TPTP Problem File: SLH0249^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Query_Optimization/0013_IKKBZ_Optimality/prob_05696_258855__15991866_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1532 ( 475 unt; 252 typ; 0 def)
% Number of atoms : 4429 (1544 equ; 0 cnn)
% Maximal formula atoms : 20 ( 3 avg)
% Number of connectives : 15699 ( 502 ~; 106 |; 536 &;12428 @)
% ( 0 <=>;2127 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 8 avg)
% Number of types : 35 ( 34 usr)
% Number of type conns : 797 ( 797 >; 0 *; 0 +; 0 <<)
% Number of symbols : 219 ( 218 usr; 18 con; 0-4 aty)
% Number of variables : 4369 ( 85 ^;3802 !; 482 ?;4369 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:08:34.440
%------------------------------------------------------------------------------
% Could-be-implicit typings (34)
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member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
member_set_list_a: set_list_a > set_set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
member1816616512716248880od_a_a: set_Product_prod_a_a > set_se5735800977113168103od_a_a > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__b_J,type,
member_set_b: set_b > set_set_b > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_T,type,
t: pre_pr7278220950009878019t_unit ).
thf(sy_v_t1____,type,
t1: dtree_list_a_b ).
thf(sy_v_t2____,type,
t2: dtree_list_a_b ).
thf(sy_v_v1,type,
v1: list_a ).
thf(sy_v_v2,type,
v2: list_a ).
% Relevant facts (1279)
thf(fact_0_False,axiom,
t1 != t2 ).
% False
thf(fact_1_assms_I5_J,axiom,
v1 != v2 ).
% assms(5)
thf(fact_2__092_060open_062set_Av2_A_092_060inter_062_Adlverts_At1_A_061_A_123_125_092_060close_062,axiom,
( ( inf_inf_set_a @ ( set_a2 @ v2 ) @ ( list_dlverts_a_b @ t1 ) )
= bot_bot_set_a ) ).
% \<open>set v2 \<inter> dlverts t1 = {}\<close>
thf(fact_3_loopfree_Oadj__not__same,axiom,
! [A: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ ( arcs_ends_a_b @ t ) ) ).
% loopfree.adj_not_same
thf(fact_4_reach__t1,axiom,
! [X: a] :
( ( member_a @ X @ ( set_a2 @ v1 ) )
=> ! [Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ t1 ) ) ) ) ).
% reach_t1
thf(fact_5_reachable1__not__reverse,axiom,
! [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y2 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ).
% reachable1_not_reverse
thf(fact_6_reachable1__from__outside__dom,axiom,
! [X2: a,Y2: a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ( ( member_a @ Y2 @ ( set_a2 @ Ys ) )
=> ? [X3: a,X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Ys ) )
& ~ ( member_a @ X3 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X4 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% reachable1_from_outside_dom
thf(fact_7_loopfree_Oloopfree__digraph__axioms,axiom,
loopfree_digraph_a_b @ t ).
% loopfree.loopfree_digraph_axioms
thf(fact_8_nomulti_Onomulti__digraph__axioms,axiom,
nomulti_digraph_a_b @ t ).
% nomulti.nomulti_digraph_axioms
thf(fact_9_source__nmem__k__nh,axiom,
! [V: a,W: b > real,K: real] :
~ ( member_a @ V @ ( graph_3921080825633621230od_a_b @ t @ W @ V @ K ) ) ).
% source_nmem_k_nh
thf(fact_10_cycle__free,axiom,
~ ? [X_1: list_b] : ( arc_pre_cycle_a_b @ t @ X_1 ) ).
% cycle_free
thf(fact_11_before__ArcI,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ S1 ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ S2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% before_ArcI
thf(fact_12_reachable1__in__verts_I2_J,axiom,
! [U: a,V: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% reachable1_in_verts(2)
thf(fact_13_reachable1__in__verts_I1_J,axiom,
! [U: a,V: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% reachable1_in_verts(1)
thf(fact_14_scc__of__eq,axiom,
! [U: a,V: a] :
( ( member_a @ U @ ( digrap2937667069914300949of_a_b @ t @ V ) )
=> ( ( digrap2937667069914300949of_a_b @ t @ U )
= ( digrap2937667069914300949of_a_b @ t @ V ) ) ) ).
% scc_of_eq
thf(fact_15_reachable1__reachable__trans,axiom,
! [U: a,V: a,W: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( reachable_a_b @ t @ V @ W )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% reachable1_reachable_trans
thf(fact_16_reachable__reachable1__trans,axiom,
! [U: a,V: a,W: a] :
( ( reachable_a_b @ t @ U @ V )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ V @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% reachable_reachable1_trans
thf(fact_17_adj__in__verts_I2_J,axiom,
! [U: a,V: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% adj_in_verts(2)
thf(fact_18_adj__in__verts_I1_J,axiom,
! [U: a,V: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% adj_in_verts(1)
thf(fact_19_reachable__trans,axiom,
! [U: a,V: a,W: a] :
( ( reachable_a_b @ t @ U @ V )
=> ( ( reachable_a_b @ t @ V @ W )
=> ( reachable_a_b @ t @ U @ W ) ) ) ).
% reachable_trans
thf(fact_20_non__empty,axiom,
( ( pre_ve642382030648772252t_unit @ t )
!= bot_bot_set_a ) ).
% non_empty
thf(fact_21_reachable__in__verts_I1_J,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% reachable_in_verts(1)
thf(fact_22_reachable__in__verts_I2_J,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% reachable_in_verts(2)
thf(fact_23_in__scc__of__self,axiom,
! [U: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( member_a @ U @ ( digrap2937667069914300949of_a_b @ t @ U ) ) ) ).
% in_scc_of_self
thf(fact_24_k__nh__reachable,axiom,
! [U: a,W: b > real,V: a,K: real] :
( ( member_a @ U @ ( graph_3921080825633621230od_a_b @ t @ W @ V @ K ) )
=> ( reachable_a_b @ t @ V @ U ) ) ).
% k_nh_reachable
thf(fact_25_reachable__via__child__impl__same,axiom,
! [X2: a,V: a,Y2: a,U: a] :
( ( reachable_a_b @ t @ X2 @ V )
=> ( ( reachable_a_b @ t @ Y2 @ V )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ X2 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ Y2 ) @ ( arcs_ends_a_b @ t ) )
=> ( X2 = Y2 ) ) ) ) ) ).
% reachable_via_child_impl_same
thf(fact_26_reachable__adj__trans,axiom,
! [A: a,B: a,C: a] :
( ( reachable_a_b @ t @ A @ B )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ C ) @ ( arcs_ends_a_b @ t ) )
=> ( reachable_a_b @ t @ A @ C ) ) ) ).
% reachable_adj_trans
thf(fact_27_adj__reachable__trans,axiom,
! [A: a,B: a,C: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( arcs_ends_a_b @ t ) )
=> ( ( reachable_a_b @ t @ B @ C )
=> ( reachable_a_b @ t @ A @ C ) ) ) ).
% adj_reachable_trans
thf(fact_28_scc__of__empty__conv,axiom,
! [U: a] :
( ( ( digrap2937667069914300949of_a_b @ t @ U )
= bot_bot_set_a )
= ( ~ ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% scc_of_empty_conv
thf(fact_29_reachable__induct,axiom,
! [U: a,V: a,P: a > $o] :
( ( reachable_a_b @ t @ U @ V )
=> ( ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( P @ U ) )
=> ( ! [X4: a,Y3: a] :
( ( reachable_a_b @ t @ U @ X4 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( P @ X4 )
=> ( P @ Y3 ) ) ) )
=> ( P @ V ) ) ) ) ).
% reachable_induct
thf(fact_30_converse__reachable__induct,axiom,
! [U: a,V: a,P: a > $o] :
( ( reachable_a_b @ t @ U @ V )
=> ( ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( P @ V ) )
=> ( ! [X4: a,Y3: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( reachable_a_b @ t @ Y3 @ V )
=> ( ( P @ Y3 )
=> ( P @ X4 ) ) ) )
=> ( P @ U ) ) ) ) ).
% converse_reachable_induct
thf(fact_31_converse__reachable__cases,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
=> ( ( ( U = V )
=> ~ ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) )
=> ~ ! [W2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ W2 ) @ ( arcs_ends_a_b @ t ) )
=> ~ ( reachable_a_b @ t @ W2 @ V ) ) ) ) ).
% converse_reachable_cases
thf(fact_32__092_060open_062dlverts_At1_A_092_060inter_062_Adlverts_At2_A_061_A_123_125_092_060close_062,axiom,
( ( inf_inf_set_a @ ( list_dlverts_a_b @ t1 ) @ ( list_dlverts_a_b @ t2 ) )
= bot_bot_set_a ) ).
% \<open>dlverts t1 \<inter> dlverts t2 = {}\<close>
thf(fact_33_reachable__refl,axiom,
! [V: a] :
( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( reachable_a_b @ t @ V @ V ) ) ).
% reachable_refl
thf(fact_34_reachable__adjI,axiom,
! [U: a,V: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ t ) )
=> ( reachable_a_b @ t @ U @ V ) ) ).
% reachable_adjI
thf(fact_35_reachable__neq__reachable1,axiom,
! [V: a,W: a] :
( ( reachable_a_b @ t @ V @ W )
=> ( ( V != W )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ V @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% reachable_neq_reachable1
thf(fact_36_reachable1__reachable,axiom,
! [V: a,W: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ V @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( reachable_a_b @ t @ V @ W ) ) ).
% reachable1_reachable
thf(fact_37_reachable__arc__trans,axiom,
! [U: a,V: a,E: b,W: a] :
( ( reachable_a_b @ t @ U @ V )
=> ( ( wf_arc_a_b @ t @ E @ ( product_Pair_a_a @ V @ W ) )
=> ( reachable_a_b @ t @ U @ W ) ) ) ).
% reachable_arc_trans
thf(fact_38_directed__tree_Obefore_Ocong,axiom,
iKKBZ_7682935289300565975re_a_b = iKKBZ_7682935289300565975re_a_b ).
% directed_tree.before.cong
thf(fact_39_closed__w__imp__cycle,axiom,
! [P2: list_b] :
( ( arc_wf_closed_w_a_b @ t @ P2 )
=> ? [X_12: list_b] : ( arc_pre_cycle_a_b @ t @ X_12 ) ) ).
% closed_w_imp_cycle
thf(fact_40_before__def,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
= ( ( iKKBZ_4622586873178280511rm_a_b @ t @ S1 )
& ( iKKBZ_4622586873178280511rm_a_b @ t @ S2 )
& ( ( inf_inf_set_a @ ( set_a2 @ S1 ) @ ( set_a2 @ S2 ) )
= bot_bot_set_a )
& ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ S1 ) )
& ? [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ S2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y4 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% before_def
thf(fact_41__092_060open_062set_Av1_A_092_060subseteq_062_Adlverts_At1_092_060close_062,axiom,
ord_less_eq_set_a @ ( set_a2 @ v1 ) @ ( list_dlverts_a_b @ t1 ) ).
% \<open>set v1 \<subseteq> dlverts t1\<close>
thf(fact_42_scc__of__in__sccs__verts,axiom,
! [U: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( member_set_a @ ( digrap2937667069914300949of_a_b @ t @ U ) @ ( digrap2871191568752656621ts_a_b @ t ) ) ) ).
% scc_of_in_sccs_verts
thf(fact_43_merge__in__verts,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( graph_2957805489637798020ts_a_b @ t ) )
=> ( member_a @ X2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% merge_in_verts
thf(fact_44_merging__empty,axiom,
( ( graph_2957805489637798020ts_a_b @ t )
= bot_bot_set_a ) ).
% merging_empty
thf(fact_45_mem__Collect__eq,axiom,
! [A: product_prod_a_a,P: product_prod_a_a > $o] :
( ( member1426531477525435216od_a_a @ A @ ( collec3336397797384452498od_a_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
! [A: b,P: b > $o] :
( ( member_b @ A @ ( collect_b @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A2: set_Product_prod_a_a] :
( ( collec3336397797384452498od_a_a
@ ^ [X5: product_prod_a_a] : ( member1426531477525435216od_a_a @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X5: list_a] : ( member_list_a @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X5: set_a] : ( member_set_a @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X5: b] : ( member_b @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X5: a] : ( member_a @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_55_inf__bot__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_56_inf__bot__left,axiom,
! [X2: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X2 )
= bot_bot_set_b ) ).
% inf_bot_left
thf(fact_57_inf__bot__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_58_inf__bot__right,axiom,
! [X2: set_b] :
( ( inf_inf_set_b @ X2 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% inf_bot_right
thf(fact_59_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X2 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_60_boolean__algebra_Oconj__zero__left,axiom,
! [X2: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ X2 )
= bot_bot_set_b ) ).
% boolean_algebra.conj_zero_left
thf(fact_61_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_62_boolean__algebra_Oconj__zero__right,axiom,
! [X2: set_b] :
( ( inf_inf_set_b @ X2 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% boolean_algebra.conj_zero_right
thf(fact_63_in__sccs__verts__conv__reachable,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) )
= ( ( S != bot_bot_set_a )
& ! [X5: a] :
( ( member_a @ X5 @ S )
=> ! [Y4: a] :
( ( member_a @ Y4 @ S )
=> ( reachable_a_b @ t @ X5 @ Y4 ) ) )
& ! [X5: a] :
( ( member_a @ X5 @ S )
=> ! [V2: a] :
( ~ ( member_a @ V2 @ S )
=> ( ~ ( reachable_a_b @ t @ X5 @ V2 )
| ~ ( reachable_a_b @ t @ V2 @ X5 ) ) ) ) ) ) ).
% in_sccs_verts_conv_reachable
thf(fact_64_sccs__verts__disjoint,axiom,
! [S: set_a,T: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( ( member_set_a @ T @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( ( S != T )
=> ( ( inf_inf_set_a @ S @ T )
= bot_bot_set_a ) ) ) ) ).
% sccs_verts_disjoint
thf(fact_65_before__conform1I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ S1 ) ) ).
% before_conform1I
thf(fact_66_before__conform2I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ S2 ) ) ).
% before_conform2I
thf(fact_67_inf__right__idem,axiom,
! [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Y2 )
= ( inf_inf_set_a @ X2 @ Y2 ) ) ).
% inf_right_idem
thf(fact_68_inf_Oright__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ B )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.right_idem
thf(fact_69_inf__left__idem,axiom,
! [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ X2 @ Y2 ) )
= ( inf_inf_set_a @ X2 @ Y2 ) ) ).
% inf_left_idem
thf(fact_70_inf_Oleft__idem,axiom,
! [A: set_a,B: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B ) )
= ( inf_inf_set_a @ A @ B ) ) ).
% inf.left_idem
thf(fact_71_inf__idem,axiom,
! [X2: set_a] :
( ( inf_inf_set_a @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_72_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_73_sccs__verts__subsets,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( ord_less_eq_set_a @ S @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% sccs_verts_subsets
thf(fact_74_inf_Obounded__iff,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
= ( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_75_inf_Obounded__iff,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ ( inf_inf_set_b @ B @ C ) )
= ( ( ord_less_eq_set_b @ A @ B )
& ( ord_less_eq_set_b @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_76_le__inf__iff,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z ) )
= ( ( ord_less_eq_set_a @ X2 @ Y2 )
& ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_77_le__inf__iff,axiom,
! [X2: set_b,Y2: set_b,Z: set_b] :
( ( ord_less_eq_set_b @ X2 @ ( inf_inf_set_b @ Y2 @ Z ) )
= ( ( ord_less_eq_set_b @ X2 @ Y2 )
& ( ord_less_eq_set_b @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_78_last__merge__is__merge,axiom,
! [Y2: a] :
( ( member_a @ Y2 @ ( graph_2659413520663303054ts_a_b @ t ) )
=> ( member_a @ Y2 @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ).
% last_merge_is_merge
thf(fact_79_is__chain_H__def,axiom,
( ( graph_8150681439568091980in_a_b @ t )
= ( ( graph_2957805489637798020ts_a_b @ t )
= bot_bot_set_a ) ) ).
% is_chain'_def
thf(fact_80_last__merge__alt,axiom,
! [X2: a] :
( ( member_a @ X2 @ ( graph_2659413520663303054ts_a_b @ t ) )
=> ! [Z2: a] :
( ( ( reachable_a_b @ t @ X2 @ Z2 )
& ( Z2 != X2 ) )
=> ~ ( member_a @ Z2 @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ) ).
% last_merge_alt
thf(fact_81_directed__tree_Oseq__conform_Ocong,axiom,
iKKBZ_4622586873178280511rm_a_b = iKKBZ_4622586873178280511rm_a_b ).
% directed_tree.seq_conform.cong
thf(fact_82_inf_OcoboundedI2,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_83_inf_OcoboundedI2,axiom,
! [B: set_b,C: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_84_inf_OcoboundedI1,axiom,
! [A: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_85_inf_OcoboundedI1,axiom,
! [A: set_b,C: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ C )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_86_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B2: set_a,A3: set_a] :
( ( inf_inf_set_a @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_87_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_b
= ( ^ [B2: set_b,A3: set_b] :
( ( inf_inf_set_b @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_88_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
( ( inf_inf_set_a @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_89_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B2: set_b] :
( ( inf_inf_set_b @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_90_inf_Ocobounded2,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_91_inf_Ocobounded2,axiom,
! [A: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_92_inf_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_93_inf_Ocobounded1,axiom,
! [A: set_b,B: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_94_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B2: set_a] :
( A3
= ( inf_inf_set_a @ A3 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_95_inf_Oorder__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A3: set_b,B2: set_b] :
( A3
= ( inf_inf_set_b @ A3 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_96_inf__greatest,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ X2 @ Z )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_97_inf__greatest,axiom,
! [X2: set_b,Y2: set_b,Z: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y2 )
=> ( ( ord_less_eq_set_b @ X2 @ Z )
=> ( ord_less_eq_set_b @ X2 @ ( inf_inf_set_b @ Y2 @ Z ) ) ) ) ).
% inf_greatest
thf(fact_98_inf_OboundedI,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ A @ C )
=> ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_99_inf_OboundedI,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ A @ C )
=> ( ord_less_eq_set_b @ A @ ( inf_inf_set_b @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_100_inf_OboundedE,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B @ C ) )
=> ~ ( ( ord_less_eq_set_a @ A @ B )
=> ~ ( ord_less_eq_set_a @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_101_inf_OboundedE,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ ( inf_inf_set_b @ B @ C ) )
=> ~ ( ( ord_less_eq_set_b @ A @ B )
=> ~ ( ord_less_eq_set_b @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_102_inf__absorb2,axiom,
! [Y2: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( ( inf_inf_set_a @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_103_inf__absorb2,axiom,
! [Y2: set_b,X2: set_b] :
( ( ord_less_eq_set_b @ Y2 @ X2 )
=> ( ( inf_inf_set_b @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_104_inf__absorb1,axiom,
! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( inf_inf_set_a @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_105_inf__absorb1,axiom,
! [X2: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y2 )
=> ( ( inf_inf_set_b @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_106_inf_Oabsorb2,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( inf_inf_set_a @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_107_inf_Oabsorb2,axiom,
! [B: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( inf_inf_set_b @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_108_inf_Oabsorb1,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( inf_inf_set_a @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_109_inf_Oabsorb1,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( inf_inf_set_b @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_110_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X5: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X5 @ Y4 )
= X5 ) ) ) ).
% le_iff_inf
thf(fact_111_le__iff__inf,axiom,
( ord_less_eq_set_b
= ( ^ [X5: set_b,Y4: set_b] :
( ( inf_inf_set_b @ X5 @ Y4 )
= X5 ) ) ) ).
% le_iff_inf
thf(fact_112_inf__unique,axiom,
! [F: set_a > set_a > set_a,X2: set_a,Y2: set_a] :
( ! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y3 ) @ X4 )
=> ( ! [X4: set_a,Y3: set_a] : ( ord_less_eq_set_a @ ( F @ X4 @ Y3 ) @ Y3 )
=> ( ! [X4: set_a,Y3: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ( ord_less_eq_set_a @ X4 @ Z3 )
=> ( ord_less_eq_set_a @ X4 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_113_inf__unique,axiom,
! [F: set_b > set_b > set_b,X2: set_b,Y2: set_b] :
( ! [X4: set_b,Y3: set_b] : ( ord_less_eq_set_b @ ( F @ X4 @ Y3 ) @ X4 )
=> ( ! [X4: set_b,Y3: set_b] : ( ord_less_eq_set_b @ ( F @ X4 @ Y3 ) @ Y3 )
=> ( ! [X4: set_b,Y3: set_b,Z3: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ( ord_less_eq_set_b @ X4 @ Z3 )
=> ( ord_less_eq_set_b @ X4 @ ( F @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf_set_b @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_114_inf_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( inf_inf_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ A @ B ) ) ).
% inf.orderI
thf(fact_115_inf_OorderI,axiom,
! [A: set_b,B: set_b] :
( ( A
= ( inf_inf_set_b @ A @ B ) )
=> ( ord_less_eq_set_b @ A @ B ) ) ).
% inf.orderI
thf(fact_116_inf_OorderE,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( A
= ( inf_inf_set_a @ A @ B ) ) ) ).
% inf.orderE
thf(fact_117_inf_OorderE,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( A
= ( inf_inf_set_b @ A @ B ) ) ) ).
% inf.orderE
thf(fact_118_le__infI2,axiom,
! [B: set_a,X2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_119_le__infI2,axiom,
! [B: set_b,X2: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B @ X2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_120_le__infI1,axiom,
! [A: set_a,X2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_121_le__infI1,axiom,
! [A: set_b,X2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ X2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_122_inf__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_123_inf__mono,axiom,
! [A: set_b,C: set_b,B: set_b,D: set_b] :
( ( ord_less_eq_set_b @ A @ C )
=> ( ( ord_less_eq_set_b @ B @ D )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ ( inf_inf_set_b @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_124_le__infI,axiom,
! [X2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X2 @ A )
=> ( ( ord_less_eq_set_a @ X2 @ B )
=> ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_125_le__infI,axiom,
! [X2: set_b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ X2 @ A )
=> ( ( ord_less_eq_set_b @ X2 @ B )
=> ( ord_less_eq_set_b @ X2 @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% le_infI
thf(fact_126_le__infE,axiom,
! [X2: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X2 @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X2 @ A )
=> ~ ( ord_less_eq_set_a @ X2 @ B ) ) ) ).
% le_infE
thf(fact_127_le__infE,axiom,
! [X2: set_b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ X2 @ ( inf_inf_set_b @ A @ B ) )
=> ~ ( ( ord_less_eq_set_b @ X2 @ A )
=> ~ ( ord_less_eq_set_b @ X2 @ B ) ) ) ).
% le_infE
thf(fact_128_inf__le2,axiom,
! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_129_inf__le2,axiom,
! [X2: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_130_inf__le1,axiom,
! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_131_inf__le1,axiom,
! [X2: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_132_inf__sup__ord_I1_J,axiom,
! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_133_inf__sup__ord_I1_J,axiom,
! [X2: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_134_inf__sup__ord_I2_J,axiom,
! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_135_inf__sup__ord_I2_J,axiom,
! [X2: set_b,Y2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_136_inf__left__commute,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z ) )
= ( inf_inf_set_a @ Y2 @ ( inf_inf_set_a @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_137_inf_Oleft__commute,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( inf_inf_set_a @ B @ ( inf_inf_set_a @ A @ C ) )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_138_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_a,K: set_a,B: set_a,A: set_a] :
( ( B3
= ( inf_inf_set_a @ K @ B ) )
=> ( ( inf_inf_set_a @ A @ B3 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_139_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_a,K: set_a,A: set_a,B: set_a] :
( ( A2
= ( inf_inf_set_a @ K @ A ) )
=> ( ( inf_inf_set_a @ A2 @ B )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_140_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X5: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X5 ) ) ) ).
% inf_commute
thf(fact_141_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A3: set_a,B2: set_a] : ( inf_inf_set_a @ B2 @ A3 ) ) ) ).
% inf.commute
thf(fact_142_inf__assoc,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Z )
= ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z ) ) ) ).
% inf_assoc
thf(fact_143_inf_Oassoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B ) @ C )
= ( inf_inf_set_a @ A @ ( inf_inf_set_a @ B @ C ) ) ) ).
% inf.assoc
thf(fact_144_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X5: set_a,Y4: set_a] : ( inf_inf_set_a @ Y4 @ X5 ) ) ) ).
% inf_sup_aci(1)
thf(fact_145_inf__sup__aci_I2_J,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X2 @ Y2 ) @ Z )
= ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_146_inf__sup__aci_I3_J,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ Y2 @ Z ) )
= ( inf_inf_set_a @ Y2 @ ( inf_inf_set_a @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_147_inf__sup__aci_I4_J,axiom,
! [X2: set_a,Y2: set_a] :
( ( inf_inf_set_a @ X2 @ ( inf_inf_set_a @ X2 @ Y2 ) )
= ( inf_inf_set_a @ X2 @ Y2 ) ) ).
% inf_sup_aci(4)
thf(fact_148_merge__in__supergraph,axiom,
! [C2: pre_pr7278220950009878019t_unit,X2: a] :
( ( shorte3657265928840388360ph_a_b @ C2 @ t )
=> ( ( member_a @ X2 @ ( graph_2957805489637798020ts_a_b @ C2 ) )
=> ( member_a @ X2 @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ) ).
% merge_in_supergraph
thf(fact_149_Int__subset__iff,axiom,
! [C2: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( ( ord_less_eq_set_a @ C2 @ A2 )
& ( ord_less_eq_set_a @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_150_Int__subset__iff,axiom,
! [C2: set_b,A2: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A2 @ B3 ) )
= ( ( ord_less_eq_set_b @ C2 @ A2 )
& ( ord_less_eq_set_b @ C2 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_151_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_152_empty__subsetI,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% empty_subsetI
thf(fact_153_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_154_subset__empty,axiom,
! [A2: set_b] :
( ( ord_less_eq_set_b @ A2 @ bot_bot_set_b )
= ( A2 = bot_bot_set_b ) ) ).
% subset_empty
thf(fact_155_reachable__induce__subgraphD,axiom,
! [S: set_a,U: a,V: a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ U @ V )
=> ( ( ord_less_eq_set_a @ S @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( reachable_a_b @ t @ U @ V ) ) ) ).
% reachable_induce_subgraphD
thf(fact_156_dominates__induce__ss,axiom,
! [U: a,V: a,S: set_a,T: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) ) )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ ( digrap7873285959652527175ph_a_b @ t @ T ) ) ) ) ) ).
% dominates_induce_ss
thf(fact_157_scc__disj,axiom,
! [C: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ t ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ t ) )
=> ( ( C != D )
=> ( ( inf_inf_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D ) )
= bot_bot_set_a ) ) ) ) ).
% scc_disj
thf(fact_158_pre__digraph_Oconverse__reachable__cases,axiom,
! [G: pre_pr7278220950009878019t_unit,U: a,V: a] :
( ( reachable_a_b @ G @ U @ V )
=> ( ( ( U = V )
=> ~ ( member_a @ U @ ( pre_ve642382030648772252t_unit @ G ) ) )
=> ~ ! [W2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ W2 ) @ ( arcs_ends_a_b @ G ) )
=> ~ ( reachable_a_b @ G @ W2 @ V ) ) ) ) ).
% pre_digraph.converse_reachable_cases
thf(fact_159_pre__digraph_Oconverse__reachable__cases,axiom,
! [G: pre_pr2882871181989701257t_unit,U: list_a,V: list_a] :
( ( reachable_list_a_b @ G @ U @ V )
=> ( ( ( U = V )
=> ~ ( member_list_a @ U @ ( pre_ve1830060048215441954t_unit @ G ) ) )
=> ~ ! [W2: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ U @ W2 ) @ ( arcs_ends_list_a_b @ G ) )
=> ~ ( reachable_list_a_b @ G @ W2 @ V ) ) ) ) ).
% pre_digraph.converse_reachable_cases
thf(fact_160_before__arc__to__hd,axiom,
! [Xs: list_a,Ys: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ Xs @ Ys )
=> ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ ( hd_a @ Ys ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% before_arc_to_hd
thf(fact_161_verts__reachable__connected,axiom,
( ( ( pre_ve642382030648772252t_unit @ t )
!= bot_bot_set_a )
=> ( ! [X4: a] :
( ( member_a @ X4 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ! [Xa: a] :
( ( member_a @ Xa @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( reachable_a_b @ t @ X4 @ Xa ) ) )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ) ).
% verts_reachable_connected
thf(fact_162_sccs__verts__conv__scc__of,axiom,
( ( digrap2871191568752656621ts_a_b @ t )
= ( image_a_set_a @ ( digrap2937667069914300949of_a_b @ t ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% sccs_verts_conv_scc_of
thf(fact_163_connected,axiom,
digrap8783888973171253482ed_a_b @ t ).
% connected
thf(fact_164_empty__iff,axiom,
! [C: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ C @ bot_bo3357376287454694259od_a_a ) ).
% empty_iff
thf(fact_165_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_166_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_167_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_168_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_169_all__not__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ! [X5: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ X5 @ A2 ) )
= ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% all_not_in_conv
thf(fact_170_all__not__in__conv,axiom,
! [A2: set_list_a] :
( ( ! [X5: list_a] :
~ ( member_list_a @ X5 @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_171_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X5: set_a] :
~ ( member_set_a @ X5 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_172_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X5: a] :
~ ( member_a @ X5 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_173_all__not__in__conv,axiom,
! [A2: set_b] :
( ( ! [X5: b] :
~ ( member_b @ X5 @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% all_not_in_conv
thf(fact_174_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X5: a] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_175_Collect__empty__eq,axiom,
! [P: b > $o] :
( ( ( collect_b @ P )
= bot_bot_set_b )
= ( ! [X5: b] :
~ ( P @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_176_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X5: a] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_177_empty__Collect__eq,axiom,
! [P: b > $o] :
( ( bot_bot_set_b
= ( collect_b @ P ) )
= ( ! [X5: b] :
~ ( P @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_178_image__eqI,axiom,
! [B: b,F: b > b,X2: b,A2: set_b] :
( ( B
= ( F @ X2 ) )
=> ( ( member_b @ X2 @ A2 )
=> ( member_b @ B @ ( image_b_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_179_image__eqI,axiom,
! [B: a,F: b > a,X2: b,A2: set_b] :
( ( B
= ( F @ X2 ) )
=> ( ( member_b @ X2 @ A2 )
=> ( member_a @ B @ ( image_b_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_180_image__eqI,axiom,
! [B: b,F: a > b,X2: a,A2: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A2 )
=> ( member_b @ B @ ( image_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_181_image__eqI,axiom,
! [B: a,F: a > a,X2: a,A2: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A2 )
=> ( member_a @ B @ ( image_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_182_image__eqI,axiom,
! [B: b,F: list_a > b,X2: list_a,A2: set_list_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_list_a @ X2 @ A2 )
=> ( member_b @ B @ ( image_list_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_183_image__eqI,axiom,
! [B: a,F: list_a > a,X2: list_a,A2: set_list_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_list_a @ X2 @ A2 )
=> ( member_a @ B @ ( image_list_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_184_image__eqI,axiom,
! [B: b,F: set_a > b,X2: set_a,A2: set_set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_a @ X2 @ A2 )
=> ( member_b @ B @ ( image_set_a_b @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_185_image__eqI,axiom,
! [B: a,F: set_a > a,X2: set_a,A2: set_set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_set_a @ X2 @ A2 )
=> ( member_a @ B @ ( image_set_a_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_186_image__eqI,axiom,
! [B: list_a,F: b > list_a,X2: b,A2: set_b] :
( ( B
= ( F @ X2 ) )
=> ( ( member_b @ X2 @ A2 )
=> ( member_list_a @ B @ ( image_b_list_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_187_image__eqI,axiom,
! [B: set_a,F: b > set_a,X2: b,A2: set_b] :
( ( B
= ( F @ X2 ) )
=> ( ( member_b @ X2 @ A2 )
=> ( member_set_a @ B @ ( image_b_set_a @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_188_subsetI,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ( member1426531477525435216od_a_a @ X4 @ B3 ) )
=> ( ord_le746702958409616551od_a_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_189_subsetI,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ A2 )
=> ( member_list_a @ X4 @ B3 ) )
=> ( ord_le8861187494160871172list_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_190_subsetI,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( member_set_a @ X4 @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_191_subsetI,axiom,
! [A2: set_a,B3: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_a @ X4 @ B3 ) )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_192_subsetI,axiom,
! [A2: set_b,B3: set_b] :
( ! [X4: b] :
( ( member_b @ X4 @ A2 )
=> ( member_b @ X4 @ B3 ) )
=> ( ord_less_eq_set_b @ A2 @ B3 ) ) ).
% subsetI
thf(fact_193_subset__antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_194_subset__antisym,axiom,
! [A2: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_195_IntI,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ A2 )
=> ( ( member1426531477525435216od_a_a @ C @ B3 )
=> ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_196_IntI,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ A2 )
=> ( ( member_list_a @ C @ B3 )
=> ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_197_IntI,axiom,
! [C: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ( member_set_a @ C @ B3 )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_198_IntI,axiom,
! [C: b,A2: set_b,B3: set_b] :
( ( member_b @ C @ A2 )
=> ( ( member_b @ C @ B3 )
=> ( member_b @ C @ ( inf_inf_set_b @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_199_IntI,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B3 )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_200_Int__iff,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B3 ) )
= ( ( member1426531477525435216od_a_a @ C @ A2 )
& ( member1426531477525435216od_a_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_201_Int__iff,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B3 ) )
= ( ( member_list_a @ C @ A2 )
& ( member_list_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_202_Int__iff,axiom,
! [C: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B3 ) )
= ( ( member_set_a @ C @ A2 )
& ( member_set_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_203_Int__iff,axiom,
! [C: b,A2: set_b,B3: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A2 @ B3 ) )
= ( ( member_b @ C @ A2 )
& ( member_b @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_204_Int__iff,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_205_subgraph__no__last__merge__chain,axiom,
! [C2: pre_pr7278220950009878019t_unit] :
( ( shorte3657265928840388360ph_a_b @ C2 @ t )
=> ( graph_8150681439568091980in_a_b @ C2 ) ) ).
% subgraph_no_last_merge_chain
thf(fact_206_in__sccs__subset__imp__eq,axiom,
! [C: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ t ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ t ) )
=> ( ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D ) )
=> ( C = D ) ) ) ) ).
% in_sccs_subset_imp_eq
thf(fact_207_reachable__induce__ss,axiom,
! [S: set_a,U: a,V: a,T: set_a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ U @ V )
=> ( ( ord_less_eq_set_a @ S @ T )
=> ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ t @ T ) @ U @ V ) ) ) ).
% reachable_induce_ss
thf(fact_208_in__verts__sccsD__sccs,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( member6939884229742472986t_unit @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ ( digraph_pre_sccs_a_b @ t ) ) ) ).
% in_verts_sccsD_sccs
thf(fact_209_in__sccs__verts__conv,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) )
= ( member6939884229742472986t_unit @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ ( digraph_pre_sccs_a_b @ t ) ) ) ).
% in_sccs_verts_conv
thf(fact_210_image__empty,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a] :
( ( image_7466199892558553556_set_a @ F @ bot_bo1839476491465656141t_unit )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_211_image__empty,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit] :
( ( image_6801035452528096924t_unit @ F @ bot_bot_set_set_a )
= bot_bo1839476491465656141t_unit ) ).
% image_empty
thf(fact_212_image__empty,axiom,
! [F: list_a > set_a] :
( ( image_list_a_set_a @ F @ bot_bot_set_list_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_213_image__empty,axiom,
! [F: a > set_a] :
( ( image_a_set_a @ F @ bot_bot_set_a )
= bot_bot_set_set_a ) ).
% image_empty
thf(fact_214_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_215_image__empty,axiom,
! [F: a > b] :
( ( image_a_b @ F @ bot_bot_set_a )
= bot_bot_set_b ) ).
% image_empty
thf(fact_216_image__empty,axiom,
! [F: b > a] :
( ( image_b_a @ F @ bot_bot_set_b )
= bot_bot_set_a ) ).
% image_empty
thf(fact_217_image__empty,axiom,
! [F: b > b] :
( ( image_b_b @ F @ bot_bot_set_b )
= bot_bot_set_b ) ).
% image_empty
thf(fact_218_empty__is__image,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( bot_bot_set_set_a
= ( image_7466199892558553556_set_a @ F @ A2 ) )
= ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% empty_is_image
thf(fact_219_empty__is__image,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a] :
( ( bot_bo1839476491465656141t_unit
= ( image_6801035452528096924t_unit @ F @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% empty_is_image
thf(fact_220_empty__is__image,axiom,
! [F: list_a > set_a,A2: set_list_a] :
( ( bot_bot_set_set_a
= ( image_list_a_set_a @ F @ A2 ) )
= ( A2 = bot_bot_set_list_a ) ) ).
% empty_is_image
thf(fact_221_empty__is__image,axiom,
! [F: a > set_a,A2: set_a] :
( ( bot_bot_set_set_a
= ( image_a_set_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_222_empty__is__image,axiom,
! [F: a > a,A2: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_223_empty__is__image,axiom,
! [F: b > a,A2: set_b] :
( ( bot_bot_set_a
= ( image_b_a @ F @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_224_empty__is__image,axiom,
! [F: a > b,A2: set_a] :
( ( bot_bot_set_b
= ( image_a_b @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_225_empty__is__image,axiom,
! [F: b > b,A2: set_b] :
( ( bot_bot_set_b
= ( image_b_b @ F @ A2 ) )
= ( A2 = bot_bot_set_b ) ) ).
% empty_is_image
thf(fact_226_image__is__empty,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( ( image_7466199892558553556_set_a @ F @ A2 )
= bot_bot_set_set_a )
= ( A2 = bot_bo1839476491465656141t_unit ) ) ).
% image_is_empty
thf(fact_227_image__is__empty,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a] :
( ( ( image_6801035452528096924t_unit @ F @ A2 )
= bot_bo1839476491465656141t_unit )
= ( A2 = bot_bot_set_set_a ) ) ).
% image_is_empty
thf(fact_228_image__is__empty,axiom,
! [F: list_a > set_a,A2: set_list_a] :
( ( ( image_list_a_set_a @ F @ A2 )
= bot_bot_set_set_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% image_is_empty
thf(fact_229_image__is__empty,axiom,
! [F: a > set_a,A2: set_a] :
( ( ( image_a_set_a @ F @ A2 )
= bot_bot_set_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_230_image__is__empty,axiom,
! [F: a > a,A2: set_a] :
( ( ( image_a_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_231_image__is__empty,axiom,
! [F: b > a,A2: set_b] :
( ( ( image_b_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_232_image__is__empty,axiom,
! [F: a > b,A2: set_a] :
( ( ( image_a_b @ F @ A2 )
= bot_bot_set_b )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_233_image__is__empty,axiom,
! [F: b > b,A2: set_b] :
( ( ( image_b_b @ F @ A2 )
= bot_bot_set_b )
= ( A2 = bot_bot_set_b ) ) ).
% image_is_empty
thf(fact_234_spanning__tree__imp__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5718416180170401981ee_a_b @ H @ t )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ).
% spanning_tree_imp_connected
thf(fact_235_connected__spanning__imp__connected,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digraph_spanning_a_b @ H @ t )
=> ( ( digrap8783888973171253482ed_a_b @ H )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ) ).
% connected_spanning_imp_connected
thf(fact_236_imageI,axiom,
! [X2: b,A2: set_b,F: b > b] :
( ( member_b @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( image_b_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_237_imageI,axiom,
! [X2: b,A2: set_b,F: b > a] :
( ( member_b @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( image_b_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_238_imageI,axiom,
! [X2: a,A2: set_a,F: a > b] :
( ( member_a @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( image_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_239_imageI,axiom,
! [X2: a,A2: set_a,F: a > a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_240_imageI,axiom,
! [X2: list_a,A2: set_list_a,F: list_a > b] :
( ( member_list_a @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( image_list_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_241_imageI,axiom,
! [X2: list_a,A2: set_list_a,F: list_a > a] :
( ( member_list_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( image_list_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_242_imageI,axiom,
! [X2: set_a,A2: set_set_a,F: set_a > b] :
( ( member_set_a @ X2 @ A2 )
=> ( member_b @ ( F @ X2 ) @ ( image_set_a_b @ F @ A2 ) ) ) ).
% imageI
thf(fact_243_imageI,axiom,
! [X2: set_a,A2: set_set_a,F: set_a > a] :
( ( member_set_a @ X2 @ A2 )
=> ( member_a @ ( F @ X2 ) @ ( image_set_a_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_244_imageI,axiom,
! [X2: b,A2: set_b,F: b > list_a] :
( ( member_b @ X2 @ A2 )
=> ( member_list_a @ ( F @ X2 ) @ ( image_b_list_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_245_imageI,axiom,
! [X2: b,A2: set_b,F: b > set_a] :
( ( member_b @ X2 @ A2 )
=> ( member_set_a @ ( F @ X2 ) @ ( image_b_set_a @ F @ A2 ) ) ) ).
% imageI
thf(fact_246_image__iff,axiom,
! [Z: pre_pr7278220950009878019t_unit,F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a] :
( ( member6939884229742472986t_unit @ Z @ ( image_6801035452528096924t_unit @ F @ A2 ) )
= ( ? [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
& ( Z
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_247_image__iff,axiom,
! [Z: set_a,F: a > set_a,A2: set_a] :
( ( member_set_a @ Z @ ( image_a_set_a @ F @ A2 ) )
= ( ? [X5: a] :
( ( member_a @ X5 @ A2 )
& ( Z
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_248_image__iff,axiom,
! [Z: set_a,F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( member_set_a @ Z @ ( image_7466199892558553556_set_a @ F @ A2 ) )
= ( ? [X5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X5 @ A2 )
& ( Z
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_249_image__iff,axiom,
! [Z: set_a,F: list_a > set_a,A2: set_list_a] :
( ( member_set_a @ Z @ ( image_list_a_set_a @ F @ A2 ) )
= ( ? [X5: list_a] :
( ( member_list_a @ X5 @ A2 )
& ( Z
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_250_image__iff,axiom,
! [Z: a,F: b > a,A2: set_b] :
( ( member_a @ Z @ ( image_b_a @ F @ A2 ) )
= ( ? [X5: b] :
( ( member_b @ X5 @ A2 )
& ( Z
= ( F @ X5 ) ) ) ) ) ).
% image_iff
thf(fact_251_bex__imageD,axiom,
! [F: a > set_a,A2: set_a,P: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( image_a_set_a @ F @ A2 ) )
& ( P @ X ) )
=> ? [X4: a] :
( ( member_a @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_252_bex__imageD,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,P: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( image_7466199892558553556_set_a @ F @ A2 ) )
& ( P @ X ) )
=> ? [X4: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_253_bex__imageD,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,P: pre_pr7278220950009878019t_unit > $o] :
( ? [X: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X @ ( image_6801035452528096924t_unit @ F @ A2 ) )
& ( P @ X ) )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_254_bex__imageD,axiom,
! [F: list_a > set_a,A2: set_list_a,P: set_a > $o] :
( ? [X: set_a] :
( ( member_set_a @ X @ ( image_list_a_set_a @ F @ A2 ) )
& ( P @ X ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_255_bex__imageD,axiom,
! [F: b > a,A2: set_b,P: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( image_b_a @ F @ A2 ) )
& ( P @ X ) )
=> ? [X4: b] :
( ( member_b @ X4 @ A2 )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_256_image__cong,axiom,
! [M: set_pr5411798346947241657t_unit,N: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > set_a,G2: pre_pr7278220950009878019t_unit > set_a] :
( ( M = N )
=> ( ! [X4: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X4 @ N )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( image_7466199892558553556_set_a @ F @ M )
= ( image_7466199892558553556_set_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_257_image__cong,axiom,
! [M: set_list_a,N: set_list_a,F: list_a > set_a,G2: list_a > set_a] :
( ( M = N )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ N )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( image_list_a_set_a @ F @ M )
= ( image_list_a_set_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_258_image__cong,axiom,
! [M: set_set_a,N: set_set_a,F: set_a > pre_pr7278220950009878019t_unit,G2: set_a > pre_pr7278220950009878019t_unit] :
( ( M = N )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ N )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( image_6801035452528096924t_unit @ F @ M )
= ( image_6801035452528096924t_unit @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_259_image__cong,axiom,
! [M: set_b,N: set_b,F: b > a,G2: b > a] :
( ( M = N )
=> ( ! [X4: b] :
( ( member_b @ X4 @ N )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( image_b_a @ F @ M )
= ( image_b_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_260_image__cong,axiom,
! [M: set_a,N: set_a,F: a > set_a,G2: a > set_a] :
( ( M = N )
=> ( ! [X4: a] :
( ( member_a @ X4 @ N )
=> ( ( F @ X4 )
= ( G2 @ X4 ) ) )
=> ( ( image_a_set_a @ F @ M )
= ( image_a_set_a @ G2 @ N ) ) ) ) ).
% image_cong
thf(fact_261_ball__imageD,axiom,
! [F: a > set_a,A2: set_a,P: set_a > $o] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ ( image_a_set_a @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X: a] :
( ( member_a @ X @ A2 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_262_ball__imageD,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,P: set_a > $o] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X @ A2 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_263_ball__imageD,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,P: pre_pr7278220950009878019t_unit > $o] :
( ! [X4: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X4 @ ( image_6801035452528096924t_unit @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X: set_a] :
( ( member_set_a @ X @ A2 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_264_ball__imageD,axiom,
! [F: list_a > set_a,A2: set_list_a,P: set_a > $o] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ ( image_list_a_set_a @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X: list_a] :
( ( member_list_a @ X @ A2 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_265_ball__imageD,axiom,
! [F: b > a,A2: set_b,P: a > $o] :
( ! [X4: a] :
( ( member_a @ X4 @ ( image_b_a @ F @ A2 ) )
=> ( P @ X4 ) )
=> ! [X: b] :
( ( member_b @ X @ A2 )
=> ( P @ ( F @ X ) ) ) ) ).
