TPTP Problem File: SLH0250^1.p
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- 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_03688_165792__15850660_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1520 ( 508 unt; 254 typ; 0 def)
% Number of atoms : 3866 (1336 equ; 0 cnn)
% Maximal formula atoms : 20 ( 3 avg)
% Number of connectives : 15890 ( 577 ~; 93 |; 403 &;13012 @)
% ( 0 <=>;1805 =>; 0 <=; 0 <~>)
% Maximal formula depth : 32 ( 8 avg)
% Number of types : 51 ( 50 usr)
% Number of type conns : 480 ( 480 >; 0 *; 0 +; 0 <<)
% Number of symbols : 205 ( 204 usr; 29 con; 0-5 aty)
% Number of variables : 4075 ( 75 ^;3567 !; 433 ?;4075 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 16:08:13.243
%------------------------------------------------------------------------------
% Could-be-implicit typings (50)
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% Explicit typings (204)
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_001tf__b,type,
produc5719641485658034180_a_b_b: produc6499617310964463488_a_b_b > b ).
thf(sy_c_Set_OCollect_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
collec2944820760411501129st_a_b: ( dtree_list_a_b > $o ) > set_dtree_list_a_b ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OCollect_001tf__b,type,
collect_b: ( b > $o ) > set_b ).
thf(sy_c_Set_Oimage_001t__List__Olist_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_list_a_set_a: ( list_a > set_a ) > set_list_a > set_set_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
image_5965465251548763643st_a_b: ( produc6499617310964463488_a_b_b > dtree_list_a_b ) > set_Pr3443975907877334966_a_b_b > set_dtree_list_a_b ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J_001tf__b,type,
image_4684437738885282872_b_b_b: ( produc6499617310964463488_a_b_b > b ) > set_Pr3443975907877334966_a_b_b > set_b ).
thf(sy_c_Set_Oimage_001tf__a_001t__Set__Oset_Itf__a_J,type,
image_a_set_a: ( a > set_a ) > set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_a ).
thf(sy_c_Set_Oinsert_001tf__b,type,
insert_b: b > set_b > set_b ).
thf(sy_c_Shortest__Path__Tree_Opre__digraph_Oleaf_001tf__a_001tf__b,type,
shorte1213025427933718126af_a_b: pre_pr7278220950009878019t_unit > a > $o ).
thf(sy_c_Shortest__Path__Tree_Osubgraph_001tf__a_001tf__b,type,
shorte3657265928840388360ph_a_b: pre_pr7278220950009878019t_unit > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Sublist_Osublist_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
sublis1540088274494349249st_a_b: list_dtree_list_a_b > list_dtree_list_a_b > $o ).
thf(sy_c_Sublist_Osublist_001t__List__Olist_Itf__a_J,type,
sublist_list_a: list_list_a > list_list_a > $o ).
thf(sy_c_Sublist_Osublist_001t__Set__Oset_Itf__a_J,type,
sublist_set_a: list_set_a > list_set_a > $o ).
thf(sy_c_Sublist_Osublist_001tf__a,type,
sublist_a: list_a > list_a > $o ).
thf(sy_c_Sublist_Osublist_001tf__b,type,
sublist_b: list_b > list_b > $o ).
thf(sy_c_Transitive__Closure_Otrancl_001tf__a,type,
transitive_trancl_a: set_Product_prod_a_a > set_Product_prod_a_a ).
thf(sy_c_Vertex__Walk_Ovpath_001t__List__Olist_Itf__a_J_001tf__b,type,
vertex6060786982766068989st_a_b: list_list_a > pre_pr2882871181989701257t_unit > $o ).
thf(sy_c_Vertex__Walk_Ovpath_001tf__a_001tf__b,type,
vertex_vpath_a_b: list_a > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_Vertex__Walk_Ovwalk_001t__List__Olist_Itf__a_J_001tf__b,type,
vertex2966258834163962945st_a_b: list_list_a > pre_pr2882871181989701257t_unit > $o ).
thf(sy_c_Vertex__Walk_Ovwalk_001tf__a_001tf__b,type,
vertex_vwalk_a_b: list_a > pre_pr7278220950009878019t_unit > $o ).
thf(sy_c_member_001t__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J,type,
member551035911493665803st_a_b: dtree_list_a_b > set_dtree_list_a_b > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__b_J,type,
member_list_b: list_b > set_list_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__List__Olist_Itf__a_J_Mtf__b_J_Mtf__b_J,type,
member4695696432722591383_a_b_b: produc6499617310964463488_a_b_b > set_Pr3443975907877334966_a_b_b > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
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_Itf__a_J,type,
member_set_a: set_a > set_set_a > $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_e2____,type,
e2: b ).
thf(sy_v_e____,type,
e: b ).
thf(sy_v_r,type,
r: list_a ).
thf(sy_v_r1____,type,
r1: list_a ).
thf(sy_v_r2____,type,
r2: a ).
thf(sy_v_rank,type,
rank: list_a > real ).
thf(sy_v_t,type,
t2: dtree_list_a_b ).
thf(sy_v_t1,type,
t1: dtree_list_a_b ).
thf(sy_v_t2____,type,
t22: dtree_list_a_b ).
thf(sy_v_ta____,type,
ta: dtree_list_a_b ).
thf(sy_v_v____,type,
v: list_a ).
thf(sy_v_xs,type,
xs: fset_P2153231429829016240_a_b_b ).
% Relevant facts (1265)
thf(fact_0_merge1_Ocases,axiom,
! [X: dtree_list_a_b] :
~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ).
% merge1.cases
thf(fact_1__092_060open_062r2_A_092_060in_062_Aset_Ar1_A_092_060union_062_Aset_A_Idtree_Oroot_At_J_092_060close_062,axiom,
member_a @ r2 @ ( sup_sup_set_a @ ( set_a2 @ r1 ) @ ( set_a2 @ ( root_list_a_b @ ta ) ) ) ).
% \<open>r2 \<in> set r1 \<union> set (dtree.root t)\<close>
thf(fact_2_r2__def_I2_J,axiom,
member1426531477525435216od_a_a @ ( product_Pair_a_a @ r2 @ ( hd_a @ ( root_list_a_b @ t1 ) ) ) @ ( arcs_ends_a_b @ t ) ).
% r2_def(2)
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_loopfree_Oloopfree__digraph__axioms,axiom,
loopfree_digraph_a_b @ t ).
% loopfree.loopfree_digraph_axioms
thf(fact_5_nomulti_Onomulti__digraph__axioms,axiom,
nomulti_digraph_a_b @ t ).
% nomulti.nomulti_digraph_axioms
thf(fact_6_eq,axiom,
( ( node_list_a_b @ r @ xs )
= ( node_list_a_b @ ( append_a @ r1 @ ( root_list_a_b @ ta ) ) @ ( sucs_list_a_b @ ta ) ) ) ).
% eq
thf(fact_7_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_8_before__arc__to__hd,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ Xs2 @ Ys )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ ( hd_a @ Ys ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% before_arc_to_hd
thf(fact_9_before__ArcI,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ S1 @ S2 )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ S1 ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ S2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% before_ArcI
thf(fact_10_cycle__free,axiom,
~ ? [X_1: list_b] : ( arc_pre_cycle_a_b @ t @ X_1 ) ).
% cycle_free
thf(fact_11_forward__arc__to__head_H,axiom,
! [Ys: list_a,X: a,Y: a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( member_a @ Y @ ( set_a2 @ Ys ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( Y
= ( hd_a @ Ys ) ) ) ) ) ) ).
% forward_arc_to_head'
thf(fact_12_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_13_reachable1__from__outside__dom,axiom,
! [X: a,Y: a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( member_a @ Y @ ( set_a2 @ Ys ) )
=> ? [X3: a,X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Ys ) )
& ~ ( member_a @ X3 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ X2 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% reachable1_from_outside_dom
thf(fact_14__092_060open_062dtree_Oroot_At1_A_092_060noteq_062_A_091_093_092_060close_062,axiom,
( ( root_list_a_b @ t1 )
!= nil_a ) ).
% \<open>dtree.root t1 \<noteq> []\<close>
thf(fact_15_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_16_forward__split,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ Xs2 @ Ys ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs2 ) ) ).
% forward_split
thf(fact_17_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_18_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_19_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_20_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_21_reachable1__not__reverse,axiom,
! [X: a,Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ).
% reachable1_not_reverse
thf(fact_22_move__mid__backward__if__noarc,axiom,
! [U2: list_a,V2: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ U2 @ V2 )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ V2 @ ( append_a @ Bs @ Cs ) ) ) ) ) ) ) ).
% move_mid_backward_if_noarc
thf(fact_23_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_24_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_25_move__mid__forward__if__noarc,axiom,
! [As: list_a,U2: list_a,Bs: list_a,Cs: list_a] :
( ( As != nil_a )
=> ( ~ ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ U2 ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ Bs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ 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_26_reachable1__append__old__if__arc,axiom,
! [Xs2: list_a,Ys: list_a,Z: a,Y: a] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ~ ( member_a @ Z @ ( set_a2 @ Xs2 ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs2 )
=> ( ( member_a @ Y @ ( set_a2 @ ( append_a @ Xs2 @ Ys ) ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% reachable1_append_old_if_arc
thf(fact_27_forward__app,axiom,
! [S1: list_a,S2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ S1 )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ S2 )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ S1 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ ( hd_a @ S2 ) ) @ ( arcs_ends_a_b @ t ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ S1 @ S2 ) ) ) ) ) ).
% forward_app
thf(fact_28_hd__reachable1__from__outside_H,axiom,
! [X: a,Y: a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( member_a @ Y @ ( set_a2 @ Ys ) )
=> ( ? [X2: a] : ( member_a @ X2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ ( hd_a @ Ys ) ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ).
% hd_reachable1_from_outside'
thf(fact_29_r2__def_I1_J,axiom,
member_a @ r2 @ ( sup_sup_set_a @ ( set_a2 @ r1 ) @ ( iKKBZ_6987179986356532253ts_a_b @ ta @ ( hd_a @ ( root_list_a_b @ t1 ) ) ) ) ).
% r2_def(1)
thf(fact_30_directed__tree_Obefore_Ocong,axiom,
iKKBZ_7682935289300565975re_a_b = iKKBZ_7682935289300565975re_a_b ).
% directed_tree.before.cong
thf(fact_31_directed__tree_Oforward_Ocong,axiom,
iKKBZ_4778857019735642799rd_a_b = iKKBZ_4778857019735642799rd_a_b ).
% directed_tree.forward.cong
thf(fact_32_closed__w__imp__cycle,axiom,
! [P: list_b] :
( ( arc_wf_closed_w_a_b @ t @ P )
=> ? [X_12: list_b] : ( arc_pre_cycle_a_b @ t @ X_12 ) ) ).
% closed_w_imp_cycle
thf(fact_33__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062r2_O_A_092_060lbrakk_062r2_A_092_060in_062_Aset_Ar1_A_092_060union_062_Apath__lverts_At_A_Ihd_A_Idtree_Oroot_At1_J_J_059_Ar2_A_092_060rightarrow_062_092_060_094bsub_062T_092_060_094esub_062_Ahd_A_Idtree_Oroot_At1_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [R2: a] :
( ( member_a @ R2 @ ( sup_sup_set_a @ ( set_a2 @ r1 ) @ ( iKKBZ_6987179986356532253ts_a_b @ ta @ ( hd_a @ ( root_list_a_b @ t1 ) ) ) ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ R2 @ ( hd_a @ ( root_list_a_b @ t1 ) ) ) @ ( arcs_ends_a_b @ t ) ) ) ).
% \<open>\<And>thesis. (\<And>r2. \<lbrakk>r2 \<in> set r1 \<union> path_lverts t (hd (dtree.root t1)); r2 \<rightarrow>\<^bsub>T\<^esub> hd (dtree.root t1)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_34_dtree_Ocollapse,axiom,
! [Dtree: dtree_list_a_b] :
( ( node_list_a_b @ ( root_list_a_b @ Dtree ) @ ( sucs_list_a_b @ Dtree ) )
= Dtree ) ).
% dtree.collapse
thf(fact_35_hd__append2,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( Xs2 != nil_a )
=> ( ( hd_a @ ( append_a @ Xs2 @ Ys ) )
= ( hd_a @ Xs2 ) ) ) ).
% hd_append2
thf(fact_36_hd__append2,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( Xs2 != nil_b )
=> ( ( hd_b @ ( append_b @ Xs2 @ Ys ) )
= ( hd_b @ Xs2 ) ) ) ).
% hd_append2
thf(fact_37_set__append,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( set_b2 @ ( append_b @ Xs2 @ Ys ) )
= ( sup_sup_set_b @ ( set_b2 @ Xs2 ) @ ( set_b2 @ Ys ) ) ) ).
% set_append
thf(fact_38_set__append,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( set_list_a2 @ ( append_list_a @ Xs2 @ Ys ) )
= ( sup_sup_set_list_a @ ( set_list_a2 @ Xs2 ) @ ( set_list_a2 @ Ys ) ) ) ).
% set_append
thf(fact_39_set__append,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( set_a2 @ ( append_a @ Xs2 @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Ys ) ) ) ).
% set_append
thf(fact_40_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 ) )
=> ~ ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Bs ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ As ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ).
% no_back_reach1_if_fwd_dstct
thf(fact_41_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_42_merge__in__verts,axiom,
! [X: a] :
( ( member_a @ X @ ( graph_2957805489637798020ts_a_b @ t ) )
=> ( member_a @ X @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% merge_in_verts
thf(fact_43_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_44_append_Oright__neutral,axiom,
! [A: list_b] :
( ( append_b @ A @ nil_b )
= A ) ).
% append.right_neutral
thf(fact_45_append__Nil2,axiom,
! [Xs2: list_a] :
( ( append_a @ Xs2 @ nil_a )
= Xs2 ) ).
% append_Nil2
thf(fact_46_append__Nil2,axiom,
! [Xs2: list_b] :
( ( append_b @ Xs2 @ nil_b )
= Xs2 ) ).
% append_Nil2
thf(fact_47_append__self__conv,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_48_append__self__conv,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( ( append_b @ Xs2 @ Ys )
= Xs2 )
= ( Ys = nil_b ) ) ).
% append_self_conv
thf(fact_49_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_50_self__append__conv,axiom,
! [Y: list_b,Ys: list_b] :
( ( Y
= ( append_b @ Y @ Ys ) )
= ( Ys = nil_b ) ) ).
% self_append_conv
thf(fact_51_distinct__mid__unique2,axiom,
! [Xs2: list_a,U2: list_a,Ys: list_a,As: list_a,Bs: list_a] :
( ( distinct_a @ ( append_a @ Xs2 @ ( append_a @ U2 @ Ys ) ) )
=> ( ( U2 != nil_a )
=> ( ( ( append_a @ Xs2 @ ( append_a @ U2 @ Ys ) )
= ( append_a @ As @ ( append_a @ U2 @ Bs ) ) )
=> ( Ys = Bs ) ) ) ) ).
% distinct_mid_unique2
thf(fact_52_distinct__mid__unique2,axiom,
! [Xs2: list_b,U2: list_b,Ys: list_b,As: list_b,Bs: list_b] :
( ( distinct_b @ ( append_b @ Xs2 @ ( append_b @ U2 @ Ys ) ) )
=> ( ( U2 != nil_b )
=> ( ( ( append_b @ Xs2 @ ( append_b @ U2 @ Ys ) )
= ( append_b @ As @ ( append_b @ U2 @ Bs ) ) )
=> ( Ys = Bs ) ) ) ) ).
% distinct_mid_unique2
thf(fact_53_mem__Collect__eq,axiom,
! [A: dtree_list_a_b,P2: dtree_list_a_b > $o] :
( ( member551035911493665803st_a_b @ A @ ( collec2944820760411501129st_a_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A: list_a,P2: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_55_mem__Collect__eq,axiom,
! [A: set_a,P2: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A: b,P2: b > $o] :
( ( member_b @ A @ ( collect_b @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_57_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A2: set_dtree_list_a_b] :
( ( collec2944820760411501129st_a_b
@ ^ [X5: dtree_list_a_b] : ( member551035911493665803st_a_b @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_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_60_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_61_Collect__mem__eq,axiom,
! [A2: set_b] :
( ( collect_b
@ ^ [X5: b] : ( member_b @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X5: a] : ( member_a @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_63_distinct__mid__unique1,axiom,
! [Xs2: list_a,U2: list_a,Ys: list_a,As: list_a,Bs: list_a] :
( ( distinct_a @ ( append_a @ Xs2 @ ( append_a @ U2 @ Ys ) ) )
=> ( ( U2 != nil_a )
=> ( ( ( append_a @ Xs2 @ ( append_a @ U2 @ Ys ) )
= ( append_a @ As @ ( append_a @ U2 @ Bs ) ) )
=> ( As = Xs2 ) ) ) ) ).
% distinct_mid_unique1
thf(fact_64_distinct__mid__unique1,axiom,
! [Xs2: list_b,U2: list_b,Ys: list_b,As: list_b,Bs: list_b] :
( ( distinct_b @ ( append_b @ Xs2 @ ( append_b @ U2 @ Ys ) ) )
=> ( ( U2 != nil_b )
=> ( ( ( append_b @ Xs2 @ ( append_b @ U2 @ Ys ) )
= ( append_b @ As @ ( append_b @ U2 @ Bs ) ) )
=> ( As = Xs2 ) ) ) ) ).
% distinct_mid_unique1
thf(fact_65_dtree_Oinject,axiom,
! [X1: list_a,X22: fset_P2153231429829016240_a_b_b,Y1: list_a,Y2: fset_P2153231429829016240_a_b_b] :
( ( ( node_list_a_b @ X1 @ X22 )
= ( node_list_a_b @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% dtree.inject
thf(fact_66_same__append__eq,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= ( append_a @ Xs2 @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_67_same__append__eq,axiom,
! [Xs2: list_b,Ys: list_b,Zs: list_b] :
( ( ( append_b @ Xs2 @ Ys )
= ( append_b @ Xs2 @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_68_append__same__eq,axiom,
! [Ys: list_a,Xs2: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs2 )
= ( append_a @ Zs @ Xs2 ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_69_append__same__eq,axiom,
! [Ys: list_b,Xs2: list_b,Zs: list_b] :
( ( ( append_b @ Ys @ Xs2 )
= ( append_b @ Zs @ Xs2 ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_70_append__assoc,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs2 @ Ys ) @ Zs )
= ( append_a @ Xs2 @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_71_append__assoc,axiom,
! [Xs2: list_b,Ys: list_b,Zs: list_b] :
( ( append_b @ ( append_b @ Xs2 @ Ys ) @ Zs )
= ( append_b @ Xs2 @ ( append_b @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_72_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_73_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_74_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 ) )
=> ~ ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Bs ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ As ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% no_back_arc_if_fwd_dstct
thf(fact_75_append__is__Nil__conv,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= nil_a )
= ( ( Xs2 = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_76_append__is__Nil__conv,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( ( append_b @ Xs2 @ Ys )
= nil_b )
= ( ( Xs2 = nil_b )
& ( Ys = nil_b ) ) ) ).
% append_is_Nil_conv
thf(fact_77_Nil__is__append__conv,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_78_Nil__is__append__conv,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( nil_b
= ( append_b @ Xs2 @ Ys ) )
= ( ( Xs2 = nil_b )
& ( Ys = nil_b ) ) ) ).
% Nil_is_append_conv
thf(fact_79_self__append__conv2,axiom,
! [Y: list_a,Xs2: list_a] :
( ( Y
= ( append_a @ Xs2 @ Y ) )
= ( Xs2 = nil_a ) ) ).
% self_append_conv2
thf(fact_80_self__append__conv2,axiom,
! [Y: list_b,Xs2: list_b] :
( ( Y
= ( append_b @ Xs2 @ Y ) )
= ( Xs2 = nil_b ) ) ).
% self_append_conv2
thf(fact_81_append__self__conv2,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_a ) ) ).
% append_self_conv2
thf(fact_82_append__self__conv2,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( ( append_b @ Xs2 @ Ys )
= Ys )
= ( Xs2 = nil_b ) ) ).
% append_self_conv2
thf(fact_83_last__merge__is__merge,axiom,
! [Y: a] :
( ( member_a @ Y @ ( graph_2659413520663303054ts_a_b @ t ) )
=> ( member_a @ Y @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ).
% last_merge_is_merge
thf(fact_84_distinct_Osimps_I1_J,axiom,
distinct_a @ nil_a ).
% distinct.simps(1)
thf(fact_85_distinct_Osimps_I1_J,axiom,
distinct_b @ nil_b ).
% distinct.simps(1)
thf(fact_86_dverts__mset_Ocases,axiom,
! [X: dtree_list_a_b] :
~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ).
% dverts_mset.cases
thf(fact_87_dtree_Oexhaust,axiom,
! [Y: dtree_list_a_b] :
~ ! [X12: list_a,X23: fset_P2153231429829016240_a_b_b] :
( Y
!= ( node_list_a_b @ X12 @ X23 ) ) ).
% dtree.exhaust
thf(fact_88_append__eq__append__conv2,axiom,
! [Xs2: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us: list_a] :
( ( ( Xs2
= ( append_a @ Zs @ Us ) )
& ( ( append_a @ Us @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs2 @ Us )
= Zs )
& ( Ys
= ( append_a @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_89_append__eq__append__conv2,axiom,
! [Xs2: list_b,Ys: list_b,Zs: list_b,Ts: list_b] :
( ( ( append_b @ Xs2 @ Ys )
= ( append_b @ Zs @ Ts ) )
= ( ? [Us: list_b] :
( ( ( Xs2
= ( append_b @ Zs @ Us ) )
& ( ( append_b @ Us @ Ys )
= Ts ) )
| ( ( ( append_b @ Xs2 @ Us )
= Zs )
& ( Ys
= ( append_b @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_90_append__eq__appendI,axiom,
! [Xs2: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us2: list_a] :
( ( ( append_a @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us2 ) )
=> ( ( append_a @ Xs2 @ Ys )
= ( append_a @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_91_append__eq__appendI,axiom,
! [Xs2: list_b,Xs1: list_b,Zs: list_b,Ys: list_b,Us2: list_b] :
( ( ( append_b @ Xs2 @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_b @ Xs1 @ Us2 ) )
=> ( ( append_b @ Xs2 @ Ys )
= ( append_b @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_92_eq__Nil__appendI,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_93_eq__Nil__appendI,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( Xs2 = Ys )
=> ( Xs2
= ( append_b @ nil_b @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_94_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_95_append_Oleft__neutral,axiom,
! [A: list_b] :
( ( append_b @ nil_b @ A )
= A ) ).
% append.left_neutral
thf(fact_96_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_97_append__Nil,axiom,
! [Ys: list_b] :
( ( append_b @ nil_b @ Ys )
= Ys ) ).
% append_Nil
thf(fact_98_dtree_Osel_I1_J,axiom,
! [X1: list_a,X22: fset_P2153231429829016240_a_b_b] :
( ( root_list_a_b @ ( node_list_a_b @ X1 @ X22 ) )
= X1 ) ).
% dtree.sel(1)
thf(fact_99_dtree_Osel_I2_J,axiom,
! [X1: list_a,X22: fset_P2153231429829016240_a_b_b] :
( ( sucs_list_a_b @ ( node_list_a_b @ X1 @ X22 ) )
= X22 ) ).
% dtree.sel(2)
thf(fact_100_dtree_Oexpand,axiom,
! [Dtree: dtree_list_a_b,Dtree2: dtree_list_a_b] :
( ( ( ( root_list_a_b @ Dtree )
= ( root_list_a_b @ Dtree2 ) )
& ( ( sucs_list_a_b @ Dtree )
= ( sucs_list_a_b @ Dtree2 ) ) )
=> ( Dtree = Dtree2 ) ) ).
% dtree.expand
thf(fact_101_list_Oset__sel_I1_J,axiom,
! [A: list_dtree_list_a_b] :
( ( A != nil_dtree_list_a_b )
=> ( member551035911493665803st_a_b @ ( hd_dtree_list_a_b @ A ) @ ( set_dtree_list_a_b2 @ A ) ) ) ).
% list.set_sel(1)
thf(fact_102_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_103_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_104_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_105_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_106_hd__in__set,axiom,
! [Xs2: list_dtree_list_a_b] :
( ( Xs2 != nil_dtree_list_a_b )
=> ( member551035911493665803st_a_b @ ( hd_dtree_list_a_b @ Xs2 ) @ ( set_dtree_list_a_b2 @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_107_hd__in__set,axiom,
! [Xs2: list_set_a] :
( ( Xs2 != nil_set_a )
=> ( member_set_a @ ( hd_set_a @ Xs2 ) @ ( set_set_a2 @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_108_hd__in__set,axiom,
! [Xs2: list_b] :
( ( Xs2 != nil_b )
=> ( member_b @ ( hd_b @ Xs2 ) @ ( set_b2 @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_109_hd__in__set,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( member_a @ ( hd_a @ Xs2 ) @ ( set_a2 @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_110_hd__in__set,axiom,
! [Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( member_list_a @ ( hd_list_a @ Xs2 ) @ ( set_list_a2 @ Xs2 ) ) ) ).
% hd_in_set
thf(fact_111_hd__append,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( Xs2 = nil_a )
=> ( ( hd_a @ ( append_a @ Xs2 @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs2 != nil_a )
=> ( ( hd_a @ ( append_a @ Xs2 @ Ys ) )
= ( hd_a @ Xs2 ) ) ) ) ).
% hd_append
thf(fact_112_hd__append,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( ( Xs2 = nil_b )
=> ( ( hd_b @ ( append_b @ Xs2 @ Ys ) )
= ( hd_b @ Ys ) ) )
& ( ( Xs2 != nil_b )
=> ( ( hd_b @ ( append_b @ Xs2 @ Ys ) )
= ( hd_b @ Xs2 ) ) ) ) ).
% hd_append
thf(fact_113_longest__common__prefix,axiom,
! [Xs2: list_a,Ys: list_a] :
? [Ps: list_a,Xs3: list_a,Ys2: list_a] :
( ( Xs2
= ( append_a @ Ps @ Xs3 ) )
& ( Ys
= ( append_a @ Ps @ Ys2 ) )
& ( ( Xs3 = nil_a )
| ( Ys2 = nil_a )
| ( ( hd_a @ Xs3 )
!= ( hd_a @ Ys2 ) ) ) ) ).
% longest_common_prefix
thf(fact_114_longest__common__prefix,axiom,
! [Xs2: list_b,Ys: list_b] :
? [Ps: list_b,Xs3: list_b,Ys2: list_b] :
( ( Xs2
= ( append_b @ Ps @ Xs3 ) )
& ( Ys
= ( append_b @ Ps @ Ys2 ) )
& ( ( Xs3 = nil_b )
| ( Ys2 = nil_b )
| ( ( hd_b @ Xs3 )
!= ( hd_b @ Ys2 ) ) ) ) ).
% longest_common_prefix
thf(fact_115_dtree_Oexhaust__sel,axiom,
! [Dtree: dtree_list_a_b] :
( Dtree
= ( node_list_a_b @ ( root_list_a_b @ Dtree ) @ ( sucs_list_a_b @ Dtree ) ) ) ).
% dtree.exhaust_sel
thf(fact_116_dverts__reachable1__if__dom__children__aux__root,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,R0: list_a,X6: set_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) ) )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( set_a2 @ R0 ) @ X6 ) @ ( iKKBZ_6987179986356532253ts_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ ( hd_a @ X2 ) ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ ( hd_a @ X2 ) ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ X6 )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ R0 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ R3 )
=> ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ R3 ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ R0 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ).
% dverts_reachable1_if_dom_children_aux_root
thf(fact_117_dverts__reachable1__if__dom__children__aux,axiom,
! [T1: dtree_list_a_b,R0: list_a,X6: set_a,V: list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ T1 ) )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( set_a2 @ R0 ) @ X6 ) @ ( iKKBZ_6987179986356532253ts_a_b @ T1 @ ( hd_a @ X2 ) ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ ( hd_a @ X2 ) ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ X6 )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ R0 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ T1 ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ X2 ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ V ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ R0 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% dverts_reachable1_if_dom_children_aux
thf(fact_118_merge__in__supergraph,axiom,
! [C2: pre_pr7278220950009878019t_unit,X: a] :
( ( shorte3657265928840388360ph_a_b @ C2 @ t )
=> ( ( member_a @ X @ ( graph_2957805489637798020ts_a_b @ C2 ) )
=> ( member_a @ X @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ) ).
% merge_in_supergraph
thf(fact_119_no__back__reach1__if__fwd__dstct__bs,axiom,
! [As: list_a,Bs: list_list_a,V2: list_a,Cs: list_a,Xs2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ ( concat_a @ Bs ) @ ( append_a @ V2 @ Cs ) ) ) )
=> ( ( distinct_a @ ( append_a @ As @ ( append_a @ ( concat_a @ Bs ) @ ( append_a @ V2 @ Cs ) ) ) )
=> ( ( member_list_a @ Xs2 @ ( set_list_a2 @ Bs ) )
=> ~ ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ V2 ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ).
% no_back_reach1_if_fwd_dstct_bs
thf(fact_120_in__set__inner__verts__appendI__l,axiom,
! [U: a,P: list_b,Q: list_b] :
( ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ P ) ) )
=> ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ ( append_b @ P @ Q ) ) ) ) ) ).
% in_set_inner_verts_appendI_l
thf(fact_121_in__set__inner__verts__appendI__r,axiom,
! [U: a,Q: list_b,P: list_b] :
( ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ Q ) ) )
=> ( member_a @ U @ ( set_a2 @ ( pre_inner_verts_a_b @ t @ ( append_b @ P @ Q ) ) ) ) ) ).
% in_set_inner_verts_appendI_r
thf(fact_122_seq__conform__if__before,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ Xs2 @ Ys )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ ( append_a @ Xs2 @ Ys ) ) ) ).
% seq_conform_if_before
thf(fact_123_hd__reach__all__forward,axiom,
! [Xs2: list_a,X: a] :
( ( member_a @ ( hd_a @ Xs2 ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs2 )
=> ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ( reachable_a_b @ t @ ( hd_a @ Xs2 ) @ X ) ) ) ) ).
% hd_reach_all_forward
thf(fact_124_set__union,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( set_list_a2 @ ( union_list_a @ Xs2 @ Ys ) )
= ( sup_sup_set_list_a @ ( set_list_a2 @ Xs2 ) @ ( set_list_a2 @ Ys ) ) ) ).
% set_union
thf(fact_125_set__union,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( set_a2 @ ( union_a @ Xs2 @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Ys ) ) ) ).
% set_union
thf(fact_126_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_127_hd__in__verts__if__forward,axiom,
! [X: a,Y: a,Xs2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( cons_a @ X @ ( cons_a @ Y @ Xs2 ) ) )
=> ( member_a @ ( hd_a @ ( cons_a @ X @ ( cons_a @ Y @ Xs2 ) ) ) @ ( pre_ve642382030648772252t_unit @ t ) ) ) ).
% hd_in_verts_if_forward
thf(fact_128_forward__arcs_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X2: a] :
( X
!= ( cons_a @ X2 @ nil_a ) )
=> ~ ! [X2: a,V3: a,Va: list_a] :
( X
!= ( cons_a @ X2 @ ( cons_a @ V3 @ Va ) ) ) ) ) ).
% forward_arcs.cases
thf(fact_129_no__back__arcs_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [X2: a,Xs: list_a] :
( X
!= ( cons_a @ X2 @ Xs ) ) ) ).
% no_back_arcs.cases
thf(fact_130_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_131_T_Overts__distinct,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( dverts_list_a_b @ ta ) )
=> ( distinct_a @ V ) ) ).
% T.verts_distinct
thf(fact_132_non__empty,axiom,
( ( pre_ve642382030648772252t_unit @ t )
!= bot_bot_set_a ) ).
% non_empty
thf(fact_133_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_134_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_135_merging__empty,axiom,
( ( graph_2957805489637798020ts_a_b @ t )
= bot_bot_set_a ) ).
% merging_empty
thf(fact_136_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_137_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_138_T_Overts__forward,axiom,
! [X4: list_a] :
( ( member_list_a @ X4 @ ( dverts_list_a_b @ ta ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ X4 ) ) ).
% T.verts_forward
thf(fact_139_T_Overts__conform,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( dverts_list_a_b @ ta ) )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ V ) ) ).
% T.verts_conform
thf(fact_140_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_141_list_Oinject,axiom,
! [X21: a,X222: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X222 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_142_list_Oinject,axiom,
! [X21: b,X222: list_b,Y21: b,Y22: list_b] :
( ( ( cons_b @ X21 @ X222 )
= ( cons_b @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_143_reachable__via__child__impl__same,axiom,
! [X: a,V: a,Y: a,U: a] :
( ( reachable_a_b @ t @ X @ V )
=> ( ( reachable_a_b @ t @ Y @ V )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ X ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( X = Y ) ) ) ) ) ).
% reachable_via_child_impl_same
thf(fact_144_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_145_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_146__092_060open_062dtree_Oroot_At1_A_092_060in_062_Adverts_At_092_060close_062,axiom,
member_list_a @ ( root_list_a_b @ t1 ) @ ( dverts_list_a_b @ ta ) ).
% \<open>dtree.root t1 \<in> dverts t\<close>
thf(fact_147_forward__single,axiom,
! [X: a] : ( iKKBZ_4778857019735642799rd_a_b @ t @ ( cons_a @ X @ nil_a ) ) ).
% forward_single
thf(fact_148_seq__conform__single,axiom,
! [X: a] : ( iKKBZ_4622586873178280511rm_a_b @ t @ ( cons_a @ X @ nil_a ) ) ).
% seq_conform_single
thf(fact_149_seq__conform__if__dstnct__fwd,axiom,
! [Xs2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs2 )
=> ( ( distinct_a @ Xs2 )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs2 ) ) ) ).
% seq_conform_if_dstnct_fwd
thf(fact_150_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 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ S )
=> ( reachable_a_b @ t @ X5 @ Y3 ) ) )
& ! [X5: a] :
( ( member_a @ X5 @ S )
=> ! [V4: a] :
( ~ ( member_a @ V4 @ S )
=> ( ~ ( reachable_a_b @ t @ X5 @ V4 )
| ~ ( reachable_a_b @ t @ V4 @ X5 ) ) ) ) ) ) ).
% in_sccs_verts_conv_reachable
thf(fact_151_nonempty__notin__distinct__prefix,axiom,
! [As: list_b,Bs: list_b,V2: list_b,Cs: list_b,As2: list_list_b] :
( ( distinct_b @ ( append_b @ As @ ( append_b @ Bs @ ( append_b @ V2 @ Cs ) ) ) )
=> ( ( ( concat_b @ As2 )
= As )
=> ( ( V2 != nil_b )
=> ~ ( member_list_b @ V2 @ ( set_list_b2 @ As2 ) ) ) ) ) ).
% nonempty_notin_distinct_prefix
thf(fact_152_nonempty__notin__distinct__prefix,axiom,
! [As: list_a,Bs: list_a,V2: list_a,Cs: list_a,As2: list_list_a] :
( ( distinct_a @ ( append_a @ As @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) )
=> ( ( ( concat_a @ As2 )
= As )
=> ( ( V2 != nil_a )
=> ~ ( member_list_a @ V2 @ ( set_list_a2 @ As2 ) ) ) ) ) ).
% nonempty_notin_distinct_prefix
thf(fact_153_reachable__induct,axiom,
! [U: a,V: a,P2: a > $o] :
( ( reachable_a_b @ t @ U @ V )
=> ( ( ( member_a @ U @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( P2 @ U ) )
=> ( ! [X2: a,Y4: a] :
( ( reachable_a_b @ t @ U @ X2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( P2 @ X2 )
=> ( P2 @ Y4 ) ) ) )
=> ( P2 @ V ) ) ) ) ).
% reachable_induct
thf(fact_154_converse__reachable__induct,axiom,
! [U: a,V: a,P2: a > $o] :
( ( reachable_a_b @ t @ U @ V )
=> ( ( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( P2 @ V ) )
=> ( ! [X2: a,Y4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y4 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( reachable_a_b @ t @ Y4 @ V )
=> ( ( P2 @ Y4 )
=> ( P2 @ X2 ) ) ) )
=> ( P2 @ U ) ) ) ) ).
% converse_reachable_induct
thf(fact_155_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_156_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_157_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_158_last__merge__alt,axiom,
! [X: a] :
( ( member_a @ X @ ( graph_2659413520663303054ts_a_b @ t ) )
=> ! [Z2: a] :
( ( ( reachable_a_b @ t @ X @ Z2 )
& ( Z2 != X ) )
=> ~ ( member_a @ Z2 @ ( graph_2957805489637798020ts_a_b @ t ) ) ) ) ).
% last_merge_alt
thf(fact_159_hd__reach__all__forward_H_H,axiom,
! [X: a,Y: a,Xs2: list_a,Z: a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( cons_a @ X @ ( cons_a @ Y @ Xs2 ) ) )
=> ( ( member_a @ Z @ ( set_a2 @ ( cons_a @ X @ ( cons_a @ Y @ Xs2 ) ) ) )
=> ( reachable_a_b @ t @ ( hd_a @ ( cons_a @ X @ ( cons_a @ Y @ Xs2 ) ) ) @ Z ) ) ) ).
% hd_reach_all_forward''
thf(fact_160_concat__append,axiom,
! [Xs2: list_list_b,Ys: list_list_b] :
( ( concat_b @ ( append_list_b @ Xs2 @ Ys ) )
= ( append_b @ ( concat_b @ Xs2 ) @ ( concat_b @ Ys ) ) ) ).
% concat_append
thf(fact_161_concat__append,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( concat_a @ ( append_list_a @ Xs2 @ Ys ) )
= ( append_a @ ( concat_a @ Xs2 ) @ ( concat_a @ Ys ) ) ) ).
% concat_append
thf(fact_162_distinct__union,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( distinct_a @ ( union_a @ Xs2 @ Ys ) )
= ( distinct_a @ Ys ) ) ).
% distinct_union
thf(fact_163_set__empty,axiom,
! [Xs2: list_b] :
( ( ( set_b2 @ Xs2 )
= bot_bot_set_b )
= ( Xs2 = nil_b ) ) ).
% set_empty
thf(fact_164_set__empty,axiom,
! [Xs2: list_a] :
( ( ( set_a2 @ Xs2 )
= bot_bot_set_a )
= ( Xs2 = nil_a ) ) ).
% set_empty
thf(fact_165_set__empty,axiom,
! [Xs2: list_list_a] :
( ( ( set_list_a2 @ Xs2 )
= bot_bot_set_list_a )
= ( Xs2 = nil_list_a ) ) ).
% set_empty
thf(fact_166_set__empty2,axiom,
! [Xs2: list_b] :
( ( bot_bot_set_b
= ( set_b2 @ Xs2 ) )
= ( Xs2 = nil_b ) ) ).
% set_empty2
thf(fact_167_set__empty2,axiom,
! [Xs2: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs2 ) )
= ( Xs2 = nil_a ) ) ).
% set_empty2
thf(fact_168_set__empty2,axiom,
! [Xs2: list_list_a] :
( ( bot_bot_set_list_a
= ( set_list_a2 @ Xs2 ) )
= ( Xs2 = nil_list_a ) ) ).
% set_empty2
thf(fact_169_append1__eq__conv,axiom,
! [Xs2: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs2 @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs2 = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_170_append1__eq__conv,axiom,
! [Xs2: list_b,X: b,Ys: list_b,Y: b] :
( ( ( append_b @ Xs2 @ ( cons_b @ X @ nil_b ) )
= ( append_b @ Ys @ ( cons_b @ Y @ nil_b ) ) )
= ( ( Xs2 = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_171_concat__eq__Nil__conv,axiom,
! [Xss: list_list_b] :
( ( ( concat_b @ Xss )
= nil_b )
= ( ! [X5: list_b] :
( ( member_list_b @ X5 @ ( set_list_b2 @ Xss ) )
=> ( X5 = nil_b ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_172_concat__eq__Nil__conv,axiom,
! [Xss: list_list_a] :
( ( ( concat_a @ Xss )
= nil_a )
= ( ! [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xss ) )
=> ( X5 = nil_a ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_173_Nil__eq__concat__conv,axiom,
! [Xss: list_list_b] :
( ( nil_b
= ( concat_b @ Xss ) )
= ( ! [X5: list_b] :
( ( member_list_b @ X5 @ ( set_list_b2 @ Xss ) )
=> ( X5 = nil_b ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_174_Nil__eq__concat__conv,axiom,
! [Xss: list_list_a] :
( ( nil_a
= ( concat_a @ Xss ) )
= ( ! [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xss ) )
=> ( X5 = nil_a ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_175_reachable__refl,axiom,
! [V: a] :
( ( member_a @ V @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( reachable_a_b @ t @ V @ V ) ) ).
% reachable_refl
thf(fact_176_inner__verts__Nil,axiom,
( ( pre_inner_verts_a_b @ t @ nil_b )
= nil_a ) ).
% inner_verts_Nil
thf(fact_177_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_178_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_179_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_180_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_181_is__chain_H__def,axiom,
( ( graph_8150681439568091980in_a_b @ t )
= ( ( graph_2957805489637798020ts_a_b @ t )
= bot_bot_set_a ) ) ).
% is_chain'_def
thf(fact_182_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_183_not__Cons__self2,axiom,
! [X: a,Xs2: list_a] :
( ( cons_a @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_184_not__Cons__self2,axiom,
! [X: b,Xs2: list_b] :
( ( cons_b @ X @ Xs2 )
!= Xs2 ) ).
% not_Cons_self2
thf(fact_185_directed__tree_Oseq__conform_Ocong,axiom,
iKKBZ_4622586873178280511rm_a_b = iKKBZ_4622586873178280511rm_a_b ).
% directed_tree.seq_conform.cong
thf(fact_186_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss2: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss2 ) )
=> ~ ! [X2: a,Xs: list_a,Xss2: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X2 @ Xs ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_187_transpose_Ocases,axiom,
! [X: list_list_b] :
( ( X != nil_list_b )
=> ( ! [Xss2: list_list_b] :
( X
!= ( cons_list_b @ nil_b @ Xss2 ) )
=> ~ ! [X2: b,Xs: list_b,Xss2: list_list_b] :
( X
!= ( cons_list_b @ ( cons_b @ X2 @ Xs ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_188_concat_Osimps_I1_J,axiom,
( ( concat_a @ nil_list_a )
= nil_a ) ).
% concat.simps(1)
thf(fact_189_concat_Osimps_I1_J,axiom,
( ( concat_b @ nil_list_b )
= nil_b ) ).
