(*  Title:      HOL/BNF/Tools/bnf_def.ML

     Author:     Dmitriy Traytel, TU Muenchen

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   2012

 Definition of bounded natural functors.

 *)

 signature BNF_DEF =

 sig

   type bnf

   type nonemptiness_witness = {I: int list, wit: term, prop: thm list}

   val morph_bnf: morphism -> bnf -> bnf

   val eq_bnf: bnf * bnf -> bool

   val bnf_of: Proof.context -> string -> bnf option

   val register_bnf: string -> (bnf * local_theory) -> (bnf * local_theory)

   val name_of_bnf: bnf -> binding

   val T_of_bnf: bnf -> typ

   val live_of_bnf: bnf -> int

   val lives_of_bnf: bnf -> typ list

   val dead_of_bnf: bnf -> int

   val deads_of_bnf: bnf -> typ list

   val nwits_of_bnf: bnf -> int

   val mapN: string

   val relN: string

   val setN: string

   val mk_setN: int -> string

   val map_of_bnf: bnf -> term

   val sets_of_bnf: bnf -> term list

   val rel_of_bnf: bnf -> term

   val mk_T_of_bnf: typ list -> typ list -> bnf -> typ

   val mk_bd_of_bnf: typ list -> typ list -> bnf -> term

   val mk_map_of_bnf: typ list -> typ list -> typ list -> bnf -> term

   val mk_rel_of_bnf: typ list -> typ list -> typ list -> bnf -> term

   val mk_sets_of_bnf: typ list list -> typ list list -> bnf -> term list

   val mk_wits_of_bnf: typ list list -> typ list list -> bnf -> (int list * term) list

   val bd_Card_order_of_bnf: bnf -> thm

   val bd_Cinfinite_of_bnf: bnf -> thm

   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_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

   val in_rel_of_bnf: bnf -> thm

   val map_comp0_of_bnf: bnf -> thm

   val map_comp_of_bnf: bnf -> thm

   val map_cong0_of_bnf: bnf -> thm

   val map_cong_of_bnf: bnf -> thm

   val map_def_of_bnf: bnf -> thm

   val map_id0_of_bnf: bnf -> thm

   val map_id_of_bnf: bnf -> thm

   val map_transfer_of_bnf: bnf -> thm

   val map_wppull_of_bnf: bnf -> thm

   val map_wpull_of_bnf: bnf -> thm

   val rel_def_of_bnf: bnf -> thm

   val rel_Grp_of_bnf: bnf -> thm

   val rel_OO_of_bnf: bnf -> thm

   val rel_OO_Grp_of_bnf: bnf -> thm

   val rel_cong_of_bnf: bnf -> thm

   val rel_conversep_of_bnf: bnf -> thm

   val rel_mono_of_bnf: bnf -> thm

   val rel_mono_strong_of_bnf: bnf -> thm

   val rel_eq_of_bnf: bnf -> thm

   val rel_flip_of_bnf: bnf -> thm

   val set_bd_of_bnf: bnf -> thm list

   val set_defs_of_bnf: bnf -> thm list

   val set_map0_of_bnf: bnf -> thm list

   val set_map_of_bnf: bnf -> thm list

   val wit_thms_of_bnf: bnf -> thm list

   val wit_thmss_of_bnf: bnf -> thm list list

   val mk_map: int -> typ list -> typ list -> term -> term

   val mk_rel: int -> typ list -> typ list -> term -> term

   val build_map: Proof.context -> (typ * typ -> term) -> typ * typ -> term

   val build_rel: Proof.context -> (typ * typ -> term) -> typ * typ -> term

   val flatten_type_args_of_bnf: bnf -> 'a -> 'a list -> 'a list

   val map_flattened_map_args: Proof.context -> string -> (term list -> 'a list) -> term list ->

     'a list

   val mk_witness: int list * term -> thm list -> nonemptiness_witness

   val minimize_wits: (''a list * 'b) list -> (''a list * 'b) list

   val wits_of_bnf: bnf -> nonemptiness_witness list

   val zip_axioms: 'a -> 'a -> 'a -> 'a list -> 'a -> 'a -> 'a list -> 'a -> 'a -> 'a list

   datatype const_policy = Dont_Inline | Hardly_Inline | Smart_Inline | Do_Inline

   datatype fact_policy = Dont_Note | Note_Some | Note_All

   val bnf_note_all: bool Config.T

   val bnf_timing: bool Config.T

   val user_policy: fact_policy -> Proof.context -> fact_policy

   val note_bnf_thms: fact_policy -> (binding -> binding) -> binding -> bnf -> Proof.context ->

     Proof.context

   val print_bnfs: Proof.context -> unit

   val prepare_def: const_policy -> (Proof.context -> fact_policy) -> (binding -> binding) ->

     (Proof.context -> 'a -> typ) -> (Proof.context -> 'b -> term) -> typ list option ->

     binding -> binding -> binding list ->

     (((((binding * 'a) * 'b) * 'b list) * 'b) * 'b list) * 'b option -> Proof.context ->

     string * term list *

     ((thm list -> {context: Proof.context, prems: thm list} -> tactic) option * term list list) *

     ((thm list -> thm list list) -> thm list list -> Proof.context -> bnf * local_theory) *

     local_theory * thm list

   val bnf_def: const_policy -> (Proof.context -> fact_policy) -> (binding -> binding) ->

     ({prems: thm list, context: Proof.context} -> tactic) list ->

     ({prems: thm list, context: Proof.context} -> tactic) -> typ list option -> binding ->

     binding -> binding list ->

     (((((binding * typ) * term) * term list) * term) * term list) * term option ->

     local_theory -> bnf * local_theory

 end;

 structure BNF_Def : BNF_DEF =

 struct

 open BNF_Util

 open BNF_Tactics

 open BNF_Def_Tactics

 val fundefcong_attrs = @{attributes [fundef_cong]};

 type axioms = {

   map_id0: thm,

   map_comp0: thm,

   map_cong0: thm,

   set_map0: thm list,

   bd_card_order: thm,

   bd_cinfinite: thm,

   set_bd: thm list,

   map_wpull: thm,

   rel_OO_Grp: thm

 };

 fun mk_axioms' ((((((((id, comp), cong), map), c_o), cinf), set_bd), wpull), rel) =

   {map_id0 = id, map_comp0 = comp, map_cong0 = cong, set_map0 = map, bd_card_order = c_o,

    bd_cinfinite = cinf, set_bd = set_bd, map_wpull = wpull, rel_OO_Grp = rel};

 fun dest_cons [] = raise List.Empty

   | dest_cons (x :: xs) = (x, xs);

 fun mk_axioms n thms = thms

   |> map the_single

   |> dest_cons

   ||>> dest_cons

   ||>> dest_cons

   ||>> chop n

   ||>> dest_cons

   ||>> dest_cons

   ||>> chop n

   ||>> dest_cons

   ||> the_single

   |> mk_axioms';

 fun zip_axioms mid mcomp mcong smap bdco bdinf sbd wpull rel =

   [mid, mcomp, mcong] @ smap @ [bdco, bdinf] @ sbd @ [wpull, rel];

 fun dest_axioms {map_id0, map_comp0, map_cong0, set_map0, bd_card_order, bd_cinfinite, set_bd,

   map_wpull, rel_OO_Grp} =

   zip_axioms map_id0 map_comp0 map_cong0 set_map0 bd_card_order bd_cinfinite set_bd map_wpull

     rel_OO_Grp;

 fun map_axioms f {map_id0, map_comp0, map_cong0, set_map0, bd_card_order, bd_cinfinite, set_bd,

   map_wpull, rel_OO_Grp} =

   {map_id0 = f map_id0,

     map_comp0 = f map_comp0,

     map_cong0 = f map_cong0,

     set_map0 = map f set_map0,

     bd_card_order = f bd_card_order,

     bd_cinfinite = f bd_cinfinite,

     set_bd = map f set_bd,

     map_wpull = f map_wpull,

     rel_OO_Grp = f rel_OO_Grp};

 val morph_axioms = map_axioms o Morphism.thm;

 type defs = {

   map_def: thm,

   set_defs: thm list,

   rel_def: thm

 }

 fun mk_defs map sets rel = {map_def = map, set_defs = sets, rel_def = rel};

 fun map_defs f {map_def, set_defs, rel_def} =

   {map_def = f map_def, set_defs = map f set_defs, rel_def = f rel_def};

 val morph_defs = map_defs o Morphism.thm;

