(*  Title:      HOL/Codatatype/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 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 setN: string

   val relN: string

   val predN: string

   val mk_setN: int -> string

   val rel_unfoldN: string

   val pred_unfoldN: string

   val map_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_pred_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_natural_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_comp'_of_bnf: BNF -> thm

   val map_comp_of_bnf: BNF -> thm

   val map_cong_of_bnf: BNF -> thm

   val map_def_of_bnf: BNF -> thm

   val map_id'_of_bnf: BNF -> thm

   val map_id_of_bnf: BNF -> thm

   val map_wppull_of_bnf: BNF -> thm

   val map_wpull_of_bnf: BNF -> thm

   val pred_def_of_bnf: BNF -> thm

   val rel_Gr_of_bnf: BNF -> thm

   val rel_Id_of_bnf: BNF -> thm

   val rel_O_of_bnf: BNF -> thm

   val rel_cong_of_bnf: BNF -> thm

   val rel_converse_of_bnf: BNF -> thm

   val rel_def_of_bnf: BNF -> thm

   val rel_mono_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 sets_of_bnf: BNF -> term list

   val wit_thms_of_bnf: BNF -> thm list

   val wit_thmss_of_bnf: BNF -> thm list 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

   datatype const_policy = Dont_Inline | Hardly_Inline | Smart_Inline | Do_Inline

   datatype fact_policy =

     Derive_Some_Facts | Derive_All_Facts | Derive_All_Facts_Note_Most | Note_All_Facts_and_Axioms

   val bnf_note_all: bool Config.T

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

   val print_bnfs: Proof.context -> unit

   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 * term) * term list) * term) * term list -> local_theory ->

     BNF * local_theory

   val filter_refl: thm list -> thm list

   val bnf_def_cmd: (((binding * string) * string list) * string) * string list -> local_theory ->

     Proof.state

 end;

 structure BNF_Def : BNF_DEF =

 struct

 open BNF_Util

 open BNF_Def_Tactics

 type axioms = {

   map_id: thm,

   map_comp: thm,

   map_cong: thm,

   set_natural: thm list,

   bd_card_order: thm,

   bd_cinfinite: thm,

   set_bd: thm list,

   in_bd: thm,

   map_wpull: thm

 };

 fun mk_axioms' ((((((((id, comp), cong), nat), c_o), cinf), set_bd), in_bd), wpull) =

   {map_id = id, map_comp = comp, map_cong = cong, set_natural = nat, bd_card_order = c_o,

    bd_cinfinite = cinf, set_bd = set_bd, in_bd = in_bd, map_wpull = wpull};

 fun dest_cons [] = raise 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 dest_axioms {map_id, map_comp, map_cong, set_natural,

   bd_card_order, bd_cinfinite, set_bd, in_bd, map_wpull} =

   [map_id, map_comp, map_cong] @ set_natural @ [bd_card_order, bd_cinfinite] @

   set_bd @ [in_bd, map_wpull];

 fun map_axioms f

   {map_id = map_id, map_comp = map_comp, map_cong = map_cong, set_natural = set_natural,

    bd_card_order = bd_card_order, bd_cinfinite = bd_cinfinite,

    set_bd = set_bd, in_bd = in_bd, map_wpull = map_wpull} =

   {map_id = f map_id,

    map_comp = f map_comp,

    map_cong = f map_cong,

    set_natural = map f set_natural,

    bd_card_order = f bd_card_order,

    bd_cinfinite = f bd_cinfinite,

    set_bd = map f set_bd,

    in_bd = f in_bd,

    map_wpull = f map_wpull};

 val morph_axioms = map_axioms o Morphism.thm;

 type defs = {

   map_def: thm,

   set_defs: thm list,

   rel_def: thm,

   pred_def: thm

 }

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

 fun map_defs f {map_def = map, set_defs = sets, rel_def = rel, pred_def = pred} =

