src/HOL/Tools/BNF/bnf_lfp_tactics.ML
author haftmann
Sat Mar 22 08:37:43 2014 +0100 (2014-03-22)
changeset 56248 67dc9549fa15
parent 56237 69a9dfe71aed
child 56262 251f60be62a7
permissions -rw-r--r--
generalized and strengthened cong rules on compound operators, similar to 1ed737a98198
     1 (*  Title:      HOL/Tools/BNF/bnf_lfp_tactics.ML
     2     Author:     Dmitriy Traytel, TU Muenchen
     3     Author:     Andrei Popescu, TU Muenchen
     4     Copyright   2012
     5 
     6 Tactics for the datatype construction.
     7 *)
     8 
     9 signature BNF_LFP_TACTICS =
    10 sig
    11   val mk_alg_min_alg_tac: int -> thm -> thm list -> thm -> thm -> thm list list -> thm list ->
    12     thm list -> tactic
    13   val mk_alg_not_empty_tac: Proof.context -> thm -> thm list -> thm list -> tactic
    14   val mk_alg_select_tac: Proof.context -> thm -> tactic
    15   val mk_alg_set_tac: thm -> tactic
    16   val mk_bd_card_order_tac: thm list -> tactic
    17   val mk_bd_limit_tac: int -> thm -> tactic
    18   val mk_card_of_min_alg_tac: thm -> thm -> thm -> thm -> thm -> tactic
    19   val mk_copy_tac: int -> thm -> thm -> thm list -> thm list list -> tactic
    20   val mk_ctor_induct_tac: Proof.context -> int -> thm list list -> thm -> thm list -> thm ->
    21     thm list -> thm list -> thm list -> tactic
    22   val mk_ctor_induct2_tac: Proof.context -> ctyp option list -> cterm option list -> thm ->
    23     thm list -> tactic
    24   val mk_ctor_set_tac: thm -> thm -> thm list -> tactic
    25   val mk_ctor_rel_tac: Proof.context -> thm list -> int -> thm -> thm -> thm -> thm -> thm list ->
    26     thm -> thm -> thm list -> thm list -> thm list list -> tactic
    27   val mk_dtor_o_ctor_tac: thm -> thm -> thm -> thm -> thm list -> tactic
    28   val mk_init_ex_mor_tac: Proof.context -> thm -> thm -> thm list -> thm -> thm -> thm -> thm ->
    29     tactic
    30   val mk_init_induct_tac: int -> thm -> thm -> thm list -> thm list -> tactic
    31   val mk_init_unique_mor_tac: int -> thm -> thm -> thm list -> thm list -> thm list -> thm list ->
    32     thm list -> tactic
    33   val mk_fold_unique_mor_tac: thm list -> thm list -> thm list -> thm -> thm -> thm -> tactic
    34   val mk_fold_transfer_tac: Proof.context -> int -> thm -> thm list -> thm list -> tactic
    35   val mk_least_min_alg_tac: thm -> thm -> tactic
    36   val mk_le_rel_OO_tac: Proof.context -> int -> thm -> thm list -> thm list -> thm list ->
    37     thm list -> tactic
    38   val mk_map_comp0_tac: thm list -> thm list -> thm -> int -> tactic
    39   val mk_map_id0_tac: thm list -> thm -> tactic
    40   val mk_map_tac: int -> int -> thm -> thm -> thm -> tactic
    41   val mk_ctor_map_unique_tac: Proof.context -> thm -> thm list -> tactic
    42   val mk_mcong_tac: Proof.context -> (int -> tactic) -> thm list list list -> thm list ->
    43     thm list -> tactic
    44   val mk_min_algs_card_of_tac: ctyp -> cterm -> int -> thm -> thm list -> thm list -> thm -> thm ->
    45     thm -> thm -> thm -> thm -> thm -> tactic
    46   val mk_min_algs_least_tac: ctyp -> cterm -> thm -> thm list -> thm list -> tactic
    47   val mk_min_algs_mono_tac: Proof.context -> thm -> tactic
    48   val mk_min_algs_tac: thm -> thm list -> tactic
    49   val mk_mor_Abs_tac: Proof.context -> cterm option list -> thm list -> thm list -> thm list ->
    50     thm list -> tactic
    51   val mk_mor_Rep_tac: Proof.context -> int -> thm list -> thm list -> thm list -> thm -> thm list ->
    52     thm list list -> tactic
    53   val mk_mor_UNIV_tac: int -> thm list -> thm -> tactic
    54   val mk_mor_comp_tac: thm -> thm list list -> thm list -> tactic
    55   val mk_mor_convol_tac: 'a list -> thm -> tactic
    56   val mk_mor_elim_tac: thm -> tactic
    57   val mk_mor_incl_tac: thm -> thm list -> tactic
    58   val mk_mor_fold_tac: ctyp -> cterm -> thm list -> thm -> thm -> tactic
    59   val mk_mor_select_tac: thm -> thm -> thm -> thm -> thm -> thm -> thm list -> thm list list ->
    60     thm list -> tactic
    61   val mk_mor_str_tac: 'a list -> thm -> tactic
    62   val mk_rel_induct_tac: Proof.context -> thm list -> int -> thm -> int list -> thm list ->
    63     thm list -> tactic
    64   val mk_rec_tac: Proof.context -> thm list -> thm -> thm list -> tactic
    65   val mk_rec_unique_mor_tac: Proof.context -> thm list -> thm list -> thm -> tactic
    66   val mk_set_bd_tac: Proof.context -> int -> (int -> tactic) -> thm -> thm list list -> thm list ->
    67     int -> tactic
    68   val mk_set_nat_tac: Proof.context -> int -> (int -> tactic) -> thm list list -> thm list ->
    69     cterm list -> thm list -> int -> tactic
    70   val mk_set_map0_tac: thm -> tactic
    71   val mk_set_tac: thm -> tactic
    72   val mk_wit_tac: Proof.context -> int -> thm list -> thm list -> tactic
    73 end;
    74 
    75 structure BNF_LFP_Tactics : BNF_LFP_TACTICS =
    76 struct
    77 
    78 open BNF_Tactics
    79 open BNF_LFP_Util
    80 open BNF_Util
    81 
    82 val fst_snd_convs = @{thms fst_conv snd_conv};
