(* Title: HOL/BNF/Tools/bnf_gfp_tactics.ML
Author: Dmitriy Traytel, TU Muenchen
Author: Andrei Popescu, TU Muenchen
Author: Jasmin Blanchette, TU Muenchen
Copyright 2012
Tactics for the codatatype construction.
*)
signature BNF_GFP_TACTICS =
sig
val mk_Lev_sbd_tac: cterm option list -> thm list -> thm list -> thm list list -> tactic
val mk_bd_card_order_tac: thm -> tactic
val mk_bd_cinfinite_tac: thm -> tactic
val mk_bis_Gr_tac: thm -> thm list -> thm list -> thm list -> thm list ->
{prems: 'a, context: Proof.context} -> tactic
val mk_bis_O_tac: int -> thm -> thm list -> thm list -> tactic
val mk_bis_Union_tac: thm -> thm list -> {prems: 'a, context: Proof.context} -> tactic
val mk_bis_converse_tac: int -> thm -> thm list -> thm list -> tactic
val mk_bis_srel_tac: int -> thm -> thm list -> thm list -> thm list -> thm list list -> tactic
val mk_carT_set_tac: int -> int -> thm -> thm -> thm -> thm ->
{prems: 'a, context: Proof.context} -> tactic
val mk_card_of_carT_tac: int -> thm list -> thm -> thm -> thm -> thm -> thm -> thm list -> tactic
val mk_coalgT_tac: int -> thm list -> thm list -> thm list list ->
{prems: 'a, context: Proof.context} -> tactic
val mk_coalg_final_tac: int -> thm -> thm list -> thm list -> thm list list -> thm list list ->
tactic
val mk_coalg_set_tac: thm -> tactic
val mk_coalg_thePull_tac: int -> thm -> thm list -> thm list list -> (int -> tactic) list ->
{prems: 'a, context: Proof.context} -> tactic
val mk_coind_wit_tac: thm -> thm list -> thm list -> thm list ->
{prems: 'a, context: Proof.context} -> tactic
val mk_col_bd_tac: int -> int -> cterm option list -> thm list -> thm list -> thm -> thm ->
thm list list -> tactic
val mk_col_natural_tac: cterm option list -> thm list -> thm list -> thm list -> thm list list ->
{prems: 'a, context: Proof.context} -> tactic
val mk_congruent_str_final_tac: int -> thm -> thm -> thm -> thm list -> tactic
val mk_corec_tac: int -> thm list -> thm -> thm -> thm list ->
{prems: 'a, context: Proof.context} -> tactic
val mk_dtor_coinduct_tac: int -> int list -> thm -> thm -> tactic
val mk_dtor_srel_tac: thm list -> int -> thm -> thm -> thm -> thm -> thm list -> thm -> thm ->
thm list -> thm list -> thm list list -> tactic
val mk_dtor_strong_coinduct_tac: int list -> ctyp option list -> cterm option list -> thm ->
thm -> thm -> tactic
val mk_dtor_o_ctor_tac: thm -> thm -> thm -> thm -> thm list ->
{prems: 'a, context: Proof.context} -> tactic
val mk_equiv_lsbis_tac: thm -> thm -> thm -> thm -> thm -> thm -> tactic
val mk_hset_minimal_tac: int -> thm list -> thm -> {prems: 'a, context: Proof.context} -> tactic
val mk_hset_rec_minimal_tac: int -> cterm option list -> thm list -> thm list ->
{prems: 'a, context: Proof.context} -> tactic
val mk_in_bd_tac: thm -> thm list -> thm -> thm -> thm -> thm -> thm list -> thm -> thm -> thm ->
thm -> thm -> thm -> tactic
val mk_incl_lsbis_tac: int -> int -> thm -> tactic
val mk_isNode_hset_tac: int -> thm -> thm list -> {prems: 'a, context: Proof.context} -> tactic
val mk_length_Lev'_tac: thm -> tactic
val mk_length_Lev_tac: cterm option list -> thm list -> thm list -> tactic
val mk_map_comp_tac: int -> int -> thm list -> thm list -> thm list -> thm -> tactic
val mk_mcong_tac: int -> (int -> tactic) -> thm list -> thm list -> thm list -> thm list list ->
thm list list -> thm list list list -> tactic
val mk_map_id_tac: thm list -> thm -> thm -> tactic
val mk_map_tac: int -> int -> ctyp option -> thm -> thm -> thm -> tactic
val mk_dtor_map_unique_tac: thm -> thm list -> {prems: 'a, context: Proof.context} -> tactic
val mk_mor_Abs_tac: thm list -> thm list -> {prems: 'a, context: Proof.context} -> tactic
val mk_mor_Rep_tac: int -> thm list -> thm list -> thm list -> thm list list -> thm list ->
thm list -> {prems: 'a, context: Proof.context} -> tactic
val mk_mor_T_final_tac: thm -> thm list -> thm list -> tactic
val mk_mor_UNIV_tac: thm list -> thm -> tactic
val mk_mor_beh_tac: int -> thm -> thm -> thm list -> thm list -> thm list -> thm list ->
thm list list -> thm list list -> thm list -> thm list -> thm list -> thm list -> thm list ->
thm list -> thm list -> thm list -> thm list list -> thm list list list -> thm list list list ->
thm list list list -> thm list list -> thm list list -> thm list -> thm list -> thm list ->
{prems: 'a, context: Proof.context} -> tactic
val mk_mor_comp_tac: thm -> thm list -> thm list -> thm list -> tactic
val mk_mor_elim_tac: thm -> tactic
val mk_mor_hset_rec_tac: int -> int -> cterm option list -> int -> thm list -> thm list ->
thm list -> thm list list -> thm list list -> tactic
val mk_mor_hset_tac: thm -> thm -> tactic
val mk_mor_incl_tac: thm -> thm list -> tactic
val mk_mor_str_tac: 'a list -> thm -> tactic
val mk_mor_sum_case_tac: 'a list -> thm -> tactic
val mk_mor_thePull_fst_tac: int -> thm -> thm list -> thm list -> (int -> tactic) list ->
{prems: thm list, context: Proof.context} -> tactic
val mk_mor_thePull_snd_tac: int -> thm -> thm list -> thm list -> (int -> tactic) list ->
{prems: thm list, context: Proof.context} -> tactic
val mk_mor_thePull_pick_tac: thm -> thm list -> thm list ->
{prems: 'a, context: Proof.context} -> tactic
val mk_mor_unfold_tac: int -> thm -> thm list -> thm list -> thm list -> thm list -> thm list ->
thm list -> tactic
val mk_prefCl_Lev_tac: cterm option list -> thm list -> thm list -> tactic
val mk_pickWP_assms_tac: thm list -> thm list -> thm -> (int -> tactic)
val mk_pick_col_tac: int -> int -> cterm option list -> thm list -> thm list -> thm list ->
thm list list -> thm list -> (int -> tactic) list -> {prems: 'a, context: Proof.context} ->
tactic
val mk_raw_coind_tac: thm -> thm -> thm -> thm -> thm -> thm -> thm -> thm -> thm -> thm list ->
thm list -> thm list -> thm -> thm list -> tactic
val mk_rv_last_tac: ctyp option list -> cterm option list -> thm list -> thm list -> tactic
val mk_sbis_lsbis_tac: thm list -> thm -> thm -> tactic
val mk_set_Lev_tac: cterm option list -> thm list -> thm list -> thm list -> thm list ->
thm list list -> tactic
val mk_set_bd_tac: thm -> thm -> thm -> tactic
val mk_set_hset_incl_hset_tac: int -> thm list -> thm -> int -> tactic
val mk_set_image_Lev_tac: cterm option list -> thm list -> thm list -> thm list -> thm list ->
thm list list -> thm list list -> tactic
val mk_set_incl_hin_tac: thm list -> tactic
val mk_set_incl_hset_tac: thm -> thm -> tactic
val mk_set_le_tac: int -> thm -> thm list -> thm list list -> tactic
val mk_set_natural_tac: thm -> thm -> tactic
val mk_set_rv_Lev_tac: int -> cterm option list -> thm list -> thm list -> thm list -> thm list ->
thm list list -> thm list list -> tactic
val mk_set_simp_tac: int -> thm -> thm -> thm list -> tactic
val mk_srel_coinduct_tac: 'a list -> thm -> thm -> tactic
val mk_srel_strong_coinduct_tac: int -> ctyp option list -> cterm option list -> thm ->
thm list -> thm list -> tactic
val mk_strT_hset_tac: int -> int -> int -> ctyp option list -> ctyp option list ->
cterm option list -> thm list -> thm list -> thm list -> thm list -> thm list list ->
thm list list -> thm list list -> thm -> thm list list -> tactic
val mk_unfold_unique_mor_tac: thm list -> thm -> thm -> thm list -> tactic
val mk_unique_mor_tac: thm list -> thm -> tactic
val mk_wit_tac: int -> thm list -> thm list -> thm list -> thm list ->
{prems: 'a, context: Proof.context} -> tactic
val mk_wpull_tac: int -> thm -> thm -> thm -> thm -> thm -> thm list -> thm list -> tactic
end;
structure BNF_GFP_Tactics : BNF_GFP_TACTICS =
struct
open BNF_Tactics
open BNF_Util
open BNF_FP
open BNF_GFP_Util
val fst_convol_fun_cong_sym = @{thm fst_convol} RS fun_cong RS sym;
val list_inject_iffD1 = @{thm list.inject[THEN iffD1]};
val nat_induct = @{thm nat_induct};
val o_apply_trans_sym = o_apply RS trans RS sym;
val ord_eq_le_trans = @{thm ord_eq_le_trans};
val ord_eq_le_trans_trans_fun_cong_image_id_id_apply =
@{thm ord_eq_le_trans[OF trans[OF fun_cong[OF image_id] id_apply]]};
val ordIso_ordLeq_trans = @{thm ordIso_ordLeq_trans};
val snd_convol_fun_cong_sym = @{thm snd_convol} RS fun_cong RS sym;
val sum_case_weak_cong = @{thm sum_case_weak_cong};
val trans_fun_cong_image_id_id_apply = @{thm trans[OF fun_cong[OF image_id] id_apply]};
fun mk_coalg_set_tac coalg_def =
dtac (coalg_def RS iffD1) 1 THEN
REPEAT_DETERM (etac conjE 1) THEN
EVERY' [dtac @{thm rev_bspec}, atac] 1 THEN
REPEAT_DETERM (eresolve_tac [CollectE, conjE] 1) THEN atac 1;
fun mk_mor_elim_tac mor_def =
(dtac (subst OF [mor_def]) THEN'
REPEAT o etac conjE THEN'
TRY o rtac @{thm image_subsetI} THEN'
etac bspec THEN'
atac) 1;
fun mk_mor_incl_tac mor_def map_id's =
(stac mor_def THEN'
rtac conjI THEN'
CONJ_WRAP' (K (EVERY' [rtac ballI, etac set_mp, stac @{thm id_apply}, atac]))
map_id's THEN'
CONJ_WRAP' (fn thm =>
(EVERY' [rtac ballI, rtac (thm RS trans), rtac sym, rtac (@{thm id_apply} RS arg_cong)]))
map_id's) 1;
fun mk_mor_comp_tac mor_def mor_images morEs map_comp_ids =
let
fun fbetw_tac image = EVERY' [rtac ballI, stac o_apply, etac image, etac image, atac];
fun mor_tac ((mor_image, morE), map_comp_id) =
EVERY' [rtac ballI, stac o_apply, rtac trans, rtac (map_comp_id RS sym), rtac trans,
etac (morE RS arg_cong), atac, etac morE, etac mor_image, atac];
in
(stac mor_def THEN' rtac conjI THEN'
CONJ_WRAP' fbetw_tac mor_images THEN'
