(*  Title:      HOL/Codatatype/Tools/bnf_comp.ML

     Author:     Dmitriy Traytel, TU Muenchen

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   2012

 Composition of bounded natural functors.

 *)

 signature BNF_COMP =

 sig

   type unfold_thms

   val empty_unfold: unfold_thms

   val map_unfolds_of: unfold_thms -> thm list

   val set_unfoldss_of: unfold_thms -> thm list list

   val rel_unfolds_of: unfold_thms -> thm list

   val pred_unfolds_of: unfold_thms -> thm list

   val bnf_of_typ: BNF_Def.const_policy -> (binding -> binding) ->

     ((string * sort) list list -> (string * sort) list) -> typ -> unfold_thms * Proof.context ->

     (BNF_Def.BNF * (typ list * typ list)) * (unfold_thms * Proof.context)

   val default_comp_sort: (string * sort) list list -> (string * sort) list

   val normalize_bnfs: (int -> binding -> binding) -> ''a list list -> ''a list ->

     (''a list list -> ''a list) -> BNF_Def.BNF list -> unfold_thms -> Proof.context ->

     (int list list * ''a list) * (BNF_Def.BNF list * (unfold_thms * Proof.context))

   val seal_bnf: unfold_thms -> binding -> typ list -> BNF_Def.BNF -> Proof.context ->

     (BNF_Def.BNF * typ list) * local_theory

 end;

 structure BNF_Comp : BNF_COMP =

 struct

 open BNF_Def

 open BNF_Util

 open BNF_Tactics

 open BNF_Comp_Tactics

 type unfold_thms = {

   map_unfolds: thm list,

   set_unfoldss: thm list list,

   rel_unfolds: thm list,

   pred_unfolds: thm list

 };

 fun add_to_thms thms NONE = thms

   | add_to_thms thms (SOME new) = if Thm.is_reflexive new then thms else insert Thm.eq_thm new thms;

 fun adds_to_thms thms NONE = thms

   | adds_to_thms thms (SOME news) = insert (eq_set Thm.eq_thm) (filter_refl news) thms;

 fun mk_unfold_thms maps setss rels preds =

   {map_unfolds = maps, set_unfoldss = setss, rel_unfolds = rels, pred_unfolds = preds};

 val empty_unfold = mk_unfold_thms [] [] [] [];

 fun add_to_unfold_opt map_opt sets_opt rel_opt pred_opt

   {map_unfolds = maps, set_unfoldss = setss, rel_unfolds = rels, pred_unfolds = preds} = {

     map_unfolds = add_to_thms maps map_opt,

     set_unfoldss = adds_to_thms setss sets_opt,

     rel_unfolds = add_to_thms rels rel_opt,

     pred_unfolds = add_to_thms preds pred_opt};

 fun add_to_unfold map sets rel pred =

   add_to_unfold_opt (SOME map) (SOME sets) (SOME rel) (SOME pred);

 val map_unfolds_of = #map_unfolds;

 val set_unfoldss_of = #set_unfoldss;

 val rel_unfolds_of = #rel_unfolds;

 val pred_unfolds_of = #pred_unfolds;

 val bdTN = "bdT";

 fun mk_killN n = "_kill" ^ string_of_int n;

 fun mk_liftN n = "_lift" ^ string_of_int n;

 fun mk_permuteN src dest =

   "_permute_" ^ implode (map string_of_int src) ^ "_" ^ implode (map string_of_int dest);

 val no_thm = refl;

 val Collect_split_box_equals = box_equals RS @{thm Collect_split_cong};

 val abs_pred_sym = sym RS @{thm abs_pred_def};

 val abs_pred_sym_pred_abs = abs_pred_sym RS @{thm pred_def_abs};

 (*copied from Envir.expand_term_free*)

 fun expand_term_const defs =

   let

     val eqs = map ((fn ((x, U), u) => (x, (U, u))) o apfst dest_Const) defs;

     val get = fn Const (x, _) => AList.lookup (op =) eqs x | _ => NONE;

   in Envir.expand_term get end;

 fun clean_compose_bnf const_policy qualify b outer inners (unfold, lthy) =

   let

     val olive = live_of_bnf outer;

     val onwits = nwits_of_bnf outer;

     val odead = dead_of_bnf outer;

     val inner = hd inners;

     val ilive = live_of_bnf inner;

     val ideads = map dead_of_bnf inners;

     val inwitss = map nwits_of_bnf inners;

     (* TODO: check olive = length inners > 0,

                    forall inner from inners. ilive = live,

                    forall inner from inners. idead = dead  *)

     val (oDs, lthy1) = apfst (map TFree)

       (Variable.invent_types (replicate odead HOLogic.typeS) lthy);

     val (Dss, lthy2) = apfst (map (map TFree))

         (fold_map Variable.invent_types (map (fn n => replicate n HOLogic.typeS) ideads) lthy1);

     val (Ass, lthy3) = apfst (replicate ilive o map TFree)