    83 val ord_eq_le_trans = @{thm ord_eq_le_trans};
    84 val subset_trans = @{thm subset_trans};
    85 val trans_fun_cong_image_id_id_apply = @{thm trans[OF fun_cong[OF image_id] id_apply]};
    86 val rev_bspec = Drule.rotate_prems 1 bspec;
    87 val Un_cong = @{thm arg_cong2[of _ _ _ _ "op \<union>"]};
    88 val relChainD = @{thm iffD2[OF meta_eq_to_obj_eq[OF relChain_def]]};
    89 
    90 fun mk_alg_set_tac alg_def =
    91   EVERY' [dtac (alg_def RS iffD1), REPEAT_DETERM o etac conjE, etac bspec, rtac CollectI,
    92    REPEAT_DETERM o (rtac (subset_UNIV RS conjI) ORELSE' etac conjI), atac] 1;
    93 
    94 fun mk_alg_not_empty_tac ctxt alg_set alg_sets wits =
    95   (EVERY' [rtac notI, hyp_subst_tac ctxt, ftac alg_set] THEN'
    96   REPEAT_DETERM o FIRST'
    97     [EVERY' [rtac @{thm subset_emptyI}, eresolve_tac wits],
    98     EVERY' [rtac subsetI, rtac FalseE, eresolve_tac wits],
    99     EVERY' [rtac subsetI, dresolve_tac wits, hyp_subst_tac ctxt,
   100       FIRST' (map (fn thm => rtac thm THEN' atac) alg_sets)]] THEN'
   101   etac @{thm emptyE}) 1;
   102 
   103 fun mk_mor_elim_tac mor_def =
   104   (dtac (mor_def RS iffD1) THEN'
   105   REPEAT o etac conjE THEN'
   106   TRY o rtac @{thm image_subsetI} THEN'
   107   etac bspec THEN'
   108   atac) 1;
   109 
   110 fun mk_mor_incl_tac mor_def map_ids =
   111   (rtac (mor_def RS iffD2) THEN'
   112   rtac conjI THEN'
   113   CONJ_WRAP' (K (EVERY' [rtac ballI, etac set_mp, etac (id_apply RS @{thm ssubst_mem})]))
   114     map_ids THEN'
   115   CONJ_WRAP' (fn thm =>
   116     (EVERY' [rtac ballI, rtac trans, rtac id_apply, stac thm, rtac refl])) map_ids) 1;
   117 
   118 fun mk_mor_comp_tac mor_def set_maps map_comp_ids =
   119   let
   120     val fbetw_tac =
   121       EVERY' [rtac ballI, rtac (o_apply RS @{thm ssubst_mem}), etac bspec, etac bspec, atac];
   122     fun mor_tac (set_map, map_comp_id) =
   123       EVERY' [rtac ballI, rtac (o_apply RS trans), rtac trans,
   124         rtac trans, dtac rev_bspec, atac, etac arg_cong,
   125          REPEAT o eresolve_tac [CollectE, conjE], etac bspec, rtac CollectI] THEN'
   126       CONJ_WRAP' (fn thm =>
   127         FIRST' [rtac subset_UNIV,
   128           (EVERY' [rtac ord_eq_le_trans, rtac thm, rtac @{thm image_subsetI},
   129             etac bspec, etac set_mp, atac])]) set_map THEN'
   130       rtac (map_comp_id RS arg_cong);
   131   in
   132     (dtac (mor_def RS iffD1) THEN' dtac (mor_def RS iffD1) THEN' rtac (mor_def RS iffD2) THEN'
   133     REPEAT o etac conjE THEN'
   134     rtac conjI THEN'
   135     CONJ_WRAP' (K fbetw_tac) set_maps THEN'
   136     CONJ_WRAP' mor_tac (set_maps ~~ map_comp_ids)) 1
   137   end;
   138 
   139 fun mk_mor_str_tac ks mor_def =
   140   (rtac (mor_def RS iffD2) THEN' rtac conjI THEN'
   141   CONJ_WRAP' (K (EVERY' [rtac ballI, rtac UNIV_I])) ks THEN'
   142   CONJ_WRAP' (K (EVERY' [rtac ballI, rtac refl])) ks) 1;
   143 
   144 fun mk_mor_convol_tac ks mor_def =
   145   (rtac (mor_def RS iffD2) THEN' rtac conjI THEN'
   146   CONJ_WRAP' (K (EVERY' [rtac ballI, rtac UNIV_I])) ks THEN'
   147   CONJ_WRAP' (K (EVERY' [rtac ballI, rtac trans, rtac @{thm fst_convol'}, rtac o_apply])) ks) 1;
   148 
   149 fun mk_mor_UNIV_tac m morEs mor_def =
   150   let
   151     val n = length morEs;
   152     fun mor_tac morE = EVERY' [rtac @{thm ext}, rtac trans, rtac o_apply, rtac trans, etac morE,
   153       rtac CollectI, CONJ_WRAP' (K (rtac subset_UNIV)) (1 upto m + n),
   154       rtac sym, rtac o_apply];
   155   in
   156     EVERY' [rtac iffI, CONJ_WRAP' mor_tac morEs,
   157     rtac (mor_def RS iffD2), rtac conjI, CONJ_WRAP' (K (rtac ballI THEN' rtac UNIV_I)) morEs,
   158     REPEAT_DETERM o etac conjE, REPEAT_DETERM_N n o dtac (@{thm fun_eq_iff} RS iffD1),
   159     CONJ_WRAP' (K (EVERY' [rtac ballI, REPEAT_DETERM o etac allE, rtac trans,
   160       etac (o_apply RS sym RS trans), rtac o_apply])) morEs] 1
   161   end;
   162 
   163 fun mk_copy_tac m alg_def mor_def alg_sets set_mapss =
   164   let
   165     val n = length alg_sets;
   166     fun set_tac thm =
   167       EVERY' [rtac ord_eq_le_trans, rtac thm, rtac subset_trans, etac @{thm image_mono},
   168         rtac equalityD1, etac @{thm bij_betw_imageE}];
   169     val alg_tac =
   170       CONJ_WRAP' (fn (set_maps, alg_set) =>
   171         EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac set_mp,
   172           rtac equalityD1, etac @{thm bij_betw_imageE[OF bij_betw_the_inv_into]},
   173           rtac imageI, etac alg_set, EVERY' (map set_tac (drop m set_maps))])
   174       (set_mapss ~~ alg_sets);
   175 
   176     val mor_tac = rtac conjI THEN' CONJ_WRAP' (K (etac @{thm bij_betwE})) alg_sets THEN'
   177       CONJ_WRAP' (fn (set_maps, alg_set) =>
   178         EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, conjE],
   179           etac @{thm f_the_inv_into_f_bij_betw}, etac alg_set,
   180           EVERY' (map set_tac (drop m set_maps))])
   181       (set_mapss ~~ alg_sets);
   182   in
   183     (REPEAT_DETERM_N n o rtac exI THEN' rtac conjI THEN'