CONJ_WRAP' mor_tac ((mor_images ~~ morEs) ~~ map_comp_ids)) 1
end;
fun mk_mor_UNIV_tac morEs mor_def =
let
val n = length morEs;
fun mor_tac morE = EVERY' [rtac ext, rtac trans, rtac o_apply, rtac trans, etac morE,
rtac UNIV_I, rtac sym, rtac o_apply];
in
EVERY' [rtac iffI, CONJ_WRAP' mor_tac morEs,
stac mor_def, rtac conjI, CONJ_WRAP' (K (rtac ballI THEN' rtac UNIV_I)) morEs,
CONJ_WRAP' (fn i =>
EVERY' [dtac (mk_conjunctN n i), rtac ballI, etac @{thm pointfreeE}]) (1 upto n)] 1
end;
fun mk_mor_str_tac ks mor_UNIV =
(stac mor_UNIV THEN' CONJ_WRAP' (K (rtac refl)) ks) 1;
fun mk_mor_sum_case_tac ks mor_UNIV =
(stac mor_UNIV THEN' CONJ_WRAP' (K (rtac @{thm sum_case_comp_Inl[symmetric]})) ks) 1;
fun mk_set_incl_hset_tac def rec_Suc =
EVERY' (stac def ::
map rtac [@{thm incl_UNION_I}, UNIV_I, @{thm ord_le_eq_trans}, @{thm Un_upper1},
sym, rec_Suc]) 1;
fun mk_set_hset_incl_hset_tac n defs rec_Suc i =
EVERY' (map (TRY oo stac) defs @
map rtac [@{thm UN_least}, subsetI, @{thm UN_I}, UNIV_I, set_mp, equalityD2, rec_Suc, UnI2,
mk_UnIN n i] @
[etac @{thm UN_I}, atac]) 1;
fun mk_set_incl_hin_tac incls =
if null incls then rtac subset_UNIV 1
else EVERY' [rtac subsetI, rtac CollectI,
CONJ_WRAP' (fn incl => EVERY' [rtac subset_trans, etac incl, atac]) incls] 1;
fun mk_hset_rec_minimal_tac m cts rec_0s rec_Sucs {context = ctxt, prems = _} =
EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn thm => EVERY'
[rtac ord_eq_le_trans, rtac thm, rtac @{thm empty_subsetI}]) rec_0s,
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn rec_Suc => EVERY'
[rtac ord_eq_le_trans, rtac rec_Suc,
if m = 0 then K all_tac
else (rtac @{thm Un_least} THEN' Goal.assume_rule_tac ctxt),
CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least}))
(K (EVERY' [rtac @{thm UN_least}, REPEAT_DETERM o eresolve_tac [allE, conjE],
rtac subset_trans, atac, Goal.assume_rule_tac ctxt])) rec_0s])
rec_Sucs] 1;
fun mk_hset_minimal_tac n hset_defs hset_rec_minimal {context = ctxt, prems = _} =
(CONJ_WRAP' (fn def => (EVERY' [rtac ord_eq_le_trans, rtac def,
rtac @{thm UN_least}, rtac rev_mp, rtac hset_rec_minimal,
EVERY' (replicate ((n + 1) * n) (Goal.assume_rule_tac ctxt)), rtac impI,
REPEAT_DETERM o eresolve_tac [allE, conjE], atac])) hset_defs) 1
fun mk_mor_hset_rec_tac m n cts j rec_0s rec_Sucs morEs set_naturalss coalg_setss =
EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn thm => EVERY' (map rtac [impI, thm RS trans, thm RS sym])) rec_0s,
REPEAT_DETERM o rtac allI,
CONJ_WRAP'
(fn (rec_Suc, (morE, ((passive_set_naturals, active_set_naturals), coalg_sets))) =>
EVERY' [rtac impI, rtac (rec_Suc RS trans), rtac (rec_Suc RS trans RS sym),
if m = 0 then K all_tac
else EVERY' [rtac @{thm Un_cong}, rtac box_equals,
rtac (nth passive_set_naturals (j - 1) RS sym),
rtac trans_fun_cong_image_id_id_apply, etac (morE RS arg_cong), atac],
CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_cong}))
(fn (i, (set_natural, coalg_set)) =>
EVERY' [rtac sym, rtac trans, rtac (refl RSN (2, @{thm UN_cong})),
etac (morE RS sym RS arg_cong RS trans), atac, rtac set_natural,
rtac (@{thm UN_simps(10)} RS trans), rtac (refl RS @{thm UN_cong}),
ftac coalg_set, atac, dtac set_mp, atac, rtac mp, rtac (mk_conjunctN n i),
REPEAT_DETERM o etac allE, atac, atac])
(rev ((1 upto n) ~~ (active_set_naturals ~~ coalg_sets)))])
(rec_Sucs ~~ (morEs ~~ (map (chop m) set_naturalss ~~ map (drop m) coalg_setss)))] 1;
fun mk_mor_hset_tac hset_def mor_hset_rec =
EVERY' [rtac (hset_def RS trans), rtac (refl RS @{thm UN_cong} RS trans), etac mor_hset_rec,
atac, atac, rtac (hset_def RS sym)] 1
fun mk_bis_srel_tac m bis_def srel_O_Grs map_comps map_congs set_naturalss =
let
val n = length srel_O_Grs;
val thms = ((1 upto n) ~~ map_comps ~~ map_congs ~~ set_naturalss ~~ srel_O_Grs);
fun mk_if_tac ((((i, map_comp), map_cong), set_naturals), srel_O_Gr) =
EVERY' [rtac allI, rtac allI, rtac impI, dtac (mk_conjunctN n i),
etac allE, etac allE, etac impE, atac, etac bexE, etac conjE,
rtac (srel_O_Gr RS equalityD2 RS set_mp),
rtac @{thm relcompI}, rtac @{thm converseI},
EVERY' (map (fn thm =>
EVERY' [rtac @{thm GrI}, REPEAT_DETERM o eresolve_tac [CollectE, conjE],
rtac CollectI,
CONJ_WRAP' (fn (i, thm) =>
if i <= m
then EVERY' [rtac ord_eq_le_trans, rtac thm, rtac subset_trans,
etac @{thm image_mono}, rtac @{thm image_subsetI}, etac @{thm diagI}]
else EVERY' [rtac ord_eq_le_trans, rtac trans, rtac thm,
rtac trans_fun_cong_image_id_id_apply, atac])
(1 upto (m + n) ~~ set_naturals),
rtac trans, rtac trans, rtac map_comp, rtac map_cong, REPEAT_DETERM_N m o rtac thm,
REPEAT_DETERM_N n o rtac (@{thm o_id} RS fun_cong), atac])
@{thms fst_diag_id snd_diag_id})];
fun mk_only_if_tac ((((i, map_comp), map_cong), set_naturals), srel_O_Gr) =
EVERY' [dtac (mk_conjunctN n i), rtac allI, rtac allI, rtac impI,
etac allE, etac allE, etac impE, atac,
dtac (srel_O_Gr RS equalityD1 RS set_mp),
REPEAT_DETERM o eresolve_tac [CollectE, @{thm relcompE}, @{thm converseE}],
REPEAT_DETERM o eresolve_tac [@{thm GrE}, exE, conjE],
REPEAT_DETERM o dtac Pair_eqD,
REPEAT_DETERM o etac conjE,
hyp_subst_tac,
REPEAT_DETERM o eresolve_tac [CollectE, conjE],
rtac bexI, rtac conjI, rtac trans, rtac map_comp,
REPEAT_DETERM_N m o stac @{thm id_o},
REPEAT_DETERM_N n o stac @{thm o_id},
etac sym, rtac trans, rtac map_comp,
REPEAT_DETERM_N m o stac @{thm id_o},
REPEAT_DETERM_N n o stac @{thm o_id},
rtac trans, rtac map_cong,
REPEAT_DETERM_N m o EVERY' [rtac @{thm diagE'}, etac set_mp, atac],
REPEAT_DETERM_N n o rtac refl,
etac sym, rtac CollectI,
CONJ_WRAP' (fn (i, thm) =>
if i <= m
then EVERY' [rtac ord_eq_le_trans, rtac thm, rtac @{thm image_subsetI},
rtac @{thm diag_fst}, etac set_mp, atac]
else EVERY' [rtac ord_eq_le_trans, rtac trans, rtac thm,
rtac trans_fun_cong_image_id_id_apply, atac])
(1 upto (m + n) ~~ set_naturals)];
in
EVERY' [rtac (bis_def RS trans),
rtac iffI, etac conjE, etac conjI, CONJ_WRAP' mk_if_tac thms,
etac conjE, etac conjI, CONJ_WRAP' mk_only_if_tac thms] 1
end;
fun mk_bis_converse_tac m bis_srel srel_congs srel_converses =
EVERY' [stac bis_srel, dtac (bis_srel RS iffD1),
REPEAT_DETERM o etac conjE, rtac conjI,
CONJ_WRAP' (K (EVERY' [rtac @{thm converse_shift}, etac subset_trans,
rtac equalityD2, rtac @{thm converse_Times}])) srel_congs,
CONJ_WRAP' (fn (srel_cong, srel_converse) =>
EVERY' [rtac allI, rtac allI, rtac impI, rtac @{thm set_mp[OF equalityD2]},
rtac (srel_cong RS trans),
REPEAT_DETERM_N m o rtac @{thm diag_converse},
REPEAT_DETERM_N (length srel_congs) o rtac refl,
rtac srel_converse,
REPEAT_DETERM o etac allE,
rtac @{thm converseI}, etac mp, etac @{thm converseD}]) (srel_congs ~~ srel_converses)] 1;
fun mk_bis_O_tac m bis_srel srel_congs srel_Os =
EVERY' [stac bis_srel, REPEAT_DETERM o dtac (bis_srel RS iffD1),
REPEAT_DETERM o etac conjE, rtac conjI,
CONJ_WRAP' (K (EVERY' [etac @{thm relcomp_subset_Sigma}, atac])) srel_congs,
CONJ_WRAP' (fn (srel_cong, srel_O) =>
EVERY' [rtac allI, rtac allI, rtac impI, rtac @{thm set_mp[OF equalityD2]},
rtac (srel_cong RS trans),
REPEAT_DETERM_N m o rtac @{thm diag_Comp},
REPEAT_DETERM_N (length srel_congs) o rtac refl,
rtac srel_O,
etac @{thm relcompE},
REPEAT_DETERM o dtac Pair_eqD,
etac conjE, hyp_subst_tac,
REPEAT_DETERM o etac allE, rtac @{thm relcompI},
etac mp, atac, etac mp, atac]) (srel_congs ~~ srel_Os)] 1;
fun mk_bis_Gr_tac bis_srel srel_Grs mor_images morEs coalg_ins
{context = ctxt, prems = _} =
unfold_thms_tac ctxt (bis_srel :: @{thm diag_Gr} :: srel_Grs) THEN
EVERY' [rtac conjI,
CONJ_WRAP' (fn thm => rtac (@{thm Gr_incl} RS ssubst) THEN' etac thm) mor_images,
CONJ_WRAP' (fn (coalg_in, morE) =>
EVERY' [rtac allI, rtac allI, rtac impI, rtac @{thm GrI}, etac coalg_in,
etac @{thm GrD1}, etac (morE RS trans), etac @{thm GrD1},
etac (@{thm GrD2} RS arg_cong)]) (coalg_ins ~~ morEs)] 1;
fun mk_bis_Union_tac bis_def in_monos {context = ctxt, prems = _} =
let
val n = length in_monos;
val ks = 1 upto n;
in
unfold_thms_tac ctxt [bis_def] THEN
EVERY' [rtac conjI,
CONJ_WRAP' (fn i =>
EVERY' [rtac @{thm UN_least}, dtac bspec, atac,
dtac conjunct1, etac (mk_conjunctN n i)]) ks,
CONJ_WRAP' (fn (i, in_mono) =>
EVERY' [rtac allI, rtac allI, rtac impI, etac @{thm UN_E}, dtac bspec, atac,
dtac conjunct2, dtac (mk_conjunctN n i), etac allE, etac allE, dtac mp,
atac, etac bexE, rtac bexI, atac, rtac in_mono,
REPEAT_DETERM_N n o etac @{thm incl_UNION_I[OF _ subset_refl]},
atac]) (ks ~~ in_monos)] 1
end;
fun mk_sbis_lsbis_tac lsbis_defs bis_Union bis_cong =
let
val n = length lsbis_defs;
in
EVERY' [rtac (Thm.permute_prems 0 1 bis_cong), EVERY' (map rtac lsbis_defs),
rtac bis_Union, rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, conjE, exE],
hyp_subst_tac, etac bis_cong, EVERY' (map (rtac o mk_nth_conv n) (1 upto n))] 1
end;
fun mk_incl_lsbis_tac n i lsbis_def =
EVERY' [rtac @{thm xt1(3)}, rtac lsbis_def, rtac @{thm incl_UNION_I}, rtac CollectI,
REPEAT_DETERM_N n o rtac exI, rtac conjI, rtac refl, atac, rtac equalityD2,
rtac (mk_nth_conv n i)] 1;
fun mk_equiv_lsbis_tac sbis_lsbis lsbis_incl incl_lsbis bis_diag bis_converse bis_O =
EVERY' [rtac (@{thm equiv_def} RS iffD2),
rtac conjI, rtac (@{thm refl_on_def} RS iffD2),
rtac conjI, rtac lsbis_incl, rtac ballI, rtac set_mp,
rtac incl_lsbis, rtac bis_diag, atac, etac @{thm diagI},
rtac conjI, rtac (@{thm sym_def} RS iffD2),
rtac allI, rtac allI, rtac impI, rtac set_mp,
