(* Title: HOL/Codatatype/Tools/bnf_fp_sugar_tactics.ML
Author: Jasmin Blanchette, TU Muenchen
Copyright 2012
Tactics for the LFP/GFP sugar.
*)
signature BNF_FP_SUGAR_TACTICS =
sig
val mk_case_tac: Proof.context -> int -> int -> int -> thm -> thm -> thm -> tactic
val mk_coiter_like_tac: thm list -> thm list -> thm -> thm -> thm -> Proof.context -> tactic
val mk_exhaust_tac: Proof.context -> int -> thm list -> thm -> thm -> tactic
val mk_fld_iff_unf_tac: Proof.context -> ctyp option list -> cterm -> cterm -> thm -> thm ->
tactic
val mk_half_distinct_tac: Proof.context -> thm -> thm list -> tactic
val mk_induct_tac: Proof.context -> int list -> int list list -> (int * int) list list list ->
thm list -> thm -> thm list -> thm list list -> tactic
val mk_inject_tac: Proof.context -> thm -> thm -> tactic
val mk_iter_like_tac: thm list -> thm list -> thm list -> thm -> thm -> Proof.context -> tactic
end;
structure BNF_FP_Sugar_Tactics : BNF_FP_SUGAR_TACTICS =
struct
open BNF_Tactics
open BNF_Util
open BNF_FP_Util
val meta_mp = @{thm meta_mp};
val meta_spec = @{thm meta_spec};
fun smash_spurious_fs lthy thm =
let
val spurious_fs =
Term.add_vars (prop_of thm) []
|> filter (fn (_, Type (@{type_name fun}, [_, T'])) => T' <> HOLogic.boolT | _ => false);
val cxs =
map (fn s as (_, T) =>
(certify lthy (Var s), certify lthy (id_abs (domain_type T)))) spurious_fs;
in
Drule.cterm_instantiate cxs thm
end;
val smash_spurious_fs_tac = PRIMITIVE o smash_spurious_fs;
fun mk_case_tac ctxt n k m case_def ctr_def unf_fld =
Local_Defs.unfold_tac ctxt [case_def, ctr_def, unf_fld] 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 fld_iff_unf sumEN' =
Local_Defs.unfold_tac ctxt (fld_iff_unf :: ctr_defs) THEN rtac sumEN' 1 THEN
Local_Defs.unfold_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_fld_iff_unf_tac ctxt cTs cfld cunf fld_unf unf_fld =
(rtac iffI THEN'
EVERY' (map3 (fn cTs => fn cx => fn th =>
dtac (Drule.instantiate' cTs [NONE, NONE, SOME cx] arg_cong) THEN'
SELECT_GOAL (Local_Defs.unfold_tac ctxt [th]) THEN'
atac) [rev cTs, cTs] [cunf, cfld] [unf_fld, fld_unf])) 1;
fun mk_half_distinct_tac ctxt fld_inject ctr_defs =
Local_Defs.unfold_tac ctxt (fld_inject :: @{thms sum.inject} @ ctr_defs) THEN
rtac @{thm sum.distinct(1)} 1;
fun mk_inject_tac ctxt ctr_def fld_inject =
Local_Defs.unfold_tac ctxt [ctr_def] THEN rtac (fld_inject RS ssubst) 1 THEN
Local_Defs.unfold_tac ctxt @{thms sum.inject Pair_eq conj_assoc} THEN rtac refl 1;
val iter_like_thms =
@{thms case_unit comp_def convol_def id_apply map_pair_def sum.simps(5,6) sum_map.simps
split_conv};
fun mk_iter_like_tac pre_map_defs map_ids iter_like_defs fld_iter_like ctr_def ctxt =
Local_Defs.unfold_tac ctxt (ctr_def :: fld_iter_like :: iter_like_defs @ pre_map_defs @ map_ids @
iter_like_thms) THEN Local_Defs.unfold_tac ctxt @{thms id_def} THEN rtac refl 1;
val coiter_like_ss = ss_only @{thms if_True if_False};
