(* Title: HOL/Tools/BNF/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 sumprod_thms_map: thm list
val sumprod_thms_set: thm list
val sumprod_thms_rel: thm list
val mk_coinduct_tac: Proof.context -> thm list -> int -> int list -> thm -> thm list ->
thm list -> thm list -> thm list -> thm list -> thm list list -> thm list list list ->
thm list list list -> tactic
val mk_corec_tac: thm list -> thm list -> thm -> 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_iff_tac: thm list -> thm list -> thm list -> Proof.context -> tactic
val mk_disc_map_iff_tac: Proof.context -> cterm -> thm -> thm list -> thm list -> tactic
val mk_exhaust_tac: Proof.context -> int -> thm list -> thm -> thm -> tactic
val mk_half_distinct_tac: Proof.context -> thm -> 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 -> thm list -> thm list list -> tactic
val mk_inject_tac: Proof.context -> thm -> thm -> thm -> tactic
val mk_rec_tac: thm list -> thm list -> thm list -> thm -> thm -> thm -> thm -> Proof.context ->
tactic
val mk_rel_coinduct0_tac: Proof.context -> thm -> cterm list -> thm list -> thm list ->
thm list list -> thm list list -> thm list list -> thm list -> thm list -> thm list ->
thm list -> thm list -> tactic
val mk_sel_map_tac: Proof.context -> cterm -> thm -> thm list -> thm list -> thm list -> tactic
val mk_sel_set_tac: Proof.context -> cterm -> thm -> thm list -> thm list -> thm list -> tactic
val mk_set_empty_tac: Proof.context -> cterm -> thm -> thm list -> thm list -> tactic
end;
structure BNF_FP_Def_Sugar_Tactics : BNF_FP_DEF_SUGAR_TACTICS =
struct
open Ctr_Sugar_Util
open BNF_Tactics
open BNF_Util
open BNF_FP_Util
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 simp_thms' = @{thms simp_thms(6,7,8,11,12,15,16,22,24)};
val sumprod_thms_map = @{thms id_apply map_prod_simp prod.case sum.case map_sum.simps};
val sumprod_thms_set =
@{thms UN_empty UN_insert Un_empty_left Un_empty_right Un_iff UN_simps(10) UN_iff
Union_Un_distrib image_iff o_apply map_prod_simp
mem_Collect_eq prod_set_simps map_sum.simps sum_set_simps};
val sumprod_thms_rel = @{thms rel_sum_simps rel_prod_apply prod.inject id_apply conj_assoc};
fun hhf_concl_conv cv ctxt ct =
(case Thm.term_of ct of
Const (@{const_name Pure.all}, _) $ Abs _ =>
Conv.arg_conv (Conv.abs_conv (hhf_concl_conv cv o snd) ctxt) ct
| _ => Conv.concl_conv ~1 cv ct);
fun co_induct_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);
fun mk_cfp (f as (_, T)) =
(certify ctxt (Var f), certify ctxt (mk_proj T (num_binder_types T) k));
val cfps = map mk_cfp fs;
in
Drule.cterm_instantiate cfps thm
end;
val co_induct_inst_as_projs_tac = PRIMITIVE oo co_induct_inst_as_projs;
fun mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor sumEN' =
unfold_thms_tac ctxt (ctor_iff_dtor :: ctr_defs) THEN HEADGOAL (rtac sumEN') THEN
HEADGOAL (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)));
fun mk_ctor_iff_dtor_tac ctxt cTs cctor cdtor ctor_dtor dtor_ctor =
HEADGOAL (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]));
fun mk_half_distinct_tac ctxt ctor_inject abs_inject ctr_defs =
unfold_thms_tac ctxt (ctor_inject :: abs_inject :: @{thms sum.inject} @ ctr_defs) THEN
HEADGOAL (rtac @{thm sum.distinct(1)});
fun mk_inject_tac ctxt ctr_def ctor_inject abs_inject =
unfold_thms_tac ctxt [ctr_def] THEN
HEADGOAL (rtac (ctor_inject RS ssubst)) THEN
unfold_thms_tac ctxt (abs_inject :: @{thms sum.inject prod.inject conj_assoc}) THEN
HEADGOAL (rtac refl);
val rec_unfold_thms =
@{thms comp_def convol_def fst_conv id_def case_prod_Pair_iden snd_conv split_conv
case_unit_Unity} @ sumprod_thms_map;
fun mk_rec_tac pre_map_defs map_idents rec_defs ctor_rec fp_abs_inverse abs_inverse ctr_def ctxt =
