theory Perm
imports "Nominal2_Atoms"
begin

ML {*
  open Datatype_Aux; (* typ_of_dtyp, DtRec, ... *)
  fun permute ty = Const (@{const_name permute}, @{typ perm} --> ty --> ty);
  val minus_perm = Const (@{const_name minus}, @{typ perm} --> @{typ perm});
*}

ML {*

(* TODO: full_name can be obtained from new_type_names with Datatype *)
fun define_raw_perms new_type_names full_tnames thy =
let
  val {descr, induct, ...} = Datatype.the_info thy (hd full_tnames);
  (* TODO: [] should be the sorts that we'll take from the specification *)
  val sorts = [];
  fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);
  val perm_names' = Datatype_Prop.indexify_names (map (fn (i, _) =>
    "permute_" ^ name_of_typ (nth_dtyp i)) descr);
  val perm_types = map (fn (i, _) =>
    let val T = nth_dtyp i
    in @{typ perm} --> T --> T end) descr;
  val perm_indnames = Datatype_Prop.make_tnames (map body_type perm_types);
  val perm_names_types' = perm_names' ~~ perm_types;
  val pi = Free ("pi", @{typ perm});
  fun perm_eq_constr i (cname, dts) =
    let
      val Ts = map (typ_of_dtyp descr sorts) dts;
      val names = Name.variant_list ["pi"] (Datatype_Prop.make_tnames Ts);
      val args = map Free (names ~~ Ts);
      val c = Const (cname, Ts ---> (nth_dtyp i));
      fun perm_arg (dt, x) =
        let val T = type_of x
        in
          if is_rec_type dt then
            let val (Us, _) = strip_type T
            in list_abs (map (pair "x") Us,
              Free (nth perm_names_types' (body_index dt)) $ pi $
                list_comb (x, map (fn (i, U) =>
                  (permute U) $ (minus_perm $ pi) $ Bound i)
                  ((length Us - 1 downto 0) ~~ Us)))
            end
          else (permute T) $ pi $ x
        end;
    in
      (Attrib.empty_binding, HOLogic.mk_Trueprop (HOLogic.mk_eq
        (Free (nth perm_names_types' i) $
           Free ("pi", @{typ perm}) $ list_comb (c, args),
         list_comb (c, map perm_arg (dts ~~ args)))))
    end;
    fun perm_eq (i, (_, _, constrs)) = map (perm_eq_constr i) constrs;
    val perm_eqs = maps perm_eq descr;
    val lthy =
      Theory_Target.instantiation (full_tnames, [], @{sort pt}) thy;
    (* TODO: Use the version of prmrec that gives the names explicitely. *)
    val ((_, perm_ldef), lthy') =
      Primrec.add_primrec
        (map (fn s => (Binding.name s, NONE, NoSyn)) perm_names') perm_eqs lthy;
    val perm_frees =
      (distinct (op =)) (map (fst o strip_comb o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) perm_ldef);
    val perm_empty_thms =
      let
        val gl =
          HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
            (map (fn ((perm, T), x) => HOLogic.mk_eq
                (perm $ @{term "0 :: perm"} $ Free (x, T),
                 Free (x, T)))
             (perm_frees ~~
              map body_type perm_types ~~ perm_indnames)));
        fun tac {context = ctxt, ...} =
          EVERY [
            indtac induct perm_indnames 1,
            ALLGOALS (asm_full_simp_tac (@{simpset} addsimps perm_ldef))
          ];
      in
        (List.take (split_conj_thm (Goal.prove lthy' perm_indnames [] gl tac), length new_type_names))
      end;
    val add_perm = @{term "op + :: (perm \<Rightarrow> perm \<Rightarrow> perm)"}
    val pi1 = Free ("pi1", @{typ perm});
    val pi2 = Free ("pi2", @{typ perm});
    val perm_append_thms =
       List.take ((split_conj_thm
         (Goal.prove lthy' ("pi1" :: "pi2" :: perm_indnames) []
            (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
               (map (fn ((perm, T), x) =>
                   let
                     val lhs = perm $ (add_perm $ pi1 $ pi2) $ Free (x, T)
                     val rhs = perm $ pi1 $ (perm $ pi2 $ Free (x, T))
                   in HOLogic.mk_eq (lhs, rhs)
                   end)
                 (perm_frees ~~
                  map body_type perm_types ~~ perm_indnames))))
            (fn _ => EVERY [indtac induct perm_indnames 1,
               ALLGOALS (asm_full_simp_tac (@{simpset} addsimps perm_ldef))]))),
          length new_type_names);
    fun tac ctxt perm_thms =
      (Class.intro_classes_tac []) THEN (ALLGOALS (
        simp_tac (@{simpset} addsimps perm_thms
      )));
    fun morphism phi = map (Morphism.thm phi);
  in
    Class_Target.prove_instantiation_exit_result morphism tac (perm_empty_thms @ perm_append_thms) lthy'
  end

*}

(* Test
atom_decl name

datatype rtrm1 =
  rVr1 "name"
| rAp1 "rtrm1" "rtrm1 list"
| rLm1 "name" "rtrm1"
| rLt1 "bp" "rtrm1" "rtrm1"
and bp =
  BUnit
| BVr "name"
| BPr "bp" "bp"


setup {* snd o define_raw_perms ["rtrm1", "bp"] ["Perm.rtrm1", "Perm.bp"] *}
print_theorems
*)

end