   184     rtac (alg_def RS iffD2) THEN' alg_tac THEN' rtac (mor_def RS iffD2) THEN' mor_tac) 1
   185   end;
   186 
   187 fun mk_bd_limit_tac n bd_Cinfinite =
   188   EVERY' [REPEAT_DETERM o etac conjE, rtac rev_mp, rtac @{thm Cinfinite_limit_finite},
   189     REPEAT_DETERM_N n o rtac @{thm finite.insertI}, rtac @{thm finite.emptyI},
   190     REPEAT_DETERM_N n o etac @{thm insert_subsetI}, rtac @{thm empty_subsetI},
   191     rtac bd_Cinfinite, rtac impI, etac bexE, rtac bexI,
   192     CONJ_WRAP' (fn i =>
   193       EVERY' [etac bspec, REPEAT_DETERM_N i o rtac @{thm insertI2}, rtac @{thm insertI1}])
   194       (0 upto n - 1),
   195     atac] 1;
   196 
   197 fun mk_min_algs_tac worel in_congs =
   198   let
   199     val minG_tac = EVERY' [rtac @{thm SUP_cong}, rtac refl, dtac bspec, atac, etac arg_cong];
   200     fun minH_tac thm =
   201       EVERY' [rtac Un_cong, minG_tac, rtac @{thm image_cong}, rtac thm,
   202         REPEAT_DETERM_N (length in_congs) o minG_tac, rtac refl];
   203   in
   204     (rtac (worel RS (@{thm wo_rel.worec_fixpoint} RS fun_cong)) THEN' rtac iffD2 THEN'
   205     rtac meta_eq_to_obj_eq THEN' rtac (worel RS @{thm wo_rel.adm_wo_def}) THEN'
   206     REPEAT_DETERM_N 3 o rtac allI THEN' rtac impI THEN'
   207     CONJ_WRAP_GEN' (EVERY' [rtac Pair_eqI, rtac conjI]) minH_tac in_congs) 1
   208   end;
   209 
   210 fun mk_min_algs_mono_tac ctxt min_algs = EVERY' [rtac relChainD, rtac allI, rtac allI, rtac impI,
   211   rtac @{thm case_split}, rtac @{thm xt1(3)}, rtac min_algs, etac @{thm FieldI2}, rtac subsetI,
   212   rtac UnI1, rtac @{thm UN_I}, etac @{thm underS_I}, atac, atac, rtac equalityD1,
   213   dtac @{thm notnotD}, hyp_subst_tac ctxt, rtac refl] 1;
   214 
   215 fun mk_min_algs_card_of_tac cT ct m worel min_algs in_bds bd_Card_order bd_Cnotzero
   216   suc_Card_order suc_Cinfinite suc_Cnotzero suc_Asuc Asuc_Cinfinite =
   217   let
   218     val induct = worel RS
   219       Drule.instantiate' [SOME cT] [NONE, SOME ct] @{thm well_order_induct_imp};
   220     val src = 1 upto m + 1;
   221     val dest = (m + 1) :: (1 upto m);
   222     val absorbAs_tac = if m = 0 then K (all_tac)
   223       else EVERY' [rtac @{thm ordIso_transitive}, rtac @{thm csum_cong1},
   224         rtac @{thm ordIso_transitive},
   225         BNF_Tactics.mk_rotate_eq_tac (rtac @{thm ordIso_refl} THEN'
   226           FIRST' [rtac @{thm card_of_Card_order}, rtac @{thm Card_order_csum},
   227             rtac @{thm Card_order_cexp}])
   228         @{thm ordIso_transitive} @{thm csum_assoc} @{thm csum_com} @{thm csum_cong}
   229         src dest,
   230         rtac @{thm csum_absorb1}, rtac Asuc_Cinfinite, rtac ctrans, rtac @{thm ordLeq_csum1},
   231         FIRST' [rtac @{thm Card_order_csum}, rtac @{thm card_of_Card_order}],
   232         rtac @{thm ordLeq_cexp1}, rtac suc_Cnotzero, rtac @{thm Card_order_csum}];
   233 
   234     val minG_tac = EVERY' [rtac @{thm UNION_Cinfinite_bound}, rtac @{thm ordLess_imp_ordLeq},
   235       rtac @{thm ordLess_transitive}, rtac @{thm card_of_underS}, rtac suc_Card_order,
   236       atac, rtac suc_Asuc, rtac ballI, etac allE, dtac mp, etac @{thm underS_E},
   237       dtac mp, etac @{thm underS_Field}, REPEAT o etac conjE, atac, rtac Asuc_Cinfinite]
   238 
   239     fun mk_minH_tac (min_alg, in_bd) = EVERY' [rtac @{thm ordIso_ordLeq_trans},
   240       rtac @{thm card_of_ordIso_subst}, etac min_alg, rtac @{thm Un_Cinfinite_bound},
   241       minG_tac, rtac ctrans, rtac @{thm card_of_image}, rtac ctrans, rtac in_bd, rtac ctrans,
   242       rtac @{thm cexp_mono1}, rtac @{thm csum_mono1},
   243       REPEAT_DETERM_N m o rtac @{thm csum_mono2},
   244       CONJ_WRAP_GEN' (rtac @{thm csum_cinfinite_bound}) (K minG_tac) min_algs,
   245       REPEAT_DETERM o FIRST'
   246         [rtac @{thm card_of_Card_order}, rtac @{thm Card_order_csum},
   247         rtac Asuc_Cinfinite, rtac bd_Card_order],
   248       rtac @{thm ordIso_ordLeq_trans}, rtac @{thm cexp_cong1}, absorbAs_tac,
   249       rtac @{thm csum_absorb1}, rtac Asuc_Cinfinite, rtac @{thm ctwo_ordLeq_Cinfinite},
   250       rtac Asuc_Cinfinite, rtac bd_Card_order,
   251       rtac @{thm ordIso_imp_ordLeq}, rtac @{thm cexp_cprod_ordLeq},
   252       resolve_tac @{thms Card_order_csum Card_order_ctwo}, rtac suc_Cinfinite,
   253       rtac bd_Cnotzero, rtac @{thm cardSuc_ordLeq}, rtac bd_Card_order, rtac Asuc_Cinfinite];
   254   in
   255     (rtac induct THEN'
   256     rtac impI THEN'
   257     CONJ_WRAP' mk_minH_tac (min_algs ~~ in_bds)) 1
   258   end;
   259 
   260 fun mk_min_algs_least_tac cT ct worel min_algs alg_sets =
   261   let
   262     val induct = worel RS
   263       Drule.instantiate' [SOME cT] [NONE, SOME ct] @{thm well_order_induct_imp};
   264 
   265     val minG_tac = EVERY' [rtac @{thm UN_least}, etac allE, dtac mp, etac @{thm underS_E},
   266       dtac mp, etac @{thm underS_Field}, REPEAT_DETERM o etac conjE, atac];
   267 
   268     fun mk_minH_tac (min_alg, alg_set) = EVERY' [rtac ord_eq_le_trans, etac min_alg,