unfold_thms_tac ctxt (ctr_def :: ctor_rec :: fp_abs_inverse :: abs_inverse :: rec_defs @
pre_map_defs @ map_idents @ rec_unfold_thms) THEN HEADGOAL (rtac refl);
val corec_unfold_thms = @{thms id_def} @ sumprod_thms_map;
fun mk_corec_tac corec_defs map_idents ctor_dtor_corec pre_map_def abs_inverse ctr_def ctxt =
let
val ss = ss_only (pre_map_def :: abs_inverse :: map_idents @ corec_unfold_thms @
@{thms o_apply vimage2p_def if_True if_False}) ctxt;
in
unfold_thms_tac ctxt (ctr_def :: corec_defs) THEN
HEADGOAL (rtac (ctor_dtor_corec RS trans) THEN' asm_simp_tac ss) THEN_MAYBE
HEADGOAL (rtac refl ORELSE' rtac (@{thm unit_eq} RS arg_cong))
end;
fun mk_disc_corec_iff_tac case_splits' corecs discs ctxt =
EVERY (map3 (fn case_split_tac => fn corec_thm => fn disc =>
HEADGOAL case_split_tac THEN unfold_thms_tac ctxt [corec_thm] THEN
HEADGOAL (asm_simp_tac (ss_only basic_simp_thms ctxt)) THEN
(if is_refl disc then all_tac else HEADGOAL (rtac disc)))
(map rtac case_splits' @ [K all_tac]) corecs discs);
fun mk_disc_map_iff_tac ctxt ct exhaust discs maps =
TRYALL Goal.conjunction_tac THEN
ALLGOALS (rtac (cterm_instantiate_pos [SOME ct] exhaust) THEN_ALL_NEW
REPEAT_DETERM o hyp_subst_tac ctxt) THEN
unfold_thms_tac ctxt maps THEN
unfold_thms_tac ctxt (map (fn thm => thm RS @{thm iffD2[OF eq_False]}
handle THM _ => thm RS @{thm iffD2[OF eq_True]}) discs) THEN
ALLGOALS (rtac refl ORELSE' rtac TrueI);
fun solve_prem_prem_tac ctxt =
REPEAT o (eresolve_tac @{thms bexE rev_bexI} ORELSE' rtac @{thm rev_bexI[OF UNIV_I]} ORELSE'
hyp_subst_tac ctxt 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 fp_abs_inverses abs_inverses set_maps
pre_set_defs =
HEADGOAL (EVERY' (maps (fn kk => [select_prem_tac nn (dtac meta_spec) kk, etac meta_mp,
SELECT_GOAL (unfold_thms_tac ctxt (pre_set_defs @ fp_abs_inverses @ abs_inverses @ set_maps @
sumprod_thms_set)),
solve_prem_prem_tac ctxt]) (rev kks)));
fun mk_induct_discharge_prem_tac ctxt nn n fp_abs_inverses abs_inverses set_maps pre_set_defs m k
kks =
let val r = length kks in
HEADGOAL (EVERY' [select_prem_tac n (rotate_tac 1) k, rotate_tac ~1, hyp_subst_tac ctxt,
REPEAT_DETERM_N m o (dtac meta_spec THEN' rotate_tac ~1)]) THEN
EVERY [REPEAT_DETERM_N r
(HEADGOAL (rotate_tac ~1 THEN' dtac meta_mp THEN' rotate_tac 1) THEN prefer_tac 2),
if r > 0 then ALLGOALS (Goal.norm_hhf_tac ctxt) else all_tac, HEADGOAL atac,
mk_induct_leverage_prem_prems_tac ctxt nn kks fp_abs_inverses abs_inverses set_maps
pre_set_defs]
end;
fun mk_induct_tac ctxt nn ns mss kkss ctr_defs ctor_induct' fp_abs_inverses abs_inverses set_maps
pre_set_defss =
let val n = Integer.sum ns in
unfold_thms_tac ctxt ctr_defs THEN HEADGOAL (rtac ctor_induct') THEN
co_induct_inst_as_projs_tac ctxt 0 THEN
EVERY (map4 (EVERY oooo map3 o
mk_induct_discharge_prem_tac ctxt nn n fp_abs_inverses abs_inverses set_maps)
pre_set_defss mss (unflat mss (1 upto n)) kkss)
end;
fun mk_coinduct_same_ctr_tac ctxt rel_eqs pre_rel_def fp_abs_inverse abs_inverse dtor_ctor ctr_def
discs sels =
hyp_subst_tac ctxt 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 :: fp_abs_inverse :: abs_inverse :: dtor_ctor ::
sels @ sumprod_thms_rel @ @{thms o_apply vimage2p_def})) THEN'
(atac ORELSE' REPEAT o etac conjE THEN'
full_simp_tac (ss_only (no_refl discs @ rel_eqs @ more_simp_thms) ctxt) THEN'
REPEAT o etac conjE THEN_MAYBE' REPEAT o hyp_subst_tac ctxt THEN'
REPEAT o (resolve_tac [refl, conjI] ORELSE' atac));
fun mk_coinduct_distinct_ctrs_tac ctxt discs discs' =
let
val discs'' = map (perhaps (try (fn th => th RS @{thm notnotD}))) (discs @ discs')
|> distinct Thm.eq_thm_prop;
in
