(* Title: HOL/Tools/BNF/bnf_fp_def_sugar_tactics.ML
Author: Jasmin Blanchette, TU Muenchen
Author: Martin Desharnais, TU Muenchen
Copyright 2012, 2013, 2014
Tactics for datatype and codatatype sugar.
*)
signature BNF_FP_DEF_SUGAR_TACTICS =
sig
val sumprod_thms_rel: thm list
val co_induct_inst_as_projs_tac: Proof.context -> int -> tactic
val mk_case_transfer_tac: Proof.context -> thm -> thm list -> tactic
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_corec_disc_iff_tac: thm list -> thm list -> thm list -> Proof.context -> tactic
val mk_co_rec_o_map_tac: Proof.context -> thm -> thm list -> thm list -> thm list -> thm -> thm ->
thm Seq.seq
val mk_corec_transfer_tac: Proof.context -> cterm list -> cterm list -> thm list -> thm list ->
thm list -> thm list -> thm list -> ''a list -> ''a list list -> ''a list list list list ->
''a list list list list -> tactic
val mk_ctor_iff_dtor_tac: Proof.context -> ctyp option list -> cterm -> cterm -> thm -> thm ->
tactic
val mk_ctr_transfer_tac: Proof.context -> thm list -> thm list -> tactic
val mk_disc_transfer_tac: Proof.context -> thm -> thm -> 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_map_tac: Proof.context -> thm list -> thm -> thm -> thm list -> thm list -> thm list ->
tactic
val mk_map_disc_iff_tac: Proof.context -> cterm -> thm -> thm list -> thm list -> tactic
val mk_map_sel_tac: Proof.context -> cterm -> thm -> thm list -> thm list -> thm list ->
thm list -> tactic
val mk_rec_tac: thm list -> thm list -> thm list -> thm -> thm -> thm -> thm -> Proof.context ->
tactic
val mk_rec_transfer_tac: Proof.context -> int -> int list -> cterm list -> cterm list ->
term list list list list -> thm list -> thm list -> thm list -> thm list -> tactic
val mk_rel_tac: Proof.context -> thm list -> thm -> thm -> thm list -> thm list -> thm list ->
tactic
val mk_rel_case_tac: Proof.context -> cterm -> cterm -> thm -> thm list -> thm list -> thm list ->
thm list -> thm list -> 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 -> thm list -> tactic
val mk_rel_induct0_tac: Proof.context -> thm -> thm list -> cterm list -> thm list ->
thm list list -> thm list -> thm list -> thm list -> thm list -> tactic
val mk_rel_sel_tac: Proof.context -> cterm -> cterm -> thm -> thm list -> thm list -> thm list ->
thm list -> thm list -> thm list -> tactic
val mk_sel_transfer_tac: Proof.context -> int -> thm list -> thm -> tactic
val mk_set0_tac: Proof.context -> thm list -> thm list -> thm -> thm list -> thm list ->
thm list -> thm list -> thm list -> tactic
val mk_set_cases_tac: Proof.context -> cterm -> thm list -> thm -> thm list -> tactic
val mk_set_induct0_tac: Proof.context -> cterm list -> thm list -> thm list -> thm list ->
thm list -> thm list -> thm list -> thm list -> tactic
val mk_set_intros_tac: Proof.context -> thm list -> tactic
