(* Title: HOL/Tools/BNF/bnf_fp_util.ML Author: Dmitriy Traytel, TU Muenchen Author: Jasmin Blanchette, TU Muenchen Author: Martin Desharnais, TU Muenchen Author: Stefan Berghofer, TU Muenchen Copyright 2012, 2013, 2014
Shared library for the datatype and codatatype constructions.
*)
signature BNF_FP_UTIL = sig
exception EMPTY_DATATYPE ofstring
val morph_fp_result: morphism -> fp_result -> fp_result
val time: Proof.context -> Timer.real_timer -> string -> Timer.real_timer
val fixpoint: ('a * 'a -> bool) -> ('a list -> 'a list) -> 'a list -> 'a list
val IITN: string val LevN: string val algN: string val behN: string val bisN: string val carTN: string val caseN: string val coN: string val coinductN: string val coinduct_strongN: string val corecN: string val corec_discN: string val corec_disc_iffN: string val ctorN: string val ctor_dtorN: string val ctor_exhaustN: string val ctor_induct2N: string val ctor_inductN: string val ctor_injectN: string val ctor_foldN: string val ctor_fold_o_mapN: string val ctor_fold_transferN: string val ctor_fold_uniqueN: string val ctor_mapN: string val ctor_map_uniqueN: string val ctor_recN: string val ctor_rec_o_mapN: string val ctor_rec_transferN: string val ctor_rec_uniqueN: string val ctor_relN: string val ctor_rel_inductN: string val ctor_set_inclN: string val ctor_set_set_inclN: string val dtorN: string val dtor_coinductN: string val dtor_corecN: string val dtor_corec_o_mapN: string val dtor_corec_transferN: string val dtor_corec_uniqueN: string val dtor_ctorN: string val dtor_exhaustN: string val dtor_injectN: string val dtor_mapN: string val dtor_map_coinductN: string val dtor_map_coinduct_strongN: string val dtor_map_uniqueN: string val dtor_relN: string val dtor_rel_coinductN: string val dtor_set_inclN: string val dtor_set_set_inclN: string val dtor_coinduct_strongN: string val dtor_unfoldN: string val dtor_unfold_o_mapN: string val dtor_unfold_transferN: string val dtor_unfold_uniqueN: string val exhaustN: string val colN: string val inductN: string val injectN: string val isNodeN: string val lsbisN: string val mapN: string val map_uniqueN: string val min_algN: string val morN: string val nchotomyN: string val recN: string val rel_casesN: string val rel_coinductN: string val rel_inductN: string val rel_injectN: string val rel_introsN: string val rel_distinctN: string val rel_selN: string val rvN: string val corec_selN: string val set_inclN: string val set_set_inclN: string val setN: string val simpsN: string val strTN: string val str_initN: string val sum_bdN: string val sum_bdTN: string val uniqueN: string
(* TODO: Don't index set facts. Isabelle packages traditionally generate uniform names. *) val mk_ctor_setN: int -> string val mk_dtor_setN: int -> string val mk_dtor_set_inductN: int -> string val mk_set_inductN: int -> string
val co_prefix: BNF_Util.fp_kind -> string
val split_conj_thm: thm -> thm list val split_conj_prems: int -> thm -> thm
val mk_sumTN: typ list -> typ val mk_sumTN_balanced: typ list -> typ val mk_tupleT_balanced: typ list -> typ val mk_sumprodT_balanced: typ listlist -> typ
val mk_proj: typ -> int -> int -> term
val mk_convol: term * term -> term val mk_rel_prod: term -> term -> term val mk_rel_sum: term -> term -> term