       (Variable.invent_types (replicate ilive HOLogic.typeS) lthy2);

     val As = if ilive > 0 then hd Ass else [];

     val Ass_repl = replicate olive As;

     val (Bs, _(*lthy4*)) = apfst (map TFree)

       (Variable.invent_types (replicate ilive HOLogic.typeS) lthy3);

     val Bss_repl = replicate olive Bs;

     val (((fs', Asets), xs), _(*names_lthy*)) = lthy

       |> apfst snd o mk_Frees' "f" (map2 (curry (op -->)) As Bs)

       ||>> mk_Frees "A" (map (HOLogic.mk_setT) As)

       ||>> mk_Frees "x" As;

     val CAs = map3 mk_T_of_bnf Dss Ass_repl inners;

     val CCA = mk_T_of_bnf oDs CAs outer;

     val CBs = map3 mk_T_of_bnf Dss Bss_repl inners;

     val outer_sets = mk_sets_of_bnf (replicate olive oDs) (replicate olive CAs) outer;

     val inner_setss = map3 mk_sets_of_bnf (map (replicate ilive) Dss) (replicate olive Ass) inners;

     val inner_bds = map3 mk_bd_of_bnf Dss Ass_repl inners;

     val outer_bd = mk_bd_of_bnf oDs CAs outer;

     (*%f1 ... fn. outer.map (inner_1.map f1 ... fn) ... (inner_m.map f1 ... fn)*)

     val mapx = fold_rev Term.abs fs'

       (Term.list_comb (mk_map_of_bnf oDs CAs CBs outer,

         map2 (fn Ds => (fn f => Term.list_comb (f, map Bound ((ilive - 1) downto 0))) o

           mk_map_of_bnf Ds As Bs) Dss inners));

     (*Union o collect {outer.set_1 ... outer.set_m} o outer.map inner_1.set_i ... inner_m.set_i*)

     (*Union o collect {image inner_1.set_i o outer.set_1 ... image inner_m.set_i o outer.set_m}*)

     fun mk_set i =

       let

         val (setTs, T) = `(replicate olive o HOLogic.mk_setT) (nth As i);

         val outer_set = mk_collect

           (mk_sets_of_bnf (replicate olive oDs) (replicate olive setTs) outer)

           (mk_T_of_bnf oDs setTs outer --> HOLogic.mk_setT T);

         val inner_sets = map (fn sets => nth sets i) inner_setss;

         val outer_map = mk_map_of_bnf oDs CAs setTs outer;

         val map_inner_sets = Term.list_comb (outer_map, inner_sets);

         val collect_image = mk_collect

           (map2 (fn f => fn set => HOLogic.mk_comp (mk_image f, set)) inner_sets outer_sets)

           (CCA --> HOLogic.mk_setT T);

       in

         (Library.foldl1 HOLogic.mk_comp [mk_Union T, outer_set, map_inner_sets],

         HOLogic.mk_comp (mk_Union T, collect_image))

       end;

     val (sets, sets_alt) = map_split mk_set (0 upto ilive - 1);

     (*(inner_1.bd +c ... +c inner_m.bd) *c outer.bd*)

     val bd = Term.absdummy CCA (mk_cprod (Library.foldr1 (uncurry mk_csum) inner_bds) outer_bd);

     fun map_id_tac {context = ctxt, ...} =

       let

         (*order the theorems by reverse size to prevent bad interaction with nonconfluent rewrite

           rules*)

         val thms = (map map_id_of_bnf inners

           |> map (`(Term.size_of_term o Thm.prop_of))

           |> sort (rev_order o int_ord o pairself fst)

           |> map snd) @ [map_id_of_bnf outer];

       in

         (EVERY' (map (fn thm => subst_tac ctxt [thm]) thms) THEN' rtac refl) 1

       end;

     fun map_comp_tac _ =

       mk_comp_map_comp_tac (map_comp_of_bnf outer) (map_cong_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_cong_of_bnf outer)

         (collect_set_natural_of_bnf outer)

         (map ((fn thms => nth thms i) o set_natural_of_bnf) inners);

     val set_natural_tacs = map mk_single_set_natural_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);

     fun bd_cinfinite_tac _ =

       mk_comp_bd_cinfinite_tac (bd_cinfinite_of_bnf inner) (bd_cinfinite_of_bnf outer);

     val set_alt_thms =

       if ! quick_and_dirty then

         replicate ilive no_thm

       else

         map (fn goal =>

           Skip_Proof.prove lthy [] [] goal

             (fn {context, ...} => (mk_comp_set_alt_tac context (collect_set_natural_of_bnf outer)))

           |> Thm.close_derivation)