   269       rtac @{thm Un_least}, minG_tac, rtac @{thm image_subsetI},
   270       REPEAT_DETERM o eresolve_tac [CollectE, conjE], etac alg_set,
   271       REPEAT_DETERM o (etac subset_trans THEN' minG_tac)];
   272   in
   273     (rtac induct THEN'
   274     rtac impI THEN'
   275     CONJ_WRAP' mk_minH_tac (min_algs ~~ alg_sets)) 1
   276   end;
   277 
   278 fun mk_alg_min_alg_tac m alg_def min_alg_defs bd_limit bd_Cinfinite
   279     set_bdss min_algs min_alg_monos =
   280   let
   281     val n = length min_algs;
   282     fun mk_cardSuc_UNION_tac set_bds (mono, def) = EVERY'
   283       [rtac bexE, rtac @{thm cardSuc_UNION_Cinfinite}, rtac bd_Cinfinite, rtac mono,
   284        etac (def RSN (2, @{thm subset_trans[OF _ equalityD1]})), resolve_tac set_bds];
   285     fun mk_conjunct_tac (set_bds, (min_alg, min_alg_def)) =
   286       EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, conjE],
   287         EVERY' (map (mk_cardSuc_UNION_tac set_bds) (min_alg_monos ~~ min_alg_defs)), rtac bexE,
   288         rtac bd_limit, REPEAT_DETERM_N (n - 1) o etac conjI, atac,
   289         rtac (min_alg_def RS @{thm set_mp[OF equalityD2]}),
   290         rtac @{thm UN_I}, REPEAT_DETERM_N (m + 3 * n) o etac thin_rl, atac, rtac set_mp,
   291         rtac equalityD2, rtac min_alg, atac, rtac UnI2, rtac @{thm image_eqI}, rtac refl,
   292         rtac CollectI, REPEAT_DETERM_N m o dtac asm_rl, REPEAT_DETERM_N n o etac thin_rl,
   293         REPEAT_DETERM o etac conjE,
   294         CONJ_WRAP' (K (FIRST' [atac,
   295           EVERY' [etac subset_trans, rtac subsetI, rtac @{thm UN_I},
   296             etac @{thm underS_I}, atac, atac]]))
   297           set_bds];
   298   in
   299     (rtac (alg_def RS iffD2) THEN'
   300     CONJ_WRAP' mk_conjunct_tac (set_bdss ~~ (min_algs ~~ min_alg_defs))) 1
   301   end;
   302 
   303 fun mk_card_of_min_alg_tac min_alg_def card_of suc_Card_order suc_Asuc Asuc_Cinfinite =
   304   EVERY' [rtac @{thm ordIso_ordLeq_trans}, rtac (min_alg_def RS @{thm card_of_ordIso_subst}),
   305     rtac @{thm UNION_Cinfinite_bound}, rtac @{thm ordIso_ordLeq_trans},
   306     rtac @{thm card_of_Field_ordIso}, rtac suc_Card_order, rtac @{thm ordLess_imp_ordLeq},
   307     rtac suc_Asuc, rtac ballI, dtac rev_mp, rtac card_of,  REPEAT_DETERM o etac conjE, atac,
   308     rtac Asuc_Cinfinite] 1;
   309 
   310 fun mk_least_min_alg_tac min_alg_def least =
   311   EVERY' [rtac (min_alg_def RS ord_eq_le_trans), rtac @{thm UN_least}, dtac least, dtac mp, atac,
   312     REPEAT_DETERM o etac conjE, atac] 1;
   313 
   314 fun mk_alg_select_tac ctxt Abs_inverse =
   315   EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac ctxt] 1 THEN
   316   unfold_thms_tac ctxt (Abs_inverse :: fst_snd_convs) THEN atac 1;
   317 
   318 fun mk_mor_select_tac mor_def mor_cong mor_comp mor_incl_min_alg alg_def alg_select alg_sets
   319     set_maps str_init_defs =
   320   let
   321     val n = length alg_sets;
   322     val fbetw_tac =
   323       CONJ_WRAP' (K (EVERY' [rtac ballI, etac rev_bspec, etac CollectE, atac])) alg_sets;
   324     val mor_tac =
   325       CONJ_WRAP' (fn thm => EVERY' [rtac ballI, rtac thm]) str_init_defs;
   326     fun alg_epi_tac ((alg_set, str_init_def), set_map) =
   327       EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac CollectI,
   328         rtac ballI, ftac (alg_select RS bspec), rtac (str_init_def RS @{thm ssubst_mem}),
   329         etac alg_set, REPEAT_DETERM o EVERY' [rtac ord_eq_le_trans, resolve_tac set_map,
   330           rtac subset_trans, etac @{thm image_mono}, rtac @{thm image_Collect_subsetI}, etac bspec,
   331           atac]];
   332   in
   333     EVERY' [rtac mor_cong, REPEAT_DETERM_N n o (rtac sym THEN' rtac @{thm comp_id}),
   334       rtac (Thm.permute_prems 0 1 mor_comp), etac (Thm.permute_prems 0 1 mor_comp),
   335       rtac (mor_def RS iffD2), rtac conjI, fbetw_tac, mor_tac, rtac mor_incl_min_alg,
   336       rtac (alg_def RS iffD2), CONJ_WRAP' alg_epi_tac ((alg_sets ~~ str_init_defs) ~~ set_maps)] 1
   337   end;
   338 
   339 fun mk_init_ex_mor_tac ctxt Abs_inverse copy card_of_min_algs mor_Rep mor_comp mor_select mor_incl =
   340   let
   341     val n = length card_of_min_algs;
   342   in
   343     EVERY' [Method.insert_tac (map (fn thm => thm RS @{thm ex_bij_betw}) card_of_min_algs),
   344       REPEAT_DETERM o etac exE, rtac rev_mp, rtac copy, REPEAT_DETERM_N n o atac,
   345       rtac impI, REPEAT_DETERM o eresolve_tac [exE, conjE], REPEAT_DETERM_N n o rtac exI,
   346       rtac mor_comp, rtac mor_Rep, rtac mor_select, rtac CollectI, REPEAT_DETERM o rtac exI,
   347       rtac conjI, rtac refl, atac,
   348       SELECT_GOAL (unfold_thms_tac ctxt (Abs_inverse :: fst_snd_convs)),
   349       etac mor_comp, rtac mor_incl, REPEAT_DETERM_N n o rtac subset_UNIV] 1
   350   end;
   351 
   352 fun mk_init_unique_mor_tac m
   353     alg_def alg_min_alg least_min_algs in_monos alg_sets morEs map_cong0s =
   354   let
   355     val n = length least_min_algs;
   356     val ks = (1 upto n);
   357 
   358     fun mor_tac morE in_mono = EVERY' [etac morE, rtac set_mp, rtac in_mono,
