(* Title: HOL/Tools/BNF/bnf_lift.ML
Author: Julian Biendarra, TU Muenchen
Author: Dmitriy Traytel, ETH Zurich
Copyright 2015
Lifting of BNFs through typedefs.
*)
signature BNF_LIFT =
sig
datatype lift_bnf_option =
Plugins_Option of Proof.context -> Plugin_Name.filter
| No_Warn_Wits
val copy_bnf:
(((lift_bnf_option list * (binding option * (string * sort option)) list) *
string) * thm option) * (binding * binding * binding) ->
local_theory -> local_theory
val copy_bnf_cmd:
(((lift_bnf_option list * (binding option * (string * string option)) list) *
string) * (Facts.ref * Token.src list) option) * (binding * binding * binding) ->
local_theory -> local_theory
val lift_bnf:
((((lift_bnf_option list * (binding option * (string * sort option)) list) *
string) * term list option) * thm option) * (binding * binding * binding) ->
({context: Proof.context, prems: thm list} -> tactic) list ->
local_theory -> local_theory
val lift_bnf_cmd:
((((lift_bnf_option list * (binding option * (string * string option)) list) *
string) * string list) * (Facts.ref * Token.src list) option) *
(binding * binding * binding) -> local_theory -> Proof.state
end
structure BNF_Lift : BNF_LIFT =
struct
open Ctr_Sugar_Tactics
open BNF_Util
open BNF_Comp
open BNF_Def
(* typedef_bnf *)
datatype lift_bnf_option =
Plugins_Option of Proof.context -> Plugin_Name.filter
| No_Warn_Wits;
fun typedef_bnf thm wits specs map_b rel_b pred_b opts lthy =
let
val plugins =
get_first (fn Plugins_Option f => SOME (f lthy) | _ => NONE) (rev opts)
|> the_default Plugin_Name.default_filter;
val no_warn_wits = exists (fn No_Warn_Wits => true | _ => false) opts;
(* extract Rep Abs F RepT AbsT *)
val (_, [Rep_G, Abs_G, F]) = Term.strip_comb (HOLogic.dest_Trueprop (Thm.prop_of thm));
val typ_Abs_G = dest_funT (fastype_of Abs_G);
val RepT = fst typ_Abs_G; (* F *)
val AbsT = snd typ_Abs_G; (* G *)
val AbsT_name = fst (dest_Type AbsT);
val tvs = AbsT |> dest_Type |> snd |> map (fst o dest_TVar);
val alpha0s = map (TFree o snd) specs;
val _ = length tvs = length alpha0s orelse
error ("Expected " ^ string_of_int (length tvs) ^ " type argument(s) to " ^ quote AbsT_name);
(* instantiate the new type variables newtvs to oldtvs *)
val subst = subst_TVars (tvs ~~ alpha0s);
val typ_subst = typ_subst_TVars (tvs ~~ alpha0s);
val Rep_G = subst Rep_G;
val Abs_G = subst Abs_G;
val F = subst F;
val RepT = typ_subst RepT;
val AbsT = typ_subst AbsT;
fun flatten_tyargs Ass =
map dest_TFree alpha0s
|> filter (fn T => exists (fn Ts => member (op =) Ts T) Ass);
val Ds0 = filter (is_none o fst) specs |> map snd;
(* get the bnf for RepT *)
val ((bnf, (deads, alphas)),((_, unfolds), lthy)) =
bnf_of_typ true Dont_Inline (Binding.qualify true AbsT_name) flatten_tyargs []
Ds0 RepT ((empty_comp_cache, empty_unfolds), lthy);
val set_bs =
map (fn T => find_index (fn U => T = U) alpha0s) alphas
|> map (the_default Binding.empty o fst o nth specs);
val _ = (case alphas of [] => error "No live variables" | _ => ());
val defs = #map_unfolds unfolds @ flat (#set_unfoldss unfolds) @ #rel_unfolds unfolds;
(* number of live variables *)
val lives = length alphas;
(* state the three required properties *)
val sorts = map Type.sort_of_atyp alphas;
val names_lthy = fold Variable.declare_typ (alphas @ deads) lthy;
val (alphas', names_lthy) = mk_TFrees' sorts names_lthy;
val (betas, names_lthy) = mk_TFrees' sorts names_lthy;
val map_F = mk_map_of_bnf deads alphas betas bnf;
val (typ_fs, typ_aF) = fastype_of map_F |> strip_typeN lives ||> domain_type;
