(* Title: HOL/Codatatype/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_natural_tac: Proof.context -> thm list -> tactic
val mk_id': thm -> thm
val mk_comp': thm -> thm
val mk_in_mono_tac: int -> tactic
val mk_map_wppull_tac: thm -> thm -> thm -> thm -> thm list -> tactic
val mk_set_natural': thm -> thm
val mk_rel_Gr_tac: thm list -> thm -> thm -> thm -> thm -> thm -> thm -> thm list ->
{prems: thm list, context: Proof.context} -> tactic
val mk_rel_Id_tac: int -> thm -> thm -> {prems: 'a, context: Proof.context} -> tactic
val mk_rel_O_tac: thm list -> thm -> thm -> thm -> thm -> thm list ->
{prems: thm list, context: Proof.context} -> tactic
val mk_in_rel_tac: thm list -> int -> {prems: 'b, context: Proof.context} -> tactic
val mk_rel_converse_tac: thm -> tactic
val mk_rel_converse_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
val set_mp = @{thm set_mp};
fun mk_id' id = mk_trans (fun_cong OF [id]) @{thm id_apply};
fun mk_comp' comp = @{thm o_eq_dest_lhs} OF [mk_sym comp];
fun mk_set_natural' set_natural = set_natural RS @{thm pointfreeE};
fun mk_in_mono_tac n = if n = 0 then rtac @{thm 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_natural_tac ctxt set_naturals =
substs_tac ctxt (@{thms collect_o image_insert image_empty} @ set_naturals) 1 THEN rtac refl 1;
fun mk_map_wppull_tac map_id map_cong map_wpull map_comp set_naturals =
if null set_naturals 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_naturals,
CONJ_WRAP' (fn thm => EVERY' [rtac (map_comp RS trans), rtac (map_comp RS trans),
rtac (map_comp RS trans RS sym), rtac map_cong,
REPEAT_DETERM_N (length set_naturals) 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_Gr_tac rel_O_Grs map_id map_cong map_wpull in_cong map_id' map_comp set_naturals
{context = ctxt, prems = _} =
let
val n = length set_naturals;
in
if null set_naturals then
unfold_defs_tac ctxt rel_O_Grs THEN EVERY' [rtac @{thm Gr_UNIV_id}, rtac map_id] 1
else unfold_defs_tac ctxt (@{thm Gr_def} :: rel_O_Grs) THEN
EVERY' [rtac equalityI, rtac subsetI,
REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE, @{thm relcompE}, @{thm converseE}],
REPEAT_DETERM o dtac @{thm Pair_eq[THEN subst, of "%x. x"]},
REPEAT_DETERM o etac conjE, hyp_subst_tac,
rtac CollectI, rtac exI, rtac conjI, stac @{thm Pair_eq}, rtac conjI, rtac refl,
rtac sym, rtac trans, rtac map_comp, rtac map_cong,
REPEAT_DETERM_N n o EVERY' [dtac @{thm set_rev_mp}, atac,
REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
rtac (o_apply RS trans), rtac (@{thm fst_conv} RS arg_cong RS trans),
rtac (@{thm snd_conv} RS sym)],
rtac CollectI,
CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS @{thm ord_eq_le_trans}),
rtac @{thm image_subsetI}, dtac @{thm set_rev_mp}, atac,
REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
stac @{thm fst_conv}, atac]) set_naturals,
rtac @{thm subrelI}, etac CollectE, REPEAT_DETERM o eresolve_tac [exE, conjE],
REPEAT_DETERM o dtac @{thm Pair_eq[THEN subst, of "%x. x"]},
REPEAT_DETERM o etac conjE, hyp_subst_tac,
rtac allE, rtac subst, rtac @{thm wpull_def}, rtac map_wpull,
REPEAT_DETERM_N n o rtac @{thm wpull_Gr}, etac allE, etac impE, rtac conjI, atac,
rtac conjI, REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac CollectI,
CONJ_WRAP' (fn thm => EVERY' [rtac (thm RS @{thm ord_eq_le_trans}),
rtac @{thm image_mono}, atac]) set_naturals,
rtac sym, rtac map_id', REPEAT_DETERM o eresolve_tac [bexE, conjE],
rtac @{thm relcompI}, rtac @{thm converseI},
REPEAT_DETERM_N 2 o EVERY' [rtac CollectI, rtac exI,
rtac conjI, stac @{thm Pair_eq}, rtac conjI, rtac refl, etac sym,
etac @{thm set_rev_mp}, rtac equalityD1, rtac in_cong,
REPEAT_DETERM_N n o rtac @{thm Gr_def}]] 1
end;
fun mk_rel_Id_tac n rel_Gr map_id {context = ctxt, prems = _} =
unfold_defs_tac ctxt [rel_Gr, @{thm Id_alt}] THEN
subst_tac ctxt [map_id] 1 THEN
(if n = 0 then rtac refl 1
else EVERY' [rtac @{thm arg_cong2[of _ _ _ _ Gr]},
rtac equalityI, rtac @{thm subset_UNIV}, rtac subsetI, rtac CollectI,
CONJ_WRAP' (K (rtac @{thm subset_UNIV})) (1 upto n), rtac refl] 1);
fun mk_rel_mono_tac rel_O_Grs in_mono {context = ctxt, prems = _} =
unfold_defs_tac ctxt rel_O_Grs THEN
EVERY' [rtac @{thm relcomp_mono}, rtac @{thm iffD2[OF converse_mono]},