rtac incl_lsbis, rtac bis_converse, rtac sbis_lsbis, etac @{thm converseI},
rtac (@{thm trans_def} RS iffD2),
rtac allI, rtac allI, rtac allI, rtac impI, rtac impI, rtac set_mp,
rtac incl_lsbis, rtac bis_O, rtac sbis_lsbis, rtac sbis_lsbis,
etac @{thm relcompI}, atac] 1;
fun mk_coalgT_tac m defs strT_defs set_naturalss {context = ctxt, prems = _} =
let
val n = length strT_defs;
val ks = 1 upto n;
fun coalg_tac (i, ((passive_sets, active_sets), def)) =
EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE],
hyp_subst_tac, rtac (def RS trans RS @{thm ssubst_mem}), etac (arg_cong RS trans),
rtac (mk_sum_casesN n i), rtac CollectI,
EVERY' (map (fn thm => EVERY' [rtac conjI, rtac (thm RS ord_eq_le_trans),
etac ((trans OF [@{thm image_id} RS fun_cong, @{thm id_apply}]) RS ord_eq_le_trans)])
passive_sets),
CONJ_WRAP' (fn (i, thm) => EVERY' [rtac (thm RS ord_eq_le_trans),
rtac @{thm image_subsetI}, rtac CollectI, rtac exI, rtac exI, rtac conjI, rtac refl,
rtac conjI,
rtac conjI, etac @{thm empty_Shift}, dtac set_rev_mp,
etac equalityD1, etac CollectD,
rtac conjI, etac @{thm Shift_clists},
rtac conjI, etac @{thm Shift_prefCl},
rtac conjI, rtac ballI,
rtac conjI, dtac @{thm iffD1[OF ball_conj_distrib]}, dtac conjunct1,
SELECT_GOAL (unfold_thms_tac ctxt @{thms Succ_Shift shift_def}),
etac bspec, etac @{thm ShiftD},
CONJ_WRAP' (fn i => EVERY' [rtac ballI, etac CollectE, dtac @{thm ShiftD},
dtac bspec, etac thin_rl, atac, dtac conjunct2, dtac (mk_conjunctN n i),
dtac bspec, rtac CollectI, etac @{thm set_mp[OF equalityD1[OF Succ_Shift]]},
REPEAT_DETERM o eresolve_tac [exE, conjE], rtac exI,
rtac conjI, rtac (@{thm shift_def} RS fun_cong RS trans),
rtac (@{thm append_Cons} RS sym RS arg_cong RS trans), atac,
REPEAT_DETERM_N m o (rtac conjI THEN' atac),
CONJ_WRAP' (K (EVERY' [etac trans, rtac @{thm Collect_cong},
rtac @{thm eqset_imp_iff}, rtac sym, rtac trans, rtac @{thm Succ_Shift},
rtac (@{thm append_Cons} RS sym RS arg_cong)])) ks]) ks,
rtac allI, rtac impI, REPEAT_DETERM o eresolve_tac [allE, impE],
etac @{thm not_in_Shift}, rtac trans, rtac (@{thm shift_def} RS fun_cong), atac,
dtac bspec, atac, dtac conjunct2, dtac (mk_conjunctN n i), dtac bspec,
etac @{thm set_mp[OF equalityD1]}, atac,
REPEAT_DETERM o eresolve_tac [exE, conjE], rtac exI,
rtac conjI, rtac (@{thm shift_def} RS fun_cong RS trans),
etac (@{thm append_Nil} RS sym RS arg_cong RS trans),
REPEAT_DETERM_N m o (rtac conjI THEN' atac),
CONJ_WRAP' (K (EVERY' [etac trans, rtac @{thm Collect_cong},
rtac @{thm eqset_imp_iff}, rtac sym, rtac trans, rtac @{thm Succ_Shift},
rtac (@{thm append_Nil} RS sym RS arg_cong)])) ks]) (ks ~~ active_sets)];
in
unfold_thms_tac ctxt defs THEN
CONJ_WRAP' coalg_tac (ks ~~ (map (chop m) set_naturalss ~~ strT_defs)) 1
end;
fun mk_card_of_carT_tac m isNode_defs sbd_sbd
sbd_card_order sbd_Card_order sbd_Cinfinite sbd_Cnotzero in_sbds =
let
val n = length isNode_defs;
in
EVERY' [rtac (Thm.permute_prems 0 1 ctrans),
rtac @{thm card_of_Sigma_ordLeq_Cinfinite}, rtac @{thm Cinfinite_cexp},
if m = 0 then rtac @{thm ordLeq_refl} else rtac @{thm ordLeq_csum2},
rtac @{thm Card_order_ctwo}, rtac @{thm Cinfinite_cexp},
rtac @{thm ctwo_ordLeq_Cinfinite}, rtac sbd_Cinfinite, rtac sbd_Cinfinite,
rtac ctrans, rtac @{thm card_of_diff},
rtac ordIso_ordLeq_trans, rtac @{thm card_of_Field_ordIso},
rtac @{thm Card_order_cpow}, rtac ordIso_ordLeq_trans,
rtac @{thm cpow_cexp_ctwo}, rtac ctrans, rtac @{thm cexp_mono1_Cnotzero},
if m = 0 then rtac @{thm ordLeq_refl} else rtac @{thm ordLeq_csum2},
rtac @{thm Card_order_ctwo}, rtac @{thm ctwo_Cnotzero}, rtac @{thm Card_order_clists},
rtac @{thm cexp_mono2_Cnotzero}, rtac ordIso_ordLeq_trans,
rtac @{thm clists_Cinfinite},
if n = 1 then rtac sbd_Cinfinite else rtac (sbd_Cinfinite RS @{thm Cinfinite_csum1}),
rtac ordIso_ordLeq_trans, rtac sbd_sbd, rtac @{thm infinite_ordLeq_cexp},
rtac sbd_Cinfinite,
if m = 0 then rtac @{thm ctwo_Cnotzero} else rtac @{thm csum_Cnotzero2[OF ctwo_Cnotzero]},
rtac @{thm Cnotzero_clists},
rtac ballI, rtac ordIso_ordLeq_trans, rtac @{thm card_of_Func_Ffunc},
rtac ordIso_ordLeq_trans, rtac @{thm Func_cexp},
rtac ctrans, rtac @{thm cexp_mono},
rtac @{thm ordLeq_ordIso_trans},
CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1
(sbd_Cinfinite RS @{thm Cinfinite_cexp[OF ordLeq_csum2[OF Card_order_ctwo]]}
RSN (3, @{thm Un_Cinfinite_bound}))))
(fn thm => EVERY' [rtac ctrans, rtac @{thm card_of_image}, rtac thm]) (rev in_sbds),
rtac @{thm cexp_cong1_Cnotzero}, rtac @{thm csum_cong1},
REPEAT_DETERM_N m o rtac @{thm csum_cong2},
CONJ_WRAP_GEN' (rtac @{thm csum_cong})
(K (rtac (sbd_Card_order RS @{thm card_of_Field_ordIso}))) in_sbds,
rtac sbd_Card_order,
rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero},
rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero},
rtac @{thm ordLeq_ordIso_trans}, etac @{thm clists_bound},
rtac @{thm clists_Cinfinite}, TRY o rtac @{thm Cinfinite_csum1}, rtac sbd_Cinfinite,
rtac disjI2, rtac @{thm cone_ordLeq_cexp}, rtac @{thm cone_ordLeq_cexp},
rtac ctrans, rtac @{thm cone_ordLeq_ctwo}, rtac @{thm ordLeq_csum2},
rtac @{thm Card_order_ctwo}, rtac FalseE, etac @{thm cpow_clists_czero}, atac,
rtac @{thm card_of_Card_order},
rtac ordIso_ordLeq_trans, rtac @{thm cexp_cprod},
rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero},
rtac ordIso_ordLeq_trans, rtac @{thm cexp_cong2_Cnotzero},
rtac @{thm ordIso_transitive}, rtac @{thm cprod_cong2}, rtac sbd_sbd,
rtac @{thm cprod_infinite}, rtac sbd_Cinfinite,
rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero}, rtac @{thm Card_order_cprod},
rtac ctrans, rtac @{thm cexp_mono1_Cnotzero},
rtac ordIso_ordLeq_trans, rtac @{thm ordIso_transitive}, rtac @{thm csum_cong1},
rtac @{thm ordIso_transitive},
REPEAT_DETERM_N m o rtac @{thm csum_cong2},
rtac sbd_sbd,
BNF_Tactics.mk_rotate_eq_tac (rtac @{thm ordIso_refl} THEN'
FIRST' [rtac @{thm card_of_Card_order},
rtac @{thm Card_order_csum}, rtac sbd_Card_order])
@{thm ordIso_transitive} @{thm csum_assoc} @{thm csum_com} @{thm csum_cong}
(1 upto m + 1) (m + 1 :: (1 upto m)),
if m = 0 then K all_tac else EVERY' [rtac @{thm ordIso_transitive}, rtac @{thm csum_assoc}],
rtac @{thm csum_com}, rtac @{thm csum_cexp'}, rtac sbd_Cinfinite,
if m = 0 then rtac @{thm Card_order_ctwo} else rtac @{thm Card_order_csum},
if m = 0 then rtac @{thm ordLeq_refl} else rtac @{thm ordLeq_csum2},
rtac @{thm Card_order_ctwo},
rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero}, rtac sbd_Card_order,
rtac ordIso_ordLeq_trans, rtac @{thm cexp_cprod_ordLeq},
if m = 0 then rtac @{thm ctwo_Cnotzero} else rtac @{thm csum_Cnotzero2[OF ctwo_Cnotzero]},
rtac sbd_Cinfinite, rtac sbd_Cnotzero, rtac @{thm ordLeq_refl}, rtac sbd_Card_order,
rtac @{thm cexp_mono2_Cnotzero}, rtac @{thm infinite_ordLeq_cexp},
rtac sbd_Cinfinite,
if m = 0 then rtac @{thm ctwo_Cnotzero} else rtac @{thm csum_Cnotzero2[OF ctwo_Cnotzero]},
rtac sbd_Cnotzero,
rtac @{thm card_of_mono1}, rtac subsetI,
REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE, @{thm prod_caseE}], hyp_subst_tac,
rtac @{thm SigmaI}, rtac @{thm DiffI}, rtac set_mp, rtac equalityD2,
rtac (@{thm cpow_def} RS arg_cong RS trans), rtac (@{thm Pow_def} RS arg_cong RS trans),
rtac @{thm Field_card_of}, rtac CollectI, atac, rtac notI, etac @{thm singletonE},
hyp_subst_tac, etac @{thm emptyE}, rtac (@{thm Ffunc_def} RS equalityD2 RS set_mp),
rtac CollectI, rtac conjI, rtac ballI, dtac bspec, etac thin_rl, atac, dtac conjunct1,
CONJ_WRAP_GEN' (etac disjE) (fn (i, def) => EVERY'
[rtac (mk_UnIN n i), dtac (def RS iffD1),
REPEAT_DETERM o eresolve_tac [exE, conjE], rtac @{thm image_eqI}, atac, rtac CollectI,
REPEAT_DETERM_N m o (rtac conjI THEN' atac),
CONJ_WRAP' (K (EVERY' [etac ord_eq_le_trans, rtac subset_trans,
rtac subset_UNIV, rtac equalityD2, rtac @{thm Field_card_order},
rtac sbd_card_order])) isNode_defs]) (1 upto n ~~ isNode_defs),
atac] 1
end;
fun mk_carT_set_tac n i carT_def strT_def isNode_def set_natural {context = ctxt, prems = _}=
EVERY' [dtac (carT_def RS equalityD1 RS set_mp),
REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE],
dtac Pair_eqD,
etac conjE, hyp_subst_tac,
dtac (isNode_def RS iffD1),
REPEAT_DETERM o eresolve_tac [exE, conjE],
rtac (equalityD2 RS set_mp),
rtac (strT_def RS arg_cong RS trans),
etac (arg_cong RS trans),
fo_rtac (mk_sum_casesN n i RS arg_cong RS trans) ctxt,
rtac set_natural, rtac imageI, etac (equalityD2 RS set_mp), rtac CollectI,
etac @{thm prefCl_Succ}, atac] 1;
fun mk_strT_hset_tac n m j arg_cong_cTs cTs cts carT_defs strT_defs isNode_defs
set_incl_hsets set_hset_incl_hsetss coalg_setss carT_setss coalgT set_naturalss =
let
val set_naturals = map (fn xs => nth xs (j - 1)) set_naturalss;
val ks = 1 upto n;
fun base_tac (i, (cT, (strT_def, (set_incl_hset, set_natural)))) =
CONJ_WRAP' (fn (i', (carT_def, isNode_def)) => rtac impI THEN' etac conjE THEN'
(if i = i'
then EVERY' [rtac @{thm xt1(4)}, rtac set_incl_hset,
rtac (strT_def RS arg_cong RS trans), etac (arg_cong RS trans),
rtac (Thm.permute_prems 0 1 (set_natural RS box_equals)),
rtac (trans OF [@{thm image_id} RS fun_cong, @{thm id_apply}]),
rtac (mk_sum_casesN n i RS (Drule.instantiate' [cT] [] arg_cong) RS sym)]
else EVERY' [dtac (carT_def RS equalityD1 RS set_mp),
REPEAT_DETERM o eresolve_tac [CollectE, exE], etac conjE,