% concat.simps(1)
thf(fact_190_dtree_Oset__intros_I1_J,axiom,
! [X1: list_a,X22: fset_P2153231429829016240_a_b_b] : ( member_list_a @ X1 @ ( dverts_list_a_b @ ( node_list_a_b @ X1 @ X22 ) ) ) ).
% dtree.set_intros(1)
thf(fact_191_dtree_Oset__sel_I1_J,axiom,
! [A: dtree_list_a_b] : ( member_list_a @ ( root_list_a_b @ A ) @ ( dverts_list_a_b @ A ) ) ).
% dtree.set_sel(1)
thf(fact_192_set__ConsD,axiom,
! [Y: dtree_list_a_b,X: dtree_list_a_b,Xs2: list_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ Y @ ( set_dtree_list_a_b2 @ ( cons_dtree_list_a_b @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member551035911493665803st_a_b @ Y @ ( set_dtree_list_a_b2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_193_set__ConsD,axiom,
! [Y: set_a,X: set_a,Xs2: list_set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_set_a @ Y @ ( set_set_a2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_194_set__ConsD,axiom,
! [Y: list_a,X: list_a,Xs2: list_list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_list_a @ Y @ ( set_list_a2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_195_set__ConsD,axiom,
! [Y: a,X: a,Xs2: list_a] :
( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_a @ Y @ ( set_a2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_196_set__ConsD,axiom,
! [Y: b,X: b,Xs2: list_b] :
( ( member_b @ Y @ ( set_b2 @ ( cons_b @ X @ Xs2 ) ) )
=> ( ( Y = X )
| ( member_b @ Y @ ( set_b2 @ Xs2 ) ) ) ) ).
% set_ConsD
thf(fact_197_list_Oset__cases,axiom,
! [E: dtree_list_a_b,A: list_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ E @ ( set_dtree_list_a_b2 @ A ) )
=> ( ! [Z22: list_dtree_list_a_b] :
( A
!= ( cons_dtree_list_a_b @ E @ Z22 ) )
=> ~ ! [Z1: dtree_list_a_b,Z22: list_dtree_list_a_b] :
( ( A
= ( cons_dtree_list_a_b @ Z1 @ Z22 ) )
=> ~ ( member551035911493665803st_a_b @ E @ ( set_dtree_list_a_b2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_198_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_199_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_200_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_201_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_202_list_Oset__intros_I1_J,axiom,
! [X21: dtree_list_a_b,X222: list_dtree_list_a_b] : ( member551035911493665803st_a_b @ X21 @ ( set_dtree_list_a_b2 @ ( cons_dtree_list_a_b @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_203_list_Oset__intros_I1_J,axiom,
! [X21: set_a,X222: list_set_a] : ( member_set_a @ X21 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_204_list_Oset__intros_I1_J,axiom,
! [X21: list_a,X222: list_list_a] : ( member_list_a @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_205_list_Oset__intros_I1_J,axiom,
! [X21: a,X222: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_206_list_Oset__intros_I1_J,axiom,
! [X21: b,X222: list_b] : ( member_b @ X21 @ ( set_b2 @ ( cons_b @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_207_list_Oset__intros_I2_J,axiom,
! [Y: dtree_list_a_b,X222: list_dtree_list_a_b,X21: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ Y @ ( set_dtree_list_a_b2 @ X222 ) )
=> ( member551035911493665803st_a_b @ Y @ ( set_dtree_list_a_b2 @ ( cons_dtree_list_a_b @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_208_list_Oset__intros_I2_J,axiom,
! [Y: set_a,X222: list_set_a,X21: set_a] :
( ( member_set_a @ Y @ ( set_set_a2 @ X222 ) )
=> ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_209_list_Oset__intros_I2_J,axiom,
! [Y: list_a,X222: list_list_a,X21: list_a] :
( ( member_list_a @ Y @ ( set_list_a2 @ X222 ) )
=> ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_210_list_Oset__intros_I2_J,axiom,
! [Y: a,X222: list_a,X21: a] :
( ( member_a @ Y @ ( set_a2 @ X222 ) )
=> ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_211_list_Oset__intros_I2_J,axiom,
! [Y: b,X222: list_b,X21: b] :
( ( member_b @ Y @ ( set_b2 @ X222 ) )
=> ( member_b @ Y @ ( set_b2 @ ( cons_b @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_212_append__Cons,axiom,
! [X: a,Xs2: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X @ Xs2 ) @ Ys )
= ( cons_a @ X @ ( append_a @ Xs2 @ Ys ) ) ) ).
% append_Cons
thf(fact_213_append__Cons,axiom,
! [X: b,Xs2: list_b,Ys: list_b] :
( ( append_b @ ( cons_b @ X @ Xs2 ) @ Ys )
= ( cons_b @ X @ ( append_b @ Xs2 @ Ys ) ) ) ).
% append_Cons
thf(fact_214_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys: list_a,Xs2: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys )
=> ( ( Xs2
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X @ Xs2 )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_215_Cons__eq__appendI,axiom,
! [X: b,Xs1: list_b,Ys: list_b,Xs2: list_b,Zs: list_b] :
( ( ( cons_b @ X @ Xs1 )
= Ys )
=> ( ( Xs2
= ( append_b @ Xs1 @ Zs ) )
=> ( ( cons_b @ X @ Xs2 )
= ( append_b @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_216_list_Odistinct_I1_J,axiom,
! [X21: a,X222: list_a] :
( nil_a
!= ( cons_a @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_217_list_Odistinct_I1_J,axiom,
! [X21: b,X222: list_b] :
( nil_b
!= ( cons_b @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_218_list_OdiscI,axiom,
! [List: list_a,X21: a,X222: list_a] :
( ( List
= ( cons_a @ X21 @ X222 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_219_list_OdiscI,axiom,
! [List: list_b,X21: b,X222: list_b] :
( ( List
= ( cons_b @ X21 @ X222 ) )
=> ( List != nil_b ) ) ).
% list.discI
thf(fact_220_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X223: list_a] :
( Y
!= ( cons_a @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_221_list_Oexhaust,axiom,
! [Y: list_b] :
( ( Y != nil_b )
=> ~ ! [X212: b,X223: list_b] :
( Y
!= ( cons_b @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_222_neq__Nil__conv,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
= ( ? [Y3: a,Ys3: list_a] :
( Xs2
= ( cons_a @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_223_neq__Nil__conv,axiom,
! [Xs2: list_b] :
( ( Xs2 != nil_b )
= ( ? [Y3: b,Ys3: list_b] :
( Xs2
= ( cons_b @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_224_list__induct2_H,axiom,
! [P2: list_a > list_a > $o,Xs2: list_a,Ys: list_a] :
( ( P2 @ nil_a @ nil_a )
=> ( ! [X2: a,Xs: list_a] : ( P2 @ ( cons_a @ X2 @ Xs ) @ nil_a )
=> ( ! [Y4: a,Ys4: list_a] : ( P2 @ nil_a @ ( cons_a @ Y4 @ Ys4 ) )
=> ( ! [X2: a,Xs: list_a,Y4: a,Ys4: list_a] :
( ( P2 @ Xs @ Ys4 )
=> ( P2 @ ( cons_a @ X2 @ Xs ) @ ( cons_a @ Y4 @ Ys4 ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_225_list__induct2_H,axiom,
! [P2: list_a > list_b > $o,Xs2: list_a,Ys: list_b] :
( ( P2 @ nil_a @ nil_b )
=> ( ! [X2: a,Xs: list_a] : ( P2 @ ( cons_a @ X2 @ Xs ) @ nil_b )
=> ( ! [Y4: b,Ys4: list_b] : ( P2 @ nil_a @ ( cons_b @ Y4 @ Ys4 ) )
=> ( ! [X2: a,Xs: list_a,Y4: b,Ys4: list_b] :
( ( P2 @ Xs @ Ys4 )
=> ( P2 @ ( cons_a @ X2 @ Xs ) @ ( cons_b @ Y4 @ Ys4 ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_226_list__induct2_H,axiom,
! [P2: list_b > list_a > $o,Xs2: list_b,Ys: list_a] :
( ( P2 @ nil_b @ nil_a )
=> ( ! [X2: b,Xs: list_b] : ( P2 @ ( cons_b @ X2 @ Xs ) @ nil_a )
=> ( ! [Y4: a,Ys4: list_a] : ( P2 @ nil_b @ ( cons_a @ Y4 @ Ys4 ) )
=> ( ! [X2: b,Xs: list_b,Y4: a,Ys4: list_a] :
( ( P2 @ Xs @ Ys4 )
=> ( P2 @ ( cons_b @ X2 @ Xs ) @ ( cons_a @ Y4 @ Ys4 ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_227_list__induct2_H,axiom,
! [P2: list_b > list_b > $o,Xs2: list_b,Ys: list_b] :
( ( P2 @ nil_b @ nil_b )
=> ( ! [X2: b,Xs: list_b] : ( P2 @ ( cons_b @ X2 @ Xs ) @ nil_b )
=> ( ! [Y4: b,Ys4: list_b] : ( P2 @ nil_b @ ( cons_b @ Y4 @ Ys4 ) )
=> ( ! [X2: b,Xs: list_b,Y4: b,Ys4: list_b] :
( ( P2 @ Xs @ Ys4 )
=> ( P2 @ ( cons_b @ X2 @ Xs ) @ ( cons_b @ Y4 @ Ys4 ) ) )
=> ( P2 @ Xs2 @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_228_list__nonempty__induct,axiom,
! [Xs2: list_a,P2: list_a > $o] :
( ( Xs2 != nil_a )
=> ( ! [X2: a] : ( P2 @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs: list_a] :
( ( Xs != nil_a )
=> ( ( P2 @ Xs )
=> ( P2 @ ( cons_a @ X2 @ Xs ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_229_list__nonempty__induct,axiom,
! [Xs2: list_b,P2: list_b > $o] :
( ( Xs2 != nil_b )
=> ( ! [X2: b] : ( P2 @ ( cons_b @ X2 @ nil_b ) )
=> ( ! [X2: b,Xs: list_b] :
( ( Xs != nil_b )
=> ( ( P2 @ Xs )
=> ( P2 @ ( cons_b @ X2 @ Xs ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% list_nonempty_induct
thf(fact_230_distinct__length__2__or__more,axiom,
! [A: a,B: a,Xs2: list_a] :
( ( distinct_a @ ( cons_a @ A @ ( cons_a @ B @ Xs2 ) ) )
= ( ( A != B )
& ( distinct_a @ ( cons_a @ A @ Xs2 ) )
& ( distinct_a @ ( cons_a @ B @ Xs2 ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_231_distinct__length__2__or__more,axiom,
! [A: b,B: b,Xs2: list_b] :
( ( distinct_b @ ( cons_b @ A @ ( cons_b @ B @ Xs2 ) ) )
= ( ( A != B )
& ( distinct_b @ ( cons_b @ A @ Xs2 ) )
& ( distinct_b @ ( cons_b @ B @ Xs2 ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_232_list_Osel_I1_J,axiom,
! [X21: a,X222: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_233_list_Osel_I1_J,axiom,
! [X21: b,X222: list_b] :
( ( hd_b @ ( cons_b @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_234_hd__concat,axiom,
! [Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( ( hd_list_a @ Xs2 )
!= nil_a )
=> ( ( hd_a @ ( concat_a @ Xs2 ) )
= ( hd_a @ ( hd_list_a @ Xs2 ) ) ) ) ) ).
% hd_concat
thf(fact_235_hd__concat,axiom,
! [Xs2: list_list_b] :
( ( Xs2 != nil_list_b )
=> ( ( ( hd_list_b @ Xs2 )
!= nil_b )
=> ( ( hd_b @ ( concat_b @ Xs2 ) )
= ( hd_b @ ( hd_list_b @ Xs2 ) ) ) ) ) ).
% hd_concat
thf(fact_236_split__list__first__prop__iff,axiom,
! [Xs2: list_list_a,P2: list_a > $o] :
( ( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs2 ) )
& ( P2 @ X5 ) ) )
= ( ? [Ys3: list_list_a,X5: list_a] :
( ? [Zs2: list_list_a] :
( Xs2
= ( append_list_a @ Ys3 @ ( cons_list_a @ X5 @ Zs2 ) ) )
& ( P2 @ X5 )
& ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( set_list_a2 @ Ys3 ) )
=> ~ ( P2 @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_237_split__list__first__prop__iff,axiom,
! [Xs2: list_a,P2: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
& ( P2 @ X5 ) ) )
= ( ? [Ys3: list_a,X5: a] :
( ? [Zs2: list_a] :
( Xs2
= ( append_a @ Ys3 @ ( cons_a @ X5 @ Zs2 ) ) )
& ( P2 @ X5 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Ys3 ) )
=> ~ ( P2 @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_238_split__list__first__prop__iff,axiom,
! [Xs2: list_b,P2: b > $o] :
( ( ? [X5: b] :
( ( member_b @ X5 @ ( set_b2 @ Xs2 ) )
& ( P2 @ X5 ) ) )
= ( ? [Ys3: list_b,X5: b] :
( ? [Zs2: list_b] :
( Xs2
= ( append_b @ Ys3 @ ( cons_b @ X5 @ Zs2 ) ) )
& ( P2 @ X5 )
& ! [Y3: b] :
( ( member_b @ Y3 @ ( set_b2 @ Ys3 ) )
=> ~ ( P2 @ Y3 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_239_split__list__last__prop__iff,axiom,
! [Xs2: list_list_a,P2: list_a > $o] :
( ( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs2 ) )
& ( P2 @ X5 ) ) )
= ( ? [Ys3: list_list_a,X5: list_a,Zs2: list_list_a] :
( ( Xs2
= ( append_list_a @ Ys3 @ ( cons_list_a @ X5 @ Zs2 ) ) )
& ( P2 @ X5 )
& ! [Y3: list_a] :
( ( member_list_a @ Y3 @ ( set_list_a2 @ Zs2 ) )
=> ~ ( P2 @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_240_split__list__last__prop__iff,axiom,
! [Xs2: list_a,P2: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
& ( P2 @ X5 ) ) )
= ( ? [Ys3: list_a,X5: a,Zs2: list_a] :
( ( Xs2
= ( append_a @ Ys3 @ ( cons_a @ X5 @ Zs2 ) ) )
& ( P2 @ X5 )
& ! [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Zs2 ) )
=> ~ ( P2 @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_241_split__list__last__prop__iff,axiom,
! [Xs2: list_b,P2: b > $o] :
( ( ? [X5: b] :
( ( member_b @ X5 @ ( set_b2 @ Xs2 ) )
& ( P2 @ X5 ) ) )
= ( ? [Ys3: list_b,X5: b,Zs2: list_b] :
( ( Xs2
= ( append_b @ Ys3 @ ( cons_b @ X5 @ Zs2 ) ) )
& ( P2 @ X5 )
& ! [Y3: b] :
( ( member_b @ Y3 @ ( set_b2 @ Zs2 ) )
=> ~ ( P2 @ Y3 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_242_in__set__conv__decomp__first,axiom,
! [X: dtree_list_a_b,Xs2: list_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
= ( ? [Ys3: list_dtree_list_a_b,Zs2: list_dtree_list_a_b] :
( ( Xs2
= ( append2129087805494049561st_a_b @ Ys3 @ ( cons_dtree_list_a_b @ X @ Zs2 ) ) )
& ~ ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_243_in__set__conv__decomp__first,axiom,
! [X: set_a,Xs2: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs2 ) )
= ( ? [Ys3: list_set_a,Zs2: list_set_a] :
( ( Xs2
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs2 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_244_in__set__conv__decomp__first,axiom,
! [X: list_a,Xs2: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
= ( ? [Ys3: list_list_a,Zs2: list_list_a] :
( ( Xs2
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs2 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_245_in__set__conv__decomp__first,axiom,
! [X: a,Xs2: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
= ( ? [Ys3: list_a,Zs2: list_a] :
( ( Xs2
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_246_in__set__conv__decomp__first,axiom,
! [X: b,Xs2: list_b] :
( ( member_b @ X @ ( set_b2 @ Xs2 ) )
= ( ? [Ys3: list_b,Zs2: list_b] :
( ( Xs2
= ( append_b @ Ys3 @ ( cons_b @ X @ Zs2 ) ) )
& ~ ( member_b @ X @ ( set_b2 @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_247_in__set__conv__decomp__last,axiom,
! [X: dtree_list_a_b,Xs2: list_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
= ( ? [Ys3: list_dtree_list_a_b,Zs2: list_dtree_list_a_b] :
( ( Xs2
= ( append2129087805494049561st_a_b @ Ys3 @ ( cons_dtree_list_a_b @ X @ Zs2 ) ) )
& ~ ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_248_in__set__conv__decomp__last,axiom,
! [X: set_a,Xs2: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs2 ) )
= ( ? [Ys3: list_set_a,Zs2: list_set_a] :
( ( Xs2
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs2 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_249_in__set__conv__decomp__last,axiom,
! [X: list_a,Xs2: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
= ( ? [Ys3: list_list_a,Zs2: list_list_a] :
( ( Xs2
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs2 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_250_in__set__conv__decomp__last,axiom,
! [X: a,Xs2: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
= ( ? [Ys3: list_a,Zs2: list_a] :
( ( Xs2
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_251_in__set__conv__decomp__last,axiom,
! [X: b,Xs2: list_b] :
( ( member_b @ X @ ( set_b2 @ Xs2 ) )
= ( ? [Ys3: list_b,Zs2: list_b] :
( ( Xs2
= ( append_b @ Ys3 @ ( cons_b @ X @ Zs2 ) ) )
& ~ ( member_b @ X @ ( set_b2 @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_252_split__list__first__propE,axiom,
! [Xs2: list_list_a,P2: list_a > $o] :
( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys4: list_list_a,X2: list_a] :
( ? [Zs3: list_list_a] :
( Xs2
= ( append_list_a @ Ys4 @ ( cons_list_a @ X2 @ Zs3 ) ) )
=> ( ( P2 @ X2 )
=> ~ ! [Xa2: list_a] :
( ( member_list_a @ Xa2 @ ( set_list_a2 @ Ys4 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_253_split__list__first__propE,axiom,
! [Xs2: list_a,P2: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys4: list_a,X2: a] :
( ? [Zs3: list_a] :
( Xs2
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) )
=> ( ( P2 @ X2 )
=> ~ ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys4 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_254_split__list__first__propE,axiom,
! [Xs2: list_b,P2: b > $o] :
( ? [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys4: list_b,X2: b] :
( ? [Zs3: list_b] :
( Xs2
= ( append_b @ Ys4 @ ( cons_b @ X2 @ Zs3 ) ) )
=> ( ( P2 @ X2 )
=> ~ ! [Xa2: b] :
( ( member_b @ Xa2 @ ( set_b2 @ Ys4 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_255_split__list__last__propE,axiom,
! [Xs2: list_list_a,P2: list_a > $o] :
( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys4: list_list_a,X2: list_a,Zs3: list_list_a] :
( ( Xs2
= ( append_list_a @ Ys4 @ ( cons_list_a @ X2 @ Zs3 ) ) )
=> ( ( P2 @ X2 )
=> ~ ! [Xa2: list_a] :
( ( member_list_a @ Xa2 @ ( set_list_a2 @ Zs3 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_256_split__list__last__propE,axiom,
! [Xs2: list_a,P2: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys4: list_a,X2: a,Zs3: list_a] :
( ( Xs2
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) )
=> ( ( P2 @ X2 )
=> ~ ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Zs3 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_257_split__list__last__propE,axiom,
! [Xs2: list_b,P2: b > $o] :
( ? [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys4: list_b,X2: b,Zs3: list_b] :
( ( Xs2
= ( append_b @ Ys4 @ ( cons_b @ X2 @ Zs3 ) ) )
=> ( ( P2 @ X2 )
=> ~ ! [Xa2: b] :
( ( member_b @ Xa2 @ ( set_b2 @ Zs3 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_258_split__list__first__prop,axiom,
! [Xs2: list_list_a,P2: list_a > $o] :
( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ? [Ys4: list_list_a,X2: list_a] :
( ? [Zs3: list_list_a] :
( Xs2
= ( append_list_a @ Ys4 @ ( cons_list_a @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Xa2: list_a] :
( ( member_list_a @ Xa2 @ ( set_list_a2 @ Ys4 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_259_split__list__first__prop,axiom,
! [Xs2: list_a,P2: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ? [Ys4: list_a,X2: a] :
( ? [Zs3: list_a] :
( Xs2
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys4 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_260_split__list__first__prop,axiom,
! [Xs2: list_b,P2: b > $o] :
( ? [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ? [Ys4: list_b,X2: b] :
( ? [Zs3: list_b] :
( Xs2
= ( append_b @ Ys4 @ ( cons_b @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Xa2: b] :
( ( member_b @ Xa2 @ ( set_b2 @ Ys4 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_261_split__list__last__prop,axiom,
! [Xs2: list_list_a,P2: list_a > $o] :
( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ? [Ys4: list_list_a,X2: list_a,Zs3: list_list_a] :
( ( Xs2
= ( append_list_a @ Ys4 @ ( cons_list_a @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Xa2: list_a] :
( ( member_list_a @ Xa2 @ ( set_list_a2 @ Zs3 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_262_split__list__last__prop,axiom,
! [Xs2: list_a,P2: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ? [Ys4: list_a,X2: a,Zs3: list_a] :
( ( Xs2
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Zs3 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_263_split__list__last__prop,axiom,
! [Xs2: list_b,P2: b > $o] :
( ? [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ? [Ys4: list_b,X2: b,Zs3: list_b] :
( ( Xs2
= ( append_b @ Ys4 @ ( cons_b @ X2 @ Zs3 ) ) )
& ( P2 @ X2 )
& ! [Xa2: b] :
( ( member_b @ Xa2 @ ( set_b2 @ Zs3 ) )
=> ~ ( P2 @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_264_in__set__conv__decomp,axiom,
! [X: dtree_list_a_b,Xs2: list_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
= ( ? [Ys3: list_dtree_list_a_b,Zs2: list_dtree_list_a_b] :
( Xs2
= ( append2129087805494049561st_a_b @ Ys3 @ ( cons_dtree_list_a_b @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_265_in__set__conv__decomp,axiom,
! [X: set_a,Xs2: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs2 ) )
= ( ? [Ys3: list_set_a,Zs2: list_set_a] :
( Xs2
= ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_266_in__set__conv__decomp,axiom,
! [X: list_a,Xs2: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
= ( ? [Ys3: list_list_a,Zs2: list_list_a] :
( Xs2
= ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_267_in__set__conv__decomp,axiom,
! [X: a,Xs2: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
= ( ? [Ys3: list_a,Zs2: list_a] :
( Xs2
= ( append_a @ Ys3 @ ( cons_a @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_268_in__set__conv__decomp,axiom,
! [X: b,Xs2: list_b] :
( ( member_b @ X @ ( set_b2 @ Xs2 ) )
= ( ? [Ys3: list_b,Zs2: list_b] :
( Xs2
= ( append_b @ Ys3 @ ( cons_b @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_269_append__Cons__eq__iff,axiom,
! [X: dtree_list_a_b,Xs2: list_dtree_list_a_b,Ys: list_dtree_list_a_b,Xs4: list_dtree_list_a_b,Ys5: list_dtree_list_a_b] :
( ~ ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
=> ( ~ ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Ys ) )
=> ( ( ( append2129087805494049561st_a_b @ Xs2 @ ( cons_dtree_list_a_b @ X @ Ys ) )
= ( append2129087805494049561st_a_b @ Xs4 @ ( cons_dtree_list_a_b @ X @ Ys5 ) ) )
= ( ( Xs2 = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_270_append__Cons__eq__iff,axiom,
! [X: set_a,Xs2: list_set_a,Ys: list_set_a,Xs4: list_set_a,Ys5: list_set_a] :
( ~ ( member_set_a @ X @ ( set_set_a2 @ Xs2 ) )
=> ( ~ ( member_set_a @ X @ ( set_set_a2 @ Ys ) )
=> ( ( ( append_set_a @ Xs2 @ ( cons_set_a @ X @ Ys ) )
= ( append_set_a @ Xs4 @ ( cons_set_a @ X @ Ys5 ) ) )
= ( ( Xs2 = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_271_append__Cons__eq__iff,axiom,
! [X: list_a,Xs2: list_list_a,Ys: list_list_a,Xs4: list_list_a,Ys5: list_list_a] :
( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
=> ( ~ ( member_list_a @ X @ ( set_list_a2 @ Ys ) )
=> ( ( ( append_list_a @ Xs2 @ ( cons_list_a @ X @ Ys ) )
= ( append_list_a @ Xs4 @ ( cons_list_a @ X @ Ys5 ) ) )
= ( ( Xs2 = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_272_append__Cons__eq__iff,axiom,
! [X: a,Xs2: list_a,Ys: list_a,Xs4: list_a,Ys5: list_a] :
( ~ ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
=> ( ( ( append_a @ Xs2 @ ( cons_a @ X @ Ys ) )
= ( append_a @ Xs4 @ ( cons_a @ X @ Ys5 ) ) )
= ( ( Xs2 = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_273_append__Cons__eq__iff,axiom,
! [X: b,Xs2: list_b,Ys: list_b,Xs4: list_b,Ys5: list_b] :
( ~ ( member_b @ X @ ( set_b2 @ Xs2 ) )
=> ( ~ ( member_b @ X @ ( set_b2 @ Ys ) )
=> ( ( ( append_b @ Xs2 @ ( cons_b @ X @ Ys ) )
= ( append_b @ Xs4 @ ( cons_b @ X @ Ys5 ) ) )
= ( ( Xs2 = Xs4 )
& ( Ys = Ys5 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_274_split__list__propE,axiom,
! [Xs2: list_list_a,P2: list_a > $o] :
( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys4: list_list_a,X2: list_a] :
( ? [Zs3: list_list_a] :
( Xs2
= ( append_list_a @ Ys4 @ ( cons_list_a @ X2 @ Zs3 ) ) )
=> ~ ( P2 @ X2 ) ) ) ).
% split_list_propE
thf(fact_275_split__list__propE,axiom,
! [Xs2: list_a,P2: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys4: list_a,X2: a] :
( ? [Zs3: list_a] :
( Xs2
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) )
=> ~ ( P2 @ X2 ) ) ) ).
% split_list_propE
thf(fact_276_split__list__propE,axiom,
! [Xs2: list_b,P2: b > $o] :
( ? [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ~ ! [Ys4: list_b,X2: b] :
( ? [Zs3: list_b] :
( Xs2
= ( append_b @ Ys4 @ ( cons_b @ X2 @ Zs3 ) ) )
=> ~ ( P2 @ X2 ) ) ) ).
% split_list_propE
thf(fact_277_split__list__first,axiom,
! [X: dtree_list_a_b,Xs2: list_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
=> ? [Ys4: list_dtree_list_a_b,Zs3: list_dtree_list_a_b] :
( ( Xs2
= ( append2129087805494049561st_a_b @ Ys4 @ ( cons_dtree_list_a_b @ X @ Zs3 ) ) )
& ~ ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_278_split__list__first,axiom,
! [X: set_a,Xs2: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs2 ) )
=> ? [Ys4: list_set_a,Zs3: list_set_a] :
( ( Xs2
= ( append_set_a @ Ys4 @ ( cons_set_a @ X @ Zs3 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_279_split__list__first,axiom,
! [X: list_a,Xs2: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
=> ? [Ys4: list_list_a,Zs3: list_list_a] :
( ( Xs2
= ( append_list_a @ Ys4 @ ( cons_list_a @ X @ Zs3 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_280_split__list__first,axiom,
! [X: a,Xs2: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ? [Ys4: list_a,Zs3: list_a] :
( ( Xs2
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_281_split__list__first,axiom,
! [X: b,Xs2: list_b] :
( ( member_b @ X @ ( set_b2 @ Xs2 ) )
=> ? [Ys4: list_b,Zs3: list_b] :
( ( Xs2
= ( append_b @ Ys4 @ ( cons_b @ X @ Zs3 ) ) )
& ~ ( member_b @ X @ ( set_b2 @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_282_split__list__prop,axiom,
! [Xs2: list_list_a,P2: list_a > $o] :
( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ? [Ys4: list_list_a,X2: list_a] :
( ? [Zs3: list_list_a] :
( Xs2
= ( append_list_a @ Ys4 @ ( cons_list_a @ X2 @ Zs3 ) ) )
& ( P2 @ X2 ) ) ) ).
% split_list_prop
thf(fact_283_split__list__prop,axiom,
! [Xs2: list_a,P2: a > $o] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ? [Ys4: list_a,X2: a] :
( ? [Zs3: list_a] :
( Xs2
= ( append_a @ Ys4 @ ( cons_a @ X2 @ Zs3 ) ) )
& ( P2 @ X2 ) ) ) ).
% split_list_prop
thf(fact_284_split__list__prop,axiom,
! [Xs2: list_b,P2: b > $o] :
( ? [X4: b] :
( ( member_b @ X4 @ ( set_b2 @ Xs2 ) )
& ( P2 @ X4 ) )
=> ? [Ys4: list_b,X2: b] :
( ? [Zs3: list_b] :
( Xs2
= ( append_b @ Ys4 @ ( cons_b @ X2 @ Zs3 ) ) )
& ( P2 @ X2 ) ) ) ).
% split_list_prop
thf(fact_285_split__list__last,axiom,
! [X: dtree_list_a_b,Xs2: list_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
=> ? [Ys4: list_dtree_list_a_b,Zs3: list_dtree_list_a_b] :
( ( Xs2
= ( append2129087805494049561st_a_b @ Ys4 @ ( cons_dtree_list_a_b @ X @ Zs3 ) ) )
& ~ ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_286_split__list__last,axiom,
! [X: set_a,Xs2: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs2 ) )
=> ? [Ys4: list_set_a,Zs3: list_set_a] :
( ( Xs2
= ( append_set_a @ Ys4 @ ( cons_set_a @ X @ Zs3 ) ) )
& ~ ( member_set_a @ X @ ( set_set_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_287_split__list__last,axiom,
! [X: list_a,Xs2: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
=> ? [Ys4: list_list_a,Zs3: list_list_a] :
( ( Xs2
= ( append_list_a @ Ys4 @ ( cons_list_a @ X @ Zs3 ) ) )
& ~ ( member_list_a @ X @ ( set_list_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_288_split__list__last,axiom,
! [X: a,Xs2: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ? [Ys4: list_a,Zs3: list_a] :
( ( Xs2
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs3 ) ) )
& ~ ( member_a @ X @ ( set_a2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_289_split__list__last,axiom,
! [X: b,Xs2: list_b] :
( ( member_b @ X @ ( set_b2 @ Xs2 ) )
=> ? [Ys4: list_b,Zs3: list_b] :
( ( Xs2
= ( append_b @ Ys4 @ ( cons_b @ X @ Zs3 ) ) )
& ~ ( member_b @ X @ ( set_b2 @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_290_split__list,axiom,
! [X: dtree_list_a_b,Xs2: list_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
=> ? [Ys4: list_dtree_list_a_b,Zs3: list_dtree_list_a_b] :
( Xs2
= ( append2129087805494049561st_a_b @ Ys4 @ ( cons_dtree_list_a_b @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_291_split__list,axiom,
! [X: set_a,Xs2: list_set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs2 ) )
=> ? [Ys4: list_set_a,Zs3: list_set_a] :
( Xs2
= ( append_set_a @ Ys4 @ ( cons_set_a @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_292_split__list,axiom,
! [X: list_a,Xs2: list_list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
=> ? [Ys4: list_list_a,Zs3: list_list_a] :
( Xs2
= ( append_list_a @ Ys4 @ ( cons_list_a @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_293_split__list,axiom,
! [X: a,Xs2: list_a] :
( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ? [Ys4: list_a,Zs3: list_a] :
( Xs2
= ( append_a @ Ys4 @ ( cons_a @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_294_split__list,axiom,
! [X: b,Xs2: list_b] :
( ( member_b @ X @ ( set_b2 @ Xs2 ) )
=> ? [Ys4: list_b,Zs3: list_b] :
( Xs2
= ( append_b @ Ys4 @ ( cons_b @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_295_rev__induct,axiom,
! [P2: list_a > $o,Xs2: list_a] :
( ( P2 @ nil_a )
=> ( ! [X2: a,Xs: list_a] :
( ( P2 @ Xs )
=> ( P2 @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) ) )
=> ( P2 @ Xs2 ) ) ) ).
% rev_induct
thf(fact_296_rev__induct,axiom,
! [P2: list_b > $o,Xs2: list_b] :
( ( P2 @ nil_b )
=> ( ! [X2: b,Xs: list_b] :
( ( P2 @ Xs )
=> ( P2 @ ( append_b @ Xs @ ( cons_b @ X2 @ nil_b ) ) ) )
=> ( P2 @ Xs2 ) ) ) ).
% rev_induct
thf(fact_297_rev__exhaust,axiom,
! [Xs2: list_a] :
( ( Xs2 != nil_a )
=> ~ ! [Ys4: list_a,Y4: a] :
( Xs2
!= ( append_a @ Ys4 @ ( cons_a @ Y4 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_298_rev__exhaust,axiom,
! [Xs2: list_b] :
( ( Xs2 != nil_b )
=> ~ ! [Ys4: list_b,Y4: b] :
( Xs2
!= ( append_b @ Ys4 @ ( cons_b @ Y4 @ nil_b ) ) ) ) ).
% rev_exhaust
thf(fact_299_Cons__eq__append__conv,axiom,
! [X: a,Xs2: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs2 )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs2 )
= Zs ) )
| ? [Ys6: list_a] :
( ( ( cons_a @ X @ Ys6 )
= Ys )
& ( Xs2
= ( append_a @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_300_Cons__eq__append__conv,axiom,
! [X: b,Xs2: list_b,Ys: list_b,Zs: list_b] :
( ( ( cons_b @ X @ Xs2 )
= ( append_b @ Ys @ Zs ) )
= ( ( ( Ys = nil_b )
& ( ( cons_b @ X @ Xs2 )
= Zs ) )
| ? [Ys6: list_b] :
( ( ( cons_b @ X @ Ys6 )
= Ys )
& ( Xs2
= ( append_b @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_301_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X: a,Xs2: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X @ Xs2 ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X @ Xs2 ) ) )
| ? [Ys6: list_a] :
( ( Ys
= ( cons_a @ X @ Ys6 ) )
& ( ( append_a @ Ys6 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_302_append__eq__Cons__conv,axiom,
! [Ys: list_b,Zs: list_b,X: b,Xs2: list_b] :
( ( ( append_b @ Ys @ Zs )
= ( cons_b @ X @ Xs2 ) )
= ( ( ( Ys = nil_b )
& ( Zs
= ( cons_b @ X @ Xs2 ) ) )
| ? [Ys6: list_b] :
( ( Ys
= ( cons_b @ X @ Ys6 ) )
& ( ( append_b @ Ys6 @ Zs )
= Xs2 ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_303_rev__nonempty__induct,axiom,
! [Xs2: list_a,P2: list_a > $o] :
( ( Xs2 != nil_a )
=> ( ! [X2: a] : ( P2 @ ( cons_a @ X2 @ nil_a ) )
=> ( ! [X2: a,Xs: list_a] :
( ( Xs != nil_a )
=> ( ( P2 @ Xs )
=> ( P2 @ ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_304_rev__nonempty__induct,axiom,
! [Xs2: list_b,P2: list_b > $o] :
( ( Xs2 != nil_b )
=> ( ! [X2: b] : ( P2 @ ( cons_b @ X2 @ nil_b ) )
=> ( ! [X2: b,Xs: list_b] :
( ( Xs != nil_b )
=> ( ( P2 @ Xs )
=> ( P2 @ ( append_b @ Xs @ ( cons_b @ X2 @ nil_b ) ) ) ) )
=> ( P2 @ Xs2 ) ) ) ) ).
% rev_nonempty_induct
thf(fact_305_distinct_Osimps_I2_J,axiom,
! [X: dtree_list_a_b,Xs2: list_dtree_list_a_b] :
( ( distin2097911376254939835st_a_b @ ( cons_dtree_list_a_b @ X @ Xs2 ) )
= ( ~ ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
& ( distin2097911376254939835st_a_b @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_306_distinct_Osimps_I2_J,axiom,
! [X: set_a,Xs2: list_set_a] :
( ( distinct_set_a @ ( cons_set_a @ X @ Xs2 ) )
= ( ~ ( member_set_a @ X @ ( set_set_a2 @ Xs2 ) )
& ( distinct_set_a @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_307_distinct_Osimps_I2_J,axiom,
! [X: list_a,Xs2: list_list_a] :
( ( distinct_list_a @ ( cons_list_a @ X @ Xs2 ) )
= ( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
& ( distinct_list_a @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_308_distinct_Osimps_I2_J,axiom,
! [X: a,Xs2: list_a] :
( ( distinct_a @ ( cons_a @ X @ Xs2 ) )
= ( ~ ( member_a @ X @ ( set_a2 @ Xs2 ) )
& ( distinct_a @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_309_distinct_Osimps_I2_J,axiom,
! [X: b,Xs2: list_b] :
( ( distinct_b @ ( cons_b @ X @ Xs2 ) )
= ( ~ ( member_b @ X @ ( set_b2 @ Xs2 ) )
& ( distinct_b @ Xs2 ) ) ) ).
% distinct.simps(2)
thf(fact_310_distinct__singleton,axiom,
! [X: a] : ( distinct_a @ ( cons_a @ X @ nil_a ) ) ).
% distinct_singleton
thf(fact_311_distinct__singleton,axiom,
! [X: b] : ( distinct_b @ ( cons_b @ X @ nil_b ) ) ).
% distinct_singleton
thf(fact_312_empty__set,axiom,
( bot_bot_set_b
= ( set_b2 @ nil_b ) ) ).
% empty_set
thf(fact_313_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_314_empty__set,axiom,
( bot_bot_set_list_a
= ( set_list_a2 @ nil_list_a ) ) ).
% empty_set
thf(fact_315_not__distinct__conv__prefix,axiom,
! [As: list_dtree_list_a_b] :
( ( ~ ( distin2097911376254939835st_a_b @ As ) )
= ( ? [Xs5: list_dtree_list_a_b,Y3: dtree_list_a_b,Ys3: list_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ Y3 @ ( set_dtree_list_a_b2 @ Xs5 ) )
& ( distin2097911376254939835st_a_b @ Xs5 )
& ( As
= ( append2129087805494049561st_a_b @ Xs5 @ ( cons_dtree_list_a_b @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_316_not__distinct__conv__prefix,axiom,
! [As: list_set_a] :
( ( ~ ( distinct_set_a @ As ) )
= ( ? [Xs5: list_set_a,Y3: set_a,Ys3: list_set_a] :
( ( member_set_a @ Y3 @ ( set_set_a2 @ Xs5 ) )
& ( distinct_set_a @ Xs5 )
& ( As
= ( append_set_a @ Xs5 @ ( cons_set_a @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_317_not__distinct__conv__prefix,axiom,
! [As: list_list_a] :
( ( ~ ( distinct_list_a @ As ) )
= ( ? [Xs5: list_list_a,Y3: list_a,Ys3: list_list_a] :
( ( member_list_a @ Y3 @ ( set_list_a2 @ Xs5 ) )
& ( distinct_list_a @ Xs5 )
& ( As
= ( append_list_a @ Xs5 @ ( cons_list_a @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_318_not__distinct__conv__prefix,axiom,
! [As: list_a] :
( ( ~ ( distinct_a @ As ) )
= ( ? [Xs5: list_a,Y3: a,Ys3: list_a] :
( ( member_a @ Y3 @ ( set_a2 @ Xs5 ) )
& ( distinct_a @ Xs5 )
& ( As
= ( append_a @ Xs5 @ ( cons_a @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_319_not__distinct__conv__prefix,axiom,
! [As: list_b] :
( ( ~ ( distinct_b @ As ) )
= ( ? [Xs5: list_b,Y3: b,Ys3: list_b] :
( ( member_b @ Y3 @ ( set_b2 @ Xs5 ) )
& ( distinct_b @ Xs5 )
& ( As
= ( append_b @ Xs5 @ ( cons_b @ Y3 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_320_not__distinct__decomp,axiom,
! [Ws: list_a] :
( ~ ( distinct_a @ Ws )
=> ? [Xs: list_a,Ys4: list_a,Zs3: list_a,Y4: a] :
( Ws
= ( append_a @ Xs @ ( append_a @ ( cons_a @ Y4 @ nil_a ) @ ( append_a @ Ys4 @ ( append_a @ ( cons_a @ Y4 @ nil_a ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_321_not__distinct__decomp,axiom,
! [Ws: list_b] :
( ~ ( distinct_b @ Ws )
=> ? [Xs: list_b,Ys4: list_b,Zs3: list_b,Y4: b] :
( Ws
= ( append_b @ Xs @ ( append_b @ ( cons_b @ Y4 @ nil_b ) @ ( append_b @ Ys4 @ ( append_b @ ( cons_b @ Y4 @ nil_b ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_322_path__lverts__empty__if__roothd,axiom,
! [T: dtree_list_a_b] :
( ( ( root_list_a_b @ T )
!= nil_a )
=> ( ( iKKBZ_6987179986356532253ts_a_b @ T @ ( hd_a @ ( root_list_a_b @ T ) ) )
= bot_bot_set_a ) ) ).
% path_lverts_empty_if_roothd
thf(fact_323_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_324_dlverts__reachable1__if__dom__children__aux,axiom,
! [T1: dtree_list_a_b,R3: list_a,X6: set_a,Y: a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ T1 ) )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( set_a2 @ R3 ) @ X6 ) @ ( iKKBZ_6987179986356532253ts_a_b @ T1 @ ( hd_a @ X2 ) ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ ( hd_a @ X2 ) ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ X6 )
=> ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ T1 ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ X2 ) )
=> ( ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ).
% dlverts_reachable1_if_dom_children_aux
thf(fact_325_hd__reachable1__from__outside,axiom,
! [X: a,Y: a,Ys: list_a,Xs2: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ( ( member_a @ Y @ ( set_a2 @ Ys ) )
=> ( ? [X2: a] : ( member_a @ X2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ ( hd_a @ Ys ) ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% hd_reachable1_from_outside
thf(fact_326_reachable1__append__old__if__arcU,axiom,
! [Xs2: list_a,Ys: list_a,U2: list_a,Z: a,Y: a] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ Xs2 ) )
= bot_bot_set_a )
=> ( ( member_a @ Z @ ( set_a2 @ U2 ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs2 )
=> ( ( member_a @ Y @ ( set_a2 @ ( append_a @ Xs2 @ Ys ) ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Z @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ) ).
