(*  Title:      nominal_dt_rawfuns.ML+
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    Author:     Cezary Kaliszyk+
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    Author:     Christian Urban+
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  Definitions of the raw fv and fv_bn functions+
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*)+
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signature NOMINAL_DT_RAWFUNS =+
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sig+
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  (* info of binding functions *)+
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  type bn_info = (term * int * (int * term option) list list) list+
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  (* binding modes and binding clauses *)+
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  datatype bmode = Lst | Res | Set+
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  datatype bclause = BC of bmode * (term option * int) list * int list+
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+
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  val is_atom: Proof.context -> typ -> bool+
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  val is_atom_set: Proof.context -> typ -> bool+
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  val is_atom_fset: Proof.context -> typ -> bool+
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  val is_atom_list: Proof.context -> typ -> bool+
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  val mk_atom_set: term -> term+
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  val mk_atom_fset: term -> term+
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+
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+
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  val setify: Proof.context -> term -> term+
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  val listify: Proof.context -> term -> term+
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+
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  val define_raw_fvs: Datatype_Aux.descr -> (string * sort) list -> bn_info ->+
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    bclause list list list -> Proof.context -> term list * term list * thm list * local_theory+
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end+
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structure Nominal_Dt_RawFuns: NOMINAL_DT_RAWFUNS =+
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struct+
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type bn_info = (term * int * (int * term option) list list) list+
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+
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datatype bmode = Lst | Res | Set+
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datatype bclause = BC of bmode * (term option * int) list * int list+
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+
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(* testing for concrete atom types *)+
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fun is_atom ctxt ty =+
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  Sign.of_sort (ProofContext.theory_of ctxt) (ty, @{sort at_base})+
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+
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fun is_atom_set ctxt (Type ("fun", [t, @{typ bool}])) = is_atom ctxt t+
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  | is_atom_set _ _ = false;+
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fun is_atom_fset ctxt (Type (@{type_name "fset"}, [t])) = is_atom ctxt t+
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  | is_atom_fset _ _ = false;+
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+
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fun is_atom_list ctxt (Type (@{type_name "list"}, [t])) = is_atom ctxt t+
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  | is_atom_list _ _ = false+
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(* functions for producing sets, fsets and lists of general atom type+
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   out from concrete atom types *)+
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fun mk_atom_set t =+
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let+
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  val ty = fastype_of t;+
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  val atom_ty = HOLogic.dest_setT ty --> @{typ "atom"};+
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  val img_ty = atom_ty --> ty --> @{typ "atom set"};+
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in+
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  Const (@{const_name image}, img_ty) $ mk_atom_ty atom_ty t+
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end+
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fun mk_atom_fset t =+
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let+
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  val ty = fastype_of t;+
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  val atom_ty = dest_fsetT ty --> @{typ "atom"};+
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  val fmap_ty = atom_ty --> ty --> @{typ "atom fset"};+
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  val fset_to_set = @{term "fset_to_set :: atom fset => atom set"}+
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in+
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  fset_to_set $ (Const (@{const_name fmap}, fmap_ty) $ Const (@{const_name atom}, atom_ty) $ t)+
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end+
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fun mk_atom_list t =+
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let+
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  val ty = fastype_of t;+
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  val atom_ty = dest_listT ty --> @{typ atom};+
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  val map_ty = atom_ty --> ty --> @{typ "atom list"};+
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in+
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  Const (@{const_name map}, map_ty) $ mk_atom_ty atom_ty t+
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end+
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+
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(* functions that coerces singletons, sets and fsets of concrete atoms+
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   into sets of general atoms *)+
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fun setify ctxt t =+
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let+
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  val ty = fastype_of t;+
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in+
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  if is_atom ctxt ty+
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    then  HOLogic.mk_set @{typ atom} [mk_atom t]+
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  else if is_atom_set ctxt ty+
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    then mk_atom_set t+
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  else if is_atom_fset ctxt ty+
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    then mk_atom_fset t+
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  else raise TERM ("setify", [t])+
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end+
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+
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(* functions that coerces singletons and lists of concrete atoms+
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   into lists of general atoms  *)+
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fun listify ctxt t =+
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let+
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  val ty = fastype_of t;+
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in+
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  if is_atom ctxt ty+
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    then HOLogic.mk_list @{typ atom} [mk_atom t]+
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  else if is_atom_list ctxt ty+
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    then mk_atom_set t+
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  else raise TERM ("listify", [t])+
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end+
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+
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(* coerces a list into a set *)+
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fun to_set t =+
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  if fastype_of t = @{typ "atom list"}+
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  then @{term "set::atom list => atom set"} $ t+
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  else t+
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(** functions that construct the equations for fv and fv_bn **)+
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fun mk_fv_rhs lthy fv_map fv_bn_map args (BC (_, binders, bodies)) =+
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let+
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  fun mk_fv_body fv_map args i = +
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  let+
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    val arg = nth args i+
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    val ty = fastype_of arg+
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  in+
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    case AList.lookup (op=) fv_map ty of+
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      NONE => mk_supp arg+
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    | SOME fv => fv $ arg+