 type facts = {

   bd_Card_order: thm,

   bd_Cinfinite: thm,

   bd_Cnotzero: thm,

   collect_set_map: thm lazy,

   in_bd: thm lazy,

   in_cong: thm lazy,

   in_mono: thm lazy,

   in_rel: thm lazy,

   map_comp: thm lazy,

   map_cong: thm lazy,

   map_id: thm lazy,

   map_transfer: thm lazy,

   map_wppull: thm lazy,

   rel_eq: thm lazy,

   rel_flip: thm lazy,

   set_map: thm lazy list,

   rel_cong: thm lazy,

   rel_mono: thm lazy,

   rel_mono_strong: thm lazy,

   rel_Grp: thm lazy,

   rel_conversep: thm lazy,

   rel_OO: thm lazy

 };

 fun mk_facts bd_Card_order bd_Cinfinite bd_Cnotzero collect_set_map in_bd in_cong in_mono in_rel

     map_comp map_cong map_id map_transfer map_wppull rel_eq rel_flip set_map rel_cong rel_mono

     rel_mono_strong rel_Grp rel_conversep rel_OO = {

   bd_Card_order = bd_Card_order,

   bd_Cinfinite = bd_Cinfinite,

   bd_Cnotzero = bd_Cnotzero,

   collect_set_map = collect_set_map,

   in_bd = in_bd,

   in_cong = in_cong,

   in_mono = in_mono,

   in_rel = in_rel,

   map_comp = map_comp,

   map_cong = map_cong,

   map_id = map_id,

   map_transfer = map_transfer,

   map_wppull = map_wppull,

   rel_eq = rel_eq,

   rel_flip = rel_flip,

   set_map = set_map,

   rel_cong = rel_cong,

   rel_mono = rel_mono,

   rel_mono_strong = rel_mono_strong,

   rel_Grp = rel_Grp,

   rel_conversep = rel_conversep,

   rel_OO = rel_OO};

 fun map_facts f {

   bd_Card_order,

   bd_Cinfinite,

   bd_Cnotzero,

   collect_set_map,

   in_bd,

   in_cong,

   in_mono,

   in_rel,

   map_comp,

   map_cong,

   map_id,

   map_transfer,

   map_wppull,

   rel_eq,

   rel_flip,

   set_map,

   rel_cong,

   rel_mono,

   rel_mono_strong,

   rel_Grp,

   rel_conversep,

   rel_OO} =

   {bd_Card_order = f bd_Card_order,

     bd_Cinfinite = f bd_Cinfinite,

     bd_Cnotzero = f bd_Cnotzero,

     collect_set_map = Lazy.map f collect_set_map,

     in_bd = Lazy.map f in_bd,

     in_cong = Lazy.map f in_cong,

     in_mono = Lazy.map f in_mono,

     in_rel = Lazy.map f in_rel,

     map_comp = Lazy.map f map_comp,

     map_cong = Lazy.map f map_cong,

     map_id = Lazy.map f map_id,

     map_transfer = Lazy.map f map_transfer,

     map_wppull = Lazy.map f map_wppull,

     rel_eq = Lazy.map f rel_eq,

     rel_flip = Lazy.map f rel_flip,

     set_map = map (Lazy.map f) set_map,

     rel_cong = Lazy.map f rel_cong,

     rel_mono = Lazy.map f rel_mono,

     rel_mono_strong = Lazy.map f rel_mono_strong,

     rel_Grp = Lazy.map f rel_Grp,

     rel_conversep = Lazy.map f rel_conversep,

     rel_OO = Lazy.map f rel_OO};

 val morph_facts = map_facts o Morphism.thm;

 type nonemptiness_witness = {

   I: int list,

   wit: term,

   prop: thm list

 };

 fun mk_witness (I, wit) prop = {I = I, wit = wit, prop = prop};

 fun map_witness f g {I, wit, prop} = {I = I, wit = f wit, prop = map g prop};

 fun morph_witness phi = map_witness (Morphism.term phi) (Morphism.thm phi);

 datatype bnf = BNF of {

   name: binding,

   T: typ,

   live: int,

   lives: typ list, (*source type variables of map*)

   lives': typ list, (*target type variables of map*)

   dead: int,

   deads: typ list,

   map: term,

   sets: term list,

   bd: term,

   axioms: axioms,

   defs: defs,

   facts: facts,

   nwits: int,

   wits: nonemptiness_witness list,

   rel: term

 };

 (* getters *)

 fun rep_bnf (BNF bnf) = bnf;

 val name_of_bnf = #name o rep_bnf;

 val T_of_bnf = #T o rep_bnf;

 fun mk_T_of_bnf Ds Ts bnf =

   let val bnf_rep = rep_bnf bnf

   in Term.typ_subst_atomic ((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts)) (#T bnf_rep) end;

 val live_of_bnf = #live o rep_bnf;

 val lives_of_bnf = #lives o rep_bnf;

 val dead_of_bnf = #dead o rep_bnf;

 val deads_of_bnf = #deads o rep_bnf;

 val axioms_of_bnf = #axioms o rep_bnf;

 val facts_of_bnf = #facts o rep_bnf;

 val nwits_of_bnf = #nwits o rep_bnf;

 val wits_of_bnf = #wits o rep_bnf;

 fun flatten_type_args_of_bnf bnf dead_x xs =

   let

     val Type (_, Ts) = T_of_bnf bnf;

     val lives = lives_of_bnf bnf;

     val deads = deads_of_bnf bnf;

   in

     permute_like (op =) (deads @ lives) Ts (replicate (length deads) dead_x @ xs)

   end;

 (*terms*)

 val map_of_bnf = #map o rep_bnf;

 val sets_of_bnf = #sets o rep_bnf;

 fun mk_map_of_bnf Ds Ts Us bnf =

   let val bnf_rep = rep_bnf bnf;

   in

     Term.subst_atomic_types

       ((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts) @ (#lives' bnf_rep ~~ Us)) (#map bnf_rep)

   end;

 fun mk_sets_of_bnf Dss Tss bnf =

   let val bnf_rep = rep_bnf bnf;

   in

     map2 (fn (Ds, Ts) => Term.subst_atomic_types

       ((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts))) (Dss ~~ Tss) (#sets bnf_rep)

   end;

 val bd_of_bnf = #bd o rep_bnf;

 fun mk_bd_of_bnf Ds Ts bnf =

   let val bnf_rep = rep_bnf bnf;

   in Term.subst_atomic_types ((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts)) (#bd bnf_rep) end;

 fun mk_wits_of_bnf Dss Tss bnf =

   let

     val bnf_rep = rep_bnf bnf;

     val wits = map (fn x => (#I x, #wit x)) (#wits bnf_rep);

   in

     map2 (fn (Ds, Ts) => apsnd (Term.subst_atomic_types

       ((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts)))) (Dss ~~ Tss) wits

   end;

 val rel_of_bnf = #rel o rep_bnf;

 fun mk_rel_of_bnf Ds Ts Us bnf =

   let val bnf_rep = rep_bnf bnf;

   in

     Term.subst_atomic_types

       ((#deads bnf_rep ~~ Ds) @ (#lives bnf_rep ~~ Ts) @ (#lives' bnf_rep ~~ Us)) (#rel bnf_rep)

   end;

 (*thms*)

 val bd_card_order_of_bnf = #bd_card_order o #axioms o rep_bnf;

 val bd_cinfinite_of_bnf = #bd_cinfinite o #axioms o rep_bnf;