% reachable1_append_old_if_arcU
thf(fact_327_not__reachable1__append__if__not__old,axiom,
! [U2: list_a,B: list_a,X: list_a] :
( ~ ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ U2 ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ B ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ X ) )
= bot_bot_set_a )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ X )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ X ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ B ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ~ ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ U2 ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ ( append_a @ X @ B ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% not_reachable1_append_if_not_old
thf(fact_328_no__back__arcs_Oelims_I1_J,axiom,
! [X: list_a,Y: $o] :
( ( ( iKKBZ_7773321254043928001cs_a_b @ t @ X )
= Y )
=> ( ( ( X = nil_a )
=> ~ Y )
=> ~ ! [X2: a,Xs: list_a] :
( ( X
= ( cons_a @ X2 @ Xs ) )
=> ( Y
= ( ~ ( ~ ? [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Xs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X2 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs ) ) ) ) ) ) ) ).
% no_back_arcs.elims(1)
thf(fact_329_no__back__arcs_Oelims_I2_J,axiom,
! [X: list_a] :
( ( iKKBZ_7773321254043928001cs_a_b @ t @ X )
=> ( ( X != nil_a )
=> ~ ! [X2: a,Xs: list_a] :
( ( X
= ( cons_a @ X2 @ Xs ) )
=> ~ ( ~ ? [Y5: a] :
( ( member_a @ Y5 @ ( set_a2 @ Xs ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y5 @ X2 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs ) ) ) ) ) ).
% no_back_arcs.elims(2)
thf(fact_330_dverts__reachable1__if__dom__children,axiom,
! [T1: dtree_list_a_b,V: list_a] :
( ( iKKBZ_3908525916494739553en_a_b @ T1 @ t )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) )
=> ( ( V
!= ( root_list_a_b @ T1 ) )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ T1 ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ X2 ) )
=> ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ V ) )
=> ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ) ) ).
% dverts_reachable1_if_dom_children
thf(fact_331_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 ) )
& ? [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ S2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y3 ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% before_def
thf(fact_332_forward__arc__to__head,axiom,
! [Ys: list_a,Xs2: list_a,X: a,Y: a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Ys )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ( ( member_a @ Y @ ( set_a2 @ Ys ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( Y
= ( hd_a @ Ys ) ) ) ) ) ) ) ).
% forward_arc_to_head
thf(fact_333_no__back__arcs_Osimps_I1_J,axiom,
iKKBZ_7773321254043928001cs_a_b @ t @ nil_a ).
% no_back_arcs.simps(1)
thf(fact_334_no__back__arcs__single,axiom,
! [X: a] : ( iKKBZ_7773321254043928001cs_a_b @ t @ ( cons_a @ X @ nil_a ) ) ).
% no_back_arcs_single
thf(fact_335_sccs__verts__disjoint,axiom,
! [S: set_a,T2: set_a] :
( ( member_set_a @ S @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( ( member_set_a @ T2 @ ( digrap2871191568752656621ts_a_b @ t ) )
=> ( ( S != T2 )
=> ( ( inf_inf_set_a @ S @ T2 )
= bot_bot_set_a ) ) ) ) ).
% sccs_verts_disjoint
thf(fact_336_no__back__arcs_Oelims_I3_J,axiom,
! [X: list_a] :
( ~ ( iKKBZ_7773321254043928001cs_a_b @ t @ X )
=> ~ ! [X2: a,Xs: list_a] :
( ( X
= ( 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.elims(3)
thf(fact_337_no__back__arcs_Osimps_I2_J,axiom,
! [X: a,Xs2: list_a] :
( ( iKKBZ_7773321254043928001cs_a_b @ t @ ( cons_a @ X @ Xs2 ) )
= ( ~ ? [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ Xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs2 ) ) ) ).
% no_back_arcs.simps(2)
thf(fact_338_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 )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ S1 ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ S2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ S1 @ S2 ) ) ) ) ) ) ).
% forward_app'
thf(fact_339_move__mid__backward__if__noarc_H,axiom,
! [U2: list_a,V2: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ U2 ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ V2 )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ V2 ) )
= bot_bot_set_a )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ V2 @ ( append_a @ Bs @ Cs ) ) ) ) ) ) ) ) ) ).
% move_mid_backward_if_noarc'
thf(fact_340_distinct__append,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( distinct_b @ ( append_b @ Xs2 @ Ys ) )
= ( ( distinct_b @ Xs2 )
& ( distinct_b @ Ys )
& ( ( inf_inf_set_b @ ( set_b2 @ Xs2 ) @ ( set_b2 @ Ys ) )
= bot_bot_set_b ) ) ) ).
% distinct_append
thf(fact_341_distinct__append,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( distinct_a @ ( append_a @ Xs2 @ Ys ) )
= ( ( distinct_a @ Xs2 )
& ( distinct_a @ Ys )
& ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a ) ) ) ).
% distinct_append
thf(fact_342_distinct__append,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( distinct_list_a @ ( append_list_a @ Xs2 @ Ys ) )
= ( ( distinct_list_a @ Xs2 )
& ( distinct_list_a @ Ys )
& ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs2 ) @ ( set_list_a2 @ Ys ) )
= bot_bot_set_list_a ) ) ) ).
% distinct_append
thf(fact_343_inner__verts__singleton,axiom,
! [X: b] :
( ( pre_inner_verts_a_b @ t @ ( cons_b @ X @ nil_b ) )
= nil_a ) ).
% inner_verts_singleton
thf(fact_344_directed__tree_Ono__back__arcs_Ocong,axiom,
iKKBZ_7773321254043928001cs_a_b = iKKBZ_7773321254043928001cs_a_b ).
% directed_tree.no_back_arcs.cong
thf(fact_345_wf__list__verts_Ocases,axiom,
! [X: list_P1396940483166286381od_a_a] :
( ( X != nil_Product_prod_a_a )
=> ~ ! [V3: a,E2: a,Xs: list_P1396940483166286381od_a_a] :
( X
!= ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ V3 @ E2 ) @ Xs ) ) ) ).
% wf_list_verts.cases
thf(fact_346_wf__list__verts_Ocases,axiom,
! [X: list_P5058367588434897935st_b_a] :
( ( X != nil_Pr3326166910223580559st_b_a )
=> ~ ! [V3: a,E2: produc1943741644644106336st_b_a,Xs: list_P5058367588434897935st_b_a] :
( X
!= ( cons_P4566545638659916095st_b_a @ ( produc7119031474978700025st_b_a @ V3 @ E2 ) @ Xs ) ) ) ).
% wf_list_verts.cases
thf(fact_347_wf__list__verts_Ocases,axiom,
! [X: list_P4861094521484316518st_b_a] :
( ( X != nil_Pr9111545067797741414st_b_a )
=> ~ ! [V3: list_b,E2: a,Xs: list_P4861094521484316518st_b_a] :
( X
!= ( cons_P1383153349113754390st_b_a @ ( produc4145578316043568848st_b_a @ V3 @ E2 ) @ Xs ) ) ) ).
% wf_list_verts.cases
thf(fact_348_wf__list__verts_Ocases,axiom,
! [X: list_P321204300973800749list_a] :
( ( X != nil_Pr3188421586756112173list_a )
=> ~ ! [V3: list_a,E2: list_a,Xs: list_P321204300973800749list_a] :
( X
!= ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ V3 @ E2 ) @ Xs ) ) ) ).
% wf_list_verts.cases
thf(fact_349_wf__list__verts_Ocases,axiom,
! [X: list_P7479555583871319568_a_b_b] :
( ( X != nil_Pr1282242650802307322_a_b_b )
=> ~ ! [V3: dtree_list_a_b,E2: b,Xs: list_P7479555583871319568_a_b_b] :
( X
!= ( cons_P433472372565078346_a_b_b @ ( produc7704165765595008946_a_b_b @ V3 @ E2 ) @ Xs ) ) ) ).
% wf_list_verts.cases
thf(fact_350_dtree__from__list_Ocases,axiom,
! [X: produc5160346394017861590od_a_a] :
( ! [R: a] :
( X
!= ( produc6572230313500880070od_a_a @ R @ nil_Product_prod_a_a ) )
=> ~ ! [R: a,V3: a,E2: a,Xs: list_P1396940483166286381od_a_a] :
( X
!= ( produc6572230313500880070od_a_a @ R @ ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ V3 @ E2 ) @ Xs ) ) ) ) ).
% dtree_from_list.cases
thf(fact_351_dtree__from__list_Ocases,axiom,
! [X: produc4855257872370660280st_b_a] :
( ! [R: a] :
( X
!= ( produc6798837171946177832st_b_a @ R @ nil_Pr3326166910223580559st_b_a ) )
=> ~ ! [R: a,V3: a,E2: produc1943741644644106336st_b_a,Xs: list_P5058367588434897935st_b_a] :
( X
!= ( produc6798837171946177832st_b_a @ R @ ( cons_P4566545638659916095st_b_a @ ( produc7119031474978700025st_b_a @ V3 @ E2 ) @ Xs ) ) ) ) ).
% dtree_from_list.cases
thf(fact_352_dtree__from__list_Ocases,axiom,
! [X: produc1407386371851737800st_b_a] :
( ! [R: list_b] :
( X
!= ( produc1203288445310824248st_b_a @ R @ nil_Pr9111545067797741414st_b_a ) )
=> ~ ! [R: list_b,V3: list_b,E2: a,Xs: list_P4861094521484316518st_b_a] :
( X
!= ( produc1203288445310824248st_b_a @ R @ ( cons_P1383153349113754390st_b_a @ ( produc4145578316043568848st_b_a @ V3 @ E2 ) @ Xs ) ) ) ) ).
% dtree_from_list.cases
thf(fact_353_dtree__from__list_Ocases,axiom,
! [X: produc4823114226738795600list_a] :
( ! [R: list_a] :
( X
!= ( produc3757311814099817152list_a @ R @ nil_Pr3188421586756112173list_a ) )
=> ~ ! [R: list_a,V3: list_a,E2: list_a,Xs: list_P321204300973800749list_a] :
( X
!= ( produc3757311814099817152list_a @ R @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ V3 @ E2 ) @ Xs ) ) ) ) ).
% dtree_from_list.cases
thf(fact_354_dtree__from__list_Ocases,axiom,
! [X: produc4426939369905041893_a_b_b] :
( ! [R: dtree_list_a_b] :
( X
!= ( produc737586152522436509_a_b_b @ R @ nil_Pr1282242650802307322_a_b_b ) )
=> ~ ! [R: dtree_list_a_b,V3: dtree_list_a_b,E2: b,Xs: list_P7479555583871319568_a_b_b] :
( X
!= ( produc737586152522436509_a_b_b @ R @ ( cons_P433472372565078346_a_b_b @ ( produc7704165765595008946_a_b_b @ V3 @ E2 ) @ Xs ) ) ) ) ).
% dtree_from_list.cases
thf(fact_355_concat_Osimps_I2_J,axiom,
! [X: list_b,Xs2: list_list_b] :
( ( concat_b @ ( cons_list_b @ X @ Xs2 ) )
= ( append_b @ X @ ( concat_b @ Xs2 ) ) ) ).
% concat.simps(2)
thf(fact_356_concat_Osimps_I2_J,axiom,
! [X: list_a,Xs2: list_list_a] :
( ( concat_a @ ( cons_list_a @ X @ Xs2 ) )
= ( append_a @ X @ ( concat_a @ Xs2 ) ) ) ).
% concat.simps(2)
thf(fact_357_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,Xs: list_b,Xs3: list_b,Xss22: list_list_b] :
( ( Xss
= ( append_list_b @ Xss1 @ ( cons_list_b @ ( append_b @ Xs @ Xs3 ) @ Xss22 ) ) )
& ( Ys
= ( append_b @ ( concat_b @ Xss1 ) @ Xs ) )
& ( Zs
= ( append_b @ Xs3 @ ( concat_b @ Xss22 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_358_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,Xs: list_a,Xs3: list_a,Xss22: list_list_a] :
( ( Xss
= ( append_list_a @ Xss1 @ ( cons_list_a @ ( append_a @ Xs @ Xs3 ) @ Xss22 ) ) )
& ( Ys
= ( append_a @ ( concat_a @ Xss1 ) @ Xs ) )
& ( Zs
= ( append_a @ Xs3 @ ( concat_a @ Xss22 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_359_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,Xs5: list_a,Xs6: list_a,Xss23: list_list_a] :
( ( Xss
= ( append_list_a @ Xss12 @ ( cons_list_a @ ( append_a @ Xs5 @ Xs6 ) @ Xss23 ) ) )
& ( Ys
= ( append_a @ ( concat_a @ Xss12 ) @ Xs5 ) )
& ( Zs
= ( append_a @ Xs6 @ ( concat_a @ Xss23 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_360_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,Xs5: list_b,Xs6: list_b,Xss23: list_list_b] :
( ( Xss
= ( append_list_b @ Xss12 @ ( cons_list_b @ ( append_b @ Xs5 @ Xs6 ) @ Xss23 ) ) )
& ( Ys
= ( append_b @ ( concat_b @ Xss12 ) @ Xs5 ) )
& ( Zs
= ( append_b @ Xs6 @ ( concat_b @ Xss23 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_361_distinct__concat,axiom,
! [Xs2: list_list_a] :
( ( distinct_list_a @ Xs2 )
=> ( ! [Ys4: list_a] :
( ( member_list_a @ Ys4 @ ( set_list_a2 @ Xs2 ) )
=> ( distinct_a @ Ys4 ) )
=> ( ! [Ys4: list_a,Zs3: list_a] :
( ( member_list_a @ Ys4 @ ( set_list_a2 @ Xs2 ) )
=> ( ( member_list_a @ Zs3 @ ( set_list_a2 @ Xs2 ) )
=> ( ( Ys4 != Zs3 )
=> ( ( inf_inf_set_a @ ( set_a2 @ Ys4 ) @ ( set_a2 @ Zs3 ) )
= bot_bot_set_a ) ) ) )
=> ( distinct_a @ ( concat_a @ Xs2 ) ) ) ) ) ).
% distinct_concat
thf(fact_362_distinct__concat,axiom,
! [Xs2: list_list_list_a] :
( ( distinct_list_list_a @ Xs2 )
=> ( ! [Ys4: list_list_a] :
( ( member_list_list_a @ Ys4 @ ( set_list_list_a2 @ Xs2 ) )
=> ( distinct_list_a @ Ys4 ) )
=> ( ! [Ys4: list_list_a,Zs3: list_list_a] :
( ( member_list_list_a @ Ys4 @ ( set_list_list_a2 @ Xs2 ) )
=> ( ( member_list_list_a @ Zs3 @ ( set_list_list_a2 @ Xs2 ) )
=> ( ( Ys4 != Zs3 )
=> ( ( inf_inf_set_list_a @ ( set_list_a2 @ Ys4 ) @ ( set_list_a2 @ Zs3 ) )
= bot_bot_set_list_a ) ) ) )
=> ( distinct_list_a @ ( concat_list_a @ Xs2 ) ) ) ) ) ).
% distinct_concat
thf(fact_363_no__back__before__aux,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs2 )
=> ( ( iKKBZ_4622586873178280511rm_a_b @ t @ Ys )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ Ys ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( arcs_ends_a_b @ t ) ) ) )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ ( append_a @ Xs2 @ Ys ) ) ) ) ) ) ).
% no_back_before_aux
thf(fact_364_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_365_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_366_T_Odverts__reach1__in__dverts__root,axiom,
! [T1: dtree_list_a_b,V: list_a] :
( ( is_subtree_list_a_b @ T1 @ ta )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ta ) )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) ) ) ) ) ).
% T.dverts_reach1_in_dverts_root
thf(fact_367_T_Odverts__reach1__in__dverts__r,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,V: list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ ta )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ta ) )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ R3 ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) ) ) ) ) ) ).
% T.dverts_reach1_in_dverts_r
thf(fact_368_nempty__inter__notin__dverts,axiom,
! [V: list_a,T: dtree_list_a_b] :
( ( V != nil_a )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ V ) @ ( list_dlverts_a_b @ T ) )
= bot_bot_set_a )
=> ~ ( member_list_a @ V @ ( dverts_list_a_b @ T ) ) ) ) ).
% nempty_inter_notin_dverts
thf(fact_369_T_Odverts__reach1__in__dlverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a] :
( ( is_subtree_list_a_b @ T1 @ ta )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( 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 ) ) ) ) ) ) ).
% T.dverts_reach1_in_dlverts
thf(fact_370_T_Odverts__reach1__in__dverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a,V22: list_a] :
( ( is_subtree_list_a_b @ T1 @ ta )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( member_a @ Y @ ( set_a2 @ V22 ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ ta ) )
=> ( member_list_a @ V22 @ ( dverts_list_a_b @ T1 ) ) ) ) ) ) ) ) ).
% T.dverts_reach1_in_dverts
thf(fact_371_T_Oarc__to__dverts__in__subtree,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: a,Y: a,V: list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ ta )
=> ( ( member_a @ X @ ( set_a2 @ R3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ Y @ ( set_a2 @ V ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ta ) )
=> ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) ) ) ) ) ) ) ) ).
% T.arc_to_dverts_in_subtree
thf(fact_372_cas_Ocases,axiom,
! [X: produc7945266988514096265st_b_a] :
( ! [U3: a,V3: a] :
( X
!= ( produc7119031474978700025st_b_a @ U3 @ ( produc4145578316043568848st_b_a @ nil_b @ V3 ) ) )
=> ~ ! [U3: a,E2: b,Es: list_b,V3: a] :
( X
!= ( produc7119031474978700025st_b_a @ U3 @ ( produc4145578316043568848st_b_a @ ( cons_b @ E2 @ Es ) @ V3 ) ) ) ) ).
% cas.cases
thf(fact_373_no__back__insert,axiom,
! [X: a,Xs2: list_a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ ( cons_a @ X @ Xs2 ) )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs2 ) ) ).
% no_back_insert
thf(fact_374_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_375_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_376_no__back__arcs__alt__aux2,axiom,
! [Xs2: list_a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs2 )
=> ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs2 ) ) ).
% no_back_arcs_alt_aux2
thf(fact_377_no__back__arcs__alt,axiom,
! [Xs2: list_a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs2 )
= ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs2 ) ) ).
% no_back_arcs_alt
thf(fact_378_ex__subtree__if__in__lverts,axiom,
! [V: a,T1: dtree_list_a_b] :
( ( member_a @ V @ ( list_dlverts_a_b @ T1 ) )
=> ? [T22: dtree_list_a_b] :
( ( is_subtree_list_a_b @ T22 @ T1 )
& ( member_a @ V @ ( set_a2 @ ( root_list_a_b @ T22 ) ) ) ) ) ).
% ex_subtree_if_in_lverts
thf(fact_379_T_Odverts__same__if__set__subtree,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,V22: list_a] :
( ( is_subtree_list_a_b @ T1 @ ta )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V22 ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ ta ) )
=> ( V1 = V22 ) ) ) ) ) ) ).
% T.dverts_same_if_set_subtree
thf(fact_380_T_Overts__distinct__subtree,axiom,
! [Tn: dtree_list_a_b,V: list_a] :
( ( is_subtree_list_a_b @ Tn @ ta )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ Tn ) )
=> ( distinct_a @ V ) ) ) ).
% T.verts_distinct_subtree
thf(fact_381_no__back__single,axiom,
! [X: a] : ( iKKBZ_3684931046464919648ck_a_b @ t @ ( cons_a @ X @ nil_a ) ) ).
% no_back_single
thf(fact_382_no__back__if__distinct__forward,axiom,
! [Xs2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs2 )
=> ( ( distinct_a @ Xs2 )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs2 ) ) ) ).
% no_back_if_distinct_forward
thf(fact_383_no__back__before,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( iKKBZ_7682935289300565975re_a_b @ t @ Xs2 @ Ys )
=> ( iKKBZ_3684931046464919648ck_a_b @ t @ ( append_a @ Xs2 @ Ys ) ) ) ).
% no_back_before
thf(fact_384_seq__conform__alt,axiom,
! [Xs2: list_a] :
( ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs2 )
= ( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs2 )
& ( iKKBZ_3684931046464919648ck_a_b @ t @ Xs2 ) ) ) ).
% seq_conform_alt
thf(fact_385_T_Odlverts__reach__in__dlverts,axiom,
! [X: a,Y: a,T1: dtree_list_a_b] :
( ( reachable_a_b @ t @ X @ Y )
=> ( ( is_subtree_list_a_b @ T1 @ ta )
=> ( ( member_a @ X @ ( list_dlverts_a_b @ T1 ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ).
% T.dlverts_reach_in_dlverts
thf(fact_386_T_Overts__conform__subtree,axiom,
! [Tn: dtree_list_a_b,V: list_a] :
( ( is_subtree_list_a_b @ Tn @ ta )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ Tn ) )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ V ) ) ) ).
% T.verts_conform_subtree
thf(fact_387_no__arc__fst__if__no__back,axiom,
! [X: a,Xs2: list_a,Y: a] :
( ( iKKBZ_3684931046464919648ck_a_b @ t @ ( cons_a @ X @ Xs2 ) )
=> ( ( member_a @ Y @ ( set_a2 @ Xs2 ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ ( arcs_ends_a_b @ t ) ) ) ) ).
% no_arc_fst_if_no_back
thf(fact_388_T_Odverts__reach__in__dverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a,V22: list_a] :
( ( is_subtree_list_a_b @ T1 @ ta )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( reachable_a_b @ t @ X @ Y )
=> ( ( member_a @ Y @ ( set_a2 @ V22 ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ ta ) )
=> ( member_list_a @ V22 @ ( dverts_list_a_b @ T1 ) ) ) ) ) ) ) ) ).
% T.dverts_reach_in_dverts
thf(fact_389_T_Odlverts__arc__in__dlverts,axiom,
! [T1: dtree_list_a_b,X: a,Y: a] :
( ( is_subtree_list_a_b @ T1 @ ta )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ X @ ( list_dlverts_a_b @ T1 ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ).
% T.dlverts_arc_in_dlverts
thf(fact_390_T_Odverts__arc__in__dverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a,V22: list_a] :
( ( is_subtree_list_a_b @ T1 @ ta )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ Y @ ( set_a2 @ V22 ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ ta ) )
=> ( member_list_a @ V22 @ ( dverts_list_a_b @ T1 ) ) ) ) ) ) ) ) ).
% T.dverts_arc_in_dverts
thf(fact_391_T_Odverts__reach__in__dlverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a] :
( ( is_subtree_list_a_b @ T1 @ ta )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( reachable_a_b @ t @ X @ Y )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ) ).
% T.dverts_reach_in_dlverts
thf(fact_392_T_Odlverts__reach1__in__dlverts,axiom,
! [X: a,Y: a,T1: dtree_list_a_b] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( is_subtree_list_a_b @ T1 @ ta )
=> ( ( member_a @ X @ ( list_dlverts_a_b @ T1 ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ).
% T.dlverts_reach1_in_dlverts
thf(fact_393_T_Oarc__in__dlverts__subtree,axiom,
! [Tn: dtree_list_a_b,R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: a,Y: a] :
( ( is_subtree_list_a_b @ Tn @ ta )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ Tn )
=> ( ( member_a @ X @ ( set_a2 @ R3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ ( node_list_a_b @ R3 @ Xs2 ) ) ) ) ) ) ) ).
% T.arc_in_dlverts_subtree
thf(fact_394_T_Oarc__in__dlverts,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: a,Y: a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ ta )
=> ( ( member_a @ X @ ( set_a2 @ R3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ ( node_list_a_b @ R3 @ Xs2 ) ) ) ) ) ) ).
% T.arc_in_dlverts
thf(fact_395_T_Odverts__arc__in__dlverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a] :
( ( is_subtree_list_a_b @ T1 @ ta )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ) ).
% T.dverts_arc_in_dlverts
thf(fact_396_subtree__eq__if__trans__eq2,axiom,
! [T1: dtree_list_a_b,T23: dtree_list_a_b,T3: dtree_list_a_b] :
( ( is_subtree_list_a_b @ T1 @ T23 )
=> ( ( is_subtree_list_a_b @ T23 @ T3 )
=> ( ( T1 = T3 )
=> ( T23 = T3 ) ) ) ) ).
% subtree_eq_if_trans_eq2
thf(fact_397_subtree__eq__if__trans__eq1,axiom,
! [T1: dtree_list_a_b,T23: dtree_list_a_b,T3: dtree_list_a_b] :
( ( is_subtree_list_a_b @ T1 @ T23 )
=> ( ( is_subtree_list_a_b @ T23 @ T3 )
=> ( ( T1 = T3 )
=> ( T1 = T23 ) ) ) ) ).
% subtree_eq_if_trans_eq1
thf(fact_398_subtree__antisym,axiom,
! [T1: dtree_list_a_b,T23: dtree_list_a_b] :
( ( is_subtree_list_a_b @ T1 @ T23 )
=> ( ( is_subtree_list_a_b @ T23 @ T1 )
=> ( T1 = T23 ) ) ) ).
% subtree_antisym
thf(fact_399_subtree__trans,axiom,
! [X: dtree_list_a_b,Y: dtree_list_a_b,Z: dtree_list_a_b] :
( ( is_subtree_list_a_b @ X @ Y )
=> ( ( is_subtree_list_a_b @ Y @ Z )
=> ( is_subtree_list_a_b @ X @ Z ) ) ) ).
% subtree_trans
thf(fact_400_self__subtree,axiom,
! [T: dtree_list_a_b] : ( is_subtree_list_a_b @ T @ T ) ).
% self_subtree
thf(fact_401_directed__tree_Ono__back_Ocong,axiom,
iKKBZ_3684931046464919648ck_a_b = iKKBZ_3684931046464919648ck_a_b ).
% directed_tree.no_back.cong
thf(fact_402_insert__between_Ocases,axiom,
! [X: produc9008341577332299707st_a_b] :
~ ! [V3: list_a,E2: b,X2: list_a,Y4: list_a,R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( produc1848684973559390389st_a_b @ V3 @ ( produc5064203622704112514st_a_b @ E2 @ ( produc673257793671328980st_a_b @ X2 @ ( produc148520996349637281st_a_b @ Y4 @ ( node_list_a_b @ R @ Xs ) ) ) ) ) ) ).
% insert_between.cases
thf(fact_403_combine_Ocases,axiom,
! [X: produc7147531718898801626st_a_b] :
~ ! [X2: list_a,Y4: list_a,R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( produc673257793671328980st_a_b @ X2 @ ( produc148520996349637281st_a_b @ Y4 @ ( node_list_a_b @ R @ Xs ) ) ) ) ).
% combine.cases
thf(fact_404_map__tailrec__rev_Ocases,axiom,
! [X: produc1473018763691903991list_a] :
( ! [F: a > a,Bs2: list_a] :
( X
!= ( produc8643929849434629545list_a @ F @ ( produc6837034575241423639list_a @ nil_a @ Bs2 ) ) )
=> ~ ! [F: a > a,A3: a,As3: list_a,Bs2: list_a] :
( X
!= ( produc8643929849434629545list_a @ F @ ( produc6837034575241423639list_a @ ( cons_a @ A3 @ As3 ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_405_dtail_Ocases,axiom,
! [X: produc1920479565126685823list_a] :
~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b,Def: b > list_a] :
( X
!= ( produc2621617146629198007list_a @ ( node_list_a_b @ R @ Xs ) @ Def ) ) ).
% dtail.cases
thf(fact_406_is__subtree_Ocases,axiom,
! [X: produc1510363273921914569st_a_b] :
~ ! [X2: dtree_list_a_b,R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( produc783528831147138817st_a_b @ X2 @ ( node_list_a_b @ R @ Xs ) ) ) ).
% is_subtree.cases
thf(fact_407_subtree__root__if__dverts,axiom,
! [X: list_a,T: dtree_list_a_b] :
( ( member_list_a @ X @ ( dverts_list_a_b @ T ) )
=> ? [Xs: fset_P2153231429829016240_a_b_b] : ( is_subtree_list_a_b @ ( node_list_a_b @ X @ Xs ) @ T ) ) ).
% subtree_root_if_dverts
thf(fact_408_subtree__root__if__dlverts,axiom,
! [X: a,T: dtree_list_a_b] :
( ( member_a @ X @ ( list_dlverts_a_b @ T ) )
=> ? [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R @ Xs ) @ T )
& ( member_a @ X @ ( set_a2 @ R ) ) ) ) ).
% subtree_root_if_dlverts
thf(fact_409_successively_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P3: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P3 @ nil_a ) )
=> ( ! [P3: a > a > $o,X2: a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X2 @ nil_a ) ) )
=> ~ ! [P3: a > a > $o,X2: a,Y4: a,Xs: list_a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X2 @ ( cons_a @ Y4 @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_410_successively_Ocases,axiom,
! [X: produc5185152304234826110list_b] :
( ! [P3: b > b > $o] :
( X
!= ( produc8193136575784045678list_b @ P3 @ nil_b ) )
=> ( ! [P3: b > b > $o,X2: b] :
( X
!= ( produc8193136575784045678list_b @ P3 @ ( cons_b @ X2 @ nil_b ) ) )
=> ~ ! [P3: b > b > $o,X2: b,Y4: b,Xs: list_b] :
( X
!= ( produc8193136575784045678list_b @ P3 @ ( cons_b @ X2 @ ( cons_b @ Y4 @ Xs ) ) ) ) ) ) ).
% successively.cases
thf(fact_411_sorted__wrt_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P3: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P3 @ nil_a ) )
=> ~ ! [P3: a > a > $o,X2: a,Ys4: list_a] :
( X
!= ( produc8111569692950616493list_a @ P3 @ ( cons_a @ X2 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_412_sorted__wrt_Ocases,axiom,
! [X: produc5185152304234826110list_b] :
( ! [P3: b > b > $o] :
( X
!= ( produc8193136575784045678list_b @ P3 @ nil_b ) )
=> ~ ! [P3: b > b > $o,X2: b,Ys4: list_b] :
( X
!= ( produc8193136575784045678list_b @ P3 @ ( cons_b @ X2 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_413_wf__dlverts_Ocases,axiom,
! [X: dtree_list_a_b] :
~ ! [R: list_a,Xs: fset_P2153231429829016240_a_b_b] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ).
% wf_dlverts.cases
thf(fact_414_dverts__nempty,axiom,
! [T: dtree_list_a_b] :
( ( dverts_list_a_b @ T )
!= bot_bot_set_list_a ) ).
% dverts_nempty
thf(fact_415_list__in__verts__iff__lverts,axiom,
! [X: a,T: dtree_list_a_b] :
( ( member_a @ X @ ( list_dlverts_a_b @ T ) )
= ( ? [X5: list_a] :
( ( member_list_a @ X5 @ ( dverts_list_a_b @ T ) )
& ( member_a @ X @ ( set_a2 @ X5 ) ) ) ) ) ).
% list_in_verts_iff_lverts
thf(fact_416_list__in__verts__if__lverts,axiom,
! [X: a,T: dtree_list_a_b] :
( ( member_a @ X @ ( list_dlverts_a_b @ T ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ T ) )
& ( member_a @ X @ ( set_a2 @ X2 ) ) ) ) ).
% list_in_verts_if_lverts
thf(fact_417_lverts__if__in__verts,axiom,
! [V: list_a,T: dtree_list_a_b,X: a] :
( ( member_list_a @ V @ ( dverts_list_a_b @ T ) )
=> ( ( member_a @ X @ ( set_a2 @ V ) )
=> ( member_a @ X @ ( list_dlverts_a_b @ T ) ) ) ) ).
% lverts_if_in_verts
thf(fact_418_dlverts__nempty__aux,axiom,
! [T: dtree_list_a_b] :
( ~ ( member_list_a @ nil_a @ ( dverts_list_a_b @ T ) )
=> ( ( list_dlverts_a_b @ T )
!= bot_bot_set_a ) ) ).
% dlverts_nempty_aux
thf(fact_419_nempty__root__in__lverts,axiom,
! [T: dtree_list_a_b] :
( ( ( root_list_a_b @ T )
!= nil_a )
=> ( member_a @ ( hd_a @ ( root_list_a_b @ T ) ) @ ( list_dlverts_a_b @ T ) ) ) ).
% nempty_root_in_lverts
thf(fact_420_to__list__tree__dom__iff,axiom,
! [X: a,Y: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
= ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ nil_a ) @ ( cons_a @ Y @ nil_a ) ) @ ( arcs_ends_list_a_b @ ( direct3773525127397338803ee_a_b @ t ) ) ) ) ).
% to_list_tree_dom_iff
thf(fact_421_forward__arcs_Oelims_I1_J,axiom,
! [X: list_a,Y: $o] :
( ( ( iKKBZ_4180558001818622352cs_a_b @ t @ X )
= Y )
=> ( ( ( X = nil_a )
=> ~ Y )
=> ( ( ? [X2: a] :
( X
= ( cons_a @ X2 @ nil_a ) )
=> ~ Y )
=> ~ ! [X2: a,V3: a,Va: list_a] :
( ( X
= ( cons_a @ X2 @ ( cons_a @ V3 @ Va ) ) )
=> ( Y
= ( ~ ( ? [Y3: a] :
( ( member_a @ Y3 @ ( set_a2 @ ( cons_a @ V3 @ Va ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y3 @ X2 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ V3 @ Va ) ) ) ) ) ) ) ) ) ).
% forward_arcs.elims(1)
thf(fact_422_forward__arcs_Oelims_I2_J,axiom,
! [X: list_a] :
( ( iKKBZ_4180558001818622352cs_a_b @ t @ X )
=> ( ( X != nil_a )
=> ( ! [X2: a] :
( X
!= ( cons_a @ X2 @ nil_a ) )
=> ~ ! [X2: a,V3: a,Va: list_a] :
( ( X
= ( cons_a @ X2 @ ( cons_a @ V3 @ Va ) ) )
=> ~ ( ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ ( cons_a @ V3 @ Va ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ V3 @ Va ) ) ) ) ) ) ) ).
% forward_arcs.elims(2)
thf(fact_423_vwalk__wf__digraph__consI,axiom,
! [P: list_a,A: a] :
( ( vertex_vwalk_a_b @ P @ t )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( hd_a @ P ) ) @ ( arcs_ends_a_b @ t ) )
=> ( vertex_vwalk_a_b @ ( cons_a @ A @ P ) @ t ) ) ) ).
% vwalk_wf_digraph_consI
thf(fact_424_arc__to__lst__if__forward,axiom,
! [X: a,Xs2: list_a,Y: a,Ys: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ ( cons_a @ X @ Xs2 ) ) )
=> ( ( Xs2
= ( cons_a @ Y @ Ys ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ X ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% arc_to_lst_if_forward
thf(fact_425_Int__Un__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( sup_sup_set_a @ T2 @ ( inf_inf_set_a @ S @ T2 ) )
= T2 ) ).
% Int_Un_eq(4)
thf(fact_426_Int__Un__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ S @ ( inf_inf_set_a @ S @ T2 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_427_forward__arcs__split,axiom,
! [Ys: list_a,Xs2: list_a] :
( ( iKKBZ_4180558001818622352cs_a_b @ t @ ( append_a @ Ys @ Xs2 ) )
=> ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs2 ) ) ).
% forward_arcs_split
thf(fact_428_forward__arcs_Osimps_I1_J,axiom,
iKKBZ_4180558001818622352cs_a_b @ t @ nil_a ).
% forward_arcs.simps(1)
thf(fact_429_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X5: a] :
~ ( P2 @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_430_empty__Collect__eq,axiom,
! [P2: list_a > $o] :
( ( bot_bot_set_list_a
= ( collect_list_a @ P2 ) )
= ( ! [X5: list_a] :
~ ( P2 @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_431_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X5: a] :
~ ( P2 @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_432_Collect__empty__eq,axiom,
! [P2: list_a > $o] :
( ( ( collect_list_a @ P2 )
= bot_bot_set_list_a )
= ( ! [X5: list_a] :
~ ( P2 @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_433_all__not__in__conv,axiom,
! [A2: set_dtree_list_a_b] :
( ( ! [X5: dtree_list_a_b] :
~ ( member551035911493665803st_a_b @ X5 @ A2 ) )
= ( A2 = bot_bo798015271861357502st_a_b ) ) ).
% all_not_in_conv
thf(fact_434_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_435_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_436_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_437_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_438_empty__iff,axiom,
! [C: dtree_list_a_b] :
~ ( member551035911493665803st_a_b @ C @ bot_bo798015271861357502st_a_b ) ).
% empty_iff
thf(fact_439_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_440_empty__iff,axiom,
! [C: b] :
~ ( member_b @ C @ bot_bot_set_b ) ).
% empty_iff
thf(fact_441_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_442_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_443_Int__iff,axiom,
! [C: dtree_list_a_b,A2: set_dtree_list_a_b,B2: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ ( inf_in3355993651213403836st_a_b @ A2 @ B2 ) )
= ( ( member551035911493665803st_a_b @ C @ A2 )
& ( member551035911493665803st_a_b @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_444_Int__iff,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B2 ) )
= ( ( member_list_a @ C @ A2 )
& ( member_list_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_445_Int__iff,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) )
= ( ( member_set_a @ C @ A2 )
& ( member_set_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_446_Int__iff,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A2 @ B2 ) )
= ( ( member_b @ C @ A2 )
& ( member_b @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_447_Int__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ( member_a @ C @ B2 ) ) ) ).
% Int_iff
thf(fact_448_IntI,axiom,
! [C: dtree_list_a_b,A2: set_dtree_list_a_b,B2: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ A2 )
=> ( ( member551035911493665803st_a_b @ C @ B2 )
=> ( member551035911493665803st_a_b @ C @ ( inf_in3355993651213403836st_a_b @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_449_IntI,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A2 )
=> ( ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_450_IntI,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_451_IntI,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ A2 )
=> ( ( member_b @ C @ B2 )
=> ( member_b @ C @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_452_IntI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% IntI
thf(fact_453_Un__iff,axiom,
! [C: dtree_list_a_b,A2: set_dtree_list_a_b,B2: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ ( sup_su5199045856643764822st_a_b @ A2 @ B2 ) )
= ( ( member551035911493665803st_a_b @ C @ A2 )
| ( member551035911493665803st_a_b @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_454_Un__iff,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( sup_sup_set_list_a @ A2 @ B2 ) )
= ( ( member_list_a @ C @ A2 )
| ( member_list_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_455_Un__iff,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) )
= ( ( member_set_a @ C @ A2 )
| ( member_set_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_456_Un__iff,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( sup_sup_set_b @ A2 @ B2 ) )
= ( ( member_b @ C @ A2 )
| ( member_b @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_457_Un__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
| ( member_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_458_UnCI,axiom,
! [C: dtree_list_a_b,B2: set_dtree_list_a_b,A2: set_dtree_list_a_b] :
( ( ~ ( member551035911493665803st_a_b @ C @ B2 )
=> ( member551035911493665803st_a_b @ C @ A2 ) )
=> ( member551035911493665803st_a_b @ C @ ( sup_su5199045856643764822st_a_b @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_459_UnCI,axiom,
! [C: list_a,B2: set_list_a,A2: set_list_a] :
( ( ~ ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ A2 ) )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_460_UnCI,axiom,
! [C: set_a,B2: set_set_a,A2: set_set_a] :
( ( ~ ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ A2 ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_461_UnCI,axiom,
! [C: b,B2: set_b,A2: set_b] :
( ( ~ ( member_b @ C @ B2 )
=> ( member_b @ C @ A2 ) )
=> ( member_b @ C @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_462_UnCI,axiom,
! [C: a,B2: set_a,A2: set_a] :
( ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ A2 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_463_rev__rev__ident,axiom,
! [Xs2: list_a] :
( ( rev_a @ ( rev_a @ Xs2 ) )
= Xs2 ) ).
% rev_rev_ident
thf(fact_464_rev__is__rev__conv,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( rev_a @ Xs2 )
= ( rev_a @ Ys ) )
= ( Xs2 = Ys ) ) ).
% rev_is_rev_conv
thf(fact_465_forward__cons,axiom,
! [X: a,Xs2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ ( cons_a @ X @ Xs2 ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ Xs2 ) ) ) ).
% forward_cons
thf(fact_466_forward__arcs__alt,axiom,
! [Xs2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ Xs2 )
= ( iKKBZ_4180558001818622352cs_a_b @ t @ ( rev_a @ Xs2 ) ) ) ).
% forward_arcs_alt
thf(fact_467_forward__arcs__alt_H,axiom,
! [Xs2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ Xs2 ) )
= ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs2 ) ) ).
% forward_arcs_alt'
thf(fact_468_forward__arcs__alt__aux2,axiom,
! [Xs2: list_a] :
( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( rev_a @ Xs2 ) )
=> ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs2 ) ) ).
% forward_arcs_alt_aux2
thf(fact_469_forward__arcs_Osimps_I2_J,axiom,
! [X: a] : ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ X @ nil_a ) ) ).
% forward_arcs.simps(2)
thf(fact_470_forward__arcs__single,axiom,
! [X: a] : ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ X @ nil_a ) ) ).
% forward_arcs_single
thf(fact_471_seq__conform__def,axiom,
! [Xs2: list_a] :
( ( iKKBZ_4622586873178280511rm_a_b @ t @ Xs2 )
= ( ( iKKBZ_4180558001818622352cs_a_b @ t @ ( rev_a @ Xs2 ) )
& ( iKKBZ_7773321254043928001cs_a_b @ t @ Xs2 ) ) ) ).
% seq_conform_def
thf(fact_472_forward__arcs_Oelims_I3_J,axiom,
! [X: list_a] :
( ~ ( iKKBZ_4180558001818622352cs_a_b @ t @ X )
=> ~ ! [X2: a,V3: a,Va: list_a] :
( ( X
= ( cons_a @ X2 @ ( cons_a @ V3 @ Va ) ) )
=> ( ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ ( cons_a @ V3 @ Va ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X2 ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ V3 @ Va ) ) ) ) ) ).
% forward_arcs.elims(3)
thf(fact_473_forward__arcs_Osimps_I3_J,axiom,
! [X: a,V: a,Va2: list_a] :
( ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ X @ ( cons_a @ V @ Va2 ) ) )
= ( ? [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ ( cons_a @ V @ Va2 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ X ) @ ( arcs_ends_a_b @ t ) ) )
& ( iKKBZ_4180558001818622352cs_a_b @ t @ ( cons_a @ V @ Va2 ) ) ) ) ).
