TPTP Problem File: ITP051^1.p
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%------------------------------------------------------------------------------
% File : ITP051^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer EdmondsKarp_Termination_Abstract problem prob_318__7590952_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_318__7590952_1 [Des21]
% Status : ContradictoryAxioms
% Rating : 0.40 v8.2.0, 0.23 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 430 ( 161 unt; 81 typ; 0 def)
% Number of atoms : 836 ( 202 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 2328 ( 57 ~; 4 |; 50 &;1885 @)
% ( 0 <=>; 332 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 285 ( 285 >; 0 *; 0 +; 0 <<)
% Number of symbols : 73 ( 71 usr; 7 con; 0-4 aty)
% Number of variables : 694 ( 62 ^; 594 !; 38 ?; 694 :)
% SPC : TH0_CAX_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:31:25.892
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_li664300135at_nat: $tType ).
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__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (71)
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_001tf__b,type,
edmond1517640973ysis_b: ( product_prod_nat_nat > b ) > $o ).
thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis__defs_OekMeasure_001tf__a,type,
edmond1022345716sure_a: ( product_prod_nat_nat > a ) > nat > nat > nat ).
thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis__defs_OekMeasure_001tf__b,type,
edmond1022345717sure_b: ( product_prod_nat_nat > b ) > nat > nat > nat ).
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_OspEdges_001tf__b,type,
edmond475474836dges_b: ( product_prod_nat_nat > b ) > 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_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis__defs_OuE_001tf__b,type,
edmond771116671s_uE_b: ( product_prod_nat_nat > b ) > set_Pr1986765409at_nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
finite83082927at_nat: set_li664300135at_nat > 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_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
finite_card_set_nat: set_set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
finite1701894793at_nat: set_se1612935105at_nat > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
finite1299096496at_nat: set_li664300135at_nat > $o ).
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_OFinite__Graph_001tf__b,type,
finite_Graph_b: ( product_prod_nat_nat > b ) > $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_OE_001tf__b,type,
e_b: ( product_prod_nat_nat > b ) > 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_OV_001tf__b,type,
v_b: ( product_prod_nat_nat > b ) > 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_Oadjacent__nodes_001tf__b,type,
adjacent_nodes_b: ( product_prod_nat_nat > b ) > nat > set_nat ).
thf(sy_c_Graph_OGraph_Oconnected_001tf__a,type,
connected_a: ( product_prod_nat_nat > a ) > nat > nat > $o ).
thf(sy_c_Graph_OGraph_Oconnected_001tf__b,type,
connected_b: ( product_prod_nat_nat > b ) > nat > nat > $o ).
thf(sy_c_Graph_OGraph_Odist_001tf__a,type,
dist_a: ( product_prod_nat_nat > a ) > nat > nat > nat > $o ).
thf(sy_c_Graph_OGraph_Odist_001tf__b,type,
dist_b: ( product_prod_nat_nat > b ) > nat > nat > nat > $o ).
thf(sy_c_Graph_OGraph_Oincoming_H_001tf__a,type,
incoming_a: ( product_prod_nat_nat > a ) > set_nat > set_Pr1986765409at_nat ).
thf(sy_c_Graph_OGraph_Oincoming_H_001tf__b,type,
incoming_b: ( product_prod_nat_nat > b ) > 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_OisPath_001tf__b,type,
isPath_b: ( product_prod_nat_nat > b ) > 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_OisShortestPath_001tf__b,type,
isShortestPath_b: ( product_prod_nat_nat > b ) > nat > list_P559422087at_nat > nat > $o ).
thf(sy_c_Graph_OGraph_OisSimplePath_001tf__a,type,
isSimplePath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
thf(sy_c_Graph_OGraph_OisSimplePath_001tf__b,type,
isSimplePath_b: ( product_prod_nat_nat > b ) > nat > list_P559422087at_nat > nat > $o ).
thf(sy_c_Graph_OGraph_Omin__dist_001tf__a,type,
min_dist_a: ( product_prod_nat_nat > a ) > nat > nat > nat ).
thf(sy_c_Graph_OGraph_Omin__dist_001tf__b,type,
min_dist_b: ( product_prod_nat_nat > b ) > nat > nat > nat ).
thf(sy_c_Graph_OGraph_Ooutgoing_H_001tf__a,type,
outgoing_a: ( product_prod_nat_nat > a ) > set_nat > set_Pr1986765409at_nat ).
thf(sy_c_Graph_OGraph_Ooutgoing_H_001tf__b,type,
outgoing_b: ( product_prod_nat_nat > b ) > set_nat > set_Pr1986765409at_nat ).
thf(sy_c_Graph_OGraph_OreachableNodes_001tf__a,type,
reachableNodes_a: ( product_prod_nat_nat > a ) > nat > set_nat ).
thf(sy_c_Graph_OGraph_OreachableNodes_001tf__b,type,
reachableNodes_b: ( product_prod_nat_nat > b ) > nat > set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: 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_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le116442893at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o ).
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_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le1613022364et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
ord_le2096002913at_nat: set_se1612935105at_nat > set_se1612935105at_nat > $o ).
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_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
collec1606769740at_nat: ( set_Pr1986765409at_nat > $o ) > set_se1612935105at_nat ).
thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member1608759472at_nat: list_P559422087at_nat > set_li664300135at_nat > $o ).
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_c_H,type,
c2: product_prod_nat_nat > b ).
thf(sy_v_edges,type,
edges: set_Pr1986765409at_nat ).
thf(sy_v_p,type,
p: list_P559422087at_nat ).
thf(sy_v_p_H____,type,
p2: list_P559422087at_nat ).
thf(sy_v_s,type,
s: nat ).
thf(sy_v_t,type,
t: nat ).
% Relevant facts (348)
thf(fact_0_ek__analysis__axioms,axiom,
edmond1517640972ysis_a @ c ).
% ek_analysis_axioms
thf(fact_1_LENP,axiom,
( ( size_s1990949619at_nat @ p )
= ( min_dist_a @ c @ s @ t ) ) ).
% LENP
thf(fact_2_LENP_H,axiom,
( ( size_s1990949619at_nat @ p2 )
= ( min_dist_b @ c2 @ s @ t ) ) ).
% LENP'
thf(fact_3_ek__analysis__defs_OekMeasure_Ocong,axiom,
edmond1022345717sure_b = edmond1022345717sure_b ).
% ek_analysis_defs.ekMeasure.cong
thf(fact_4_ek__analysis__defs_OekMeasure_Ocong,axiom,
edmond1022345716sure_a = edmond1022345716sure_a ).
% ek_analysis_defs.ekMeasure.cong
thf(fact_5_g_H_Oek__analysis__axioms,axiom,
edmond1517640973ysis_b @ c2 ).
% g'.ek_analysis_axioms
thf(fact_6_SHORTER,axiom,
ord_less_nat @ ( min_dist_b @ c2 @ s @ t ) @ ( min_dist_a @ c @ s @ t ) ).
% SHORTER
thf(fact_7_ek__analysis__defs_OuE_Ocong,axiom,
edmond771116671s_uE_b = edmond771116671s_uE_b ).
% ek_analysis_defs.uE.cong
thf(fact_8_ek__analysis__defs_OuE_Ocong,axiom,
edmond771116670s_uE_a = edmond771116670s_uE_a ).
% ek_analysis_defs.uE.cong
thf(fact_9_ek__analysis__defs_OspEdges_Ocong,axiom,
edmond475474835dges_a = edmond475474835dges_a ).
% ek_analysis_defs.spEdges.cong
thf(fact_10_ek__analysis__defs_OspEdges_Ocong,axiom,
edmond475474836dges_b = edmond475474836dges_b ).
% ek_analysis_defs.spEdges.cong
thf(fact_11_g_H_OFinite__Graph__axioms,axiom,
finite_Graph_b @ c2 ).
% g'.Finite_Graph_axioms
thf(fact_12_Finite__Graph__axioms,axiom,
finite_Graph_a @ c ).
% Finite_Graph_axioms
thf(fact_13__092_060open_062length_Ap_A_060_Alength_Ap_H_092_060close_062,axiom,
ord_less_nat @ ( size_s1990949619at_nat @ p ) @ ( size_s1990949619at_nat @ p2 ) ).