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  end  +
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  fun mk_fv_binder lthy fv_bn_map args (bn_option, i) = +
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  let+
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    val arg = nth args i+
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  in+
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    case bn_option of+
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      NONE => (setify lthy arg, @{term "{}::atom set"})+
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    | SOME bn => (to_set (bn $ arg), the (AList.lookup (op=) fv_bn_map bn) $ arg)+
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  end  +
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  val t1 = map (mk_fv_body fv_map args) bodies+
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  val (t2, t3) = split_list (map (mk_fv_binder lthy fv_bn_map args) binders)+
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in +
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  fold_union (mk_diff (fold_union t1, fold_union t2)::t3)+
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end+
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(* in case of fv_bn we have to treat the case special, where an+
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   "empty" binding clause is given *)+
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fun mk_fv_bn_rhs lthy fv_map fv_bn_map bn_args args bclause =+
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let+
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  fun mk_fv_bn_body fv_map fv_bn_map bn_args args i = +
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  let+
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    val arg = nth args i+
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    val ty = fastype_of arg+
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  in+
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    case AList.lookup (op=) bn_args i of+
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      NONE => (case (AList.lookup (op=) fv_map ty) of+
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                 NONE => mk_supp arg+
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               | SOME fv => fv $ arg)+
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    | SOME (NONE) => @{term "{}::atom set"}+
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    | SOME (SOME bn) => the (AList.lookup (op=) fv_bn_map bn) $ arg+
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  end  +
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in+
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  case bclause of+
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    BC (_, [], bodies) => fold_union (map (mk_fv_bn_body fv_map fv_bn_map bn_args args) bodies)+
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  | _ => mk_fv_rhs lthy fv_map fv_bn_map args bclause+
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end+
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fun mk_fv_eq lthy fv_map fv_bn_map (constr, ty, arg_tys, _) bclauses = +
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let+
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  val arg_names = Datatype_Prop.make_tnames arg_tys+
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  val args = map Free (arg_names ~~ arg_tys)+
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  val fv = the (AList.lookup (op=) fv_map ty)+
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  val lhs = fv $ list_comb (constr, args)+
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  val rhs_trms = map (mk_fv_rhs lthy fv_map fv_bn_map args) bclauses+
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  val rhs = fold_union rhs_trms+
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in+
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  HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))+
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end+
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fun mk_fv_bn_eq lthy bn_trm fv_map fv_bn_map (bn_args, (constr, _, arg_tys, _)) bclauses =+
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let+
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  val arg_names = Datatype_Prop.make_tnames arg_tys+
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  val args = map Free (arg_names ~~ arg_tys)+
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  val fv_bn = the (AList.lookup (op=) fv_bn_map bn_trm)+
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  val lhs = fv_bn $ list_comb (constr, args)+
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  val rhs_trms = map (mk_fv_bn_rhs lthy fv_map fv_bn_map bn_args args) bclauses+
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  val rhs = fold_union rhs_trms+
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in+
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  HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))+
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end+
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fun mk_fv_bn_eqs lthy fv_map fv_bn_map constrs_info bclausesss (bn_trm, bn_n, bn_argss) = +
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let+
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  val nth_constrs_info = nth constrs_info bn_n+
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  val nth_bclausess = nth bclausesss bn_n+
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in+
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  map2 (mk_fv_bn_eq lthy bn_trm fv_map fv_bn_map) (bn_argss ~~ nth_constrs_info) nth_bclausess+
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end+
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fun define_raw_fvs descr sorts bn_info bclausesss lthy =+
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let+
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  val fv_names = prefix_dt_names descr sorts "fv_"+
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  val fv_arg_tys = all_dtyps descr sorts+
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  val fv_tys = map (fn ty => ty --> @{typ "atom set"}) fv_arg_tys;+
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  val fv_frees = map Free (fv_names ~~ fv_tys);+
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  val fv_map = fv_arg_tys ~~ fv_frees+
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  val (bns, bn_tys) = split_list (map (fn (bn, i, _) => (bn, i)) bn_info)+
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  val bn_names = map (fn bn => Long_Name.base_name (fst (dest_Const bn))) bns+
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  val fv_bn_names = map (prefix "fv_") bn_names+
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  val fv_bn_arg_tys = map (fn i => nth_dtyp descr sorts i) bn_tys+
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  val fv_bn_tys = map (fn ty => ty --> @{typ "atom set"}) fv_bn_arg_tys+
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  val fv_bn_frees = map Free (fv_bn_names ~~ fv_bn_tys)+
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  val fv_bn_map = bns ~~ fv_bn_frees+
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  val constrs_info = all_dtyp_constrs_types descr sorts+
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  val fv_eqs = map2 (map2 (mk_fv_eq lthy fv_map fv_bn_map)) constrs_info bclausesss +
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  val fv_bn_eqs = map (mk_fv_bn_eqs lthy fv_map fv_bn_map constrs_info bclausesss) bn_info+
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  +
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  val all_fun_names = map (fn s => (Binding.name s, NONE, NoSyn)) (fv_names @ fv_bn_names)+
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  val all_fun_eqs = map (pair Attrib.empty_binding) (flat fv_eqs @ flat fv_bn_eqs)+
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  fun pat_completeness_auto lthy =+
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    Pat_Completeness.pat_completeness_tac lthy 1+
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      THEN auto_tac (clasimpset_of lthy)+
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  fun prove_termination lthy =+
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    Function.prove_termination NONE+
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      (Lexicographic_Order.lexicographic_order_tac true lthy) lthy+
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  val (_, lthy') = Function.add_function all_fun_names all_fun_eqs+
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    Function_Common.default_config pat_completeness_auto lthy+
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  val (info, lthy'') = prove_termination (Local_Theory.restore lthy')+
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  val {fs, simps, ...} = info;+
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  val morphism = ProofContext.export_morphism lthy'' lthy+
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  val fs_exp = map (Morphism.term morphism) fs+
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  val (fv_frees_exp, fv_bns_exp) = chop (length fv_frees) fs_exp+
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  val simps_exp = Morphism.fact morphism (the simps)+
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in+
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  (fv_frees_exp, fv_bns_exp, simps_exp, lthy'')+
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end+
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end (* structure *)+
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