% ball_imageD
thf(fact_266_rev__image__eqI,axiom,
! [X2: b,A2: set_b,B: b,F: b > b] :
( ( member_b @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_b @ B @ ( image_b_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_267_rev__image__eqI,axiom,
! [X2: b,A2: set_b,B: a,F: b > a] :
( ( member_b @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_b_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_268_rev__image__eqI,axiom,
! [X2: a,A2: set_a,B: b,F: a > b] :
( ( member_a @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_b @ B @ ( image_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_269_rev__image__eqI,axiom,
! [X2: a,A2: set_a,B: a,F: a > a] :
( ( member_a @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_270_rev__image__eqI,axiom,
! [X2: list_a,A2: set_list_a,B: b,F: list_a > b] :
( ( member_list_a @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_b @ B @ ( image_list_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_271_rev__image__eqI,axiom,
! [X2: list_a,A2: set_list_a,B: a,F: list_a > a] :
( ( member_list_a @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_list_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_272_rev__image__eqI,axiom,
! [X2: set_a,A2: set_set_a,B: b,F: set_a > b] :
( ( member_set_a @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_b @ B @ ( image_set_a_b @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_273_rev__image__eqI,axiom,
! [X2: set_a,A2: set_set_a,B: a,F: set_a > a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_set_a_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_274_rev__image__eqI,axiom,
! [X2: b,A2: set_b,B: list_a,F: b > list_a] :
( ( member_b @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_list_a @ B @ ( image_b_list_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_275_rev__image__eqI,axiom,
! [X2: b,A2: set_b,B: set_a,F: b > set_a] :
( ( member_b @ X2 @ A2 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_set_a @ B @ ( image_b_set_a @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_276_image__mono,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > set_a] :
( ( ord_le8200006823705900825t_unit @ A2 @ B3 )
=> ( ord_le3724670747650509150_set_a @ ( image_7466199892558553556_set_a @ F @ A2 ) @ ( image_7466199892558553556_set_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_277_image__mono,axiom,
! [A2: set_set_a,B3: set_set_a,F: set_a > pre_pr7278220950009878019t_unit] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ ( image_6801035452528096924t_unit @ F @ B3 ) ) ) ).
% image_mono
thf(fact_278_image__mono,axiom,
! [A2: set_list_a,B3: set_list_a,F: list_a > set_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ord_le3724670747650509150_set_a @ ( image_list_a_set_a @ F @ A2 ) @ ( image_list_a_set_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_279_image__mono,axiom,
! [A2: set_a,B3: set_a,F: a > set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_280_image__mono,axiom,
! [A2: set_a,B3: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_281_image__mono,axiom,
! [A2: set_a,B3: set_a,F: a > b] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_b @ ( image_a_b @ F @ A2 ) @ ( image_a_b @ F @ B3 ) ) ) ).
% image_mono
thf(fact_282_image__mono,axiom,
! [A2: set_b,B3: set_b,F: b > a] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A2 ) @ ( image_b_a @ F @ B3 ) ) ) ).
% image_mono
thf(fact_283_image__mono,axiom,
! [A2: set_b,B3: set_b,F: b > b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A2 ) @ ( image_b_b @ F @ B3 ) ) ) ).
% image_mono
thf(fact_284_image__subsetI,axiom,
! [A2: set_b,F: b > a,B3: set_a] :
( ! [X4: b] :
( ( member_b @ X4 @ A2 )
=> ( member_a @ ( F @ X4 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_b_a @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_285_image__subsetI,axiom,
! [A2: set_a,F: a > a,B3: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_a @ ( F @ X4 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_286_image__subsetI,axiom,
! [A2: set_b,F: b > b,B3: set_b] :
( ! [X4: b] :
( ( member_b @ X4 @ A2 )
=> ( member_b @ ( F @ X4 ) @ B3 ) )
=> ( ord_less_eq_set_b @ ( image_b_b @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_287_image__subsetI,axiom,
! [A2: set_a,F: a > b,B3: set_b] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_b @ ( F @ X4 ) @ B3 ) )
=> ( ord_less_eq_set_b @ ( image_a_b @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_288_image__subsetI,axiom,
! [A2: set_b,F: b > list_a,B3: set_list_a] :
( ! [X4: b] :
( ( member_b @ X4 @ A2 )
=> ( member_list_a @ ( F @ X4 ) @ B3 ) )
=> ( ord_le8861187494160871172list_a @ ( image_b_list_a @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_289_image__subsetI,axiom,
! [A2: set_b,F: b > set_a,B3: set_set_a] :
( ! [X4: b] :
( ( member_b @ X4 @ A2 )
=> ( member_set_a @ ( F @ X4 ) @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_b_set_a @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_290_image__subsetI,axiom,
! [A2: set_a,F: a > list_a,B3: set_list_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_list_a @ ( F @ X4 ) @ B3 ) )
=> ( ord_le8861187494160871172list_a @ ( image_a_list_a @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_291_image__subsetI,axiom,
! [A2: set_a,F: a > set_a,B3: set_set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( member_set_a @ ( F @ X4 ) @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_292_image__subsetI,axiom,
! [A2: set_list_a,F: list_a > a,B3: set_a] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ A2 )
=> ( member_a @ ( F @ X4 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_list_a_a @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_293_image__subsetI,axiom,
! [A2: set_set_a,F: set_a > a,B3: set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( member_a @ ( F @ X4 ) @ B3 ) )
=> ( ord_less_eq_set_a @ ( image_set_a_a @ F @ A2 ) @ B3 ) ) ).
% image_subsetI
thf(fact_294_subset__imageE,axiom,
! [B3: set_set_a,F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
=> ~ ! [C3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ C3 @ A2 )
=> ( B3
!= ( image_7466199892558553556_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_295_subset__imageE,axiom,
! [B3: set_pr5411798346947241657t_unit,F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a] :
( ( ord_le8200006823705900825t_unit @ B3 @ ( image_6801035452528096924t_unit @ F @ A2 ) )
=> ~ ! [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A2 )
=> ( B3
!= ( image_6801035452528096924t_unit @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_296_subset__imageE,axiom,
! [B3: set_set_a,F: list_a > set_a,A2: set_list_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_list_a_set_a @ F @ A2 ) )
=> ~ ! [C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C3 @ A2 )
=> ( B3
!= ( image_list_a_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_297_subset__imageE,axiom,
! [B3: set_set_a,F: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_a_set_a @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B3
!= ( image_a_set_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_298_subset__imageE,axiom,
! [B3: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B3
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_299_subset__imageE,axiom,
! [B3: set_a,F: b > a,A2: set_b] :
( ( ord_less_eq_set_a @ B3 @ ( image_b_a @ F @ A2 ) )
=> ~ ! [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A2 )
=> ( B3
!= ( image_b_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_300_subset__imageE,axiom,
! [B3: set_b,F: a > b,A2: set_a] :
( ( ord_less_eq_set_b @ B3 @ ( image_a_b @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B3
!= ( image_a_b @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_301_subset__imageE,axiom,
! [B3: set_b,F: b > b,A2: set_b] :
( ( ord_less_eq_set_b @ B3 @ ( image_b_b @ F @ A2 ) )
=> ~ ! [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A2 )
=> ( B3
!= ( image_b_b @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_302_image__subset__iff,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,B3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ B3 )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
=> ( member6939884229742472986t_unit @ ( F @ X5 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_303_image__subset__iff,axiom,
! [F: a > set_a,A2: set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ A2 ) @ B3 )
= ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( member_set_a @ ( F @ X5 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_304_image__subset__iff,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_7466199892558553556_set_a @ F @ A2 ) @ B3 )
= ( ! [X5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X5 @ A2 )
=> ( member_set_a @ ( F @ X5 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_305_image__subset__iff,axiom,
! [F: list_a > set_a,A2: set_list_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( image_list_a_set_a @ F @ A2 ) @ B3 )
= ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A2 )
=> ( member_set_a @ ( F @ X5 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_306_image__subset__iff,axiom,
! [F: b > a,A2: set_b,B3: set_a] :
( ( ord_less_eq_set_a @ ( image_b_a @ F @ A2 ) @ B3 )
= ( ! [X5: b] :
( ( member_b @ X5 @ A2 )
=> ( member_a @ ( F @ X5 ) @ B3 ) ) ) ) ).
% image_subset_iff
thf(fact_307_subset__image__iff,axiom,
! [B3: set_set_a,F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
= ( ? [AA: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ AA @ A2 )
& ( B3
= ( image_7466199892558553556_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_308_subset__image__iff,axiom,
! [B3: set_pr5411798346947241657t_unit,F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a] :
( ( ord_le8200006823705900825t_unit @ B3 @ ( image_6801035452528096924t_unit @ F @ A2 ) )
= ( ? [AA: set_set_a] :
( ( ord_le3724670747650509150_set_a @ AA @ A2 )
& ( B3
= ( image_6801035452528096924t_unit @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_309_subset__image__iff,axiom,
! [B3: set_set_a,F: list_a > set_a,A2: set_list_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_list_a_set_a @ F @ A2 ) )
= ( ? [AA: set_list_a] :
( ( ord_le8861187494160871172list_a @ AA @ A2 )
& ( B3
= ( image_list_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_310_subset__image__iff,axiom,
! [B3: set_set_a,F: a > set_a,A2: set_a] :
( ( ord_le3724670747650509150_set_a @ B3 @ ( image_a_set_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B3
= ( image_a_set_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_311_subset__image__iff,axiom,
! [B3: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B3
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_312_subset__image__iff,axiom,
! [B3: set_a,F: b > a,A2: set_b] :
( ( ord_less_eq_set_a @ B3 @ ( image_b_a @ F @ A2 ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A2 )
& ( B3
= ( image_b_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_313_subset__image__iff,axiom,
! [B3: set_b,F: a > b,A2: set_a] :
( ( ord_less_eq_set_b @ B3 @ ( image_a_b @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B3
= ( image_a_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_314_subset__image__iff,axiom,
! [B3: set_b,F: b > b,A2: set_b] :
( ( ord_less_eq_set_b @ B3 @ ( image_b_b @ F @ A2 ) )
= ( ? [AA: set_b] :
( ( ord_less_eq_set_b @ AA @ A2 )
& ( B3
= ( image_b_b @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_315_image__Int__subset,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit] : ( ord_le3724670747650509150_set_a @ ( image_7466199892558553556_set_a @ F @ ( inf_in1092213268631476299t_unit @ A2 @ B3 ) ) @ ( inf_inf_set_set_a @ ( image_7466199892558553556_set_a @ F @ A2 ) @ ( image_7466199892558553556_set_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_316_image__Int__subset,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,B3: set_set_a] : ( ord_le8200006823705900825t_unit @ ( image_6801035452528096924t_unit @ F @ ( inf_inf_set_set_a @ A2 @ B3 ) ) @ ( inf_in1092213268631476299t_unit @ ( image_6801035452528096924t_unit @ F @ A2 ) @ ( image_6801035452528096924t_unit @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_317_image__Int__subset,axiom,
! [F: list_a > set_a,A2: set_list_a,B3: set_list_a] : ( ord_le3724670747650509150_set_a @ ( image_list_a_set_a @ F @ ( inf_inf_set_list_a @ A2 @ B3 ) ) @ ( inf_inf_set_set_a @ ( image_list_a_set_a @ F @ A2 ) @ ( image_list_a_set_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_318_image__Int__subset,axiom,
! [F: a > set_a,A2: set_a,B3: set_a] : ( ord_le3724670747650509150_set_a @ ( image_a_set_a @ F @ ( inf_inf_set_a @ A2 @ B3 ) ) @ ( inf_inf_set_set_a @ ( image_a_set_a @ F @ A2 ) @ ( image_a_set_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_319_image__Int__subset,axiom,
! [F: b > a,A2: set_b,B3: set_b] : ( ord_less_eq_set_a @ ( image_b_a @ F @ ( inf_inf_set_b @ A2 @ B3 ) ) @ ( inf_inf_set_a @ ( image_b_a @ F @ A2 ) @ ( image_b_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_320_image__Int__subset,axiom,
! [F: a > a,A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( image_a_a @ F @ ( inf_inf_set_a @ A2 @ B3 ) ) @ ( inf_inf_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_321_image__Int__subset,axiom,
! [F: a > b,A2: set_a,B3: set_a] : ( ord_less_eq_set_b @ ( image_a_b @ F @ ( inf_inf_set_a @ A2 @ B3 ) ) @ ( inf_inf_set_b @ ( image_a_b @ F @ A2 ) @ ( image_a_b @ F @ B3 ) ) ) ).
% image_Int_subset
thf(fact_322_emptyE,axiom,
! [A: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ A @ bot_bo3357376287454694259od_a_a ) ).
% emptyE
thf(fact_323_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_324_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_325_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_326_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_327_equals0D,axiom,
! [A2: set_Product_prod_a_a,A: product_prod_a_a] :
( ( A2 = bot_bo3357376287454694259od_a_a )
=> ~ ( member1426531477525435216od_a_a @ A @ A2 ) ) ).
% equals0D
thf(fact_328_equals0D,axiom,
! [A2: set_list_a,A: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_329_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_330_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_331_equals0D,axiom,
! [A2: set_b,A: b] :
( ( A2 = bot_bot_set_b )
=> ~ ( member_b @ A @ A2 ) ) ).
% equals0D
thf(fact_332_equals0I,axiom,
! [A2: set_Product_prod_a_a] :
( ! [Y3: product_prod_a_a] :
~ ( member1426531477525435216od_a_a @ Y3 @ A2 )
=> ( A2 = bot_bo3357376287454694259od_a_a ) ) ).
% equals0I
thf(fact_333_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y3: list_a] :
~ ( member_list_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_334_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y3: set_a] :
~ ( member_set_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_335_equals0I,axiom,
! [A2: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_336_equals0I,axiom,
! [A2: set_b] :
( ! [Y3: b] :
~ ( member_b @ Y3 @ A2 )
=> ( A2 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_337_ex__in__conv,axiom,
! [A2: set_Product_prod_a_a] :
( ( ? [X5: product_prod_a_a] : ( member1426531477525435216od_a_a @ X5 @ A2 ) )
= ( A2 != bot_bo3357376287454694259od_a_a ) ) ).
% ex_in_conv
thf(fact_338_ex__in__conv,axiom,
! [A2: set_list_a] :
( ( ? [X5: list_a] : ( member_list_a @ X5 @ A2 ) )
= ( A2 != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_339_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X5: set_a] : ( member_set_a @ X5 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_340_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X5: a] : ( member_a @ X5 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_341_ex__in__conv,axiom,
! [A2: set_b] :
( ( ? [X5: b] : ( member_b @ X5 @ A2 ) )
= ( A2 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_342_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_343_bot__set__def,axiom,
( bot_bot_set_b
= ( collect_b @ bot_bot_b_o ) ) ).
% bot_set_def
thf(fact_344_in__mono,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,X2: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B3 )
=> ( ( member1426531477525435216od_a_a @ X2 @ A2 )
=> ( member1426531477525435216od_a_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_345_in__mono,axiom,
! [A2: set_list_a,B3: set_list_a,X2: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( member_list_a @ X2 @ A2 )
=> ( member_list_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_346_in__mono,axiom,
! [A2: set_set_a,B3: set_set_a,X2: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( member_set_a @ X2 @ A2 )
=> ( member_set_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_347_in__mono,axiom,
! [A2: set_a,B3: set_a,X2: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_348_in__mono,axiom,
! [A2: set_b,B3: set_b,X2: b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( member_b @ X2 @ A2 )
=> ( member_b @ X2 @ B3 ) ) ) ).
% in_mono
thf(fact_349_subsetD,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,C: product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ A2 @ B3 )
=> ( ( member1426531477525435216od_a_a @ C @ A2 )
=> ( member1426531477525435216od_a_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_350_subsetD,axiom,
! [A2: set_list_a,B3: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_351_subsetD,axiom,
! [A2: set_set_a,B3: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_352_subsetD,axiom,
! [A2: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_353_subsetD,axiom,
! [A2: set_b,B3: set_b,C: b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( member_b @ C @ A2 )
=> ( member_b @ C @ B3 ) ) ) ).
% subsetD
thf(fact_354_equalityE,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_355_equalityE,axiom,
! [A2: set_b,B3: set_b] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_b @ A2 @ B3 )
=> ~ ( ord_less_eq_set_b @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_356_subset__eq,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A4 )
=> ( member1426531477525435216od_a_a @ X5 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_357_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B4: set_list_a] :
! [X5: list_a] :
( ( member_list_a @ X5 @ A4 )
=> ( member_list_a @ X5 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_358_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [X5: set_a] :
( ( member_set_a @ X5 @ A4 )
=> ( member_set_a @ X5 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_359_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X5: a] :
( ( member_a @ X5 @ A4 )
=> ( member_a @ X5 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_360_subset__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B4: set_b] :
! [X5: b] :
( ( member_b @ X5 @ A4 )
=> ( member_b @ X5 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_361_equalityD1,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_362_equalityD1,axiom,
! [A2: set_b,B3: set_b] :
( ( A2 = B3 )
=> ( ord_less_eq_set_b @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_363_equalityD2,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_364_equalityD2,axiom,
! [A2: set_b,B3: set_b] :
( ( A2 = B3 )
=> ( ord_less_eq_set_b @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_365_subset__iff,axiom,
( ord_le746702958409616551od_a_a
= ( ^ [A4: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
! [T2: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ T2 @ A4 )
=> ( member1426531477525435216od_a_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_366_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B4: set_list_a] :
! [T2: list_a] :
( ( member_list_a @ T2 @ A4 )
=> ( member_list_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_367_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [T2: set_a] :
( ( member_set_a @ T2 @ A4 )
=> ( member_set_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_368_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A4 )
=> ( member_a @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_369_subset__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B4: set_b] :
! [T2: b] :
( ( member_b @ T2 @ A4 )
=> ( member_b @ T2 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_370_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_371_subset__refl,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ A2 @ A2 ) ).
% subset_refl
thf(fact_372_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_373_Collect__mono,axiom,
! [P: b > $o,Q: b > $o] :
( ! [X4: b] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) ) ) ).
% Collect_mono
thf(fact_374_subset__trans,axiom,
! [A2: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_375_subset__trans,axiom,
! [A2: set_b,B3: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ C2 )
=> ( ord_less_eq_set_b @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_376_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z4: set_a] : ( Y5 = Z4 ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_377_set__eq__subset,axiom,
( ( ^ [Y5: set_b,Z4: set_b] : ( Y5 = Z4 ) )
= ( ^ [A4: set_b,B4: set_b] :
( ( ord_less_eq_set_b @ A4 @ B4 )
& ( ord_less_eq_set_b @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_378_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X5: a] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_379_Collect__mono__iff,axiom,
! [P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ ( collect_b @ P ) @ ( collect_b @ Q ) )
= ( ! [X5: b] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_380_IntE,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B3 ) )
=> ~ ( ( member1426531477525435216od_a_a @ C @ A2 )
=> ~ ( member1426531477525435216od_a_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_381_IntE,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B3 ) )
=> ~ ( ( member_list_a @ C @ A2 )
=> ~ ( member_list_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_382_IntE,axiom,
! [C: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B3 ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ~ ( member_set_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_383_IntE,axiom,
! [C: b,A2: set_b,B3: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A2 @ B3 ) )
=> ~ ( ( member_b @ C @ A2 )
=> ~ ( member_b @ C @ B3 ) ) ) ).
% IntE
thf(fact_384_IntE,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B3 ) ) ) ).
% IntE
thf(fact_385_IntD1,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B3 ) )
=> ( member1426531477525435216od_a_a @ C @ A2 ) ) ).
% IntD1
thf(fact_386_IntD1,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B3 ) )
=> ( member_list_a @ C @ A2 ) ) ).
% IntD1
thf(fact_387_IntD1,axiom,
! [C: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B3 ) )
=> ( member_set_a @ C @ A2 ) ) ).
% IntD1
thf(fact_388_IntD1,axiom,
! [C: b,A2: set_b,B3: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A2 @ B3 ) )
=> ( member_b @ C @ A2 ) ) ).
% IntD1
thf(fact_389_IntD1,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_390_IntD2,axiom,
! [C: product_prod_a_a,A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ C @ ( inf_in8905007599844390133od_a_a @ A2 @ B3 ) )
=> ( member1426531477525435216od_a_a @ C @ B3 ) ) ).
% IntD2
thf(fact_391_IntD2,axiom,
! [C: list_a,A2: set_list_a,B3: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B3 ) )
=> ( member_list_a @ C @ B3 ) ) ).
% IntD2
thf(fact_392_IntD2,axiom,
! [C: set_a,A2: set_set_a,B3: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B3 ) )
=> ( member_set_a @ C @ B3 ) ) ).
% IntD2
thf(fact_393_IntD2,axiom,
! [C: b,A2: set_b,B3: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A2 @ B3 ) )
=> ( member_b @ C @ B3 ) ) ).
% IntD2
thf(fact_394_IntD2,axiom,
! [C: a,A2: set_a,B3: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B3 ) )
=> ( member_a @ C @ B3 ) ) ).
% IntD2
thf(fact_395_Int__assoc,axiom,
! [A2: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C2 ) ) ) ).
% Int_assoc
thf(fact_396_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_397_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B4: set_a] : ( inf_inf_set_a @ B4 @ A4 ) ) ) ).
% Int_commute
thf(fact_398_Int__left__absorb,axiom,
! [A2: set_a,B3: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B3 ) )
= ( inf_inf_set_a @ A2 @ B3 ) ) ).
% Int_left_absorb
thf(fact_399_Int__left__commute,axiom,
! [A2: set_a,B3: set_a,C2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B3 @ C2 ) )
= ( inf_inf_set_a @ B3 @ ( inf_inf_set_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_400_nomulti__digraph_Onomulti__digraph,axiom,
! [G: pre_pr7278220950009878019t_unit] :
( ( nomulti_digraph_a_b @ G )
=> ( nomulti_digraph_a_b @ G ) ) ).
% nomulti_digraph.nomulti_digraph
thf(fact_401_wf__digraph_Oarc_Ocong,axiom,
wf_arc_a_b = wf_arc_a_b ).
% wf_digraph.arc.cong
thf(fact_402_Int__emptyI,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ~ ( member1426531477525435216od_a_a @ X4 @ B3 ) )
=> ( ( inf_in8905007599844390133od_a_a @ A2 @ B3 )
= bot_bo3357376287454694259od_a_a ) ) ).
% Int_emptyI
thf(fact_403_Int__emptyI,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ A2 )
=> ~ ( member_list_a @ X4 @ B3 ) )
=> ( ( inf_inf_set_list_a @ A2 @ B3 )
= bot_bot_set_list_a ) ) ).
% Int_emptyI
thf(fact_404_Int__emptyI,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ~ ( member_set_a @ X4 @ B3 ) )
=> ( ( inf_inf_set_set_a @ A2 @ B3 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_405_Int__emptyI,axiom,
! [A2: set_a,B3: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ~ ( member_a @ X4 @ B3 ) )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_406_Int__emptyI,axiom,
! [A2: set_b,B3: set_b] :
( ! [X4: b] :
( ( member_b @ X4 @ A2 )
=> ~ ( member_b @ X4 @ B3 ) )
=> ( ( inf_inf_set_b @ A2 @ B3 )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_407_disjoint__iff,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
( ( ( inf_in8905007599844390133od_a_a @ A2 @ B3 )
= bot_bo3357376287454694259od_a_a )
= ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ A2 )
=> ~ ( member1426531477525435216od_a_a @ X5 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_408_disjoint__iff,axiom,
! [A2: set_list_a,B3: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B3 )
= bot_bot_set_list_a )
= ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A2 )
=> ~ ( member_list_a @ X5 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_409_disjoint__iff,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B3 )
= bot_bot_set_set_a )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
=> ~ ( member_set_a @ X5 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_410_disjoint__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ~ ( member_a @ X5 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_411_disjoint__iff,axiom,
! [A2: set_b,B3: set_b] :
( ( ( inf_inf_set_b @ A2 @ B3 )
= bot_bot_set_b )
= ( ! [X5: b] :
( ( member_b @ X5 @ A2 )
=> ~ ( member_b @ X5 @ B3 ) ) ) ) ).
% disjoint_iff
thf(fact_412_Int__empty__left,axiom,
! [B3: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B3 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_413_Int__empty__left,axiom,
! [B3: set_b] :
( ( inf_inf_set_b @ bot_bot_set_b @ B3 )
= bot_bot_set_b ) ).
% Int_empty_left
thf(fact_414_Int__empty__right,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% Int_empty_right
thf(fact_415_Int__empty__right,axiom,
! [A2: set_b] :
( ( inf_inf_set_b @ A2 @ bot_bot_set_b )
= bot_bot_set_b ) ).
% Int_empty_right
thf(fact_416_disjoint__iff__not__equal,axiom,
! [A2: set_a,B3: set_a] :
( ( ( inf_inf_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ B3 )
=> ( X5 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_417_disjoint__iff__not__equal,axiom,
! [A2: set_b,B3: set_b] :
( ( ( inf_inf_set_b @ A2 @ B3 )
= bot_bot_set_b )
= ( ! [X5: b] :
( ( member_b @ X5 @ A2 )
=> ! [Y4: b] :
( ( member_b @ Y4 @ B3 )
=> ( X5 != Y4 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_418_Int__mono,axiom,
! [A2: set_a,C2: set_a,B3: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ ( inf_inf_set_a @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_419_Int__mono,axiom,
! [A2: set_b,C2: set_b,B3: set_b,D2: set_b] :
( ( ord_less_eq_set_b @ A2 @ C2 )
=> ( ( ord_less_eq_set_b @ B3 @ D2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B3 ) @ ( inf_inf_set_b @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_420_Int__lower1,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_421_Int__lower1,axiom,
! [A2: set_b,B3: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B3 ) @ A2 ) ).
% Int_lower1
thf(fact_422_Int__lower2,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_423_Int__lower2,axiom,
! [A2: set_b,B3: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_424_Int__absorb1,axiom,
! [B3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_425_Int__absorb1,axiom,
! [B3: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B3 @ A2 )
=> ( ( inf_inf_set_b @ A2 @ B3 )
= B3 ) ) ).
% Int_absorb1
thf(fact_426_Int__absorb2,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( inf_inf_set_a @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_427_Int__absorb2,axiom,
! [A2: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( inf_inf_set_b @ A2 @ B3 )
= A2 ) ) ).
% Int_absorb2
thf(fact_428_Int__greatest,axiom,
! [C2: set_a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B3 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_429_Int__greatest,axiom,
! [C2: set_b,A2: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ C2 @ A2 )
=> ( ( ord_less_eq_set_b @ C2 @ B3 )
=> ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A2 @ B3 ) ) ) ) ).
% Int_greatest
thf(fact_430_Int__Collect__mono,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
( ( ord_le746702958409616551od_a_a @ A2 @ B3 )
=> ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le746702958409616551od_a_a @ ( inf_in8905007599844390133od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P ) ) @ ( inf_in8905007599844390133od_a_a @ B3 @ ( collec3336397797384452498od_a_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_431_Int__Collect__mono,axiom,
! [A2: set_list_a,B3: set_list_a,P: list_a > $o,Q: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A2 @ B3 )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ ( collect_list_a @ P ) ) @ ( inf_inf_set_list_a @ B3 @ ( collect_list_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_432_Int__Collect__mono,axiom,
! [A2: set_set_a,B3: set_set_a,P: set_a > $o,Q: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ ( collect_set_a @ P ) ) @ ( inf_inf_set_set_a @ B3 @ ( collect_set_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_433_Int__Collect__mono,axiom,
! [A2: set_a,B3: set_a,P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P ) ) @ ( inf_inf_set_a @ B3 @ ( collect_a @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_434_Int__Collect__mono,axiom,
! [A2: set_b,B3: set_b,P: b > $o,Q: b > $o] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ! [X4: b] :
( ( member_b @ X4 @ A2 )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ ( collect_b @ P ) ) @ ( inf_inf_set_b @ B3 @ ( collect_b @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_435_reachable__in__vertsE,axiom,
! [G: pre_pr7278220950009878019t_unit,U: a,V: a] :
( ( reachable_a_b @ G @ U @ V )
=> ~ ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ G ) )
=> ~ ( member_a @ V @ ( pre_ve642382030648772252t_unit @ G ) ) ) ) ).
% reachable_in_vertsE
thf(fact_436_reachable__in__vertsE,axiom,
! [G: pre_pr2882871181989701257t_unit,U: list_a,V: list_a] :
( ( reachable_list_a_b @ G @ U @ V )
=> ~ ( ( member_list_a @ U @ ( pre_ve1830060048215441954t_unit @ G ) )
=> ~ ( member_list_a @ V @ ( pre_ve1830060048215441954t_unit @ G ) ) ) ) ).
% reachable_in_vertsE
thf(fact_437_loopfree__digraph_Oadj__not__same,axiom,
! [G: pre_pr2882871181989701257t_unit,A: list_a] :
( ( loopfr7852502256416881111st_a_b @ G )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A @ A ) @ ( arcs_ends_list_a_b @ G ) ) ) ).
% loopfree_digraph.adj_not_same
thf(fact_438_loopfree__digraph_Oadj__not__same,axiom,
! [G: pre_pr7278220950009878019t_unit,A: a] :
( ( loopfree_digraph_a_b @ G )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ ( arcs_ends_a_b @ G ) ) ) ).
% loopfree_digraph.adj_not_same
thf(fact_439_euler__imp__connected,axiom,
! [U: a,P2: list_b,V: a] :
( ( pre_euler_trail_a_b @ t @ U @ P2 @ V )
=> ( digrap8783888973171253482ed_a_b @ t ) ) ).
% euler_imp_connected
thf(fact_440_hd__reachable1__from__outside,axiom,
! [X2: a,Y2: a,Ys: list_a,Xs: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( member_a @ Y2 @ ( set_a2 @ Ys ) )
=> ( ? [X4: a] : ( member_a @ X4 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ ( hd_a @ Ys ) ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% hd_reachable1_from_outside
thf(fact_441_induce__subgraph__verts,axiom,
! [G: pre_pr2882871181989701257t_unit,Vs: set_list_a] :
( ( pre_ve1830060048215441954t_unit @ ( digrap21804061584661953st_a_b @ G @ Vs ) )
= Vs ) ).
% induce_subgraph_verts
thf(fact_442_induce__subgraph__verts,axiom,
! [G: pre_pr7278220950009878019t_unit,Vs: set_a] :
( ( pre_ve642382030648772252t_unit @ ( digrap7873285959652527175ph_a_b @ G @ Vs ) )
= Vs ) ).
% induce_subgraph_verts
thf(fact_443_to__list__tree__disjoint__verts,axiom,
! [U: list_a,V: list_a] :
( ( member_list_a @ U @ ( pre_ve1830060048215441954t_unit @ ( direct3773525127397338803ee_a_b @ t ) ) )
=> ( ( member_list_a @ V @ ( pre_ve1830060048215441954t_unit @ ( direct3773525127397338803ee_a_b @ t ) ) )
=> ( ( U != V )
=> ( ( inf_inf_set_a @ ( set_a2 @ U ) @ ( set_a2 @ V ) )
= bot_bot_set_a ) ) ) ) ).
% to_list_tree_disjoint_verts
thf(fact_444_forward__arc__to__head,axiom,
! [Ys: list_a,Xs: list_a,X2: a,Y2: a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( member_a @ Y2 @ ( set_a2 @ Ys ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( arcs_ends_a_b @ t ) )
=> ( Y2
= ( hd_a @ Ys ) ) ) ) ) ) ) ).
% forward_arc_to_head
thf(fact_445_hd__reachable1__from__outside_H,axiom,
! [X2: a,Y2: a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ( ( member_a @ Y2 @ ( set_a2 @ Ys ) )
=> ( ? [X4: a] : ( member_a @ X4 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ ( hd_a @ Ys ) ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ).
% hd_reachable1_from_outside'
thf(fact_446_induce__eq__iff__induced,axiom,
! [H: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ H @ t )
=> ( ( digrap7873285959652527175ph_a_b @ t @ ( pre_ve642382030648772252t_unit @ H ) )
= H ) ) ).
% induce_eq_iff_induced
thf(fact_447_hd__reach__all__forward,axiom,
! [Xs: list_a,X2: a] :
( ( member_a @ ( hd_a @ Xs ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( reachable_a_b @ t @ ( hd_a @ Xs ) @ X2 ) ) ) ) ).
% hd_reach_all_forward
thf(fact_448_forward__arc__to__head_H,axiom,
! [Ys: list_a,X2: a,Y2: a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ( ( member_a @ Y2 @ ( set_a2 @ Ys ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( arcs_ends_a_b @ t ) )
=> ( Y2
= ( hd_a @ Ys ) ) ) ) ) ) ).
% forward_arc_to_head'
thf(fact_449_induced__subgraph__refl,axiom,
digrap5251062021860773499ph_a_b @ t @ t ).
% induced_subgraph_refl
thf(fact_450_before__forward1I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ S1 ) ) ).
% before_forward1I
thf(fact_451_before__forward2I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ S2 ) ) ).
% before_forward2I
thf(fact_452_in__sccs__imp__induced,axiom,
! [C: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ t ) )
=> ( digrap5251062021860773499ph_a_b @ C @ t ) ) ).
% in_sccs_imp_induced
thf(fact_453_in__sccs__vertsI__sccs,axiom,
! [S: set_a] :
( ( member_set_a @ S @ ( image_7466199892558553556_set_a @ pre_ve642382030648772252t_unit @ ( digraph_pre_sccs_a_b @ t ) ) )
=> ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) ) ) ).
% in_sccs_vertsI_sccs
thf(fact_454_sccs__verts__conv,axiom,
( ( digrap2871191568752656621ts_a_b @ t )
= ( image_7466199892558553556_set_a @ pre_ve642382030648772252t_unit @ ( digraph_pre_sccs_a_b @ t ) ) ) ).
% sccs_verts_conv
thf(fact_455_sccs__conv__sccs__verts,axiom,
( ( digraph_pre_sccs_a_b @ t )
= ( image_6801035452528096924t_unit @ ( digrap7873285959652527175ph_a_b @ t ) @ ( digrap2871191568752656621ts_a_b @ t ) ) ) ).
% sccs_conv_sccs_verts
thf(fact_456_induced__induce,axiom,
! [Vs: set_a] :
( ( ord_less_eq_set_a @ Vs @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( digrap5251062021860773499ph_a_b @ ( digrap7873285959652527175ph_a_b @ t @ Vs ) @ t ) ) ).
% induced_induce
thf(fact_457_directed__tree_Oforward_Ocong,axiom,
iKKBZ_4778857019735642799rd_a_b = iKKBZ_4778857019735642799rd_a_b ).
% directed_tree.forward.cong
thf(fact_458_induced__eq__verts__imp__eq,axiom,
! [G: pre_pr2882871181989701257t_unit,H: pre_pr2882871181989701257t_unit,G3: pre_pr2882871181989701257t_unit] :
( ( digrap536511840712062453st_a_b @ G @ H )
=> ( ( digrap536511840712062453st_a_b @ G3 @ H )
=> ( ( ( pre_ve1830060048215441954t_unit @ G )
= ( pre_ve1830060048215441954t_unit @ G3 ) )
=> ( G = G3 ) ) ) ) ).
% induced_eq_verts_imp_eq
thf(fact_459_induced__eq__verts__imp__eq,axiom,
! [G: pre_pr7278220950009878019t_unit,H: pre_pr7278220950009878019t_unit,G3: pre_pr7278220950009878019t_unit] :
( ( digrap5251062021860773499ph_a_b @ G @ H )
=> ( ( digrap5251062021860773499ph_a_b @ G3 @ H )
=> ( ( ( pre_ve642382030648772252t_unit @ G )
= ( pre_ve642382030648772252t_unit @ G3 ) )
=> ( G = G3 ) ) ) ) ).
% induced_eq_verts_imp_eq
thf(fact_460_pre__digraph_Oin__sccs__imp__induced,axiom,
! [C: pre_pr7278220950009878019t_unit,G: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ G ) )
=> ( digrap5251062021860773499ph_a_b @ C @ G ) ) ).
% pre_digraph.in_sccs_imp_induced
thf(fact_461_pre__digraph_Osccs_Ocong,axiom,
digraph_pre_sccs_a_b = digraph_pre_sccs_a_b ).
% pre_digraph.sccs.cong
thf(fact_462_pre__digraph_Osccs__verts_Ocong,axiom,
digrap2871191568752656621ts_a_b = digrap2871191568752656621ts_a_b ).
% pre_digraph.sccs_verts.cong
thf(fact_463_pre__digraph_Oscc__of_Ocong,axiom,
digrap2937667069914300949of_a_b = digrap2937667069914300949of_a_b ).
% pre_digraph.scc_of.cong
thf(fact_464_dominates__induce__subgraphD,axiom,
! [U: list_a,V: list_a,G: pre_pr2882871181989701257t_unit,S: set_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ U @ V ) @ ( arcs_ends_list_a_b @ ( digrap21804061584661953st_a_b @ G @ S ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ U @ V ) @ ( arcs_ends_list_a_b @ G ) ) ) ).
% dominates_induce_subgraphD
thf(fact_465_dominates__induce__subgraphD,axiom,
! [U: a,V: a,G: pre_pr7278220950009878019t_unit,S: set_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ ( digrap7873285959652527175ph_a_b @ G @ S ) ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ G ) ) ) ).
% dominates_induce_subgraphD
thf(fact_466_pre__digraph_Oin__sccs__subset__imp__eq,axiom,
! [C: pre_pr2882871181989701257t_unit,G: pre_pr2882871181989701257t_unit,D: pre_pr2882871181989701257t_unit] :
( ( member8552421979260422176t_unit @ C @ ( digrap8621149594610875153st_a_b @ G ) )
=> ( ( member8552421979260422176t_unit @ D @ ( digrap8621149594610875153st_a_b @ G ) )
=> ( ( ord_le8861187494160871172list_a @ ( pre_ve1830060048215441954t_unit @ C ) @ ( pre_ve1830060048215441954t_unit @ D ) )
=> ( C = D ) ) ) ) ).
% pre_digraph.in_sccs_subset_imp_eq
thf(fact_467_pre__digraph_Oin__sccs__subset__imp__eq,axiom,
! [C: pre_pr7278220950009878019t_unit,G: pre_pr7278220950009878019t_unit,D: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ C @ ( digraph_pre_sccs_a_b @ G ) )
=> ( ( member6939884229742472986t_unit @ D @ ( digraph_pre_sccs_a_b @ G ) )
=> ( ( ord_less_eq_set_a @ ( pre_ve642382030648772252t_unit @ C ) @ ( pre_ve642382030648772252t_unit @ D ) )
=> ( C = D ) ) ) ) ).
% pre_digraph.in_sccs_subset_imp_eq
thf(fact_468_not__reachable1__append__if__not__old,axiom,
! [U2: list_a,B: list_a,X2: list_a] :
( ~ ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ U2 ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ B ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ X2 ) )
= bot_bot_set_a )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ X2 )
=> ( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ X2 ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ B ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ~ ? [X: a] :
( ( member_a @ X @ ( set_a2 @ U2 ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ ( append_a @ X2 @ B ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% not_reachable1_append_if_not_old
thf(fact_469_reachable1__append__old__if__arcU,axiom,
! [Xs: list_a,Ys: list_a,U2: list_a,Z: a,Y2: a] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ Xs ) )
= bot_bot_set_a )
=> ( ( member_a @ Z @ ( set_a2 @ U2 ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( member_a @ Y2 @ ( set_a2 @ ( append_a @ Xs @ Ys ) ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ Y2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ) ).
% reachable1_append_old_if_arcU
thf(fact_470_forward__app_H,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ S1 )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ S2 )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ S1 ) @ ( set_a2 @ S2 ) )
= bot_bot_set_a )
=> ( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ S1 ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ S2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ S1 @ S2 ) ) ) ) ) ) ).
% forward_app'
thf(fact_471_move__mid__backward__if__noarc_H,axiom,
! [U2: list_a,V3: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ U2 ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ V3 )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ V3 ) )
= bot_bot_set_a )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ V3 @ ( append_a @ Bs @ Cs ) ) ) ) ) ) ) ) ) ).
% move_mid_backward_if_noarc'
thf(fact_472_reachable1__append__old__if__arc,axiom,
! [Xs: list_a,Ys: list_a,Z: a,Y2: a] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ~ ( member_a @ Z @ ( set_a2 @ Xs ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( member_a @ Y2 @ ( set_a2 @ ( append_a @ Xs @ Ys ) ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ Y2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% reachable1_append_old_if_arc
thf(fact_473_forward__app,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ S1 )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ S2 )
=> ( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ S1 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ ( hd_a @ S2 ) ) @ ( arcs_ends_a_b @ t ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ S1 @ S2 ) ) ) ) ) ).
% forward_app
thf(fact_474_to__list__tree__union__verts__eq,axiom,
( ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ set_a2 @ ( pre_ve1830060048215441954t_unit @ ( direct3773525127397338803ee_a_b @ t ) ) ) )
= ( pre_ve642382030648772252t_unit @ t ) ) ).
% to_list_tree_union_verts_eq
thf(fact_475_forward__split,axiom,
! [Xs: list_a,Ys: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ Xs @ Ys ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs ) ) ).
% forward_split
thf(fact_476_order__refl,axiom,
! [X2: set_a] : ( ord_less_eq_set_a @ X2 @ X2 ) ).
% order_refl
thf(fact_477_order__refl,axiom,
! [X2: set_b] : ( ord_less_eq_set_b @ X2 @ X2 ) ).
% order_refl
thf(fact_478_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_479_dual__order_Orefl,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ A @ A ) ).
% dual_order.refl
thf(fact_480_move__mid__backward__if__noarc,axiom,
! [U2: list_a,V3: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ U2 @ V3 )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ V3 @ ( append_a @ Bs @ Cs ) ) ) ) ) ) ) ).
% move_mid_backward_if_noarc
thf(fact_481_seq__conform__if__before,axiom,
! [Xs: list_a,Ys: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ Xs @ Ys )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ ( append_a @ Xs @ Ys ) ) ) ).
% seq_conform_if_before
thf(fact_482_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z4: set_a] : ( Y5 = Z4 ) )
= ( ^ [X5: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_483_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_b,Z4: set_b] : ( Y5 = Z4 ) )
= ( ^ [X5: set_b,Y4: set_b] :
( ( ord_less_eq_set_b @ X5 @ Y4 )
& ( ord_less_eq_set_b @ Y4 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_484_ord__eq__le__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_485_ord__eq__le__trans,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( A = B )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_486_ord__le__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_487_ord__le__eq__trans,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_set_b @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_488_order__antisym,axiom,
! [X2: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_489_order__antisym,axiom,
! [X2: set_b,Y2: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y2 )
=> ( ( ord_less_eq_set_b @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% order_antisym
thf(fact_490_order_Otrans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A @ C ) ) ) ).
% order.trans
thf(fact_491_order_Otrans,axiom,
! [A: set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ord_less_eq_set_b @ A @ C ) ) ) ).
% order.trans
thf(fact_492_order__trans,axiom,
! [X2: set_a,Y2: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y2 )
=> ( ( ord_less_eq_set_a @ Y2 @ Z )
=> ( ord_less_eq_set_a @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_493_order__trans,axiom,
! [X2: set_b,Y2: set_b,Z: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y2 )
=> ( ( ord_less_eq_set_b @ Y2 @ Z )
=> ( ord_less_eq_set_b @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_494_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_a,Z4: set_a] : ( Y5 = Z4 ) )
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A3 )
& ( ord_less_eq_set_a @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_495_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_b,Z4: set_b] : ( Y5 = Z4 ) )
= ( ^ [A3: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ B2 @ A3 )
& ( ord_less_eq_set_b @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_496_dual__order_Oantisym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_497_dual__order_Oantisym,axiom,
! [B: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( ord_less_eq_set_b @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_498_dual__order_Otrans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_499_dual__order_Otrans,axiom,
! [B: set_b,A: set_b,C: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( ord_less_eq_set_b @ C @ B )
=> ( ord_less_eq_set_b @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_500_antisym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_501_antisym,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_502_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_a,Z4: set_a] : ( Y5 = Z4 ) )
= ( ^ [A3: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B2 )
& ( ord_less_eq_set_a @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_503_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_b,Z4: set_b] : ( Y5 = Z4 ) )
= ( ^ [A3: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A3 @ B2 )
& ( ord_less_eq_set_b @ B2 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_504_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_505_order__subst1,axiom,
! [A: set_a,F: set_b > set_a,B: set_b,C: set_b] :
( ( ord_less_eq_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X4: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_506_order__subst1,axiom,
! [A: set_b,F: set_a > set_b,B: set_a,C: set_a] :
( ( ord_less_eq_set_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_507_order__subst1,axiom,
! [A: set_b,F: set_b > set_b,B: set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X4: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_508_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_509_order__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_b,C: set_b] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_eq_set_b @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_510_order__subst2,axiom,
! [A: set_b,B: set_b,F: set_b > set_a,C: set_a] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_set_a @ ( F @ B ) @ C )
=> ( ! [X4: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_511_order__subst2,axiom,
! [A: set_b,B: set_b,F: set_b > set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ord_less_eq_set_b @ ( F @ B ) @ C )
=> ( ! [X4: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_512_order__eq__refl,axiom,
! [X2: set_a,Y2: set_a] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_a @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_513_order__eq__refl,axiom,
! [X2: set_b,Y2: set_b] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_b @ X2 @ Y2 ) ) ).
