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"] *} *)

    Type

  | KPi "rty" "name" "rkind"

and rty =

    TConst "ident"

  | TApp "rty" "rtrm"

  | TPi "rty" "name" "rty"

and rtrm =

    Const "ident"

  | Var "name"

  | App "rtrm" "rtrm"

  | Lam "rty" "name" "rtrm"

ML {*

val thy1 = @{theory};

val info = Datatype.the_info @{theory} "Lift.rkind"

val number = 3;

val bvs = []

val bv_simps = []

val ntnames = [@{binding kind}, @{binding ty}, @{binding trm}]

  rVr5 "name"

| rAp5 "rtrm5" "rtrm5"

| rLt5 "rlts" "rtrm5" --"bind (bv5 lts) in (rtrm5)"

and rlts =

  rLnil

| rLcons "name" "rtrm5" "rlts"

primrec

  rbv5

where

  "rbv5 rLnil = {}"

| "rbv5 (rLcons n t ltl) = {atom n} \<union> (rbv5 ltl)"

val thy1 = @{theory};

val info = Datatype.the_info @{theory} "Lift.rtrm5"

val number = 2;

val bvs = [(@{term rbv5}, 1)]

val bv_simps = @{thms rbv5.simps}

val ntnames = [@{binding trm5}, @{binding lts}]

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_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;

  fn (bv_t, i) => prove_const_rsp Binding.empty [bv_t]

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 thy3 = Local_Theory.exit_global lthy10;

(* TODO: fix this hack... *)

(*val thy4 = define_lifted_perms ["Term1.trm1"] [("permute_trm1", @{term "permute :: perm \<Rightarrow> rtrm1 \<Rightarrow> rtrm1"})]

  @{thms permute_rtrm1_permute_bp_zero permute_rtrm1_permute_bp_append}*)

val lthy11 = Theory_Target.init NONE thy3;

val lift_induct = snd (Quotient_Tacs.lifted_attrib (Context.Proof lthy11, induct));

val lthy12 = snd (Local_Theory.note ((@{binding lift_induct}, []), [lift_induct]) lthy11);

*}

setup {* fn _ => Local_Theory.exit_global lthy12 *}

thm lift_induct

end