(* Title: HOL/BNF/Tools/bnf_fp_def_sugar_tactics.ML
Author: Jasmin Blanchette, TU Muenchen
Copyright 2012
Tactics for datatype and codatatype sugar.
*)
signature BNF_FP_DEF_SUGAR_TACTICS =
sig
val sum_prod_thms_map: thm list
val sum_prod_thms_set: thm list
val sum_prod_thms_rel: thm list
val mk_case_tac: Proof.context -> int -> int -> int -> thm -> thm -> thm -> tactic
val mk_coinduct_tac: Proof.context -> thm list -> int -> int list -> thm -> thm list ->
thm list -> thm list -> thm list list -> thm list list list -> thm list list list -> tactic
val mk_corec_like_tac: thm list -> thm list -> thm list -> thm list -> thm list -> thm -> thm ->
thm -> Proof.context -> tactic
val mk_ctor_iff_dtor_tac: Proof.context -> ctyp option list -> cterm -> cterm -> thm -> thm ->
tactic
val mk_disc_corec_like_iff_tac: thm list -> thm list -> thm list -> Proof.context -> tactic
val mk_exhaust_tac: Proof.context -> int -> thm list -> thm -> thm -> tactic
val mk_half_distinct_tac: Proof.context -> thm -> thm list -> tactic
val mk_induct_tac: Proof.context -> int -> int list -> int list list -> int list list list ->
thm list -> thm -> thm list -> thm list list -> tactic
val mk_inject_tac: Proof.context -> thm -> thm -> tactic
val mk_rec_like_tac: thm list -> thm list -> thm list -> thm list -> thm -> thm -> Proof.context
-> tactic
end;
structure BNF_FP_Def_Sugar_Tactics : BNF_FP_DEF_SUGAR_TACTICS =
struct
open BNF_Tactics
open BNF_Util
open BNF_FP
val basic_simp_thms = @{thms simp_thms(7,8,12,14,22,24)};
val more_simp_thms = basic_simp_thms @ @{thms simp_thms(11,15,16,21)};
val sum_prod_thms_map = @{thms id_apply map_pair_simp prod.cases sum.cases sum_map.simps};
val sum_prod_thms_set0 =
@{thms SUP_empty Sup_empty Sup_insert UN_insert Un_empty_left Un_empty_right Un_iff
Union_Un_distrib collect_def[abs_def] image_def o_apply map_pair_simp
mem_Collect_eq mem_UN_compreh_eq prod_set_simps sum_map.simps sum_set_simps};
val sum_prod_thms_set = @{thms UN_compreh_eq_eq} @ sum_prod_thms_set0;
val sum_prod_thms_rel = @{thms prod_rel_simp sum_rel_simps};
val ss_if_True_False = simpset_of (ss_only @{thms if_True if_False} @{context});
fun mk_proj T k =
let val binders = binder_types T in
fold_rev (fn T => fn t => Abs (Name.uu, T, t)) binders (Bound (length binders - k))
end;
fun hhf_concl_conv cv ctxt ct =
(case Thm.term_of ct of
Const (@{const_name all}, _) $ Abs _ =>
Conv.arg_conv (Conv.abs_conv (hhf_concl_conv cv o snd) ctxt) ct
| _ => Conv.concl_conv ~1 cv ct);
fun inst_as_projs ctxt k thm =
let
val fs =
Term.add_vars (prop_of thm) []
|> filter (fn (_, Type (@{type_name fun}, [_, T'])) => T' <> HOLogic.boolT | _ => false);
val cfs =
map (fn f as (_, T) => (certify ctxt (Var f), certify ctxt (mk_proj T k))) fs;
in
Drule.cterm_instantiate cfs thm
end;
val inst_as_projs_tac = PRIMITIVE oo inst_as_projs;
fun mk_case_tac ctxt n k m case_def ctr_def dtor_ctor =