   {map_def = f map, set_defs = List.map f sets, rel_def = f rel, pred_def = f pred};

 val morph_defs = map_defs o Morphism.thm;

 type facts = {

   bd_Card_order: thm,

   bd_Cinfinite: thm,

   bd_Cnotzero: thm,

   collect_set_natural: thm lazy,

   in_cong: thm lazy,

   in_mono: thm lazy,

   in_rel: thm lazy,

   map_comp': thm lazy,

   map_id': thm lazy,

   map_wppull: thm lazy,

   rel_cong: thm lazy,

   rel_mono: thm lazy,

   rel_Id: thm lazy,

   rel_Gr: thm lazy,

   rel_converse: thm lazy,

   rel_O: thm lazy,

   set_natural': thm lazy list

 };

 fun mk_facts bd_Card_order bd_Cinfinite bd_Cnotzero

     collect_set_natural in_cong in_mono in_rel map_comp' map_id' map_wppull

     rel_cong rel_mono rel_Id rel_Gr rel_converse rel_O set_natural' = {

   bd_Card_order = bd_Card_order,

   bd_Cinfinite = bd_Cinfinite,

   bd_Cnotzero = bd_Cnotzero,

   collect_set_natural = collect_set_natural,

   in_cong = in_cong,

   in_mono = in_mono,

   in_rel = in_rel,

   map_comp' = map_comp',

   map_id' = map_id',

   map_wppull = map_wppull,

   rel_cong = rel_cong,

   rel_mono = rel_mono,

   rel_Id = rel_Id,

   rel_Gr = rel_Gr,

   rel_converse = rel_converse,

   rel_O = rel_O,

   set_natural' = set_natural'};

 fun map_facts f {

   bd_Card_order,

   bd_Cinfinite,

   bd_Cnotzero,

   collect_set_natural,

   in_cong,

   in_mono,

   in_rel,

   map_comp',

   map_id',

   map_wppull,

   rel_cong,

   rel_mono,

   rel_Id,

   rel_Gr,

   rel_converse,

   rel_O,

   set_natural'} =

   {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,

     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_id' = Lazy.map f map_id',

     map_wppull = Lazy.map f map_wppull,

     rel_cong = Lazy.map f rel_cong,

     rel_mono = Lazy.map f rel_mono,

     rel_Id = Lazy.map f rel_Id,

     rel_Gr = Lazy.map f rel_Gr,

     rel_converse = Lazy.map f rel_converse,

     rel_O = Lazy.map f rel_O,

     set_natural' = map (Lazy.map f) set_natural'};

 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, only for composition*)

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

   dead: int,

   deads: typ list, (*only for composition*)

   map: term,

   sets: term list,

   bd: term,

   axioms: axioms,

   defs: defs,

   facts: facts,

   nwits: int,

   wits: nonemptiness_witness list,

   rel: term,

   pred: 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;

 (*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;

 val pred_of_bnf = #pred o rep_bnf;

 fun mk_pred_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)) (#pred 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_natural_of_bnf = Lazy.force o #collect_set_natural 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;

 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_id_of_bnf = #map_id o #axioms o rep_bnf;

 val map_id'_of_bnf = Lazy.force o #map_id' o #facts o rep_bnf;

 val map_comp_of_bnf = #map_comp o #axioms o rep_bnf;

 val map_comp'_of_bnf = Lazy.force o #map_comp' o #facts o rep_bnf;

 val map_cong_of_bnf = #map_cong o #axioms 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 pred_def_of_bnf = #pred_def o #defs 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_def_of_bnf = #rel_def o #defs o rep_bnf;

 val rel_Id_of_bnf = Lazy.force o #rel_Id o #facts o rep_bnf;

 val rel_Gr_of_bnf = Lazy.force o #rel_Gr o #facts o rep_bnf;

 val rel_converse_of_bnf = Lazy.force o #rel_converse o #facts o rep_bnf;

 val rel_O_of_bnf = Lazy.force o #rel_O 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 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 pred =

   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, pred = pred};

 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, pred = pred}) =

   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, pred = Morphism.term phi pred};

 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);

 );

 val bnf_of = Symtab.lookup o Data.get o Context.Proof;

 (* 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_relT (instA, instB)))

         Vartab.empty;

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

 fun normalize_pred ctxt instTs instA instB pred =

   let

     val thy = Proof_Context.theory_of ctxt;

     val tyenv =

       Sign.typ_match thy (fastype_of pred,

         Library.foldr (op -->) (instTs, instA --> instB --> HOLogic.boolT)) Vartab.empty;

   in Envir.subst_term (tyenv, Vartab.empty) pred end;