val mk_set_sel_tac: Proof.context -> cterm -> thm -> thm list -> 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 case_sum_transfer = @{thm case_sum_transfer};
val case_sum_transfer_eq = @{thm case_sum_transfer[of "op =" _ "op =", simplified sum.rel_eq]};
val case_prod_transfer = @{thm case_prod_transfer};
val case_prod_transfer_eq = @{thm case_prod_transfer[of "op =" "op =", simplified prod.rel_eq]};
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_rel = @{thms rel_sum_simps rel_prod_inject prod.inject id_apply conj_assoc};
val basic_sumprod_thms_set =
@{thms UN_empty UN_insert UN_iff Un_empty_left Un_empty_right Un_iff Union_Un_distrib o_apply
map_prod_simp mem_Collect_eq prod_set_simps map_sum.simps sum_set_simps};
val sumprod_thms_set = @{thms UN_simps(10) image_iff} @ basic_sumprod_thms_set;
fun is_def_looping def =
(case Thm.prop_of def of
Const (@{const_name Pure.eq}, _) $ lhs $ rhs => Term.exists_subterm (curry (op aconv) lhs) rhs
| _ => false);
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 (Thm.prop_of thm) []
|> filter (fn (_, Type (@{type_name fun}, [_, T'])) => T' <> HOLogic.boolT | _ => false);
fun mk_inst (xi, T) = (xi, Thm.cterm_of ctxt (mk_proj T (num_binder_types T) k));
in
infer_instantiate ctxt (map mk_inst fs) thm
end;
val co_induct_inst_as_projs_tac = PRIMITIVE oo co_induct_inst_as_projs;
fun mk_case_transfer_tac ctxt rel_case cases =
let val n = length (tl (Thm.prems_of rel_case)) in
REPEAT_DETERM (HEADGOAL (rtac ctxt rel_funI)) THEN
HEADGOAL (etac ctxt rel_case) THEN
ALLGOALS (hyp_subst_tac ctxt) THEN
unfold_thms_tac ctxt cases THEN
ALLGOALS (fn k => (select_prem_tac ctxt n (dtac ctxt asm_rl) k) k) THEN
ALLGOALS (REPEAT_DETERM o (rotate_tac ~1 THEN' dtac ctxt rel_funD THEN'
(assume_tac ctxt THEN' etac ctxt thin_rl ORELSE' rtac ctxt refl)) THEN' assume_tac ctxt)
end;
fun mk_ctr_transfer_tac ctxt rel_intros rel_eqs =
HEADGOAL Goal.conjunction_tac THEN
ALLGOALS (REPEAT o (resolve_tac ctxt (rel_funI :: rel_intros) THEN'
TRY o (REPEAT_DETERM1 o (SELECT_GOAL (unfold_thms_tac ctxt rel_eqs) THEN'
(assume_tac ctxt ORELSE' hyp_subst_tac ctxt THEN' rtac ctxt refl)))));
fun mk_disc_transfer_tac ctxt rel_sel exhaust_disc distinct_disc =
let
fun last_disc_tac iffD =
HEADGOAL (rtac ctxt (rotate_prems ~1 exhaust_disc) THEN' assume_tac ctxt THEN'
REPEAT_DETERM o (rotate_tac ~1 THEN' dtac ctxt (rotate_prems 1 iffD) THEN'
assume_tac ctxt THEN' rotate_tac ~1 THEN'
etac ctxt (rotate_prems 1 notE) THEN' eresolve_tac ctxt distinct_disc));
in
HEADGOAL Goal.conjunction_tac THEN
REPEAT_DETERM (HEADGOAL (rtac ctxt rel_funI THEN' dtac ctxt (rel_sel RS iffD1) THEN'
REPEAT_DETERM o (etac ctxt conjE) THEN' (assume_tac ctxt ORELSE' rtac ctxt iffI))) THEN
TRY (last_disc_tac iffD2) THEN TRY (last_disc_tac iffD1)
end;
fun mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor sumEN' =