val typ_pairs = map HOLogic.mk_prodT (alphas ~~ alphas');
val typ_subst_pair = typ_subst_atomic (alphas ~~ typ_pairs);
val typ_pair = typ_subst_pair RepT;
val subst_b = subst_atomic_types (alphas ~~ betas);
val subst_a' = subst_atomic_types (alphas ~~ alphas');
val subst_pair = subst_atomic_types (alphas ~~ typ_pairs);
val aF_set = F;
val bF_set = subst_b F;
val aF_set' = subst_a' F;
val pairF_set = subst_pair F;
val map_F_fst = mk_map_of_bnf deads typ_pairs alphas bnf;
val map_F_snd = mk_map_of_bnf deads typ_pairs alphas' bnf;
val wits_F = mk_wits_of_bnf
(replicate (nwits_of_bnf bnf) deads)
(replicate (nwits_of_bnf bnf) alphas) bnf;
(* val map_closed_F = @{term "\<And>f x. x \<in> F \<Longrightarrow> map_F f x \<in> F"}; *)
val (var_fs, names_lthy) = mk_Frees "f" typ_fs names_lthy;
val (var_x, names_lthy) = mk_Frees "x" [typ_aF] names_lthy |>> the_single;
val mem_x = HOLogic.mk_Trueprop (HOLogic.mk_mem (var_x, aF_set));
val map_f = list_comb (map_F, var_fs);
val mem_map = HOLogic.mk_Trueprop (HOLogic.mk_mem (map_f $ var_x, bF_set));
val imp_map = Logic.mk_implies (mem_x, mem_map);
val map_closed_F = fold_rev Logic.all var_fs (Logic.all var_x imp_map);
(* val zip_closed_F = @{term "\<And>z. map_F fst z \<in> F \<Longrightarrow> map_F snd z \<in> F \<Longrightarrow> z \<in> F"}; *)
val (var_zs, names_lthy) = mk_Frees "z" [typ_pair] names_lthy;
val (pairs, names_lthy) = mk_Frees "tmp" typ_pairs names_lthy;
val var_z = hd var_zs;
val fsts = map (fst o Term.strip_comb o HOLogic.mk_fst) pairs;
val snds = map (fst o Term.strip_comb o HOLogic.mk_snd) pairs;
val map_fst = list_comb (list_comb (map_F_fst, fsts), var_zs);
val mem_map_fst = HOLogic.mk_Trueprop (HOLogic.mk_mem (map_fst, aF_set));
val map_snd = list_comb (list_comb (map_F_snd, snds), var_zs);
val mem_map_snd = HOLogic.mk_Trueprop (HOLogic.mk_mem (map_snd, aF_set'));
val mem_z = HOLogic.mk_Trueprop (HOLogic.mk_mem (var_z, pairF_set));
val imp_zip = Logic.mk_implies (mem_map_fst, Logic.mk_implies (mem_map_snd, mem_z));
val zip_closed_F = Logic.all var_z imp_zip;
(* val wit_closed_F = @{term "wit_F a \<in> F"}; *)
val (var_as, names_lthy) = mk_Frees "a" alphas names_lthy;
val (var_bs, _) = mk_Frees "a" alphas names_lthy;
fun binder_types_until_eq V T =
let
fun strip (TU as Type ("fun", [T, U])) = if V = TU then [] else T :: strip U
| strip T = if V = T then [] else
error ("Bad type for witness: " ^ quote (Syntax.string_of_typ lthy T));
in strip T end;
val Iwits = the_default wits_F (Option.map (map (`(map (fn T =>
find_index (fn U => T = U) alphas) o binder_types_until_eq RepT o fastype_of))) wits);
val wit_closed_Fs =
Iwits |> map (fn (I, wit_F) =>
let
val vars = map (nth var_as) I;
val wit_a = list_comb (wit_F, vars);
in fold_rev Logic.all vars (HOLogic.mk_Trueprop (HOLogic.mk_mem (wit_a, aF_set))) end);
val mk_wit_goals = mk_wit_goals var_as var_bs
(mk_sets_of_bnf (replicate lives deads) (replicate lives alphas) bnf);
val goals = [map_closed_F, zip_closed_F] @ wit_closed_Fs @
(case wits of NONE => [] | _ => maps mk_wit_goals Iwits);
val lost_wits = filter_out (fn (J, _) => exists (fn (I, _) => I = J) Iwits) wits_F;
val _ =
if null lost_wits orelse no_warn_wits then ()
else
lost_wits
|> map (Syntax.pretty_typ lthy o fastype_of o snd)
|> Pretty.big_list
"The following types of nonemptiness witnesses of the raw type's BNF were lost:"
|> (fn pt => Pretty.chunks [pt,
Pretty.para "You can specify a liftable witness (e.g., a term of one of the above types\
\ that satisfies the typedef's invariant)\
\ using the annotation [wits: <term>]."])
