TPTP Problem File: ITP050^1.p
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%------------------------------------------------------------------------------
% File : ITP050^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer EdmondsKarp_Termination_Abstract problem prob_276__7590636_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : EdmondsKarp_Termination_Abstract/prob_276__7590636_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.31 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 408 ( 123 unt; 54 typ; 0 def)
% Number of atoms : 1021 ( 212 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 2902 ( 62 ~; 3 |; 49 &;2262 @)
% ( 0 <=>; 526 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 179 ( 179 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 45 usr; 5 con; 0-4 aty)
% Number of variables : 1010 ( 80 ^; 915 !; 15 ?;1010 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:31:07.662
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_se1612935105at_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P559422087at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1986765409at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (45)
thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis_001tf__a,type,
edmond1517640972ysis_a: ( product_prod_nat_nat > a ) > $o ).
thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis__defs_OspEdges_001tf__a,type,
edmond475474835dges_a: ( product_prod_nat_nat > a ) > nat > nat > set_Pr1986765409at_nat ).
thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis__defs_OuE_001tf__a,type,
edmond771116670s_uE_a: ( product_prod_nat_nat > a ) > set_Pr1986765409at_nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite447719721at_nat: set_Pr1986765409at_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite772653738at_nat: set_Pr1986765409at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite2012248349et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
finite1457549322at_nat: set_se1612935105at_nat > $o ).
thf(sy_c_Graph_OFinite__Graph_001tf__a,type,
finite_Graph_a: ( product_prod_nat_nat > a ) > $o ).
thf(sy_c_Graph_OGraph_OE_001tf__a,type,
e_a: ( product_prod_nat_nat > a ) > set_Pr1986765409at_nat ).
thf(sy_c_Graph_OGraph_OV_001tf__a,type,
v_a: ( product_prod_nat_nat > a ) > set_nat ).
thf(sy_c_Graph_OGraph_Oadjacent__nodes_001tf__a,type,
adjacent_nodes_a: ( product_prod_nat_nat > a ) > nat > set_nat ).
thf(sy_c_Graph_OGraph_Oincoming_001tf__a,type,
incoming_a: ( product_prod_nat_nat > a ) > nat > set_Pr1986765409at_nat ).
thf(sy_c_Graph_OGraph_Oincoming_H_001tf__a,type,
incoming_a2: ( product_prod_nat_nat > a ) > set_nat > set_Pr1986765409at_nat ).
thf(sy_c_Graph_OGraph_OisPath_001tf__a,type,
isPath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
thf(sy_c_Graph_OGraph_OisShortestPath_001tf__a,type,
isShortestPath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
thf(sy_c_Graph_OGraph_Ooutgoing_001tf__a,type,
outgoing_a: ( product_prod_nat_nat > a ) > nat > set_Pr1986765409at_nat ).
thf(sy_c_Graph_OGraph_Ooutgoing_H_001tf__a,type,
outgoing_a2: ( product_prod_nat_nat > a ) > set_nat > set_Pr1986765409at_nat ).
thf(sy_c_Graph_OGraph_OpathVertices,type,
pathVertices: nat > list_P559422087at_nat > list_nat ).
thf(sy_c_Graph_OGraph_OreachableNodes_001tf__a,type,
reachableNodes_a: ( product_prod_nat_nat > a ) > nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inf_in586391887at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > set_Pr1986765409at_nat ).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
cons_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
cons_P66992567at_nat: product_prod_nat_nat > list_P559422087at_nat > list_P559422087at_nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
nil_Pr1308055047at_nat: list_P559422087at_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_Pr2131844118at_nat: list_P559422087at_nat > set_Pr1986765409at_nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s1990949619at_nat: list_P559422087at_nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le841296385at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member298845450at_nat: set_Pr1986765409at_nat > set_se1612935105at_nat > $o ).
thf(sy_v_c,type,
c: product_prod_nat_nat > a ).
thf(sy_v_edges,type,
edges: set_Pr1986765409at_nat ).
thf(sy_v_p,type,
p: list_P559422087at_nat ).
thf(sy_v_s,type,
s: nat ).
thf(sy_v_t,type,
t: nat ).
% Relevant facts (353)
thf(fact_0_SP,axiom,
isShortestPath_a @ c @ s @ p @ t ).
% SP
thf(fact_1_SP__EDGES,axiom,
ord_le841296385at_nat @ edges @ ( set_Pr2131844118at_nat @ p ) ).
% SP_EDGES
thf(fact_2_ek__analysis__axioms,axiom,
edmond1517640972ysis_a @ c ).
% ek_analysis_axioms
thf(fact_3_incoming_H__edges,axiom,
! [U: set_nat] : ( ord_le841296385at_nat @ ( incoming_a2 @ c @ U ) @ ( e_a @ c ) ) ).
% incoming'_edges
thf(fact_4_outgoing_H__edges,axiom,
! [U: set_nat] : ( ord_le841296385at_nat @ ( outgoing_a2 @ c @ U ) @ ( e_a @ c ) ) ).
% outgoing'_edges
thf(fact_5_E__ss__uE,axiom,
ord_le841296385at_nat @ ( e_a @ c ) @ ( edmond771116670s_uE_a @ c ) ).
% E_ss_uE
thf(fact_6_incoming__edges,axiom,
! [U2: nat] : ( ord_le841296385at_nat @ ( incoming_a @ c @ U2 ) @ ( e_a @ c ) ) ).
% incoming_edges
thf(fact_7_outgoing__edges,axiom,
! [U2: nat] : ( ord_le841296385at_nat @ ( outgoing_a @ c @ U2 ) @ ( e_a @ c ) ) ).
% outgoing_edges
thf(fact_8_Finite__Graph__axioms,axiom,
finite_Graph_a @ c ).
% Finite_Graph_axioms
thf(fact_9_subsetI,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ! [X: product_prod_nat_nat] :
( ( member701585322at_nat @ X @ A )
=> ( member701585322at_nat @ X @ B ) )
=> ( ord_le841296385at_nat @ A @ B ) ) ).
% subsetI
thf(fact_10_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_11_subset__antisym,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ( ord_le841296385at_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_12_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_13_order__refl,axiom,
! [X2: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_14_order__refl,axiom,
! [X2: set_nat] : ( ord_less_eq_set_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_15_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_16_SV,axiom,
member_nat @ s @ ( v_a @ c ) ).
% SV
thf(fact_17_spEdges__ss__E,axiom,
ord_le841296385at_nat @ ( edmond475474835dges_a @ c @ s @ t ) @ ( e_a @ c ) ).
% spEdges_ss_E
thf(fact_18_isPath__edgeset,axiom,
! [U2: nat,P: list_P559422087at_nat,V: nat,E: product_prod_nat_nat] :
( ( isPath_a @ c @ U2 @ P @ V )
=> ( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
=> ( member701585322at_nat @ E @ ( e_a @ c ) ) ) ) ).
% isPath_edgeset
thf(fact_19_shortestPath__is__path,axiom,
! [U2: nat,P: list_P559422087at_nat,V: nat] :
( ( isShortestPath_a @ c @ U2 @ P @ V )
=> ( isPath_a @ c @ U2 @ P @ V ) ) ).