unfold_thms_tac ctxt (ctor_iff_dtor :: ctr_defs) THEN HEADGOAL (rtac ctxt sumEN') THEN
HEADGOAL (EVERY' (maps (fn k => [select_prem_tac ctxt n (rotate_tac 1) k,
REPEAT_DETERM o dtac ctxt meta_spec, etac ctxt meta_mp, assume_tac ctxt]) (1 upto n)));
fun mk_ctor_iff_dtor_tac ctxt cTs cctor cdtor ctor_dtor dtor_ctor =
HEADGOAL (rtac ctxt iffI THEN'
EVERY' (@{map 3} (fn cTs => fn cx => fn th =>
dtac ctxt (Thm.instantiate' cTs [NONE, NONE, SOME cx] arg_cong) THEN'
SELECT_GOAL (unfold_thms_tac ctxt [th]) THEN'
assume_tac ctxt) [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 ctxt @{thm sum.distinct(1)});
fun mk_inject_tac ctxt ctr_def ctor_inject abs_inject =
unfold_thms_tac ctxt [ctr_def] THEN
HEADGOAL (rtac ctxt (ctor_inject RS ssubst)) THEN
unfold_thms_tac ctxt (abs_inject :: @{thms sum.inject prod.inject conj_assoc}) THEN
HEADGOAL (rtac ctxt 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_co_rec_o_map_tac ctxt co_rec_def pre_map_defs map_ident0s abs_inverses xtor_co_rec_o_map =
let
val rec_o_map_simps = @{thms o_def[abs_def] id_def case_prod_app case_sum_map_sum map_sum.simps
case_prod_map_prod id_bnf_def map_prod_simp map_sum_if_distrib_then map_sum_if_distrib_else
if_distrib[THEN sym]};
in
HEADGOAL (subst_tac ctxt (SOME [1, 2]) [co_rec_def] THEN'
rtac ctxt (xtor_co_rec_o_map RS trans) THEN'
CONVERSION Thm.eta_long_conversion THEN'
asm_simp_tac (ss_only (pre_map_defs @ distinct Thm.eq_thm_prop (map_ident0s @ abs_inverses) @
rec_o_map_simps) ctxt))
end;
fun mk_rec_tac pre_map_defs map_ident0s rec_defs ctor_rec fp_abs_inverse abs_inverse ctr_def ctxt =
HEADGOAL ((if is_def_looping ctr_def then subst_tac ctxt NONE
else SELECT_GOAL o unfold_thms_tac ctxt) [ctr_def]) THEN
unfold_thms_tac ctxt (ctor_rec :: fp_abs_inverse :: abs_inverse :: rec_defs @
pre_map_defs @ map_ident0s @ rec_unfold_thms) THEN HEADGOAL (rtac ctxt refl);
fun mk_rec_transfer_tac ctxt nn ns actives passives xssss rec_defs ctor_rec_transfers rel_pre_T_defs
rel_eqs =
let
val ctor_rec_transfers' =
map (infer_instantiate' ctxt (map SOME (passives @ actives))) ctor_rec_transfers;
val total_n = Integer.sum ns;
val True = @{term True};
in
HEADGOAL Goal.conjunction_tac THEN
EVERY (map (fn ctor_rec_transfer =>
REPEAT_DETERM (HEADGOAL (rtac ctxt rel_funI)) THEN
unfold_thms_tac ctxt rec_defs THEN
HEADGOAL (etac ctxt (mk_rel_funDN_rotated (nn + 1) ctor_rec_transfer)) THEN
unfold_thms_tac ctxt rel_pre_T_defs THEN
EVERY (fst (@{fold_map 2} (fn k => fn xsss => fn acc =>
rpair (k + acc)
(HEADGOAL (rtac ctxt (mk_rel_funDN_rotated 2 @{thm comp_transfer})) THEN
HEADGOAL (rtac ctxt @{thm vimage2p_rel_fun}) THEN
unfold_thms_tac ctxt rel_eqs THEN
EVERY (@{map 2} (fn n => fn xss =>
REPEAT_DETERM (HEADGOAL (resolve_tac ctxt
[mk_rel_funDN 2 case_sum_transfer_eq, mk_rel_funDN 2 case_sum_transfer])) THEN
