theory Parser

imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Perm" "Fv" "Rsp" "Lift"

begin

section{* Interface for nominal_datatype *}

text {*

Nominal-Datatype-part:

1nd Arg: (string list * binding * mixfix * (binding * typ list * mixfix) list) list

         ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^   ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

               type(s) to be defined             constructors list

               (ty args, name, syn)              (name, typs, syn)

Binder-Function-part:

2rd Arg: (binding * typ option * mixfix) list 

         ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^    

            binding function(s)           

              to be defined               

            (name, type, syn)             

3th Arg:  term list 

          ^^^^^^^^^

          the equations of the binding functions

          (Trueprop equations)

*}

ML {*

*}

text {*****************************************************}

ML {* 

(* nominal datatype parser *)

local

  structure P = OuterParse

  fun tuple ((x, y, z), u) = (x, y, z, u)

  fun tswap (((x, y), z), u) = (x, y, u, z)

in

val _ = OuterKeyword.keyword "bind"

val anno_typ = Scan.option (P.name --| P.$$$ "::") -- P.typ

(* binding specification *)

(* maybe use and_list *)

val bind_parser = 

  P.enum "," ((P.$$$ "bind" |-- P.term) -- (P.$$$ "in" |-- P.name) >> swap)

val constr_parser =

  P.binding -- Scan.repeat anno_typ

(* datatype parser *)

val dt_parser =

  (P.type_args -- P.binding -- P.opt_mixfix >> P.triple1) -- 

    (P.$$$ "=" |-- P.enum1 "|" (constr_parser -- bind_parser -- P.opt_mixfix >> tswap)) >> tuple

(* function equation parser *)

val fun_parser = 

  Scan.optional (P.$$$ "binder" |-- P.fixes -- SpecParse.where_alt_specs) ([],[])

(* main parser *)

val main_parser =

  (P.and_list1 dt_parser) -- fun_parser >> P.triple2

end

*}

(* adds "_raw" to the end of constants and types *)

ML {*

fun add_raw s = s ^ "_raw"

fun add_raws ss = map add_raw ss

fun raw_bind bn = Binding.suffix_name "_raw" bn

fun replace_str ss s = 

  case (AList.lookup (op=) ss s) of 

     SOME s' => s'

   | NONE => s

fun replace_typ ty_ss (Type (a, Ts)) = Type (replace_str ty_ss a, map (replace_typ ty_ss) Ts)

  | replace_typ ty_ss T = T  

fun raw_dts ty_ss dts =

let

  fun raw_dts_aux1 (bind, tys, mx) =

    (raw_bind bind, map (replace_typ ty_ss) tys, mx)

  fun raw_dts_aux2 (ty_args, bind, mx, constrs) =

    (ty_args, raw_bind bind, mx, map raw_dts_aux1 constrs)

in

  map raw_dts_aux2 dts

end

fun replace_aterm trm_ss (Const (a, T)) = Const (replace_str trm_ss a, T)

  | replace_aterm trm_ss (Free (a, T)) = Free (replace_str trm_ss a, T)

  | replace_aterm trm_ss trm = trm

fun replace_term trm_ss ty_ss trm =

  trm |> Term.map_aterms (replace_aterm trm_ss) |> map_types (replace_typ ty_ss) 

*}

ML {*

fun get_cnstrs dts =

  map (fn (_, _, _, constrs) => constrs) dts

fun get_typed_cnstrs dts =

  flat (map (fn (_, bn, _, constrs) => 

   (map (fn (bn', _, _) => (Binding.name_of bn, Binding.name_of bn')) constrs)) dts)

fun get_cnstr_strs dts =

  map (fn (bn, _, _) => Binding.name_of bn) (flat (get_cnstrs dts))

fun get_bn_fun_strs bn_funs =

  map (fn (bn_fun, _, _) => Binding.name_of bn_fun) bn_funs

*}

ML {*

fun rawify_dts dt_names dts dts_env =

let

  val raw_dts = raw_dts dts_env dts

  val raw_dt_names = add_raws dt_names

in

  (raw_dt_names, raw_dts)

end 

*}

ML {*

fun rawify_bn_funs dts_env cnstrs_env bn_fun_env bn_funs bn_eqs =

let

  val bn_funs' = map (fn (bn, ty, mx) => 

    (raw_bind bn, replace_typ dts_env ty, mx)) bn_funs

  val bn_eqs' = map (fn (attr, trm) => 

    (attr, replace_term (cnstrs_env @ bn_fun_env) dts_env trm)) bn_eqs

in

  (bn_funs', bn_eqs') 

end 

*}

ML {* 

fun rawify_binds dts_env cnstrs_env bn_fun_env binds =

  map (map (map (map (fn (opt_trm, i, j) => 

    (Option.map (apfst (replace_term (cnstrs_env @ bn_fun_env) dts_env)) opt_trm, i, j))))) binds