rtac @{thm Gr_mono}, rtac in_mono, REPEAT_DETERM o atac,
rtac @{thm Gr_mono}, rtac in_mono, REPEAT_DETERM o atac] 1;
fun mk_rel_converse_le_tac rel_O_Grs rel_Id map_cong map_comp set_naturals
{context = ctxt, prems = _} =
let
val n = length set_naturals;
in
if null set_naturals then
unfold_defs_tac ctxt [rel_Id] THEN rtac equalityD2 1 THEN rtac @{thm converse_Id} 1
else unfold_defs_tac ctxt (@{thm Gr_def} :: rel_O_Grs) THEN
EVERY' [rtac @{thm subrelI},
REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE, @{thm relcompE}, @{thm converseE}],
REPEAT_DETERM o dtac @{thm Pair_eq[THEN subst, of "%x. x"]},
REPEAT_DETERM o etac conjE, hyp_subst_tac, rtac @{thm converseI},
rtac @{thm relcompI}, rtac @{thm converseI},
EVERY' (map (fn thm => EVERY' [rtac CollectI, rtac exI,
rtac conjI, stac @{thm Pair_eq}, rtac conjI, rtac refl, rtac trans,
rtac map_cong, 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_rel}]) set_naturals]) [@{thm snd_fst_flip}, @{thm fst_snd_flip}])] 1
end;
fun mk_rel_converse_tac le_converse =
EVERY' [rtac equalityI, rtac le_converse, rtac @{thm xt1(6)}, rtac @{thm converse_shift},
rtac le_converse, REPEAT_DETERM o stac @{thm converse_converse}, rtac subset_refl] 1;
fun mk_rel_O_tac rel_O_Grs rel_Id map_cong map_wppull map_comp set_naturals
{context = ctxt, prems = _} =
let
val n = length set_naturals;
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_naturals;
in
if null set_naturals then unfold_defs_tac ctxt [rel_Id] THEN rtac (@{thm Id_O_R} RS sym) 1
else unfold_defs_tac ctxt (@{thm Gr_def} :: rel_O_Grs) THEN
EVERY' [rtac equalityI, rtac @{thm subrelI},
REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE, @{thm relcompE}, @{thm converseE}],
REPEAT_DETERM o dtac @{thm Pair_eq[THEN subst, of "%x. x"]},
REPEAT_DETERM o etac conjE, hyp_subst_tac,
rtac @{thm relcompI}, rtac @{thm relcompI}, rtac @{thm converseI},
rtac CollectI, rtac exI, rtac conjI, stac @{thm Pair_eq}, rtac conjI, rtac refl,
rtac sym, rtac trans, rtac map_comp, rtac sym, rtac map_cong,
REPEAT_DETERM_N n o rtac @{thm fst_fstO},
in_tac @{thm fstO_in},
rtac CollectI, rtac exI, rtac conjI, stac @{thm Pair_eq}, rtac conjI, rtac refl,
rtac sym, rtac trans, rtac map_comp, rtac map_cong,
REPEAT_DETERM_N n o EVERY' [rtac trans, rtac o_apply, rtac ballE, rtac subst,
rtac @{thm csquare_def}, rtac @{thm csquare_fstO_sndO}, atac, etac notE,
etac set_mp, atac],
in_tac @{thm fstO_in},
rtac @{thm relcompI}, rtac @{thm converseI},
rtac CollectI, rtac exI, rtac conjI, stac @{thm Pair_eq}, rtac conjI, rtac refl,
rtac sym, rtac trans, rtac map_comp, rtac map_cong,
REPEAT_DETERM_N n o rtac o_apply,
in_tac @{thm sndO_in},
rtac CollectI, rtac exI, rtac conjI, stac @{thm Pair_eq}, rtac conjI, rtac refl,
rtac sym, rtac trans, rtac map_comp, rtac sym, rtac map_cong,
REPEAT_DETERM_N n o rtac @{thm snd_sndO},
in_tac @{thm sndO_in},
rtac @{thm subrelI},
REPEAT_DETERM o eresolve_tac [CollectE, @{thm relcompE}, @{thm converseE}],
REPEAT_DETERM o eresolve_tac [exE, conjE],
REPEAT_DETERM o dtac @{thm Pair_eq[THEN subst, of "%x. x"]},
REPEAT_DETERM o etac conjE, hyp_subst_tac,
rtac allE, rtac subst, rtac @{thm wppull_def}, rtac map_wppull,
CONJ_WRAP' (K (rtac @{thm wppull_fstO_sndO})) set_naturals,
etac allE, etac impE, etac conjI, etac conjI, atac,
REPEAT_DETERM o eresolve_tac [bexE, conjE],
rtac @{thm relcompI}, rtac @{thm converseI},
EVERY' (map (fn thm => EVERY' [rtac CollectI, rtac exI,
rtac conjI, stac @{thm Pair_eq}, rtac conjI, rtac refl, rtac sym, rtac trans,
rtac trans, rtac map_cong, REPEAT_DETERM_N n o rtac thm,
rtac (map_comp RS sym), atac, atac]) [@{thm fst_fstO}, @{thm snd_sndO}])] 1
end;
fun mk_in_rel_tac rel_O_Grs m {context = ctxt, prems = _} =
let
val ls' = replicate (Int.max (1, m)) ();
in
unfold_defs_tac ctxt (rel_O_Grs @
@{thms Gr_def converse_unfold relcomp_unfold mem_Collect_eq prod.cases Pair_eq}) THEN
EVERY' [rtac iffI, REPEAT_DETERM o eresolve_tac [exE, conjE], hyp_subst_tac, rtac exI,
rtac conjI, CONJ_WRAP' (K atac) ls', rtac conjI, rtac refl, rtac refl,
REPEAT_DETERM o eresolve_tac [exE, conjE], rtac exI, rtac conjI,
REPEAT_DETERM_N 2 o EVERY' [rtac exI, rtac conjI, etac @{thm conjI[OF refl sym]},
CONJ_WRAP' (K atac) ls']] 1
end;
end;