HEADGOAL (select_prem_tac ctxt total_n (dtac ctxt asm_rl) (acc + n)) THEN
HEADGOAL (SELECT_GOAL (HEADGOAL
(REPEAT_DETERM o (assume_tac ctxt ORELSE' resolve_tac ctxt
[mk_rel_funDN 1 case_prod_transfer_eq,
mk_rel_funDN 1 case_prod_transfer,
rel_funI]) THEN_ALL_NEW
(Subgoal.FOCUS (fn {prems, ...} =>
let val thm = prems
|> permute_like (op =) (True :: flat xss) (True :: flat_rec_arg_args xss)
|> Library.foldl1 (fn (acc, elem) => elem RS (acc RS rel_funD))
in HEADGOAL (rtac ctxt thm) end) ctxt)))))
(1 upto k) xsss)))
ns xssss 0)))
ctor_rec_transfers')
end;
val corec_unfold_thms = @{thms id_def} @ sumprod_thms_map;
fun mk_corec_tac corec_defs map_ident0s ctor_dtor_corec pre_map_def abs_inverse ctr_def ctxt =
let
val ss = ss_only (pre_map_def :: abs_inverse :: map_ident0s @ 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 ctxt (ctor_dtor_corec RS trans) THEN' asm_simp_tac ss) THEN_MAYBE
HEADGOAL (rtac ctxt refl ORELSE' rtac ctxt (@{thm unit_eq} RS arg_cong))
end;
fun mk_corec_disc_iff_tac case_splits' corecs discs ctxt =
EVERY (@{map 3} (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 ctxt disc)))
(map (rtac ctxt) case_splits' @ [K all_tac]) corecs discs);
fun mk_corec_transfer_tac ctxt actives passives type_definitions corec_defs dtor_corec_transfers
rel_pre_T_defs rel_eqs pgs pss qssss gssss =
let
val num_pgs = length pgs;
fun prem_no_of x = 1 + find_index (curry (op =) x) pgs;
val Inl_Inr_Pair_tac = REPEAT_DETERM o (resolve_tac ctxt
[mk_rel_funDN 1 @{thm Inl_transfer},
mk_rel_funDN 1 @{thm Inl_transfer[of "op =" "op =", simplified sum.rel_eq]},
mk_rel_funDN 1 @{thm Inr_transfer},
mk_rel_funDN 1 @{thm Inr_transfer[of "op =" "op =", simplified sum.rel_eq]},
mk_rel_funDN 2 @{thm Pair_transfer},
mk_rel_funDN 2 @{thm Pair_transfer[of "op =" "op =", simplified prod.rel_eq]}]);
fun mk_unfold_If_tac total pos =
HEADGOAL (Inl_Inr_Pair_tac THEN'
rtac ctxt (mk_rel_funDN 3 @{thm If_transfer}) THEN'
select_prem_tac ctxt total (dtac ctxt asm_rl) pos THEN'
dtac ctxt rel_funD THEN' assume_tac ctxt THEN' assume_tac ctxt);
fun mk_unfold_Inl_Inr_Pair_tac total pos =
HEADGOAL (Inl_Inr_Pair_tac THEN'
select_prem_tac ctxt total (dtac ctxt asm_rl) pos THEN'
dtac ctxt rel_funD THEN' assume_tac ctxt THEN' assume_tac ctxt);
fun mk_unfold_arg_tac qs gs =
EVERY (map (mk_unfold_If_tac num_pgs o prem_no_of) qs) THEN
EVERY (map (mk_unfold_Inl_Inr_Pair_tac num_pgs o prem_no_of) gs);
fun mk_unfold_ctr_tac type_definition qss gss =
HEADGOAL (rtac ctxt (mk_rel_funDN 1 (@{thm Abs_transfer} OF
[type_definition, type_definition])) THEN' Inl_Inr_Pair_tac) THEN
(case (qss, gss) of
([], []) => HEADGOAL (rtac ctxt refl)
| _ => EVERY (map2 mk_unfold_arg_tac qss gss));
fun mk_unfold_type_tac type_definition ps qsss gsss =
let
val p_tacs = map (mk_unfold_If_tac num_pgs o prem_no_of) ps;
val qg_tacs = map2 (mk_unfold_ctr_tac type_definition) qsss gsss;