*}

ML {*

fun find [] _ = error ("cannot find element")

  | find ((x, z)::xs) y = if (Long_Name.base_name x) = y then z else find xs y

*}

ML {*

fun prep_bn dt_names dts eqs = 

let

  fun aux eq = 

  let

    val (lhs, rhs) = eq

      |> strip_qnt_body "all" 

      |> HOLogic.dest_Trueprop

      |> HOLogic.dest_eq

    val (bn_fun, [cnstr]) = strip_comb lhs

    val (_, ty) = dest_Free bn_fun

    val (ty_name, _) = dest_Type (domain_type ty)

    val dt_index = find_index (fn x => x = ty_name) dt_names

    val (cnstr_head, cnstr_args) = strip_comb cnstr

    val included = map (fn i => length (cnstr_args) - i - 1) (loose_bnos rhs)

  in

    (dt_index, (bn_fun, (cnstr_head, included)))

  end 

  fun order dts i ts = 

  let

    val dt = nth dts i

    val cts = map (fn (x, _, _) => Binding.name_of x) ((fn (_, _, _, x) => x) dt)

    val ts' = map (fn (x, y) => (fst (dest_Const x), y)) ts

  in

    map (find ts') cts

  end

  val unordered = AList.group (op=) (map aux eqs)

  val unordered' = map (fn (x, y) =>  (x, AList.group (op=) y)) unordered

  val ordered = map (fn (x, y) => (x, map (fn (v, z) => (v, order dts x z)) y)) unordered' 

in

  ordered

end

*}

ML {* 

fun add_primrec_wrapper funs eqs lthy = 

  if null funs then (([], []), lthy)

  else 

   let 

     val eqs' = map (fn (_, eq) => (Attrib.empty_binding, eq)) eqs

     val funs' = map (fn (bn, ty, mx) => (bn, SOME ty, mx)) funs

   in 

     Primrec.add_primrec funs' eqs' lthy

   end

*}

ML {*

fun add_datatype_wrapper dt_names dts =

let

  val conf = Datatype.default_config

in

  Local_Theory.theory_result (Datatype.add_datatype conf dt_names dts)

end

*}

ML {* 

fun raw_nominal_decls dts bn_funs bn_eqs binds lthy =

let

  val thy = ProofContext.theory_of lthy

  val thy_name = Context.theory_name thy

  val dt_names = map (fn (_, s, _, _) => Binding.name_of s) dts

  val dt_full_names = map (Long_Name.qualify thy_name) dt_names 

  val dt_full_names' = add_raws dt_full_names

  val dts_env = dt_full_names ~~ dt_full_names'

  val cnstrs = get_cnstr_strs dts

  val cnstrs_ty = get_typed_cnstrs dts

  val cnstrs_full_names = map (Long_Name.qualify thy_name) cnstrs

  val cnstrs_full_names' = map (fn (x, y) => Long_Name.qualify thy_name 

    (Long_Name.qualify (add_raw x) (add_raw y))) cnstrs_ty

  val cnstrs_env = cnstrs_full_names ~~ cnstrs_full_names'

  val bn_fun_strs = get_bn_fun_strs bn_funs

  val bn_fun_strs' = add_raws bn_fun_strs

  val bn_fun_env = bn_fun_strs ~~ bn_fun_strs'

  val bn_fun_full_env = map (pairself (Long_Name.qualify thy_name)) 

    (bn_fun_strs ~~ bn_fun_strs')

  val (raw_dt_names, raw_dts) = rawify_dts dt_names dts dts_env

  val (raw_bn_funs, raw_bn_eqs) = rawify_bn_funs dts_env cnstrs_env bn_fun_env bn_funs bn_eqs 

  val raw_binds = rawify_binds dts_env cnstrs_env bn_fun_full_env binds 

  val raw_bns = prep_bn dt_full_names' raw_dts (map snd raw_bn_eqs)