val Inl_const: typ -> typ -> term val Inr_const: typ -> typ -> term val mk_tuple_balanced: term list -> term val mk_tuple1_balanced: typ list -> term list -> term val abs_curried_balanced: typ list -> term -> term val mk_tupled_fun: term -> term -> term list -> term
val mk_case_sum: term * term -> term val mk_case_sumN: term list -> term val mk_case_absumprod: typ -> term -> term list -> term listlist -> term listlist -> term
val mk_Inl: typ -> term -> term val mk_Inr: typ -> term -> term val mk_sumprod_balanced: typ -> int -> int -> term list -> term val mk_absumprod: typ -> term -> int -> int -> term list -> term
val dest_sumT: typ -> typ * typ val dest_sumTN_balanced: int -> typ -> typ list val dest_tupleT_balanced: int -> typ -> typ list val dest_absumprodT: typ -> typ -> int -> int list -> typ -> typ listlist
val If_const: typ -> term
val mk_Field: term -> term val mk_If: term -> term -> term -> term
val mk_absumprodE: thm -> int list -> thm
val mk_sum_caseN: int -> int -> thm val mk_sum_caseN_balanced: int -> int -> thm
val mk_sum_Cinfinite: thm list -> thm val mk_sum_card_order: thm list -> thm val mk_sum_Cinfinite_regularCard: (thm * thm) list -> thm * thm
val force_typ: Proof.context -> typ -> term -> term
val mk_xtor_rel_co_induct_thm: BNF_Util.fp_kind -> term list -> term list -> term list ->
term list -> term list -> term list -> term list -> term list ->
({prems: thm list, context: Proof.context} -> tactic) -> Proof.context -> thm val mk_xtor_co_iter_transfer_thms: BNF_Util.fp_kind -> term list -> term list -> term list ->
term list -> term list -> term list -> term list ->
({prems: thm list, context: Proof.context} -> tactic) -> Proof.context -> thm list val mk_xtor_co_iter_o_map_thms: BNF_Util.fp_kind -> bool -> int -> thm -> thm list -> thm list ->
thm list -> thm list -> thm list val derive_xtor_co_recs: BNF_Util.fp_kind -> binding list -> (typ list -> typ list) ->
(typ listlist * typ list) -> BNF_Def.bnf list -> term list -> term list ->
thm -> thm list -> thm list -> thm list -> thm list ->
(BNF_Comp.absT_info * BNF_Comp.absT_info) optionlist ->
local_theory ->
(term list * (thm list * thm * thm list * thm list)) * local_theory val raw_qualify: (binding -> binding) -> binding -> binding -> binding
val fixpoint_bnf: bool -> (binding -> binding) ->
(binding list -> (string * sort) list -> typ list * typ listlist -> BNF_Def.bnf list ->
BNF_Comp.absT_info list -> local_theory -> 'a) ->
binding list -> (string * sort) list -> (string * sort) list -> (string * sort) list ->
typ list -> BNF_Comp.comp_cache -> local_theory ->
((BNF_Def.bnf list * BNF_Comp.absT_info list) * BNF_Comp.comp_cache) * 'a end;
structure BNF_FP_Util : BNF_FP_UTIL = struct
open Ctr_Sugar open BNF_Comp open BNF_Def open BNF_Util open BNF_FP_Util_Tactics
val mk_sumprodT_balanced = mk_sumTN_balanced o map mk_tupleT_balanced;
fun mk_proj T n k = letval (binders, _) = strip_typeN n T in
fold_rev (fn T => fn t => Abs (Name.uu, T, t)) binders (Bound (n - k - 1)) end;
fun mk_convol (f, g) = let val (fU, fTU) = `range_type (fastype_of f); val ((gT, gU), gTU) = `dest_funT (fastype_of g); val convolT = fTU --> gTU --> gT --> HOLogic.mk_prodT (fU, gU); inConst (\<^const_name>\<open>convol\<close>, convolT) $ f $ g end;