   359       REPEAT_DETERM_N n o rtac @{thm Collect_restrict}, rtac CollectI,
   360       REPEAT_DETERM_N (m + n) o (TRY o rtac conjI THEN' atac)];
   361     fun cong_tac map_cong0 = EVERY' [rtac (map_cong0 RS arg_cong),
   362       REPEAT_DETERM_N m o rtac refl,
   363       REPEAT_DETERM_N n o (etac @{thm prop_restrict} THEN' atac)];
   364 
   365     fun mk_alg_tac (alg_set, (in_mono, (morE, map_cong0))) = EVERY' [rtac ballI, rtac CollectI,
   366       REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac conjI, rtac (alg_min_alg RS alg_set),
   367       REPEAT_DETERM_N n o (etac subset_trans THEN' rtac @{thm Collect_restrict}),
   368       rtac trans, mor_tac morE in_mono,
   369       rtac trans, cong_tac map_cong0,
   370       rtac sym, mor_tac morE in_mono];
   371 
   372     fun mk_unique_tac (k, least_min_alg) =
   373       select_prem_tac n (etac @{thm prop_restrict}) k THEN' rtac least_min_alg THEN'
   374       rtac (alg_def RS iffD2) THEN'
   375       CONJ_WRAP' mk_alg_tac (alg_sets ~~ (in_monos ~~ (morEs ~~ map_cong0s)));
   376   in
   377     CONJ_WRAP' mk_unique_tac (ks ~~ least_min_algs) 1
   378   end;
   379 
   380 fun mk_init_induct_tac m alg_def alg_min_alg least_min_algs alg_sets =
   381   let
   382     val n = length least_min_algs;
   383 
   384     fun mk_alg_tac alg_set = EVERY' [rtac ballI, rtac CollectI,
   385       REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac conjI, rtac (alg_min_alg RS alg_set),
   386       REPEAT_DETERM_N n o (etac subset_trans THEN' rtac @{thm Collect_restrict}),
   387       rtac mp, etac bspec, rtac CollectI,
   388       REPEAT_DETERM_N m o (rtac conjI THEN' atac),
   389       CONJ_WRAP' (K (etac subset_trans THEN' rtac @{thm Collect_restrict})) alg_sets,
   390       CONJ_WRAP' (K (rtac ballI THEN' etac @{thm prop_restrict} THEN' atac)) alg_sets];
   391 
   392     fun mk_induct_tac least_min_alg =
   393       rtac ballI THEN' etac @{thm prop_restrict} THEN' rtac least_min_alg THEN'
   394       rtac (alg_def RS iffD2) THEN'
   395       CONJ_WRAP' mk_alg_tac alg_sets;
   396   in
   397     CONJ_WRAP' mk_induct_tac least_min_algs 1
   398   end;
   399 
   400 fun mk_mor_Rep_tac ctxt m defs Reps Abs_inverses alg_min_alg alg_sets set_mapss =
   401   unfold_thms_tac ctxt (@{thm o_apply} :: defs) THEN
   402   EVERY' [rtac conjI,
   403     CONJ_WRAP' (fn thm => rtac ballI THEN' rtac thm) Reps,
   404     CONJ_WRAP' (fn (Abs_inverse, (set_maps, alg_set)) =>
   405       EVERY' [rtac ballI, rtac Abs_inverse, rtac (alg_min_alg RS alg_set),
   406         EVERY' (map2 (fn Rep => fn set_map =>
   407           EVERY' [rtac (set_map RS ord_eq_le_trans), rtac @{thm image_subsetI}, rtac Rep])
   408         Reps (drop m set_maps))])
   409     (Abs_inverses ~~ (set_mapss ~~ alg_sets))] 1;
   410 
   411 fun mk_mor_Abs_tac ctxt cts defs Abs_inverses map_comp_ids map_congLs =
   412   unfold_thms_tac ctxt (@{thm o_apply} :: defs) THEN
   413   EVERY' [rtac conjI,
   414     CONJ_WRAP' (K (rtac ballI THEN' rtac UNIV_I)) Abs_inverses,
   415     CONJ_WRAP' (fn (ct, thm) =>
   416       EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, conjE],
   417         rtac (thm RS (cterm_instantiate_pos [NONE, NONE, ct] arg_cong) RS sym),
   418         EVERY' (map (fn Abs_inverse =>
   419           EVERY' [rtac (o_apply RS trans RS ballI), etac (set_mp RS Abs_inverse), atac])
   420         Abs_inverses)])
   421     (cts ~~ map2 mk_trans map_comp_ids map_congLs)] 1;;
   422 
   423 fun mk_mor_fold_tac cT ct fold_defs ex_mor mor =
   424   (EVERY' (map stac fold_defs) THEN' EVERY' [rtac rev_mp, rtac ex_mor, rtac impI] THEN'
   425   REPEAT_DETERM_N (length fold_defs) o etac exE THEN'
   426   rtac (Drule.instantiate' [SOME cT] [SOME ct] @{thm someI}) THEN' etac mor) 1;
   427 
   428 fun mk_fold_unique_mor_tac type_defs init_unique_mors Reps mor_comp mor_Abs mor_fold =
   429   let
   430     fun mk_unique type_def =
   431       EVERY' [rtac @{thm surj_fun_eq}, rtac (type_def RS @{thm type_definition.Abs_image}),
   432         rtac ballI, resolve_tac init_unique_mors,
   433         EVERY' (map (fn thm => atac ORELSE' rtac thm) Reps),
   434         rtac mor_comp, rtac mor_Abs, atac,
   435         rtac mor_comp, rtac mor_Abs, rtac mor_fold];
   436   in
   437     CONJ_WRAP' mk_unique type_defs 1
   438   end;
   439 
   440 fun mk_dtor_o_ctor_tac dtor_def foldx map_comp_id map_cong0L ctor_o_folds =
   441   EVERY' [rtac @{thm ext}, rtac trans, rtac o_apply, rtac (dtor_def RS fun_cong RS trans),
   442     rtac trans, rtac foldx, rtac trans, rtac map_comp_id, rtac trans, rtac map_cong0L,
   443     EVERY' (map (fn thm => rtac ballI THEN' rtac (trans OF [thm RS fun_cong, id_apply]))
   444       ctor_o_folds),
   445     rtac sym, rtac id_apply] 1;
   446 
   447 fun mk_rec_tac ctxt rec_defs foldx fst_recs =
   448   unfold_thms_tac ctxt
   449     (rec_defs @ map (fn thm => thm RS @{thm convol_expand_snd}) fst_recs) THEN
   450   EVERY' [rtac trans, rtac o_apply, rtac trans, rtac (foldx RS @{thm arg_cong[of _ _ snd]}),
