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(* Title: HOL/Tools/BNF/bnf_lfp_countable.ML
Author: Jasmin Blanchette, TU Muenchen
Copyright 2014
Countability tactic for BNF datatypes.
*)
signature BNF_LFP_COUNTABLE =
sig
val derive_encode_injectives_thms: Proof.context -> string list -> thm list
val countable_datatype_tac: Proof.context -> tactic
end;
structure BNF_LFP_Countable : BNF_LFP_COUNTABLE =
struct
open BNF_FP_Rec_Sugar_Util
open BNF_Def
open BNF_Util
open BNF_Tactics
open BNF_FP_Util
open BNF_FP_Def_Sugar
val countableS = \<^sort>\<open>countable\<close>;
fun nchotomy_tac ctxt nchotomy =
HEADGOAL (resolve_tac ctxt [nchotomy RS @{thm all_reg[rotated]}] THEN'
REPEAT_ALL_NEW (resolve_tac ctxt [allI, impI] ORELSE' eresolve_tac ctxt [exE, disjE]));
fun meta_spec_mp_tac _ 0 = K all_tac
| meta_spec_mp_tac ctxt depth =
dtac ctxt meta_spec THEN' meta_spec_mp_tac ctxt (depth - 1) THEN'
dtac ctxt meta_mp THEN' assume_tac ctxt;
fun use_induction_hypothesis_tac ctxt =
DEEPEN (1, 64 (* large number *))
(fn depth => meta_spec_mp_tac ctxt depth THEN' etac ctxt allE THEN' etac ctxt impE THEN'
assume_tac ctxt THEN' assume_tac ctxt) 0;
val same_ctr_simps = @{thms sum_encode_eq prod_encode_eq sum.inject prod.inject to_nat_split
id_apply snd_conv simp_thms};
val distinct_ctrs_simps = @{thms sum_encode_eq sum.inject sum.distinct simp_thms};
fun same_ctr_tac ctxt injects recs map_congs' inj_map_strongs' =
HEADGOAL (asm_full_simp_tac
(ss_only (injects @ recs @ map_congs' @ same_ctr_simps) ctxt) THEN_MAYBE'
TRY o REPEAT_ALL_NEW (rtac ctxt conjI) THEN_ALL_NEW
REPEAT_ALL_NEW (eresolve_tac ctxt (conjE :: inj_map_strongs')) THEN_ALL_NEW
(assume_tac ctxt ORELSE' use_induction_hypothesis_tac ctxt));
fun distinct_ctrs_tac ctxt recs =
HEADGOAL (asm_full_simp_tac (ss_only (recs @ distinct_ctrs_simps) ctxt));
fun mk_encode_injective_tac ctxt n nchotomy injects recs map_comps' inj_map_strongs' =
let val ks = 1 upto n in
EVERY (maps (fn k => nchotomy_tac ctxt nchotomy :: map (fn k' =>
if k = k' then same_ctr_tac ctxt injects recs map_comps' inj_map_strongs'
else distinct_ctrs_tac ctxt recs) ks) ks)
end;
fun mk_encode_injectives_tac ctxt ns induct nchotomys injectss recss map_comps' inj_map_strongs' =
HEADGOAL (rtac ctxt induct) THEN
EVERY (@{map 4} (fn n => fn nchotomy => fn injects => fn recs =>
mk_encode_injective_tac ctxt n nchotomy injects recs map_comps' inj_map_strongs')
ns nchotomys injectss recss);
fun endgame_tac ctxt encode_injectives =
unfold_thms_tac ctxt @{thms inj_on_def ball_UNIV} THEN
ALLGOALS (rtac ctxt exI THEN' rtac ctxt allI THEN' resolve_tac ctxt encode_injectives);
fun encode_sumN n k t =
Balanced_Tree.access {init = t,
left = fn t => \<^const>\<open>sum_encode\<close> $ (@{const Inl (nat, nat)} $ t),
right = fn t => \<^const>\<open>sum_encode\<close> $ (@{const Inr (nat, nat)} $ t)}
n k;
fun encode_tuple [] = \<^term>\<open>0 :: nat\<close>
| encode_tuple ts =
Balanced_Tree.make (fn (t, u) => \<^const>\<open>prod_encode\<close> $ (@{const Pair (nat, nat)} $ u $ t)) ts;
fun mk_encode_funs ctxt fpTs ns ctrss0 recs0 =
let
val thy = Proof_Context.theory_of ctxt;
fun check_countable T =
Sign.of_sort thy (T, countableS) orelse
raise TYPE ("Type is not of sort " ^ Syntax.string_of_sort ctxt countableS, [T], []);
fun mk_to_nat_checked T =
Const (\<^const_name>\<open>to_nat\<close>, tap check_countable T --> HOLogic.natT);
val nn = length ns;
val recs as rec1 :: _ = map2 (mk_co_rec thy Least_FP (replicate nn HOLogic.natT)) fpTs recs0;
val arg_Ts = binder_fun_types (fastype_of rec1);
val arg_Tss = Library.unflat ctrss0 arg_Ts;
fun mk_U (Type (\<^type_name>\<open>prod\<close>, [T1, T2])) =
if member (op =) fpTs T1 then T2 else HOLogic.mk_prodT (mk_U T1, mk_U T2)