% order_eq_refl
thf(fact_514_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_515_ord__eq__le__subst,axiom,
! [A: set_b,F: set_a > set_b,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_516_ord__eq__le__subst,axiom,
! [A: set_a,F: set_b > set_a,B: set_b,C: set_b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X4: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_517_ord__eq__le__subst,axiom,
! [A: set_b,F: set_b > set_b,B: set_b,C: set_b] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_set_b @ B @ C )
=> ( ! [X4: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_518_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_519_ord__le__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_b,C: set_b] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_520_ord__le__eq__subst,axiom,
! [A: set_b,B: set_b,F: set_b > set_a,C: set_a] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ord_less_eq_set_a @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_521_ord__le__eq__subst,axiom,
! [A: set_b,B: set_b,F: set_b > set_b,C: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_b,Y3: set_b] :
( ( ord_less_eq_set_b @ X4 @ Y3 )
=> ( ord_less_eq_set_b @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_set_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_522_order__antisym__conv,axiom,
! [Y2: set_a,X2: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( ( ord_less_eq_set_a @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_523_order__antisym__conv,axiom,
! [Y2: set_b,X2: set_b] :
( ( ord_less_eq_set_b @ Y2 @ X2 )
=> ( ( ord_less_eq_set_b @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% order_antisym_conv
thf(fact_524_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_525_bot_Oextremum,axiom,
! [A: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A ) ).
% bot.extremum
thf(fact_526_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_527_bot_Oextremum__unique,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
= ( A = bot_bot_set_b ) ) ).
% bot.extremum_unique
thf(fact_528_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_529_bot_Oextremum__uniqueI,axiom,
! [A: set_b] :
( ( ord_less_eq_set_b @ A @ bot_bot_set_b )
=> ( A = bot_bot_set_b ) ) ).
% bot.extremum_uniqueI
thf(fact_530_no__back__before__aux,axiom,
! [Xs: list_a,Ys: list_a] :
( ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs )
=> ( ( iKKBZ_4622586873178280511rm_a_b @ t @ Ys )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ ( append_a @ Xs @ Ys ) ) ) ) ) ) ).
% no_back_before_aux
thf(fact_531_Sup__empty,axiom,
( ( comple2307003614231284044_set_b @ bot_bot_set_set_b )
= bot_bot_set_b ) ).
% Sup_empty
thf(fact_532_Sup__empty,axiom,
( ( comple2307003609928055243_set_a @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% Sup_empty
thf(fact_533_ccSup__empty,axiom,
( ( comple2307003614231284044_set_b @ bot_bot_set_set_b )
= bot_bot_set_b ) ).
% ccSup_empty
thf(fact_534_ccSup__empty,axiom,
( ( comple2307003609928055243_set_a @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% ccSup_empty
thf(fact_535_move__mid__forward__if__noarc,axiom,
! [As: list_a,U2: list_a,Bs: list_a,Cs: list_a] :
( ( As != nil_a )
=> ( ~ ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ U2 ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Bs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ Cs ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ Bs @ ( append_a @ U2 @ Cs ) ) ) ) ) ) ) ).
% move_mid_forward_if_noarc
thf(fact_536_no__back__reach1__if__fwd__dstct,axiom,
! [As: list_a,Bs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ Bs ) )
=> ( ( distinct_a @ ( append_a @ As @ Bs ) )
=> ~ ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Bs ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ As ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ).
% no_back_reach1_if_fwd_dstct
thf(fact_537_ball__UN,axiom,
! [B3: a > set_a,A2: set_a,P: a > $o] :
( ( ! [X5: a] :
( ( member_a @ X5 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B3 @ A2 ) ) )
=> ( P @ X5 ) ) )
= ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( B3 @ X5 ) )
=> ( P @ Y4 ) ) ) ) ) ).
% ball_UN
thf(fact_538_ball__UN,axiom,
! [B3: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,P: a > $o] :
( ( ! [X5: a] :
( ( member_a @ X5 @ ( comple2307003609928055243_set_a @ ( image_7466199892558553556_set_a @ B3 @ A2 ) ) )
=> ( P @ X5 ) ) )
= ( ! [X5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X5 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( B3 @ X5 ) )
=> ( P @ Y4 ) ) ) ) ) ).
% ball_UN
thf(fact_539_ball__UN,axiom,
! [B3: list_a > set_a,A2: set_list_a,P: a > $o] :
( ( ! [X5: a] :
( ( member_a @ X5 @ ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ B3 @ A2 ) ) )
=> ( P @ X5 ) ) )
= ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( B3 @ X5 ) )
=> ( P @ Y4 ) ) ) ) ) ).
% ball_UN
thf(fact_540_bex__UN,axiom,
! [B3: a > set_a,A2: set_a,P: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B3 @ A2 ) ) )
& ( P @ X5 ) ) )
= ( ? [X5: a] :
( ( member_a @ X5 @ A2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ ( B3 @ X5 ) )
& ( P @ Y4 ) ) ) ) ) ).
% bex_UN
thf(fact_541_bex__UN,axiom,
! [B3: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,P: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( comple2307003609928055243_set_a @ ( image_7466199892558553556_set_a @ B3 @ A2 ) ) )
& ( P @ X5 ) ) )
= ( ? [X5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X5 @ A2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ ( B3 @ X5 ) )
& ( P @ Y4 ) ) ) ) ) ).
% bex_UN
thf(fact_542_bex__UN,axiom,
! [B3: list_a > set_a,A2: set_list_a,P: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ B3 @ A2 ) ) )
& ( P @ X5 ) ) )
= ( ? [X5: list_a] :
( ( member_list_a @ X5 @ A2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ ( B3 @ X5 ) )
& ( P @ Y4 ) ) ) ) ) ).
% bex_UN
thf(fact_543_distinct__mid__unique2,axiom,
! [Xs: list_a,U2: list_a,Ys: list_a,As: list_a,Bs: list_a] :
( ( distinct_a @ ( append_a @ Xs @ ( append_a @ U2 @ Ys ) ) )
=> ( ( U2 != nil_a )
=> ( ( ( append_a @ Xs @ ( append_a @ U2 @ Ys ) )
= ( append_a @ As @ ( append_a @ U2 @ Bs ) ) )
=> ( Ys = Bs ) ) ) ) ).
% distinct_mid_unique2
thf(fact_544_distinct__mid__unique2,axiom,
! [Xs: list_b,U2: list_b,Ys: list_b,As: list_b,Bs: list_b] :
( ( distinct_b @ ( append_b @ Xs @ ( append_b @ U2 @ Ys ) ) )
=> ( ( U2 != nil_b )
=> ( ( ( append_b @ Xs @ ( append_b @ U2 @ Ys ) )
= ( append_b @ As @ ( append_b @ U2 @ Bs ) ) )
=> ( Ys = Bs ) ) ) ) ).
% distinct_mid_unique2
thf(fact_545_distinct__mid__unique1,axiom,
! [Xs: list_a,U2: list_a,Ys: list_a,As: list_a,Bs: list_a] :
( ( distinct_a @ ( append_a @ Xs @ ( append_a @ U2 @ Ys ) ) )
=> ( ( U2 != nil_a )
=> ( ( ( append_a @ Xs @ ( append_a @ U2 @ Ys ) )
= ( append_a @ As @ ( append_a @ U2 @ Bs ) ) )
=> ( As = Xs ) ) ) ) ).
% distinct_mid_unique1
thf(fact_546_distinct__mid__unique1,axiom,
! [Xs: list_b,U2: list_b,Ys: list_b,As: list_b,Bs: list_b] :
( ( distinct_b @ ( append_b @ Xs @ ( append_b @ U2 @ Ys ) ) )
=> ( ( U2 != nil_b )
=> ( ( ( append_b @ Xs @ ( append_b @ U2 @ Ys ) )
= ( append_b @ As @ ( append_b @ U2 @ Bs ) ) )
=> ( As = Xs ) ) ) ) ).
% distinct_mid_unique1
thf(fact_547_before__no__back1I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ S1 ) ) ).
% before_no_back1I
thf(fact_548_before__no__back2I,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ S2 ) ) ).
% before_no_back2I
thf(fact_549_Union__iff,axiom,
! [A2: product_prod_a_a,C2: set_se5735800977113168103od_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( comple8421679170691845492od_a_a @ C2 ) )
= ( ? [X5: set_Product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X5 @ C2 )
& ( member1426531477525435216od_a_a @ A2 @ X5 ) ) ) ) ).
% Union_iff
thf(fact_550_Union__iff,axiom,
! [A2: list_a,C2: set_set_list_a] :
( ( member_list_a @ A2 @ ( comple6928918032620976721list_a @ C2 ) )
= ( ? [X5: set_list_a] :
( ( member_set_list_a @ X5 @ C2 )
& ( member_list_a @ A2 @ X5 ) ) ) ) ).
% Union_iff
thf(fact_551_Union__iff,axiom,
! [A2: set_a,C2: set_set_set_a] :
( ( member_set_a @ A2 @ ( comple3958522678809307947_set_a @ C2 ) )
= ( ? [X5: set_set_a] :
( ( member_set_set_a @ X5 @ C2 )
& ( member_set_a @ A2 @ X5 ) ) ) ) ).
% Union_iff
thf(fact_552_Union__iff,axiom,
! [A2: b,C2: set_set_b] :
( ( member_b @ A2 @ ( comple2307003614231284044_set_b @ C2 ) )
= ( ? [X5: set_b] :
( ( member_set_b @ X5 @ C2 )
& ( member_b @ A2 @ X5 ) ) ) ) ).
% Union_iff
thf(fact_553_Union__iff,axiom,
! [A2: a,C2: set_set_a] :
( ( member_a @ A2 @ ( comple2307003609928055243_set_a @ C2 ) )
= ( ? [X5: set_a] :
( ( member_set_a @ X5 @ C2 )
& ( member_a @ A2 @ X5 ) ) ) ) ).
% Union_iff
thf(fact_554_UnionI,axiom,
! [X6: set_Product_prod_a_a,C2: set_se5735800977113168103od_a_a,A2: product_prod_a_a] :
( ( member1816616512716248880od_a_a @ X6 @ C2 )
=> ( ( member1426531477525435216od_a_a @ A2 @ X6 )
=> ( member1426531477525435216od_a_a @ A2 @ ( comple8421679170691845492od_a_a @ C2 ) ) ) ) ).
% UnionI
thf(fact_555_UnionI,axiom,
! [X6: set_list_a,C2: set_set_list_a,A2: list_a] :
( ( member_set_list_a @ X6 @ C2 )
=> ( ( member_list_a @ A2 @ X6 )
=> ( member_list_a @ A2 @ ( comple6928918032620976721list_a @ C2 ) ) ) ) ).
% UnionI
thf(fact_556_UnionI,axiom,
! [X6: set_set_a,C2: set_set_set_a,A2: set_a] :
( ( member_set_set_a @ X6 @ C2 )
=> ( ( member_set_a @ A2 @ X6 )
=> ( member_set_a @ A2 @ ( comple3958522678809307947_set_a @ C2 ) ) ) ) ).
% UnionI
thf(fact_557_UnionI,axiom,
! [X6: set_b,C2: set_set_b,A2: b] :
( ( member_set_b @ X6 @ C2 )
=> ( ( member_b @ A2 @ X6 )
=> ( member_b @ A2 @ ( comple2307003614231284044_set_b @ C2 ) ) ) ) ).
% UnionI
thf(fact_558_UnionI,axiom,
! [X6: set_a,C2: set_set_a,A2: a] :
( ( member_set_a @ X6 @ C2 )
=> ( ( member_a @ A2 @ X6 )
=> ( member_a @ A2 @ ( comple2307003609928055243_set_a @ C2 ) ) ) ) ).
% UnionI
thf(fact_559_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_a,P: a > $o] :
( ( ! [X5: a] :
( ( member_a @ X5 @ ( comple2307003609928055243_set_a @ A2 ) )
=> ( P @ X5 ) ) )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ X5 )
=> ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_560_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_a,P: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( comple2307003609928055243_set_a @ A2 ) )
& ( P @ X5 ) ) )
= ( ? [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ X5 )
& ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_561_seq__conform__if__dstnct__fwd,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( distinct_a @ Xs )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs ) ) ) ).
% seq_conform_if_dstnct_fwd
thf(fact_562_no__back__if__distinct__forward,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
=> ( ( distinct_a @ Xs )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs ) ) ) ).
% no_back_if_distinct_forward
thf(fact_563_no__back__before,axiom,
! [Xs: list_a,Ys: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ Xs @ Ys )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ ( append_a @ Xs @ Ys ) ) ) ).
% no_back_before
thf(fact_564_seq__conform__alt,axiom,
! [Xs: list_a] :
( ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs )
= ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
& ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs ) ) ) ).
% seq_conform_alt
thf(fact_565_to__list__tree__nempty,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( pre_ve1830060048215441954t_unit @ ( direct3773525127397338803ee_a_b @ t ) ) )
=> ( V != nil_a ) ) ).
% to_list_tree_nempty
thf(fact_566_no__back__arc__if__fwd__dstct,axiom,
! [As: list_a,Bs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ Bs ) )
=> ( ( distinct_a @ ( append_a @ As @ Bs ) )
=> ~ ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Bs ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ As ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% no_back_arc_if_fwd_dstct
thf(fact_567_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_b] :
( ( bot_bot_set_b
= ( comple2307003614231284044_set_b @ A2 ) )
= ( ! [X5: set_b] :
( ( member_set_b @ X5 @ A2 )
=> ( X5 = bot_bot_set_b ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_568_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ A2 ) )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
=> ( X5 = bot_bot_set_a ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_569_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_b] :
( ( ( comple2307003614231284044_set_b @ A2 )
= bot_bot_set_b )
= ( ! [X5: set_b] :
( ( member_set_b @ X5 @ A2 )
=> ( X5 = bot_bot_set_b ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_570_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_a] :
( ( ( comple2307003609928055243_set_a @ A2 )
= bot_bot_set_a )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
=> ( X5 = bot_bot_set_a ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_571_UN__ball__bex__simps_I4_J,axiom,
! [B3: a > set_a,A2: set_a,P: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B3 @ A2 ) ) )
& ( P @ X5 ) ) )
= ( ? [X5: a] :
( ( member_a @ X5 @ A2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ ( B3 @ X5 ) )
& ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_572_UN__ball__bex__simps_I4_J,axiom,
! [B3: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,P: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( comple2307003609928055243_set_a @ ( image_7466199892558553556_set_a @ B3 @ A2 ) ) )
& ( P @ X5 ) ) )
= ( ? [X5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X5 @ A2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ ( B3 @ X5 ) )
& ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_573_UN__ball__bex__simps_I4_J,axiom,
! [B3: list_a > set_a,A2: set_list_a,P: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ B3 @ A2 ) ) )
& ( P @ X5 ) ) )
= ( ? [X5: list_a] :
( ( member_list_a @ X5 @ A2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ ( B3 @ X5 ) )
& ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_574_UN__ball__bex__simps_I2_J,axiom,
! [B3: a > set_a,A2: set_a,P: a > $o] :
( ( ! [X5: a] :
( ( member_a @ X5 @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B3 @ A2 ) ) )
=> ( P @ X5 ) ) )
= ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( B3 @ X5 ) )
=> ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_575_UN__ball__bex__simps_I2_J,axiom,
! [B3: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,P: a > $o] :
( ( ! [X5: a] :
( ( member_a @ X5 @ ( comple2307003609928055243_set_a @ ( image_7466199892558553556_set_a @ B3 @ A2 ) ) )
=> ( P @ X5 ) ) )
= ( ! [X5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X5 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( B3 @ X5 ) )
=> ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_576_UN__ball__bex__simps_I2_J,axiom,
! [B3: list_a > set_a,A2: set_list_a,P: a > $o] :
( ( ! [X5: a] :
( ( member_a @ X5 @ ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ B3 @ A2 ) ) )
=> ( P @ X5 ) ) )
= ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( B3 @ X5 ) )
=> ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_577_directed__tree_Ono__back_Ocong,axiom,
iKKBZ_3684931046464919648ck_a_b = iKKBZ_3684931046464919648ck_a_b ).
% directed_tree.no_back.cong
thf(fact_578_Sup_OSUP__cong,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,C2: pre_pr7278220950009878019t_unit > set_a,D2: pre_pr7278220950009878019t_unit > set_a,Sup: set_set_a > set_a] :
( ( A2 = B3 )
=> ( ! [X4: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X4 @ B3 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_7466199892558553556_set_a @ C2 @ A2 ) )
= ( Sup @ ( image_7466199892558553556_set_a @ D2 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_579_Sup_OSUP__cong,axiom,
! [A2: set_list_a,B3: set_list_a,C2: list_a > set_a,D2: list_a > set_a,Sup: set_set_a > set_a] :
( ( A2 = B3 )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ B3 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_list_a_set_a @ C2 @ A2 ) )
= ( Sup @ ( image_list_a_set_a @ D2 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_580_Sup_OSUP__cong,axiom,
! [A2: set_set_a,B3: set_set_a,C2: set_a > pre_pr7278220950009878019t_unit,D2: set_a > pre_pr7278220950009878019t_unit,Sup: set_pr5411798346947241657t_unit > pre_pr7278220950009878019t_unit] :
( ( A2 = B3 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ B3 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_6801035452528096924t_unit @ C2 @ A2 ) )
= ( Sup @ ( image_6801035452528096924t_unit @ D2 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_581_Sup_OSUP__cong,axiom,
! [A2: set_b,B3: set_b,C2: b > a,D2: b > a,Sup: set_a > a] :
( ( A2 = B3 )
=> ( ! [X4: b] :
( ( member_b @ X4 @ B3 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_b_a @ C2 @ A2 ) )
= ( Sup @ ( image_b_a @ D2 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_582_Sup_OSUP__cong,axiom,
! [A2: set_a,B3: set_a,C2: a > set_a,D2: a > set_a,Sup: set_set_a > set_a] :
( ( A2 = B3 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B3 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_a_set_a @ C2 @ A2 ) )
= ( Sup @ ( image_a_set_a @ D2 @ B3 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_583_Inf_OINF__cong,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,C2: pre_pr7278220950009878019t_unit > set_a,D2: pre_pr7278220950009878019t_unit > set_a,Inf: set_set_a > set_a] :
( ( A2 = B3 )
=> ( ! [X4: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X4 @ B3 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_7466199892558553556_set_a @ C2 @ A2 ) )
= ( Inf @ ( image_7466199892558553556_set_a @ D2 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_584_Inf_OINF__cong,axiom,
! [A2: set_list_a,B3: set_list_a,C2: list_a > set_a,D2: list_a > set_a,Inf: set_set_a > set_a] :
( ( A2 = B3 )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ B3 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_list_a_set_a @ C2 @ A2 ) )
= ( Inf @ ( image_list_a_set_a @ D2 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_585_Inf_OINF__cong,axiom,
! [A2: set_set_a,B3: set_set_a,C2: set_a > pre_pr7278220950009878019t_unit,D2: set_a > pre_pr7278220950009878019t_unit,Inf: set_pr5411798346947241657t_unit > pre_pr7278220950009878019t_unit] :
( ( A2 = B3 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ B3 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_6801035452528096924t_unit @ C2 @ A2 ) )
= ( Inf @ ( image_6801035452528096924t_unit @ D2 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_586_Inf_OINF__cong,axiom,
! [A2: set_b,B3: set_b,C2: b > a,D2: b > a,Inf: set_a > a] :
( ( A2 = B3 )
=> ( ! [X4: b] :
( ( member_b @ X4 @ B3 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_b_a @ C2 @ A2 ) )
= ( Inf @ ( image_b_a @ D2 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_587_Inf_OINF__cong,axiom,
! [A2: set_a,B3: set_a,C2: a > set_a,D2: a > set_a,Inf: set_set_a > set_a] :
( ( A2 = B3 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B3 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_a_set_a @ C2 @ A2 ) )
= ( Inf @ ( image_a_set_a @ D2 @ B3 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_588_UnionE,axiom,
! [A2: product_prod_a_a,C2: set_se5735800977113168103od_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ ( comple8421679170691845492od_a_a @ C2 ) )
=> ~ ! [X7: set_Product_prod_a_a] :
( ( member1426531477525435216od_a_a @ A2 @ X7 )
=> ~ ( member1816616512716248880od_a_a @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_589_UnionE,axiom,
! [A2: list_a,C2: set_set_list_a] :
( ( member_list_a @ A2 @ ( comple6928918032620976721list_a @ C2 ) )
=> ~ ! [X7: set_list_a] :
( ( member_list_a @ A2 @ X7 )
=> ~ ( member_set_list_a @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_590_UnionE,axiom,
! [A2: set_a,C2: set_set_set_a] :
( ( member_set_a @ A2 @ ( comple3958522678809307947_set_a @ C2 ) )
=> ~ ! [X7: set_set_a] :
( ( member_set_a @ A2 @ X7 )
=> ~ ( member_set_set_a @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_591_UnionE,axiom,
! [A2: b,C2: set_set_b] :
( ( member_b @ A2 @ ( comple2307003614231284044_set_b @ C2 ) )
=> ~ ! [X7: set_b] :
( ( member_b @ A2 @ X7 )
=> ~ ( member_set_b @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_592_UnionE,axiom,
! [A2: a,C2: set_set_a] :
( ( member_a @ A2 @ ( comple2307003609928055243_set_a @ C2 ) )
=> ~ ! [X7: set_a] :
( ( member_a @ A2 @ X7 )
=> ~ ( member_set_a @ X7 @ C2 ) ) ) ).
% UnionE
thf(fact_593_list__empty__if__subset__dsjnt,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ Ys ) )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ Ys ) )
= bot_bot_set_list_a )
=> ( Xs = nil_list_a ) ) ) ).
% list_empty_if_subset_dsjnt
thf(fact_594_list__empty__if__subset__dsjnt,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( Xs = nil_a ) ) ) ).
% list_empty_if_subset_dsjnt
thf(fact_595_list__empty__if__subset__dsjnt,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ Ys ) )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ Ys ) )
= bot_bot_set_b )
=> ( Xs = nil_b ) ) ) ).
% list_empty_if_subset_dsjnt
thf(fact_596_Sup__eqI,axiom,
! [A2: set_set_b,X2: set_b] :
( ! [Y3: set_b] :
( ( member_set_b @ Y3 @ A2 )
=> ( ord_less_eq_set_b @ Y3 @ X2 ) )
=> ( ! [Y3: set_b] :
( ! [Z2: set_b] :
( ( member_set_b @ Z2 @ A2 )
=> ( ord_less_eq_set_b @ Z2 @ Y3 ) )
=> ( ord_less_eq_set_b @ X2 @ Y3 ) )
=> ( ( comple2307003614231284044_set_b @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_597_Sup__eqI,axiom,
! [A2: set_set_a,X2: set_a] :
( ! [Y3: set_a] :
( ( member_set_a @ Y3 @ A2 )
=> ( ord_less_eq_set_a @ Y3 @ X2 ) )
=> ( ! [Y3: set_a] :
( ! [Z2: set_a] :
( ( member_set_a @ Z2 @ A2 )
=> ( ord_less_eq_set_a @ Z2 @ Y3 ) )
=> ( ord_less_eq_set_a @ X2 @ Y3 ) )
=> ( ( comple2307003609928055243_set_a @ A2 )
= X2 ) ) ) ).
% Sup_eqI
thf(fact_598_Sup__mono,axiom,
! [A2: set_set_b,B3: set_set_b] :
( ! [A5: set_b] :
( ( member_set_b @ A5 @ A2 )
=> ? [X: set_b] :
( ( member_set_b @ X @ B3 )
& ( ord_less_eq_set_b @ A5 @ X ) ) )
=> ( ord_less_eq_set_b @ ( comple2307003614231284044_set_b @ A2 ) @ ( comple2307003614231284044_set_b @ B3 ) ) ) ).
% Sup_mono
thf(fact_599_Sup__mono,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ! [A5: set_a] :
( ( member_set_a @ A5 @ A2 )
=> ? [X: set_a] :
( ( member_set_a @ X @ B3 )
& ( ord_less_eq_set_a @ A5 @ X ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B3 ) ) ) ).
% Sup_mono
thf(fact_600_Sup__least,axiom,
! [A2: set_set_b,Z: set_b] :
( ! [X4: set_b] :
( ( member_set_b @ X4 @ A2 )
=> ( ord_less_eq_set_b @ X4 @ Z ) )
=> ( ord_less_eq_set_b @ ( comple2307003614231284044_set_b @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_601_Sup__least,axiom,
! [A2: set_set_a,Z: set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ( ord_less_eq_set_a @ X4 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_602_Sup__upper,axiom,
! [X2: set_b,A2: set_set_b] :
( ( member_set_b @ X2 @ A2 )
=> ( ord_less_eq_set_b @ X2 @ ( comple2307003614231284044_set_b @ A2 ) ) ) ).
% Sup_upper
thf(fact_603_Sup__upper,axiom,
! [X2: set_a,A2: set_set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ X2 @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% Sup_upper
thf(fact_604_Sup__le__iff,axiom,
! [A2: set_set_b,B: set_b] :
( ( ord_less_eq_set_b @ ( comple2307003614231284044_set_b @ A2 ) @ B )
= ( ! [X5: set_b] :
( ( member_set_b @ X5 @ A2 )
=> ( ord_less_eq_set_b @ X5 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_605_Sup__le__iff,axiom,
! [A2: set_set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ B )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
=> ( ord_less_eq_set_a @ X5 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_606_Sup__upper2,axiom,
! [U: set_b,A2: set_set_b,V: set_b] :
( ( member_set_b @ U @ A2 )
=> ( ( ord_less_eq_set_b @ V @ U )
=> ( ord_less_eq_set_b @ V @ ( comple2307003614231284044_set_b @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_607_Sup__upper2,axiom,
! [U: set_a,A2: set_set_a,V: set_a] :
( ( member_set_a @ U @ A2 )
=> ( ( ord_less_eq_set_a @ V @ U )
=> ( ord_less_eq_set_a @ V @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_608_SUP__cong,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_pr5411798346947241657t_unit,C2: pre_pr7278220950009878019t_unit > set_a,D2: pre_pr7278220950009878019t_unit > set_a] :
( ( A2 = B3 )
=> ( ! [X4: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X4 @ B3 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_7466199892558553556_set_a @ C2 @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_7466199892558553556_set_a @ D2 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_609_SUP__cong,axiom,
! [A2: set_Product_prod_a_a,B3: set_Product_prod_a_a,C2: product_prod_a_a > set_a,D2: product_prod_a_a > set_a] :
( ( A2 = B3 )
=> ( ! [X4: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X4 @ B3 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_9052089385058188540_set_a @ C2 @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_9052089385058188540_set_a @ D2 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_610_SUP__cong,axiom,
! [A2: set_list_a,B3: set_list_a,C2: list_a > set_a,D2: list_a > set_a] :
( ( A2 = B3 )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ B3 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ C2 @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ D2 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_611_SUP__cong,axiom,
! [A2: set_set_a,B3: set_set_a,C2: set_a > set_a,D2: set_a > set_a] :
( ( A2 = B3 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ B3 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ C2 @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ D2 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_612_SUP__cong,axiom,
! [A2: set_b,B3: set_b,C2: b > set_a,D2: b > set_a] :
( ( A2 = B3 )
=> ( ! [X4: b] :
( ( member_b @ X4 @ B3 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_b_set_a @ C2 @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_b_set_a @ D2 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_613_SUP__cong,axiom,
! [A2: set_a,B3: set_a,C2: a > set_a,D2: a > set_a] :
( ( A2 = B3 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ B3 )
=> ( ( C2 @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ C2 @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ D2 @ B3 ) ) ) ) ) ).
% SUP_cong
thf(fact_614_empty__Union__conv,axiom,
! [A2: set_set_b] :
( ( bot_bot_set_b
= ( comple2307003614231284044_set_b @ A2 ) )
= ( ! [X5: set_b] :
( ( member_set_b @ X5 @ A2 )
=> ( X5 = bot_bot_set_b ) ) ) ) ).
% empty_Union_conv
thf(fact_615_empty__Union__conv,axiom,
! [A2: set_set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ A2 ) )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
=> ( X5 = bot_bot_set_a ) ) ) ) ).
% empty_Union_conv
thf(fact_616_Union__empty__conv,axiom,
! [A2: set_set_b] :
( ( ( comple2307003614231284044_set_b @ A2 )
= bot_bot_set_b )
= ( ! [X5: set_b] :
( ( member_set_b @ X5 @ A2 )
=> ( X5 = bot_bot_set_b ) ) ) ) ).
% Union_empty_conv
thf(fact_617_Union__empty__conv,axiom,
! [A2: set_set_a] :
( ( ( comple2307003609928055243_set_a @ A2 )
= bot_bot_set_a )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
=> ( X5 = bot_bot_set_a ) ) ) ) ).
% Union_empty_conv
thf(fact_618_Union__empty,axiom,
( ( comple2307003614231284044_set_b @ bot_bot_set_set_b )
= bot_bot_set_b ) ).
% Union_empty
thf(fact_619_Union__empty,axiom,
( ( comple2307003609928055243_set_a @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% Union_empty
thf(fact_620_Union__mono,axiom,
! [A2: set_set_b,B3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B3 )
=> ( ord_less_eq_set_b @ ( comple2307003614231284044_set_b @ A2 ) @ ( comple2307003614231284044_set_b @ B3 ) ) ) ).
% Union_mono
thf(fact_621_Union__mono,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B3 ) ) ) ).
% Union_mono
thf(fact_622_Union__least,axiom,
! [A2: set_set_b,C2: set_b] :
( ! [X7: set_b] :
( ( member_set_b @ X7 @ A2 )
=> ( ord_less_eq_set_b @ X7 @ C2 ) )
=> ( ord_less_eq_set_b @ ( comple2307003614231284044_set_b @ A2 ) @ C2 ) ) ).
% Union_least
thf(fact_623_Union__least,axiom,
! [A2: set_set_a,C2: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A2 )
=> ( ord_less_eq_set_a @ X7 @ C2 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ C2 ) ) ).
% Union_least
thf(fact_624_Union__upper,axiom,
! [B3: set_b,A2: set_set_b] :
( ( member_set_b @ B3 @ A2 )
=> ( ord_less_eq_set_b @ B3 @ ( comple2307003614231284044_set_b @ A2 ) ) ) ).
% Union_upper
thf(fact_625_Union__upper,axiom,
! [B3: set_a,A2: set_set_a] :
( ( member_set_a @ B3 @ A2 )
=> ( ord_less_eq_set_a @ B3 @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% Union_upper
thf(fact_626_Union__subsetI,axiom,
! [A2: set_set_b,B3: set_set_b] :
( ! [X4: set_b] :
( ( member_set_b @ X4 @ A2 )
=> ? [Y: set_b] :
( ( member_set_b @ Y @ B3 )
& ( ord_less_eq_set_b @ X4 @ Y ) ) )
=> ( ord_less_eq_set_b @ ( comple2307003614231284044_set_b @ A2 ) @ ( comple2307003614231284044_set_b @ B3 ) ) ) ).
% Union_subsetI
thf(fact_627_Union__subsetI,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ! [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
=> ? [Y: set_a] :
( ( member_set_a @ Y @ B3 )
& ( ord_less_eq_set_a @ X4 @ Y ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B3 ) ) ) ).
% Union_subsetI
thf(fact_628_less__eq__Sup,axiom,
! [A2: set_set_b,U: set_b] :
( ! [V4: set_b] :
( ( member_set_b @ V4 @ A2 )
=> ( ord_less_eq_set_b @ U @ V4 ) )
=> ( ( A2 != bot_bot_set_set_b )
=> ( ord_less_eq_set_b @ U @ ( comple2307003614231284044_set_b @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_629_less__eq__Sup,axiom,
! [A2: set_set_a,U: set_a] :
( ! [V4: set_a] :
( ( member_set_a @ V4 @ A2 )
=> ( ord_less_eq_set_a @ U @ V4 ) )
=> ( ( A2 != bot_bot_set_set_a )
=> ( ord_less_eq_set_a @ U @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% less_eq_Sup
thf(fact_630_SUP__eq,axiom,
! [A2: set_b,B3: set_b,F: b > set_b,G2: b > set_b] :
( ! [I: b] :
( ( member_b @ I @ A2 )
=> ? [X: b] :
( ( member_b @ X @ B3 )
& ( ord_less_eq_set_b @ ( F @ I ) @ ( G2 @ X ) ) ) )
=> ( ! [J: b] :
( ( member_b @ J @ B3 )
=> ? [X: b] :
( ( member_b @ X @ A2 )
& ( ord_less_eq_set_b @ ( G2 @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003614231284044_set_b @ ( image_b_set_b @ F @ A2 ) )
= ( comple2307003614231284044_set_b @ ( image_b_set_b @ G2 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_631_SUP__eq,axiom,
! [A2: set_b,B3: set_a,F: b > set_b,G2: a > set_b] :
( ! [I: b] :
( ( member_b @ I @ A2 )
=> ? [X: a] :
( ( member_a @ X @ B3 )
& ( ord_less_eq_set_b @ ( F @ I ) @ ( G2 @ X ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B3 )
=> ? [X: b] :
( ( member_b @ X @ A2 )
& ( ord_less_eq_set_b @ ( G2 @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003614231284044_set_b @ ( image_b_set_b @ F @ A2 ) )
= ( comple2307003614231284044_set_b @ ( image_a_set_b @ G2 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_632_SUP__eq,axiom,
! [A2: set_a,B3: set_b,F: a > set_b,G2: b > set_b] :
( ! [I: a] :
( ( member_a @ I @ A2 )
=> ? [X: b] :
( ( member_b @ X @ B3 )
& ( ord_less_eq_set_b @ ( F @ I ) @ ( G2 @ X ) ) ) )
=> ( ! [J: b] :
( ( member_b @ J @ B3 )
=> ? [X: a] :
( ( member_a @ X @ A2 )
& ( ord_less_eq_set_b @ ( G2 @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003614231284044_set_b @ ( image_a_set_b @ F @ A2 ) )
= ( comple2307003614231284044_set_b @ ( image_b_set_b @ G2 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_633_SUP__eq,axiom,
! [A2: set_a,B3: set_a,F: a > set_b,G2: a > set_b] :
( ! [I: a] :
( ( member_a @ I @ A2 )
=> ? [X: a] :
( ( member_a @ X @ B3 )
& ( ord_less_eq_set_b @ ( F @ I ) @ ( G2 @ X ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B3 )
=> ? [X: a] :
( ( member_a @ X @ A2 )
& ( ord_less_eq_set_b @ ( G2 @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003614231284044_set_b @ ( image_a_set_b @ F @ A2 ) )
= ( comple2307003614231284044_set_b @ ( image_a_set_b @ G2 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_634_SUP__eq,axiom,
! [A2: set_b,B3: set_b,F: b > set_a,G2: b > set_a] :
( ! [I: b] :
( ( member_b @ I @ A2 )
=> ? [X: b] :
( ( member_b @ X @ B3 )
& ( ord_less_eq_set_a @ ( F @ I ) @ ( G2 @ X ) ) ) )
=> ( ! [J: b] :
( ( member_b @ J @ B3 )
=> ? [X: b] :
( ( member_b @ X @ A2 )
& ( ord_less_eq_set_a @ ( G2 @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_b_set_a @ F @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_b_set_a @ G2 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_635_SUP__eq,axiom,
! [A2: set_b,B3: set_a,F: b > set_a,G2: a > set_a] :
( ! [I: b] :
( ( member_b @ I @ A2 )
=> ? [X: a] :
( ( member_a @ X @ B3 )
& ( ord_less_eq_set_a @ ( F @ I ) @ ( G2 @ X ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B3 )
=> ? [X: b] :
( ( member_b @ X @ A2 )
& ( ord_less_eq_set_a @ ( G2 @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_b_set_a @ F @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ G2 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_636_SUP__eq,axiom,
! [A2: set_a,B3: set_b,F: a > set_a,G2: b > set_a] :
( ! [I: a] :
( ( member_a @ I @ A2 )
=> ? [X: b] :
( ( member_b @ X @ B3 )
& ( ord_less_eq_set_a @ ( F @ I ) @ ( G2 @ X ) ) ) )
=> ( ! [J: b] :
( ( member_b @ J @ B3 )
=> ? [X: a] :
( ( member_a @ X @ A2 )
& ( ord_less_eq_set_a @ ( G2 @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_b_set_a @ G2 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_637_SUP__eq,axiom,
! [A2: set_a,B3: set_a,F: a > set_a,G2: a > set_a] :
( ! [I: a] :
( ( member_a @ I @ A2 )
=> ? [X: a] :
( ( member_a @ X @ B3 )
& ( ord_less_eq_set_a @ ( F @ I ) @ ( G2 @ X ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B3 )
=> ? [X: a] :
( ( member_a @ X @ A2 )
& ( ord_less_eq_set_a @ ( G2 @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ A2 ) )
= ( comple2307003609928055243_set_a @ ( image_a_set_a @ G2 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_638_SUP__eq,axiom,
! [A2: set_list_a,B3: set_b,F: list_a > set_b,G2: b > set_b] :
( ! [I: list_a] :
( ( member_list_a @ I @ A2 )
=> ? [X: b] :
( ( member_b @ X @ B3 )
& ( ord_less_eq_set_b @ ( F @ I ) @ ( G2 @ X ) ) ) )
=> ( ! [J: b] :
( ( member_b @ J @ B3 )
=> ? [X: list_a] :
( ( member_list_a @ X @ A2 )
& ( ord_less_eq_set_b @ ( G2 @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003614231284044_set_b @ ( image_list_a_set_b @ F @ A2 ) )
= ( comple2307003614231284044_set_b @ ( image_b_set_b @ G2 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_639_SUP__eq,axiom,
! [A2: set_list_a,B3: set_a,F: list_a > set_b,G2: a > set_b] :
( ! [I: list_a] :
( ( member_list_a @ I @ A2 )
=> ? [X: a] :
( ( member_a @ X @ B3 )
& ( ord_less_eq_set_b @ ( F @ I ) @ ( G2 @ X ) ) ) )
=> ( ! [J: a] :
( ( member_a @ J @ B3 )
=> ? [X: list_a] :
( ( member_list_a @ X @ A2 )
& ( ord_less_eq_set_b @ ( G2 @ J ) @ ( F @ X ) ) ) )
=> ( ( comple2307003614231284044_set_b @ ( image_list_a_set_b @ F @ A2 ) )
= ( comple2307003614231284044_set_b @ ( image_a_set_b @ G2 @ B3 ) ) ) ) ) ).
% SUP_eq
thf(fact_640_Sup__subset__mono,axiom,
! [A2: set_set_b,B3: set_set_b] :
( ( ord_le3795704787696855135_set_b @ A2 @ B3 )
=> ( ord_less_eq_set_b @ ( comple2307003614231284044_set_b @ A2 ) @ ( comple2307003614231284044_set_b @ B3 ) ) ) ).
% Sup_subset_mono
thf(fact_641_Sup__subset__mono,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B3 ) ) ) ).
% Sup_subset_mono
thf(fact_642_SUP__eq__const,axiom,
! [I2: set_pr5411798346947241657t_unit,F: pre_pr7278220950009878019t_unit > set_a,X2: set_a] :
( ( I2 != bot_bo1839476491465656141t_unit )
=> ( ! [I: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ I @ I2 )
=> ( ( F @ I )
= X2 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_7466199892558553556_set_a @ F @ I2 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_643_SUP__eq__const,axiom,
! [I2: set_Product_prod_a_a,F: product_prod_a_a > set_a,X2: set_a] :
( ( I2 != bot_bo3357376287454694259od_a_a )
=> ( ! [I: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ I @ I2 )
=> ( ( F @ I )
= X2 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_9052089385058188540_set_a @ F @ I2 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_644_SUP__eq__const,axiom,
! [I2: set_list_a,F: list_a > set_a,X2: set_a] :
( ( I2 != bot_bot_set_list_a )
=> ( ! [I: list_a] :
( ( member_list_a @ I @ I2 )
=> ( ( F @ I )
= X2 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ F @ I2 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_645_SUP__eq__const,axiom,
! [I2: set_set_a,F: set_a > set_a,X2: set_a] :
( ( I2 != bot_bot_set_set_a )
=> ( ! [I: set_a] :
( ( member_set_a @ I @ I2 )
=> ( ( F @ I )
= X2 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ I2 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_646_SUP__eq__const,axiom,
! [I2: set_a,F: a > set_a,X2: set_a] :
( ( I2 != bot_bot_set_a )
=> ( ! [I: a] :
( ( member_a @ I @ I2 )
=> ( ( F @ I )
= X2 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ I2 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_647_SUP__eq__const,axiom,
! [I2: set_b,F: b > set_a,X2: set_a] :
( ( I2 != bot_bot_set_b )
=> ( ! [I: b] :
( ( member_b @ I @ I2 )
=> ( ( F @ I )
= X2 ) )
=> ( ( comple2307003609928055243_set_a @ ( image_b_set_a @ F @ I2 ) )
= X2 ) ) ) ).
% SUP_eq_const
thf(fact_648_Union__disjoint,axiom,
! [C2: set_set_b,A2: set_b] :
( ( ( inf_inf_set_b @ ( comple2307003614231284044_set_b @ C2 ) @ A2 )
= bot_bot_set_b )
= ( ! [X5: set_b] :
( ( member_set_b @ X5 @ C2 )
=> ( ( inf_inf_set_b @ X5 @ A2 )
= bot_bot_set_b ) ) ) ) ).
% Union_disjoint
thf(fact_649_Union__disjoint,axiom,
! [C2: set_set_a,A2: set_a] :
( ( ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ C2 ) @ A2 )
= bot_bot_set_a )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ C2 )
=> ( ( inf_inf_set_a @ X5 @ A2 )
= bot_bot_set_a ) ) ) ) ).
% Union_disjoint
thf(fact_650_Union__Int__subset,axiom,
! [A2: set_set_b,B3: set_set_b] : ( ord_less_eq_set_b @ ( comple2307003614231284044_set_b @ ( inf_inf_set_set_b @ A2 @ B3 ) ) @ ( inf_inf_set_b @ ( comple2307003614231284044_set_b @ A2 ) @ ( comple2307003614231284044_set_b @ B3 ) ) ) ).
% Union_Int_subset
thf(fact_651_Union__Int__subset,axiom,
! [A2: set_set_a,B3: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A2 @ B3 ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B3 ) ) ) ).
% Union_Int_subset
thf(fact_652_SUP__eq__iff,axiom,
! [I2: set_a,C: set_b,F: a > set_b] :
( ( I2 != bot_bot_set_a )
=> ( ! [I: a] :
( ( member_a @ I @ I2 )
=> ( ord_less_eq_set_b @ C @ ( F @ I ) ) )
=> ( ( ( comple2307003614231284044_set_b @ ( image_a_set_b @ F @ I2 ) )
= C )
= ( ! [X5: a] :
( ( member_a @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_653_SUP__eq__iff,axiom,
! [I2: set_b,C: set_b,F: b > set_b] :
( ( I2 != bot_bot_set_b )
=> ( ! [I: b] :
( ( member_b @ I @ I2 )
=> ( ord_less_eq_set_b @ C @ ( F @ I ) ) )
=> ( ( ( comple2307003614231284044_set_b @ ( image_b_set_b @ F @ I2 ) )
= C )
= ( ! [X5: b] :
( ( member_b @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_654_SUP__eq__iff,axiom,
! [I2: set_a,C: set_a,F: a > set_a] :
( ( I2 != bot_bot_set_a )
=> ( ! [I: a] :
( ( member_a @ I @ I2 )
=> ( ord_less_eq_set_a @ C @ ( F @ I ) ) )
=> ( ( ( comple2307003609928055243_set_a @ ( image_a_set_a @ F @ I2 ) )
= C )
= ( ! [X5: a] :
( ( member_a @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_655_SUP__eq__iff,axiom,
! [I2: set_b,C: set_a,F: b > set_a] :
( ( I2 != bot_bot_set_b )
=> ( ! [I: b] :
( ( member_b @ I @ I2 )
=> ( ord_less_eq_set_a @ C @ ( F @ I ) ) )
=> ( ( ( comple2307003609928055243_set_a @ ( image_b_set_a @ F @ I2 ) )
= C )
= ( ! [X5: b] :
( ( member_b @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_656_SUP__eq__iff,axiom,
! [I2: set_list_a,C: set_b,F: list_a > set_b] :
( ( I2 != bot_bot_set_list_a )
=> ( ! [I: list_a] :
( ( member_list_a @ I @ I2 )
=> ( ord_less_eq_set_b @ C @ ( F @ I ) ) )
=> ( ( ( comple2307003614231284044_set_b @ ( image_list_a_set_b @ F @ I2 ) )
= C )
= ( ! [X5: list_a] :
( ( member_list_a @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_657_SUP__eq__iff,axiom,
! [I2: set_set_a,C: set_b,F: set_a > set_b] :
( ( I2 != bot_bot_set_set_a )
=> ( ! [I: set_a] :
( ( member_set_a @ I @ I2 )
=> ( ord_less_eq_set_b @ C @ ( F @ I ) ) )
=> ( ( ( comple2307003614231284044_set_b @ ( image_set_a_set_b @ F @ I2 ) )
= C )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_658_SUP__eq__iff,axiom,
! [I2: set_list_a,C: set_a,F: list_a > set_a] :
( ( I2 != bot_bot_set_list_a )
=> ( ! [I: list_a] :
( ( member_list_a @ I @ I2 )
=> ( ord_less_eq_set_a @ C @ ( F @ I ) ) )
=> ( ( ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ F @ I2 ) )
= C )
= ( ! [X5: list_a] :
( ( member_list_a @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_659_SUP__eq__iff,axiom,
! [I2: set_set_a,C: set_a,F: set_a > set_a] :
( ( I2 != bot_bot_set_set_a )
=> ( ! [I: set_a] :
( ( member_set_a @ I @ I2 )
=> ( ord_less_eq_set_a @ C @ ( F @ I ) ) )
=> ( ( ( comple2307003609928055243_set_a @ ( image_set_a_set_a @ F @ I2 ) )
= C )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_660_SUP__eq__iff,axiom,
! [I2: set_Product_prod_a_a,C: set_b,F: product_prod_a_a > set_b] :
( ( I2 != bot_bo3357376287454694259od_a_a )
=> ( ! [I: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ I @ I2 )
=> ( ord_less_eq_set_b @ C @ ( F @ I ) ) )
=> ( ( ( comple2307003614231284044_set_b @ ( image_9052089389361417341_set_b @ F @ I2 ) )
= C )
= ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_661_SUP__eq__iff,axiom,
! [I2: set_Product_prod_a_a,C: set_a,F: product_prod_a_a > set_a] :
( ( I2 != bot_bo3357376287454694259od_a_a )
=> ( ! [I: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ I @ I2 )
=> ( ord_less_eq_set_a @ C @ ( F @ I ) ) )
=> ( ( ( comple2307003609928055243_set_a @ ( image_9052089385058188540_set_a @ F @ I2 ) )
= C )
= ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ I2 )
=> ( ( F @ X5 )
= C ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_662_Sup__inter__less__eq,axiom,
! [A2: set_set_b,B3: set_set_b] : ( ord_less_eq_set_b @ ( comple2307003614231284044_set_b @ ( inf_inf_set_set_b @ A2 @ B3 ) ) @ ( inf_inf_set_b @ ( comple2307003614231284044_set_b @ A2 ) @ ( comple2307003614231284044_set_b @ B3 ) ) ) ).