% forward_arcs.simps(3)
thf(fact_474_hd__reach__all__forward__arcs,axiom,
! [Xs2: list_a,X: a] :
( ( member_a @ ( hd_a @ ( rev_a @ Xs2 ) ) @ ( pre_ve642382030648772252t_unit @ t ) )
=> ( ( iKKBZ_4180558001818622352cs_a_b @ t @ Xs2 )
=> ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ( reachable_a_b @ t @ ( hd_a @ ( rev_a @ Xs2 ) ) @ X ) ) ) ) ).
% hd_reach_all_forward_arcs
thf(fact_475_Un__empty,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_476_Un__empty,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ( sup_sup_set_list_a @ A2 @ B2 )
= bot_bot_set_list_a )
= ( ( A2 = bot_bot_set_list_a )
& ( B2 = bot_bot_set_list_a ) ) ) ).
% Un_empty
thf(fact_477_set__rev,axiom,
! [Xs2: list_a] :
( ( set_a2 @ ( rev_a @ Xs2 ) )
= ( set_a2 @ Xs2 ) ) ).
% set_rev
thf(fact_478_set__rev,axiom,
! [Xs2: list_list_a] :
( ( set_list_a2 @ ( rev_list_a @ Xs2 ) )
= ( set_list_a2 @ Xs2 ) ) ).
% set_rev
thf(fact_479_rev__append,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( rev_a @ ( append_a @ Xs2 @ Ys ) )
= ( append_a @ ( rev_a @ Ys ) @ ( rev_a @ Xs2 ) ) ) ).
% rev_append
thf(fact_480_rev__append,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( rev_b @ ( append_b @ Xs2 @ Ys ) )
= ( append_b @ ( rev_b @ Ys ) @ ( rev_b @ Xs2 ) ) ) ).
% rev_append
thf(fact_481_rev__is__Nil__conv,axiom,
! [Xs2: list_a] :
( ( ( rev_a @ Xs2 )
= nil_a )
= ( Xs2 = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_482_rev__is__Nil__conv,axiom,
! [Xs2: list_b] :
( ( ( rev_b @ Xs2 )
= nil_b )
= ( Xs2 = nil_b ) ) ).
% rev_is_Nil_conv
thf(fact_483_Nil__is__rev__conv,axiom,
! [Xs2: list_a] :
( ( nil_a
= ( rev_a @ Xs2 ) )
= ( Xs2 = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_484_Nil__is__rev__conv,axiom,
! [Xs2: list_b] :
( ( nil_b
= ( rev_b @ Xs2 ) )
= ( Xs2 = nil_b ) ) ).
% Nil_is_rev_conv
thf(fact_485_Un__Int__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_486_Un__Int__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Un_Int_eq(2)
thf(fact_487_Un__Int__eq_I3_J,axiom,
! [S: set_a,T2: set_a] :
( ( inf_inf_set_a @ S @ ( sup_sup_set_a @ S @ T2 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_488_Un__Int__eq_I4_J,axiom,
! [T2: set_a,S: set_a] :
( ( inf_inf_set_a @ T2 @ ( sup_sup_set_a @ S @ T2 ) )
= T2 ) ).
% Un_Int_eq(4)
thf(fact_489_Int__Un__eq_I1_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_490_Int__Un__eq_I2_J,axiom,
! [S: set_a,T2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ S @ T2 ) @ T2 )
= T2 ) ).
% Int_Un_eq(2)
thf(fact_491_distinct__rev,axiom,
! [Xs2: list_a] :
( ( distinct_a @ ( rev_a @ Xs2 ) )
= ( distinct_a @ Xs2 ) ) ).
% distinct_rev
thf(fact_492_rev__singleton__conv,axiom,
! [Xs2: list_a,X: a] :
( ( ( rev_a @ Xs2 )
= ( cons_a @ X @ nil_a ) )
= ( Xs2
= ( cons_a @ X @ nil_a ) ) ) ).
% rev_singleton_conv
thf(fact_493_rev__singleton__conv,axiom,
! [Xs2: list_b,X: b] :
( ( ( rev_b @ Xs2 )
= ( cons_b @ X @ nil_b ) )
= ( Xs2
= ( cons_b @ X @ nil_b ) ) ) ).
% rev_singleton_conv
thf(fact_494_singleton__rev__conv,axiom,
! [X: a,Xs2: list_a] :
( ( ( cons_a @ X @ nil_a )
= ( rev_a @ Xs2 ) )
= ( ( cons_a @ X @ nil_a )
= Xs2 ) ) ).
% singleton_rev_conv
thf(fact_495_singleton__rev__conv,axiom,
! [X: b,Xs2: list_b] :
( ( ( cons_b @ X @ nil_b )
= ( rev_b @ Xs2 ) )
= ( ( cons_b @ X @ nil_b )
= Xs2 ) ) ).
% singleton_rev_conv
thf(fact_496_rev__eq__Cons__iff,axiom,
! [Xs2: list_a,Y: a,Ys: list_a] :
( ( ( rev_a @ Xs2 )
= ( cons_a @ Y @ Ys ) )
= ( Xs2
= ( append_a @ ( rev_a @ Ys ) @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_497_rev__eq__Cons__iff,axiom,
! [Xs2: list_b,Y: b,Ys: list_b] :
( ( ( rev_b @ Xs2 )
= ( cons_b @ Y @ Ys ) )
= ( Xs2
= ( append_b @ ( rev_b @ Ys ) @ ( cons_b @ Y @ nil_b ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_498_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_499_wf__list__lverts_Ocases,axiom,
! [X: list_P4861094521484316518st_b_a] :
( ( X != nil_Pr9111545067797741414st_b_a )
=> ~ ! [V3: list_b,E2: a,Xs: list_P4861094521484316518st_b_a] :
( X
!= ( cons_P1383153349113754390st_b_a @ ( produc4145578316043568848st_b_a @ V3 @ E2 ) @ Xs ) ) ) ).
% wf_list_lverts.cases
thf(fact_500_wf__list__lverts_Ocases,axiom,
! [X: list_P321204300973800749list_a] :
( ( X != nil_Pr3188421586756112173list_a )
=> ~ ! [V3: list_a,E2: list_a,Xs: list_P321204300973800749list_a] :
( X
!= ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ V3 @ E2 ) @ Xs ) ) ) ).
% wf_list_lverts.cases
thf(fact_501_separate__P_Ocases,axiom,
! [X: produc3286415118216283229list_a] :
~ ! [P3: a > $o,Acc: list_a,Xs: list_a] :
( X
!= ( produc8731264218526379663list_a @ P3 @ ( produc6837034575241423639list_a @ Acc @ Xs ) ) ) ).
% separate_P.cases
thf(fact_502_directed__tree_Oforward__arcs_Ocong,axiom,
iKKBZ_4180558001818622352cs_a_b = iKKBZ_4180558001818622352cs_a_b ).
% directed_tree.forward_arcs.cong
thf(fact_503_rev__swap,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ( rev_a @ Xs2 )
= Ys )
= ( Xs2
= ( rev_a @ Ys ) ) ) ).
% rev_swap
thf(fact_504_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_505_rev_Osimps_I1_J,axiom,
( ( rev_b @ nil_b )
= nil_b ) ).
% rev.simps(1)
thf(fact_506_rev_Osimps_I2_J,axiom,
! [X: a,Xs2: list_a] :
( ( rev_a @ ( cons_a @ X @ Xs2 ) )
= ( append_a @ ( rev_a @ Xs2 ) @ ( cons_a @ X @ nil_a ) ) ) ).
% rev.simps(2)
thf(fact_507_rev_Osimps_I2_J,axiom,
! [X: b,Xs2: list_b] :
( ( rev_b @ ( cons_b @ X @ Xs2 ) )
= ( append_b @ ( rev_b @ Xs2 ) @ ( cons_b @ X @ nil_b ) ) ) ).
% rev.simps(2)
thf(fact_508_ex__in__conv,axiom,
! [A2: set_dtree_list_a_b] :
( ( ? [X5: dtree_list_a_b] : ( member551035911493665803st_a_b @ X5 @ A2 ) )
= ( A2 != bot_bo798015271861357502st_a_b ) ) ).
% ex_in_conv
thf(fact_509_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_510_ex__in__conv,axiom,
! [A2: set_b] :
( ( ? [X5: b] : ( member_b @ X5 @ A2 ) )
= ( A2 != bot_bot_set_b ) ) ).
% ex_in_conv
thf(fact_511_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X5: a] : ( member_a @ X5 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_512_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_513_equals0I,axiom,
! [A2: set_dtree_list_a_b] :
( ! [Y4: dtree_list_a_b] :
~ ( member551035911493665803st_a_b @ Y4 @ A2 )
=> ( A2 = bot_bo798015271861357502st_a_b ) ) ).
% equals0I
thf(fact_514_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y4: set_a] :
~ ( member_set_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_515_equals0I,axiom,
! [A2: set_b] :
( ! [Y4: b] :
~ ( member_b @ Y4 @ A2 )
=> ( A2 = bot_bot_set_b ) ) ).
% equals0I
thf(fact_516_equals0I,axiom,
! [A2: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_517_equals0I,axiom,
! [A2: set_list_a] :
( ! [Y4: list_a] :
~ ( member_list_a @ Y4 @ A2 )
=> ( A2 = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_518_equals0D,axiom,
! [A2: set_dtree_list_a_b,A: dtree_list_a_b] :
( ( A2 = bot_bo798015271861357502st_a_b )
=> ~ ( member551035911493665803st_a_b @ A @ A2 ) ) ).
% equals0D
thf(fact_519_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_520_equals0D,axiom,
! [A2: set_b,A: b] :
( ( A2 = bot_bot_set_b )
=> ~ ( member_b @ A @ A2 ) ) ).
% equals0D
thf(fact_521_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_522_equals0D,axiom,
! [A2: set_list_a,A: list_a] :
( ( A2 = bot_bot_set_list_a )
=> ~ ( member_list_a @ A @ A2 ) ) ).
% equals0D
thf(fact_523_emptyE,axiom,
! [A: dtree_list_a_b] :
~ ( member551035911493665803st_a_b @ A @ bot_bo798015271861357502st_a_b ) ).
% emptyE
thf(fact_524_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_525_emptyE,axiom,
! [A: b] :
~ ( member_b @ A @ bot_bot_set_b ) ).
% emptyE
thf(fact_526_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_527_emptyE,axiom,
! [A: list_a] :
~ ( member_list_a @ A @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_528_Int__left__commute,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( inf_inf_set_a @ B2 @ ( inf_inf_set_a @ A2 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_529_Int__left__absorb,axiom,
! [A2: set_a,B2: set_a] :
( ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( inf_inf_set_a @ A2 @ B2 ) ) ).
% Int_left_absorb
thf(fact_530_Int__commute,axiom,
( inf_inf_set_a
= ( ^ [A4: set_a,B3: set_a] : ( inf_inf_set_a @ B3 @ A4 ) ) ) ).
% Int_commute
thf(fact_531_Int__absorb,axiom,
! [A2: set_a] :
( ( inf_inf_set_a @ A2 @ A2 )
= A2 ) ).
% Int_absorb
thf(fact_532_Int__assoc,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) ) ) ).
% Int_assoc
thf(fact_533_IntD2,axiom,
! [C: dtree_list_a_b,A2: set_dtree_list_a_b,B2: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ ( inf_in3355993651213403836st_a_b @ A2 @ B2 ) )
=> ( member551035911493665803st_a_b @ C @ B2 ) ) ).
% IntD2
thf(fact_534_IntD2,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B2 ) )
=> ( member_list_a @ C @ B2 ) ) ).
% IntD2
thf(fact_535_IntD2,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C @ B2 ) ) ).
% IntD2
thf(fact_536_IntD2,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A2 @ B2 ) )
=> ( member_b @ C @ B2 ) ) ).
% IntD2
thf(fact_537_IntD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ B2 ) ) ).
% IntD2
thf(fact_538_IntD1,axiom,
! [C: dtree_list_a_b,A2: set_dtree_list_a_b,B2: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ ( inf_in3355993651213403836st_a_b @ A2 @ B2 ) )
=> ( member551035911493665803st_a_b @ C @ A2 ) ) ).
% IntD1
thf(fact_539_IntD1,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B2 ) )
=> ( member_list_a @ C @ A2 ) ) ).
% IntD1
thf(fact_540_IntD1,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C @ A2 ) ) ).
% IntD1
thf(fact_541_IntD1,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A2 @ B2 ) )
=> ( member_b @ C @ A2 ) ) ).
% IntD1
thf(fact_542_IntD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% IntD1
thf(fact_543_IntE,axiom,
! [C: dtree_list_a_b,A2: set_dtree_list_a_b,B2: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ ( inf_in3355993651213403836st_a_b @ A2 @ B2 ) )
=> ~ ( ( member551035911493665803st_a_b @ C @ A2 )
=> ~ ( member551035911493665803st_a_b @ C @ B2 ) ) ) ).
% IntE
thf(fact_544_IntE,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( inf_inf_set_list_a @ A2 @ B2 ) )
=> ~ ( ( member_list_a @ C @ A2 )
=> ~ ( member_list_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_545_IntE,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( inf_inf_set_set_a @ A2 @ B2 ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ~ ( member_set_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_546_IntE,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( inf_inf_set_b @ A2 @ B2 ) )
=> ~ ( ( member_b @ C @ A2 )
=> ~ ( member_b @ C @ B2 ) ) ) ).
% IntE
thf(fact_547_IntE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( inf_inf_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ~ ( member_a @ C @ B2 ) ) ) ).
% IntE
thf(fact_548_Un__left__commute,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) )
= ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_549_Un__left__absorb,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_550_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A4 ) ) ) ).
% Un_commute
thf(fact_551_Un__absorb,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_552_Un__assoc,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_553_ball__Un,axiom,
! [A2: set_a,B2: set_a,P2: a > $o] :
( ( ! [X5: a] :
( ( member_a @ X5 @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ( P2 @ X5 ) ) )
= ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ( P2 @ X5 ) )
& ! [X5: a] :
( ( member_a @ X5 @ B2 )
=> ( P2 @ X5 ) ) ) ) ).
% ball_Un
thf(fact_554_bex__Un,axiom,
! [A2: set_a,B2: set_a,P2: a > $o] :
( ( ? [X5: a] :
( ( member_a @ X5 @ ( sup_sup_set_a @ A2 @ B2 ) )
& ( P2 @ X5 ) ) )
= ( ? [X5: a] :
( ( member_a @ X5 @ A2 )
& ( P2 @ X5 ) )
| ? [X5: a] :
( ( member_a @ X5 @ B2 )
& ( P2 @ X5 ) ) ) ) ).
% bex_Un
thf(fact_555_UnI2,axiom,
! [C: dtree_list_a_b,B2: set_dtree_list_a_b,A2: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ B2 )
=> ( member551035911493665803st_a_b @ C @ ( sup_su5199045856643764822st_a_b @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_556_UnI2,axiom,
! [C: list_a,B2: set_list_a,A2: set_list_a] :
( ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_557_UnI2,axiom,
! [C: set_a,B2: set_set_a,A2: set_set_a] :
( ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_558_UnI2,axiom,
! [C: b,B2: set_b,A2: set_b] :
( ( member_b @ C @ B2 )
=> ( member_b @ C @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_559_UnI2,axiom,
! [C: a,B2: set_a,A2: set_a] :
( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_560_UnI1,axiom,
! [C: dtree_list_a_b,A2: set_dtree_list_a_b,B2: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ A2 )
=> ( member551035911493665803st_a_b @ C @ ( sup_su5199045856643764822st_a_b @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_561_UnI1,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ ( sup_sup_set_list_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_562_UnI1,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_563_UnI1,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ A2 )
=> ( member_b @ C @ ( sup_sup_set_b @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_564_UnI1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_565_UnE,axiom,
! [C: dtree_list_a_b,A2: set_dtree_list_a_b,B2: set_dtree_list_a_b] :
( ( member551035911493665803st_a_b @ C @ ( sup_su5199045856643764822st_a_b @ A2 @ B2 ) )
=> ( ~ ( member551035911493665803st_a_b @ C @ A2 )
=> ( member551035911493665803st_a_b @ C @ B2 ) ) ) ).
% UnE
thf(fact_566_UnE,axiom,
! [C: list_a,A2: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( sup_sup_set_list_a @ A2 @ B2 ) )
=> ( ~ ( member_list_a @ C @ A2 )
=> ( member_list_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_567_UnE,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) )
=> ( ~ ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_568_UnE,axiom,
! [C: b,A2: set_b,B2: set_b] :
( ( member_b @ C @ ( sup_sup_set_b @ A2 @ B2 ) )
=> ( ~ ( member_b @ C @ A2 )
=> ( member_b @ C @ B2 ) ) ) ).
% UnE
thf(fact_569_UnE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ~ ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_570_disjoint__iff__not__equal,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ! [Y3: a] :
( ( member_a @ Y3 @ B2 )
=> ( X5 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_571_disjoint__iff__not__equal,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B2 )
= bot_bot_set_list_a )
= ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A2 )
=> ! [Y3: list_a] :
( ( member_list_a @ Y3 @ B2 )
=> ( X5 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_572_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_573_Int__empty__right,axiom,
! [A2: set_list_a] :
( ( inf_inf_set_list_a @ A2 @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% Int_empty_right
thf(fact_574_Int__empty__left,axiom,
! [B2: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ B2 )
= bot_bot_set_a ) ).
% Int_empty_left
thf(fact_575_Int__empty__left,axiom,
! [B2: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ B2 )
= bot_bot_set_list_a ) ).
% Int_empty_left
thf(fact_576_disjoint__iff,axiom,
! [A2: set_dtree_list_a_b,B2: set_dtree_list_a_b] :
( ( ( inf_in3355993651213403836st_a_b @ A2 @ B2 )
= bot_bo798015271861357502st_a_b )
= ( ! [X5: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X5 @ A2 )
=> ~ ( member551035911493665803st_a_b @ X5 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_577_disjoint__iff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ A2 )
=> ~ ( member_set_a @ X5 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_578_disjoint__iff,axiom,
! [A2: set_b,B2: set_b] :
( ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b )
= ( ! [X5: b] :
( ( member_b @ X5 @ A2 )
=> ~ ( member_b @ X5 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_579_disjoint__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ! [X5: a] :
( ( member_a @ X5 @ A2 )
=> ~ ( member_a @ X5 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_580_disjoint__iff,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ( ( inf_inf_set_list_a @ A2 @ B2 )
= bot_bot_set_list_a )
= ( ! [X5: list_a] :
( ( member_list_a @ X5 @ A2 )
=> ~ ( member_list_a @ X5 @ B2 ) ) ) ) ).
% disjoint_iff
thf(fact_581_Int__emptyI,axiom,
! [A2: set_dtree_list_a_b,B2: set_dtree_list_a_b] :
( ! [X2: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X2 @ A2 )
=> ~ ( member551035911493665803st_a_b @ X2 @ B2 ) )
=> ( ( inf_in3355993651213403836st_a_b @ A2 @ B2 )
= bot_bo798015271861357502st_a_b ) ) ).
% Int_emptyI
thf(fact_582_Int__emptyI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ~ ( member_set_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_set_a @ A2 @ B2 )
= bot_bot_set_set_a ) ) ).
% Int_emptyI
thf(fact_583_Int__emptyI,axiom,
! [A2: set_b,B2: set_b] :
( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ~ ( member_b @ X2 @ B2 ) )
=> ( ( inf_inf_set_b @ A2 @ B2 )
= bot_bot_set_b ) ) ).
% Int_emptyI
thf(fact_584_Int__emptyI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ~ ( member_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= bot_bot_set_a ) ) ).
% Int_emptyI
thf(fact_585_Int__emptyI,axiom,
! [A2: set_list_a,B2: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ~ ( member_list_a @ X2 @ B2 ) )
=> ( ( inf_inf_set_list_a @ A2 @ B2 )
= bot_bot_set_list_a ) ) ).
% Int_emptyI
thf(fact_586_Un__empty__right,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_587_Un__empty__right,axiom,
! [A2: set_list_a] :
( ( sup_sup_set_list_a @ A2 @ bot_bot_set_list_a )
= A2 ) ).
% Un_empty_right
thf(fact_588_Un__empty__left,axiom,
! [B2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_589_Un__empty__left,axiom,
! [B2: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_590_Un__Int__distrib2,axiom,
! [B2: set_a,C2: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ B2 @ C2 ) @ A2 )
= ( inf_inf_set_a @ ( sup_sup_set_a @ B2 @ A2 ) @ ( sup_sup_set_a @ C2 @ A2 ) ) ) ).
% Un_Int_distrib2
thf(fact_591_Int__Un__distrib2,axiom,
! [B2: set_a,C2: set_a,A2: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A2 )
= ( sup_sup_set_a @ ( inf_inf_set_a @ B2 @ A2 ) @ ( inf_inf_set_a @ C2 @ A2 ) ) ) ).
% Int_Un_distrib2
thf(fact_592_Un__Int__distrib,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ A2 @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ A2 @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_593_Int__Un__distrib,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ A2 @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_594_Un__Int__crazy,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ B2 @ C2 ) ) @ ( inf_inf_set_a @ C2 @ A2 ) )
= ( inf_inf_set_a @ ( inf_inf_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ B2 @ C2 ) ) @ ( sup_sup_set_a @ C2 @ A2 ) ) ) ).
% Un_Int_crazy
thf(fact_595_mid__ranks__ge__if__reach1,axiom,
! [Y6: set_list_a,U2: list_a,As: list_a,Bs: list_a,V2: list_a,Cs: list_a,Bs3: 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 @ V2 @ Cs ) ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) )
=> ( ( ( concat_a @ Bs3 )
= Bs )
=> ( ( ( concat_a @ Cs2 )
= Cs )
=> ( ( ( set_list_a2 @ ( append_list_a @ As2 @ ( cons_list_a @ U2 @ ( append_list_a @ Bs3 @ ( cons_list_a @ V2 @ Cs2 ) ) ) ) )
= Y6 )
=> ( ! [Xs: list_a] :
( ( member_list_a @ Xs @ Y6 )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ~ ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ U2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ( Xs != U2 )
=> ( ord_less_eq_set_a @ ( Rank @ ( rev_a @ V2 ) ) @ ( Rank @ ( rev_a @ Xs ) ) ) ) ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Bs3 ) )
=> ( ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ U2 ) )
& ? [Xb: a] :
( ( member_a @ Xb @ ( set_a2 @ X4 ) )
& ( 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 @ V2 ) ) @ ( Rank @ ( rev_a @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% mid_ranks_ge_if_reach1
thf(fact_596_mid__ranks__ge__if__reach1,axiom,
! [Y6: set_list_a,U2: list_a,As: list_a,Bs: list_a,V2: list_a,Cs: list_a,Bs3: list_list_a,Cs2: list_list_a,As2: list_list_a,Rank: list_a > real] :
( ~ ( member_list_a @ nil_a @ Y6 )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( distinct_a @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) )
=> ( ( ( concat_a @ Bs3 )
= Bs )
=> ( ( ( concat_a @ Cs2 )
= Cs )
=> ( ( ( set_list_a2 @ ( append_list_a @ As2 @ ( cons_list_a @ U2 @ ( append_list_a @ Bs3 @ ( cons_list_a @ V2 @ Cs2 ) ) ) ) )
= Y6 )
=> ( ! [Xs: list_a] :
( ( member_list_a @ Xs @ Y6 )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ~ ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ U2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ( Xs != U2 )
=> ( ord_less_eq_real @ ( Rank @ ( rev_a @ V2 ) ) @ ( Rank @ ( rev_a @ Xs ) ) ) ) ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Bs3 ) )
=> ( ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ U2 ) )
& ? [Xb: a] :
( ( member_a @ Xb @ ( set_a2 @ X4 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ Xb ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ord_less_eq_real @ ( Rank @ ( rev_a @ V2 ) ) @ ( Rank @ ( rev_a @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% mid_ranks_ge_if_reach1
thf(fact_597_mid__ranks__ge__if__reach1,axiom,
! [Y6: set_list_a,U2: list_a,As: list_a,Bs: list_a,V2: list_a,Cs: list_a,Bs3: 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 @ V2 @ Cs ) ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) )
=> ( ( ( concat_a @ Bs3 )
= Bs )
=> ( ( ( concat_a @ Cs2 )
= Cs )
=> ( ( ( set_list_a2 @ ( append_list_a @ As2 @ ( cons_list_a @ U2 @ ( append_list_a @ Bs3 @ ( cons_list_a @ V2 @ Cs2 ) ) ) ) )
= Y6 )
=> ( ! [Xs: list_a] :
( ( member_list_a @ Xs @ Y6 )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ~ ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ U2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ( Xs != U2 )
=> ( ord_less_eq_set_b @ ( Rank @ ( rev_a @ V2 ) ) @ ( Rank @ ( rev_a @ Xs ) ) ) ) ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Bs3 ) )
=> ( ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ U2 ) )
& ? [Xb: a] :
( ( member_a @ Xb @ ( set_a2 @ X4 ) )
& ( 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 @ V2 ) ) @ ( Rank @ ( rev_a @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% mid_ranks_ge_if_reach1
thf(fact_598_bs__ranks__only__ge__r,axiom,
! [Y6: set_list_a,As: list_a,U2: list_a,Bs: list_a,V2: list_a,Cs: list_a,Bs3: 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 @ V2 @ Cs ) ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) )
=> ( ( As = nil_a )
=> ( ( ( concat_a @ Bs3 )
= Bs )
=> ( ( ( concat_a @ Cs2 )
= Cs )
=> ( ( ( set_list_a2 @ ( cons_list_a @ U2 @ ( append_list_a @ Bs3 @ ( cons_list_a @ V2 @ Cs2 ) ) ) )
= Y6 )
=> ( ! [Xs: list_a] :
( ( member_list_a @ Xs @ Y6 )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ~ ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ U2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ( Xs != U2 )
=> ( ord_less_eq_set_a @ ( Rank @ ( rev_a @ V2 ) ) @ ( Rank @ ( rev_a @ Xs ) ) ) ) ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Bs3 ) )
=> ( ord_less_eq_set_a @ ( Rank @ ( rev_a @ V2 ) ) @ ( Rank @ ( rev_a @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% bs_ranks_only_ge_r
thf(fact_599_bs__ranks__only__ge__r,axiom,
! [Y6: set_list_a,As: list_a,U2: list_a,Bs: list_a,V2: list_a,Cs: list_a,Bs3: list_list_a,Cs2: list_list_a,Rank: list_a > real] :
( ~ ( member_list_a @ nil_a @ Y6 )
=> ( ( distinct_a @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) )
=> ( ( As = nil_a )
=> ( ( ( concat_a @ Bs3 )
= Bs )
=> ( ( ( concat_a @ Cs2 )
= Cs )
=> ( ( ( set_list_a2 @ ( cons_list_a @ U2 @ ( append_list_a @ Bs3 @ ( cons_list_a @ V2 @ Cs2 ) ) ) )
= Y6 )
=> ( ! [Xs: list_a] :
( ( member_list_a @ Xs @ Y6 )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ~ ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ U2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ( Xs != U2 )
=> ( ord_less_eq_real @ ( Rank @ ( rev_a @ V2 ) ) @ ( Rank @ ( rev_a @ Xs ) ) ) ) ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Bs3 ) )
=> ( ord_less_eq_real @ ( Rank @ ( rev_a @ V2 ) ) @ ( Rank @ ( rev_a @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% bs_ranks_only_ge_r
thf(fact_600_bs__ranks__only__ge__r,axiom,
! [Y6: set_list_a,As: list_a,U2: list_a,Bs: list_a,V2: list_a,Cs: list_a,Bs3: 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 @ V2 @ Cs ) ) ) ) )
=> ( ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) )
=> ( ( As = nil_a )
=> ( ( ( concat_a @ Bs3 )
= Bs )
=> ( ( ( concat_a @ Cs2 )
= Cs )
=> ( ( ( set_list_a2 @ ( cons_list_a @ U2 @ ( append_list_a @ Bs3 @ ( cons_list_a @ V2 @ Cs2 ) ) ) )
= Y6 )
=> ( ! [Xs: list_a] :
( ( member_list_a @ Xs @ Y6 )
=> ( ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
& ~ ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ U2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ( Xs != U2 )
=> ( ord_less_eq_set_b @ ( Rank @ ( rev_a @ V2 ) ) @ ( Rank @ ( rev_a @ Xs ) ) ) ) ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Bs3 ) )
=> ( ord_less_eq_set_b @ ( Rank @ ( rev_a @ V2 ) ) @ ( Rank @ ( rev_a @ X4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% bs_ranks_only_ge_r
thf(fact_601_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_602_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_603_vpathI,axiom,
! [P: list_a,G: pre_pr7278220950009878019t_unit] :
( ( vertex_vwalk_a_b @ P @ G )
=> ( ( distinct_a @ P )
=> ( vertex_vpath_a_b @ P @ G ) ) ) ).
% vpathI
thf(fact_604_vwalk__consI,axiom,
! [P: list_a,G: pre_pr7278220950009878019t_unit,A: a] :
( ( vertex_vwalk_a_b @ P @ G )
=> ( ( member_a @ A @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( hd_a @ P ) ) @ ( arcs_ends_a_b @ G ) )
=> ( vertex_vwalk_a_b @ ( cons_a @ A @ P ) @ G ) ) ) ) ).
% vwalk_consI
thf(fact_605_vwalk__consI,axiom,
! [P: list_list_a,G: pre_pr2882871181989701257t_unit,A: list_a] :
( ( vertex2966258834163962945st_a_b @ P @ G )
=> ( ( member_list_a @ A @ ( pre_ve1830060048215441954t_unit @ G ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A @ ( hd_list_a @ P ) ) @ ( arcs_ends_list_a_b @ G ) )
=> ( vertex2966258834163962945st_a_b @ ( cons_list_a @ A @ P ) @ G ) ) ) ) ).
% vwalk_consI
thf(fact_606_vwalk__consE,axiom,
! [A: list_a,P: list_list_a,G: pre_pr2882871181989701257t_unit] :
( ( vertex2966258834163962945st_a_b @ ( cons_list_a @ A @ P ) @ G )
=> ( ( P != nil_list_a )
=> ~ ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A @ ( hd_list_a @ P ) ) @ ( arcs_ends_list_a_b @ G ) )
=> ~ ( vertex2966258834163962945st_a_b @ P @ G ) ) ) ) ).
% vwalk_consE
thf(fact_607_vwalk__consE,axiom,
! [A: a,P: list_a,G: pre_pr7278220950009878019t_unit] :
( ( vertex_vwalk_a_b @ ( cons_a @ A @ P ) @ G )
=> ( ( P != nil_a )
=> ~ ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ ( hd_a @ P ) ) @ ( arcs_ends_a_b @ G ) )
=> ~ ( vertex_vwalk_a_b @ P @ G ) ) ) ) ).
% vwalk_consE
thf(fact_608_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_609_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_610_empty__subsetI,axiom,
! [A2: set_list_a] : ( ord_le8861187494160871172list_a @ bot_bot_set_list_a @ A2 ) ).
% empty_subsetI
thf(fact_611_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_612_empty__subsetI,axiom,
! [A2: set_b] : ( ord_less_eq_set_b @ bot_bot_set_b @ A2 ) ).
% empty_subsetI
thf(fact_613_subset__empty,axiom,
! [A2: set_list_a] :
( ( ord_le8861187494160871172list_a @ A2 @ bot_bot_set_list_a )
= ( A2 = bot_bot_set_list_a ) ) ).
% subset_empty
thf(fact_614_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_615_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_616_Int__subset__iff,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) )
= ( ( ord_less_eq_set_a @ C2 @ A2 )
& ( ord_less_eq_set_a @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_617_Int__subset__iff,axiom,
! [C2: set_b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A2 @ B2 ) )
= ( ( ord_less_eq_set_b @ C2 @ A2 )
& ( ord_less_eq_set_b @ C2 @ B2 ) ) ) ).
% Int_subset_iff
thf(fact_618_Un__subset__iff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_a @ A2 @ C2 )
& ( ord_less_eq_set_a @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_619_Un__subset__iff,axiom,
! [A2: set_b,B2: set_b,C2: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_b @ A2 @ C2 )
& ( ord_less_eq_set_b @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_620_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_621_to__list__tree__single,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( pre_ve1830060048215441954t_unit @ ( direct3773525127397338803ee_a_b @ t ) ) )
=> ? [X2: a] :
( ( V
= ( cons_a @ X2 @ nil_a ) )
& ( member_a @ X2 @ ( pre_ve642382030648772252t_unit @ t ) ) ) ) ).
% to_list_tree_single
thf(fact_622_subset__code_I1_J,axiom,
! [Xs2: list_dtree_list_a_b,B2: set_dtree_list_a_b] :
( ( ord_le7599451563663638410st_a_b @ ( set_dtree_list_a_b2 @ Xs2 ) @ B2 )
= ( ! [X5: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X5 @ ( set_dtree_list_a_b2 @ Xs2 ) )
=> ( member551035911493665803st_a_b @ X5 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_623_subset__code_I1_J,axiom,
! [Xs2: list_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs2 ) @ B2 )
= ( ! [X5: set_a] :
( ( member_set_a @ X5 @ ( set_set_a2 @ Xs2 ) )
=> ( member_set_a @ X5 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_624_subset__code_I1_J,axiom,
! [Xs2: list_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs2 ) @ B2 )
= ( ! [X5: list_a] :
( ( member_list_a @ X5 @ ( set_list_a2 @ Xs2 ) )
=> ( member_list_a @ X5 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_625_subset__code_I1_J,axiom,
! [Xs2: list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ B2 )
= ( ! [X5: a] :
( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
=> ( member_a @ X5 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_626_subset__code_I1_J,axiom,
! [Xs2: list_b,B2: set_b] :
( ( ord_less_eq_set_b @ ( set_b2 @ Xs2 ) @ B2 )
= ( ! [X5: b] :
( ( member_b @ X5 @ ( set_b2 @ Xs2 ) )
=> ( member_b @ X5 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_627_Int__mono,axiom,
! [A2: set_a,C2: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ ( inf_inf_set_a @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_628_Int__mono,axiom,
! [A2: set_b,C2: set_b,B2: set_b,D: set_b] :
( ( ord_less_eq_set_b @ A2 @ C2 )
=> ( ( ord_less_eq_set_b @ B2 @ D )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ ( inf_inf_set_b @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_629_Int__lower1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_630_Int__lower1,axiom,
! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ A2 ) ).
% Int_lower1
thf(fact_631_Int__lower2,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_632_Int__lower2,axiom,
! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ B2 ) ).
% Int_lower2
thf(fact_633_Int__absorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_634_Int__absorb1,axiom,
! [B2: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( inf_inf_set_b @ A2 @ B2 )
= B2 ) ) ).
% Int_absorb1
thf(fact_635_Int__absorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( inf_inf_set_a @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_636_Int__absorb2,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( inf_inf_set_b @ A2 @ B2 )
= A2 ) ) ).
% Int_absorb2
thf(fact_637_Int__greatest,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A2 )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ ( inf_inf_set_a @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_638_Int__greatest,axiom,
! [C2: set_b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ C2 @ A2 )
=> ( ( ord_less_eq_set_b @ C2 @ B2 )
=> ( ord_less_eq_set_b @ C2 @ ( inf_inf_set_b @ A2 @ B2 ) ) ) ) ).
% Int_greatest
thf(fact_639_Int__Collect__mono,axiom,
! [A2: set_dtree_list_a_b,B2: set_dtree_list_a_b,P2: dtree_list_a_b > $o,Q2: dtree_list_a_b > $o] :
( ( ord_le7599451563663638410st_a_b @ A2 @ B2 )
=> ( ! [X2: dtree_list_a_b] :
( ( member551035911493665803st_a_b @ X2 @ A2 )
=> ( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_le7599451563663638410st_a_b @ ( inf_in3355993651213403836st_a_b @ A2 @ ( collec2944820760411501129st_a_b @ P2 ) ) @ ( inf_in3355993651213403836st_a_b @ B2 @ ( collec2944820760411501129st_a_b @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_640_Int__Collect__mono,axiom,
! [A2: set_list_a,B2: set_list_a,P2: list_a > $o,Q2: list_a > $o] :
( ( ord_le8861187494160871172list_a @ A2 @ B2 )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ A2 )
=> ( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_le8861187494160871172list_a @ ( inf_inf_set_list_a @ A2 @ ( collect_list_a @ P2 ) ) @ ( inf_inf_set_list_a @ B2 @ ( collect_list_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_641_Int__Collect__mono,axiom,
! [A2: set_set_a,B2: set_set_a,P2: set_a > $o,Q2: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( inf_inf_set_set_a @ A2 @ ( collect_set_a @ P2 ) ) @ ( inf_inf_set_set_a @ B2 @ ( collect_set_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_642_Int__Collect__mono,axiom,
! [A2: set_a,B2: set_a,P2: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A2 @ ( collect_a @ P2 ) ) @ ( inf_inf_set_a @ B2 @ ( collect_a @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_643_Int__Collect__mono,axiom,
! [A2: set_b,B2: set_b,P2: b > $o,Q2: b > $o] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ! [X2: b] :
( ( member_b @ X2 @ A2 )
=> ( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A2 @ ( collect_b @ P2 ) ) @ ( inf_inf_set_b @ B2 @ ( collect_b @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_644_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B3: set_a] :
( ( sup_sup_set_a @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_645_subset__Un__eq,axiom,
( ord_less_eq_set_b
= ( ^ [A4: set_b,B3: set_b] :
( ( sup_sup_set_b @ A4 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_646_subset__UnE,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ~ ! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ A2 )
=> ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ B2 )
=> ( C2
!= ( sup_sup_set_a @ A5 @ B4 ) ) ) ) ) ).
% subset_UnE
thf(fact_647_subset__UnE,axiom,
! [C2: set_b,A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ C2 @ ( sup_sup_set_b @ A2 @ B2 ) )
=> ~ ! [A5: set_b] :
( ( ord_less_eq_set_b @ A5 @ A2 )
=> ! [B4: set_b] :
( ( ord_less_eq_set_b @ B4 @ B2 )
=> ( C2
!= ( sup_sup_set_b @ A5 @ B4 ) ) ) ) ) ).
% subset_UnE
thf(fact_648_Un__absorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_649_Un__absorb2,axiom,
! [B2: set_b,A2: set_b] :
( ( ord_less_eq_set_b @ B2 @ A2 )
=> ( ( sup_sup_set_b @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_650_Un__absorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_651_Un__absorb1,axiom,
! [A2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ B2 )
=> ( ( sup_sup_set_b @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_652_Un__upper2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_653_Un__upper2,axiom,
! [B2: set_b,A2: set_b] : ( ord_less_eq_set_b @ B2 @ ( sup_sup_set_b @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_654_Un__upper1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_655_Un__upper1,axiom,
! [A2: set_b,B2: set_b] : ( ord_less_eq_set_b @ A2 @ ( sup_sup_set_b @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_656_Un__least,axiom,
! [A2: set_a,C2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_657_Un__least,axiom,
! [A2: set_b,C2: set_b,B2: set_b] :
( ( ord_less_eq_set_b @ A2 @ C2 )
=> ( ( ord_less_eq_set_b @ B2 @ C2 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_658_Un__mono,axiom,
! [A2: set_a,C2: set_a,B2: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_659_Un__mono,axiom,
! [A2: set_b,C2: set_b,B2: set_b,D: set_b] :
( ( ord_less_eq_set_b @ A2 @ C2 )
=> ( ( ord_less_eq_set_b @ B2 @ D )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A2 @ B2 ) @ ( sup_sup_set_b @ C2 @ D ) ) ) ) ).
% Un_mono
thf(fact_660_set__subset__Cons,axiom,
! [Xs2: list_list_a,X: list_a] : ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs2 ) @ ( set_list_a2 @ ( cons_list_a @ X @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_661_set__subset__Cons,axiom,
! [Xs2: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ ( cons_a @ X @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_662_set__subset__Cons,axiom,
! [Xs2: list_b,X: b] : ( ord_less_eq_set_b @ ( set_b2 @ Xs2 ) @ ( set_b2 @ ( cons_b @ X @ Xs2 ) ) ) ).
% set_subset_Cons
thf(fact_663_Un__Int__assoc__eq,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ( sup_sup_set_a @ ( inf_inf_set_a @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) ) )
= ( ord_less_eq_set_a @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_664_Un__Int__assoc__eq,axiom,
! [A2: set_b,B2: set_b,C2: set_b] :
( ( ( sup_sup_set_b @ ( inf_inf_set_b @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_b @ A2 @ ( sup_sup_set_b @ B2 @ C2 ) ) )
= ( ord_less_eq_set_b @ C2 @ A2 ) ) ).
% Un_Int_assoc_eq
thf(fact_665_dverts__subtree__subset,axiom,
! [X: dtree_list_a_b,Y: dtree_list_a_b] :
( ( is_subtree_list_a_b @ X @ Y )
=> ( ord_le8861187494160871172list_a @ ( dverts_list_a_b @ X ) @ ( dverts_list_a_b @ Y ) ) ) ).
% dverts_subtree_subset
thf(fact_666_subtree__in__dlverts,axiom,
! [T1: dtree_list_a_b,T23: dtree_list_a_b] :
( ( is_subtree_list_a_b @ T1 @ T23 )
=> ( ord_less_eq_set_a @ ( list_dlverts_a_b @ T1 ) @ ( list_dlverts_a_b @ T23 ) ) ) ).
% subtree_in_dlverts
thf(fact_667_path__lverts__subset__dlverts,axiom,
! [T: dtree_list_a_b,X: a] : ( ord_less_eq_set_a @ ( iKKBZ_6987179986356532253ts_a_b @ T @ X ) @ ( list_dlverts_a_b @ T ) ) ).
% path_lverts_subset_dlverts
thf(fact_668_list__empty__if__subset__dsjnt,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs2 ) @ ( set_list_a2 @ Ys ) )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs2 ) @ ( set_list_a2 @ Ys ) )
= bot_bot_set_list_a )
=> ( Xs2 = nil_list_a ) ) ) ).
% list_empty_if_subset_dsjnt
thf(fact_669_list__empty__if__subset__dsjnt,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Ys ) )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( Xs2 = nil_a ) ) ) ).
% list_empty_if_subset_dsjnt
thf(fact_670_list__empty__if__subset__dsjnt,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( ord_less_eq_set_b @ ( set_b2 @ Xs2 ) @ ( set_b2 @ Ys ) )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ Xs2 ) @ ( set_b2 @ Ys ) )
= bot_bot_set_b )
=> ( Xs2 = nil_b ) ) ) ).
% list_empty_if_subset_dsjnt
thf(fact_671_vwalk__to__vpath_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ~ ! [X2: a,Xs: list_a] :
( X
!= ( cons_a @ X2 @ Xs ) ) ) ).