% shortestPath_is_path
thf(fact_20_adjacent__nodes__ss__V,axiom,
! [U2: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_a @ c @ U2 ) @ ( v_a @ c ) ) ).
% adjacent_nodes_ss_V
thf(fact_21_ek__analysis__def,axiom,
edmond1517640972ysis_a = finite_Graph_a ).
% ek_analysis_def
thf(fact_22_ek__analysis_Ointro,axiom,
! [C: product_prod_nat_nat > a] :
( ( finite_Graph_a @ C )
=> ( edmond1517640972ysis_a @ C ) ) ).
% ek_analysis.intro
thf(fact_23_ek__analysis_Oaxioms,axiom,
! [C: product_prod_nat_nat > a] :
( ( edmond1517640972ysis_a @ C )
=> ( finite_Graph_a @ C ) ) ).
% ek_analysis.axioms
thf(fact_24_ek__analysis__defs_OspEdges_Ocong,axiom,
edmond475474835dges_a = edmond475474835dges_a ).
% ek_analysis_defs.spEdges.cong
thf(fact_25_ek__analysis__defs_OuE_Ocong,axiom,
edmond771116670s_uE_a = edmond771116670s_uE_a ).
% ek_analysis_defs.uE.cong
thf(fact_26_ek__analysis_OspEdges__ss__E,axiom,
! [C: product_prod_nat_nat > a,S: nat,T: nat] :
( ( edmond1517640972ysis_a @ C )
=> ( ord_le841296385at_nat @ ( edmond475474835dges_a @ C @ S @ T ) @ ( e_a @ C ) ) ) ).
% ek_analysis.spEdges_ss_E
thf(fact_27_ek__analysis_OE__ss__uE,axiom,
! [C: product_prod_nat_nat > a] :
( ( edmond1517640972ysis_a @ C )
=> ( ord_le841296385at_nat @ ( e_a @ C ) @ ( edmond771116670s_uE_a @ C ) ) ) ).
% ek_analysis.E_ss_uE
thf(fact_28_dual__order_Oantisym,axiom,
! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ B2 @ A2 )
=> ( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_29_dual__order_Oantisym,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_30_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_31_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] : Y = Z )
= ( ^ [A3: set_Pr1986765409at_nat,B3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ B3 @ A3 )
& ( ord_le841296385at_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_32_dual__order_Oeq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : Y = Z )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A3 )
& ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_33_dual__order_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : Y = Z )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_34_dual__order_Otrans,axiom,
! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ B2 @ A2 )
=> ( ( ord_le841296385at_nat @ C @ B2 )
=> ( ord_le841296385at_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_35_dual__order_Otrans,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_36_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_37_linorder__wlog,axiom,
! [P2: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_38_dual__order_Orefl,axiom,
! [A2: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_39_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_40_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_41_order__trans,axiom,
! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat,Z2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y2 )
=> ( ( ord_le841296385at_nat @ Y2 @ Z2 )
=> ( ord_le841296385at_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_42_order__trans,axiom,
! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_43_order__trans,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_44_order__class_Oorder_Oantisym,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( ord_le841296385at_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_45_order__class_Oorder_Oantisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_46_order__class_Oorder_Oantisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_47_ord__le__eq__trans,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_le841296385at_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_48_ord__le__eq__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_49_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_50_ord__eq__le__trans,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( A2 = B2 )
=> ( ( ord_le841296385at_nat @ B2 @ C )
=> ( ord_le841296385at_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_51_ord__eq__le__trans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_52_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_53_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] : Y = Z )
= ( ^ [A3: set_Pr1986765409at_nat,B3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A3 @ B3 )
& ( ord_le841296385at_nat @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_54_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : Y = Z )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_55_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : Y = Z )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_56_antisym__conv,axiom,
! [Y2: set_Pr1986765409at_nat,X2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ Y2 @ X2 )
=> ( ( ord_le841296385at_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv
thf(fact_57_antisym__conv,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv
thf(fact_58_antisym__conv,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ).
% antisym_conv
thf(fact_59_le__cases3,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_60_order_Otrans,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( ord_le841296385at_nat @ B2 @ C )
=> ( ord_le841296385at_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_61_order_Otrans,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_62_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_63_le__cases,axiom,
! [X2: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% le_cases
thf(fact_64_eq__refl,axiom,
! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( X2 = Y2 )
=> ( ord_le841296385at_nat @ X2 @ Y2 ) ) ).
% eq_refl
thf(fact_65_eq__refl,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_set_nat @ X2 @ Y2 ) ) ).
% eq_refl
thf(fact_66_eq__refl,axiom,
! [X2: nat,Y2: nat] :
( ( X2 = Y2 )
=> ( ord_less_eq_nat @ X2 @ Y2 ) ) ).
% eq_refl
thf(fact_67_linear,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ).
% linear
thf(fact_68_antisym,axiom,
! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y2 )
=> ( ( ord_le841296385at_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% antisym
thf(fact_69_antisym,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% antisym
thf(fact_70_antisym,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ).
% antisym
thf(fact_71_eq__iff,axiom,
( ( ^ [Y: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] : Y = Z )
= ( ^ [X3: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X3 @ Y3 )
& ( ord_le841296385at_nat @ Y3 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_72_eq__iff,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : Y = Z )
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
& ( ord_less_eq_set_nat @ Y3 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_73_eq__iff,axiom,
( ( ^ [Y: nat,Z: nat] : Y = Z )
= ( ^ [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_74_ord__le__eq__subst,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_Pr1986765409at_nat,Y4: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X @ Y4 )
=> ( ord_le841296385at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le841296385at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_75_ord__le__eq__subst,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_nat,C: set_nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_Pr1986765409at_nat,Y4: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_76_ord__le__eq__subst,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > nat,C: nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_Pr1986765409at_nat,Y4: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_77_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_le841296385at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le841296385at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_78_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_79_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_80_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le841296385at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le841296385at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_81_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_82_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_83_ord__eq__le__subst,axiom,
! [A2: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le841296385at_nat @ B2 @ C )
=> ( ! [X: set_Pr1986765409at_nat,Y4: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X @ Y4 )
=> ( ord_le841296385at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le841296385at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_84_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_Pr1986765409at_nat > set_nat,B2: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le841296385at_nat @ B2 @ C )
=> ( ! [X: set_Pr1986765409at_nat,Y4: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_85_ord__eq__le__subst,axiom,
! [A2: nat,F: set_Pr1986765409at_nat > nat,B2: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_le841296385at_nat @ B2 @ C )
=> ( ! [X: set_Pr1986765409at_nat,Y4: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_86_ord__eq__le__subst,axiom,
! [A2: set_Pr1986765409at_nat,F: set_nat > set_Pr1986765409at_nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_le841296385at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le841296385at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_87_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_88_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_89_ord__eq__le__subst,axiom,