% \<open>length p < length p'\<close>
thf(fact_14_uE__eq,axiom,
( ( edmond771116671s_uE_b @ c2 )
= ( edmond771116670s_uE_a @ c ) ) ).
% uE_eq
thf(fact_15_SP,axiom,
isShortestPath_a @ c @ s @ p @ t ).
% SP
thf(fact_16_CONN2,axiom,
connected_b @ c2 @ s @ t ).
% CONN2
thf(fact_17_SV,axiom,
member_nat @ s @ ( v_a @ c ) ).
% SV
thf(fact_18_P_H,axiom,
isPath_b @ c2 @ s @ p2 @ t ).
% P'
thf(fact_19_ek__analysis__def,axiom,
edmond1517640972ysis_a = finite_Graph_a ).
% ek_analysis_def
thf(fact_20_ek__analysis__def,axiom,
edmond1517640973ysis_b = finite_Graph_b ).
% ek_analysis_def
thf(fact_21_ek__analysis_Ointro,axiom,
! [C: product_prod_nat_nat > a] :
( ( finite_Graph_a @ C )
=> ( edmond1517640972ysis_a @ C ) ) ).
% ek_analysis.intro
thf(fact_22_ek__analysis_Ointro,axiom,
! [C: product_prod_nat_nat > b] :
( ( finite_Graph_b @ C )
=> ( edmond1517640973ysis_b @ 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_Oaxioms,axiom,
! [C: product_prod_nat_nat > b] :
( ( edmond1517640973ysis_b @ C )
=> ( finite_Graph_b @ C ) ) ).
% ek_analysis.axioms
thf(fact_25__092_060open_062_092_060not_062_Ag_H_Oconnected_As_At_A_092_060Longrightarrow_062_Ag_H_OekMeasure_A_060_AekMeasure_092_060close_062,axiom,
( ~ ( connected_b @ c2 @ s @ t )
=> ( ord_less_nat @ ( edmond1022345717sure_b @ c2 @ s @ t ) @ ( edmond1022345716sure_a @ c @ s @ t ) ) ) ).
% \<open>\<not> g'.connected s t \<Longrightarrow> g'.ekMeasure < ekMeasure\<close>
thf(fact_26__092_060open_062g_H_Odist_As_A_Ig_H_Omin__dist_As_At_J_At_092_060close_062,axiom,
dist_b @ c2 @ s @ ( min_dist_b @ c2 @ s @ t ) @ t ).
% \<open>g'.dist s (g'.min_dist s t) t\<close>
thf(fact_27__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062p_H_O_A_092_060lbrakk_062g_H_OisPath_As_Ap_H_At_059_Alength_Ap_H_A_061_Ag_H_Omin__dist_As_At_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [P: list_P559422087at_nat] :
( ( isPath_b @ c2 @ s @ P @ t )
=> ( ( size_s1990949619at_nat @ P )
!= ( min_dist_b @ c2 @ s @ t ) ) ) ).
% \<open>\<And>thesis. (\<And>p'. \<lbrakk>g'.isPath s p' t; length p' = g'.min_dist s t\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_28_CONN,axiom,
connected_a @ c @ s @ t ).
% CONN
thf(fact_29_min__dist__less,axiom,
! [Src: nat,V: nat,D: nat,D2: nat] :
( ( connected_a @ c @ Src @ V )
=> ( ( ( min_dist_a @ c @ Src @ V )
= D )
=> ( ( ord_less_nat @ D2 @ D )
=> ? [V2: nat] :
( ( connected_a @ c @ Src @ V2 )
& ( ( min_dist_a @ c @ Src @ V2 )
= D2 ) ) ) ) ) ).
% min_dist_less
thf(fact_30_g_H_Omin__dist__less,axiom,
! [Src: nat,V: nat,D: nat,D2: nat] :
( ( connected_b @ c2 @ Src @ V )
=> ( ( ( min_dist_b @ c2 @ Src @ V )
= D )
=> ( ( ord_less_nat @ D2 @ D )
=> ? [V2: nat] :
( ( connected_b @ c2 @ Src @ V2 )
& ( ( min_dist_b @ c2 @ Src @ V2 )
= D2 ) ) ) ) ) ).
% g'.min_dist_less
thf(fact_31_length__induct,axiom,
! [P2: list_P559422087at_nat > $o,Xs: list_P559422087at_nat] :
( ! [Xs2: list_P559422087at_nat] :
( ! [Ys: list_P559422087at_nat] :
( ( ord_less_nat @ ( size_s1990949619at_nat @ Ys ) @ ( size_s1990949619at_nat @ Xs2 ) )
=> ( P2 @ Ys ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_32_finite__spEdges,axiom,
finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) ).
% finite_spEdges
thf(fact_33_g_H_Ofinite__spEdges,axiom,
finite772653738at_nat @ ( edmond475474836dges_b @ c2 @ s @ t ) ).
% g'.finite_spEdges
thf(fact_34_min__dist__z,axiom,
! [V: nat] :
( ( min_dist_a @ c @ V @ V )
= zero_zero_nat ) ).
% min_dist_z
thf(fact_35_g_H_Omin__dist__z,axiom,
! [V: nat] :
( ( min_dist_b @ c2 @ V @ V )
= zero_zero_nat ) ).
% g'.min_dist_z
thf(fact_36_g_H_Oconnected__def,axiom,
! [U: nat,V: nat] :
( ( connected_b @ c2 @ U @ V )
= ( ? [P3: list_P559422087at_nat] : ( isPath_b @ c2 @ U @ P3 @ V ) ) ) ).
% g'.connected_def
thf(fact_37_connected__inV__iff,axiom,
! [U: nat,V: nat] :
( ( connected_a @ c @ U @ V )
=> ( ( member_nat @ V @ ( v_a @ c ) )
= ( member_nat @ U @ ( v_a @ c ) ) ) ) ).
% connected_inV_iff
thf(fact_38_g_H_Oconnected__by__dist,axiom,
! [V: nat,V3: nat] :
( ( connected_b @ c2 @ V @ V3 )
= ( ? [D3: nat] : ( dist_b @ c2 @ V @ D3 @ V3 ) ) ) ).
% g'.connected_by_dist
thf(fact_39_obtain__shortest__path,axiom,
! [U: nat,V: nat] :
( ( connected_a @ c @ U @ V )
=> ~ ! [P4: list_P559422087at_nat] :
~ ( isShortestPath_a @ c @ U @ P4 @ V ) ) ).
% obtain_shortest_path
thf(fact_40_g_H_OisPath__distD,axiom,
! [U: nat,P5: list_P559422087at_nat,V: nat] :
( ( isPath_b @ c2 @ U @ P5 @ V )
=> ( dist_b @ c2 @ U @ ( size_s1990949619at_nat @ P5 ) @ V ) ) ).
% g'.isPath_distD
thf(fact_41_g_H_Odist__def,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist_b @ c2 @ V @ D @ V3 )
= ( ? [P3: list_P559422087at_nat] :
( ( isPath_b @ c2 @ V @ P3 @ V3 )
& ( ( size_s1990949619at_nat @ P3 )
= D ) ) ) ) ).
% g'.dist_def
thf(fact_42_g_H_Omin__dist__is__dist,axiom,
! [V: nat,V3: nat] :
( ( connected_b @ c2 @ V @ V3 )
=> ( dist_b @ c2 @ V @ ( min_dist_b @ c2 @ V @ V3 ) @ V3 ) ) ).
% g'.min_dist_is_dist
thf(fact_43_g_H_Oconnected__refl,axiom,
! [V: nat] : ( connected_b @ c2 @ V @ V ) ).
% g'.connected_refl
thf(fact_44_connected__refl,axiom,
! [V: nat] : ( connected_a @ c @ V @ V ) ).
% connected_refl
thf(fact_45_g_H_Odist__z__iff,axiom,
! [V: nat,V3: nat] :
( ( dist_b @ c2 @ V @ zero_zero_nat @ V3 )
= ( V3 = V ) ) ).
% g'.dist_z_iff
thf(fact_46_g_H_Odist__z,axiom,
! [V: nat] : ( dist_b @ c2 @ V @ zero_zero_nat @ V ) ).
% g'.dist_z
thf(fact_47_g_H_Oconnected__distI,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist_b @ c2 @ V @ D @ V3 )
=> ( connected_b @ c2 @ V @ V3 ) ) ).