% Sup_inter_less_eq
thf(fact_663_Sup__inter__less__eq,axiom,
! [A2: set_set_a,B3: set_set_a] : ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ ( inf_inf_set_set_a @ A2 @ B3 ) ) @ ( inf_inf_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B3 ) ) ) ).
% Sup_inter_less_eq
thf(fact_664_distinct__append,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( distinct_list_a @ ( append_list_a @ Xs @ Ys ) )
= ( ( distinct_list_a @ Xs )
& ( distinct_list_a @ Ys )
& ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ Ys ) )
= bot_bot_set_list_a ) ) ) ).
% distinct_append
thf(fact_665_distinct__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_a @ ( append_a @ Xs @ Ys ) )
= ( ( distinct_a @ Xs )
& ( distinct_a @ Ys )
& ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a ) ) ) ).
% distinct_append
thf(fact_666_distinct__append,axiom,
! [Xs: list_b,Ys: list_b] :
( ( distinct_b @ ( append_b @ Xs @ Ys ) )
= ( ( distinct_b @ Xs )
& ( distinct_b @ Ys )
& ( ( inf_inf_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ Ys ) )
= bot_bot_set_b ) ) ) ).
% distinct_append
thf(fact_667_hd__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_668_hd__append2,axiom,
! [Xs: list_b,Ys: list_b] :
( ( Xs != nil_b )
=> ( ( hd_b @ ( append_b @ Xs @ Ys ) )
= ( hd_b @ Xs ) ) ) ).
% hd_append2
thf(fact_669_set__empty,axiom,
! [Xs: list_list_a] :
( ( ( set_list_a2 @ Xs )
= bot_bot_set_list_a )
= ( Xs = nil_list_a ) ) ).
% set_empty
thf(fact_670_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_671_set__empty,axiom,
! [Xs: list_b] :
( ( ( set_b2 @ Xs )
= bot_bot_set_b )
= ( Xs = nil_b ) ) ).
% set_empty
thf(fact_672_set__empty2,axiom,
! [Xs: list_list_a] :
( ( bot_bot_set_list_a
= ( set_list_a2 @ Xs ) )
= ( Xs = nil_list_a ) ) ).
% set_empty2
thf(fact_673_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_674_set__empty2,axiom,
! [Xs: list_b] :
( ( bot_bot_set_b
= ( set_b2 @ Xs ) )
= ( Xs = nil_b ) ) ).
% set_empty2
thf(fact_675_to__list__tree__single,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( pre_ve1830060048215441954t_unit @ ( direct3773525127397338803ee_a_b @ t ) ) )
=> ? [X4: a] :
( ( V
= ( cons_a @ X4 @ nil_a ) )
& ( member_a @ X4 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% to_list_tree_single
thf(fact_676_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_677_append_Oright__neutral,axiom,
! [A: list_b] :
( ( append_b @ A @ nil_b )
= A ) ).
% append.right_neutral
thf(fact_678_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_679_append__Nil2,axiom,
! [Xs: list_b] :
( ( append_b @ Xs @ nil_b )
= Xs ) ).
% append_Nil2
thf(fact_680_forward__arcs_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ( ! [X4: a] :
( X2
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,V4: a,Va: list_a] :
( X2
!= ( cons_a @ X4 @ ( cons_a @ V4 @ Va ) ) ) ) ) ).
% forward_arcs.cases
thf(fact_681_no__back__arcs_Ocases,axiom,
! [X2: list_a] :
( ( X2 != nil_a )
=> ~ ! [X4: a,Xs2: list_a] :
( X2
!= ( cons_a @ X4 @ Xs2 ) ) ) ).
% no_back_arcs.cases
thf(fact_682_no__back__insert,axiom,
! [X2: a,Xs: list_a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ ( cons_a @ X2 @ Xs ) )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs ) ) ).
% no_back_insert
thf(fact_683_append_Oassoc,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( append_a @ ( append_a @ A @ B ) @ C )
= ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_684_append_Oassoc,axiom,
! [A: list_b,B: list_b,C: list_b] :
( ( append_b @ ( append_b @ A @ B ) @ C )
= ( append_b @ A @ ( append_b @ B @ C ) ) ) ).
% append.assoc
thf(fact_685_append__assoc,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_686_append__assoc,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_b] :
( ( append_b @ ( append_b @ Xs @ Ys ) @ Zs )
= ( append_b @ Xs @ ( append_b @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_687_append__same__eq,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_688_append__same__eq,axiom,
! [Ys: list_b,Xs: list_b,Zs: list_b] :
( ( ( append_b @ Ys @ Xs )
= ( append_b @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_689_same__append__eq,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_690_same__append__eq,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_b] :
( ( ( append_b @ Xs @ Ys )
= ( append_b @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_691_forward__single,axiom,
! [X2: a] : ( iKKBZ_4778857019735642799rd_a_b @ t @ ( cons_a @ X2 @ nil_a ) ) ).
% forward_single
thf(fact_692_seq__conform__single,axiom,
! [X2: a] : ( iKKBZ_4622586873178280511rm_a_b @ t @ ( cons_a @ X2 @ nil_a ) ) ).
% seq_conform_single
thf(fact_693_no__back__single,axiom,
! [X2: a] : ( iKKBZ_3684931046464919648ck_a_b @ t @ ( cons_a @ X2 @ nil_a ) ) ).
% no_back_single
thf(fact_694_hd__in__verts__if__forward,axiom,
! [X2: a,Y2: a,Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs ) ) )
=> ( member_a @ ( hd_a @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs ) ) ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% hd_in_verts_if_forward
thf(fact_695_hd__reach__all__forward_H_H,axiom,
! [X2: a,Y2: a,Xs: list_a,Z: a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs ) ) )
=> ( ( member_a @ Z @ ( set_a2 @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs ) ) ) )
=> ( reachable_a_b @ t @ ( hd_a @ ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs ) ) ) @ Z ) ) ) ).
% hd_reach_all_forward''
thf(fact_696_no__arc__fst__if__no__back,axiom,
! [X2: a,Xs: list_a,Y2: a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ ( cons_a @ X2 @ Xs ) )
=> ( ( member_a @ Y2 @ ( set_a2 @ Xs ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y2 @ X2 ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% no_arc_fst_if_no_back
thf(fact_697_to__list__tree__dom__iff,axiom,
! [X2: a,Y2: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y2 ) @ ( arcs_ends_a_b @ t ) )
= ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X2 @ nil_a ) @ ( cons_a @ Y2 @ nil_a ) ) @ ( arcs_ends_list_a_b @ ( direct3773525127397338803ee_a_b @ t ) ) ) ) ).
% to_list_tree_dom_iff
thf(fact_698_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_699_append__is__Nil__conv,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( append_b @ Xs @ Ys )
= nil_b )
= ( ( Xs = nil_b )
& ( Ys = nil_b ) ) ) ).
% append_is_Nil_conv
thf(fact_700_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_701_Nil__is__append__conv,axiom,
! [Xs: list_b,Ys: list_b] :
( ( nil_b
= ( append_b @ Xs @ Ys ) )
= ( ( Xs = nil_b )
& ( Ys = nil_b ) ) ) ).
% Nil_is_append_conv
thf(fact_702_self__append__conv2,axiom,
! [Y2: list_a,Xs: list_a] :
( ( Y2
= ( append_a @ Xs @ Y2 ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_703_self__append__conv2,axiom,
! [Y2: list_b,Xs: list_b] :
( ( Y2
= ( append_b @ Xs @ Y2 ) )
= ( Xs = nil_b ) ) ).
% self_append_conv2
thf(fact_704_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_705_append__self__conv2,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( append_b @ Xs @ Ys )
= Ys )
= ( Xs = nil_b ) ) ).
% append_self_conv2
thf(fact_706_self__append__conv,axiom,
! [Y2: list_a,Ys: list_a] :
( ( Y2
= ( append_a @ Y2 @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_707_self__append__conv,axiom,
! [Y2: list_b,Ys: list_b] :
( ( Y2
= ( append_b @ Y2 @ Ys ) )
= ( Ys = nil_b ) ) ).
% self_append_conv
thf(fact_708_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_709_append__self__conv,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( append_b @ Xs @ Ys )
= Xs )
= ( Ys = nil_b ) ) ).
% append_self_conv
thf(fact_710_append1__eq__conv,axiom,
! [Xs: list_a,X2: a,Ys: list_a,Y2: a] :
( ( ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y2 @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_711_append1__eq__conv,axiom,
! [Xs: list_b,X2: b,Ys: list_b,Y2: b] :
( ( ( append_b @ Xs @ ( cons_b @ X2 @ nil_b ) )
= ( append_b @ Ys @ ( cons_b @ Y2 @ nil_b ) ) )
= ( ( Xs = Ys )
& ( X2 = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_712_list_Osel_I1_J,axiom,
! [X21: a,X22: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_713_list_Osel_I1_J,axiom,
! [X21: b,X22: list_b] :
( ( hd_b @ ( cons_b @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_714_list_Oset__intros_I2_J,axiom,
! [Y2: product_prod_a_a,X22: list_P1396940483166286381od_a_a,X21: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ X22 ) )
=> ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_715_list_Oset__intros_I2_J,axiom,
! [Y2: set_a,X22: list_set_a,X21: set_a] :
( ( member_set_a @ Y2 @ ( set_set_a2 @ X22 ) )
=> ( member_set_a @ Y2 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_716_list_Oset__intros_I2_J,axiom,
! [Y2: list_a,X22: list_list_a,X21: list_a] :
( ( member_list_a @ Y2 @ ( set_list_a2 @ X22 ) )
=> ( member_list_a @ Y2 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_717_list_Oset__intros_I2_J,axiom,
! [Y2: a,X22: list_a,X21: a] :
( ( member_a @ Y2 @ ( set_a2 @ X22 ) )
=> ( member_a @ Y2 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_718_list_Oset__intros_I2_J,axiom,
! [Y2: b,X22: list_b,X21: b] :
( ( member_b @ Y2 @ ( set_b2 @ X22 ) )
=> ( member_b @ Y2 @ ( set_b2 @ ( cons_b @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_719_list_Oset__intros_I1_J,axiom,
! [X21: product_prod_a_a,X22: list_P1396940483166286381od_a_a] : ( member1426531477525435216od_a_a @ X21 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_720_list_Oset__intros_I1_J,axiom,
! [X21: set_a,X22: list_set_a] : ( member_set_a @ X21 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_721_list_Oset__intros_I1_J,axiom,
! [X21: list_a,X22: list_list_a] : ( member_list_a @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_722_list_Oset__intros_I1_J,axiom,
! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_723_list_Oset__intros_I1_J,axiom,
! [X21: b,X22: list_b] : ( member_b @ X21 @ ( set_b2 @ ( cons_b @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_724_list_Oset__cases,axiom,
! [E: product_prod_a_a,A: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ E @ ( set_Product_prod_a_a2 @ A ) )
=> ( ! [Z22: list_P1396940483166286381od_a_a] :
( A
!= ( cons_P7316939126706565853od_a_a @ E @ Z22 ) )
=> ~ ! [Z1: product_prod_a_a,Z22: list_P1396940483166286381od_a_a] :
( ( A
= ( cons_P7316939126706565853od_a_a @ Z1 @ Z22 ) )
=> ~ ( member1426531477525435216od_a_a @ E @ ( set_Product_prod_a_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_725_list_Oset__cases,axiom,
! [E: set_a,A: list_set_a] :
( ( member_set_a @ E @ ( set_set_a2 @ A ) )
=> ( ! [Z22: list_set_a] :
( A
!= ( cons_set_a @ E @ Z22 ) )
=> ~ ! [Z1: set_a,Z22: list_set_a] :
( ( A
= ( cons_set_a @ Z1 @ Z22 ) )
=> ~ ( member_set_a @ E @ ( set_set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_726_list_Oset__cases,axiom,
! [E: list_a,A: list_list_a] :
( ( member_list_a @ E @ ( set_list_a2 @ A ) )
=> ( ! [Z22: list_list_a] :
( A
!= ( cons_list_a @ E @ Z22 ) )
=> ~ ! [Z1: list_a,Z22: list_list_a] :
( ( A
= ( cons_list_a @ Z1 @ Z22 ) )
=> ~ ( member_list_a @ E @ ( set_list_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_727_list_Oset__cases,axiom,
! [E: a,A: list_a] :
( ( member_a @ E @ ( set_a2 @ A ) )
=> ( ! [Z22: list_a] :
( A
!= ( cons_a @ E @ Z22 ) )
=> ~ ! [Z1: a,Z22: list_a] :
( ( A
= ( cons_a @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_728_list_Oset__cases,axiom,
! [E: b,A: list_b] :
( ( member_b @ E @ ( set_b2 @ A ) )
=> ( ! [Z22: list_b] :
( A
!= ( cons_b @ E @ Z22 ) )
=> ~ ! [Z1: b,Z22: list_b] :
( ( A
= ( cons_b @ Z1 @ Z22 ) )
=> ~ ( member_b @ E @ ( set_b2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_729_set__ConsD,axiom,
! [Y2: product_prod_a_a,X2: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_730_set__ConsD,axiom,
! [Y2: set_a,X2: set_a,Xs: list_set_a] :
( ( member_set_a @ Y2 @ ( set_set_a2 @ ( cons_set_a @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_set_a @ Y2 @ ( set_set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_731_set__ConsD,axiom,
! [Y2: list_a,X2: list_a,Xs: list_list_a] :
( ( member_list_a @ Y2 @ ( set_list_a2 @ ( cons_list_a @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_list_a @ Y2 @ ( set_list_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_732_set__ConsD,axiom,
! [Y2: a,X2: a,Xs: list_a] :
( ( member_a @ Y2 @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_a @ Y2 @ ( set_a2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_733_set__ConsD,axiom,
! [Y2: b,X2: b,Xs: list_b] :
( ( member_b @ Y2 @ ( set_b2 @ ( cons_b @ X2 @ Xs ) ) )
=> ( ( Y2 = X2 )
| ( member_b @ Y2 @ ( set_b2 @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_734_Cons__eq__appendI,axiom,
! [X2: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X2 @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_735_Cons__eq__appendI,axiom,
! [X2: b,Xs1: list_b,Ys: list_b,Xs: list_b,Zs: list_b] :
( ( ( cons_b @ X2 @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_b @ Xs1 @ Zs ) )
=> ( ( cons_b @ X2 @ Xs )
= ( append_b @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_736_append__Cons,axiom,
! [X2: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X2 @ Xs ) @ Ys )
= ( cons_a @ X2 @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_737_append__Cons,axiom,
! [X2: b,Xs: list_b,Ys: list_b] :
( ( append_b @ ( cons_b @ X2 @ Xs ) @ Ys )
= ( cons_b @ X2 @ ( append_b @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_738_split__list,axiom,
! [X2: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys2: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( Xs
= ( append5335208819046833346od_a_a @ Ys2 @ ( cons_P7316939126706565853od_a_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_739_split__list,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs2: list_set_a] :
( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_740_split__list,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_741_split__list,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_742_split__list,axiom,
! [X2: b,Xs: list_b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
=> ? [Ys2: list_b,Zs2: list_b] :
( Xs
= ( append_b @ Ys2 @ ( cons_b @ X2 @ Zs2 ) ) ) ) ).
% split_list
thf(fact_743_split__list__last,axiom,
! [X2: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys2: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys2 @ ( cons_P7316939126706565853od_a_a @ X2 @ Zs2 ) ) )
& ~ ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_744_split__list__last,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X2 @ Zs2 ) ) )
& ~ ( member_set_a @ X2 @ ( set_set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_745_split__list__last,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs2 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_746_split__list__last,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_747_split__list__last,axiom,
! [X2: b,Xs: list_b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
=> ? [Ys2: list_b,Zs2: list_b] :
( ( Xs
= ( append_b @ Ys2 @ ( cons_b @ X2 @ Zs2 ) ) )
& ~ ( member_b @ X2 @ ( set_b2 @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_748_split__list__prop,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ? [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_list_a,X4: list_a] :
( ? [Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X4 @ Zs2 ) ) )
& ( P @ X4 ) ) ) ).
% split_list_prop
thf(fact_749_split__list__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P @ X4 ) ) ) ).
% split_list_prop
thf(fact_750_split__list__prop,axiom,
! [Xs: list_b,P: b > $o] :
( ? [X: b] :
( ( member_b @ X @ ( set_b2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_b,X4: b] :
( ? [Zs2: list_b] :
( Xs
= ( append_b @ Ys2 @ ( cons_b @ X4 @ Zs2 ) ) )
& ( P @ X4 ) ) ) ).
% split_list_prop
thf(fact_751_split__list__first,axiom,
! [X2: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ? [Ys2: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys2 @ ( cons_P7316939126706565853od_a_a @ X2 @ Zs2 ) ) )
& ~ ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_752_split__list__first,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ? [Ys2: list_set_a,Zs2: list_set_a] :
( ( Xs
= ( append_set_a @ Ys2 @ ( cons_set_a @ X2 @ Zs2 ) ) )
& ~ ( member_set_a @ X2 @ ( set_set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_753_split__list__first,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ? [Ys2: list_list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X2 @ Zs2 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_754_split__list__first,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ? [Ys2: list_a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X2 @ Zs2 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_755_split__list__first,axiom,
! [X2: b,Xs: list_b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
=> ? [Ys2: list_b,Zs2: list_b] :
( ( Xs
= ( append_b @ Ys2 @ ( cons_b @ X2 @ Zs2 ) ) )
& ~ ( member_b @ X2 @ ( set_b2 @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_756_split__list__propE,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ? [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_list_a,X4: list_a] :
( ? [Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X4 @ Zs2 ) ) )
=> ~ ( P @ X4 ) ) ) ).
% split_list_propE
thf(fact_757_split__list__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
=> ~ ( P @ X4 ) ) ) ).
% split_list_propE
thf(fact_758_split__list__propE,axiom,
! [Xs: list_b,P: b > $o] :
( ? [X: b] :
( ( member_b @ X @ ( set_b2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_b,X4: b] :
( ? [Zs2: list_b] :
( Xs
= ( append_b @ Ys2 @ ( cons_b @ X4 @ Zs2 ) ) )
=> ~ ( P @ X4 ) ) ) ).
% split_list_propE
thf(fact_759_append__Cons__eq__iff,axiom,
! [X2: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a,Xs3: list_P1396940483166286381od_a_a,Ys3: list_P1396940483166286381od_a_a] :
( ~ ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ~ ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Ys ) )
=> ( ( ( append5335208819046833346od_a_a @ Xs @ ( cons_P7316939126706565853od_a_a @ X2 @ Ys ) )
= ( append5335208819046833346od_a_a @ Xs3 @ ( cons_P7316939126706565853od_a_a @ X2 @ Ys3 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys3 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_760_append__Cons__eq__iff,axiom,
! [X2: set_a,Xs: list_set_a,Ys: list_set_a,Xs3: list_set_a,Ys3: list_set_a] :
( ~ ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ( ~ ( member_set_a @ X2 @ ( set_set_a2 @ Ys ) )
=> ( ( ( append_set_a @ Xs @ ( cons_set_a @ X2 @ Ys ) )
= ( append_set_a @ Xs3 @ ( cons_set_a @ X2 @ Ys3 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys3 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_761_append__Cons__eq__iff,axiom,
! [X2: list_a,Xs: list_list_a,Ys: list_list_a,Xs3: list_list_a,Ys3: list_list_a] :
( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Ys ) )
=> ( ( ( append_list_a @ Xs @ ( cons_list_a @ X2 @ Ys ) )
= ( append_list_a @ Xs3 @ ( cons_list_a @ X2 @ Ys3 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys3 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_762_append__Cons__eq__iff,axiom,
! [X2: a,Xs: list_a,Ys: list_a,Xs3: list_a,Ys3: list_a] :
( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ~ ( member_a @ X2 @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs @ ( cons_a @ X2 @ Ys ) )
= ( append_a @ Xs3 @ ( cons_a @ X2 @ Ys3 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys3 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_763_append__Cons__eq__iff,axiom,
! [X2: b,Xs: list_b,Ys: list_b,Xs3: list_b,Ys3: list_b] :
( ~ ( member_b @ X2 @ ( set_b2 @ Xs ) )
=> ( ~ ( member_b @ X2 @ ( set_b2 @ Ys ) )
=> ( ( ( append_b @ Xs @ ( cons_b @ X2 @ Ys ) )
= ( append_b @ Xs3 @ ( cons_b @ X2 @ Ys3 ) ) )
= ( ( Xs = Xs3 )
& ( Ys = Ys3 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_764_in__set__conv__decomp,axiom,
! [X2: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys4: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( Xs
= ( append5335208819046833346od_a_a @ Ys4 @ ( cons_P7316939126706565853od_a_a @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_765_in__set__conv__decomp,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys4: list_set_a,Zs3: list_set_a] :
( Xs
= ( append_set_a @ Ys4 @ ( cons_set_a @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_766_in__set__conv__decomp,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
= ( ? [Ys4: list_list_a,Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys4 @ ( cons_list_a @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_767_in__set__conv__decomp,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs3: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_768_in__set__conv__decomp,axiom,
! [X2: b,Xs: list_b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
= ( ? [Ys4: list_b,Zs3: list_b] :
( Xs
= ( append_b @ Ys4 @ ( cons_b @ X2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_769_split__list__last__prop,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ? [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_list_a,X4: list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa2: list_a] :
( ( member_list_a @ Xa2 @ ( set_list_a2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_770_split__list__last__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_a,X4: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_771_split__list__last__prop,axiom,
! [Xs: list_b,P: b > $o] :
( ? [X: b] :
( ( member_b @ X @ ( set_b2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_b,X4: b,Zs2: list_b] :
( ( Xs
= ( append_b @ Ys2 @ ( cons_b @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa2: b] :
( ( member_b @ Xa2 @ ( set_b2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_772_split__list__first__prop,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ? [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_list_a,X4: list_a] :
( ? [Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa2: list_a] :
( ( member_list_a @ Xa2 @ ( set_list_a2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_773_split__list__first__prop,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_774_split__list__first__prop,axiom,
! [Xs: list_b,P: b > $o] :
( ? [X: b] :
( ( member_b @ X @ ( set_b2 @ Xs ) )
& ( P @ X ) )
=> ? [Ys2: list_b,X4: b] :
( ? [Zs2: list_b] :
( Xs
= ( append_b @ Ys2 @ ( cons_b @ X4 @ Zs2 ) ) )
& ( P @ X4 )
& ! [Xa2: b] :
( ( member_b @ Xa2 @ ( set_b2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_775_split__list__last__propE,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ? [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_list_a,X4: list_a,Zs2: list_list_a] :
( ( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: list_a] :
( ( member_list_a @ Xa2 @ ( set_list_a2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_776_split__list__last__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_a,X4: a,Zs2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_777_split__list__last__propE,axiom,
! [Xs: list_b,P: b > $o] :
( ? [X: b] :
( ( member_b @ X @ ( set_b2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_b,X4: b,Zs2: list_b] :
( ( Xs
= ( append_b @ Ys2 @ ( cons_b @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: b] :
( ( member_b @ Xa2 @ ( set_b2 @ Zs2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_778_split__list__first__propE,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ? [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_list_a,X4: list_a] :
( ? [Zs2: list_list_a] :
( Xs
= ( append_list_a @ Ys2 @ ( cons_list_a @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: list_a] :
( ( member_list_a @ Xa2 @ ( set_list_a2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_779_split__list__first__propE,axiom,
! [Xs: list_a,P: a > $o] :
( ? [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_a,X4: a] :
( ? [Zs2: list_a] :
( Xs
= ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_780_split__list__first__propE,axiom,
! [Xs: list_b,P: b > $o] :
( ? [X: b] :
( ( member_b @ X @ ( set_b2 @ Xs ) )
& ( P @ X ) )
=> ~ ! [Ys2: list_b,X4: b] :
( ? [Zs2: list_b] :
( Xs
= ( append_b @ Ys2 @ ( cons_b @ X4 @ Zs2 ) ) )
=> ( ( P @ X4 )
=> ~ ! [Xa2: b] :
( ( member_b @ Xa2 @ ( set_b2 @ Ys2 ) )
=> ~ ( P @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_781_in__set__conv__decomp__last,axiom,
! [X2: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys4: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys4 @ ( cons_P7316939126706565853od_a_a @ X2 @ Zs3 ) ) )
& ~ ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_782_in__set__conv__decomp__last,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys4: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys4 @ ( cons_set_a @ X2 @ Zs3 ) ) )
& ~ ( member_set_a @ X2 @ ( set_set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_783_in__set__conv__decomp__last,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
= ( ? [Ys4: list_list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys4 @ ( cons_list_a @ X2 @ Zs3 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_784_in__set__conv__decomp__last,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_785_in__set__conv__decomp__last,axiom,
! [X2: b,Xs: list_b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
= ( ? [Ys4: list_b,Zs3: list_b] :
( ( Xs
= ( append_b @ Ys4 @ ( cons_b @ X2 @ Zs3 ) ) )
& ~ ( member_b @ X2 @ ( set_b2 @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_786_in__set__conv__decomp__first,axiom,
! [X2: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Xs ) )
= ( ? [Ys4: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
( ( Xs
= ( append5335208819046833346od_a_a @ Ys4 @ ( cons_P7316939126706565853od_a_a @ X2 @ Zs3 ) ) )
& ~ ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_787_in__set__conv__decomp__first,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
= ( ? [Ys4: list_set_a,Zs3: list_set_a] :
( ( Xs
= ( append_set_a @ Ys4 @ ( cons_set_a @ X2 @ Zs3 ) ) )
& ~ ( member_set_a @ X2 @ ( set_set_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_788_in__set__conv__decomp__first,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
= ( ? [Ys4: list_list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys4 @ ( cons_list_a @ X2 @ Zs3 ) ) )
& ~ ( member_list_a @ X2 @ ( set_list_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_789_in__set__conv__decomp__first,axiom,
! [X2: a,Xs: list_a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
= ( ? [Ys4: list_a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) )
& ~ ( member_a @ X2 @ ( set_a2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_790_in__set__conv__decomp__first,axiom,
! [X2: b,Xs: list_b] :
( ( member_b @ X2 @ ( set_b2 @ Xs ) )
= ( ? [Ys4: list_b,Zs3: list_b] :
( ( Xs
= ( append_b @ Ys4 @ ( cons_b @ X2 @ Zs3 ) ) )
& ~ ( member_b @ X2 @ ( set_b2 @ Ys4 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_791_split__list__last__prop__iff,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
& ( P @ X5 ) ) )
= ( ? [Ys4: list_list_a,X5: list_a,Zs3: list_list_a] :
( ( Xs
= ( append_list_a @ Ys4 @ ( cons_list_a @ X5 @ Zs3 ) ) )
& ( P @ X5 )
& ! [Y4: list_a] :
( ( member_list_a @ Y4 @ ( set_list_a2 @ Zs3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_792_split__list__last__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) ) )
= ( ? [Ys4: list_a,X5: a,Zs3: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ X5 @ Zs3 ) ) )
& ( P @ X5 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Zs3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_793_split__list__last__prop__iff,axiom,
! [Xs: list_b,P: b > $o] :
( ( ? [X5: b] :
( ( member_b @ X5 @ ( set_b2 @ Xs ) )
& ( P @ X5 ) ) )
= ( ? [Ys4: list_b,X5: b,Zs3: list_b] :
( ( Xs
= ( append_b @ Ys4 @ ( cons_b @ X5 @ Zs3 ) ) )
& ( P @ X5 )
& ! [Y4: b] :
( ( member_b @ Y4 @ ( set_b2 @ Zs3 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_794_split__list__first__prop__iff,axiom,
! [Xs: list_list_a,P: list_a > $o] :
( ( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
& ( P @ X5 ) ) )
= ( ? [Ys4: list_list_a,X5: list_a] :
( ? [Zs3: list_list_a] :
( Xs
= ( append_list_a @ Ys4 @ ( cons_list_a @ X5 @ Zs3 ) ) )
& ( P @ X5 )
& ! [Y4: list_a] :
( ( member_list_a @ Y4 @ ( set_list_a2 @ Ys4 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_795_split__list__first__prop__iff,axiom,
! [Xs: list_a,P: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
& ( P @ X5 ) ) )
= ( ? [Ys4: list_a,X5: a] :
( ? [Zs3: list_a] :
( Xs
= ( append_a @ Ys4 @ ( cons_a @ X5 @ Zs3 ) ) )
& ( P @ X5 )
& ! [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Ys4 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_796_split__list__first__prop__iff,axiom,
! [Xs: list_b,P: b > $o] :
( ( ? [X5: b] :
( ( member_b @ X5 @ ( set_b2 @ Xs ) )
& ( P @ X5 ) ) )
= ( ? [Ys4: list_b,X5: b] :
( ? [Zs3: list_b] :
( Xs
= ( append_b @ Ys4 @ ( cons_b @ X5 @ Zs3 ) ) )
& ( P @ X5 )
& ! [Y4: b] :
( ( member_b @ Y4 @ ( set_b2 @ Ys4 ) )
=> ~ ( P @ Y4 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_797_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X4: a] : ( P @ ( cons_a @ X4 @ nil_a ) )
=> ( ! [X4: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X4 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_798_rev__nonempty__induct,axiom,
! [Xs: list_b,P: list_b > $o] :
( ( Xs != nil_b )
=> ( ! [X4: b] : ( P @ ( cons_b @ X4 @ nil_b ) )
=> ( ! [X4: b,Xs2: list_b] :
( ( Xs2 != nil_b )
=> ( ( P @ Xs2 )
=> ( P @ ( append_b @ Xs2 @ ( cons_b @ X4 @ nil_b ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_799_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X2: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X2 @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X2 @ Xs ) ) )
| ? [Ys5: list_a] :
( ( Ys
= ( cons_a @ X2 @ Ys5 ) )
& ( ( append_a @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_800_append__eq__Cons__conv,axiom,
! [Ys: list_b,Zs: list_b,X2: b,Xs: list_b] :
( ( ( append_b @ Ys @ Zs )
= ( cons_b @ X2 @ Xs ) )
= ( ( ( Ys = nil_b )
& ( Zs
= ( cons_b @ X2 @ Xs ) ) )
| ? [Ys5: list_b] :
( ( Ys
= ( cons_b @ X2 @ Ys5 ) )
& ( ( append_b @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_801_Cons__eq__append__conv,axiom,
! [X2: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X2 @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X2 @ Xs )
= Zs ) )
| ? [Ys5: list_a] :
( ( ( cons_a @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_802_Cons__eq__append__conv,axiom,
! [X2: b,Xs: list_b,Ys: list_b,Zs: list_b] :
( ( ( cons_b @ X2 @ Xs )
= ( append_b @ Ys @ Zs ) )
= ( ( ( Ys = nil_b )
& ( ( cons_b @ X2 @ Xs )
= Zs ) )
| ? [Ys5: list_b] :
( ( ( cons_b @ X2 @ Ys5 )
= Ys )
& ( Xs
= ( append_b @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_803_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y3: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y3 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_804_rev__exhaust,axiom,
! [Xs: list_b] :
( ( Xs != nil_b )
=> ~ ! [Ys2: list_b,Y3: b] :
( Xs
!= ( append_b @ Ys2 @ ( cons_b @ Y3 @ nil_b ) ) ) ) ).
% rev_exhaust
thf(fact_805_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X4: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X4 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_806_rev__induct,axiom,
! [P: list_b > $o,Xs: list_b] :
( ( P @ nil_b )
=> ( ! [X4: b,Xs2: list_b] :
( ( P @ Xs2 )
=> ( P @ ( append_b @ Xs2 @ ( cons_b @ X4 @ nil_b ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_807_set__subset__Cons,axiom,
! [Xs: list_list_a,X2: list_a] : ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ ( cons_list_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_808_set__subset__Cons,axiom,
! [Xs: list_a,X2: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_809_set__subset__Cons,axiom,
! [Xs: list_b,X2: b] : ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ ( cons_b @ X2 @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_810_distinct_Osimps_I2_J,axiom,
! [X2: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
( ( distin132333870042060960od_a_a @ ( cons_P7316939126706565853od_a_a @ X2 @ Xs ) )
= ( ~ ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Xs ) )
& ( distin132333870042060960od_a_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_811_distinct_Osimps_I2_J,axiom,
! [X2: set_a,Xs: list_set_a] :
( ( distinct_set_a @ ( cons_set_a @ X2 @ Xs ) )
= ( ~ ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
& ( distinct_set_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_812_distinct_Osimps_I2_J,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( distinct_list_a @ ( cons_list_a @ X2 @ Xs ) )
= ( ~ ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
& ( distinct_list_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_813_distinct_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( distinct_a @ ( cons_a @ X2 @ Xs ) )
= ( ~ ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ( distinct_a @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_814_distinct_Osimps_I2_J,axiom,
! [X2: b,Xs: list_b] :
( ( distinct_b @ ( cons_b @ X2 @ Xs ) )
= ( ~ ( member_b @ X2 @ ( set_b2 @ Xs ) )
& ( distinct_b @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_815_not__distinct__conv__prefix,axiom,
! [As: list_P1396940483166286381od_a_a] :
( ( ~ ( distin132333870042060960od_a_a @ As ) )
= ( ? [Xs4: list_P1396940483166286381od_a_a,Y4: product_prod_a_a,Ys4: list_P1396940483166286381od_a_a] :
( ( member1426531477525435216od_a_a @ Y4 @ ( set_Product_prod_a_a2 @ Xs4 ) )
& ( distin132333870042060960od_a_a @ Xs4 )
& ( As
= ( append5335208819046833346od_a_a @ Xs4 @ ( cons_P7316939126706565853od_a_a @ Y4 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_816_not__distinct__conv__prefix,axiom,
! [As: list_set_a] :
( ( ~ ( distinct_set_a @ As ) )
= ( ? [Xs4: list_set_a,Y4: set_a,Ys4: list_set_a] :
( ( member_set_a @ Y4 @ ( set_set_a2 @ Xs4 ) )
& ( distinct_set_a @ Xs4 )
& ( As
= ( append_set_a @ Xs4 @ ( cons_set_a @ Y4 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_817_not__distinct__conv__prefix,axiom,
! [As: list_list_a] :
( ( ~ ( distinct_list_a @ As ) )
= ( ? [Xs4: list_list_a,Y4: list_a,Ys4: list_list_a] :
( ( member_list_a @ Y4 @ ( set_list_a2 @ Xs4 ) )
& ( distinct_list_a @ Xs4 )
& ( As
= ( append_list_a @ Xs4 @ ( cons_list_a @ Y4 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_818_not__distinct__conv__prefix,axiom,
! [As: list_a] :
( ( ~ ( distinct_a @ As ) )
= ( ? [Xs4: list_a,Y4: a,Ys4: list_a] :
( ( member_a @ Y4 @ ( set_a2 @ Xs4 ) )
& ( distinct_a @ Xs4 )
& ( As
= ( append_a @ Xs4 @ ( cons_a @ Y4 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_819_not__distinct__conv__prefix,axiom,
! [As: list_b] :
( ( ~ ( distinct_b @ As ) )
= ( ? [Xs4: list_b,Y4: b,Ys4: list_b] :
( ( member_b @ Y4 @ ( set_b2 @ Xs4 ) )
& ( distinct_b @ Xs4 )
& ( As
= ( append_b @ Xs4 @ ( cons_b @ Y4 @ Ys4 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_820_not__distinct__decomp,axiom,
! [Ws: list_a] :
( ~ ( distinct_a @ Ws )
=> ? [Xs2: list_a,Ys2: list_a,Zs2: list_a,Y3: a] :
( Ws
= ( append_a @ Xs2 @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ ( append_a @ Ys2 @ ( append_a @ ( cons_a @ Y3 @ nil_a ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_821_not__distinct__decomp,axiom,
! [Ws: list_b] :
( ~ ( distinct_b @ Ws )
=> ? [Xs2: list_b,Ys2: list_b,Zs2: list_b,Y3: b] :
( Ws
= ( append_b @ Xs2 @ ( append_b @ ( cons_b @ Y3 @ nil_b ) @ ( append_b @ Ys2 @ ( append_b @ ( cons_b @ Y3 @ nil_b ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_822_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_823_append__eq__appendI,axiom,
! [Xs: list_b,Xs1: list_b,Zs: list_b,Ys: list_b,Us: list_b] :
( ( ( append_b @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_b @ Xs1 @ Us ) )
=> ( ( append_b @ Xs @ Ys )
= ( append_b @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_824_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_825_append__eq__append__conv2,axiom,
! [Xs: list_b,Ys: list_b,Zs: list_b,Ts: list_b] :
( ( ( append_b @ Xs @ Ys )
= ( append_b @ Zs @ Ts ) )
= ( ? [Us2: list_b] :
( ( ( Xs
= ( append_b @ Zs @ Us2 ) )
& ( ( append_b @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_b @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_b @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_826_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_827_eq__Nil__appendI,axiom,
! [Xs: list_b,Ys: list_b] :
( ( Xs = Ys )
=> ( Xs
= ( append_b @ nil_b @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_828_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_829_append_Oleft__neutral,axiom,
! [A: list_b] :
( ( append_b @ nil_b @ A )
= A ) ).
% append.left_neutral
thf(fact_830_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_831_append__Nil,axiom,
! [Ys: list_b] :
( ( append_b @ nil_b @ Ys )
= Ys ) ).
% append_Nil
thf(fact_832_subset__code_I1_J,axiom,
! [Xs: list_P1396940483166286381od_a_a,B3: set_Product_prod_a_a] :
( ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Xs ) @ B3 )
= ( ! [X5: product_prod_a_a] :
( ( member1426531477525435216od_a_a @ X5 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( member1426531477525435216od_a_a @ X5 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_833_subset__code_I1_J,axiom,
! [Xs: list_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B3 )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( set_set_a2 @ Xs ) )
=> ( member_set_a @ X5 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_834_subset__code_I1_J,axiom,
! [Xs: list_list_a,B3: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B3 )
= ( ! [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X5 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_835_subset__code_I1_J,axiom,
! [Xs: list_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B3 )
= ( ! [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs ) )
=> ( member_a @ X5 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_836_subset__code_I1_J,axiom,
! [Xs: list_b,B3: set_b] :
( ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ B3 )
= ( ! [X5: b] :
( ( member_b @ X5 @ ( set_b2 @ Xs ) )
=> ( member_b @ X5 @ B3 ) ) ) ) ).
% subset_code(1)
thf(fact_837_empty__set,axiom,
( bot_bot_set_list_a
= ( set_list_a2 @ nil_list_a ) ) ).
% empty_set
thf(fact_838_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_839_empty__set,axiom,
( bot_bot_set_b
= ( set_b2 @ nil_b ) ) ).
% empty_set
thf(fact_840_list_Oset__sel_I1_J,axiom,
! [A: list_P1396940483166286381od_a_a] :
( ( A != nil_Product_prod_a_a )
=> ( member1426531477525435216od_a_a @ ( hd_Product_prod_a_a @ A ) @ ( set_Product_prod_a_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_841_list_Oset__sel_I1_J,axiom,
! [A: list_set_a] :
( ( A != nil_set_a )
=> ( member_set_a @ ( hd_set_a @ A ) @ ( set_set_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_842_list_Oset__sel_I1_J,axiom,
! [A: list_a] :
( ( A != nil_a )
=> ( member_a @ ( hd_a @ A ) @ ( set_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_843_list_Oset__sel_I1_J,axiom,
! [A: list_list_a] :
( ( A != nil_list_a )
=> ( member_list_a @ ( hd_list_a @ A ) @ ( set_list_a2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_844_list_Oset__sel_I1_J,axiom,
! [A: list_b] :
( ( A != nil_b )
=> ( member_b @ ( hd_b @ A ) @ ( set_b2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_845_hd__in__set,axiom,
! [Xs: list_P1396940483166286381od_a_a] :
( ( Xs != nil_Product_prod_a_a )
=> ( member1426531477525435216od_a_a @ ( hd_Product_prod_a_a @ Xs ) @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_846_hd__in__set,axiom,
! [Xs: list_set_a] :
( ( Xs != nil_set_a )
=> ( member_set_a @ ( hd_set_a @ Xs ) @ ( set_set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_847_hd__in__set,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( member_a @ ( hd_a @ Xs ) @ ( set_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_848_hd__in__set,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( member_list_a @ ( hd_list_a @ Xs ) @ ( set_list_a2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_849_hd__in__set,axiom,
! [Xs: list_b] :
( ( Xs != nil_b )
=> ( member_b @ ( hd_b @ Xs ) @ ( set_b2 @ Xs ) ) ) ).
% hd_in_set
thf(fact_850_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs5: list_a,Ys6: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs5 ) )
& ( Ys
= ( append_a @ Ps @ Ys6 ) )
& ( ( Xs5 = nil_a )
| ( Ys6 = nil_a )
| ( ( hd_a @ Xs5 )
!= ( hd_a @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_851_longest__common__prefix,axiom,
! [Xs: list_b,Ys: list_b] :
? [Ps: list_b,Xs5: list_b,Ys6: list_b] :
( ( Xs
= ( append_b @ Ps @ Xs5 ) )
& ( Ys
= ( append_b @ Ps @ Ys6 ) )
& ( ( Xs5 = nil_b )
| ( Ys6 = nil_b )
| ( ( hd_b @ Xs5 )
!= ( hd_b @ Ys6 ) ) ) ) ).
% longest_common_prefix
thf(fact_852_hd__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_853_hd__append,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( Xs = nil_b )
=> ( ( hd_b @ ( append_b @ Xs @ Ys ) )
= ( hd_b @ Ys ) ) )
& ( ( Xs != nil_b )
=> ( ( hd_b @ ( append_b @ Xs @ Ys ) )
= ( hd_b @ Xs ) ) ) ) ).
% hd_append
thf(fact_854_loopfree_OvpathI__arc,axiom,
! [A: a,B: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( arcs_ends_a_b @ t ) )
=> ( vertex_vpath_a_b @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) @ t ) ) ).
% loopfree.vpathI_arc
thf(fact_855_no__back__arcs_Oelims_I1_J,axiom,
! [X2: list_a,Y2: $o] :
( ( ( iKKBZ_7773321254043928001cs_a_b @ t @ X2 )
= Y2 )
=> ( ( ( X2 = nil_a )
=> ~ Y2 )
=> ~ ! [X4: a,Xs2: list_a] :
( ( X2
= ( cons_a @ X4 @ Xs2 ) )
=> ( Y2
= ( ~ ( ~ ? [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs2 ) ) ) ) ) ) ) ).
% no_back_arcs.elims(1)
thf(fact_856_no__back__arcs_Oelims_I2_J,axiom,
! [X2: list_a] :
( ( iKKBZ_7773321254043928001cs_a_b @ t @ X2 )
=> ( ( X2 != nil_a )
=> ~ ! [X4: a,Xs2: list_a] :
( ( X2
= ( cons_a @ X4 @ Xs2 ) )
=> ~ ( ~ ? [Y: a] :
( ( member_a @ Y @ ( set_a2 @ Xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X4 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs2 ) ) ) ) ) ).
% no_back_arcs.elims(2)
thf(fact_857_no__back__arcs_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( iKKBZ_7773321254043928001cs_a_b @ t @ ( cons_a @ X2 @ Xs ) )
= ( ~ ? [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ Xs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X2 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs ) ) ) ).
% no_back_arcs.simps(2)
thf(fact_858_no__back__arcs_Oelims_I3_J,axiom,
! [X2: list_a] :
( ~ ( iKKBZ_7773321254043928001cs_a_b @ t @ X2 )
=> ~ ! [X4: a,Xs2: list_a] :
( ( X2
= ( cons_a @ X4 @ Xs2 ) )
=> ( ~ ? [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X4 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs2 ) ) ) ) ).