fun mk_rel_prod R S = let val ((A1, A2), RT) = `dest_pred2T (fastype_of R); val ((B1, B2), ST) = `dest_pred2T (fastype_of S); val rel_prodT = RT --> ST --> mk_pred2T (HOLogic.mk_prodT (A1, B1)) (HOLogic.mk_prodT (A2, B2)); inConst (\<^const_name>\<open>rel_prod\<close>, rel_prodT) $ R $ S end;
fun mk_rel_sum R S = let val ((A1, A2), RT) = `dest_pred2T (fastype_of R); val ((B1, B2), ST) = `dest_pred2T (fastype_of S); val rel_sumT = RT --> ST --> mk_pred2T (mk_sumT (A1, B1)) (mk_sumT (A2, B2)); inConst (\<^const_name>\<open>rel_sum\<close>, rel_sumT) $ R $ S end;
fun Inl_const LT RT = Const (\<^const_name>\<open>Inl\<close>, LT --> mk_sumT (LT, RT)); fun mk_Inl RT t = Inl_const (fastype_of t) RT $ t;
fun Inr_const LT RT = Const (\<^const_name>\<open>Inr\<close>, RT --> mk_sumT (LT, RT)); fun mk_Inr LT t = Inr_const LT (fastype_of t) $ t;
fun mk_prod1 bound_Ts (t, u) =
HOLogic.pair_const (fastype_of1 (bound_Ts, t)) (fastype_of1 (bound_Ts, u)) $ t $ u;
fun If_const T = Const (\<^const_name>\<open>If\<close>, HOLogic.boolT --> T --> T --> T); fun mk_If p t f = letval T = fastype_of t in If_const T $ p $ t $ f end;
fun mk_Field r = letval T = fst (dest_relT (fastype_of r)); inConst (\<^const_name>\<open>Field\<close>, mk_relT (T, T) --> HOLogic.mk_setT T) $ r end;
(*dangerous; use with monotonic, converging functions only!*) fun fixpoint eq f X = if subset eq (f X, X) then X else fixpoint eq f (f X);
fun split_conj_prems limit th = let fun split n i th = if i = n then th else split n (i + 1) (conjI RSN (i, th)) handle THM _ => th; in split limit 1 th end;
fun mk_obj_sumEN_balanced n =
Balanced_Tree.make (fn (thm1, thm2) => thm1 RSN (1, thm2 RSN (2, @{thm obj_sumE_f})))
(replicate n asm_rl);
fun mk_tupled_allIN_balanced 0 = @{thm unit_all_impI}
| mk_tupled_allIN_balanced n = let val (tfrees, _) = BNF_Util.mk_TFrees n \<^context>; val T = mk_tupleT_balanced tfrees; in
@{thm asm_rl[of"\x. P x \ Q x" for P Q]}
|> Thm.instantiate' [SOME (Thm.ctyp_of \<^context> T)] []
|> Simplifier.rewrite_goals_rule \<^context> @{thms split_paired_All[THEN eq_reflection]}
|> (fn thm => impI RS funpow n (fn th => allI RS th) thm)
|> Thm.varifyT_global end;
fun mk_absumprodE type_definition ms = letval n = length ms in
mk_obj_sumEN_balanced n OFmap mk_tupled_allIN_balanced ms RS
(type_definition RS @{thm type_copy_obj_one_point_absE}) end;
fun mk_sum_caseN 1 1 = refl
| mk_sum_caseN _ 1 = @{thm sum.case(1)}
| mk_sum_caseN 2 2 = @{thm sum.case(2)}
| mk_sum_caseN n k = trans OF [@{thm case_sum_step(2)}, mk_sum_caseN (n - 1) (k - 1)];
fun mk_sum_step base step thm = if Thm.eq_thm_prop (thm, refl) then base else trans OF [step, thm];
fun mk_sum_caseN_balanced 1 1 = refl
| mk_sum_caseN_balanced n k =
Balanced_Tree.access {left = mk_sum_step @{thm sum.case(1)} @{thm case_sum_step(1)},
right = mk_sum_step @{thm sum.case(2)} @{thm case_sum_step(2)}, init = refl} n k;
fun mk_sum_Cinfinite [thm] = thm
| mk_sum_Cinfinite (thm :: thms) = @{thm Cinfinite_csum_weak} OF [thm, mk_sum_Cinfinite thms];
fun mk_sum_card_order [thm] = thm
| mk_sum_card_order (thm :: thms) = @{thm card_order_csum} OF [thm, mk_sum_card_order thms];
fun mk_sum_Cinfinite_regularCard [x] = x
| mk_sum_Cinfinite_regularCard ((cinf, thm) :: thms) = letval (cinf_sum, thm_sum) = mk_sum_Cinfinite_regularCard thms in (