(*val _ = tracing (cat_lines (map PolyML.makestring raw_bns))*)

in

  lthy 

  |> add_datatype_wrapper raw_dt_names raw_dts 

  ||>> add_primrec_wrapper raw_bn_funs raw_bn_eqs

  ||>> pair raw_binds

  ||>> pair raw_bns

end

*}

lemma equivp_hack: "equivp x"

sorry

ML {*

fun equivp_hack ctxt rel =

let

  val thy = ProofContext.theory_of ctxt

  val ty = domain_type (fastype_of rel)

  val cty = ctyp_of thy ty

  val ct = cterm_of thy rel

in

  Drule.instantiate' [SOME cty] [SOME ct] @{thm equivp_hack}

end

*}

(* These 4 are not needed any more *)

ML {* val cheat_fv_rsp = Unsynchronized.ref false *}

ML {* val cheat_fv_eqvt = Unsynchronized.ref false *}

ML {* val cheat_alpha_eqvt = Unsynchronized.ref false *}

ML {* val cheat_alpha_bn_rsp = Unsynchronized.ref false *}

ML {*

fun nominal_datatype2 dts bn_funs bn_eqs binds lthy =

let

  val _ = tracing "Raw declarations";

  val thy = ProofContext.theory_of lthy

  val thy_name = Context.theory_name thy

  val ((((raw_dt_names, (raw_bn_funs_loc, raw_bn_eqs_loc)), raw_binds), raw_bns), lthy2) =

    raw_nominal_decls dts bn_funs bn_eqs binds lthy

  val morphism_2_1 = ProofContext.export_morphism lthy2 lthy

  val raw_bns_exp = map (apsnd (map (apfst (Morphism.term morphism_2_1)))) raw_bns;

  val bn_funs_decls = flat (map (fn (ith, l) => map (fn (bn, data) => (bn, ith, data)) l) raw_bns_exp);

  val raw_bn_funs = map (Morphism.term morphism_2_1) raw_bn_funs_loc

  val raw_bn_eqs = ProofContext.export lthy2 lthy raw_bn_eqs_loc

  val dtinfo = Datatype.the_info (ProofContext.theory_of lthy2) (hd raw_dt_names);

  val descr = #descr dtinfo;

  val sorts = #sorts dtinfo;

  val all_typs = map (fn i => typ_of_dtyp descr sorts (DtRec i)) (map fst descr)

  val all_full_tnames = map (fn (_, (n, _, _)) => n) descr;

  val dtinfos = map (Datatype.the_info (ProofContext.theory_of lthy2)) all_full_tnames;

  val rel_dtinfos = List.take (dtinfos, (length dts));

  val inject = flat (map #inject dtinfos);

  val distincts = flat (map #distinct dtinfos);

  val rel_distinct = map #distinct rel_dtinfos;

  val induct = #induct dtinfo;

  val inducts = #inducts dtinfo;

  val exhausts = map #exhaust dtinfos;

  val _ = tracing "Defining permutations, fv and alpha";

  val ((raw_perm_def, raw_perm_simps, perms), lthy3) =

    Local_Theory.theory_result (define_raw_perms dtinfo (length dts)) lthy2;

  val raw_binds_flat = map (map flat) raw_binds;

  val ((((fv_ts, ordered_fv_ts), fv_def), ((alpha_ts, alpha_intros), (alpha_cases, alpha_induct))), lthy4) =

    define_fv_alpha_export dtinfo raw_binds_flat bn_funs_decls lthy3;

  val (fv_ts_nobn, fv_ts_bn) = chop (length perms) fv_ts;

  val (alpha_ts_nobn, alpha_ts_bn) = chop (length perms) alpha_ts

  val alpha_inducts = Project_Rule.projects lthy4 (1 upto (length dts)) alpha_induct;

  val dts_names = map (fn (i, (s, _, _)) => (s, i)) (#descr dtinfo);

  val bn_tys = map (domain_type o fastype_of) raw_bn_funs;

  val bn_nos = map (dtyp_no_of_typ dts_names) bn_tys;

  val bns = raw_bn_funs ~~ bn_nos;

  val rel_dists = flat (map (distinct_rel lthy4 alpha_cases)

