(* Title: HOL/BNF/Tools/bnf_def_tactics.ML
Author: Dmitriy Traytel, TU Muenchen
Author: Jasmin Blanchette, TU Muenchen
Copyright 2012
Tactics for definition of bounded natural functors.
*)
signature BNF_DEF_TACTICS =
sig
val mk_collect_set_map_tac: thm list -> tactic
val mk_map_id': thm -> thm
val mk_map_comp': thm -> thm
val mk_map_cong_tac: Proof.context -> thm -> tactic
val mk_in_mono_tac: int -> tactic
val mk_map_wppull_tac: thm -> thm -> thm -> thm -> thm list -> tactic
val mk_set_map': thm -> thm
val mk_rel_Grp_tac: thm list -> thm -> thm -> thm -> thm -> thm list ->
{prems: thm list, context: Proof.context} -> tactic
val mk_rel_eq_tac: int -> thm -> thm -> thm -> tactic
val mk_rel_OO_tac: thm list -> thm -> thm -> thm -> thm -> thm list ->
{prems: thm list, context: Proof.context} -> tactic
val mk_in_rel_tac: thm -> {prems: 'a, context: Proof.context} -> tactic
val mk_rel_conversep_tac: thm -> thm -> tactic
val mk_rel_conversep_le_tac: thm list -> thm -> thm -> thm -> thm list ->
{prems: thm list, context: Proof.context} -> tactic
val mk_rel_mono_tac: thm list -> thm -> {prems: 'a, context: Proof.context} -> tactic
end;
structure BNF_Def_Tactics : BNF_DEF_TACTICS =
struct
open BNF_Util
open BNF_Tactics
fun mk_map_id' id = mk_trans (fun_cong OF [id]) @{thm id_apply};
fun mk_map_comp' comp = @{thm o_eq_dest_lhs} OF [mk_sym comp];
fun mk_map_cong_tac ctxt cong0 =
(hyp_subst_tac ctxt THEN' rtac cong0 THEN'
REPEAT_DETERM o (dtac meta_spec THEN' etac meta_mp THEN' atac)) 1;
fun mk_set_map' set_map = set_map RS @{thm pointfreeE};
fun mk_in_mono_tac n = if n = 0 then rtac subset_UNIV 1
else (rtac subsetI THEN'
rtac CollectI) 1 THEN
REPEAT_DETERM (eresolve_tac [CollectE, conjE] 1) THEN
REPEAT_DETERM_N (n - 1)
((rtac conjI THEN' etac subset_trans THEN' atac) 1) THEN
(etac subset_trans THEN' atac) 1;
fun mk_collect_set_map_tac set_maps =
(rtac (@{thm collect_o} RS trans) THEN' rtac @{thm arg_cong[of _ _ collect]} THEN'
EVERY' (map (fn set_map =>
rtac (mk_trans @{thm image_insert} @{thm arg_cong2[of _ _ _ _ insert]}) THEN'
rtac set_map) set_maps) THEN'
rtac @{thm image_empty}) 1;
fun mk_map_wppull_tac map_id map_cong0 map_wpull map_comp set_maps =
if null set_maps then
EVERY' [rtac @{thm wppull_id}, rtac map_wpull, rtac map_id, rtac map_id] 1
else EVERY' [REPEAT_DETERM o etac conjE, REPEAT_DETERM o dtac @{thm wppull_thePull},
REPEAT_DETERM o etac exE, rtac @{thm wpull_wppull}, rtac map_wpull,
REPEAT_DETERM o rtac @{thm wpull_thePull}, rtac ballI,
REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac conjI, rtac CollectI,
CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS @{thm ord_eq_le_trans}),
rtac @{thm image_subsetI}, rtac conjunct1, etac bspec, etac set_mp, atac])
set_maps,
CONJ_WRAP' (fn thm => EVERY' [rtac (map_comp RS trans), rtac (map_comp RS trans),
rtac (map_comp RS trans RS sym), rtac map_cong0,
REPEAT_DETERM_N (length set_maps) o EVERY' [rtac (o_apply RS trans),