   451     rtac @{thm snd_convol'}] 1;
   452 
   453 fun mk_rec_unique_mor_tac ctxt rec_defs fst_recs fold_unique_mor =
   454   unfold_thms_tac ctxt
   455     (rec_defs @ map (fn thm => thm RS @{thm convol_expand_snd'}) fst_recs) THEN
   456   etac fold_unique_mor 1;
   457 
   458 fun mk_ctor_induct_tac ctxt m set_mapss init_induct morEs mor_Abs Rep_invs Abs_invs Reps =
   459   let
   460     val n = length set_mapss;
   461     val ks = 1 upto n;
   462 
   463     fun mk_IH_tac Rep_inv Abs_inv set_map =
   464       DETERM o EVERY' [dtac meta_mp, rtac (Rep_inv RS arg_cong RS iffD1), etac bspec,
   465         dtac set_rev_mp, rtac equalityD1, rtac set_map, etac imageE,
   466         hyp_subst_tac ctxt, rtac (Abs_inv RS @{thm ssubst_mem}), etac set_mp, atac, atac];
   467 
   468     fun mk_closed_tac (k, (morE, set_maps)) =
   469       EVERY' [select_prem_tac n (dtac asm_rl) k, rtac ballI, rtac impI,
   470         rtac (mor_Abs RS morE RS arg_cong RS iffD2), atac,
   471         REPEAT_DETERM o eresolve_tac [CollectE, conjE], dtac @{thm meta_spec},
   472         EVERY' (map3 mk_IH_tac Rep_invs Abs_invs (drop m set_maps)), atac];
   473 
   474     fun mk_induct_tac (Rep, Rep_inv) =
   475       EVERY' [rtac (Rep_inv RS arg_cong RS iffD1), etac (Rep RSN (2, bspec))];
   476   in
   477     (rtac mp THEN' rtac impI THEN'
   478     DETERM o CONJ_WRAP_GEN' (etac conjE THEN' rtac conjI) mk_induct_tac (Reps ~~ Rep_invs) THEN'
   479     rtac init_induct THEN'
   480     DETERM o CONJ_WRAP' mk_closed_tac (ks ~~ (morEs ~~ set_mapss))) 1
   481   end;
   482 
   483 fun mk_ctor_induct2_tac ctxt cTs cts ctor_induct weak_ctor_inducts =
   484   let
   485     val n = length weak_ctor_inducts;
   486     val ks = 1 upto n;
   487     fun mk_inner_induct_tac induct i =
   488       EVERY' [rtac allI, fo_rtac induct ctxt,
   489         select_prem_tac n (dtac @{thm meta_spec2}) i,
   490         REPEAT_DETERM_N n o
   491           EVERY' [dtac meta_mp THEN_ALL_NEW Goal.norm_hhf_tac ctxt,
   492             REPEAT_DETERM o dtac @{thm meta_spec}, etac (spec RS meta_mp), atac],
   493         atac];
   494   in
   495     EVERY' [rtac rev_mp, rtac (Drule.instantiate' cTs cts ctor_induct),
   496       EVERY' (map2 mk_inner_induct_tac weak_ctor_inducts ks), rtac impI,
   497       REPEAT_DETERM o eresolve_tac [conjE, allE],
   498       CONJ_WRAP' (K atac) ks] 1
   499   end;
   500 
   501 fun mk_map_tac m n foldx map_comp_id map_cong0 =
   502   EVERY' [rtac @{thm ext}, rtac trans, rtac o_apply, rtac trans, rtac foldx, rtac trans,
   503     rtac o_apply,
   504     rtac trans, rtac (map_comp_id RS arg_cong), rtac trans, rtac (map_cong0 RS arg_cong),
   505     REPEAT_DETERM_N m o rtac refl,
   506     REPEAT_DETERM_N n o (EVERY' (map rtac [trans, o_apply, id_apply])),
   507     rtac sym, rtac o_apply] 1;
   508 
   509 fun mk_ctor_map_unique_tac ctxt fold_unique sym_map_comps =
   510   rtac fold_unique 1 THEN
   511   unfold_thms_tac ctxt (sym_map_comps @ @{thms comp_assoc id_comp comp_id}) THEN
   512   ALLGOALS atac;
   513 
   514 fun mk_set_tac foldx = EVERY' [rtac @{thm ext}, rtac trans, rtac o_apply,
   515   rtac trans, rtac foldx, rtac sym, rtac o_apply] 1;
   516 
   517 fun mk_ctor_set_tac set set_map set_maps =
   518   let
   519     val n = length set_maps;
   520     fun mk_UN thm = rtac (thm RS @{thm arg_cong[of _ _ Union]} RS trans) THEN'
   521       rtac @{thm Union_image_eq};
   522   in
   523     EVERY' [rtac (set RS @{thm comp_eq_dest} RS trans), rtac Un_cong,
   524       rtac (trans OF [set_map, trans_fun_cong_image_id_id_apply]),
   525       REPEAT_DETERM_N (n - 1) o rtac Un_cong,
   526       EVERY' (map mk_UN set_maps)] 1
   527   end;
   528 
   529 fun mk_set_nat_tac ctxt m induct_tac set_mapss ctor_maps csets ctor_sets i =
   530   let
   531     val n = length ctor_maps;
   532 
   533     fun useIH set_nat = EVERY' [rtac trans, rtac @{thm image_UN}, rtac trans, rtac @{thm SUP_cong},
   534       rtac refl, Goal.assume_rule_tac ctxt, rtac sym, rtac trans, rtac @{thm SUP_cong},
   535       rtac set_nat, rtac refl, rtac @{thm UN_simps(10)}];
   536 
   537     fun mk_set_nat cset ctor_map ctor_set set_nats =
   538       EVERY' [rtac trans, rtac @{thm image_cong}, rtac ctor_set, rtac refl,
   539         rtac sym, rtac (trans OF [ctor_map RS HOL_arg_cong cset, ctor_set RS trans]),
   540         rtac sym, EVERY' (map rtac [trans, @{thm image_Un}, Un_cong]),
   541         rtac sym, rtac (nth set_nats (i - 1)),
   542         REPEAT_DETERM_N (n - 1) o EVERY' (map rtac [trans, @{thm image_Un}, Un_cong]),
   543         EVERY' (map useIH (drop m set_nats))];
   544   in
   545     (induct_tac THEN' EVERY' (map4 mk_set_nat csets ctor_maps ctor_sets set_mapss)) 1
   546   end;
   547 
   548 fun mk_set_bd_tac ctxt m induct_tac bd_Cinfinite set_bdss ctor_sets i =
   549   let
   550     val n = length ctor_sets;
   551 
   552     fun useIH set_bd = EVERY' [rtac @{thm UNION_Cinfinite_bound}, rtac set_bd, rtac ballI,
   553       Goal.assume_rule_tac ctxt, rtac bd_Cinfinite];
   554 
   555     fun mk_set_nat ctor_set set_bds =