         (map2 (curry (HOLogic.mk_Trueprop o HOLogic.mk_eq)) sets sets_alt);

     fun map_cong_tac _ =

       mk_comp_map_cong_tac set_alt_thms (map_cong_of_bnf outer) (map map_cong_of_bnf inners);

     val set_bd_tacs =

       if ! quick_and_dirty then

         replicate (length set_alt_thms) (K all_tac)

       else

         let

           val outer_set_bds = set_bd_of_bnf outer;

           val inner_set_bdss = map set_bd_of_bnf inners;

           val inner_bd_Card_orders = map bd_Card_order_of_bnf inners;

           fun single_set_bd_thm i j =

             @{thm comp_single_set_bd} OF [nth inner_bd_Card_orders j, nth (nth inner_set_bdss j) i,

               nth outer_set_bds j]

           val single_set_bd_thmss =

             map ((fn f => map f (0 upto olive - 1)) o single_set_bd_thm) (0 upto ilive - 1);

         in

           map2 (fn set_alt => fn single_set_bds => fn {context, ...} =>

             mk_comp_set_bd_tac context set_alt single_set_bds)

           set_alt_thms single_set_bd_thmss

         end;

     val in_alt_thm =

       let

         val inx = mk_in Asets sets CCA;

         val in_alt = mk_in (map2 (mk_in Asets) inner_setss CAs) outer_sets CCA;

         val goal = fold_rev Logic.all Asets (mk_Trueprop_eq (inx, in_alt));

       in

         Skip_Proof.prove lthy [] [] goal

           (fn {context, ...} => mk_comp_in_alt_tac context set_alt_thms)

         |> Thm.close_derivation

       end;

     fun in_bd_tac _ =

       mk_comp_in_bd_tac in_alt_thm (map in_bd_of_bnf inners) (in_bd_of_bnf outer)

         (map bd_Cinfinite_of_bnf inners) (bd_Card_order_of_bnf outer);

     fun map_wpull_tac _ =

       mk_map_wpull_tac in_alt_thm (map map_wpull_of_bnf inners) (map_wpull_of_bnf outer);

     val tacs = [map_id_tac, map_comp_tac, map_cong_tac] @ set_natural_tacs @ [bd_card_order_tac,

       bd_cinfinite_tac] @ set_bd_tacs @ [in_bd_tac, map_wpull_tac];

     val outer_wits = mk_wits_of_bnf (replicate onwits oDs) (replicate onwits CAs) outer;

     val inner_witss = map (map (fn (I, wit) => Term.list_comb (wit, map (nth xs) I)))

       (map3 (fn Ds => fn n => mk_wits_of_bnf (replicate n Ds) (replicate n As))

         Dss inwitss inners);

     val inner_witsss = map (map (nth inner_witss) o fst) outer_wits;

     val wits = (inner_witsss, (map (single o snd) outer_wits))

       |-> map2 (fold (map_product (fn iwit => fn owit => owit $ iwit)))

       |> flat

       |> map (`(fn t => Term.add_frees t []))

       |> minimize_wits

       |> map (fn (frees, t) => fold absfree frees t);

     fun wit_tac {context = ctxt, ...} =

       mk_comp_wit_tac ctxt (wit_thms_of_bnf outer) (collect_set_natural_of_bnf outer)

         (maps wit_thms_of_bnf inners);

     val (bnf', lthy') =

       bnf_def const_policy (K Derive_Some_Facts) qualify tacs wit_tac (SOME (oDs @ flat Dss))

         ((((b, mapx), sets), bd), wits) lthy;

     val outer_rel_Gr = rel_Gr_of_bnf outer RS sym;

     val outer_rel_cong = rel_cong_of_bnf outer;

     val rel_unfold_thm =

       trans OF [rel_def_of_bnf bnf',

         trans OF [in_alt_thm RS @{thm subst_rel_def},

           trans OF [@{thm arg_cong2[of _ _ _ _ relcomp]} OF

             [trans OF [outer_rel_Gr RS @{thm arg_cong[of _ _ converse]},

               rel_converse_of_bnf outer RS sym], outer_rel_Gr],

             trans OF [rel_O_of_bnf outer RS sym, outer_rel_cong OF

               (map (fn bnf => rel_def_of_bnf bnf RS sym) inners)]]]];

     val pred_unfold_thm = Collect_split_box_equals OF [rel_unfold_thm,

       pred_def_of_bnf bnf' RS abs_pred_sym,

         trans OF [outer_rel_cong OF (map (fn bnf => pred_def_of_bnf bnf RS abs_pred_sym) inners),

           pred_def_of_bnf outer RS abs_pred_sym]];

     val unfold' = add_to_unfold (map_def_of_bnf bnf') (set_defs_of_bnf bnf') rel_unfold_thm

       pred_unfold_thm unfold;

   in

     (bnf', (unfold', lthy'))

   end;

 (* Killing live variables *)

 fun kill_bnf qualify n bnf (unfold, lthy) = if n = 0 then (bnf, (unfold, lthy)) else

   let

     val b = Binding.suffix_name (mk_killN n) (name_of_bnf bnf);

     val live = live_of_bnf bnf;

     val dead = dead_of_bnf bnf;

     val nwits = nwits_of_bnf bnf;

     (* TODO: check 0 < n <= live *)

     val (Ds, lthy1) = apfst (map TFree)