dtac conjunct2, dtac Pair_eqD, etac conjE,
hyp_subst_tac, dtac (isNode_def RS iffD1),
REPEAT_DETERM o eresolve_tac [exE, conjE],
rtac (mk_InN_not_InM i i' RS notE), etac (sym RS trans), atac]))
(ks ~~ (carT_defs ~~ isNode_defs));
fun step_tac (i, (coalg_sets, (carT_sets, set_hset_incl_hsets))) =
dtac (mk_conjunctN n i) THEN'
CONJ_WRAP' (fn (coalg_set, (carT_set, set_hset_incl_hset)) =>
EVERY' [rtac impI, etac conjE, etac impE, rtac conjI,
rtac (coalgT RS coalg_set RS set_mp), atac, etac carT_set, atac, atac,
etac (@{thm shift_def} RS fun_cong RS trans), etac subset_trans,
rtac set_hset_incl_hset, etac carT_set, atac, atac])
(coalg_sets ~~ (carT_sets ~~ set_hset_incl_hsets));
in
EVERY' [rtac (Drule.instantiate' cTs cts @{thm list.induct}),
REPEAT_DETERM o rtac allI, rtac impI,
CONJ_WRAP' base_tac
(ks ~~ (arg_cong_cTs ~~ (strT_defs ~~ (set_incl_hsets ~~ set_naturals)))),
REPEAT_DETERM o rtac allI, rtac impI,
REPEAT_DETERM o eresolve_tac [allE, impE], etac @{thm ShiftI},
CONJ_WRAP' (fn i => dtac (mk_conjunctN n i) THEN' rtac (mk_sumEN n) THEN'
CONJ_WRAP_GEN' (K all_tac) step_tac
(ks ~~ (drop m coalg_setss ~~ (carT_setss ~~ set_hset_incl_hsetss)))) ks] 1
end;
fun mk_isNode_hset_tac n isNode_def strT_hsets {context = ctxt, prems = _} =
let
val m = length strT_hsets;
in
if m = 0 then atac 1
else (unfold_thms_tac ctxt [isNode_def] THEN
EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE],
rtac exI, rtac conjI, atac,
CONJ_WRAP' (fn (thm, i) => if i > m then atac
else EVERY' [rtac (thm RS subset_trans), atac, rtac conjI, atac, atac, atac])
(strT_hsets @ (replicate n mp) ~~ (1 upto (m + n)))] 1)
end;
fun mk_Lev_sbd_tac cts Lev_0s Lev_Sucs to_sbdss =
let
val n = length Lev_0s;
val ks = 1 upto n;
in
EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn Lev_0 =>
EVERY' (map rtac [ord_eq_le_trans, Lev_0, @{thm Nil_clists}])) Lev_0s,
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn (Lev_Suc, to_sbds) =>
EVERY' [rtac ord_eq_le_trans, rtac Lev_Suc,
CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least}))
(fn (i, to_sbd) => EVERY' [rtac subsetI,
REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
rtac @{thm Cons_clists}, rtac (mk_InN_Field n i), etac to_sbd,
etac set_rev_mp, REPEAT_DETERM o etac allE,
etac (mk_conjunctN n i)])
(rev (ks ~~ to_sbds))])
(Lev_Sucs ~~ to_sbdss)] 1
end;
fun mk_length_Lev_tac cts Lev_0s Lev_Sucs =
let
val n = length Lev_0s;
val ks = n downto 1;
in
EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn Lev_0 =>
EVERY' [rtac impI, dtac (Lev_0 RS equalityD1 RS set_mp),
etac @{thm singletonE}, etac ssubst, rtac @{thm list.size(3)}]) Lev_0s,
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn Lev_Suc =>
EVERY' [rtac impI, dtac (Lev_Suc RS equalityD1 RS set_mp),
CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
(fn i => EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
rtac trans, rtac @{thm length_Cons}, rtac @{thm arg_cong[of _ _ Suc]},
REPEAT_DETERM o etac allE, dtac (mk_conjunctN n i), etac mp, atac]) ks])
Lev_Sucs] 1
end;
fun mk_length_Lev'_tac length_Lev =
EVERY' [ftac length_Lev, etac ssubst, atac] 1;
fun mk_prefCl_Lev_tac cts Lev_0s Lev_Sucs =
let
val n = length Lev_0s;
val ks = n downto 1;
in
EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn Lev_0 =>
EVERY' [rtac impI, etac conjE, dtac (Lev_0 RS equalityD1 RS set_mp),
etac @{thm singletonE}, hyp_subst_tac, dtac @{thm prefix_Nil[THEN subst, of "%x. x"]},
hyp_subst_tac, rtac @{thm set_mp[OF equalityD2[OF trans[OF arg_cong[OF list.size(3)]]]]},
rtac Lev_0, rtac @{thm singletonI}]) Lev_0s,
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn (Lev_0, Lev_Suc) =>
EVERY' [rtac impI, etac conjE, dtac (Lev_Suc RS equalityD1 RS set_mp),
CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
(fn i => EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
dtac @{thm prefix_Cons[THEN subst, of "%x. x"]}, etac disjE, hyp_subst_tac,
rtac @{thm set_mp[OF equalityD2[OF trans[OF arg_cong[OF list.size(3)]]]]},
rtac Lev_0, rtac @{thm singletonI},
REPEAT_DETERM o eresolve_tac [exE, conjE], hyp_subst_tac,
rtac @{thm set_mp[OF equalityD2[OF trans[OF arg_cong[OF length_Cons]]]]},
rtac Lev_Suc, rtac (mk_UnIN n i), rtac CollectI, REPEAT_DETERM o rtac exI, rtac conjI,
rtac refl, etac conjI, REPEAT_DETERM o etac allE, dtac (mk_conjunctN n i),
etac mp, etac conjI, atac]) ks])
(Lev_0s ~~ Lev_Sucs)] 1
end;
fun mk_rv_last_tac cTs cts rv_Nils rv_Conss =
let
val n = length rv_Nils;
val ks = 1 upto n;
in
EVERY' [rtac (Drule.instantiate' cTs cts @{thm list.induct}),
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn rv_Cons =>
CONJ_WRAP' (fn (i, rv_Nil) => (EVERY' [rtac exI,
rtac (@{thm append_Nil} RS arg_cong RS trans),
rtac (rv_Cons RS trans), rtac (mk_sum_casesN n i RS arg_cong RS trans), rtac rv_Nil]))
(ks ~~ rv_Nils))
rv_Conss,
REPEAT_DETERM o rtac allI, rtac (mk_sumEN n),
EVERY' (map (fn i =>
CONJ_WRAP' (fn rv_Cons => EVERY' [REPEAT_DETERM o etac allE, dtac (mk_conjunctN n i),
CONJ_WRAP' (fn i' => EVERY' [dtac (mk_conjunctN n i'), etac exE, rtac exI,
rtac (@{thm append_Cons} RS arg_cong RS trans),
rtac (rv_Cons RS trans), etac (sum_case_weak_cong RS arg_cong RS trans),
rtac (mk_sum_casesN n i RS arg_cong RS trans), atac])
ks])
rv_Conss)
ks)] 1
end;
fun mk_set_rv_Lev_tac m cts Lev_0s Lev_Sucs rv_Nils rv_Conss coalg_setss from_to_sbdss =
let
val n = length Lev_0s;
val ks = 1 upto n;
in
EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn (i, ((Lev_0, rv_Nil), coalg_sets)) =>
EVERY' [rtac impI, REPEAT_DETERM o etac conjE,
dtac (Lev_0 RS equalityD1 RS set_mp), etac @{thm singletonE}, etac ssubst,
rtac (rv_Nil RS arg_cong RS iffD2),
rtac (mk_sum_casesN n i RS iffD2),
CONJ_WRAP' (fn thm => etac thm THEN' atac) (take m coalg_sets)])
(ks ~~ ((Lev_0s ~~ rv_Nils) ~~ coalg_setss)),
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn ((Lev_Suc, rv_Cons), (from_to_sbds, coalg_sets)) =>
EVERY' [rtac impI, etac conjE, dtac (Lev_Suc RS equalityD1 RS set_mp),
CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
(fn (i, (from_to_sbd, coalg_set)) =>
EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
rtac (rv_Cons RS arg_cong RS iffD2),
rtac (mk_sum_casesN n i RS arg_cong RS trans RS iffD2),
etac (from_to_sbd RS arg_cong), REPEAT_DETERM o etac allE,
dtac (mk_conjunctN n i), etac mp, etac conjI, etac set_rev_mp,
etac coalg_set, atac])
(rev (ks ~~ (from_to_sbds ~~ drop m coalg_sets)))])
((Lev_Sucs ~~ rv_Conss) ~~ (from_to_sbdss ~~ coalg_setss))] 1
end;
fun mk_set_Lev_tac cts Lev_0s Lev_Sucs rv_Nils rv_Conss from_to_sbdss =
let
val n = length Lev_0s;
val ks = 1 upto n;
in
EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn ((i, (Lev_0, Lev_Suc)), rv_Nil) =>
EVERY' [rtac impI, dtac (Lev_0 RS equalityD1 RS set_mp),
etac @{thm singletonE}, hyp_subst_tac,
CONJ_WRAP' (fn i' => rtac impI THEN' dtac (sym RS trans) THEN' rtac rv_Nil THEN'
(if i = i'
then EVERY' [dtac (mk_InN_inject n i), hyp_subst_tac,
CONJ_WRAP' (fn (i'', Lev_0'') =>
EVERY' [rtac impI, rtac @{thm ssubst_mem[OF append_Nil]},
rtac (Lev_Suc RS equalityD2 RS set_mp), rtac (mk_UnIN n i''),
rtac CollectI, REPEAT_DETERM o rtac exI, rtac conjI, rtac refl,
etac conjI, rtac (Lev_0'' RS equalityD2 RS set_mp),
rtac @{thm singletonI}])
(ks ~~ Lev_0s)]
else etac (mk_InN_not_InM i' i RS notE)))
ks])
((ks ~~ (Lev_0s ~~ Lev_Sucs)) ~~ rv_Nils),
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn ((Lev_Suc, rv_Cons), from_to_sbds) =>
EVERY' [rtac impI, dtac (Lev_Suc RS equalityD1 RS set_mp),
CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
(fn (i, from_to_sbd) =>
EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
CONJ_WRAP' (fn i' => rtac impI THEN'
CONJ_WRAP' (fn i'' =>
EVERY' [rtac impI, rtac (Lev_Suc RS equalityD2 RS set_mp),
rtac @{thm ssubst_mem[OF append_Cons]}, rtac (mk_UnIN n i),
rtac CollectI, REPEAT_DETERM o rtac exI, rtac conjI, rtac refl,
rtac conjI, atac, dtac (sym RS trans RS sym),
rtac (rv_Cons RS trans), rtac (mk_sum_casesN n i RS trans),
etac (from_to_sbd RS arg_cong), REPEAT_DETERM o etac allE,
dtac (mk_conjunctN n i), dtac mp, atac,
dtac (mk_conjunctN n i'), dtac mp, atac,
dtac (mk_conjunctN n i''), etac mp, atac])
ks)
ks])
(rev (ks ~~ from_to_sbds))])
((Lev_Sucs ~~ rv_Conss) ~~ from_to_sbdss)] 1
end;
fun mk_set_image_Lev_tac cts Lev_0s Lev_Sucs rv_Nils rv_Conss from_to_sbdss to_sbd_injss =
let
val n = length Lev_0s;
val ks = 1 upto n;
in
EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn ((i, (Lev_0, Lev_Suc)), rv_Nil) =>
EVERY' [rtac impI, dtac (Lev_0 RS equalityD1 RS set_mp),
etac @{thm singletonE}, hyp_subst_tac,
CONJ_WRAP' (fn i' => rtac impI THEN'
CONJ_WRAP' (fn i'' => rtac impI THEN' dtac (sym RS trans) THEN' rtac rv_Nil THEN'
(if i = i''
then EVERY' [dtac @{thm ssubst_mem[OF sym[OF append_Nil]]},
dtac (Lev_Suc RS equalityD1 RS set_mp), dtac (mk_InN_inject n i),
hyp_subst_tac,
CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
(fn k => REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE] THEN'
dtac list_inject_iffD1 THEN' etac conjE THEN'
(if k = i'
then EVERY' [dtac (mk_InN_inject n k), hyp_subst_tac, etac imageI]
else etac (mk_InN_not_InM i' k RS notE)))
(rev ks)]
else etac (mk_InN_not_InM i'' i RS notE)))
ks)
ks])
((ks ~~ (Lev_0s ~~ Lev_Sucs)) ~~ rv_Nils),
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn ((Lev_Suc, rv_Cons), (from_to_sbds, to_sbd_injs)) =>