   556       EVERY' [rtac @{thm ordIso_ordLeq_trans}, rtac @{thm card_of_ordIso_subst}, rtac ctor_set,
   557         rtac (bd_Cinfinite RSN (3, @{thm Un_Cinfinite_bound})), rtac (nth set_bds (i - 1)),
   558         REPEAT_DETERM_N (n - 1) o rtac (bd_Cinfinite RSN (3, @{thm Un_Cinfinite_bound})),
   559         EVERY' (map useIH (drop m set_bds))];
   560   in
   561     (induct_tac THEN' EVERY' (map2 mk_set_nat ctor_sets set_bdss)) 1
   562   end;
   563 
   564 fun mk_mcong_tac ctxt induct_tac set_setsss map_cong0s ctor_maps =
   565   let
   566     fun use_asm thm = EVERY' [etac bspec, etac set_rev_mp, rtac thm];
   567 
   568     fun useIH set_sets = EVERY' [rtac mp, Goal.assume_rule_tac ctxt,
   569       CONJ_WRAP' (fn thm =>
   570         EVERY' [rtac ballI, etac bspec, etac set_rev_mp, etac thm]) set_sets];
   571 
   572     fun mk_map_cong0 ctor_map map_cong0 set_setss =
   573       EVERY' [rtac impI, REPEAT_DETERM o etac conjE,
   574         rtac trans, rtac ctor_map, rtac trans, rtac (map_cong0 RS arg_cong),
   575         EVERY' (map use_asm (map hd set_setss)),
   576         EVERY' (map useIH (transpose (map tl set_setss))),
   577         rtac sym, rtac ctor_map];
   578   in
   579     (induct_tac THEN' EVERY' (map3 mk_map_cong0 ctor_maps map_cong0s set_setsss)) 1
   580   end;
   581 
   582 fun mk_le_rel_OO_tac ctxt m induct ctor_nchotomys ctor_Irels rel_mono_strongs rel_OOs =
   583   EVERY' (rtac induct ::
   584   map4 (fn nchotomy => fn Irel => fn rel_mono => fn rel_OO =>
   585     EVERY' [rtac impI, etac (nchotomy RS @{thm nchotomy_relcomppE}),
   586       REPEAT_DETERM_N 2 o dtac (Irel RS iffD1), rtac (Irel RS iffD2),
   587       rtac rel_mono, rtac (rel_OO RS @{thm predicate2_eqD} RS iffD2),
   588       rtac @{thm relcomppI}, atac, atac,
   589       REPEAT_DETERM_N m o EVERY' [rtac ballI, rtac ballI, rtac impI, atac],
   590       REPEAT_DETERM_N (length rel_OOs) o
   591         EVERY' [rtac ballI, rtac ballI, Goal.assume_rule_tac ctxt]])
   592   ctor_nchotomys ctor_Irels rel_mono_strongs rel_OOs) 1;
   593 
   594 (* BNF tactics *)
   595 
   596 fun mk_map_id0_tac map_id0s unique =
   597   (rtac sym THEN' rtac unique THEN'
   598   EVERY' (map (fn thm =>
   599     EVERY' [rtac trans, rtac @{thm id_comp}, rtac trans, rtac sym, rtac @{thm comp_id},
   600       rtac (thm RS sym RS arg_cong)]) map_id0s)) 1;
   601 
   602 fun mk_map_comp0_tac map_comp0s ctor_maps unique iplus1 =
   603   let
   604     val i = iplus1 - 1;
   605     val unique' = Thm.permute_prems 0 i unique;
   606     val map_comp0s' = drop i map_comp0s @ take i map_comp0s;
   607     val ctor_maps' = drop i ctor_maps @ take i ctor_maps;
   608     fun mk_comp comp simp =
   609       EVERY' [rtac @{thm ext}, rtac trans, rtac o_apply, rtac trans, rtac o_apply,
   610         rtac trans, rtac (simp RS arg_cong), rtac trans, rtac simp,
   611         rtac trans, rtac (comp RS arg_cong), rtac sym, rtac o_apply];
   612   in
   613     (rtac sym THEN' rtac unique' THEN' EVERY' (map2 mk_comp map_comp0s' ctor_maps')) 1
   614   end;
   615 
   616 fun mk_set_map0_tac set_nat =
   617   EVERY' (map rtac [@{thm ext}, trans, o_apply, sym, trans, o_apply, set_nat]) 1;
   618 
   619 fun mk_bd_card_order_tac bd_card_orders =
   620   CONJ_WRAP_GEN' (rtac @{thm card_order_csum}) rtac bd_card_orders 1;
   621 
   622 fun mk_wit_tac ctxt n ctor_set wit =
   623   REPEAT_DETERM (atac 1 ORELSE
   624     EVERY' [dtac set_rev_mp, rtac equalityD1, resolve_tac ctor_set,
   625     REPEAT_DETERM o
   626       (TRY o REPEAT_DETERM o etac UnE THEN' TRY o etac @{thm UN_E} THEN'
   627         (eresolve_tac wit ORELSE'
   628         (dresolve_tac wit THEN'
   629           (etac FalseE ORELSE'
   630           EVERY' [hyp_subst_tac ctxt, dtac set_rev_mp, rtac equalityD1, resolve_tac ctor_set,
   631             REPEAT_DETERM_N n o etac UnE]))))] 1);
   632 
   633 fun mk_ctor_rel_tac ctxt in_Irels i in_rel map_comp0 map_cong0 ctor_map ctor_sets ctor_inject
   634   ctor_dtor set_map0s ctor_set_incls ctor_set_set_inclss =
   635   let
   636     val m = length ctor_set_incls;
   637     val n = length ctor_set_set_inclss;
   638 
   639     val (passive_set_map0s, active_set_map0s) = chop m set_map0s;
   640     val in_Irel = nth in_Irels (i - 1);
   641     val le_arg_cong_ctor_dtor = ctor_dtor RS arg_cong RS ord_eq_le_trans;
   642     val eq_arg_cong_ctor_dtor = ctor_dtor RS arg_cong RS trans;
   643     val if_tac =
   644       EVERY' [dtac (in_Irel RS iffD1), REPEAT_DETERM o eresolve_tac [exE, conjE, CollectE],
   645         rtac (in_rel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
   646         EVERY' (map2 (fn set_map0 => fn ctor_set_incl =>
   647           EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac set_map0,
   648             rtac ord_eq_le_trans, rtac trans_fun_cong_image_id_id_apply,
   649             rtac (ctor_set_incl RS subset_trans), etac le_arg_cong_ctor_dtor])
   650         passive_set_map0s ctor_set_incls),
   651         CONJ_WRAP' (fn (in_Irel, (set_map0, ctor_set_set_incls)) =>