       (Variable.invent_types (replicate dead HOLogic.typeS) lthy);

     val ((killedAs, As), lthy2) = apfst (`(take n) o map TFree)

       (Variable.invent_types (replicate live HOLogic.typeS) lthy1);

     val (Bs, _(*lthy3*)) = apfst (append killedAs o map TFree)

       (Variable.invent_types (replicate (live - n) HOLogic.typeS) lthy2);

     val ((Asets, lives), _(*names_lthy*)) = lthy

       |> mk_Frees "A" (map (HOLogic.mk_setT) (drop n As))

       ||>> mk_Frees "x" (drop n As);

     val xs = map (fn T => HOLogic.choice_const T $ absdummy T @{term True}) killedAs @ lives;

     val T = mk_T_of_bnf Ds As bnf;

     (*bnf.map id ... id*)

     val mapx = Term.list_comb (mk_map_of_bnf Ds As Bs bnf, map HOLogic.id_const killedAs);

     val bnf_sets = mk_sets_of_bnf (replicate live Ds) (replicate live As) bnf;

     val sets = drop n bnf_sets;

     (*(|UNIV :: A1 set| +c ... +c |UNIV :: An set|) *c bnf.bd*)

     val bnf_bd = mk_bd_of_bnf Ds As bnf;

     val bd = mk_cprod

       (Library.foldr1 (uncurry mk_csum) (map (mk_card_of o HOLogic.mk_UNIV) killedAs)) bnf_bd;

     fun map_id_tac _ = rtac (map_id_of_bnf bnf) 1;

     fun map_comp_tac {context, ...} =

       Local_Defs.unfold_tac context ((map_comp_of_bnf bnf RS sym) :: @{thms o_assoc id_o o_id}) THEN

       rtac refl 1;

     fun map_cong_tac {context, ...} =

       mk_kill_map_cong_tac context n (live - n) (map_cong_of_bnf bnf);

     val set_natural_tacs = map (fn thm => fn _ => rtac thm 1) (drop n (set_natural_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 =

       map (fn thm => fn _ => mk_kill_set_bd_tac (bd_Card_order_of_bnf bnf) thm)

         (drop n (set_bd_of_bnf bnf));

     val in_alt_thm =

       let

         val inx = mk_in Asets sets T;

         val in_alt = mk_in (map HOLogic.mk_UNIV killedAs @ Asets) bnf_sets T;

         val goal = fold_rev Logic.all Asets (mk_Trueprop_eq (inx, in_alt));

       in

         Skip_Proof.prove lthy [] [] goal (K kill_in_alt_tac) |> Thm.close_derivation

       end;

     fun in_bd_tac _ =

       mk_kill_in_bd_tac n (live > n) in_alt_thm (in_bd_of_bnf bnf) (bd_Card_order_of_bnf bnf)

         (bd_Cinfinite_of_bnf bnf) (bd_Cnotzero_of_bnf bnf);

     fun map_wpull_tac _ = mk_map_wpull_tac in_alt_thm [] (map_wpull_of_bnf bnf);

     val tacs = [map_id_tac, map_comp_tac, map_cong_tac] @ set_natural_tacs @ [bd_card_order_tac,

       bd_cinfinite_tac] @ set_bd_tacs @ [in_bd_tac, map_wpull_tac];

     val bnf_wits = mk_wits_of_bnf (replicate nwits Ds) (replicate nwits As) bnf;

     val wits = map (fn t => fold absfree (Term.add_frees t []) t)

       (map (fn (I, wit) => Term.list_comb (wit, map (nth xs) I)) bnf_wits);

     fun wit_tac _ = mk_simple_wit_tac (wit_thms_of_bnf bnf);

     val (bnf', lthy') =

       bnf_def Smart_Inline (K Derive_Some_Facts) qualify tacs wit_tac (SOME (killedAs @ Ds))

         ((((b, mapx), sets), Term.absdummy T bd), wits) lthy;

     val rel_Gr = rel_Gr_of_bnf bnf RS sym;

     val rel_unfold_thm =

       trans OF [rel_def_of_bnf bnf',

         trans OF [in_alt_thm RS @{thm subst_rel_def},

           trans OF [@{thm arg_cong2[of _ _ _ _ relcomp]} OF

             [trans OF [rel_Gr RS @{thm arg_cong[of _ _ converse]}, rel_converse_of_bnf bnf RS sym],

               rel_Gr],

             trans OF [rel_O_of_bnf bnf RS sym, rel_cong_of_bnf bnf OF

               (replicate n @{thm trans[OF Gr_UNIV_id[OF refl] Id_alt[symmetric]]} @

                replicate (live - n) @{thm Gr_fst_snd})]]]];

     val pred_unfold_thm = Collect_split_box_equals OF

       [Local_Defs.unfold lthy' @{thms Id_def'} rel_unfold_thm, pred_def_of_bnf bnf' RS abs_pred_sym,

         pred_def_of_bnf bnf RS abs_pred_sym];

     val unfold' = add_to_unfold (map_def_of_bnf bnf') (set_defs_of_bnf bnf') rel_unfold_thm

       pred_unfold_thm unfold;

   in

     (bnf', (unfold', lthy'))

   end;