fun mk_unfold_ty [] [qg_tac] = qg_tac
| mk_unfold_ty (p_tac :: p_tacs) (qg_tac :: qg_tacs) =
p_tac THEN qg_tac THEN mk_unfold_ty p_tacs qg_tacs
in
HEADGOAL (rtac ctxt rel_funI) THEN mk_unfold_ty p_tacs qg_tacs
end;
fun mk_unfold_corec_type_tac dtor_corec_transfer corec_def =
let
val active :: actives' = actives;
val dtor_corec_transfer' =
infer_instantiate' ctxt
(SOME active :: map SOME passives @ map SOME actives') dtor_corec_transfer;
in
HEADGOAL Goal.conjunction_tac THEN REPEAT_DETERM (HEADGOAL (rtac ctxt rel_funI)) THEN
unfold_thms_tac ctxt [corec_def] THEN
HEADGOAL (etac ctxt (mk_rel_funDN_rotated (1 + length actives) dtor_corec_transfer')) THEN
unfold_thms_tac ctxt (rel_pre_T_defs @ rel_eqs)
end;
fun mk_unfold_prop_tac dtor_corec_transfer corec_def =
mk_unfold_corec_type_tac dtor_corec_transfer corec_def THEN
EVERY (@{map 4} mk_unfold_type_tac type_definitions pss qssss gssss);
in
HEADGOAL Goal.conjunction_tac THEN
EVERY (map2 mk_unfold_prop_tac dtor_corec_transfers corec_defs)
end;
fun solve_prem_prem_tac ctxt =
REPEAT o (eresolve_tac ctxt @{thms bexE rev_bexI} ORELSE'
rtac ctxt @{thm rev_bexI[OF UNIV_I]} ORELSE' hyp_subst_tac ctxt ORELSE'
resolve_tac ctxt @{thms disjI1 disjI2}) THEN'
(rtac ctxt refl ORELSE' assume_tac ctxt ORELSE' rtac ctxt @{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 ctxt nn (dtac ctxt meta_spec) kk,
etac ctxt 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 ctxt n (rotate_tac 1) k, rotate_tac ~1, hyp_subst_tac ctxt,
REPEAT_DETERM_N m o (dtac ctxt meta_spec THEN' rotate_tac ~1)]) THEN
EVERY [REPEAT_DETERM_N r
(HEADGOAL (rotate_tac ~1 THEN' dtac ctxt meta_mp THEN' rotate_tac 1) THEN prefer_tac 2),
if r > 0 then ALLGOALS (Goal.norm_hhf_tac ctxt) else all_tac, HEADGOAL (assume_tac ctxt),
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
(if exists is_def_looping ctr_defs then
EVERY (map (fn def => HEADGOAL (subst_asm_tac ctxt NONE [def])) ctr_defs)
else
unfold_thms_tac ctxt ctr_defs) THEN
HEADGOAL (rtac ctxt ctor_induct') THEN co_induct_inst_as_projs_tac ctxt 0 THEN
EVERY (@{map 4} (EVERY oooo @{map 3} 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'
(assume_tac ctxt ORELSE' REPEAT o etac ctxt conjE THEN'
full_simp_tac (ss_only (no_refl discs @ rel_eqs @ more_simp_thms) ctxt) THEN'
REPEAT o etac ctxt conjE THEN_MAYBE' REPEAT o hyp_subst_tac ctxt THEN'
REPEAT o (resolve_tac ctxt [refl, conjI] ORELSE' assume_tac ctxt));
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 ctxt 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 ctxt allI, rtac ctxt allI, rtac ctxt impI,
select_prem_tac ctxt nn (dtac ctxt meta_spec) kk, dtac ctxt meta_spec, dtac ctxt meta_mp,
assume_tac ctxt, rtac ctxt exhaust, K (co_induct_inst_as_projs_tac ctxt 0),
hyp_subst_tac ctxt] @
@{map 4} (fn k => fn ctr_def => fn discs => fn sels =>