% no_back_arcs.elims(3)
thf(fact_859_forward__arcs_Oelims_I1_J,axiom,
! [X2: list_a,Y2: $o] :
( ( ( iKKBZ_4180558001818622352cs_a_b @ t @ X2 )
= Y2 )
=> ( ( ( X2 = nil_a )
=> ~ Y2 )
=> ( ( ? [X4: a] :
( X2
= ( cons_a @ X4 @ nil_a ) )
=> ~ Y2 )
=> ~ ! [X4: a,V4: a,Va: list_a] :
( ( X2
= ( cons_a @ X4 @ ( cons_a @ V4 @ Va ) ) )
=> ( Y2
= ( ~ ( ? [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ ( cons_a @ V4 @ Va ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X4 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ V4 @ Va ) ) ) ) ) ) ) ) ) ).
% forward_arcs.elims(1)
thf(fact_860_forward__arcs__split,axiom,
! [Ys: list_a,Xs: list_a] :
( ( iKKBZ_4180558001818622352cs_a_b @ t @ ( append_a @ Ys @ Xs ) )
=> ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs ) ) ).
% forward_arcs_split
thf(fact_861_forward__arcs_Osimps_I1_J,axiom,
iKKBZ_4180558001818622352cs_a_b @ t @ nil_a ).
% forward_arcs.simps(1)
thf(fact_862_no__back__arcs_Osimps_I1_J,axiom,
iKKBZ_7773321254043928001cs_a_b @ t @ nil_a ).
% no_back_arcs.simps(1)
thf(fact_863_no__back__arcs__alt,axiom,
! [Xs: list_a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs )
= ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs ) ) ).
% no_back_arcs_alt
thf(fact_864_no__back__arcs__alt__aux2,axiom,
! [Xs: list_a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs )
=> ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs ) ) ).
% no_back_arcs_alt_aux2
thf(fact_865_forward__arcs__single,axiom,
! [X2: a] : ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ X2 @ nil_a ) ) ).
% forward_arcs_single
thf(fact_866_forward__arcs_Osimps_I2_J,axiom,
! [X2: a] : ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ X2 @ nil_a ) ) ).
% forward_arcs.simps(2)
thf(fact_867_no__back__arcs__single,axiom,
! [X2: a] : ( iKKBZ_7773321254043928001cs_a_b @ t @ ( cons_a @ X2 @ nil_a ) ) ).
% no_back_arcs_single
thf(fact_868_forward__arcs_Oelims_I3_J,axiom,
! [X2: list_a] :
( ~ ( iKKBZ_4180558001818622352cs_a_b @ t @ X2 )
=> ~ ! [X4: a,V4: a,Va: list_a] :
( ( X2
= ( cons_a @ X4 @ ( cons_a @ V4 @ Va ) ) )
=> ( ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ ( cons_a @ V4 @ Va ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X4 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ V4 @ Va ) ) ) ) ) ).
% forward_arcs.elims(3)
thf(fact_869_forward__arcs_Osimps_I3_J,axiom,
! [X2: a,V: a,Va2: list_a] :
( ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ X2 @ ( cons_a @ V @ Va2 ) ) )
= ( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ ( cons_a @ V @ Va2 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ X2 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ V @ Va2 ) ) ) ) ).
% forward_arcs.simps(3)
thf(fact_870_forward__arcs_Oelims_I2_J,axiom,
! [X2: list_a] :
( ( iKKBZ_4180558001818622352cs_a_b @ t @ X2 )
=> ( ( X2 != nil_a )
=> ( ! [X4: a] :
( X2
!= ( cons_a @ X4 @ nil_a ) )
=> ~ ! [X4: a,V4: a,Va: list_a] :
( ( X2
= ( cons_a @ X4 @ ( cons_a @ V4 @ Va ) ) )
=> ~ ( ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ ( cons_a @ V4 @ Va ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X4 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ V4 @ Va ) ) ) ) ) ) ) ).
% forward_arcs.elims(2)
thf(fact_871_directed__tree_Ono__back__arcs_Ocong,axiom,
iKKBZ_7773321254043928001cs_a_b = iKKBZ_7773321254043928001cs_a_b ).
% directed_tree.no_back_arcs.cong
thf(fact_872_directed__tree_Oforward__arcs_Ocong,axiom,
iKKBZ_4180558001818622352cs_a_b = iKKBZ_4180558001818622352cs_a_b ).
% directed_tree.forward_arcs.cong
thf(fact_873_loopfree__digraph_OvpathI__arc,axiom,
! [G: pre_pr2882871181989701257t_unit,A: list_a,B: list_a] :
( ( loopfr7852502256416881111st_a_b @ G )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A @ B ) @ ( arcs_ends_list_a_b @ G ) )
=> ( vertex6060786982766068989st_a_b @ ( cons_list_a @ A @ ( cons_list_a @ B @ nil_list_a ) ) @ G ) ) ) ).
% loopfree_digraph.vpathI_arc
thf(fact_874_loopfree__digraph_OvpathI__arc,axiom,
! [G: pre_pr7278220950009878019t_unit,A: a,B: a] :
( ( loopfree_digraph_a_b @ G )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( arcs_ends_a_b @ G ) )
=> ( vertex_vpath_a_b @ ( cons_a @ A @ ( cons_a @ B @ nil_a ) ) @ G ) ) ) ).
% loopfree_digraph.vpathI_arc
thf(fact_875_no__back__reach1__if__fwd__dstct__bs,axiom,
! [As: list_a,Bs: list_list_a,V3: list_a,Cs: list_a,Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ ( concat_a @ Bs ) @ ( append_a @ V3 @ Cs ) ) ) )
=> ( ( distinct_a @ ( append_a @ As @ ( append_a @ ( concat_a @ Bs ) @ ( append_a @ V3 @ Cs ) ) ) )
=> ( ( member_list_a @ Xs @ ( set_list_a2 @ Bs ) )
=> ~ ? [X: a] :
( ( member_a @ X @ ( set_a2 @ V3 ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Xs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ).
% no_back_reach1_if_fwd_dstct_bs
thf(fact_876_hd__reach__all__forward__arcs,axiom,
! [Xs: list_a,X2: a] :
( ( member_a @ ( hd_a @ ( rev_a @ Xs ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( reachable_a_b @ t @ ( hd_a @ ( rev_a @ Xs ) ) @ X2 ) ) ) ) ).
% hd_reach_all_forward_arcs
thf(fact_877_arc__to__lst__if__forward,axiom,
! [X2: a,Xs: list_a,Y2: a,Ys: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ ( cons_a @ X2 @ Xs ) ) )
=> ( ( Xs
= ( cons_a @ Y2 @ Ys ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ X2 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% arc_to_lst_if_forward
thf(fact_878_vwalk__wf__digraph__consI,axiom,
! [P2: list_a,A: a] :
( ( vertex_vwalk_a_b @ P2 @ t )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( hd_a @ P2 ) ) @ ( arcs_ends_a_b @ t ) )
=> ( vertex_vwalk_a_b @ ( cons_a @ A @ P2 ) @ t ) ) ) ).
% vwalk_wf_digraph_consI
thf(fact_879_forward__cons,axiom,
! [X2: a,Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ ( cons_a @ X2 @ Xs ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ Xs ) ) ) ).
% forward_cons
thf(fact_880_forward__arcs__alt,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs )
= ( iKKBZ_4180558001818622352cs_a_b @ t @ ( rev_a @ Xs ) ) ) ).
% forward_arcs_alt
thf(fact_881_forward__arcs__alt_H,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ Xs ) )
= ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs ) ) ).
% forward_arcs_alt'
thf(fact_882_forward__arcs__alt__aux2,axiom,
! [Xs: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ Xs ) )
=> ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs ) ) ).
% forward_arcs_alt_aux2
thf(fact_883_nonempty__notin__distinct__prefix,axiom,
! [As: list_b,Bs: list_b,V3: list_b,Cs: list_b,As2: list_list_b] :
( ( distinct_b @ ( append_b @ As @ ( append_b @ Bs @ ( append_b @ V3 @ Cs ) ) ) )
=> ( ( ( concat_b @ As2 )
= As )
=> ( ( V3 != nil_b )
=> ~ ( member_list_b @ V3 @ ( set_list_b2 @ As2 ) ) ) ) ) ).
% nonempty_notin_distinct_prefix
thf(fact_884_nonempty__notin__distinct__prefix,axiom,
! [As: list_a,Bs: list_a,V3: list_a,Cs: list_a,As2: list_list_a] :
( ( distinct_a @ ( append_a @ As @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) )
=> ( ( ( concat_a @ As2 )
= As )
=> ( ( V3 != nil_a )
=> ~ ( member_list_a @ V3 @ ( set_list_a2 @ As2 ) ) ) ) ) ).
% nonempty_notin_distinct_prefix
thf(fact_885_seq__conform__def,axiom,
! [Xs: list_a] :
( ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs )
= ( ( iKKBZ_4180558001818622352cs_a_b @ t @ ( rev_a @ Xs ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs ) ) ) ).
% seq_conform_def
thf(fact_886_set__rev,axiom,
! [Xs: list_a] :
( ( set_a2 @ ( rev_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_rev
thf(fact_887_set__rev,axiom,
! [Xs: list_list_a] :
( ( set_list_a2 @ ( rev_list_a @ Xs ) )
= ( set_list_a2 @ Xs ) ) ).
% set_rev
thf(fact_888_set__rev,axiom,
! [Xs: list_b] :
( ( set_b2 @ ( rev_b @ Xs ) )
= ( set_b2 @ Xs ) ) ).
% set_rev
thf(fact_889_rev__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( rev_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( rev_a @ Ys ) @ ( rev_a @ Xs ) ) ) ).
% rev_append
thf(fact_890_rev__append,axiom,
! [Xs: list_b,Ys: list_b] :
( ( rev_b @ ( append_b @ Xs @ Ys ) )
= ( append_b @ ( rev_b @ Ys ) @ ( rev_b @ Xs ) ) ) ).
% rev_append
thf(fact_891_concat__append,axiom,
! [Xs: list_list_b,Ys: list_list_b] :
( ( concat_b @ ( append_list_b @ Xs @ Ys ) )
= ( append_b @ ( concat_b @ Xs ) @ ( concat_b @ Ys ) ) ) ).
% concat_append
thf(fact_892_concat__append,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( concat_a @ ( append_list_a @ Xs @ Ys ) )
= ( append_a @ ( concat_a @ Xs ) @ ( concat_a @ Ys ) ) ) ).
% concat_append
thf(fact_893_mid__ranks__ge__if__reach1,axiom,
! [Y6: set_list_a,U2: list_a,As: list_a,Bs: list_a,V3: list_a,Cs: list_a,Bs2: list_list_a,Cs2: list_list_a,As2: list_list_a,Rank: list_a > set_a] :
( ~ ( member_list_a @ nil_a @ Y6 )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( distinct_a @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
=> ( ( ( concat_a @ Bs2 )
= Bs )
=> ( ( ( concat_a @ Cs2 )
= Cs )
=> ( ( ( set_list_a2 @ ( append_list_a @ As2 @ ( cons_list_a @ U2 @ ( append_list_a @ Bs2 @ ( cons_list_a @ V3 @ Cs2 ) ) ) ) )
= Y6 )
=> ( ! [Xs2: list_a] :
( ( member_list_a @ Xs2 @ Y6 )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ~ ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ U2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ( Xs2 != U2 )
=> ( ord_less_eq_set_a @ ( Rank @ ( rev_a @ V3 ) ) @ ( Rank @ ( rev_a @ Xs2 ) ) ) ) ) )
=> ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Bs2 ) )
=> ( ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ U2 ) )
& ? [Xb: a] :
( ( member_a @ Xb @ ( set_a2 @ X ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ Xb ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ord_less_eq_set_a @ ( Rank @ ( rev_a @ V3 ) ) @ ( Rank @ ( rev_a @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% mid_ranks_ge_if_reach1
thf(fact_894_mid__ranks__ge__if__reach1,axiom,
! [Y6: set_list_a,U2: list_a,As: list_a,Bs: list_a,V3: list_a,Cs: list_a,Bs2: list_list_a,Cs2: list_list_a,As2: list_list_a,Rank: list_a > set_b] :
( ~ ( member_list_a @ nil_a @ Y6 )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( distinct_a @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
=> ( ( ( concat_a @ Bs2 )
= Bs )
=> ( ( ( concat_a @ Cs2 )
= Cs )
=> ( ( ( set_list_a2 @ ( append_list_a @ As2 @ ( cons_list_a @ U2 @ ( append_list_a @ Bs2 @ ( cons_list_a @ V3 @ Cs2 ) ) ) ) )
= Y6 )
=> ( ! [Xs2: list_a] :
( ( member_list_a @ Xs2 @ Y6 )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ~ ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ U2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ( Xs2 != U2 )
=> ( ord_less_eq_set_b @ ( Rank @ ( rev_a @ V3 ) ) @ ( Rank @ ( rev_a @ Xs2 ) ) ) ) ) )
=> ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Bs2 ) )
=> ( ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ U2 ) )
& ? [Xb: a] :
( ( member_a @ Xb @ ( set_a2 @ X ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ Xb ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ord_less_eq_set_b @ ( Rank @ ( rev_a @ V3 ) ) @ ( Rank @ ( rev_a @ X ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% mid_ranks_ge_if_reach1
thf(fact_895_bs__ranks__only__ge__r,axiom,
! [Y6: set_list_a,As: list_a,U2: list_a,Bs: list_a,V3: list_a,Cs: list_a,Bs2: list_list_a,Cs2: list_list_a,Rank: list_a > set_a] :
( ~ ( member_list_a @ nil_a @ Y6 )
=> ( ( distinct_a @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
=> ( ( As = nil_a )
=> ( ( ( concat_a @ Bs2 )
= Bs )
=> ( ( ( concat_a @ Cs2 )
= Cs )
=> ( ( ( set_list_a2 @ ( cons_list_a @ U2 @ ( append_list_a @ Bs2 @ ( cons_list_a @ V3 @ Cs2 ) ) ) )
= Y6 )
=> ( ! [Xs2: list_a] :
( ( member_list_a @ Xs2 @ Y6 )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ~ ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ U2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ( Xs2 != U2 )
=> ( ord_less_eq_set_a @ ( Rank @ ( rev_a @ V3 ) ) @ ( Rank @ ( rev_a @ Xs2 ) ) ) ) ) )
=> ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Bs2 ) )
=> ( ord_less_eq_set_a @ ( Rank @ ( rev_a @ V3 ) ) @ ( Rank @ ( rev_a @ X ) ) ) ) ) ) ) ) ) ) ) ) ).
% bs_ranks_only_ge_r
thf(fact_896_bs__ranks__only__ge__r,axiom,
! [Y6: set_list_a,As: list_a,U2: list_a,Bs: list_a,V3: list_a,Cs: list_a,Bs2: list_list_a,Cs2: list_list_a,Rank: list_a > set_b] :
( ~ ( member_list_a @ nil_a @ Y6 )
=> ( ( distinct_a @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
=> ( ( As = nil_a )
=> ( ( ( concat_a @ Bs2 )
= Bs )
=> ( ( ( concat_a @ Cs2 )
= Cs )
=> ( ( ( set_list_a2 @ ( cons_list_a @ U2 @ ( append_list_a @ Bs2 @ ( cons_list_a @ V3 @ Cs2 ) ) ) )
= Y6 )
=> ( ! [Xs2: list_a] :
( ( member_list_a @ Xs2 @ Y6 )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ~ ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ U2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ( Xs2 != U2 )
=> ( ord_less_eq_set_b @ ( Rank @ ( rev_a @ V3 ) ) @ ( Rank @ ( rev_a @ Xs2 ) ) ) ) ) )
=> ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Bs2 ) )
=> ( ord_less_eq_set_b @ ( Rank @ ( rev_a @ V3 ) ) @ ( Rank @ ( rev_a @ X ) ) ) ) ) ) ) ) ) ) ) ) ).
% bs_ranks_only_ge_r
thf(fact_897_rev__eq__Cons__iff,axiom,
! [Xs: list_a,Y2: a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( cons_a @ Y2 @ Ys ) )
= ( Xs
= ( append_a @ ( rev_a @ Ys ) @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_898_rev__eq__Cons__iff,axiom,
! [Xs: list_b,Y2: b,Ys: list_b] :
( ( ( rev_b @ Xs )
= ( cons_b @ Y2 @ Ys ) )
= ( Xs
= ( append_b @ ( rev_b @ Ys ) @ ( cons_b @ Y2 @ nil_b ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_899_vwalk__Cons__Cons,axiom,
! [U: a,V: a,Ws: list_a] :
( ( vertex_vwalk_a_b @ ( cons_a @ U @ ( cons_a @ V @ Ws ) ) @ t )
= ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ t ) )
& ( vertex_vwalk_a_b @ ( cons_a @ V @ Ws ) @ t ) ) ) ).
% vwalk_Cons_Cons
thf(fact_900_concat__eq__appendD,axiom,
! [Xss: list_list_b,Ys: list_b,Zs: list_b] :
( ( ( concat_b @ Xss )
= ( append_b @ Ys @ Zs ) )
=> ( ( Xss != nil_list_b )
=> ? [Xss1: list_list_b,Xs2: list_b,Xs5: list_b,Xss2: list_list_b] :
( ( Xss
= ( append_list_b @ Xss1 @ ( cons_list_b @ ( append_b @ Xs2 @ Xs5 ) @ Xss2 ) ) )
& ( Ys
= ( append_b @ ( concat_b @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_b @ Xs5 @ ( concat_b @ Xss2 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_901_concat__eq__appendD,axiom,
! [Xss: list_list_a,Ys: list_a,Zs: list_a] :
( ( ( concat_a @ Xss )
= ( append_a @ Ys @ Zs ) )
=> ( ( Xss != nil_list_a )
=> ? [Xss1: list_list_a,Xs2: list_a,Xs5: list_a,Xss2: list_list_a] :
( ( Xss
= ( append_list_a @ Xss1 @ ( cons_list_a @ ( append_a @ Xs2 @ Xs5 ) @ Xss2 ) ) )
& ( Ys
= ( append_a @ ( concat_a @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append_a @ Xs5 @ ( concat_a @ Xss2 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_902_concat_Osimps_I2_J,axiom,
! [X2: list_b,Xs: list_list_b] :
( ( concat_b @ ( cons_list_b @ X2 @ Xs ) )
= ( append_b @ X2 @ ( concat_b @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_903_concat_Osimps_I2_J,axiom,
! [X2: list_a,Xs: list_list_a] :
( ( concat_a @ ( cons_list_a @ X2 @ Xs ) )
= ( append_a @ X2 @ ( concat_a @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_904_concat__eq__append__conv,axiom,
! [Xss: list_list_a,Ys: list_a,Zs: list_a] :
( ( ( concat_a @ Xss )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Xss = nil_list_a )
=> ( ( Ys = nil_a )
& ( Zs = nil_a ) ) )
& ( ( Xss != nil_list_a )
=> ? [Xss12: list_list_a,Xs4: list_a,Xs6: list_a,Xss22: list_list_a] :
( ( Xss
= ( append_list_a @ Xss12 @ ( cons_list_a @ ( append_a @ Xs4 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_a @ ( concat_a @ Xss12 ) @ Xs4 ) )
& ( Zs
= ( append_a @ Xs6 @ ( concat_a @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_905_concat__eq__append__conv,axiom,
! [Xss: list_list_b,Ys: list_b,Zs: list_b] :
( ( ( concat_b @ Xss )
= ( append_b @ Ys @ Zs ) )
= ( ( ( Xss = nil_list_b )
=> ( ( Ys = nil_b )
& ( Zs = nil_b ) ) )
& ( ( Xss != nil_list_b )
=> ? [Xss12: list_list_b,Xs4: list_b,Xs6: list_b,Xss22: list_list_b] :
( ( Xss
= ( append_list_b @ Xss12 @ ( cons_list_b @ ( append_b @ Xs4 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append_b @ ( concat_b @ Xss12 ) @ Xs4 ) )
& ( Zs
= ( append_b @ Xs6 @ ( concat_b @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_906_hd__concat,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( ( hd_list_a @ Xs )
!= nil_a )
=> ( ( hd_a @ ( concat_a @ Xs ) )
= ( hd_a @ ( hd_list_a @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_907_hd__concat,axiom,
! [Xs: list_list_b] :
( ( Xs != nil_list_b )
=> ( ( ( hd_list_b @ Xs )
!= nil_b )
=> ( ( hd_b @ ( concat_b @ Xs ) )
= ( hd_b @ ( hd_list_b @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_908_rev_Osimps_I2_J,axiom,
! [X2: a,Xs: list_a] :
( ( rev_a @ ( cons_a @ X2 @ Xs ) )
= ( append_a @ ( rev_a @ Xs ) @ ( cons_a @ X2 @ nil_a ) ) ) ).
% rev.simps(2)
thf(fact_909_rev_Osimps_I2_J,axiom,
! [X2: b,Xs: list_b] :
( ( rev_b @ ( cons_b @ X2 @ Xs ) )
= ( append_b @ ( rev_b @ Xs ) @ ( cons_b @ X2 @ nil_b ) ) ) ).
% rev.simps(2)
thf(fact_910_distinct__concat,axiom,
! [Xs: list_list_list_a] :
( ( distinct_list_list_a @ Xs )
=> ( ! [Ys2: list_list_a] :
( ( member_list_list_a @ Ys2 @ ( set_list_list_a2 @ Xs ) )
=> ( distinct_list_a @ Ys2 ) )
=> ( ! [Ys2: list_list_a,Zs2: list_list_a] :
( ( member_list_list_a @ Ys2 @ ( set_list_list_a2 @ Xs ) )
=> ( ( member_list_list_a @ Zs2 @ ( set_list_list_a2 @ Xs ) )
=> ( ( Ys2 != Zs2 )
=> ( ( inf_inf_set_list_a @ ( set_list_a2 @ Ys2 ) @ ( set_list_a2 @ Zs2 ) )
= bot_bot_set_list_a ) ) ) )
=> ( distinct_list_a @ ( concat_list_a @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_911_distinct__concat,axiom,
! [Xs: list_list_a] :
( ( distinct_list_a @ Xs )
=> ( ! [Ys2: list_a] :
( ( member_list_a @ Ys2 @ ( set_list_a2 @ Xs ) )
=> ( distinct_a @ Ys2 ) )
=> ( ! [Ys2: list_a,Zs2: list_a] :
( ( member_list_a @ Ys2 @ ( set_list_a2 @ Xs ) )
=> ( ( member_list_a @ Zs2 @ ( set_list_a2 @ Xs ) )
=> ( ( Ys2 != Zs2 )
=> ( ( inf_inf_set_a @ ( set_a2 @ Ys2 ) @ ( set_a2 @ Zs2 ) )
= bot_bot_set_a ) ) ) )
=> ( distinct_a @ ( concat_a @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_912_distinct__concat,axiom,
! [Xs: list_list_b] :
( ( distinct_list_b @ Xs )
=> ( ! [Ys2: list_b] :
( ( member_list_b @ Ys2 @ ( set_list_b2 @ Xs ) )
=> ( distinct_b @ Ys2 ) )
=> ( ! [Ys2: list_b,Zs2: list_b] :
( ( member_list_b @ Ys2 @ ( set_list_b2 @ Xs ) )
=> ( ( member_list_b @ Zs2 @ ( set_list_b2 @ Xs ) )
=> ( ( Ys2 != Zs2 )
=> ( ( inf_inf_set_b @ ( set_b2 @ Ys2 ) @ ( set_b2 @ Zs2 ) )
= bot_bot_set_b ) ) ) )
=> ( distinct_b @ ( concat_b @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_913_vwalk__singleton,axiom,
! [U: a,G: pre_pr7278220950009878019t_unit] :
( ( vertex_vwalk_a_b @ ( cons_a @ U @ nil_a ) @ G )
= ( member_a @ U @ ( pre_ve642382030648772252t_unit @ G ) ) ) ).
% vwalk_singleton
thf(fact_914_vwalk__singleton,axiom,
! [U: list_a,G: pre_pr2882871181989701257t_unit] :
( ( vertex2966258834163962945st_a_b @ ( cons_list_a @ U @ nil_list_a ) @ G )
= ( member_list_a @ U @ ( pre_ve1830060048215441954t_unit @ G ) ) ) ).
% vwalk_singleton
thf(fact_915_concat__all__sublist2,axiom,
! [As: list_list_a,U2: list_list_a,Bs: list_list_a,Cs: list_list_list_a,Ds: list_list_list_a,Y6: set_list_list_a] :
( ( distinct_list_a @ ( append_list_a @ As @ ( append_list_a @ U2 @ Bs ) ) )
=> ( ( ( append_list_a @ ( concat_list_a @ Cs ) @ ( append_list_a @ U2 @ ( concat_list_a @ Ds ) ) )
= ( append_list_a @ As @ ( append_list_a @ U2 @ Bs ) ) )
=> ( ( U2 != nil_list_a )
=> ( ( ( set_list_list_a2 @ ( append_list_list_a @ Cs @ ( cons_list_list_a @ U2 @ Ds ) ) )
= Y6 )
=> ? [X7: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ X7 @ Y6 )
& ( ( set_list_a2 @ Bs )
= ( comple6928918032620976721list_a @ ( image_432481560377026271list_a @ set_list_a2 @ X7 ) ) )
& ! [X: list_list_a] :
( ( member_list_list_a @ X @ X7 )
=> ( sublist_list_a @ X @ Bs ) ) ) ) ) ) ) ).
% concat_all_sublist2
thf(fact_916_concat__all__sublist2,axiom,
! [As: list_b,U2: list_b,Bs: list_b,Cs: list_list_b,Ds: list_list_b,Y6: set_list_b] :
( ( distinct_b @ ( append_b @ As @ ( append_b @ U2 @ Bs ) ) )
=> ( ( ( append_b @ ( concat_b @ Cs ) @ ( append_b @ U2 @ ( concat_b @ Ds ) ) )
= ( append_b @ As @ ( append_b @ U2 @ Bs ) ) )
=> ( ( U2 != nil_b )
=> ( ( ( set_list_b2 @ ( append_list_b @ Cs @ ( cons_list_b @ U2 @ Ds ) ) )
= Y6 )
=> ? [X7: set_list_b] :
( ( ord_le8932221534207217157list_b @ X7 @ Y6 )
& ( ( set_b2 @ Bs )
= ( comple2307003614231284044_set_b @ ( image_list_b_set_b @ set_b2 @ X7 ) ) )
& ! [X: list_b] :
( ( member_list_b @ X @ X7 )
=> ( sublist_b @ X @ Bs ) ) ) ) ) ) ) ).
% concat_all_sublist2
thf(fact_917_concat__all__sublist2,axiom,
! [As: list_a,U2: list_a,Bs: list_a,Cs: list_list_a,Ds: list_list_a,Y6: set_list_a] :
( ( distinct_a @ ( append_a @ As @ ( append_a @ U2 @ Bs ) ) )
=> ( ( ( append_a @ ( concat_a @ Cs ) @ ( append_a @ U2 @ ( concat_a @ Ds ) ) )
= ( append_a @ As @ ( append_a @ U2 @ Bs ) ) )
=> ( ( U2 != nil_a )
=> ( ( ( set_list_a2 @ ( append_list_a @ Cs @ ( cons_list_a @ U2 @ Ds ) ) )
= Y6 )
=> ? [X7: set_list_a] :
( ( ord_le8861187494160871172list_a @ X7 @ Y6 )
& ( ( set_a2 @ Bs )
= ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ set_a2 @ X7 ) ) )
& ! [X: list_a] :
( ( member_list_a @ X @ X7 )
=> ( sublist_a @ X @ Bs ) ) ) ) ) ) ) ).
% concat_all_sublist2
thf(fact_918_concat__all__sublist1,axiom,
! [As: list_list_a,U2: list_list_a,Bs: list_list_a,Cs: list_list_list_a,Ds: list_list_list_a,Y6: set_list_list_a] :
( ( distinct_list_a @ ( append_list_a @ As @ ( append_list_a @ U2 @ Bs ) ) )
=> ( ( ( append_list_a @ ( concat_list_a @ Cs ) @ ( append_list_a @ U2 @ ( concat_list_a @ Ds ) ) )
= ( append_list_a @ As @ ( append_list_a @ U2 @ Bs ) ) )
=> ( ( U2 != nil_list_a )
=> ( ( ( set_list_list_a2 @ ( append_list_list_a @ Cs @ ( cons_list_list_a @ U2 @ Ds ) ) )
= Y6 )
=> ? [X7: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ X7 @ Y6 )
& ( ( set_list_a2 @ As )
= ( comple6928918032620976721list_a @ ( image_432481560377026271list_a @ set_list_a2 @ X7 ) ) )
& ! [X: list_list_a] :
( ( member_list_list_a @ X @ X7 )
=> ( sublist_list_a @ X @ As ) ) ) ) ) ) ) ).
% concat_all_sublist1
thf(fact_919_concat__all__sublist1,axiom,
! [As: list_b,U2: list_b,Bs: list_b,Cs: list_list_b,Ds: list_list_b,Y6: set_list_b] :
( ( distinct_b @ ( append_b @ As @ ( append_b @ U2 @ Bs ) ) )
=> ( ( ( append_b @ ( concat_b @ Cs ) @ ( append_b @ U2 @ ( concat_b @ Ds ) ) )
= ( append_b @ As @ ( append_b @ U2 @ Bs ) ) )
=> ( ( U2 != nil_b )
=> ( ( ( set_list_b2 @ ( append_list_b @ Cs @ ( cons_list_b @ U2 @ Ds ) ) )
= Y6 )
=> ? [X7: set_list_b] :
( ( ord_le8932221534207217157list_b @ X7 @ Y6 )
& ( ( set_b2 @ As )
= ( comple2307003614231284044_set_b @ ( image_list_b_set_b @ set_b2 @ X7 ) ) )
& ! [X: list_b] :
( ( member_list_b @ X @ X7 )
=> ( sublist_b @ X @ As ) ) ) ) ) ) ) ).
% concat_all_sublist1
thf(fact_920_concat__all__sublist1,axiom,
! [As: list_a,U2: list_a,Bs: list_a,Cs: list_list_a,Ds: list_list_a,Y6: set_list_a] :
( ( distinct_a @ ( append_a @ As @ ( append_a @ U2 @ Bs ) ) )
=> ( ( ( append_a @ ( concat_a @ Cs ) @ ( append_a @ U2 @ ( concat_a @ Ds ) ) )
= ( append_a @ As @ ( append_a @ U2 @ Bs ) ) )
=> ( ( U2 != nil_a )
=> ( ( ( set_list_a2 @ ( append_list_a @ Cs @ ( cons_list_a @ U2 @ Ds ) ) )
= Y6 )
=> ? [X7: set_list_a] :
( ( ord_le8861187494160871172list_a @ X7 @ Y6 )
& ( ( set_a2 @ As )
= ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ set_a2 @ X7 ) ) )
& ! [X: list_a] :
( ( member_list_a @ X @ X7 )
=> ( sublist_a @ X @ As ) ) ) ) ) ) ) ).
% concat_all_sublist1
thf(fact_921_vwalk__consI,axiom,
! [P2: list_a,G: pre_pr7278220950009878019t_unit,A: a] :
( ( vertex_vwalk_a_b @ P2 @ G )
=> ( ( member_a @ A @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( hd_a @ P2 ) ) @ ( arcs_ends_a_b @ G ) )
=> ( vertex_vwalk_a_b @ ( cons_a @ A @ P2 ) @ G ) ) ) ) ).
% vwalk_consI
thf(fact_922_vwalk__consI,axiom,
! [P2: list_list_a,G: pre_pr2882871181989701257t_unit,A: list_a] :
( ( vertex2966258834163962945st_a_b @ P2 @ G )
=> ( ( member_list_a @ A @ ( pre_ve1830060048215441954t_unit @ G ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A @ ( hd_list_a @ P2 ) ) @ ( arcs_ends_list_a_b @ G ) )
=> ( vertex2966258834163962945st_a_b @ ( cons_list_a @ A @ P2 ) @ G ) ) ) ) ).
% vwalk_consI
thf(fact_923_vwalk__consE,axiom,
! [A: list_a,P2: list_list_a,G: pre_pr2882871181989701257t_unit] :
( ( vertex2966258834163962945st_a_b @ ( cons_list_a @ A @ P2 ) @ G )
=> ( ( P2 != nil_list_a )
=> ~ ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A @ ( hd_list_a @ P2 ) ) @ ( arcs_ends_list_a_b @ G ) )
=> ~ ( vertex2966258834163962945st_a_b @ P2 @ G ) ) ) ) ).
% vwalk_consE
thf(fact_924_vwalk__consE,axiom,
! [A: a,P2: list_a,G: pre_pr7278220950009878019t_unit] :
( ( vertex_vwalk_a_b @ ( cons_a @ A @ P2 ) @ G )
=> ( ( P2 != nil_a )
=> ~ ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( hd_a @ P2 ) ) @ ( arcs_ends_a_b @ G ) )
=> ~ ( vertex_vwalk_a_b @ P2 @ G ) ) ) ) ).
% vwalk_consE
thf(fact_925_sublist__app__l,axiom,
! [Ys: list_a,Cs: list_a,Xs: list_a] :
( ( sublist_a @ Ys @ Cs )
=> ( sublist_a @ Ys @ ( append_a @ Xs @ Cs ) ) ) ).
% sublist_app_l
thf(fact_926_sublist__app__l,axiom,
! [Ys: list_b,Cs: list_b,Xs: list_b] :
( ( sublist_b @ Ys @ Cs )
=> ( sublist_b @ Ys @ ( append_b @ Xs @ Cs ) ) ) ).
% sublist_app_l
thf(fact_927_sublist__set__concat__or__cases__aux1,axiom,
! [Ys: list_b,As: list_b,U2: list_b,Cs: list_b,Xs: list_list_b] :
( ( ( sublist_b @ Ys @ As )
| ( sublist_b @ Ys @ U2 )
| ( sublist_b @ Ys @ Cs ) )
=> ( ( sublist_b @ Ys @ ( append_b @ As @ ( append_b @ U2 @ ( concat_b @ ( rev_list_b @ Xs ) ) ) ) )
| ( sublist_b @ Ys @ Cs ) ) ) ).
% sublist_set_concat_or_cases_aux1
thf(fact_928_sublist__set__concat__or__cases__aux1,axiom,
! [Ys: list_a,As: list_a,U2: list_a,Cs: list_a,Xs: list_list_a] :
( ( ( sublist_a @ Ys @ As )
| ( sublist_a @ Ys @ U2 )
| ( sublist_a @ Ys @ Cs ) )
=> ( ( sublist_a @ Ys @ ( append_a @ As @ ( append_a @ U2 @ ( concat_a @ ( rev_list_a @ Xs ) ) ) ) )
| ( sublist_a @ Ys @ Cs ) ) ) ).
% sublist_set_concat_or_cases_aux1
thf(fact_929_concat__all__sublist__rev,axiom,
! [Xs: list_list_a,X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( sublist_a @ X @ ( concat_a @ ( rev_list_a @ Xs ) ) ) ) ).
% concat_all_sublist_rev
thf(fact_930_concat__all__sublist,axiom,
! [Xs: list_list_a,X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( sublist_a @ X @ ( concat_a @ Xs ) ) ) ).
% concat_all_sublist
thf(fact_931_sublist__exists__append,axiom,
! [X2: list_b,Xs: list_list_b,B: list_b,Ys: list_b] :
( ? [X: list_b] :
( ( member_list_b @ X @ ( set_list_b2 @ ( append_list_b @ ( cons_list_b @ X2 @ Xs ) @ ( cons_list_b @ B @ nil_list_b ) ) ) )
& ( sublist_b @ Ys @ X ) )
=> ? [X4: list_b] :
( ( member_list_b @ X4 @ ( set_list_b2 @ ( append_list_b @ Xs @ ( cons_list_b @ ( append_b @ X2 @ B ) @ nil_list_b ) ) ) )
& ( sublist_b @ Ys @ X4 ) ) ) ).
% sublist_exists_append
thf(fact_932_sublist__exists__append,axiom,
! [X2: list_a,Xs: list_list_a,B: list_a,Ys: list_a] :
( ? [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ ( append_list_a @ ( cons_list_a @ X2 @ Xs ) @ ( cons_list_a @ B @ nil_list_a ) ) ) )
& ( sublist_a @ Ys @ X ) )
=> ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ ( append_list_a @ Xs @ ( cons_list_a @ ( append_a @ X2 @ B ) @ nil_list_a ) ) ) )
& ( sublist_a @ Ys @ X4 ) ) ) ).
% sublist_exists_append
thf(fact_933_sublist__set__concat__cases,axiom,
! [X2: list_a,Xs: list_list_a,B: list_a,Ys: list_a] :
( ? [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ ( append_list_a @ ( cons_list_a @ X2 @ Xs ) @ ( cons_list_a @ B @ nil_list_a ) ) ) )
& ( sublist_a @ Ys @ X ) )
=> ( ( sublist_a @ Ys @ ( concat_a @ ( rev_list_a @ Xs ) ) )
| ( sublist_a @ Ys @ X2 )
| ( sublist_a @ Ys @ B ) ) ) ).
% sublist_set_concat_cases
thf(fact_934_sublist__split__concat_H,axiom,
! [Acc: list_list_b,As: list_list_b,X2: list_b,Bs: list_list_b,Ys: list_b,Cs: list_b] :
( ? [X: list_b] :
( ( member_list_b @ X @ ( set_list_b2 @ ( append_list_b @ Acc @ ( append_list_b @ As @ ( cons_list_b @ X2 @ Bs ) ) ) ) )
& ( ( sublist_b @ Ys @ X )
| ( sublist_b @ Ys @ Cs ) ) )
=> ( ? [X4: list_b] :
( ( member_list_b @ X4 @ ( set_list_b2 @ ( append_list_b @ ( rev_list_b @ Acc ) @ ( append_list_b @ As @ ( cons_list_b @ X2 @ nil_list_b ) ) ) ) )
& ( sublist_b @ Ys @ X4 ) )
| ( sublist_b @ Ys @ ( append_b @ ( concat_b @ Bs ) @ Cs ) ) ) ) ).
% sublist_split_concat'
thf(fact_935_sublist__split__concat_H,axiom,
! [Acc: list_list_a,As: list_list_a,X2: list_a,Bs: list_list_a,Ys: list_a,Cs: list_a] :
( ? [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ ( append_list_a @ Acc @ ( append_list_a @ As @ ( cons_list_a @ X2 @ Bs ) ) ) ) )
& ( ( sublist_a @ Ys @ X )
| ( sublist_a @ Ys @ Cs ) ) )
=> ( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ ( append_list_a @ ( rev_list_a @ Acc ) @ ( append_list_a @ As @ ( cons_list_a @ X2 @ nil_list_a ) ) ) ) )
& ( sublist_a @ Ys @ X4 ) )
| ( sublist_a @ Ys @ ( append_a @ ( concat_a @ Bs ) @ Cs ) ) ) ) ).
% sublist_split_concat'
thf(fact_936_sublist__split__concat,axiom,
! [A: list_b,Acc: list_list_b,As: list_list_b,X2: list_b,Bs: list_list_b,Ys: list_b,Cs: list_b] :
( ( member_list_b @ A @ ( set_list_b2 @ ( append_list_b @ Acc @ ( append_list_b @ As @ ( cons_list_b @ X2 @ Bs ) ) ) ) )
=> ( ( sublist_b @ Ys @ A )
=> ( ? [X4: list_b] :
( ( member_list_b @ X4 @ ( set_list_b2 @ ( append_list_b @ ( rev_list_b @ Acc ) @ ( append_list_b @ As @ ( cons_list_b @ X2 @ nil_list_b ) ) ) ) )
& ( sublist_b @ Ys @ X4 ) )
| ( sublist_b @ Ys @ ( append_b @ ( concat_b @ Bs ) @ Cs ) ) ) ) ) ).
% sublist_split_concat
thf(fact_937_sublist__split__concat,axiom,
! [A: list_a,Acc: list_list_a,As: list_list_a,X2: list_a,Bs: list_list_a,Ys: list_a,Cs: list_a] :
( ( member_list_a @ A @ ( set_list_a2 @ ( append_list_a @ Acc @ ( append_list_a @ As @ ( cons_list_a @ X2 @ Bs ) ) ) ) )
=> ( ( sublist_a @ Ys @ A )
=> ( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ ( append_list_a @ ( rev_list_a @ Acc ) @ ( append_list_a @ As @ ( cons_list_a @ X2 @ nil_list_a ) ) ) ) )
& ( sublist_a @ Ys @ X4 ) )
| ( sublist_a @ Ys @ ( append_a @ ( concat_a @ Bs ) @ Cs ) ) ) ) ) ).
% sublist_split_concat
thf(fact_938_sublist__set__concat__or__cases__aux2,axiom,
! [X2: list_b,Xs: list_list_b,B: list_b,Ys: list_b,As: list_b,U2: list_b] :
( ? [X: list_b] :
( ( member_list_b @ X @ ( set_list_b2 @ ( append_list_b @ ( cons_list_b @ X2 @ Xs ) @ ( cons_list_b @ B @ nil_list_b ) ) ) )
& ( sublist_b @ Ys @ X ) )
=> ( ( sublist_b @ Ys @ ( append_b @ As @ ( append_b @ U2 @ ( concat_b @ ( rev_list_b @ Xs ) ) ) ) )
| ( sublist_b @ Ys @ X2 )
| ( sublist_b @ Ys @ B ) ) ) ).
% sublist_set_concat_or_cases_aux2
thf(fact_939_sublist__set__concat__or__cases__aux2,axiom,
! [X2: list_a,Xs: list_list_a,B: list_a,Ys: list_a,As: list_a,U2: list_a] :
( ? [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ ( append_list_a @ ( cons_list_a @ X2 @ Xs ) @ ( cons_list_a @ B @ nil_list_a ) ) ) )
& ( sublist_a @ Ys @ X ) )
=> ( ( sublist_a @ Ys @ ( append_a @ As @ ( append_a @ U2 @ ( concat_a @ ( rev_list_a @ Xs ) ) ) ) )
| ( sublist_a @ Ys @ X2 )
| ( sublist_a @ Ys @ B ) ) ) ).
% sublist_set_concat_or_cases_aux2
thf(fact_940_sublist__set__concat__or__cases,axiom,
! [Ys: list_b,As: list_b,U2: list_b,X2: list_b,Xs: list_list_b,B: list_b,Cs: list_b] :
( ( ( sublist_b @ Ys @ As )
| ( sublist_b @ Ys @ U2 )
| ? [X: list_b] :
( ( member_list_b @ X @ ( set_list_b2 @ ( append_list_b @ ( cons_list_b @ X2 @ Xs ) @ ( cons_list_b @ B @ nil_list_b ) ) ) )
& ( sublist_b @ Ys @ X ) )
| ( sublist_b @ Ys @ Cs ) )
=> ( ( sublist_b @ Ys @ ( append_b @ As @ ( append_b @ U2 @ ( concat_b @ ( rev_list_b @ Xs ) ) ) ) )
| ( sublist_b @ Ys @ X2 )
| ? [X4: list_b] :
( ( member_list_b @ X4 @ ( set_list_b2 @ ( cons_list_b @ B @ nil_list_b ) ) )
& ( sublist_b @ Ys @ X4 ) )
| ( sublist_b @ Ys @ Cs ) ) ) ).
% sublist_set_concat_or_cases
thf(fact_941_sublist__set__concat__or__cases,axiom,
! [Ys: list_a,As: list_a,U2: list_a,X2: list_a,Xs: list_list_a,B: list_a,Cs: list_a] :
( ( ( sublist_a @ Ys @ As )
| ( sublist_a @ Ys @ U2 )
| ? [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ ( append_list_a @ ( cons_list_a @ X2 @ Xs ) @ ( cons_list_a @ B @ nil_list_a ) ) ) )
& ( sublist_a @ Ys @ X ) )
| ( sublist_a @ Ys @ Cs ) )
=> ( ( sublist_a @ Ys @ ( append_a @ As @ ( append_a @ U2 @ ( concat_a @ ( rev_list_a @ Xs ) ) ) ) )
| ( sublist_a @ Ys @ X2 )
| ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ ( cons_list_a @ B @ nil_list_a ) ) )
& ( sublist_a @ Ys @ X4 ) )
| ( sublist_a @ Ys @ Cs ) ) ) ).
% sublist_set_concat_or_cases
thf(fact_942_sublist__app,axiom,
! [A2: list_a,B3: list_a,C2: list_a] :
( ( sublist_a @ ( append_a @ A2 @ B3 ) @ C2 )
=> ( ( sublist_a @ A2 @ C2 )
& ( sublist_a @ B3 @ C2 ) ) ) ).
% sublist_app
thf(fact_943_sublist__app,axiom,
! [A2: list_b,B3: list_b,C2: list_b] :
( ( sublist_b @ ( append_b @ A2 @ B3 ) @ C2 )
=> ( ( sublist_b @ A2 @ C2 )
& ( sublist_b @ B3 @ C2 ) ) ) ).
% sublist_app
thf(fact_944_fst__sublist__if__not__snd__sublist,axiom,
! [Xs: list_a,Ys: list_a,A2: list_a,B3: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ A2 @ B3 ) )
=> ( ~ ( sublist_a @ B3 @ Ys )
=> ? [As3: list_a,Bs3: list_a] :
( ( ( append_a @ As3 @ Bs3 )
= Xs )
& ( ( append_a @ Bs3 @ Ys )
= B3 ) ) ) ) ).