 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 unfold_defs_tac lthy defs mk_tac context = Local_Defs.unfold_tac lthy defs THEN mk_tac context;

 (* 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 predN = "pred";

 val rel_unfoldN = relN ^ "_unfold";

 val pred_unfoldN = predN ^ "_unfold";

 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_naturalN = "collect_set_natural";

 val in_bdN = "in_bd";

 val in_monoN = "in_mono";

 val in_relN = "in_rel";

 val map_idN = "map_id";

 val map_id'N = "map_id'";

 val map_compN = "map_comp";

 val map_comp'N = "map_comp'";

 val map_congN = "map_cong";

 val map_wpullN = "map_wpull";

 val rel_IdN = "rel_Id";

 val rel_GrN = "rel_Gr";

 val rel_converseN = "rel_converse";

 val rel_monoN = "rel_mono"

 val rel_ON = "rel_comp";

 val set_naturalN = "set_natural";

 val set_natural'N = "set_natural'";

 val set_bdN = "set_bd";

 datatype const_policy = Dont_Inline | Hardly_Inline | Smart_Inline | Do_Inline;

 datatype fact_policy =

   Derive_Some_Facts | Derive_All_Facts | Derive_All_Facts_Note_Most | Note_All_Facts_and_Axioms;

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

 fun user_policy policy ctxt =

   if Config.get ctxt bnf_note_all then Note_All_Facts_and_Axioms else policy;

 val smart_max_inline_size = 25; (*FUDGE*)

 val no_def = Drule.reflexive_thm;

 val no_fact = refl;

 fun is_reflexive th =

   let val t = Thm.prop_of th;

   in

     op aconv (Logic.dest_equals t)

     handle TERM _ => op aconv (HOLogic.dest_eq (HOLogic.dest_Trueprop t))

       handle TERM _ => false

   end;

 val filter_refl = filter_out is_reflexive;

 (* Define new BNFs *)

 fun prepare_def const_policy mk_fact_policy qualify prep_term Ds_opt

   ((((raw_b, raw_map), raw_sets), raw_bd_Abs), raw_wits) no_defs_lthy =

   let

     val fact_policy = mk_fact_policy no_defs_lthy;

     val b = qualify raw_b;

     val live = length raw_sets;

     val nwits = length raw_wits;

     val map_rhs = prep_term no_defs_lthy raw_map;

     val set_rhss = map (prep_term no_defs_lthy) raw_sets;

     val (bd_rhsT, bd_rhs) = (case prep_term no_defs_lthy raw_bd_Abs of

       Abs (_, T, t) => (T, t)

     | _ => error "Bad bound constant");

     val wit_rhss = map (prep_term no_defs_lthy) raw_wits;

     fun err T =

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

         " as unnamed BNF");

     val (b, key) =

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

         (case bd_rhsT of

           Type (C, Ts) => if forall (is_some o try dest_TFree) Ts

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

         | T => err T)

       else (b, Local_Theory.full_name no_defs_lthy b);

     val map_bind_def = (fn () => Binding.suffix_name ("_" ^ mapN) b, map_rhs);

     val set_binds_defs =

       let

         val bs = if live = 1 then [fn () => Binding.suffix_name ("_" ^ setN) b]

           else map (fn i => fn () => Binding.suffix_name ("_" ^ mk_setN i) b) (1 upto live)

       in map2 pair bs set_rhss end;

     val bd_bind_def = (fn () => Binding.suffix_name ("_" ^ bdN) b, bd_rhs);

     val wit_binds_defs =

       let

         val bs = if nwits = 1 then [fn () => Binding.suffix_name ("_" ^ witN) b]

           else map (fn i => fn () => Binding.suffix_name ("_" ^ mk_witN i) b) (1 upto nwits);

       in map2 pair bs wit_rhss end;

     fun maybe_define needed_for_extra_facts (b, rhs) lthy =

       let

         val inline =

           (not needed_for_extra_facts orelse fact_policy = Derive_Some_Facts) 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, no_def), 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 lthy0 lthy = lthy |> not (pointer_eq (lthy0, lthy)) ? Local_Theory.restore;

     val (((((bnf_map_term, raw_map_def),

       (bnf_set_terms, raw_set_defs)),

       (bnf_bd_term, raw_bd_def)),

       (bnf_wit_terms, raw_wit_defs)), (lthy', lthy)) =

         no_defs_lthy

         |> maybe_define false map_bind_def

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

         ||>> maybe_define false bd_bind_def

         ||>> apfst split_list o fold_map (maybe_define false) wit_binds_defs

         ||> `(maybe_restore no_defs_lthy);