 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_map_of_bnf = Lazy.force o #collect_set_map o #facts o rep_bnf;

 val in_bd_of_bnf = Lazy.force o #in_bd o #facts 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;

 val in_rel_of_bnf = Lazy.force o #in_rel o #facts o rep_bnf;

 val map_def_of_bnf = #map_def o #defs o rep_bnf;

 val map_id0_of_bnf = #map_id0 o #axioms o rep_bnf;

 val map_id_of_bnf = Lazy.force o #map_id o #facts o rep_bnf;

 val map_comp0_of_bnf = #map_comp0 o #axioms o rep_bnf;

 val map_comp_of_bnf = Lazy.force o #map_comp o #facts o rep_bnf;

 val map_cong0_of_bnf = #map_cong0 o #axioms o rep_bnf;

 val map_cong_of_bnf = Lazy.force o #map_cong o #facts o rep_bnf;

 val map_transfer_of_bnf = Lazy.force o #map_transfer o #facts o rep_bnf;

 val map_wppull_of_bnf = Lazy.force o #map_wppull o #facts o rep_bnf;

 val map_wpull_of_bnf = #map_wpull o #axioms o rep_bnf;

 val rel_def_of_bnf = #rel_def o #defs o rep_bnf;

 val rel_eq_of_bnf = Lazy.force o #rel_eq o #facts o rep_bnf;

 val rel_flip_of_bnf = Lazy.force o #rel_flip 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_map0_of_bnf = #set_map0 o #axioms o rep_bnf;

 val set_map_of_bnf = map Lazy.force o #set_map o #facts o rep_bnf;

 val rel_cong_of_bnf = Lazy.force o #rel_cong o #facts o rep_bnf;

 val rel_mono_of_bnf = Lazy.force o #rel_mono o #facts o rep_bnf;

 val rel_mono_strong_of_bnf = Lazy.force o #rel_mono_strong o #facts o rep_bnf;

 val rel_Grp_of_bnf = Lazy.force o #rel_Grp o #facts o rep_bnf;

 val rel_conversep_of_bnf = Lazy.force o #rel_conversep o #facts o rep_bnf;

 val rel_OO_of_bnf = Lazy.force o #rel_OO o #facts o rep_bnf;

 val rel_OO_Grp_of_bnf = #rel_OO_Grp o #axioms o rep_bnf;

 val wit_thms_of_bnf = maps #prop o wits_of_bnf;

 val wit_thmss_of_bnf = map #prop o wits_of_bnf;

 fun mk_bnf name T live lives lives' dead deads map sets bd axioms defs facts wits rel =

   BNF {name = name, T = T,

        live = live, lives = lives, lives' = lives', dead = dead, deads = deads,

        map = map, sets = sets, bd = bd,

        axioms = axioms, defs = defs, facts = facts,

        nwits = length wits, wits = wits, rel = rel};

 fun morph_bnf phi (BNF {name = name, T = T, live = live, lives = lives, lives' = lives',

   dead = dead, deads = deads, map = map, sets = sets, bd = bd,

   axioms = axioms, defs = defs, facts = facts,

   nwits = nwits, wits = wits, rel = rel}) =

   BNF {name = Morphism.binding phi name, T = Morphism.typ phi T,

     live = live, lives = List.map (Morphism.typ phi) lives,

     lives' = List.map (Morphism.typ phi) lives',

     dead = dead, deads = List.map (Morphism.typ phi) deads,

     map = Morphism.term phi map, sets = List.map (Morphism.term phi) sets,

     bd = Morphism.term phi bd,

     axioms = morph_axioms phi axioms,

     defs = morph_defs phi defs,

     facts = morph_facts phi facts,

     nwits = nwits,

     wits = List.map (morph_witness phi) wits,

     rel = Morphism.term phi rel};

 fun eq_bnf (BNF {T = T1, live = live1, dead = dead1, ...},

   BNF {T = T2, live = live2, dead = dead2, ...}) =

   Type.could_unify (T1, T2) andalso live1 = live2 andalso dead1 = dead2;

 structure Data = Generic_Data

 (

   type T = bnf Symtab.table;

   val empty = Symtab.empty;

   val extend = I;

   val merge = Symtab.merge eq_bnf;

 );

 fun bnf_of ctxt =

   Symtab.lookup (Data.get (Context.Proof ctxt))

   #> Option.map (morph_bnf (Morphism.thm_morphism (Thm.transfer (Proof_Context.theory_of ctxt))));

 (* Utilities *)

 fun normalize_set insts instA set =

   let

     val (T, T') = dest_funT (fastype_of set);

     val A = fst (Term.dest_TVar (HOLogic.dest_setT T'));

     val params = Term.add_tvar_namesT T [];

   in Term.subst_TVars ((A :: params) ~~ (instA :: insts)) set end;

 fun normalize_rel ctxt instTs instA instB rel =

   let

     val thy = Proof_Context.theory_of ctxt;

     val tyenv =

       Sign.typ_match thy (fastype_of rel, Library.foldr (op -->) (instTs, mk_pred2T instA instB))

         Vartab.empty;

   in Envir.subst_term (tyenv, Vartab.empty) rel end

   handle Type.TYPE_MATCH => error "Bad relator";

 fun normalize_wit insts CA As wit =

   let

     fun strip_param (Ts, T as Type (@{type_name fun}, [T1, T2])) =

         if Type.raw_instance (CA, T) then (Ts, T) else strip_param (T1 :: Ts, T2)

       | strip_param x = x;

     val (Ts, T) = strip_param ([], fastype_of wit);

     val subst = Term.add_tvar_namesT T [] ~~ insts;

     fun find y = find_index (fn x => x = y) As;

   in

     (map (find o Term.typ_subst_TVars subst) (rev Ts), Term.subst_TVars subst wit)

   end;

 fun minimize_wits wits =

  let

    fun minimize done [] = done

      | minimize done ((I, wit) :: todo) =

        if exists (fn (J, _) => subset (op =) (J, I)) (done @ todo)

        then minimize done todo

        else minimize ((I, wit) :: done) todo;

  in minimize [] wits end;

 fun mk_map live Ts Us t =

   let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in

     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t

   end;

 fun mk_rel live Ts Us t =

   let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in

     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t

   end;

 fun build_map_or_rel mk const of_bnf dest ctxt build_simple =

   let

     fun build (TU as (T, U)) =

       if T = U then

         const T

       else

         (case TU of

           (Type (s, Ts), Type (s', Us)) =>

           if s = s' then

             let

               val bnf = the (bnf_of ctxt s);

               val live = live_of_bnf bnf;

               val mapx = mk live Ts Us (of_bnf bnf);

               val TUs' = map dest (fst (strip_typeN live (fastype_of mapx)));

             in Term.list_comb (mapx, map build TUs') end

           else

             build_simple TU

         | _ => build_simple TU);

   in build end;

 val build_map = build_map_or_rel mk_map HOLogic.id_const map_of_bnf dest_funT;

 val build_rel = build_map_or_rel mk_rel HOLogic.eq_const rel_of_bnf dest_pred2T;

 fun map_flattened_map_args ctxt s map_args fs =

   let

     val flat_fs = flatten_type_args_of_bnf (the (bnf_of ctxt s)) Term.dummy fs;

     val flat_fs' = map_args flat_fs;

   in

     permute_like (op aconv) flat_fs fs flat_fs'

   end;

 (* Names *)

 val mapN = "map";

 val setN = "set";

 fun mk_setN i = setN ^ nonzero_string_of_int i;

 val bdN = "bd";

 val witN = "wit";

 fun mk_witN i = witN ^ nonzero_string_of_int i;

 val relN = "rel";

 val bd_card_orderN = "bd_card_order";

 val bd_cinfiniteN = "bd_cinfinite";

 val bd_Card_orderN = "bd_Card_order";

 val bd_CinfiniteN = "bd_Cinfinite";

 val bd_CnotzeroN = "bd_Cnotzero";

 val collect_set_mapN = "collect_set_map";

 val in_bdN = "in_bd";

 val in_monoN = "in_mono";

 val in_relN = "in_rel";

 val map_id0N = "map_id0";

 val map_idN = "map_id";

 val map_comp0N = "map_comp0";

 val map_compN = "map_comp";

 val map_cong0N = "map_cong0";

 val map_congN = "map_cong";

 val map_transferN = "map_transfer";

 val map_wpullN = "map_wpull";

 val rel_eqN = "rel_eq";

 val rel_flipN = "rel_flip";

 val set_map0N = "set_map0";

 val set_mapN = "set_map";

 val set_bdN = "set_bd";

 val rel_GrpN = "rel_Grp";

 val rel_conversepN = "rel_conversep";

 val rel_monoN = "rel_mono"

 val rel_mono_strongN = "rel_mono_strong"

 val rel_comppN = "rel_compp";

 val rel_compp_GrpN = "rel_compp_Grp";

 datatype const_policy = Dont_Inline | Hardly_Inline | Smart_Inline | Do_Inline;

 datatype fact_policy = Dont_Note | Note_Some | Note_All;

 val bnf_note_all = Attrib.setup_config_bool @{binding bnf_note_all} (K false);

 val bnf_timing = Attrib.setup_config_bool @{binding bnf_timing} (K false);

 fun user_policy policy ctxt = if Config.get ctxt bnf_note_all then Note_All else policy;

 val smart_max_inline_size = 25; (*FUDGE*)

 fun note_bnf_thms fact_policy qualify' bnf_b bnf =

   let

     val axioms = axioms_of_bnf bnf;

     val facts = facts_of_bnf bnf;

     val wits = wits_of_bnf bnf;

     val qualify =

       let val (_, qs, _) = Binding.dest bnf_b;

       in fold_rev (fn (s, mand) => Binding.qualify mand s) qs #> qualify' end;