val coiter_like_thms = @{thms id_apply map_pair_def sum_map.simps prod.cases};
fun mk_coiter_like_tac coiter_like_defs map_ids fld_unf_coiter_like pre_map_def ctr_def ctxt =
Local_Defs.unfold_tac ctxt (ctr_def :: coiter_like_defs) THEN
subst_tac ctxt [fld_unf_coiter_like] 1 THEN asm_simp_tac coiter_like_ss 1 THEN
Local_Defs.unfold_tac ctxt (pre_map_def :: coiter_like_thms @ map_ids) THEN
Local_Defs.unfold_tac ctxt @{thms id_def} THEN
TRY ((rtac refl ORELSE' subst_tac ctxt @{thms unit_eq} THEN' rtac refl) 1);
fun mk_induct_prelude_tac ctxt ctr_defs fld_induct' =
Local_Defs.unfold_tac ctxt ctr_defs THEN rtac fld_induct' 1 THEN smash_spurious_fs_tac ctxt;
fun mk_induct_prepare_prem_tac n m k =
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;
fun mk_induct_prepare_prem_prems_tac 0 = all_tac
| mk_induct_prepare_prem_prems_tac r =
REPEAT_DETERM_N r ((rotate_tac ~1) 1 THEN dtac meta_mp 1 THEN
defer_tac 2 THEN PRIMITIVE (Thm.permute_prems 0 ~1) THEN rotate_tac 1 1) THEN
PRIMITIVE Raw_Simplifier.norm_hhf;
val induct_prem_prem_thms =
@{thms SUP_empty Sup_empty Sup_insert UN_compreh_bex UN_insert Un_assoc[symmetric] Un_empty_left
Un_empty_right Union_Un_distrib collect_def[abs_def] fst_conv image_def o_apply snd_conv
snd_prod_fun sum.cases sup_bot_right fst_map_pair map_pair_simp sum_map.simps};
(* These rules interfere with the "set_natural'" properties of "sum" and "prod", so we explicitly
delay them. *)
val induct_prem_prem_thms_delayed =
@{thms fsts_def[abs_def] snds_def[abs_def] sum_setl_def[abs_def] sum_setr_def[abs_def]};
(* TODO: Get rid of the "blast_tac" *)
fun mk_induct_discharge_prem_prems_tac ctxt ppis set_natural's pre_set_defs =
EVERY' (maps (fn (pp, i) =>
[(* ### select_prem_tac pp (dtac meta_spec) i, *) dtac meta_spec, rotate_tac ~1, etac meta_mp,
SELECT_GOAL (Local_Defs.unfold_tac ctxt pre_set_defs), (* ### why on a line of its own? *)
SELECT_GOAL (Local_Defs.unfold_tac ctxt (set_natural's @ induct_prem_prem_thms)),
SELECT_GOAL (Local_Defs.unfold_tac ctxt
(induct_prem_prem_thms_delayed @ induct_prem_prem_thms)),
TRY o rtac (mk_UnIN pp i), (*#####*)
atac ORELSE'
rtac @{thm singletonI} ORELSE'
(REPEAT_DETERM o (SELECT_GOAL (Local_Defs.unfold_tac ctxt @{thms Union_iff bex_simps(6)}) THEN'
etac @{thm induct_set_step}) THEN'
(atac ORELSE' blast_tac ctxt))]) (rev ppis)) 1;
fun mk_induct_discharge_prem_tac ctxt n set_natural's pre_set_defs m k ppis =
EVERY [mk_induct_prepare_prem_tac n m k,
mk_induct_prepare_prem_prems_tac (length ppis), atac 1,
mk_induct_discharge_prem_prems_tac ctxt ppis set_natural's pre_set_defs];
fun mk_induct_tac ctxt ns mss ppisss ctr_defs fld_induct' set_natural's pre_set_defss =
let val n = Integer.sum ns in
mk_induct_prelude_tac ctxt ctr_defs fld_induct' THEN
EVERY (map4 (EVERY oooo map3 o mk_induct_discharge_prem_tac ctxt n set_natural's)
pre_set_defss mss (unflat mss (1 upto n)) ppisss)
end;
end;