     (*transforms defined frees into consts (and more)*)

     val phi = Proof_Context.export_morphism lthy 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_wit_defs = map (Morphism.thm phi) raw_wit_defs;

     val one_step_defs = filter_refl (bnf_map_def :: bnf_bd_def :: bnf_set_defs @ bnf_wit_defs);

     val _ = case map_filter (try dest_Free)

         (bnf_map_term :: bnf_set_terms @ [bnf_bd_term] @ bnf_wit_terms) of

         [] => ()

       | frees => Proof_Display.print_consts true lthy (K false) frees;

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

     val bdT = Morphism.typ phi bd_rhsT;

     val bnf_bd =

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

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

     (*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*)

     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' t =

       Term.subst_atomic_types ((deads ~~ Ds) @ (alphas ~~ As')) t;

     fun mk_bnf_T As' T =

       Term.typ_subst_atomic ((deads ~~ Ds) @ (alphas ~~ As')) T;

     val (setRTs, RTs) = map_split (`HOLogic.mk_setT o HOLogic.mk_prodT) (As' ~~ Bs');

     val setRTsAsCs = map (HOLogic.mk_setT o HOLogic.mk_prodT) (As' ~~ Cs);

     val setRTsBsCs = map (HOLogic.mk_setT o HOLogic.mk_prodT) (Bs' ~~ Cs);

     val setRT's = map (HOLogic.mk_setT o HOLogic.mk_prodT) (Bs' ~~ As');

     val self_setRTs = map (HOLogic.mk_setT o HOLogic.mk_prodT) (As' ~~ As');

     val QTs = map2 (fn T => fn U => T --> U --> HOLogic.boolT) As' Bs';

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

     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 ((((((((((((((((((((((((fs, fs_copy), gs), hs), (x, x')), (y, y')), (z, z')), zs), As),

       As_copy), Xs), B1s), B2s), f1s), f2s), e1s), e2s), p1s), p2s), bs),

       (Rs, Rs')), Rs_copy), Ss), (Qs, Qs')), _) = lthy'

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

       ||>> 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 (apfst (op ~~) oo mk_Frees' "x") CA'

       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "y") CB'

       ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "z") CRs'

       ||>> mk_Frees "z" As'

       ||>> 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" setRTs

       ||>> mk_Frees "R" setRTs

       ||>> mk_Frees "S" setRTsBsCs

       ||>> mk_Frees' "Q" QTs;

     val goal_map_id =

       let

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

       in

         HOLogic.mk_Trueprop

           (HOLogic.mk_eq (bnf_map_app_id, HOLogic.id_const CA'))

       end;

     val goal_map_comp =

       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;

     val goal_map_cong =

       let

         fun mk_prem 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 prems = map4 mk_prem zs bnf_sets_As fs fs_copy;

         val eq = HOLogic.mk_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, HOLogic.mk_Trueprop eq))

       end;

     val goal_set_naturals =

       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 goal_card_order_bd = HOLogic.mk_Trueprop (mk_card_order bnf_bd_As);

     val goal_cinfinite_bd = HOLogic.mk_Trueprop (mk_cinfinite bnf_bd_As);

     val goal_set_bds =

       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 goal_in_bd =

       let

         val bd = mk_cexp

           (if live = 0 then ctwo

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

           bnf_bd_As;

       in

         fold_rev Logic.all As

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

       end;

     val goal_map_wpull =

       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 CB1 = mk_bnf_T B1Ts CA;

         val CB2 = mk_bnf_T B2Ts 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 goals =

       [goal_map_id, goal_map_comp, goal_map_cong] @ goal_set_naturals @

       [goal_card_order_bd, goal_cinfinite_bd] @ goal_set_bds @

       [goal_in_bd, goal_map_wpull];