% g'.connected_distI
thf(fact_48_g_H_Ofinite__uE,axiom,
finite772653738at_nat @ ( edmond771116671s_uE_b @ c2 ) ).
% g'.finite_uE
thf(fact_49_finite__uE,axiom,
finite772653738at_nat @ ( edmond771116670s_uE_a @ c ) ).
% finite_uE
thf(fact_50_g_H_Omin__dist__z__iff,axiom,
! [V: nat,V3: nat] :
( ( connected_b @ c2 @ V @ V3 )
=> ( ( ( min_dist_b @ c2 @ V @ V3 )
= zero_zero_nat )
= ( V3 = V ) ) ) ).
% g'.min_dist_z_iff
thf(fact_51_mem__Collect__eq,axiom,
! [A: set_Pr1986765409at_nat,P2: set_Pr1986765409at_nat > $o] :
( ( member298845450at_nat @ A @ ( collec1606769740at_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_52_mem__Collect__eq,axiom,
! [A: set_nat,P2: set_nat > $o] :
( ( member_set_nat @ A @ ( collect_set_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_53_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
( ( member701585322at_nat @ A @ ( collec7649004at_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_54_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_55_Collect__mem__eq,axiom,
! [A2: set_se1612935105at_nat] :
( ( collec1606769740at_nat
@ ^ [X: set_Pr1986765409at_nat] : ( member298845450at_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_56_Collect__mem__eq,axiom,
! [A2: set_set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( member_set_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_57_Collect__mem__eq,axiom,
! [A2: set_Pr1986765409at_nat] :
( ( collec7649004at_nat
@ ^ [X: product_prod_nat_nat] : ( member701585322at_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_Collect__cong,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P2 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_60_Collect__cong,axiom,
! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X2: product_prod_nat_nat] :
( ( P2 @ X2 )
= ( Q @ X2 ) )
=> ( ( collec7649004at_nat @ P2 )
= ( collec7649004at_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_61_min__dist__z__iff,axiom,
! [V: nat,V3: nat] :
( ( connected_a @ c @ V @ V3 )
=> ( ( ( min_dist_a @ c @ V @ V3 )
= zero_zero_nat )
= ( V3 = V ) ) ) ).
% min_dist_z_iff
thf(fact_62_ek__analysis_Ofinite__uE,axiom,
! [C: product_prod_nat_nat > b] :
( ( edmond1517640973ysis_b @ C )
=> ( finite772653738at_nat @ ( edmond771116671s_uE_b @ C ) ) ) ).
% ek_analysis.finite_uE
thf(fact_63_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_64_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_65_ek__analysis_Ofinite__spEdges,axiom,
! [C: product_prod_nat_nat > b,S: nat,T: nat] :
( ( edmond1517640973ysis_b @ C )
=> ( finite772653738at_nat @ ( edmond475474836dges_b @ C @ S @ T ) ) ) ).
% ek_analysis.finite_spEdges
thf(fact_66_neq__if__length__neq,axiom,
! [Xs: list_P559422087at_nat,Ys2: list_P559422087at_nat] :
( ( ( size_s1990949619at_nat @ Xs )
!= ( size_s1990949619at_nat @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_67_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_P559422087at_nat] :
( ( size_s1990949619at_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_68_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_69_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_70_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_71_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_72_Graph_OisShortestPath__min__dist__def,axiom,
( isShortestPath_a
= ( ^ [C2: product_prod_nat_nat > a,U2: nat,P3: list_P559422087at_nat,V4: nat] :
( ( isPath_a @ C2 @ U2 @ P3 @ V4 )
& ( ( size_s1990949619at_nat @ P3 )
= ( min_dist_a @ C2 @ U2 @ V4 ) ) ) ) ) ).
% Graph.isShortestPath_min_dist_def
thf(fact_73_Graph_OisShortestPath__min__dist__def,axiom,
( isShortestPath_b
= ( ^ [C2: product_prod_nat_nat > b,U2: nat,P3: list_P559422087at_nat,V4: nat] :
( ( isPath_b @ C2 @ U2 @ P3 @ V4 )
& ( ( size_s1990949619at_nat @ P3 )
= ( min_dist_b @ C2 @ U2 @ V4 ) ) ) ) ) ).
% Graph.isShortestPath_min_dist_def
thf(fact_74_g_H_OisShortestPath__min__dist__def,axiom,
! [U: nat,P5: list_P559422087at_nat,V: nat] :
( ( isShortestPath_b @ c2 @ U @ P5 @ V )
= ( ( isPath_b @ c2 @ U @ P5 @ V )
& ( ( size_s1990949619at_nat @ P5 )
= ( min_dist_b @ c2 @ U @ V ) ) ) ) ).
% g'.isShortestPath_min_dist_def
thf(fact_75_g_H_Omin__distI2,axiom,
! [V: nat,V3: nat,Q: nat > $o] :
( ( connected_b @ c2 @ V @ V3 )
=> ( ! [D4: nat] :
( ( dist_b @ c2 @ V @ D4 @ V3 )
=> ( ! [D5: nat] :
( ( dist_b @ c2 @ V @ D5 @ V3 )
=> ( ord_less_eq_nat @ D4 @ D5 ) )
=> ( Q @ D4 ) ) )
=> ( Q @ ( min_dist_b @ c2 @ V @ V3 ) ) ) ) ).
% g'.min_distI2
thf(fact_76_isShortestPath__length__less__V,axiom,
! [S: nat,P5: list_P559422087at_nat,T: nat] :
( ( member_nat @ S @ ( v_a @ c ) )
=> ( ( isShortestPath_a @ c @ S @ P5 @ T )
=> ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_a @ c ) ) ) ) ) ).
% isShortestPath_length_less_V
thf(fact_77_min__dist__less__V,axiom,
! [S: nat,T: nat] :
( ( member_nat @ S @ ( v_a @ c ) )
=> ( ( connected_a @ c @ S @ T )
=> ( ord_less_nat @ ( min_dist_a @ c @ S @ T ) @ ( finite_card_nat @ ( v_a @ c ) ) ) ) ) ).
% min_dist_less_V
thf(fact_78_connected__def,axiom,
! [U: nat,V: nat] :
( ( connected_a @ c @ U @ V )
= ( ? [P3: list_P559422087at_nat] : ( isPath_a @ c @ U @ P3 @ V ) ) ) ).
% connected_def
thf(fact_79_shortestPath__is__path,axiom,
! [U: nat,P5: list_P559422087at_nat,V: nat] :
( ( isShortestPath_a @ c @ U @ P5 @ V )
=> ( isPath_a @ c @ U @ P5 @ V ) ) ).
% shortestPath_is_path
thf(fact_80_g_H_OshortestPath__is__path,axiom,
! [U: nat,P5: list_P559422087at_nat,V: nat] :
( ( isShortestPath_b @ c2 @ U @ P5 @ V )
=> ( isPath_b @ c2 @ U @ P5 @ V ) ) ).
% g'.shortestPath_is_path
thf(fact_81_g_H_Oobtain__shortest__path,axiom,
! [U: nat,V: nat] :
( ( connected_b @ c2 @ U @ V )
=> ~ ! [P4: list_P559422087at_nat] :
~ ( isShortestPath_b @ c2 @ U @ P4 @ V ) ) ).
% g'.obtain_shortest_path
thf(fact_82_g_H_Omin__dist__le,axiom,
! [Src: nat,V: nat,D2: nat] :
( ( connected_b @ c2 @ Src @ V )
=> ( ( ord_less_eq_nat @ D2 @ ( min_dist_b @ c2 @ Src @ V ) )
=> ? [V2: nat] :
( ( connected_b @ c2 @ Src @ V2 )
& ( ( min_dist_b @ c2 @ Src @ V2 )
= D2 ) ) ) ) ).
% g'.min_dist_le
thf(fact_83_g_H_Omin__distI__eq,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist_b @ c2 @ V @ D @ V3 )
=> ( ! [D6: nat] :
( ( dist_b @ c2 @ V @ D6 @ V3 )
=> ( ord_less_eq_nat @ D @ D6 ) )
=> ( ( min_dist_b @ c2 @ V @ V3 )
= D ) ) ) ).