! [A2: set_Pr1986765409at_nat,F: nat > set_Pr1986765409at_nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le841296385at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le841296385at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_90_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_91_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_92_order__subst2,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( ord_le841296385at_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_Pr1986765409at_nat,Y4: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X @ Y4 )
=> ( ord_le841296385at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le841296385at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_93_order__subst2,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_nat,C: set_nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_Pr1986765409at_nat,Y4: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_94_order__subst2,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > nat,C: nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_Pr1986765409at_nat,Y4: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_95_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_le841296385at_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_le841296385at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le841296385at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_96_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_97_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_98_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_le841296385at_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le841296385at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le841296385at_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_99_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_100_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_101_mem__Collect__eq,axiom,
! [A2: nat,P2: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_102_mem__Collect__eq,axiom,
! [A2: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
( ( member701585322at_nat @ A2 @ ( collec7649004at_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_103_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_104_Collect__mem__eq,axiom,
! [A: set_Pr1986765409at_nat] :
( ( collec7649004at_nat
@ ^ [X3: product_prod_nat_nat] : ( member701585322at_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_105_order__subst1,axiom,
! [A2: set_Pr1986765409at_nat,F: set_Pr1986765409at_nat > set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le841296385at_nat @ B2 @ C )
=> ( ! [X: set_Pr1986765409at_nat,Y4: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X @ Y4 )
=> ( ord_le841296385at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le841296385at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_106_order__subst1,axiom,
! [A2: set_Pr1986765409at_nat,F: set_nat > set_Pr1986765409at_nat,B2: set_nat,C: set_nat] :
( ( ord_le841296385at_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_le841296385at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le841296385at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_107_order__subst1,axiom,
! [A2: set_Pr1986765409at_nat,F: nat > set_Pr1986765409at_nat,B2: nat,C: nat] :
( ( ord_le841296385at_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_le841296385at_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_le841296385at_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_108_order__subst1,axiom,
! [A2: set_nat,F: set_Pr1986765409at_nat > set_nat,B2: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le841296385at_nat @ B2 @ C )
=> ( ! [X: set_Pr1986765409at_nat,Y4: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_109_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_110_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_set_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_111_order__subst1,axiom,
! [A2: nat,F: set_Pr1986765409at_nat > nat,B2: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_le841296385at_nat @ B2 @ C )
=> ( ! [X: set_Pr1986765409at_nat,Y4: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_112_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ! [X: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_113_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y4: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_114_Collect__mono__iff,axiom,
! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( ord_le841296385at_nat @ ( collec7649004at_nat @ P2 ) @ ( collec7649004at_nat @ Q ) )
= ( ! [X3: product_prod_nat_nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_115_Collect__mono__iff,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
= ( ! [X3: nat] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_116_set__eq__subset,axiom,
( ( ^ [Y: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] : Y = Z )
= ( ^ [A5: set_Pr1986765409at_nat,B5: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A5 @ B5 )
& ( ord_le841296385at_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_117_set__eq__subset,axiom,
( ( ^ [Y: set_nat,Z: set_nat] : Y = Z )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_118_subset__trans,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,C2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ( ord_le841296385at_nat @ B @ C2 )
=> ( ord_le841296385at_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_119_subset__trans,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_120_Collect__mono,axiom,
! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X: product_prod_nat_nat] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_le841296385at_nat @ ( collec7649004at_nat @ P2 ) @ ( collec7649004at_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_121_Collect__mono,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P2 @ X )
=> ( Q @ X ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_122_subset__refl,axiom,
! [A: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ A @ A ) ).
% subset_refl
thf(fact_123_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_124_subset__iff,axiom,
( ord_le841296385at_nat
= ( ^ [A5: set_Pr1986765409at_nat,B5: set_Pr1986765409at_nat] :
! [T2: product_prod_nat_nat] :
( ( member701585322at_nat @ T2 @ A5 )
=> ( member701585322at_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_125_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_126_equalityD2,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( A = B )
=> ( ord_le841296385at_nat @ B @ A ) ) ).
% equalityD2
thf(fact_127_equalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% equalityD2
thf(fact_128_equalityD1,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( A = B )
=> ( ord_le841296385at_nat @ A @ B ) ) ).
% equalityD1
thf(fact_129_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_130_subset__eq,axiom,
( ord_le841296385at_nat
= ( ^ [A5: set_Pr1986765409at_nat,B5: set_Pr1986765409at_nat] :
! [X3: product_prod_nat_nat] :
( ( member701585322at_nat @ X3 @ A5 )
=> ( member701585322at_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_131_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_132_equalityE,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( A = B )
=> ~ ( ( ord_le841296385at_nat @ A @ B )
=> ~ ( ord_le841296385at_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_133_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_134_subsetD,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,C: product_prod_nat_nat] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ( member701585322at_nat @ C @ A )
=> ( member701585322at_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_135_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_136_in__mono,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,X2: product_prod_nat_nat] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ( member701585322at_nat @ X2 @ A )
=> ( member701585322at_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_137_in__mono,axiom,
! [A: set_nat,B: set_nat,X2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) ) ) ).
% in_mono
thf(fact_138_reachable__ss__V,axiom,
! [S: nat] :
( ( member_nat @ S @ ( v_a @ c ) )
=> ( ord_less_eq_set_nat @ ( reachableNodes_a @ c @ S ) @ ( v_a @ c ) ) ) ).
% reachable_ss_V
thf(fact_139_transfer__path,axiom,
! [P: list_P559422087at_nat,C3: product_prod_nat_nat > a,U2: nat,V: nat] :
( ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ ( set_Pr2131844118at_nat @ P ) @ ( e_a @ c ) ) @ ( e_a @ C3 ) )
=> ( ( isPath_a @ c @ U2 @ P @ V )
=> ( isPath_a @ C3 @ U2 @ P @ V ) ) ) ).
% transfer_path
thf(fact_140_card__spEdges__le,axiom,
ord_less_eq_nat @ ( finite447719721at_nat @ ( edmond475474835dges_a @ c @ s @ t ) ) @ ( finite447719721at_nat @ ( edmond771116670s_uE_a @ c ) ) ).
% card_spEdges_le
thf(fact_141_Finite__Graph__EI,axiom,
( ( finite772653738at_nat @ ( e_a @ c ) )
=> ( finite_Graph_a @ c ) ) ).
% Finite_Graph_EI
thf(fact_142_Graph_Oincoming_H__edges,axiom,
! [C: product_prod_nat_nat > a,U: set_nat] : ( ord_le841296385at_nat @ ( incoming_a2 @ C @ U ) @ ( e_a @ C ) ) ).
% Graph.incoming'_edges
thf(fact_143_Graph_Ooutgoing_H__edges,axiom,
! [C: product_prod_nat_nat > a,U: set_nat] : ( ord_le841296385at_nat @ ( outgoing_a2 @ C @ U ) @ ( e_a @ C ) ) ).
% Graph.outgoing'_edges
thf(fact_144_Graph_Oincoming__edges,axiom,
! [C: product_prod_nat_nat > a,U2: nat] : ( ord_le841296385at_nat @ ( incoming_a @ C @ U2 ) @ ( e_a @ C ) ) ).
% Graph.incoming_edges
thf(fact_145_Graph_Ooutgoing__edges,axiom,
! [C: product_prod_nat_nat > a,U2: nat] : ( ord_le841296385at_nat @ ( outgoing_a @ C @ U2 ) @ ( e_a @ C ) ) ).