    (rel_distinct ~~ alpha_ts_nobn));

  val rel_dists_bn = flat (map (distinct_rel lthy4 alpha_cases)

    ((map (fn i => nth rel_distinct i) bn_nos) ~~ alpha_ts_bn))

  val alpha_eq_iff = build_alpha_inj alpha_intros (inject @ distincts) alpha_cases lthy4

  val _ = tracing "Proving equivariance";

  val (bv_eqvts, lthy5) = fold_map (build_bv_eqvt (raw_bn_eqs @ raw_perm_def) inducts) bns lthy4;

  val fv_eqvt_tac =

    if !cheat_fv_eqvt then (fn _ => fn _ => Skip_Proof.cheat_tac thy)

    else build_eqvts_tac induct ((flat (map snd bv_eqvts)) @ fv_def @ raw_perm_def) lthy5

  val (fv_eqvts, lthy6) = build_eqvts Binding.empty fv_ts_nobn fv_eqvt_tac lthy5;

  val (fv_bn_eqvts, lthy6a) =

    if fv_ts_bn = [] then ([], lthy6) else

    fold_map (build_bv_eqvt ((flat (map snd bv_eqvts)) @ fv_def @ raw_perm_def) inducts)

      (fv_ts_bn ~~ bn_nos) lthy6;

  val raw_fv_bv_eqvt = flat (map snd bv_eqvts) @ (snd fv_eqvts) @ flat (map snd fv_bn_eqvts)

  fun alpha_eqvt_tac' _ =

    if !cheat_alpha_eqvt then Skip_Proof.cheat_tac thy

    else alpha_eqvt_tac alpha_induct (raw_perm_def @ alpha_eq_iff @ raw_fv_bv_eqvt) lthy6a 1

  val alpha_eqvt = build_alpha_eqvts alpha_ts alpha_eqvt_tac' lthy6a;

  val _ = tracing "Proving equivalence";

  val (rfv_ts_nobn, rfv_ts_bn) = chop (length perms) ordered_fv_ts;

  val fv_alpha_all = combine_fv_alpha_bns (rfv_ts_nobn, rfv_ts_bn) (alpha_ts_nobn, alpha_ts_bn) bn_nos;

  val reflps = build_alpha_refl fv_alpha_all alpha_ts induct alpha_eq_iff lthy6a;

  val alpha_equivp =

    if !cheat_equivp then map (equivp_hack lthy6a) alpha_ts_nobn

    else build_equivps alpha_ts reflps alpha_induct

      inject alpha_eq_iff distincts alpha_cases alpha_eqvt lthy6a;

  val qty_binds = map (fn (_, b, _, _) => b) dts;

  val qty_names = map Name.of_binding qty_binds;

  val qty_full_names = map (Long_Name.qualify thy_name) qty_names

  val lthy7 = define_quotient_type

    (map (fn ((b, t), alpha) => (([], b, NoSyn), (t, alpha))) ((qty_binds ~~ all_typs) ~~ alpha_ts_nobn))

    (ALLGOALS (resolve_tac alpha_equivp)) lthy6a;

  val const_names = map Name.of_binding (flat (map (fn (_, _, _, t) => map (fn (b, _, _) => b) t) dts));

  val raw_consts =

    flat (map (fn (i, (_, _, l)) =>

      map (fn (cname, dts) =>

        Const (cname, map (typ_of_dtyp descr sorts) dts --->

          typ_of_dtyp descr sorts (DtRec i))) l) descr);

  val (consts, const_defs, lthy8) = quotient_lift_consts_export (const_names ~~ raw_consts) lthy7;

  (* Could be done nicer *)

  val q_tys = distinct (op =) (map (snd o strip_type o fastype_of) consts);

  val _ = tracing "Proving respects";

  val (bns_rsp_pre, lthy9) = fold_map (

    fn (bn_t, i) => prove_const_rsp Binding.empty [bn_t]