% g'.min_distI_eq
thf(fact_84_g_H_Omin__dist__minD,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist_b @ c2 @ V @ D @ V3 )
=> ( ord_less_eq_nat @ ( min_dist_b @ c2 @ V @ V3 ) @ D ) ) ).
% g'.min_dist_minD
thf(fact_85_min__dist__le,axiom,
! [Src: nat,V: nat,D2: nat] :
( ( connected_a @ c @ Src @ V )
=> ( ( ord_less_eq_nat @ D2 @ ( min_dist_a @ c @ Src @ V ) )
=> ? [V2: nat] :
( ( connected_a @ c @ Src @ V2 )
& ( ( min_dist_a @ c @ Src @ V2 )
= D2 ) ) ) ) ).
% min_dist_le
thf(fact_86_isShortestPath__def,axiom,
! [U: nat,P5: list_P559422087at_nat,V: nat] :
( ( isShortestPath_a @ c @ U @ P5 @ V )
= ( ( isPath_a @ c @ U @ P5 @ V )
& ! [P6: list_P559422087at_nat] :
( ( isPath_a @ c @ U @ P6 @ V )
=> ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P5 ) @ ( size_s1990949619at_nat @ P6 ) ) ) ) ) ).
% isShortestPath_def
thf(fact_87_isShortestPath__min__dist__def,axiom,
! [U: nat,P5: list_P559422087at_nat,V: nat] :
( ( isShortestPath_a @ c @ U @ P5 @ V )
= ( ( isPath_a @ c @ U @ P5 @ V )
& ( ( size_s1990949619at_nat @ P5 )
= ( min_dist_a @ c @ U @ V ) ) ) ) ).
% isShortestPath_min_dist_def
thf(fact_88_g_H_OisShortestPath__def,axiom,
! [U: nat,P5: list_P559422087at_nat,V: nat] :
( ( isShortestPath_b @ c2 @ U @ P5 @ V )
= ( ( isPath_b @ c2 @ U @ P5 @ V )
& ! [P6: list_P559422087at_nat] :
( ( isPath_b @ c2 @ U @ P6 @ V )
=> ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P5 ) @ ( size_s1990949619at_nat @ P6 ) ) ) ) ) ).
% g'.isShortestPath_def
thf(fact_89_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_90_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_91_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_92_zero__le,axiom,
! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% zero_le
thf(fact_93_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_94_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_95_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_96_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_97_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_98_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K: nat,B: nat] :
( ( P2 @ K )
=> ( ! [Y: nat] :
( ( P2 @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X2: nat] :
( ( P2 @ X2 )
& ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_99_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_100_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_101_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_102_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_103_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_104_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_105_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_106_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_107_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_108_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_109_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P2 @ I3 ) )
& ( P2 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_110_Graph_Omin__dist__le,axiom,
! [C: product_prod_nat_nat > b,Src: nat,V: nat,D2: nat] :
( ( connected_b @ C @ Src @ V )
=> ( ( ord_less_eq_nat @ D2 @ ( min_dist_b @ C @ Src @ V ) )
=> ? [V2: nat] :
( ( connected_b @ C @ Src @ V2 )
& ( ( min_dist_b @ C @ Src @ V2 )
= D2 ) ) ) ) ).
% Graph.min_dist_le
thf(fact_111_Graph_Omin__dist__le,axiom,
! [C: product_prod_nat_nat > a,Src: nat,V: nat,D2: nat] :
( ( connected_a @ C @ Src @ V )
=> ( ( ord_less_eq_nat @ D2 @ ( min_dist_a @ C @ Src @ V ) )
=> ? [V2: nat] :
( ( connected_a @ C @ Src @ V2 )
& ( ( min_dist_a @ C @ Src @ V2 )
= D2 ) ) ) ) ).
% Graph.min_dist_le
thf(fact_112_Graph_Omin__dist__minD,axiom,
! [C: product_prod_nat_nat > a,V: nat,D: nat,V3: nat] :
( ( dist_a @ C @ V @ D @ V3 )
=> ( ord_less_eq_nat @ ( min_dist_a @ C @ V @ V3 ) @ D ) ) ).
% Graph.min_dist_minD
thf(fact_113_Graph_Omin__dist__minD,axiom,
! [C: product_prod_nat_nat > b,V: nat,D: nat,V3: nat] :
( ( dist_b @ C @ V @ D @ V3 )
=> ( ord_less_eq_nat @ ( min_dist_b @ C @ V @ V3 ) @ D ) ) ).
% Graph.min_dist_minD
thf(fact_114_Graph_Omin__distI__eq,axiom,
! [C: product_prod_nat_nat > a,V: nat,D: nat,V3: nat] :
( ( dist_a @ C @ V @ D @ V3 )
=> ( ! [D6: nat] :
( ( dist_a @ C @ V @ D6 @ V3 )
=> ( ord_less_eq_nat @ D @ D6 ) )
=> ( ( min_dist_a @ C @ V @ V3 )
= D ) ) ) ).
% Graph.min_distI_eq
thf(fact_115_Graph_Omin__distI__eq,axiom,
! [C: product_prod_nat_nat > b,V: nat,D: nat,V3: nat] :
( ( dist_b @ C @ V @ D @ V3 )
=> ( ! [D6: nat] :
( ( dist_b @ C @ V @ D6 @ V3 )
=> ( ord_less_eq_nat @ D @ D6 ) )
=> ( ( min_dist_b @ C @ V @ V3 )
= D ) ) ) ).
% Graph.min_distI_eq
thf(fact_116_Graph_Omin__distI2,axiom,
! [C: product_prod_nat_nat > b,V: nat,V3: nat,Q: nat > $o] :
( ( connected_b @ C @ V @ V3 )
=> ( ! [D4: nat] :
( ( dist_b @ C @ V @ D4 @ V3 )
=> ( ! [D5: nat] :
( ( dist_b @ C @ V @ D5 @ V3 )
=> ( ord_less_eq_nat @ D4 @ D5 ) )
=> ( Q @ D4 ) ) )
=> ( Q @ ( min_dist_b @ C @ V @ V3 ) ) ) ) ).
% Graph.min_distI2
thf(fact_117_Graph_Omin__distI2,axiom,
! [C: product_prod_nat_nat > a,V: nat,V3: nat,Q: nat > $o] :
( ( connected_a @ C @ V @ V3 )
=> ( ! [D4: nat] :
( ( dist_a @ C @ V @ D4 @ V3 )
=> ( ! [D5: nat] :
( ( dist_a @ C @ V @ D5 @ V3 )
=> ( ord_less_eq_nat @ D4 @ D5 ) )
=> ( Q @ D4 ) ) )
=> ( Q @ ( min_dist_a @ C @ V @ V3 ) ) ) ) ).
% Graph.min_distI2
thf(fact_118_Graph_OisShortestPath__def,axiom,
( isShortestPath_a
= ( ^ [C2: product_prod_nat_nat > a,U2: nat,P3: list_P559422087at_nat,V4: nat] :
( ( isPath_a @ C2 @ U2 @ P3 @ V4 )
& ! [P6: list_P559422087at_nat] :
( ( isPath_a @ C2 @ U2 @ P6 @ V4 )
=> ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P3 ) @ ( size_s1990949619at_nat @ P6 ) ) ) ) ) ) ).
% Graph.isShortestPath_def
thf(fact_119_Graph_OisShortestPath__def,axiom,
( isShortestPath_b
= ( ^ [C2: product_prod_nat_nat > b,U2: nat,P3: list_P559422087at_nat,V4: nat] :
( ( isPath_b @ C2 @ U2 @ P3 @ V4 )
& ! [P6: list_P559422087at_nat] :
( ( isPath_b @ C2 @ U2 @ P6 @ V4 )
=> ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P3 ) @ ( size_s1990949619at_nat @ P6 ) ) ) ) ) ) ).
% Graph.isShortestPath_def
thf(fact_120_zero__reorient,axiom,
! [X3: nat] :
( ( zero_zero_nat = X3 )
= ( X3 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_121_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_122_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_123_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_124_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_125_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_126_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_127_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_128_linorder__neqE__nat,axiom,
! [X3: nat,Y3: nat] :
( ( X3 != Y3 )
=> ( ~ ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_129_size__neq__size__imp__neq,axiom,
! [X3: list_P559422087at_nat,Y3: list_P559422087at_nat] :
( ( ( size_s1990949619at_nat @ X3 )
!= ( size_s1990949619at_nat @ Y3 ) )
=> ( X3 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_130_Graph_OisPath_Ocong,axiom,
isPath_b = isPath_b ).