|> Pretty.string_of
|> warning;
fun after_qed ([map_closed_thm] :: [zip_closed_thm] :: wit_thmss) lthy =
let
val (wit_closed_thms, wit_thms) =
(case wits of
NONE => (map the_single wit_thmss, wit_thms_of_bnf bnf)
| _ => chop (length wit_closed_Fs) (map the_single wit_thmss))
(* construct map set bd rel wit *)
(* val map_G = @{term "\<lambda>f. Abs_G o map_F f o Rep_G"}; *)
val Abs_Gb = subst_b Abs_G;
val map_G =
fold_rev HOLogic.tupled_lambda var_fs
(HOLogic.mk_comp (HOLogic.mk_comp (Abs_Gb, map_f), Rep_G));
(* val sets_G = [@{term "set_F o Rep_G"}]; *)
val sets_F = mk_sets_of_bnf (replicate lives deads) (replicate lives alphas) bnf;
val sets_G = map (fn set_F => HOLogic.mk_comp (set_F, Rep_G)) sets_F;
(* val bd_G = @{term "bd_F"}; *)
val bd_F = mk_bd_of_bnf deads alphas bnf;
val bd_G = bd_F;
(* val rel_G = @{term "\<lambda>R. BNF_Def.vimage2p Rep_G Rep_G (rel_F R)"}; *)
val rel_F = mk_rel_of_bnf deads alphas betas bnf;
val (typ_Rs, _) = strip_typeN lives (fastype_of rel_F);
val (var_Rs, names_lthy) = mk_Frees "R" typ_Rs lthy;
val Rep_Gb = subst_b Rep_G;
val rel_G = fold_rev absfree (map dest_Free var_Rs)
(mk_vimage2p Rep_G Rep_Gb $ list_comb (rel_F, var_Rs));
(* val pred_G = @{term "\<lambda>P. pred_F P o Rep_G"}; *)
val pred_F = mk_pred_of_bnf deads alphas bnf;
val (typ_Ps, _) = strip_typeN lives (fastype_of pred_F);
val (var_Ps, names_lthy) = mk_Frees "P" typ_Ps names_lthy;
val pred_G = fold_rev absfree (map dest_Free var_Ps)
(HOLogic.mk_comp (list_comb (pred_F, var_Ps), Rep_G));
(* val wits_G = [@{term "Abs_G o wit_F"}]; *)
val (var_as, _) = mk_Frees "a" alphas names_lthy;
val wits_G =
map (fn (I, wit_F) =>
let
val vs = map (nth var_as) I;
in fold_rev absfree (map dest_Free vs) (Abs_G $ (list_comb (wit_F, vs))) end)
Iwits;
(* tactics *)
val Rep_thm = thm RS @{thm type_definition.Rep};
val Abs_inverse_thm = thm RS @{thm type_definition.Abs_inverse};
val Abs_inject_thm = thm RS @{thm type_definition.Abs_inject};
val Rep_cases_thm = thm RS @{thm type_definition.Rep_cases};
val Rep_inverse_thm = thm RS @{thm type_definition.Rep_inverse};
fun map_id0_tac ctxt =
HEADGOAL (EVERY' [rtac ctxt ext,
SELECT_GOAL (unfold_thms_tac ctxt [map_id0_of_bnf bnf, id_apply, o_apply,
Rep_inverse_thm]),
rtac ctxt refl]);
fun map_comp0_tac ctxt =
HEADGOAL (EVERY' [rtac ctxt ext,
SELECT_GOAL (unfold_thms_tac ctxt [map_comp0_of_bnf bnf, o_apply,
Rep_thm RS (map_closed_thm RS Abs_inverse_thm)]),
rtac ctxt refl]);
fun map_cong0_tac ctxt =
HEADGOAL (EVERY' ([SELECT_GOAL (unfold_thms_tac ctxt [o_apply]),
rtac ctxt (([Rep_thm RS map_closed_thm, Rep_thm RS map_closed_thm] MRS
Abs_inject_thm) RS iffD2),
rtac ctxt (map_cong0_of_bnf bnf)] @ replicate lives (Goal.assume_rule_tac ctxt)));
val set_map0s_tac =
map (fn set_map => fn ctxt =>
HEADGOAL (EVERY' [rtac ctxt ext,
SELECT_GOAL (unfold_thms_tac ctxt [set_map, o_apply,
Rep_thm RS (map_closed_thm RS Abs_inverse_thm)]),
rtac ctxt refl]))
(set_map_of_bnf bnf);
fun card_order_bd_tac ctxt = HEADGOAL (rtac ctxt (bd_card_order_of_bnf bnf));
fun cinfinite_bd_tac ctxt = HEADGOAL (rtac ctxt (bd_cinfinite_of_bnf bnf));