% vwalk_to_vpath.cases
thf(fact_672_vwalk__to__vpath_Ocases,axiom,
! [X: list_b] :
( ( X != nil_b )
=> ~ ! [X2: b,Xs: list_b] :
( X
!= ( cons_b @ X2 @ Xs ) ) ) ).
% vwalk_to_vpath.cases
thf(fact_673_vwalkI__append__l,axiom,
! [P: list_a,Q: list_a,G: pre_pr7278220950009878019t_unit] :
( ( P != nil_a )
=> ( ( vertex_vwalk_a_b @ ( append_a @ P @ Q ) @ G )
=> ( vertex_vwalk_a_b @ P @ G ) ) ) ).
% vwalkI_append_l
thf(fact_674_vwalkI__append__r,axiom,
! [Q: list_a,P: list_a,G: pre_pr7278220950009878019t_unit] :
( ( Q != nil_a )
=> ( ( vertex_vwalk_a_b @ ( append_a @ P @ Q ) @ G )
=> ( vertex_vwalk_a_b @ Q @ G ) ) ) ).
% vwalkI_append_r
thf(fact_675_vwalk__verts__in__verts,axiom,
! [P: list_a,G: pre_pr7278220950009878019t_unit,U: a] :
( ( vertex_vwalk_a_b @ P @ G )
=> ( ( member_a @ U @ ( set_a2 @ P ) )
=> ( member_a @ U @ ( pre_ve642382030648772252t_unit @ G ) ) ) ) ).
% vwalk_verts_in_verts
thf(fact_676_vwalk__verts__in__verts,axiom,
! [P: list_list_a,G: pre_pr2882871181989701257t_unit,U: list_a] :
( ( vertex2966258834163962945st_a_b @ P @ G )
=> ( ( member_list_a @ U @ ( set_list_a2 @ P ) )
=> ( member_list_a @ U @ ( pre_ve1830060048215441954t_unit @ G ) ) ) ) ).
% vwalk_verts_in_verts
thf(fact_677_vpathE,axiom,
! [P: list_a,G: pre_pr7278220950009878019t_unit] :
( ( vertex_vpath_a_b @ P @ G )
=> ~ ( ( vertex_vwalk_a_b @ P @ G )
=> ~ ( distinct_a @ P ) ) ) ).
% vpathE
thf(fact_678_vpath__def,axiom,
( vertex_vpath_a_b
= ( ^ [P4: list_a,G2: pre_pr7278220950009878019t_unit] :
( ( vertex_vwalk_a_b @ P4 @ G2 )
& ( distinct_a @ P4 ) ) ) ) ).
% vpath_def
thf(fact_679_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_680_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_681_vwalk__induct,axiom,
! [P: list_a,G: pre_pr7278220950009878019t_unit,P2: list_a > $o] :
( ( vertex_vwalk_a_b @ P @ G )
=> ( ! [U3: a] :
( ( member_a @ U3 @ ( pre_ve642382030648772252t_unit @ G ) )
=> ( P2 @ ( cons_a @ U3 @ nil_a ) ) )
=> ( ! [U3: a,V3: a,Es: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U3 @ V3 ) @ ( arcs_ends_a_b @ G ) )
=> ( ( P2 @ ( cons_a @ V3 @ Es ) )
=> ( P2 @ ( cons_a @ U3 @ ( cons_a @ V3 @ Es ) ) ) ) )
=> ( P2 @ P ) ) ) ) ).
% vwalk_induct
thf(fact_682_vwalk__induct,axiom,
! [P: list_list_a,G: pre_pr2882871181989701257t_unit,P2: list_list_a > $o] :
( ( vertex2966258834163962945st_a_b @ P @ G )
=> ( ! [U3: list_a] :
( ( member_list_a @ U3 @ ( pre_ve1830060048215441954t_unit @ G ) )
=> ( P2 @ ( cons_list_a @ U3 @ nil_list_a ) ) )
=> ( ! [U3: list_a,V3: list_a,Es: list_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ U3 @ V3 ) @ ( arcs_ends_list_a_b @ G ) )
=> ( ( P2 @ ( cons_list_a @ V3 @ Es ) )
=> ( P2 @ ( cons_list_a @ U3 @ ( cons_list_a @ V3 @ Es ) ) ) ) )
=> ( P2 @ P ) ) ) ) ).
% vwalk_induct
thf(fact_683_inf__sup__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_684_sup__inf__absorb,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_685_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_686_sup__bot__left,axiom,
! [X: fset_P2153231429829016240_a_b_b] :
( ( sup_su5028961100972612572_a_b_b @ bot_bo2248824169281960260_a_b_b @ X )
= X ) ).
% sup_bot_left
thf(fact_687_sup__bot__left,axiom,
! [X: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ X )
= X ) ).
% sup_bot_left
thf(fact_688_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_689_sup__bot__right,axiom,
! [X: fset_P2153231429829016240_a_b_b] :
( ( sup_su5028961100972612572_a_b_b @ X @ bot_bo2248824169281960260_a_b_b )
= X ) ).
% sup_bot_right
thf(fact_690_sup__bot__right,axiom,
! [X: set_list_a] :
( ( sup_sup_set_list_a @ X @ bot_bot_set_list_a )
= X ) ).
% sup_bot_right
thf(fact_691_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_692_bot__eq__sup__iff,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] :
( ( bot_bo2248824169281960260_a_b_b
= ( sup_su5028961100972612572_a_b_b @ X @ Y ) )
= ( ( X = bot_bo2248824169281960260_a_b_b )
& ( Y = bot_bo2248824169281960260_a_b_b ) ) ) ).
% bot_eq_sup_iff
thf(fact_693_bot__eq__sup__iff,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( bot_bot_set_list_a
= ( sup_sup_set_list_a @ X @ Y ) )
= ( ( X = bot_bot_set_list_a )
& ( Y = bot_bot_set_list_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_694_inf__right__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_right_idem
thf(fact_695_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_696_inf__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_left_idem
thf(fact_697_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_698_inf__idem,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ X )
= X ) ).
% inf_idem
thf(fact_699_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_700_sup_Oright__idem,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ B )
= ( sup_sup_set_a @ A @ B ) ) ).
% sup.right_idem
thf(fact_701_sup__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_702_sup_Oleft__idem,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% sup.left_idem
thf(fact_703_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_704_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_705_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_706_inf_Obounded__iff,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B @ C ) )
= ( ( ord_less_eq_real @ A @ B )
& ( ord_less_eq_real @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_707_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_708_le__inf__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( ( ord_less_eq_set_a @ X @ Y )
& ( ord_less_eq_set_a @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_709_le__inf__iff,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z ) )
= ( ( ord_less_eq_real @ X @ Y )
& ( ord_less_eq_real @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_710_le__inf__iff,axiom,
! [X: set_b,Y: set_b,Z: set_b] :
( ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) )
= ( ( ord_less_eq_set_b @ X @ Y )
& ( ord_less_eq_set_b @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_711_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_a @ X @ Z )
& ( ord_less_eq_set_a @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_712_le__sup__iff,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ X @ Y ) @ Z )
= ( ( ord_less_eq_real @ X @ Z )
& ( ord_less_eq_real @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_713_le__sup__iff,axiom,
! [X: set_b,Y: set_b,Z: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_b @ X @ Z )
& ( ord_less_eq_set_b @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_714_sup_Obounded__iff,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
= ( ( ord_less_eq_set_a @ B @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_715_sup_Obounded__iff,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B @ C ) @ A )
= ( ( ord_less_eq_real @ B @ A )
& ( ord_less_eq_real @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_716_sup_Obounded__iff,axiom,
! [B: set_b,C: set_b,A: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ B @ C ) @ A )
= ( ( ord_less_eq_set_b @ B @ A )
& ( ord_less_eq_set_b @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_717_inf__bot__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_718_inf__bot__right,axiom,
! [X: fset_P2153231429829016240_a_b_b] :
( ( inf_in525273840840435522_a_b_b @ X @ bot_bo2248824169281960260_a_b_b )
= bot_bo2248824169281960260_a_b_b ) ).
% inf_bot_right
thf(fact_719_inf__bot__right,axiom,
! [X: set_list_a] :
( ( inf_inf_set_list_a @ X @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% inf_bot_right
thf(fact_720_inf__bot__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_721_inf__bot__left,axiom,
! [X: fset_P2153231429829016240_a_b_b] :
( ( inf_in525273840840435522_a_b_b @ bot_bo2248824169281960260_a_b_b @ X )
= bot_bo2248824169281960260_a_b_b ) ).
% inf_bot_left
thf(fact_722_inf__bot__left,axiom,
! [X: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X )
= bot_bot_set_list_a ) ).
% inf_bot_left
thf(fact_723_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_724_sup__bot_Oright__neutral,axiom,
! [A: fset_P2153231429829016240_a_b_b] :
( ( sup_su5028961100972612572_a_b_b @ A @ bot_bo2248824169281960260_a_b_b )
= A ) ).
% sup_bot.right_neutral
thf(fact_725_sup__bot_Oright__neutral,axiom,
! [A: set_list_a] :
( ( sup_sup_set_list_a @ A @ bot_bot_set_list_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_726_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B ) )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_727_sup__bot_Oneutr__eq__iff,axiom,
! [A: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( bot_bo2248824169281960260_a_b_b
= ( sup_su5028961100972612572_a_b_b @ A @ B ) )
= ( ( A = bot_bo2248824169281960260_a_b_b )
& ( B = bot_bo2248824169281960260_a_b_b ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_728_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_list_a,B: set_list_a] :
( ( bot_bot_set_list_a
= ( sup_sup_set_list_a @ A @ B ) )
= ( ( A = bot_bot_set_list_a )
& ( B = bot_bot_set_list_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_729_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_730_sup__bot_Oleft__neutral,axiom,
! [A: fset_P2153231429829016240_a_b_b] :
( ( sup_su5028961100972612572_a_b_b @ bot_bo2248824169281960260_a_b_b @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_731_sup__bot_Oleft__neutral,axiom,
! [A: set_list_a] :
( ( sup_sup_set_list_a @ bot_bot_set_list_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_732_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_733_sup__bot_Oeq__neutr__iff,axiom,
! [A: fset_P2153231429829016240_a_b_b,B: fset_P2153231429829016240_a_b_b] :
( ( ( sup_su5028961100972612572_a_b_b @ A @ B )
= bot_bo2248824169281960260_a_b_b )
= ( ( A = bot_bo2248824169281960260_a_b_b )
& ( B = bot_bo2248824169281960260_a_b_b ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_734_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_list_a,B: set_list_a] :
( ( ( sup_sup_set_list_a @ A @ B )
= bot_bot_set_list_a )
= ( ( A = bot_bot_set_list_a )
& ( B = bot_bot_set_list_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_735_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_736_sup__eq__bot__iff,axiom,
! [X: fset_P2153231429829016240_a_b_b,Y: fset_P2153231429829016240_a_b_b] :
( ( ( sup_su5028961100972612572_a_b_b @ X @ Y )
= bot_bo2248824169281960260_a_b_b )
= ( ( X = bot_bo2248824169281960260_a_b_b )
& ( Y = bot_bo2248824169281960260_a_b_b ) ) ) ).
% sup_eq_bot_iff
thf(fact_737_sup__eq__bot__iff,axiom,
! [X: set_list_a,Y: set_list_a] :
( ( ( sup_sup_set_list_a @ X @ Y )
= bot_bot_set_list_a )
= ( ( X = bot_bot_set_list_a )
& ( Y = bot_bot_set_list_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_738_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_739_bot__set__def,axiom,
( bot_bot_set_list_a
= ( collect_list_a @ bot_bot_list_a_o ) ) ).
% bot_set_def
thf(fact_740_inf__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_741_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_742_inf__commute,axiom,
( inf_inf_set_a
= ( ^ [X5: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X5 ) ) ) ).
% inf_commute
thf(fact_743_inf_Ocommute,axiom,
( inf_inf_set_a
= ( ^ [A6: set_a,B5: set_a] : ( inf_inf_set_a @ B5 @ A6 ) ) ) ).
% inf.commute
thf(fact_744_inf__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_745_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_746_inf__sup__aci_I1_J,axiom,
( inf_inf_set_a
= ( ^ [X5: set_a,Y3: set_a] : ( inf_inf_set_a @ Y3 @ X5 ) ) ) ).
% inf_sup_aci(1)
thf(fact_747_inf__sup__aci_I2_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X @ Y ) @ Z )
= ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_748_inf__sup__aci_I3_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ Y @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_749_inf__sup__aci_I4_J,axiom,
! [X: set_a,Y: set_a] :
( ( inf_inf_set_a @ X @ ( inf_inf_set_a @ X @ Y ) )
= ( inf_inf_set_a @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_750_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_751_sup_Oleft__commute,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sup.left_commute
thf(fact_752_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X5: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X5 ) ) ) ).
% sup_commute
thf(fact_753_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A6: set_a,B5: set_a] : ( sup_sup_set_a @ B5 @ A6 ) ) ) ).
% sup.commute
thf(fact_754_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_755_sup_Oassoc,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% sup.assoc
thf(fact_756_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X5: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X5 ) ) ) ).
% inf_sup_aci(5)
thf(fact_757_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_758_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_759_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_760_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_761_inf_OcoboundedI2,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_762_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_763_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_764_inf_OcoboundedI1,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_765_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_766_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A6: set_a] :
( ( inf_inf_set_a @ A6 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_767_inf_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A6: real] :
( ( inf_inf_real @ A6 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_768_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_b
= ( ^ [B5: set_b,A6: set_b] :
( ( inf_inf_set_b @ A6 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_769_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
( ( inf_inf_set_a @ A6 @ B5 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_770_inf_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [A6: real,B5: real] :
( ( inf_inf_real @ A6 @ B5 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_771_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B5: set_b] :
( ( inf_inf_set_b @ A6 @ B5 )
= A6 ) ) ) ).
% inf.absorb_iff1
thf(fact_772_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_773_inf_Ocobounded2,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_774_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_775_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_776_inf_Ocobounded1,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_777_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_778_inf_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
( A6
= ( inf_inf_set_a @ A6 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_779_inf_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [A6: real,B5: real] :
( A6
= ( inf_inf_real @ A6 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_780_inf_Oorder__iff,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B5: set_b] :
( A6
= ( inf_inf_set_b @ A6 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_781_inf__greatest,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Z )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_782_inf__greatest,axiom,
! [X: real,Y: real,Z: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Z )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_783_inf__greatest,axiom,
! [X: set_b,Y: set_b,Z: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( ord_less_eq_set_b @ X @ Z )
=> ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_784_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_785_inf_OboundedI,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ A @ C )
=> ( ord_less_eq_real @ A @ ( inf_inf_real @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_786_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_787_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_788_inf_OboundedE,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B @ C ) )
=> ~ ( ( ord_less_eq_real @ A @ B )
=> ~ ( ord_less_eq_real @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_789_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_790_inf__absorb2,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( inf_inf_set_a @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_791_inf__absorb2,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( inf_inf_real @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_792_inf__absorb2,axiom,
! [Y: set_b,X: set_b] :
( ( ord_less_eq_set_b @ Y @ X )
=> ( ( inf_inf_set_b @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_793_inf__absorb1,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( inf_inf_set_a @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_794_inf__absorb1,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( inf_inf_real @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_795_inf__absorb1,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( inf_inf_set_b @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_796_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_797_inf_Oabsorb2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( inf_inf_real @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_798_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_799_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_800_inf_Oabsorb1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( inf_inf_real @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_801_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_802_le__iff__inf,axiom,
( ord_less_eq_set_a
= ( ^ [X5: set_a,Y3: set_a] :
( ( inf_inf_set_a @ X5 @ Y3 )
= X5 ) ) ) ).
% le_iff_inf
thf(fact_803_le__iff__inf,axiom,
( ord_less_eq_real
= ( ^ [X5: real,Y3: real] :
( ( inf_inf_real @ X5 @ Y3 )
= X5 ) ) ) ).
% le_iff_inf
thf(fact_804_le__iff__inf,axiom,
( ord_less_eq_set_b
= ( ^ [X5: set_b,Y3: set_b] :
( ( inf_inf_set_b @ X5 @ Y3 )
= X5 ) ) ) ).
% le_iff_inf
thf(fact_805_inf__unique,axiom,
! [F2: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F2 @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ ( F2 @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y4 )
=> ( ( ord_less_eq_set_a @ X2 @ Z3 )
=> ( ord_less_eq_set_a @ X2 @ ( F2 @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_set_a @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_806_inf__unique,axiom,
! [F2: real > real > real,X: real,Y: real] :
( ! [X2: real,Y4: real] : ( ord_less_eq_real @ ( F2 @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: real,Y4: real] : ( ord_less_eq_real @ ( F2 @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: real,Y4: real,Z3: real] :
( ( ord_less_eq_real @ X2 @ Y4 )
=> ( ( ord_less_eq_real @ X2 @ Z3 )
=> ( ord_less_eq_real @ X2 @ ( F2 @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_real @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_807_inf__unique,axiom,
! [F2: set_b > set_b > set_b,X: set_b,Y: set_b] :
( ! [X2: set_b,Y4: set_b] : ( ord_less_eq_set_b @ ( F2 @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: set_b,Y4: set_b] : ( ord_less_eq_set_b @ ( F2 @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: set_b,Y4: set_b,Z3: set_b] :
( ( ord_less_eq_set_b @ X2 @ Y4 )
=> ( ( ord_less_eq_set_b @ X2 @ Z3 )
=> ( ord_less_eq_set_b @ X2 @ ( F2 @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_set_b @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_808_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_809_inf_OorderI,axiom,
! [A: real,B: real] :
( ( A
= ( inf_inf_real @ A @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% inf.orderI
thf(fact_810_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_811_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_812_inf_OorderE,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( A
= ( inf_inf_real @ A @ B ) ) ) ).
% inf.orderE
thf(fact_813_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_814_le__infI2,axiom,
! [B: set_a,X: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_815_le__infI2,axiom,
! [B: real,X: real,A: real] :
( ( ord_less_eq_real @ B @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_816_le__infI2,axiom,
! [B: set_b,X: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B @ X )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ X ) ) ).
% le_infI2
thf(fact_817_le__infI1,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_818_le__infI1,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_eq_real @ A @ X )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_819_le__infI1,axiom,
! [A: set_b,X: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ X )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ X ) ) ).
% le_infI1
thf(fact_820_inf__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ B ) @ ( inf_inf_set_a @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_821_inf__mono,axiom,
! [A: real,C: real,B: real,D2: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ B @ D2 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ ( inf_inf_real @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_822_inf__mono,axiom,
! [A: set_b,C: set_b,B: set_b,D2: set_b] :
( ( ord_less_eq_set_b @ A @ C )
=> ( ( ord_less_eq_set_b @ B @ D2 )
=> ( ord_less_eq_set_b @ ( inf_inf_set_b @ A @ B ) @ ( inf_inf_set_b @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_823_le__infI,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% le_infI
thf(fact_824_le__infI,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ( ord_less_eq_real @ X @ B )
=> ( ord_less_eq_real @ X @ ( inf_inf_real @ A @ B ) ) ) ) ).
% le_infI
thf(fact_825_le__infI,axiom,
! [X: set_b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ X @ A )
=> ( ( ord_less_eq_set_b @ X @ B )
=> ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ A @ B ) ) ) ) ).
% le_infI
thf(fact_826_le__infE,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ ( inf_inf_set_a @ A @ B ) )
=> ~ ( ( ord_less_eq_set_a @ X @ A )
=> ~ ( ord_less_eq_set_a @ X @ B ) ) ) ).
% le_infE
thf(fact_827_le__infE,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_eq_real @ X @ ( inf_inf_real @ A @ B ) )
=> ~ ( ( ord_less_eq_real @ X @ A )
=> ~ ( ord_less_eq_real @ X @ B ) ) ) ).
% le_infE
thf(fact_828_le__infE,axiom,
! [X: set_b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ X @ ( inf_inf_set_b @ A @ B ) )
=> ~ ( ( ord_less_eq_set_b @ X @ A )
=> ~ ( ord_less_eq_set_b @ X @ B ) ) ) ).
% le_infE
thf(fact_829_inf__le2,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_830_inf__le2,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_831_inf__le2,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_832_inf__le1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_833_inf__le1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_834_inf__le1,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_835_inf__sup__ord_I1_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_836_inf__sup__ord_I1_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_837_inf__sup__ord_I1_J,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_838_inf__sup__ord_I2_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ ( inf_inf_set_a @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_839_inf__sup__ord_I2_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ ( inf_inf_real @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_840_inf__sup__ord_I2_J,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ ( inf_inf_set_b @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_841_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_842_inf__sup__ord_I4_J,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_843_inf__sup__ord_I4_J,axiom,
! [Y: set_b,X: set_b] : ( ord_less_eq_set_b @ Y @ ( sup_sup_set_b @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_844_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_845_inf__sup__ord_I3_J,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sup_sup_real @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_846_inf__sup__ord_I3_J,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ X @ ( sup_sup_set_b @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_847_le__supE,axiom,
! [A: set_a,B: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A @ X )
=> ~ ( ord_less_eq_set_a @ B @ X ) ) ) ).
% le_supE
thf(fact_848_le__supE,axiom,
! [A: real,B: real,X: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_real @ A @ X )
=> ~ ( ord_less_eq_real @ B @ X ) ) ) ).
% le_supE
thf(fact_849_le__supE,axiom,
! [A: set_b,B: set_b,X: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_b @ A @ X )
=> ~ ( ord_less_eq_set_b @ B @ X ) ) ) ).
% le_supE
thf(fact_850_le__supI,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_851_le__supI,axiom,
! [A: real,X: real,B: real] :
( ( ord_less_eq_real @ A @ X )
=> ( ( ord_less_eq_real @ B @ X )
=> ( ord_less_eq_real @ ( sup_sup_real @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_852_le__supI,axiom,
! [A: set_b,X: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ X )
=> ( ( ord_less_eq_set_b @ B @ X )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_853_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_854_sup__ge1,axiom,
! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sup_sup_real @ X @ Y ) ) ).
% sup_ge1
thf(fact_855_sup__ge1,axiom,
! [X: set_b,Y: set_b] : ( ord_less_eq_set_b @ X @ ( sup_sup_set_b @ X @ Y ) ) ).
% sup_ge1
thf(fact_856_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_857_sup__ge2,axiom,
! [Y: real,X: real] : ( ord_less_eq_real @ Y @ ( sup_sup_real @ X @ Y ) ) ).
% sup_ge2
thf(fact_858_sup__ge2,axiom,
! [Y: set_b,X: set_b] : ( ord_less_eq_set_b @ Y @ ( sup_sup_set_b @ X @ Y ) ) ).
% sup_ge2
thf(fact_859_le__supI1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI1
thf(fact_860_le__supI1,axiom,
! [X: real,A: real,B: real] :
( ( ord_less_eq_real @ X @ A )
=> ( ord_less_eq_real @ X @ ( sup_sup_real @ A @ B ) ) ) ).
% le_supI1
thf(fact_861_le__supI1,axiom,
! [X: set_b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ X @ A )
=> ( ord_less_eq_set_b @ X @ ( sup_sup_set_b @ A @ B ) ) ) ).
% le_supI1
thf(fact_862_le__supI2,axiom,
! [X: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI2
thf(fact_863_le__supI2,axiom,
! [X: real,B: real,A: real] :
( ( ord_less_eq_real @ X @ B )
=> ( ord_less_eq_real @ X @ ( sup_sup_real @ A @ B ) ) ) ).
% le_supI2
thf(fact_864_le__supI2,axiom,
! [X: set_b,B: set_b,A: set_b] :
( ( ord_less_eq_set_b @ X @ B )
=> ( ord_less_eq_set_b @ X @ ( sup_sup_set_b @ A @ B ) ) ) ).
% le_supI2
thf(fact_865_sup_Omono,axiom,
! [C: set_a,A: set_a,D2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ D2 @ B )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D2 ) @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_866_sup_Omono,axiom,
! [C: real,A: real,D2: real,B: real] :
( ( ord_less_eq_real @ C @ A )
=> ( ( ord_less_eq_real @ D2 @ B )
=> ( ord_less_eq_real @ ( sup_sup_real @ C @ D2 ) @ ( sup_sup_real @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_867_sup_Omono,axiom,
! [C: set_b,A: set_b,D2: set_b,B: set_b] :
( ( ord_less_eq_set_b @ C @ A )
=> ( ( ord_less_eq_set_b @ D2 @ B )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ C @ D2 ) @ ( sup_sup_set_b @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_868_sup__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_869_sup__mono,axiom,
! [A: real,C: real,B: real,D2: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ B @ D2 )
=> ( ord_less_eq_real @ ( sup_sup_real @ A @ B ) @ ( sup_sup_real @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_870_sup__mono,axiom,
! [A: set_b,C: set_b,B: set_b,D2: set_b] :
( ( ord_less_eq_set_b @ A @ C )
=> ( ( ord_less_eq_set_b @ B @ D2 )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ A @ B ) @ ( sup_sup_set_b @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_871_sup__least,axiom,
! [Y: set_a,X: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_872_sup__least,axiom,
! [Y: real,X: real,Z: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ Z @ X )
=> ( ord_less_eq_real @ ( sup_sup_real @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_873_sup__least,axiom,
! [Y: set_b,X: set_b,Z: set_b] :
( ( ord_less_eq_set_b @ Y @ X )
=> ( ( ord_less_eq_set_b @ Z @ X )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_874_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X5: set_a,Y3: set_a] :
( ( sup_sup_set_a @ X5 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_875_le__iff__sup,axiom,
( ord_less_eq_real
= ( ^ [X5: real,Y3: real] :
( ( sup_sup_real @ X5 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_876_le__iff__sup,axiom,
( ord_less_eq_set_b
= ( ^ [X5: set_b,Y3: set_b] :
( ( sup_sup_set_b @ X5 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_877_sup_OorderE,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( A
= ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.orderE
thf(fact_878_sup_OorderE,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( A
= ( sup_sup_real @ A @ B ) ) ) ).
% sup.orderE
thf(fact_879_sup_OorderE,axiom,
! [B: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( A
= ( sup_sup_set_b @ A @ B ) ) ) ).
% sup.orderE
thf(fact_880_sup_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% sup.orderI
thf(fact_881_sup_OorderI,axiom,
! [A: real,B: real] :
( ( A
= ( sup_sup_real @ A @ B ) )
=> ( ord_less_eq_real @ B @ A ) ) ).
% sup.orderI
thf(fact_882_sup_OorderI,axiom,
! [A: set_b,B: set_b] :
( ( A
= ( sup_sup_set_b @ A @ B ) )
=> ( ord_less_eq_set_b @ B @ A ) ) ).
% sup.orderI
thf(fact_883_sup__unique,axiom,
! [F2: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X2 @ ( F2 @ X2 @ Y4 ) )
=> ( ! [X2: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ ( F2 @ X2 @ Y4 ) )
=> ( ! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X2 )
=> ( ( ord_less_eq_set_a @ Z3 @ X2 )
=> ( ord_less_eq_set_a @ ( F2 @ Y4 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_884_sup__unique,axiom,
! [F2: real > real > real,X: real,Y: real] :
( ! [X2: real,Y4: real] : ( ord_less_eq_real @ X2 @ ( F2 @ X2 @ Y4 ) )
=> ( ! [X2: real,Y4: real] : ( ord_less_eq_real @ Y4 @ ( F2 @ X2 @ Y4 ) )
=> ( ! [X2: real,Y4: real,Z3: real] :
( ( ord_less_eq_real @ Y4 @ X2 )
=> ( ( ord_less_eq_real @ Z3 @ X2 )
=> ( ord_less_eq_real @ ( F2 @ Y4 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_real @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_885_sup__unique,axiom,
! [F2: set_b > set_b > set_b,X: set_b,Y: set_b] :
( ! [X2: set_b,Y4: set_b] : ( ord_less_eq_set_b @ X2 @ ( F2 @ X2 @ Y4 ) )
=> ( ! [X2: set_b,Y4: set_b] : ( ord_less_eq_set_b @ Y4 @ ( F2 @ X2 @ Y4 ) )
=> ( ! [X2: set_b,Y4: set_b,Z3: set_b] :
( ( ord_less_eq_set_b @ Y4 @ X2 )
=> ( ( ord_less_eq_set_b @ Z3 @ X2 )
=> ( ord_less_eq_set_b @ ( F2 @ Y4 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_b @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_886_sup_Oabsorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_887_sup_Oabsorb1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( sup_sup_real @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_888_sup_Oabsorb1,axiom,
! [B: set_b,A: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( sup_sup_set_b @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_889_sup_Oabsorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_890_sup_Oabsorb2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( sup_sup_real @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_891_sup_Oabsorb2,axiom,
! [A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ A @ B )
=> ( ( sup_sup_set_b @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_892_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_893_sup__absorb1,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( sup_sup_real @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_894_sup__absorb1,axiom,
! [Y: set_b,X: set_b] :
( ( ord_less_eq_set_b @ Y @ X )
=> ( ( sup_sup_set_b @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_895_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_896_sup__absorb2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( sup_sup_real @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_897_sup__absorb2,axiom,
! [X: set_b,Y: set_b] :
( ( ord_less_eq_set_b @ X @ Y )
=> ( ( sup_sup_set_b @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_898_sup_OboundedE,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B @ A )
=> ~ ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_899_sup_OboundedE,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ ( sup_sup_real @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_real @ B @ A )
=> ~ ( ord_less_eq_real @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_900_sup_OboundedE,axiom,
! [B: set_b,C: set_b,A: set_b] :
( ( ord_less_eq_set_b @ ( sup_sup_set_b @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_b @ B @ A )
=> ~ ( ord_less_eq_set_b @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_901_sup_OboundedI,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_902_sup_OboundedI,axiom,
! [B: real,A: real,C: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( ord_less_eq_real @ C @ A )
=> ( ord_less_eq_real @ ( sup_sup_real @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_903_sup_OboundedI,axiom,
! [B: set_b,A: set_b,C: set_b] :
( ( ord_less_eq_set_b @ B @ A )
=> ( ( ord_less_eq_set_b @ C @ A )
=> ( ord_less_eq_set_b @ ( sup_sup_set_b @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_904_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A6: set_a] :
( A6
= ( sup_sup_set_a @ A6 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_905_sup_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A6: real] :
( A6
= ( sup_sup_real @ A6 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_906_sup_Oorder__iff,axiom,
( ord_less_eq_set_b
= ( ^ [B5: set_b,A6: set_b] :
( A6
= ( sup_sup_set_b @ A6 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_907_sup_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded1
thf(fact_908_sup_Ocobounded1,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ A @ ( sup_sup_real @ A @ B ) ) ).
% sup.cobounded1
thf(fact_909_sup_Ocobounded1,axiom,
! [A: set_b,B: set_b] : ( ord_less_eq_set_b @ A @ ( sup_sup_set_b @ A @ B ) ) ).
% sup.cobounded1
thf(fact_910_sup_Ocobounded2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded2
thf(fact_911_sup_Ocobounded2,axiom,
! [B: real,A: real] : ( ord_less_eq_real @ B @ ( sup_sup_real @ A @ B ) ) ).
% sup.cobounded2
thf(fact_912_sup_Ocobounded2,axiom,
! [B: set_b,A: set_b] : ( ord_less_eq_set_b @ B @ ( sup_sup_set_b @ A @ B ) ) ).
% sup.cobounded2
thf(fact_913_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A6: set_a] :
( ( sup_sup_set_a @ A6 @ B5 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_914_sup_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [B5: real,A6: real] :
( ( sup_sup_real @ A6 @ B5 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_915_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_b
= ( ^ [B5: set_b,A6: set_b] :
( ( sup_sup_set_b @ A6 @ B5 )
= A6 ) ) ) ).
% sup.absorb_iff1
thf(fact_916_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A6: set_a,B5: set_a] :
( ( sup_sup_set_a @ A6 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_917_sup_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [A6: real,B5: real] :
( ( sup_sup_real @ A6 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_918_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_b
= ( ^ [A6: set_b,B5: set_b] :
( ( sup_sup_set_b @ A6 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_919_sup_OcoboundedI1,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_920_sup_OcoboundedI1,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ A )
=> ( ord_less_eq_real @ C @ ( sup_sup_real @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_921_sup_OcoboundedI1,axiom,
! [C: set_b,A: set_b,B: set_b] :
( ( ord_less_eq_set_b @ C @ A )
=> ( ord_less_eq_set_b @ C @ ( sup_sup_set_b @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_922_sup_OcoboundedI2,axiom,
! [C: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_923_sup_OcoboundedI2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ C @ ( sup_sup_real @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_924_sup_OcoboundedI2,axiom,
! [C: set_b,B: set_b,A: set_b] :
( ( ord_less_eq_set_b @ C @ B )
=> ( ord_less_eq_set_b @ C @ ( sup_sup_set_b @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_925_sup__inf__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ Z ) @ X )
= ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ X ) @ ( sup_sup_set_a @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_926_sup__inf__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_927_inf__sup__distrib2,axiom,
! [Y: set_a,Z: set_a,X: set_a] :
( ( inf_inf_set_a @ ( sup_sup_set_a @ Y @ Z ) @ X )
= ( sup_sup_set_a @ ( inf_inf_set_a @ Y @ X ) @ ( inf_inf_set_a @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_928_inf__sup__distrib1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_929_distrib__imp2,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X2 @ ( inf_inf_set_a @ Y4 @ Z3 ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X2 @ Y4 ) @ ( sup_sup_set_a @ X2 @ Z3 ) ) )
=> ( ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_930_distrib__imp1,axiom,
! [X: set_a,Y: set_a,Z: set_a] :
( ! [X2: set_a,Y4: set_a,Z3: set_a] :
( ( inf_inf_set_a @ X2 @ ( sup_sup_set_a @ Y4 @ Z3 ) )
= ( sup_sup_set_a @ ( inf_inf_set_a @ X2 @ Y4 ) @ ( inf_inf_set_a @ X2 @ Z3 ) ) )
=> ( ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) )
= ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_931_distrib__sup__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ ( inf_inf_set_a @ Y @ Z ) ) @ ( inf_inf_set_a @ ( sup_sup_set_a @ X @ Y ) @ ( sup_sup_set_a @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_932_distrib__sup__le,axiom,
! [X: real,Y: real,Z: real] : ( ord_less_eq_real @ ( sup_sup_real @ X @ ( inf_inf_real @ Y @ Z ) ) @ ( inf_inf_real @ ( sup_sup_real @ X @ Y ) @ ( sup_sup_real @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_933_distrib__sup__le,axiom,
! [X: set_b,Y: set_b,Z: set_b] : ( ord_less_eq_set_b @ ( sup_sup_set_b @ X @ ( inf_inf_set_b @ Y @ Z ) ) @ ( inf_inf_set_b @ ( sup_sup_set_b @ X @ Y ) @ ( sup_sup_set_b @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_934_distrib__inf__le,axiom,
! [X: set_a,Y: set_a,Z: set_a] : ( ord_less_eq_set_a @ ( sup_sup_set_a @ ( inf_inf_set_a @ X @ Y ) @ ( inf_inf_set_a @ X @ Z ) ) @ ( inf_inf_set_a @ X @ ( sup_sup_set_a @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_935_distrib__inf__le,axiom,
! [X: real,Y: real,Z: real] : ( ord_less_eq_real @ ( sup_sup_real @ ( inf_inf_real @ X @ Y ) @ ( inf_inf_real @ X @ Z ) ) @ ( inf_inf_real @ X @ ( sup_sup_real @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_936_distrib__inf__le,axiom,
! [X: set_b,Y: set_b,Z: set_b] : ( ord_less_eq_set_b @ ( sup_sup_set_b @ ( inf_inf_set_b @ X @ Y ) @ ( inf_inf_set_b @ X @ Z ) ) @ ( inf_inf_set_b @ X @ ( sup_sup_set_b @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_937_T_Odistint__verts__singleton__subtree,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ta )
=> ( distinct_a @ ( append_a @ R3 @ ( root_list_a_b @ T1 ) ) ) ) ).
% T.distint_verts_singleton_subtree
thf(fact_938_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ X @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_939_boolean__algebra_Oconj__zero__right,axiom,
! [X: set_list_a] :
( ( inf_inf_set_list_a @ X @ bot_bot_set_list_a )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_940_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_941_boolean__algebra_Oconj__zero__left,axiom,
! [X: set_list_a] :
( ( inf_inf_set_list_a @ bot_bot_set_list_a @ X )
= bot_bot_set_list_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_942_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_943_sublist__split__concat_H,axiom,
! [Acc2: list_list_b,As: list_list_b,X: list_b,Bs: list_list_b,Ys: list_b,Cs: list_b] :
( ? [X4: list_b] :
( ( member_list_b @ X4 @ ( set_list_b2 @ ( append_list_b @ Acc2 @ ( append_list_b @ As @ ( cons_list_b @ X @ Bs ) ) ) ) )
& ( ( sublist_b @ Ys @ X4 )
| ( sublist_b @ Ys @ Cs ) ) )
=> ( ? [X2: list_b] :
( ( member_list_b @ X2 @ ( set_list_b2 @ ( append_list_b @ ( rev_list_b @ Acc2 ) @ ( append_list_b @ As @ ( cons_list_b @ X @ nil_list_b ) ) ) ) )
& ( sublist_b @ Ys @ X2 ) )
| ( sublist_b @ Ys @ ( append_b @ ( concat_b @ Bs ) @ Cs ) ) ) ) ).
% sublist_split_concat'
thf(fact_944_sublist__split__concat_H,axiom,
! [Acc2: list_list_a,As: list_list_a,X: list_a,Bs: list_list_a,Ys: list_a,Cs: list_a] :
( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ ( append_list_a @ Acc2 @ ( append_list_a @ As @ ( cons_list_a @ X @ Bs ) ) ) ) )
& ( ( sublist_a @ Ys @ X4 )
| ( sublist_a @ Ys @ Cs ) ) )
=> ( ? [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ ( append_list_a @ ( rev_list_a @ Acc2 ) @ ( append_list_a @ As @ ( cons_list_a @ X @ nil_list_a ) ) ) ) )
& ( sublist_a @ Ys @ X2 ) )
| ( sublist_a @ Ys @ ( append_a @ ( concat_a @ Bs ) @ Cs ) ) ) ) ).
% sublist_split_concat'
thf(fact_945_sublist__app__l,axiom,
! [Ys: list_a,Cs: list_a,Xs2: list_a] :
( ( sublist_a @ Ys @ Cs )
=> ( sublist_a @ Ys @ ( append_a @ Xs2 @ Cs ) ) ) ).
% sublist_app_l
thf(fact_946_sublist__app__l,axiom,
! [Ys: list_b,Cs: list_b,Xs2: list_b] :
( ( sublist_b @ Ys @ Cs )
=> ( sublist_b @ Ys @ ( append_b @ Xs2 @ Cs ) ) ) ).
% sublist_app_l
thf(fact_947_concat__all__sublist,axiom,
! [Xs2: list_list_a,X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs2 ) )
=> ( sublist_a @ X4 @ ( concat_a @ Xs2 ) ) ) ).
% concat_all_sublist
thf(fact_948_normalize1_Ocases,axiom,
! [X: dtree_list_a_b] :
( ! [R: list_a,T12: dtree_list_a_b,E2: b] :
( X
!= ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T12 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X4: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X4 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [R: list_a] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ) ) ).
% normalize1.cases
thf(fact_949_sublist__set__concat__or__cases__aux1,axiom,
! [Ys: list_b,As: list_b,U2: list_b,Cs: list_b,Xs2: 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 @ Xs2 ) ) ) ) )
| ( sublist_b @ Ys @ Cs ) ) ) ).
% sublist_set_concat_or_cases_aux1
thf(fact_950_sublist__set__concat__or__cases__aux1,axiom,
! [Ys: list_a,As: list_a,U2: list_a,Cs: list_a,Xs2: 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 @ Xs2 ) ) ) ) )
| ( sublist_a @ Ys @ Cs ) ) ) ).
% sublist_set_concat_or_cases_aux1
thf(fact_951_concat__all__sublist__rev,axiom,
! [Xs2: list_list_a,X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ Xs2 ) )
=> ( sublist_a @ X4 @ ( concat_a @ ( rev_list_a @ Xs2 ) ) ) ) ).
% concat_all_sublist_rev
thf(fact_952_reachable__induce__ss,axiom,
! [S: set_a,U: a,V: a,T2: set_a] :
( ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ t @ S ) @ U @ V )
=> ( ( ord_less_eq_set_a @ S @ T2 )
=> ( reachable_a_b @ ( digrap7873285959652527175ph_a_b @ t @ T2 ) @ U @ V ) ) ) ).
% reachable_induce_ss
thf(fact_953_dominates__induce__ss,axiom,
! [U: a,V: a,S: set_a,T2: 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 @ T2 )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V ) @ ( arcs_ends_a_b @ ( digrap7873285959652527175ph_a_b @ t @ T2 ) ) ) ) ) ).
% dominates_induce_ss
thf(fact_954_sublist__exists__append,axiom,
! [X: list_b,Xs2: list_list_b,B: list_b,Ys: list_b] :
( ? [X4: list_b] :
( ( member_list_b @ X4 @ ( set_list_b2 @ ( append_list_b @ ( cons_list_b @ X @ Xs2 ) @ ( cons_list_b @ B @ nil_list_b ) ) ) )
& ( sublist_b @ Ys @ X4 ) )
=> ? [X2: list_b] :
( ( member_list_b @ X2 @ ( set_list_b2 @ ( append_list_b @ Xs2 @ ( cons_list_b @ ( append_b @ X @ B ) @ nil_list_b ) ) ) )
& ( sublist_b @ Ys @ X2 ) ) ) ).
% sublist_exists_append
thf(fact_955_sublist__exists__append,axiom,
! [X: list_a,Xs2: list_list_a,B: list_a,Ys: list_a] :
( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ ( append_list_a @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ B @ nil_list_a ) ) ) )
& ( sublist_a @ Ys @ X4 ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ ( append_list_a @ Xs2 @ ( cons_list_a @ ( append_a @ X @ B ) @ nil_list_a ) ) ) )
& ( sublist_a @ Ys @ X2 ) ) ) ).
% sublist_exists_append
thf(fact_956_sublist__set__concat__cases,axiom,
! [X: list_a,Xs2: list_list_a,B: list_a,Ys: list_a] :
( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ ( append_list_a @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ B @ nil_list_a ) ) ) )
& ( sublist_a @ Ys @ X4 ) )
=> ( ( sublist_a @ Ys @ ( concat_a @ ( rev_list_a @ Xs2 ) ) )
| ( sublist_a @ Ys @ X )
| ( sublist_a @ Ys @ B ) ) ) ).
