fun mk_cphi f map z set set' = certify lthy
-              (Term.absfree (dest_Free z) (mk_set_natural f map z set set'));
+              (Term.absfree (dest_Free z) (mk_set_map f map z set set'));
 
             val csetss = map (map (certify lthy)) setss_by_range';
 
@@ -1547,10 +1547,10 @@
             val goals =
               map3 (fn f => fn sets => fn sets' =>
                 HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
-                  (map4 (mk_set_natural f) fs_maps Izs sets sets')))
+                  (map4 (mk_set_map f) fs_maps Izs sets sets')))
                   fs setss_by_range setss_by_range';
 
-            fun mk_tac induct = mk_set_nat_tac m (rtac induct) set_natural'ss ctor_map_thms;
+            fun mk_tac induct = mk_set_nat_tac m (rtac induct) set_map'ss ctor_map_thms;
             val thms =
               map5 (fn goal => fn csets => fn ctor_sets => fn induct => fn i =>
                 singleton (Proof_Context.export names_lthy lthy)
@@ -1693,7 +1693,7 @@
         val map_comp_tacs =
           map2 (K oo mk_map_comp_tac map_comp's ctor_map_thms) ctor_map_unique_thms ks;
         val map_cong0_tacs = map (mk_map_cong0_tac m) map_cong0_thms;
-        val set_nat_tacss = map (map (K o mk_set_natural_tac)) (transpose set_natural_thmss);
+        val set_nat_tacss = map (map (K o mk_set_map_tac)) (transpose set_map_thmss);
         val bd_co_tacs = replicate n (K (mk_bd_card_order_tac bd_card_orders));
         val bd_cinf_tacs = replicate n (K (rtac (bd_Cinfinite RS conjunct1) 1));
         val set_bd_tacss = map (map (fn thm => K (rtac thm 1))) (transpose set_bd_thmss);
@@ -1780,13 +1780,13 @@
           in
             map12 (fn i => fn goal => fn in_srel => fn map_comp => fn map_cong0 =>
               fn ctor_map => fn ctor_sets => fn ctor_inject => fn ctor_dtor =>
-              fn set_naturals => fn ctor_set_incls => fn ctor_set_set_inclss =>
+              fn set_maps => fn ctor_set_incls => fn ctor_set_set_inclss =>
               Goal.prove_sorry lthy [] [] goal
                (K (mk_ctor_srel_tac in_Isrels i in_srel map_comp map_cong0 ctor_map ctor_sets
-                 ctor_inject ctor_dtor set_naturals ctor_set_incls ctor_set_set_inclss))
+                 ctor_inject ctor_dtor set_maps ctor_set_incls ctor_set_set_inclss))
               |> Thm.close_derivation)
             ks goals in_srels map_comp's map_cong0s folded_ctor_map_thms folded_ctor_set_thmss'
-              ctor_inject_thms ctor_dtor_thms set_natural'ss ctor_set_incl_thmss
+              ctor_inject_thms ctor_dtor_thms set_map'ss ctor_set_incl_thmss
               ctor_set_set_incl_thmsss
           end;
 
--- a/src/HOL/BNF/Tools/bnf_lfp_tactics.ML	Wed Apr 24 17:03:43 2013 +0200
+++ b/src/HOL/BNF/Tools/bnf_lfp_tactics.ML	Wed Apr 24 17:47:22 2013 +0200
@@ -71,7 +71,7 @@
     {prems: 'a, context: Proof.context} -> tactic
   val mk_set_nat_tac: int -> (int -> tactic) -> thm list list -> thm list -> cterm list ->
     thm list -> int -> {prems: 'a, context: Proof.context} -> tactic
-  val mk_set_natural_tac: thm -> tactic
+  val mk_set_map_tac: thm -> tactic
   val mk_set_tac: thm -> tactic
   val mk_wit_tac: int -> thm list -> thm list -> tactic
   val mk_wpull_tac: thm -> tactic
@@ -119,39 +119,39 @@
   CONJ_WRAP' (fn thm =>
    (EVERY' [rtac ballI, rtac trans, rtac id_apply, stac thm, rtac refl])) map_id's) 1;
 