% fst_sublist_if_not_snd_sublist
thf(fact_945_fst__sublist__if__not__snd__sublist,axiom,
! [Xs: list_b,Ys: list_b,A2: list_b,B3: list_b] :
( ( ( append_b @ Xs @ Ys )
= ( append_b @ A2 @ B3 ) )
=> ( ~ ( sublist_b @ B3 @ Ys )
=> ? [As3: list_b,Bs3: list_b] :
( ( ( append_b @ As3 @ Bs3 )
= Xs )
& ( ( append_b @ Bs3 @ Ys )
= B3 ) ) ) ) ).
% fst_sublist_if_not_snd_sublist
thf(fact_946_sublist__set__elem,axiom,
! [Xs: list_P1396940483166286381od_a_a,A2: list_P1396940483166286381od_a_a,B3: list_P1396940483166286381od_a_a,X2: product_prod_a_a] :
( ( sublis713342594703834650od_a_a @ Xs @ ( append5335208819046833346od_a_a @ A2 @ B3 ) )
=> ( ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ Xs ) )
=> ( ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ A2 ) )
| ( member1426531477525435216od_a_a @ X2 @ ( set_Product_prod_a_a2 @ B3 ) ) ) ) ) ).
% sublist_set_elem
thf(fact_947_sublist__set__elem,axiom,
! [Xs: list_set_a,A2: list_set_a,B3: list_set_a,X2: set_a] :
( ( sublist_set_a @ Xs @ ( append_set_a @ A2 @ B3 ) )
=> ( ( member_set_a @ X2 @ ( set_set_a2 @ Xs ) )
=> ( ( member_set_a @ X2 @ ( set_set_a2 @ A2 ) )
| ( member_set_a @ X2 @ ( set_set_a2 @ B3 ) ) ) ) ) ).
% sublist_set_elem
thf(fact_948_sublist__set__elem,axiom,
! [Xs: list_a,A2: list_a,B3: list_a,X2: a] :
( ( sublist_a @ Xs @ ( append_a @ A2 @ B3 ) )
=> ( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( ( member_a @ X2 @ ( set_a2 @ A2 ) )
| ( member_a @ X2 @ ( set_a2 @ B3 ) ) ) ) ) ).
% sublist_set_elem
thf(fact_949_sublist__set__elem,axiom,
! [Xs: list_list_a,A2: list_list_a,B3: list_list_a,X2: list_a] :
( ( sublist_list_a @ Xs @ ( append_list_a @ A2 @ B3 ) )
=> ( ( member_list_a @ X2 @ ( set_list_a2 @ Xs ) )
=> ( ( member_list_a @ X2 @ ( set_list_a2 @ A2 ) )
| ( member_list_a @ X2 @ ( set_list_a2 @ B3 ) ) ) ) ) ).
% sublist_set_elem
thf(fact_950_sublist__set__elem,axiom,
! [Xs: list_b,A2: list_b,B3: list_b,X2: b] :
( ( sublist_b @ Xs @ ( append_b @ A2 @ B3 ) )
=> ( ( member_b @ X2 @ ( set_b2 @ Xs ) )
=> ( ( member_b @ X2 @ ( set_b2 @ A2 ) )
| ( member_b @ X2 @ ( set_b2 @ B3 ) ) ) ) ) ).
% sublist_set_elem
thf(fact_951_sublist__Cons,axiom,
! [A2: a,B3: list_a,C2: list_a] :
( ( sublist_a @ ( cons_a @ A2 @ B3 ) @ C2 )
=> ( ( sublist_a @ ( cons_a @ A2 @ nil_a ) @ C2 )
& ( sublist_a @ B3 @ C2 ) ) ) ).
% sublist_Cons
thf(fact_952_sublist__Cons,axiom,
! [A2: b,B3: list_b,C2: list_b] :
( ( sublist_b @ ( cons_b @ A2 @ B3 ) @ C2 )
=> ( ( sublist_b @ ( cons_b @ A2 @ nil_b ) @ C2 )
& ( sublist_b @ B3 @ C2 ) ) ) ).
% sublist_Cons
thf(fact_953_sublist__behind__if__nbefore,axiom,
! [U2: list_list_a,Xs: list_list_a,V3: list_list_a] :
( ( sublist_list_a @ U2 @ Xs )
=> ( ( sublist_list_a @ V3 @ Xs )
=> ( ~ ? [As3: list_list_a,Bs3: list_list_a,Cs3: list_list_a] :
( ( append_list_a @ As3 @ ( append_list_a @ U2 @ ( append_list_a @ Bs3 @ ( append_list_a @ V3 @ Cs3 ) ) ) )
= Xs )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ U2 ) @ ( set_list_a2 @ V3 ) )
= bot_bot_set_list_a )
=> ? [As3: list_list_a,Bs3: list_list_a,Cs3: list_list_a] :
( ( append_list_a @ As3 @ ( append_list_a @ V3 @ ( append_list_a @ Bs3 @ ( append_list_a @ U2 @ Cs3 ) ) ) )
= Xs ) ) ) ) ) ).
% sublist_behind_if_nbefore
thf(fact_954_sublist__behind__if__nbefore,axiom,
! [U2: list_a,Xs: list_a,V3: list_a] :
( ( sublist_a @ U2 @ Xs )
=> ( ( sublist_a @ V3 @ Xs )
=> ( ~ ? [As3: list_a,Bs3: list_a,Cs3: list_a] :
( ( append_a @ As3 @ ( append_a @ U2 @ ( append_a @ Bs3 @ ( append_a @ V3 @ Cs3 ) ) ) )
= Xs )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ V3 ) )
= bot_bot_set_a )
=> ? [As3: list_a,Bs3: list_a,Cs3: list_a] :
( ( append_a @ As3 @ ( append_a @ V3 @ ( append_a @ Bs3 @ ( append_a @ U2 @ Cs3 ) ) ) )
= Xs ) ) ) ) ) ).
% sublist_behind_if_nbefore
thf(fact_955_sublist__behind__if__nbefore,axiom,
! [U2: list_b,Xs: list_b,V3: list_b] :
( ( sublist_b @ U2 @ Xs )
=> ( ( sublist_b @ V3 @ Xs )
=> ( ~ ? [As3: list_b,Bs3: list_b,Cs3: list_b] :
( ( append_b @ As3 @ ( append_b @ U2 @ ( append_b @ Bs3 @ ( append_b @ V3 @ Cs3 ) ) ) )
= Xs )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ U2 ) @ ( set_b2 @ V3 ) )
= bot_bot_set_b )
=> ? [As3: list_b,Bs3: list_b,Cs3: list_b] :
( ( append_b @ As3 @ ( append_b @ V3 @ ( append_b @ Bs3 @ ( append_b @ U2 @ Cs3 ) ) ) )
= Xs ) ) ) ) ) ).
% sublist_behind_if_nbefore
thf(fact_956_sublist__snd__if__fst__dsjnt,axiom,
! [U2: list_list_a,V3: list_list_a,B3: list_list_a] :
( ( sublist_list_a @ U2 @ ( append_list_a @ V3 @ B3 ) )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ U2 ) @ ( set_list_a2 @ V3 ) )
= bot_bot_set_list_a )
=> ( sublist_list_a @ U2 @ B3 ) ) ) ).
% sublist_snd_if_fst_dsjnt
thf(fact_957_sublist__snd__if__fst__dsjnt,axiom,
! [U2: list_a,V3: list_a,B3: list_a] :
( ( sublist_a @ U2 @ ( append_a @ V3 @ B3 ) )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ V3 ) )
= bot_bot_set_a )
=> ( sublist_a @ U2 @ B3 ) ) ) ).
% sublist_snd_if_fst_dsjnt
thf(fact_958_sublist__snd__if__fst__dsjnt,axiom,
! [U2: list_b,V3: list_b,B3: list_b] :
( ( sublist_b @ U2 @ ( append_b @ V3 @ B3 ) )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ U2 ) @ ( set_b2 @ V3 ) )
= bot_bot_set_b )
=> ( sublist_b @ U2 @ B3 ) ) ) ).
% sublist_snd_if_fst_dsjnt
thf(fact_959_sublist__fst__if__snd__dsjnt,axiom,
! [U2: list_list_a,B3: list_list_a,V3: list_list_a] :
( ( sublist_list_a @ U2 @ ( append_list_a @ B3 @ V3 ) )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ U2 ) @ ( set_list_a2 @ V3 ) )
= bot_bot_set_list_a )
=> ( sublist_list_a @ U2 @ B3 ) ) ) ).
% sublist_fst_if_snd_dsjnt
thf(fact_960_sublist__fst__if__snd__dsjnt,axiom,
! [U2: list_a,B3: list_a,V3: list_a] :
( ( sublist_a @ U2 @ ( append_a @ B3 @ V3 ) )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ V3 ) )
= bot_bot_set_a )
=> ( sublist_a @ U2 @ B3 ) ) ) ).
% sublist_fst_if_snd_dsjnt
thf(fact_961_sublist__fst__if__snd__dsjnt,axiom,
! [U2: list_b,B3: list_b,V3: list_b] :
( ( sublist_b @ U2 @ ( append_b @ B3 @ V3 ) )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ U2 ) @ ( set_b2 @ V3 ) )
= bot_bot_set_b )
=> ( sublist_b @ U2 @ B3 ) ) ) ).
% sublist_fst_if_snd_dsjnt
thf(fact_962_empty__if__sublist__dsjnt,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( sublist_list_a @ Xs @ Ys )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ Ys ) )
= bot_bot_set_list_a )
=> ( Xs = nil_list_a ) ) ) ).
% empty_if_sublist_dsjnt
thf(fact_963_empty__if__sublist__dsjnt,axiom,
! [Xs: list_a,Ys: list_a] :
( ( sublist_a @ Xs @ Ys )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( Xs = nil_a ) ) ) ).
% empty_if_sublist_dsjnt
thf(fact_964_empty__if__sublist__dsjnt,axiom,
! [Xs: list_b,Ys: list_b] :
( ( sublist_b @ Xs @ Ys )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ Ys ) )
= bot_bot_set_b )
=> ( Xs = nil_b ) ) ) ).
% empty_if_sublist_dsjnt
thf(fact_965_sublists__preserv__move__VY__all,axiom,
! [Y6: set_list_list_a,U2: list_list_a,V3: list_list_a,As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_list_a @ ( set_list_a2 @ X4 ) @ ( set_list_a2 @ Xa ) )
= bot_bot_set_list_a ) ) ) )
=> ( ( member_list_list_a @ U2 @ Y6 )
=> ( ( member_list_list_a @ V3 @ Y6 )
=> ( ( U2 != nil_list_a )
=> ( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ( sublist_list_a @ X4 @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V3 @ Cs ) ) ) ) ) )
=> ! [X: list_list_a] :
( ( member_list_list_a @ X @ Y6 )
=> ( sublist_list_a @ X @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ V3 @ ( append_list_a @ Bs @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_VY_all
thf(fact_966_sublists__preserv__move__VY__all,axiom,
! [Y6: set_list_a,U2: list_a,V3: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_a @ ( set_a2 @ X4 ) @ ( set_a2 @ Xa ) )
= bot_bot_set_a ) ) ) )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( member_list_a @ V3 @ Y6 )
=> ( ( U2 != nil_a )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ( sublist_a @ X4 @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) ) )
=> ! [X: list_a] :
( ( member_list_a @ X @ Y6 )
=> ( sublist_a @ X @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ V3 @ ( append_a @ Bs @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_VY_all
thf(fact_967_sublists__preserv__move__VY__all,axiom,
! [Y6: set_list_b,U2: list_b,V3: list_b,As: list_b,Bs: list_b,Cs: list_b] :
( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ! [Xa: list_b] :
( ( member_list_b @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_b @ ( set_b2 @ X4 ) @ ( set_b2 @ Xa ) )
= bot_bot_set_b ) ) ) )
=> ( ( member_list_b @ U2 @ Y6 )
=> ( ( member_list_b @ V3 @ Y6 )
=> ( ( U2 != nil_b )
=> ( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ( sublist_b @ X4 @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V3 @ Cs ) ) ) ) ) )
=> ! [X: list_b] :
( ( member_list_b @ X @ Y6 )
=> ( sublist_b @ X @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ V3 @ ( append_b @ Bs @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_VY_all
thf(fact_968_sublists__preserv__move__UY__all,axiom,
! [Y6: set_list_list_a,U2: list_list_a,V3: list_list_a,As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_list_a @ ( set_list_a2 @ X4 ) @ ( set_list_a2 @ Xa ) )
= bot_bot_set_list_a ) ) ) )
=> ( ( member_list_list_a @ U2 @ Y6 )
=> ( ( member_list_list_a @ V3 @ Y6 )
=> ( ( V3 != nil_list_a )
=> ( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ( sublist_list_a @ X4 @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V3 @ Cs ) ) ) ) ) )
=> ! [X: list_list_a] :
( ( member_list_list_a @ X @ Y6 )
=> ( sublist_list_a @ X @ ( append_list_a @ As @ ( append_list_a @ Bs @ ( append_list_a @ U2 @ ( append_list_a @ V3 @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_UY_all
thf(fact_969_sublists__preserv__move__UY__all,axiom,
! [Y6: set_list_a,U2: list_a,V3: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_a @ ( set_a2 @ X4 ) @ ( set_a2 @ Xa ) )
= bot_bot_set_a ) ) ) )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( member_list_a @ V3 @ Y6 )
=> ( ( V3 != nil_a )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ( sublist_a @ X4 @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) ) )
=> ! [X: list_a] :
( ( member_list_a @ X @ Y6 )
=> ( sublist_a @ X @ ( append_a @ As @ ( append_a @ Bs @ ( append_a @ U2 @ ( append_a @ V3 @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_UY_all
thf(fact_970_sublists__preserv__move__UY__all,axiom,
! [Y6: set_list_b,U2: list_b,V3: list_b,As: list_b,Bs: list_b,Cs: list_b] :
( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ! [Xa: list_b] :
( ( member_list_b @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_b @ ( set_b2 @ X4 ) @ ( set_b2 @ Xa ) )
= bot_bot_set_b ) ) ) )
=> ( ( member_list_b @ U2 @ Y6 )
=> ( ( member_list_b @ V3 @ Y6 )
=> ( ( V3 != nil_b )
=> ( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ( sublist_b @ X4 @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V3 @ Cs ) ) ) ) ) )
=> ! [X: list_b] :
( ( member_list_b @ X @ Y6 )
=> ( sublist_b @ X @ ( append_b @ As @ ( append_b @ Bs @ ( append_b @ U2 @ ( append_b @ V3 @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_UY_all
thf(fact_971_sublists__preserv__move__VY,axiom,
! [Y6: set_list_list_a,Xs: list_list_a,U2: list_list_a,V3: list_list_a,As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_list_a @ ( set_list_a2 @ X4 ) @ ( set_list_a2 @ Xa ) )
= bot_bot_set_list_a ) ) ) )
=> ( ( member_list_list_a @ Xs @ Y6 )
=> ( ( member_list_list_a @ U2 @ Y6 )
=> ( ( member_list_list_a @ V3 @ Y6 )
=> ( ( U2 != nil_list_a )
=> ( ( sublist_list_a @ Xs @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V3 @ Cs ) ) ) ) )
=> ( sublist_list_a @ Xs @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ V3 @ ( append_list_a @ Bs @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_VY
thf(fact_972_sublists__preserv__move__VY,axiom,
! [Y6: set_list_a,Xs: list_a,U2: list_a,V3: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_a @ ( set_a2 @ X4 ) @ ( set_a2 @ Xa ) )
= bot_bot_set_a ) ) ) )
=> ( ( member_list_a @ Xs @ Y6 )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( member_list_a @ V3 @ Y6 )
=> ( ( U2 != nil_a )
=> ( ( sublist_a @ Xs @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
=> ( sublist_a @ Xs @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ V3 @ ( append_a @ Bs @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_VY
thf(fact_973_sublists__preserv__move__VY,axiom,
! [Y6: set_list_b,Xs: list_b,U2: list_b,V3: list_b,As: list_b,Bs: list_b,Cs: list_b] :
( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ! [Xa: list_b] :
( ( member_list_b @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_b @ ( set_b2 @ X4 ) @ ( set_b2 @ Xa ) )
= bot_bot_set_b ) ) ) )
=> ( ( member_list_b @ Xs @ Y6 )
=> ( ( member_list_b @ U2 @ Y6 )
=> ( ( member_list_b @ V3 @ Y6 )
=> ( ( U2 != nil_b )
=> ( ( sublist_b @ Xs @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V3 @ Cs ) ) ) ) )
=> ( sublist_b @ Xs @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ V3 @ ( append_b @ Bs @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_VY
thf(fact_974_sublists__preserv__move__UY,axiom,
! [Y6: set_list_list_a,Xs: list_list_a,U2: list_list_a,V3: list_list_a,As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_list_a @ ( set_list_a2 @ X4 ) @ ( set_list_a2 @ Xa ) )
= bot_bot_set_list_a ) ) ) )
=> ( ( member_list_list_a @ Xs @ Y6 )
=> ( ( member_list_list_a @ U2 @ Y6 )
=> ( ( member_list_list_a @ V3 @ Y6 )
=> ( ( V3 != nil_list_a )
=> ( ( sublist_list_a @ Xs @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V3 @ Cs ) ) ) ) )
=> ( sublist_list_a @ Xs @ ( append_list_a @ As @ ( append_list_a @ Bs @ ( append_list_a @ U2 @ ( append_list_a @ V3 @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_UY
thf(fact_975_sublists__preserv__move__UY,axiom,
! [Y6: set_list_a,Xs: list_a,U2: list_a,V3: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_a @ ( set_a2 @ X4 ) @ ( set_a2 @ Xa ) )
= bot_bot_set_a ) ) ) )
=> ( ( member_list_a @ Xs @ Y6 )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( member_list_a @ V3 @ Y6 )
=> ( ( V3 != nil_a )
=> ( ( sublist_a @ Xs @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
=> ( sublist_a @ Xs @ ( append_a @ As @ ( append_a @ Bs @ ( append_a @ U2 @ ( append_a @ V3 @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_UY
thf(fact_976_sublists__preserv__move__UY,axiom,
! [Y6: set_list_b,Xs: list_b,U2: list_b,V3: list_b,As: list_b,Bs: list_b,Cs: list_b] :
( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ! [Xa: list_b] :
( ( member_list_b @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_b @ ( set_b2 @ X4 ) @ ( set_b2 @ Xa ) )
= bot_bot_set_b ) ) ) )
=> ( ( member_list_b @ Xs @ Y6 )
=> ( ( member_list_b @ U2 @ Y6 )
=> ( ( member_list_b @ V3 @ Y6 )
=> ( ( V3 != nil_b )
=> ( ( sublist_b @ Xs @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V3 @ Cs ) ) ) ) )
=> ( sublist_b @ Xs @ ( append_b @ As @ ( append_b @ Bs @ ( append_b @ U2 @ ( append_b @ V3 @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_UY
thf(fact_977_sublists__preserv__move__V,axiom,
! [Xs: list_list_a,U2: list_list_a,V3: list_list_a,As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ U2 ) )
= bot_bot_set_list_a )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ V3 ) )
= bot_bot_set_list_a )
=> ( ( U2 != nil_list_a )
=> ( ( sublist_list_a @ Xs @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V3 @ Cs ) ) ) ) )
=> ( sublist_list_a @ Xs @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ V3 @ ( append_list_a @ Bs @ Cs ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_V
thf(fact_978_sublists__preserv__move__V,axiom,
! [Xs: list_a,U2: list_a,V3: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ U2 ) )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ V3 ) )
= bot_bot_set_a )
=> ( ( U2 != nil_a )
=> ( ( sublist_a @ Xs @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
=> ( sublist_a @ Xs @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ V3 @ ( append_a @ Bs @ Cs ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_V
thf(fact_979_sublists__preserv__move__V,axiom,
! [Xs: list_b,U2: list_b,V3: list_b,As: list_b,Bs: list_b,Cs: list_b] :
( ( ( inf_inf_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ U2 ) )
= bot_bot_set_b )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ V3 ) )
= bot_bot_set_b )
=> ( ( U2 != nil_b )
=> ( ( sublist_b @ Xs @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V3 @ Cs ) ) ) ) )
=> ( sublist_b @ Xs @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ V3 @ ( append_b @ Bs @ Cs ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_V
thf(fact_980_sublists__preserv__move__U,axiom,
! [Xs: list_list_a,U2: list_list_a,V3: list_list_a,As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ U2 ) )
= bot_bot_set_list_a )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ V3 ) )
= bot_bot_set_list_a )
=> ( ( V3 != nil_list_a )
=> ( ( sublist_list_a @ Xs @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V3 @ Cs ) ) ) ) )
=> ( sublist_list_a @ Xs @ ( append_list_a @ As @ ( append_list_a @ Bs @ ( append_list_a @ U2 @ ( append_list_a @ V3 @ Cs ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_U
thf(fact_981_sublists__preserv__move__U,axiom,
! [Xs: list_a,U2: list_a,V3: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ U2 ) )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ V3 ) )
= bot_bot_set_a )
=> ( ( V3 != nil_a )
=> ( ( sublist_a @ Xs @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
=> ( sublist_a @ Xs @ ( append_a @ As @ ( append_a @ Bs @ ( append_a @ U2 @ ( append_a @ V3 @ Cs ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_U
thf(fact_982_sublists__preserv__move__U,axiom,
! [Xs: list_b,U2: list_b,V3: list_b,As: list_b,Bs: list_b,Cs: list_b] :
( ( ( inf_inf_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ U2 ) )
= bot_bot_set_b )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ V3 ) )
= bot_bot_set_b )
=> ( ( V3 != nil_b )
=> ( ( sublist_b @ Xs @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V3 @ Cs ) ) ) ) )
=> ( sublist_b @ Xs @ ( append_b @ As @ ( append_b @ Bs @ ( append_b @ U2 @ ( append_b @ V3 @ Cs ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_U
thf(fact_983_sublist__before__if__mid,axiom,
! [U2: list_list_a,A2: list_list_a,V3: list_list_a,B3: list_list_a,Xs: list_list_a] :
( ( sublist_list_a @ U2 @ ( append_list_a @ A2 @ V3 ) )
=> ( ( ( append_list_a @ A2 @ ( append_list_a @ V3 @ B3 ) )
= Xs )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ U2 ) @ ( set_list_a2 @ V3 ) )
= bot_bot_set_list_a )
=> ( ( U2 != nil_list_a )
=> ? [As3: list_list_a,Bs3: list_list_a,Cs3: list_list_a] :
( ( append_list_a @ As3 @ ( append_list_a @ U2 @ ( append_list_a @ Bs3 @ ( append_list_a @ V3 @ Cs3 ) ) ) )
= Xs ) ) ) ) ) ).
% sublist_before_if_mid
thf(fact_984_sublist__before__if__mid,axiom,
! [U2: list_a,A2: list_a,V3: list_a,B3: list_a,Xs: list_a] :
( ( sublist_a @ U2 @ ( append_a @ A2 @ V3 ) )
=> ( ( ( append_a @ A2 @ ( append_a @ V3 @ B3 ) )
= Xs )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ V3 ) )
= bot_bot_set_a )
=> ( ( U2 != nil_a )
=> ? [As3: list_a,Bs3: list_a,Cs3: list_a] :
( ( append_a @ As3 @ ( append_a @ U2 @ ( append_a @ Bs3 @ ( append_a @ V3 @ Cs3 ) ) ) )
= Xs ) ) ) ) ) ).
% sublist_before_if_mid
thf(fact_985_sublist__before__if__mid,axiom,
! [U2: list_b,A2: list_b,V3: list_b,B3: list_b,Xs: list_b] :
( ( sublist_b @ U2 @ ( append_b @ A2 @ V3 ) )
=> ( ( ( append_b @ A2 @ ( append_b @ V3 @ B3 ) )
= Xs )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ U2 ) @ ( set_b2 @ V3 ) )
= bot_bot_set_b )
=> ( ( U2 != nil_b )
=> ? [As3: list_b,Bs3: list_b,Cs3: list_b] :
( ( append_b @ As3 @ ( append_b @ U2 @ ( append_b @ Bs3 @ ( append_b @ V3 @ Cs3 ) ) ) )
= Xs ) ) ) ) ) ).
% sublist_before_if_mid
thf(fact_986_sublist__Y__cases__UV,axiom,
! [Y6: set_list_list_a,U2: list_list_a,V3: list_list_a,As: list_list_a,Bs: list_list_a,Cs: list_list_a,Xs: list_list_a] :
( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_list_a @ ( set_list_a2 @ X4 ) @ ( set_list_a2 @ Xa ) )
= bot_bot_set_list_a ) ) ) )
=> ( ( member_list_list_a @ U2 @ Y6 )
=> ( ( member_list_list_a @ V3 @ Y6 )
=> ( ( U2 != nil_list_a )
=> ( ( V3 != nil_list_a )
=> ( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ( sublist_list_a @ X4 @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V3 @ Cs ) ) ) ) ) )
=> ( ( member_list_list_a @ Xs @ Y6 )
=> ( ( sublist_list_a @ Xs @ As )
| ( sublist_list_a @ Xs @ Bs )
| ( sublist_list_a @ Xs @ Cs )
| ( U2 = Xs )
| ( V3 = Xs ) ) ) ) ) ) ) ) ) ).
% sublist_Y_cases_UV
thf(fact_987_sublist__Y__cases__UV,axiom,
! [Y6: set_list_a,U2: list_a,V3: list_a,As: list_a,Bs: list_a,Cs: list_a,Xs: list_a] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_a @ ( set_a2 @ X4 ) @ ( set_a2 @ Xa ) )
= bot_bot_set_a ) ) ) )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( member_list_a @ V3 @ Y6 )
=> ( ( U2 != nil_a )
=> ( ( V3 != nil_a )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ( sublist_a @ X4 @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) ) )
=> ( ( member_list_a @ Xs @ Y6 )
=> ( ( sublist_a @ Xs @ As )
| ( sublist_a @ Xs @ Bs )
| ( sublist_a @ Xs @ Cs )
| ( U2 = Xs )
| ( V3 = Xs ) ) ) ) ) ) ) ) ) ).
% sublist_Y_cases_UV
thf(fact_988_sublist__Y__cases__UV,axiom,
! [Y6: set_list_b,U2: list_b,V3: list_b,As: list_b,Bs: list_b,Cs: list_b,Xs: list_b] :
( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ! [Xa: list_b] :
( ( member_list_b @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_b @ ( set_b2 @ X4 ) @ ( set_b2 @ Xa ) )
= bot_bot_set_b ) ) ) )
=> ( ( member_list_b @ U2 @ Y6 )
=> ( ( member_list_b @ V3 @ Y6 )
=> ( ( U2 != nil_b )
=> ( ( V3 != nil_b )
=> ( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ( sublist_b @ X4 @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V3 @ Cs ) ) ) ) ) )
=> ( ( member_list_b @ Xs @ Y6 )
=> ( ( sublist_b @ Xs @ As )
| ( sublist_b @ Xs @ Bs )
| ( sublist_b @ Xs @ Cs )
| ( U2 = Xs )
| ( V3 = Xs ) ) ) ) ) ) ) ) ) ).
% sublist_Y_cases_UV
thf(fact_989_sublist__not__mid,axiom,
! [U2: list_list_a,A2: list_list_a,V3: list_list_a,B3: list_list_a] :
( ( sublist_list_a @ U2 @ ( append_list_a @ ( append_list_a @ A2 @ V3 ) @ B3 ) )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ U2 ) @ ( set_list_a2 @ V3 ) )
= bot_bot_set_list_a )
=> ( ( V3 != nil_list_a )
=> ( ( sublist_list_a @ U2 @ A2 )
| ( sublist_list_a @ U2 @ B3 ) ) ) ) ) ).
% sublist_not_mid
thf(fact_990_sublist__not__mid,axiom,
! [U2: list_a,A2: list_a,V3: list_a,B3: list_a] :
( ( sublist_a @ U2 @ ( append_a @ ( append_a @ A2 @ V3 ) @ B3 ) )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ V3 ) )
= bot_bot_set_a )
=> ( ( V3 != nil_a )
=> ( ( sublist_a @ U2 @ A2 )
| ( sublist_a @ U2 @ B3 ) ) ) ) ) ).
% sublist_not_mid
thf(fact_991_sublist__not__mid,axiom,
! [U2: list_b,A2: list_b,V3: list_b,B3: list_b] :
( ( sublist_b @ U2 @ ( append_b @ ( append_b @ A2 @ V3 ) @ B3 ) )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ U2 ) @ ( set_b2 @ V3 ) )
= bot_bot_set_b )
=> ( ( V3 != nil_b )
=> ( ( sublist_b @ U2 @ A2 )
| ( sublist_b @ U2 @ B3 ) ) ) ) ) ).
% sublist_not_mid
thf(fact_992_subset__snd__if__hd__notin__fst,axiom,
! [Ys: list_P1396940483166286381od_a_a,V3: list_P1396940483166286381od_a_a,B3: list_P1396940483166286381od_a_a] :
( ( sublis713342594703834650od_a_a @ Ys @ ( append5335208819046833346od_a_a @ V3 @ B3 ) )
=> ( ~ ( member1426531477525435216od_a_a @ ( hd_Product_prod_a_a @ Ys ) @ ( set_Product_prod_a_a2 @ V3 ) )
=> ( ( Ys != nil_Product_prod_a_a )
=> ( ord_le746702958409616551od_a_a @ ( set_Product_prod_a_a2 @ Ys ) @ ( set_Product_prod_a_a2 @ B3 ) ) ) ) ) ).
% subset_snd_if_hd_notin_fst
thf(fact_993_subset__snd__if__hd__notin__fst,axiom,
! [Ys: list_set_a,V3: list_set_a,B3: list_set_a] :
( ( sublist_set_a @ Ys @ ( append_set_a @ V3 @ B3 ) )
=> ( ~ ( member_set_a @ ( hd_set_a @ Ys ) @ ( set_set_a2 @ V3 ) )
=> ( ( Ys != nil_set_a )
=> ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Ys ) @ ( set_set_a2 @ B3 ) ) ) ) ) ).
% subset_snd_if_hd_notin_fst
thf(fact_994_subset__snd__if__hd__notin__fst,axiom,
! [Ys: list_list_a,V3: list_list_a,B3: list_list_a] :
( ( sublist_list_a @ Ys @ ( append_list_a @ V3 @ B3 ) )
=> ( ~ ( member_list_a @ ( hd_list_a @ Ys ) @ ( set_list_a2 @ V3 ) )
=> ( ( Ys != nil_list_a )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ys ) @ ( set_list_a2 @ B3 ) ) ) ) ) ).
% subset_snd_if_hd_notin_fst
thf(fact_995_subset__snd__if__hd__notin__fst,axiom,
! [Ys: list_a,V3: list_a,B3: list_a] :
( ( sublist_a @ Ys @ ( append_a @ V3 @ B3 ) )
=> ( ~ ( member_a @ ( hd_a @ Ys ) @ ( set_a2 @ V3 ) )
=> ( ( Ys != nil_a )
=> ( ord_less_eq_set_a @ ( set_a2 @ Ys ) @ ( set_a2 @ B3 ) ) ) ) ) ).
% subset_snd_if_hd_notin_fst
thf(fact_996_subset__snd__if__hd__notin__fst,axiom,
! [Ys: list_b,V3: list_b,B3: list_b] :
( ( sublist_b @ Ys @ ( append_b @ V3 @ B3 ) )
=> ( ~ ( member_b @ ( hd_b @ Ys ) @ ( set_b2 @ V3 ) )
=> ( ( Ys != nil_b )
=> ( ord_less_eq_set_b @ ( set_b2 @ Ys ) @ ( set_b2 @ B3 ) ) ) ) ) ).
% subset_snd_if_hd_notin_fst
thf(fact_997_vwalkI__append__l,axiom,
! [P2: list_a,Q2: list_a,G: pre_pr7278220950009878019t_unit] :
( ( P2 != nil_a )
=> ( ( vertex_vwalk_a_b @ ( append_a @ P2 @ Q2 ) @ G )
=> ( vertex_vwalk_a_b @ P2 @ G ) ) ) ).
% vwalkI_append_l
thf(fact_998_vwalkI__append__r,axiom,
! [Q2: list_a,P2: list_a,G: pre_pr7278220950009878019t_unit] :
( ( Q2 != nil_a )
=> ( ( vertex_vwalk_a_b @ ( append_a @ P2 @ Q2 ) @ G )
=> ( vertex_vwalk_a_b @ Q2 @ G ) ) ) ).
% vwalkI_append_r
thf(fact_999_vwalk__verts__in__verts,axiom,
! [P2: list_a,G: pre_pr7278220950009878019t_unit,U: a] :
( ( vertex_vwalk_a_b @ P2 @ G )
=> ( ( member_a @ U @ ( set_a2 @ P2 ) )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ G ) ) ) ) ).
% vwalk_verts_in_verts
thf(fact_1000_vwalk__verts__in__verts,axiom,
! [P2: list_list_a,G: pre_pr2882871181989701257t_unit,U: list_a] :
( ( vertex2966258834163962945st_a_b @ P2 @ G )
=> ( ( member_list_a @ U @ ( set_list_a2 @ P2 ) )
=> ( member_list_a @ U @ ( pre_ve1830060048215441954t_unit @ G ) ) ) ) ).
% vwalk_verts_in_verts
thf(fact_1001_vpath__self,axiom,
! [U: a,G: pre_pr7278220950009878019t_unit] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( vertex_vpath_a_b @ ( cons_a @ U @ nil_a ) @ G ) ) ).
% vpath_self
thf(fact_1002_vpath__self,axiom,
! [U: list_a,G: pre_pr2882871181989701257t_unit] :
( ( member_list_a @ U @ ( pre_ve1830060048215441954t_unit @ G ) )
=> ( vertex6060786982766068989st_a_b @ ( cons_list_a @ U @ nil_list_a ) @ G ) ) ).
% vpath_self
thf(fact_1003_vwalk__induct,axiom,
! [P2: list_a,G: pre_pr7278220950009878019t_unit,P: list_a > $o] :
( ( vertex_vwalk_a_b @ P2 @ G )
=> ( ! [U3: a] :
( ( member_a @ U3 @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( P @ ( cons_a @ U3 @ nil_a ) ) )
=> ( ! [U3: a,V4: a,Es: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U3 @ V4 ) @ ( arcs_ends_a_b @ G ) )
=> ( ( P @ ( cons_a @ V4 @ Es ) )
=> ( P @ ( cons_a @ U3 @ ( cons_a @ V4 @ Es ) ) ) ) )
=> ( P @ P2 ) ) ) ) ).
% vwalk_induct
thf(fact_1004_vwalk__induct,axiom,
! [P2: list_list_a,G: pre_pr2882871181989701257t_unit,P: list_list_a > $o] :
( ( vertex2966258834163962945st_a_b @ P2 @ G )
=> ( ! [U3: list_a] :
( ( member_list_a @ U3 @ ( pre_ve1830060048215441954t_unit @ G ) )
=> ( P @ ( cons_list_a @ U3 @ nil_list_a ) ) )
=> ( ! [U3: list_a,V4: list_a,Es: list_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ U3 @ V4 ) @ ( arcs_ends_list_a_b @ G ) )
=> ( ( P @ ( cons_list_a @ V4 @ Es ) )
=> ( P @ ( cons_list_a @ U3 @ ( cons_list_a @ V4 @ Es ) ) ) ) )
=> ( P @ P2 ) ) ) ) ).
% vwalk_induct
thf(fact_1005_concat__split__mid,axiom,
! [Y6: set_list_list_a,U2: list_list_a,As: list_list_a,Bs: list_list_a] :
( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_list_a @ ( set_list_a2 @ X4 ) @ ( set_list_a2 @ Xa ) )
= bot_bot_set_list_a ) ) ) )
=> ( ( finite1660835950917165235list_a @ Y6 )
=> ( ( member_list_list_a @ U2 @ Y6 )
=> ( ( distinct_list_a @ ( append_list_a @ As @ ( append_list_a @ U2 @ Bs ) ) )
=> ( ( ( set_list_a2 @ ( append_list_a @ As @ ( append_list_a @ U2 @ Bs ) ) )
= ( comple6928918032620976721list_a @ ( image_432481560377026271list_a @ set_list_a2 @ Y6 ) ) )
=> ( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ( sublist_list_a @ X4 @ ( append_list_a @ As @ ( append_list_a @ U2 @ Bs ) ) ) )
=> ( ( U2 != nil_list_a )
=> ? [Cs3: list_list_list_a] :
( ( ( concat_list_a @ Cs3 )
= As )
& ? [Ds2: list_list_list_a] :
( ( ( concat_list_a @ Ds2 )
= Bs )
& ( ( set_list_list_a2 @ ( append_list_list_a @ Cs3 @ ( cons_list_list_a @ U2 @ Ds2 ) ) )
= Y6 )
& ( distinct_list_list_a @ ( append_list_list_a @ Cs3 @ ( cons_list_list_a @ U2 @ Ds2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% concat_split_mid
thf(fact_1006_concat__split__mid,axiom,
! [Y6: set_list_b,U2: list_b,As: list_b,Bs: list_b] :
( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ! [Xa: list_b] :
( ( member_list_b @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_b @ ( set_b2 @ X4 ) @ ( set_b2 @ Xa ) )
= bot_bot_set_b ) ) ) )
=> ( ( finite_finite_list_b @ Y6 )
=> ( ( member_list_b @ U2 @ Y6 )
=> ( ( distinct_b @ ( append_b @ As @ ( append_b @ U2 @ Bs ) ) )
=> ( ( ( set_b2 @ ( append_b @ As @ ( append_b @ U2 @ Bs ) ) )
= ( comple2307003614231284044_set_b @ ( image_list_b_set_b @ set_b2 @ Y6 ) ) )
=> ( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ( sublist_b @ X4 @ ( append_b @ As @ ( append_b @ U2 @ Bs ) ) ) )
=> ( ( U2 != nil_b )
=> ? [Cs3: list_list_b] :
( ( ( concat_b @ Cs3 )
= As )
& ? [Ds2: list_list_b] :
( ( ( concat_b @ Ds2 )
= Bs )
& ( ( set_list_b2 @ ( append_list_b @ Cs3 @ ( cons_list_b @ U2 @ Ds2 ) ) )
= Y6 )
& ( distinct_list_b @ ( append_list_b @ Cs3 @ ( cons_list_b @ U2 @ Ds2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% concat_split_mid
thf(fact_1007_concat__split__mid,axiom,
! [Y6: set_list_a,U2: list_a,As: list_a,Bs: list_a] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_a @ ( set_a2 @ X4 ) @ ( set_a2 @ Xa ) )
= bot_bot_set_a ) ) ) )
=> ( ( finite_finite_list_a @ Y6 )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( distinct_a @ ( append_a @ As @ ( append_a @ U2 @ Bs ) ) )
=> ( ( ( set_a2 @ ( append_a @ As @ ( append_a @ U2 @ Bs ) ) )
= ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ set_a2 @ Y6 ) ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ( sublist_a @ X4 @ ( append_a @ As @ ( append_a @ U2 @ Bs ) ) ) )
=> ( ( U2 != nil_a )
=> ? [Cs3: list_list_a] :
( ( ( concat_a @ Cs3 )
= As )
& ? [Ds2: list_list_a] :
( ( ( concat_a @ Ds2 )
= Bs )
& ( ( set_list_a2 @ ( append_list_a @ Cs3 @ ( cons_list_a @ U2 @ Ds2 ) ) )
= Y6 )
& ( distinct_list_a @ ( append_list_a @ Cs3 @ ( cons_list_a @ U2 @ Ds2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% concat_split_mid
thf(fact_1008_concat__split__UV,axiom,
! [Y6: set_list_list_a,U2: list_list_a,V3: list_list_a,As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_list_a @ ( set_list_a2 @ X4 ) @ ( set_list_a2 @ Xa ) )
= bot_bot_set_list_a ) ) ) )
=> ( ( finite1660835950917165235list_a @ Y6 )
=> ( ( member_list_list_a @ U2 @ Y6 )
=> ( ( member_list_list_a @ V3 @ Y6 )
=> ( ( distinct_list_a @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V3 @ Cs ) ) ) ) )
=> ( ( ( set_list_a2 @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V3 @ Cs ) ) ) ) )
= ( comple6928918032620976721list_a @ ( image_432481560377026271list_a @ set_list_a2 @ Y6 ) ) )
=> ( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ( sublist_list_a @ X4 @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V3 @ Cs ) ) ) ) ) )
=> ( ( U2 != nil_list_a )
=> ( ( V3 != nil_list_a )
=> ? [As4: list_list_list_a] :
( ( ( concat_list_a @ As4 )
= As )
& ? [Bs4: list_list_list_a] :
( ( ( concat_list_a @ Bs4 )
= Bs )
& ? [Cs4: list_list_list_a] :
( ( ( concat_list_a @ Cs4 )
= Cs )
& ( ( set_list_list_a2 @ ( append_list_list_a @ As4 @ ( cons_list_list_a @ U2 @ ( append_list_list_a @ Bs4 @ ( cons_list_list_a @ V3 @ Cs4 ) ) ) ) )
= Y6 )
& ( distinct_list_list_a @ ( append_list_list_a @ As4 @ ( cons_list_list_a @ U2 @ ( append_list_list_a @ Bs4 @ ( cons_list_list_a @ V3 @ Cs4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% concat_split_UV
thf(fact_1009_concat__split__UV,axiom,
! [Y6: set_list_b,U2: list_b,V3: list_b,As: list_b,Bs: list_b,Cs: list_b] :
( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ! [Xa: list_b] :
( ( member_list_b @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_b @ ( set_b2 @ X4 ) @ ( set_b2 @ Xa ) )
= bot_bot_set_b ) ) ) )
=> ( ( finite_finite_list_b @ Y6 )
=> ( ( member_list_b @ U2 @ Y6 )
=> ( ( member_list_b @ V3 @ Y6 )
=> ( ( distinct_b @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V3 @ Cs ) ) ) ) )
=> ( ( ( set_b2 @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V3 @ Cs ) ) ) ) )
= ( comple2307003614231284044_set_b @ ( image_list_b_set_b @ set_b2 @ Y6 ) ) )
=> ( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ( sublist_b @ X4 @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V3 @ Cs ) ) ) ) ) )
=> ( ( U2 != nil_b )
=> ( ( V3 != nil_b )
=> ? [As4: list_list_b] :
( ( ( concat_b @ As4 )
= As )
& ? [Bs4: list_list_b] :
( ( ( concat_b @ Bs4 )
= Bs )
& ? [Cs4: list_list_b] :
( ( ( concat_b @ Cs4 )
= Cs )
& ( ( set_list_b2 @ ( append_list_b @ As4 @ ( cons_list_b @ U2 @ ( append_list_b @ Bs4 @ ( cons_list_b @ V3 @ Cs4 ) ) ) ) )
= Y6 )
& ( distinct_list_b @ ( append_list_b @ As4 @ ( cons_list_b @ U2 @ ( append_list_b @ Bs4 @ ( cons_list_b @ V3 @ Cs4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% concat_split_UV
thf(fact_1010_concat__split__UV,axiom,
! [Y6: set_list_a,U2: list_a,V3: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_a @ ( set_a2 @ X4 ) @ ( set_a2 @ Xa ) )
= bot_bot_set_a ) ) ) )
=> ( ( finite_finite_list_a @ Y6 )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( member_list_a @ V3 @ Y6 )
=> ( ( distinct_a @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
=> ( ( ( set_a2 @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
= ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ set_a2 @ Y6 ) ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ( sublist_a @ X4 @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) ) )
=> ( ( U2 != nil_a )
=> ( ( V3 != nil_a )
=> ? [As4: list_list_a] :
( ( ( concat_a @ As4 )
= As )
& ? [Bs4: list_list_a] :
( ( ( concat_a @ Bs4 )
= Bs )
& ? [Cs4: list_list_a] :
( ( ( concat_a @ Cs4 )
= Cs )
& ( ( set_list_a2 @ ( append_list_a @ As4 @ ( cons_list_a @ U2 @ ( append_list_a @ Bs4 @ ( cons_list_a @ V3 @ Cs4 ) ) ) ) )
= Y6 )
& ( distinct_list_a @ ( append_list_a @ As4 @ ( cons_list_a @ U2 @ ( append_list_a @ Bs4 @ ( cons_list_a @ V3 @ Cs4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% concat_split_UV
thf(fact_1011_mid__all__sublists__set2,axiom,
! [Y6: set_list_list_a,U2: list_list_a,As: list_list_a,Bs: list_list_a] :
( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_list_a @ ( set_list_a2 @ X4 ) @ ( set_list_a2 @ Xa ) )
= bot_bot_set_list_a ) ) ) )
=> ( ( finite1660835950917165235list_a @ Y6 )
=> ( ( member_list_list_a @ U2 @ Y6 )
=> ( ( distinct_list_a @ ( append_list_a @ As @ ( append_list_a @ U2 @ Bs ) ) )
=> ( ( ( set_list_a2 @ ( append_list_a @ As @ ( append_list_a @ U2 @ Bs ) ) )
= ( comple6928918032620976721list_a @ ( image_432481560377026271list_a @ set_list_a2 @ Y6 ) ) )
=> ( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ( sublist_list_a @ X4 @ ( append_list_a @ As @ ( append_list_a @ U2 @ Bs ) ) ) )
=> ( ( U2 != nil_list_a )
=> ? [X7: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ X7 @ Y6 )
& ( ( set_list_a2 @ Bs )
= ( comple6928918032620976721list_a @ ( image_432481560377026271list_a @ set_list_a2 @ X7 ) ) )
& ! [X: list_list_a] :
( ( member_list_list_a @ X @ X7 )
=> ( sublist_list_a @ X @ Bs ) ) ) ) ) ) ) ) ) ) ).
