src/HOL/Codatatype/Tools/bnf_gfp_tactics.ML
author blanchet
Fri Sep 21 15:53:29 2012 +0200 (2012-09-21)
changeset 49504 df9b897fb254
parent 49501 acc9635a644a
child 49506 aeb0cc8889f2
permissions -rw-r--r--
renamed "iter"/"coiter" to "fold"/"unfold" (cf. Wadler)
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(*  Title:      HOL/Codatatype/Tools/bnf_gfp_tactics.ML
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    Author:     Dmitriy Traytel, TU Muenchen
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    Author:     Andrei Popescu, TU Muenchen
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    Author:     Jasmin Blanchette, TU Muenchen
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    Copyright   2012
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Tactics for the codatatype construction.
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*)
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signature BNF_GFP_TACTICS =
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sig
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  val mk_Lev_sbd_tac: cterm option list -> thm list -> thm list -> thm list list -> tactic
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  val mk_bd_card_order_tac: thm -> tactic
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  val mk_bd_cinfinite_tac: thm -> tactic
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  val mk_bis_Gr_tac: thm -> thm list -> thm list -> thm list -> thm list ->
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    {prems: 'a, context: Proof.context} -> tactic
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  val mk_bis_O_tac: int -> thm -> thm list -> thm list -> tactic
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  val mk_bis_Union_tac: thm -> thm list -> {prems: 'a, context: Proof.context} -> tactic
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  val mk_bis_converse_tac: int -> thm -> thm list -> thm list -> tactic
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  val mk_bis_rel_tac: int -> thm -> thm list -> thm list -> thm list -> thm list list -> tactic
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  val mk_carT_set_tac: int -> int -> thm -> thm -> thm -> thm ->
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    {prems: 'a, context: Proof.context} -> tactic
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  val mk_card_of_carT_tac: int -> thm list -> thm -> thm -> thm -> thm -> thm -> thm list -> tactic
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  val mk_coalgT_tac: int -> thm list -> thm list -> thm list list ->
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    {prems: 'a, context: Proof.context} -> tactic
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  val mk_coalg_final_tac: int -> thm -> thm list -> thm list -> thm list list -> thm list list ->
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    tactic
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  val mk_coalg_set_tac: thm -> tactic
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  val mk_coalg_thePull_tac: int -> thm -> thm list -> thm list list -> (int -> tactic) list ->
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    {prems: 'a, context: Proof.context} -> tactic
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  val mk_coind_wit_tac: thm -> thm list -> thm list -> thm list ->
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    {prems: 'a, context: Proof.context} -> tactic
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  val mk_col_bd_tac: int -> int -> cterm option list -> thm list -> thm list -> thm -> thm ->
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    thm list list -> tactic
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  val mk_col_natural_tac: cterm option list -> thm list -> thm list -> thm list -> thm list list ->
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    {prems: 'a, context: Proof.context} -> tactic
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  val mk_congruent_str_final_tac: int -> thm -> thm -> thm -> thm list -> tactic
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  val mk_corec_tac: int -> thm list -> thm -> thm -> thm list ->
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    {prems: 'a, context: Proof.context} -> tactic
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  val mk_dtor_coinduct_tac: int -> int list -> thm -> thm -> tactic
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  val mk_dtor_strong_coinduct_tac: int list -> ctyp option list -> cterm option list -> thm -> thm ->
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    thm -> tactic
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  val mk_dtor_o_ctor_tac: thm -> thm -> thm -> thm -> thm list ->
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    {prems: 'a, context: Proof.context} -> tactic
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  val mk_equiv_lsbis_tac: thm -> thm -> thm -> thm -> thm -> thm -> tactic
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  val mk_hset_minimal_tac: int -> thm list -> thm -> {prems: 'a, context: Proof.context} -> tactic
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  val mk_hset_rec_minimal_tac: int -> cterm option list -> thm list -> thm list ->
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    {prems: 'a, context: Proof.context} -> tactic
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  val mk_in_bd_tac: thm -> thm list -> thm -> thm -> thm -> thm -> thm list -> thm -> thm -> thm ->
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    thm -> thm -> thm -> tactic
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  val mk_incl_lsbis_tac: int -> int -> thm -> tactic
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  val mk_isNode_hset_tac: int -> thm -> thm list -> {prems: 'a, context: Proof.context} -> tactic
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  val mk_length_Lev'_tac: thm -> tactic
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  val mk_length_Lev_tac: cterm option list -> thm list -> thm list -> tactic
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  val mk_map_comp_tac: int -> int -> thm list -> thm list -> thm list -> thm -> tactic
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  val mk_mcong_tac: int -> (int -> tactic) -> thm list -> thm list -> thm list -> thm list list ->
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    thm list list -> thm list list list -> tactic
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  val mk_map_id_tac: thm list -> thm -> thm -> tactic
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  val mk_map_tac: int -> int -> ctyp option -> thm -> thm -> thm -> tactic
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  val mk_map_unique_tac: thm -> thm list -> {prems: 'a, context: Proof.context} -> tactic
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  val mk_mor_Abs_tac: thm list -> thm list -> {prems: 'a, context: Proof.context} -> tactic
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  val mk_mor_Rep_tac: int -> thm list -> thm list -> thm list -> thm list list -> thm list ->
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    thm list -> {prems: 'a, context: Proof.context} -> tactic
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  val mk_mor_T_final_tac: thm -> thm list -> thm list -> tactic
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  val mk_mor_UNIV_tac: thm list -> thm -> tactic
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  val mk_mor_beh_tac: int -> thm -> thm -> thm list -> thm list -> thm list -> thm list ->
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    thm list list -> thm list list -> thm list -> thm list -> thm list -> thm list -> thm list ->
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    thm list -> thm list -> thm list -> thm list list -> thm list list list -> thm list list list ->
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    thm list list list -> thm list list -> thm list list -> thm list -> thm list -> thm list ->
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    {prems: 'a, context: Proof.context} -> tactic
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  val mk_mor_comp_tac: thm -> thm list -> thm list -> thm list -> tactic
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  val mk_mor_elim_tac: thm -> tactic
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  val mk_mor_hset_rec_tac: int -> int -> cterm option list -> int -> thm list -> thm list ->
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    thm list -> thm list list -> thm list list -> tactic
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  val mk_mor_hset_tac: thm -> thm -> tactic
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  val mk_mor_incl_tac: thm -> thm list -> tactic
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  val mk_mor_str_tac: 'a list -> thm -> tactic
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  val mk_mor_sum_case_tac: 'a list -> thm -> tactic
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  val mk_mor_thePull_fst_tac: int -> thm -> thm list -> thm list -> (int -> tactic) list ->
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    {prems: thm list, context: Proof.context} -> tactic
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  val mk_mor_thePull_snd_tac: int -> thm -> thm list -> thm list -> (int -> tactic) list ->
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    {prems: thm list, context: Proof.context} -> tactic
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  val mk_mor_thePull_pick_tac: thm -> thm list -> thm list ->
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    {prems: 'a, context: Proof.context} -> tactic
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  val mk_mor_unfold_tac: int -> thm -> thm list -> thm list -> thm list -> thm list -> thm list ->
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    thm list -> tactic
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  val mk_prefCl_Lev_tac: cterm option list -> thm list -> thm list -> tactic
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  val mk_pickWP_assms_tac: thm list -> thm list -> thm -> (int -> tactic)
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  val mk_pick_col_tac: int -> int -> cterm option list -> thm list -> thm list -> thm list ->
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    thm list list -> thm list -> (int -> tactic) list -> {prems: 'a, context: Proof.context} ->
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    tactic
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  val mk_raw_coind_tac: thm -> thm -> thm -> thm -> thm -> thm -> thm -> thm -> thm -> thm list ->
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    thm list -> thm list -> thm -> thm list -> tactic
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  val mk_rel_coinduct_tac: 'a list -> thm -> thm -> tactic
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  val mk_rel_strong_coinduct_tac: int -> ctyp option list -> cterm option list -> thm -> thm list ->
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    thm list -> tactic
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  val mk_rel_simp_tac: thm list -> int -> thm -> thm -> thm -> thm -> thm list -> thm -> thm ->
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    thm list -> thm list -> thm list list -> tactic
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  val mk_rv_last_tac: ctyp option list -> cterm option list -> thm list -> thm list -> tactic
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  val mk_sbis_lsbis_tac: thm list -> thm -> thm -> tactic
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  val mk_set_Lev_tac: cterm option list -> thm list -> thm list -> thm list -> thm list ->
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    thm list list -> tactic
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  val mk_set_bd_tac: thm -> thm -> thm -> tactic
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  val mk_set_hset_incl_hset_tac: int -> thm list -> thm -> int -> tactic
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  val mk_set_image_Lev_tac: cterm option list -> thm list -> thm list -> thm list -> thm list ->
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    thm list list -> thm list list -> tactic
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  val mk_set_incl_hin_tac: thm list -> tactic
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  val mk_set_incl_hset_tac: thm -> thm -> tactic
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  val mk_set_le_tac: int -> thm -> thm list -> thm list list -> tactic
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  val mk_set_natural_tac: thm -> thm -> tactic
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  val mk_set_rv_Lev_tac: int -> cterm option list -> thm list -> thm list -> thm list -> thm list ->
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    thm list list -> thm list list -> tactic
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  val mk_set_simp_tac: int -> thm -> thm -> thm list -> tactic
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  val mk_strT_hset_tac: int -> int -> int -> ctyp option list -> ctyp option list ->
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    cterm option list -> thm list -> thm list -> thm list -> thm list -> thm list list ->
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    thm list list -> thm list list -> thm -> thm list list -> tactic
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  val mk_unfold_unique_mor_tac: thm list -> thm -> thm -> thm list -> tactic
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  val mk_unique_mor_tac: thm list -> thm -> tactic
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  val mk_wit_tac: int -> thm list -> thm list -> thm list -> thm list ->
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    {prems: 'a, context: Proof.context} -> tactic
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  val mk_wpull_tac: int -> thm -> thm -> thm -> thm -> thm -> thm list -> thm list -> tactic
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end;
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structure BNF_GFP_Tactics : BNF_GFP_TACTICS =
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struct
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open BNF_Tactics
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open BNF_Util
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open BNF_FP
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open BNF_GFP_Util
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val fst_convol_fun_cong_sym = @{thm fst_convol} RS fun_cong RS sym;
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val list_inject_iffD1 = @{thm list.inject[THEN iffD1]};
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val nat_induct = @{thm nat_induct};
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val o_apply_trans_sym = o_apply RS trans RS sym;
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val ord_eq_le_trans = @{thm ord_eq_le_trans};
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val ord_eq_le_trans_trans_fun_cong_image_id_id_apply =
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  @{thm ord_eq_le_trans[OF trans[OF fun_cong[OF image_id] id_apply]]};
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val ordIso_ordLeq_trans = @{thm ordIso_ordLeq_trans};
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val snd_convol_fun_cong_sym = @{thm snd_convol} RS fun_cong RS sym;
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val sum_case_weak_cong = @{thm sum_case_weak_cong};
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val trans_fun_cong_image_id_id_apply = @{thm trans[OF fun_cong[OF image_id] id_apply]};
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fun mk_coalg_set_tac coalg_def =
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  dtac (coalg_def RS iffD1) 1 THEN
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  REPEAT_DETERM (etac conjE 1) THEN
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  EVERY' [dtac @{thm rev_bspec}, atac] 1 THEN