EVERY' ([rtac ctxt 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 ctxt dtor_coinduct' THEN'
EVERY' (@{map 10} (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_map_tac ctxt abs_inverses pre_map_def map_ctor live_nesting_map_id0s ctr_defs'
extra_unfolds =
TRYALL Goal.conjunction_tac THEN
unfold_thms_tac ctxt (pre_map_def :: map_ctor :: abs_inverses @ live_nesting_map_id0s @
ctr_defs' @ extra_unfolds @ sumprod_thms_map @
@{thms o_apply id_apply id_o o_id}) THEN
ALLGOALS (rtac ctxt refl);
fun mk_map_disc_iff_tac ctxt ct exhaust discs maps =
TRYALL Goal.conjunction_tac THEN
ALLGOALS (rtac ctxt (infer_instantiate' ctxt [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 eqFalseI
handle THM _ => thm RS eqTrueI) discs) THEN
ALLGOALS (rtac ctxt refl ORELSE' rtac ctxt TrueI);
fun mk_map_sel_tac ctxt ct exhaust discs maps sels map_id0s =
TRYALL Goal.conjunction_tac THEN
ALLGOALS (rtac ctxt (infer_instantiate' ctxt [SOME ct] exhaust) THEN_ALL_NEW
REPEAT_DETERM o hyp_subst_tac ctxt) THEN
unfold_thms_tac ctxt ((discs RL [eqTrueI, eqFalseI]) @
@{thms not_True_eq_False not_False_eq_True}) THEN
TRYALL (etac ctxt FalseE ORELSE' etac ctxt @{thm TrueE}) THEN
unfold_thms_tac ctxt (@{thm id_apply} :: maps @ sels @ map_id0s) THEN
ALLGOALS (rtac ctxt refl);
fun mk_rel_tac ctxt abs_inverses pre_rel_def rel_ctor live_nesting_rel_eqs ctr_defs' extra_unfolds =
TRYALL Goal.conjunction_tac THEN
unfold_thms_tac ctxt (pre_rel_def :: rel_ctor :: abs_inverses @ live_nesting_rel_eqs @ ctr_defs' @
extra_unfolds @ sumprod_thms_rel @ @{thms vimage2p_def o_apply sum.inject
sum.distinct(1)[THEN eq_False[THEN iffD2]] not_False_eq_True}) THEN
ALLGOALS (resolve_tac ctxt [TrueI, refl]);
fun mk_rel_case_tac ctxt ct1 ct2 exhaust injects rel_injects distincts rel_distincts rel_eqs =
HEADGOAL (rtac ctxt (infer_instantiate' ctxt [SOME ct1] exhaust) THEN_ALL_NEW
rtac ctxt (infer_instantiate' ctxt [SOME ct2] exhaust) THEN_ALL_NEW
hyp_subst_tac ctxt) THEN
unfold_thms_tac ctxt (rel_eqs @ injects @ rel_injects @
@{thms conj_imp_eq_imp_imp simp_thms(6) True_implies_equals} @
map (fn thm => thm RS eqFalseI) (distincts @ rel_distincts) @
map (fn thm => thm RS eqTrueI) rel_injects) THEN
TRYALL (assume_tac ctxt ORELSE' etac ctxt FalseE ORELSE'
(REPEAT_DETERM o dtac ctxt meta_spec THEN'
TRY o filter_prems_tac ctxt
(forall (curry (op <>) (HOLogic.mk_Trueprop @{term False})) o Logic.strip_imp_prems) THEN'
REPEAT_DETERM o (dtac ctxt meta_mp THEN' rtac ctxt refl) THEN'
(assume_tac ctxt ORELSE' Goal.assume_rule_tac ctxt)));
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 nesting_rel_eqs =
rtac ctxt dtor_rel_coinduct 1 THEN
EVERY (@{map 11} (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 ctxt exhaust THEN_ALL_NEW (rtac ctxt exhaust THEN_ALL_NEW
(dtac ctxt (rotate_prems ~1 (infer_instantiate' ctxt [NONE, NONE, NONE, NONE, SOME ct]
@{thm arg_cong2} RS iffD1)) THEN'
