(* Title: HOL/Tools/BNF/bnf_fp_n2m.ML
Author: Dmitriy Traytel, TU Muenchen
Copyright 2013
Flattening of nested to mutual (co)recursion.
*)
signature BNF_FP_N2M =
sig
val construct_mutualized_fp: BNF_Util.fp_kind -> typ list -> BNF_FP_Def_Sugar.fp_sugar list ->
binding list -> (string * sort) list -> typ list * typ list list -> BNF_Def.bnf list ->
BNF_Comp.absT_info list -> local_theory -> BNF_FP_Util.fp_result * local_theory
end;
structure BNF_FP_N2M : BNF_FP_N2M =
struct
open BNF_Def
open BNF_Util
open BNF_Comp
open BNF_FP_Util
open BNF_FP_Def_Sugar
open BNF_Tactics
open BNF_FP_N2M_Tactics
fun force_typ ctxt T =
map_types Type_Infer.paramify_vars
#> Type.constraint T
#> Syntax.check_term ctxt
#> singleton (Variable.polymorphic ctxt);
fun mk_prod_map f g =
let
val ((fAT, fBT), fT) = `dest_funT (fastype_of f);
val ((gAT, gBT), gT) = `dest_funT (fastype_of g);
in
Const (@{const_name map_pair},
fT --> gT --> HOLogic.mk_prodT (fAT, gAT) --> HOLogic.mk_prodT (fBT, gBT)) $ f $ g
end;
fun mk_sum_map f g =
let
val ((fAT, fBT), fT) = `dest_funT (fastype_of f);
val ((gAT, gBT), gT) = `dest_funT (fastype_of g);
in
Const (@{const_name sum_map}, fT --> gT --> mk_sumT (fAT, gAT) --> mk_sumT (fBT, gBT)) $ f $ g
end;
fun construct_mutualized_fp fp fpTs (fp_sugars : fp_sugar list) bs resBs (resDs, Dss) bnfs
(absT_infos : absT_info list) lthy =
let
fun of_fp_res get =
map (fn {fp_res, fp_res_index, ...} => nth (get fp_res) fp_res_index) fp_sugars;
fun mk_co_algT T U = fp_case fp (T --> U) (U --> T);
fun co_swap pair = fp_case fp I swap pair;
val mk_co_comp = HOLogic.mk_comp o co_swap;
val dest_co_algT = co_swap o dest_funT;
val co_alg_argT = fp_case fp range_type domain_type;
val co_alg_funT = fp_case fp domain_type range_type;
val mk_co_product = curry (fp_case fp mk_convol mk_case_sum);
val mk_map_co_product = fp_case fp mk_prod_map mk_sum_map;
val co_proj1_const = fp_case fp (fst_const o fst) (uncurry Inl_const o dest_sumT o snd);
val mk_co_productT = curry (fp_case fp HOLogic.mk_prodT mk_sumT);
val dest_co_productT = fp_case fp HOLogic.dest_prodT dest_sumT;
val rewrite_comp_comp = fp_case fp @{thm rewriteL_comp_comp} @{thm rewriteR_comp_comp};
val fp_absT_infos = map #absT_info fp_sugars;
val fp_bnfs = of_fp_res #bnfs;
val pre_bnfs = map #pre_bnf fp_sugars;
val nesty_bnfss = map (fn sugar => #nested_bnfs sugar @ #nesting_bnfs sugar) fp_sugars;
val fp_nesty_bnfss = fp_bnfs :: nesty_bnfss;
val fp_nesty_bnfs = distinct (op = o pairself T_of_bnf) (flat fp_nesty_bnfss);
val fp_absTs = map #absT fp_absT_infos;
val fp_repTs = map #repT fp_absT_infos;
val fp_abss = map #abs fp_absT_infos;
val fp_reps = map #rep fp_absT_infos;
val fp_type_definitions = map #type_definition fp_absT_infos;
val absTs = map #absT absT_infos;
val repTs = map #repT absT_infos;
val absTs' = map (Logic.type_map (singleton (Variable.polymorphic lthy))) absTs;
val repTs' = map (Logic.type_map (singleton (Variable.polymorphic lthy))) repTs;
val abss = map #abs absT_infos;