| mk_U (Type (s, Ts)) = Type (s, map mk_U Ts)
| mk_U T = T;
fun mk_nat (j, T) =
if T = HOLogic.natT then
SOME (Bound j)
else if member (op =) fpTs T then
NONE
else if exists_subtype_in fpTs T then
let val U = mk_U T in
SOME (mk_to_nat_checked U $ (build_map ctxt [] [] (snd_const o fst) (T, U) $ Bound j))
end
else
SOME (mk_to_nat_checked T $ Bound j);
fun mk_arg n (k, arg_T) =
let
val bound_Ts = rev (binder_types arg_T);
val nats = map_filter mk_nat (tag_list 0 bound_Ts);
in
fold (fn T => fn t => Abs (Name.uu, T, t)) bound_Ts (encode_sumN n k (encode_tuple nats))
end;
val argss = map2 (map o mk_arg) ns (map (tag_list 1) arg_Tss);
in
map (fn recx => Term.list_comb (recx, flat argss)) recs
end;
fun derive_encode_injectives_thms _ [] = []
| derive_encode_injectives_thms ctxt fpT_names0 =
let
fun not_datatype_name s =
error (quote s ^ " is not a datatype");
fun not_mutually_recursive ss = error (commas ss ^ " are not mutually recursive datatypes");
fun lfp_sugar_of s =
(case fp_sugar_of ctxt s of
SOME (fp_sugar as {fp = Least_FP, fp_co_induct_sugar = SOME _, ...}) => fp_sugar
| _ => not_datatype_name s);
val fpTs0 as Type (_, var_As) :: _ =
map (#T o lfp_sugar_of o fst o dest_Type) (#Ts (#fp_res (lfp_sugar_of (hd fpT_names0))));
val fpT_names = map (fst o dest_Type) fpTs0;
val (As_names, _) = Variable.variant_fixes (map (fn TVar ((s, _), _) => s) var_As) ctxt;
val As =
map2 (fn s => fn TVar (_, S) => TFree (s, union (op =) countableS S))
As_names var_As;
val fpTs = map (fn s => Type (s, As)) fpT_names;
val _ = subset (op =) (fpT_names0, fpT_names) orelse not_mutually_recursive fpT_names0;
fun mk_conjunct fpT x encode_fun =
HOLogic.all_const fpT $ Abs (Name.uu, fpT,
HOLogic.mk_imp (HOLogic.mk_eq (encode_fun $ x, encode_fun $ Bound 0),
HOLogic.eq_const fpT $ x $ Bound 0));
val fp_sugars as
{fp_nesting_bnfs, fp_co_induct_sugar = SOME {common_co_inducts = induct :: _, ...},
...} :: _ =
map (the o fp_sugar_of ctxt o fst o dest_Type) fpTs0;
val ctr_sugars = map (#ctr_sugar o #fp_ctr_sugar) fp_sugars;
val ctrss0 = map #ctrs ctr_sugars;
val ns = map length ctrss0;
val recs0 = map (#co_rec o the o #fp_co_induct_sugar) fp_sugars;
val nchotomys = map #nchotomy ctr_sugars;
val injectss = map #injects ctr_sugars;
val rec_thmss = map (#co_rec_thms o the o #fp_co_induct_sugar) fp_sugars;
val map_comps' = map (unfold_thms ctxt @{thms comp_def} o map_comp_of_bnf) fp_nesting_bnfs;
val inj_map_strongs' = map (Thm.permute_prems 0 ~1 o inj_map_strong_of_bnf) fp_nesting_bnfs;
val (xs, names_ctxt) = ctxt |> mk_Frees "x" fpTs;
val conjuncts = @{map 3} mk_conjunct fpTs xs (mk_encode_funs ctxt fpTs ns ctrss0 recs0);
val goal = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj conjuncts);
in
Goal.prove (*no sorry*) ctxt [] [] goal (fn {context = ctxt, prems = _} =>
mk_encode_injectives_tac ctxt ns induct nchotomys injectss rec_thmss map_comps'
inj_map_strongs')
|> HOLogic.conj_elims ctxt
|> Proof_Context.export names_ctxt ctxt
|> map (Thm.close_derivation \<^here>)
end;
fun get_countable_goal_type_name (\<^const>\<open>Trueprop\<close> $ (Const (\<^const_name>\<open>Ex\<close>, _)
$ Abs (_, Type (_, [Type (s, _), _]), Const (\<^const_name>\<open>inj_on\<close>, _) $ Bound 0
$ Const (\<^const_name>\<open>top\<close>, _)))) = s
| get_countable_goal_type_name _ = error "Wrong goal format for datatype countability tactic";
fun core_countable_datatype_tac ctxt st =
let val T_names = map get_countable_goal_type_name (Thm.prems_of st) in
endgame_tac ctxt (derive_encode_injectives_thms ctxt T_names) st
end;
fun countable_datatype_tac ctxt =
TRY (Class.intro_classes_tac ctxt []) THEN core_countable_datatype_tac ctxt;
end;
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