   in

     (if fact_policy = Note_All then

       let

         val witNs = if length wits = 1 then [witN] else map mk_witN (1 upto length wits);

         val notes =

           [(bd_card_orderN, [#bd_card_order axioms]),

             (bd_cinfiniteN, [#bd_cinfinite axioms]),

             (bd_Card_orderN, [#bd_Card_order facts]),

             (bd_CinfiniteN, [#bd_Cinfinite facts]),

             (bd_CnotzeroN, [#bd_Cnotzero facts]),

             (collect_set_mapN, [Lazy.force (#collect_set_map facts)]),

             (in_bdN, [Lazy.force (#in_bd facts)]),

             (in_monoN, [Lazy.force (#in_mono facts)]),

             (in_relN, [Lazy.force (#in_rel facts)]),

             (map_comp0N, [#map_comp0 axioms]),

             (map_id0N, [#map_id0 axioms]),

             (map_transferN, [Lazy.force (#map_transfer facts)]),

             (map_wpullN, [#map_wpull axioms]),

             (rel_mono_strongN, [Lazy.force (#rel_mono_strong facts)]),

             (set_map0N, #set_map0 axioms),

             (set_bdN, #set_bd axioms)] @

             (witNs ~~ wit_thmss_of_bnf bnf)

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

               ((qualify (Binding.qualify true (Binding.name_of bnf_b) (Binding.name thmN)), []),

               [(thms, [])]));

         in

           Local_Theory.notes notes #> snd

         end

       else

         I)

     #> (if fact_policy <> Dont_Note then

         let

           val notes =

             [(map_compN, [Lazy.force (#map_comp facts)], []),

             (map_cong0N, [#map_cong0 axioms], []),

             (map_congN, [Lazy.force (#map_cong facts)], fundefcong_attrs),

             (map_idN, [Lazy.force (#map_id facts)], []),

             (rel_comppN, [Lazy.force (#rel_OO facts)], []),

             (rel_compp_GrpN, no_refl [#rel_OO_Grp axioms], []),

             (rel_conversepN, [Lazy.force (#rel_conversep facts)], []),

             (rel_eqN, [Lazy.force (#rel_eq facts)], []),

             (rel_flipN, [Lazy.force (#rel_flip facts)], []),

             (rel_GrpN, [Lazy.force (#rel_Grp facts)], []),

             (rel_monoN, [Lazy.force (#rel_mono facts)], []),

             (set_mapN, map Lazy.force (#set_map facts), [])]

             |> filter_out (null o #2)

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

               ((qualify (Binding.qualify true (Binding.name_of bnf_b) (Binding.name thmN)),

                 attrs), [(thms, [])]));

         in

           Local_Theory.notes notes #> snd

         end

       else

         I)

   end;

 (* Define new BNFs *)

 fun prepare_def const_policy mk_fact_policy qualify prep_typ prep_term Ds_opt map_b rel_b set_bs

   ((((((raw_bnf_b, raw_bnf_T), raw_map), raw_sets), raw_bd), raw_wits), raw_rel_opt)

   no_defs_lthy =

   let

     val fact_policy = mk_fact_policy no_defs_lthy;

     val bnf_b = qualify raw_bnf_b;

     val live = length raw_sets;

     val T_rhs = prep_typ no_defs_lthy raw_bnf_T;

     val map_rhs = prep_term no_defs_lthy raw_map;

     val set_rhss = map (prep_term no_defs_lthy) raw_sets;

     val bd_rhs = prep_term no_defs_lthy raw_bd;

     fun err T =

       error ("Trying to register the type " ^ quote (Syntax.string_of_typ no_defs_lthy T) ^

         " as unnamed BNF");

     val (bnf_b, key) =

       if Binding.eq_name (bnf_b, Binding.empty) then

         (case T_rhs of

           Type (C, Ts) => if forall (can dest_TFree) Ts

             then (Binding.qualified_name C, C) else err T_rhs

         | T => err T)

       else (bnf_b, Local_Theory.full_name no_defs_lthy bnf_b);

     val def_qualify = Binding.conceal o Binding.qualify false (Binding.name_of bnf_b);

     fun mk_prefix_binding pre = Binding.prefix_name (pre ^ "_") bnf_b;

     fun maybe_define user_specified (b, rhs) lthy =

       let

         val inline =

           (user_specified orelse fact_policy = Dont_Note) andalso

           (case const_policy of

             Dont_Inline => false

           | Hardly_Inline => Term.is_Free rhs orelse Term.is_Const rhs

           | Smart_Inline => Term.size_of_term rhs <= smart_max_inline_size

           | Do_Inline => true)

       in

         if inline then

           ((rhs, Drule.reflexive_thm), lthy)

         else

           let val b = b () in

             apfst (apsnd snd) (Local_Theory.define ((b, NoSyn), ((Thm.def_binding b, []), rhs))

               lthy)

           end

       end;

     fun maybe_restore lthy_old lthy =

       lthy |> not (pointer_eq (lthy_old, lthy)) ? Local_Theory.restore;

     val map_bind_def =

       (fn () => def_qualify (if Binding.is_empty map_b then mk_prefix_binding mapN else map_b),

          map_rhs);

     val set_binds_defs =

       let

         fun set_name i get_b =

           (case try (nth set_bs) (i - 1) of

             SOME b => if Binding.is_empty b then get_b else K b

           | NONE => get_b) #> def_qualify;

         val bs = if live = 1 then [set_name 1 (fn () => mk_prefix_binding setN)]

           else map (fn i => set_name i (fn () => mk_prefix_binding (mk_setN i))) (1 upto live);

       in bs ~~ set_rhss end;

     val bd_bind_def = (fn () => def_qualify (mk_prefix_binding bdN), bd_rhs);

     val ((((bnf_map_term, raw_map_def),

       (bnf_set_terms, raw_set_defs)),

       (bnf_bd_term, raw_bd_def)), (lthy, lthy_old)) =

         no_defs_lthy

         |> maybe_define true map_bind_def

         ||>> apfst split_list o fold_map (maybe_define true) set_binds_defs

         ||>> maybe_define true bd_bind_def

         ||> `(maybe_restore no_defs_lthy);

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val bnf_map_def = Morphism.thm phi raw_map_def;

     val bnf_set_defs = map (Morphism.thm phi) raw_set_defs;

     val bnf_bd_def = Morphism.thm phi raw_bd_def;

     val bnf_map = Morphism.term phi bnf_map_term;

     (*TODO: handle errors*)

     (*simple shape analysis of a map function*)

     val ((alphas, betas), (CA, _)) =

       fastype_of bnf_map

       |> strip_typeN live

       |>> map_split dest_funT

       ||> dest_funT

       handle TYPE _ => error "Bad map function";

     val CA_params = map TVar (Term.add_tvarsT CA []);

     val bnf_T = Morphism.typ phi T_rhs;

     val bad_args = Term.add_tfreesT bnf_T [];

     val _ = if null bad_args then () else error ("Locally fixed type arguments " ^

       commas_quote (map (Syntax.string_of_typ no_defs_lthy o TFree) bad_args));

     val bnf_sets = map2 (normalize_set CA_params) alphas (map (Morphism.term phi) bnf_set_terms);

     val bnf_bd =

       Term.subst_TVars (Term.add_tvar_namesT bnf_T [] ~~ CA_params) (Morphism.term phi bnf_bd_term);

     (*TODO: assert Ds = (TVars of bnf_map) \ (alphas @ betas) as sets*)

     val deads = (case Ds_opt of

       NONE => subtract (op =) (alphas @ betas) (map TVar (Term.add_tvars bnf_map []))

     | SOME Ds => map (Morphism.typ phi) Ds);

     val dead = length deads;

     (*TODO: further checks of type of bnf_map*)

     (*TODO: check types of bnf_sets*)

     (*TODO: check type of bnf_bd*)

     (*TODO: check type of bnf_rel*)

     val ((((((((((As', Bs'), Cs), Ds), B1Ts), B2Ts), domTs), ranTs), ranTs'), ranTs''),