rtac (o_apply RS trans RS sym), rtac (o_apply RS trans), rtac thm,
rtac conjunct2, etac bspec, etac set_mp, atac]]) [conjunct1, conjunct2]] 1;
fun mk_rel_Grp_tac rel_OO_Grps map_id map_cong0 map_id' map_comp set_maps
{context = ctxt, prems = _} =
let
val n = length set_maps;
in
if null set_maps then
unfold_thms_tac ctxt ((map_id RS @{thm Grp_UNIV_id}) :: rel_OO_Grps) THEN
rtac @{thm Grp_UNIV_idI[OF refl]} 1
else unfold_thms_tac ctxt rel_OO_Grps THEN
EVERY' [rtac @{thm antisym}, rtac @{thm predicate2I},
REPEAT_DETERM o
eresolve_tac [CollectE, exE, conjE, @{thm GrpE}, @{thm relcomppE}, @{thm conversepE}],
hyp_subst_tac ctxt, rtac @{thm GrpI}, rtac trans, rtac map_comp, rtac map_cong0,
REPEAT_DETERM_N n o EVERY' [rtac @{thm Collect_split_Grp_eqD}, etac @{thm set_mp}, atac],
rtac CollectI,
CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS @{thm ord_eq_le_trans}),
rtac @{thm image_subsetI}, rtac @{thm Collect_split_Grp_inD}, etac @{thm set_mp}, atac])
set_maps,
rtac @{thm predicate2I}, REPEAT_DETERM o eresolve_tac [@{thm GrpE}, exE, conjE],
hyp_subst_tac ctxt,
rtac @{thm relcomppI}, rtac @{thm conversepI},
EVERY' (map2 (fn convol => fn map_id =>
EVERY' [rtac @{thm GrpI}, rtac (box_equals OF [map_cong0, map_comp RS sym, map_id]),
REPEAT_DETERM_N n o rtac (convol RS fun_cong),
REPEAT_DETERM o eresolve_tac [CollectE, conjE],
rtac CollectI,
CONJ_WRAP' (fn thm =>
EVERY' [rtac @{thm ord_eq_le_trans}, rtac thm, rtac @{thm image_subsetI},
rtac @{thm convol_mem_GrpI[OF refl]}, etac set_mp, atac])
set_maps])
@{thms fst_convol snd_convol} [map_id', refl])] 1
end;
fun mk_rel_eq_tac n rel_Grp rel_cong map_id =
(EVERY' (rtac (rel_cong RS trans) :: replicate n (rtac @{thm eq_alt})) THEN'
rtac (rel_Grp RSN (2, @{thm box_equals[OF _ sym sym[OF eq_alt]]})) THEN'
(if n = 0 then rtac refl
else EVERY' [rtac @{thm arg_cong2[of _ _ _ _ "Grp"]},
rtac @{thm equalityI}, rtac subset_UNIV, rtac subsetI, rtac CollectI,
CONJ_WRAP' (K (rtac subset_UNIV)) (1 upto n), rtac map_id])) 1;
fun mk_rel_mono_tac rel_OO_Grps in_mono {context = ctxt, prems = _} =
unfold_thms_tac ctxt rel_OO_Grps THEN
EVERY' [rtac @{thm relcompp_mono}, rtac @{thm iffD2[OF conversep_mono]},
rtac @{thm Grp_mono}, rtac in_mono, REPEAT_DETERM o etac @{thm Collect_split_mono},
rtac @{thm Grp_mono}, rtac in_mono, REPEAT_DETERM o etac @{thm Collect_split_mono}] 1;
fun mk_rel_conversep_le_tac rel_OO_Grps rel_eq map_cong0 map_comp set_maps
{context = ctxt, prems = _} =
let
val n = length set_maps;
in
if null set_maps then rtac (rel_eq RS @{thm leq_conversepI}) 1
else unfold_thms_tac ctxt (rel_OO_Grps) THEN
EVERY' [rtac @{thm predicate2I},
REPEAT_DETERM o
eresolve_tac [CollectE, exE, conjE, @{thm GrpE}, @{thm relcomppE}, @{thm conversepE}],
hyp_subst_tac ctxt, rtac @{thm conversepI}, rtac @{thm relcomppI}, rtac @{thm conversepI},
EVERY' (map (fn thm => EVERY' [rtac @{thm GrpI}, rtac sym, rtac trans,
rtac map_cong0, REPEAT_DETERM_N n o rtac thm,