 (* Adding dummy live variables *)

 fun lift_bnf qualify n bnf (unfold, lthy) = if n = 0 then (bnf, (unfold, lthy)) else

   let

     val b = Binding.suffix_name (mk_liftN n) (name_of_bnf bnf);

     val live = live_of_bnf bnf;

     val dead = dead_of_bnf bnf;

     val nwits = nwits_of_bnf bnf;

     (* TODO: check 0 < n *)

     val (Ds, lthy1) = apfst (map TFree)

       (Variable.invent_types (replicate dead HOLogic.typeS) lthy);

     val ((newAs, As), lthy2) = apfst (chop n o map TFree)

       (Variable.invent_types (replicate (n + live) HOLogic.typeS) lthy1);

     val ((newBs, Bs), _(*lthy3*)) = apfst (chop n o map TFree)

       (Variable.invent_types (replicate (n + live) HOLogic.typeS) lthy2);

     val (Asets, _(*names_lthy*)) = lthy

       |> mk_Frees "A" (map (HOLogic.mk_setT) (newAs @ As));

     val T = mk_T_of_bnf Ds As bnf;

     (*%f1 ... fn. bnf.map*)

     val mapx =

       fold_rev Term.absdummy (map2 (curry (op -->)) newAs newBs) (mk_map_of_bnf Ds As Bs bnf);

     val bnf_sets = mk_sets_of_bnf (replicate live Ds) (replicate live As) bnf;

     val sets = map (fn A => absdummy T (HOLogic.mk_set A [])) newAs @ bnf_sets;

     val bd = mk_bd_of_bnf Ds As bnf;

     fun map_id_tac _ = rtac (map_id_of_bnf bnf) 1;

     fun map_comp_tac {context, ...} =

       Local_Defs.unfold_tac context ((map_comp_of_bnf bnf RS sym) :: @{thms o_assoc id_o o_id}) THEN

       rtac refl 1;

     fun map_cong_tac {context, ...} =

       rtac (map_cong_of_bnf bnf) 1 THEN REPEAT_DETERM_N live (Goal.assume_rule_tac context 1);

     val set_natural_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);

     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 =

       if ! quick_and_dirty then

         replicate (n + live) (K all_tac)

       else

         replicate n (K (mk_lift_set_bd_tac (bd_Card_order_of_bnf bnf))) @

         (map (fn thm => fn _ => rtac thm 1) (set_bd_of_bnf bnf));

     val in_alt_thm =

       let

         val inx = mk_in Asets sets T;

         val in_alt = mk_in (drop n Asets) bnf_sets T;

         val goal = fold_rev Logic.all Asets (mk_Trueprop_eq (inx, in_alt));

       in

         Skip_Proof.prove lthy [] [] goal (K lift_in_alt_tac) |> Thm.close_derivation

       end;

     fun in_bd_tac _ = mk_lift_in_bd_tac n in_alt_thm (in_bd_of_bnf bnf) (bd_Card_order_of_bnf bnf);

     fun map_wpull_tac _ = mk_map_wpull_tac in_alt_thm [] (map_wpull_of_bnf bnf);

     val tacs = [map_id_tac, map_comp_tac, map_cong_tac] @ set_natural_tacs @ [bd_card_order_tac,

       bd_cinfinite_tac] @ set_bd_tacs @ [in_bd_tac, map_wpull_tac];

     val wits = map snd (mk_wits_of_bnf (replicate nwits Ds) (replicate nwits As) bnf);

     fun wit_tac _ = mk_simple_wit_tac (wit_thms_of_bnf bnf);

     val (bnf', lthy') =

       bnf_def Smart_Inline (K Derive_Some_Facts) qualify tacs wit_tac (SOME Ds)

         ((((b, mapx), sets), Term.absdummy T bd), wits) lthy;

     val rel_unfold_thm =

       trans OF [rel_def_of_bnf bnf',

         trans OF [in_alt_thm RS @{thm subst_rel_def}, rel_def_of_bnf bnf RS sym]];

     val pred_unfold_thm = Collect_split_box_equals OF [rel_unfold_thm,

       pred_def_of_bnf bnf' RS abs_pred_sym, pred_def_of_bnf bnf RS abs_pred_sym];

     val unfold' = add_to_unfold (map_def_of_bnf bnf') (set_defs_of_bnf bnf') rel_unfold_thm

       pred_unfold_thm unfold;

   in

     (bnf', (unfold', lthy'))

   end;

 (* Changing the order of live variables *)

 fun permute_bnf qualify src dest bnf (unfold, lthy) = if src = dest then (bnf, (unfold, lthy)) else

   let

     val b = Binding.suffix_name (mk_permuteN src dest) (name_of_bnf bnf);

     val live = live_of_bnf bnf;

     val dead = dead_of_bnf bnf;

     val nwits = nwits_of_bnf bnf;

     fun permute xs = mk_permute src dest xs;

     fun permute_rev xs = mk_permute dest src xs;

     val (Ds, lthy1) = apfst (map TFree)