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  REPEAT_DETERM (eresolve_tac [CollectE, conjE] 1) THEN atac 1;
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fun mk_mor_elim_tac mor_def =
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  (dtac (subst OF [mor_def]) THEN'
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  REPEAT o etac conjE THEN'
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  TRY o rtac @{thm image_subsetI} THEN'
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  etac bspec THEN'
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  atac) 1;
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fun mk_mor_incl_tac mor_def map_id's =
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  (stac mor_def THEN'
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  rtac conjI THEN'
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  CONJ_WRAP' (K (EVERY' [rtac ballI, etac set_mp, stac @{thm id_apply}, atac]))
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    map_id's THEN'
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  CONJ_WRAP' (fn thm =>
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   (EVERY' [rtac ballI, rtac (thm RS trans), rtac sym, rtac (@{thm id_apply} RS arg_cong)]))
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  map_id's) 1;
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fun mk_mor_comp_tac mor_def mor_images morEs map_comp_ids =
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  let
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    fun fbetw_tac image = EVERY' [rtac ballI, stac o_apply, etac image, etac image, atac];
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    fun mor_tac ((mor_image, morE), map_comp_id) =
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      EVERY' [rtac ballI, stac o_apply, rtac trans, rtac (map_comp_id RS sym), rtac trans,
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        etac (morE RS arg_cong), atac, etac morE, etac mor_image, atac];
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  in
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    (stac mor_def THEN' rtac conjI THEN'
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    CONJ_WRAP' fbetw_tac mor_images THEN'
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    CONJ_WRAP' mor_tac ((mor_images ~~ morEs) ~~ map_comp_ids)) 1
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  end;
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fun mk_mor_UNIV_tac morEs mor_def =
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  let
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    val n = length morEs;
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    fun mor_tac morE = EVERY' [rtac ext, rtac trans, rtac o_apply, rtac trans, etac morE,
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      rtac UNIV_I, rtac sym, rtac o_apply];
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  in
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    EVERY' [rtac iffI, CONJ_WRAP' mor_tac morEs,
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    stac mor_def, rtac conjI, CONJ_WRAP' (K (rtac ballI THEN' rtac UNIV_I)) morEs,
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    CONJ_WRAP' (fn i =>
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      EVERY' [dtac (mk_conjunctN n i), rtac ballI, etac @{thm pointfreeE}]) (1 upto n)] 1
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  end;
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fun mk_mor_str_tac ks mor_UNIV =
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  (stac mor_UNIV THEN' CONJ_WRAP' (K (rtac refl)) ks) 1;
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fun mk_mor_sum_case_tac ks mor_UNIV =
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  (stac mor_UNIV THEN' CONJ_WRAP' (K (rtac @{thm sum_case_comp_Inl[symmetric]})) ks) 1;
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fun mk_set_incl_hset_tac def rec_Suc =
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  EVERY' (stac def ::
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    map rtac [@{thm incl_UNION_I}, UNIV_I, @{thm ord_le_eq_trans}, @{thm Un_upper1},
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      sym, rec_Suc]) 1;
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fun mk_set_hset_incl_hset_tac n defs rec_Suc i =
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  EVERY' (map (TRY oo stac) defs @
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    map rtac [@{thm UN_least}, subsetI, @{thm UN_I}, UNIV_I, set_mp, equalityD2, rec_Suc, UnI2,
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      mk_UnIN n i] @
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    [etac @{thm UN_I}, atac]) 1;
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fun mk_set_incl_hin_tac incls =
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  if null incls then rtac subset_UNIV 1
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  else EVERY' [rtac subsetI, rtac CollectI,
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    CONJ_WRAP' (fn incl => EVERY' [rtac subset_trans, etac incl, atac]) incls] 1;
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fun mk_hset_rec_minimal_tac m cts rec_0s rec_Sucs {context = ctxt, prems = _} =
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  EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
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    REPEAT_DETERM o rtac allI,
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    CONJ_WRAP' (fn thm => EVERY'
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      [rtac ord_eq_le_trans, rtac thm, rtac @{thm empty_subsetI}]) rec_0s,
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    REPEAT_DETERM o rtac allI,
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    CONJ_WRAP' (fn rec_Suc => EVERY'
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      [rtac ord_eq_le_trans, rtac rec_Suc,
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        if m = 0 then K all_tac
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        else (rtac @{thm Un_least} THEN' Goal.assume_rule_tac ctxt),
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        CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least}))
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          (K (EVERY' [rtac @{thm UN_least}, REPEAT_DETERM o eresolve_tac [allE, conjE],
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            rtac subset_trans, atac, Goal.assume_rule_tac ctxt])) rec_0s])
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      rec_Sucs] 1;
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fun mk_hset_minimal_tac n hset_defs hset_rec_minimal {context = ctxt, prems = _} =
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  (CONJ_WRAP' (fn def => (EVERY' [rtac ord_eq_le_trans, rtac def,
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    rtac @{thm UN_least}, rtac rev_mp, rtac hset_rec_minimal,
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    EVERY' (replicate ((n + 1) * n) (Goal.assume_rule_tac ctxt)), rtac impI,
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    REPEAT_DETERM o eresolve_tac [allE, conjE], atac])) hset_defs) 1
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fun mk_mor_hset_rec_tac m n cts j rec_0s rec_Sucs morEs set_naturalss coalg_setss =
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  EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
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    REPEAT_DETERM o rtac allI,
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    CONJ_WRAP' (fn thm => EVERY' (map rtac [impI, thm RS trans, thm RS sym])) rec_0s,
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    REPEAT_DETERM o rtac allI,
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    CONJ_WRAP'
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      (fn (rec_Suc, (morE, ((passive_set_naturals, active_set_naturals), coalg_sets))) =>
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        EVERY' [rtac impI, rtac (rec_Suc RS trans), rtac (rec_Suc RS trans RS sym),
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   241
          if m = 0 then K all_tac
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   242
          else EVERY' [rtac @{thm Un_cong}, rtac box_equals,
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   243
            rtac (nth passive_set_naturals (j - 1) RS sym),
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            rtac trans_fun_cong_image_id_id_apply, etac (morE RS arg_cong), atac],
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   245
          CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_cong}))
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   246
            (fn (i, (set_natural, coalg_set)) =>
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   247
              EVERY' [rtac sym, rtac trans, rtac (refl RSN (2, @{thm UN_cong})),
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   248
                etac (morE RS sym RS arg_cong RS trans), atac, rtac set_natural,
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   249
                rtac (@{thm UN_simps(10)} RS trans), rtac (refl RS @{thm UN_cong}),
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                ftac coalg_set, atac, dtac set_mp, atac, rtac mp, rtac (mk_conjunctN n i),
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   251
                REPEAT_DETERM o etac allE, atac, atac])
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            (rev ((1 upto n) ~~ (active_set_naturals ~~ coalg_sets)))])
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      (rec_Sucs ~~ (morEs ~~ (map (chop m) set_naturalss ~~ map (drop m) coalg_setss)))] 1;
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   254
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fun mk_mor_hset_tac hset_def mor_hset_rec =
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   256
  EVERY' [rtac (hset_def RS trans), rtac (refl RS @{thm UN_cong} RS trans), etac mor_hset_rec,
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   257
    atac, atac, rtac (hset_def RS sym)] 1
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   258
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   259
fun mk_bis_rel_tac m bis_def rel_O_Grs map_comps map_congs set_naturalss =
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  let
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    val n = length rel_O_Grs;
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    val thms = ((1 upto n) ~~ map_comps ~~ map_congs ~~ set_naturalss ~~ rel_O_Grs);
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   263
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   264
    fun mk_if_tac ((((i, map_comp), map_cong), set_naturals), rel_O_Gr) =
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   265
      EVERY' [rtac allI, rtac allI, rtac impI, dtac (mk_conjunctN n i),
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   266
        etac allE, etac allE, etac impE, atac, etac bexE, etac conjE,
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        rtac (rel_O_Gr RS equalityD2 RS set_mp),
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   268
        rtac @{thm relcompI}, rtac @{thm converseI},
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   269
        EVERY' (map (fn thm =>
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          EVERY' [rtac @{thm GrI}, REPEAT_DETERM o eresolve_tac [CollectE, conjE],
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   271
            rtac CollectI,
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   272
            CONJ_WRAP' (fn (i, thm) =>
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   273
              if i <= m
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              then EVERY' [rtac ord_eq_le_trans, rtac thm, rtac subset_trans,
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   275
                etac @{thm image_mono}, rtac @{thm image_subsetI}, etac @{thm diagI}]
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   276
              else EVERY' [rtac ord_eq_le_trans, rtac trans, rtac thm,
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   277
                rtac trans_fun_cong_image_id_id_apply, atac])
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   278
            (1 upto (m + n) ~~ set_naturals),
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   279
            rtac trans, rtac trans, rtac map_comp, rtac map_cong, REPEAT_DETERM_N m o rtac thm,
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   280
            REPEAT_DETERM_N n o rtac (@{thm o_id} RS fun_cong), atac])
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   281
          @{thms fst_diag_id snd_diag_id})];
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   282
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   283
    fun mk_only_if_tac ((((i, map_comp), map_cong), set_naturals), rel_O_Gr) =
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   284
      EVERY' [dtac (mk_conjunctN n i), rtac allI, rtac allI, rtac impI,
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   285
        etac allE, etac allE, etac impE, atac,
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   286
        dtac (rel_O_Gr RS equalityD1 RS set_mp),
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   287
        REPEAT_DETERM o eresolve_tac [CollectE, @{thm relcompE}, @{thm converseE}],
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        REPEAT_DETERM o eresolve_tac [@{thm GrE}, exE, conjE],
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        REPEAT_DETERM o dtac Pair_eqD,
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        REPEAT_DETERM o etac conjE,
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   291
        hyp_subst_tac,
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   292
        REPEAT_DETERM o eresolve_tac [CollectE, conjE],
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   293
        rtac bexI, rtac conjI, rtac trans, rtac map_comp,
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   294
        REPEAT_DETERM_N m o stac @{thm id_o},
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   295
        REPEAT_DETERM_N n o stac @{thm o_id},
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   296
        etac sym, rtac trans, rtac map_comp,
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   297
        REPEAT_DETERM_N m o stac @{thm id_o},
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   298
        REPEAT_DETERM_N n o stac @{thm o_id},
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   299
        rtac trans, rtac map_cong,
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   300
        REPEAT_DETERM_N m o EVERY' [rtac @{thm diagE'}, etac set_mp, atac],
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   301
        REPEAT_DETERM_N n o rtac refl,
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   302
        etac sym, rtac CollectI,
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   303
        CONJ_WRAP' (fn (i, thm) =>
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   304
          if i <= m
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   305
          then EVERY' [rtac ord_eq_le_trans, rtac thm, rtac @{thm image_subsetI},
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   306
            rtac @{thm diag_fst}, etac set_mp, atac]
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   307
          else EVERY' [rtac ord_eq_le_trans, rtac trans, rtac thm,
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   308
            rtac trans_fun_cong_image_id_id_apply, atac])
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   309
        (1 upto (m + n) ~~ set_naturals)];
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   310
  in
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   311
    EVERY' [rtac (bis_def RS trans),
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   312
      rtac iffI, etac conjE, etac conjI, CONJ_WRAP' mk_if_tac thms,
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   313