EVERY' [rtac impI, dtac (Lev_Suc RS equalityD1 RS set_mp),
CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
(fn (i, (from_to_sbd, to_sbd_inj)) =>
REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE] THEN' hyp_subst_tac THEN'
CONJ_WRAP' (fn i' => rtac impI THEN'
dtac @{thm ssubst_mem[OF sym[OF append_Cons]]} THEN'
dtac (Lev_Suc RS equalityD1 RS set_mp) THEN'
CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE)) (fn k =>
REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE] THEN'
dtac list_inject_iffD1 THEN' etac conjE THEN'
(if k = i
then EVERY' [dtac (mk_InN_inject n i),
dtac (Thm.permute_prems 0 2 (to_sbd_inj RS iffD1)),
atac, atac, hyp_subst_tac] THEN'
CONJ_WRAP' (fn i'' =>
EVERY' [rtac impI, dtac (sym RS trans),
rtac (rv_Cons RS trans), rtac (mk_sum_casesN n i RS arg_cong RS trans),
etac (from_to_sbd RS arg_cong),
REPEAT_DETERM o etac allE,
dtac (mk_conjunctN n i), dtac mp, atac,
dtac (mk_conjunctN n i'), dtac mp, atac,
dtac (mk_conjunctN n i''), etac mp, etac sym])
ks
else etac (mk_InN_not_InM i k RS notE)))
(rev ks))
ks)
(rev (ks ~~ (from_to_sbds ~~ to_sbd_injs)))])
((Lev_Sucs ~~ rv_Conss) ~~ (from_to_sbdss ~~ to_sbd_injss))] 1
end;
fun mk_mor_beh_tac m mor_def mor_cong beh_defs carT_defs strT_defs isNode_defs
to_sbd_injss from_to_sbdss Lev_0s Lev_Sucs rv_Nils rv_Conss Lev_sbds length_Levs length_Lev's
prefCl_Levs rv_lastss set_rv_Levsss set_Levsss set_image_Levsss set_naturalss coalg_setss
map_comp_ids map_congs map_arg_congs {context = ctxt, prems = _} =
let
val n = length beh_defs;
val ks = 1 upto n;
fun fbetw_tac (i, (carT_def, (isNode_def, (Lev_0, (rv_Nil, (Lev_sbd,
((length_Lev, length_Lev'), (prefCl_Lev, (rv_lasts, (set_naturals,
(coalg_sets, (set_rv_Levss, (set_Levss, set_image_Levss))))))))))))) =
EVERY' [rtac ballI, rtac (carT_def RS equalityD2 RS set_mp),
rtac CollectI, REPEAT_DETERM o rtac exI, rtac conjI, rtac refl, rtac conjI,
rtac conjI,
rtac @{thm UN_I}, rtac UNIV_I, rtac (Lev_0 RS equalityD2 RS set_mp),
rtac @{thm singletonI},
rtac conjI,
rtac @{thm UN_least}, rtac Lev_sbd,
rtac conjI,
rtac @{thm prefCl_UN}, rtac ssubst, rtac @{thm PrefCl_def}, REPEAT_DETERM o rtac allI,
rtac impI, etac conjE, rtac exI, rtac conjI, rtac @{thm ord_le_eq_trans},
etac @{thm prefix_length_le}, etac length_Lev, rtac prefCl_Lev, etac conjI, atac,
rtac conjI,
rtac ballI, etac @{thm UN_E}, rtac conjI,
if n = 1 then K all_tac else rtac (mk_sumEN n),
EVERY' (map6 (fn i => fn isNode_def => fn set_naturals =>
fn set_rv_Levs => fn set_Levs => fn set_image_Levs =>
EVERY' [rtac (mk_disjIN n i), rtac (isNode_def RS ssubst),
rtac exI, rtac conjI,
(if n = 1 then rtac @{thm if_P} THEN' etac length_Lev'
else rtac (@{thm if_P} RS arg_cong RS trans) THEN' etac length_Lev' THEN'
etac (sum_case_weak_cong RS trans) THEN' rtac (mk_sum_casesN n i)),
EVERY' (map2 (fn set_natural => fn set_rv_Lev =>
EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac (set_natural RS trans),
rtac trans_fun_cong_image_id_id_apply,
etac set_rv_Lev, TRY o atac, etac conjI, atac])
(take m set_naturals) set_rv_Levs),
CONJ_WRAP' (fn (set_natural, (set_Lev, set_image_Lev)) =>
EVERY' [rtac (set_natural RS trans), rtac equalityI, rtac @{thm image_subsetI},
rtac CollectI, rtac @{thm SuccI}, rtac @{thm UN_I}, rtac UNIV_I, etac set_Lev,
if n = 1 then rtac refl else atac, atac, rtac subsetI,
REPEAT_DETERM o eresolve_tac [CollectE, @{thm SuccE}, @{thm UN_E}],
rtac set_image_Lev, atac, dtac length_Lev, hyp_subst_tac, dtac length_Lev',
etac @{thm set_mp[OF equalityD1[OF arg_cong[OF length_append_singleton]]]},
if n = 1 then rtac refl else atac])
(drop m set_naturals ~~ (set_Levs ~~ set_image_Levs))])
ks isNode_defs set_naturalss set_rv_Levss set_Levss set_image_Levss),
CONJ_WRAP' (fn (i, (rv_last, (isNode_def, (set_naturals,
(set_rv_Levs, (set_Levs, set_image_Levs)))))) =>
EVERY' [rtac ballI,
REPEAT_DETERM o eresolve_tac [CollectE, @{thm SuccE}, @{thm UN_E}],
rtac (rev_mp OF [rv_last, impI]), etac exE, rtac (isNode_def RS ssubst),
rtac exI, rtac conjI,
(if n = 1 then rtac @{thm if_P} THEN' etac length_Lev'
else rtac (@{thm if_P} RS trans) THEN' etac length_Lev' THEN'
etac (sum_case_weak_cong RS trans) THEN' rtac (mk_sum_casesN n i)),
EVERY' (map2 (fn set_natural => fn set_rv_Lev =>
EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac (set_natural RS trans),
rtac trans_fun_cong_image_id_id_apply,
etac set_rv_Lev, TRY o atac, etac conjI, atac])
(take m set_naturals) set_rv_Levs),
CONJ_WRAP' (fn (set_natural, (set_Lev, set_image_Lev)) =>
EVERY' [rtac (set_natural RS trans), rtac equalityI, rtac @{thm image_subsetI},
rtac CollectI, rtac @{thm SuccI}, rtac @{thm UN_I}, rtac UNIV_I, etac set_Lev,
if n = 1 then rtac refl else atac, atac, rtac subsetI,
REPEAT_DETERM o eresolve_tac [CollectE, @{thm SuccE}, @{thm UN_E}],
REPEAT_DETERM_N 4 o etac thin_rl,
rtac set_image_Lev,
atac, dtac length_Lev, hyp_subst_tac, dtac length_Lev',
etac @{thm set_mp[OF equalityD1[OF arg_cong[OF length_append_singleton]]]},
if n = 1 then rtac refl else atac])
(drop m set_naturals ~~ (set_Levs ~~ set_image_Levs))])
(ks ~~ (rv_lasts ~~ (isNode_defs ~~ (set_naturalss ~~
(set_rv_Levss ~~ (set_Levss ~~ set_image_Levss)))))),
(**)
rtac allI, rtac impI, rtac @{thm if_not_P}, rtac notI,
etac notE, etac @{thm UN_I[OF UNIV_I]},
(*root isNode*)
rtac (isNode_def RS ssubst), rtac exI, rtac conjI, rtac (@{thm if_P} RS trans),
rtac length_Lev', rtac (Lev_0 RS equalityD2 RS set_mp), rtac @{thm singletonI},
CONVERSION (Conv.top_conv
(K (Conv.try_conv (Conv.rewr_conv (rv_Nil RS eq_reflection)))) ctxt),
if n = 1 then rtac refl else rtac (mk_sum_casesN n i),
EVERY' (map2 (fn set_natural => fn coalg_set =>
EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac (set_natural RS trans),
rtac trans_fun_cong_image_id_id_apply, etac coalg_set, atac])
(take m set_naturals) (take m coalg_sets)),
CONJ_WRAP' (fn (set_natural, (set_Lev, set_image_Lev)) =>
EVERY' [rtac (set_natural RS trans), rtac equalityI, rtac @{thm image_subsetI},
rtac CollectI, rtac @{thm SuccI}, rtac @{thm UN_I}, rtac UNIV_I, rtac set_Lev,
rtac (Lev_0 RS equalityD2 RS set_mp), rtac @{thm singletonI}, rtac rv_Nil,
atac, rtac subsetI,
REPEAT_DETERM o eresolve_tac [CollectE, @{thm SuccE}, @{thm UN_E}],
rtac set_image_Lev, rtac (Lev_0 RS equalityD2 RS set_mp),
rtac @{thm singletonI}, dtac length_Lev',
etac @{thm set_mp[OF equalityD1[OF arg_cong[OF
trans[OF length_append_singleton arg_cong[of _ _ Suc, OF list.size(3)]]]]]},
rtac rv_Nil])
(drop m set_naturals ~~ (nth set_Levss (i - 1) ~~ nth set_image_Levss (i - 1)))];
fun mor_tac (i, (strT_def, (((Lev_0, Lev_Suc), (rv_Nil, rv_Cons)),
((map_comp_id, (map_cong, map_arg_cong)), (length_Lev', (from_to_sbds, to_sbd_injs)))))) =
EVERY' [rtac ballI, rtac sym, rtac trans, rtac strT_def,
rtac (@{thm if_P} RS
(if n = 1 then map_arg_cong else sum_case_weak_cong) RS trans),
rtac (@{thm list.size(3)} RS arg_cong RS trans RS equalityD2 RS set_mp),
rtac Lev_0, rtac @{thm singletonI},
CONVERSION (Conv.top_conv
(K (Conv.try_conv (Conv.rewr_conv (rv_Nil RS eq_reflection)))) ctxt),
if n = 1 then K all_tac
else (rtac (sum_case_weak_cong RS trans) THEN'
rtac (mk_sum_casesN n i) THEN' rtac (mk_sum_casesN n i RS trans)),
rtac (map_comp_id RS trans), rtac (map_cong OF replicate m refl),
EVERY' (map3 (fn i' => fn to_sbd_inj => fn from_to_sbd =>
DETERM o EVERY' [rtac trans, rtac o_apply, rtac Pair_eqI, rtac conjI,
rtac trans, rtac @{thm Shift_def},
rtac equalityI, rtac subsetI, etac thin_rl, etac thin_rl,
REPEAT_DETERM o eresolve_tac [CollectE, @{thm UN_E}], dtac length_Lev', dtac asm_rl,
etac thin_rl, dtac @{thm set_rev_mp[OF _ equalityD1]},
rtac (@{thm length_Cons} RS arg_cong RS trans), rtac Lev_Suc,
CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE)) (fn i'' =>
EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE],
dtac list_inject_iffD1, etac conjE,
if i' = i'' then EVERY' [dtac (mk_InN_inject n i'),
dtac (Thm.permute_prems 0 2 (to_sbd_inj RS iffD1)),
atac, atac, hyp_subst_tac, etac @{thm UN_I[OF UNIV_I]}]
else etac (mk_InN_not_InM i' i'' RS notE)])
(rev ks),
rtac @{thm UN_least}, rtac subsetI, rtac CollectI, rtac @{thm UN_I[OF UNIV_I]},
rtac (Lev_Suc RS equalityD2 RS set_mp), rtac (mk_UnIN n i'), rtac CollectI,
REPEAT_DETERM o rtac exI, rtac conjI, rtac refl, etac conjI, atac,
rtac trans, rtac @{thm shift_def}, rtac ssubst, rtac @{thm fun_eq_iff}, rtac allI,
rtac @{thm if_cong}, rtac (@{thm length_Cons} RS arg_cong RS trans), rtac iffI,
dtac asm_rl, dtac asm_rl, dtac asm_rl,
dtac (Lev_Suc RS equalityD1 RS set_mp),
CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE)) (fn i'' =>
EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE],
dtac list_inject_iffD1, etac conjE,
if i' = i'' then EVERY' [dtac (mk_InN_inject n i'),
dtac (Thm.permute_prems 0 2 (to_sbd_inj RS iffD1)),
atac, atac, hyp_subst_tac, atac]
else etac (mk_InN_not_InM i' i'' RS notE)])
(rev ks),
rtac (Lev_Suc RS equalityD2 RS set_mp), rtac (mk_UnIN n i'), rtac CollectI,
REPEAT_DETERM o rtac exI, rtac conjI, rtac refl, etac conjI, atac,
CONVERSION (Conv.top_conv
(K (Conv.try_conv (Conv.rewr_conv (rv_Cons RS eq_reflection)))) ctxt),
if n = 1 then K all_tac
else rtac sum_case_weak_cong THEN' rtac (mk_sum_casesN n i' RS trans),
SELECT_GOAL (unfold_thms_tac ctxt [from_to_sbd]), rtac refl,
rtac refl])