% mid_all_sublists_set2
thf(fact_1012_mid__all__sublists__set2,axiom,
! [Y6: set_list_b,U2: list_b,As: list_b,Bs: list_b] :
( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ! [Xa: list_b] :
( ( member_list_b @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_b @ ( set_b2 @ X4 ) @ ( set_b2 @ Xa ) )
= bot_bot_set_b ) ) ) )
=> ( ( finite_finite_list_b @ Y6 )
=> ( ( member_list_b @ U2 @ Y6 )
=> ( ( distinct_b @ ( append_b @ As @ ( append_b @ U2 @ Bs ) ) )
=> ( ( ( set_b2 @ ( append_b @ As @ ( append_b @ U2 @ Bs ) ) )
= ( comple2307003614231284044_set_b @ ( image_list_b_set_b @ set_b2 @ Y6 ) ) )
=> ( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ( sublist_b @ X4 @ ( append_b @ As @ ( append_b @ U2 @ Bs ) ) ) )
=> ( ( U2 != nil_b )
=> ? [X7: set_list_b] :
( ( ord_le8932221534207217157list_b @ X7 @ Y6 )
& ( ( set_b2 @ Bs )
= ( comple2307003614231284044_set_b @ ( image_list_b_set_b @ set_b2 @ X7 ) ) )
& ! [X: list_b] :
( ( member_list_b @ X @ X7 )
=> ( sublist_b @ X @ Bs ) ) ) ) ) ) ) ) ) ) ).
% mid_all_sublists_set2
thf(fact_1013_mid__all__sublists__set2,axiom,
! [Y6: set_list_a,U2: list_a,As: list_a,Bs: list_a] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_a @ ( set_a2 @ X4 ) @ ( set_a2 @ Xa ) )
= bot_bot_set_a ) ) ) )
=> ( ( finite_finite_list_a @ Y6 )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( distinct_a @ ( append_a @ As @ ( append_a @ U2 @ Bs ) ) )
=> ( ( ( set_a2 @ ( append_a @ As @ ( append_a @ U2 @ Bs ) ) )
= ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ set_a2 @ Y6 ) ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ( sublist_a @ X4 @ ( append_a @ As @ ( append_a @ U2 @ Bs ) ) ) )
=> ( ( U2 != nil_a )
=> ? [X7: set_list_a] :
( ( ord_le8861187494160871172list_a @ X7 @ Y6 )
& ( ( set_a2 @ Bs )
= ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ set_a2 @ X7 ) ) )
& ! [X: list_a] :
( ( member_list_a @ X @ X7 )
=> ( sublist_a @ X @ Bs ) ) ) ) ) ) ) ) ) ) ).
% mid_all_sublists_set2
thf(fact_1014_mid__all__sublists__set1,axiom,
! [Y6: set_list_list_a,U2: list_list_a,As: list_list_a,Bs: list_list_a] :
( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_list_a @ ( set_list_a2 @ X4 ) @ ( set_list_a2 @ Xa ) )
= bot_bot_set_list_a ) ) ) )
=> ( ( finite1660835950917165235list_a @ Y6 )
=> ( ( member_list_list_a @ U2 @ Y6 )
=> ( ( distinct_list_a @ ( append_list_a @ As @ ( append_list_a @ U2 @ Bs ) ) )
=> ( ( ( set_list_a2 @ ( append_list_a @ As @ ( append_list_a @ U2 @ Bs ) ) )
= ( comple6928918032620976721list_a @ ( image_432481560377026271list_a @ set_list_a2 @ Y6 ) ) )
=> ( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ( sublist_list_a @ X4 @ ( append_list_a @ As @ ( append_list_a @ U2 @ Bs ) ) ) )
=> ( ( U2 != nil_list_a )
=> ? [X7: set_list_list_a] :
( ( ord_le8488217952732425610list_a @ X7 @ Y6 )
& ( ( set_list_a2 @ As )
= ( comple6928918032620976721list_a @ ( image_432481560377026271list_a @ set_list_a2 @ X7 ) ) )
& ! [X: list_list_a] :
( ( member_list_list_a @ X @ X7 )
=> ( sublist_list_a @ X @ As ) ) ) ) ) ) ) ) ) ) ).
% mid_all_sublists_set1
thf(fact_1015_mid__all__sublists__set1,axiom,
! [Y6: set_list_b,U2: list_b,As: list_b,Bs: list_b] :
( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ! [Xa: list_b] :
( ( member_list_b @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_b @ ( set_b2 @ X4 ) @ ( set_b2 @ Xa ) )
= bot_bot_set_b ) ) ) )
=> ( ( finite_finite_list_b @ Y6 )
=> ( ( member_list_b @ U2 @ Y6 )
=> ( ( distinct_b @ ( append_b @ As @ ( append_b @ U2 @ Bs ) ) )
=> ( ( ( set_b2 @ ( append_b @ As @ ( append_b @ U2 @ Bs ) ) )
= ( comple2307003614231284044_set_b @ ( image_list_b_set_b @ set_b2 @ Y6 ) ) )
=> ( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ( sublist_b @ X4 @ ( append_b @ As @ ( append_b @ U2 @ Bs ) ) ) )
=> ( ( U2 != nil_b )
=> ? [X7: set_list_b] :
( ( ord_le8932221534207217157list_b @ X7 @ Y6 )
& ( ( set_b2 @ As )
= ( comple2307003614231284044_set_b @ ( image_list_b_set_b @ set_b2 @ X7 ) ) )
& ! [X: list_b] :
( ( member_list_b @ X @ X7 )
=> ( sublist_b @ X @ As ) ) ) ) ) ) ) ) ) ) ).
% mid_all_sublists_set1
thf(fact_1016_mid__all__sublists__set1,axiom,
! [Y6: set_list_a,U2: list_a,As: list_a,Bs: list_a] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_a @ ( set_a2 @ X4 ) @ ( set_a2 @ Xa ) )
= bot_bot_set_a ) ) ) )
=> ( ( finite_finite_list_a @ Y6 )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( distinct_a @ ( append_a @ As @ ( append_a @ U2 @ Bs ) ) )
=> ( ( ( set_a2 @ ( append_a @ As @ ( append_a @ U2 @ Bs ) ) )
= ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ set_a2 @ Y6 ) ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ( sublist_a @ X4 @ ( append_a @ As @ ( append_a @ U2 @ Bs ) ) ) )
=> ( ( U2 != nil_a )
=> ? [X7: set_list_a] :
( ( ord_le8861187494160871172list_a @ X7 @ Y6 )
& ( ( set_a2 @ As )
= ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ set_a2 @ X7 ) ) )
& ! [X: list_a] :
( ( member_list_a @ X @ X7 )
=> ( sublist_a @ X @ As ) ) ) ) ) ) ) ) ) ) ).
% mid_all_sublists_set1
thf(fact_1017_sublist__append__rightI,axiom,
! [Xs: list_a,Ss: list_a] : ( sublist_a @ Xs @ ( append_a @ Xs @ Ss ) ) ).
% sublist_append_rightI
thf(fact_1018_sublist__append__rightI,axiom,
! [Xs: list_b,Ss: list_b] : ( sublist_b @ Xs @ ( append_b @ Xs @ Ss ) ) ).
% sublist_append_rightI
thf(fact_1019_List_Ofinite__set,axiom,
! [Xs: list_list_a] : ( finite_finite_list_a @ ( set_list_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_1020_List_Ofinite__set,axiom,
! [Xs: list_a] : ( finite_finite_a @ ( set_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_1021_List_Ofinite__set,axiom,
! [Xs: list_b] : ( finite_finite_b @ ( set_b2 @ Xs ) ) ).
% List.finite_set
thf(fact_1022_sublist__appendI,axiom,
! [Xs: list_a,Ps2: list_a,Ss: list_a] : ( sublist_a @ Xs @ ( append_a @ Ps2 @ ( append_a @ Xs @ Ss ) ) ) ).
% sublist_appendI
thf(fact_1023_sublist__appendI,axiom,
! [Xs: list_b,Ps2: list_b,Ss: list_b] : ( sublist_b @ Xs @ ( append_b @ Ps2 @ ( append_b @ Xs @ Ss ) ) ) ).
% sublist_appendI
thf(fact_1024_sublist__append__leftI,axiom,
! [Xs: list_a,Ps2: list_a] : ( sublist_a @ Xs @ ( append_a @ Ps2 @ Xs ) ) ).
% sublist_append_leftI
thf(fact_1025_sublist__append__leftI,axiom,
! [Xs: list_b,Ps2: list_b] : ( sublist_b @ Xs @ ( append_b @ Ps2 @ Xs ) ) ).
% sublist_append_leftI
thf(fact_1026_finite__list,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ? [Xs2: list_list_a] :
( ( set_list_a2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_1027_finite__list,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ? [Xs2: list_a] :
( ( set_a2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_1028_finite__list,axiom,
! [A2: set_b] :
( ( finite_finite_b @ A2 )
=> ? [Xs2: list_b] :
( ( set_b2 @ Xs2 )
= A2 ) ) ).
% finite_list
thf(fact_1029_finite__distinct__list,axiom,
! [A2: set_list_a] :
( ( finite_finite_list_a @ A2 )
=> ? [Xs2: list_list_a] :
( ( ( set_list_a2 @ Xs2 )
= A2 )
& ( distinct_list_a @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_1030_finite__distinct__list,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ? [Xs2: list_a] :
( ( ( set_a2 @ Xs2 )
= A2 )
& ( distinct_a @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_1031_finite__distinct__list,axiom,
! [A2: set_b] :
( ( finite_finite_b @ A2 )
=> ? [Xs2: list_b] :
( ( ( set_b2 @ Xs2 )
= A2 )
& ( distinct_b @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_1032_valid__UV__lists__argmin__ex__noP,axiom,
! [Xs: set_list_a,U2: list_list_a,V3: list_list_a,Cost: list_list_a > real] :
( ( finite_finite_list_a @ Xs )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ U2 ) @ ( set_list_a2 @ V3 ) )
= bot_bot_set_list_a )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ U2 ) @ Xs )
=> ( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ V3 ) @ Xs )
=> ( ( distinct_list_a @ U2 )
=> ( ( distinct_list_a @ V3 )
=> ? [As4: list_list_a,Bs4: list_list_a,Cs4: list_list_a] :
( ( ( set_list_a2 @ ( append_list_a @ As4 @ ( append_list_a @ U2 @ ( append_list_a @ Bs4 @ ( append_list_a @ V3 @ Cs4 ) ) ) ) )
= Xs )
& ( distinct_list_a @ ( append_list_a @ As4 @ ( append_list_a @ U2 @ ( append_list_a @ Bs4 @ ( append_list_a @ V3 @ Cs4 ) ) ) ) )
& ! [As5: list_list_a,Bs5: list_list_a,Cs5: list_list_a] :
( ( ( ( set_list_a2 @ ( append_list_a @ As5 @ ( append_list_a @ U2 @ ( append_list_a @ Bs5 @ ( append_list_a @ V3 @ Cs5 ) ) ) ) )
= Xs )
& ( distinct_list_a @ ( append_list_a @ As5 @ ( append_list_a @ U2 @ ( append_list_a @ Bs5 @ ( append_list_a @ V3 @ Cs5 ) ) ) ) ) )
=> ( ord_less_eq_real @ ( Cost @ ( append_list_a @ As4 @ ( append_list_a @ U2 @ ( append_list_a @ Bs4 @ ( append_list_a @ V3 @ Cs4 ) ) ) ) ) @ ( Cost @ ( append_list_a @ As5 @ ( append_list_a @ U2 @ ( append_list_a @ Bs5 @ ( append_list_a @ V3 @ Cs5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% valid_UV_lists_argmin_ex_noP
thf(fact_1033_valid__UV__lists__argmin__ex__noP,axiom,
! [Xs: set_a,U2: list_a,V3: list_a,Cost: list_a > real] :
( ( finite_finite_a @ Xs )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ V3 ) )
= bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ U2 ) @ Xs )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ V3 ) @ Xs )
=> ( ( distinct_a @ U2 )
=> ( ( distinct_a @ V3 )
=> ? [As4: list_a,Bs4: list_a,Cs4: list_a] :
( ( ( set_a2 @ ( append_a @ As4 @ ( append_a @ U2 @ ( append_a @ Bs4 @ ( append_a @ V3 @ Cs4 ) ) ) ) )
= Xs )
& ( distinct_a @ ( append_a @ As4 @ ( append_a @ U2 @ ( append_a @ Bs4 @ ( append_a @ V3 @ Cs4 ) ) ) ) )
& ! [As5: list_a,Bs5: list_a,Cs5: list_a] :
( ( ( ( set_a2 @ ( append_a @ As5 @ ( append_a @ U2 @ ( append_a @ Bs5 @ ( append_a @ V3 @ Cs5 ) ) ) ) )
= Xs )
& ( distinct_a @ ( append_a @ As5 @ ( append_a @ U2 @ ( append_a @ Bs5 @ ( append_a @ V3 @ Cs5 ) ) ) ) ) )
=> ( ord_less_eq_real @ ( Cost @ ( append_a @ As4 @ ( append_a @ U2 @ ( append_a @ Bs4 @ ( append_a @ V3 @ Cs4 ) ) ) ) ) @ ( Cost @ ( append_a @ As5 @ ( append_a @ U2 @ ( append_a @ Bs5 @ ( append_a @ V3 @ Cs5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% valid_UV_lists_argmin_ex_noP
thf(fact_1034_valid__UV__lists__argmin__ex__noP,axiom,
! [Xs: set_b,U2: list_b,V3: list_b,Cost: list_b > real] :
( ( finite_finite_b @ Xs )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ U2 ) @ ( set_b2 @ V3 ) )
= bot_bot_set_b )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ U2 ) @ Xs )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ V3 ) @ Xs )
=> ( ( distinct_b @ U2 )
=> ( ( distinct_b @ V3 )
=> ? [As4: list_b,Bs4: list_b,Cs4: list_b] :
( ( ( set_b2 @ ( append_b @ As4 @ ( append_b @ U2 @ ( append_b @ Bs4 @ ( append_b @ V3 @ Cs4 ) ) ) ) )
= Xs )
& ( distinct_b @ ( append_b @ As4 @ ( append_b @ U2 @ ( append_b @ Bs4 @ ( append_b @ V3 @ Cs4 ) ) ) ) )
& ! [As5: list_b,Bs5: list_b,Cs5: list_b] :
( ( ( ( set_b2 @ ( append_b @ As5 @ ( append_b @ U2 @ ( append_b @ Bs5 @ ( append_b @ V3 @ Cs5 ) ) ) ) )
= Xs )
& ( distinct_b @ ( append_b @ As5 @ ( append_b @ U2 @ ( append_b @ Bs5 @ ( append_b @ V3 @ Cs5 ) ) ) ) ) )
=> ( ord_less_eq_real @ ( Cost @ ( append_b @ As4 @ ( append_b @ U2 @ ( append_b @ Bs4 @ ( append_b @ V3 @ Cs4 ) ) ) ) ) @ ( Cost @ ( append_b @ As5 @ ( append_b @ U2 @ ( append_b @ Bs5 @ ( append_b @ V3 @ Cs5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% valid_UV_lists_argmin_ex_noP
thf(fact_1035_sublist__def,axiom,
( sublist_a
= ( ^ [Xs4: list_a,Ys4: list_a] :
? [Ps3: list_a,Ss2: list_a] :
( Ys4
= ( append_a @ Ps3 @ ( append_a @ Xs4 @ Ss2 ) ) ) ) ) ).
% sublist_def
thf(fact_1036_sublist__def,axiom,
( sublist_b
= ( ^ [Xs4: list_b,Ys4: list_b] :
? [Ps3: list_b,Ss2: list_b] :
( Ys4
= ( append_b @ Ps3 @ ( append_b @ Xs4 @ Ss2 ) ) ) ) ) ).
% sublist_def
thf(fact_1037_set__mono__sublist,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( sublist_list_a @ Xs @ Ys )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ ( set_list_a2 @ Ys ) ) ) ).
% set_mono_sublist
thf(fact_1038_set__mono__sublist,axiom,
! [Xs: list_a,Ys: list_a] :
( ( sublist_a @ Xs @ Ys )
=> ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).
% set_mono_sublist
thf(fact_1039_set__mono__sublist,axiom,
! [Xs: list_b,Ys: list_b] :
( ( sublist_b @ Xs @ Ys )
=> ( ord_less_eq_set_b @ ( set_b2 @ Xs ) @ ( set_b2 @ Ys ) ) ) ).
% set_mono_sublist
thf(fact_1040_list__of__sublist__concat__eq,axiom,
! [Y6: set_list_list_a,Xs: list_list_a] :
( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_list_a @ ( set_list_a2 @ X4 ) @ ( set_list_a2 @ Xa ) )
= bot_bot_set_list_a ) ) ) )
=> ( ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ( sublist_list_a @ X4 @ Xs ) )
=> ( ( distinct_list_a @ Xs )
=> ( ( ( set_list_a2 @ Xs )
= ( comple6928918032620976721list_a @ ( image_432481560377026271list_a @ set_list_a2 @ Y6 ) ) )
=> ( ( finite1660835950917165235list_a @ Y6 )
=> ? [Ys2: list_list_list_a] :
( ( ( set_list_list_a2 @ Ys2 )
= Y6 )
& ( ( concat_list_a @ Ys2 )
= Xs )
& ( distinct_list_list_a @ Ys2 ) ) ) ) ) ) ) ).
% list_of_sublist_concat_eq
thf(fact_1041_list__of__sublist__concat__eq,axiom,
! [Y6: set_list_b,Xs: list_b] :
( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ! [Xa: list_b] :
( ( member_list_b @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_b @ ( set_b2 @ X4 ) @ ( set_b2 @ Xa ) )
= bot_bot_set_b ) ) ) )
=> ( ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ( sublist_b @ X4 @ Xs ) )
=> ( ( distinct_b @ Xs )
=> ( ( ( set_b2 @ Xs )
= ( comple2307003614231284044_set_b @ ( image_list_b_set_b @ set_b2 @ Y6 ) ) )
=> ( ( finite_finite_list_b @ Y6 )
=> ? [Ys2: list_list_b] :
( ( ( set_list_b2 @ Ys2 )
= Y6 )
& ( ( concat_b @ Ys2 )
= Xs )
& ( distinct_list_b @ Ys2 ) ) ) ) ) ) ) ).
% list_of_sublist_concat_eq
thf(fact_1042_list__of__sublist__concat__eq,axiom,
! [Y6: set_list_a,Xs: list_a] :
( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ Y6 )
=> ( ( X4 = Xa )
| ( ( inf_inf_set_a @ ( set_a2 @ X4 ) @ ( set_a2 @ Xa ) )
= bot_bot_set_a ) ) ) )
=> ( ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ( sublist_a @ X4 @ Xs ) )
=> ( ( distinct_a @ Xs )
=> ( ( ( set_a2 @ Xs )
= ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ set_a2 @ Y6 ) ) )
=> ( ( finite_finite_list_a @ Y6 )
=> ? [Ys2: list_list_a] :
( ( ( set_list_a2 @ Ys2 )
= Y6 )
& ( ( concat_a @ Ys2 )
= Xs )
& ( distinct_list_a @ Ys2 ) ) ) ) ) ) ) ).
% list_of_sublist_concat_eq
thf(fact_1043_finite__UN,axiom,
! [A2: set_a,B3: a > set_b] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_b @ ( comple2307003614231284044_set_b @ ( image_a_set_b @ B3 @ A2 ) ) )
= ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( finite_finite_b @ ( B3 @ X5 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1044_finite__UN,axiom,
! [A2: set_b,B3: b > set_b] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_b @ ( comple2307003614231284044_set_b @ ( image_b_set_b @ B3 @ A2 ) ) )
= ( ! [X5: b] :
( ( member_b @ X5 @ A2 )
=> ( finite_finite_b @ ( B3 @ X5 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1045_finite__UN,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: pre_pr7278220950009878019t_unit > set_a] :
( ( finite8852549406693098522t_unit @ A2 )
=> ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_7466199892558553556_set_a @ B3 @ A2 ) ) )
= ( ! [X5: pre_pr7278220950009878019t_unit] :
( ( member6939884229742472986t_unit @ X5 @ A2 )
=> ( finite_finite_a @ ( B3 @ X5 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1046_finite__UN,axiom,
! [A2: set_list_a,B3: list_a > set_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_list_a_set_a @ B3 @ A2 ) ) )
= ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A2 )
=> ( finite_finite_a @ ( B3 @ X5 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1047_finite__UN,axiom,
! [A2: set_a,B3: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_a_set_a @ B3 @ A2 ) ) )
= ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( finite_finite_a @ ( B3 @ X5 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1048_finite__UN,axiom,
! [A2: set_b,B3: b > set_a] :
( ( finite_finite_b @ A2 )
=> ( ( finite_finite_a @ ( comple2307003609928055243_set_a @ ( image_b_set_a @ B3 @ A2 ) ) )
= ( ! [X5: b] :
( ( member_b @ X5 @ A2 )
=> ( finite_finite_a @ ( B3 @ X5 ) ) ) ) ) ) ).
% finite_UN
thf(fact_1049_Finite__Set_Ofinite__Union,axiom,
! [A2: set_set_b] :
( ( finite_finite_set_b @ A2 )
=> ( ! [M2: set_b] :
( ( member_set_b @ M2 @ A2 )
=> ( finite_finite_b @ M2 ) )
=> ( finite_finite_b @ ( comple2307003614231284044_set_b @ A2 ) ) ) ) ).
% Finite_Set.finite_Union
thf(fact_1050_Finite__Set_Ofinite__Union,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ! [M2: set_a] :
( ( member_set_a @ M2 @ A2 )
=> ( finite_finite_a @ M2 ) )
=> ( finite_finite_a @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% Finite_Set.finite_Union
thf(fact_1051_finite__Int,axiom,
! [F2: set_b,G: set_b] :
( ( ( finite_finite_b @ F2 )
| ( finite_finite_b @ G ) )
=> ( finite_finite_b @ ( inf_inf_set_b @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_1052_finite__Int,axiom,
! [F2: set_a,G: set_a] :
( ( ( finite_finite_a @ F2 )
| ( finite_finite_a @ G ) )
=> ( finite_finite_a @ ( inf_inf_set_a @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_1053_finite__imageI,axiom,
! [F2: set_pr5411798346947241657t_unit,H2: pre_pr7278220950009878019t_unit > set_a] :
( ( finite8852549406693098522t_unit @ F2 )
=> ( finite_finite_set_a @ ( image_7466199892558553556_set_a @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_1054_finite__imageI,axiom,
! [F2: set_set_a,H2: set_a > pre_pr7278220950009878019t_unit] :
( ( finite_finite_set_a @ F2 )
=> ( finite8852549406693098522t_unit @ ( image_6801035452528096924t_unit @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_1055_finite__imageI,axiom,
! [F2: set_list_a,H2: list_a > set_a] :
( ( finite_finite_list_a @ F2 )
=> ( finite_finite_set_a @ ( image_list_a_set_a @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_1056_finite__imageI,axiom,
! [F2: set_a,H2: a > set_a] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_set_a @ ( image_a_set_a @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_1057_finite__imageI,axiom,
! [F2: set_a,H2: a > a] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_a @ ( image_a_a @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_1058_finite__imageI,axiom,
! [F2: set_a,H2: a > b] :
( ( finite_finite_a @ F2 )
=> ( finite_finite_b @ ( image_a_b @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_1059_finite__imageI,axiom,
! [F2: set_b,H2: b > a] :
( ( finite_finite_b @ F2 )
=> ( finite_finite_a @ ( image_b_a @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_1060_finite__imageI,axiom,
! [F2: set_b,H2: b > b] :
( ( finite_finite_b @ F2 )
=> ( finite_finite_b @ ( image_b_b @ H2 @ F2 ) ) ) ).
% finite_imageI
thf(fact_1061_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ( ord_less_eq_set_a @ X4 @ A )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X4 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1062_finite__has__minimal2,axiom,
! [A2: set_set_b,A: set_b] :
( ( finite_finite_set_b @ A2 )
=> ( ( member_set_b @ A @ A2 )
=> ? [X4: set_b] :
( ( member_set_b @ X4 @ A2 )
& ( ord_less_eq_set_b @ X4 @ A )
& ! [Xa2: set_b] :
( ( member_set_b @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_b @ Xa2 @ X4 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_1063_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ( ord_less_eq_set_a @ A @ X4 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ X4 @ Xa2 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1064_finite__has__maximal2,axiom,
! [A2: set_set_b,A: set_b] :
( ( finite_finite_set_b @ A2 )
=> ( ( member_set_b @ A @ A2 )
=> ? [X4: set_b] :
( ( member_set_b @ X4 @ A2 )
& ( ord_less_eq_set_b @ A @ X4 )
& ! [Xa2: set_b] :
( ( member_set_b @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_b @ X4 @ Xa2 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_1065_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_1066_infinite__imp__nonempty,axiom,
! [S: set_b] :
( ~ ( finite_finite_b @ S )
=> ( S != bot_bot_set_b ) ) ).
% infinite_imp_nonempty
thf(fact_1067_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_1068_finite_OemptyI,axiom,
finite_finite_b @ bot_bot_set_b ).
% finite.emptyI
thf(fact_1069_all__subset__image,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B4 @ A2 )
=> ( P @ ( image_7466199892558553556_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1070_all__subset__image,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,P: set_pr5411798346947241657t_unit > $o] :
( ( ! [B4: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ B4 @ ( image_6801035452528096924t_unit @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ A2 )
=> ( P @ ( image_6801035452528096924t_unit @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1071_all__subset__image,axiom,
! [F: list_a > set_a,A2: set_list_a,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_list_a_set_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_list_a] :
( ( ord_le8861187494160871172list_a @ B4 @ A2 )
=> ( P @ ( image_list_a_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1072_all__subset__image,axiom,
! [F: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ B4 @ ( image_a_set_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( P @ ( image_a_set_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1073_all__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1074_all__subset__image,axiom,
! [F: b > a,A2: set_b,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ ( image_b_a @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_b] :
( ( ord_less_eq_set_b @ B4 @ A2 )
=> ( P @ ( image_b_a @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1075_all__subset__image,axiom,
! [F: a > b,A2: set_a,P: set_b > $o] :
( ( ! [B4: set_b] :
( ( ord_less_eq_set_b @ B4 @ ( image_a_b @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ A2 )
=> ( P @ ( image_a_b @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1076_all__subset__image,axiom,
! [F: b > b,A2: set_b,P: set_b > $o] :
( ( ! [B4: set_b] :
( ( ord_less_eq_set_b @ B4 @ ( image_b_b @ F @ A2 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_b] :
( ( ord_less_eq_set_b @ B4 @ A2 )
=> ( P @ ( image_b_b @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_1077_rev__finite__subset,axiom,
! [B3: set_a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1078_rev__finite__subset,axiom,
! [B3: set_b,A2: set_b] :
( ( finite_finite_b @ B3 )
=> ( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( finite_finite_b @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_1079_infinite__super,axiom,
! [S: set_a,T: set_a] :
( ( ord_less_eq_set_a @ S @ T )
=> ( ~ ( finite_finite_a @ S )
=> ~ ( finite_finite_a @ T ) ) ) ).
% infinite_super
thf(fact_1080_infinite__super,axiom,
! [S: set_b,T: set_b] :
( ( ord_less_eq_set_b @ S @ T )
=> ( ~ ( finite_finite_b @ S )
=> ~ ( finite_finite_b @ T ) ) ) ).
% infinite_super
thf(fact_1081_finite__subset,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( finite_finite_a @ B3 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_1082_finite__subset,axiom,
! [A2: set_b,B3: set_b] :
( ( ord_less_eq_set_b @ A2 @ B3 )
=> ( ( finite_finite_b @ B3 )
=> ( finite_finite_b @ A2 ) ) ) ).
% finite_subset
thf(fact_1083_finite__UnionD,axiom,
! [A2: set_set_b] :
( ( finite_finite_b @ ( comple2307003614231284044_set_b @ A2 ) )
=> ( finite_finite_set_b @ A2 ) ) ).
% finite_UnionD
thf(fact_1084_finite__UnionD,axiom,
! [A2: set_set_a] :
( ( finite_finite_a @ ( comple2307003609928055243_set_a @ A2 ) )
=> ( finite_finite_set_a @ A2 ) ) ).
% finite_UnionD
thf(fact_1085_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ X4 @ Xa2 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1086_finite__has__maximal,axiom,
! [A2: set_set_b] :
( ( finite_finite_set_b @ A2 )
=> ( ( A2 != bot_bot_set_set_b )
=> ? [X4: set_b] :
( ( member_set_b @ X4 @ A2 )
& ! [Xa2: set_b] :
( ( member_set_b @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_b @ X4 @ Xa2 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_1087_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A2 )
& ! [Xa2: set_a] :
( ( member_set_a @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_a @ Xa2 @ X4 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1088_finite__has__minimal,axiom,
! [A2: set_set_b] :
( ( finite_finite_set_b @ A2 )
=> ( ( A2 != bot_bot_set_set_b )
=> ? [X4: set_b] :
( ( member_set_b @ X4 @ A2 )
& ! [Xa2: set_b] :
( ( member_set_b @ Xa2 @ A2 )
=> ( ( ord_less_eq_set_b @ Xa2 @ X4 )
=> ( X4 = Xa2 ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_1089_finite__surj,axiom,
! [A2: set_pr5411798346947241657t_unit,B3: set_set_a,F: pre_pr7278220950009878019t_unit > set_a] :
( ( finite8852549406693098522t_unit @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
=> ( finite_finite_set_a @ B3 ) ) ) ).
% finite_surj
thf(fact_1090_finite__surj,axiom,
! [A2: set_set_a,B3: set_pr5411798346947241657t_unit,F: set_a > pre_pr7278220950009878019t_unit] :
( ( finite_finite_set_a @ A2 )
=> ( ( ord_le8200006823705900825t_unit @ B3 @ ( image_6801035452528096924t_unit @ F @ A2 ) )
=> ( finite8852549406693098522t_unit @ B3 ) ) ) ).
% finite_surj
thf(fact_1091_finite__surj,axiom,
! [A2: set_list_a,B3: set_set_a,F: list_a > set_a] :
( ( finite_finite_list_a @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ ( image_list_a_set_a @ F @ A2 ) )
=> ( finite_finite_set_a @ B3 ) ) ) ).
% finite_surj
thf(fact_1092_finite__surj,axiom,
! [A2: set_a,B3: set_set_a,F: a > set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ ( image_a_set_a @ F @ A2 ) )
=> ( finite_finite_set_a @ B3 ) ) ) ).
% finite_surj
thf(fact_1093_finite__surj,axiom,
! [A2: set_a,B3: set_a,F: a > a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A2 ) )
=> ( finite_finite_a @ B3 ) ) ) ).
% finite_surj
thf(fact_1094_finite__surj,axiom,
! [A2: set_b,B3: set_a,F: b > a] :
( ( finite_finite_b @ A2 )
=> ( ( ord_less_eq_set_a @ B3 @ ( image_b_a @ F @ A2 ) )
=> ( finite_finite_a @ B3 ) ) ) ).
% finite_surj
thf(fact_1095_finite__surj,axiom,
! [A2: set_a,B3: set_b,F: a > b] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_b @ B3 @ ( image_a_b @ F @ A2 ) )
=> ( finite_finite_b @ B3 ) ) ) ).
% finite_surj
thf(fact_1096_finite__surj,axiom,
! [A2: set_b,B3: set_b,F: b > b] :
( ( finite_finite_b @ A2 )
=> ( ( ord_less_eq_set_b @ B3 @ ( image_b_b @ F @ A2 ) )
=> ( finite_finite_b @ B3 ) ) ) ).
% finite_surj
thf(fact_1097_finite__subset__image,axiom,
! [B3: set_set_a,F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit] :
( ( finite_finite_set_a @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
=> ? [C3: set_pr5411798346947241657t_unit] :
( ( ord_le8200006823705900825t_unit @ C3 @ A2 )
& ( finite8852549406693098522t_unit @ C3 )
& ( B3
= ( image_7466199892558553556_set_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1098_finite__subset__image,axiom,
! [B3: set_pr5411798346947241657t_unit,F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a] :
( ( finite8852549406693098522t_unit @ B3 )
=> ( ( ord_le8200006823705900825t_unit @ B3 @ ( image_6801035452528096924t_unit @ F @ A2 ) )
=> ? [C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ A2 )
& ( finite_finite_set_a @ C3 )
& ( B3
= ( image_6801035452528096924t_unit @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1099_finite__subset__image,axiom,
! [B3: set_set_a,F: list_a > set_a,A2: set_list_a] :
( ( finite_finite_set_a @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ ( image_list_a_set_a @ F @ A2 ) )
=> ? [C3: set_list_a] :
( ( ord_le8861187494160871172list_a @ C3 @ A2 )
& ( finite_finite_list_a @ C3 )
& ( B3
= ( image_list_a_set_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1100_finite__subset__image,axiom,
! [B3: set_set_a,F: a > set_a,A2: set_a] :
( ( finite_finite_set_a @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ ( image_a_set_a @ F @ A2 ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
& ( finite_finite_a @ C3 )
& ( B3
= ( image_a_set_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1101_finite__subset__image,axiom,
! [B3: set_a,F: a > a,A2: set_a] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ ( image_a_a @ F @ A2 ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
& ( finite_finite_a @ C3 )
& ( B3
= ( image_a_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1102_finite__subset__image,axiom,
! [B3: set_a,F: b > a,A2: set_b] :
( ( finite_finite_a @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ ( image_b_a @ F @ A2 ) )
=> ? [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A2 )
& ( finite_finite_b @ C3 )
& ( B3
= ( image_b_a @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1103_finite__subset__image,axiom,
! [B3: set_b,F: a > b,A2: set_a] :
( ( finite_finite_b @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ ( image_a_b @ F @ A2 ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
& ( finite_finite_a @ C3 )
& ( B3
= ( image_a_b @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1104_finite__subset__image,axiom,
! [B3: set_b,F: b > b,A2: set_b] :
( ( finite_finite_b @ B3 )
=> ( ( ord_less_eq_set_b @ B3 @ ( image_b_b @ F @ A2 ) )
=> ? [C3: set_b] :
( ( ord_less_eq_set_b @ C3 @ A2 )
& ( finite_finite_b @ C3 )
& ( B3
= ( image_b_b @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_1105_ex__finite__subset__image,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,P: set_set_a > $o] :
( ( ? [B4: set_set_a] :
( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ ( image_7466199892558553556_set_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ B4 )
& ( ord_le8200006823705900825t_unit @ B4 @ A2 )
& ( P @ ( image_7466199892558553556_set_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1106_ex__finite__subset__image,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,P: set_pr5411798346947241657t_unit > $o] :
( ( ? [B4: set_pr5411798346947241657t_unit] :
( ( finite8852549406693098522t_unit @ B4 )
& ( ord_le8200006823705900825t_unit @ B4 @ ( image_6801035452528096924t_unit @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_set_a] :
( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A2 )
& ( P @ ( image_6801035452528096924t_unit @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1107_ex__finite__subset__image,axiom,
! [F: list_a > set_a,A2: set_list_a,P: set_set_a > $o] :
( ( ? [B4: set_set_a] :
( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ ( image_list_a_set_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_list_a] :
( ( finite_finite_list_a @ B4 )
& ( ord_le8861187494160871172list_a @ B4 @ A2 )
& ( P @ ( image_list_a_set_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1108_ex__finite__subset__image,axiom,
! [F: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ? [B4: set_set_a] :
( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ ( image_a_set_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 )
& ( P @ ( image_a_set_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1109_ex__finite__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 )
& ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1110_ex__finite__subset__image,axiom,
! [F: b > a,A2: set_b,P: set_a > $o] :
( ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_b_a @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 )
& ( P @ ( image_b_a @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1111_ex__finite__subset__image,axiom,
! [F: a > b,A2: set_a,P: set_b > $o] :
( ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_a_b @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_a] :
( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 )
& ( P @ ( image_a_b @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1112_ex__finite__subset__image,axiom,
! [F: b > b,A2: set_b,P: set_b > $o] :
( ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_b_b @ F @ A2 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set_b] :
( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 )
& ( P @ ( image_b_b @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_1113_all__finite__subset__image,axiom,
! [F: pre_pr7278220950009878019t_unit > set_a,A2: set_pr5411798346947241657t_unit,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ ( image_7466199892558553556_set_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_pr5411798346947241657t_unit] :
( ( ( finite8852549406693098522t_unit @ B4 )
& ( ord_le8200006823705900825t_unit @ B4 @ A2 ) )
=> ( P @ ( image_7466199892558553556_set_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1114_all__finite__subset__image,axiom,
! [F: set_a > pre_pr7278220950009878019t_unit,A2: set_set_a,P: set_pr5411798346947241657t_unit > $o] :
( ( ! [B4: set_pr5411798346947241657t_unit] :
( ( ( finite8852549406693098522t_unit @ B4 )
& ( ord_le8200006823705900825t_unit @ B4 @ ( image_6801035452528096924t_unit @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_set_a] :
( ( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A2 ) )
=> ( P @ ( image_6801035452528096924t_unit @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1115_all__finite__subset__image,axiom,
! [F: list_a > set_a,A2: set_list_a,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ ( image_list_a_set_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_list_a] :
( ( ( finite_finite_list_a @ B4 )
& ( ord_le8861187494160871172list_a @ B4 @ A2 ) )
=> ( P @ ( image_list_a_set_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1116_all__finite__subset__image,axiom,
! [F: a > set_a,A2: set_a,P: set_set_a > $o] :
( ( ! [B4: set_set_a] :
( ( ( finite_finite_set_a @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ ( image_a_set_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 ) )
=> ( P @ ( image_a_set_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1117_all__finite__subset__image,axiom,
! [F: a > a,A2: set_a,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_a_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 ) )
=> ( P @ ( image_a_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1118_all__finite__subset__image,axiom,
! [F: b > a,A2: set_b,P: set_a > $o] :
( ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ ( image_b_a @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 ) )
=> ( P @ ( image_b_a @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1119_all__finite__subset__image,axiom,
! [F: a > b,A2: set_a,P: set_b > $o] :
( ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_a_b @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_a] :
( ( ( finite_finite_a @ B4 )
& ( ord_less_eq_set_a @ B4 @ A2 ) )
=> ( P @ ( image_a_b @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1120_all__finite__subset__image,axiom,
! [F: b > b,A2: set_b,P: set_b > $o] :
( ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ ( image_b_b @ F @ A2 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set_b] :
( ( ( finite_finite_b @ B4 )
& ( ord_less_eq_set_b @ B4 @ A2 ) )
=> ( P @ ( image_b_b @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_1121_ex__leaf,axiom,
( ( finite_finite_a @ ( pre_ve642382030648772252t_unit @ t ) )
=> ? [X4: a] :
( ( member_a @ X4 @ ( pre_ve642382030648772252t_unit @ t ) )
& ( shorte1213025427933718126af_a_b @ t @ X4 ) ) ) ).
% ex_leaf
thf(fact_1122_finite__subset__Union,axiom,
! [A2: set_b,B5: set_set_b] :
( ( finite_finite_b @ A2 )
=> ( ( ord_less_eq_set_b @ A2 @ ( comple2307003614231284044_set_b @ B5 ) )
=> ~ ! [F3: set_set_b] :
( ( finite_finite_set_b @ F3 )
=> ( ( ord_le3795704787696855135_set_b @ F3 @ B5 )
=> ~ ( ord_less_eq_set_b @ A2 @ ( comple2307003614231284044_set_b @ F3 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1123_finite__subset__Union,axiom,
! [A2: set_a,B5: set_set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( comple2307003609928055243_set_a @ B5 ) )
=> ~ ! [F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ B5 )
=> ~ ( ord_less_eq_set_a @ A2 @ ( comple2307003609928055243_set_a @ F3 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_1124_directed__tree_Oforward__split,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,Xs: list_a,Ys: list_a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ T @ ( append_a @ Xs @ Ys ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ T @ Xs ) ) ) ).
% directed_tree.forward_split
thf(fact_1125_directed__tree_Oforward__arcs__split,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,Ys: list_a,Xs: list_a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( ( iKKBZ_4180558001818622352cs_a_b @ T @ ( append_a @ Ys @ Xs ) )
=> ( iKKBZ_4180558001818622352cs_a_b @ T @ Xs ) ) ) ).
% directed_tree.forward_arcs_split
thf(fact_1126_directed__tree_Oforward__arcs_Osimps_I1_J,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( iKKBZ_4180558001818622352cs_a_b @ T @ nil_a ) ) ).
% directed_tree.forward_arcs.simps(1)
thf(fact_1127_directed__tree_Ono__back__insert,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,X2: a,Xs: list_a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( ( iKKBZ_3684931046464919648ck_a_b @ T @ ( cons_a @ X2 @ Xs ) )
=> ( iKKBZ_3684931046464919648ck_a_b @ T @ Xs ) ) ) ).
% directed_tree.no_back_insert
thf(fact_1128_directed__tree_Ono__back__arcs_Osimps_I1_J,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( iKKBZ_7773321254043928001cs_a_b @ T @ nil_a ) ) ).
% directed_tree.no_back_arcs.simps(1)
thf(fact_1129_directed__tree_Obefore__forward2I,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,S1: list_a,S2: list_a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( ( iKKBZ_7682935289300565975re_a_b @ T @ S1 @ S2 )
=> ( iKKBZ_4778857019735642799rd_a_b @ T @ S2 ) ) ) ).
% directed_tree.before_forward2I
thf(fact_1130_directed__tree_Obefore__forward1I,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,S1: list_a,S2: list_a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( ( iKKBZ_7682935289300565975re_a_b @ T @ S1 @ S2 )
=> ( iKKBZ_4778857019735642799rd_a_b @ T @ S1 ) ) ) ).
% directed_tree.before_forward1I
thf(fact_1131_directed__tree_Obefore__conform1I,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,S1: list_a,S2: list_a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( ( iKKBZ_7682935289300565975re_a_b @ T @ S1 @ S2 )
=> ( iKKBZ_4622586873178280511rm_a_b @ T @ S1 ) ) ) ).
% directed_tree.before_conform1I
thf(fact_1132_directed__tree_Obefore__conform2I,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,S1: list_a,S2: list_a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( ( iKKBZ_7682935289300565975re_a_b @ T @ S1 @ S2 )
=> ( iKKBZ_4622586873178280511rm_a_b @ T @ S2 ) ) ) ).
% directed_tree.before_conform2I
thf(fact_1133_directed__tree_Obefore__no__back2I,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,S1: list_a,S2: list_a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( ( iKKBZ_7682935289300565975re_a_b @ T @ S1 @ S2 )
=> ( iKKBZ_3684931046464919648ck_a_b @ T @ S2 ) ) ) ).
% directed_tree.before_no_back2I
thf(fact_1134_directed__tree_Obefore__no__back1I,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,S1: list_a,S2: list_a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( ( iKKBZ_7682935289300565975re_a_b @ T @ S1 @ S2 )
=> ( iKKBZ_3684931046464919648ck_a_b @ T @ S1 ) ) ) ).
% directed_tree.before_no_back1I
thf(fact_1135_directed__tree_Ono__back__arcs__alt,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,Xs: list_a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( ( iKKBZ_3684931046464919648ck_a_b @ T @ Xs )
= ( iKKBZ_7773321254043928001cs_a_b @ T @ Xs ) ) ) ).
% directed_tree.no_back_arcs_alt
thf(fact_1136_directed__tree_Ono__back__arcs__alt__aux2,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,Xs: list_a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( ( iKKBZ_3684931046464919648ck_a_b @ T @ Xs )
=> ( iKKBZ_7773321254043928001cs_a_b @ T @ Xs ) ) ) ).
% directed_tree.no_back_arcs_alt_aux2
thf(fact_1137_directed__tree_Oforward__single,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,X2: a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( iKKBZ_4778857019735642799rd_a_b @ T @ ( cons_a @ X2 @ nil_a ) ) ) ).
% directed_tree.forward_single
thf(fact_1138_directed__tree_Oforward__cons,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,X2: a,Xs: list_a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ T @ ( rev_a @ ( cons_a @ X2 @ Xs ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ T @ ( rev_a @ Xs ) ) ) ) ).
% directed_tree.forward_cons
thf(fact_1139_directed__tree_Oseq__conform__single,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,X2: a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( iKKBZ_4622586873178280511rm_a_b @ T @ ( cons_a @ X2 @ nil_a ) ) ) ).
% directed_tree.seq_conform_single
thf(fact_1140_directed__tree_Oforward__arcs__single,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,X2: a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( iKKBZ_4180558001818622352cs_a_b @ T @ ( cons_a @ X2 @ nil_a ) ) ) ).
% directed_tree.forward_arcs_single
thf(fact_1141_directed__tree_Oforward__arcs_Osimps_I2_J,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,X2: a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( iKKBZ_4180558001818622352cs_a_b @ T @ ( cons_a @ X2 @ nil_a ) ) ) ).
% directed_tree.forward_arcs.simps(2)
thf(fact_1142_directed__tree_Ono__back__single,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,X2: a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( iKKBZ_3684931046464919648ck_a_b @ T @ ( cons_a @ X2 @ nil_a ) ) ) ).
% directed_tree.no_back_single
thf(fact_1143_directed__tree_Ono__back__arcs__single,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,X2: a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( iKKBZ_7773321254043928001cs_a_b @ T @ ( cons_a @ X2 @ nil_a ) ) ) ).