       (Ts, T)) = lthy

       |> mk_TFrees live

       ||>> mk_TFrees live

       ||>> mk_TFrees live

       ||>> mk_TFrees dead

       ||>> mk_TFrees live

       ||>> mk_TFrees live

       ||>> mk_TFrees live

       ||>> mk_TFrees live

       ||>> mk_TFrees live

       ||>> mk_TFrees live

       ||> fst o mk_TFrees 1

       ||> the_single

       ||> `(replicate live);

     fun mk_bnf_map As' Bs' =

       Term.subst_atomic_types ((deads ~~ Ds) @ (alphas ~~ As') @ (betas ~~ Bs')) bnf_map;

     fun mk_bnf_t As' = Term.subst_atomic_types ((deads ~~ Ds) @ (alphas ~~ As'));

     fun mk_bnf_T As' = Term.typ_subst_atomic ((deads ~~ Ds) @ (alphas ~~ As'));

     val RTs = map HOLogic.mk_prodT (As' ~~ Bs');

     val pred2RTs = map2 mk_pred2T As' Bs';

     val pred2RTsAsCs = map2 mk_pred2T As' Cs;

     val pred2RTsBsCs = map2 mk_pred2T Bs' Cs;

     val pred2RT's = map2 mk_pred2T Bs' As';

     val self_pred2RTs = map2 mk_pred2T As' As';

     val transfer_domRTs = map2 mk_pred2T As' B1Ts;

     val transfer_ranRTs = map2 mk_pred2T Bs' B2Ts;

     val CA' = mk_bnf_T As' CA;

     val CB' = mk_bnf_T Bs' CA;

     val CC' = mk_bnf_T Cs CA;

     val CRs' = mk_bnf_T RTs CA;

     val CB1 = mk_bnf_T B1Ts CA;

     val CB2 = mk_bnf_T B2Ts CA;

     val bnf_map_AsAs = mk_bnf_map As' As';

     val bnf_map_AsBs = mk_bnf_map As' Bs';

     val bnf_map_AsCs = mk_bnf_map As' Cs;

     val bnf_map_BsCs = mk_bnf_map Bs' Cs;

     val bnf_sets_As = map (mk_bnf_t As') bnf_sets;

     val bnf_sets_Bs = map (mk_bnf_t Bs') bnf_sets;

     val bnf_bd_As = mk_bnf_t As' bnf_bd;

     val pre_names_lthy = lthy;

     val ((((((((((((((((((((((((fs, gs), hs), x), y), zs), ys), As),

       As_copy), Xs), B1s), B2s), f1s), f2s), e1s), e2s), p1s), p2s), bs), (Rs, Rs')), Rs_copy), Ss),

       transfer_domRs), transfer_ranRs), names_lthy) = pre_names_lthy

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

       ||>> mk_Frees "g" (map2 (curry op -->) Bs' Cs)

       ||>> mk_Frees "h" (map2 (curry op -->) As' Ts)

       ||>> yield_singleton (mk_Frees "x") CA'

       ||>> yield_singleton (mk_Frees "y") CB'

       ||>> mk_Frees "z" As'

       ||>> mk_Frees "y" Bs'

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

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

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

       ||>> mk_Frees "B1" (map HOLogic.mk_setT B1Ts)

       ||>> mk_Frees "B2" (map HOLogic.mk_setT B2Ts)

       ||>> mk_Frees "f1" (map2 (curry op -->) B1Ts ranTs)

       ||>> mk_Frees "f2" (map2 (curry op -->) B2Ts ranTs)

       ||>> mk_Frees "e1" (map2 (curry op -->) B1Ts ranTs')

       ||>> mk_Frees "e2" (map2 (curry op -->) B2Ts ranTs'')

       ||>> mk_Frees "p1" (map2 (curry op -->) domTs B1Ts)

       ||>> mk_Frees "p2" (map2 (curry op -->) domTs B2Ts)

       ||>> mk_Frees "b" As'

       ||>> mk_Frees' "R" pred2RTs

       ||>> mk_Frees "R" pred2RTs

       ||>> mk_Frees "S" pred2RTsBsCs

       ||>> mk_Frees "R" transfer_domRTs

       ||>> mk_Frees "S" transfer_ranRTs;

     val fs_copy = map2 (retype_free o fastype_of) fs gs;

     val x_copy = retype_free CA' y;

     val setRs =

       map3 (fn R => fn T => fn U =>

           HOLogic.Collect_const (HOLogic.mk_prodT (T, U)) $ HOLogic.mk_split R) Rs As' Bs';

     (*Grp (in (Collect (split R1) .. Collect (split Rn))) (map fst .. fst)^--1 OO

       Grp (in (Collect (split R1) .. Collect (split Rn))) (map snd .. snd)*)

     val OO_Grp =

       let

         val map1 = Term.list_comb (mk_bnf_map RTs As', map fst_const RTs);

         val map2 = Term.list_comb (mk_bnf_map RTs Bs', map snd_const RTs);

         val bnf_in = mk_in setRs (map (mk_bnf_t RTs) bnf_sets) CRs';

       in

         mk_rel_compp (mk_conversep (mk_Grp bnf_in map1), mk_Grp bnf_in map2)

       end;

     val rel_rhs = (case raw_rel_opt of

         NONE => fold_rev Term.absfree Rs' OO_Grp

       | SOME raw_rel => prep_term no_defs_lthy raw_rel);

     val rel_bind_def =

       (fn () => def_qualify (if Binding.is_empty rel_b then mk_prefix_binding relN else rel_b),

          rel_rhs);

     val wit_rhss =

       if null raw_wits then

         [fold_rev Term.absdummy As' (Term.list_comb (bnf_map_AsAs,

           map2 (fn T => fn i => Term.absdummy T (Bound i)) As' (live downto 1)) $

           Const (@{const_name undefined}, CA'))]

       else map (prep_term no_defs_lthy) raw_wits;

     val nwits = length wit_rhss;

     val wit_binds_defs =

       let

         val bs = if nwits = 1 then [fn () => def_qualify (mk_prefix_binding witN)]

           else map (fn i => fn () => def_qualify (mk_prefix_binding (mk_witN i))) (1 upto nwits);

       in bs ~~ wit_rhss end;

     val (((bnf_rel_term, raw_rel_def), (bnf_wit_terms, raw_wit_defs)), (lthy, lthy_old)) =

       lthy

       |> maybe_define (is_some raw_rel_opt) rel_bind_def

       ||>> apfst split_list o fold_map (maybe_define (not (null raw_wits))) wit_binds_defs

       ||> `(maybe_restore lthy);

     val phi = Proof_Context.export_morphism lthy_old lthy;

     val bnf_rel_def = Morphism.thm phi raw_rel_def;

     val bnf_rel = Morphism.term phi bnf_rel_term;

     fun mk_bnf_rel RTs CA' CB' = normalize_rel lthy RTs CA' CB' bnf_rel;

     val rel = mk_bnf_rel pred2RTs CA' CB';

     val relAsAs = mk_bnf_rel self_pred2RTs CA' CA';

     val bnf_wit_defs = map (Morphism.thm phi) raw_wit_defs;

     val bnf_wits = map (normalize_wit CA_params CA alphas o Morphism.term phi) bnf_wit_terms;

     val bnf_wit_As = map (apsnd (mk_bnf_t As')) bnf_wits;

     val map_id0_goal =

       let val bnf_map_app_id = Term.list_comb (bnf_map_AsAs, map HOLogic.id_const As') in

         mk_Trueprop_eq (bnf_map_app_id, HOLogic.id_const CA')

       end;

     val map_comp0_goal =

       let

         val bnf_map_app_comp = Term.list_comb (bnf_map_AsCs, map2 (curry HOLogic.mk_comp) gs fs);

         val comp_bnf_map_app = HOLogic.mk_comp

           (Term.list_comb (bnf_map_BsCs, gs), Term.list_comb (bnf_map_AsBs, fs));

       in

         fold_rev Logic.all (fs @ gs) (mk_Trueprop_eq (bnf_map_app_comp, comp_bnf_map_app))

       end;

     fun mk_map_cong_prem x z set f f_copy =

       Logic.all z (Logic.mk_implies

         (HOLogic.mk_Trueprop (HOLogic.mk_mem (z, set $ x)),

         mk_Trueprop_eq (f $ z, f_copy $ z)));

     val map_cong0_goal =

       let

         val prems = map4 (mk_map_cong_prem x) zs bnf_sets_As fs fs_copy;

         val eq = mk_Trueprop_eq (Term.list_comb (bnf_map_AsBs, fs) $ x,

           Term.list_comb (bnf_map_AsBs, fs_copy) $ x);

       in

         fold_rev Logic.all (x :: fs @ fs_copy) (Logic.list_implies (prems, eq))

       end;

     val set_map0s_goal =

       let

         fun mk_goal setA setB f =

           let

             val set_comp_map =

               HOLogic.mk_comp (setB, Term.list_comb (bnf_map_AsBs, fs));

             val image_comp_set = HOLogic.mk_comp (mk_image f, setA);