     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 wit_goalss = map mk_wit_goals bnf_wit_As;

     fun after_qed thms lthy =

       let

         val (axioms, 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_lazy f = if fact_policy <> Derive_Some_Facts then Lazy.value (f ()) else Lazy.lazy f;

         fun mk_collect_set_natural () =

           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

             Skip_Proof.prove lthy [] [] goal

               (fn {context = ctxt, ...} => mk_collect_set_natural_tac ctxt (#set_natural axioms))

             |> Thm.close_derivation

           end;

         val collect_set_natural = mk_lazy mk_collect_set_natural;

         fun mk_in_mono () =

           let

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

             val goal_in_mono =

               fold_rev Logic.all (As @ As_copy)

                 (Logic.list_implies (prems_mono, HOLogic.mk_Trueprop

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

           in

             Skip_Proof.prove lthy [] [] goal_in_mono (K (mk_in_mono_tac live))

             |> Thm.close_derivation

           end;

         val in_mono = mk_lazy mk_in_mono;

         fun mk_in_cong () =

           let

             val prems_cong = map2 (HOLogic.mk_Trueprop oo curry HOLogic.mk_eq) As As_copy;

             val goal_in_cong =

               fold_rev Logic.all (As @ As_copy)

                 (Logic.list_implies (prems_cong, HOLogic.mk_Trueprop

                   (HOLogic.mk_eq (mk_in As bnf_sets_As CA', mk_in As_copy bnf_sets_As CA'))));

           in

             Skip_Proof.prove lthy [] [] goal_in_cong (K ((TRY o hyp_subst_tac THEN' rtac refl) 1))

             |> Thm.close_derivation

           end;

         val in_cong = mk_lazy mk_in_cong;

         val map_id' = mk_lazy (fn () => mk_id' (#map_id axioms));

         val map_comp' = mk_lazy (fn () => mk_comp' (#map_comp axioms));

         val set_natural' =

           map (fn thm => mk_lazy (fn () => mk_set_natural' thm)) (#set_natural axioms);

         (* relator *)

         (*%R1 .. Rn. Gr (in R1 .. Rn) (map fst .. fst)^-1 O Gr (in R1 .. Rn) (map snd .. snd)*)

         val rel_rhs =

           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 Rs (map (mk_bnf_t RTs) bnf_sets) CRs';

           in

             fold_rev Term.absfree Rs'

               (mk_rel_comp (mk_converse (mk_Gr bnf_in map1), mk_Gr bnf_in map2))

           end;

         val rel_bind_def = (fn () => Binding.suffix_name ("_" ^ relN) b, rel_rhs);

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

           lthy

           |> maybe_define true rel_bind_def

           ||> `(maybe_restore lthy);

         (*transforms defined frees into consts*)

         val phi = Proof_Context.export_morphism lthy_old lthy;

         val bnf_rel = Morphism.term phi bnf_rel_term;

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

         val relAsBs = mk_bnf_rel setRTs CA' CB';

         val bnf_rel_def = Morphism.thm phi raw_rel_def;

         val rel_def_unabs =

           if fact_policy <> Derive_Some_Facts then

             mk_unabs_def live (bnf_rel_def RS meta_eq_to_obj_eq)

           else

             no_fact;

         val pred_rhs = fold absfree (y' :: x' :: rev Qs') (HOLogic.mk_mem (HOLogic.mk_prod (x, y),

           Term.list_comb (relAsBs, map3 (fn Q => fn T => fn U =>

             HOLogic.Collect_const (HOLogic.mk_prodT (T, U)) $ HOLogic.mk_split Q)

             Qs As' Bs')));

         val pred_bind_def = (fn () => Binding.suffix_name ("_" ^ predN) b, pred_rhs);

         val ((bnf_pred_term, raw_pred_def), (lthy, lthy_old)) =

           lthy

           |> maybe_define true pred_bind_def

           ||> `(maybe_restore lthy);

         (*transforms defined frees into consts*)

         val phi = Proof_Context.export_morphism lthy_old lthy;

         val bnf_pred = Morphism.term phi bnf_pred_term;

         fun mk_bnf_pred QTs CA' CB' = normalize_pred lthy QTs CA' CB' bnf_pred;

         val pred = mk_bnf_pred QTs CA' CB';

         val bnf_pred_def = Morphism.thm phi raw_pred_def;

         val pred_def_unabs =

           if fact_policy <> Derive_Some_Facts then

             mk_unabs_def (live + 2) (bnf_pred_def RS meta_eq_to_obj_eq)

           else

             no_fact;