      etac conjE, etac conjI, CONJ_WRAP' mk_only_if_tac thms] 1
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   314
  end;
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   315
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   316
fun mk_bis_converse_tac m bis_rel rel_congs rel_converses =
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  EVERY' [stac bis_rel, dtac (bis_rel RS iffD1),
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   318
    REPEAT_DETERM o etac conjE, rtac conjI,
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   319
    CONJ_WRAP' (K (EVERY' [rtac @{thm converse_shift}, etac subset_trans,
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   320
      rtac equalityD2, rtac @{thm converse_Times}])) rel_congs,
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   321
    CONJ_WRAP' (fn (rel_cong, rel_converse) =>
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   322
      EVERY' [rtac allI, rtac allI, rtac impI, rtac @{thm set_mp[OF equalityD2]},
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   323
        rtac (rel_cong RS trans),
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   324
        REPEAT_DETERM_N m o rtac @{thm diag_converse},
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   325
        REPEAT_DETERM_N (length rel_congs) o rtac refl,
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   326
        rtac rel_converse,
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   327
        REPEAT_DETERM o etac allE,
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   328
        rtac @{thm converseI}, etac mp, etac @{thm converseD}]) (rel_congs ~~ rel_converses)] 1;
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   329
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   330
fun mk_bis_O_tac m bis_rel rel_congs rel_Os =
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   331
  EVERY' [stac bis_rel, REPEAT_DETERM o dtac (bis_rel RS iffD1),
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   332
    REPEAT_DETERM o etac conjE, rtac conjI,
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   333
    CONJ_WRAP' (K (EVERY' [etac @{thm relcomp_subset_Sigma}, atac])) rel_congs,
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   334
    CONJ_WRAP' (fn (rel_cong, rel_O) =>
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   335
      EVERY' [rtac allI, rtac allI, rtac impI, rtac @{thm set_mp[OF equalityD2]},
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   336
        rtac (rel_cong RS trans),
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   337
        REPEAT_DETERM_N m o rtac @{thm diag_Comp},
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   338
        REPEAT_DETERM_N (length rel_congs) o rtac refl,
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   339
        rtac rel_O,
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   340
        etac @{thm relcompE},
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   341
        REPEAT_DETERM o dtac Pair_eqD,
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   342
        etac conjE, hyp_subst_tac,
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   343
        REPEAT_DETERM o etac allE, rtac @{thm relcompI},
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   344
        etac mp, atac, etac mp, atac]) (rel_congs ~~ rel_Os)] 1;
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   345
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   346
fun mk_bis_Gr_tac bis_rel rel_Grs mor_images morEs coalg_ins
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   347
  {context = ctxt, prems = _} =
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   348
  unfold_thms_tac ctxt (bis_rel :: @{thm diag_Gr} :: rel_Grs) THEN
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   349
  EVERY' [rtac conjI,
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   350
    CONJ_WRAP' (fn thm => rtac (@{thm Gr_incl} RS ssubst) THEN' etac thm) mor_images,
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   351
    CONJ_WRAP' (fn (coalg_in, morE) =>
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   352
      EVERY' [rtac allI, rtac allI, rtac impI, rtac @{thm GrI}, etac coalg_in,
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   353
        etac @{thm GrD1}, etac (morE RS trans), etac @{thm GrD1},
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   354
        etac (@{thm GrD2} RS arg_cong)]) (coalg_ins ~~ morEs)] 1;
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   355
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   356
fun mk_bis_Union_tac bis_def in_monos {context = ctxt, prems = _} =
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   357
  let
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   358
    val n = length in_monos;
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   359
    val ks = 1 upto n;
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   360
  in
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   361
    unfold_thms_tac ctxt [bis_def] THEN
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   362
    EVERY' [rtac conjI,
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   363
      CONJ_WRAP' (fn i =>
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   364
        EVERY' [rtac @{thm UN_least}, dtac bspec, atac,
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   365
          dtac conjunct1, etac (mk_conjunctN n i)]) ks,
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   366
      CONJ_WRAP' (fn (i, in_mono) =>
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   367
        EVERY' [rtac allI, rtac allI, rtac impI, etac @{thm UN_E}, dtac bspec, atac,
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   368
          dtac conjunct2, dtac (mk_conjunctN n i), etac allE, etac allE, dtac mp,
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   369
          atac, etac bexE, rtac bexI, atac, rtac in_mono,
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   370
          REPEAT_DETERM_N n o etac @{thm incl_UNION_I[OF _ subset_refl]},
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   371
          atac]) (ks ~~ in_monos)] 1
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   372
  end;
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   373
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   374
fun mk_sbis_lsbis_tac lsbis_defs bis_Union bis_cong =
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   375
  let
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   376
    val n = length lsbis_defs;
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   377
  in
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   378
    EVERY' [rtac (Thm.permute_prems 0 1 bis_cong), EVERY' (map rtac lsbis_defs),
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   379
      rtac bis_Union, rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, conjE, exE],
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   380
      hyp_subst_tac, etac bis_cong, EVERY' (map (rtac o mk_nth_conv n) (1 upto n))] 1
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   381
  end;
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   382
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   383
fun mk_incl_lsbis_tac n i lsbis_def =
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   384
  EVERY' [rtac @{thm xt1(3)}, rtac lsbis_def, rtac @{thm incl_UNION_I}, rtac CollectI,
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   385
    REPEAT_DETERM_N n o rtac exI, rtac conjI, rtac refl, atac, rtac equalityD2,
blanchet@48975
   386
    rtac (mk_nth_conv n i)] 1;
blanchet@48975
   387
blanchet@48975
   388
fun mk_equiv_lsbis_tac sbis_lsbis lsbis_incl incl_lsbis bis_diag bis_converse bis_O =
traytel@49488
   389
  EVERY' [rtac (@{thm equiv_def} RS iffD2),
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   390
traytel@49488
   391
    rtac conjI, rtac (@{thm refl_on_def} RS iffD2),
blanchet@49305
   392
    rtac conjI, rtac lsbis_incl, rtac ballI, rtac set_mp,
blanchet@48975
   393
    rtac incl_lsbis, rtac bis_diag, atac, etac @{thm diagI},
blanchet@48975
   394
traytel@49488
   395
    rtac conjI, rtac (@{thm sym_def} RS iffD2),
blanchet@49305
   396
    rtac allI, rtac allI, rtac impI, rtac set_mp,
blanchet@48975
   397
    rtac incl_lsbis, rtac bis_converse, rtac sbis_lsbis, etac @{thm converseI},
blanchet@48975
   398
traytel@49488
   399
    rtac (@{thm trans_def} RS iffD2),
blanchet@49305
   400
    rtac allI, rtac allI, rtac allI, rtac impI, rtac impI, rtac set_mp,
blanchet@48975
   401
    rtac incl_lsbis, rtac bis_O, rtac sbis_lsbis, rtac sbis_lsbis,
blanchet@48975
   402
    etac @{thm relcompI}, atac] 1;
blanchet@48975
   403
blanchet@48975
   404
fun mk_coalgT_tac m defs strT_defs set_naturalss {context = ctxt, prems = _} =
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   405
  let
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   406
    val n = length strT_defs;
blanchet@48975
   407
    val ks = 1 upto n;
blanchet@48975
   408
    fun coalg_tac (i, ((passive_sets, active_sets), def)) =
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   409
      EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE],
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   410
        hyp_subst_tac, rtac (def RS trans RS @{thm ssubst_mem}), etac (arg_cong RS trans),
blanchet@48975
   411
        rtac (mk_sum_casesN n i), rtac CollectI,
blanchet@49305
   412
        EVERY' (map (fn thm => EVERY' [rtac conjI, rtac (thm RS ord_eq_le_trans),
blanchet@49305
   413
          etac ((trans OF [@{thm image_id} RS fun_cong, @{thm id_apply}]) RS ord_eq_le_trans)])
blanchet@49305
   414
          passive_sets),
blanchet@49305
   415
        CONJ_WRAP' (fn (i, thm) => EVERY' [rtac (thm RS ord_eq_le_trans),
blanchet@48975
   416
          rtac @{thm image_subsetI}, rtac CollectI, rtac exI, rtac exI, rtac conjI, rtac refl,
blanchet@48975
   417
          rtac conjI,
blanchet@49305
   418
          rtac conjI, etac @{thm empty_Shift}, dtac set_rev_mp,
blanchet@48975
   419
            etac equalityD1, etac CollectD,
blanchet@48975
   420
          rtac conjI, etac @{thm Shift_clists},
blanchet@48975
   421
          rtac conjI, etac @{thm Shift_prefCl},
blanchet@48975
   422
          rtac conjI, rtac ballI,
blanchet@48975
   423
            rtac conjI, dtac @{thm iffD1[OF ball_conj_distrib]}, dtac conjunct1,
blanchet@49504
   424
            SELECT_GOAL (unfold_thms_tac ctxt @{thms Succ_Shift shift_def}),
blanchet@48975
   425
            etac bspec, etac @{thm ShiftD},
blanchet@48975
   426
            CONJ_WRAP' (fn i => EVERY' [rtac ballI, etac CollectE, dtac @{thm ShiftD},
blanchet@48975
   427
              dtac bspec, etac thin_rl, atac, dtac conjunct2, dtac (mk_conjunctN n i),
blanchet@48975
   428
              dtac bspec, rtac CollectI, etac @{thm set_mp[OF equalityD1[OF Succ_Shift]]},
blanchet@48975
   429
              REPEAT_DETERM o eresolve_tac [exE, conjE], rtac exI,
blanchet@48975
   430
              rtac conjI, rtac (@{thm shift_def} RS fun_cong RS trans),
blanchet@48975
   431
              rtac (@{thm append_Cons} RS sym RS arg_cong RS trans), atac,
blanchet@48975
   432
              REPEAT_DETERM_N m o (rtac conjI THEN' atac),
blanchet@48975
   433
              CONJ_WRAP' (K (EVERY' [etac trans, rtac @{thm Collect_cong},
blanchet@48975
   434
                rtac @{thm eqset_imp_iff}, rtac sym, rtac trans, rtac @{thm Succ_Shift},
blanchet@48975
   435
                rtac (@{thm append_Cons} RS sym RS arg_cong)])) ks]) ks,
blanchet@48975
   436
          rtac allI, rtac impI, REPEAT_DETERM o eresolve_tac [allE, impE],
blanchet@48975
   437
          etac @{thm not_in_Shift}, rtac trans, rtac (@{thm shift_def} RS fun_cong), atac,
blanchet@48975
   438
          dtac bspec, atac, dtac conjunct2, dtac (mk_conjunctN n i), dtac bspec,
blanchet@48975
   439
          etac @{thm set_mp[OF equalityD1]}, atac,
blanchet@48975
   440
          REPEAT_DETERM o eresolve_tac [exE, conjE], rtac exI,
blanchet@48975
   441
          rtac conjI, rtac (@{thm shift_def} RS fun_cong RS trans),
blanchet@48975
   442
          etac (@{thm append_Nil} RS sym RS arg_cong RS trans),
blanchet@48975
   443
          REPEAT_DETERM_N m o (rtac conjI THEN' atac),
blanchet@48975
   444
          CONJ_WRAP' (K (EVERY' [etac trans, rtac @{thm Collect_cong},
blanchet@48975
   445
            rtac @{thm eqset_imp_iff}, rtac sym, rtac trans, rtac @{thm Succ_Shift},
blanchet@48975
   446
            rtac (@{thm append_Nil} RS sym RS arg_cong)])) ks]) (ks ~~ active_sets)];
blanchet@48975
   447
  in
blanchet@49504
   448
    unfold_thms_tac ctxt defs THEN
blanchet@48975
   449
    CONJ_WRAP' coalg_tac (ks ~~ (map (chop m) set_naturalss ~~ strT_defs)) 1
blanchet@48975
   450
  end;
blanchet@48975
   451
blanchet@48975
   452
fun mk_card_of_carT_tac m isNode_defs sbd_sbd
blanchet@48975
   453
  sbd_card_order sbd_Card_order sbd_Cinfinite sbd_Cnotzero in_sbds =
blanchet@48975
   454
  let
blanchet@48975
   455
    val n = length isNode_defs;
blanchet@48975
   456
  in
blanchet@48975
   457
    EVERY' [rtac (Thm.permute_prems 0 1 ctrans),
blanchet@48975
   458
      rtac @{thm card_of_Sigma_ordLeq_Cinfinite}, rtac @{thm Cinfinite_cexp},
blanchet@48975
   459
      if m = 0 then rtac @{thm ordLeq_refl} else rtac @{thm ordLeq_csum2},
blanchet@48975
   460
      rtac @{thm Card_order_ctwo}, rtac @{thm Cinfinite_cexp},
blanchet@48975
   461
      rtac @{thm ctwo_ordLeq_Cinfinite}, rtac sbd_Cinfinite, rtac sbd_Cinfinite,
blanchet@48975
   462
      rtac ctrans, rtac @{thm card_of_diff},
blanchet@49305
   463
      rtac ordIso_ordLeq_trans, rtac @{thm card_of_Field_ordIso},
blanchet@49305
   464
      rtac @{thm Card_order_cpow}, rtac ordIso_ordLeq_trans,
blanchet@48975
   465
      rtac @{thm cpow_cexp_ctwo}, rtac ctrans, rtac @{thm cexp_mono1_Cnotzero},
blanchet@48975
   466
      if m = 0 then rtac @{thm ordLeq_refl} else rtac @{thm ordLeq_csum2},
blanchet@48975
   467
      rtac @{thm Card_order_ctwo}, rtac @{thm ctwo_Cnotzero}, rtac @{thm Card_order_clists},
blanchet@49305
   468
      rtac @{thm cexp_mono2_Cnotzero}, rtac ordIso_ordLeq_trans,
blanchet@48975
   469
      rtac @{thm clists_Cinfinite},
blanchet@48975
   470
      if n = 1 then rtac sbd_Cinfinite else rtac (sbd_Cinfinite RS @{thm Cinfinite_csum1}),
blanchet@49305
   471
      rtac ordIso_ordLeq_trans, rtac sbd_sbd, rtac @{thm infinite_ordLeq_cexp},
blanchet@48975
   472
      rtac sbd_Cinfinite,
blanchet@48975
   473
      if m = 0 then rtac @{thm ctwo_Cnotzero} else rtac @{thm csum_Cnotzero2[OF ctwo_Cnotzero]},
blanchet@48975
   474
      rtac @{thm Cnotzero_clists},
blanchet@49305
   475
      rtac ballI, rtac ordIso_ordLeq_trans, rtac @{thm card_of_Func_Ffunc},
blanchet@49305
   476
      rtac ordIso_ordLeq_trans, rtac @{thm Func_cexp},
blanchet@48975
   477
      rtac ctrans, rtac @{thm cexp_mono},
blanchet@48975
   478
      rtac @{thm ordLeq_ordIso_trans},
blanchet@48975
   479
      CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1
blanchet@48975
   480
          (sbd_Cinfinite RS @{thm Cinfinite_cexp[OF ordLeq_csum2[OF Card_order_ctwo]]}
blanchet@48975
   481
        RSN (3, @{thm Un_Cinfinite_bound}))))
blanchet@48975
   482
        (fn thm => EVERY' [rtac ctrans, rtac @{thm card_of_image}, rtac thm]) (rev in_sbds),
blanchet@48975
   483
      rtac @{thm cexp_cong1_Cnotzero}, rtac @{thm csum_cong1},
blanchet@48975
   484
      REPEAT_DETERM_N m o rtac @{thm csum_cong2},
blanchet@48975
   485
      CONJ_WRAP_GEN' (rtac @{thm csum_cong})
blanchet@48975
   486
        (K (rtac (sbd_Card_order RS @{thm card_of_Field_ordIso}))) in_sbds,
blanchet@48975
   487
      rtac sbd_Card_order,
blanchet@48975
   488
      rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero},
blanchet@48975
   489
      rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero},
blanchet@48975
   490
      rtac @{thm ordLeq_ordIso_trans}, etac @{thm clists_bound},
blanchet@48975
   491
      rtac @{thm clists_Cinfinite}, TRY o rtac @{thm Cinfinite_csum1}, rtac sbd_Cinfinite,
blanchet@48975
   492
      rtac disjI2, rtac @{thm cone_ordLeq_cexp}, rtac @{thm cone_ordLeq_cexp},
blanchet@48975
   493
      rtac ctrans, rtac @{thm cone_ordLeq_ctwo}, rtac @{thm ordLeq_csum2},