unfold_thms_tac ctxt [case_def, ctr_def, dtor_ctor] THEN
(rtac (mk_sum_casesN_balanced n k RS ssubst) THEN'
REPEAT_DETERM_N (Int.max (0, m - 1)) o rtac (@{thm split} RS ssubst) THEN'
rtac refl) 1;
fun mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor sumEN' =
unfold_thms_tac ctxt (ctor_iff_dtor :: ctr_defs) THEN rtac sumEN' 1 THEN
unfold_thms_tac ctxt @{thms all_prod_eq} THEN
EVERY' (maps (fn k => [select_prem_tac n (rotate_tac 1) k, REPEAT_DETERM o dtac meta_spec,
etac meta_mp, atac]) (1 upto n)) 1;
fun mk_ctor_iff_dtor_tac ctxt cTs cctor cdtor ctor_dtor dtor_ctor =
(rtac iffI THEN'
EVERY' (map3 (fn cTs => fn cx => fn th =>
dtac (Drule.instantiate' cTs [NONE, NONE, SOME cx] arg_cong) THEN'
SELECT_GOAL (unfold_thms_tac ctxt [th]) THEN'
atac) [rev cTs, cTs] [cdtor, cctor] [dtor_ctor, ctor_dtor])) 1;
fun mk_half_distinct_tac ctxt ctor_inject ctr_defs =
unfold_thms_tac ctxt (ctor_inject :: @{thms sum.inject} @ ctr_defs) THEN
rtac @{thm sum.distinct(1)} 1;
fun mk_inject_tac ctxt ctr_def ctor_inject =
unfold_thms_tac ctxt [ctr_def] THEN rtac (ctor_inject RS ssubst) 1 THEN
unfold_thms_tac ctxt @{thms sum.inject Pair_eq conj_assoc} THEN rtac refl 1;
(*TODO: Try "sum_prod_thms_map" here, enriched with a few theorems*)
val rec_like_unfold_thms =
@{thms comp_def convol_def fst_conv id_def map_pair_simp prod_case_Pair_iden snd_conv split_conv
sum.simps(5,6) sum_map.simps unit_case_Unity};
fun mk_rec_like_tac pre_map_defs map_comp's map_ids'' rec_like_defs ctor_rec_like ctr_def ctxt =
unfold_thms_tac ctxt (ctr_def :: ctor_rec_like :: rec_like_defs @ pre_map_defs @ map_comp's @
map_ids'' @ rec_like_unfold_thms) THEN rtac refl 1;
(*TODO: sum_case_if needed?*)
val corec_like_unfold_thms =
@{thms id_def ident_o_ident sum_case_if sum_case_o_inj} @ sum_prod_thms_map;
fun mk_corec_like_tac corec_like_defs map_comps'' map_comp's map_ids'' map_if_distribs
ctor_dtor_corec_like pre_map_def ctr_def ctxt =
unfold_thms_tac ctxt (ctr_def :: corec_like_defs) THEN
(rtac (ctor_dtor_corec_like RS trans) THEN'
asm_simp_tac (put_simpset ss_if_True_False ctxt)) 1 THEN_MAYBE
(unfold_thms_tac ctxt (pre_map_def :: map_comp's @ map_comps'' @ map_ids'' @ map_if_distribs @
corec_like_unfold_thms) THEN
(rtac refl ORELSE' rtac (@{thm unit_eq} RS arg_cong)) 1);
fun mk_disc_corec_like_iff_tac case_splits' corec_likes discs ctxt =
EVERY (map3 (fn case_split_tac => fn corec_like_thm => fn disc =>
case_split_tac 1 THEN unfold_thms_tac ctxt [corec_like_thm] THEN
asm_simp_tac (ss_only basic_simp_thms ctxt) 1 THEN
(if is_refl disc then all_tac else rtac disc 1))
(map rtac case_splits' @ [K all_tac]) corec_likes discs);
val solve_prem_prem_tac =
REPEAT o (eresolve_tac @{thms bexE rev_bexI} ORELSE' rtac @{thm rev_bexI[OF UNIV_I]} ORELSE'
hyp_subst_tac ORELSE' resolve_tac @{thms disjI1 disjI2}) THEN'