% Graph.isPath.cong
thf(fact_131_Graph_OisPath_Ocong,axiom,
isPath_a = isPath_a ).
% Graph.isPath.cong
thf(fact_132_Graph_OV_Ocong,axiom,
v_a = v_a ).
% Graph.V.cong
thf(fact_133_Graph_OV_Ocong,axiom,
v_b = v_b ).
% Graph.V.cong
thf(fact_134_Graph_Omin__dist_Ocong,axiom,
min_dist_a = min_dist_a ).
% Graph.min_dist.cong
thf(fact_135_Graph_Omin__dist_Ocong,axiom,
min_dist_b = min_dist_b ).
% Graph.min_dist.cong
thf(fact_136_Graph_Oconnected_Ocong,axiom,
connected_b = connected_b ).
% Graph.connected.cong
thf(fact_137_Graph_Oconnected_Ocong,axiom,
connected_a = connected_a ).
% Graph.connected.cong
thf(fact_138_Graph_Oconnected__refl,axiom,
! [C: product_prod_nat_nat > b,V: nat] : ( connected_b @ C @ V @ V ) ).
% Graph.connected_refl
thf(fact_139_Graph_Oconnected__refl,axiom,
! [C: product_prod_nat_nat > a,V: nat] : ( connected_a @ C @ V @ V ) ).
% Graph.connected_refl
thf(fact_140_Graph_Odist_Ocong,axiom,
dist_b = dist_b ).
% Graph.dist.cong
thf(fact_141_Graph_Odist_Ocong,axiom,
dist_a = dist_a ).
% Graph.dist.cong
thf(fact_142_Graph_OisShortestPath_Ocong,axiom,
isShortestPath_a = isShortestPath_a ).
% Graph.isShortestPath.cong
thf(fact_143_Graph_OisShortestPath_Ocong,axiom,
isShortestPath_b = isShortestPath_b ).
% Graph.isShortestPath.cong
thf(fact_144_Finite__Graph_Omin__dist__less__V,axiom,
! [C: product_prod_nat_nat > b,S: nat,T: nat] :
( ( finite_Graph_b @ C )
=> ( ( member_nat @ S @ ( v_b @ C ) )
=> ( ( connected_b @ C @ S @ T )
=> ( ord_less_nat @ ( min_dist_b @ C @ S @ T ) @ ( finite_card_nat @ ( v_b @ C ) ) ) ) ) ) ).
% Finite_Graph.min_dist_less_V
thf(fact_145_Finite__Graph_Omin__dist__less__V,axiom,
! [C: product_prod_nat_nat > a,S: nat,T: nat] :
( ( finite_Graph_a @ C )
=> ( ( member_nat @ S @ ( v_a @ C ) )
=> ( ( connected_a @ C @ S @ T )
=> ( ord_less_nat @ ( min_dist_a @ C @ S @ T ) @ ( finite_card_nat @ ( v_a @ C ) ) ) ) ) ) ).
% Finite_Graph.min_dist_less_V
thf(fact_146_Finite__Graph_OisShortestPath__length__less__V,axiom,
! [C: product_prod_nat_nat > a,S: nat,P5: list_P559422087at_nat,T: nat] :
( ( finite_Graph_a @ C )
=> ( ( member_nat @ S @ ( v_a @ C ) )
=> ( ( isShortestPath_a @ C @ S @ P5 @ T )
=> ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_a @ C ) ) ) ) ) ) ).
% Finite_Graph.isShortestPath_length_less_V
thf(fact_147_Finite__Graph_OisShortestPath__length__less__V,axiom,
! [C: product_prod_nat_nat > b,S: nat,P5: list_P559422087at_nat,T: nat] :
( ( finite_Graph_b @ C )
=> ( ( member_nat @ S @ ( v_b @ C ) )
=> ( ( isShortestPath_b @ C @ S @ P5 @ T )
=> ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_b @ C ) ) ) ) ) ) ).
% Finite_Graph.isShortestPath_length_less_V
thf(fact_148_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_149_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_150_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_151_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_152_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_153_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_154_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_155_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_156_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_157_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_158_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_159_Graph_Omin__dist__z,axiom,
! [C: product_prod_nat_nat > a,V: nat] :
( ( min_dist_a @ C @ V @ V )
= zero_zero_nat ) ).
% Graph.min_dist_z
thf(fact_160_Graph_Omin__dist__z,axiom,
! [C: product_prod_nat_nat > b,V: nat] :
( ( min_dist_b @ C @ V @ V )
= zero_zero_nat ) ).
% Graph.min_dist_z
thf(fact_161_Graph_Odist__z,axiom,
! [C: product_prod_nat_nat > b,V: nat] : ( dist_b @ C @ V @ zero_zero_nat @ V ) ).
% Graph.dist_z
thf(fact_162_Graph_Odist__z,axiom,
! [C: product_prod_nat_nat > a,V: nat] : ( dist_a @ C @ V @ zero_zero_nat @ V ) ).
% Graph.dist_z
thf(fact_163_Graph_Odist__z__iff,axiom,
! [C: product_prod_nat_nat > b,V: nat,V3: nat] :
( ( dist_b @ C @ V @ zero_zero_nat @ V3 )
= ( V3 = V ) ) ).
% Graph.dist_z_iff
thf(fact_164_Graph_Odist__z__iff,axiom,
! [C: product_prod_nat_nat > a,V: nat,V3: nat] :
( ( dist_a @ C @ V @ zero_zero_nat @ V3 )
= ( V3 = V ) ) ).
% Graph.dist_z_iff
thf(fact_165_Graph_Oconnected__def,axiom,
( connected_b
= ( ^ [C2: product_prod_nat_nat > b,U2: nat,V4: nat] :
? [P3: list_P559422087at_nat] : ( isPath_b @ C2 @ U2 @ P3 @ V4 ) ) ) ).
% Graph.connected_def
thf(fact_166_Graph_Oconnected__def,axiom,
( connected_a
= ( ^ [C2: product_prod_nat_nat > a,U2: nat,V4: nat] :
? [P3: list_P559422087at_nat] : ( isPath_a @ C2 @ U2 @ P3 @ V4 ) ) ) ).
% Graph.connected_def
thf(fact_167_Graph_Oconnected__inV__iff,axiom,
! [C: product_prod_nat_nat > b,U: nat,V: nat] :
( ( connected_b @ C @ U @ V )
=> ( ( member_nat @ V @ ( v_b @ C ) )
= ( member_nat @ U @ ( v_b @ C ) ) ) ) ).
% Graph.connected_inV_iff
thf(fact_168_Graph_Oconnected__inV__iff,axiom,
! [C: product_prod_nat_nat > a,U: nat,V: nat] :
( ( connected_a @ C @ U @ V )
=> ( ( member_nat @ V @ ( v_a @ C ) )
= ( member_nat @ U @ ( v_a @ C ) ) ) ) ).
% Graph.connected_inV_iff
thf(fact_169_Graph_OshortestPath__is__path,axiom,
! [C: product_prod_nat_nat > a,U: nat,P5: list_P559422087at_nat,V: nat] :
( ( isShortestPath_a @ C @ U @ P5 @ V )
=> ( isPath_a @ C @ U @ P5 @ V ) ) ).
% Graph.shortestPath_is_path
thf(fact_170_Graph_OshortestPath__is__path,axiom,
! [C: product_prod_nat_nat > b,U: nat,P5: list_P559422087at_nat,V: nat] :
( ( isShortestPath_b @ C @ U @ P5 @ V )
=> ( isPath_b @ C @ U @ P5 @ V ) ) ).
% Graph.shortestPath_is_path
thf(fact_171_Graph_Oconnected__distI,axiom,
! [C: product_prod_nat_nat > b,V: nat,D: nat,V3: nat] :
( ( dist_b @ C @ V @ D @ V3 )
=> ( connected_b @ C @ V @ V3 ) ) ).
% Graph.connected_distI
thf(fact_172_Graph_Oconnected__distI,axiom,
! [C: product_prod_nat_nat > a,V: nat,D: nat,V3: nat] :
( ( dist_a @ C @ V @ D @ V3 )
=> ( connected_a @ C @ V @ V3 ) ) ).