ks to_sbd_injs from_to_sbds)];
in
(rtac mor_cong THEN'
EVERY' (map (fn thm => rtac (thm RS ext)) beh_defs) THEN'
stac mor_def THEN' rtac conjI THEN'
CONJ_WRAP' fbetw_tac
(ks ~~ (carT_defs ~~ (isNode_defs ~~ (Lev_0s ~~ (rv_Nils ~~ (Lev_sbds ~~
((length_Levs ~~ length_Lev's) ~~ (prefCl_Levs ~~ (rv_lastss ~~
(set_naturalss ~~ (coalg_setss ~~
(set_rv_Levsss ~~ (set_Levsss ~~ set_image_Levsss))))))))))))) THEN'
CONJ_WRAP' mor_tac
(ks ~~ (strT_defs ~~ (((Lev_0s ~~ Lev_Sucs) ~~ (rv_Nils ~~ rv_Conss)) ~~
((map_comp_ids ~~ (map_congs ~~ map_arg_congs)) ~~
(length_Lev's ~~ (from_to_sbdss ~~ to_sbd_injss))))))) 1
end;
fun mk_congruent_str_final_tac m lsbisE map_comp_id map_cong equiv_LSBISs =
EVERY' [rtac @{thm congruentI}, dtac lsbisE,
REPEAT_DETERM o eresolve_tac [CollectE, conjE, bexE], rtac (o_apply RS trans),
etac (sym RS arg_cong RS trans), rtac (map_comp_id RS trans),
rtac (map_cong RS trans), REPEAT_DETERM_N m o rtac refl,
EVERY' (map (fn equiv_LSBIS =>
EVERY' [rtac @{thm equiv_proj}, rtac equiv_LSBIS, etac set_mp, atac])
equiv_LSBISs), rtac sym, rtac (o_apply RS trans),
etac (sym RS arg_cong RS trans), rtac map_comp_id] 1;
fun mk_coalg_final_tac m coalg_def congruent_str_finals equiv_LSBISs set_naturalss coalgT_setss =
EVERY' [stac coalg_def,
CONJ_WRAP' (fn ((set_naturals, coalgT_sets), (equiv_LSBIS, congruent_str_final)) =>
EVERY' [rtac @{thm univ_preserves}, rtac equiv_LSBIS, rtac congruent_str_final,
rtac ballI, rtac @{thm ssubst_mem}, rtac o_apply, rtac CollectI,
EVERY' (map2 (fn set_natural => fn coalgT_set =>
EVERY' [rtac conjI, rtac (set_natural RS ord_eq_le_trans),
rtac ord_eq_le_trans_trans_fun_cong_image_id_id_apply,
etac coalgT_set])
(take m set_naturals) (take m coalgT_sets)),
CONJ_WRAP' (fn (equiv_LSBIS, (set_natural, coalgT_set)) =>
EVERY' [rtac (set_natural RS ord_eq_le_trans),
rtac @{thm image_subsetI}, rtac ssubst, rtac @{thm proj_in_iff},
rtac equiv_LSBIS, etac set_rev_mp, etac coalgT_set])
(equiv_LSBISs ~~ drop m (set_naturals ~~ coalgT_sets))])
((set_naturalss ~~ coalgT_setss) ~~ (equiv_LSBISs ~~ congruent_str_finals))] 1;
fun mk_mor_T_final_tac mor_def congruent_str_finals equiv_LSBISs =
EVERY' [stac mor_def, rtac conjI,
CONJ_WRAP' (fn equiv_LSBIS =>
EVERY' [rtac ballI, rtac ssubst, rtac @{thm proj_in_iff}, rtac equiv_LSBIS, atac])
equiv_LSBISs,
CONJ_WRAP' (fn (equiv_LSBIS, congruent_str_final) =>
EVERY' [rtac ballI, rtac sym, rtac trans, rtac @{thm univ_commute}, rtac equiv_LSBIS,
rtac congruent_str_final, atac, rtac o_apply])
(equiv_LSBISs ~~ congruent_str_finals)] 1;
fun mk_mor_Rep_tac m defs Reps Abs_inverses coalg_final_setss map_comp_ids map_congLs
{context = ctxt, prems = _} =
unfold_thms_tac ctxt defs THEN
EVERY' [rtac conjI,
CONJ_WRAP' (fn thm => rtac ballI THEN' rtac thm) Reps,
CONJ_WRAP' (fn (Rep, ((map_comp_id, map_congL), coalg_final_sets)) =>
EVERY' [rtac ballI, rtac (map_comp_id RS trans), rtac map_congL,
EVERY' (map2 (fn Abs_inverse => fn coalg_final_set =>
EVERY' [rtac ballI, rtac (o_apply RS trans), rtac Abs_inverse,
etac set_rev_mp, rtac coalg_final_set, rtac Rep])
Abs_inverses (drop m coalg_final_sets))])
(Reps ~~ ((map_comp_ids ~~ map_congLs) ~~ coalg_final_setss))] 1;
fun mk_mor_Abs_tac defs Abs_inverses {context = ctxt, prems = _} =
unfold_thms_tac ctxt defs THEN
EVERY' [rtac conjI,
CONJ_WRAP' (K (rtac ballI THEN' rtac UNIV_I)) Abs_inverses,
CONJ_WRAP' (fn thm => rtac ballI THEN' etac (thm RS arg_cong RS sym)) Abs_inverses] 1;
fun mk_mor_unfold_tac m mor_UNIV dtor_defs unfold_defs Abs_inverses morEs map_comp_ids map_congs =
EVERY' [rtac iffD2, rtac mor_UNIV,
CONJ_WRAP' (fn ((Abs_inverse, morE), ((dtor_def, unfold_def), (map_comp_id, map_cong))) =>
EVERY' [rtac ext, rtac (o_apply RS trans RS sym), rtac (dtor_def RS trans),
rtac (unfold_def RS arg_cong RS trans), rtac (Abs_inverse RS arg_cong RS trans),
rtac (morE RS arg_cong RS trans), rtac (map_comp_id RS trans),
rtac (o_apply RS trans RS sym), rtac map_cong,
REPEAT_DETERM_N m o rtac refl,
EVERY' (map (fn thm => rtac (thm RS trans) THEN' rtac (o_apply RS sym)) unfold_defs)])
((Abs_inverses ~~ morEs) ~~ ((dtor_defs ~~ unfold_defs) ~~ (map_comp_ids ~~ map_congs)))] 1;
fun mk_raw_coind_tac bis_def bis_cong bis_O bis_converse bis_Gr tcoalg coalgT mor_T_final
sbis_lsbis lsbis_incls incl_lsbiss equiv_LSBISs mor_Rep Rep_injects =
let
val n = length Rep_injects;
in
EVERY' [rtac rev_mp, ftac (bis_def RS iffD1),
REPEAT_DETERM o etac conjE, rtac bis_cong, rtac bis_O, rtac bis_converse,
rtac bis_Gr, rtac tcoalg, rtac mor_Rep, rtac bis_O, atac, rtac bis_Gr, rtac tcoalg,
rtac mor_Rep, REPEAT_DETERM_N n o etac @{thm relImage_Gr},
rtac impI, rtac rev_mp, rtac bis_cong, rtac bis_O, rtac bis_Gr, rtac coalgT,
rtac mor_T_final, rtac bis_O, rtac sbis_lsbis, rtac bis_converse, rtac bis_Gr, rtac coalgT,
rtac mor_T_final, EVERY' (map (fn thm => rtac (thm RS @{thm relInvImage_Gr})) lsbis_incls),
rtac impI,
CONJ_WRAP' (fn (Rep_inject, (equiv_LSBIS , (incl_lsbis, lsbis_incl))) =>
EVERY' [rtac subset_trans, rtac @{thm relInvImage_UNIV_relImage}, rtac subset_trans,
rtac @{thm relInvImage_mono}, rtac subset_trans, etac incl_lsbis,
rtac ord_eq_le_trans, rtac @{thm sym[OF relImage_relInvImage]},
rtac @{thm xt1(3)}, rtac @{thm Sigma_cong},
rtac @{thm proj_image}, rtac @{thm proj_image}, rtac lsbis_incl,
rtac subset_trans, rtac @{thm relImage_mono}, rtac incl_lsbis, atac,
rtac @{thm relImage_proj}, rtac equiv_LSBIS, rtac @{thm relInvImage_diag},
rtac Rep_inject])
(Rep_injects ~~ (equiv_LSBISs ~~ (incl_lsbiss ~~ lsbis_incls)))] 1
end;
fun mk_unique_mor_tac raw_coinds bis =
CONJ_WRAP' (fn raw_coind =>
EVERY' [rtac impI, rtac (bis RS raw_coind RS set_mp RS @{thm IdD}), atac,
etac conjunct1, atac, etac conjunct2, rtac @{thm image2_eqI}, rtac refl, rtac refl, atac])
raw_coinds 1;
fun mk_unfold_unique_mor_tac raw_coinds bis mor unfold_defs =
CONJ_WRAP' (fn (raw_coind, unfold_def) =>
EVERY' [rtac ext, etac (bis RS raw_coind RS set_mp RS @{thm IdD}), rtac mor,
rtac @{thm image2_eqI}, rtac refl, rtac (unfold_def RS arg_cong RS trans),
rtac (o_apply RS sym), rtac UNIV_I]) (raw_coinds ~~ unfold_defs) 1;
fun mk_dtor_o_ctor_tac ctor_def unfold map_comp_id map_congL unfold_o_dtors
{context = ctxt, prems = _} =
unfold_thms_tac ctxt [ctor_def] THEN EVERY' [rtac ext, rtac trans, rtac o_apply,
rtac trans, rtac unfold, rtac trans, rtac map_comp_id, rtac trans, rtac map_congL,
EVERY' (map (fn thm =>
rtac ballI THEN' rtac (trans OF [thm RS fun_cong, @{thm id_apply}])) unfold_o_dtors),
rtac sym, rtac @{thm id_apply}] 1;
fun mk_corec_tac m corec_defs unfold map_cong corec_Inls {context = ctxt, prems = _} =
unfold_thms_tac ctxt corec_defs THEN EVERY' [rtac trans, rtac (o_apply RS arg_cong),
rtac trans, rtac unfold, fo_rtac (@{thm sum.cases(2)} RS arg_cong RS trans) ctxt, rtac map_cong,
REPEAT_DETERM_N m o rtac refl,
EVERY' (map (fn thm => rtac @{thm sum_case_expand_Inr} THEN' rtac thm) corec_Inls)] 1;
fun mk_srel_coinduct_tac ks raw_coind bis_srel =
EVERY' [rtac rev_mp, rtac raw_coind, rtac ssubst, rtac bis_srel, rtac conjI,
CONJ_WRAP' (K (rtac @{thm ord_le_eq_trans[OF subset_UNIV UNIV_Times_UNIV[THEN sym]]})) ks,
CONJ_WRAP' (K (EVERY' [rtac allI, rtac allI, rtac impI,
REPEAT_DETERM o etac allE, etac mp, etac CollectE, etac @{thm splitD}])) ks,
rtac impI, REPEAT_DETERM o etac conjE,
CONJ_WRAP' (K (EVERY' [rtac impI, rtac @{thm IdD}, etac set_mp,
rtac CollectI, etac @{thm prod_caseI}])) ks] 1;
fun mk_srel_strong_coinduct_tac m cTs cts srel_coinduct srel_monos srel_Ids =
EVERY' [rtac rev_mp, rtac (Drule.instantiate' cTs cts srel_coinduct),
EVERY' (map2 (fn srel_mono => fn srel_Id =>
EVERY' [REPEAT_DETERM o resolve_tac [allI, impI], REPEAT_DETERM o etac allE,
etac disjE, etac mp, atac, hyp_subst_tac, rtac (srel_mono RS set_mp),
REPEAT_DETERM_N m o rtac @{thm subset_refl},
REPEAT_DETERM_N (length srel_monos) o rtac @{thm Id_subset},
rtac (srel_Id RS equalityD2 RS set_mp), rtac @{thm IdI}])
srel_monos srel_Ids),
rtac impI, REPEAT_DETERM o etac conjE,
CONJ_WRAP' (K (rtac impI THEN' etac mp THEN' etac disjI1)) srel_Ids] 1;
fun mk_dtor_coinduct_tac m ks raw_coind bis_def =
let
val n = length ks;
in
EVERY' [rtac rev_mp, rtac raw_coind, rtac ssubst, rtac bis_def, rtac conjI,
CONJ_WRAP' (K (rtac @{thm ord_le_eq_trans[OF subset_UNIV UNIV_Times_UNIV[THEN sym]]})) ks,
CONJ_WRAP' (fn i => EVERY' [select_prem_tac n (dtac asm_rl) i, REPEAT_DETERM o rtac allI,
rtac impI, REPEAT_DETERM o dtac @{thm meta_spec}, etac CollectE, etac @{thm meta_impE},
atac, etac exE, etac conjE, etac conjE, rtac bexI, rtac conjI,
etac @{thm fst_conv[THEN subst]}, etac @{thm snd_conv[THEN subst]},
rtac CollectI, REPEAT_DETERM_N m o (rtac conjI THEN' rtac subset_UNIV),
CONJ_WRAP' (fn i' => EVERY' [rtac subsetI, rtac CollectI, dtac (mk_conjunctN n i'),
REPEAT_DETERM o etac allE, etac mp, rtac @{thm ssubst_mem[OF pair_collapse]}, atac])
ks])
ks,
rtac impI,
CONJ_WRAP' (fn i => EVERY' [rtac impI, dtac (mk_conjunctN n i),
rtac @{thm subst[OF pair_in_Id_conv]}, etac set_mp,
rtac CollectI, etac (refl RSN (2, @{thm subst_Pair}))]) ks] 1
end;
fun mk_dtor_strong_coinduct_tac ks cTs cts dtor_coinduct bis_def bis_diag =
EVERY' [rtac rev_mp, rtac (Drule.instantiate' cTs cts dtor_coinduct),
EVERY' (map (fn i =>
EVERY' [etac disjE, REPEAT_DETERM o dtac @{thm meta_spec}, etac @{thm meta_mp},
atac, rtac rev_mp, rtac subst, rtac bis_def, rtac bis_diag,