val reps = map #rep absT_infos;
val abs_inverses = map #abs_inverse absT_infos;
val type_definitions = map #type_definition absT_infos;
val n = length bnfs;
val deads = fold (union (op =)) Dss resDs;
val As = subtract (op =) deads (map TFree resBs);
val names_lthy = fold Variable.declare_typ (As @ deads) lthy;
val m = length As;
val live = m + n;
val ((Xs, Bs), names_lthy) = names_lthy
|> mk_TFrees n
||>> mk_TFrees m;
val allAs = As @ Xs;
val allBs = Bs @ Xs;
val phiTs = map2 mk_pred2T As Bs;
val thetaBs = As ~~ Bs;
val fpTs' = map (Term.typ_subst_atomic thetaBs) fpTs;
val fold_thetaAs = Xs ~~ fpTs;
val fold_thetaBs = Xs ~~ fpTs';
val rec_theta = Xs ~~ map2 mk_co_productT fpTs Xs;
val pre_phiTs = map2 mk_pred2T fpTs fpTs';
val ((ctors, dtors), (xtor's, xtors)) =
let
val ctors = map2 (force_typ names_lthy o (fn T => dummyT --> T)) fpTs (of_fp_res #ctors);
val dtors = map2 (force_typ names_lthy o (fn T => T --> dummyT)) fpTs (of_fp_res #dtors);
in
((ctors, dtors), `(map (Term.subst_atomic_types thetaBs)) (fp_case fp ctors dtors))
end;
val absATs = map (domain_type o fastype_of) ctors;
val absBTs = map (Term.typ_subst_atomic thetaBs) absATs;
val xTs = map (domain_type o fastype_of) xtors;
val yTs = map (domain_type o fastype_of) xtor's;
val absAs = map3 (fn Ds => mk_abs o mk_T_of_bnf Ds allAs) Dss bnfs abss;
val absBs = map3 (fn Ds => mk_abs o mk_T_of_bnf Ds allBs) Dss bnfs abss;
val fp_repAs = map2 mk_rep absATs fp_reps;
val fp_repBs = map2 mk_rep absBTs fp_reps;
val (((((phis, phis'), pre_phis), xs), ys), names_lthy) = names_lthy
|> mk_Frees' "R" phiTs
||>> mk_Frees "S" pre_phiTs
||>> mk_Frees "x" xTs
||>> mk_Frees "y" yTs;
val rels =
let
fun find_rel T As Bs = fp_nesty_bnfss
|> map (filter_out (curry (op = o pairself name_of_bnf) BNF_Comp.DEADID_bnf))
|> get_first (find_first (fn bnf => Type.could_unify (T_of_bnf bnf, T)))
|> Option.map (fn bnf =>
let val live = live_of_bnf bnf;
in (mk_rel live As Bs (rel_of_bnf bnf), live) end)
|> the_default (HOLogic.eq_const T, 0);
fun mk_rel (T as Type (_, Ts)) (Type (_, Us)) =
let
val (rel, live) = find_rel T Ts Us;
val (Ts', Us') = fastype_of rel |> strip_typeN live |> fst |> map_split dest_pred2T;
val rels = map2 mk_rel Ts' Us';
in
Term.list_comb (rel, rels)
end
| mk_rel (T as TFree _) _ = (nth phis (find_index (curry op = T) As)
handle General.Subscript => HOLogic.eq_const T)
| mk_rel _ _ = raise Fail "fpTs contains schematic type variables";
in
map2 (fold_rev Term.absfree phis' oo mk_rel) fpTs fpTs'
end;
val pre_rels = map2 (fn Ds => mk_rel_of_bnf Ds (As @ fpTs) (Bs @ fpTs')) Dss bnfs;
val rel_unfolds = maps (no_refl o single o rel_def_of_bnf) pre_bnfs;
val rel_xtor_co_inducts = of_fp_res (split_conj_thm o #rel_xtor_co_induct_thm)
|> map (unfold_thms lthy (id_apply :: rel_unfolds));
val rel_defs = map rel_def_of_bnf bnfs;
val rel_monos = map rel_mono_of_bnf bnfs;
fun cast castA castB pre_rel =
let
val castAB = mk_vimage2p (Term.subst_atomic_types fold_thetaAs castA)