           in

             fold_rev Logic.all fs (mk_Trueprop_eq (set_comp_map, image_comp_set))

           end;

       in

         map3 mk_goal bnf_sets_As bnf_sets_Bs fs

       end;

     val card_order_bd_goal = HOLogic.mk_Trueprop (mk_card_order bnf_bd_As);

     val cinfinite_bd_goal = HOLogic.mk_Trueprop (mk_cinfinite bnf_bd_As);

     val set_bds_goal =

       let

         fun mk_goal set =

           Logic.all x (HOLogic.mk_Trueprop (mk_ordLeq (mk_card_of (set $ x)) bnf_bd_As));

       in

         map mk_goal bnf_sets_As

       end;

     val map_wpull_goal =

       let

         val prems = map HOLogic.mk_Trueprop

           (map8 mk_wpull Xs B1s B2s f1s f2s (replicate live NONE) p1s p2s);

         val CX = mk_bnf_T domTs CA;

         val bnf_sets_CX = map2 (normalize_set (map (mk_bnf_T domTs) CA_params)) domTs bnf_sets;

         val bnf_sets_CB1 = map2 (normalize_set (map (mk_bnf_T B1Ts) CA_params)) B1Ts bnf_sets;

         val bnf_sets_CB2 = map2 (normalize_set (map (mk_bnf_T B2Ts) CA_params)) B2Ts bnf_sets;

         val bnf_map_app_f1 = Term.list_comb (mk_bnf_map B1Ts ranTs, f1s);

         val bnf_map_app_f2 = Term.list_comb (mk_bnf_map B2Ts ranTs, f2s);

         val bnf_map_app_p1 = Term.list_comb (mk_bnf_map domTs B1Ts, p1s);

         val bnf_map_app_p2 = Term.list_comb (mk_bnf_map domTs B2Ts, p2s);

         val map_wpull = mk_wpull (mk_in Xs bnf_sets_CX CX)

           (mk_in B1s bnf_sets_CB1 CB1) (mk_in B2s bnf_sets_CB2 CB2)

           bnf_map_app_f1 bnf_map_app_f2 NONE bnf_map_app_p1 bnf_map_app_p2;

       in

         fold_rev Logic.all (Xs @ B1s @ B2s @ f1s @ f2s @ p1s @ p2s)

           (Logic.list_implies (prems, HOLogic.mk_Trueprop map_wpull))

       end;

     val rel_OO_Grp_goal = fold_rev Logic.all Rs (mk_Trueprop_eq (Term.list_comb (rel, Rs), OO_Grp));

     val goals = zip_axioms map_id0_goal map_comp0_goal map_cong0_goal set_map0s_goal

       card_order_bd_goal cinfinite_bd_goal set_bds_goal map_wpull_goal rel_OO_Grp_goal;

     fun mk_wit_goals (I, wit) =

       let

         val xs = map (nth bs) I;

         fun wit_goal i =

           let

             val z = nth zs i;

             val set_wit = nth bnf_sets_As i $ Term.list_comb (wit, xs);

             val concl = HOLogic.mk_Trueprop

               (if member (op =) I i then HOLogic.mk_eq (z, nth bs i)

               else @{term False});

           in

             fold_rev Logic.all (z :: xs)

               (Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (z, set_wit)), concl))

           end;

       in

         map wit_goal (0 upto live - 1)

       end;

     val trivial_wit_tac = mk_trivial_wit_tac bnf_wit_defs;

     val wit_goalss =

       (if null raw_wits then SOME trivial_wit_tac else NONE, map mk_wit_goals bnf_wit_As);

     fun after_qed mk_wit_thms thms lthy =

       let

         val (axioms, nontriv_wit_thms) = apfst (mk_axioms live) (chop (length goals) thms);

         val bd_Card_order = #bd_card_order axioms RS @{thm conjunct2[OF card_order_on_Card_order]};

         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_map () =

           let

             val defT = mk_bnf_T Ts CA --> HOLogic.mk_setT T;

             val collect_map = HOLogic.mk_comp

               (mk_collect (map (mk_bnf_t Ts) bnf_sets) defT,

               Term.list_comb (mk_bnf_map As' Ts, hs));

             val image_collect = mk_collect

               (map2 (fn h => fn set => HOLogic.mk_comp (mk_image h, set)) hs bnf_sets_As)

               defT;

             (*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_map_tac (#set_map0 axioms)))

             |> Thm.close_derivation

           end;

         val collect_set_map = Lazy.lazy mk_collect_set_map;

         fun mk_in_mono () =

           let

             val prems_mono = map2 (HOLogic.mk_Trueprop oo mk_leq) As As_copy;

             val in_mono_goal =

               fold_rev Logic.all (As @ As_copy)

                 (Logic.list_implies (prems_mono, HOLogic.mk_Trueprop

                   (mk_leq (mk_in As bnf_sets_As CA') (mk_in As_copy bnf_sets_As CA'))));

           in

             Goal.prove_sorry lthy [] [] in_mono_goal (K (mk_in_mono_tac live))

             |> Thm.close_derivation

           end;

         val in_mono = Lazy.lazy mk_in_mono;

         fun mk_in_cong () =

           let

             val prems_cong = map2 (curry mk_Trueprop_eq) As As_copy;

             val in_cong_goal =

               fold_rev Logic.all (As @ As_copy)

                 (Logic.list_implies (prems_cong,

                   mk_Trueprop_eq (mk_in As bnf_sets_As CA', mk_in As_copy bnf_sets_As CA')));

           in

             Goal.prove_sorry lthy [] [] in_cong_goal

               (K ((TRY o hyp_subst_tac lthy THEN' rtac refl) 1))

             |> Thm.close_derivation

           end;

         val in_cong = Lazy.lazy mk_in_cong;

         val map_id = Lazy.lazy (fn () => mk_map_id (#map_id0 axioms));

         val map_comp = Lazy.lazy (fn () => mk_map_comp (#map_comp0 axioms));

         fun mk_map_cong () =

           let

             val prem0 = mk_Trueprop_eq (x, x_copy);

             val prems = map4 (mk_map_cong_prem x_copy) zs bnf_sets_As fs fs_copy;

             val eq = mk_Trueprop_eq (Term.list_comb (bnf_map_AsBs, fs) $ x,

               Term.list_comb (bnf_map_AsBs, fs_copy) $ x_copy);

             val goal = fold_rev Logic.all (x :: x_copy :: fs @ fs_copy)

               (Logic.list_implies (prem0 :: prems, eq));

           in

             Goal.prove_sorry lthy [] [] goal (fn _ => mk_map_cong_tac lthy (#map_cong0 axioms))

             |> Thm.close_derivation

           end;

         val map_cong = Lazy.lazy mk_map_cong;

         val set_map = map (fn thm => Lazy.lazy (fn () => mk_set_map thm)) (#set_map0 axioms);

         val wit_thms =

           if null nontriv_wit_thms then mk_wit_thms (map Lazy.force set_map) else nontriv_wit_thms;

         fun mk_in_bd () =

           let

             val bdT = fst (dest_relT (fastype_of bnf_bd_As));

             val bdTs = replicate live bdT;

             val bd_bnfT = mk_bnf_T bdTs CA;

             val surj_imp_ordLeq_inst = (if live = 0 then TrueI else

               let

                 val ranTs = map (fn AT => mk_sumT (AT, HOLogic.unitT)) As';

                 val funTs = map (fn T => bdT --> T) ranTs;

                 val ran_bnfT = mk_bnf_T ranTs CA;

                 val (revTs, Ts) = `rev (bd_bnfT :: funTs);

                 val cTs = map (SOME o certifyT lthy) [ran_bnfT, Library.foldr1 HOLogic.mk_prodT Ts];

                 val tinst = fold (fn T => fn t => HOLogic.mk_split (Term.absdummy T t)) (tl revTs)

                   (Term.absdummy (hd revTs) (Term.list_comb (mk_bnf_map bdTs ranTs,

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

                 val cts = [NONE, SOME (certify lthy tinst)];

               in

                 Drule.instantiate' cTs cts @{thm surj_imp_ordLeq}

               end);

             val bd = mk_cexp

               (if live = 0 then ctwo

                 else mk_csum (Library.foldr1 (uncurry mk_csum) (map mk_card_of As)) ctwo)

               (mk_csum bnf_bd_As (mk_card_of (HOLogic.mk_UNIV bd_bnfT)));

             val in_bd_goal =

               fold_rev Logic.all As

                 (HOLogic.mk_Trueprop (mk_ordLeq (mk_card_of (mk_in As bnf_sets_As CA')) bd));

           in

             Goal.prove_sorry lthy [] [] in_bd_goal

               (mk_in_bd_tac live surj_imp_ordLeq_inst

                 (Lazy.force map_comp) (Lazy.force map_id) (#map_cong0 axioms)