rtac impI, dtac conjunct2, dtac (mk_conjunctN (length ks) i), REPEAT_DETERM o etac allE,
etac impE, etac @{thm diag_UNIV_I}, REPEAT_DETERM o eresolve_tac [bexE, conjE, CollectE],
rtac exI, rtac conjI, etac conjI, atac,
CONJ_WRAP' (K (EVERY' [REPEAT_DETERM o resolve_tac [allI, impI],
rtac disjI2, rtac @{thm diagE}, etac set_mp, atac])) ks])
ks),
rtac impI, REPEAT_DETERM o etac conjE,
CONJ_WRAP' (K (rtac impI THEN' etac mp THEN' etac disjI1)) ks] 1;
fun mk_map_tac m n cT unfold map_comp' map_cong =
EVERY' [rtac ext, rtac (o_apply RS trans RS sym), rtac (o_apply RS trans RS sym),
rtac (unfold RS trans), rtac (Thm.permute_prems 0 1 (map_comp' RS box_equals)), rtac map_cong,
REPEAT_DETERM_N m o rtac (@{thm id_o} RS fun_cong),
REPEAT_DETERM_N n o rtac (@{thm o_id} RS fun_cong),
rtac (o_apply RS (Drule.instantiate' [cT] [] arg_cong) RS sym)] 1;
fun mk_set_le_tac n hset_minimal set_hsets set_hset_hsetss =
EVERY' [rtac hset_minimal,
REPEAT_DETERM_N n o rtac @{thm Un_upper1},
REPEAT_DETERM_N n o
EVERY' (map3 (fn i => fn set_hset => fn set_hset_hsets =>
EVERY' [rtac subsetI, rtac @{thm UnI2}, rtac (mk_UnIN n i), etac @{thm UN_I},
etac UnE, etac set_hset, REPEAT_DETERM_N (n - 1) o etac UnE,
EVERY' (map (fn thm => EVERY' [etac @{thm UN_E}, etac thm, atac]) set_hset_hsets)])
(1 upto n) set_hsets set_hset_hsetss)] 1;
fun mk_set_simp_tac n set_le set_incl_hset set_hset_incl_hsets =
EVERY' [rtac equalityI, rtac set_le, rtac @{thm Un_least}, rtac set_incl_hset,
REPEAT_DETERM_N (n - 1) o rtac @{thm Un_least},
EVERY' (map (fn thm => rtac @{thm UN_least} THEN' etac thm) set_hset_incl_hsets)] 1;
fun mk_map_id_tac maps unfold_unique unfold_dtor =
EVERY' [rtac (unfold_unique RS trans), EVERY' (map (fn thm => rtac (thm RS sym)) maps),
rtac unfold_dtor] 1;
fun mk_map_comp_tac m n maps map_comps map_congs unfold_unique =
EVERY' [rtac unfold_unique,
EVERY' (map3 (fn map_thm => fn map_comp => fn map_cong =>
EVERY' (map rtac
([@{thm o_assoc} RS trans,
@{thm arg_cong2[of _ _ _ _ "op o"]} OF [map_comp RS sym, refl] RS trans,
@{thm o_assoc} RS trans RS sym,
@{thm arg_cong2[of _ _ _ _ "op o"]} OF [map_thm, refl] RS trans,
@{thm o_assoc} RS sym RS trans, map_thm RS arg_cong RS trans, @{thm o_assoc} RS trans,
@{thm arg_cong2[of _ _ _ _ "op o"]} OF [map_comp RS sym, refl] RS trans,
ext, o_apply RS trans, o_apply RS trans RS sym, map_cong] @
replicate m (@{thm id_o} RS fun_cong) @
replicate n (@{thm o_id} RS fun_cong))))
maps map_comps map_congs)] 1;
fun mk_mcong_tac m coinduct_tac map_comp's dtor_maps map_congs set_naturalss set_hsetss
set_hset_hsetsss =
let
val n = length map_comp's;
val ks = 1 upto n;
in
EVERY' ([rtac rev_mp,
coinduct_tac] @
maps (fn (((((map_comp'_trans, dtor_maps_trans), map_cong), set_naturals), set_hsets),
set_hset_hsetss) =>
[REPEAT_DETERM o eresolve_tac [exE, conjE], hyp_subst_tac, rtac exI, rtac conjI, rtac conjI,
rtac map_comp'_trans, rtac sym, rtac dtor_maps_trans, rtac map_cong,
REPEAT_DETERM_N m o (rtac o_apply_trans_sym THEN' rtac @{thm id_apply}),
REPEAT_DETERM_N n o rtac fst_convol_fun_cong_sym,
rtac map_comp'_trans, rtac sym, rtac dtor_maps_trans, rtac map_cong,
EVERY' (maps (fn set_hset =>
[rtac o_apply_trans_sym, rtac (@{thm id_apply} RS trans), etac CollectE,
REPEAT_DETERM o etac conjE, etac bspec, etac set_hset]) set_hsets),
REPEAT_DETERM_N n o rtac snd_convol_fun_cong_sym,
CONJ_WRAP' (fn (set_natural, set_hset_hsets) =>
EVERY' [REPEAT_DETERM o rtac allI, rtac impI, rtac @{thm image_convolD},
etac set_rev_mp, rtac ord_eq_le_trans, rtac set_natural,
rtac @{thm image_mono}, rtac subsetI, rtac CollectI, etac CollectE,
REPEAT_DETERM o etac conjE,
CONJ_WRAP' (fn set_hset_hset =>
EVERY' [rtac ballI, etac bspec, etac set_hset_hset, atac]) set_hset_hsets])
(drop m set_naturals ~~ set_hset_hsetss)])
(map (fn th => th RS trans) map_comp's ~~ map (fn th => th RS trans) dtor_maps ~~
map_congs ~~ set_naturalss ~~ set_hsetss ~~ set_hset_hsetsss) @
[rtac impI,
CONJ_WRAP' (fn k =>
EVERY' [rtac impI, dtac (mk_conjunctN n k), etac mp, rtac exI, rtac conjI, etac CollectI,
rtac conjI, rtac refl, rtac refl]) ks]) 1
end
fun mk_dtor_map_unique_tac unfold_unique map_comps {context = ctxt, prems = _} =
rtac unfold_unique 1 THEN
unfold_thms_tac ctxt (map (fn thm => thm RS sym) map_comps @ @{thms o_assoc id_o o_id}) THEN
ALLGOALS (etac sym);
fun mk_col_natural_tac cts rec_0s rec_Sucs dtor_maps set_naturalss
{context = ctxt, prems = _} =
let
val n = length dtor_maps;
in
EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
REPEAT_DETERM o rtac allI, SELECT_GOAL (unfold_thms_tac ctxt rec_0s),
CONJ_WRAP' (K (rtac @{thm image_empty})) rec_0s,
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn (rec_Suc, (dtor_map, set_nats)) => EVERY'
[SELECT_GOAL (unfold_thms_tac ctxt
(rec_Suc :: dtor_map :: set_nats @ @{thms image_Un image_UN UN_simps(10)})),
rtac @{thm Un_cong}, rtac refl,
CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_cong}))
(fn i => EVERY' [rtac @{thm UN_cong[OF refl]},
REPEAT_DETERM o etac allE, etac (mk_conjunctN n i)]) (n downto 1)])
(rec_Sucs ~~ (dtor_maps ~~ set_naturalss))] 1
end;
fun mk_set_natural_tac hset_def col_natural =
EVERY' (map rtac [ext, (o_apply RS trans), (hset_def RS trans), sym,
(o_apply RS trans), (@{thm image_cong} OF [hset_def, refl] RS trans),
(@{thm image_UN} RS trans), (refl RS @{thm UN_cong}), col_natural]) 1;
fun mk_col_bd_tac m j cts rec_0s rec_Sucs sbd_Card_order sbd_Cinfinite set_sbdss =
let
val n = length rec_0s;
in
EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn rec_0 => EVERY' (map rtac [ordIso_ordLeq_trans,
@{thm card_of_ordIso_subst}, rec_0, @{thm Card_order_empty}, sbd_Card_order])) rec_0s,
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn (rec_Suc, set_sbds) => EVERY'
[rtac ordIso_ordLeq_trans, rtac @{thm card_of_ordIso_subst}, rtac rec_Suc,
rtac (sbd_Cinfinite RSN (3, @{thm Un_Cinfinite_bound})), rtac (nth set_sbds (j - 1)),
REPEAT_DETERM_N (n - 1) o rtac (sbd_Cinfinite RSN (3, @{thm Un_Cinfinite_bound})),
EVERY' (map2 (fn i => fn set_sbd => EVERY' [rtac @{thm UNION_Cinfinite_bound},
rtac set_sbd, rtac ballI, REPEAT_DETERM o etac allE,
etac (mk_conjunctN n i), rtac sbd_Cinfinite]) (1 upto n) (drop m set_sbds))])
(rec_Sucs ~~ set_sbdss)] 1
end;
fun mk_set_bd_tac sbd_Cinfinite hset_def col_bd =
EVERY' (map rtac [ordIso_ordLeq_trans, @{thm card_of_ordIso_subst}, hset_def,
ctrans, @{thm UNION_Cinfinite_bound}, ordIso_ordLeq_trans, @{thm card_of_nat},
@{thm natLeq_ordLeq_cinfinite}, sbd_Cinfinite, ballI, col_bd, sbd_Cinfinite,
ctrans, @{thm infinite_ordLeq_cexp}, sbd_Cinfinite, @{thm cexp_ordLeq_ccexp}]) 1;
fun mk_in_bd_tac isNode_hset isNode_hsets carT_def card_of_carT mor_image Rep_inverse mor_hsets
sbd_Cnotzero sbd_Card_order mor_Rep coalgT mor_T_final tcoalg =
let
val n = length isNode_hsets;
val in_hin_tac = rtac CollectI THEN'
CONJ_WRAP' (fn mor_hset => EVERY' (map etac
[mor_hset OF [coalgT, mor_T_final] RS sym RS ord_eq_le_trans,
arg_cong RS sym RS ord_eq_le_trans,
mor_hset OF [tcoalg, mor_Rep, UNIV_I] RS ord_eq_le_trans])) mor_hsets;
in
EVERY' [rtac (Thm.permute_prems 0 1 @{thm ordLeq_transitive}), rtac ctrans,
rtac @{thm card_of_image}, rtac ordIso_ordLeq_trans,
rtac @{thm card_of_ordIso_subst}, rtac @{thm sym[OF proj_image]}, rtac ctrans,
rtac @{thm card_of_image}, rtac ctrans, rtac card_of_carT, rtac @{thm cexp_mono2_Cnotzero},
rtac @{thm cexp_ordLeq_ccexp}, rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero},
rtac @{thm Cnotzero_cexp}, rtac sbd_Cnotzero, rtac sbd_Card_order,
rtac @{thm card_of_mono1}, rtac subsetI, rtac @{thm image_eqI}, rtac sym,
rtac Rep_inverse, REPEAT_DETERM o eresolve_tac [CollectE, conjE],
rtac set_mp, rtac equalityD2, rtac @{thm sym[OF proj_image]}, rtac imageE,
rtac set_rev_mp, rtac mor_image, rtac mor_Rep, rtac UNIV_I, rtac equalityD2,
rtac @{thm proj_image}, rtac @{thm image_eqI}, atac,
ftac (carT_def RS equalityD1 RS set_mp),
REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
rtac (carT_def RS equalityD2 RS set_mp), rtac CollectI, REPEAT_DETERM o rtac exI,
rtac conjI, rtac refl, rtac conjI, etac conjI, etac conjI, etac conjI, rtac conjI,
rtac ballI, dtac bspec, atac, REPEAT_DETERM o etac conjE, rtac conjI,
CONJ_WRAP_GEN' (etac disjE) (fn (i, isNode_hset) =>
EVERY' [rtac (mk_disjIN n i), rtac isNode_hset, atac, atac, atac, in_hin_tac])
(1 upto n ~~ isNode_hsets),
CONJ_WRAP' (fn isNode_hset =>
EVERY' [rtac ballI, rtac isNode_hset, atac, ftac CollectD, etac @{thm SuccD},
etac bspec, atac, in_hin_tac])
isNode_hsets,
atac, rtac isNode_hset, atac, atac, atac, in_hin_tac] 1
end;
fun mk_bd_card_order_tac sbd_card_order =
EVERY' (map rtac [@{thm card_order_ccexp}, sbd_card_order, sbd_card_order]) 1;
fun mk_bd_cinfinite_tac sbd_Cinfinite =
EVERY' (map rtac [@{thm cinfinite_ccexp}, @{thm ctwo_ordLeq_Cinfinite},
sbd_Cinfinite, sbd_Cinfinite]) 1;
fun mk_pickWP_assms_tac set_incl_hsets set_incl_hins map_eq =
let
val m = length set_incl_hsets;
in
EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac CollectI,
EVERY' (map (fn thm => rtac conjI THEN' etac (thm RS @{thm subset_trans})) set_incl_hsets),
CONJ_WRAP' (fn thm => rtac thm THEN' REPEAT_DETERM_N m o atac) set_incl_hins,
REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac CollectI,
EVERY' (map (fn thm => rtac conjI THEN' etac (thm RS @{thm subset_trans})) set_incl_hsets),