@{thm Cinfinite_csum_weak} OF [cinf, cinf_sum],
@{thm regularCard_csum} OF [cinf, cinf_sum, thm, thm_sum]
) end;
fun mk_xtor_rel_co_induct_thm fp pre_rels pre_phis rels phis xs ys xtors xtor's tac lthy = let val pre_relphis = map (fn rel => Term.list_comb (rel, phis @ pre_phis)) pre_rels; val relphis = map (fn rel => Term.list_comb (rel, phis)) rels; fun mk_xtor fp' xtor x = if fp = fp'then xtor $ x else x; val dtor = mk_xtor Greatest_FP; val ctor = mk_xtor Least_FP; fun flip f x y = if fp = Greatest_FP then f y x else f x y;
fun mk_xtor_co_iter_transfer_thms fp pre_rels pre_iphis pre_ophis rels phis un_folds un_folds' tac lthy = let val pre_relphis = map (fn rel => Term.list_comb (rel, phis @ pre_iphis)) pre_rels; val relphis = map (fn rel => Term.list_comb (rel, phis)) rels; fun flip f x y = if fp = Greatest_FP then f y x else f x y;
fun mk_xtor_co_iter_o_map_thms fp is_rec m un_fold_unique xtor_maps xtor_un_folds sym_map_comps
map_cong0s = let val n = length sym_map_comps; val rewrite_comp_comp2 = case_fp fp @{thm rewriteR_comp_comp2} @{thm rewriteL_comp_comp2}; val rewrite_comp_comp = case_fp fp @{thm rewriteR_comp_comp} @{thm rewriteL_comp_comp}; val map_cong_passive_args1 = replicate m (case_fp fp @{thm id_comp} @{thm comp_id} RS fun_cong); val map_cong_active_args1 = replicate n (if is_rec then case_fp fp @{thm convol_o} @{thm o_case_sum} RS fun_cong else refl); val map_cong_passive_args2 = replicate m (case_fp fp @{thm comp_id} @{thm id_comp} RS fun_cong); val map_cong_active_args2 = replicate n (if is_rec then case_fp fp @{thm map_prod_o_convol_id} @{thm case_sum_o_map_sum_id} else case_fp fp @{thm id_comp} @{thm comp_id} RS fun_cong); fun mk_map_congs passive active = map (fn thm => thm OF (passive @ active) RS @{thm ext}) map_cong0s; val map_cong1s = mk_map_congs map_cong_passive_args1 map_cong_active_args1; val map_cong2s = mk_map_congs map_cong_passive_args2 map_cong_active_args2;
fun mk_rewrites map_congs = map2 (fn sym_map_comp => fn map_cong =>
mk_trans sym_map_comp map_cong RS rewrite_comp_comp) sym_map_comps map_congs; val rewrite1s = mk_rewrites map_cong1s; val rewrite2s = mk_rewrites map_cong2s; val unique_prems =
@{map 4} (fn xtor_map => fn un_fold => fn rewrite1 => fn rewrite2 =>
mk_trans (rewrite_comp_comp2 OF [xtor_map, un_fold])
(mk_trans rewrite1 (mk_sym rewrite2)))
xtor_maps xtor_un_folds rewrite1s rewrite2s; in
split_conj_thm (un_fold_unique OFmap (case_fp fp I mk_sym) unique_prems) end;
fun force_typ ctxt T =
Term.map_types Type_Infer.paramify_vars
#> Type.constraint T
#> Syntax.check_term ctxt
#> singleton (Variable.polymorphic ctxt);
fun absT_info_encodeT thy (SOME (src : absT_info, dst : absT_info)) src_absT = let val src_repT = mk_repT (#absT src) (#repT src) src_absT; val dst_absT = mk_absT thy (#repT dst) (#absT dst) src_repT; in
dst_absT end
| absT_info_encodeT _ NONE T = T;
fun absT_info_decodeT thy = absT_info_encodeT thy o Option.map swap;