-fun mk_mor_comp_tac mor_def set_natural's map_comp_ids =
+fun mk_mor_comp_tac mor_def set_map's map_comp_ids =
   let
     val fbetw_tac = EVERY' [rtac ballI, stac o_apply, etac bspec, etac bspec, atac];
-    fun mor_tac (set_natural', map_comp_id) =
+    fun mor_tac (set_map', map_comp_id) =
       EVERY' [rtac ballI, stac o_apply, rtac trans,
         rtac trans, dtac @{thm rev_bspec}, atac, etac arg_cong,
          REPEAT o eresolve_tac [CollectE, conjE], etac bspec, rtac CollectI] THEN'
       CONJ_WRAP' (fn thm =>
         FIRST' [rtac subset_UNIV,
           (EVERY' [rtac ord_eq_le_trans, rtac thm, rtac @{thm image_subsetI},
-            etac bspec, etac set_mp, atac])]) set_natural' THEN'
+            etac bspec, etac set_mp, atac])]) set_map' THEN'
       rtac (map_comp_id RS arg_cong);
   in
     (dtac (mor_def RS subst) THEN' dtac (mor_def RS subst) THEN' stac mor_def THEN'
     REPEAT o etac conjE THEN'
     rtac conjI THEN'
-    CONJ_WRAP' (K fbetw_tac) set_natural's THEN'
-    CONJ_WRAP' mor_tac (set_natural's ~~ map_comp_ids)) 1
+    CONJ_WRAP' (K fbetw_tac) set_map's THEN'
+    CONJ_WRAP' mor_tac (set_map's ~~ map_comp_ids)) 1
   end;
 
-fun mk_mor_inv_tac alg_def mor_def set_natural's morEs map_comp_ids map_cong0Ls =
+fun mk_mor_inv_tac alg_def mor_def set_map's morEs map_comp_ids map_cong0Ls =
   let
     val fbetw_tac = EVERY' [rtac ballI, etac set_mp, etac imageI];
-    fun Collect_tac set_natural' =
+    fun Collect_tac set_map' =
       CONJ_WRAP' (fn thm =>
         FIRST' [rtac subset_UNIV,
           (EVERY' [rtac ord_eq_le_trans, rtac thm, rtac subset_trans,
-            etac @{thm image_mono}, atac])]) set_natural';
-    fun mor_tac (set_natural', ((morE, map_comp_id), map_cong0L)) =
+            etac @{thm image_mono}, atac])]) set_map';
+    fun mor_tac (set_map', ((morE, map_comp_id), map_cong0L)) =
       EVERY' [rtac ballI, ftac @{thm rev_bspec}, atac,
          REPEAT o eresolve_tac [CollectE, conjE], rtac sym, rtac trans, rtac sym,
-         etac @{thm inverE}, etac bspec, rtac CollectI, Collect_tac set_natural',
-         rtac trans, etac (morE RS arg_cong), rtac CollectI, Collect_tac set_natural',
+         etac @{thm inverE}, etac bspec, rtac CollectI, Collect_tac set_map',
+         rtac trans, etac (morE RS arg_cong), rtac CollectI, Collect_tac set_map',
          rtac trans, rtac (map_comp_id RS arg_cong), rtac (map_cong0L RS arg_cong),
          REPEAT_DETERM_N (length morEs) o
            (EVERY' [rtac subst, rtac @{thm inver_pointfree}, etac @{thm inver_mono}, atac])];
@@ -161,8 +161,8 @@
     dtac (alg_def RS iffD1) THEN'
     REPEAT o etac conjE THEN'
     rtac conjI THEN'
-    CONJ_WRAP' (K fbetw_tac) set_natural's THEN'
-    CONJ_WRAP' mor_tac (set_natural's ~~ (morEs ~~ map_comp_ids ~~ map_cong0Ls))) 1
+    CONJ_WRAP' (K fbetw_tac) set_map's THEN'
+    CONJ_WRAP' mor_tac (set_map's ~~ (morEs ~~ map_comp_ids ~~ map_cong0Ls))) 1
   end;
 
 fun mk_mor_str_tac ks mor_def =
@@ -208,7 +208,7 @@
     (rtac iffI THEN' if_tac THEN' only_if_tac) 1
   end;
 
-fun mk_copy_str_tac set_natural's alg_def alg_sets =
+fun mk_copy_str_tac set_map's alg_def alg_sets =
   let
     val n = length alg_sets;
     val bij_betw_inv_tac =
@@ -222,13 +222,13 @@
         EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac set_mp,
           rtac equalityD1, etac @{thm bij_betw_imageE}, rtac imageI, etac thm,
           REPEAT_DETERM o rtac subset_UNIV, REPEAT_DETERM_N n o (set_tac thms)])
-      (set_natural's ~~ alg_sets);
+      (set_map's ~~ alg_sets);
   in
     (rtac rev_mp THEN' DETERM o bij_betw_inv_tac THEN' rtac impI THEN'
     stac alg_def THEN' copy_str_tac) 1
   end;
 