       (Variable.invent_types (replicate dead HOLogic.typeS) lthy);

     val (As, lthy2) = apfst (map TFree)

       (Variable.invent_types (replicate live HOLogic.typeS) lthy1);

     val (Bs, _(*lthy3*)) = apfst (map TFree)

       (Variable.invent_types (replicate live HOLogic.typeS) lthy2);

     val (Asets, _(*names_lthy*)) = lthy

       |> mk_Frees "A" (map (HOLogic.mk_setT) (permute As));

     val T = mk_T_of_bnf Ds As bnf;

     (*%f(1) ... f(n). bnf.map f\<sigma>(1) ... f\<sigma>(n)*)

     val mapx = fold_rev Term.absdummy (permute (map2 (curry op -->) As Bs))

       (Term.list_comb (mk_map_of_bnf Ds As Bs bnf,

         permute_rev (map Bound ((live - 1) downto 0))));

     val bnf_sets = mk_sets_of_bnf (replicate live Ds) (replicate live As) bnf;

     val sets = permute bnf_sets;

     val bd = mk_bd_of_bnf Ds As bnf;

     fun map_id_tac _ = rtac (map_id_of_bnf bnf) 1;

     fun map_comp_tac _ = rtac (map_comp_of_bnf bnf) 1;

     fun map_cong_tac {context, ...} =

       rtac (map_cong_of_bnf bnf) 1 THEN REPEAT_DETERM_N live (Goal.assume_rule_tac context 1);

     val set_natural_tacs = permute (map (fn thm => fn _ => rtac thm 1) (set_natural_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));

     val in_alt_thm =

       let

         val inx = mk_in Asets sets T;

         val in_alt = mk_in (permute_rev Asets) bnf_sets T;

         val goal = fold_rev Logic.all Asets (mk_Trueprop_eq (inx, in_alt));

       in

         Skip_Proof.prove lthy [] [] goal (K (mk_permute_in_alt_tac src dest))

         |> Thm.close_derivation

       end;

     fun in_bd_tac _ =

       mk_permute_in_bd_tac src dest in_alt_thm (in_bd_of_bnf bnf) (bd_Card_order_of_bnf bnf);

     fun map_wpull_tac _ = mk_map_wpull_tac in_alt_thm [] (map_wpull_of_bnf bnf);

     val tacs = [map_id_tac, map_comp_tac, map_cong_tac] @ set_natural_tacs @ [bd_card_order_tac,

       bd_cinfinite_tac] @ set_bd_tacs @ [in_bd_tac, map_wpull_tac];

     val wits = map snd (mk_wits_of_bnf (replicate nwits Ds) (replicate nwits As) bnf);

     fun wit_tac _ = mk_simple_wit_tac (wit_thms_of_bnf bnf);

     val (bnf', lthy') =

       bnf_def Smart_Inline (K Derive_Some_Facts) qualify tacs wit_tac (SOME Ds)

         ((((b, mapx), sets), Term.absdummy T bd), wits) lthy;

     val rel_unfold_thm =

       trans OF [rel_def_of_bnf bnf',

         trans OF [in_alt_thm RS @{thm subst_rel_def}, rel_def_of_bnf bnf RS sym]];

     val pred_unfold_thm = Collect_split_box_equals OF [rel_unfold_thm,

       pred_def_of_bnf bnf' RS abs_pred_sym, pred_def_of_bnf bnf RS abs_pred_sym];

     val unfold' = add_to_unfold (map_def_of_bnf bnf') (set_defs_of_bnf bnf') rel_unfold_thm

       pred_unfold_thm unfold;

   in

     (bnf', (unfold', lthy'))

   end;

 (* Composition pipeline *)

 fun permute_and_kill qualify n src dest bnf =

   bnf

   |> permute_bnf qualify src dest

   #> uncurry (kill_bnf qualify n);

 fun lift_and_permute qualify n src dest bnf =

   bnf

   |> lift_bnf qualify n

   #> uncurry (permute_bnf qualify src dest);

 fun normalize_bnfs qualify Ass Ds sort bnfs unfold lthy =

   let

     val before_kill_src = map (fn As => 0 upto (length As - 1)) Ass;

     val kill_poss = map (find_indices Ds) Ass;

     val live_poss = map2 (subtract (op =)) kill_poss before_kill_src;

     val before_kill_dest = map2 append kill_poss live_poss;

     val kill_ns = map length kill_poss;

     val (inners', (unfold', lthy')) =

       fold_map5 (fn i => permute_and_kill (qualify i))