% sublist_set_concat_cases
thf(fact_957_sublist__set__concat__or__cases,axiom,
! [Ys: list_b,As: list_b,U2: list_b,X: list_b,Xs2: list_list_b,B: list_b,Cs: list_b] :
( ( ( sublist_b @ Ys @ As )
| ( sublist_b @ Ys @ U2 )
| ? [X4: list_b] :
( ( member_list_b @ X4 @ ( set_list_b2 @ ( append_list_b @ ( cons_list_b @ X @ Xs2 ) @ ( cons_list_b @ B @ nil_list_b ) ) ) )
& ( sublist_b @ Ys @ X4 ) )
| ( sublist_b @ Ys @ Cs ) )
=> ( ( sublist_b @ Ys @ ( append_b @ As @ ( append_b @ U2 @ ( concat_b @ ( rev_list_b @ Xs2 ) ) ) ) )
| ( sublist_b @ Ys @ X )
| ? [X2: list_b] :
( ( member_list_b @ X2 @ ( set_list_b2 @ ( cons_list_b @ B @ nil_list_b ) ) )
& ( sublist_b @ Ys @ X2 ) )
| ( sublist_b @ Ys @ Cs ) ) ) ).
% sublist_set_concat_or_cases
thf(fact_958_sublist__set__concat__or__cases,axiom,
! [Ys: list_a,As: list_a,U2: list_a,X: list_a,Xs2: list_list_a,B: list_a,Cs: list_a] :
( ( ( sublist_a @ Ys @ As )
| ( sublist_a @ Ys @ U2 )
| ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ ( append_list_a @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ B @ nil_list_a ) ) ) )
& ( sublist_a @ Ys @ X4 ) )
| ( sublist_a @ Ys @ Cs ) )
=> ( ( sublist_a @ Ys @ ( append_a @ As @ ( append_a @ U2 @ ( concat_a @ ( rev_list_a @ Xs2 ) ) ) ) )
| ( sublist_a @ Ys @ X )
| ? [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ ( cons_list_a @ B @ nil_list_a ) ) )
& ( sublist_a @ Ys @ X2 ) )
| ( sublist_a @ Ys @ Cs ) ) ) ).
% sublist_set_concat_or_cases
thf(fact_959_sublist__set__concat__or__cases__aux2,axiom,
! [X: list_b,Xs2: list_list_b,B: list_b,Ys: list_b,As: list_b,U2: list_b] :
( ? [X4: list_b] :
( ( member_list_b @ X4 @ ( set_list_b2 @ ( append_list_b @ ( cons_list_b @ X @ Xs2 ) @ ( cons_list_b @ B @ nil_list_b ) ) ) )
& ( sublist_b @ Ys @ X4 ) )
=> ( ( sublist_b @ Ys @ ( append_b @ As @ ( append_b @ U2 @ ( concat_b @ ( rev_list_b @ Xs2 ) ) ) ) )
| ( sublist_b @ Ys @ X )
| ( sublist_b @ Ys @ B ) ) ) ).
% sublist_set_concat_or_cases_aux2
thf(fact_960_sublist__set__concat__or__cases__aux2,axiom,
! [X: list_a,Xs2: list_list_a,B: list_a,Ys: list_a,As: list_a,U2: list_a] :
( ? [X4: list_a] :
( ( member_list_a @ X4 @ ( set_list_a2 @ ( append_list_a @ ( cons_list_a @ X @ Xs2 ) @ ( cons_list_a @ B @ nil_list_a ) ) ) )
& ( sublist_a @ Ys @ X4 ) )
=> ( ( sublist_a @ Ys @ ( append_a @ As @ ( append_a @ U2 @ ( concat_a @ ( rev_list_a @ Xs2 ) ) ) ) )
| ( sublist_a @ Ys @ X )
| ( sublist_a @ Ys @ B ) ) ) ).
% sublist_set_concat_or_cases_aux2
thf(fact_961_sublist__split__concat,axiom,
! [A: list_b,Acc2: list_list_b,As: list_list_b,X: list_b,Bs: list_list_b,Ys: list_b,Cs: list_b] :
( ( member_list_b @ A @ ( set_list_b2 @ ( append_list_b @ Acc2 @ ( append_list_b @ As @ ( cons_list_b @ X @ Bs ) ) ) ) )
=> ( ( sublist_b @ Ys @ A )
=> ( ? [X2: list_b] :
( ( member_list_b @ X2 @ ( set_list_b2 @ ( append_list_b @ ( rev_list_b @ Acc2 ) @ ( append_list_b @ As @ ( cons_list_b @ X @ nil_list_b ) ) ) ) )
& ( sublist_b @ Ys @ X2 ) )
| ( sublist_b @ Ys @ ( append_b @ ( concat_b @ Bs ) @ Cs ) ) ) ) ) ).
% sublist_split_concat
thf(fact_962_sublist__split__concat,axiom,
! [A: list_a,Acc2: list_list_a,As: list_list_a,X: list_a,Bs: list_list_a,Ys: list_a,Cs: list_a] :
( ( member_list_a @ A @ ( set_list_a2 @ ( append_list_a @ Acc2 @ ( append_list_a @ As @ ( cons_list_a @ X @ Bs ) ) ) ) )
=> ( ( sublist_a @ Ys @ A )
=> ( ? [X2: list_a] :
( ( member_list_a @ X2 @ ( set_list_a2 @ ( append_list_a @ ( rev_list_a @ Acc2 ) @ ( append_list_a @ As @ ( cons_list_a @ X @ nil_list_a ) ) ) ) )
& ( sublist_a @ Ys @ X2 ) )
| ( sublist_a @ Ys @ ( append_a @ ( concat_a @ Bs ) @ Cs ) ) ) ) ) ).
% sublist_split_concat
thf(fact_963_path__lverts_Ocases,axiom,
! [X: produc6499617306661234687_a_b_a] :
( ! [R: list_a,T4: dtree_list_a_b,E2: b,X2: a] :
( X
!= ( produc7704165765595008945_a_b_a @ ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T4 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) @ X2 ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X4: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X4 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [R: list_a,X2: a] :
( X
!= ( produc7704165765595008945_a_b_a @ ( node_list_a_b @ R @ Xs ) @ X2 ) ) ) ) ).
% path_lverts.cases
thf(fact_964_fst__sublist__if__not__snd__sublist,axiom,
! [Xs2: list_a,Ys: list_a,A2: list_a,B2: list_a] :
( ( ( append_a @ Xs2 @ Ys )
= ( append_a @ A2 @ B2 ) )
=> ( ~ ( sublist_a @ B2 @ Ys )
=> ? [As3: list_a,Bs2: list_a] :
( ( ( append_a @ As3 @ Bs2 )
= Xs2 )
& ( ( append_a @ Bs2 @ Ys )
= B2 ) ) ) ) ).
% fst_sublist_if_not_snd_sublist
thf(fact_965_fst__sublist__if__not__snd__sublist,axiom,
! [Xs2: list_b,Ys: list_b,A2: list_b,B2: list_b] :
( ( ( append_b @ Xs2 @ Ys )
= ( append_b @ A2 @ B2 ) )
=> ( ~ ( sublist_b @ B2 @ Ys )
=> ? [As3: list_b,Bs2: list_b] :
( ( ( append_b @ As3 @ Bs2 )
= Xs2 )
& ( ( append_b @ Bs2 @ Ys )
= B2 ) ) ) ) ).
% fst_sublist_if_not_snd_sublist
thf(fact_966_sublist__app,axiom,
! [A2: list_a,B2: list_a,C2: list_a] :
( ( sublist_a @ ( append_a @ A2 @ B2 ) @ C2 )
=> ( ( sublist_a @ A2 @ C2 )
& ( sublist_a @ B2 @ C2 ) ) ) ).
% sublist_app
thf(fact_967_sublist__app,axiom,
! [A2: list_b,B2: list_b,C2: list_b] :
( ( sublist_b @ ( append_b @ A2 @ B2 ) @ C2 )
=> ( ( sublist_b @ A2 @ C2 )
& ( sublist_b @ B2 @ C2 ) ) ) ).
% sublist_app
thf(fact_968_dtree__to__list_Ocases,axiom,
! [X: dtree_list_a_b] :
( ! [R: list_a,T4: dtree_list_a_b,E2: b] :
( X
!= ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T4 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X4: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X4 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [R: list_a] :
( X
!= ( node_list_a_b @ R @ Xs ) ) ) ) ).
% dtree_to_list.cases
thf(fact_969_singleton__uneq,axiom,
! [R3: list_a,T: dtree_list_a_b,E: b] :
( ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) )
!= T ) ).
% singleton_uneq
thf(fact_970_sublist__set__elem,axiom,
! [Xs2: list_dtree_list_a_b,A2: list_dtree_list_a_b,B2: list_dtree_list_a_b,X: dtree_list_a_b] :
( ( sublis1540088274494349249st_a_b @ Xs2 @ ( append2129087805494049561st_a_b @ A2 @ B2 ) )
=> ( ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ Xs2 ) )
=> ( ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ A2 ) )
| ( member551035911493665803st_a_b @ X @ ( set_dtree_list_a_b2 @ B2 ) ) ) ) ) ).
% sublist_set_elem
thf(fact_971_sublist__set__elem,axiom,
! [Xs2: list_set_a,A2: list_set_a,B2: list_set_a,X: set_a] :
( ( sublist_set_a @ Xs2 @ ( append_set_a @ A2 @ B2 ) )
=> ( ( member_set_a @ X @ ( set_set_a2 @ Xs2 ) )
=> ( ( member_set_a @ X @ ( set_set_a2 @ A2 ) )
| ( member_set_a @ X @ ( set_set_a2 @ B2 ) ) ) ) ) ).
% sublist_set_elem
thf(fact_972_sublist__set__elem,axiom,
! [Xs2: list_b,A2: list_b,B2: list_b,X: b] :
( ( sublist_b @ Xs2 @ ( append_b @ A2 @ B2 ) )
=> ( ( member_b @ X @ ( set_b2 @ Xs2 ) )
=> ( ( member_b @ X @ ( set_b2 @ A2 ) )
| ( member_b @ X @ ( set_b2 @ B2 ) ) ) ) ) ).
% sublist_set_elem
thf(fact_973_sublist__set__elem,axiom,
! [Xs2: list_a,A2: list_a,B2: list_a,X: a] :
( ( sublist_a @ Xs2 @ ( append_a @ A2 @ B2 ) )
=> ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
=> ( ( member_a @ X @ ( set_a2 @ A2 ) )
| ( member_a @ X @ ( set_a2 @ B2 ) ) ) ) ) ).
% sublist_set_elem
thf(fact_974_sublist__set__elem,axiom,
! [Xs2: list_list_a,A2: list_list_a,B2: list_list_a,X: list_a] :
( ( sublist_list_a @ Xs2 @ ( append_list_a @ A2 @ B2 ) )
=> ( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
=> ( ( member_list_a @ X @ ( set_list_a2 @ A2 ) )
| ( member_list_a @ X @ ( set_list_a2 @ B2 ) ) ) ) ) ).
% sublist_set_elem
thf(fact_975_sublist__Cons,axiom,
! [A2: a,B2: list_a,C2: list_a] :
( ( sublist_a @ ( cons_a @ A2 @ B2 ) @ C2 )
=> ( ( sublist_a @ ( cons_a @ A2 @ nil_a ) @ C2 )
& ( sublist_a @ B2 @ C2 ) ) ) ).
% sublist_Cons
thf(fact_976_sublist__Cons,axiom,
! [A2: b,B2: list_b,C2: list_b] :
( ( sublist_b @ ( cons_b @ A2 @ B2 ) @ C2 )
=> ( ( sublist_b @ ( cons_b @ A2 @ nil_b ) @ C2 )
& ( sublist_b @ B2 @ C2 ) ) ) ).
% sublist_Cons
thf(fact_977_singleton__uneq_H,axiom,
! [R3: list_a,T: dtree_list_a_b,E: b,V: list_a] :
( ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) )
!= ( node_list_a_b @ V @ ( sucs_list_a_b @ T ) ) ) ).
% singleton_uneq'
thf(fact_978_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_979_boolean__algebra__cancel_Oinf2,axiom,
! [B2: set_a,K: set_a,B: set_a,A: set_a] :
( ( B2
= ( inf_inf_set_a @ K @ B ) )
=> ( ( inf_inf_set_a @ A @ B2 )
= ( inf_inf_set_a @ K @ ( inf_inf_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_980_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_a,K: set_a,A: set_a,B: set_a] :
( ( A2
= ( sup_sup_set_a @ K @ A ) )
=> ( ( sup_sup_set_a @ A2 @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_981_boolean__algebra__cancel_Osup2,axiom,
! [B2: set_a,K: set_a,B: set_a,A: set_a] :
( ( B2
= ( sup_sup_set_a @ K @ B ) )
=> ( ( sup_sup_set_a @ A @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_982_sublist__fst__if__snd__dsjnt,axiom,
! [U2: list_b,B2: list_b,V2: list_b] :
( ( sublist_b @ U2 @ ( append_b @ B2 @ V2 ) )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ U2 ) @ ( set_b2 @ V2 ) )
= bot_bot_set_b )
=> ( sublist_b @ U2 @ B2 ) ) ) ).
% sublist_fst_if_snd_dsjnt
thf(fact_983_sublist__fst__if__snd__dsjnt,axiom,
! [U2: list_a,B2: list_a,V2: list_a] :
( ( sublist_a @ U2 @ ( append_a @ B2 @ V2 ) )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ V2 ) )
= bot_bot_set_a )
=> ( sublist_a @ U2 @ B2 ) ) ) ).
% sublist_fst_if_snd_dsjnt
thf(fact_984_sublist__fst__if__snd__dsjnt,axiom,
! [U2: list_list_a,B2: list_list_a,V2: list_list_a] :
( ( sublist_list_a @ U2 @ ( append_list_a @ B2 @ V2 ) )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ U2 ) @ ( set_list_a2 @ V2 ) )
= bot_bot_set_list_a )
=> ( sublist_list_a @ U2 @ B2 ) ) ) ).
% sublist_fst_if_snd_dsjnt
thf(fact_985_sublist__snd__if__fst__dsjnt,axiom,
! [U2: list_b,V2: list_b,B2: list_b] :
( ( sublist_b @ U2 @ ( append_b @ V2 @ B2 ) )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ U2 ) @ ( set_b2 @ V2 ) )
= bot_bot_set_b )
=> ( sublist_b @ U2 @ B2 ) ) ) ).
% sublist_snd_if_fst_dsjnt
thf(fact_986_sublist__snd__if__fst__dsjnt,axiom,
! [U2: list_a,V2: list_a,B2: list_a] :
( ( sublist_a @ U2 @ ( append_a @ V2 @ B2 ) )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ V2 ) )
= bot_bot_set_a )
=> ( sublist_a @ U2 @ B2 ) ) ) ).
% sublist_snd_if_fst_dsjnt
thf(fact_987_sublist__snd__if__fst__dsjnt,axiom,
! [U2: list_list_a,V2: list_list_a,B2: list_list_a] :
( ( sublist_list_a @ U2 @ ( append_list_a @ V2 @ B2 ) )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ U2 ) @ ( set_list_a2 @ V2 ) )
= bot_bot_set_list_a )
=> ( sublist_list_a @ U2 @ B2 ) ) ) ).
% sublist_snd_if_fst_dsjnt
thf(fact_988_sublist__behind__if__nbefore,axiom,
! [U2: list_b,Xs2: list_b,V2: list_b] :
( ( sublist_b @ U2 @ Xs2 )
=> ( ( sublist_b @ V2 @ Xs2 )
=> ( ~ ? [As3: list_b,Bs2: list_b,Cs3: list_b] :
( ( append_b @ As3 @ ( append_b @ U2 @ ( append_b @ Bs2 @ ( append_b @ V2 @ Cs3 ) ) ) )
= Xs2 )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ U2 ) @ ( set_b2 @ V2 ) )
= bot_bot_set_b )
=> ? [As3: list_b,Bs2: list_b,Cs3: list_b] :
( ( append_b @ As3 @ ( append_b @ V2 @ ( append_b @ Bs2 @ ( append_b @ U2 @ Cs3 ) ) ) )
= Xs2 ) ) ) ) ) ).
% sublist_behind_if_nbefore
thf(fact_989_sublist__behind__if__nbefore,axiom,
! [U2: list_a,Xs2: list_a,V2: list_a] :
( ( sublist_a @ U2 @ Xs2 )
=> ( ( sublist_a @ V2 @ Xs2 )
=> ( ~ ? [As3: list_a,Bs2: list_a,Cs3: list_a] :
( ( append_a @ As3 @ ( append_a @ U2 @ ( append_a @ Bs2 @ ( append_a @ V2 @ Cs3 ) ) ) )
= Xs2 )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ V2 ) )
= bot_bot_set_a )
=> ? [As3: list_a,Bs2: list_a,Cs3: list_a] :
( ( append_a @ As3 @ ( append_a @ V2 @ ( append_a @ Bs2 @ ( append_a @ U2 @ Cs3 ) ) ) )
= Xs2 ) ) ) ) ) ).
% sublist_behind_if_nbefore
thf(fact_990_sublist__behind__if__nbefore,axiom,
! [U2: list_list_a,Xs2: list_list_a,V2: list_list_a] :
( ( sublist_list_a @ U2 @ Xs2 )
=> ( ( sublist_list_a @ V2 @ Xs2 )
=> ( ~ ? [As3: list_list_a,Bs2: list_list_a,Cs3: list_list_a] :
( ( append_list_a @ As3 @ ( append_list_a @ U2 @ ( append_list_a @ Bs2 @ ( append_list_a @ V2 @ Cs3 ) ) ) )
= Xs2 )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ U2 ) @ ( set_list_a2 @ V2 ) )
= bot_bot_set_list_a )
=> ? [As3: list_list_a,Bs2: list_list_a,Cs3: list_list_a] :
( ( append_list_a @ As3 @ ( append_list_a @ V2 @ ( append_list_a @ Bs2 @ ( append_list_a @ U2 @ Cs3 ) ) ) )
= Xs2 ) ) ) ) ) ).
% sublist_behind_if_nbefore
thf(fact_991_empty__if__sublist__dsjnt,axiom,
! [Xs2: list_b,Ys: list_b] :
( ( sublist_b @ Xs2 @ Ys )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ Xs2 ) @ ( set_b2 @ Ys ) )
= bot_bot_set_b )
=> ( Xs2 = nil_b ) ) ) ).
% empty_if_sublist_dsjnt
thf(fact_992_empty__if__sublist__dsjnt,axiom,
! [Xs2: list_a,Ys: list_a] :
( ( sublist_a @ Xs2 @ Ys )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ Ys ) )
= bot_bot_set_a )
=> ( Xs2 = nil_a ) ) ) ).
% empty_if_sublist_dsjnt
thf(fact_993_empty__if__sublist__dsjnt,axiom,
! [Xs2: list_list_a,Ys: list_list_a] :
( ( sublist_list_a @ Xs2 @ Ys )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs2 ) @ ( set_list_a2 @ Ys ) )
= bot_bot_set_list_a )
=> ( Xs2 = nil_list_a ) ) ) ).
% empty_if_sublist_dsjnt
thf(fact_994_single__subtree__root__dverts,axiom,
! [V22: list_a,T23: dtree_list_a_b,E22: b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V22 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T1 )
=> ( member_list_a @ V22 @ ( dverts_list_a_b @ T1 ) ) ) ).
% single_subtree_root_dverts
thf(fact_995_combine__uneq,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) )
!= ( node_list_a_b @ ( append_a @ R3 @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) ) ).
% combine_uneq
thf(fact_996_path__lverts_Osimps_I2_J,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,X: a,R3: list_a] :
( ! [X2: produc6499617310964463488_a_b_b] :
( Xs2
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( ( member_a @ X @ ( set_a2 @ R3 ) )
=> ( ( iKKBZ_6987179986356532253ts_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ X )
= bot_bot_set_a ) )
& ( ~ ( member_a @ X @ ( set_a2 @ R3 ) )
=> ( ( iKKBZ_6987179986356532253ts_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ X )
= ( set_a2 @ R3 ) ) ) ) ) ).
% path_lverts.simps(2)
thf(fact_997_sublist__not__mid,axiom,
! [U2: list_b,A2: list_b,V2: list_b,B2: list_b] :
( ( sublist_b @ U2 @ ( append_b @ ( append_b @ A2 @ V2 ) @ B2 ) )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ U2 ) @ ( set_b2 @ V2 ) )
= bot_bot_set_b )
=> ( ( V2 != nil_b )
=> ( ( sublist_b @ U2 @ A2 )
| ( sublist_b @ U2 @ B2 ) ) ) ) ) ).
% sublist_not_mid
thf(fact_998_sublist__not__mid,axiom,
! [U2: list_a,A2: list_a,V2: list_a,B2: list_a] :
( ( sublist_a @ U2 @ ( append_a @ ( append_a @ A2 @ V2 ) @ B2 ) )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ V2 ) )
= bot_bot_set_a )
=> ( ( V2 != nil_a )
=> ( ( sublist_a @ U2 @ A2 )
| ( sublist_a @ U2 @ B2 ) ) ) ) ) ).
% sublist_not_mid
thf(fact_999_sublist__not__mid,axiom,
! [U2: list_list_a,A2: list_list_a,V2: list_list_a,B2: list_list_a] :
( ( sublist_list_a @ U2 @ ( append_list_a @ ( append_list_a @ A2 @ V2 ) @ B2 ) )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ U2 ) @ ( set_list_a2 @ V2 ) )
= bot_bot_set_list_a )
=> ( ( V2 != nil_list_a )
=> ( ( sublist_list_a @ U2 @ A2 )
| ( sublist_list_a @ U2 @ B2 ) ) ) ) ) ).
% sublist_not_mid
thf(fact_1000_sublist__Y__cases__UV,axiom,
! [Y6: set_list_b,U2: list_b,V2: list_b,As: list_b,Bs: list_b,Cs: list_b,Xs2: list_b] :
( ! [X2: list_b] :
( ( member_list_b @ X2 @ Y6 )
=> ! [Xa: list_b] :
( ( member_list_b @ Xa @ Y6 )
=> ( ( X2 = Xa )
| ( ( inf_inf_set_b @ ( set_b2 @ X2 ) @ ( set_b2 @ Xa ) )
= bot_bot_set_b ) ) ) )
=> ( ( member_list_b @ U2 @ Y6 )
=> ( ( member_list_b @ V2 @ Y6 )
=> ( ( U2 != nil_b )
=> ( ( V2 != nil_b )
=> ( ! [X2: list_b] :
( ( member_list_b @ X2 @ Y6 )
=> ( sublist_b @ X2 @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V2 @ Cs ) ) ) ) ) )
=> ( ( member_list_b @ Xs2 @ Y6 )
=> ( ( sublist_b @ Xs2 @ As )
| ( sublist_b @ Xs2 @ Bs )
| ( sublist_b @ Xs2 @ Cs )
| ( U2 = Xs2 )
| ( V2 = Xs2 ) ) ) ) ) ) ) ) ) ).
% sublist_Y_cases_UV
thf(fact_1001_sublist__Y__cases__UV,axiom,
! [Y6: set_list_a,U2: list_a,V2: list_a,As: list_a,Bs: list_a,Cs: list_a,Xs2: list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ Y6 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ Y6 )
=> ( ( X2 = Xa )
| ( ( inf_inf_set_a @ ( set_a2 @ X2 ) @ ( set_a2 @ Xa ) )
= bot_bot_set_a ) ) ) )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( member_list_a @ V2 @ Y6 )
=> ( ( U2 != nil_a )
=> ( ( V2 != nil_a )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ Y6 )
=> ( sublist_a @ X2 @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) ) )
=> ( ( member_list_a @ Xs2 @ Y6 )
=> ( ( sublist_a @ Xs2 @ As )
| ( sublist_a @ Xs2 @ Bs )
| ( sublist_a @ Xs2 @ Cs )
| ( U2 = Xs2 )
| ( V2 = Xs2 ) ) ) ) ) ) ) ) ) ).
% sublist_Y_cases_UV
thf(fact_1002_sublist__Y__cases__UV,axiom,
! [Y6: set_list_list_a,U2: list_list_a,V2: list_list_a,As: list_list_a,Bs: list_list_a,Cs: list_list_a,Xs2: list_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ Y6 )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ Y6 )
=> ( ( X2 = Xa )
| ( ( inf_inf_set_list_a @ ( set_list_a2 @ X2 ) @ ( set_list_a2 @ Xa ) )
= bot_bot_set_list_a ) ) ) )
=> ( ( member_list_list_a @ U2 @ Y6 )
=> ( ( member_list_list_a @ V2 @ Y6 )
=> ( ( U2 != nil_list_a )
=> ( ( V2 != nil_list_a )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ Y6 )
=> ( sublist_list_a @ X2 @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V2 @ Cs ) ) ) ) ) )
=> ( ( member_list_list_a @ Xs2 @ Y6 )
=> ( ( sublist_list_a @ Xs2 @ As )
| ( sublist_list_a @ Xs2 @ Bs )
| ( sublist_list_a @ Xs2 @ Cs )
| ( U2 = Xs2 )
| ( V2 = Xs2 ) ) ) ) ) ) ) ) ) ).
% sublist_Y_cases_UV
thf(fact_1003_sublist__before__if__mid,axiom,
! [U2: list_b,A2: list_b,V2: list_b,B2: list_b,Xs2: list_b] :
( ( sublist_b @ U2 @ ( append_b @ A2 @ V2 ) )
=> ( ( ( append_b @ A2 @ ( append_b @ V2 @ B2 ) )
= Xs2 )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ U2 ) @ ( set_b2 @ V2 ) )
= bot_bot_set_b )
=> ( ( U2 != nil_b )
=> ? [As3: list_b,Bs2: list_b,Cs3: list_b] :
( ( append_b @ As3 @ ( append_b @ U2 @ ( append_b @ Bs2 @ ( append_b @ V2 @ Cs3 ) ) ) )
= Xs2 ) ) ) ) ) ).
% sublist_before_if_mid
thf(fact_1004_sublist__before__if__mid,axiom,
! [U2: list_a,A2: list_a,V2: list_a,B2: list_a,Xs2: list_a] :
( ( sublist_a @ U2 @ ( append_a @ A2 @ V2 ) )
=> ( ( ( append_a @ A2 @ ( append_a @ V2 @ B2 ) )
= Xs2 )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ U2 ) @ ( set_a2 @ V2 ) )
= bot_bot_set_a )
=> ( ( U2 != nil_a )
=> ? [As3: list_a,Bs2: list_a,Cs3: list_a] :
( ( append_a @ As3 @ ( append_a @ U2 @ ( append_a @ Bs2 @ ( append_a @ V2 @ Cs3 ) ) ) )
= Xs2 ) ) ) ) ) ).
% sublist_before_if_mid
thf(fact_1005_sublist__before__if__mid,axiom,
! [U2: list_list_a,A2: list_list_a,V2: list_list_a,B2: list_list_a,Xs2: list_list_a] :
( ( sublist_list_a @ U2 @ ( append_list_a @ A2 @ V2 ) )
=> ( ( ( append_list_a @ A2 @ ( append_list_a @ V2 @ B2 ) )
= Xs2 )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ U2 ) @ ( set_list_a2 @ V2 ) )
= bot_bot_set_list_a )
=> ( ( U2 != nil_list_a )
=> ? [As3: list_list_a,Bs2: list_list_a,Cs3: list_list_a] :
( ( append_list_a @ As3 @ ( append_list_a @ U2 @ ( append_list_a @ Bs2 @ ( append_list_a @ V2 @ Cs3 ) ) ) )
= Xs2 ) ) ) ) ) ).
% sublist_before_if_mid
thf(fact_1006_sublists__preserv__move__U,axiom,
! [Xs2: list_b,U2: list_b,V2: list_b,As: list_b,Bs: list_b,Cs: list_b] :
( ( ( inf_inf_set_b @ ( set_b2 @ Xs2 ) @ ( set_b2 @ U2 ) )
= bot_bot_set_b )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ Xs2 ) @ ( set_b2 @ V2 ) )
= bot_bot_set_b )
=> ( ( V2 != nil_b )
=> ( ( sublist_b @ Xs2 @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V2 @ Cs ) ) ) ) )
=> ( sublist_b @ Xs2 @ ( append_b @ As @ ( append_b @ Bs @ ( append_b @ U2 @ ( append_b @ V2 @ Cs ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_U
thf(fact_1007_sublists__preserv__move__U,axiom,
! [Xs2: list_a,U2: list_a,V2: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ U2 ) )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ V2 ) )
= bot_bot_set_a )
=> ( ( V2 != nil_a )
=> ( ( sublist_a @ Xs2 @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) )
=> ( sublist_a @ Xs2 @ ( append_a @ As @ ( append_a @ Bs @ ( append_a @ U2 @ ( append_a @ V2 @ Cs ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_U
thf(fact_1008_sublists__preserv__move__U,axiom,
! [Xs2: list_list_a,U2: list_list_a,V2: list_list_a,As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs2 ) @ ( set_list_a2 @ U2 ) )
= bot_bot_set_list_a )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs2 ) @ ( set_list_a2 @ V2 ) )
= bot_bot_set_list_a )
=> ( ( V2 != nil_list_a )
=> ( ( sublist_list_a @ Xs2 @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V2 @ Cs ) ) ) ) )
=> ( sublist_list_a @ Xs2 @ ( append_list_a @ As @ ( append_list_a @ Bs @ ( append_list_a @ U2 @ ( append_list_a @ V2 @ Cs ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_U
thf(fact_1009_sublists__preserv__move__V,axiom,
! [Xs2: list_b,U2: list_b,V2: list_b,As: list_b,Bs: list_b,Cs: list_b] :
( ( ( inf_inf_set_b @ ( set_b2 @ Xs2 ) @ ( set_b2 @ U2 ) )
= bot_bot_set_b )
=> ( ( ( inf_inf_set_b @ ( set_b2 @ Xs2 ) @ ( set_b2 @ V2 ) )
= bot_bot_set_b )
=> ( ( U2 != nil_b )
=> ( ( sublist_b @ Xs2 @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V2 @ Cs ) ) ) ) )
=> ( sublist_b @ Xs2 @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ V2 @ ( append_b @ Bs @ Cs ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_V
thf(fact_1010_sublists__preserv__move__V,axiom,
! [Xs2: list_a,U2: list_a,V2: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ U2 ) )
= bot_bot_set_a )
=> ( ( ( inf_inf_set_a @ ( set_a2 @ Xs2 ) @ ( set_a2 @ V2 ) )
= bot_bot_set_a )
=> ( ( U2 != nil_a )
=> ( ( sublist_a @ Xs2 @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) )
=> ( sublist_a @ Xs2 @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ V2 @ ( append_a @ Bs @ Cs ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_V
thf(fact_1011_sublists__preserv__move__V,axiom,
! [Xs2: list_list_a,U2: list_list_a,V2: list_list_a,As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs2 ) @ ( set_list_a2 @ U2 ) )
= bot_bot_set_list_a )
=> ( ( ( inf_inf_set_list_a @ ( set_list_a2 @ Xs2 ) @ ( set_list_a2 @ V2 ) )
= bot_bot_set_list_a )
=> ( ( U2 != nil_list_a )
=> ( ( sublist_list_a @ Xs2 @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V2 @ Cs ) ) ) ) )
=> ( sublist_list_a @ Xs2 @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ V2 @ ( append_list_a @ Bs @ Cs ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_V
thf(fact_1012_sublists__preserv__move__UY,axiom,
! [Y6: set_list_b,Xs2: list_b,U2: list_b,V2: list_b,As: list_b,Bs: list_b,Cs: list_b] :
( ! [X2: list_b] :
( ( member_list_b @ X2 @ Y6 )
=> ! [Xa: list_b] :
( ( member_list_b @ Xa @ Y6 )
=> ( ( X2 = Xa )
| ( ( inf_inf_set_b @ ( set_b2 @ X2 ) @ ( set_b2 @ Xa ) )
= bot_bot_set_b ) ) ) )
=> ( ( member_list_b @ Xs2 @ Y6 )
=> ( ( member_list_b @ U2 @ Y6 )
=> ( ( member_list_b @ V2 @ Y6 )
=> ( ( V2 != nil_b )
=> ( ( sublist_b @ Xs2 @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V2 @ Cs ) ) ) ) )
=> ( sublist_b @ Xs2 @ ( append_b @ As @ ( append_b @ Bs @ ( append_b @ U2 @ ( append_b @ V2 @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_UY
thf(fact_1013_sublists__preserv__move__UY,axiom,
! [Y6: set_list_a,Xs2: list_a,U2: list_a,V2: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ Y6 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ Y6 )
=> ( ( X2 = Xa )
| ( ( inf_inf_set_a @ ( set_a2 @ X2 ) @ ( set_a2 @ Xa ) )
= bot_bot_set_a ) ) ) )
=> ( ( member_list_a @ Xs2 @ Y6 )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( member_list_a @ V2 @ Y6 )
=> ( ( V2 != nil_a )
=> ( ( sublist_a @ Xs2 @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) )
=> ( sublist_a @ Xs2 @ ( append_a @ As @ ( append_a @ Bs @ ( append_a @ U2 @ ( append_a @ V2 @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_UY
thf(fact_1014_sublists__preserv__move__UY,axiom,
! [Y6: set_list_list_a,Xs2: list_list_a,U2: list_list_a,V2: list_list_a,As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ Y6 )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ Y6 )
=> ( ( X2 = Xa )
| ( ( inf_inf_set_list_a @ ( set_list_a2 @ X2 ) @ ( set_list_a2 @ Xa ) )
= bot_bot_set_list_a ) ) ) )
=> ( ( member_list_list_a @ Xs2 @ Y6 )
=> ( ( member_list_list_a @ U2 @ Y6 )
=> ( ( member_list_list_a @ V2 @ Y6 )
=> ( ( V2 != nil_list_a )
=> ( ( sublist_list_a @ Xs2 @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V2 @ Cs ) ) ) ) )
=> ( sublist_list_a @ Xs2 @ ( append_list_a @ As @ ( append_list_a @ Bs @ ( append_list_a @ U2 @ ( append_list_a @ V2 @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_UY
thf(fact_1015_sublists__preserv__move__VY,axiom,
! [Y6: set_list_b,Xs2: list_b,U2: list_b,V2: list_b,As: list_b,Bs: list_b,Cs: list_b] :
( ! [X2: list_b] :
( ( member_list_b @ X2 @ Y6 )
=> ! [Xa: list_b] :
( ( member_list_b @ Xa @ Y6 )
=> ( ( X2 = Xa )
| ( ( inf_inf_set_b @ ( set_b2 @ X2 ) @ ( set_b2 @ Xa ) )
= bot_bot_set_b ) ) ) )
=> ( ( member_list_b @ Xs2 @ Y6 )
=> ( ( member_list_b @ U2 @ Y6 )
=> ( ( member_list_b @ V2 @ Y6 )
=> ( ( U2 != nil_b )
=> ( ( sublist_b @ Xs2 @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V2 @ Cs ) ) ) ) )
=> ( sublist_b @ Xs2 @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ V2 @ ( append_b @ Bs @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_VY
thf(fact_1016_sublists__preserv__move__VY,axiom,
! [Y6: set_list_a,Xs2: list_a,U2: list_a,V2: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ Y6 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ Y6 )
=> ( ( X2 = Xa )
| ( ( inf_inf_set_a @ ( set_a2 @ X2 ) @ ( set_a2 @ Xa ) )
= bot_bot_set_a ) ) ) )
=> ( ( member_list_a @ Xs2 @ Y6 )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( member_list_a @ V2 @ Y6 )
=> ( ( U2 != nil_a )
=> ( ( sublist_a @ Xs2 @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) )
=> ( sublist_a @ Xs2 @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ V2 @ ( append_a @ Bs @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_VY
thf(fact_1017_sublists__preserv__move__VY,axiom,
! [Y6: set_list_list_a,Xs2: list_list_a,U2: list_list_a,V2: list_list_a,As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ Y6 )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ Y6 )
=> ( ( X2 = Xa )
| ( ( inf_inf_set_list_a @ ( set_list_a2 @ X2 ) @ ( set_list_a2 @ Xa ) )
= bot_bot_set_list_a ) ) ) )
=> ( ( member_list_list_a @ Xs2 @ Y6 )
=> ( ( member_list_list_a @ U2 @ Y6 )
=> ( ( member_list_list_a @ V2 @ Y6 )
=> ( ( U2 != nil_list_a )
=> ( ( sublist_list_a @ Xs2 @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V2 @ Cs ) ) ) ) )
=> ( sublist_list_a @ Xs2 @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ V2 @ ( append_list_a @ Bs @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_VY
thf(fact_1018_sublists__preserv__move__UY__all,axiom,
! [Y6: set_list_b,U2: list_b,V2: list_b,As: list_b,Bs: list_b,Cs: list_b] :
( ! [X2: list_b] :
( ( member_list_b @ X2 @ Y6 )
=> ! [Xa: list_b] :
( ( member_list_b @ Xa @ Y6 )
=> ( ( X2 = Xa )
| ( ( inf_inf_set_b @ ( set_b2 @ X2 ) @ ( set_b2 @ Xa ) )
= bot_bot_set_b ) ) ) )
=> ( ( member_list_b @ U2 @ Y6 )
=> ( ( member_list_b @ V2 @ Y6 )
=> ( ( V2 != nil_b )
=> ( ! [X2: list_b] :
( ( member_list_b @ X2 @ Y6 )
=> ( sublist_b @ X2 @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V2 @ Cs ) ) ) ) ) )
=> ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ( sublist_b @ X4 @ ( append_b @ As @ ( append_b @ Bs @ ( append_b @ U2 @ ( append_b @ V2 @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_UY_all
thf(fact_1019_sublists__preserv__move__UY__all,axiom,
! [Y6: set_list_a,U2: list_a,V2: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ Y6 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ Y6 )
=> ( ( X2 = Xa )
| ( ( inf_inf_set_a @ ( set_a2 @ X2 ) @ ( set_a2 @ Xa ) )
= bot_bot_set_a ) ) ) )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( member_list_a @ V2 @ Y6 )
=> ( ( V2 != nil_a )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ Y6 )
=> ( sublist_a @ X2 @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ( sublist_a @ X4 @ ( append_a @ As @ ( append_a @ Bs @ ( append_a @ U2 @ ( append_a @ V2 @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_UY_all
thf(fact_1020_sublists__preserv__move__UY__all,axiom,
! [Y6: set_list_list_a,U2: list_list_a,V2: list_list_a,As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ Y6 )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ Y6 )
=> ( ( X2 = Xa )
| ( ( inf_inf_set_list_a @ ( set_list_a2 @ X2 ) @ ( set_list_a2 @ Xa ) )
= bot_bot_set_list_a ) ) ) )
=> ( ( member_list_list_a @ U2 @ Y6 )
=> ( ( member_list_list_a @ V2 @ Y6 )
=> ( ( V2 != nil_list_a )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ Y6 )
=> ( sublist_list_a @ X2 @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V2 @ Cs ) ) ) ) ) )
=> ! [X4: list_list_a] :
( ( member_list_list_a @ X4 @ Y6 )
=> ( sublist_list_a @ X4 @ ( append_list_a @ As @ ( append_list_a @ Bs @ ( append_list_a @ U2 @ ( append_list_a @ V2 @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_UY_all
thf(fact_1021_sublists__preserv__move__VY__all,axiom,
! [Y6: set_list_b,U2: list_b,V2: list_b,As: list_b,Bs: list_b,Cs: list_b] :
( ! [X2: list_b] :
( ( member_list_b @ X2 @ Y6 )
=> ! [Xa: list_b] :
( ( member_list_b @ Xa @ Y6 )
=> ( ( X2 = Xa )
| ( ( inf_inf_set_b @ ( set_b2 @ X2 ) @ ( set_b2 @ Xa ) )
= bot_bot_set_b ) ) ) )
=> ( ( member_list_b @ U2 @ Y6 )
=> ( ( member_list_b @ V2 @ Y6 )
=> ( ( U2 != nil_b )
=> ( ! [X2: list_b] :
( ( member_list_b @ X2 @ Y6 )
=> ( sublist_b @ X2 @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ Bs @ ( append_b @ V2 @ Cs ) ) ) ) ) )
=> ! [X4: list_b] :
( ( member_list_b @ X4 @ Y6 )
=> ( sublist_b @ X4 @ ( append_b @ As @ ( append_b @ U2 @ ( append_b @ V2 @ ( append_b @ Bs @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_VY_all
thf(fact_1022_sublists__preserv__move__VY__all,axiom,
! [Y6: set_list_a,U2: list_a,V2: list_a,As: list_a,Bs: list_a,Cs: list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ Y6 )
=> ! [Xa: list_a] :
( ( member_list_a @ Xa @ Y6 )
=> ( ( X2 = Xa )
| ( ( inf_inf_set_a @ ( set_a2 @ X2 ) @ ( set_a2 @ Xa ) )
= bot_bot_set_a ) ) ) )
=> ( ( member_list_a @ U2 @ Y6 )
=> ( ( member_list_a @ V2 @ Y6 )
=> ( ( U2 != nil_a )
=> ( ! [X2: list_a] :
( ( member_list_a @ X2 @ Y6 )
=> ( sublist_a @ X2 @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ Bs @ ( append_a @ V2 @ Cs ) ) ) ) ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ Y6 )
=> ( sublist_a @ X4 @ ( append_a @ As @ ( append_a @ U2 @ ( append_a @ V2 @ ( append_a @ Bs @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_VY_all
thf(fact_1023_sublists__preserv__move__VY__all,axiom,
! [Y6: set_list_list_a,U2: list_list_a,V2: list_list_a,As: list_list_a,Bs: list_list_a,Cs: list_list_a] :
( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ Y6 )
=> ! [Xa: list_list_a] :
( ( member_list_list_a @ Xa @ Y6 )
=> ( ( X2 = Xa )
| ( ( inf_inf_set_list_a @ ( set_list_a2 @ X2 ) @ ( set_list_a2 @ Xa ) )
= bot_bot_set_list_a ) ) ) )
=> ( ( member_list_list_a @ U2 @ Y6 )
=> ( ( member_list_list_a @ V2 @ Y6 )
=> ( ( U2 != nil_list_a )
=> ( ! [X2: list_list_a] :
( ( member_list_list_a @ X2 @ Y6 )
=> ( sublist_list_a @ X2 @ ( append_list_a @ As @ ( append_list_a @ U2 @ ( append_list_a @ Bs @ ( append_list_a @ V2 @ Cs ) ) ) ) ) )
=> ! [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 @ V2 @ ( append_list_a @ Bs @ Cs ) ) ) ) ) ) ) ) ) ) ) ).