-fun mk_copy_alg_tac set_natural's alg_sets mor_def iso_alt copy_str =
+fun mk_copy_alg_tac set_map's alg_sets mor_def iso_alt copy_str =
   let
     val n = length alg_sets;
     val fbetw_tac = CONJ_WRAP' (K (etac @{thm bij_betwE})) alg_sets;
@@ -240,7 +240,7 @@
       CONJ_WRAP' (fn (thms, thm) =>
         EVERY' [rtac ballI, etac CollectE, etac @{thm inverE}, etac thm,
           REPEAT_DETERM o rtac subset_UNIV, REPEAT_DETERM_N n o (set_tac thms)])
-      (set_natural's ~~ alg_sets);
+      (set_map's ~~ alg_sets);
   in
     (rtac (iso_alt RS @{thm ssubst[of _ _ "%x. x"]}) THEN'
     etac copy_str THEN' REPEAT_DETERM o atac THEN'
@@ -389,24 +389,24 @@
   unfold_thms_tac ctxt (Abs_inverse :: fst_snd_convs) THEN atac 1;
 
 fun mk_mor_select_tac mor_def mor_cong mor_comp mor_incl_min_alg alg_def alg_select
-    alg_sets set_natural's str_init_defs =
+    alg_sets set_map's str_init_defs =
   let
     val n = length alg_sets;
     val fbetw_tac =
       CONJ_WRAP' (K (EVERY' [rtac ballI, etac @{thm rev_bspec}, etac CollectE, atac])) alg_sets;
     val mor_tac =
       CONJ_WRAP' (fn thm => EVERY' [rtac ballI, rtac thm]) str_init_defs;
-    fun alg_epi_tac ((alg_set, str_init_def), set_natural') =
+    fun alg_epi_tac ((alg_set, str_init_def), set_map') =
       EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac CollectI,
         rtac ballI, ftac (alg_select RS bspec), stac str_init_def, etac alg_set,
         REPEAT_DETERM o FIRST' [rtac subset_UNIV,
-          EVERY' [rtac ord_eq_le_trans, resolve_tac set_natural', rtac subset_trans,
+          EVERY' [rtac ord_eq_le_trans, resolve_tac set_map', rtac subset_trans,
             etac @{thm image_mono}, rtac @{thm image_Collect_subsetI}, etac bspec, atac]]];
   in
     (rtac mor_cong THEN' REPEAT_DETERM_N n o (rtac sym THEN' rtac @{thm o_id}) THEN'
     rtac (Thm.permute_prems 0 1 mor_comp) THEN' etac (Thm.permute_prems 0 1 mor_comp) THEN'
     stac mor_def THEN' rtac conjI THEN' fbetw_tac THEN' mor_tac THEN' rtac mor_incl_min_alg THEN'
-    stac alg_def THEN' CONJ_WRAP' alg_epi_tac ((alg_sets ~~ str_init_defs) ~~ set_natural's)) 1
+    stac alg_def THEN' CONJ_WRAP' alg_epi_tac ((alg_sets ~~ str_init_defs) ~~ set_map's)) 1
   end;
 
 fun mk_init_ex_mor_tac Abs_inverse copy_alg_ex alg_min_alg card_of_min_algs
@@ -532,21 +532,21 @@
     (rec_defs @ map (fn thm => thm RS @{thm convol_expand_snd'}) fst_recs) THEN
   etac fold_unique_mor 1;
 
-fun mk_ctor_induct_tac m set_natural'ss init_induct morEs mor_Abs Rep_invs Abs_invs Reps =
+fun mk_ctor_induct_tac m set_map'ss init_induct morEs mor_Abs Rep_invs Abs_invs Reps =
   let
-    val n = length set_natural'ss;
+    val n = length set_map'ss;
     val ks = 1 upto n;
 
-    fun mk_IH_tac Rep_inv Abs_inv set_natural' =
+    fun mk_IH_tac Rep_inv Abs_inv set_map' =
       DETERM o EVERY' [dtac meta_mp, rtac (Rep_inv RS arg_cong RS subst), etac bspec,
-        dtac set_rev_mp, rtac equalityD1, rtac set_natural', etac imageE,
+        dtac set_rev_mp, rtac equalityD1, rtac set_map', etac imageE,
         hyp_subst_tac, rtac (Abs_inv RS ssubst), etac set_mp, atac, atac];
 