         (if length bnfs = 1 then [0] else (1 upto length bnfs))

         kill_ns before_kill_src before_kill_dest bnfs (unfold, lthy);

     val Ass' = map2 (map o nth) Ass live_poss;

     val As = sort Ass';

     val after_lift_dest = replicate (length Ass') (0 upto (length As - 1));

     val old_poss = map (map (fn x => find_index (fn y => x = y) As)) Ass';

     val new_poss = map2 (subtract (op =)) old_poss after_lift_dest;

     val after_lift_src = map2 append new_poss old_poss;

     val lift_ns = map (fn xs => length As - length xs) Ass';

   in

     ((kill_poss, As), fold_map5 (fn i => lift_and_permute (qualify i))

       (if length bnfs = 1 then [0] else (1 upto length bnfs))

       lift_ns after_lift_src after_lift_dest inners' (unfold', lthy'))

   end;

 fun default_comp_sort Ass =

   Library.sort (Term_Ord.typ_ord o pairself TFree) (fold (fold (insert (op =))) Ass []);

 fun compose_bnf const_policy qualify sort outer inners oDs Dss tfreess (unfold, lthy) =

   let

     val b = name_of_bnf outer;

     val Ass = map (map Term.dest_TFree) tfreess;

     val Ds = fold (fold Term.add_tfreesT) (oDs :: Dss) [];

     val ((kill_poss, As), (inners', (unfold', lthy'))) =

       normalize_bnfs qualify Ass Ds sort inners unfold lthy;

     val Ds = oDs @ flat (map3 (append oo map o nth) tfreess kill_poss Dss);

     val As = map TFree As;

   in

     apfst (rpair (Ds, As))

       (clean_compose_bnf const_policy (qualify 0) b outer inners' (unfold', lthy'))

   end;

 (* Hide the type of the bound (optimization) and unfold the definitions (nicer to the user) *)

 fun seal_bnf unfold b Ds bnf lthy =

   let

     val live = live_of_bnf bnf;

     val nwits = nwits_of_bnf bnf;

     val (As, lthy1) = apfst (map TFree)

       (Variable.invent_types (replicate live HOLogic.typeS) (fold Variable.declare_typ Ds lthy));

     val (Bs, _) = apfst (map TFree)

       (Variable.invent_types (replicate live HOLogic.typeS) lthy1);

     val map_unfolds = filter_refl (map_unfolds_of unfold);

     val set_unfoldss = map filter_refl (set_unfoldss_of unfold);

     val expand_maps = fold expand_term_const (map (single o Logic.dest_equals o Thm.prop_of)

       map_unfolds);

     val expand_sets = fold expand_term_const (map (map (Logic.dest_equals o Thm.prop_of))

       set_unfoldss);

     val unfold_maps = fold (Local_Defs.unfold lthy o single) map_unfolds;

     val unfold_sets = fold (Local_Defs.unfold lthy) set_unfoldss;

     val unfold_defs = unfold_sets o unfold_maps;

     val bnf_map = expand_maps (mk_map_of_bnf Ds As Bs bnf);

     val bnf_sets = map (expand_maps o expand_sets)

       (mk_sets_of_bnf (replicate live Ds) (replicate live As) bnf);

     val bnf_bd = mk_bd_of_bnf Ds As bnf;

     val T = mk_T_of_bnf Ds As bnf;

     (*bd should only depend on dead type variables!*)

     val bd_repT = fst (dest_relT (fastype_of bnf_bd));

     val bdT_bind = Binding.suffix_name ("_" ^ bdTN) b;

     val params = fold Term.add_tfreesT Ds [];

     val deads = map TFree params;

     val ((bdT_name, (bdT_glob_info, bdT_loc_info)), lthy) =

       typedef false NONE (bdT_bind, params, NoSyn)

         (HOLogic.mk_UNIV bd_repT) NONE (EVERY' [rtac exI, rtac UNIV_I] 1) lthy;

     val bnf_bd' = mk_dir_image bnf_bd

       (Const (#Abs_name bdT_glob_info, bd_repT --> Type (bdT_name, deads)))

     val Abs_bdT_inj = mk_Abs_inj_thm (#Abs_inject bdT_loc_info);

     val Abs_bdT_bij = mk_Abs_bij_thm lthy Abs_bdT_inj (#Abs_cases bdT_loc_info);

     val bd_ordIso = @{thm dir_image} OF [Abs_bdT_inj, bd_Card_order_of_bnf bnf];

     val bd_card_order =

       @{thm card_order_dir_image} OF [Abs_bdT_bij, bd_card_order_of_bnf bnf];

     val bd_cinfinite =

       (@{thm Cinfinite_cong} OF [bd_ordIso, bd_Cinfinite_of_bnf bnf]) RS conjunct1;

     val set_bds =

       map (fn thm => @{thm ordLeq_ordIso_trans} OF [thm, bd_ordIso]) (set_bd_of_bnf bnf);

     val in_bd =

       @{thm ordLeq_ordIso_trans} OF [in_bd_of_bnf bnf,

         @{thm cexp_cong2_Cnotzero} OF [bd_ordIso, if live = 0 then

           @{thm ctwo_Cnotzero} else @{thm ctwo_Cnotzero} RS @{thm csum_Cnotzero2},

             bd_Card_order_of_bnf bnf]];

     fun mk_tac thm {context = ctxt, prems = _} = (rtac (unfold_defs thm) THEN'