% Graph.connected_distI
thf(fact_173_Graph_Oconnected__by__dist,axiom,
( connected_b
= ( ^ [C2: product_prod_nat_nat > b,V4: nat,V5: nat] :
? [D3: nat] : ( dist_b @ C2 @ V4 @ D3 @ V5 ) ) ) ).
% Graph.connected_by_dist
thf(fact_174_Graph_Oconnected__by__dist,axiom,
( connected_a
= ( ^ [C2: product_prod_nat_nat > a,V4: nat,V5: nat] :
? [D3: nat] : ( dist_a @ C2 @ V4 @ D3 @ V5 ) ) ) ).
% Graph.connected_by_dist
thf(fact_175_Graph_Oobtain__shortest__path,axiom,
! [C: product_prod_nat_nat > a,U: nat,V: nat] :
( ( connected_a @ C @ U @ V )
=> ~ ! [P4: list_P559422087at_nat] :
~ ( isShortestPath_a @ C @ U @ P4 @ V ) ) ).
% Graph.obtain_shortest_path
thf(fact_176_Graph_Oobtain__shortest__path,axiom,
! [C: product_prod_nat_nat > b,U: nat,V: nat] :
( ( connected_b @ C @ U @ V )
=> ~ ! [P4: list_P559422087at_nat] :
~ ( isShortestPath_b @ C @ U @ P4 @ V ) ) ).
% Graph.obtain_shortest_path
thf(fact_177_Graph_Omin__dist__z__iff,axiom,
! [C: product_prod_nat_nat > b,V: nat,V3: nat] :
( ( connected_b @ C @ V @ V3 )
=> ( ( ( min_dist_b @ C @ V @ V3 )
= zero_zero_nat )
= ( V3 = V ) ) ) ).
% Graph.min_dist_z_iff
thf(fact_178_Graph_Omin__dist__z__iff,axiom,
! [C: product_prod_nat_nat > a,V: nat,V3: nat] :
( ( connected_a @ C @ V @ V3 )
=> ( ( ( min_dist_a @ C @ V @ V3 )
= zero_zero_nat )
= ( V3 = V ) ) ) ).
% Graph.min_dist_z_iff
thf(fact_179_Graph_Omin__dist__less,axiom,
! [C: product_prod_nat_nat > b,Src: nat,V: nat,D: nat,D2: nat] :
( ( connected_b @ C @ Src @ V )
=> ( ( ( min_dist_b @ C @ Src @ V )
= D )
=> ( ( ord_less_nat @ D2 @ D )
=> ? [V2: nat] :
( ( connected_b @ C @ Src @ V2 )
& ( ( min_dist_b @ C @ Src @ V2 )
= D2 ) ) ) ) ) ).
% Graph.min_dist_less
thf(fact_180_Graph_Omin__dist__less,axiom,
! [C: product_prod_nat_nat > a,Src: nat,V: nat,D: nat,D2: nat] :
( ( connected_a @ C @ Src @ V )
=> ( ( ( min_dist_a @ C @ Src @ V )
= D )
=> ( ( ord_less_nat @ D2 @ D )
=> ? [V2: nat] :
( ( connected_a @ C @ Src @ V2 )
& ( ( min_dist_a @ C @ Src @ V2 )
= D2 ) ) ) ) ) ).
% Graph.min_dist_less
thf(fact_181_Graph_Omin__dist__is__dist,axiom,
! [C: product_prod_nat_nat > b,V: nat,V3: nat] :
( ( connected_b @ C @ V @ V3 )
=> ( dist_b @ C @ V @ ( min_dist_b @ C @ V @ V3 ) @ V3 ) ) ).
% Graph.min_dist_is_dist
thf(fact_182_Graph_Omin__dist__is__dist,axiom,
! [C: product_prod_nat_nat > a,V: nat,V3: nat] :
( ( connected_a @ C @ V @ V3 )
=> ( dist_a @ C @ V @ ( min_dist_a @ C @ V @ V3 ) @ V3 ) ) ).
% Graph.min_dist_is_dist
thf(fact_183_Graph_OisPath__distD,axiom,
! [C: product_prod_nat_nat > b,U: nat,P5: list_P559422087at_nat,V: nat] :
( ( isPath_b @ C @ U @ P5 @ V )
=> ( dist_b @ C @ U @ ( size_s1990949619at_nat @ P5 ) @ V ) ) ).
% Graph.isPath_distD
thf(fact_184_Graph_OisPath__distD,axiom,
! [C: product_prod_nat_nat > a,U: nat,P5: list_P559422087at_nat,V: nat] :
( ( isPath_a @ C @ U @ P5 @ V )
=> ( dist_a @ C @ U @ ( size_s1990949619at_nat @ P5 ) @ V ) ) ).
% Graph.isPath_distD
thf(fact_185_Graph_Odist__def,axiom,
( dist_b
= ( ^ [C2: product_prod_nat_nat > b,V4: nat,D3: nat,V5: nat] :
? [P3: list_P559422087at_nat] :
( ( isPath_b @ C2 @ V4 @ P3 @ V5 )
& ( ( size_s1990949619at_nat @ P3 )
= D3 ) ) ) ) ).
% Graph.dist_def
thf(fact_186_Graph_Odist__def,axiom,
( dist_a
= ( ^ [C2: product_prod_nat_nat > a,V4: nat,D3: nat,V5: nat] :
? [P3: list_P559422087at_nat] :
( ( isPath_a @ C2 @ V4 @ P3 @ V5 )
& ( ( size_s1990949619at_nat @ P3 )
= D3 ) ) ) ) ).
% Graph.dist_def
thf(fact_187_card_Oinfinite,axiom,
! [A2: set_li664300135at_nat] :
( ~ ( finite1299096496at_nat @ A2 )
=> ( ( finite83082927at_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_188_card_Oinfinite,axiom,
! [A2: set_se1612935105at_nat] :
( ~ ( finite1457549322at_nat @ A2 )
=> ( ( finite1701894793at_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_189_card_Oinfinite,axiom,
! [A2: set_set_nat] :
( ~ ( finite2012248349et_nat @ A2 )
=> ( ( finite_card_set_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_190_card_Oinfinite,axiom,
! [A2: set_Pr1986765409at_nat] :
( ~ ( finite772653738at_nat @ A2 )
=> ( ( finite447719721at_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_191_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_192_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_193_g_H_Ocard__spEdges__le,axiom,
ord_less_eq_nat @ ( finite447719721at_nat @ ( edmond475474836dges_b @ c2 @ s @ t ) ) @ ( finite447719721at_nat @ ( edmond771116671s_uE_b @ c2 ) ) ).
% g'.card_spEdges_le
thf(fact_194_simplePath__length__less__V,axiom,
! [U: nat,P5: list_P559422087at_nat,V: nat] :
( ( member_nat @ U @ ( v_a @ c ) )
=> ( ( isSimplePath_a @ c @ U @ P5 @ V )
=> ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_a @ c ) ) ) ) ) ).
% simplePath_length_less_V
thf(fact_195_card__ge__0__finite,axiom,
! [A2: set_li664300135at_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite83082927at_nat @ A2 ) )
=> ( finite1299096496at_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_196_card__ge__0__finite,axiom,
! [A2: set_se1612935105at_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite1701894793at_nat @ A2 ) )
=> ( finite1457549322at_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_197_card__ge__0__finite,axiom,
! [A2: set_set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A2 ) )
=> ( finite2012248349et_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_198_card__ge__0__finite,axiom,
! [A2: set_Pr1986765409at_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite447719721at_nat @ A2 ) )
=> ( finite772653738at_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_199_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_200_min__distI2,axiom,
! [V: nat,V3: nat,Q: nat > $o] :
( ( connected_a @ c @ V @ V3 )
=> ( ! [D4: nat] :
( ( dist_a @ c @ V @ D4 @ V3 )
=> ( ! [D5: nat] :
( ( dist_a @ c @ V @ D5 @ V3 )
=> ( ord_less_eq_nat @ D4 @ D5 ) )
=> ( Q @ D4 ) ) )
=> ( Q @ ( min_dist_a @ c @ V @ V3 ) ) ) ) ).