-    fun mk_closed_tac (k, (morE, set_natural's)) =
+    fun mk_closed_tac (k, (morE, set_map's)) =
       EVERY' [select_prem_tac n (dtac asm_rl) k, rtac ballI, rtac impI,
         rtac (mor_Abs RS morE RS arg_cong RS ssubst), atac,
         REPEAT_DETERM o eresolve_tac [CollectE, conjE], dtac @{thm meta_spec},
-        EVERY' (map3 mk_IH_tac Rep_invs Abs_invs (drop m set_natural's)), atac];
+        EVERY' (map3 mk_IH_tac Rep_invs Abs_invs (drop m set_map's)), atac];
 
     fun mk_induct_tac (Rep, Rep_inv) =
       EVERY' [rtac (Rep_inv RS arg_cong RS subst), etac (Rep RSN (2, bspec))];
@@ -554,7 +554,7 @@
     (rtac mp THEN' rtac impI THEN'
     DETERM o CONJ_WRAP_GEN' (etac conjE THEN' rtac conjI) mk_induct_tac (Reps ~~ Rep_invs) THEN'
     rtac init_induct THEN'
-    DETERM o CONJ_WRAP' mk_closed_tac (ks ~~ (morEs ~~ set_natural'ss))) 1
+    DETERM o CONJ_WRAP' mk_closed_tac (ks ~~ (morEs ~~ set_map'ss))) 1
   end;
 
 fun mk_ctor_induct2_tac cTs cts ctor_induct weak_ctor_inducts {context = ctxt, prems = _} =
@@ -599,19 +599,19 @@
 fun mk_set_tac foldx = EVERY' [rtac ext, rtac trans, rtac o_apply,
   rtac trans, rtac foldx, rtac sym, rtac o_apply] 1;
 
-fun mk_ctor_set_tac set set_natural' set_natural's =
+fun mk_ctor_set_tac set set_map' set_map's =
   let
-    val n = length set_natural's;
+    val n = length set_map's;
     fun mk_UN thm = rtac (thm RS @{thm arg_cong[of _ _ Union]} RS trans) THEN'
       rtac @{thm Union_image_eq};
   in
     EVERY' [rtac (set RS @{thm pointfreeE} RS trans), rtac @{thm Un_cong},
-      rtac (trans OF [set_natural', trans_fun_cong_image_id_id_apply]),
+      rtac (trans OF [set_map', trans_fun_cong_image_id_id_apply]),
       REPEAT_DETERM_N (n - 1) o rtac @{thm Un_cong},
-      EVERY' (map mk_UN set_natural's)] 1
+      EVERY' (map mk_UN set_map's)] 1
   end;
 
-fun mk_set_nat_tac m induct_tac set_natural'ss
+fun mk_set_nat_tac m induct_tac set_map'ss
     ctor_maps csets ctor_sets i {context = ctxt, prems = _} =
   let
     val n = length ctor_maps;
@@ -628,7 +628,7 @@
         REPEAT_DETERM_N (n - 1) o EVERY' (map rtac (trans :: @{thms image_Un Un_cong})),
         EVERY' (map useIH (drop m set_nats))];
   in
-    (induct_tac THEN' EVERY' (map4 mk_set_nat csets ctor_maps ctor_sets set_natural'ss)) 1
+    (induct_tac THEN' EVERY' (map4 mk_set_nat csets ctor_maps ctor_sets set_map'ss)) 1
   end;
 
 fun mk_set_bd_tac m induct_tac bd_Cinfinite set_bdss ctor_sets i {context = ctxt, prems = _} =
@@ -747,7 +747,7 @@
     (rtac sym THEN' rtac unique' THEN' EVERY' (map2 mk_comp map_comps' ctor_maps')) 1
   end;
 
-fun mk_set_natural_tac set_nat =
+fun mk_set_map_tac set_nat =
   EVERY' (map rtac [ext, trans, o_apply, sym, trans, o_apply, set_nat]) 1;
 
 fun mk_in_bd_tac sum_Card_order sucbd_Cnotzero incl card_of_min_alg =
@@ -776,31 +776,31 @@
             REPEAT_DETERM_N n o etac UnE]))))] 1);
 
 fun mk_ctor_srel_tac in_Isrels i in_srel map_comp map_cong0 ctor_map ctor_sets ctor_inject
-  ctor_dtor set_naturals ctor_set_incls ctor_set_set_inclss =
+  ctor_dtor set_maps ctor_set_incls ctor_set_set_inclss =
   let
     val m = length ctor_set_incls;
     val n = length ctor_set_set_inclss;
 