                 (map Lazy.force set_map) (#set_bd axioms) (#bd_card_order axioms)

                 bd_Card_order bd_Cinfinite bd_Cnotzero)

             |> Thm.close_derivation

           end;

         val in_bd = Lazy.lazy mk_in_bd;

         fun mk_map_wppull () =

           let

             val prems = if live = 0 then [] else

               [HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

                 (map8 mk_wpull Xs B1s B2s f1s f2s (map SOME (e1s ~~ e2s)) p1s p2s))];

             val CX = mk_bnf_T domTs CA;

             val bnf_sets_CX =

               map2 (normalize_set (map (mk_bnf_T domTs) CA_params)) domTs bnf_sets;

             val bnf_sets_CB1 =

               map2 (normalize_set (map (mk_bnf_T B1Ts) CA_params)) B1Ts bnf_sets;

             val bnf_sets_CB2 =

               map2 (normalize_set (map (mk_bnf_T B2Ts) CA_params)) B2Ts bnf_sets;

             val bnf_map_app_f1 = Term.list_comb (mk_bnf_map B1Ts ranTs, f1s);

             val bnf_map_app_f2 = Term.list_comb (mk_bnf_map B2Ts ranTs, f2s);

             val bnf_map_app_e1 = Term.list_comb (mk_bnf_map B1Ts ranTs', e1s);

             val bnf_map_app_e2 = Term.list_comb (mk_bnf_map B2Ts ranTs'', e2s);

             val bnf_map_app_p1 = Term.list_comb (mk_bnf_map domTs B1Ts, p1s);

             val bnf_map_app_p2 = Term.list_comb (mk_bnf_map domTs B2Ts, p2s);

             val concl = mk_wpull (mk_in Xs bnf_sets_CX CX)

               (mk_in B1s bnf_sets_CB1 CB1) (mk_in B2s bnf_sets_CB2 CB2)

               bnf_map_app_f1 bnf_map_app_f2 (SOME (bnf_map_app_e1, bnf_map_app_e2))

               bnf_map_app_p1 bnf_map_app_p2;

             val goal =

               fold_rev Logic.all (Xs @ B1s @ B2s @ f1s @ f2s @ e1s @ e2s @ p1s @ p2s)

                 (Logic.list_implies (prems, HOLogic.mk_Trueprop concl))

           in

             Goal.prove_sorry lthy [] [] goal

               (fn _ => mk_map_wppull_tac (#map_id0 axioms) (#map_cong0 axioms)

                 (#map_wpull axioms) (Lazy.force map_comp) (map Lazy.force set_map))

             |> Thm.close_derivation

           end;

         val map_wppull = Lazy.lazy mk_map_wppull;

         val rel_OO_Grp = #rel_OO_Grp axioms;

         val rel_OO_Grps = no_refl [rel_OO_Grp];

         fun mk_rel_Grp () =

           let

             val lhs = Term.list_comb (rel, map2 mk_Grp As fs);

             val rhs = mk_Grp (mk_in As bnf_sets_As CA') (Term.list_comb (bnf_map_AsBs, fs));

             val goal = fold_rev Logic.all (As @ fs) (mk_Trueprop_eq (lhs, rhs));

           in

             Goal.prove_sorry lthy [] [] goal

               (mk_rel_Grp_tac rel_OO_Grps (#map_id0 axioms) (#map_cong0 axioms) (Lazy.force map_id)

                 (Lazy.force map_comp) (map Lazy.force set_map))

             |> Thm.close_derivation

           end;

         val rel_Grp = Lazy.lazy mk_rel_Grp;

         fun mk_rel_prems f = map2 (HOLogic.mk_Trueprop oo f) Rs Rs_copy

         fun mk_rel_concl f = HOLogic.mk_Trueprop

           (f (Term.list_comb (rel, Rs), Term.list_comb (rel, Rs_copy)));

         fun mk_rel_mono () =

           let

             val mono_prems = mk_rel_prems mk_leq;

             val mono_concl = mk_rel_concl (uncurry mk_leq);

           in

             Goal.prove_sorry lthy [] []

               (fold_rev Logic.all (Rs @ Rs_copy) (Logic.list_implies (mono_prems, mono_concl)))

               (K (mk_rel_mono_tac rel_OO_Grps (Lazy.force in_mono)))

             |> Thm.close_derivation

           end;

         fun mk_rel_cong () =

           let

             val cong_prems = mk_rel_prems (curry HOLogic.mk_eq);

             val cong_concl = mk_rel_concl HOLogic.mk_eq;

           in

             Goal.prove_sorry lthy [] []

               (fold_rev Logic.all (Rs @ Rs_copy) (Logic.list_implies (cong_prems, cong_concl)))

               (fn _ => (TRY o hyp_subst_tac lthy THEN' rtac refl) 1)

             |> Thm.close_derivation

           end;

         val rel_mono = Lazy.lazy mk_rel_mono;

         val rel_cong = Lazy.lazy mk_rel_cong;

         fun mk_rel_eq () =

           Goal.prove_sorry lthy [] []

             (mk_Trueprop_eq (Term.list_comb (relAsAs, map HOLogic.eq_const As'),

               HOLogic.eq_const CA'))

             (K (mk_rel_eq_tac live (Lazy.force rel_Grp) (Lazy.force rel_cong) (#map_id0 axioms)))

           |> Thm.close_derivation;

         val rel_eq = Lazy.lazy mk_rel_eq;

         fun mk_rel_conversep () =

           let

             val relBsAs = mk_bnf_rel pred2RT's CB' CA';

             val lhs = Term.list_comb (relBsAs, map mk_conversep Rs);

             val rhs = mk_conversep (Term.list_comb (rel, Rs));

             val le_goal = fold_rev Logic.all Rs (HOLogic.mk_Trueprop (mk_leq lhs rhs));

             val le_thm = Goal.prove_sorry lthy [] [] le_goal

               (mk_rel_conversep_le_tac rel_OO_Grps (Lazy.force rel_eq) (#map_cong0 axioms)

                 (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

             Goal.prove_sorry lthy [] [] goal

               (K (mk_rel_conversep_tac le_thm (Lazy.force rel_mono)))

             |> Thm.close_derivation

           end;

         val rel_conversep = Lazy.lazy mk_rel_conversep;

         fun mk_rel_OO () =

           let

             val relAsCs = mk_bnf_rel pred2RTsAsCs CA' CC';

             val relBsCs = mk_bnf_rel pred2RTsBsCs CB' CC';

             val lhs = Term.list_comb (relAsCs, map2 (curry mk_rel_compp) Rs Ss);

             val rhs = mk_rel_compp (Term.list_comb (rel, Rs), Term.list_comb (relBsCs, Ss));

             val goal = fold_rev Logic.all (Rs @ Ss) (mk_Trueprop_eq (lhs, rhs));

           in

             Goal.prove_sorry lthy [] [] goal

               (mk_rel_OO_tac rel_OO_Grps (Lazy.force rel_eq) (#map_cong0 axioms)

                 (Lazy.force map_wppull) (Lazy.force map_comp) (map Lazy.force set_map))

             |> Thm.close_derivation

           end;

         val rel_OO = Lazy.lazy mk_rel_OO;

         fun mk_in_rel () = trans OF [rel_OO_Grp, @{thm OO_Grp_alt}] RS @{thm predicate2_eqD};

         val in_rel = Lazy.lazy mk_in_rel;

         fun mk_rel_flip () =

           let

             val rel_conversep_thm = Lazy.force rel_conversep;

             val cts = map (SOME o certify lthy) Rs;

             val rel_conversep_thm' = cterm_instantiate_pos cts rel_conversep_thm;

           in

             unfold_thms lthy @{thms conversep_iff} (rel_conversep_thm' RS @{thm predicate2_eqD})

             |> singleton (Proof_Context.export names_lthy pre_names_lthy)

           end;

         val rel_flip = Lazy.lazy mk_rel_flip;

         fun mk_rel_mono_strong () =

           let

             fun mk_prem setA setB R S a b =

               HOLogic.mk_Trueprop

                 (mk_Ball (setA $ x) (Term.absfree (dest_Free a)

                   (mk_Ball (setB $ y) (Term.absfree (dest_Free b)