% Graph.outgoing_edges
thf(fact_146_Graph_OisPath__edgeset,axiom,
! [C: product_prod_nat_nat > a,U2: nat,P: list_P559422087at_nat,V: nat,E: product_prod_nat_nat] :
( ( isPath_a @ C @ U2 @ P @ V )
=> ( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
=> ( member701585322at_nat @ E @ ( e_a @ C ) ) ) ) ).
% Graph.isPath_edgeset
thf(fact_147_isPath__ex__edge2,axiom,
! [U2: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
( ( isPath_a @ c @ U2 @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( V1 != V )
=> ? [V2: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ V1 @ V2 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% isPath_ex_edge2
thf(fact_148_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_149_IntI,axiom,
! [C: product_prod_nat_nat,A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ C @ A )
=> ( ( member701585322at_nat @ C @ B )
=> ( member701585322at_nat @ C @ ( inf_in586391887at_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_150_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_151_Int__iff,axiom,
! [C: product_prod_nat_nat,A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ C @ ( inf_in586391887at_nat @ A @ B ) )
= ( ( member701585322at_nat @ C @ A )
& ( member701585322at_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_152_isPath__ex__edge1,axiom,
! [U2: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
( ( isPath_a @ c @ U2 @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( U1 != U2 )
=> ? [U22: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ U22 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% isPath_ex_edge1
thf(fact_153_reachableNodes__append__edge,axiom,
! [U2: nat,S: nat,V: nat] :
( ( member_nat @ U2 @ ( reachableNodes_a @ c @ S ) )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U2 @ V ) @ ( e_a @ c ) )
=> ( member_nat @ V @ ( reachableNodes_a @ c @ S ) ) ) ) ).
% reachableNodes_append_edge
thf(fact_154_Int__subset__iff,axiom,
! [C2: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ C2 @ ( inf_in586391887at_nat @ A @ B ) )
= ( ( ord_le841296385at_nat @ C2 @ A )
& ( ord_le841296385at_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_155_Int__subset__iff,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( ord_less_eq_set_nat @ C2 @ A )
& ( ord_less_eq_set_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_156_finite__E,axiom,
finite772653738at_nat @ ( e_a @ c ) ).
% finite_E
thf(fact_157_finite__uE,axiom,
finite772653738at_nat @ ( edmond771116670s_uE_a @ c ) ).
% finite_uE
thf(fact_158_finite__spEdges,axiom,
finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) ).
% finite_spEdges
thf(fact_159_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_160_IntE,axiom,
! [C: product_prod_nat_nat,A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ C @ ( inf_in586391887at_nat @ A @ B ) )
=> ~ ( ( member701585322at_nat @ C @ A )
=> ~ ( member701585322at_nat @ C @ B ) ) ) ).
% IntE
thf(fact_161_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_162_IntD1,axiom,
! [C: product_prod_nat_nat,A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ C @ ( inf_in586391887at_nat @ A @ B ) )
=> ( member701585322at_nat @ C @ A ) ) ).
% IntD1
thf(fact_163_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_164_IntD2,axiom,
! [C: product_prod_nat_nat,A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ C @ ( inf_in586391887at_nat @ A @ B ) )
=> ( member701585322at_nat @ C @ B ) ) ).
% IntD2
thf(fact_165_Graph_OreachableNodes__append__edge,axiom,
! [U2: nat,C: product_prod_nat_nat > a,S: nat,V: nat] :
( ( member_nat @ U2 @ ( reachableNodes_a @ C @ S ) )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U2 @ V ) @ ( e_a @ C ) )
=> ( member_nat @ V @ ( reachableNodes_a @ C @ S ) ) ) ) ).
% Graph.reachableNodes_append_edge
thf(fact_166_Graph_OreachableNodes_Ocong,axiom,
reachableNodes_a = reachableNodes_a ).
% Graph.reachableNodes.cong
thf(fact_167_Graph_Oadjacent__nodes_Ocong,axiom,
adjacent_nodes_a = adjacent_nodes_a ).
% Graph.adjacent_nodes.cong
thf(fact_168_Graph_Oreachable__ss__V,axiom,
! [S: nat,C: product_prod_nat_nat > a] :
( ( member_nat @ S @ ( v_a @ C ) )
=> ( ord_less_eq_set_nat @ ( reachableNodes_a @ C @ S ) @ ( v_a @ C ) ) ) ).
% Graph.reachable_ss_V
thf(fact_169_Int__mono,axiom,
! [A: set_Pr1986765409at_nat,C2: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,D: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ C2 )
=> ( ( ord_le841296385at_nat @ B @ D )
=> ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A @ B ) @ ( inf_in586391887at_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_170_Int__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C2 @ D ) ) ) ) ).
% Int_mono
thf(fact_171_Int__lower1,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_172_Int__lower1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_173_Int__lower2,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_174_Int__lower2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_175_Int__absorb1,axiom,
! [B: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ B @ A )
=> ( ( inf_in586391887at_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_176_Int__absorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_177_Int__absorb2,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ( inf_in586391887at_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_178_Int__absorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_179_Int__greatest,axiom,
! [C2: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ C2 @ A )
=> ( ( ord_le841296385at_nat @ C2 @ B )
=> ( ord_le841296385at_nat @ C2 @ ( inf_in586391887at_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_180_Int__greatest,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_181_Int__Collect__mono,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ! [X: product_prod_nat_nat] :
( ( member701585322at_nat @ X @ A )
=> ( ( P2 @ X )
=> ( Q @ X ) ) )
=> ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A @ ( collec7649004at_nat @ P2 ) ) @ ( inf_in586391887at_nat @ B @ ( collec7649004at_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_182_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( ( P2 @ X )
=> ( Q @ X ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P2 ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_183_Graph_Oadjacent__nodes__ss__V,axiom,
! [C: product_prod_nat_nat > a,U2: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_a @ C @ U2 ) @ ( v_a @ C ) ) ).
% Graph.adjacent_nodes_ss_V
thf(fact_184_Graph_OisPath__ex__edge2,axiom,
! [C: product_prod_nat_nat > a,U2: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
( ( isPath_a @ C @ U2 @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( V1 != V )
=> ? [V2: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ V1 @ V2 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% Graph.isPath_ex_edge2
thf(fact_185_Graph_OisPath__ex__edge1,axiom,
! [C: product_prod_nat_nat > a,U2: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
( ( isPath_a @ C @ U2 @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( U1 != U2 )
=> ? [U22: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ U22 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% Graph.isPath_ex_edge1
thf(fact_186_Graph_OFinite__Graph__EI,axiom,
! [C: product_prod_nat_nat > a] :
( ( finite772653738at_nat @ ( e_a @ C ) )
=> ( finite_Graph_a @ C ) ) ).
% Graph.Finite_Graph_EI
thf(fact_187_Finite__Graph_Ofinite__E,axiom,
! [C: product_prod_nat_nat > a] :
( ( finite_Graph_a @ C )
=> ( finite772653738at_nat @ ( e_a @ C ) ) ) ).
% Finite_Graph.finite_E
thf(fact_188_ek__analysis_Ofinite__spEdges,axiom,
! [C: product_prod_nat_nat > a,S: nat,T: nat] :
( ( edmond1517640972ysis_a @ C )
=> ( finite772653738at_nat @ ( edmond475474835dges_a @ C @ S @ T ) ) ) ).