       SOLVE o REPEAT_DETERM o (atac ORELSE' Goal.assume_rule_tac ctxt)) 1;

     val tacs =

       map mk_tac ([map_id_of_bnf bnf, map_comp_of_bnf bnf, map_cong_of_bnf bnf] @

         set_natural_of_bnf bnf) @

       map K [rtac bd_card_order 1, rtac bd_cinfinite 1] @

       map mk_tac (set_bds @ [in_bd, map_wpull_of_bnf bnf]);

     val bnf_wits = map snd (mk_wits_of_bnf (replicate nwits Ds) (replicate nwits As) bnf);

     fun wit_tac _ = mk_simple_wit_tac (map unfold_defs (wit_thms_of_bnf bnf));

     val (bnf', lthy') = bnf_def Hardly_Inline (K Derive_All_Facts) I tacs wit_tac (SOME deads)

       ((((b, bnf_map), bnf_sets), Term.absdummy T bnf_bd'), bnf_wits) lthy;

     val defs' = filter_refl (map_def_of_bnf bnf' :: set_defs_of_bnf bnf');

     val unfold_defs' = unfold_defs o Local_Defs.unfold lthy' defs';

     val rel_def = unfold_defs' (rel_def_of_bnf bnf');

     val rel_unfold = Local_Defs.unfold lthy'

       (map unfold_defs (filter_refl (rel_unfolds_of unfold))) rel_def;

     val unfold_defs'' = unfold_defs' o Local_Defs.unfold lthy' (filter_refl [rel_def_of_bnf bnf']);

     val pred_def = unfold_defs'' (pred_def_of_bnf bnf' RS abs_pred_sym_pred_abs);

     val pred_unfold = Local_Defs.unfold lthy'

       (map unfold_defs (filter_refl (pred_unfolds_of unfold))) pred_def;

     val notes =

       [(rel_unfoldN, [rel_unfold]),

       (pred_unfoldN, [pred_unfold])]

       |> map (fn (thmN, thms) =>

         ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]));

   in

     ((bnf', deads), lthy' |> Local_Theory.notes notes |> snd)

   end;

 fun bnf_of_typ _ _ _ (T as TFree _) (unfold, lthy) =

     ((Basic_BNFs.ID_bnf, ([], [T])), (add_to_unfold_opt NONE NONE

       (SOME Basic_BNFs.ID_rel_def) (SOME Basic_BNFs.ID_pred_def) unfold, lthy))

   | bnf_of_typ _ _ _ (TVar _) _ = error "Unexpected schematic variable"

   | bnf_of_typ const_policy qualify' sort (T as Type (C, Ts)) (unfold, lthy) =

     let

       val tfrees = Term.add_tfreesT T [];

       val bnf_opt = if null tfrees then NONE else bnf_of lthy C;

     in

       (case bnf_opt of

         NONE => ((Basic_BNFs.DEADID_bnf, ([T], [])), (unfold, lthy))

       | SOME bnf =>

         if forall (can Term.dest_TFree) Ts andalso length Ts = length tfrees then

           let

             val T' = T_of_bnf bnf;

             val deads = deads_of_bnf bnf;

             val lives = lives_of_bnf bnf;

             val tvars' = Term.add_tvarsT T' [];

             val deads_lives =

               pairself (map (Term.typ_subst_TVars (map fst tvars' ~~ map TFree tfrees)))

                 (deads, lives);

             val rel_def = rel_def_of_bnf bnf;

             val unfold' = add_to_unfold_opt NONE NONE (SOME (rel_def RS sym))

               (SOME (Local_Defs.unfold lthy [rel_def] (pred_def_of_bnf bnf) RS sym)) unfold;

           in ((bnf, deads_lives), (unfold', lthy)) end

         else

           let

             val name = Long_Name.base_name C;

             fun qualify i =

               let val namei = name ^ nonzero_string_of_int i;

               in qualify' o Binding.qualify true namei end;

             val odead = dead_of_bnf bnf;

             val olive = live_of_bnf bnf;

             val oDs_pos = find_indices [TFree ("dead", [])] (snd (Term.dest_Type

               (mk_T_of_bnf (replicate odead (TFree ("dead", []))) (replicate olive dummyT) bnf)));

             val oDs = map (nth Ts) oDs_pos;

             val Ts' = map (nth Ts) (subtract (op =) oDs_pos (0 upto length Ts - 1));

             val ((inners, (Dss, Ass)), (unfold', lthy')) =

               apfst (apsnd split_list o split_list)

                 (fold_map2 (fn i => bnf_of_typ Smart_Inline (qualify i) sort)

                 (if length Ts' = 1 then [0] else (1 upto length Ts')) Ts' (unfold, lthy));

           in

             compose_bnf const_policy qualify sort bnf inners oDs Dss Ass (unfold', lthy')

           end)

     end;

 end;