rtac (map_comp RS sym), rtac CollectI,
CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS @{thm ord_eq_le_trans}),
etac @{thm flip_pred}]) set_maps]) [@{thm snd_fst_flip}, @{thm fst_snd_flip}])] 1
end;
fun mk_rel_conversep_tac le_conversep rel_mono =
EVERY' [rtac @{thm antisym}, rtac le_conversep, rtac @{thm xt1(6)}, rtac @{thm conversep_shift},
rtac le_conversep, rtac @{thm iffD2[OF conversep_mono]}, rtac rel_mono,
REPEAT_DETERM o rtac @{thm eq_refl[OF sym[OF conversep_conversep]]}] 1;
fun mk_rel_OO_tac rel_OO_Grs rel_eq map_cong0 map_wppull map_comp set_maps
{context = ctxt, prems = _} =
let
val n = length set_maps;
fun in_tac nthO_in = rtac CollectI THEN'
CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS @{thm ord_eq_le_trans}),
rtac @{thm image_subsetI}, rtac nthO_in, etac set_mp, atac]) set_maps;
in
if null set_maps then rtac (rel_eq RS @{thm eq_OOI}) 1
else unfold_thms_tac ctxt rel_OO_Grs THEN
EVERY' [rtac @{thm antisym}, rtac @{thm predicate2I},
REPEAT_DETERM o
eresolve_tac [CollectE, exE, conjE, @{thm GrpE}, @{thm relcomppE}, @{thm conversepE}],
hyp_subst_tac ctxt,
rtac @{thm relcomppI}, rtac @{thm relcomppI}, rtac @{thm conversepI}, rtac @{thm GrpI},
rtac trans, rtac map_comp, rtac sym, rtac map_cong0,
REPEAT_DETERM_N n o rtac @{thm fst_fstOp},
in_tac @{thm fstOp_in},
rtac @{thm GrpI}, rtac trans, rtac map_comp, rtac map_cong0,
REPEAT_DETERM_N n o EVERY' [rtac trans, rtac o_apply,
rtac ballE, rtac subst,
rtac @{thm csquare_def}, rtac @{thm csquare_fstOp_sndOp}, atac, etac notE,
etac set_mp, atac],
in_tac @{thm fstOp_in},
rtac @{thm relcomppI}, rtac @{thm conversepI}, rtac @{thm GrpI},
rtac trans, rtac map_comp, rtac map_cong0,
REPEAT_DETERM_N n o rtac o_apply,
in_tac @{thm sndOp_in},
rtac @{thm GrpI}, rtac trans, rtac map_comp, rtac sym, rtac map_cong0,
REPEAT_DETERM_N n o rtac @{thm snd_sndOp},
in_tac @{thm sndOp_in},
rtac @{thm predicate2I},
REPEAT_DETERM o eresolve_tac [@{thm relcomppE}, @{thm conversepE}, @{thm GrpE}],
hyp_subst_tac ctxt,
rtac allE, rtac subst, rtac @{thm wppull_def}, rtac map_wppull,
CONJ_WRAP' (K (rtac @{thm wppull_fstOp_sndOp})) set_maps,
etac allE, etac impE, etac conjI, etac conjI, etac sym,
REPEAT_DETERM o eresolve_tac [bexE, conjE],
rtac @{thm relcomppI}, rtac @{thm conversepI},
EVERY' (map (fn thm => EVERY' [rtac @{thm GrpI}, rtac trans,
rtac trans, rtac map_cong0, REPEAT_DETERM_N n o rtac thm,
rtac (map_comp RS sym), atac, atac]) [@{thm fst_fstOp}, @{thm snd_sndOp}])] 1
end;
fun mk_in_rel_tac rel_OO_Gr {context = ctxt, prems = _} =
EVERY' [rtac (rel_OO_Gr RS fun_cong RS fun_cong RS trans), rtac iffI,
REPEAT_DETERM o eresolve_tac [@{thm GrpE}, @{thm relcomppE}, @{thm conversepE}],
hyp_subst_tac ctxt, rtac exI, rtac conjI, atac, rtac conjI, rtac refl, rtac refl,
REPEAT_DETERM o eresolve_tac [exE, conjE], rtac @{thm relcomppI}, rtac @{thm conversepI},
etac @{thm GrpI}, atac, etac @{thm GrpI}, atac] 1;
end;