% ek_analysis.finite_spEdges
thf(fact_189_ek__analysis_Ofinite__uE,axiom,
! [C: product_prod_nat_nat > a] :
( ( edmond1517640972ysis_a @ C )
=> ( finite772653738at_nat @ ( edmond771116670s_uE_a @ C ) ) ) ).
% ek_analysis.finite_uE
thf(fact_190_Graph_Otransfer__path,axiom,
! [P: list_P559422087at_nat,C: product_prod_nat_nat > a,C3: product_prod_nat_nat > a,U2: nat,V: nat] :
( ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ ( set_Pr2131844118at_nat @ P ) @ ( e_a @ C ) ) @ ( e_a @ C3 ) )
=> ( ( isPath_a @ C @ U2 @ P @ V )
=> ( isPath_a @ C3 @ U2 @ P @ V ) ) ) ).
% Graph.transfer_path
thf(fact_191_Graph_OE_Ocong,axiom,
e_a = e_a ).
% Graph.E.cong
thf(fact_192_Graph_OisPath_Ocong,axiom,
isPath_a = isPath_a ).
% Graph.isPath.cong
thf(fact_193_Graph_OV_Ocong,axiom,
v_a = v_a ).
% Graph.V.cong
thf(fact_194_ek__analysis_Ocard__spEdges__le,axiom,
! [C: product_prod_nat_nat > a,S: nat,T: nat] :
( ( edmond1517640972ysis_a @ C )
=> ( ord_less_eq_nat @ ( finite447719721at_nat @ ( edmond475474835dges_a @ C @ S @ T ) ) @ ( finite447719721at_nat @ ( edmond771116670s_uE_a @ C ) ) ) ) ).
% ek_analysis.card_spEdges_le
thf(fact_195_Graph_OisShortestPath_Ocong,axiom,
isShortestPath_a = isShortestPath_a ).
% Graph.isShortestPath.cong
thf(fact_196_Graph_Ooutgoing_Ocong,axiom,
outgoing_a = outgoing_a ).
% Graph.outgoing.cong
thf(fact_197_Graph_Oincoming_Ocong,axiom,
incoming_a = incoming_a ).
% Graph.incoming.cong
thf(fact_198_Graph_Ooutgoing_H_Ocong,axiom,
outgoing_a2 = outgoing_a2 ).
% Graph.outgoing'.cong
thf(fact_199_Graph_Oincoming_H_Ocong,axiom,
incoming_a2 = incoming_a2 ).
% Graph.incoming'.cong
thf(fact_200_Graph_OshortestPath__is__path,axiom,
! [C: product_prod_nat_nat > a,U2: nat,P: list_P559422087at_nat,V: nat] :
( ( isShortestPath_a @ C @ U2 @ P @ V )
=> ( isPath_a @ C @ U2 @ P @ V ) ) ).
% Graph.shortestPath_is_path
thf(fact_201_finite__Int,axiom,
! [F2: set_Pr1986765409at_nat,G: set_Pr1986765409at_nat] :
( ( ( finite772653738at_nat @ F2 )
| ( finite772653738at_nat @ G ) )
=> ( finite772653738at_nat @ ( inf_in586391887at_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_202_finite__Int,axiom,
! [F2: set_nat,G: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).
% finite_Int
thf(fact_203_List_Ofinite__set,axiom,
! [Xs: list_P559422087at_nat] : ( finite772653738at_nat @ ( set_Pr2131844118at_nat @ Xs ) ) ).
% List.finite_set
thf(fact_204_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_205_le__inf__iff,axiom,
! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat,Z2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ ( inf_in586391887at_nat @ Y2 @ Z2 ) )
= ( ( ord_le841296385at_nat @ X2 @ Y2 )
& ( ord_le841296385at_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_206_le__inf__iff,axiom,
! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z2 ) )
= ( ( ord_less_eq_set_nat @ X2 @ Y2 )
& ( ord_less_eq_set_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_207_le__inf__iff,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z2 ) )
= ( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_208_inf_Obounded__iff,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ ( inf_in586391887at_nat @ B2 @ C ) )
= ( ( ord_le841296385at_nat @ A2 @ B2 )
& ( ord_le841296385at_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_209_inf_Obounded__iff,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
= ( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_210_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_211_card__mono,axiom,
! [B: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ B )
=> ( ( ord_le841296385at_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite447719721at_nat @ A ) @ ( finite447719721at_nat @ B ) ) ) ) ).
% card_mono
thf(fact_212_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_213_card__seteq,axiom,
! [B: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ B )
=> ( ( ord_le841296385at_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite447719721at_nat @ B ) @ ( finite447719721at_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_214_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_215_Efin__imp__Vfin,axiom,
( ( finite772653738at_nat @ ( e_a @ c ) )
=> ( finite_finite_nat @ ( v_a @ c ) ) ) ).
% Efin_imp_Vfin
thf(fact_216_finite__V,axiom,
finite_finite_nat @ ( v_a @ c ) ).
% finite_V
thf(fact_217_adjacent__nodes__finite,axiom,
! [U2: nat] : ( finite_finite_nat @ ( adjacent_nodes_a @ c @ U2 ) ) ).
% adjacent_nodes_finite
thf(fact_218_Vfin__imp__Efin,axiom,
( ( finite_finite_nat @ ( v_a @ c ) )
=> ( finite772653738at_nat @ ( e_a @ c ) ) ) ).
% Vfin_imp_Efin
thf(fact_219_finite__incoming,axiom,
! [U2: nat] :
( ( finite_finite_nat @ ( v_a @ c ) )
=> ( finite772653738at_nat @ ( incoming_a @ c @ U2 ) ) ) ).
% finite_incoming
thf(fact_220_finite__outgoing,axiom,
! [U2: nat] :
( ( finite_finite_nat @ ( v_a @ c ) )
=> ( finite772653738at_nat @ ( outgoing_a @ c @ U2 ) ) ) ).
% finite_outgoing
thf(fact_221_finite__incoming_H,axiom,
! [U: set_nat] :
( ( finite_finite_nat @ ( v_a @ c ) )
=> ( finite772653738at_nat @ ( incoming_a2 @ c @ U ) ) ) ).
% finite_incoming'
thf(fact_222_finite__outgoing_H,axiom,
! [U: set_nat] :
( ( finite_finite_nat @ ( v_a @ c ) )
=> ( finite772653738at_nat @ ( outgoing_a2 @ c @ U ) ) ) ).
% finite_outgoing'
thf(fact_223_Finite__Graph__def,axiom,
( finite_Graph_a
= ( ^ [C4: product_prod_nat_nat > a] : ( finite_finite_nat @ ( v_a @ C4 ) ) ) ) ).
% Finite_Graph_def
thf(fact_224_Finite__Graph_Ointro,axiom,
! [C: product_prod_nat_nat > a] :
( ( finite_finite_nat @ ( v_a @ C ) )
=> ( finite_Graph_a @ C ) ) ).
% Finite_Graph.intro
thf(fact_225_Finite__Graph_Ofinite__V,axiom,
! [C: product_prod_nat_nat > a] :
( ( finite_Graph_a @ C )
=> ( finite_finite_nat @ ( v_a @ C ) ) ) ).