   652           EVERY' [rtac ord_eq_le_trans, rtac set_map0, rtac @{thm image_subsetI}, rtac CollectI,
   653             rtac @{thm case_prodI}, rtac (in_Irel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
   654             CONJ_WRAP' (fn thm =>
   655               EVERY' (map etac [thm RS subset_trans, le_arg_cong_ctor_dtor]))
   656             ctor_set_set_incls,
   657             rtac conjI, rtac refl, rtac refl])
   658         (in_Irels ~~ (active_set_map0s ~~ ctor_set_set_inclss)),
   659         CONJ_WRAP' (fn conv =>
   660           EVERY' [rtac trans, rtac map_comp0, rtac trans, rtac map_cong0,
   661           REPEAT_DETERM_N m o rtac @{thm fun_cong[OF comp_id]},
   662           REPEAT_DETERM_N n o EVERY' (map rtac [trans, o_apply, conv]),
   663           rtac (ctor_inject RS iffD1), rtac trans, rtac sym, rtac ctor_map,
   664           etac eq_arg_cong_ctor_dtor])
   665         fst_snd_convs];
   666     val only_if_tac =
   667       EVERY' [dtac (in_rel RS iffD1), REPEAT_DETERM o eresolve_tac [exE, conjE, CollectE],
   668         rtac (in_Irel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
   669         CONJ_WRAP' (fn (ctor_set, passive_set_map0) =>
   670           EVERY' [rtac ord_eq_le_trans, rtac ctor_set, rtac @{thm Un_least},
   671             rtac ord_eq_le_trans, rtac @{thm box_equals[OF _ refl]},
   672             rtac passive_set_map0, rtac trans_fun_cong_image_id_id_apply, atac,
   673             CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least}))
   674               (fn (active_set_map0, in_Irel) => EVERY' [rtac ord_eq_le_trans,
   675                 rtac @{thm SUP_cong[OF _ refl]}, rtac active_set_map0, rtac @{thm UN_least},
   676                 dtac set_rev_mp, etac @{thm image_mono}, etac imageE,
   677                 dtac @{thm ssubst_mem[OF pair_collapse]},
   678                 REPEAT_DETERM o eresolve_tac (CollectE :: conjE ::
   679                   @{thms case_prodE iffD1[OF Pair_eq, elim_format]}),
   680                 hyp_subst_tac ctxt,
   681                 dtac (in_Irel RS iffD1), dtac @{thm someI_ex}, REPEAT_DETERM o etac conjE,
   682                 TRY o
   683                   EVERY' [dtac (Thm.permute_prems 0 1 @{thm ssubst_mem}), atac, hyp_subst_tac ctxt],
   684                 REPEAT_DETERM o eresolve_tac [CollectE, conjE], atac])
   685             (rev (active_set_map0s ~~ in_Irels))])
   686         (ctor_sets ~~ passive_set_map0s),
   687         rtac conjI,
   688         REPEAT_DETERM_N 2 o EVERY' [rtac trans, rtac ctor_map, rtac (ctor_inject RS iffD2),
   689           rtac trans, rtac map_comp0, rtac trans, rtac map_cong0,
   690           REPEAT_DETERM_N m o rtac @{thm fun_cong[OF comp_id]},
   691           EVERY' (map (fn in_Irel => EVERY' [rtac trans, rtac o_apply, dtac set_rev_mp, atac,
   692             dtac @{thm ssubst_mem[OF pair_collapse]},
   693             REPEAT_DETERM o
   694               eresolve_tac (CollectE :: conjE :: @{thms case_prodE iffD1[OF Pair_eq, elim_format]}),
   695             hyp_subst_tac ctxt,
   696             dtac (in_Irel RS iffD1), dtac @{thm someI_ex}, REPEAT_DETERM o etac conjE, atac])
   697           in_Irels),
   698           atac]]
   699   in
   700     EVERY' [rtac iffI, if_tac, only_if_tac] 1
   701   end;
   702 
   703 fun mk_rel_induct_tac ctxt IHs m ctor_induct2 ks ctor_rels rel_mono_strongs =
   704   let val n = length ks;
   705   in
   706     unfold_tac ctxt @{thms le_fun_def le_bool_def all_simps(1,2)[symmetric]} THEN
   707     EVERY' [REPEAT_DETERM o rtac allI, rtac ctor_induct2,
   708       EVERY' (map3 (fn IH => fn ctor_rel => fn rel_mono_strong =>
   709         EVERY' [rtac impI, dtac (ctor_rel RS iffD1), rtac (IH RS @{thm spec2} RS mp),
   710           etac rel_mono_strong,
   711           REPEAT_DETERM_N m o rtac @{thm ballI[OF ballI[OF imp_refl]]},
   712           EVERY' (map (fn j =>
   713             EVERY' [select_prem_tac n (dtac asm_rl) j, rtac @{thm ballI[OF ballI]},
   714               Goal.assume_rule_tac ctxt]) ks)])
   715       IHs ctor_rels rel_mono_strongs)] 1
   716   end;
   717 
   718 fun mk_fold_transfer_tac ctxt m ctor_rel_induct map_transfers folds =
   719   let
   720     val n = length map_transfers;
   721   in
   722     unfold_thms_tac ctxt
   723       @{thms rel_fun_def_butlast all_conj_distrib[symmetric] imp_conjR[symmetric]} THEN
   724     unfold_thms_tac ctxt @{thms rel_fun_iff_leq_vimage2p} THEN
   725     HEADGOAL (EVERY'
   726       [REPEAT_DETERM o resolve_tac [allI, impI], rtac ctor_rel_induct,
   727       EVERY' (map (fn map_transfer => EVERY'
   728         [REPEAT_DETERM o resolve_tac [allI, impI, @{thm vimage2pI}],
   729         SELECT_GOAL (unfold_thms_tac ctxt folds),
   730         etac @{thm predicate2D_vimage2p},
   731         rtac (funpow (m + n + 1) (fn thm => thm RS @{thm rel_funD}) map_transfer),
   732         REPEAT_DETERM_N m o rtac @{thm id_transfer},
   733         REPEAT_DETERM_N n o rtac @{thm vimage2p_rel_fun},
   734         atac])
   735       map_transfers)])
   736   end;
   737 
   738 end;