theory Lift
imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm" "Fv" "Rsp" "../Attic/Prove"
begin

atom_decl name
atom_decl ident

datatype rtrm2 =
  rVr2 "name"
| rAp2 "rtrm2" "rtrm2"
| rLt2 "ras" "rtrm2" --"bind (bv2 l) in (r)"
and ras =
  rAs "name" "rtrm2"

primrec rbv2 where "rbv2 (rAs x t) = {atom x}"

ML {*
val thy1 = @{theory};
val info = Datatype.the_info @{theory} "Lift.rtrm2"
val number = 2; (* Number of defined types, rest are unfoldings *)
val binds = [[[[]], [[], []], [[], [(SOME @{term rbv2}, 0)]]],
             [[[], []]]  (*, [[], [[], []]] *) ];
val bvs = [(@{term rbv2}, 1)] (* Which type it operates on *)
val bv_simps = @{thms rbv2.simps}

val ntnames = [@{binding trm2}, @{binding as}]
val ncnames = ["Vr2", "Ap2", "Lt2", "As"]




val descr = #descr info;
val sorts = #sorts info;
val nos = map fst descr
val all_typs = map (fn i => typ_of_dtyp descr sorts (DtRec i)) nos
val typs = List.take (all_typs, number)
val all_full_tnames = map (fn (_, (n, _, _)) => n) descr;
val full_tnames = List.take (all_full_tnames, number);
val induct = #induct info;
val inducts = #inducts info;
val infos = map (Datatype.the_info thy1) all_full_tnames;
val inject = flat (map #inject infos);
val distinct = flat (map #distinct infos);
val ((raw_perm_def, raw_perm_simps, perms), thy2) = define_raw_perms info number thy1;
val lthy1 = Theory_Target.init NONE thy2
val (((fv_ts_loc, fv_def_loc), alpha), lthy2) = define_fv_alpha info binds lthy1;
val alpha_ts_loc = #preds alpha
val alpha_intros = #intrs alpha
val alpha_cases = #elims alpha
val alpha_induct_loc = #induct alpha
val [alpha_induct] = ProofContext.export lthy2 lthy1 [alpha_induct_loc]
(* TODO replace when inducts is provided by the 2 lines below: *)
val alpha_inducts = Project_Rule.projects lthy2 (1 upto number) alpha_induct
(*val alpha_inducts_loc = #inducts alpha
val alpha_inducts = ProofContext.export lthy2 lthy1 alpha_inducts_loc*)
val alpha_inj_loc = build_alpha_inj alpha_intros (inject @ distinct) alpha_cases lthy2
val alpha_inj = ProofContext.export lthy2 lthy1 alpha_inj_loc
val fv_def = ProofContext.export lthy2 lthy1 fv_def_loc
val morphism = ProofContext.export_morphism lthy2 lthy1
val fv_ts = map (Morphism.term morphism) fv_ts_loc
val alpha_ts = map (Morphism.term morphism) alpha_ts_loc
fun build_bv_eqvt (t, n) =
  build_eqvts Binding.empty [t] [nth perms n]
    (bv_simps @ raw_perm_def) (nth inducts n)
val (bv_eqvts, lthy3) = fold_map build_bv_eqvt bvs lthy2;
val (fv_eqvts, lthy4) = build_eqvts Binding.empty fv_ts_loc perms (fv_def_loc @ raw_perm_def) induct lthy3;
val alpha_eqvt_loc = build_alpha_eqvts alpha_ts_loc perms (raw_perm_def @ alpha_inj_loc) alpha_induct_loc lthy4;
val alpha_eqvt = ProofContext.export lthy4 lthy1 alpha_eqvt_loc
val alpha_equivp_loc = build_equivps alpha_ts_loc induct alpha_induct_loc inject alpha_inj_loc distinct alpha_cases alpha_eqvt_loc lthy4;
val alpha_equivp = ProofContext.export lthy4 lthy1 alpha_equivp_loc
val lthy5 = define_quotient_type
  (map (fn ((b, t), alpha) => (([], b, NoSyn), (t, alpha))) ((ntnames ~~ typs) ~~ alpha_ts))
  (ALLGOALS (resolve_tac alpha_equivp)) lthy4;
val 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 (csdefl, lthy6) = fold_map Quotient_Def.quotient_lift_const (ncnames ~~ consts) lthy5;
val (cs, def) = split_list csdefl;
val ((_, fv_rsp), lthy7) = prove_const_rsp Binding.empty fv_ts
  (fn _ => fvbv_rsp_tac alpha_induct fv_def 1) lthy6
val (bvs_rsp', lthy8) = fold_map (
  fn (bv_t, i) => prove_const_rsp Binding.empty [bv_t]
    (fn _ => fvbv_rsp_tac (nth alpha_inducts i) bv_simps 1)) bvs lthy7
val bvs_rsp = flat (map snd bvs_rsp')
val (const_rsps, lthy9) = fold_map (fn cnst => prove_const_rsp Binding.empty [cnst]
  (fn _ => constr_rsp_tac alpha_inj (fv_rsp @ bvs_rsp) alpha_equivp 1)) consts lthy8
val (perms_rsp, lthy10) = prove_const_rsp Binding.empty perms
  (fn _ => asm_simp_tac (HOL_ss addsimps alpha_eqvt) 1) lthy9;
val lift_induct = snd (Quotient_Tacs.lifted_attrib (Context.Proof lthy1, induct));
val lthy11 = snd (Local_Theory.note ((@{binding lift_induct}, []), [lift_induct]) lthy10)
*}

setup {* fn _ => Local_Theory.exit_global lthy11 *}
thm lift_induct

end