blanchet@48975
   494
      rtac @{thm Card_order_ctwo}, rtac FalseE, etac @{thm cpow_clists_czero}, atac,
blanchet@48975
   495
      rtac @{thm card_of_Card_order},
blanchet@49305
   496
      rtac ordIso_ordLeq_trans, rtac @{thm cexp_cprod},
blanchet@48975
   497
      rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero},
blanchet@49305
   498
      rtac ordIso_ordLeq_trans, rtac @{thm cexp_cong2_Cnotzero},
blanchet@48975
   499
      rtac @{thm ordIso_transitive}, rtac @{thm cprod_cong2}, rtac sbd_sbd,
blanchet@48975
   500
      rtac @{thm cprod_infinite}, rtac sbd_Cinfinite,
blanchet@48975
   501
      rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero}, rtac @{thm Card_order_cprod},
blanchet@48975
   502
      rtac ctrans, rtac @{thm cexp_mono1_Cnotzero},
blanchet@49305
   503
      rtac ordIso_ordLeq_trans, rtac @{thm ordIso_transitive}, rtac @{thm csum_cong1},
blanchet@48975
   504
      rtac @{thm ordIso_transitive},
blanchet@48975
   505
      REPEAT_DETERM_N m o rtac @{thm csum_cong2},
blanchet@48975
   506
      rtac sbd_sbd,
blanchet@48975
   507
      BNF_Tactics.mk_rotate_eq_tac (rtac @{thm ordIso_refl} THEN'
blanchet@48975
   508
        FIRST' [rtac @{thm card_of_Card_order},
blanchet@48975
   509
          rtac @{thm Card_order_csum}, rtac sbd_Card_order])
blanchet@48975
   510
        @{thm ordIso_transitive} @{thm csum_assoc} @{thm csum_com} @{thm csum_cong}
blanchet@48975
   511
        (1 upto m + 1) (m + 1 :: (1 upto m)),
blanchet@48975
   512
      if m = 0 then K all_tac else EVERY' [rtac @{thm ordIso_transitive}, rtac @{thm csum_assoc}],
blanchet@48975
   513
      rtac @{thm csum_com}, rtac @{thm csum_cexp'}, rtac sbd_Cinfinite,
blanchet@48975
   514
      if m = 0 then rtac @{thm Card_order_ctwo} else rtac @{thm Card_order_csum},
blanchet@48975
   515
      if m = 0 then rtac @{thm ordLeq_refl} else rtac @{thm ordLeq_csum2},
blanchet@48975
   516
      rtac @{thm Card_order_ctwo},
blanchet@48975
   517
      rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero}, rtac sbd_Card_order,
blanchet@49305
   518
      rtac ordIso_ordLeq_trans, rtac @{thm cexp_cprod_ordLeq},
blanchet@48975
   519
      if m = 0 then rtac @{thm ctwo_Cnotzero} else rtac @{thm csum_Cnotzero2[OF ctwo_Cnotzero]},
blanchet@48975
   520
      rtac sbd_Cinfinite, rtac sbd_Cnotzero, rtac @{thm ordLeq_refl}, rtac sbd_Card_order,
blanchet@48975
   521
      rtac @{thm cexp_mono2_Cnotzero}, rtac @{thm infinite_ordLeq_cexp},
blanchet@48975
   522
      rtac sbd_Cinfinite,
blanchet@48975
   523
      if m = 0 then rtac @{thm ctwo_Cnotzero} else rtac @{thm csum_Cnotzero2[OF ctwo_Cnotzero]},
blanchet@48975
   524
      rtac sbd_Cnotzero,
blanchet@48975
   525
      rtac @{thm card_of_mono1}, rtac subsetI,
blanchet@48975
   526
      REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE, @{thm prod_caseE}], hyp_subst_tac,
blanchet@49305
   527
      rtac @{thm SigmaI}, rtac @{thm DiffI}, rtac set_mp, rtac equalityD2,
blanchet@48975
   528
      rtac (@{thm cpow_def} RS arg_cong RS trans), rtac (@{thm Pow_def} RS arg_cong RS trans),
blanchet@48975
   529
      rtac @{thm Field_card_of}, rtac CollectI, atac, rtac notI, etac @{thm singletonE},
blanchet@49305
   530
      hyp_subst_tac, etac @{thm emptyE}, rtac (@{thm Ffunc_def} RS equalityD2 RS set_mp),
blanchet@48975
   531
      rtac CollectI, rtac conjI, rtac ballI, dtac bspec, etac thin_rl, atac, dtac conjunct1,
blanchet@48975
   532
      CONJ_WRAP_GEN' (etac disjE) (fn (i, def) => EVERY'
traytel@49488
   533
        [rtac (mk_UnIN n i), dtac (def RS iffD1),
blanchet@48975
   534
        REPEAT_DETERM o eresolve_tac [exE, conjE], rtac @{thm image_eqI}, atac, rtac CollectI,
blanchet@48975
   535
        REPEAT_DETERM_N m o (rtac conjI THEN' atac),
blanchet@49305
   536
        CONJ_WRAP' (K (EVERY' [etac ord_eq_le_trans, rtac subset_trans,
traytel@49490
   537
          rtac subset_UNIV, rtac equalityD2, rtac @{thm Field_card_order},
blanchet@48975
   538
          rtac sbd_card_order])) isNode_defs]) (1 upto n ~~ isNode_defs),
blanchet@48975
   539
      atac] 1
blanchet@48975
   540
  end;
blanchet@48975
   541
blanchet@48975
   542
fun mk_carT_set_tac n i carT_def strT_def isNode_def set_natural {context = ctxt, prems = _}=
blanchet@49305
   543
  EVERY' [dtac (carT_def RS equalityD1 RS set_mp),
blanchet@48975
   544
    REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE],
traytel@49488
   545
    dtac Pair_eqD,
blanchet@48975
   546
    etac conjE, hyp_subst_tac,
traytel@49488
   547
    dtac (isNode_def RS iffD1),
blanchet@48975
   548
    REPEAT_DETERM o eresolve_tac [exE, conjE],
blanchet@49305
   549
    rtac (equalityD2 RS set_mp),
blanchet@48975
   550
    rtac (strT_def RS arg_cong RS trans),
blanchet@48975
   551
    etac (arg_cong RS trans),
blanchet@48975
   552
    fo_rtac (mk_sum_casesN n i RS arg_cong RS trans) ctxt,
blanchet@49305
   553
    rtac set_natural, rtac imageI, etac (equalityD2 RS set_mp), rtac CollectI,
blanchet@48975
   554
    etac @{thm prefCl_Succ}, atac] 1;
blanchet@48975
   555
blanchet@48975
   556
fun mk_strT_hset_tac n m j arg_cong_cTs cTs cts carT_defs strT_defs isNode_defs
blanchet@48975
   557
  set_incl_hsets set_hset_incl_hsetss coalg_setss carT_setss coalgT set_naturalss =
blanchet@48975
   558
  let
blanchet@48975
   559
    val set_naturals = map (fn xs => nth xs (j - 1)) set_naturalss;
blanchet@48975
   560
    val ks = 1 upto n;
blanchet@48975
   561
    fun base_tac (i, (cT, (strT_def, (set_incl_hset, set_natural)))) =
blanchet@48975
   562
      CONJ_WRAP' (fn (i', (carT_def, isNode_def)) => rtac impI THEN' etac conjE THEN'
blanchet@48975
   563
        (if i = i'
blanchet@48975
   564
        then EVERY' [rtac @{thm xt1(4)}, rtac set_incl_hset,
blanchet@48975
   565
          rtac (strT_def RS arg_cong RS trans), etac (arg_cong RS trans),
blanchet@48975
   566
          rtac (Thm.permute_prems 0 1 (set_natural RS box_equals)),
blanchet@48975
   567
          rtac (trans OF [@{thm image_id} RS fun_cong, @{thm id_apply}]),
blanchet@48975
   568
          rtac (mk_sum_casesN n i RS (Drule.instantiate' [cT] [] arg_cong) RS sym)]
blanchet@49305
   569
        else EVERY' [dtac (carT_def RS equalityD1 RS set_mp),
blanchet@48975
   570
          REPEAT_DETERM o eresolve_tac [CollectE, exE], etac conjE,
traytel@49488
   571
          dtac conjunct2, dtac Pair_eqD, etac conjE,
traytel@49488
   572
          hyp_subst_tac, dtac (isNode_def RS iffD1),
blanchet@48975
   573
          REPEAT_DETERM o eresolve_tac [exE, conjE],
blanchet@48975
   574
          rtac (mk_InN_not_InM i i' RS notE), etac (sym RS trans), atac]))
blanchet@48975
   575
      (ks ~~ (carT_defs ~~ isNode_defs));
blanchet@48975
   576
    fun step_tac (i, (coalg_sets, (carT_sets, set_hset_incl_hsets))) =
blanchet@48975
   577
      dtac (mk_conjunctN n i) THEN'
blanchet@48975
   578
      CONJ_WRAP' (fn (coalg_set, (carT_set, set_hset_incl_hset)) =>
blanchet@48975
   579
        EVERY' [rtac impI, etac conjE, etac impE, rtac conjI,
blanchet@49305
   580
          rtac (coalgT RS coalg_set RS set_mp), atac, etac carT_set, atac, atac,
blanchet@48975
   581
          etac (@{thm shift_def} RS fun_cong RS trans), etac subset_trans,
blanchet@48975
   582
          rtac set_hset_incl_hset, etac carT_set, atac, atac])
blanchet@48975
   583
      (coalg_sets ~~ (carT_sets ~~ set_hset_incl_hsets));
blanchet@48975
   584
  in
blanchet@48975
   585
    EVERY' [rtac (Drule.instantiate' cTs cts @{thm list.induct}),
blanchet@48975
   586
      REPEAT_DETERM o rtac allI, rtac impI,
blanchet@48975
   587
      CONJ_WRAP' base_tac
blanchet@48975
   588
        (ks ~~ (arg_cong_cTs ~~ (strT_defs ~~ (set_incl_hsets ~~ set_naturals)))),
blanchet@48975
   589
      REPEAT_DETERM o rtac allI, rtac impI,
blanchet@48975
   590
      REPEAT_DETERM o eresolve_tac [allE, impE], etac @{thm ShiftI},
blanchet@48975
   591
      CONJ_WRAP' (fn i => dtac (mk_conjunctN n i) THEN' rtac (mk_sumEN n) THEN'
blanchet@48975
   592
        CONJ_WRAP_GEN' (K all_tac) step_tac
blanchet@48975
   593
          (ks ~~ (drop m coalg_setss ~~ (carT_setss ~~ set_hset_incl_hsetss)))) ks] 1
blanchet@48975
   594
  end;
blanchet@48975
   595
blanchet@48975
   596
fun mk_isNode_hset_tac n isNode_def strT_hsets {context = ctxt, prems = _} =
blanchet@48975
   597
  let
blanchet@48975
   598
    val m = length strT_hsets;
blanchet@48975
   599
  in
blanchet@48975
   600
    if m = 0 then atac 1
blanchet@49504
   601
    else (unfold_thms_tac ctxt [isNode_def] THEN
blanchet@48975
   602
      EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE],
blanchet@48975
   603
        rtac exI, rtac conjI, atac,
blanchet@48975
   604
        CONJ_WRAP' (fn (thm, i) =>  if i > m then atac
blanchet@48975
   605
          else EVERY' [rtac (thm RS subset_trans), atac, rtac conjI, atac, atac, atac])
blanchet@48975
   606
        (strT_hsets @ (replicate n mp) ~~ (1 upto (m + n)))] 1)
blanchet@48975
   607
  end;
blanchet@48975
   608
blanchet@48975
   609
fun mk_Lev_sbd_tac cts Lev_0s Lev_Sucs to_sbdss =
blanchet@48975
   610
  let
blanchet@48975
   611
    val n = length Lev_0s;
blanchet@48975
   612
    val ks = 1 upto n;
blanchet@48975
   613
  in
blanchet@49305
   614
    EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
blanchet@48975
   615
      REPEAT_DETERM o rtac allI,
blanchet@48975
   616
      CONJ_WRAP' (fn Lev_0 =>
blanchet@49305
   617
        EVERY' (map rtac [ord_eq_le_trans, Lev_0, @{thm Nil_clists}])) Lev_0s,
blanchet@48975
   618
      REPEAT_DETERM o rtac allI,
blanchet@48975
   619
      CONJ_WRAP' (fn (Lev_Suc, to_sbds) =>
blanchet@49305
   620
        EVERY' [rtac ord_eq_le_trans, rtac Lev_Suc,
blanchet@48975
   621
          CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least}))
blanchet@48975
   622
            (fn (i, to_sbd) => EVERY' [rtac subsetI,
blanchet@48975
   623
              REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
blanchet@48975
   624
              rtac @{thm Cons_clists}, rtac (mk_InN_Field n i), etac to_sbd,
blanchet@49305
   625
              etac set_rev_mp, REPEAT_DETERM o etac allE,
blanchet@48975
   626
              etac (mk_conjunctN n i)])
blanchet@48975
   627
          (rev (ks ~~ to_sbds))])
blanchet@48975
   628
      (Lev_Sucs ~~ to_sbdss)] 1
blanchet@48975
   629
  end;
blanchet@48975
   630
blanchet@48975
   631
fun mk_length_Lev_tac cts Lev_0s Lev_Sucs =
blanchet@48975
   632
  let
blanchet@48975
   633
    val n = length Lev_0s;
blanchet@48975
   634
    val ks = n downto 1;
blanchet@48975
   635
  in
blanchet@49305
   636
    EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
blanchet@48975
   637
      REPEAT_DETERM o rtac allI,
blanchet@48975
   638
      CONJ_WRAP' (fn Lev_0 =>
blanchet@49305
   639
        EVERY' [rtac impI, dtac (Lev_0 RS equalityD1 RS set_mp),
blanchet@48975
   640
          etac @{thm singletonE}, etac ssubst, rtac @{thm list.size(3)}]) Lev_0s,
blanchet@48975
   641
      REPEAT_DETERM o rtac allI,
blanchet@48975
   642
      CONJ_WRAP' (fn Lev_Suc =>
blanchet@49305
   643
        EVERY' [rtac impI, dtac (Lev_Suc RS equalityD1 RS set_mp),
blanchet@48975
   644
          CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
blanchet@48975
   645
            (fn i => EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
blanchet@48975
   646
              rtac trans, rtac @{thm length_Cons}, rtac @{thm arg_cong[of _ _ Suc]},
blanchet@48975
   647
              REPEAT_DETERM o etac allE, dtac (mk_conjunctN n i), etac mp, atac]) ks])
blanchet@48975
   648
      Lev_Sucs] 1
blanchet@48975
   649
  end;
blanchet@48975
   650
blanchet@48975
   651
fun mk_length_Lev'_tac length_Lev =
blanchet@48975
   652
  EVERY' [ftac length_Lev, etac ssubst, atac] 1;
blanchet@48975
   653
blanchet@48975
   654
fun mk_prefCl_Lev_tac cts Lev_0s Lev_Sucs =
blanchet@48975
   655
  let
blanchet@48975
   656
    val n = length Lev_0s;
blanchet@48975
   657
    val ks = n downto 1;
blanchet@48975
   658
  in
blanchet@49305
   659
    EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
blanchet@48975
   660
      REPEAT_DETERM o rtac allI,
blanchet@48975
   661
      CONJ_WRAP' (fn Lev_0 =>
blanchet@49305
   662
        EVERY' [rtac impI, etac conjE, dtac (Lev_0 RS equalityD1 RS set_mp),
Christian@49091
   663
          etac @{thm singletonE}, hyp_subst_tac, dtac @{thm prefix_Nil[THEN subst, of "%x. x"]},
blanchet@48975
   664
          hyp_subst_tac, rtac @{thm set_mp[OF equalityD2[OF trans[OF arg_cong[OF list.size(3)]]]]},
blanchet@48975
   665
          rtac Lev_0, rtac @{thm singletonI}]) Lev_0s,
blanchet@48975
   666
      REPEAT_DETERM o rtac allI,
blanchet@48975
   667
      CONJ_WRAP' (fn (Lev_0, Lev_Suc) =>
blanchet@49305
   668
        EVERY' [rtac impI, etac conjE, dtac (Lev_Suc RS equalityD1 RS set_mp),
blanchet@48975
   669
          CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
blanchet@48975
   670
            (fn i => EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
Christian@49091
   671
              dtac @{thm prefix_Cons[THEN subst, of "%x. x"]}, etac disjE, hyp_subst_tac,
blanchet@48975
   672
              rtac @{thm set_mp[OF equalityD2[OF trans[OF arg_cong[OF list.size(3)]]]]},
blanchet@48975
   673
              rtac Lev_0, rtac @{thm singletonI},
blanchet@48975
   674
              REPEAT_DETERM o eresolve_tac [exE, conjE], hyp_subst_tac,
blanchet@48975
   675
              rtac @{thm set_mp[OF equalityD2[OF trans[OF arg_cong[OF length_Cons]]]]},
blanchet@49366
   676
              rtac Lev_Suc, rtac (mk_UnIN n i), rtac CollectI, REPEAT_DETERM o rtac exI, rtac conjI,
blanchet@48975
   677
              rtac refl, etac conjI, REPEAT_DETERM o etac allE, dtac (mk_conjunctN n i),
blanchet@48975
   678
              etac mp, etac conjI, atac]) ks])
blanchet@48975
   679
      (Lev_0s ~~ Lev_Sucs)] 1
blanchet@48975
   680
  end;
blanchet@48975
   681
blanchet@48975
   682
fun mk_rv_last_tac cTs cts rv_Nils rv_Conss =
blanchet@48975
   683
  let
blanchet@48975
   684
    val n = length rv_Nils;
blanchet@48975
   685
    val ks = 1 upto n;
blanchet@48975
   686
  in
blanchet@48975
   687
    EVERY' [rtac (Drule.instantiate' cTs cts @{thm list.induct}),
blanchet@48975
   688
      REPEAT_DETERM o rtac allI,
blanchet@48975
   689
      CONJ_WRAP' (fn rv_Cons =>
blanchet@48975
   690
        CONJ_WRAP' (fn (i, rv_Nil) => (EVERY' [rtac exI,
blanchet@48975
   691
          rtac (@{thm append_Nil} RS arg_cong RS trans),
blanchet@48975
   692
          rtac (rv_Cons RS trans), rtac (mk_sum_casesN n i RS arg_cong RS trans), rtac rv_Nil]))
blanchet@48975
   693
        (ks ~~ rv_Nils))
blanchet@48975
   694
      rv_Conss,
blanchet@48975
   695
      REPEAT_DETERM o rtac allI, rtac (mk_sumEN n),
blanchet@48975
   696
      EVERY' (map (fn i =>
blanchet@48975
   697
        CONJ_WRAP' (fn rv_Cons => EVERY' [REPEAT_DETERM o etac allE, dtac (mk_conjunctN n i),
blanchet@48975
   698
          CONJ_WRAP' (fn i' => EVERY' [dtac (mk_conjunctN n i'), etac exE, rtac exI,
blanchet@48975
   699
            rtac (@{thm append_Cons} RS arg_cong RS trans),
blanchet@49328
   700
            rtac (rv_Cons RS trans), etac (sum_case_weak_cong RS arg_cong RS trans),
blanchet@48975
   701
            rtac (mk_sum_casesN n i RS arg_cong RS trans), atac])
blanchet@48975
   702
          ks])
blanchet@48975
   703
        rv_Conss)
blanchet@48975
   704
      ks)] 1
blanchet@48975
   705
  end;
blanchet@48975
   706
blanchet@48975
   707
fun mk_set_rv_Lev_tac m cts Lev_0s Lev_Sucs rv_Nils rv_Conss coalg_setss from_to_sbdss =
blanchet@48975
   708
  let
blanchet@48975
   709
    val n = length Lev_0s;
blanchet@48975
   710
    val ks = 1 upto n;
blanchet@48975
   711
  in
blanchet@49305
   712
    EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
blanchet@48975
   713
      REPEAT_DETERM o rtac allI,
blanchet@48975
   714
      CONJ_WRAP' (fn (i, ((Lev_0, rv_Nil), coalg_sets)) =>
blanchet@48975
   715
        EVERY' [rtac impI, REPEAT_DETERM o etac conjE,
blanchet@49305
   716
          dtac (Lev_0 RS equalityD1 RS set_mp), etac @{thm singletonE}, etac ssubst,
traytel@49488
   717
          rtac (rv_Nil RS arg_cong RS iffD2),
traytel@49488
   718
          rtac (mk_sum_casesN n i RS iffD2),
blanchet@48975
   719