fun absT_info_encode thy fp (opt as SOME (src : absT_info, dst : absT_info)) t = let val co_alg_funT = case_fp fp domain_type range_type; fun co_swap pair = case_fp fp I swap pair; val mk_co_comp = curry (HOLogic.mk_comp o co_swap); val mk_co_abs = case_fp fp mk_abs mk_rep; val mk_co_rep = case_fp fp mk_rep mk_abs; val co_abs = case_fp fp #abs #rep; val co_rep = case_fp fp #rep #abs; val src_absT = co_alg_funT (fastype_of t); val dst_absT = absT_info_encodeT thy opt src_absT; val co_src_abs = mk_co_abs src_absT (co_abs src); val co_dst_rep = mk_co_rep dst_absT (co_rep dst); in
mk_co_comp (mk_co_comp t co_src_abs) co_dst_rep end
| absT_info_encode _ _ NONE t = t;
fun absT_info_decode thy fp = absT_info_encode thy fp o Option.map swap;
fun mk_xtor_un_fold_xtor_thms ctxt fp un_folds xtors xtor_un_fold_unique map_id0s
absT_info_opts = let val thy = Proof_Context.theory_of ctxt; fun mk_goal un_fold = let val rhs = list_comb (un_fold, @{map 2} (absT_info_encode thy fp) absT_info_opts xtors); val T = range_type (fastype_of rhs); in
HOLogic.mk_eq (HOLogic.id_const T, rhs) end; val goal = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map mk_goal un_folds)); fun mk_inverses NONE = []
| mk_inverses (SOME (src, dst)) =
[#type_definition dst RS @{thm type_definition.Abs_inverse[OF _ UNIV_I]},
#type_definition src RS @{thm type_definition.Rep_inverse}]; val inverses = maps mk_inverses absT_info_opts; in
Goal.prove_sorry ctxt [] [] goal (fn {context = ctxt, prems = _} =>
mk_xtor_un_fold_xtor_tac ctxt xtor_un_fold_unique map_id0s inverses)
|> split_conj_thm |> map mk_sym end;
fun derive_xtor_co_recs fp bs mk_Ts (Dss, resDs) pre_bnfs xtors0 un_folds0
xtor_un_fold_unique xtor_un_folds xtor_un_fold_transfers xtor_maps xtor_rels
absT_info_opts lthy = let val thy = Proof_Context.theory_of lthy; fun co_swap pair = case_fp fp I swap pair; val mk_co_comp = curry (HOLogic.mk_comp o co_swap); fun mk_co_algT T U = case_fp fp (T --> U) (U --> T); val co_alg_funT = case_fp fp domain_type range_type; val mk_co_product = curry (case_fp fp mk_convol mk_case_sum); val co_proj1_const = case_fp fp fst_const (uncurry Inl_const o dest_sumT) o co_alg_funT; val co_proj2_const = case_fp fp snd_const (uncurry Inr_const o dest_sumT) o co_alg_funT; val mk_co_productT = curry (case_fp fp HOLogic.mk_prodT mk_sumT); val rewrite_comp_comp = case_fp fp @{thm rewriteL_comp_comp} @{thm rewriteR_comp_comp};
val n = length pre_bnfs; val live = live_of_bnf (hd pre_bnfs); val m = live - n; val ks = 1 upto n;
val map_id0s = map map_id0_of_bnf pre_bnfs; val map_comps = map map_comp_of_bnf pre_bnfs; val map_cong0s = map map_cong0_of_bnf pre_bnfs; val map_transfers = map map_transfer_of_bnf pre_bnfs; val sym_map_comp0s = map (mk_sym o map_comp0_of_bnf) pre_bnfs;
val deads = fold (union (op =)) Dss resDs; val ((((As, Bs), Xs), Ys), names_lthy) = lthy
|> fold Variable.declare_typ deads
|> mk_TFrees m
||>> mk_TFrees m
||>> mk_TFrees n
||>> mk_TFrees n;
val XFTs = @{map 2} (fn Ds => mk_T_of_bnf Ds (As @ Xs)) Dss pre_bnfs; val co_algXFTs = @{map 2} mk_co_algT XFTs Xs; val Ts = mk_Ts As; val un_foldTs = @{map 2} (fn T => fn X => co_algXFTs ---> mk_co_algT T X) Ts Xs; val un_folds = @{map 2} (force_typ names_lthy) un_foldTs un_folds0; val ABs = As ~~ Bs; val XYs = Xs ~~ Ys;
val co_rec_ids = @{map 2} (mk_co_comp o co_proj1_const) co_rec_algXs co_rec_spec0s; val co_rec_specs = @{map 2} (mk_co_comp o co_proj2_const) co_rec_algXs co_rec_spec0s;