CONJ_WRAP' (fn thm => rtac thm THEN' REPEAT_DETERM_N m o atac) set_incl_hins,
REPEAT_DETERM o eresolve_tac [CollectE, conjE], etac map_eq]
end;
fun mk_coalg_thePull_tac m coalg_def map_wpulls set_naturalss pickWP_assms_tacs
{context = ctxt, prems = _} =
unfold_thms_tac ctxt [coalg_def] THEN
CONJ_WRAP' (fn (map_wpull, (pickWP_assms_tac, set_naturals)) =>
EVERY' [rtac ballI, dtac @{thm set_mp[OF equalityD1[OF thePull_def]]},
REPEAT_DETERM o eresolve_tac [CollectE, @{thm prod_caseE}],
hyp_subst_tac, rtac rev_mp, rtac (map_wpull RS @{thm pickWP(1)}),
EVERY' (map (etac o mk_conjunctN m) (1 upto m)),
pickWP_assms_tac,
SELECT_GOAL (unfold_thms_tac ctxt @{thms o_apply prod.cases}), rtac impI,
REPEAT_DETERM o eresolve_tac [CollectE, conjE],
rtac CollectI,
REPEAT_DETERM_N m o (rtac conjI THEN' rtac subset_UNIV),
CONJ_WRAP' (fn set_natural =>
EVERY' [rtac ord_eq_le_trans, rtac trans, rtac set_natural,
rtac trans_fun_cong_image_id_id_apply, atac])
(drop m set_naturals)])
(map_wpulls ~~ (pickWP_assms_tacs ~~ set_naturalss)) 1;
fun mk_mor_thePull_nth_tac conv pick m mor_def map_wpulls map_comps pickWP_assms_tacs
{context = ctxt, prems = _} =
let
val n = length map_comps;
in
unfold_thms_tac ctxt [mor_def] THEN
EVERY' [rtac conjI,
CONJ_WRAP' (K (rtac ballI THEN' rtac UNIV_I)) (1 upto n),
CONJ_WRAP' (fn (map_wpull, (pickWP_assms_tac, map_comp)) =>
EVERY' [rtac ballI, dtac @{thm set_mp[OF equalityD1[OF thePull_def]]},
REPEAT_DETERM o eresolve_tac [CollectE, @{thm prod_caseE}, conjE],
hyp_subst_tac,
SELECT_GOAL (unfold_thms_tac ctxt @{thms o_apply prod.cases}),
rtac (map_comp RS trans),
SELECT_GOAL (unfold_thms_tac ctxt (conv :: @{thms o_id id_o})),
rtac (map_wpull RS pick), REPEAT_DETERM_N m o atac,
pickWP_assms_tac])
(map_wpulls ~~ (pickWP_assms_tacs ~~ map_comps))] 1
end;
val mk_mor_thePull_fst_tac = mk_mor_thePull_nth_tac @{thm fst_conv} @{thm pickWP(2)};
val mk_mor_thePull_snd_tac = mk_mor_thePull_nth_tac @{thm snd_conv} @{thm pickWP(3)};
fun mk_mor_thePull_pick_tac mor_def unfolds map_comps {context = ctxt, prems = _} =
unfold_thms_tac ctxt [mor_def, @{thm thePull_def}] THEN rtac conjI 1 THEN
CONJ_WRAP' (K (rtac ballI THEN' rtac UNIV_I)) unfolds 1 THEN
CONJ_WRAP' (fn (unfold, map_comp) =>
EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, @{thm prod_caseE}, conjE],
hyp_subst_tac,
SELECT_GOAL (unfold_thms_tac ctxt (unfold :: map_comp :: @{thms comp_def id_def})),
rtac refl])
(unfolds ~~ map_comps) 1;
fun mk_pick_col_tac m j cts rec_0s rec_Sucs unfolds set_naturalss map_wpulls pickWP_assms_tacs
{context = ctxt, prems = _} =
let
val n = length rec_0s;
val ks = n downto 1;
in
EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn thm => EVERY'
[rtac impI, rtac ord_eq_le_trans, rtac thm, rtac @{thm empty_subsetI}]) rec_0s,
REPEAT_DETERM o rtac allI,
CONJ_WRAP' (fn (rec_Suc, ((unfold, set_naturals), (map_wpull, pickWP_assms_tac))) =>
EVERY' [rtac impI, dtac @{thm set_mp[OF equalityD1[OF thePull_def]]},
REPEAT_DETERM o eresolve_tac [CollectE, @{thm prod_caseE}],
hyp_subst_tac, rtac rev_mp, rtac (map_wpull RS @{thm pickWP(1)}),
EVERY' (map (etac o mk_conjunctN m) (1 upto m)),
pickWP_assms_tac,
rtac impI, REPEAT_DETERM o eresolve_tac [CollectE, conjE],
rtac ord_eq_le_trans, rtac rec_Suc,
rtac @{thm Un_least},
SELECT_GOAL (unfold_thms_tac ctxt [unfold, nth set_naturals (j - 1),
@{thm prod.cases}]),
etac ord_eq_le_trans_trans_fun_cong_image_id_id_apply,
CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least})) (fn (i, set_natural) =>
EVERY' [rtac @{thm UN_least},
SELECT_GOAL (unfold_thms_tac ctxt [unfold, set_natural, @{thm prod.cases}]),
etac imageE, hyp_subst_tac, REPEAT_DETERM o etac allE,
dtac (mk_conjunctN n i), etac mp, etac set_mp, atac])
(ks ~~ rev (drop m set_naturals))])
(rec_Sucs ~~ ((unfolds ~~ set_naturalss) ~~ (map_wpulls ~~ pickWP_assms_tacs)))] 1
end;
fun mk_wpull_tac m coalg_thePull mor_thePull_fst mor_thePull_snd mor_thePull_pick
mor_unique pick_cols hset_defs =
EVERY' [rtac (@{thm wpull_def} RS iffD2), REPEAT_DETERM o rtac allI, rtac impI,
REPEAT_DETERM o etac conjE, rtac bexI, rtac conjI,
rtac box_equals, rtac mor_unique,
rtac coalg_thePull, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
rtac mor_thePull_pick, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
rtac mor_thePull_fst, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
rtac @{thm set_mp[OF equalityD2[OF thePull_def]]}, rtac CollectI,
rtac @{thm prod_caseI}, etac conjI, etac conjI, atac, rtac o_apply, rtac @{thm fst_conv},
rtac box_equals, rtac mor_unique,
rtac coalg_thePull, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
rtac mor_thePull_pick, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
rtac mor_thePull_snd, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
rtac @{thm set_mp[OF equalityD2[OF thePull_def]]}, rtac CollectI,
rtac @{thm prod_caseI}, etac conjI, etac conjI, atac, rtac o_apply, rtac @{thm snd_conv},
rtac CollectI,
CONJ_WRAP' (fn (pick, def) =>
EVERY' [rtac (def RS ord_eq_le_trans), rtac @{thm UN_least},
rtac pick, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
rtac @{thm set_mp[OF equalityD2[OF thePull_def]]}, rtac CollectI,
rtac @{thm prod_caseI}, etac conjI, etac conjI, atac])
(pick_cols ~~ hset_defs)] 1;
fun mk_wit_tac n dtor_ctors set_simp wit coind_wits {context = ctxt, prems = _} =
ALLGOALS (TRY o (eresolve_tac coind_wits THEN' rtac refl)) THEN
REPEAT_DETERM (atac 1 ORELSE
EVERY' [dtac set_rev_mp, rtac equalityD1, resolve_tac set_simp,
K (unfold_thms_tac ctxt dtor_ctors),
REPEAT_DETERM_N n o etac UnE,
REPEAT_DETERM o
(TRY o REPEAT_DETERM o etac UnE THEN' TRY o etac @{thm UN_E} THEN'
(eresolve_tac wit ORELSE'
(dresolve_tac wit THEN'
(etac FalseE ORELSE'
EVERY' [hyp_subst_tac, dtac set_rev_mp, rtac equalityD1, resolve_tac set_simp,
K (unfold_thms_tac ctxt dtor_ctors), REPEAT_DETERM_N n o etac UnE]))))] 1);
fun mk_coind_wit_tac induct unfolds set_nats wits {context = ctxt, prems = _} =
rtac induct 1 THEN ALLGOALS (TRY o rtac impI THEN' TRY o hyp_subst_tac) THEN
unfold_thms_tac ctxt (unfolds @ set_nats @ @{thms image_id id_apply}) THEN
ALLGOALS (REPEAT_DETERM o etac imageE THEN' TRY o hyp_subst_tac) THEN
ALLGOALS (TRY o
FIRST' [rtac TrueI, rtac refl, etac (refl RSN (2, mp)), dresolve_tac wits THEN' etac FalseE])
fun mk_dtor_srel_tac in_Jsrels i in_srel map_comp map_cong dtor_map dtor_sets dtor_inject dtor_ctor
set_naturals set_incls set_set_inclss =
let
val m = length set_incls;
val n = length set_set_inclss;
val (passive_set_naturals, active_set_naturals) = chop m set_naturals;
val in_Jsrel = nth in_Jsrels (i - 1);
val if_tac =
EVERY' [dtac (in_Jsrel RS iffD1), REPEAT_DETERM o eresolve_tac [exE, conjE, CollectE],
rtac (in_srel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
EVERY' (map2 (fn set_natural => fn set_incl =>
EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac set_natural,
rtac ord_eq_le_trans, rtac trans_fun_cong_image_id_id_apply,
etac (set_incl RS @{thm subset_trans})])
passive_set_naturals set_incls),
CONJ_WRAP' (fn (in_Jsrel, (set_natural, set_set_incls)) =>
EVERY' [rtac ord_eq_le_trans, rtac set_natural, rtac @{thm image_subsetI},
rtac (in_Jsrel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
CONJ_WRAP' (fn thm => etac (thm RS @{thm subset_trans}) THEN' atac) set_set_incls,
rtac conjI, rtac refl, rtac refl])
(in_Jsrels ~~ (active_set_naturals ~~ set_set_inclss)),
CONJ_WRAP' (fn conv =>
EVERY' [rtac trans, rtac map_comp, rtac trans, rtac map_cong,
REPEAT_DETERM_N m o rtac @{thm fun_cong[OF o_id]},
REPEAT_DETERM_N n o EVERY' (map rtac [trans, o_apply, conv]),
rtac trans, rtac sym, rtac dtor_map, rtac (dtor_inject RS iffD2), atac])
@{thms fst_conv snd_conv}];
val only_if_tac =
EVERY' [dtac (in_srel RS iffD1), REPEAT_DETERM o eresolve_tac [exE, conjE, CollectE],
rtac (in_Jsrel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
CONJ_WRAP' (fn (set_simp, passive_set_natural) =>
EVERY' [rtac ord_eq_le_trans, rtac set_simp, rtac @{thm Un_least},
rtac ord_eq_le_trans, rtac box_equals, rtac passive_set_natural,
rtac (dtor_ctor RS sym RS arg_cong), rtac trans_fun_cong_image_id_id_apply, atac,
CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least}))
(fn (active_set_natural, in_Jsrel) => EVERY' [rtac ord_eq_le_trans,
rtac @{thm UN_cong[OF _ refl]}, rtac @{thm box_equals[OF _ _ refl]},
rtac active_set_natural, rtac (dtor_ctor RS sym RS arg_cong), rtac @{thm UN_least},
dtac set_rev_mp, etac @{thm image_mono}, etac imageE,
dtac @{thm ssubst_mem[OF pair_collapse]}, dtac (in_Jsrel RS iffD1),
dtac @{thm someI_ex}, REPEAT_DETERM o etac conjE,
dtac (Thm.permute_prems 0 1 @{thm ssubst_mem}), atac,
hyp_subst_tac, REPEAT_DETERM o eresolve_tac [CollectE, conjE], atac])
(rev (active_set_naturals ~~ in_Jsrels))])
(dtor_sets ~~ passive_set_naturals),
rtac conjI,
REPEAT_DETERM_N 2 o EVERY'[rtac (dtor_inject RS iffD1), rtac trans, rtac dtor_map,
rtac box_equals, rtac map_comp, rtac (dtor_ctor RS sym RS arg_cong), rtac trans,
rtac map_cong, REPEAT_DETERM_N m o rtac @{thm fun_cong[OF o_id]},
EVERY' (map (fn in_Jsrel => EVERY' [rtac trans, rtac o_apply, dtac set_rev_mp, atac,
dtac @{thm ssubst_mem[OF pair_collapse]}, dtac (in_Jsrel RS iffD1),
dtac @{thm someI_ex}, REPEAT_DETERM o etac conjE, atac]) in_Jsrels),
atac]]
in
EVERY' [rtac iffI, if_tac, only_if_tac] 1
end;
end;