         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 CB1 = mk_bnf_T B1Ts CA;

             val CB2 = mk_bnf_T B2Ts 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

             Skip_Proof.prove lthy [] [] goal

               (fn _ => mk_map_wppull_tac (#map_id axioms) (#map_cong axioms)

                 (#map_wpull axioms) (Lazy.force map_comp') (map Lazy.force set_natural'))

             |> Thm.close_derivation

           end;

         val map_wppull = mk_lazy mk_map_wppull;

         fun mk_rel_Gr () =

           let

             val lhs = Term.list_comb (relAsBs, map2 mk_Gr As fs);

             val rhs = mk_Gr (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

             Skip_Proof.prove lthy [] [] goal

               (mk_rel_Gr_tac bnf_rel_def (#map_id axioms) (#map_cong axioms)

                 (#map_wpull axioms) (Lazy.force in_cong) (Lazy.force map_id')

                 (Lazy.force map_comp') (map Lazy.force set_natural'))

             |> Thm.close_derivation

           end;

         val rel_Gr = mk_lazy mk_rel_Gr;

         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 (relAsBs, Rs), Term.list_comb (relAsBs, Rs_copy)));

         fun mk_rel_mono () =

           let

             val mono_prems = mk_rel_prems mk_subset;

             val mono_concl = mk_rel_concl (uncurry mk_subset);

           in

             Skip_Proof.prove lthy [] []

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

               (mk_rel_mono_tac bnf_rel_def (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

             Skip_Proof.prove lthy [] []

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

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

             |> Thm.close_derivation

           end;

         val rel_mono = mk_lazy mk_rel_mono;

         val rel_cong = mk_lazy mk_rel_cong;

         fun mk_rel_Id () =

           let val relAsAs = mk_bnf_rel self_setRTs CA' CA' in

             Skip_Proof.prove lthy [] []

               (HOLogic.mk_Trueprop

                 (HOLogic.mk_eq (Term.list_comb (relAsAs, map Id_const As'), Id_const CA')))

               (mk_rel_Id_tac live (Lazy.force rel_Gr) (#map_id axioms))

             |> Thm.close_derivation

           end;

         val rel_Id = mk_lazy mk_rel_Id;

         fun mk_rel_converse () =

           let

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

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

             val rhs = mk_converse (Term.list_comb (relAsBs, Rs));

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

             val le_thm = Skip_Proof.prove lthy [] [] le_goal

               (mk_rel_converse_le_tac bnf_rel_def (Lazy.force rel_Id) (#map_cong axioms)

                 (Lazy.force map_comp') (map Lazy.force set_natural'))

               |> Thm.close_derivation

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

           in

             Skip_Proof.prove lthy [] [] goal (fn _ => mk_rel_converse_tac le_thm)

             |> Thm.close_derivation

           end;

         val rel_converse = mk_lazy mk_rel_converse;

         fun mk_rel_O () =

           let

             val relAsCs = mk_bnf_rel setRTsAsCs CA' CC';

             val relBsCs = mk_bnf_rel setRTsBsCs CB' CC';

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

             val rhs = mk_rel_comp (Term.list_comb (relAsBs, Rs), Term.list_comb (relBsCs, Ss));

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

           in

             Skip_Proof.prove lthy [] [] goal

               (mk_rel_O_tac bnf_rel_def (Lazy.force rel_Id) (#map_cong axioms)

                 (Lazy.force map_wppull) (Lazy.force map_comp') (map Lazy.force set_natural'))

             |> Thm.close_derivation

           end;

         val rel_O = mk_lazy mk_rel_O;

         fun mk_in_rel () =

           let

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

             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 map_fst_eq = HOLogic.mk_eq (map1 $ z, x);

             val map_snd_eq = HOLogic.mk_eq (map2 $ z, y);

             val lhs = HOLogic.mk_mem (HOLogic.mk_prod (x, y), Term.list_comb (relAsBs, Rs));

             val rhs =

               HOLogic.mk_exists (fst z', snd z', HOLogic.mk_conj (HOLogic.mk_mem (z, bnf_in),

                 HOLogic.mk_conj (map_fst_eq, map_snd_eq)));

             val goal =

               fold_rev Logic.all (x :: y :: Rs) (mk_Trueprop_eq (lhs, rhs));