          CONJ_WRAP' (fn thm => etac thm THEN' atac) (take m coalg_sets)])
blanchet@48975
   720
      (ks ~~ ((Lev_0s ~~ rv_Nils) ~~ coalg_setss)),
blanchet@48975
   721
      REPEAT_DETERM o rtac allI,
blanchet@48975
   722
      CONJ_WRAP' (fn ((Lev_Suc, rv_Cons), (from_to_sbds, coalg_sets)) =>
blanchet@49305
   723
        EVERY' [rtac impI, etac conjE, dtac (Lev_Suc RS equalityD1 RS set_mp),
blanchet@48975
   724
          CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
blanchet@48975
   725
            (fn (i, (from_to_sbd, coalg_set)) =>
blanchet@48975
   726
              EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
traytel@49488
   727
              rtac (rv_Cons RS arg_cong RS iffD2),
traytel@49488
   728
              rtac (mk_sum_casesN n i RS arg_cong RS trans RS iffD2),
blanchet@48975
   729
              etac (from_to_sbd RS arg_cong), REPEAT_DETERM o etac allE,
blanchet@49305
   730
              dtac (mk_conjunctN n i), etac mp, etac conjI, etac set_rev_mp,
blanchet@48975
   731
              etac coalg_set, atac])
blanchet@48975
   732
          (rev (ks ~~ (from_to_sbds ~~ drop m coalg_sets)))])
blanchet@48975
   733
      ((Lev_Sucs ~~ rv_Conss) ~~ (from_to_sbdss ~~ coalg_setss))] 1
blanchet@48975
   734
  end;
blanchet@48975
   735
blanchet@48975
   736
fun mk_set_Lev_tac cts Lev_0s Lev_Sucs rv_Nils rv_Conss from_to_sbdss =
blanchet@48975
   737
  let
blanchet@48975
   738
    val n = length Lev_0s;
blanchet@48975
   739
    val ks = 1 upto n;
blanchet@48975
   740
  in
blanchet@49305
   741
    EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
blanchet@48975
   742
      REPEAT_DETERM o rtac allI,
blanchet@48975
   743
      CONJ_WRAP' (fn ((i, (Lev_0, Lev_Suc)), rv_Nil) =>
blanchet@49305
   744
        EVERY' [rtac impI, dtac (Lev_0 RS equalityD1 RS set_mp),
blanchet@48975
   745
          etac @{thm singletonE}, hyp_subst_tac,
blanchet@48975
   746
          CONJ_WRAP' (fn i' => rtac impI THEN' dtac (sym RS trans) THEN' rtac rv_Nil THEN'
blanchet@48975
   747
            (if i = i'
blanchet@48975
   748
            then EVERY' [dtac (mk_InN_inject n i), hyp_subst_tac,
blanchet@48975
   749
              CONJ_WRAP' (fn (i'', Lev_0'') =>
blanchet@48975
   750
                EVERY' [rtac impI, rtac @{thm ssubst_mem[OF append_Nil]},
blanchet@49366
   751
                  rtac (Lev_Suc RS equalityD2 RS set_mp), rtac (mk_UnIN n i''),
blanchet@48975
   752
                  rtac CollectI, REPEAT_DETERM o rtac exI, rtac conjI, rtac refl,
blanchet@49305
   753
                  etac conjI, rtac (Lev_0'' RS equalityD2 RS set_mp),
blanchet@48975
   754
                  rtac @{thm singletonI}])
blanchet@48975
   755
              (ks ~~ Lev_0s)]
blanchet@48975
   756
            else etac (mk_InN_not_InM i' i RS notE)))
blanchet@48975
   757
          ks])
blanchet@48975
   758
      ((ks ~~ (Lev_0s ~~ Lev_Sucs)) ~~ rv_Nils),
blanchet@48975
   759
      REPEAT_DETERM o rtac allI,
blanchet@48975
   760
      CONJ_WRAP' (fn ((Lev_Suc, rv_Cons), from_to_sbds) =>
blanchet@49305
   761
        EVERY' [rtac impI, dtac (Lev_Suc RS equalityD1 RS set_mp),
blanchet@48975
   762
          CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
blanchet@48975
   763
            (fn (i, from_to_sbd) =>
blanchet@48975
   764
              EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
blanchet@48975
   765
                CONJ_WRAP' (fn i' => rtac impI THEN'
blanchet@48975
   766
                  CONJ_WRAP' (fn i'' =>
blanchet@49305
   767
                    EVERY' [rtac impI, rtac (Lev_Suc RS equalityD2 RS set_mp),
blanchet@49366
   768
                      rtac @{thm ssubst_mem[OF append_Cons]}, rtac (mk_UnIN n i),
blanchet@48975
   769
                      rtac CollectI, REPEAT_DETERM o rtac exI, rtac conjI, rtac refl,
blanchet@48975
   770
                      rtac conjI, atac, dtac (sym RS trans RS sym),
blanchet@48975
   771
                      rtac (rv_Cons RS trans), rtac (mk_sum_casesN n i RS trans),
blanchet@48975
   772
                      etac (from_to_sbd RS arg_cong), REPEAT_DETERM o etac allE,
blanchet@48975
   773
                      dtac (mk_conjunctN n i), dtac mp, atac,
blanchet@48975
   774
                      dtac (mk_conjunctN n i'), dtac mp, atac,
blanchet@48975
   775
                      dtac (mk_conjunctN n i''), etac mp, atac])
blanchet@48975
   776
                  ks)
blanchet@48975
   777
                ks])
blanchet@48975
   778
          (rev (ks ~~ from_to_sbds))])
blanchet@48975
   779
      ((Lev_Sucs ~~ rv_Conss) ~~ from_to_sbdss)] 1
blanchet@48975
   780
  end;
blanchet@48975
   781
blanchet@48975
   782
fun mk_set_image_Lev_tac cts Lev_0s Lev_Sucs rv_Nils rv_Conss from_to_sbdss to_sbd_injss =
blanchet@48975
   783
  let
blanchet@48975
   784
    val n = length Lev_0s;
blanchet@48975
   785
    val ks = 1 upto n;
blanchet@48975
   786
  in
blanchet@49305
   787
    EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
blanchet@48975
   788
      REPEAT_DETERM o rtac allI,
blanchet@48975
   789
      CONJ_WRAP' (fn ((i, (Lev_0, Lev_Suc)), rv_Nil) =>
blanchet@49305
   790
        EVERY' [rtac impI, dtac (Lev_0 RS equalityD1 RS set_mp),
blanchet@48975
   791
          etac @{thm singletonE}, hyp_subst_tac,
blanchet@48975
   792
          CONJ_WRAP' (fn i' => rtac impI THEN'
blanchet@48975
   793
            CONJ_WRAP' (fn i'' => rtac impI  THEN' dtac (sym RS trans) THEN' rtac rv_Nil THEN'
blanchet@48975
   794
              (if i = i''
blanchet@48975
   795
              then EVERY' [dtac @{thm ssubst_mem[OF sym[OF append_Nil]]},
blanchet@49305
   796
                dtac (Lev_Suc RS equalityD1 RS set_mp), dtac (mk_InN_inject n i),
blanchet@48975
   797
                hyp_subst_tac,
blanchet@48975
   798
                CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
blanchet@48975
   799
                  (fn k => REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE] THEN'
traytel@49488
   800
                    dtac list_inject_iffD1 THEN' etac conjE THEN'
blanchet@48975
   801
                    (if k = i'
blanchet@48975
   802
                    then EVERY' [dtac (mk_InN_inject n k), hyp_subst_tac, etac imageI]
blanchet@48975
   803
                    else etac (mk_InN_not_InM i' k RS notE)))
blanchet@48975
   804
                (rev ks)]
blanchet@48975
   805
              else etac (mk_InN_not_InM i'' i RS notE)))
blanchet@48975
   806
            ks)
blanchet@48975
   807
          ks])
blanchet@48975
   808
      ((ks ~~ (Lev_0s ~~ Lev_Sucs)) ~~ rv_Nils),
blanchet@48975
   809
      REPEAT_DETERM o rtac allI,
blanchet@48975
   810
      CONJ_WRAP' (fn ((Lev_Suc, rv_Cons), (from_to_sbds, to_sbd_injs)) =>
blanchet@49305
   811
        EVERY' [rtac impI, dtac (Lev_Suc RS equalityD1 RS set_mp),
blanchet@48975
   812
          CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE))
blanchet@48975
   813
            (fn (i, (from_to_sbd, to_sbd_inj)) =>
blanchet@48975
   814
              REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE] THEN' hyp_subst_tac THEN'
blanchet@48975
   815
              CONJ_WRAP' (fn i' => rtac impI THEN'
blanchet@48975
   816
                dtac @{thm ssubst_mem[OF sym[OF append_Cons]]} THEN'
blanchet@49305
   817
                dtac (Lev_Suc RS equalityD1 RS set_mp) THEN'
blanchet@48975
   818
                CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE)) (fn k =>
blanchet@48975
   819
                  REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE] THEN'
traytel@49488
   820
                  dtac list_inject_iffD1 THEN' etac conjE THEN'
blanchet@48975
   821
                  (if k = i
blanchet@48975
   822
                  then EVERY' [dtac (mk_InN_inject n i),
traytel@49488
   823
                    dtac (Thm.permute_prems 0 2 (to_sbd_inj RS iffD1)),
blanchet@48975
   824
                    atac, atac, hyp_subst_tac] THEN'
blanchet@48975
   825
                    CONJ_WRAP' (fn i'' =>
blanchet@48975
   826
                      EVERY' [rtac impI, dtac (sym RS trans),
blanchet@48975
   827
                        rtac (rv_Cons RS trans), rtac (mk_sum_casesN n i RS arg_cong RS trans),
blanchet@48975
   828
                        etac (from_to_sbd RS arg_cong),
blanchet@48975
   829
                        REPEAT_DETERM o etac allE,
blanchet@48975
   830
                        dtac (mk_conjunctN n i), dtac mp, atac,
blanchet@48975
   831
                        dtac (mk_conjunctN n i'), dtac mp, atac,
blanchet@48975
   832
                        dtac (mk_conjunctN n i''), etac mp, etac sym])
blanchet@48975
   833
                    ks
blanchet@48975
   834
                  else etac (mk_InN_not_InM i k RS notE)))
blanchet@48975
   835
                (rev ks))
blanchet@48975
   836
              ks)
blanchet@48975
   837
          (rev (ks ~~ (from_to_sbds ~~ to_sbd_injs)))])
blanchet@48975
   838
      ((Lev_Sucs ~~ rv_Conss) ~~ (from_to_sbdss ~~ to_sbd_injss))] 1
blanchet@48975
   839
  end;
blanchet@48975
   840
blanchet@48975
   841
fun mk_mor_beh_tac m mor_def mor_cong beh_defs carT_defs strT_defs isNode_defs
blanchet@48975
   842
  to_sbd_injss from_to_sbdss Lev_0s Lev_Sucs rv_Nils rv_Conss Lev_sbds length_Levs length_Lev's
blanchet@48975
   843
  prefCl_Levs rv_lastss set_rv_Levsss set_Levsss set_image_Levsss set_naturalss coalg_setss
blanchet@48975
   844
  map_comp_ids map_congs map_arg_congs {context = ctxt, prems = _} =
blanchet@48975
   845
  let
blanchet@48975
   846
    val n = length beh_defs;
blanchet@48975
   847
    val ks = 1 upto n;
blanchet@48975
   848
blanchet@48975
   849
    fun fbetw_tac (i, (carT_def, (isNode_def, (Lev_0, (rv_Nil, (Lev_sbd,
blanchet@48975
   850
      ((length_Lev, length_Lev'), (prefCl_Lev, (rv_lasts, (set_naturals,
blanchet@48975
   851
        (coalg_sets, (set_rv_Levss, (set_Levss, set_image_Levss))))))))))))) =
blanchet@49305
   852
      EVERY' [rtac ballI, rtac (carT_def RS equalityD2 RS set_mp),
blanchet@48975
   853
        rtac CollectI, REPEAT_DETERM o rtac exI, rtac conjI, rtac refl, rtac conjI,
blanchet@48975
   854
        rtac conjI,
blanchet@49305
   855
          rtac @{thm UN_I}, rtac UNIV_I, rtac (Lev_0 RS equalityD2 RS set_mp),
blanchet@48975
   856
          rtac @{thm singletonI},
blanchet@48975
   857
        rtac conjI,
blanchet@48975
   858
          rtac @{thm UN_least}, rtac Lev_sbd,
blanchet@48975
   859
        rtac conjI,
blanchet@48975
   860
          rtac @{thm prefCl_UN}, rtac ssubst, rtac @{thm PrefCl_def}, REPEAT_DETERM o rtac allI,
blanchet@48975
   861
          rtac impI, etac conjE, rtac exI, rtac conjI, rtac @{thm ord_le_eq_trans},
Christian@49091
   862
          etac @{thm prefix_length_le}, etac length_Lev, rtac prefCl_Lev, etac conjI, atac,
blanchet@48975
   863
        rtac conjI,
blanchet@48975
   864
          rtac ballI, etac @{thm UN_E}, rtac conjI,
blanchet@48975
   865
          if n = 1 then K all_tac else rtac (mk_sumEN n),
blanchet@48975
   866
          EVERY' (map6 (fn i => fn isNode_def => fn set_naturals =>
blanchet@48975
   867
            fn set_rv_Levs => fn set_Levs => fn set_image_Levs =>
blanchet@48975
   868
            EVERY' [rtac (mk_disjIN n i), rtac (isNode_def RS ssubst),
blanchet@48975
   869
              rtac exI, rtac conjI,
blanchet@48975
   870
              (if n = 1 then rtac @{thm if_P} THEN' etac length_Lev'
blanchet@48975
   871
              else rtac (@{thm if_P} RS arg_cong RS trans) THEN' etac length_Lev' THEN'
blanchet@49328
   872
                etac (sum_case_weak_cong RS trans) THEN' rtac (mk_sum_casesN n i)),
blanchet@48975
   873
              EVERY' (map2 (fn set_natural => fn set_rv_Lev =>
blanchet@49305
   874
                EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac (set_natural RS trans),
blanchet@49305
   875
                  rtac trans_fun_cong_image_id_id_apply,
blanchet@48975
   876
                  etac set_rv_Lev, TRY o atac, etac conjI, atac])
blanchet@48975
   877
              (take m set_naturals) set_rv_Levs),
blanchet@48975
   878
              CONJ_WRAP' (fn (set_natural, (set_Lev, set_image_Lev)) =>
blanchet@48975
   879
                EVERY' [rtac (set_natural RS trans), rtac equalityI, rtac @{thm image_subsetI},
blanchet@48975
   880
                  rtac CollectI, rtac @{thm SuccI}, rtac @{thm UN_I}, rtac UNIV_I, etac set_Lev,
blanchet@48975
   881
                  if n = 1 then rtac refl else atac, atac, rtac subsetI,
blanchet@48975
   882
                  REPEAT_DETERM o eresolve_tac [CollectE, @{thm SuccE}, @{thm UN_E}],
blanchet@48975
   883
                  rtac set_image_Lev, atac, dtac length_Lev, hyp_subst_tac, dtac length_Lev',
blanchet@48975
   884
                  etac @{thm set_mp[OF equalityD1[OF arg_cong[OF length_append_singleton]]]},
blanchet@48975
   885
                  if n = 1 then rtac refl else atac])
blanchet@48975
   886
              (drop m set_naturals ~~ (set_Levs ~~ set_image_Levs))])
blanchet@48975
   887
          ks isNode_defs set_naturalss set_rv_Levss set_Levss set_image_Levss),
blanchet@48975
   888
          CONJ_WRAP' (fn (i, (rv_last, (isNode_def, (set_naturals,
blanchet@48975
   889
            (set_rv_Levs, (set_Levs, set_image_Levs)))))) =>
blanchet@48975
   890
            EVERY' [rtac ballI,
blanchet@48975
   891
              REPEAT_DETERM o eresolve_tac [CollectE, @{thm SuccE}, @{thm UN_E}],
blanchet@48975
   892
              rtac (rev_mp OF [rv_last, impI]), etac exE, rtac (isNode_def RS ssubst),
blanchet@48975
   893
              rtac exI, rtac conjI,
blanchet@48975
   894
              (if n = 1 then rtac @{thm if_P} THEN' etac length_Lev'
blanchet@48975
   895
              else rtac (@{thm if_P} RS trans) THEN' etac length_Lev' THEN'
blanchet@49328
   896
                etac (sum_case_weak_cong RS trans) THEN' rtac (mk_sum_casesN n i)),
blanchet@48975
   897
              EVERY' (map2 (fn set_natural => fn set_rv_Lev =>
blanchet@49305
   898
                EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac (set_natural RS trans),
blanchet@49305
   899
                  rtac trans_fun_cong_image_id_id_apply,
blanchet@48975
   900
                  etac set_rv_Lev, TRY o atac, etac conjI, atac])
blanchet@48975
   901
              (take m set_naturals) set_rv_Levs),
blanchet@48975
   902
              CONJ_WRAP' (fn (set_natural, (set_Lev, set_image_Lev)) =>
blanchet@48975
   903
                EVERY' [rtac (set_natural RS trans), rtac equalityI, rtac @{thm image_subsetI},
blanchet@48975
   904
                  rtac CollectI, rtac @{thm SuccI}, rtac @{thm UN_I}, rtac UNIV_I, etac set_Lev,
blanchet@48975
   905
                  if n = 1 then rtac refl else atac, atac, rtac subsetI,
blanchet@48975
   906
                  REPEAT_DETERM o eresolve_tac [CollectE, @{thm SuccE}, @{thm UN_E}],
blanchet@48975
   907
                  REPEAT_DETERM_N 4 o etac thin_rl,
blanchet@48975
   908
                  rtac set_image_Lev,
blanchet@48975
   909
                  atac, dtac length_Lev, hyp_subst_tac, dtac length_Lev',
blanchet@48975
   910
                  etac @{thm set_mp[OF equalityD1[OF arg_cong[OF length_append_singleton]]]},
blanchet@48975
   911
                  if n = 1 then rtac refl else atac])
blanchet@48975
   912
              (drop m set_naturals ~~ (set_Levs ~~ set_image_Levs))])
blanchet@48975
   913
          (ks ~~ (rv_lasts ~~ (isNode_defs ~~ (set_naturalss ~~
blanchet@48975
   914
            (set_rv_Levss ~~ (set_Levss ~~ set_image_Levss)))))),
blanchet@48975
   915
        (**)
blanchet@48975
   916
          rtac allI, rtac impI, rtac @{thm if_not_P}, rtac notI,
blanchet@48975
   917
          etac notE, etac @{thm UN_I[OF UNIV_I]},
blanchet@48975
   918
        (*root isNode*)
blanchet@48975
   919
          rtac (isNode_def RS ssubst), rtac exI, rtac conjI, rtac (@{thm if_P} RS trans),
blanchet@49305
   920
          rtac length_Lev', rtac (Lev_0 RS equalityD2 RS set_mp), rtac @{thm singletonI},
blanchet@48975
   921
          CONVERSION (Conv.top_conv
blanchet@48975
   922
            (K (Conv.try_conv (Conv.rewr_conv (rv_Nil RS eq_reflection)))) ctxt),
blanchet@48975
   923
          if n = 1 then rtac refl else rtac (mk_sum_casesN n i),
blanchet@48975
   924
          EVERY' (map2 (fn set_natural => fn coalg_set =>
blanchet@49305
   925
            EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac (set_natural RS trans),
blanchet@49305
   926
              rtac trans_fun_cong_image_id_id_apply, etac coalg_set, atac])
blanchet@48975
   927
            (take m set_naturals) (take m coalg_sets)),
blanchet@48975
   928
          CONJ_WRAP' (fn (set_natural, (set_Lev, set_image_Lev)) =>
blanchet@48975
   929
            EVERY' [rtac (set_natural RS trans), rtac equalityI, rtac @{thm image_subsetI},
blanchet@48975
   930
              rtac CollectI, rtac @{thm SuccI}, rtac @{thm UN_I}, rtac UNIV_I, rtac set_Lev,
blanchet@49305
   931
              rtac (Lev_0 RS equalityD2 RS set_mp), rtac @{thm singletonI}, rtac rv_Nil,
blanchet@48975
   932
              atac, rtac subsetI,
blanchet@48975
   933
              REPEAT_DETERM o eresolve_tac [CollectE, @{thm SuccE}, @{thm UN_E}],
blanchet@49305
   934
              rtac set_image_Lev, rtac (Lev_0 RS equalityD2 RS set_mp),
blanchet@48975
   935
              rtac @{thm singletonI}, dtac length_Lev',
blanchet@48975
   936
              etac @{thm set_mp[OF equalityD1[OF arg_cong[OF
blanchet@48975
   937
                trans[OF length_append_singleton arg_cong[of _ _ Suc, OF list.size(3)]]]]]},
blanchet@48975
   938
              rtac rv_Nil])
blanchet@48975
   939
          (drop m set_naturals ~~ (nth set_Levss (i - 1) ~~ nth set_image_Levss (i - 1)))];
blanchet@48975
   940
blanchet@48975
   941
    fun mor_tac (i, (strT_def, (((Lev_0, Lev_Suc), (rv_Nil, rv_Cons)),
blanchet@48975
   942
      ((map_comp_id, (map_cong, map_arg_cong)), (length_Lev', (from_to_sbds, to_sbd_injs)))))) =
blanchet@48975
   943
      EVERY' [rtac ballI, rtac sym, rtac trans, rtac strT_def,
blanchet@48975
   944
        rtac (@{thm if_P} RS
blanchet@49328
   945
          (if n = 1 then map_arg_cong else sum_case_weak_cong) RS trans),
blanchet@49305
   946
        rtac (@{thm list.size(3)} RS arg_cong RS trans RS equalityD2 RS set_mp),
blanchet@48975
   947
        rtac Lev_0, rtac @{thm singletonI},
blanchet@48975
   948
        CONVERSION (Conv.top_conv
blanchet@48975
   949
          (K (Conv.try_conv (Conv.rewr_conv (rv_Nil RS eq_reflection)))) ctxt),
blanchet@48975
   950
        if n = 1 then K all_tac
blanchet@49328
   951
        else (rtac (sum_case_weak_cong RS trans) THEN'
blanchet@48975
   952
          rtac (mk_sum_casesN n i) THEN' rtac (mk_sum_casesN n i RS trans)),
blanchet@48975
   953
        rtac (map_comp_id RS trans), rtac (map_cong OF replicate m refl),
blanchet@48975
   954
        EVERY' (map3 (fn i' => fn to_sbd_inj => fn from_to_sbd =>
traytel@49488
   955
          DETERM o EVERY' [rtac trans, rtac o_apply, rtac Pair_eqI, rtac conjI,
blanchet@48975
   956
            rtac trans, rtac @{thm Shift_def},
blanchet@48975
   957
            rtac equalityI, rtac subsetI, etac thin_rl, etac thin_rl,
blanchet@48975
   958
            REPEAT_DETERM o eresolve_tac [CollectE, @{thm UN_E}], dtac length_Lev', dtac asm_rl,
blanchet@48975
   959
            etac thin_rl, dtac @{thm set_rev_mp[OF _ equalityD1]},
blanchet@48975
   960
            rtac (@{thm length_Cons} RS arg_cong RS trans), rtac Lev_Suc,
blanchet@48975
   961
            CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE)) (fn i'' =>
blanchet@48975
   962
              EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE],
traytel@49488
   963
                dtac list_inject_iffD1, etac conjE,
blanchet@48975
   964
                if i' = i'' then EVERY' [dtac (mk_InN_inject n i'),
traytel@49488
   965
                  dtac (Thm.permute_prems 0 2 (to_sbd_inj RS iffD1)),
blanchet@48975
   966
                  atac, atac, hyp_subst_tac, etac @{thm UN_I[OF UNIV_I]}]
blanchet@48975
   967
                else etac (mk_InN_not_InM i' i'' RS notE)])
blanchet@48975
   968
            (rev ks),
blanchet@48975
   969
            rtac @{thm UN_least}, rtac subsetI, rtac CollectI, rtac @{thm UN_I[OF UNIV_I]},
blanchet@49366
   970
            rtac (Lev_Suc RS equalityD2 RS set_mp), rtac (mk_UnIN n i'), rtac CollectI,
blanchet@48975
   971
            REPEAT_DETERM o rtac exI, rtac conjI, rtac refl, etac conjI, atac,
blanchet@48975
   972
            rtac trans, rtac @{thm shift_def}, rtac ssubst, rtac @{thm fun_eq_iff}, rtac allI,
blanchet@48975
   973
            rtac @{thm if_cong}, rtac (@{thm length_Cons} RS arg_cong RS trans), rtac iffI,
blanchet@48975
   974
            dtac asm_rl, dtac asm_rl, dtac asm_rl,
blanchet@49305
   975
            dtac (Lev_Suc RS equalityD1 RS set_mp),
blanchet@48975
   976
            CONJ_WRAP_GEN' (etac (Thm.permute_prems 1 1 UnE)) (fn i'' =>
blanchet@48975
   977
              EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE],
traytel@49488
   978
                dtac list_inject_iffD1, etac conjE,
blanchet@48975
   979
                if i' = i'' then EVERY' [dtac (mk_InN_inject n i'),
traytel@49488
   980
                  dtac (Thm.permute_prems 0 2 (to_sbd_inj RS iffD1)),
blanchet@48975
   981
                  atac, atac, hyp_subst_tac, atac]
blanchet@48975
   982
                else etac (mk_InN_not_InM i' i'' RS notE)])
blanchet@48975
   983
            (rev ks),
blanchet@49366
   984
            rtac (Lev_Suc RS equalityD2 RS set_mp), rtac (mk_UnIN n i'), rtac CollectI,
blanchet@48975
   985
            REPEAT_DETERM o rtac exI, rtac conjI, rtac refl, etac conjI, atac,
blanchet@48975
   986
            CONVERSION (Conv.top_conv
blanchet@48975
   987
              (K (Conv.try_conv (Conv.rewr_conv (rv_Cons RS eq_reflection)))) ctxt),
blanchet@48975
   988
            if n = 1 then K all_tac
blanchet@49328
   989
            else rtac sum_case_weak_cong THEN' rtac (mk_sum_casesN n i' RS trans),
blanchet@49504
   990
            SELECT_GOAL (unfold_thms_tac ctxt [from_to_sbd]), rtac refl,
blanchet@48975
   991
            rtac refl])
blanchet@48975
   992
        ks to_sbd_injs from_to_sbds)];
blanchet@48975
   993
  in
blanchet@48975
   994
    (rtac mor_cong THEN'
blanchet@48975
   995
    EVERY' (map (fn thm => rtac (thm RS ext)) beh_defs) THEN'
blanchet@48975
   996
    stac mor_def THEN' rtac conjI THEN'
blanchet@48975
   997
    CONJ_WRAP' fbetw_tac
blanchet@48975
   998
      (ks ~~ (carT_defs ~~ (isNode_defs ~~ (Lev_0s ~~ (rv_Nils ~~ (Lev_sbds ~~
blanchet@48975
   999
        ((length_Levs ~~ length_Lev's) ~~ (prefCl_Levs ~~ (rv_lastss ~~
blanchet@48975
  1000
          (set_naturalss ~~ (coalg_setss ~~
blanchet@48975
  1001
            (set_rv_Levsss ~~ (set_Levsss ~~ set_image_Levsss))))))))))))) THEN'
blanchet@48975
  1002
    CONJ_WRAP' mor_tac
blanchet@48975
  1003
      (ks ~~ (strT_defs ~~ (((Lev_0s ~~ Lev_Sucs) ~~ (rv_Nils ~~ rv_Conss)) ~~
blanchet@48975
  1004
        ((map_comp_ids ~~ (map_congs ~~ map_arg_congs)) ~~
blanchet@48975
  1005
          (length_Lev's ~~ (from_to_sbdss ~~ to_sbd_injss))))))) 1
blanchet@48975
  1006
  end;
blanchet@48975
  1007
blanchet@48975
  1008
fun mk_congruent_str_final_tac m lsbisE map_comp_id map_cong equiv_LSBISs =
blanchet@48975
  1009
  EVERY' [rtac @{thm congruentI}, dtac lsbisE,
blanchet@48975
  1010
    REPEAT_DETERM o eresolve_tac [CollectE, conjE, bexE], rtac (o_apply RS trans),
blanchet@48975
  1011
    etac (sym RS arg_cong RS trans), rtac (map_comp_id RS trans),
blanchet@48975
  1012
    rtac (map_cong RS trans), REPEAT_DETERM_N m o rtac refl,
blanchet@48975
  1013
    EVERY' (map (fn equiv_LSBIS =>
blanchet@49305
  1014
      EVERY' [rtac @{thm equiv_proj}, rtac equiv_LSBIS, etac set_mp, atac])
blanchet@48975
  1015
    equiv_LSBISs), rtac sym, rtac (o_apply RS trans),
blanchet@48975
  1016
    etac (sym RS arg_cong RS trans), rtac map_comp_id] 1;
blanchet@48975
  1017
blanchet@48975
  1018
fun mk_coalg_final_tac m coalg_def congruent_str_finals equiv_LSBISs set_naturalss coalgT_setss =
blanchet@48975
  1019
  EVERY' [stac coalg_def,
blanchet@48975
  1020
    CONJ_WRAP' (fn ((set_naturals, coalgT_sets), (equiv_LSBIS, congruent_str_final)) =>
blanchet@48975
  1021
      EVERY' [rtac @{thm univ_preserves}, rtac equiv_LSBIS, rtac congruent_str_final,
blanchet@48975
  1022
        rtac ballI, rtac @{thm ssubst_mem}, rtac o_apply, rtac CollectI,
blanchet@48975
  1023
        EVERY' (map2 (fn set_natural => fn coalgT_set =>
blanchet@49305
  1024
          EVERY' [rtac conjI, rtac (set_natural RS ord_eq_le_trans),
blanchet@49305
  1025
            rtac ord_eq_le_trans_trans_fun_cong_image_id_id_apply,
blanchet@48975
  1026
            etac coalgT_set])
blanchet@48975
  1027
        (take m set_naturals) (take m coalgT_sets)),
blanchet@48975
  1028
        CONJ_WRAP' (fn (equiv_LSBIS, (set_natural, coalgT_set)) =>
blanchet@49305
  1029
          EVERY' [rtac (set_natural RS ord_eq_le_trans),
blanchet@48975
  1030
            rtac @{thm image_subsetI}, rtac ssubst, rtac @{thm proj_in_iff},
blanchet@49305
  1031
            rtac equiv_LSBIS, etac set_rev_mp, etac coalgT_set])
blanchet@48975
  1032
        (equiv_LSBISs ~~ drop m (set_naturals ~~ coalgT_sets))])
blanchet@48975
  1033
    ((set_naturalss ~~ coalgT_setss) ~~ (equiv_LSBISs ~~ congruent_str_finals))] 1;
blanchet@48975
  1034
blanchet@48975
  1035
fun mk_mor_T_final_tac mor_def congruent_str_finals equiv_LSBISs =
blanchet@48975
  1036
  EVERY' [stac mor_def, rtac conjI,
blanchet@48975
  1037
    CONJ_WRAP' (fn equiv_LSBIS =>
blanchet@48975
  1038
      EVERY' [rtac ballI, rtac ssubst, rtac @{thm proj_in_iff}, rtac equiv_LSBIS, atac])
blanchet@48975
  1039
    equiv_LSBISs,
blanchet@48975
  1040
    CONJ_WRAP' (fn (equiv_LSBIS, congruent_str_final) =>
blanchet@48975
  1041
      EVERY' [rtac ballI, rtac sym, rtac trans, rtac @{thm univ_commute}, rtac equiv_LSBIS,
blanchet@48975
  1042
        rtac congruent_str_final, atac, rtac o_apply])
blanchet@48975
  1043
    (equiv_LSBISs ~~ congruent_str_finals)] 1;
blanchet@48975
  1044
blanchet@48975
  1045
fun mk_mor_Rep_tac m defs Reps Abs_inverses coalg_final_setss map_comp_ids map_congLs
blanchet@48975
  1046
  {context = ctxt, prems = _} =
blanchet@49504
  1047
  unfold_thms_tac ctxt defs THEN
blanchet@48975
  1048
  EVERY' [rtac conjI,
blanchet@48975
  1049
    CONJ_WRAP' (fn thm => rtac ballI THEN' rtac thm) Reps,
blanchet@48975
  1050
    CONJ_WRAP' (fn (Rep, ((map_comp_id, map_congL), coalg_final_sets)) =>
blanchet@48975
  1051
      EVERY' [rtac ballI, rtac (map_comp_id RS trans), rtac map_congL,
blanchet@48975
  1052
        EVERY' (map2 (fn Abs_inverse => fn coalg_final_set =>
blanchet@48975
  1053
          EVERY' [rtac ballI, rtac (o_apply RS trans), rtac Abs_inverse,
blanchet@49305
  1054
            etac set_rev_mp, rtac coalg_final_set, rtac Rep])
blanchet@48975
  1055
        Abs_inverses (drop m coalg_final_sets))])
blanchet@48975
  1056
    (Reps ~~ ((map_comp_ids ~~ map_congLs) ~~ coalg_final_setss))] 1;
blanchet@48975
  1057
blanchet@48975
  1058
fun mk_mor_Abs_tac defs Abs_inverses {context = ctxt, prems = _} =
blanchet@49504
  1059
  unfold_thms_tac ctxt defs THEN
blanchet@48975
  1060
  EVERY' [rtac conjI,
blanchet@48975
  1061
    CONJ_WRAP' (K (rtac ballI THEN' rtac UNIV_I)) Abs_inverses,
blanchet@48975
  1062
    CONJ_WRAP' (fn thm => rtac ballI THEN' etac (thm RS arg_cong RS sym)) Abs_inverses] 1;
blanchet@48975
  1063
blanchet@49504
  1064
fun mk_mor_unfold_tac m mor_UNIV dtor_defs unfold_defs Abs_inverses morEs map_comp_ids map_congs =
traytel@49488
  1065
  EVERY' [rtac iffD2, rtac mor_UNIV,
blanchet@49504
  1066
    CONJ_WRAP' (fn ((Abs_inverse, morE), ((dtor_def, unfold_def), (map_comp_id, map_cong))) =>
blanchet@49501
  1067
      EVERY' [rtac ext, rtac (o_apply RS trans RS sym), rtac (dtor_def RS trans),
blanchet@49504
  1068
        rtac (unfold_def RS arg_cong RS trans), rtac (Abs_inverse RS arg_cong RS trans),
blanchet@48975
  1069
        rtac (morE RS arg_cong RS trans), rtac (map_comp_id RS trans),
blanchet@48975
  1070
        rtac (o_apply RS trans RS sym), rtac map_cong,
blanchet@48975
  1071
        REPEAT_DETERM_N m o rtac refl,
blanchet@49504
  1072
        EVERY' (map (fn thm => rtac (thm RS trans) THEN' rtac (o_apply RS sym)) unfold_defs)])
blanchet@49504
  1073
    ((Abs_inverses ~~ morEs) ~~ ((dtor_defs ~~ unfold_defs) ~~ (map_comp_ids ~~ map_congs)))] 1;
blanchet@48975
  1074
blanchet@48975
  1075
fun mk_raw_coind_tac bis_def bis_cong bis_O bis_converse bis_Gr tcoalg coalgT mor_T_final
blanchet@48975
  1076
  sbis_lsbis lsbis_incls incl_lsbiss equiv_LSBISs mor_Rep Rep_injects =
blanchet@48975
  1077
  let
blanchet@48975
  1078
    val n = length Rep_injects;
blanchet@48975
  1079
  in
traytel@49488
  1080
    EVERY' [rtac rev_mp, ftac (bis_def RS iffD1),
blanchet@48975
  1081
      REPEAT_DETERM o etac conjE, rtac bis_cong, rtac bis_O, rtac bis_converse,
blanchet@48975
  1082
      rtac bis_Gr, rtac tcoalg, rtac mor_Rep, rtac bis_O, atac, rtac bis_Gr, rtac tcoalg,
blanchet@48975
  1083
      rtac mor_Rep, REPEAT_DETERM_N n o etac @{thm relImage_Gr},
blanchet@48975
  1084
      rtac impI, rtac rev_mp, rtac bis_cong, rtac bis_O, rtac bis_Gr, rtac coalgT,
blanchet@48975
  1085
      rtac mor_T_final, rtac bis_O, rtac sbis_lsbis, rtac bis_converse, rtac bis_Gr, rtac coalgT,
blanchet@48975
  1086
      rtac mor_T_final, EVERY' (map (fn thm => rtac (thm RS @{thm relInvImage_Gr})) lsbis_incls),
blanchet@48975
  1087
      rtac impI,
blanchet@48975
  1088
      CONJ_WRAP' (fn (Rep_inject, (equiv_LSBIS , (incl_lsbis, lsbis_incl))) =>
blanchet@48975
  1089
        EVERY' [rtac subset_trans, rtac @{thm relInvImage_UNIV_relImage}, rtac subset_trans,
blanchet@48975
  1090
          rtac @{thm relInvImage_mono}, rtac subset_trans, etac incl_lsbis,
blanchet@49305
  1091
          rtac ord_eq_le_trans, rtac @{thm sym[OF relImage_relInvImage]},
blanchet@48975
  1092
          rtac @{thm xt1(3)}, rtac @{thm Sigma_cong},
blanchet@48975
  1093
          rtac @{thm proj_image}, rtac @{thm proj_image}, rtac lsbis_incl,
blanchet@48975
  1094
          rtac subset_trans, rtac @{thm relImage_mono}, rtac incl_lsbis, atac,
blanchet@48975
  1095
          rtac @{thm relImage_proj}, rtac equiv_LSBIS, rtac @{thm relInvImage_diag},
blanchet@48975
  1096
          rtac Rep_inject])
blanchet@48975
  1097
      (Rep_injects ~~ (equiv_LSBISs ~~ (incl_lsbiss ~~ lsbis_incls)))] 1
blanchet@48975
  1098
  end;
blanchet@48975
  1099
blanchet@48975
  1100
fun mk_unique_mor_tac raw_coinds bis =
blanchet@48975
  1101
  CONJ_WRAP' (fn raw_coind =>
blanchet@49305
  1102
    EVERY' [rtac impI, rtac (bis RS raw_coind RS set_mp RS @{thm IdD}), atac,
blanchet@48975
  1103
      etac conjunct1, atac, etac conjunct2, rtac @{thm image2_eqI}, rtac refl, rtac refl, atac])
blanchet@48975
  1104
  raw_coinds 1;
blanchet@48975
  1105
blanchet@49504
  1106
fun mk_unfold_unique_mor_tac raw_coinds bis mor unfold_defs =
blanchet@49504
  1107
  CONJ_WRAP' (fn (raw_coind, unfold_def) =>
blanchet@49305
  1108
    EVERY' [rtac ext, etac (bis RS raw_coind RS set_mp RS @{thm IdD}), rtac mor,
blanchet@49504
  1109
      rtac @{thm image2_eqI}, rtac refl, rtac (unfold_def RS arg_cong RS trans),
blanchet@49504
  1110
      rtac (o_apply RS sym), rtac UNIV_I]) (raw_coinds ~~ unfold_defs) 1;
blanchet@48975
  1111
blanchet@49504
  1112
fun mk_dtor_o_ctor_tac ctor_def unfold map_comp_id map_congL unfold_o_dtors
blanchet@48975
  1113
  {context = ctxt, prems = _} =
blanchet@49504
  1114
  unfold_thms_tac ctxt [ctor_def] THEN EVERY' [rtac ext, rtac trans, rtac o_apply,
blanchet@49504
  1115
    rtac trans, rtac unfold, rtac trans, rtac map_comp_id, rtac trans, rtac map_congL,
blanchet@48975
  1116
    EVERY' (map (fn thm =>
blanchet@49504
  1117
      rtac ballI THEN' rtac (trans OF [thm RS fun_cong, @{thm id_apply}])) unfold_o_dtors),
blanchet@48975
  1118
    rtac sym, rtac @{thm id_apply}] 1;
blanchet@48975
  1119
blanchet@49504
  1120
fun mk_corec_tac m corec_defs unfold map_cong corec_Inls {context = ctxt, prems = _} =
blanchet@49504
  1121
  unfold_thms_tac ctxt corec_defs THEN EVERY' [rtac trans, rtac (o_apply RS arg_cong),
blanchet@49504
  1122
    rtac trans, rtac unfold, fo_rtac (@{thm sum.cases(2)} RS arg_cong RS trans) ctxt, rtac map_cong,
blanchet@48975
  1123
    REPEAT_DETERM_N m o rtac refl,
blanchet@48975
  1124
    EVERY' (map (fn thm => rtac @{thm sum_case_expand_Inr} THEN' rtac thm) corec_Inls)] 1;
blanchet@48975
  1125
blanchet@48975
  1126
fun mk_rel_coinduct_tac ks raw_coind bis_rel =