% directed_tree.no_back_arcs_single
thf(fact_1144_directed__tree_Omove__mid__backward__if__noarc,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,U2: list_a,V3: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( ( iKKBZ_7682935289300565975re_a_b @ T @ U2 @ V3 )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ T @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V3 @ Cs ) ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ T @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ V3 @ ( append_a @ Bs @ Cs ) ) ) ) ) ) ) ) ).
% directed_tree.move_mid_backward_if_noarc
thf(fact_1145_directed__tree_Oforward__arcs__alt__aux2,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,Xs: list_a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ T @ ( rev_a @ Xs ) )
=> ( iKKBZ_4180558001818622352cs_a_b @ T @ Xs ) ) ) ).
% directed_tree.forward_arcs_alt_aux2
thf(fact_1146_directed__tree_Oforward__arcs__alt_H,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,Xs: list_a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ T @ ( rev_a @ Xs ) )
= ( iKKBZ_4180558001818622352cs_a_b @ T @ Xs ) ) ) ).
% directed_tree.forward_arcs_alt'
thf(fact_1147_directed__tree_Oforward__arcs__alt,axiom,
! [T: pre_pr7278220950009878019t_unit,Root: a,Xs: list_a] :
( ( shorte3810566709427824352ee_a_b @ T @ Root )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ T @ Xs )
= ( iKKBZ_4180558001818622352cs_a_b @ T @ ( rev_a @ Xs ) ) ) ) ).
% directed_tree.forward_arcs_alt
thf(fact_1148_in__set__inner__verts__appendI__l,axiom,
! [U: a,P2: list_b,Q2: list_b] :
( ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ P2 ) ) )
=> ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ ( append_b @ P2 @ Q2 ) ) ) ) ) ).
% in_set_inner_verts_appendI_l
thf(fact_1149_in__set__inner__verts__appendI__r,axiom,
! [U: a,Q2: list_b,P2: list_b] :
( ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ Q2 ) ) )
=> ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ ( append_b @ P2 @ Q2 ) ) ) ) ) ).
% in_set_inner_verts_appendI_r
thf(fact_1150_inner__verts__Nil,axiom,
( ( pre_inner_verts_a_b @ t @ nil_b )
= nil_a ) ).
% inner_verts_Nil
thf(fact_1151_inner__verts__singleton,axiom,
! [X2: b] :
( ( pre_inner_verts_a_b @ t @ ( cons_b @ X2 @ nil_b ) )
= nil_a ) ).
% inner_verts_singleton
thf(fact_1152_cas_Ocases,axiom,
! [X2: produc7945266988514096265st_b_a] :
( ! [U3: a,V4: a] :
( X2
!= ( produc7119031474978700025st_b_a @ U3 @ ( produc4145578316043568848st_b_a @ nil_b @ V4 ) ) )
=> ~ ! [U3: a,E2: b,Es: list_b,V4: a] :
( X2
!= ( produc7119031474978700025st_b_a @ U3 @ ( produc4145578316043568848st_b_a @ ( cons_b @ E2 @ Es ) @ V4 ) ) ) ) ).
% cas.cases
thf(fact_1153_trail__Nil__iff,axiom,
! [U: a,V: a] :
( ( arc_pre_trail_a_b @ t @ U @ nil_b @ V )
= ( ( U = V )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% trail_Nil_iff
thf(fact_1154_reachable1__awalk,axiom,
! [U: a,V: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
= ( ? [P3: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ P3 @ V )
& ( P3 != nil_b ) ) ) ) ).
% reachable1_awalk
thf(fact_1155_reachable1__awalkI,axiom,
! [V: a,P2: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ V @ P2 @ W )
=> ( ( P2 != nil_b )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ V @ W ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% reachable1_awalkI
thf(fact_1156_unique__awalk__All,axiom,
! [U: a,V: a] :
( ? [P4: list_b] : ( arc_pre_awalk_a_b @ t @ U @ P4 @ V )
=> ? [X4: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ X4 @ V )
& ! [Y: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ Y @ V )
=> ( Y = X4 ) ) ) ) ).
% unique_awalk_All
thf(fact_1157_awalk__ends__eqD,axiom,
! [U: a,P2: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ U )
=> ( ( arc_pre_awalk_a_b @ t @ V @ P2 @ W )
=> ( V = W ) ) ) ).
% awalk_ends_eqD
thf(fact_1158_awalk__last__in__verts,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% awalk_last_in_verts
thf(fact_1159_awalk__hd__in__verts,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% awalk_hd_in_verts
thf(fact_1160_reachable__awalkI,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( reachable_a_b @ t @ U @ V ) ) ).
% reachable_awalkI
thf(fact_1161_reachable__awalk,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
= ( ? [P3: list_b] : ( arc_pre_awalk_a_b @ t @ U @ P3 @ V ) ) ) ).
% reachable_awalk
thf(fact_1162_awalk__appendI,axiom,
! [U: a,P2: list_b,V: a,Q2: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( arc_pre_awalk_a_b @ t @ V @ Q2 @ W )
=> ( arc_pre_awalk_a_b @ t @ U @ ( append_b @ P2 @ Q2 ) @ W ) ) ) ).
% awalk_appendI
thf(fact_1163_awalk__ends,axiom,
! [U: a,P2: list_b,V: a,U4: a,V5: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( arc_pre_awalk_a_b @ t @ U4 @ P2 @ V5 )
=> ( ( ( P2 != nil_b )
& ( U = U4 )
& ( V = V5 ) )
| ( ( P2 = nil_b )
& ( U = V )
& ( U4 = V5 ) ) ) ) ) ).
% awalk_ends
thf(fact_1164_awalk__empty__ends,axiom,
! [U: a,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ nil_b @ V )
=> ( U = V ) ) ).
% awalk_empty_ends
thf(fact_1165_trail__def,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_trail_a_b @ t @ U @ P2 @ V )
= ( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
& ( distinct_b @ P2 ) ) ) ).
% trail_def
thf(fact_1166_awalk__dom__if__uneq,axiom,
! [U: a,V: a,P2: list_b] :
( ( U != V )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ? [X4: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ V ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% awalk_dom_if_uneq
thf(fact_1167_awalk__Nil__iff,axiom,
! [U: a,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ nil_b @ V )
= ( ( U = V )
& ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% awalk_Nil_iff
thf(fact_1168_arc__balancedI__trail,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_trail_a_b @ t @ U @ P2 @ V )
=> ( pre_ar5931435604406180204ed_a_b @ t @ U @ ( set_b2 @ P2 ) @ V ) ) ).
% arc_balancedI_trail
thf(fact_1169_mk__cycles__path__awalk,axiom,
! [U: a,C: list_b,N2: nat] :
( ( arc_pre_awalk_a_b @ t @ U @ C @ U )
=> ( arc_pre_awalk_a_b @ t @ U @ ( shorte6374615165232202367path_b @ N2 @ C ) @ U ) ) ).
% mk_cycles_path_awalk
thf(fact_1170_inner__verts__Cons,axiom,
! [U: a,E: b,Es2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( cons_b @ E @ Es2 ) @ V )
=> ( ( ( Es2 != nil_b )
=> ( ( pre_inner_verts_a_b @ t @ ( cons_b @ E @ Es2 ) )
= ( cons_a @ ( pre_he5236287464308401016t_unit @ t @ E ) @ ( pre_inner_verts_a_b @ t @ Es2 ) ) ) )
& ( ( Es2 = nil_b )
=> ( ( pre_inner_verts_a_b @ t @ ( cons_b @ E @ Es2 ) )
= nil_a ) ) ) ) ).
% inner_verts_Cons
thf(fact_1171_awalk__cyc__decompE_H,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
=> ~ ! [Q3: list_b,R: list_b,S3: list_b] :
( ( P2
= ( append_b @ Q3 @ ( append_b @ R @ S3 ) ) )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q3 ) )
=> ( ? [W2: a] :
( ( arc_pre_awalk_a_b @ t @ U @ Q3 @ W2 )
& ( arc_pre_awalk_a_b @ t @ W2 @ R @ W2 )
& ( arc_pre_awalk_a_b @ t @ W2 @ S3 @ V ) )
=> ~ ( arc_wf_closed_w_a_b @ t @ R ) ) ) ) ) ) ).
% awalk_cyc_decompE'
thf(fact_1172_awalk__verts__non__Nil,axiom,
! [U: a,P2: list_b] :
( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 )
!= nil_a ) ).
% awalk_verts_non_Nil
thf(fact_1173_awalk__verts__ne__eq,axiom,
! [P2: list_b,U: a,V: a] :
( ( P2 != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 )
= ( arc_pr7493981781705774526ts_a_b @ t @ V @ P2 ) ) ) ).
% awalk_verts_ne_eq
thf(fact_1174_awalk__verts__induce,axiom,
! [S: set_a] :
( ( arc_pr7493981781705774526ts_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) )
= ( arc_pr7493981781705774526ts_a_b @ t ) ) ).
% awalk_verts_induce
thf(fact_1175_hd__in__awalk__verts_I1_J,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( member_a @ U @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ).
% hd_in_awalk_verts(1)
thf(fact_1176_awhd__append,axiom,
! [U: a,P2: list_b,Q2: list_b] :
( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P2 @ Q2 ) ) )
= ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q2 ) ) @ P2 ) ) ) ).
% awhd_append
thf(fact_1177_awalk__imp__vwalk,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( vertex_vwalk_a_b @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) @ t ) ) ).
% awalk_imp_vwalk
thf(fact_1178_awalk__verts__reachable__from,axiom,
! [U: a,P2: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ( reachable_a_b @ t @ U @ W ) ) ) ).
% awalk_verts_reachable_from
thf(fact_1179_awalk__verts__reachable__to,axiom,
! [U: a,P2: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ( reachable_a_b @ t @ W @ V ) ) ) ).
% awalk_verts_reachable_to
thf(fact_1180_rotate__awalkE,axiom,
! [U: a,P2: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ U )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ~ ! [Q3: list_b,R: list_b] :
( ( P2
= ( append_b @ Q3 @ R ) )
=> ( ( arc_pre_awalk_a_b @ t @ W @ ( append_b @ R @ Q3 ) @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ W @ ( append_b @ R @ Q3 ) ) )
!= ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ) ) ) ).
% rotate_awalkE
thf(fact_1181_awalk__decomp,axiom,
! [U: a,P2: list_b,V: a,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ? [Q3: list_b,R: list_b] :
( ( P2
= ( append_b @ Q3 @ R ) )
& ( arc_pre_awalk_a_b @ t @ U @ Q3 @ W )
& ( arc_pre_awalk_a_b @ t @ W @ R @ V ) ) ) ) ).
% awalk_decomp
thf(fact_1182_awalk__verts_Osimps_I1_J,axiom,
! [U: a] :
( ( arc_pr7493981781705774526ts_a_b @ t @ U @ nil_b )
= ( cons_a @ U @ nil_a ) ) ).
% awalk_verts.simps(1)
thf(fact_1183_awalk__verts__append__distinct,axiom,
! [R2: a,P1: list_b,P22: list_b] :
( ? [X_1: a] : ( arc_pre_awalk_a_b @ t @ R2 @ ( append_b @ P1 @ P22 ) @ X_1 )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ R2 @ ( append_b @ P1 @ P22 ) ) )
=> ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ R2 @ P1 ) ) ) ) ).
% awalk_verts_append_distinct
thf(fact_1184_to__list__tree__awalk,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
= ( arc_pr6214585750886380800st_a_b @ ( direct3773525127397338803ee_a_b @ t ) @ ( cons_a @ U @ nil_a ) @ P2 @ ( cons_a @ V @ nil_a ) ) ) ).
% to_list_tree_awalk
thf(fact_1185_rotate__trailE,axiom,
! [U: a,P2: list_b,W: a] :
( ( arc_pre_trail_a_b @ t @ U @ P2 @ U )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ~ ! [Q3: list_b,R: list_b] :
( ( P2
= ( append_b @ Q3 @ R ) )
=> ( ( arc_pre_trail_a_b @ t @ W @ ( append_b @ R @ Q3 ) @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ W @ ( append_b @ R @ Q3 ) ) )
!= ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ) ) ) ).
% rotate_trailE
thf(fact_1186_rotate__trailE_H,axiom,
! [U: a,P2: list_b,W: a] :
( ( arc_pre_trail_a_b @ t @ U @ P2 @ U )
=> ( ( member_a @ W @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ~ ! [Q3: list_b] :
( ( arc_pre_trail_a_b @ t @ W @ Q3 @ W )
=> ( ( ( set_b2 @ Q3 )
= ( set_b2 @ P2 ) )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ W @ Q3 ) )
!= ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ) ) ) ).
% rotate_trailE'
thf(fact_1187_leaf__not__mem__awalk,axiom,
! [X2: a,U: a,P2: list_b,V: a] :
( ( shorte1213025427933718126af_a_b @ t @ X2 )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( V != X2 )
=> ~ ( member_a @ X2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ) ) ).
% leaf_not_mem_awalk
thf(fact_1188_distinct__verts__imp__distinct,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
=> ( distinct_b @ P2 ) ) ) ).
% distinct_verts_imp_distinct
thf(fact_1189_awalk__verts__dom__if__uneq,axiom,
! [U: a,V: a,P2: list_b] :
( ( U != V )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ? [X4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ V ) @ ( arcs_ends_a_b @ t ) )
& ( member_a @ X4 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ) ) ).
% awalk_verts_dom_if_uneq
thf(fact_1190_dominated__notin__awalk,axiom,
! [U: a,V: a,R2: a,P2: list_b] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ t ) )
=> ( ( arc_pre_awalk_a_b @ t @ R2 @ P2 @ U )
=> ~ ( member_a @ V @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ R2 @ P2 ) ) ) ) ) ).
% dominated_notin_awalk
thf(fact_1191_awalk__verts__arc2,axiom,
! [U: a,P2: list_b,V: a,E: b] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( member_b @ E @ ( set_b2 @ P2 ) )
=> ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ) ).
% awalk_verts_arc2
thf(fact_1192_awalk__verts__append3,axiom,
! [U: a,P2: list_b,E: b,Q2: list_b,R2: a,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( append_b @ P2 @ ( cons_b @ E @ Q2 ) ) @ R2 )
=> ( ( arc_pre_awalk_a_b @ t @ V @ Q2 @ R2 )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P2 @ ( cons_b @ E @ Q2 ) ) )
= ( append_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) @ ( arc_pr7493981781705774526ts_a_b @ t @ V @ Q2 ) ) ) ) ) ).
% awalk_verts_append3
thf(fact_1193_awalk__verts__subset__if__p__sub,axiom,
! [U: a,P1: list_b,V: a,P22: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ P1 @ V )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P22 @ V )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P1 ) @ ( set_b2 @ P22 ) )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P1 ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P22 ) ) ) ) ) ) ).
% awalk_verts_subset_if_p_sub
thf(fact_1194_awalk__induce,axiom,
! [U: a,P2: list_b,V: a,S: set_a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ S )
=> ( arc_pre_awalk_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ U @ P2 @ V ) ) ) ).
% awalk_induce
thf(fact_1195_awhd__of__awalk,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= U ) ) ).
% awhd_of_awalk
thf(fact_1196_awalk__cyc__decompE,axiom,
! [P2: list_b,Q2: list_b,R2: list_b,S4: list_b,U: a,V: a] :
( ( ( arc_wf4740610840468824943mp_a_b @ t @ P2 )
= ( produc305491333965050169list_b @ Q2 @ ( produc1564554178308465111list_b @ R2 @ S4 ) ) )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
=> ~ ( ( P2
= ( append_b @ Q2 @ ( append_b @ R2 @ S4 ) ) )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q2 ) )
=> ( ? [W2: a] :
( ( arc_pre_awalk_a_b @ t @ U @ Q2 @ W2 )
& ( arc_pre_awalk_a_b @ t @ W2 @ R2 @ W2 )
& ( arc_pre_awalk_a_b @ t @ W2 @ S4 @ V ) )
=> ~ ( arc_wf_closed_w_a_b @ t @ R2 ) ) ) ) ) ) ) ).
% awalk_cyc_decompE
thf(fact_1197_head__add__vert,axiom,
! [U: a] :
( ( pre_he5236287464308401016t_unit @ ( pre_add_vert_a_b @ t @ U ) )
= ( pre_he5236287464308401016t_unit @ t ) ) ).
% head_add_vert
thf(fact_1198_awalk__to__apath__induct,axiom,
! [U: a,P2: list_b,V: a,P: list_b > $o] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ! [P5: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ P5 @ V )
=> ( ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P5 ) )
=> ( P @ P5 ) ) )
=> ( ! [P5: list_b,Q3: list_b,R: list_b,S3: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ P5 @ V )
=> ( ( ( arc_wf4740610840468824943mp_a_b @ t @ P5 )
= ( produc305491333965050169list_b @ Q3 @ ( produc1564554178308465111list_b @ R @ S3 ) ) )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P5 ) )
=> ( ( P @ ( append_b @ Q3 @ S3 ) )
=> ( P @ P5 ) ) ) ) )
=> ( P @ P2 ) ) ) ) ).
% awalk_to_apath_induct
thf(fact_1199_awalk__cyc__decomp__has__prop,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
=> ( arc_wf7293661141070756729mp_a_b @ t @ P2 @ ( arc_wf4740610840468824943mp_a_b @ t @ P2 ) ) ) ) ).
% awalk_cyc_decomp_has_prop
thf(fact_1200_step__awalk__to__apath,axiom,
! [U: a,P2: list_b,V: a,Q2: list_b,R2: list_b,S4: list_b] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( ( arc_wf4740610840468824943mp_a_b @ t @ P2 )
= ( produc305491333965050169list_b @ Q2 @ ( produc1564554178308465111list_b @ R2 @ S4 ) ) )
=> ( ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
=> ( ( arc_wf446166946845163101th_a_b @ t @ P2 )
= ( arc_wf446166946845163101th_a_b @ t @ ( append_b @ Q2 @ S4 ) ) ) ) ) ) ).
% step_awalk_to_apath
thf(fact_1201_awalk__to__apath__verts__subset,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( arc_wf446166946845163101th_a_b @ t @ P2 ) ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ).
% awalk_to_apath_verts_subset
thf(fact_1202_awalk__to__apath__subset,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ord_less_eq_set_b @ ( set_b2 @ ( arc_wf446166946845163101th_a_b @ t @ P2 ) ) @ ( set_b2 @ P2 ) ) ) ).
% awalk_to_apath_subset
thf(fact_1203_not__distinct__if__head__eq__tail,axiom,
! [P2: b,U: a,E: b,R2: a,Ps2: list_b,P22: list_b,V: a] :
( ( ( pre_ta4931606617599662728t_unit @ t @ P2 )
= U )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ E )
= U )
=> ( ( arc_pre_awalk_a_b @ t @ R2 @ ( append_b @ Ps2 @ ( append_b @ ( cons_b @ P2 @ nil_b ) @ ( cons_b @ E @ P22 ) ) ) @ V )
=> ~ ( distinct_a @ ( arc_pr7493981781705774526ts_a_b @ t @ R2 @ ( append_b @ Ps2 @ ( append_b @ ( cons_b @ P2 @ nil_b ) @ ( cons_b @ E @ P22 ) ) ) ) ) ) ) ) ).
% not_distinct_if_head_eq_tail
thf(fact_1204_euler__trail__def,axiom,
! [U: a,P2: list_b,V: a] :
( ( pre_euler_trail_a_b @ t @ U @ P2 @ V )
= ( ( arc_pre_trail_a_b @ t @ U @ P2 @ V )
& ( ( set_b2 @ P2 )
= ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% euler_trail_def
thf(fact_1205_two__in__arcs__contr,axiom,
! [E1: b,E22: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( E1 != E22 )
=> ( ( pre_he5236287464308401016t_unit @ t @ E1 )
!= ( pre_he5236287464308401016t_unit @ t @ E22 ) ) ) ) ) ).
% two_in_arcs_contr
thf(fact_1206_arcs__add__vert,axiom,
! [U: a] :
( ( pre_ar1395965042833527383t_unit @ ( pre_add_vert_a_b @ t @ U ) )
= ( pre_ar1395965042833527383t_unit @ t ) ) ).
% arcs_add_vert
thf(fact_1207_tail__add__vert,axiom,
! [U: a] :
( ( pre_ta4931606617599662728t_unit @ ( pre_add_vert_a_b @ t @ U ) )
= ( pre_ta4931606617599662728t_unit @ t ) ) ).
% tail_add_vert
thf(fact_1208_head__in__verts,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% head_in_verts
thf(fact_1209_tail__in__verts,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% tail_in_verts
thf(fact_1210_loopfree_Ono__loops,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( pre_ta4931606617599662728t_unit @ t @ E )
!= ( pre_he5236287464308401016t_unit @ t @ E ) ) ) ).
% loopfree.no_loops
thf(fact_1211_nomulti_Ono__multi__alt,axiom,
! [E1: b,E22: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( E1 != E22 )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ E1 )
!= ( pre_he5236287464308401016t_unit @ t @ E22 ) )
| ( ( pre_ta4931606617599662728t_unit @ t @ E1 )
!= ( pre_ta4931606617599662728t_unit @ t @ E22 ) ) ) ) ) ) ).
% nomulti.no_multi_alt
thf(fact_1212_All__arcs__in__path,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ? [P5: list_b,U3: a,V4: a] :
( ( arc_pre_awalk_a_b @ t @ U3 @ P5 @ V4 )
& ( member_b @ E @ ( set_b2 @ P5 ) ) ) ) ).
% All_arcs_in_path
thf(fact_1213_verts__finite__imp__arcs__finite,axiom,
( ( finite_finite_a @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( finite_finite_b @ ( pre_ar1395965042833527383t_unit @ t ) ) ) ).
% verts_finite_imp_arcs_finite
thf(fact_1214_awalk__verts__arc1,axiom,
! [E: b,P2: list_b,U: a] :
( ( member_b @ E @ ( set_b2 @ P2 ) )
=> ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ).
% awalk_verts_arc1
thf(fact_1215_unique__arc_I2_J,axiom,
! [U: a,V: a] :
( ~ ? [E2: b] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_ta4931606617599662728t_unit @ t @ E2 )
= U )
& ( ( pre_he5236287464308401016t_unit @ t @ E2 )
= V ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ t ) ) ) ).
% unique_arc(2)
thf(fact_1216_unique__arc_I1_J,axiom,
! [U: a,V: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ t ) )
=> ? [X4: b] :
( ( member_b @ X4 @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_ta4931606617599662728t_unit @ t @ X4 )
= U )
& ( ( pre_he5236287464308401016t_unit @ t @ X4 )
= V )
& ! [Y: b] :
( ( ( member_b @ Y @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( pre_ta4931606617599662728t_unit @ t @ Y )
= U )
& ( ( pre_he5236287464308401016t_unit @ t @ Y )
= V ) )
=> ( Y = X4 ) ) ) ) ).
% unique_arc(1)
thf(fact_1217_in__arcs__imp__in__arcs__ends,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( pre_he5236287464308401016t_unit @ t @ E ) ) @ ( arcs_ends_a_b @ t ) ) ) ).
% in_arcs_imp_in_arcs_ends
thf(fact_1218_awalk__verts__arc1__app,axiom,
! [E: b,R2: a,P1: list_b,P22: list_b] : ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ R2 @ ( append_b @ P1 @ ( cons_b @ E @ P22 ) ) ) ) ) ).
% awalk_verts_arc1_app
thf(fact_1219_awalk__Cons__iff,axiom,
! [U: a,E: b,Es2: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( cons_b @ E @ Es2 ) @ W )
= ( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( U
= ( pre_ta4931606617599662728t_unit @ t @ E ) )
& ( arc_pre_awalk_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es2 @ W ) ) ) ).
% awalk_Cons_iff
thf(fact_1220_awalk__verts_Osimps_I2_J,axiom,
! [U: a,E: b,Es2: list_b] :
( ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( cons_b @ E @ Es2 ) )
= ( cons_a @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( arc_pr7493981781705774526ts_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es2 ) ) ) ).
% awalk_verts.simps(2)
thf(fact_1221_arcE,axiom,
! [E: b,U: a,V: a] :
( ( wf_arc_a_b @ t @ E @ ( product_Pair_a_a @ U @ V ) )
=> ~ ( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( pre_ta4931606617599662728t_unit @ t @ E )
= U )
=> ( ( pre_he5236287464308401016t_unit @ t @ E )
!= V ) ) ) ) ).
% arcE
thf(fact_1222_euler__trail__conv__connected,axiom,
! [U: a,P2: list_b,V: a] :
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( pre_euler_trail_a_b @ t @ U @ P2 @ V )
= ( ( arc_pre_trail_a_b @ t @ U @ P2 @ V )
& ( ( set_b2 @ P2 )
= ( pre_ar1395965042833527383t_unit @ t ) ) ) ) ) ).
% euler_trail_conv_connected
thf(fact_1223_awalk__verts__in__verts,axiom,
! [U: a,P2: list_b,V: a] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_a @ V @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ) ).
% awalk_verts_in_verts
thf(fact_1224_awhd__in__verts,axiom,
! [U: a,P2: list_b] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% awhd_in_verts
thf(fact_1225_arc__implies__awalk,axiom,
! [E: b] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( arc_pre_awalk_a_b @ t @ ( pre_ta4931606617599662728t_unit @ t @ E ) @ ( cons_b @ E @ nil_b ) @ ( pre_he5236287464308401016t_unit @ t @ E ) ) ) ).
% arc_implies_awalk
thf(fact_1226_trail__Cons__iff,axiom,
! [U: a,E: b,Es2: list_b,W: a] :
( ( arc_pre_trail_a_b @ t @ U @ ( cons_b @ E @ Es2 ) @ W )
= ( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( U
= ( pre_ta4931606617599662728t_unit @ t @ E ) )
& ~ ( member_b @ E @ ( set_b2 @ Es2 ) )
& ( arc_pre_trail_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es2 @ W ) ) ) ).
% trail_Cons_iff
thf(fact_1227_awalk__vertex__props,axiom,
! [U: a,P2: list_b,V: a,P: a > $o,Q: a > $o] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( P2 != nil_b )
=> ( ! [W2: a] :
( ( member_a @ W2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ( ( P @ W2 )
| ( Q @ W2 ) ) )
=> ( ( P @ U )
=> ( ( Q @ V )
=> ? [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ P2 ) )
& ( P @ ( pre_ta4931606617599662728t_unit @ t @ X4 ) )
& ( Q @ ( pre_he5236287464308401016t_unit @ t @ X4 ) ) ) ) ) ) ) ) ).
% awalk_vertex_props
thf(fact_1228_awalk__induct__raw,axiom,
! [U: a,P2: list_b,V: a,P: a > list_b > a > $o] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ! [W1: a] :
( ( member_a @ W1 @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( P @ W1 @ nil_b @ W1 ) )
=> ( ! [W1: a,W22: a,E2: b,Es: list_b] :
( ( member_b @ E2 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( arc_to_ends_a_b @ t @ E2 )
= ( product_Pair_a_a @ W1 @ W22 ) )
=> ( ( P @ W22 @ Es @ V )
=> ( P @ W1 @ ( cons_b @ E2 @ Es ) @ V ) ) ) )
=> ( P @ U @ P2 @ V ) ) ) ) ).
% awalk_induct_raw
thf(fact_1229_trail__connected,axiom,
! [U: a,P2: list_b,V: a] :
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( arc_pre_trail_a_b @ t @ U @ P2 @ V )
=> ( ( ( set_b2 @ P2 )
!= ( pre_ar1395965042833527383t_unit @ t ) )
=> ? [E2: b] :
( ( member_b @ E2 @ ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( set_b2 @ P2 ) ) )
& ( ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
| ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ) ) ) ) ).
% trail_connected
thf(fact_1230_awalk__connected,axiom,
! [U: a,P2: list_b,V: a] :
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( ( set_b2 @ P2 )
!= ( pre_ar1395965042833527383t_unit @ t ) )
=> ? [E2: b] :
( ( member_b @ E2 @ ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( set_b2 @ P2 ) ) )
& ( ( member_a @ ( pre_ta4931606617599662728t_unit @ t @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
| ( member_a @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) ) ) ) ) ) ) ).
% awalk_connected
thf(fact_1231_nomulti_Ono__multi__arcs,axiom,
! [E1: b,E22: b] :
( ( member_b @ E1 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( member_b @ E22 @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( arc_to_ends_a_b @ t @ E1 )
= ( arc_to_ends_a_b @ t @ E22 ) )
=> ( E1 = E22 ) ) ) ) ).
% nomulti.no_multi_arcs
thf(fact_1232_dominatesI,axiom,
! [A: b,U: a,V: a] :
( ( ( arc_to_ends_a_b @ t @ A )
= ( product_Pair_a_a @ U @ V ) )
=> ( ( member_b @ A @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% dominatesI
thf(fact_1233_awalk__ConsI,axiom,
! [V: a,Es2: list_b,W: a,E: b,U: a] :
( ( arc_pre_awalk_a_b @ t @ V @ Es2 @ W )
=> ( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( arc_to_ends_a_b @ t @ E )
= ( product_Pair_a_a @ U @ V ) )
=> ( arc_pre_awalk_a_b @ t @ U @ ( cons_b @ E @ Es2 ) @ W ) ) ) ) ).
% awalk_ConsI
thf(fact_1234_awalkI,axiom,
! [U: a,P2: list_b,V: a] :
( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( arc_pre_awalk_a_b @ t @ U @ P2 @ V ) ) ) ) ).
% awalkI
thf(fact_1235_cas__ends,axiom,
! [U: a,P2: list_b,V: a,U4: a,V5: a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( arc_pre_cas_a_b @ t @ U4 @ P2 @ V5 )
=> ( ( ( P2 != nil_b )
& ( U = U4 )
& ( V = V5 ) )
| ( ( P2 = nil_b )
& ( U = V )
& ( U4 = V5 ) ) ) ) ) ).
% cas_ends
thf(fact_1236_cas_Osimps_I1_J,axiom,
! [U: a,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ nil_b @ V )
= ( U = V ) ) ).
% cas.simps(1)
thf(fact_1237_awhd__if__cas,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= U ) ) ).
% awhd_if_cas
thf(fact_1238_tail__and__head__eq__impl__cas,axiom,
! [X2: a,P2: list_b,Y2: a,G3: pre_pr7278220950009878019t_unit] :
( ( arc_pre_cas_a_b @ t @ X2 @ P2 @ Y2 )
=> ( ! [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ P2 ) )
=> ( ( pre_ta4931606617599662728t_unit @ t @ X4 )
= ( pre_ta4931606617599662728t_unit @ G3 @ X4 ) ) )
=> ( ! [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ P2 ) )
=> ( ( pre_he5236287464308401016t_unit @ t @ X4 )
= ( pre_he5236287464308401016t_unit @ G3 @ X4 ) ) )
=> ( arc_pre_cas_a_b @ G3 @ X2 @ P2 @ Y2 ) ) ) ) ).
% tail_and_head_eq_impl_cas
thf(fact_1239_cas_Osimps_I2_J,axiom,
! [U: a,E: b,Es2: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ ( cons_b @ E @ Es2 ) @ V )
= ( ( ( pre_ta4931606617599662728t_unit @ t @ E )
= U )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E ) @ Es2 @ V ) ) ) ).
% cas.simps(2)
thf(fact_1240_cas_Oelims_I3_J,axiom,
! [X2: a,Xa3: list_b,Xb2: a] :
( ~ ( arc_pre_cas_a_b @ t @ X2 @ Xa3 @ Xb2 )
=> ( ( ( Xa3 = nil_b )
=> ( X2 = Xb2 ) )
=> ~ ! [E2: b,Es: list_b] :
( ( Xa3
= ( cons_b @ E2 @ Es ) )
=> ( ( ( pre_ta4931606617599662728t_unit @ t @ E2 )
= X2 )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ Es @ Xb2 ) ) ) ) ) ).
% cas.elims(3)
thf(fact_1241_cas_Oelims_I2_J,axiom,
! [X2: a,Xa3: list_b,Xb2: a] :
( ( arc_pre_cas_a_b @ t @ X2 @ Xa3 @ Xb2 )
=> ( ( ( Xa3 = nil_b )
=> ( X2 != Xb2 ) )
=> ~ ! [E2: b,Es: list_b] :
( ( Xa3
= ( cons_b @ E2 @ Es ) )
=> ~ ( ( ( pre_ta4931606617599662728t_unit @ t @ E2 )
= X2 )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ Es @ Xb2 ) ) ) ) ) ).
% cas.elims(2)
thf(fact_1242_cas_Oelims_I1_J,axiom,
! [X2: a,Xa3: list_b,Xb2: a,Y2: $o] :
( ( ( arc_pre_cas_a_b @ t @ X2 @ Xa3 @ Xb2 )
= Y2 )
=> ( ( ( Xa3 = nil_b )
=> ( Y2
= ( X2 != Xb2 ) ) )
=> ~ ! [E2: b,Es: list_b] :
( ( Xa3
= ( cons_b @ E2 @ Es ) )
=> ( Y2
= ( ~ ( ( ( pre_ta4931606617599662728t_unit @ t @ E2 )
= X2 )
& ( arc_pre_cas_a_b @ t @ ( pre_he5236287464308401016t_unit @ t @ E2 ) @ Es @ Xb2 ) ) ) ) ) ) ) ).
% cas.elims(1)
thf(fact_1243_cas__induce,axiom,
! [U: a,P2: list_b,V: a,S: set_a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ S )
=> ( arc_pre_cas_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ U @ P2 @ V ) ) ) ).
% cas_induce
thf(fact_1244_awalk__decomp__verts,axiom,
! [U: a,P2: list_b,V: a,Xs: list_a,Y2: a,Ys: list_a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 )
= ( append_a @ Xs @ ( cons_a @ Y2 @ Ys ) ) )
=> ~ ! [Q3: list_b] :
( ( arc_pre_cas_a_b @ t @ U @ Q3 @ Y2 )
=> ! [R: list_b] :
( ( arc_pre_cas_a_b @ t @ Y2 @ R @ V )
=> ( ( P2
= ( append_b @ Q3 @ R ) )
=> ( ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ Q3 )
= ( append_a @ Xs @ ( cons_a @ Y2 @ nil_a ) ) )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ Y2 @ R )
!= ( cons_a @ Y2 @ Ys ) ) ) ) ) ) ) ) ).
% awalk_decomp_verts
thf(fact_1245_awalk__def,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
= ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( arc_pre_cas_a_b @ t @ U @ P2 @ V ) ) ) ).
% awalk_def
thf(fact_1246_cas__append__if,axiom,
! [X2: a,Ps2: list_b,U: a,P2: b,V: a] :
( ( arc_pre_cas_a_b @ t @ X2 @ Ps2 @ U )
=> ( ( ( pre_ta4931606617599662728t_unit @ t @ P2 )
= U )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ P2 )
= V )
=> ( arc_pre_cas_a_b @ t @ X2 @ ( append_b @ Ps2 @ ( cons_b @ P2 @ nil_b ) ) @ V ) ) ) ) ).
% cas_append_if
thf(fact_1247_before2__def,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_1040310085189658461e2_a_b @ t @ S1 @ S2 )
= ( ( iKKBZ_4622586873178280511rm_a_b @ t @ S1 )
& ( iKKBZ_4622586873178280511rm_a_b @ t @ S2 )
& ( ( inf_inf_set_a @ ( set_a2 @ S1 ) @ ( set_a2 @ S2 ) )
= bot_bot_set_a )
& ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ S1 ) )
& ? [Y4: a] :
( ( member_a @ Y4 @ ( set_a2 @ S2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y4 ) @ ( arcs_ends_a_b @ t ) ) ) )
& ! [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ S1 ) )
=> ! [Y4: a] :
( ( member_a @ Y4 @ ( minus_minus_set_a @ ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ t ) @ ( set_a2 @ S1 ) ) @ ( set_a2 @ S2 ) ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y4 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% before2_def
thf(fact_1248_awalk__conv,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
= ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
& ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
& ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= U )
& ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= V )
& ( arc_pre_cas_a_b @ t @ U @ P2 @ V ) ) ) ).
% awalk_conv
thf(fact_1249_awalkE_H,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ~ ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= U )
=> ( ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= V )
=> ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ~ ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ) ) ) ) ) ).
% awalkE'
thf(fact_1250_awlast__append,axiom,
! [U: a,P2: list_b,Q2: list_b] :
( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P2 @ Q2 ) ) )
= ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ Q2 ) ) ) ).
% awlast_append
thf(fact_1251_awlast__if__cas,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= V ) ) ).
% awlast_if_cas
thf(fact_1252_to__list__tree__cas,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
= ( arc_pre_cas_list_a_b @ ( direct3773525127397338803ee_a_b @ t ) @ ( cons_a @ U @ nil_a ) @ P2 @ ( cons_a @ V @ nil_a ) ) ) ).
% to_list_tree_cas
thf(fact_1253_reachable__vwalk__conv,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
= ( ? [P3: list_a] :
( ( vertex_vwalk_a_b @ P3 @ t )
& ( ( hd_a @ P3 )
= U )
& ( ( last_a @ P3 )
= V ) ) ) ) ).
% reachable_vwalk_conv
thf(fact_1254_reachable__vpath__conv,axiom,
! [U: a,V: a] :
( ( reachable_a_b @ t @ U @ V )
= ( ? [P3: list_a] :
( ( vertex_vpath_a_b @ P3 @ t )
& ( ( hd_a @ P3 )
= U )
& ( ( last_a @ P3 )
= V ) ) ) ) ).
% reachable_vpath_conv
thf(fact_1255_awlast__in__verts,axiom,
! [U: a,P2: list_b] :
( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( member_a @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% awlast_in_verts
thf(fact_1256_awalkE,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ~ ( ( ord_less_eq_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( ord_less_eq_set_b @ ( set_b2 @ P2 ) @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( ( hd_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= U )
=> ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
!= V ) ) ) ) ) ) ).
% awalkE
thf(fact_1257_awalk__append__iff,axiom,
! [U: a,P2: list_b,Q2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( append_b @ P2 @ Q2 ) @ V )
= ( ( arc_pre_awalk_a_b @ t @ U @ P2 @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
& ( arc_pre_awalk_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ Q2 @ V ) ) ) ).
% awalk_append_iff
thf(fact_1258_cas__append__iff,axiom,
! [U: a,P2: list_b,Q2: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ ( append_b @ P2 @ Q2 ) @ V )
= ( ( arc_pre_cas_a_b @ t @ U @ P2 @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
& ( arc_pre_cas_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ Q2 @ V ) ) ) ).
% cas_append_iff
thf(fact_1259_awalk__verts__append2,axiom,
! [U: a,P2: list_b,Q2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ ( append_b @ P2 @ Q2 ) @ V )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P2 @ Q2 ) )
= ( append_a @ ( butlast_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( arc_pr7493981781705774526ts_a_b @ t @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ Q2 ) ) ) ) ).
% awalk_verts_append2
thf(fact_1260_awalk__del__vert,axiom,
! [U: a,P2: list_b,V: a,X2: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ~ ( member_a @ X2 @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) )
=> ( arc_pre_awalk_a_b @ ( pre_del_vert_a_b @ t @ X2 ) @ U @ P2 @ V ) ) ) ).
% awalk_del_vert
thf(fact_1261_awlast__of__awalk,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( nOMATCH_a @ ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ V )
=> ( ( last_a @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= V ) ) ) ).
% awlast_of_awalk
thf(fact_1262_head__del__vert,axiom,
! [U: a] :
( ( pre_he5236287464308401016t_unit @ ( pre_del_vert_a_b @ t @ U ) )
= ( pre_he5236287464308401016t_unit @ t ) ) ).
% head_del_vert
thf(fact_1263_tail__del__vert,axiom,
! [U: a] :
( ( pre_ta4931606617599662728t_unit @ ( pre_del_vert_a_b @ t @ U ) )
= ( pre_ta4931606617599662728t_unit @ t ) ) ).
% tail_del_vert
thf(fact_1264_ends__del__vert,axiom,
! [U: a] :
( ( arc_to_ends_a_b @ ( pre_del_vert_a_b @ t @ U ) )
= ( arc_to_ends_a_b @ t ) ) ).
% ends_del_vert
thf(fact_1265_del__vert__add__vert,axiom,
! [U: a] :
( ~ ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( pre_del_vert_a_b @ ( pre_add_vert_a_b @ t @ U ) @ U )
= t ) ) ).
% del_vert_add_vert
thf(fact_1266_arcs__del__leaf,axiom,
! [E: b,V: a] :
( ( member_b @ E @ ( pre_ar1395965042833527383t_unit @ t ) )
=> ( ( ( pre_he5236287464308401016t_unit @ t @ E )
= V )
=> ( ( shorte1213025427933718126af_a_b @ t @ V )
=> ( ( pre_ar1395965042833527383t_unit @ ( pre_del_vert_a_b @ t @ V ) )
= ( minus_minus_set_b @ ( pre_ar1395965042833527383t_unit @ t ) @ ( insert_b @ E @ bot_bot_set_b ) ) ) ) ) ) ).
% arcs_del_leaf
thf(fact_1267_awalk__verts__conv_H,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( ( P2 = nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 )
= ( cons_a @ U @ nil_a ) ) )
& ( ( P2 != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 )
= ( cons_a @ ( pre_ta4931606617599662728t_unit @ t @ ( hd_b @ P2 ) ) @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P2 ) ) ) ) ) ) ).
% awalk_verts_conv'
thf(fact_1268_awalk__verts__conv,axiom,
! [P2: list_b,U: a] :
( ( ( P2 = nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 )
= ( cons_a @ U @ nil_a ) ) )
& ( ( P2 != nil_b )
=> ( ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 )
= ( append_a @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) @ ( cons_a @ ( pre_he5236287464308401016t_unit @ t @ ( last_b @ P2 ) ) @ nil_a ) ) ) ) ) ).
% awalk_verts_conv
thf(fact_1269_set__awalk__verts__not__Nil__cas,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( P2 != nil_b )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= ( sup_sup_set_a @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P2 ) ) ) ) ) ) ).
% set_awalk_verts_not_Nil_cas
thf(fact_1270_verts__add__vert,axiom,
! [U: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_add_vert_a_b @ t @ U ) )
= ( insert_a @ U @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% verts_add_vert
thf(fact_1271_verts__del__vert,axiom,
! [U: a] :
( ( pre_ve642382030648772252t_unit @ ( pre_del_vert_a_b @ t @ U ) )
= ( minus_minus_set_a @ ( pre_ve642382030648772252t_unit @ t ) @ ( insert_a @ U @ bot_bot_set_a ) ) ) ).
% verts_del_vert
thf(fact_1272_set__awalk__verts__append,axiom,
! [U: a,P2: list_b,V: a,Q2: list_b,W: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( arc_pre_awalk_a_b @ t @ V @ Q2 @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P2 @ Q2 ) ) )
= ( sup_sup_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ V @ Q2 ) ) ) ) ) ) ).
% set_awalk_verts_append
thf(fact_1273_set__awalk__verts__append__cas,axiom,
! [U: a,P2: list_b,V: a,Q2: list_b,W: a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( arc_pre_cas_a_b @ t @ V @ Q2 @ W )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ ( append_b @ P2 @ Q2 ) ) )
= ( sup_sup_set_a @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) ) @ ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ V @ Q2 ) ) ) ) ) ) ).
% set_awalk_verts_append_cas
thf(fact_1274_connected__arcs__empty,axiom,
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( ( pre_ar1395965042833527383t_unit @ t )
= bot_bot_set_b )
=> ( ( ( pre_ve642382030648772252t_unit @ t )
!= bot_bot_set_a )
=> ~ ! [V4: a] :
( ( pre_ve642382030648772252t_unit @ t )
!= ( insert_a @ V4 @ bot_bot_set_a ) ) ) ) ) ).
% connected_arcs_empty
thf(fact_1275_connected__verts,axiom,
( ( digrap8783888973171253482ed_a_b @ t )
=> ( ( ( pre_ar1395965042833527383t_unit @ t )
!= bot_bot_set_b )
=> ( ( pre_ve642382030648772252t_unit @ t )
= ( sup_sup_set_a @ ( image_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ ( pre_ar1395965042833527383t_unit @ t ) ) @ ( image_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ ( pre_ar1395965042833527383t_unit @ t ) ) ) ) ) ) ).
% connected_verts
thf(fact_1276_set__awalk__verts,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= ( sup_sup_set_a @ ( sup_sup_set_a @ ( insert_a @ U @ bot_bot_set_a ) @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P2 ) ) ) ) ) ).
% set_awalk_verts
thf(fact_1277_set__awalk__verts__cas,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_cas_a_b @ t @ U @ P2 @ V )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= ( sup_sup_set_a @ ( sup_sup_set_a @ ( insert_a @ U @ bot_bot_set_a ) @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P2 ) ) ) ) ) ).
% set_awalk_verts_cas
thf(fact_1278_set__awalk__verts__not__Nil,axiom,
! [U: a,P2: list_b,V: a] :
( ( arc_pre_awalk_a_b @ t @ U @ P2 @ V )
=> ( ( P2 != nil_b )
=> ( ( set_a2 @ ( arc_pr7493981781705774526ts_a_b @ t @ U @ P2 ) )
= ( sup_sup_set_a @ ( set_a2 @ ( map_b_a @ ( pre_ta4931606617599662728t_unit @ t ) @ P2 ) ) @ ( set_a2 @ ( map_b_a @ ( pre_he5236287464308401016t_unit @ t ) @ P2 ) ) ) ) ) ) ).
% set_awalk_verts_not_Nil
% Conjectures (1)
thf(conj_0,conjecture,
! [X4: a] :
( ~ ( member_a @ X4 @ ( set_a2 @ v1 ) )
| ! [Xa: a] :
( ~ ( member_a @ Xa @ ( set_a2 @ v2 ) )
| ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
%------------------------------------------------------------------------------