-    val (passive_set_naturals, active_set_naturals) = chop m set_naturals;
+    val (passive_set_maps, active_set_maps) = chop m set_maps;
     val in_Isrel = nth in_Isrels (i - 1);
     val le_arg_cong_ctor_dtor = ctor_dtor RS arg_cong RS ord_eq_le_trans;
     val eq_arg_cong_ctor_dtor = ctor_dtor RS arg_cong RS trans;
     val if_tac =
       EVERY' [dtac (in_Isrel RS iffD1), REPEAT_DETERM o eresolve_tac [exE, conjE, CollectE],
         rtac (in_srel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
-        EVERY' (map2 (fn set_natural => fn ctor_set_incl =>
-          EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac set_natural,
+        EVERY' (map2 (fn set_map => fn ctor_set_incl =>
+          EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac set_map,
             rtac ord_eq_le_trans, rtac trans_fun_cong_image_id_id_apply,
             rtac (ctor_set_incl RS subset_trans), etac le_arg_cong_ctor_dtor])
-        passive_set_naturals ctor_set_incls),
-        CONJ_WRAP' (fn (in_Isrel, (set_natural, ctor_set_set_incls)) =>
-          EVERY' [rtac ord_eq_le_trans, rtac set_natural, rtac @{thm image_subsetI},
+        passive_set_maps ctor_set_incls),
+        CONJ_WRAP' (fn (in_Isrel, (set_map, ctor_set_set_incls)) =>
+          EVERY' [rtac ord_eq_le_trans, rtac set_map, rtac @{thm image_subsetI},
             rtac (in_Isrel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
             CONJ_WRAP' (fn thm =>
               EVERY' (map etac [thm RS subset_trans, le_arg_cong_ctor_dtor]))
             ctor_set_set_incls,
             rtac conjI, rtac refl, rtac refl])
-        (in_Isrels ~~ (active_set_naturals ~~ ctor_set_set_inclss)),
+        (in_Isrels ~~ (active_set_maps ~~ ctor_set_set_inclss)),
         CONJ_WRAP' (fn conv =>
           EVERY' [rtac trans, rtac map_comp, rtac trans, rtac map_cong0,
           REPEAT_DETERM_N m o rtac @{thm fun_cong[OF o_id]},
@@ -811,20 +811,20 @@
     val only_if_tac =
       EVERY' [dtac (in_srel RS iffD1), REPEAT_DETERM o eresolve_tac [exE, conjE, CollectE],
         rtac (in_Isrel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
-        CONJ_WRAP' (fn (ctor_set, passive_set_natural) =>
+        CONJ_WRAP' (fn (ctor_set, passive_set_map) =>
           EVERY' [rtac ord_eq_le_trans, rtac ctor_set, rtac @{thm Un_least},
             rtac ord_eq_le_trans, rtac @{thm box_equals[OF _ refl]},
-            rtac passive_set_natural, rtac trans_fun_cong_image_id_id_apply, atac,
+            rtac passive_set_map, rtac trans_fun_cong_image_id_id_apply, atac,
             CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least}))
-              (fn (active_set_natural, in_Isrel) => EVERY' [rtac ord_eq_le_trans,
-                rtac @{thm UN_cong[OF _ refl]}, rtac active_set_natural, rtac @{thm UN_least},
+              (fn (active_set_map, in_Isrel) => EVERY' [rtac ord_eq_le_trans,
+                rtac @{thm UN_cong[OF _ refl]}, rtac active_set_map, rtac @{thm UN_least},
                 dtac set_rev_mp, etac @{thm image_mono}, etac imageE,
                 dtac @{thm ssubst_mem[OF pair_collapse]}, dtac (in_Isrel RS iffD1),
                 dtac @{thm someI_ex}, REPEAT_DETERM o etac conjE,
                 dtac (Thm.permute_prems 0 1 @{thm ssubst_mem}), atac,
                 hyp_subst_tac, REPEAT_DETERM o eresolve_tac [CollectE, conjE], atac])
-            (rev (active_set_naturals ~~ in_Isrels))])
-        (ctor_sets ~~ passive_set_naturals),
+            (rev (active_set_maps ~~ in_Isrels))])
+        (ctor_sets ~~ passive_set_maps),
         rtac conjI,
         REPEAT_DETERM_N 2 o EVERY' [rtac trans, rtac ctor_map, rtac (ctor_inject RS iffD2),
           rtac trans, rtac map_comp, rtac trans, rtac map_cong0,