                     (HOLogic.mk_imp (R $ a $ b, S $ a $ b))))));

             val prems = HOLogic.mk_Trueprop (Term.list_comb (rel, Rs) $ x $ y) :: 

               map6 mk_prem bnf_sets_As bnf_sets_Bs Rs Rs_copy zs ys;

             val concl = HOLogic.mk_Trueprop (Term.list_comb (rel, Rs_copy) $ x $ y);

           in

             Goal.prove_sorry lthy [] []

               (fold_rev Logic.all (x :: y :: Rs @ Rs_copy) (Logic.list_implies (prems, concl)))

               (mk_rel_mono_strong_tac (Lazy.force in_rel) (map Lazy.force set_map))

             |> Thm.close_derivation

           end;

         val rel_mono_strong = Lazy.lazy mk_rel_mono_strong;

         fun mk_map_transfer () =

           let

             val rels = map2 mk_fun_rel transfer_domRs transfer_ranRs;

             val rel = mk_fun_rel

               (Term.list_comb (mk_bnf_rel transfer_domRTs CA' CB1, transfer_domRs))

               (Term.list_comb (mk_bnf_rel transfer_ranRTs CB' CB2, transfer_ranRs));

             val concl = HOLogic.mk_Trueprop

               (fold_rev mk_fun_rel rels rel $ bnf_map_AsBs $ mk_bnf_map B1Ts B2Ts);

           in

             Goal.prove_sorry lthy [] []

               (fold_rev Logic.all (transfer_domRs @ transfer_ranRs) concl)

               (mk_map_transfer_tac (Lazy.force rel_mono) (Lazy.force in_rel)

                 (map Lazy.force set_map) (#map_cong0 axioms) (Lazy.force map_comp))

             |> Thm.close_derivation

           end;

         val map_transfer = Lazy.lazy mk_map_transfer;

         val defs = mk_defs bnf_map_def bnf_set_defs bnf_rel_def;

         val facts = mk_facts bd_Card_order bd_Cinfinite bd_Cnotzero collect_set_map in_bd in_cong

           in_mono in_rel map_comp map_cong map_id map_transfer map_wppull rel_eq rel_flip set_map

           rel_cong rel_mono rel_mono_strong rel_Grp rel_conversep rel_OO;

         val wits = map2 mk_witness bnf_wits wit_thms;

         val bnf_rel =

           Term.subst_atomic_types ((Ds ~~ deads) @ (As' ~~ alphas) @ (Bs' ~~ betas)) rel;

         val bnf = mk_bnf bnf_b CA live alphas betas dead deads bnf_map bnf_sets bnf_bd axioms defs

           facts wits bnf_rel;

       in

         (bnf, lthy |> note_bnf_thms fact_policy qualify bnf_b bnf)

       end;

     val one_step_defs =

       no_reflexive (bnf_map_def :: bnf_bd_def :: bnf_set_defs @ bnf_wit_defs @ [bnf_rel_def]);

   in

     (key, goals, wit_goalss, after_qed, lthy, one_step_defs)

   end;

 fun register_bnf key (bnf, lthy) =

   (bnf, Local_Theory.declaration {syntax = false, pervasive = true}

     (fn phi => Data.map (Symtab.default (key, morph_bnf phi bnf))) lthy);

 fun bnf_def const_policy fact_policy qualify tacs wit_tac Ds map_b rel_b set_bs =

   (fn (_, goals, (triv_tac_opt, wit_goalss), after_qed, lthy, one_step_defs) =>

   let

     fun mk_wits_tac set_maps =

       K (TRYALL Goal.conjunction_tac) THEN'

       (case triv_tac_opt of

         SOME tac => tac set_maps

       | NONE => mk_unfold_thms_then_tac lthy one_step_defs wit_tac);

     val wit_goals = map Logic.mk_conjunction_balanced wit_goalss;

     fun mk_wit_thms set_maps =

       Goal.prove_sorry lthy [] [] (Logic.mk_conjunction_balanced wit_goals) (mk_wits_tac set_maps)

         |> Conjunction.elim_balanced (length wit_goals)

         |> map2 (Conjunction.elim_balanced o length) wit_goalss

         |> map (map (Thm.close_derivation o Thm.forall_elim_vars 0));

   in

     map2 (Thm.close_derivation oo Goal.prove_sorry lthy [] [])

       goals (map (mk_unfold_thms_then_tac lthy one_step_defs) tacs)

     |> (fn thms => after_qed mk_wit_thms (map single thms) lthy)

   end) oo prepare_def const_policy fact_policy qualify (K I) (K I) Ds map_b rel_b set_bs;

 val bnf_cmd = (fn (key, goals, (triv_tac_opt, wit_goalss), after_qed, lthy, defs) =>

   let

     val wit_goals = map Logic.mk_conjunction_balanced wit_goalss;

     fun mk_triv_wit_thms tac set_maps =

       Goal.prove_sorry lthy [] [] (Logic.mk_conjunction_balanced wit_goals)

         (K (TRYALL Goal.conjunction_tac) THEN' tac set_maps)

         |> Conjunction.elim_balanced (length wit_goals)

         |> map2 (Conjunction.elim_balanced o length) wit_goalss

         |> map (map (Thm.close_derivation o Thm.forall_elim_vars 0));

     val (mk_wit_thms, nontriv_wit_goals) = 

       (case triv_tac_opt of

         NONE => (fn _ => [], map (map (rpair [])) wit_goalss)

       | SOME tac => (mk_triv_wit_thms tac, []));

   in

     Proof.unfolding ([[(defs, [])]])

       (Proof.theorem NONE (snd o register_bnf key oo after_qed mk_wit_thms)

         (map (single o rpair []) goals @ nontriv_wit_goals) lthy)

   end) oo prepare_def Do_Inline (user_policy Note_Some) I Syntax.read_typ Syntax.read_term NONE

     Binding.empty Binding.empty [];

 fun print_bnfs ctxt =

   let

     fun pretty_set sets i = Pretty.block

       [Pretty.str (mk_setN (i + 1) ^ ":"), Pretty.brk 1,

           Pretty.quote (Syntax.pretty_term ctxt (nth sets i))];

     fun pretty_bnf (key, BNF {T = T, map = map, sets = sets, bd = bd,

       live = live, lives = lives, dead = dead, deads = deads, ...}) =

       Pretty.big_list

         (Pretty.string_of (Pretty.block [Pretty.str key, Pretty.str ":", Pretty.brk 1,

           Pretty.quote (Syntax.pretty_typ ctxt T)]))

         ([Pretty.block [Pretty.str "live:", Pretty.brk 1, Pretty.str (string_of_int live),

             Pretty.brk 3, Pretty.list "[" "]" (List.map (Syntax.pretty_typ ctxt) lives)],

           Pretty.block [Pretty.str "dead:", Pretty.brk 1, Pretty.str (string_of_int dead),

             Pretty.brk 3, Pretty.list "[" "]" (List.map (Syntax.pretty_typ ctxt) deads)],

           Pretty.block [Pretty.str (mapN ^ ":"), Pretty.brk 1,

             Pretty.quote (Syntax.pretty_term ctxt map)]] @

           List.map (pretty_set sets) (0 upto length sets - 1) @

           [Pretty.block [Pretty.str (bdN ^ ":"), Pretty.brk 1,

             Pretty.quote (Syntax.pretty_term ctxt bd)]]);

   in

     Pretty.big_list "BNFs:" (map pretty_bnf (Symtab.dest (Data.get (Context.Proof ctxt))))

     |> Pretty.writeln

   end;

 val _ =

   Outer_Syntax.improper_command @{command_spec "print_bnfs"}

     "print all bounded natural functors"

     (Scan.succeed (Toplevel.keep (print_bnfs o Toplevel.context_of)));

 val _ =

   Outer_Syntax.local_theory_to_proof @{command_spec "bnf"}

     "register a type as a bounded natural functor"

     (parse_opt_binding_colon -- Parse.typ --|

        (Parse.reserved "map" -- @{keyword ":"}) -- Parse.term --

        (Scan.option ((Parse.reserved "sets" -- @{keyword ":"}) |--

          Scan.repeat1 (Scan.unless (Parse.reserved "bd") Parse.term)) >> the_default []) --|

        (Parse.reserved "bd" -- @{keyword ":"}) -- Parse.term --

        (Scan.option ((Parse.reserved "wits" -- @{keyword ":"}) |--

          Scan.repeat1 (Scan.unless (Parse.reserved "rel") Parse.term)) >> the_default []) --

        Scan.option ((Parse.reserved "rel" -- @{keyword ":"}) |-- Parse.term)

        >> bnf_cmd);

 end;