      (fn _ => fvbv_rsp_tac (nth alpha_inducts i) raw_bn_eqs lthy8 1)) bns lthy8;

  val bns_rsp = flat (map snd bns_rsp_pre);

  fun fv_rsp_tac _ = if !cheat_fv_rsp then Skip_Proof.cheat_tac thy

    else fvbv_rsp_tac alpha_induct fv_def lthy8 1;

  val fv_rsps = prove_fv_rsp fv_alpha_all alpha_ts fv_rsp_tac lthy9;

  val (fv_rsp_pre, lthy10) = fold_map

    (fn fv => fn ctxt => prove_const_rsp Binding.empty [fv]

    (fn _ => asm_simp_tac (HOL_ss addsimps fv_rsps) 1) ctxt) fv_ts lthy9;

  val fv_rsp = flat (map snd fv_rsp_pre);

  val (perms_rsp, lthy11) = prove_const_rsp Binding.empty perms

    (fn _ => asm_simp_tac (HOL_ss addsimps alpha_eqvt) 1) lthy10;

  val alpha_alphabn = build_alpha_alphabn fv_alpha_all alpha_inducts alpha_eq_iff lthy11;

  fun const_rsp_tac _ = constr_rsp_tac alpha_eq_iff (fv_rsp @ bns_rsp @ reflps @ alpha_alphabn) 1

  val (const_rsps, lthy11a) = fold_map (fn cnst => prove_const_rsp Binding.empty [cnst]

    const_rsp_tac) raw_consts lthy11

  val alpha_bn_rsp_pre = flat (map (prove_alpha_bn_rsp alpha_ts_nobn alpha_inducts exhausts (alpha_eq_iff @ rel_dists @ rel_dists_bn) (alpha_equivp) lthy11a) (alpha_ts_bn ~~ bn_nos))

  val (alpha_bn_rsps, lthy12) = fold_map (fn cnst => prove_const_rsp Binding.empty [cnst]

    (fn _ => asm_simp_tac (HOL_ss addsimps alpha_bn_rsp_pre) 1)) alpha_ts_bn lthy11a

  val qfv_names = map (unsuffix "_raw" o Long_Name.base_name o fst o dest_Const) ordered_fv_ts

  val (qfv_ts, qfv_defs, lthy12a) = quotient_lift_consts_export (qfv_names ~~ ordered_fv_ts) lthy12;

  val (qfv_ts_nobn, qfv_ts_bn) = chop (length perms) qfv_ts;

  val qbn_names = map (fn (b, _ , _) => Name.of_binding b) bn_funs

  val (qbn_defs, lthy12b) = fold_map Quotient_Def.quotient_lift_const (qbn_names ~~ raw_bn_funs) lthy12a;

  val qalpha_bn_names = map (unsuffix "_raw" o Long_Name.base_name o fst o dest_Const) alpha_ts_bn

  val (qalpha_ts_bn, qbn_defs, lthy12c) = quotient_lift_consts_export (qalpha_bn_names ~~ alpha_ts_bn) lthy12b;

  val _ = tracing "Lifting permutations";

  val thy = Local_Theory.exit_global lthy12c;

  val perm_names = map (fn x => "permute_" ^ x) qty_names

  val thy' = define_lifted_perms qty_full_names (perm_names ~~ perms) raw_perm_simps thy;

  val lthy13 = Theory_Target.init NONE thy';

  val q_name = space_implode "_" qty_names;

  fun suffix_bind s = Binding.qualify true q_name (Binding.name s);

  val _ = tracing "Lifting induction";

  val constr_names = map (Long_Name.base_name o fst o dest_Const) consts;

  val q_induct = Rule_Cases.name constr_names (lift_thm lthy13 induct);

  fun note_suffix s th ctxt =

    snd (Local_Theory.note ((suffix_bind s, []), th) ctxt);

  fun note_simp_suffix s th ctxt =

    snd (Local_Theory.note ((suffix_bind s, [Attrib.internal (K Simplifier.simp_add)]), th) ctxt);

  val (_, lthy14) = Local_Theory.note ((suffix_bind "induct",

    [Attrib.internal (K (Rule_Cases.case_names constr_names))]), [Rule_Cases.name constr_names q_induct]) lthy13;

  val q_inducts = Project_Rule.projects lthy13 (1 upto (length alpha_inducts)) q_induct

  val (_, lthy14a) = Local_Theory.note ((suffix_bind "inducts", []), q_inducts) lthy14;

  val q_perm = map (lift_thm lthy14) raw_perm_def;