(rtac refl ORELSE' atac ORELSE' rtac @{thm singletonI});
fun mk_induct_leverage_prem_prems_tac ctxt nn kks set_natural's pre_set_defs =
EVERY' (maps (fn kk => [select_prem_tac nn (dtac meta_spec) kk, etac meta_mp,
SELECT_GOAL (unfold_thms_tac ctxt (pre_set_defs @ set_natural's @ sum_prod_thms_set0)),
solve_prem_prem_tac]) (rev kks)) 1;
fun mk_induct_discharge_prem_tac ctxt nn n set_natural's pre_set_defs m k kks =
let val r = length kks in
EVERY' [select_prem_tac n (rotate_tac 1) k, rotate_tac ~1, hyp_subst_tac,
REPEAT_DETERM_N m o (dtac meta_spec THEN' rotate_tac ~1)] 1 THEN
EVERY [REPEAT_DETERM_N r
(rotate_tac ~1 1 THEN dtac meta_mp 1 THEN rotate_tac 1 1 THEN prefer_tac 2),
if r > 0 then PRIMITIVE Raw_Simplifier.norm_hhf else all_tac, atac 1,
mk_induct_leverage_prem_prems_tac ctxt nn kks set_natural's pre_set_defs]
end;
fun mk_induct_tac ctxt nn ns mss kkss ctr_defs ctor_induct' set_natural's pre_set_defss =
let val n = Integer.sum ns in
unfold_thms_tac ctxt ctr_defs THEN rtac ctor_induct' 1 THEN inst_as_projs_tac ctxt 1 THEN
EVERY (map4 (EVERY oooo map3 o mk_induct_discharge_prem_tac ctxt nn n set_natural's)
pre_set_defss mss (unflat mss (1 upto n)) kkss)
end;
fun mk_coinduct_same_ctr ctxt rel_eqs pre_rel_def dtor_ctor ctr_def discs sels =
hyp_subst_tac THEN'
CONVERSION (hhf_concl_conv
(Conv.top_conv (K (Conv.try_conv (Conv.rewr_conv ctr_def))) ctxt) ctxt) THEN'
SELECT_GOAL (unfold_thms_tac ctxt (pre_rel_def :: dtor_ctor :: sels)) THEN'
SELECT_GOAL (unfold_thms_tac ctxt (pre_rel_def :: dtor_ctor :: sels @ sum_prod_thms_rel)) THEN'
(atac ORELSE' REPEAT o etac conjE THEN'
full_simp_tac
(ss_only (@{thm prod.inject} :: no_refl discs @ rel_eqs @ more_simp_thms) ctxt) THEN_MAYBE'
REPEAT o hyp_subst_tac THEN' REPEAT o rtac conjI THEN' REPEAT o rtac refl);
fun mk_coinduct_distinct_ctrs ctxt discs discs' =
hyp_subst_tac THEN' REPEAT o etac conjE THEN'
full_simp_tac (ss_only (refl :: no_refl (discs @ discs') @ basic_simp_thms) ctxt);
fun mk_coinduct_discharge_prem_tac ctxt rel_eqs' nn kk n pre_rel_def dtor_ctor exhaust ctr_defs
discss selss =
let val ks = 1 upto n in
EVERY' ([rtac allI, rtac allI, rtac impI, select_prem_tac nn (dtac meta_spec) kk, dtac
meta_spec, dtac meta_mp, atac, rtac exhaust, K (inst_as_projs_tac ctxt 1), hyp_subst_tac] @
map4 (fn k => fn ctr_def => fn discs => fn sels =>
EVERY' ([rtac exhaust, K (inst_as_projs_tac ctxt 2)] @
map2 (fn k' => fn discs' =>
if k' = k then
mk_coinduct_same_ctr ctxt rel_eqs' pre_rel_def dtor_ctor ctr_def discs sels
else
mk_coinduct_distinct_ctrs ctxt discs discs') ks discss)) ks ctr_defs discss selss)
end;
fun mk_coinduct_tac ctxt rel_eqs' nn ns dtor_coinduct' pre_rel_defs dtor_ctors exhausts ctr_defss
discsss selsss =
(rtac dtor_coinduct' THEN'
EVERY' (map8 (mk_coinduct_discharge_prem_tac ctxt rel_eqs' nn)
(1 upto nn) ns pre_rel_defs dtor_ctors exhausts ctr_defss discsss selsss)) 1;
end;