% Finite_Graph.finite_V
thf(fact_226_Finite__Graph_Oadjacent__nodes__finite,axiom,
! [C: product_prod_nat_nat > a,U2: nat] :
( ( finite_Graph_a @ C )
=> ( finite_finite_nat @ ( adjacent_nodes_a @ C @ U2 ) ) ) ).
% Finite_Graph.adjacent_nodes_finite
thf(fact_227_Graph_OEfin__imp__Vfin,axiom,
! [C: product_prod_nat_nat > a] :
( ( finite772653738at_nat @ ( e_a @ C ) )
=> ( finite_finite_nat @ ( v_a @ C ) ) ) ).
% Graph.Efin_imp_Vfin
thf(fact_228_Graph_OVfin__imp__Efin,axiom,
! [C: product_prod_nat_nat > a] :
( ( finite_finite_nat @ ( v_a @ C ) )
=> ( finite772653738at_nat @ ( e_a @ C ) ) ) ).
% Graph.Vfin_imp_Efin
thf(fact_229_Graph_Ofinite__outgoing,axiom,
! [C: product_prod_nat_nat > a,U2: nat] :
( ( finite_finite_nat @ ( v_a @ C ) )
=> ( finite772653738at_nat @ ( outgoing_a @ C @ U2 ) ) ) ).
% Graph.finite_outgoing
thf(fact_230_Graph_Ofinite__incoming,axiom,
! [C: product_prod_nat_nat > a,U2: nat] :
( ( finite_finite_nat @ ( v_a @ C ) )
=> ( finite772653738at_nat @ ( incoming_a @ C @ U2 ) ) ) ).
% Graph.finite_incoming
thf(fact_231_Graph_Ofinite__outgoing_H,axiom,
! [C: product_prod_nat_nat > a,U: set_nat] :
( ( finite_finite_nat @ ( v_a @ C ) )
=> ( finite772653738at_nat @ ( outgoing_a2 @ C @ U ) ) ) ).
% Graph.finite_outgoing'
thf(fact_232_Graph_Ofinite__incoming_H,axiom,
! [C: product_prod_nat_nat > a,U: set_nat] :
( ( finite_finite_nat @ ( v_a @ C ) )
=> ( finite772653738at_nat @ ( incoming_a2 @ C @ U ) ) ) ).
% Graph.finite_incoming'
thf(fact_233_finite__has__minimal2,axiom,
! [A: set_se1612935105at_nat,A2: set_Pr1986765409at_nat] :
( ( finite1457549322at_nat @ A )
=> ( ( member298845450at_nat @ A2 @ A )
=> ? [X: set_Pr1986765409at_nat] :
( ( member298845450at_nat @ X @ A )
& ( ord_le841296385at_nat @ X @ A2 )
& ! [Xa: set_Pr1986765409at_nat] :
( ( member298845450at_nat @ Xa @ A )
=> ( ( ord_le841296385at_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_234_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite2012248349et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ X @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_235_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ X @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_236_finite__has__maximal2,axiom,
! [A: set_se1612935105at_nat,A2: set_Pr1986765409at_nat] :
( ( finite1457549322at_nat @ A )
=> ( ( member298845450at_nat @ A2 @ A )
=> ? [X: set_Pr1986765409at_nat] :
( ( member298845450at_nat @ X @ A )
& ( ord_le841296385at_nat @ A2 @ X )
& ! [Xa: set_Pr1986765409at_nat] :
( ( member298845450at_nat @ Xa @ A )
=> ( ( ord_le841296385at_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_237_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite2012248349et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X: set_nat] :
( ( member_set_nat @ X @ A )
& ( ord_less_eq_set_nat @ A2 @ X )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_238_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X: nat] :
( ( member_nat @ X @ A )
& ( ord_less_eq_nat @ A2 @ X )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_239_inf_OcoboundedI2,axiom,
! [B2: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ B2 @ C )
=> ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_240_inf_OcoboundedI2,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_241_inf_OcoboundedI2,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_242_inf_OcoboundedI1,axiom,
! [A2: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ C )
=> ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_243_inf_OcoboundedI1,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_244_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_245_inf_Oabsorb__iff2,axiom,
( ord_le841296385at_nat
= ( ^ [B3: set_Pr1986765409at_nat,A3: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_246_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B3: set_nat,A3: set_nat] :
( ( inf_inf_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_247_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= B3 ) ) ) ).
% inf.absorb_iff2
thf(fact_248_inf_Oabsorb__iff1,axiom,
( ord_le841296385at_nat
= ( ^ [A3: set_Pr1986765409at_nat,B3: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_249_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( inf_inf_set_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_250_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( inf_inf_nat @ A3 @ B3 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_251_inf_Ocobounded2,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_252_inf_Ocobounded2,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_253_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_254_inf_Ocobounded1,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_255_inf_Ocobounded1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_256_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_257_inf_Oorder__iff,axiom,
( ord_le841296385at_nat
= ( ^ [A3: set_Pr1986765409at_nat,B3: set_Pr1986765409at_nat] :
( A3
= ( inf_in586391887at_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_258_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( A3
= ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_259_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( A3
= ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).
% inf.order_iff
thf(fact_260_inf__greatest,axiom,
! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat,Z2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y2 )
=> ( ( ord_le841296385at_nat @ X2 @ Z2 )
=> ( ord_le841296385at_nat @ X2 @ ( inf_in586391887at_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_261_inf__greatest,axiom,
! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_262_inf__greatest,axiom,
! [X2: nat,Y2: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_263_inf_OboundedI,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( ord_le841296385at_nat @ A2 @ C )
=> ( ord_le841296385at_nat @ A2 @ ( inf_in586391887at_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_264_inf_OboundedI,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_265_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_266_inf_OboundedE,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ ( inf_in586391887at_nat @ B2 @ C ) )
=> ~ ( ( ord_le841296385at_nat @ A2 @ B2 )
=> ~ ( ord_le841296385at_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_267_inf_OboundedE,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_268_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_269_inf__absorb2,axiom,
! [Y2: set_Pr1986765409at_nat,X2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ Y2 @ X2 )
=> ( ( inf_in586391887at_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_270_inf__absorb2,axiom,
! [Y2: set_nat,X2: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( inf_inf_set_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_271_inf__absorb2,axiom,
! [Y2: nat,X2: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= Y2 ) ) ).
% inf_absorb2
thf(fact_272_inf__absorb1,axiom,
! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y2 )
=> ( ( inf_in586391887at_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_273_inf__absorb1,axiom,
! [X2: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y2 )
=> ( ( inf_inf_set_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_274_inf__absorb1,axiom,
! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= X2 ) ) ).