blanchet@48975
  1127
  EVERY' [rtac rev_mp, rtac raw_coind, rtac ssubst, rtac bis_rel, rtac conjI,
blanchet@48975
  1128
    CONJ_WRAP' (K (rtac @{thm ord_le_eq_trans[OF subset_UNIV UNIV_Times_UNIV[THEN sym]]})) ks,
blanchet@48975
  1129
    CONJ_WRAP' (K (EVERY' [rtac allI, rtac allI, rtac impI,
blanchet@48975
  1130
      REPEAT_DETERM o etac allE, etac mp, etac CollectE, etac @{thm splitD}])) ks,
blanchet@48975
  1131
    rtac impI, REPEAT_DETERM o etac conjE,
blanchet@49305
  1132
    CONJ_WRAP' (K (EVERY' [rtac impI, rtac @{thm IdD}, etac set_mp,
blanchet@48975
  1133
      rtac CollectI, etac @{thm prod_caseI}])) ks] 1;
blanchet@48975
  1134
blanchet@49499
  1135
fun mk_rel_strong_coinduct_tac m cTs cts rel_coinduct rel_monos rel_Ids =
blanchet@48975
  1136
  EVERY' [rtac rev_mp, rtac (Drule.instantiate' cTs cts rel_coinduct),
blanchet@48975
  1137
    EVERY' (map2 (fn rel_mono => fn rel_Id =>
blanchet@48975
  1138
      EVERY' [REPEAT_DETERM o resolve_tac [allI, impI], REPEAT_DETERM o etac allE,
blanchet@49305
  1139
        etac disjE, etac mp, atac, hyp_subst_tac, rtac (rel_mono RS set_mp),
blanchet@48975
  1140
        REPEAT_DETERM_N m o rtac @{thm subset_refl},
blanchet@48975
  1141
        REPEAT_DETERM_N (length rel_monos) o rtac @{thm Id_subset},
blanchet@49305
  1142
        rtac (rel_Id RS equalityD2 RS set_mp), rtac @{thm IdI}])
blanchet@48975
  1143
    rel_monos rel_Ids),
blanchet@48975
  1144
    rtac impI, REPEAT_DETERM o etac conjE,
blanchet@48975
  1145
    CONJ_WRAP' (K (rtac impI THEN' etac mp THEN' etac disjI1)) rel_Ids] 1;
blanchet@48975
  1146
blanchet@49501
  1147
fun mk_dtor_coinduct_tac m ks raw_coind bis_def =
blanchet@48975
  1148
  let
blanchet@48975
  1149
    val n = length ks;
blanchet@48975
  1150
  in
blanchet@48975
  1151
    EVERY' [rtac rev_mp, rtac raw_coind, rtac ssubst, rtac bis_def, rtac conjI,
blanchet@48975
  1152
      CONJ_WRAP' (K (rtac @{thm ord_le_eq_trans[OF subset_UNIV UNIV_Times_UNIV[THEN sym]]})) ks,
blanchet@48975
  1153
      CONJ_WRAP' (fn i => EVERY' [select_prem_tac n (dtac asm_rl) i, REPEAT_DETERM o rtac allI,
blanchet@48975
  1154
        rtac impI, REPEAT_DETERM o dtac @{thm meta_spec}, etac CollectE, etac @{thm meta_impE},
blanchet@48975
  1155
        atac, etac exE, etac conjE, etac conjE, rtac bexI, rtac conjI,
blanchet@48975
  1156
        etac @{thm fst_conv[THEN subst]}, etac @{thm snd_conv[THEN subst]},
traytel@49490
  1157
        rtac CollectI, REPEAT_DETERM_N m o (rtac conjI THEN' rtac subset_UNIV),
blanchet@48975
  1158
        CONJ_WRAP' (fn i' => EVERY' [rtac subsetI, rtac CollectI, dtac (mk_conjunctN n i'),
blanchet@48975
  1159
          REPEAT_DETERM o etac allE, etac mp, rtac @{thm ssubst_mem[OF pair_collapse]}, atac])
blanchet@48975
  1160
        ks])
blanchet@48975
  1161
      ks,
blanchet@48975
  1162
      rtac impI,
blanchet@48975
  1163
      CONJ_WRAP' (fn i => EVERY' [rtac impI, dtac (mk_conjunctN n i),
blanchet@49305
  1164
        rtac @{thm subst[OF pair_in_Id_conv]}, etac set_mp,
blanchet@48975
  1165
        rtac CollectI, etac (refl RSN (2, @{thm subst_Pair}))]) ks] 1
blanchet@48975
  1166
  end;
blanchet@48975
  1167
blanchet@49501
  1168
fun mk_dtor_strong_coinduct_tac ks cTs cts dtor_coinduct bis_def bis_diag =
blanchet@49501
  1169
  EVERY' [rtac rev_mp, rtac (Drule.instantiate' cTs cts dtor_coinduct),
blanchet@48975
  1170
    EVERY' (map (fn i =>
blanchet@48975
  1171
      EVERY' [etac disjE, REPEAT_DETERM o dtac @{thm meta_spec}, etac @{thm meta_mp},
blanchet@48975
  1172
        atac, rtac rev_mp, rtac subst, rtac bis_def, rtac bis_diag,
blanchet@48975
  1173
        rtac impI, dtac conjunct2, dtac (mk_conjunctN (length ks) i), REPEAT_DETERM o etac allE,
blanchet@48975
  1174
        etac impE, etac @{thm diag_UNIV_I}, REPEAT_DETERM o eresolve_tac [bexE, conjE, CollectE],
blanchet@48975
  1175
        rtac exI, rtac conjI, etac conjI, atac,
blanchet@48975
  1176
        CONJ_WRAP' (K (EVERY' [REPEAT_DETERM o resolve_tac [allI, impI],
blanchet@49305
  1177
          rtac disjI2, rtac @{thm diagE}, etac set_mp, atac])) ks])
blanchet@48975
  1178
    ks),
blanchet@48975
  1179
    rtac impI, REPEAT_DETERM o etac conjE,
blanchet@48975
  1180
    CONJ_WRAP' (K (rtac impI THEN' etac mp THEN' etac disjI1)) ks] 1;
blanchet@48975
  1181
blanchet@49504
  1182
fun mk_map_tac m n cT unfold map_comp' map_cong =
blanchet@48975
  1183
  EVERY' [rtac ext, rtac (o_apply RS trans RS sym), rtac (o_apply RS trans RS sym),
blanchet@49504
  1184
    rtac (unfold RS trans), rtac (Thm.permute_prems 0 1 (map_comp' RS box_equals)), rtac map_cong,
blanchet@48975
  1185
    REPEAT_DETERM_N m o rtac (@{thm id_o} RS fun_cong),
blanchet@48975
  1186
    REPEAT_DETERM_N n o rtac (@{thm o_id} RS fun_cong),
blanchet@48975
  1187
    rtac (o_apply RS (Drule.instantiate' [cT] [] arg_cong) RS sym)] 1;
blanchet@48975
  1188
blanchet@48975
  1189
fun mk_set_le_tac n hset_minimal set_hsets set_hset_hsetss =
blanchet@48975
  1190
  EVERY' [rtac hset_minimal,
blanchet@48975
  1191
    REPEAT_DETERM_N n o rtac @{thm Un_upper1},
blanchet@48975
  1192
    REPEAT_DETERM_N n o
blanchet@48975
  1193
      EVERY' (map3 (fn i => fn set_hset => fn set_hset_hsets =>
blanchet@49366
  1194
        EVERY' [rtac subsetI, rtac @{thm UnI2}, rtac (mk_UnIN n i), etac @{thm UN_I},
blanchet@48975
  1195
          etac UnE, etac set_hset, REPEAT_DETERM_N (n - 1) o etac UnE,
blanchet@48975
  1196
          EVERY' (map (fn thm => EVERY' [etac @{thm UN_E}, etac thm, atac]) set_hset_hsets)])
blanchet@48975
  1197
      (1 upto n) set_hsets set_hset_hsetss)] 1;
blanchet@48975
  1198
blanchet@48975
  1199
fun mk_set_simp_tac n set_le set_incl_hset set_hset_incl_hsets =
blanchet@48975
  1200
  EVERY' [rtac equalityI, rtac set_le, rtac @{thm Un_least}, rtac set_incl_hset,
blanchet@48975
  1201
    REPEAT_DETERM_N (n - 1) o rtac @{thm Un_least},
blanchet@48975
  1202
    EVERY' (map (fn thm => rtac @{thm UN_least} THEN' etac thm) set_hset_incl_hsets)] 1;
blanchet@48975
  1203
blanchet@49504
  1204
fun mk_map_id_tac maps unfold_unique unfold_dtor =
blanchet@49504
  1205
  EVERY' [rtac (unfold_unique RS trans), EVERY' (map (fn thm => rtac (thm RS sym)) maps),
blanchet@49504
  1206
    rtac unfold_dtor] 1;
blanchet@48975
  1207
blanchet@49504
  1208
fun mk_map_comp_tac m n maps map_comps map_congs unfold_unique =
blanchet@49504
  1209
  EVERY' [rtac unfold_unique,
blanchet@48975
  1210
    EVERY' (map3 (fn map_thm => fn map_comp => fn map_cong =>
blanchet@48975
  1211
      EVERY' (map rtac
blanchet@48975
  1212
        ([@{thm o_assoc} RS trans,
blanchet@48975
  1213
        @{thm arg_cong2[of _ _ _ _ "op o"]} OF [map_comp RS sym, refl] RS trans,
blanchet@48975
  1214
        @{thm o_assoc} RS trans RS sym,
blanchet@48975
  1215
        @{thm arg_cong2[of _ _ _ _ "op o"]} OF [map_thm, refl] RS trans,
blanchet@48975
  1216
        @{thm o_assoc} RS sym RS trans, map_thm RS arg_cong RS trans, @{thm o_assoc} RS trans,
blanchet@48975
  1217
        @{thm arg_cong2[of _ _ _ _ "op o"]} OF [map_comp RS sym, refl] RS trans,
blanchet@48975
  1218
        ext, o_apply RS trans,  o_apply RS trans RS sym, map_cong] @
blanchet@48975
  1219
        replicate m (@{thm id_o} RS fun_cong) @
blanchet@48975
  1220
        replicate n (@{thm o_id} RS fun_cong))))
blanchet@48975
  1221
    maps map_comps map_congs)] 1;
blanchet@48975
  1222
blanchet@48975
  1223
fun mk_mcong_tac m coinduct_tac map_comp's map_simps map_congs set_naturalss set_hsetss
blanchet@48975
  1224
  set_hset_hsetsss =
blanchet@48975
  1225
  let
blanchet@48975
  1226
    val n = length map_comp's;
blanchet@48975
  1227
    val ks = 1 upto n;
blanchet@48975
  1228
  in
blanchet@48975
  1229
    EVERY' ([rtac rev_mp,
blanchet@48975
  1230
      coinduct_tac] @
blanchet@48975
  1231
      maps (fn (((((map_comp'_trans, map_simps_trans), map_cong), set_naturals), set_hsets),
blanchet@48975
  1232
        set_hset_hsetss) =>
blanchet@48975
  1233
        [REPEAT_DETERM o eresolve_tac [exE, conjE], hyp_subst_tac, rtac exI, rtac conjI, rtac conjI,
blanchet@48975
  1234
         rtac map_comp'_trans, rtac sym, rtac map_simps_trans, rtac map_cong,
blanchet@48975
  1235
         REPEAT_DETERM_N m o (rtac o_apply_trans_sym THEN' rtac @{thm id_apply}),
blanchet@48975
  1236
         REPEAT_DETERM_N n o rtac fst_convol_fun_cong_sym,
blanchet@48975
  1237
         rtac map_comp'_trans, rtac sym, rtac map_simps_trans, rtac map_cong,
blanchet@48975
  1238
         EVERY' (maps (fn set_hset =>
blanchet@48975
  1239
           [rtac o_apply_trans_sym, rtac (@{thm id_apply} RS trans), etac CollectE,
blanchet@48975
  1240
           REPEAT_DETERM o etac conjE, etac bspec, etac set_hset]) set_hsets),
blanchet@48975
  1241
         REPEAT_DETERM_N n o rtac snd_convol_fun_cong_sym,
blanchet@48975
  1242
         CONJ_WRAP' (fn (set_natural, set_hset_hsets) =>
blanchet@48975
  1243
           EVERY' [REPEAT_DETERM o rtac allI, rtac impI, rtac @{thm image_convolD},
blanchet@49305
  1244
             etac set_rev_mp, rtac ord_eq_le_trans, rtac set_natural,
blanchet@48975
  1245
             rtac @{thm image_mono}, rtac subsetI, rtac CollectI, etac CollectE,
blanchet@48975
  1246
             REPEAT_DETERM o etac conjE,
blanchet@48975
  1247
             CONJ_WRAP' (fn set_hset_hset =>
blanchet@48975
  1248
               EVERY' [rtac ballI, etac bspec, etac set_hset_hset, atac]) set_hset_hsets])
blanchet@48975
  1249
           (drop m set_naturals ~~ set_hset_hsetss)])
blanchet@48975
  1250
        (map (fn th => th RS trans) map_comp's ~~ map (fn th => th RS trans) map_simps ~~
blanchet@48975
  1251
          map_congs ~~ set_naturalss ~~ set_hsetss ~~ set_hset_hsetsss) @
blanchet@48975
  1252
      [rtac impI,
blanchet@48975
  1253
       CONJ_WRAP' (fn k =>
blanchet@48975
  1254
         EVERY' [rtac impI, dtac (mk_conjunctN n k), etac mp, rtac exI, rtac conjI, etac CollectI,
blanchet@48975
  1255
           rtac conjI, rtac refl, rtac refl]) ks]) 1
blanchet@48975
  1256
  end
blanchet@48975
  1257
blanchet@49504
  1258
fun mk_map_unique_tac unfold_unique map_comps {context = ctxt, prems = _} =
blanchet@49504
  1259
  rtac unfold_unique 1 THEN
blanchet@49504
  1260
  unfold_thms_tac ctxt (map (fn thm => thm RS sym) map_comps @ @{thms o_assoc id_o o_id}) THEN
blanchet@48975
  1261
  ALLGOALS (etac sym);
blanchet@48975
  1262
blanchet@48975
  1263
fun mk_col_natural_tac cts rec_0s rec_Sucs map_simps set_naturalss
blanchet@48975
  1264
  {context = ctxt, prems = _} =
blanchet@48975
  1265
  let
blanchet@48975
  1266
    val n = length map_simps;
blanchet@48975
  1267
  in
blanchet@49305
  1268
    EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
blanchet@49504
  1269
      REPEAT_DETERM o rtac allI, SELECT_GOAL (unfold_thms_tac ctxt rec_0s),
blanchet@48975
  1270
      CONJ_WRAP' (K (rtac @{thm image_empty})) rec_0s,
blanchet@48975
  1271
      REPEAT_DETERM o rtac allI,
blanchet@48975
  1272
      CONJ_WRAP' (fn (rec_Suc, (map_simp, set_nats)) => EVERY'
blanchet@49504
  1273
        [SELECT_GOAL (unfold_thms_tac ctxt
blanchet@48975
  1274
          (rec_Suc :: map_simp :: set_nats @ @{thms image_Un image_UN UN_simps(10)})),
blanchet@48975
  1275
        rtac @{thm Un_cong}, rtac refl,
blanchet@48975
  1276
        CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_cong}))
blanchet@48975
  1277
          (fn i => EVERY' [rtac @{thm UN_cong[OF refl]},
blanchet@48975
  1278
            REPEAT_DETERM o etac allE, etac (mk_conjunctN n i)]) (n downto 1)])
blanchet@48975
  1279
      (rec_Sucs ~~ (map_simps ~~ set_naturalss))] 1
blanchet@48975
  1280
  end;
blanchet@48975
  1281
blanchet@48975
  1282
fun mk_set_natural_tac hset_def col_natural =
blanchet@48975
  1283
  EVERY' (map rtac [ext, (o_apply RS trans), (hset_def RS trans), sym,
blanchet@48975
  1284
    (o_apply RS trans), (@{thm image_cong} OF [hset_def, refl] RS trans),
blanchet@48975
  1285
    (@{thm image_UN} RS trans), (refl RS @{thm UN_cong}), col_natural]) 1;
blanchet@48975
  1286
blanchet@48975
  1287
fun mk_col_bd_tac m j cts rec_0s rec_Sucs sbd_Card_order sbd_Cinfinite set_sbdss =
blanchet@48975
  1288
  let
blanchet@48975
  1289
    val n = length rec_0s;
blanchet@48975
  1290
  in
blanchet@49305
  1291
    EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
blanchet@48975
  1292
      REPEAT_DETERM o rtac allI,
blanchet@49305
  1293
      CONJ_WRAP' (fn rec_0 => EVERY' (map rtac [ordIso_ordLeq_trans,
blanchet@48975
  1294
        @{thm card_of_ordIso_subst}, rec_0, @{thm Card_order_empty}, sbd_Card_order])) rec_0s,
blanchet@48975
  1295
      REPEAT_DETERM o rtac allI,
blanchet@48975
  1296
      CONJ_WRAP' (fn (rec_Suc, set_sbds) => EVERY'
blanchet@49305
  1297
        [rtac ordIso_ordLeq_trans, rtac @{thm card_of_ordIso_subst}, rtac rec_Suc,
blanchet@48975
  1298
        rtac (sbd_Cinfinite RSN (3, @{thm Un_Cinfinite_bound})), rtac (nth set_sbds (j - 1)),
blanchet@48975
  1299
        REPEAT_DETERM_N (n - 1) o rtac (sbd_Cinfinite RSN (3, @{thm Un_Cinfinite_bound})),
blanchet@48975
  1300
        EVERY' (map2 (fn i => fn set_sbd => EVERY' [rtac @{thm UNION_Cinfinite_bound},
blanchet@48975
  1301
          rtac set_sbd, rtac ballI, REPEAT_DETERM o etac allE,
blanchet@48975
  1302
          etac (mk_conjunctN n i), rtac sbd_Cinfinite]) (1 upto n) (drop m set_sbds))])
blanchet@48975
  1303
      (rec_Sucs ~~ set_sbdss)] 1
blanchet@48975
  1304
  end;
blanchet@48975
  1305
blanchet@48975
  1306
fun mk_set_bd_tac sbd_Cinfinite hset_def col_bd =
blanchet@49305
  1307
  EVERY' (map rtac [ordIso_ordLeq_trans, @{thm card_of_ordIso_subst}, hset_def,
blanchet@49305
  1308
    ctrans, @{thm UNION_Cinfinite_bound}, ordIso_ordLeq_trans, @{thm card_of_nat},
blanchet@48975
  1309
    @{thm natLeq_ordLeq_cinfinite}, sbd_Cinfinite, ballI, col_bd, sbd_Cinfinite,
blanchet@48975
  1310
    ctrans, @{thm infinite_ordLeq_cexp}, sbd_Cinfinite, @{thm cexp_ordLeq_ccexp}]) 1;
blanchet@48975
  1311
blanchet@48975
  1312
fun mk_in_bd_tac isNode_hset isNode_hsets carT_def card_of_carT mor_image Rep_inverse mor_hsets
blanchet@48975
  1313
  sbd_Cnotzero sbd_Card_order mor_Rep coalgT mor_T_final tcoalg =
blanchet@48975
  1314
  let
blanchet@48975
  1315
    val n = length isNode_hsets;
blanchet@48975
  1316
    val in_hin_tac = rtac CollectI THEN'
blanchet@48975
  1317
      CONJ_WRAP' (fn mor_hset => EVERY' (map etac
blanchet@49305
  1318
        [mor_hset OF [coalgT, mor_T_final] RS sym RS ord_eq_le_trans,
blanchet@49305
  1319
        arg_cong RS sym RS ord_eq_le_trans,
blanchet@49305
  1320
        mor_hset OF [tcoalg, mor_Rep, UNIV_I] RS ord_eq_le_trans])) mor_hsets;
blanchet@48975
  1321
  in
blanchet@48975
  1322
    EVERY' [rtac (Thm.permute_prems 0 1 @{thm ordLeq_transitive}), rtac ctrans,
blanchet@49305
  1323
      rtac @{thm card_of_image}, rtac ordIso_ordLeq_trans,
blanchet@48975
  1324
      rtac @{thm card_of_ordIso_subst}, rtac @{thm sym[OF proj_image]}, rtac ctrans,
blanchet@48975
  1325
      rtac @{thm card_of_image}, rtac ctrans, rtac card_of_carT, rtac @{thm cexp_mono2_Cnotzero},
blanchet@48975
  1326
      rtac @{thm cexp_ordLeq_ccexp},  rtac @{thm csum_Cnotzero2}, rtac @{thm ctwo_Cnotzero},
blanchet@48975
  1327
      rtac @{thm Cnotzero_cexp}, rtac sbd_Cnotzero, rtac sbd_Card_order,
blanchet@48975
  1328
      rtac @{thm card_of_mono1}, rtac subsetI, rtac @{thm image_eqI}, rtac sym,
blanchet@48975
  1329
      rtac Rep_inverse, REPEAT_DETERM o eresolve_tac [CollectE, conjE],
blanchet@49305
  1330
      rtac set_mp, rtac equalityD2, rtac @{thm sym[OF proj_image]}, rtac imageE,
blanchet@49305
  1331
      rtac set_rev_mp, rtac mor_image, rtac mor_Rep, rtac UNIV_I, rtac equalityD2,
blanchet@48975
  1332
      rtac @{thm proj_image},  rtac @{thm image_eqI}, atac,
blanchet@49305
  1333
      ftac (carT_def RS equalityD1 RS set_mp),
blanchet@48975
  1334
      REPEAT_DETERM o eresolve_tac [CollectE, exE, conjE], hyp_subst_tac,
blanchet@49305
  1335
      rtac (carT_def RS equalityD2 RS set_mp), rtac CollectI, REPEAT_DETERM o rtac exI,
blanchet@48975
  1336
      rtac conjI, rtac refl, rtac conjI, etac conjI, etac conjI, etac conjI, rtac conjI,
blanchet@48975
  1337
      rtac ballI, dtac bspec, atac, REPEAT_DETERM o etac conjE, rtac conjI,
blanchet@48975
  1338
      CONJ_WRAP_GEN' (etac disjE) (fn (i, isNode_hset) =>
blanchet@48975