% min_distI2
thf(fact_201_g_H_OisShortestPath__alt,axiom,
! [U: nat,P5: list_P559422087at_nat,V: nat] :
( ( isShortestPath_b @ c2 @ U @ P5 @ V )
= ( ( isSimplePath_b @ c2 @ U @ P5 @ V )
& ( ( size_s1990949619at_nat @ P5 )
= ( min_dist_b @ c2 @ U @ V ) ) ) ) ).
% g'.isShortestPath_alt
thf(fact_202_g_H_Oconnected__inV__iff,axiom,
! [U: nat,V: nat] :
( ( connected_b @ c2 @ U @ V )
=> ( ( member_nat @ V @ ( v_b @ c2 ) )
= ( member_nat @ U @ ( v_b @ c2 ) ) ) ) ).
% g'.connected_inV_iff
thf(fact_203_g_H_OisSPath__pathLE,axiom,
! [S: nat,P5: list_P559422087at_nat,T: nat] :
( ( isPath_b @ c2 @ S @ P5 @ T )
=> ? [P: list_P559422087at_nat] : ( isSimplePath_b @ c2 @ S @ P @ T ) ) ).
% g'.isSPath_pathLE
thf(fact_204_isSPath__pathLE,axiom,
! [S: nat,P5: list_P559422087at_nat,T: nat] :
( ( isPath_a @ c @ S @ P5 @ T )
=> ? [P: list_P559422087at_nat] : ( isSimplePath_a @ c @ S @ P @ T ) ) ).
% isSPath_pathLE
thf(fact_205_connected__by__dist,axiom,
! [V: nat,V3: nat] :
( ( connected_a @ c @ V @ V3 )
= ( ? [D3: nat] : ( dist_a @ c @ V @ D3 @ V3 ) ) ) ).
% connected_by_dist
thf(fact_206_shortestPath__is__simple,axiom,
! [S: nat,P5: list_P559422087at_nat,T: nat] :
( ( isShortestPath_a @ c @ S @ P5 @ T )
=> ( isSimplePath_a @ c @ S @ P5 @ T ) ) ).
% shortestPath_is_simple
thf(fact_207_g_H_OshortestPath__is__simple,axiom,
! [S: nat,P5: list_P559422087at_nat,T: nat] :
( ( isShortestPath_b @ c2 @ S @ P5 @ T )
=> ( isSimplePath_b @ c2 @ S @ P5 @ T ) ) ).
% g'.shortestPath_is_simple
thf(fact_208_min__dist__minD,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist_a @ c @ V @ D @ V3 )
=> ( ord_less_eq_nat @ ( min_dist_a @ c @ V @ V3 ) @ D ) ) ).
% min_dist_minD
thf(fact_209_min__distI__eq,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist_a @ c @ V @ D @ V3 )
=> ( ! [D6: nat] :
( ( dist_a @ c @ V @ D6 @ V3 )
=> ( ord_less_eq_nat @ D @ D6 ) )
=> ( ( min_dist_a @ c @ V @ V3 )
= D ) ) ) ).
% min_distI_eq
thf(fact_210_isPath__distD,axiom,
! [U: nat,P5: list_P559422087at_nat,V: nat] :
( ( isPath_a @ c @ U @ P5 @ V )
=> ( dist_a @ c @ U @ ( size_s1990949619at_nat @ P5 ) @ V ) ) ).
% isPath_distD
thf(fact_211_dist__def,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist_a @ c @ V @ D @ V3 )
= ( ? [P3: list_P559422087at_nat] :
( ( isPath_a @ c @ V @ P3 @ V3 )
& ( ( size_s1990949619at_nat @ P3 )
= D ) ) ) ) ).
% dist_def
thf(fact_212_min__dist__is__dist,axiom,
! [V: nat,V3: nat] :
( ( connected_a @ c @ V @ V3 )
=> ( dist_a @ c @ V @ ( min_dist_a @ c @ V @ V3 ) @ V3 ) ) ).
% min_dist_is_dist
thf(fact_213_g_H_OsimplePath__length__less__V,axiom,
! [U: nat,P5: list_P559422087at_nat,V: nat] :
( ( member_nat @ U @ ( v_b @ c2 ) )
=> ( ( isSimplePath_b @ c2 @ U @ P5 @ V )
=> ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_b @ c2 ) ) ) ) ) ).
% g'.simplePath_length_less_V
thf(fact_214_g_H_Omin__dist__less__V,axiom,
! [S: nat,T: nat] :
( ( member_nat @ S @ ( v_b @ c2 ) )
=> ( ( connected_b @ c2 @ S @ T )
=> ( ord_less_nat @ ( min_dist_b @ c2 @ S @ T ) @ ( finite_card_nat @ ( v_b @ c2 ) ) ) ) ) ).
% g'.min_dist_less_V
thf(fact_215_g_H_OisShortestPath__length__less__V,axiom,
! [S: nat,P5: list_P559422087at_nat,T: nat] :
( ( member_nat @ S @ ( v_b @ c2 ) )
=> ( ( isShortestPath_b @ c2 @ S @ P5 @ T )
=> ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_b @ c2 ) ) ) ) ) ).
% g'.isShortestPath_length_less_V
thf(fact_216_isShortestPath__alt,axiom,
! [U: nat,P5: list_P559422087at_nat,V: nat] :
( ( isShortestPath_a @ c @ U @ P5 @ V )
= ( ( isSimplePath_a @ c @ U @ P5 @ V )
& ( ( size_s1990949619at_nat @ P5 )
= ( min_dist_a @ c @ U @ V ) ) ) ) ).
% isShortestPath_alt
thf(fact_217_g_H_Ofinite__V,axiom,
finite_finite_nat @ ( v_b @ c2 ) ).
% g'.finite_V
thf(fact_218_finite__V,axiom,
finite_finite_nat @ ( v_a @ c ) ).
% finite_V
thf(fact_219_dist__z__iff,axiom,
! [V: nat,V3: nat] :
( ( dist_a @ c @ V @ zero_zero_nat @ V3 )
= ( V3 = V ) ) ).
% dist_z_iff
thf(fact_220_dist__z,axiom,
! [V: nat] : ( dist_a @ c @ V @ zero_zero_nat @ V ) ).
% dist_z
thf(fact_221_connected__distI,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist_a @ c @ V @ D @ V3 )
=> ( connected_a @ c @ V @ V3 ) ) ).
% connected_distI
thf(fact_222_Veq,axiom,
( ( v_b @ c2 )
= ( v_a @ c ) ) ).
% Veq
thf(fact_223_g_H_Oadjacent__nodes__ss__V,axiom,
! [U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_b @ c2 @ U ) @ ( v_b @ c2 ) ) ).
% g'.adjacent_nodes_ss_V
thf(fact_224_adjacent__nodes__ss__V,axiom,
! [U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_a @ c @ U ) @ ( v_a @ c ) ) ).
% adjacent_nodes_ss_V
thf(fact_225_Graph_OisSimplePath_Ocong,axiom,
isSimplePath_a = isSimplePath_a ).
% Graph.isSimplePath.cong
thf(fact_226_Graph_OisSimplePath_Ocong,axiom,
isSimplePath_b = isSimplePath_b ).
% Graph.isSimplePath.cong
thf(fact_227_finite__subset,axiom,
! [A2: set_se1612935105at_nat,B2: set_se1612935105at_nat] :
( ( ord_le2096002913at_nat @ A2 @ B2 )
=> ( ( finite1457549322at_nat @ B2 )
=> ( finite1457549322at_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_228_finite__subset,axiom,
! [A2: set_set_nat,B2: set_set_nat] :
( ( ord_le1613022364et_nat @ A2 @ B2 )
=> ( ( finite2012248349et_nat @ B2 )
=> ( finite2012248349et_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_229_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_230_finite__subset,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( finite772653738at_nat @ B2 )
=> ( finite772653738at_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_231_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_232_infinite__super,axiom,
! [S2: set_Pr1986765409at_nat,T2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ S2 @ T2 )
=> ( ~ ( finite772653738at_nat @ S2 )
=> ~ ( finite772653738at_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_233_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_234_rev__finite__subset,axiom,
! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ B2 )
=> ( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( finite772653738at_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_235_card__subset__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_236_card__subset__eq,axiom,
! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ B2 )
=> ( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( ( finite447719721at_nat @ A2 )
= ( finite447719721at_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_237_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ( finite_card_nat @ B3 )
= N )
& ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_238_infinite__arbitrarily__large,axiom,
! [A2: set_Pr1986765409at_nat,N: nat] :
( ~ ( finite772653738at_nat @ A2 )
=> ? [B3: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ B3 )
& ( ( finite447719721at_nat @ B3 )
= N )
& ( ord_le841296385at_nat @ B3 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_239_Graph_OisSPath__pathLE,axiom,
! [C: product_prod_nat_nat > a,S: nat,P5: list_P559422087at_nat,T: nat] :
( ( isPath_a @ C @ S @ P5 @ T )
=> ? [P: list_P559422087at_nat] : ( isSimplePath_a @ C @ S @ P @ T ) ) ).