val set_bds_tac =
map (fn set_bd => fn ctxt =>
HEADGOAL (EVERY' [SELECT_GOAL (unfold_thms_tac ctxt [o_apply]), rtac ctxt set_bd]))
(set_bd_of_bnf bnf);
fun le_rel_OO_tac ctxt =
HEADGOAL (EVERY' [rtac ctxt @{thm vimage2p_relcompp_mono},
rtac ctxt ((rel_OO_of_bnf bnf RS sym) RS @{thm ord_eq_le_trans}),
rtac ctxt @{thm order_refl}]);
fun rel_OO_Grp_tac ctxt =
HEADGOAL (EVERY' ([SELECT_GOAL (REPEAT_DETERM (HEADGOAL (rtac ctxt ext))),
SELECT_GOAL (unfold_thms_tac ctxt [@{thm OO_Grp_alt}, mem_Collect_eq,
o_apply, @{thm vimage2p_def}, in_rel_of_bnf bnf, Bex_def, mem_Collect_eq]),
rtac ctxt iffI,
SELECT_GOAL (REPEAT_DETERM (HEADGOAL (eresolve0_tac [exE,conjE]))),
rtac ctxt (zip_closed_thm OF (replicate 2 (Rep_thm RSN (2, @{thm ssubst_mem}))) RS
Rep_cases_thm),
assume_tac ctxt,
assume_tac ctxt,
hyp_subst_tac ctxt,
SELECT_GOAL (REPEAT_DETERM (HEADGOAL (rtac ctxt exI))),
rtac ctxt conjI] @
replicate (lives - 1) (rtac ctxt conjI THEN' assume_tac ctxt) @
[assume_tac ctxt,
SELECT_GOAL (REPEAT_DETERM (HEADGOAL (rtac ctxt conjI))),
REPEAT_DETERM_N 2 o
etac ctxt (trans OF [iffD2 OF [Abs_inject_thm OF
[map_closed_thm OF [Rep_thm], Rep_thm]], Rep_inverse_thm]),
SELECT_GOAL (REPEAT_DETERM (HEADGOAL (eresolve0_tac [exE,conjE]))),
rtac ctxt exI,
rtac ctxt conjI] @
replicate (lives - 1) (rtac ctxt conjI THEN' assume_tac ctxt) @
[assume_tac ctxt,
rtac ctxt conjI,
REPEAT_DETERM_N 2 o EVERY'
[rtac ctxt (iffD1 OF [Abs_inject_thm OF [map_closed_thm OF [Rep_thm], Rep_thm]]),
etac ctxt (Rep_inverse_thm RS sym RSN (2, trans))]]));
fun pred_set_tac ctxt =
HEADGOAL (EVERY'
[rtac ctxt (pred_set_of_bnf bnf RS @{thm arg_cong[of _ _ "\<lambda>f. f o _"]} RS trans),
SELECT_GOAL (unfold_thms_tac ctxt (@{thms Ball_comp_iff conj_comp_iff})),
rtac ctxt refl]);
fun wit_tac ctxt =
HEADGOAL (EVERY'
(map (fn thm => (EVERY'
[SELECT_GOAL (unfold_thms_tac ctxt (o_apply ::
(wit_closed_thms RL [Abs_inverse_thm]))),
dtac ctxt thm, assume_tac ctxt]))
wit_thms));
val tactics = [map_id0_tac, map_comp0_tac, map_cong0_tac] @ set_map0s_tac @
[card_order_bd_tac, cinfinite_bd_tac] @ set_bds_tac @
[le_rel_OO_tac, rel_OO_Grp_tac, pred_set_tac];
val (bnf, lthy) = bnf_def Dont_Inline (user_policy Note_Some) false I
tactics wit_tac NONE map_b rel_b pred_b set_bs
(((((((Binding.empty, AbsT), map_G), sets_G), bd_G), wits_G), SOME rel_G), SOME pred_G)
lthy;
val bnf = morph_bnf_defs (Morphism.thm_morphism "BNF" (unfold_thms lthy defs)) bnf;
in
lthy |> BNF_Def.register_bnf plugins AbsT_name bnf
end
| after_qed _ _ = raise Match;
in
(goals, after_qed, defs, lthy)
end;
(* main commands *)
local
fun prepare_common prepare_name prepare_sort prepare_term prepare_thm
(((((plugins, raw_specs), raw_Tname), raw_wits), xthm_opt), (map_b, rel_b, pred_b)) lthy =
let
val Tname = prepare_name lthy raw_Tname;
val input_thm =
(case xthm_opt of
SOME xthm => prepare_thm lthy xthm
| NONE => Typedef.get_info lthy Tname |> hd |> snd |> #type_definition);
val wits = (Option.map o map) (prepare_term lthy) raw_wits;
val specs =
