(*  Title:      nominal_atoms/ML

    Authors:    Brian Huffman, Christian Urban

    Command for defining concrete atom types. 

    At the moment, only single-sorted atom types

    are supported. 

*)

signature ATOM_DECL =

sig

  val add_atom_decl: (binding * (binding option)) -> theory -> theory

end;

structure Atom_Decl :> ATOM_DECL =

struct

val simp_attr = Attrib.internal (K Simplifier.simp_add)

fun atom_decl_set (str : string) : term =

  let

    val a = Free ("a", @{typ atom});

    val s = Const (@{const_name "Sort"}, @{typ "string => atom_sort list => atom_sort"})

              $ HOLogic.mk_string str $ HOLogic.nil_const @{typ "atom_sort"};

  in

    HOLogic.mk_Collect ("a", @{typ atom}, HOLogic.mk_eq (mk_sort_of a, s))

  end

fun add_atom_decl (name : binding, arg : binding option) (thy : theory) =

  let

    val _ = Theory.requires thy "Nominal2_Base" "nominal logic";

    val str = Sign.full_name thy name;

    (* typedef *)

    val set = atom_decl_set str;

    val tac = rtac @{thm exists_eq_simple_sort} 1;

    val ((full_tname, info  as ({Rep_name, Abs_name, ...}, {type_definition, ...})), thy) =

      Typedef.add_typedef_global false NONE (name, [], NoSyn) set NONE tac thy;

    (* definition of atom and permute *)

    val newT = #abs_type (fst info);

    val RepC = Const (Rep_name, newT --> @{typ atom});

    val AbsC = Const (Abs_name, @{typ atom} --> newT);

    val a = Free ("a", newT);

    val p = Free ("p", @{typ perm});

    val atom_eqn =

      HOLogic.mk_Trueprop (HOLogic.mk_eq (mk_atom a, RepC $ a));

    val permute_eqn =

      HOLogic.mk_Trueprop (HOLogic.mk_eq

        (mk_perm p a, AbsC $ (mk_perm p (RepC $ a))));

    val atom_def_name =

      Binding.prefix_name "atom_" (Binding.suffix_name "_def" name);

    val sort_thm_name =

      Binding.prefix_name "atom_" (Binding.suffix_name "_sort" name);

    val permute_def_name =

      Binding.prefix_name "permute_" (Binding.suffix_name "_def" name);

    (* at class instance *)

    val lthy =

      Class.instantiation ([full_tname], [], @{sort at}) thy;

    val ((_, (_, permute_ldef)), lthy) =

      Specification.definition (NONE, ((permute_def_name, []), permute_eqn)) lthy;

    val ((_, (_, atom_ldef)), lthy) =

      Specification.definition (NONE, ((atom_def_name, []), atom_eqn)) lthy;

    val ctxt_thy = Proof_Context.init_global (Proof_Context.theory_of lthy);

    val permute_def = singleton (Proof_Context.export lthy ctxt_thy) permute_ldef;

    val atom_def = singleton (Proof_Context.export lthy ctxt_thy) atom_ldef;

    val class_thm = @{thm at_class} OF [type_definition, atom_def, permute_def];

    val sort_thm = @{thm at_class_sort} OF [type_definition, atom_def]

    val thy = lthy

      |> snd o (Local_Theory.note ((sort_thm_name, [simp_attr]), [sort_thm]))

      |> Class.prove_instantiation_instance (K (rtac class_thm 1))

      |> Local_Theory.exit_global;

  in

    thy

  end;

(** outer syntax **)

val _ =

  Outer_Syntax.command @{command_spec "atom_decl"}

    "declaration of a concrete atom type" 

      ((Parse.binding -- Scan.option (Args.parens (Parse.binding))) >>

        (Toplevel.print oo (Toplevel.theory o add_atom_decl)))

end;