% inf_absorb1
thf(fact_275_inf_Oabsorb2,axiom,
! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ B2 @ A2 )
=> ( ( inf_in586391887at_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_276_inf_Oabsorb2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_277_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_278_inf_Oabsorb1,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( inf_in586391887at_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_279_inf_Oabsorb1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_280_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_281_le__iff__inf,axiom,
( ord_le841296385at_nat
= ( ^ [X3: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_282_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_283_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y3: nat] :
( ( inf_inf_nat @ X3 @ Y3 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_284_inf__unique,axiom,
! [F: set_Pr1986765409at_nat > set_Pr1986765409at_nat > set_Pr1986765409at_nat,X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ! [X: set_Pr1986765409at_nat,Y4: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( F @ X @ Y4 ) @ X )
=> ( ! [X: set_Pr1986765409at_nat,Y4: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( F @ X @ Y4 ) @ Y4 )
=> ( ! [X: set_Pr1986765409at_nat,Y4: set_Pr1986765409at_nat,Z3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X @ Y4 )
=> ( ( ord_le841296385at_nat @ X @ Z3 )
=> ( ord_le841296385at_nat @ X @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_in586391887at_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_285_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X2: set_nat,Y2: set_nat] :
( ! [X: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( F @ X @ Y4 ) @ X )
=> ( ! [X: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( F @ X @ Y4 ) @ Y4 )
=> ( ! [X: set_nat,Y4: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y4 )
=> ( ( ord_less_eq_set_nat @ X @ Z3 )
=> ( ord_less_eq_set_nat @ X @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_set_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_286_inf__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y2: nat] :
( ! [X: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X @ Y4 ) @ X )
=> ( ! [X: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X @ Y4 ) @ Y4 )
=> ( ! [X: nat,Y4: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y4 )
=> ( ( ord_less_eq_nat @ X @ Z3 )
=> ( ord_less_eq_nat @ X @ ( F @ Y4 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X2 @ Y2 )
= ( F @ X2 @ Y2 ) ) ) ) ) ).
% inf_unique
thf(fact_287_inf_OorderI,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( A2
= ( inf_in586391887at_nat @ A2 @ B2 ) )
=> ( ord_le841296385at_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_288_inf_OorderI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2
= ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_289_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_290_inf_OorderE,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( A2
= ( inf_in586391887at_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_291_inf_OorderE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_292_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_293_le__infI2,axiom,
! [B2: set_Pr1986765409at_nat,X2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ B2 @ X2 )
=> ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_294_le__infI2,axiom,
! [B2: set_nat,X2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ X2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_295_le__infI2,axiom,
! [B2: nat,X2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI2
thf(fact_296_le__infI1,axiom,
! [A2: set_Pr1986765409at_nat,X2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ X2 )
=> ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_297_le__infI1,axiom,
! [A2: set_nat,X2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_298_le__infI1,axiom,
! [A2: nat,X2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X2 ) ) ).
% le_infI1
thf(fact_299_inf__mono,axiom,
! [A2: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,D2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ C )
=> ( ( ord_le841296385at_nat @ B2 @ D2 )
=> ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A2 @ B2 ) @ ( inf_in586391887at_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_300_inf__mono,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_301_inf__mono,axiom,
! [A2: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).
% inf_mono
thf(fact_302_le__infI,axiom,
! [X2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ A2 )
=> ( ( ord_le841296385at_nat @ X2 @ B2 )
=> ( ord_le841296385at_nat @ X2 @ ( inf_in586391887at_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_303_le__infI,axiom,
! [X2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ B2 )
=> ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_304_le__infI,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ B2 )
=> ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_305_le__infE,axiom,
! [X2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ ( inf_in586391887at_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le841296385at_nat @ X2 @ A2 )
=> ~ ( ord_le841296385at_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_306_le__infE,axiom,
! [X2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_set_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_307_le__infE,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X2 @ A2 )
=> ~ ( ord_less_eq_nat @ X2 @ B2 ) ) ) ).
% le_infE
thf(fact_308_inf__le2,axiom,
! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_309_inf__le2,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_310_inf__le2,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_le2
thf(fact_311_inf__le1,axiom,
! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_312_inf__le1,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_313_inf__le1,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_le1
thf(fact_314_inf__sup__ord_I1_J,axiom,
! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_315_inf__sup__ord_I1_J,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_316_inf__sup__ord_I1_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_317_inf__sup__ord_I2_J,axiom,
! [X2: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_318_inf__sup__ord_I2_J,axiom,
! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_319_inf__sup__ord_I2_J,axiom,
! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ Y2 ) ).
% inf_sup_ord(2)
thf(fact_320_rev__finite__subset,axiom,
! [B: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ B )
=> ( ( ord_le841296385at_nat @ A @ B )
=> ( finite772653738at_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_321_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_322_infinite__super,axiom,
! [S2: set_Pr1986765409at_nat,T3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ S2 @ T3 )
=> ( ~ ( finite772653738at_nat @ S2 )
=> ~ ( finite772653738at_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_323_infinite__super,axiom,
! [S2: set_nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T3 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T3 ) ) ) ).
% infinite_super
thf(fact_324_finite__subset,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ( finite772653738at_nat @ B )
=> ( finite772653738at_nat @ A ) ) ) ).
% finite_subset
thf(fact_325_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_326_finite__list,axiom,
! [A: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ A )
=> ? [Xs2: list_P559422087at_nat] :
( ( set_Pr2131844118at_nat @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_327_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_328_subset__code_I1_J,axiom,
! [Xs: list_P559422087at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ ( set_Pr2131844118at_nat @ Xs ) @ B )
= ( ! [X3: product_prod_nat_nat] :
( ( member701585322at_nat @ X3 @ ( set_Pr2131844118at_nat @ Xs ) )
=> ( member701585322at_nat @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_329_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X3 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_330_infinite__arbitrarily__large,axiom,
! [A: set_Pr1986765409at_nat,N: nat] :
( ~ ( finite772653738at_nat @ A )
=> ? [B6: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ B6 )
& ( ( finite447719721at_nat @ B6 )
= N )
& ( ord_le841296385at_nat @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_331_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B6: set_nat] :
( ( finite_finite_nat @ B6 )
& ( ( finite_card_nat @ B6 )
= N )
& ( ord_less_eq_set_nat @ B6 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_332_card__subset__eq,axiom,
! [B: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ B )
=> ( ( ord_le841296385at_nat @ A @ B )
=> ( ( ( finite447719721at_nat @ A )
= ( finite447719721at_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_333_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_334_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_Pr1986765409at_nat,C2: nat] :
( ! [G2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ G2 @ F2 )
=> ( ( finite772653738at_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite447719721at_nat @ G2 ) @ C2 ) ) )
=> ( ( finite772653738at_nat @ F2 )
& ( ord_less_eq_nat @ ( finite447719721at_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_335_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G2: set_nat] :
( ( ord_less_eq_set_nat @ G2 @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G2 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_336_pathVertices__edgeset,axiom,
! [U2: nat,P: list_P559422087at_nat,V: nat] :
( ( member_nat @ U2 @ ( v_a @ c ) )
=> ( ( isPath_a @ c @ U2 @ P @ V )
=> ( ord_less_eq_set_nat @ ( set_nat2 @ ( pathVertices @ U2 @ P ) ) @ ( v_a @ c ) ) ) ) ).