           in

             Skip_Proof.prove lthy [] [] goal (mk_in_rel_tac bnf_rel_def (length bnf_sets))

             |> Thm.close_derivation

           end;

         val in_rel = mk_lazy mk_in_rel;

         val defs = mk_defs bnf_map_def bnf_set_defs rel_def_unabs pred_def_unabs;

         val facts = mk_facts bd_Card_order bd_Cinfinite bd_Cnotzero collect_set_natural

           in_cong in_mono in_rel map_comp' map_id' map_wppull

           rel_cong rel_mono rel_Id rel_Gr rel_converse rel_O set_natural';

         val wits = map2 mk_witness bnf_wits wit_thms;

         val bnf_rel = Term.subst_atomic_types

           ((Ds ~~ deads) @ (As' ~~ alphas) @ (Bs' ~~ betas)) relAsBs;

         val bnf_pred = Term.subst_atomic_types

           ((Ds ~~ deads) @ (As' ~~ alphas) @ (Bs' ~~ betas)) pred;

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

           wits bnf_rel bnf_pred;

       in

         (bnf, lthy

           |> (if fact_policy = Note_All_Facts_and_Axioms 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_naturalN, [Lazy.force (#collect_set_natural facts)]),

                     (in_bdN, [#in_bd axioms]),

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

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

                     (map_compN, [#map_comp axioms]),

                     (map_idN, [#map_id axioms]),

                     (map_wpullN, [#map_wpull axioms]),

                     (set_naturalN, #set_natural axioms),

                     (set_bdN, #set_bd axioms)] @

                     map2 pair witNs wit_thms

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

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

                       [(thms, [])]));

                 in

                   Local_Theory.notes notes #> snd

                 end

               else

                 I)

           |> (if fact_policy = Note_All_Facts_and_Axioms orelse

                  fact_policy = Derive_All_Facts_Note_Most then

                 let

                   val notes =

                     [(map_congN, [#map_cong axioms]),

                     (rel_IdN, [Lazy.force (#rel_Id facts)]),

                     (rel_GrN, [Lazy.force (#rel_Gr facts)]),

                     (rel_converseN, [Lazy.force (#rel_converse facts)]),

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

                     (rel_ON, [Lazy.force (#rel_O facts)]),

                     (map_id'N, [Lazy.force (#map_id' facts)]),

                     (map_comp'N, [Lazy.force (#map_comp' facts)]),

                     (set_natural'N, map Lazy.force (#set_natural' facts))]

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

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

                       [(thms, [])]));

                 in

                   Local_Theory.notes notes #> snd

                 end

               else

                 I))

       end;

   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.update_new (key, morph_bnf phi bnf))) lthy);

 fun bnf_def const_policy fact_policy qualify tacs wit_tac Ds =

   (fn (_, goals, wit_goalss, after_qed, lthy, defs) =>

   let

     val wits_tac = K (TRYALL Goal.conjunction_tac) THEN' unfold_defs_tac lthy defs wit_tac;

     val wit_goals = wit_goalss |> map Logic.mk_conjunction_balanced;

     val wit_goal = Logic.mk_conjunction_balanced wit_goals;

     val wit_thms =

       Skip_Proof.prove lthy [] [] wit_goal wits_tac

       |> 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 Skip_Proof.prove lthy [] [])

       goals (map (unfold_defs_tac lthy defs) tacs)

     |> (fn thms => after_qed (map single thms @ wit_thms) lthy)

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

 val bnf_def_cmd = (fn (key, goals, wit_goals, after_qed, lthy, defs) =>

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

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

       (map (single o rpair []) goals @ map (map (rpair [])) wit_goals) lthy)) oo

   prepare_def Do_Inline (user_policy Derive_All_Facts_Note_Most) I Syntax.read_term NONE;

 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 BNFs"

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

 val _ =

   Outer_Syntax.local_theory_to_proof @{command_spec "bnf_def"} "define a BNF for an existing type"

     ((parse_opt_binding_colon -- Parse.term --

        (@{keyword "["} |-- Parse.list Parse.term --| @{keyword "]"}) -- Parse.term --

        (@{keyword "["} |-- Parse.list Parse.term --| @{keyword "]"})) >> bnf_def_cmd);

 end;