% sublists_preserv_move_VY_all
thf(fact_1024_single__subtree__child__root__dverts,axiom,
! [V22: list_a,T23: dtree_list_a_b,E22: b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V22 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T1 )
=> ( member_list_a @ ( root_list_a_b @ T23 ) @ ( dverts_list_a_b @ T1 ) ) ) ).
% single_subtree_child_root_dverts
thf(fact_1025_subset__snd__if__hd__notin__fst,axiom,
! [Ys: list_dtree_list_a_b,V2: list_dtree_list_a_b,B2: list_dtree_list_a_b] :
( ( sublis1540088274494349249st_a_b @ Ys @ ( append2129087805494049561st_a_b @ V2 @ B2 ) )
=> ( ~ ( member551035911493665803st_a_b @ ( hd_dtree_list_a_b @ Ys ) @ ( set_dtree_list_a_b2 @ V2 ) )
=> ( ( Ys != nil_dtree_list_a_b )
=> ( ord_le7599451563663638410st_a_b @ ( set_dtree_list_a_b2 @ Ys ) @ ( set_dtree_list_a_b2 @ B2 ) ) ) ) ) ).
% subset_snd_if_hd_notin_fst
thf(fact_1026_subset__snd__if__hd__notin__fst,axiom,
! [Ys: list_set_a,V2: list_set_a,B2: list_set_a] :
( ( sublist_set_a @ Ys @ ( append_set_a @ V2 @ B2 ) )
=> ( ~ ( member_set_a @ ( hd_set_a @ Ys ) @ ( set_set_a2 @ V2 ) )
=> ( ( Ys != nil_set_a )
=> ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Ys ) @ ( set_set_a2 @ B2 ) ) ) ) ) ).
% subset_snd_if_hd_notin_fst
thf(fact_1027_subset__snd__if__hd__notin__fst,axiom,
! [Ys: list_list_a,V2: list_list_a,B2: list_list_a] :
( ( sublist_list_a @ Ys @ ( append_list_a @ V2 @ B2 ) )
=> ( ~ ( member_list_a @ ( hd_list_a @ Ys ) @ ( set_list_a2 @ V2 ) )
=> ( ( Ys != nil_list_a )
=> ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Ys ) @ ( set_list_a2 @ B2 ) ) ) ) ) ).
% subset_snd_if_hd_notin_fst
thf(fact_1028_subset__snd__if__hd__notin__fst,axiom,
! [Ys: list_a,V2: list_a,B2: list_a] :
( ( sublist_a @ Ys @ ( append_a @ V2 @ B2 ) )
=> ( ~ ( member_a @ ( hd_a @ Ys ) @ ( set_a2 @ V2 ) )
=> ( ( Ys != nil_a )
=> ( ord_less_eq_set_a @ ( set_a2 @ Ys ) @ ( set_a2 @ B2 ) ) ) ) ) ).
% subset_snd_if_hd_notin_fst
thf(fact_1029_subset__snd__if__hd__notin__fst,axiom,
! [Ys: list_b,V2: list_b,B2: list_b] :
( ( sublist_b @ Ys @ ( append_b @ V2 @ B2 ) )
=> ( ~ ( member_b @ ( hd_b @ Ys ) @ ( set_b2 @ V2 ) )
=> ( ( Ys != nil_b )
=> ( ord_less_eq_set_b @ ( set_b2 @ Ys ) @ ( set_b2 @ B2 ) ) ) ) ) ).
% subset_snd_if_hd_notin_fst
thf(fact_1030_path__lverts_Osimps_I1_J,axiom,
! [X: a,R3: list_a,T: dtree_list_a_b,E: b] :
( ( ( member_a @ X @ ( set_a2 @ R3 ) )
=> ( ( iKKBZ_6987179986356532253ts_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) @ X )
= bot_bot_set_a ) )
& ( ~ ( member_a @ X @ ( set_a2 @ R3 ) )
=> ( ( iKKBZ_6987179986356532253ts_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T @ E ) @ bot_bo2248824169281960260_a_b_b ) ) @ X )
= ( sup_sup_set_a @ ( set_a2 @ R3 ) @ ( iKKBZ_6987179986356532253ts_a_b @ T @ X ) ) ) ) ) ).
% path_lverts.simps(1)
thf(fact_1031_path__lverts_Oelims,axiom,
! [X: dtree_list_a_b,Xa3: a,Y: set_a] :
( ( ( iKKBZ_6987179986356532253ts_a_b @ X @ Xa3 )
= Y )
=> ( ! [R: list_a,T4: dtree_list_a_b] :
( ? [E2: b] :
( X
= ( node_list_a_b @ R @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T4 @ E2 ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ~ ( ( ( member_a @ Xa3 @ ( set_a2 @ R ) )
=> ( Y = bot_bot_set_a ) )
& ( ~ ( member_a @ Xa3 @ ( set_a2 @ R ) )
=> ( Y
= ( sup_sup_set_a @ ( set_a2 @ R ) @ ( iKKBZ_6987179986356532253ts_a_b @ T4 @ Xa3 ) ) ) ) ) )
=> ~ ! [Xs: fset_P2153231429829016240_a_b_b] :
( ! [X4: produc6499617310964463488_a_b_b] :
( Xs
!= ( finser2303212525150181944_a_b_b @ X4 @ bot_bo2248824169281960260_a_b_b ) )
=> ! [R: list_a] :
( ( X
= ( node_list_a_b @ R @ Xs ) )
=> ~ ( ( ( member_a @ Xa3 @ ( set_a2 @ R ) )
=> ( Y = bot_bot_set_a ) )
& ( ~ ( member_a @ Xa3 @ ( set_a2 @ R ) )
=> ( Y
= ( set_a2 @ R ) ) ) ) ) ) ) ) ).
% path_lverts.elims
thf(fact_1032_path__lverts__simps1__sucs,axiom,
! [X: a,T1: dtree_list_a_b,T23: dtree_list_a_b,E22: b] :
( ~ ( member_a @ X @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
=> ( ( ( sucs_list_a_b @ T1 )
= ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T23 @ E22 ) @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( sup_sup_set_a @ ( set_a2 @ ( root_list_a_b @ T1 ) ) @ ( iKKBZ_6987179986356532253ts_a_b @ T23 @ X ) )
= ( iKKBZ_6987179986356532253ts_a_b @ T1 @ X ) ) ) ) ).
% path_lverts_simps1_sucs
thf(fact_1033_R_Odverts__reach1__in__dlverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a] :
( ( is_subtree_list_a_b @ T1 @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( 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 ) ) ) ) ) ) ).
% R.dverts_reach1_in_dlverts
thf(fact_1034_R_Overts__distinct,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( distinct_a @ V ) ) ).
% R.verts_distinct
thf(fact_1035_dom,axiom,
iKKBZ_3908525916494739553en_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ t ).
% dom
thf(fact_1036_R_Odverts__same__if__set__subtree,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,V22: list_a] :
( ( is_subtree_list_a_b @ T1 @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V22 ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( V1 = V22 ) ) ) ) ) ) ).
% R.dverts_same_if_set_subtree
thf(fact_1037_sub__t,axiom,
is_subtree_list_a_b @ ( node_list_a_b @ ( root_list_a_b @ ta ) @ ( sucs_list_a_b @ ta ) ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ).
% sub_t
thf(fact_1038_R_Overts__distinct__subtree,axiom,
! [Tn: dtree_list_a_b,V: list_a] :
( ( is_subtree_list_a_b @ Tn @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ Tn ) )
=> ( distinct_a @ V ) ) ) ).
% R.verts_distinct_subtree
thf(fact_1039_R_Overts__forward,axiom,
! [X4: list_a] :
( ( member_list_a @ X4 @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ X4 ) ) ).
% R.verts_forward
thf(fact_1040_R_Overts__conform,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ V ) ) ).
% R.verts_conform
thf(fact_1041_R_Odistint__verts__singleton__subtree,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( distinct_a @ ( append_a @ R3 @ ( root_list_a_b @ T1 ) ) ) ) ).
% R.distint_verts_singleton_subtree
thf(fact_1042_R_Odlverts__reach__in__dlverts,axiom,
! [X: a,Y: a,T1: dtree_list_a_b] :
( ( reachable_a_b @ t @ X @ Y )
=> ( ( is_subtree_list_a_b @ T1 @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_a @ X @ ( list_dlverts_a_b @ T1 ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ).
% R.dlverts_reach_in_dlverts
thf(fact_1043_R_Overts__conform__subtree,axiom,
! [Tn: dtree_list_a_b,V: list_a] :
( ( is_subtree_list_a_b @ Tn @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ Tn ) )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ V ) ) ) ).
% R.verts_conform_subtree
thf(fact_1044_R_Odverts__reach__in__dverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a,V22: list_a] :
( ( is_subtree_list_a_b @ T1 @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( reachable_a_b @ t @ X @ Y )
=> ( ( member_a @ Y @ ( set_a2 @ V22 ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( member_list_a @ V22 @ ( dverts_list_a_b @ T1 ) ) ) ) ) ) ) ) ).
% R.dverts_reach_in_dverts
thf(fact_1045_R_Odlverts__arc__in__dlverts,axiom,
! [T1: dtree_list_a_b,X: a,Y: a] :
( ( is_subtree_list_a_b @ T1 @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ X @ ( list_dlverts_a_b @ T1 ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ).
% R.dlverts_arc_in_dlverts
thf(fact_1046_R_Oarc__to__dverts__in__subtree,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: a,Y: a,V: list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_a @ X @ ( set_a2 @ R3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ Y @ ( set_a2 @ V ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) ) ) ) ) ) ) ) ).
% R.arc_to_dverts_in_subtree
thf(fact_1047_R_Odverts__arc__in__dverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a,V22: list_a] :
( ( is_subtree_list_a_b @ T1 @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ Y @ ( set_a2 @ V22 ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( member_list_a @ V22 @ ( dverts_list_a_b @ T1 ) ) ) ) ) ) ) ) ).
% R.dverts_arc_in_dverts
thf(fact_1048_R_Oarc__in__dlverts,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: a,Y: a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_a @ X @ ( set_a2 @ R3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ ( node_list_a_b @ R3 @ Xs2 ) ) ) ) ) ) ).
% R.arc_in_dlverts
thf(fact_1049_R_Oarc__in__dlverts__subtree,axiom,
! [Tn: dtree_list_a_b,R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: a,Y: a] :
( ( is_subtree_list_a_b @ Tn @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ Tn )
=> ( ( member_a @ X @ ( set_a2 @ R3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ ( node_list_a_b @ R3 @ Xs2 ) ) ) ) ) ) ) ).
% R.arc_in_dlverts_subtree
thf(fact_1050_R_Odverts__reach__in__dlverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a] :
( ( is_subtree_list_a_b @ T1 @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( reachable_a_b @ t @ X @ Y )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ) ).
% R.dverts_reach_in_dlverts
thf(fact_1051_R_Odlverts__reach1__in__dlverts,axiom,
! [X: a,Y: a,T1: dtree_list_a_b] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( is_subtree_list_a_b @ T1 @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_a @ X @ ( list_dlverts_a_b @ T1 ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ).
% R.dlverts_reach1_in_dlverts
thf(fact_1052_R_Odverts__arc__in__dlverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a] :
( ( is_subtree_list_a_b @ T1 @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ) ).
% R.dverts_arc_in_dlverts
thf(fact_1053_R_Odverts__reach1__in__dverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a,V22: list_a] :
( ( is_subtree_list_a_b @ T1 @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( member_a @ Y @ ( set_a2 @ V22 ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( member_list_a @ V22 @ ( dverts_list_a_b @ T1 ) ) ) ) ) ) ) ) ).
% R.dverts_reach1_in_dverts
thf(fact_1054_R_Odverts__reach1__in__dverts__r,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,V: list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ R3 ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) ) ) ) ) ) ).
% R.dverts_reach1_in_dverts_r
thf(fact_1055_R_Odverts__reach1__in__dverts__root,axiom,
! [T1: dtree_list_a_b,V: list_a] :
( ( is_subtree_list_a_b @ T1 @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) ) ) ) ) ).
% R.dverts_reach1_in_dverts_root
thf(fact_1056_R_Ov__in__dlverts__if__in__comb,axiom,
! [V: a,X: list_a,Y: list_a] :
( ( member_a @ V @ ( list_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
=> ( member_a @ V @ ( list_dlverts_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).
% R.v_in_dlverts_if_in_comb
thf(fact_1057_R_Ov__in__comb__if__in__dlverts,axiom,
! [V: a,X: list_a,Y: list_a] :
( ( member_a @ V @ ( list_dlverts_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( member_a @ V @ ( list_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ) ).
% R.v_in_comb_if_in_dlverts
thf(fact_1058_R_Osubtree__root__not__root,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E: b] :
( ( ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) )
= ( node_list_a_b @ R3 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( root_list_a_b @ X )
!= R3 ) ) ) ).
% R.subtree_root_not_root
thf(fact_1059_T_Ov__in__comb__if__in__dlverts,axiom,
! [V: a,X: list_a,Y: list_a] :
( ( member_a @ V @ ( list_dlverts_a_b @ ta ) )
=> ( member_a @ V @ ( list_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ ta ) ) ) ) ).
% T.v_in_comb_if_in_dlverts
thf(fact_1060_T_Ov__in__dlverts__if__in__comb,axiom,
! [V: a,X: list_a,Y: list_a] :
( ( member_a @ V @ ( list_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ ta ) ) )
=> ( member_a @ V @ ( list_dlverts_a_b @ ta ) ) ) ).
% T.v_in_dlverts_if_in_comb
thf(fact_1061_T_Oarc__uneq__if__subtree__uneq,axiom,
! [X1: dtree_list_a_b,E1: b,Xs2: fset_P2153231429829016240_a_b_b,X22: dtree_list_a_b,E22: b,R3: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X1 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( X1 != X22 )
=> ( ( ( node_list_a_b @ R3 @ Xs2 )
= ta )
=> ( E1 != E22 ) ) ) ) ) ).
% T.arc_uneq_if_subtree_uneq
thf(fact_1062_T_Osubtree__uneq__if__arc__uneq,axiom,
! [X1: dtree_list_a_b,E1: b,Xs2: fset_P2153231429829016240_a_b_b,X22: dtree_list_a_b,E22: b,R3: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X1 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( E1 != E22 )
=> ( ( ( node_list_a_b @ R3 @ Xs2 )
= ta )
=> ( X1 != X22 ) ) ) ) ) ).
% T.subtree_uneq_if_arc_uneq
thf(fact_1063_T_Osubtree__root__not__root,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E: b] :
( ( ta
= ( node_list_a_b @ R3 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( root_list_a_b @ X )
!= R3 ) ) ) ).
% T.subtree_root_not_root
thf(fact_1064_R_Osubtree__uneq__if__arc__uneq,axiom,
! [X1: dtree_list_a_b,E1: b,Xs2: fset_P2153231429829016240_a_b_b,X22: dtree_list_a_b,E22: b,R3: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X1 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( E1 != E22 )
=> ( ( ( node_list_a_b @ R3 @ Xs2 )
= ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( X1 != X22 ) ) ) ) ) ).
% R.subtree_uneq_if_arc_uneq
thf(fact_1065_R_Oarc__uneq__if__subtree__uneq,axiom,
! [X1: dtree_list_a_b,E1: b,Xs2: fset_P2153231429829016240_a_b_b,X22: dtree_list_a_b,E22: b,R3: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X1 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( X1 != X22 )
=> ( ( ( node_list_a_b @ R3 @ Xs2 )
= ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( E1 != E22 ) ) ) ) ) ).
% R.arc_uneq_if_subtree_uneq
thf(fact_1066_T_Odlverts__comb__id,axiom,
! [X: list_a,Y: list_a] :
( ( list_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ ta ) )
= ( list_dlverts_a_b @ ta ) ) ).
% T.dlverts_comb_id
thf(fact_1067_R_Odlverts__comb__id,axiom,
! [X: list_a,Y: list_a] :
( ( list_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
= ( list_dlverts_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% R.dlverts_comb_id
thf(fact_1068_R_Oarc__in__dlverts__subtree_H,axiom,
! [Tn: dtree_list_a_b] :
( ( is_subtree_list_a_b @ Tn @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ! [R4: list_a,Xs7: fset_P2153231429829016240_a_b_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R4 @ Xs7 ) @ Tn )
=> ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ R4 ) )
=> ! [Y5: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y5 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ Y5 @ ( set_a2 @ R4 ) )
| ? [Xa: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Xa @ ( fset_P9138963618725001425_a_b_b @ Xs7 ) )
& ( member_a @ Y5 @ ( list_dlverts_a_b @ ( produc5948858871325780166_a_b_b @ Xa ) ) ) ) ) ) ) ) ) ).
% R.arc_in_dlverts_subtree'
thf(fact_1069_T_Oarc__in__dlverts__subtree_H,axiom,
! [Tn: dtree_list_a_b] :
( ( is_subtree_list_a_b @ Tn @ ta )
=> ! [R4: list_a,Xs7: fset_P2153231429829016240_a_b_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R4 @ Xs7 ) @ Tn )
=> ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ R4 ) )
=> ! [Y5: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y5 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ Y5 @ ( set_a2 @ R4 ) )
| ? [Xa: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Xa @ ( fset_P9138963618725001425_a_b_b @ Xs7 ) )
& ( member_a @ Y5 @ ( list_dlverts_a_b @ ( produc5948858871325780166_a_b_b @ Xa ) ) ) ) ) ) ) ) ) ).
% T.arc_in_dlverts_subtree'
thf(fact_1070_R_Ochild__disjoint__root,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member551035911493665803st_a_b @ T1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ( inf_inf_set_a @ ( set_a2 @ R3 ) @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
= bot_bot_set_a ) ) ) ).
% R.child_disjoint_root
thf(fact_1071_R_Odistint__verts__subtree,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member551035911493665803st_a_b @ T1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( distinct_a @ ( append_a @ R3 @ ( root_list_a_b @ T1 ) ) ) ) ) ).
% R.distint_verts_subtree
thf(fact_1072_R_Oroot__not__subtree,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
( ( ( node_list_a_b @ R3 @ Xs2 )
= ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ~ ( member_list_a @ R3 @ ( dverts_list_a_b @ X ) ) ) ) ).
% R.root_not_subtree
thf(fact_1073_T_Oroot__not__subtree,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
( ( ( node_list_a_b @ R3 @ Xs2 )
= ta )
=> ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ~ ( member_list_a @ R3 @ ( dverts_list_a_b @ X ) ) ) ) ).
% T.root_not_subtree
thf(fact_1074__C0_C,axiom,
member551035911493665803st_a_b @ t1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ ta ) ) ) ).
% "0"
thf(fact_1075_T_Odistint__verts__subtree,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ ta )
=> ( ( member551035911493665803st_a_b @ T1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( distinct_a @ ( append_a @ R3 @ ( root_list_a_b @ T1 ) ) ) ) ) ).
% T.distint_verts_subtree
thf(fact_1076_T_Ochild__disjoint__root,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ ta )
=> ( ( member551035911493665803st_a_b @ T1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ( inf_inf_set_a @ ( set_a2 @ R3 ) @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
= bot_bot_set_a ) ) ) ).
% T.child_disjoint_root
thf(fact_1077_R_Odverts__child__subset,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
( ( ( node_list_a_b @ R3 @ Xs2 )
= ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ord_less_set_list_a @ ( dverts_list_a_b @ X ) @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ) ).
% R.dverts_child_subset
thf(fact_1078_R_Ochild__arc__not__subtree,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E1: b] :
( ( ( node_list_a_b @ R3 @ Xs2 )
= ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ~ ( member_b @ E1 @ ( darcs_list_a_b @ X ) ) ) ) ).
% R.child_arc_not_subtree
thf(fact_1079_R_Olist__dtree__rec__suc,axiom,
! [X: dtree_list_a_b,E: b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
=> ( list_list_dtree_a_b @ X ) ) ).
% R.list_dtree_rec_suc
thf(fact_1080_T_Olist__dtree__axioms,axiom,
list_list_dtree_a_b @ ta ).
% T.list_dtree_axioms
thf(fact_1081_T_Olist__dtree__sub,axiom,
! [X: dtree_list_a_b] :
( ( is_subtree_list_a_b @ X @ ta )
=> ( list_list_dtree_a_b @ X ) ) ).
% T.list_dtree_sub
thf(fact_1082_T_Olist__dtree__comb,axiom,
! [X: list_a,Y: list_a] : ( list_list_dtree_a_b @ ( list_combine_a_b @ X @ Y @ ta ) ) ).
% T.list_dtree_comb
thf(fact_1083_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_1084_T_Ochild__arc__not__subtree,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E1: b] :
( ( ( node_list_a_b @ R3 @ Xs2 )
= ta )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ~ ( member_b @ E1 @ ( darcs_list_a_b @ X ) ) ) ) ).
% T.child_arc_not_subtree
thf(fact_1085_T_Olist__dtree__rec,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E: b] :
( ( ( node_list_a_b @ R3 @ Xs2 )
= ta )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( list_list_dtree_a_b @ X ) ) ) ).
% T.list_dtree_rec
thf(fact_1086_T_Olist__dtree__rec__suc,axiom,
! [X: dtree_list_a_b,E: b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ ta ) ) )
=> ( list_list_dtree_a_b @ X ) ) ).
% T.list_dtree_rec_suc
thf(fact_1087_T_Odverts__child__subset,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
( ( ( node_list_a_b @ R3 @ Xs2 )
= ta )
=> ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ord_less_set_list_a @ ( dverts_list_a_b @ X ) @ ( dverts_list_a_b @ ta ) ) ) ) ).
% T.dverts_child_subset
thf(fact_1088_R_Olist__dtree__axioms,axiom,
list_list_dtree_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ).
% R.list_dtree_axioms
thf(fact_1089_R_Olist__dtree__rec,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E: b] :
( ( ( node_list_a_b @ R3 @ Xs2 )
= ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( list_list_dtree_a_b @ X ) ) ) ).
% R.list_dtree_rec
thf(fact_1090_R_Olist__dtree__sub,axiom,
! [X: dtree_list_a_b] :
( ( is_subtree_list_a_b @ X @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( list_list_dtree_a_b @ X ) ) ).
% R.list_dtree_sub
thf(fact_1091_R_Olist__dtree__comb,axiom,
! [X: list_a,Y: list_a] : ( list_list_dtree_a_b @ ( list_combine_a_b @ X @ Y @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% R.list_dtree_comb
thf(fact_1092_R_Odarcs__child__subset,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
( ( ( node_list_a_b @ R3 @ Xs2 )
= ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ord_less_set_b @ ( darcs_list_a_b @ X ) @ ( darcs_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ) ).
% R.darcs_child_subset
thf(fact_1093_R_Odarc__in__sub__if__dtail__in__sub,axiom,
! [Dt: b > list_a,E: b,V: list_a,X: dtree_list_a_b,E1: b,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b,R3: list_a] :
( ( ( dtail_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Dt @ E )
= V )
=> ( ( member_b @ E @ ( darcs_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( is_subtree_list_a_b @ T1 @ X )
=> ( ( ( node_list_a_b @ R3 @ Xs2 )
= ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) )
=> ( member_b @ E @ ( darcs_list_a_b @ X ) ) ) ) ) ) ) ) ).
% R.darc_in_sub_if_dtail_in_sub
thf(fact_1094_T_Odtail__in__dverts,axiom,
! [E: b,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ ta ) )
=> ( member_list_a @ ( dtail_list_a_b @ ta @ Def2 @ E ) @ ( dverts_list_a_b @ ta ) ) ) ).
% T.dtail_in_dverts
thf(fact_1095_T_Odtail__in__subtree__eq__subtree,axiom,
! [T1: dtree_list_a_b,E: b,Def2: b > list_a] :
( ( is_subtree_list_a_b @ T1 @ ta )
=> ( ( member_b @ E @ ( darcs_list_a_b @ T1 ) )
=> ( ( dtail_list_a_b @ ta @ Def2 @ E )
= ( dtail_list_a_b @ T1 @ Def2 @ E ) ) ) ) ).
% T.dtail_in_subtree_eq_subtree
thf(fact_1096_T_Odtail__in__subdverts,axiom,
! [E: b,X: dtree_list_a_b,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ X ) )
=> ( ( is_subtree_list_a_b @ X @ ta )
=> ( member_list_a @ ( dtail_list_a_b @ ta @ Def2 @ E ) @ ( dverts_list_a_b @ X ) ) ) ) ).
% T.dtail_in_subdverts
thf(fact_1097_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_1098_T_Oarc__in__subtree__if__tail__in__subtree,axiom,
! [Dt: b > list_a,P: b,X: dtree_list_a_b,R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,E: b] :
( ( member_list_a @ ( dtail_list_a_b @ ta @ Dt @ P ) @ ( dverts_list_a_b @ X ) )
=> ( ( member_b @ P @ ( darcs_list_a_b @ ta ) )
=> ( ( ta
= ( node_list_a_b @ R3 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( member_b @ P @ ( darcs_list_a_b @ X ) ) ) ) ) ) ).
% T.arc_in_subtree_if_tail_in_subtree
thf(fact_1099_T_Odtail__in__childverts,axiom,
! [E: b,X: dtree_list_a_b,E3: b,Xs2: fset_P2153231429829016240_a_b_b,R3: list_a,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ X ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E3 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( ( node_list_a_b @ R3 @ Xs2 )
= ta )
=> ( member_list_a @ ( dtail_list_a_b @ ta @ Def2 @ E ) @ ( dverts_list_a_b @ X ) ) ) ) ) ).
% T.dtail_in_childverts
thf(fact_1100_T_Odarcs__child__subset,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
( ( ( node_list_a_b @ R3 @ Xs2 )
= ta )
=> ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ord_less_set_b @ ( darcs_list_a_b @ X ) @ ( darcs_list_a_b @ ta ) ) ) ) ).
% T.darcs_child_subset
thf(fact_1101_T_Odarc__in__sub__if__dtail__in__sub,axiom,
! [Dt: b > list_a,E: b,V: list_a,X: dtree_list_a_b,E1: b,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b,R3: list_a] :
( ( ( dtail_list_a_b @ ta @ Dt @ E )
= V )
=> ( ( member_b @ E @ ( darcs_list_a_b @ ta ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( is_subtree_list_a_b @ T1 @ X )
=> ( ( ( node_list_a_b @ R3 @ Xs2 )
= ta )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) )
=> ( member_b @ E @ ( darcs_list_a_b @ X ) ) ) ) ) ) ) ) ).
% T.darc_in_sub_if_dtail_in_sub
thf(fact_1102_R_Odtail__in__dverts,axiom,
! [E: b,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( member_list_a @ ( dtail_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Def2 @ E ) @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).
% R.dtail_in_dverts
thf(fact_1103_R_Odtail__in__subtree__eq__subtree,axiom,
! [T1: dtree_list_a_b,E: b,Def2: b > list_a] :
( ( is_subtree_list_a_b @ T1 @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_b @ E @ ( darcs_list_a_b @ T1 ) )
=> ( ( dtail_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Def2 @ E )
= ( dtail_list_a_b @ T1 @ Def2 @ E ) ) ) ) ).
% R.dtail_in_subtree_eq_subtree
thf(fact_1104_R_Oarc__in__subtree__if__tail__in__subtree,axiom,
! [Dt: b > list_a,P: b,X: dtree_list_a_b,R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,E: b] :
( ( member_list_a @ ( dtail_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Dt @ P ) @ ( dverts_list_a_b @ X ) )
=> ( ( member_b @ P @ ( darcs_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ( ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) )
= ( node_list_a_b @ R3 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( member_b @ P @ ( darcs_list_a_b @ X ) ) ) ) ) ) ).
% R.arc_in_subtree_if_tail_in_subtree
thf(fact_1105_R_Odtail__in__childverts,axiom,
! [E: b,X: dtree_list_a_b,E3: b,Xs2: fset_P2153231429829016240_a_b_b,R3: list_a,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ X ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E3 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( ( node_list_a_b @ R3 @ Xs2 )
= ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( member_list_a @ ( dtail_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Def2 @ E ) @ ( dverts_list_a_b @ X ) ) ) ) ) ).
% R.dtail_in_childverts
thf(fact_1106_R_Odtail__in__subdverts,axiom,
! [E: b,X: dtree_list_a_b,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ X ) )
=> ( ( is_subtree_list_a_b @ X @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( member_list_a @ ( dtail_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Def2 @ E ) @ ( dverts_list_a_b @ X ) ) ) ) ).
% R.dtail_in_subdverts
thf(fact_1107__C1_Oprems_C_I3_J,axiom,
? [V3: list_a,T22: dtree_list_a_b,E23: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ r @ xs ) )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T22 ) ) ) @ ( rank @ ( rev_a @ V3 ) ) ) ) ).
% "1.prems"(3)
thf(fact_1108_R_Oout__arcs__in__subarcs__aux,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,Dt: b > list_a,E: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( ( dtail_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Dt @ E )
= R3 )
=> ( ( member_b @ E @ ( darcs_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( member_b @ E @ ( image_4684437738885282872_b_b_b @ produc5719641485658034180_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) ) ) ) ) ).
% R.out_arcs_in_subarcs_aux
thf(fact_1109_True,axiom,
ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ ta ) ) ) @ ( rank @ ( rev_a @ r1 ) ) ).
% True
thf(fact_1110_T_Odtail__root__in__set,axiom,
! [E: b,R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,Dt: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ ta ) )
=> ( ( ta
= ( node_list_a_b @ R3 @ Xs2 ) )
=> ( ( ( dtail_list_a_b @ ta @ Dt @ E )
= R3 )
=> ( member_b @ E @ ( image_4684437738885282872_b_b_b @ produc5719641485658034180_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) ) ) ) ) ).
% T.dtail_root_in_set
thf(fact_1111_T_Oout__arcs__in__subarcs__aux,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,Dt: b > list_a,E: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ ta )
=> ( ( ( dtail_list_a_b @ ta @ Dt @ E )
= R3 )
=> ( ( member_b @ E @ ( darcs_list_a_b @ ta ) )
=> ( member_b @ E @ ( image_4684437738885282872_b_b_b @ produc5719641485658034180_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) ) ) ) ) ).
% T.out_arcs_in_subarcs_aux
thf(fact_1112__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062v_At2_Ae2_O_A_092_060lbrakk_062is__subtree_A_INode_Av_A_123_124_It2_M_Ae2_J_124_125_J_A_INode_Ar_Axs_J_059_Arank_A_Irev_A_Idtree_Oroot_At2_J_J_A_060_Arank_A_Irev_Av_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [V3: list_a,T22: dtree_list_a_b] :
( ? [E23: b] : ( is_subtree_list_a_b @ ( node_list_a_b @ V3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ r @ xs ) )
=> ~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T22 ) ) ) @ ( rank @ ( rev_a @ V3 ) ) ) ) ).
% \<open>\<And>thesis. (\<And>v t2 e2. \<lbrakk>is_subtree (Node v {|(t2, e2)|}) (Node r xs); rank (rev (dtree.root t2)) < rank (rev v)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1113_T_Ocontr__before,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ta )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ( iKKBZ_7682935289300565975re_a_b @ t @ R3 @ ( root_list_a_b @ T1 ) ) ) ) ).
% T.contr_before
thf(fact_1114_R_Odtail__root__in__set,axiom,
! [E: b,R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,Dt: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ( ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) )
= ( node_list_a_b @ R3 @ Xs2 ) )
=> ( ( ( dtail_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Dt @ E )
= R3 )
=> ( member_b @ E @ ( image_4684437738885282872_b_b_b @ produc5719641485658034180_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) ) ) ) ) ).
% R.dtail_root_in_set
thf(fact_1115_T_Ocontr__forward,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ta )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ R3 @ ( root_list_a_b @ T1 ) ) ) ) ) ).
% T.contr_forward
thf(fact_1116_T_Ocontr__seq__conform,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ta )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ ( append_a @ R3 @ ( root_list_a_b @ T1 ) ) ) ) ) ).
% T.contr_seq_conform
thf(fact_1117_T_Odom__between__child__roots,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ta )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R3 ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% T.dom_between_child_roots
thf(fact_1118_T_Odom__self__contr,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ta )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ ( hd_a @ ( root_list_a_b @ T1 ) ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% T.dom_self_contr
thf(fact_1119_R_Ocontr__before,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ( iKKBZ_7682935289300565975re_a_b @ t @ R3 @ ( root_list_a_b @ T1 ) ) ) ) ).
% R.contr_before
thf(fact_1120_T_Osubtree__rank__ge__if__reach_H,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ta )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ ( dverts_list_a_b @ ta ) )
=> ( ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ X4 ) )
& ~ ? [Xb2: a] :
( ( member_a @ Xb2 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xb2 @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xb: a] :
( ( member_a @ Xb @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xb @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ( X4 != R3 ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ X4 ) ) ) ) ) ) ).
% T.subtree_rank_ge_if_reach'
thf(fact_1121_T_Osubtree__rank__ge__if__reach,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b,V: list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ta )
=> ( ( V != R3 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ta ) )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ V ) )
& ~ ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ V ) ) ) ) ) ) ) ).
% T.subtree_rank_ge_if_reach
thf(fact_1122_R_Ocontr__forward,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ R3 @ ( root_list_a_b @ T1 ) ) ) ) ) ).
% R.contr_forward
thf(fact_1123_R_Ocontr__seq__conform,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ ( append_a @ R3 @ ( root_list_a_b @ T1 ) ) ) ) ) ).
% R.contr_seq_conform
thf(fact_1124_R_Odom__between__child__roots,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R3 ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% R.dom_between_child_roots
thf(fact_1125_R_Odom__self__contr,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ ( hd_a @ ( root_list_a_b @ T1 ) ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% R.dom_self_contr
thf(fact_1126_T_Odom__sub__contr__subtree,axiom,
! [Tn: dtree_list_a_b,R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ Tn @ ta )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ Tn )
=> ( ( member551035911493665803st_a_b @ T1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ? [V5: list_a,T24: dtree_list_a_b] :
( ? [E24: b] : ( is_subtree_list_a_b @ ( node_list_a_b @ V5 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T24 @ E24 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ R3 @ Xs2 ) )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T24 ) ) ) @ ( rank @ ( rev_a @ V5 ) ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ ( hd_a @ ( root_list_a_b @ T1 ) ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ).
% T.dom_sub_contr_subtree
thf(fact_1127_T_Odom__sub__contr,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ ta )
=> ( ( member551035911493665803st_a_b @ T1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ? [V5: list_a,T24: dtree_list_a_b] :
( ? [E24: b] : ( is_subtree_list_a_b @ ( node_list_a_b @ V5 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T24 @ E24 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ R3 @ Xs2 ) )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T24 ) ) ) @ ( rank @ ( rev_a @ V5 ) ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ ( hd_a @ ( root_list_a_b @ T1 ) ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% T.dom_sub_contr
thf(fact_1128_R_Osubtree__rank__ge__if__reach,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b,V: list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( V != R3 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ V ) )
& ~ ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ V ) ) ) ) ) ) ) ).
% R.subtree_rank_ge_if_reach
thf(fact_1129_R_Osubtree__rank__ge__if__reach_H,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ X4 ) )
& ~ ? [Xb2: a] :
( ( member_a @ Xb2 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xb2 @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xb: a] :
( ( member_a @ Xb @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xb @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ( X4 != R3 ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ X4 ) ) ) ) ) ) ).
% R.subtree_rank_ge_if_reach'
thf(fact_1130_R_Odom__sub__contr__subtree,axiom,
! [Tn: dtree_list_a_b,R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ Tn @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ Tn )
=> ( ( member551035911493665803st_a_b @ T1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ? [V5: list_a,T24: dtree_list_a_b] :
( ? [E24: b] : ( is_subtree_list_a_b @ ( node_list_a_b @ V5 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T24 @ E24 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ R3 @ Xs2 ) )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T24 ) ) ) @ ( rank @ ( rev_a @ V5 ) ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ ( hd_a @ ( root_list_a_b @ T1 ) ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ).
% R.dom_sub_contr_subtree
thf(fact_1131_R_Odom__sub__contr,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member551035911493665803st_a_b @ T1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ? [V5: list_a,T24: dtree_list_a_b] :
( ? [E24: b] : ( is_subtree_list_a_b @ ( node_list_a_b @ V5 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T24 @ E24 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ R3 @ Xs2 ) )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T24 ) ) ) @ ( rank @ ( rev_a @ V5 ) ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ ( hd_a @ ( root_list_a_b @ T1 ) ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% R.dom_sub_contr
thf(fact_1132_v__def_I2_J,axiom,
ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ t22 ) ) ) @ ( rank @ ( rev_a @ v ) ) ).
% v_def(2)
thf(fact_1133_R_Onormalize1__subtree__same__hd,axiom,
! [V: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ? [T32: dtree_list_a_b,E32: b] :
( ( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T32 @ E32 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
& ( ( hd_a @ ( root_list_a_b @ T1 ) )
= ( hd_a @ ( root_list_a_b @ T32 ) ) ) )
| ? [V23: list_a] :
( ( V
= ( append_a @ V23 @ ( root_list_a_b @ T32 ) ) )
& ( ( sucs_list_a_b @ T32 )
= ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) )
& ( is_subtree_list_a_b @ ( node_list_a_b @ V23 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T32 @ E32 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T32 ) ) ) @ ( rank @ ( rev_a @ V23 ) ) ) ) ) ) ).
% R.normalize1_subtree_same_hd
thf(fact_1134_normalize1__darcs__sub,axiom,
! [T1: dtree_list_a_b] : ( ord_less_eq_set_b @ ( darcs_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) @ ( darcs_list_a_b @ T1 ) ) ).
% normalize1_darcs_sub
thf(fact_1135_normalize1__root__nempty,axiom,
! [T1: dtree_list_a_b] :
( ( ( root_list_a_b @ T1 )
!= nil_a )
=> ( ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) )
!= nil_a ) ) ).
% normalize1_root_nempty
thf(fact_1136_T_Olist__dtree__normalize1,axiom,
list_list_dtree_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ ta ) ).
% T.list_dtree_normalize1
thf(fact_1137_T_Odistinct__normalize1,axiom,
! [V: list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ ta ) )
=> ( distinct_a @ X2 ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ ta ) ) )
=> ( distinct_a @ V ) ) ) ).
% T.distinct_normalize1
thf(fact_1138_root__normalize1__eq2,axiom,
! [Xs2: fset_P2153231429829016240_a_b_b,R3: list_a] :
( ! [X2: produc6499617310964463488_a_b_b] :
( Xs2
!= ( finser2303212525150181944_a_b_b @ X2 @ bot_bo2248824169281960260_a_b_b ) )
=> ( ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R3 @ Xs2 ) ) )
= R3 ) ) ).
% root_normalize1_eq2
thf(fact_1139_normalize1__dverts__app__contr,axiom,
! [V: list_a,T1: dtree_list_a_b] :
( ( member_list_a @ V @ ( dverts_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) )
=> ? [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ T1 ) )
& ? [Xa: list_a] :
( ( member_list_a @ Xa @ ( dverts_list_a_b @ T1 ) )
& ( ( append_a @ X2 @ Xa )
= V )
& ( ord_less_real @ ( rank @ ( rev_a @ Xa ) ) @ ( rank @ ( rev_a @ X2 ) ) ) ) ) ) ) ).
% normalize1_dverts_app_contr
thf(fact_1140_root__normalize1__eq1,axiom,
! [T1: dtree_list_a_b,R3: list_a,E1: b] :
( ~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ( ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
= R3 ) ) ).
% root_normalize1_eq1
thf(fact_1141_root__normalize1__eq1_H,axiom,
! [T1: dtree_list_a_b,R3: list_a,E1: b] :
( ~ ( ord_less_eq_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ( ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
= R3 ) ) ).
% root_normalize1_eq1'
thf(fact_1142_child__contr__if__new__contr,axiom,
! [T1: dtree_list_a_b,R3: list_a] :
( ~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ? [T22: dtree_list_a_b,E23: b] :
( ( ( sucs_list_a_b @ T1 )
= ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T22 ) ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) ) ) ) ) ).
% child_contr_if_new_contr
thf(fact_1143_sub__contr__if__new__contr,axiom,
! [T1: dtree_list_a_b,R3: list_a] :
( ~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ? [V3: list_a,T22: dtree_list_a_b,E23: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T1 )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T22 ) ) ) @ ( rank @ ( rev_a @ V3 ) ) ) ) ) ) ).
% sub_contr_if_new_contr
thf(fact_1144_contr__if__normalize1__uneq,axiom,
! [T1: dtree_list_a_b] :
( ( ( ranked8905849569120154423e1_a_b @ rank @ T1 )
!= T1 )
=> ? [V3: list_a,T22: dtree_list_a_b,E23: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T1 )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T22 ) ) ) @ ( rank @ ( rev_a @ V3 ) ) ) ) ) ).
% contr_if_normalize1_uneq
thf(fact_1145_contr__before__normalize1,axiom,
! [V: list_a,T1: dtree_list_a_b,E1: b,T3: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked8905849569120154423e1_a_b @ rank @ T3 ) )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ V ) ) )
=> ? [V6: list_a,T22: dtree_list_a_b,E23: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V6 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T3 )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T22 ) ) ) @ ( rank @ ( rev_a @ V6 ) ) ) ) ) ) ).
% contr_before_normalize1
thf(fact_1146_normalize1_Osimps_I1_J,axiom,
! [T1: dtree_list_a_b,R3: list_a,E: b] :
( ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ( ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( node_list_a_b @ ( append_a @ R3 @ ( root_list_a_b @ T1 ) ) @ ( sucs_list_a_b @ T1 ) ) ) )
& ( ~ ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ( ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E ) @ bot_bo2248824169281960260_a_b_b ) ) )
= ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) @ E ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).
% normalize1.simps(1)
thf(fact_1147_R_Olist__dtree__normalize1,axiom,
list_list_dtree_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ).
% R.list_dtree_normalize1
thf(fact_1148_normalize1__dverts__contr__subtree,axiom,
! [V: list_a,T1: dtree_list_a_b] :
( ( member_list_a @ V @ ( dverts_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) )
=> ? [V23: list_a,T22: dtree_list_a_b,E23: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V23 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T22 @ E23 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T1 )
& ( ( append_a @ V23 @ ( root_list_a_b @ T22 ) )
= V )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T22 ) ) ) @ ( rank @ ( rev_a @ V23 ) ) ) ) ) ) ).