% pathVertices_edgeset
thf(fact_337_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_Pr1986765409at_nat] :
( ( ord_less_eq_nat @ N @ ( finite447719721at_nat @ S2 ) )
=> ~ ! [T4: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ T4 @ S2 )
=> ( ( ( finite447719721at_nat @ T4 )
= N )
=> ~ ( finite772653738at_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_338_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S2 ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S2 )
=> ( ( ( finite_card_nat @ T4 )
= N )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_339_isPath_Osimps_I2_J,axiom,
! [U2: nat,X2: nat,Y2: nat,P: list_P559422087at_nat,V: nat] :
( ( isPath_a @ c @ U2 @ ( cons_P66992567at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ P ) @ V )
= ( ( U2 = X2 )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ X2 @ Y2 ) @ ( e_a @ c ) )
& ( isPath_a @ c @ Y2 @ P @ V ) ) ) ).
% isPath.simps(2)
thf(fact_340_isShortestPath__def,axiom,
! [U2: nat,P: list_P559422087at_nat,V: nat] :
( ( isShortestPath_a @ c @ U2 @ P @ V )
= ( ( isPath_a @ c @ U2 @ P @ V )
& ! [P3: list_P559422087at_nat] :
( ( isPath_a @ c @ U2 @ P3 @ V )
=> ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P ) @ ( size_s1990949619at_nat @ P3 ) ) ) ) ) ).
% isShortestPath_def
thf(fact_341_list_Oinject,axiom,
! [X21: product_prod_nat_nat,X22: list_P559422087at_nat,Y21: product_prod_nat_nat,Y22: list_P559422087at_nat] :
( ( ( cons_P66992567at_nat @ X21 @ X22 )
= ( cons_P66992567at_nat @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_342_impossible__Cons,axiom,
! [Xs: list_P559422087at_nat,Ys: list_P559422087at_nat,X2: product_prod_nat_nat] :
( ( ord_less_eq_nat @ ( size_s1990949619at_nat @ Xs ) @ ( size_s1990949619at_nat @ Ys ) )
=> ( Xs
!= ( cons_P66992567at_nat @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_343_list_Oset__cases,axiom,
! [E: nat,A2: list_nat] :
( ( member_nat @ E @ ( set_nat2 @ A2 ) )
=> ( ! [Z22: list_nat] :
( A2
!= ( cons_nat @ E @ Z22 ) )
=> ~ ! [Z1: nat,Z22: list_nat] :
( ( A2
= ( cons_nat @ Z1 @ Z22 ) )
=> ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_344_list_Oset__cases,axiom,
! [E: product_prod_nat_nat,A2: list_P559422087at_nat] :
( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ A2 ) )
=> ( ! [Z22: list_P559422087at_nat] :
( A2
!= ( cons_P66992567at_nat @ E @ Z22 ) )
=> ~ ! [Z1: product_prod_nat_nat,Z22: list_P559422087at_nat] :
( ( A2
= ( cons_P66992567at_nat @ Z1 @ Z22 ) )
=> ~ ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_345_bounded__Max__nat,axiom,
! [P2: nat > $o,X2: nat,M: nat] :
( ( P2 @ X2 )
=> ( ! [X: nat] :
( ( P2 @ X )
=> ( ord_less_eq_nat @ X @ M ) )
=> ~ ! [M2: nat] :
( ( P2 @ M2 )
=> ~ ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_346_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N2: set_nat] :
? [M3: nat] :
! [X3: nat] :
( ( member_nat @ X3 @ N2 )
=> ( ord_less_eq_nat @ X3 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_347_isPath_Oelims_I3_J,axiom,
! [X2: nat,Xa2: list_P559422087at_nat,Xb: nat] :
( ~ ( isPath_a @ c @ X2 @ Xa2 @ Xb )
=> ( ( ( Xa2 = nil_Pr1308055047at_nat )
=> ( X2 = Xb ) )
=> ~ ! [X: nat,Y4: nat,P4: list_P559422087at_nat] :
( ( Xa2
= ( cons_P66992567at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ P4 ) )
=> ( ( X2 = X )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ ( e_a @ c ) )
& ( isPath_a @ c @ Y4 @ P4 @ Xb ) ) ) ) ) ).
% isPath.elims(3)
thf(fact_348_isPath_Oelims_I2_J,axiom,
! [X2: nat,Xa2: list_P559422087at_nat,Xb: nat] :
( ( isPath_a @ c @ X2 @ Xa2 @ Xb )
=> ( ( ( Xa2 = nil_Pr1308055047at_nat )
=> ( X2 != Xb ) )
=> ~ ! [X: nat,Y4: nat,P4: list_P559422087at_nat] :
( ( Xa2
= ( cons_P66992567at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ P4 ) )
=> ~ ( ( X2 = X )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ ( e_a @ c ) )
& ( isPath_a @ c @ Y4 @ P4 @ Xb ) ) ) ) ) ).
% isPath.elims(2)
thf(fact_349_isPath_Osimps_I1_J,axiom,
! [U2: nat,V: nat] :
( ( isPath_a @ c @ U2 @ nil_Pr1308055047at_nat @ V )
= ( U2 = V ) ) ).
% isPath.simps(1)
thf(fact_350_isPath__fwd__cases,axiom,
! [S: nat,P: list_P559422087at_nat,T: nat] :
( ( isPath_a @ c @ S @ P @ T )
=> ( ( ( P = nil_Pr1308055047at_nat )
=> ( T != S ) )
=> ~ ! [P5: list_P559422087at_nat,U3: nat] :
( ( P
= ( cons_P66992567at_nat @ ( product_Pair_nat_nat @ S @ U3 ) @ P5 ) )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ S @ U3 ) @ ( e_a @ c ) )
=> ~ ( isPath_a @ c @ U3 @ P5 @ T ) ) ) ) ) ).
% isPath_fwd_cases
thf(fact_351_isPath_Oelims_I1_J,axiom,
! [X2: nat,Xa2: list_P559422087at_nat,Xb: nat,Y2: $o] :
( ( ( isPath_a @ c @ X2 @ Xa2 @ Xb )
= Y2 )
=> ( ( ( Xa2 = nil_Pr1308055047at_nat )
=> ( Y2
= ( X2 != Xb ) ) )
=> ~ ! [X: nat,Y4: nat,P4: list_P559422087at_nat] :
( ( Xa2
= ( cons_P66992567at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ P4 ) )
=> ( Y2
= ( ~ ( ( X2 = X )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ ( e_a @ c ) )
& ( isPath_a @ c @ Y4 @ P4 @ Xb ) ) ) ) ) ) ) ).
% isPath.elims(1)
thf(fact_352_Graph_OisPath_Oinduct,axiom,
! [P2: nat > list_P559422087at_nat > nat > $o,A0: nat,A1: list_P559422087at_nat,A22: nat] :
( ! [U3: nat,X_1: nat] : ( P2 @ U3 @ nil_Pr1308055047at_nat @ X_1 )
=> ( ! [U3: nat,X: nat,Y4: nat,P4: list_P559422087at_nat,V3: nat] :
( ( P2 @ Y4 @ P4 @ V3 )
=> ( P2 @ U3 @ ( cons_P66992567at_nat @ ( product_Pair_nat_nat @ X @ Y4 ) @ P4 ) @ V3 ) )
=> ( P2 @ A0 @ A1 @ A22 ) ) ) ).
% Graph.isPath.induct
% Conjectures (1)
thf(conj_0,conjecture,
ord_le841296385at_nat @ edges @ ( e_a @ c ) ).
%------------------------------------------------------------------------------