(Term.subst_atomic_types fold_thetaBs castB);
in
fold_rev (fold_rev Term.absdummy) [phiTs, pre_phiTs]
(castAB $ Term.list_comb (pre_rel, map Bound (live - 1 downto 0)))
end;
val castAs = map2 (curry HOLogic.mk_comp) absAs fp_repAs;
val castBs = map2 (curry HOLogic.mk_comp) absBs fp_repBs;
val rel_xtor_co_induct_thm =
mk_rel_xtor_co_induct_thm fp (map3 cast castAs castBs pre_rels) pre_phis rels phis xs ys xtors
xtor's (mk_rel_xtor_co_induct_tactic fp abs_inverses rel_xtor_co_inducts rel_defs rel_monos)
lthy;
val rel_eqs = no_refl (map rel_eq_of_bnf fp_nesty_bnfs);
val map_id0s = no_refl (map map_id0_of_bnf bnfs);
val xtor_co_induct_thm =
(case fp of
Least_FP =>
let
val (Ps, names_lthy) = names_lthy
|> mk_Frees "P" (map (fn T => T --> HOLogic.boolT) fpTs);
fun mk_Grp_id P =
let val T = domain_type (fastype_of P);
in mk_Grp (HOLogic.Collect_const T $ P) (HOLogic.id_const T) end;
val cts = map (SOME o certify lthy) (map HOLogic.eq_const As @ map mk_Grp_id Ps);
fun mk_fp_type_copy_thms thm = map (curry op RS thm)
@{thms type_copy_Abs_o_Rep type_copy_vimage2p_Grp_Rep};
fun mk_type_copy_thms thm = map (curry op RS thm)
@{thms type_copy_Rep_o_Abs type_copy_vimage2p_Grp_Abs};
in
cterm_instantiate_pos cts rel_xtor_co_induct_thm
|> singleton (Proof_Context.export names_lthy lthy)
|> unfold_thms lthy (@{thms eq_le_Grp_id_iff all_simps(1,2)[symmetric]} @ rel_eqs)
|> funpow n (fn thm => thm RS spec)
|> unfold_thms lthy (@{thm eq_alt} :: map rel_Grp_of_bnf bnfs @ map_id0s)
|> unfold_thms lthy (@{thms vimage2p_id vimage2p_comp comp_apply comp_id
Grp_id_mono_subst eqTrueI[OF subset_UNIV] simp_thms(22)} @
maps mk_fp_type_copy_thms fp_type_definitions @
maps mk_type_copy_thms type_definitions)
|> unfold_thms lthy @{thms subset_iff mem_Collect_eq
atomize_conjL[symmetric] atomize_all[symmetric] atomize_imp[symmetric]}
end
| Greatest_FP =>
let
val cts = NONE :: map (SOME o certify lthy) (map HOLogic.eq_const As);
in
cterm_instantiate_pos cts rel_xtor_co_induct_thm
|> unfold_thms lthy (@{thms le_fun_def le_bool_def all_simps(1,2)[symmetric]} @ rel_eqs)
|> funpow (2 * n) (fn thm => thm RS spec)
|> Conv.fconv_rule (Object_Logic.atomize lthy)
|> funpow n (fn thm => thm RS mp)
end);
val fold_preTs = map2 (fn Ds => mk_T_of_bnf Ds allAs) Dss bnfs;
val rec_preTs = map (Term.typ_subst_atomic rec_theta) fold_preTs;
val fold_strTs = map2 mk_co_algT fold_preTs Xs;
val rec_strTs = map2 mk_co_algT rec_preTs Xs;
val resTs = map2 mk_co_algT fpTs Xs;
val (((fold_strs, fold_strs'), (rec_strs, rec_strs')), names_lthy) = names_lthy
|> mk_Frees' "s" fold_strTs
||>> mk_Frees' "s" rec_strTs;
val co_iters = of_fp_res #xtor_co_iterss;
val ns = map (length o #Ts o #fp_res) fp_sugars;
fun substT rho (Type (@{type_name "fun"}, [T, U])) = substT rho T --> substT rho U
| substT rho (Type (s, Ts)) = Type (s, map (typ_subst_nonatomic rho) Ts)
| substT _ T = T;
val typ_subst_nonatomic_sorted = fold_rev (typ_subst_nonatomic o single);
fun force_iter is_rec i TU TU_rec raw_iters =