assume_tac ctxt THEN' assume_tac ctxt THEN' hyp_subst_tac ctxt THEN' dtac ctxt assm THEN'
REPEAT_DETERM o etac ctxt conjE))) 1 THEN
unfold_thms_tac ctxt ((discs RL [eqTrueI, eqFalseI]) @ sels @ simp_thms') THEN
unfold_thms_tac ctxt (dtor_ctor :: rel_pre_def :: abs_inverse :: ctor_inject ::
abs_inject :: ctor_defs @ nesting_rel_eqs @ simp_thms' @
@{thms id_bnf_def rel_sum_simps rel_prod_inject vimage2p_def Inl_Inr_False
iffD2[OF eq_False Inr_not_Inl] sum.inject prod.inject}) THEN
REPEAT_DETERM (HEADGOAL ((REPEAT_DETERM o etac ctxt conjE) THEN'
(REPEAT_DETERM o rtac ctxt conjI) THEN' (rtac ctxt refl ORELSE' assume_tac ctxt))))
cts assms exhausts discss selss ctor_defss dtor_ctors ctor_injects abs_injects rel_pre_defs
abs_inverses);
fun mk_rel_induct0_tac ctxt ctor_rel_induct assms cterms exhausts ctor_defss ctor_injects
rel_pre_list_defs Abs_inverses nesting_rel_eqs =
rtac ctxt ctor_rel_induct 1 THEN EVERY (@{map 6} (fn cterm => fn exhaust => fn ctor_defs =>
fn ctor_inject => fn rel_pre_list_def => fn Abs_inverse =>
HEADGOAL (rtac ctxt exhaust THEN_ALL_NEW (rtac ctxt exhaust THEN_ALL_NEW
(rtac ctxt (infer_instantiate' ctxt (replicate 4 NONE @ [SOME cterm]) @{thm arg_cong2}
RS iffD2)
THEN' assume_tac ctxt THEN' assume_tac ctxt THEN' TRY o resolve_tac ctxt assms))) THEN
unfold_thms_tac ctxt (ctor_inject :: rel_pre_list_def :: ctor_defs @ nesting_rel_eqs @
@{thms id_bnf_def vimage2p_def}) THEN
TRYALL (hyp_subst_tac ctxt) THEN
unfold_thms_tac ctxt (Abs_inverse :: @{thms rel_sum_simps rel_prod_inject Inl_Inr_False
Inr_Inl_False sum.inject prod.inject}) THEN
TRYALL (rtac ctxt refl ORELSE' etac ctxt FalseE ORELSE'
(REPEAT_DETERM o etac ctxt conjE) THEN' assume_tac ctxt))
cterms exhausts ctor_defss ctor_injects rel_pre_list_defs Abs_inverses);
fun mk_rel_sel_tac ctxt ct1 ct2 exhaust discs sels rel_injects distincts rel_distincts rel_eqs =
HEADGOAL (rtac ctxt (infer_instantiate' ctxt [SOME ct1] exhaust) THEN_ALL_NEW
rtac ctxt (infer_instantiate' ctxt [SOME ct2] exhaust) THEN_ALL_NEW hyp_subst_tac ctxt) THEN
unfold_thms_tac ctxt (sels @ rel_injects @ rel_eqs @
@{thms simp_thms(6,7,8,11,12,15,16,21,22,24)} @ ((discs @ distincts) RL [eqTrueI, eqFalseI]) @
(rel_injects RL [eqTrueI]) @ (rel_distincts RL [eqFalseI])) THEN
TRYALL (resolve_tac ctxt [TrueI, refl]);
fun mk_sel_transfer_tac ctxt n sel_defs case_transfer =
TRYALL Goal.conjunction_tac THEN
unfold_thms_tac ctxt (map (Local_Defs.abs_def_rule ctxt) sel_defs) THEN
ALLGOALS (rtac ctxt (mk_rel_funDN n case_transfer) THEN_ALL_NEW
REPEAT_DETERM o (assume_tac ctxt ORELSE' rtac ctxt rel_funI));
fun mk_set0_tac ctxt abs_inverses pre_set_defs dtor_ctor fp_sets fp_nesting_set_maps
live_nesting_set_maps ctr_defs' extra_unfolds =
TRYALL Goal.conjunction_tac THEN
unfold_thms_tac ctxt ctr_defs' THEN
ALLGOALS (subst_tac ctxt NONE fp_sets) THEN
