(*  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 atomT = @{typ atom};

val permT = @{typ perm};

val sort_of_const = @{term sort_of};

fun atom_const T = Const (@{const_name atom}, T --> atomT);

fun permute_const T = Const (@{const_name permute}, permT --> T --> T);

fun mk_sort_of t = sort_of_const $ t;

fun mk_atom t = atom_const (fastype_of t) $ t;

fun mk_permute (p, t) = permute_const (fastype_of t) $ p $ t;

fun atom_decl_set (str : string) : term =

  let

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

    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", atomT, 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_Atoms" "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 {type_definition, Rep_name, Abs_name, ...}), thy) =

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

    (* definition of atom and permute *)

    val newT = #abs_type info;

    val RepC = Const (Rep_name, newT --> atomT);

    val AbsC = Const (Abs_name, atomT --> newT);

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

    val p = Free ("p", permT);

    val atom_eqn =

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

    val permute_eqn =

      HOLogic.mk_Trueprop (HOLogic.mk_eq

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

    val atom_def_name =

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

    val permute_def_name =

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

    (* at class instance *)

    val lthy =

      Theory_Target.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 = ProofContext.init (ProofContext.theory_of lthy);

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

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

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

    val thy = lthy

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

      |> Local_Theory.exit_global;

  in

    thy

  end;

(** outer syntax **)

local structure P = OuterParse and K = OuterKeyword in

val _ =

  OuterSyntax.command "atom_decl" "declaration of a concrete atom type" K.thy_decl

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

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

end;

end;