let
val thy = Proof_Context.theory_of lthy;
val approx_fold = un_fold_of raw_iters
|> force_typ names_lthy
(replicate (nth ns i) dummyT ---> (if is_rec then TU_rec else TU));
val subst = Term.typ_subst_atomic fold_thetaAs;
fun mk_fp_absT_repT fp_repT fp_absT = mk_absT thy fp_repT fp_absT ooo mk_repT;
val mk_fp_absT_repTs = map5 mk_fp_absT_repT fp_repTs fp_absTs absTs repTs;
val fold_preTs' = mk_fp_absT_repTs (map subst fold_preTs);
val fold_pre_deads_only_Ts =
map (typ_subst_nonatomic_sorted (map (rpair dummyT) (As @ fpTs))) fold_preTs';
val TUs = map_split dest_co_algT (binder_fun_types (fastype_of approx_fold))
|>> map subst
|> uncurry (map2 mk_co_algT);
val cands = map2 mk_co_algT fold_preTs' Xs;
val js = find_indices Type.could_unify TUs cands;
val Tpats = map (fn j => mk_co_algT (nth fold_pre_deads_only_Ts j) (nth Xs j)) js;
val iter = raw_iters |> (if is_rec then co_rec_of else un_fold_of);
in
force_typ names_lthy (Tpats ---> TU) iter
end;
fun mk_co_comp_abs_rep fp_absT absT fp_abs fp_rep abs rep t =
fp_case fp (HOLogic.mk_comp (HOLogic.mk_comp (t, mk_abs absT abs), mk_rep fp_absT fp_rep))
(HOLogic.mk_comp (mk_abs fp_absT fp_abs, HOLogic.mk_comp (mk_rep absT rep, t)));
fun mk_iter b_opt is_rec iters lthy TU =
let
val thy = Proof_Context.theory_of lthy;
val x = co_alg_argT TU;
val i = find_index (fn T => x = T) Xs;
val TUiter =
(case find_first (fn f => body_fun_type (fastype_of f) = TU) iters of
NONE => nth co_iters i
|> force_iter is_rec i
(TU |> (is_none b_opt andalso not is_rec) ? substT (fpTs ~~ Xs))
(TU |> (is_none b_opt) ? substT (map2 mk_co_productT fpTs Xs ~~ Xs))
| SOME f => f);
val TUs = binder_fun_types (fastype_of TUiter);
val iter_preTs = if is_rec then rec_preTs else fold_preTs;
val iter_strs = if is_rec then rec_strs else fold_strs;
fun mk_s TU' =
let
fun mk_absT_fp_repT repT absT = mk_absT thy repT absT ooo mk_repT;
val i = find_index (fn T => co_alg_argT TU' = T) Xs;
val fp_abs = nth fp_abss i;
val fp_rep = nth fp_reps i;
val abs = nth abss i;
val rep = nth reps i;
val sF = co_alg_funT TU';
val sF' =
mk_absT_fp_repT (nth repTs' i) (nth absTs' i) (nth fp_absTs i) (nth fp_repTs i) sF
handle Term.TYPE _ => sF;
val F = nth iter_preTs i;
val s = nth iter_strs i;
in
if sF = F then s
else if sF' = F then mk_co_comp_abs_rep sF sF' fp_abs fp_rep abs rep s
else
let
val smapT = replicate live dummyT ---> mk_co_algT sF' F;
fun hidden_to_unit t =
Term.subst_TVars (map (rpair HOLogic.unitT) (Term.add_tvar_names t [])) t;
val smap = map_of_bnf (nth bnfs i)
|> force_typ names_lthy smapT
|> hidden_to_unit;
val smap_argTs = strip_typeN live (fastype_of smap) |> fst;
fun mk_smap_arg TU =
(if domain_type TU = range_type TU then
HOLogic.id_const (domain_type TU)
else if is_rec then
let
val (TY, (U, X)) = TU |> dest_co_algT ||> dest_co_productT;
val T = mk_co_algT TY U;
in
(case try (force_typ lthy T o build_map lthy co_proj1_const o dest_funT) T of
SOME f => mk_co_product f
(fst (fst (mk_iter NONE is_rec iters lthy (mk_co_algT TY X))))