hyp_subst_tac ctxt THEN' REPEAT o etac conjE THEN'
full_simp_tac (ss_only (refl :: no_refl discs'' @ basic_simp_thms) ctxt)
end;
fun mk_coinduct_discharge_prem_tac ctxt rel_eqs' nn kk n pre_rel_def fp_abs_inverse abs_inverse
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 (co_induct_inst_as_projs_tac ctxt 0),
hyp_subst_tac ctxt] @
map4 (fn k => fn ctr_def => fn discs => fn sels =>
EVERY' ([rtac exhaust, K (co_induct_inst_as_projs_tac ctxt 1)] @
map2 (fn k' => fn discs' =>
if k' = k then
mk_coinduct_same_ctr_tac ctxt rel_eqs' pre_rel_def fp_abs_inverse abs_inverse
dtor_ctor ctr_def discs sels
else
mk_coinduct_distinct_ctrs_tac 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 fp_abs_inverses abs_inverses
dtor_ctors exhausts ctr_defss discsss selsss =
HEADGOAL (rtac dtor_coinduct' THEN'
EVERY' (map10 (mk_coinduct_discharge_prem_tac ctxt rel_eqs' nn)
(1 upto nn) ns pre_rel_defs fp_abs_inverses abs_inverses dtor_ctors exhausts ctr_defss discsss
selsss));
fun mk_rel_coinduct0_tac ctxt dtor_rel_coinduct cts assms exhausts discss selss ctor_defss
dtor_ctors ctor_injects abs_injects rel_pre_defs abs_inverses =
rtac dtor_rel_coinduct 1 THEN
EVERY (map11 (fn ct => fn assm => fn exhaust => fn discs => fn sels => fn ctor_defs =>
fn dtor_ctor => fn ctor_inject => fn abs_inject => fn rel_pre_def => fn abs_inverse =>
(rtac exhaust THEN_ALL_NEW (rtac exhaust THEN_ALL_NEW
(dtac (rotate_prems (~1) ((cterm_instantiate_pos [NONE, NONE, NONE, NONE, SOME ct] @{thm
arg_cong2}) RS iffD1)) THEN'
atac THEN' atac THEN' hyp_subst_tac ctxt THEN' dtac assm THEN'
REPEAT_DETERM o etac conjE))) 1 THEN
Local_Defs.unfold_tac ctxt ((discs RL @{thms iffD2[OF eq_True] iffD2[OF eq_False]}) @ sels
@ simp_thms') THEN
Local_Defs.unfold_tac ctxt (dtor_ctor :: rel_pre_def :: abs_inverse :: ctor_inject ::
abs_inject :: ctor_defs @ simp_thms' @ @{thms BNF_Comp.id_bnf_comp_def rel_sum_simps
rel_prod_apply vimage2p_def Inl_Inr_False iffD2[OF eq_False Inr_not_Inl] sum.inject
prod.inject}) THEN
REPEAT_DETERM (HEADGOAL ((REPEAT_DETERM o etac conjE) THEN' (REPEAT_DETERM o rtac conjI) THEN'
(rtac refl ORELSE' atac))))
cts assms exhausts discss selss ctor_defss dtor_ctors ctor_injects abs_injects rel_pre_defs
abs_inverses);
fun mk_sel_map_tac ctxt ct exhaust discs maps sels =
TRYALL Goal.conjunction_tac THEN
ALLGOALS (rtac (cterm_instantiate_pos [SOME ct] exhaust) THEN_ALL_NEW
REPEAT_DETERM o hyp_subst_tac ctxt) THEN
Local_Defs.unfold_tac ctxt ((discs RL @{thms iffD2[OF eq_True] iffD2[OF eq_False]}) @
@{thms not_True_eq_False not_False_eq_True}) THEN
TRYALL (etac FalseE ORELSE' etac @{thm TrueE}) THEN
Local_Defs.unfold_tac ctxt (maps @ sels) THEN
ALLGOALS (rtac refl);
fun mk_sel_set_tac ctxt ct exhaust discs sels sets =
TRYALL Goal.conjunction_tac THEN
ALLGOALS (rtac (cterm_instantiate_pos [SOME ct] exhaust) THEN_ALL_NEW
REPEAT_DETERM o hyp_subst_tac ctxt) THEN
Local_Defs.unfold_tac ctxt ((discs RL @{thms iffD2[OF eq_True] iffD2[OF eq_False]}) @
@{thms not_True_eq_False not_False_eq_True}) THEN
TRYALL (etac FalseE ORELSE' etac @{thm TrueE}) THEN
Local_Defs.unfold_tac ctxt (sels @ sets) THEN
ALLGOALS (
REPEAT o (resolve_tac @{thms UnI1 UnI2 imageI} ORELSE'
eresolve_tac @{thms UN_I UN_I[rotated] imageE} ORELSE'
hyp_subst_tac ctxt) THEN'
(rtac @{thm singletonI} ORELSE' atac));
fun mk_set_empty_tac ctxt ct exhaust sets discs =
TRYALL Goal.conjunction_tac THEN
ALLGOALS (rtac (cterm_instantiate_pos [SOME ct] exhaust) THEN_ALL_NEW
REPEAT_DETERM o hyp_subst_tac ctxt) THEN
unfold_thms_tac ctxt (sets @ map_filter (fn thm =>
SOME (thm RS @{thm iffD2[OF eq_False]}) handle THM _ => NONE) discs) THEN
ALLGOALS (rtac refl ORELSE' etac FalseE);
end;