  val lthy15 = note_simp_suffix "perm" q_perm lthy14a;

  val q_fv = map (lift_thm lthy15) fv_def;

  val lthy16 = note_simp_suffix "fv" q_fv lthy15;

  val q_bn = map (lift_thm lthy16) raw_bn_eqs;

  val lthy17 = note_simp_suffix "bn" q_bn lthy16;

  val _ = tracing "Lifting eq-iff";

  val eq_iff_unfolded1 = map (Local_Defs.unfold lthy17 @{thms alpha_gen2}) alpha_eq_iff

  val eq_iff_unfolded2 = map (Local_Defs.unfold lthy17 @{thms alpha_gen}) eq_iff_unfolded1

  val q_eq_iff_pre1 = map (lift_thm lthy17) eq_iff_unfolded2;

  val q_eq_iff_pre2 = map (Local_Defs.fold lthy17 @{thms alpha_gen2}) q_eq_iff_pre1

  val q_eq_iff = map (Local_Defs.fold lthy17 @{thms alpha_gen}) q_eq_iff_pre2

  val (_, lthy18) = Local_Theory.note ((suffix_bind "eq_iff", []), q_eq_iff) lthy17;

  val q_dis = map (lift_thm lthy18) rel_dists;

  val lthy19 = note_simp_suffix "distinct" q_dis lthy18;

  val q_eqvt = map (lift_thm lthy19) raw_fv_bv_eqvt;

  val (_, lthy20) = Local_Theory.note ((Binding.empty,

    [Attrib.internal (fn _ => Nominal_ThmDecls.eqvt_add)]), q_eqvt) lthy19;

  val _ = tracing "Finite Support";

  val supports = map (prove_supports lthy20 q_perm) consts;

  val fin_supp = HOLogic.conj_elims (prove_fs lthy20 q_induct supports q_tys);

  val thy3 = Local_Theory.exit_global lthy20;

  val lthy21 = Theory_Target.instantiation (qty_full_names, [], @{sort fs}) thy3;

  fun tac _ = Class.intro_classes_tac [] THEN (ALLGOALS (resolve_tac fin_supp))

  val lthy22 = Class.prove_instantiation_instance tac lthy21

  val fv_alpha_all = combine_fv_alpha_bns (qfv_ts_nobn, qfv_ts_bn) (alpha_ts_nobn, qalpha_ts_bn) bn_nos;

  val (names, supp_eq_t) = supp_eq fv_alpha_all;

  val q_supp = HOLogic.conj_elims (Goal.prove lthy22 names [] supp_eq_t (fn _ => supp_eq_tac q_induct q_fv q_perm q_eq_iff lthy22 1)) handle _ => [];

  val lthy23 = note_suffix "supp" q_supp lthy22;

in

  ((raw_dt_names, raw_bn_funs, raw_bn_eqs, raw_binds), lthy23)

end

*}

ML {* 

(* parsing the datatypes and declaring *)

(* constructors in the local theory    *)

fun prepare_dts dt_strs lthy = 

let

  val thy = ProofContext.theory_of lthy

  fun mk_type full_tname tvrs =

    Type (full_tname, map (fn a => TVar ((a, 0), [])) tvrs)

  fun prep_cnstr lthy full_tname tvs (cname, anno_tys, mx, _) =

  let

    val tys = map (Syntax.read_typ lthy o snd) anno_tys

    val ty = mk_type full_tname tvs

  in

    ((cname, tys ---> ty, mx), (cname, tys, mx))

  end

  fun prep_dt lthy (tvs, tname, mx, cnstrs) = 

  let

    val full_tname = Sign.full_name thy tname

    val (cnstrs', cnstrs'') = 

      split_list (map (prep_cnstr lthy full_tname tvs) cnstrs)

  in

    (cnstrs', (tvs, tname, mx, cnstrs''))

  end 

  val (cnstrs, dts) = 

    split_list (map (prep_dt lthy) dt_strs)

in

  lthy

  |> Local_Theory.theory (Sign.add_consts_i (flat cnstrs))

  |> pair dts

end

*}

ML {*

(* parsing the binding function specification and *)

(* declaring the functions in the local theory    *)

fun prepare_bn_funs bn_fun_strs bn_eq_strs lthy =

let