| NONE => mk_map_co_product
(build_map lthy co_proj1_const
(dest_funT (mk_co_algT (dest_co_productT TY |> fst) U)))
(HOLogic.id_const X))
end
else
fst (fst (mk_iter NONE is_rec iters lthy TU)))
val smap_args = map mk_smap_arg smap_argTs;
in
mk_co_comp_abs_rep sF sF' fp_abs fp_rep abs rep
(mk_co_comp (s, Term.list_comb (smap, smap_args)))
end
end;
val t = Term.list_comb (TUiter, map mk_s TUs);
in
(case b_opt of
NONE => ((t, Drule.dummy_thm), lthy)
| SOME b => Local_Theory.define ((b, NoSyn), ((Binding.conceal (Thm.def_binding b), []),
fold_rev Term.absfree (if is_rec then rec_strs' else fold_strs') t)) lthy |>> apsnd snd)
end;
fun mk_iters is_rec name lthy =
fold2 (fn TU => fn b => fn ((iters, defs), lthy) =>
mk_iter (SOME b) is_rec iters lthy TU |>> (fn (f, d) => (f :: iters, d :: defs)))
resTs (map (Binding.suffix_name ("_" ^ name)) bs) (([], []), lthy)
|>> apfst rev o apsnd rev;
val foldN = fp_case fp ctor_foldN dtor_unfoldN;
val recN = fp_case fp ctor_recN dtor_corecN;
val (((raw_un_folds, raw_un_fold_defs), (raw_co_recs, raw_co_rec_defs)), (lthy, raw_lthy)) =
lthy
|> mk_iters false foldN
||>> mk_iters true recN
||> `Local_Theory.restore;
val phi = Proof_Context.export_morphism raw_lthy lthy;
val un_folds = map (Morphism.term phi) raw_un_folds;
val co_recs = map (Morphism.term phi) raw_co_recs;
val (xtor_un_fold_thms, xtor_co_rec_thms) =
let
val folds = map (fn f => Term.list_comb (f, fold_strs)) raw_un_folds;
val recs = map (fn r => Term.list_comb (r, rec_strs)) raw_co_recs;
val fold_mapTs = co_swap (As @ fpTs, As @ Xs);
val rec_mapTs = co_swap (As @ fpTs, As @ map2 mk_co_productT fpTs Xs);
val pre_fold_maps =
map2 (fn Ds => fn bnf =>
Term.list_comb (uncurry (mk_map_of_bnf Ds) fold_mapTs bnf,
map HOLogic.id_const As @ folds))
Dss bnfs;
val pre_rec_maps =
map2 (fn Ds => fn bnf =>
Term.list_comb (uncurry (mk_map_of_bnf Ds) rec_mapTs bnf,
map HOLogic.id_const As @ map2 (mk_co_product o HOLogic.id_const) fpTs recs))
Dss bnfs;
fun mk_goals f xtor s smap fp_abs fp_rep abs rep =
let
val lhs = mk_co_comp (f, xtor);
val rhs = mk_co_comp (s, smap);
in
HOLogic.mk_eq (lhs,
mk_co_comp_abs_rep (co_alg_funT (fastype_of lhs)) (co_alg_funT (fastype_of rhs))
fp_abs fp_rep abs rep rhs)
end;
val fold_goals =
map8 mk_goals folds xtors fold_strs pre_fold_maps fp_abss fp_reps abss reps;
val rec_goals = map8 mk_goals recs xtors rec_strs pre_rec_maps fp_abss fp_reps abss reps;
fun mk_thms ss goals tac =
Library.foldr1 HOLogic.mk_conj goals
|> HOLogic.mk_Trueprop
|> fold_rev Logic.all ss
|> (fn goal => Goal.prove_sorry raw_lthy [] [] goal tac)
|> Thm.close_derivation
|> Morphism.thm phi
|> split_conj_thm
|> map (fn thm => thm RS @{thm comp_eq_dest});
val pre_map_defs = no_refl (map map_def_of_bnf bnfs);
val fp_pre_map_defs = no_refl (map map_def_of_bnf pre_bnfs);
val unfold_map = map (unfold_thms lthy (id_apply :: pre_map_defs));
val fp_xtor_co_iterss = of_fp_res #xtor_co_iter_thmss;
val fp_xtor_un_folds = map (mk_pointfree lthy o un_fold_of) fp_xtor_co_iterss;