unfold_thms_tac ctxt (dtor_ctor :: abs_inverses @ pre_set_defs @ fp_nesting_set_maps @
live_nesting_set_maps @ extra_unfolds @ basic_sumprod_thms_set @
@{thms UN_UN_flatten UN_Un_distrib UN_Un sup_assoc[THEN sym]}) THEN
ALLGOALS (rtac ctxt @{thm set_eqI[OF iffI]}) THEN
ALLGOALS (REPEAT_DETERM o etac ctxt UnE) THEN
ALLGOALS (REPEAT o resolve_tac ctxt @{thms UnI1 UnI2} THEN' assume_tac ctxt);
fun mk_set_sel_tac ctxt ct exhaust discs sels sets =
TRYALL Goal.conjunction_tac THEN
ALLGOALS (rtac ctxt (infer_instantiate' ctxt [SOME ct] exhaust) THEN_ALL_NEW
REPEAT_DETERM o hyp_subst_tac ctxt) THEN
unfold_thms_tac ctxt ((discs RL [eqTrueI, eqFalseI]) @
@{thms not_True_eq_False not_False_eq_True}) THEN
TRYALL (etac ctxt FalseE ORELSE' etac ctxt @{thm TrueE}) THEN
unfold_thms_tac ctxt (sels @ sets) THEN
ALLGOALS (REPEAT o (resolve_tac ctxt @{thms UnI1 UnI2 imageI} ORELSE'
eresolve_tac ctxt @{thms UN_I UN_I[rotated] imageE} ORELSE'
hyp_subst_tac ctxt) THEN'
(rtac ctxt @{thm singletonI} ORELSE' assume_tac ctxt));
fun mk_set_cases_tac ctxt ct assms exhaust sets =
HEADGOAL (rtac ctxt (infer_instantiate' ctxt [SOME ct] exhaust)
THEN_ALL_NEW hyp_subst_tac ctxt) THEN
unfold_thms_tac ctxt sets THEN
REPEAT_DETERM (HEADGOAL
(eresolve_tac ctxt @{thms FalseE emptyE singletonE UnE UN_E insertE} ORELSE'
hyp_subst_tac ctxt ORELSE'
SELECT_GOAL (SOLVE (HEADGOAL (eresolve_tac ctxt assms THEN' REPEAT_DETERM o
assume_tac ctxt)))));
fun mk_set_intros_tac ctxt sets =
TRYALL Goal.conjunction_tac THEN unfold_thms_tac ctxt sets THEN
TRYALL (REPEAT o
(resolve_tac ctxt @{thms UnI1 UnI2} ORELSE'
eresolve_tac ctxt @{thms UN_I UN_I[rotated]}) THEN'
(rtac ctxt @{thm singletonI} ORELSE' assume_tac ctxt));
fun mk_set_induct0_tac ctxt cts assms dtor_set_inducts exhausts set_pre_defs ctor_defs dtor_ctors
Abs_pre_inverses =
let
val assms_tac =
let val assms' = map (unfold_thms ctxt (@{thm id_bnf_def} :: ctor_defs)) assms in
fold (curry (op ORELSE')) (map (fn thm =>
funpow (length (Thm.prems_of thm)) (fn tac => tac THEN' assume_tac ctxt)
(rtac ctxt thm)) assms')
(etac ctxt FalseE)
end;
val exhausts' = map (fn thm => thm RS @{thm asm_rl[of "P x y" for P x y]}) exhausts
|> map2 (fn ct => infer_instantiate' ctxt [NONE, SOME ct]) cts;
in
ALLGOALS (resolve_tac ctxt dtor_set_inducts) THEN
TRYALL (resolve_tac ctxt exhausts' THEN_ALL_NEW
(resolve_tac ctxt (map (fn ct => refl RS
infer_instantiate' ctxt (replicate 4 NONE @ [SOME ct]) @{thm arg_cong2} RS iffD2) cts)
THEN' assume_tac ctxt THEN' hyp_subst_tac ctxt)) THEN
unfold_thms_tac ctxt (Abs_pre_inverses @ dtor_ctors @ set_pre_defs @ ctor_defs @
@{thms id_bnf_def o_apply sum_set_simps prod_set_simps UN_empty UN_insert Un_empty_left
Un_empty_right empty_iff singleton_iff}) THEN
REPEAT (HEADGOAL (hyp_subst_tac ctxt ORELSE'
eresolve_tac ctxt @{thms UN_E UnE singletonE} ORELSE' assms_tac))
end;
end;