val co_recN = case_fp fp ctor_recN dtor_corecN; fun co_rec_bind i = nth bs (i - 1) |> Binding.prefix_name (co_recN ^ "_"); val co_rec_def_bind = rpair [] o Binding.concealed o Thm.def_binding o co_rec_bind;
fun co_rec_spec i =
fold_rev (Term.absfree o Term.dest_Free) co_rec_ss (nth co_rec_specs (i - 1));
val ((co_rec_frees, (_, co_rec_def_frees)), (lthy, lthy_old)) =
lthy
|> (snd o Local_Theory.begin_nested)
|> fold_map (fn i =>
Local_Theory.define ((co_rec_bind i, NoSyn), (co_rec_def_bind i, co_rec_spec i))) ks
|>> apsnd split_list o split_list
||> `Local_Theory.end_nested;
val phi = Proof_Context.export_morphism lthy_old lthy; val co_rec_names = map (dest_Const_name o Morphism.term phi) co_rec_frees; val co_recs = @{map 2} (fn name => fn resT => Const (name, co_rec_argTs ---> resT)) co_rec_names co_rec_resTs; val co_rec_defs = map (fn def =>
mk_unabs_def n (HOLogic.mk_obj_eq (Morphism.thm phi def))) co_rec_def_frees;
val lthy = lthy |> Config.get lthy bnf_internals ? snd o Local_Theory.notes notes; in
((co_recs,
(xtor_co_rec_thms, xtor_co_rec_unique_thm, xtor_co_rec_o_map_thms, xtor_co_rec_transfer_thms)),
lthy) end;
fun raw_qualify extra_qualify base_b = let val qs = Binding.path_of base_b; val n = Binding.name_of base_b; in
Binding.prefix_name rawN
#> fold_rev (fn (s, mand) => Binding.qualify mand s) (qs @ [(n, true)])
#> extra_qualify #> Binding.concealed end;
fun fixpoint_bnf force_out_of_line extra_qualify construct_fp bs resBs Ds0 Xs rhsXs comp_cache0
lthy = let val time = time lthy; val timer = time (Timer.startRealTimer ());
fun flatten_tyargs Ass =
subtract (op =) Xs (filter (fn T => exists (fn Ts => member (op =) Ts T) Ass) resBs) @ Xs;
val ((bnfs, (deadss, livess)), (comp_cache_unfold_set, lthy')) =
apfst (apsnd split_list o split_list)
(@{fold_map 2}
(fn b => bnf_of_typ true Smart_Inline (raw_qualify extra_qualify b) flatten_tyargs Xs Ds0)
bs rhsXs ((comp_cache0, empty_unfolds), lthy));
fun norm_qualify i =
Binding.qualify true (Binding.name_of (nth bs (Int.max (0, i - 1))))
#> extra_qualify #> Binding.concealed;
val Ass = map (map dest_TFree) livess; val Ds' = fold (fold Term.add_tfreesT) deadss []; val Ds = union (op =) Ds' Ds0; val missing = resBs |> fold (subtract (op =)) (Ds' :: Ass); val (dead_phantoms, live_phantoms) = List.partition (member (op =) Ds0) missing; val resBs' = resBs |> fold (subtract (op =)) [dead_phantoms, Ds];
val timer = time (timer "Construction of BNFs");
val ((kill_posss, _), (bnfs', ((comp_cache', unfold_set'), lthy''))) =
normalize_bnfs norm_qualify Ass Ds (K (resBs' @ Xs)) bnfs (comp_cache_unfold_set, lthy');
fun pre_qualify b =
Binding.qualify false (Binding.name_of b)
#> extra_qualify
#> not (Config.get lthy'' bnf_internals) ? Binding.concealed;
val ((pre_bnfs, (deadss, absT_infos)), lthy''') = lthy''
|> @{fold_map 5} (fn b => seal_bnf (pre_qualify b) unfold_set' (Binding.prefix_name preN b))
bs (replicate (length rhsXs) (force_out_of_line orelse not (null live_phantoms))) Dss
all_Dss bnfs'
|>> split_list
|>> apsnd split_list;
val timer = time (timer "Normalization & sealing of BNFs");
val res = construct_fp bs resBs (map TFree dead_phantoms, deadss) pre_bnfs absT_infos lthy''';
val timer = time (timer "FP construction in total"); in
(((pre_bnfs, absT_infos), comp_cache'), res) end;
end;
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