val fp_xtor_co_recs = map (mk_pointfree lthy o co_rec_of) fp_xtor_co_iterss;
val fp_co_iter_o_mapss = of_fp_res #xtor_co_iter_o_map_thmss;
val fp_fold_o_maps = map un_fold_of fp_co_iter_o_mapss;
val fp_rec_o_maps = map co_rec_of fp_co_iter_o_mapss;
val fold_thms = fp_case fp @{thm comp_assoc} @{thm comp_assoc[symmetric]} ::
map (fn thm => thm RS rewrite_comp_comp) @{thms map_pair.comp sum_map.comp} @
@{thms id_apply comp_id id_comp map_pair.comp map_pair.id sum_map.comp sum_map.id};
val rec_thms = fold_thms @ fp_case fp
@{thms fst_convol map_pair_o_convol convol_o}
@{thms case_sum_o_inj(1) case_sum_o_sum_map o_case_sum};
val eq_thm_prop_untyped = Term.aconv_untyped o pairself Thm.full_prop_of;
val map_thms = no_refl (maps (fn bnf =>
let val map_comp0 = map_comp0_of_bnf bnf RS sym
in [map_comp0, map_comp0 RS rewrite_comp_comp, map_id0_of_bnf bnf] end) fp_nesty_bnfs) @
remove eq_thm_prop_untyped (fp_case fp @{thm comp_assoc[symmetric]} @{thm comp_assoc})
(map2 (fn thm => fn bnf =>
@{thm type_copy_map_comp0_undo} OF
(replicate 3 thm @ unfold_map [map_comp0_of_bnf bnf]) RS
rewrite_comp_comp)
type_definitions bnfs);
fun mk_Rep_o_Abs thm = thm RS @{thm type_copy_Rep_o_Abs} RS rewrite_comp_comp;
val fp_Rep_o_Abss = map mk_Rep_o_Abs fp_type_definitions;
val Rep_o_Abss = map mk_Rep_o_Abs type_definitions;
fun mk_tac defs o_map_thms xtor_thms thms {context = ctxt, prems = _} =
unfold_thms_tac ctxt (flat [thms, defs, pre_map_defs, fp_pre_map_defs,
xtor_thms, o_map_thms, map_thms, fp_Rep_o_Abss, Rep_o_Abss]) THEN
CONJ_WRAP (K (HEADGOAL (rtac refl))) bnfs;
val fold_tac = mk_tac raw_un_fold_defs fp_fold_o_maps fp_xtor_un_folds fold_thms;
val rec_tac = mk_tac raw_co_rec_defs fp_rec_o_maps fp_xtor_co_recs rec_thms;
in
(mk_thms fold_strs fold_goals fold_tac, mk_thms rec_strs rec_goals rec_tac)
end;
(* These results are half broken. This is deliberate. We care only about those fields that are
used by "primrec", "primcorecursive", and "datatype_compat". *)
val fp_res =
({Ts = fpTs,
bnfs = of_fp_res #bnfs,
dtors = dtors,
ctors = ctors,
xtor_co_iterss = transpose [un_folds, co_recs],
xtor_co_induct = xtor_co_induct_thm,
dtor_ctors = of_fp_res #dtor_ctors (*too general types*),
ctor_dtors = of_fp_res #ctor_dtors (*too general types*),
ctor_injects = of_fp_res #ctor_injects (*too general types*),
dtor_injects = of_fp_res #dtor_injects (*too general types*),
xtor_map_thms = of_fp_res #xtor_map_thms (*too general types and terms*),
xtor_set_thmss = of_fp_res #xtor_set_thmss (*too general types and terms*),
xtor_rel_thms = of_fp_res #xtor_rel_thms (*too general types and terms*),
xtor_co_iter_thmss = transpose [xtor_un_fold_thms, xtor_co_rec_thms],
xtor_co_iter_o_map_thmss = of_fp_res #xtor_co_iter_o_map_thmss
(*theorem about old constant*),
rel_xtor_co_induct_thm = rel_xtor_co_induct_thm}
|> morph_fp_result (Morphism.term_morphism "BNF" (singleton (Variable.polymorphic lthy))));
in
(fp_res, lthy)
end;
end;