% normalize1_dverts_contr_subtree
thf(fact_1149__C1_C_I1_J,axiom,
( ( node_list_a_b @ r @ xs )
= ( node_list_a_b @ v @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ t22 @ e2 ) @ bot_bo2248824169281960260_a_b_b ) ) ) ).
% "1"(1)
thf(fact_1150_R_Odistinct__normalize1,axiom,
! [V: list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( distinct_a @ X2 ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
=> ( distinct_a @ V ) ) ) ).
% R.distinct_normalize1
thf(fact_1151_v__def_I1_J,axiom,
is_subtree_list_a_b @ ( node_list_a_b @ v @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ t22 @ e2 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ r @ xs ) ).
% v_def(1)
thf(fact_1152_normalize1__dlverts__eq,axiom,
! [T1: dtree_list_a_b] :
( ( list_dlverts_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) )
= ( list_dlverts_a_b @ T1 ) ) ).
% normalize1_dlverts_eq
thf(fact_1153_T_Onormalize1__subtree__same__hd,axiom,
! [V: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked8905849569120154423e1_a_b @ rank @ ta ) )
=> ? [T32: dtree_list_a_b,E32: b] :
( ( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T32 @ E32 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ta )
& ( ( hd_a @ ( root_list_a_b @ T1 ) )
= ( hd_a @ ( root_list_a_b @ T32 ) ) ) )
| ? [V23: list_a] :
( ( V
= ( append_a @ V23 @ ( root_list_a_b @ T32 ) ) )
& ( ( sucs_list_a_b @ T32 )
= ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) )
& ( is_subtree_list_a_b @ ( node_list_a_b @ V23 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T32 @ E32 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ta )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T32 ) ) ) @ ( rank @ ( rev_a @ V23 ) ) ) ) ) ) ).
% T.normalize1_subtree_same_hd
thf(fact_1154_normalize1__hd__root__eq,axiom,
! [T1: dtree_list_a_b] :
( ( ( root_list_a_b @ T1 )
!= nil_a )
=> ( ( hd_a @ ( root_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) )
= ( hd_a @ ( root_list_a_b @ T1 ) ) ) ) ).
% normalize1_hd_root_eq
thf(fact_1155_R_Onormalize__sorted__ranks,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_normalize_a_b @ rank @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ R3 ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) ) ) ).
% R.normalize_sorted_ranks
thf(fact_1156_R_Odistinct__normalize,axiom,
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( distinct_a @ X2 ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ ( dverts_list_a_b @ ( ranked_normalize_a_b @ rank @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) )
=> ( distinct_a @ X4 ) ) ) ).
% R.distinct_normalize
thf(fact_1157_normalize__dlverts__eq,axiom,
! [T1: dtree_list_a_b] :
( ( list_dlverts_a_b @ ( ranked_normalize_a_b @ rank @ T1 ) )
= ( list_dlverts_a_b @ T1 ) ) ).
% normalize_dlverts_eq
thf(fact_1158_normalize__darcs__sub,axiom,
! [T1: dtree_list_a_b] : ( ord_less_eq_set_b @ ( darcs_list_a_b @ ( ranked_normalize_a_b @ rank @ T1 ) ) @ ( darcs_list_a_b @ T1 ) ) ).
% normalize_darcs_sub
thf(fact_1159_T_Odistinct__normalize,axiom,
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ ta ) )
=> ( distinct_a @ X2 ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ ( dverts_list_a_b @ ( ranked_normalize_a_b @ rank @ ta ) ) )
=> ( distinct_a @ X4 ) ) ) ).
% T.distinct_normalize
thf(fact_1160_T_Onormalize__sorted__ranks,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_normalize_a_b @ rank @ ta ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ R3 ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) ) ) ).
% T.normalize_sorted_ranks
thf(fact_1161_normalize__hd__root__eq,axiom,
! [T1: dtree_list_a_b] :
( ( ( root_list_a_b @ T1 )
!= nil_a )
=> ( ( hd_a @ ( root_list_a_b @ ( ranked_normalize_a_b @ rank @ T1 ) ) )
= ( hd_a @ ( root_list_a_b @ T1 ) ) ) ) ).
% normalize_hd_root_eq
thf(fact_1162_R_Odhead__in__verts__if__dtail,axiom,
! [Dt: b > list_a,P: b,X: dtree_list_a_b,R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,E: b,Dh: b > list_a] :
( ( member_list_a @ ( dtail_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Dt @ P ) @ ( dverts_list_a_b @ X ) )
=> ( ( member_b @ P @ ( darcs_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ( ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) )
= ( node_list_a_b @ R3 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( member_list_a @ ( dhead_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Dh @ P ) @ ( dverts_list_a_b @ X ) ) ) ) ) ) ).
% R.dhead_in_verts_if_dtail
thf(fact_1163_T_Odhead__unique,axiom,
! [E: b,P: b,Dh: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ ta ) )
=> ( ( member_b @ P @ ( darcs_list_a_b @ ta ) )
=> ( ( E != P )
=> ( ( dhead_list_a_b @ ta @ Dh @ E )
!= ( dhead_list_a_b @ ta @ Dh @ P ) ) ) ) ) ).
% T.dhead_unique
thf(fact_1164_T_Odhead__in__dverts,axiom,
! [E: b,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ ta ) )
=> ( member_list_a @ ( dhead_list_a_b @ ta @ Def2 @ E ) @ ( dverts_list_a_b @ ta ) ) ) ).
% T.dhead_in_dverts
thf(fact_1165_T_Odhead__not__root,axiom,
! [E: b,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ ta ) )
=> ( ( dhead_list_a_b @ ta @ Def2 @ E )
!= ( root_list_a_b @ ta ) ) ) ).
% T.dhead_not_root
thf(fact_1166_T_Odhead__in__subtree__eq__subtree,axiom,
! [T1: dtree_list_a_b,E: b,Def2: b > list_a] :
( ( is_subtree_list_a_b @ T1 @ ta )
=> ( ( member_b @ E @ ( darcs_list_a_b @ T1 ) )
=> ( ( dhead_list_a_b @ ta @ Def2 @ E )
= ( dhead_list_a_b @ T1 @ Def2 @ E ) ) ) ) ).
% T.dhead_in_subtree_eq_subtree
thf(fact_1167_T_Odhead__in__childverts,axiom,
! [E: b,X: dtree_list_a_b,E3: b,Xs2: fset_P2153231429829016240_a_b_b,R3: list_a,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ X ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E3 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( ( node_list_a_b @ R3 @ Xs2 )
= ta )
=> ( member_list_a @ ( dhead_list_a_b @ ta @ Def2 @ E ) @ ( dverts_list_a_b @ X ) ) ) ) ) ).
% T.dhead_in_childverts
thf(fact_1168_T_Odhead__in__verts__if__dtail,axiom,
! [Dt: b > list_a,P: b,X: dtree_list_a_b,R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,E: b,Dh: b > list_a] :
( ( member_list_a @ ( dtail_list_a_b @ ta @ Dt @ P ) @ ( dverts_list_a_b @ X ) )
=> ( ( member_b @ P @ ( darcs_list_a_b @ ta ) )
=> ( ( ta
= ( node_list_a_b @ R3 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( member_list_a @ ( dhead_list_a_b @ ta @ Dh @ P ) @ ( dverts_list_a_b @ X ) ) ) ) ) ) ).
% T.dhead_in_verts_if_dtail
thf(fact_1169_R_Odhead__unique,axiom,
! [E: b,P: b,Dh: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ( member_b @ P @ ( darcs_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ( E != P )
=> ( ( dhead_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Dh @ E )
!= ( dhead_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Dh @ P ) ) ) ) ) ).
% R.dhead_unique
thf(fact_1170_R_Odhead__in__dverts,axiom,
! [E: b,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( member_list_a @ ( dhead_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Def2 @ E ) @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).
% R.dhead_in_dverts
thf(fact_1171_R_Odhead__not__root,axiom,
! [E: b,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ( dhead_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Def2 @ E )
!= ( root_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ).
% R.dhead_not_root
thf(fact_1172_R_Odhead__in__subtree__eq__subtree,axiom,
! [T1: dtree_list_a_b,E: b,Def2: b > list_a] :
( ( is_subtree_list_a_b @ T1 @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( ( member_b @ E @ ( darcs_list_a_b @ T1 ) )
=> ( ( dhead_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Def2 @ E )
= ( dhead_list_a_b @ T1 @ Def2 @ E ) ) ) ) ).
% R.dhead_in_subtree_eq_subtree
thf(fact_1173_R_Odhead__in__childverts,axiom,
! [E: b,X: dtree_list_a_b,E3: b,Xs2: fset_P2153231429829016240_a_b_b,R3: list_a,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ X ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E3 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( ( node_list_a_b @ R3 @ Xs2 )
= ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( member_list_a @ ( dhead_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Def2 @ E ) @ ( dverts_list_a_b @ X ) ) ) ) ) ).
% R.dhead_in_childverts
thf(fact_1174_ex__leaf,axiom,
( ( finite_finite_a @ ( pre_ve642382030648772252t_unit @ t ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( pre_ve642382030648772252t_unit @ t ) )
& ( shorte1213025427933718126af_a_b @ t @ X2 ) ) ) ).
% ex_leaf
thf(fact_1175_sorted__ranks__if__normalize1__eq,axiom,
! [T23: dtree_list_a_b,R1: list_a,T1: dtree_list_a_b,E1: b] :
( ( wf_darcs_list_a_b @ T23 )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ R1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T23 )
=> ( ( T23
= ( ranked8905849569120154423e1_a_b @ rank @ T23 ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ R1 ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) ) ) ) ) ).
% sorted_ranks_if_normalize1_eq
thf(fact_1176_T_Owf__arcs,axiom,
wf_darcs_list_a_b @ ta ).
% T.wf_arcs
thf(fact_1177_wf__darcs__normalize1,axiom,
! [T1: dtree_list_a_b] :
( ( wf_darcs_list_a_b @ T1 )
=> ( wf_darcs_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ T1 ) ) ) ).
% wf_darcs_normalize1
thf(fact_1178_T_Owf__darcs__combine,axiom,
! [X: list_a,Y: list_a] : ( wf_darcs_list_a_b @ ( list_combine_a_b @ X @ Y @ ta ) ) ).
% T.wf_darcs_combine
thf(fact_1179_R_Owf__arcs,axiom,
wf_darcs_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ).
% R.wf_arcs
thf(fact_1180_R_Owf__darcs__combine,axiom,
! [X: list_a,Y: list_a] : ( wf_darcs_list_a_b @ ( list_combine_a_b @ X @ Y @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ).
% R.wf_darcs_combine
thf(fact_1181_normalize1__uneq__if__contr,axiom,
! [R1: list_a,T1: dtree_list_a_b,E1: b,T23: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ T23 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R1 ) ) )
=> ( ( wf_darcs_list_a_b @ T23 )
=> ( T23
!= ( ranked8905849569120154423e1_a_b @ rank @ T23 ) ) ) ) ) ).
% normalize1_uneq_if_contr
thf(fact_1182_R_Oinsert__between__wf__darcs,axiom,
! [E: b,V: list_a,X: list_a,Y: list_a] :
( ~ ( member_b @ E @ ( darcs_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( wf_darcs_list_a_b @ ( insert1898995607788287860st_a_b @ V @ E @ X @ Y @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ) ).
% R.insert_between_wf_darcs
thf(fact_1183_dom__sub__contr,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ t2 )
=> ( ( member551035911493665803st_a_b @ T1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ? [V5: list_a,T24: dtree_list_a_b] :
( ? [E24: b] : ( is_subtree_list_a_b @ ( node_list_a_b @ V5 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T24 @ E24 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ R3 @ Xs2 ) )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T24 ) ) ) @ ( rank @ ( rev_a @ V5 ) ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ ( hd_a @ ( root_list_a_b @ T1 ) ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% dom_sub_contr
thf(fact_1184_list__dtree__axioms,axiom,
list_list_dtree_a_b @ t2 ).
% list_dtree_axioms
thf(fact_1185_wf__arcs,axiom,
wf_darcs_list_a_b @ t2 ).
% wf_arcs
thf(fact_1186_dhead__unique,axiom,
! [E: b,P: b,Dh: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ t2 ) )
=> ( ( member_b @ P @ ( darcs_list_a_b @ t2 ) )
=> ( ( E != P )
=> ( ( dhead_list_a_b @ t2 @ Dh @ E )
!= ( dhead_list_a_b @ t2 @ Dh @ P ) ) ) ) ) ).
% dhead_unique
thf(fact_1187_verts__distinct,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( distinct_a @ V ) ) ).
% verts_distinct
thf(fact_1188_list__dtree__sub,axiom,
! [X: dtree_list_a_b] :
( ( is_subtree_list_a_b @ X @ t2 )
=> ( list_list_dtree_a_b @ X ) ) ).
% list_dtree_sub
thf(fact_1189_v__in__dlverts__if__in__comb,axiom,
! [V: a,X: list_a,Y: list_a] :
( ( member_a @ V @ ( list_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ t2 ) ) )
=> ( member_a @ V @ ( list_dlverts_a_b @ t2 ) ) ) ).
% v_in_dlverts_if_in_comb
thf(fact_1190_v__in__comb__if__in__dlverts,axiom,
! [V: a,X: list_a,Y: list_a] :
( ( member_a @ V @ ( list_dlverts_a_b @ t2 ) )
=> ( member_a @ V @ ( list_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ t2 ) ) ) ) ).
% v_in_comb_if_in_dlverts
thf(fact_1191_list__dtree__comb,axiom,
! [X: list_a,Y: list_a] : ( list_list_dtree_a_b @ ( list_combine_a_b @ X @ Y @ t2 ) ) ).
% list_dtree_comb
thf(fact_1192_wf__darcs__combine,axiom,
! [X: list_a,Y: list_a] : ( wf_darcs_list_a_b @ ( list_combine_a_b @ X @ Y @ t2 ) ) ).
% wf_darcs_combine
thf(fact_1193_arc__uneq__if__subtree__uneq,axiom,
! [X1: dtree_list_a_b,E1: b,Xs2: fset_P2153231429829016240_a_b_b,X22: dtree_list_a_b,E22: b,R3: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X1 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( X1 != X22 )
=> ( ( ( node_list_a_b @ R3 @ Xs2 )
= t2 )
=> ( E1 != E22 ) ) ) ) ) ).
% arc_uneq_if_subtree_uneq
thf(fact_1194_subtree__uneq__if__arc__uneq,axiom,
! [X1: dtree_list_a_b,E1: b,Xs2: fset_P2153231429829016240_a_b_b,X22: dtree_list_a_b,E22: b,R3: list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X1 @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X22 @ E22 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( E1 != E22 )
=> ( ( ( node_list_a_b @ R3 @ Xs2 )
= t2 )
=> ( X1 != X22 ) ) ) ) ) ).
% subtree_uneq_if_arc_uneq
thf(fact_1195_dverts__same__if__set__subtree,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,V22: list_a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V22 ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ t2 ) )
=> ( V1 = V22 ) ) ) ) ) ) ).
% dverts_same_if_set_subtree
thf(fact_1196_dhead__in__dverts,axiom,
! [E: b,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ t2 ) )
=> ( member_list_a @ ( dhead_list_a_b @ t2 @ Def2 @ E ) @ ( dverts_list_a_b @ t2 ) ) ) ).
% dhead_in_dverts
thf(fact_1197_dhead__not__root,axiom,
! [E: b,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ t2 ) )
=> ( ( dhead_list_a_b @ t2 @ Def2 @ E )
!= ( root_list_a_b @ t2 ) ) ) ).
% dhead_not_root
thf(fact_1198_dtail__in__dverts,axiom,
! [E: b,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ t2 ) )
=> ( member_list_a @ ( dtail_list_a_b @ t2 @ Def2 @ E ) @ ( dverts_list_a_b @ t2 ) ) ) ).
% dtail_in_dverts
thf(fact_1199_dhead__in__subtree__eq__subtree,axiom,
! [T1: dtree_list_a_b,E: b,Def2: b > list_a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_b @ E @ ( darcs_list_a_b @ T1 ) )
=> ( ( dhead_list_a_b @ t2 @ Def2 @ E )
= ( dhead_list_a_b @ T1 @ Def2 @ E ) ) ) ) ).
% dhead_in_subtree_eq_subtree
thf(fact_1200_verts__distinct__subtree,axiom,
! [Tn: dtree_list_a_b,V: list_a] :
( ( is_subtree_list_a_b @ Tn @ t2 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ Tn ) )
=> ( distinct_a @ V ) ) ) ).
% verts_distinct_subtree
thf(fact_1201_dtail__in__subtree__eq__subtree,axiom,
! [T1: dtree_list_a_b,E: b,Def2: b > list_a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_b @ E @ ( darcs_list_a_b @ T1 ) )
=> ( ( dtail_list_a_b @ t2 @ Def2 @ E )
= ( dtail_list_a_b @ T1 @ Def2 @ E ) ) ) ) ).
% dtail_in_subtree_eq_subtree
thf(fact_1202_verts__forward,axiom,
! [X4: list_a] :
( ( member_list_a @ X4 @ ( dverts_list_a_b @ t2 ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ X4 ) ) ).
% verts_forward
thf(fact_1203_list__dtree__normalize1,axiom,
list_list_dtree_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ t2 ) ).
% list_dtree_normalize1
thf(fact_1204_verts__conform,axiom,
! [V: list_a] :
( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ V ) ) ).
% verts_conform
thf(fact_1205_child__arc__not__subtree,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E1: b] :
( ( ( node_list_a_b @ R3 @ Xs2 )
= t2 )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ~ ( member_b @ E1 @ ( darcs_list_a_b @ X ) ) ) ) ).
% child_arc_not_subtree
thf(fact_1206_subtree__root__not__root,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E: b] :
( ( t2
= ( node_list_a_b @ R3 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( root_list_a_b @ X )
!= R3 ) ) ) ).
% subtree_root_not_root
thf(fact_1207_list__dtree__rec,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b,E: b] :
( ( ( node_list_a_b @ R3 @ Xs2 )
= t2 )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( list_list_dtree_a_b @ X ) ) ) ).
% list_dtree_rec
thf(fact_1208_dtail__in__subdverts,axiom,
! [E: b,X: dtree_list_a_b,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ X ) )
=> ( ( is_subtree_list_a_b @ X @ t2 )
=> ( member_list_a @ ( dtail_list_a_b @ t2 @ Def2 @ E ) @ ( dverts_list_a_b @ X ) ) ) ) ).
% dtail_in_subdverts
thf(fact_1209_list__dtree__rec__suc,axiom,
! [X: dtree_list_a_b,E: b] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ ( sucs_list_a_b @ t2 ) ) )
=> ( list_list_dtree_a_b @ X ) ) ).
% list_dtree_rec_suc
thf(fact_1210_T_Oinsert__between__wf__darcs,axiom,
! [E: b,V: list_a,X: list_a,Y: list_a] :
( ~ ( member_b @ E @ ( darcs_list_a_b @ ta ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ ta ) )
=> ( wf_darcs_list_a_b @ ( insert1898995607788287860st_a_b @ V @ E @ X @ Y @ ta ) ) ) ) ).
% T.insert_between_wf_darcs
thf(fact_1211_insert__between__wf__darcs,axiom,
! [E: b,V: list_a,X: list_a,Y: list_a] :
( ~ ( member_b @ E @ ( darcs_list_a_b @ t2 ) )
=> ( ~ ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( wf_darcs_list_a_b @ ( insert1898995607788287860st_a_b @ V @ E @ X @ Y @ t2 ) ) ) ) ).
% insert_between_wf_darcs
thf(fact_1212_dlverts__reach__in__dlverts,axiom,
! [X: a,Y: a,T1: dtree_list_a_b] :
( ( reachable_a_b @ t @ X @ Y )
=> ( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_a @ X @ ( list_dlverts_a_b @ T1 ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ).
% dlverts_reach_in_dlverts
thf(fact_1213_distinct__normalize1,axiom,
! [V: list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ t2 ) )
=> ( distinct_a @ X2 ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ ( ranked8905849569120154423e1_a_b @ rank @ t2 ) ) )
=> ( distinct_a @ V ) ) ) ).
% distinct_normalize1
thf(fact_1214_verts__conform__subtree,axiom,
! [Tn: dtree_list_a_b,V: list_a] :
( ( is_subtree_list_a_b @ Tn @ t2 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ Tn ) )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ V ) ) ) ).
% verts_conform_subtree
thf(fact_1215_distinct__normalize,axiom,
( ! [X2: list_a] :
( ( member_list_a @ X2 @ ( dverts_list_a_b @ t2 ) )
=> ( distinct_a @ X2 ) )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ ( dverts_list_a_b @ ( ranked_normalize_a_b @ rank @ t2 ) ) )
=> ( distinct_a @ X4 ) ) ) ).
% distinct_normalize
thf(fact_1216_root__not__subtree,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
( ( ( node_list_a_b @ R3 @ Xs2 )
= t2 )
=> ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ~ ( member_list_a @ R3 @ ( dverts_list_a_b @ X ) ) ) ) ).
% root_not_subtree
thf(fact_1217_dverts__reach__in__dverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a,V22: list_a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( reachable_a_b @ t @ X @ Y )
=> ( ( member_a @ Y @ ( set_a2 @ V22 ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ t2 ) )
=> ( member_list_a @ V22 @ ( dverts_list_a_b @ T1 ) ) ) ) ) ) ) ) ).
% dverts_reach_in_dverts
thf(fact_1218_dlverts__arc__in__dlverts,axiom,
! [T1: dtree_list_a_b,X: a,Y: a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ X @ ( list_dlverts_a_b @ T1 ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ).
% dlverts_arc_in_dlverts
thf(fact_1219_dhead__in__childverts,axiom,
! [E: b,X: dtree_list_a_b,E3: b,Xs2: fset_P2153231429829016240_a_b_b,R3: list_a,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ X ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E3 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( ( node_list_a_b @ R3 @ Xs2 )
= t2 )
=> ( member_list_a @ ( dhead_list_a_b @ t2 @ Def2 @ E ) @ ( dverts_list_a_b @ X ) ) ) ) ) ).
% dhead_in_childverts
thf(fact_1220_dtail__in__childverts,axiom,
! [E: b,X: dtree_list_a_b,E3: b,Xs2: fset_P2153231429829016240_a_b_b,R3: list_a,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ X ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E3 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( ( node_list_a_b @ R3 @ Xs2 )
= t2 )
=> ( member_list_a @ ( dtail_list_a_b @ t2 @ Def2 @ E ) @ ( dverts_list_a_b @ X ) ) ) ) ) ).
% dtail_in_childverts
thf(fact_1221_arc__in__subtree__if__tail__in__subtree,axiom,
! [Dt: b > list_a,P: b,X: dtree_list_a_b,R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,E: b] :
( ( member_list_a @ ( dtail_list_a_b @ t2 @ Dt @ P ) @ ( dverts_list_a_b @ X ) )
=> ( ( member_b @ P @ ( darcs_list_a_b @ t2 ) )
=> ( ( t2
= ( node_list_a_b @ R3 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( member_b @ P @ ( darcs_list_a_b @ X ) ) ) ) ) ) ).
% arc_in_subtree_if_tail_in_subtree
thf(fact_1222_dtail__root__in__set,axiom,
! [E: b,R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,Dt: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ t2 ) )
=> ( ( t2
= ( node_list_a_b @ R3 @ Xs2 ) )
=> ( ( ( dtail_list_a_b @ t2 @ Dt @ E )
= R3 )
=> ( member_b @ E @ ( image_4684437738885282872_b_b_b @ produc5719641485658034180_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) ) ) ) ) ).
% dtail_root_in_set
thf(fact_1223_dverts__arc__in__dverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a,V22: list_a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ Y @ ( set_a2 @ V22 ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ t2 ) )
=> ( member_list_a @ V22 @ ( dverts_list_a_b @ T1 ) ) ) ) ) ) ) ) ).
% dverts_arc_in_dverts
thf(fact_1224_dverts__reach__in__dlverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( reachable_a_b @ t @ X @ Y )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ) ).
% dverts_reach_in_dlverts
thf(fact_1225_dlverts__reach1__in__dlverts,axiom,
! [X: a,Y: a,T1: dtree_list_a_b] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_a @ X @ ( list_dlverts_a_b @ T1 ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ).
% dlverts_reach1_in_dlverts
thf(fact_1226_dverts__child__subset,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
( ( ( node_list_a_b @ R3 @ Xs2 )
= t2 )
=> ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ord_less_set_list_a @ ( dverts_list_a_b @ X ) @ ( dverts_list_a_b @ t2 ) ) ) ) ).
% dverts_child_subset
thf(fact_1227_darcs__child__subset,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: dtree_list_a_b] :
( ( ( node_list_a_b @ R3 @ Xs2 )
= t2 )
=> ( ( member551035911493665803st_a_b @ X @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ord_less_set_b @ ( darcs_list_a_b @ X ) @ ( darcs_list_a_b @ t2 ) ) ) ) ).
% darcs_child_subset
thf(fact_1228_darc__in__sub__if__dtail__in__sub,axiom,
! [Dt: b > list_a,E: b,V: list_a,X: dtree_list_a_b,E1: b,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b,R3: list_a] :
( ( ( dtail_list_a_b @ t2 @ Dt @ E )
= V )
=> ( ( member_b @ E @ ( darcs_list_a_b @ t2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E1 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( is_subtree_list_a_b @ T1 @ X )
=> ( ( ( node_list_a_b @ R3 @ Xs2 )
= t2 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) )
=> ( member_b @ E @ ( darcs_list_a_b @ X ) ) ) ) ) ) ) ) ).
% darc_in_sub_if_dtail_in_sub
thf(fact_1229_dhead__in__verts__if__dtail,axiom,
! [Dt: b > list_a,P: b,X: dtree_list_a_b,R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,E: b,Dh: b > list_a] :
( ( member_list_a @ ( dtail_list_a_b @ t2 @ Dt @ P ) @ ( dverts_list_a_b @ X ) )
=> ( ( member_b @ P @ ( darcs_list_a_b @ t2 ) )
=> ( ( t2
= ( node_list_a_b @ R3 @ Xs2 ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( member_list_a @ ( dhead_list_a_b @ t2 @ Dh @ P ) @ ( dverts_list_a_b @ X ) ) ) ) ) ) ).
% dhead_in_verts_if_dtail
thf(fact_1230_arc__to__dverts__in__subtree,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: a,Y: a,V: list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ t2 )
=> ( ( member_a @ X @ ( set_a2 @ R3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ Y @ ( set_a2 @ V ) )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) ) ) ) ) ) ) ) ).
% arc_to_dverts_in_subtree
thf(fact_1231_out__arcs__in__subarcs__aux,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,Dt: b > list_a,E: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ t2 )
=> ( ( ( dtail_list_a_b @ t2 @ Dt @ E )
= R3 )
=> ( ( member_b @ E @ ( darcs_list_a_b @ t2 ) )
=> ( member_b @ E @ ( image_4684437738885282872_b_b_b @ produc5719641485658034180_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) ) ) ) ) ).
% out_arcs_in_subarcs_aux
thf(fact_1232_arc__in__dlverts__subtree,axiom,
! [Tn: dtree_list_a_b,R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: a,Y: a] :
( ( is_subtree_list_a_b @ Tn @ t2 )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ Tn )
=> ( ( member_a @ X @ ( set_a2 @ R3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ ( node_list_a_b @ R3 @ Xs2 ) ) ) ) ) ) ) ).
% arc_in_dlverts_subtree
thf(fact_1233_arc__in__dlverts,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,X: a,Y: a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ t2 )
=> ( ( member_a @ X @ ( set_a2 @ R3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ ( node_list_a_b @ R3 @ Xs2 ) ) ) ) ) ) ).
% arc_in_dlverts
thf(fact_1234_dverts__arc__in__dlverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( arcs_ends_a_b @ t ) )
=> ( member_a @ Y @ ( list_dlverts_a_b @ T1 ) ) ) ) ) ) ).
% dverts_arc_in_dlverts
thf(fact_1235_dverts__reach1__in__dverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a,V22: list_a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) )
=> ( ( member_a @ Y @ ( set_a2 @ V22 ) )
=> ( ( member_list_a @ V22 @ ( dverts_list_a_b @ t2 ) )
=> ( member_list_a @ V22 @ ( dverts_list_a_b @ T1 ) ) ) ) ) ) ) ) ).
% dverts_reach1_in_dverts
thf(fact_1236_distint__verts__singleton__subtree,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( distinct_a @ ( append_a @ R3 @ ( root_list_a_b @ T1 ) ) ) ) ).
% distint_verts_singleton_subtree
thf(fact_1237_distint__verts__subtree,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ t2 )
=> ( ( member551035911493665803st_a_b @ T1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( distinct_a @ ( append_a @ R3 @ ( root_list_a_b @ T1 ) ) ) ) ) ).
% distint_verts_subtree
thf(fact_1238_dverts__reach1__in__dverts__r,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,V: list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ t2 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ R3 ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( member_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) ) ) ) ) ) ).
% dverts_reach1_in_dverts_r
thf(fact_1239_dverts__reach1__in__dverts__root,axiom,
! [T1: dtree_list_a_b,V: list_a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ V ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( member_list_a @ V @ ( dverts_list_a_b @ T1 ) ) ) ) ) ).
% dverts_reach1_in_dverts_root
thf(fact_1240_dverts__reach1__in__dlverts,axiom,
! [T1: dtree_list_a_b,V1: list_a,X: a,Y: a] :
( ( is_subtree_list_a_b @ T1 @ t2 )
=> ( ( member_list_a @ V1 @ ( dverts_list_a_b @ T1 ) )
=> ( ( member_a @ X @ ( set_a2 @ V1 ) )
=> ( ( 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 ) ) ) ) ) ) ).
% dverts_reach1_in_dlverts
thf(fact_1241_child__disjoint__root,axiom,
! [R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ t2 )
=> ( ( member551035911493665803st_a_b @ T1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ( inf_inf_set_a @ ( set_a2 @ R3 ) @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
= bot_bot_set_a ) ) ) ).
% child_disjoint_root
thf(fact_1242_arc__in__dlverts__subtree_H,axiom,
! [Tn: dtree_list_a_b] :
( ( is_subtree_list_a_b @ Tn @ t2 )
=> ! [R4: list_a,Xs7: fset_P2153231429829016240_a_b_b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R4 @ Xs7 ) @ Tn )
=> ! [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ R4 ) )
=> ! [Y5: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y5 ) @ ( arcs_ends_a_b @ t ) )
=> ( ( member_a @ Y5 @ ( set_a2 @ R4 ) )
| ? [Xa: produc6499617310964463488_a_b_b] :
( ( member4695696432722591383_a_b_b @ Xa @ ( fset_P9138963618725001425_a_b_b @ Xs7 ) )
& ( member_a @ Y5 @ ( list_dlverts_a_b @ ( produc5948858871325780166_a_b_b @ Xa ) ) ) ) ) ) ) ) ) ).
% arc_in_dlverts_subtree'
thf(fact_1243_normalize__sorted__ranks,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked_normalize_a_b @ rank @ t2 ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ R3 ) ) @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) ) ) ).
% normalize_sorted_ranks
thf(fact_1244_contr__before,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ( iKKBZ_7682935289300565975re_a_b @ t @ R3 @ ( root_list_a_b @ T1 ) ) ) ) ).
% contr_before
thf(fact_1245_contr__forward,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ( iKKBZ_4778857019735642799rd_a_b @ t @ ( append_a @ R3 @ ( root_list_a_b @ T1 ) ) ) ) ) ).
% contr_forward
thf(fact_1246_contr__seq__conform,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ( iKKBZ_4622586873178280511rm_a_b @ t @ ( append_a @ R3 @ ( root_list_a_b @ T1 ) ) ) ) ) ).
% contr_seq_conform
thf(fact_1247_dlverts__comb__id,axiom,
! [X: list_a,Y: list_a] :
( ( list_dlverts_a_b @ ( list_combine_a_b @ X @ Y @ t2 ) )
= ( list_dlverts_a_b @ t2 ) ) ).
% dlverts_comb_id
thf(fact_1248_normalize1__subtree__same__hd,axiom,
! [V: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( ranked8905849569120154423e1_a_b @ rank @ t2 ) )
=> ? [T32: dtree_list_a_b,E32: b] :
( ( ( is_subtree_list_a_b @ ( node_list_a_b @ V @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T32 @ E32 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
& ( ( hd_a @ ( root_list_a_b @ T1 ) )
= ( hd_a @ ( root_list_a_b @ T32 ) ) ) )
| ? [V23: list_a] :
( ( V
= ( append_a @ V23 @ ( root_list_a_b @ T32 ) ) )
& ( ( sucs_list_a_b @ T32 )
= ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) )
& ( is_subtree_list_a_b @ ( node_list_a_b @ V23 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T32 @ E32 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T32 ) ) ) @ ( rank @ ( rev_a @ V23 ) ) ) ) ) ) ).
% normalize1_subtree_same_hd
thf(fact_1249_dom__between__child__roots,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R3 ) )
& ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Xa ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ).
% dom_between_child_roots
thf(fact_1250_dom__self__contr,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ R3 ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ ( hd_a @ ( root_list_a_b @ T1 ) ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ).
% dom_self_contr
thf(fact_1251_subtree__rank__ge__if__reach,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b,V: list_a] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ( ( V != R3 )
=> ( ( member_list_a @ V @ ( dverts_list_a_b @ t2 ) )
=> ( ? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ V ) )
& ~ ? [Xa: a] :
( ( member_a @ Xa @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X4 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ V ) ) ) ) ) ) ) ).
% subtree_rank_ge_if_reach
thf(fact_1252_subtree__rank__ge__if__reach_H,axiom,
! [R3: list_a,T1: dtree_list_a_b,E1: b] :
( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T1 @ E1 ) @ bot_bo2248824169281960260_a_b_b ) ) @ t2 )
=> ! [X4: list_a] :
( ( member_list_a @ X4 @ ( dverts_list_a_b @ t2 ) )
=> ( ? [Xa2: a] :
( ( member_a @ Xa2 @ ( set_a2 @ X4 ) )
& ~ ? [Xb2: a] :
( ( member_a @ Xb2 @ ( set_a2 @ ( root_list_a_b @ T1 ) ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xb2 @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ? [Xb: a] :
( ( member_a @ Xb @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xb @ Xa2 ) @ ( transitive_trancl_a @ ( arcs_ends_a_b @ t ) ) ) )
& ( X4 != R3 ) )
=> ( ord_less_eq_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T1 ) ) ) @ ( rank @ ( rev_a @ X4 ) ) ) ) ) ) ).
% subtree_rank_ge_if_reach'
thf(fact_1253_dom__sub__contr__subtree,axiom,
! [Tn: dtree_list_a_b,R3: list_a,Xs2: fset_P2153231429829016240_a_b_b,T1: dtree_list_a_b] :
( ( is_subtree_list_a_b @ Tn @ t2 )
=> ( ( is_subtree_list_a_b @ ( node_list_a_b @ R3 @ Xs2 ) @ Tn )
=> ( ( member551035911493665803st_a_b @ T1 @ ( image_5965465251548763643st_a_b @ produc5948858871325780166_a_b_b @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) ) )
=> ( ? [V5: list_a,T24: dtree_list_a_b] :
( ? [E24: b] : ( is_subtree_list_a_b @ ( node_list_a_b @ V5 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ T24 @ E24 ) @ bot_bo2248824169281960260_a_b_b ) ) @ ( node_list_a_b @ R3 @ Xs2 ) )
& ( ord_less_real @ ( rank @ ( rev_a @ ( root_list_a_b @ T24 ) ) ) @ ( rank @ ( rev_a @ V5 ) ) ) )
=> ? [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ R3 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ ( hd_a @ ( root_list_a_b @ T1 ) ) ) @ ( arcs_ends_a_b @ t ) ) ) ) ) ) ) ).
% dom_sub_contr_subtree
thf(fact_1254_R_Oinsert__between__add__e__if__x__in,axiom,
! [X: list_a,V: list_a,E: b,Y: list_a] :
( ( member_list_a @ X @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ( darcs_list_a_b @ ( insert1898995607788287860st_a_b @ V @ E @ X @ Y @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
= ( insert_b @ E @ ( darcs_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ) ).
% R.insert_between_add_e_if_x_in
thf(fact_1255_R_Oinsert__between__add__v__if__x__in,axiom,
! [X: list_a,V: list_a,E: b,Y: list_a] :
( ( member_list_a @ X @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ( dverts_list_a_b @ ( insert1898995607788287860st_a_b @ V @ E @ X @ Y @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
= ( insert_list_a @ V @ ( dverts_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) ) ) ) ).
% R.insert_between_add_v_if_x_in
thf(fact_1256_T_Oinsert__between__add__v__if__x__in,axiom,
! [X: list_a,V: list_a,E: b,Y: list_a] :
( ( member_list_a @ X @ ( dverts_list_a_b @ ta ) )
=> ( ( dverts_list_a_b @ ( insert1898995607788287860st_a_b @ V @ E @ X @ Y @ ta ) )
= ( insert_list_a @ V @ ( dverts_list_a_b @ ta ) ) ) ) ).
% T.insert_between_add_v_if_x_in
thf(fact_1257_insert__between__add__v__if__x__in,axiom,
! [X: list_a,V: list_a,E: b,Y: list_a] :
( ( member_list_a @ X @ ( dverts_list_a_b @ t2 ) )
=> ( ( dverts_list_a_b @ ( insert1898995607788287860st_a_b @ V @ E @ X @ Y @ t2 ) )
= ( insert_list_a @ V @ ( dverts_list_a_b @ t2 ) ) ) ) ).
% insert_between_add_v_if_x_in
thf(fact_1258_T_Oinsert__between__add__e__if__x__in,axiom,
! [X: list_a,V: list_a,E: b,Y: list_a] :
( ( member_list_a @ X @ ( dverts_list_a_b @ ta ) )
=> ( ( darcs_list_a_b @ ( insert1898995607788287860st_a_b @ V @ E @ X @ Y @ ta ) )
= ( insert_b @ E @ ( darcs_list_a_b @ ta ) ) ) ) ).
% T.insert_between_add_e_if_x_in
thf(fact_1259_insert__between__add__e__if__x__in,axiom,
! [X: list_a,V: list_a,E: b,Y: list_a] :
( ( member_list_a @ X @ ( dverts_list_a_b @ t2 ) )
=> ( ( darcs_list_a_b @ ( insert1898995607788287860st_a_b @ V @ E @ X @ Y @ t2 ) )
= ( insert_b @ E @ ( darcs_list_a_b @ t2 ) ) ) ) ).
% insert_between_add_e_if_x_in
thf(fact_1260_R_Odhead__in__childverts__no__root,axiom,
! [E: b,X: dtree_list_a_b,E3: b,Xs2: fset_P2153231429829016240_a_b_b,R3: list_a,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ X ) )
=> ( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E3 ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ( ( node_list_a_b @ R3 @ Xs2 )
= ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) )
=> ( member_list_a @ ( dhead_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Def2 @ E ) @ ( minus_646659088055828811list_a @ ( dverts_list_a_b @ X ) @ ( insert_list_a @ ( root_list_a_b @ X ) @ bot_bot_set_list_a ) ) ) ) ) ) ).
% R.dhead_in_childverts_no_root
thf(fact_1261_R_Odhead__notin__subtree__wo__root,axiom,
! [X: dtree_list_a_b,E: b,Xs2: fset_P2153231429829016240_a_b_b,P: b,R3: list_a,Dh: b > list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ~ ( member_b @ P @ ( darcs_list_a_b @ X ) )
=> ( ( member_b @ P @ ( darcs_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) ) )
=> ( ( ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) )
= ( node_list_a_b @ R3 @ Xs2 ) )
=> ~ ( member_list_a @ ( dhead_list_a_b @ ( node_list_a_b @ r1 @ ( finser2303212525150181944_a_b_b @ ( produc7704165765595008946_a_b_b @ ta @ e ) @ bot_bo2248824169281960260_a_b_b ) ) @ Dh @ P ) @ ( minus_646659088055828811list_a @ ( dverts_list_a_b @ X ) @ ( insert_list_a @ ( root_list_a_b @ X ) @ bot_bot_set_list_a ) ) ) ) ) ) ) ).
% R.dhead_notin_subtree_wo_root
thf(fact_1262_T_Odhead__in__dverts__no__root,axiom,
! [E: b,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ ta ) )
=> ( member_list_a @ ( dhead_list_a_b @ ta @ Def2 @ E ) @ ( minus_646659088055828811list_a @ ( dverts_list_a_b @ ta ) @ ( insert_list_a @ ( root_list_a_b @ ta ) @ bot_bot_set_list_a ) ) ) ) ).
% T.dhead_in_dverts_no_root
thf(fact_1263_dhead__in__dverts__no__root,axiom,
! [E: b,Def2: b > list_a] :
( ( member_b @ E @ ( darcs_list_a_b @ t2 ) )
=> ( member_list_a @ ( dhead_list_a_b @ t2 @ Def2 @ E ) @ ( minus_646659088055828811list_a @ ( dverts_list_a_b @ t2 ) @ ( insert_list_a @ ( root_list_a_b @ t2 ) @ bot_bot_set_list_a ) ) ) ) ).
% dhead_in_dverts_no_root
thf(fact_1264_T_Odhead__notin__subtree__wo__root,axiom,
! [X: dtree_list_a_b,E: b,Xs2: fset_P2153231429829016240_a_b_b,P: b,R3: list_a,Dh: b > list_a] :
( ( member4695696432722591383_a_b_b @ ( produc7704165765595008946_a_b_b @ X @ E ) @ ( fset_P9138963618725001425_a_b_b @ Xs2 ) )
=> ( ~ ( member_b @ P @ ( darcs_list_a_b @ X ) )
=> ( ( member_b @ P @ ( darcs_list_a_b @ ta ) )
=> ( ( ta
= ( node_list_a_b @ R3 @ Xs2 ) )
=> ~ ( member_list_a @ ( dhead_list_a_b @ ta @ Dh @ P ) @ ( minus_646659088055828811list_a @ ( dverts_list_a_b @ X ) @ ( insert_list_a @ ( root_list_a_b @ X ) @ bot_bot_set_list_a ) ) ) ) ) ) ) ).
% T.dhead_notin_subtree_wo_root
% Conjectures (1)
thf(conj_0,conjecture,
? [X4: a] :
( ( member_a @ X4 @ ( set_a2 @ r ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ ( hd_a @ ( root_list_a_b @ t1 ) ) ) @ ( arcs_ends_a_b @ t ) ) ) ).
%------------------------------------------------------------------------------