map (apsnd (apsnd (the_default @{sort type} o Option.map (prepare_sort lthy)))) raw_specs;
val _ =
(case HOLogic.dest_Trueprop (Thm.prop_of input_thm) of
Const (@{const_name type_definition}, _) $ _ $ _ $ _ => ()
| _ => error "Unsupported type of a theorem: only type_definition is supported");
in
typedef_bnf input_thm wits specs map_b rel_b pred_b plugins lthy
end;
fun prepare_lift_bnf prepare_name prepare_sort prepare_term prepare_thm =
(fn (goals, after_qed, definitions, lthy) =>
lthy
|> Proof.theorem NONE after_qed (map (single o rpair []) goals)
|> Proof.refine_singleton
(Method.Basic (fn ctxt => SIMPLE_METHOD (unfold_thms_tac ctxt definitions)))
|> Proof.refine_singleton (Method.primitive_text (K I))) oo
prepare_common prepare_name prepare_sort prepare_term prepare_thm o apfst (apfst (apsnd SOME));
fun prepare_solve prepare_name prepare_typ prepare_sort prepare_thm tacs =
(fn (goals, after_qed, _, lthy) =>
lthy
|> after_qed (map2 (single oo Goal.prove lthy [] []) goals (tacs (length goals)))) oo
prepare_common prepare_name prepare_typ prepare_sort prepare_thm;
in
val lift_bnf_cmd =
prepare_lift_bnf
(fst o dest_Type oo Proof_Context.read_type_name {proper = true, strict = false})
Syntax.read_sort Syntax.read_term (singleton o Attrib.eval_thms);
fun lift_bnf args tacs =
prepare_solve (K I) (K I) (K I) (K I) (K tacs) args;
val copy_bnf =
apfst (apfst (rpair NONE))
#> prepare_solve (K I) (K I) (K I) (K I)
(fn n => replicate n (fn {context = ctxt, prems = _} => rtac ctxt UNIV_I 1));
val copy_bnf_cmd =
apfst (apfst (rpair NONE))
#> prepare_solve
(fst o dest_Type oo Proof_Context.read_type_name {proper = true, strict = false})
Syntax.read_sort Syntax.read_term (singleton o Attrib.eval_thms)
(fn n => replicate n (fn {context = ctxt, prems = _} => rtac ctxt UNIV_I 1));
end;
(* outer syntax *)
local
val parse_wits =
@{keyword "["} |-- (Parse.name --| @{keyword ":"} -- Scan.repeat Parse.term >>
(fn ("wits", Ts) => Ts
| (s, _) => error ("Unknown label " ^ quote s ^ " (expected \"wits\")"))) --|
@{keyword "]"} || Scan.succeed [];
val parse_options =
Scan.optional (@{keyword "("} |--
Parse.list1 (Parse.group (K "option")
(Plugin_Name.parse_filter >> Plugins_Option
|| Parse.reserved "no_warn_wits" >> K No_Warn_Wits))
--| @{keyword ")"}) [];
val parse_plugins =
Scan.optional (@{keyword "("} |-- Plugin_Name.parse_filter --| @{keyword ")"})
(K Plugin_Name.default_filter) >> Plugins_Option >> single;
val parse_typedef_thm = Scan.option (Parse.reserved "via" |-- Parse.xthm);
in
val _ =
Outer_Syntax.local_theory_to_proof @{command_keyword lift_bnf}
"register a subtype of a bounded natural functor (BNF) as a BNF"
((parse_options -- parse_type_args_named_constrained -- Parse.type_const -- parse_wits --
parse_typedef_thm -- parse_map_rel_pred_bindings) >> lift_bnf_cmd);
val _ =
Outer_Syntax.local_theory @{command_keyword copy_bnf}
"register a type copy of a bounded natural functor (BNF) as a BNF"
((parse_plugins -- parse_type_args_named_constrained -- Parse.type_const --
parse_typedef_thm -- parse_map_rel_pred_bindings) >> copy_bnf_cmd);
end;
end;