  1339
        EVERY' [rtac (mk_disjIN n i), rtac isNode_hset, atac, atac, atac, in_hin_tac])
blanchet@48975
  1340
      (1 upto n ~~ isNode_hsets),
blanchet@48975
  1341
      CONJ_WRAP' (fn isNode_hset =>
blanchet@48975
  1342
        EVERY' [rtac ballI, rtac isNode_hset, atac, ftac CollectD, etac @{thm SuccD},
blanchet@48975
  1343
          etac bspec, atac, in_hin_tac])
blanchet@48975
  1344
      isNode_hsets,
blanchet@48975
  1345
      atac, rtac isNode_hset, atac, atac, atac, in_hin_tac] 1
blanchet@48975
  1346
  end;
blanchet@48975
  1347
blanchet@48975
  1348
fun mk_bd_card_order_tac sbd_card_order =
blanchet@48975
  1349
  EVERY' (map rtac [@{thm card_order_ccexp}, sbd_card_order, sbd_card_order]) 1;
blanchet@48975
  1350
blanchet@48975
  1351
fun mk_bd_cinfinite_tac sbd_Cinfinite =
blanchet@48975
  1352
  EVERY' (map rtac [@{thm cinfinite_ccexp}, @{thm ctwo_ordLeq_Cinfinite},
blanchet@48975
  1353
    sbd_Cinfinite, sbd_Cinfinite]) 1;
blanchet@48975
  1354
blanchet@48975
  1355
fun mk_pickWP_assms_tac set_incl_hsets set_incl_hins map_eq =
blanchet@48975
  1356
  let
blanchet@48975
  1357
    val m = length set_incl_hsets;
blanchet@48975
  1358
  in
blanchet@48975
  1359
    EVERY' [REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac CollectI,
blanchet@48975
  1360
      EVERY' (map (fn thm => rtac conjI THEN' etac (thm RS @{thm subset_trans})) set_incl_hsets),
blanchet@48975
  1361
      CONJ_WRAP' (fn thm => rtac thm THEN' REPEAT_DETERM_N m o atac) set_incl_hins,
blanchet@48975
  1362
      REPEAT_DETERM o eresolve_tac [CollectE, conjE], rtac CollectI,
blanchet@48975
  1363
      EVERY' (map (fn thm => rtac conjI THEN' etac (thm RS @{thm subset_trans})) set_incl_hsets),
blanchet@48975
  1364
      CONJ_WRAP' (fn thm => rtac thm THEN' REPEAT_DETERM_N m o atac) set_incl_hins,
blanchet@48975
  1365
      REPEAT_DETERM o eresolve_tac [CollectE, conjE], etac map_eq]
blanchet@48975
  1366
  end;
blanchet@48975
  1367
blanchet@48975
  1368
fun mk_coalg_thePull_tac m coalg_def map_wpulls set_naturalss pickWP_assms_tacs
blanchet@48975
  1369
  {context = ctxt, prems = _} =
blanchet@49504
  1370
  unfold_thms_tac ctxt [coalg_def] THEN
blanchet@48975
  1371
  CONJ_WRAP' (fn (map_wpull, (pickWP_assms_tac, set_naturals)) =>
blanchet@48975
  1372
    EVERY' [rtac ballI, dtac @{thm set_mp[OF equalityD1[OF thePull_def]]},
blanchet@48975
  1373
      REPEAT_DETERM o eresolve_tac [CollectE, @{thm prod_caseE}],
blanchet@48975
  1374
      hyp_subst_tac, rtac rev_mp, rtac (map_wpull RS @{thm pickWP(1)}),
blanchet@48975
  1375
      EVERY' (map (etac o mk_conjunctN m) (1 upto m)),
blanchet@48975
  1376
      pickWP_assms_tac,
blanchet@49504
  1377
      SELECT_GOAL (unfold_thms_tac ctxt @{thms o_apply prod.cases}), rtac impI,
blanchet@48975
  1378
      REPEAT_DETERM o eresolve_tac [CollectE, conjE],
blanchet@48975
  1379
      rtac CollectI,
traytel@49490
  1380
      REPEAT_DETERM_N m o (rtac conjI THEN' rtac subset_UNIV),
blanchet@48975
  1381
      CONJ_WRAP' (fn set_natural =>
blanchet@49305
  1382
        EVERY' [rtac ord_eq_le_trans, rtac trans, rtac set_natural,
blanchet@49305
  1383
          rtac trans_fun_cong_image_id_id_apply, atac])
blanchet@48975
  1384
      (drop m set_naturals)])
blanchet@48975
  1385
  (map_wpulls ~~ (pickWP_assms_tacs ~~ set_naturalss)) 1;
blanchet@48975
  1386
blanchet@48975
  1387
fun mk_mor_thePull_nth_tac conv pick m mor_def map_wpulls map_comps pickWP_assms_tacs
blanchet@48975
  1388
  {context = ctxt, prems = _} =
blanchet@48975
  1389
  let
blanchet@48975
  1390
    val n = length map_comps;
blanchet@48975
  1391
  in
blanchet@49504
  1392
    unfold_thms_tac ctxt [mor_def] THEN
blanchet@48975
  1393
    EVERY' [rtac conjI,
blanchet@48975
  1394
      CONJ_WRAP' (K (rtac ballI THEN' rtac UNIV_I)) (1 upto n),
blanchet@48975
  1395
      CONJ_WRAP' (fn (map_wpull, (pickWP_assms_tac, map_comp)) =>
blanchet@48975
  1396
        EVERY' [rtac ballI, dtac @{thm set_mp[OF equalityD1[OF thePull_def]]},
blanchet@48975
  1397
          REPEAT_DETERM o eresolve_tac [CollectE, @{thm prod_caseE}, conjE],
blanchet@48975
  1398
          hyp_subst_tac,
blanchet@49504
  1399
          SELECT_GOAL (unfold_thms_tac ctxt @{thms o_apply prod.cases}),
blanchet@48975
  1400
          rtac (map_comp RS trans),
blanchet@49504
  1401
          SELECT_GOAL (unfold_thms_tac ctxt (conv :: @{thms o_id id_o})),
blanchet@48975
  1402
          rtac (map_wpull RS pick), REPEAT_DETERM_N m o atac,
blanchet@48975
  1403
          pickWP_assms_tac])
blanchet@48975
  1404
      (map_wpulls ~~ (pickWP_assms_tacs ~~ map_comps))] 1
blanchet@48975
  1405
  end;
blanchet@48975
  1406
blanchet@48975
  1407
val mk_mor_thePull_fst_tac = mk_mor_thePull_nth_tac @{thm fst_conv} @{thm pickWP(2)};
blanchet@48975
  1408
val mk_mor_thePull_snd_tac = mk_mor_thePull_nth_tac @{thm snd_conv} @{thm pickWP(3)};
blanchet@48975
  1409
blanchet@49504
  1410
fun mk_mor_thePull_pick_tac mor_def unfolds map_comps {context = ctxt, prems = _} =
blanchet@49504
  1411
  unfold_thms_tac ctxt [mor_def, @{thm thePull_def}] THEN rtac conjI 1 THEN
blanchet@49504
  1412
  CONJ_WRAP' (K (rtac ballI THEN' rtac UNIV_I)) unfolds 1 THEN
blanchet@49504
  1413
  CONJ_WRAP' (fn (unfold, map_comp) =>
blanchet@48975
  1414
    EVERY' [rtac ballI, REPEAT_DETERM o eresolve_tac [CollectE, @{thm prod_caseE}, conjE],
blanchet@48975
  1415
      hyp_subst_tac,
blanchet@49504
  1416
      SELECT_GOAL (unfold_thms_tac ctxt (unfold :: map_comp :: @{thms comp_def id_def})),
blanchet@48975
  1417
      rtac refl])
blanchet@49504
  1418
  (unfolds ~~ map_comps) 1;
blanchet@48975
  1419
blanchet@49504
  1420
fun mk_pick_col_tac m j cts rec_0s rec_Sucs unfolds set_naturalss map_wpulls pickWP_assms_tacs
blanchet@48975
  1421
  {context = ctxt, prems = _} =
blanchet@48975
  1422
  let
blanchet@48975
  1423
    val n = length rec_0s;
blanchet@48975
  1424
    val ks = n downto 1;
blanchet@48975
  1425
  in
blanchet@49305
  1426
    EVERY' [rtac (Drule.instantiate' [] cts nat_induct),
blanchet@48975
  1427
      REPEAT_DETERM o rtac allI,
blanchet@48975
  1428
      CONJ_WRAP' (fn thm => EVERY'
blanchet@49305
  1429
        [rtac impI, rtac ord_eq_le_trans, rtac thm, rtac @{thm empty_subsetI}]) rec_0s,
blanchet@48975
  1430
      REPEAT_DETERM o rtac allI,
blanchet@49504
  1431
      CONJ_WRAP' (fn (rec_Suc, ((unfold, set_naturals), (map_wpull, pickWP_assms_tac))) =>
blanchet@48975
  1432
        EVERY' [rtac impI, dtac @{thm set_mp[OF equalityD1[OF thePull_def]]},
blanchet@48975
  1433
          REPEAT_DETERM o eresolve_tac [CollectE, @{thm prod_caseE}],
blanchet@48975
  1434
          hyp_subst_tac, rtac rev_mp, rtac (map_wpull RS @{thm pickWP(1)}),
blanchet@48975
  1435
          EVERY' (map (etac o mk_conjunctN m) (1 upto m)),
blanchet@48975
  1436
          pickWP_assms_tac,
blanchet@48975
  1437
          rtac impI, REPEAT_DETERM o eresolve_tac [CollectE, conjE],
blanchet@49305
  1438
          rtac ord_eq_le_trans, rtac rec_Suc,
blanchet@48975
  1439
          rtac @{thm Un_least},
blanchet@49504
  1440
          SELECT_GOAL (unfold_thms_tac ctxt [unfold, nth set_naturals (j - 1),
blanchet@48975
  1441
            @{thm prod.cases}]),
blanchet@49305
  1442
          etac ord_eq_le_trans_trans_fun_cong_image_id_id_apply,
blanchet@48975
  1443
          CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least})) (fn (i, set_natural) =>
blanchet@48975
  1444
            EVERY' [rtac @{thm UN_least},
blanchet@49504
  1445
              SELECT_GOAL (unfold_thms_tac ctxt [unfold, set_natural, @{thm prod.cases}]),
blanchet@48975
  1446
              etac imageE, hyp_subst_tac, REPEAT_DETERM o etac allE,
blanchet@49305
  1447
              dtac (mk_conjunctN n i), etac mp, etac set_mp, atac])
blanchet@48975
  1448
          (ks ~~ rev (drop m set_naturals))])
blanchet@49504
  1449
      (rec_Sucs ~~ ((unfolds ~~ set_naturalss) ~~ (map_wpulls ~~ pickWP_assms_tacs)))] 1
blanchet@48975
  1450
  end;
blanchet@48975
  1451
blanchet@48975
  1452
fun mk_wpull_tac m coalg_thePull mor_thePull_fst mor_thePull_snd mor_thePull_pick
blanchet@48975
  1453
  mor_unique pick_cols hset_defs =
traytel@49488
  1454
  EVERY' [rtac (@{thm wpull_def} RS iffD2), REPEAT_DETERM o rtac allI, rtac impI,
blanchet@48975
  1455
    REPEAT_DETERM o etac conjE, rtac bexI, rtac conjI,
blanchet@48975
  1456
    rtac box_equals, rtac mor_unique,
blanchet@48975
  1457
    rtac coalg_thePull, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
blanchet@48975
  1458
    rtac mor_thePull_pick, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
blanchet@48975
  1459
    rtac mor_thePull_fst, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
blanchet@48975
  1460
    rtac @{thm set_mp[OF equalityD2[OF thePull_def]]}, rtac CollectI,
blanchet@48975
  1461
    rtac @{thm prod_caseI}, etac conjI, etac conjI, atac, rtac o_apply, rtac @{thm fst_conv},
blanchet@48975
  1462
    rtac box_equals, rtac mor_unique,
blanchet@48975
  1463
    rtac coalg_thePull, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
blanchet@48975
  1464
    rtac mor_thePull_pick, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
blanchet@48975
  1465
    rtac mor_thePull_snd, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
blanchet@48975
  1466
    rtac @{thm set_mp[OF equalityD2[OF thePull_def]]}, rtac CollectI,
blanchet@48975
  1467
    rtac @{thm prod_caseI}, etac conjI, etac conjI, atac, rtac o_apply, rtac @{thm snd_conv},
blanchet@48975
  1468
    rtac CollectI,
blanchet@48975
  1469
    CONJ_WRAP' (fn (pick, def) =>
blanchet@49305
  1470
      EVERY' [rtac (def RS ord_eq_le_trans), rtac @{thm UN_least},
blanchet@48975
  1471
        rtac pick, REPEAT_DETERM_N (m - 1) o etac conjI, atac,
blanchet@48975
  1472
        rtac @{thm set_mp[OF equalityD2[OF thePull_def]]}, rtac CollectI,
blanchet@48975
  1473
        rtac @{thm prod_caseI}, etac conjI, etac conjI, atac])
blanchet@48975
  1474
    (pick_cols ~~ hset_defs)] 1;
blanchet@48975
  1475
blanchet@49501
  1476
fun mk_wit_tac n dtor_ctors set_simp wit coind_wits {context = ctxt, prems = _} =
traytel@49104
  1477
  ALLGOALS (TRY o (eresolve_tac coind_wits THEN' rtac refl)) THEN
blanchet@48975
  1478
  REPEAT_DETERM (atac 1 ORELSE
blanchet@49305
  1479
    EVERY' [dtac set_rev_mp, rtac equalityD1, resolve_tac set_simp,
blanchet@49504
  1480
    K (unfold_thms_tac ctxt dtor_ctors),
blanchet@48975
  1481
    REPEAT_DETERM_N n o etac UnE,
blanchet@48975
  1482
    REPEAT_DETERM o
blanchet@48975
  1483
      (TRY o REPEAT_DETERM o etac UnE THEN' TRY o etac @{thm UN_E} THEN'
blanchet@48975
  1484
        (eresolve_tac wit ORELSE'
blanchet@48975
  1485
        (dresolve_tac wit THEN'
blanchet@48975
  1486
          (etac FalseE ORELSE'
blanchet@49305
  1487
          EVERY' [hyp_subst_tac, dtac set_rev_mp, rtac equalityD1, resolve_tac set_simp,
blanchet@49504
  1488
            K (unfold_thms_tac ctxt dtor_ctors), REPEAT_DETERM_N n o etac UnE]))))] 1);
blanchet@48975
  1489
blanchet@49504
  1490
fun mk_coind_wit_tac induct unfolds set_nats wits {context = ctxt, prems = _} =
traytel@49104
  1491
  rtac induct 1 THEN ALLGOALS (TRY o rtac impI THEN' TRY o hyp_subst_tac) THEN
blanchet@49504
  1492
  unfold_thms_tac ctxt (unfolds @ set_nats @ @{thms image_id id_apply}) THEN
traytel@49104
  1493
  ALLGOALS (REPEAT_DETERM o etac imageE THEN' TRY o hyp_subst_tac) THEN
traytel@49104
  1494
  ALLGOALS (TRY o
traytel@49104
  1495
    FIRST' [rtac TrueI, rtac refl, etac (refl RSN (2, mp)), dresolve_tac wits THEN' etac FalseE])
traytel@49104
  1496
blanchet@49501
  1497
fun mk_rel_simp_tac in_Jrels i in_rel map_comp map_cong map_simp set_simps dtor_inject dtor_ctor
blanchet@48975
  1498
  set_naturals set_incls set_set_inclss =
blanchet@48975
  1499
  let
blanchet@48975
  1500
    val m = length set_incls;
blanchet@48975
  1501
    val n = length set_set_inclss;
blanchet@48975
  1502
    val (passive_set_naturals, active_set_naturals) = chop m set_naturals;
blanchet@48975
  1503
    val in_Jrel = nth in_Jrels (i - 1);
blanchet@48975
  1504
    val if_tac =
blanchet@48975
  1505
      EVERY' [dtac (in_Jrel RS iffD1), REPEAT_DETERM o eresolve_tac [exE, conjE, CollectE],
blanchet@48975
  1506
        rtac (in_rel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
blanchet@48975
  1507
        EVERY' (map2 (fn set_natural => fn set_incl =>
blanchet@49305
  1508
          EVERY' [rtac conjI, rtac ord_eq_le_trans, rtac set_natural,
blanchet@49305
  1509
            rtac ord_eq_le_trans, rtac trans_fun_cong_image_id_id_apply,
blanchet@48975
  1510
            etac (set_incl RS @{thm subset_trans})])
blanchet@48975
  1511
        passive_set_naturals set_incls),
blanchet@48975
  1512
        CONJ_WRAP' (fn (in_Jrel, (set_natural, set_set_incls)) =>
blanchet@49305
  1513
          EVERY' [rtac ord_eq_le_trans, rtac set_natural, rtac @{thm image_subsetI},
blanchet@48975
  1514
            rtac (in_Jrel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
blanchet@48975
  1515
            CONJ_WRAP' (fn thm => etac (thm RS @{thm subset_trans}) THEN' atac) set_set_incls,
blanchet@48975
  1516
            rtac conjI, rtac refl, rtac refl])
blanchet@48975
  1517
        (in_Jrels ~~ (active_set_naturals ~~ set_set_inclss)),
blanchet@48975
  1518
        CONJ_WRAP' (fn conv =>
blanchet@48975
  1519
          EVERY' [rtac trans, rtac map_comp, rtac trans, rtac map_cong,
blanchet@48975
  1520
          REPEAT_DETERM_N m o rtac @{thm fun_cong[OF o_id]},
blanchet@48975
  1521
          REPEAT_DETERM_N n o EVERY' (map rtac [trans, o_apply, conv]),
blanchet@49501
  1522
          rtac trans, rtac sym, rtac map_simp, rtac (dtor_inject RS iffD2), atac])
blanchet@48975
  1523
        @{thms fst_conv snd_conv}];
blanchet@48975
  1524
    val only_if_tac =
blanchet@48975
  1525
      EVERY' [dtac (in_rel RS iffD1), REPEAT_DETERM o eresolve_tac [exE, conjE, CollectE],
blanchet@48975
  1526
        rtac (in_Jrel RS iffD2), rtac exI, rtac conjI, rtac CollectI,
blanchet@48975
  1527
        CONJ_WRAP' (fn (set_simp, passive_set_natural) =>
blanchet@49305
  1528
          EVERY' [rtac ord_eq_le_trans, rtac set_simp, rtac @{thm Un_least},
blanchet@49305
  1529
            rtac ord_eq_le_trans, rtac box_equals, rtac passive_set_natural,
blanchet@49501
  1530
            rtac (dtor_ctor RS sym RS arg_cong), rtac trans_fun_cong_image_id_id_apply, atac,
blanchet@48975
  1531
            CONJ_WRAP_GEN' (rtac (Thm.permute_prems 0 1 @{thm Un_least}))
blanchet@49305
  1532
              (fn (active_set_natural, in_Jrel) => EVERY' [rtac ord_eq_le_trans,
blanchet@48975
  1533
                rtac @{thm UN_cong[OF _ refl]}, rtac @{thm box_equals[OF _ _ refl]},
blanchet@49501
  1534
                rtac active_set_natural, rtac (dtor_ctor RS sym RS arg_cong), rtac @{thm UN_least},
blanchet@49305
  1535
                dtac set_rev_mp, etac @{thm image_mono}, etac imageE,
blanchet@48975
  1536
                dtac @{thm ssubst_mem[OF pair_collapse]}, dtac (in_Jrel RS iffD1),
blanchet@48975
  1537
                dtac @{thm someI_ex}, REPEAT_DETERM o etac conjE,
blanchet@48975
  1538
                dtac (Thm.permute_prems 0 1 @{thm ssubst_mem}), atac,
blanchet@48975
  1539
                hyp_subst_tac, REPEAT_DETERM o eresolve_tac [CollectE, conjE], atac])
blanchet@48975
  1540
            (rev (active_set_naturals ~~ in_Jrels))])
blanchet@48975
  1541
        (set_simps ~~ passive_set_naturals),
blanchet@48975
  1542
        rtac conjI,
blanchet@49501
  1543
        REPEAT_DETERM_N 2 o EVERY'[rtac (dtor_inject RS iffD1), rtac trans, rtac map_simp,
blanchet@49501
  1544
          rtac box_equals, rtac map_comp, rtac (dtor_ctor RS sym RS arg_cong), rtac trans,
blanchet@48975
  1545
          rtac map_cong, REPEAT_DETERM_N m o rtac @{thm fun_cong[OF o_id]},
blanchet@49305
  1546
          EVERY' (map (fn in_Jrel => EVERY' [rtac trans, rtac o_apply, dtac set_rev_mp, atac,
blanchet@48975
  1547
            dtac @{thm ssubst_mem[OF pair_collapse]}, dtac (in_Jrel RS iffD1), dtac @{thm someI_ex},
blanchet@48975
  1548
            REPEAT_DETERM o etac conjE, atac]) in_Jrels),
blanchet@48975
  1549
          atac]]
blanchet@48975
  1550
  in
blanchet@48975
  1551
    EVERY' [rtac iffI, if_tac, only_if_tac] 1
blanchet@48975
  1552
  end;
blanchet@48975
  1553
blanchet@48975
  1554
end;