% Graph.isSPath_pathLE
thf(fact_240_Graph_OisSPath__pathLE,axiom,
! [C: product_prod_nat_nat > b,S: nat,P5: list_P559422087at_nat,T: nat] :
( ( isPath_b @ C @ S @ P5 @ T )
=> ? [P: list_P559422087at_nat] : ( isSimplePath_b @ C @ S @ P @ T ) ) ).
% Graph.isSPath_pathLE
thf(fact_241_Graph_OshortestPath__is__simple,axiom,
! [C: product_prod_nat_nat > a,S: nat,P5: list_P559422087at_nat,T: nat] :
( ( isShortestPath_a @ C @ S @ P5 @ T )
=> ( isSimplePath_a @ C @ S @ P5 @ T ) ) ).
% Graph.shortestPath_is_simple
thf(fact_242_Graph_OshortestPath__is__simple,axiom,
! [C: product_prod_nat_nat > b,S: nat,P5: list_P559422087at_nat,T: nat] :
( ( isShortestPath_b @ C @ S @ P5 @ T )
=> ( isSimplePath_b @ C @ S @ P5 @ T ) ) ).
% Graph.shortestPath_is_simple
thf(fact_243_finite__maxlen,axiom,
! [M4: set_li664300135at_nat] :
( ( finite1299096496at_nat @ M4 )
=> ? [N3: nat] :
! [X4: list_P559422087at_nat] :
( ( member1608759472at_nat @ X4 @ M4 )
=> ( ord_less_nat @ ( size_s1990949619at_nat @ X4 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_244_card__psubset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_245_card__psubset,axiom,
! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ B2 )
=> ( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite447719721at_nat @ A2 ) @ ( finite447719721at_nat @ B2 ) )
=> ( ord_le116442893at_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_246_card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_247_card__mono,axiom,
! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ B2 )
=> ( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite447719721at_nat @ A2 ) @ ( finite447719721at_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_248_card__seteq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_249_card__seteq,axiom,
! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ B2 )
=> ( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite447719721at_nat @ B2 ) @ ( finite447719721at_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_250_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C3: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C3 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_251_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_Pr1986765409at_nat,C3: nat] :
( ! [G: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ G @ F2 )
=> ( ( finite772653738at_nat @ G )
=> ( ord_less_eq_nat @ ( finite447719721at_nat @ G ) @ C3 ) ) )
=> ( ( finite772653738at_nat @ F2 )
& ( ord_less_eq_nat @ ( finite447719721at_nat @ F2 ) @ C3 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_252_Finite__Graph__def,axiom,
( finite_Graph_b
= ( ^ [C2: product_prod_nat_nat > b] : ( finite_finite_nat @ ( v_b @ C2 ) ) ) ) ).
% Finite_Graph_def
thf(fact_253_Finite__Graph__def,axiom,
( finite_Graph_a
= ( ^ [C2: product_prod_nat_nat > a] : ( finite_finite_nat @ ( v_a @ C2 ) ) ) ) ).
% Finite_Graph_def
thf(fact_254_Finite__Graph_Ointro,axiom,
! [C: product_prod_nat_nat > b] :
( ( finite_finite_nat @ ( v_b @ C ) )
=> ( finite_Graph_b @ C ) ) ).
% Finite_Graph.intro
thf(fact_255_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_256_Finite__Graph_Ofinite__V,axiom,
! [C: product_prod_nat_nat > b] :
( ( finite_Graph_b @ C )
=> ( finite_finite_nat @ ( v_b @ C ) ) ) ).
% Finite_Graph.finite_V
thf(fact_257_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_258_finite__psubset__induct,axiom,
! [A2: set_Pr1986765409at_nat,P2: set_Pr1986765409at_nat > $o] :
( ( finite772653738at_nat @ A2 )
=> ( ! [A3: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ A3 )
=> ( ! [B4: set_Pr1986765409at_nat] :
( ( ord_le116442893at_nat @ B4 @ A3 )
=> ( P2 @ B4 ) )
=> ( P2 @ A3 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_259_finite__psubset__induct,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A3: set_nat] :
( ( finite_finite_nat @ A3 )
=> ( ! [B4: set_nat] :
( ( ord_less_set_nat @ B4 @ A3 )
=> ( P2 @ B4 ) )
=> ( P2 @ A3 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_260_Graph_OisShortestPath__alt,axiom,
( isShortestPath_a
= ( ^ [C2: product_prod_nat_nat > a,U2: nat,P3: list_P559422087at_nat,V4: nat] :
( ( isSimplePath_a @ C2 @ U2 @ P3 @ V4 )
& ( ( size_s1990949619at_nat @ P3 )
= ( min_dist_a @ C2 @ U2 @ V4 ) ) ) ) ) ).
% Graph.isShortestPath_alt
thf(fact_261_Graph_OisShortestPath__alt,axiom,
( isShortestPath_b
= ( ^ [C2: product_prod_nat_nat > b,U2: nat,P3: list_P559422087at_nat,V4: nat] :
( ( isSimplePath_b @ C2 @ U2 @ P3 @ V4 )
& ( ( size_s1990949619at_nat @ P3 )
= ( min_dist_b @ C2 @ U2 @ V4 ) ) ) ) ) ).
% Graph.isShortestPath_alt
thf(fact_262_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_263_ek__analysis_Ocard__spEdges__le,axiom,
! [C: product_prod_nat_nat > b,S: nat,T: nat] :
( ( edmond1517640973ysis_b @ C )
=> ( ord_less_eq_nat @ ( finite447719721at_nat @ ( edmond475474836dges_b @ C @ S @ T ) ) @ ( finite447719721at_nat @ ( edmond771116671s_uE_b @ C ) ) ) ) ).
% ek_analysis.card_spEdges_le
thf(fact_264_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_265_finite__has__minimal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite2012248349et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( ord_less_eq_set_nat @ X2 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_266_finite__has__minimal2,axiom,
! [A2: set_se1612935105at_nat,A: set_Pr1986765409at_nat] :
( ( finite1457549322at_nat @ A2 )
=> ( ( member298845450at_nat @ A @ A2 )
=> ? [X2: set_Pr1986765409at_nat] :
( ( member298845450at_nat @ X2 @ A2 )
& ( ord_le841296385at_nat @ X2 @ A )
& ! [Xa: set_Pr1986765409at_nat] :
( ( member298845450at_nat @ Xa @ A2 )
=> ( ( ord_le841296385at_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_267_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_268_finite__has__maximal2,axiom,
! [A2: set_set_nat,A: set_nat] :
( ( finite2012248349et_nat @ A2 )
=> ( ( member_set_nat @ A @ A2 )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A2 )
& ( ord_less_eq_set_nat @ A @ X2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A2 )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_269_finite__has__maximal2,axiom,
! [A2: set_se1612935105at_nat,A: set_Pr1986765409at_nat] :
( ( finite1457549322at_nat @ A2 )
=> ( ( member298845450at_nat @ A @ A2 )
=> ? [X2: set_Pr1986765409at_nat] :
( ( member298845450at_nat @ X2 @ A2 )
& ( ord_le841296385at_nat @ A @ X2 )
& ! [Xa: set_Pr1986765409at_nat] :
( ( member298845450at_nat @ Xa @ A2 )
=> ( ( ord_le841296385at_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_270_Finite__Graph_OsimplePath__length__less__V,axiom,
! [C: product_prod_nat_nat > a,U: nat,P5: list_P559422087at_nat,V: nat] :
( ( finite_Graph_a @ C )
=> ( ( member_nat @ U @ ( v_a @ C ) )
=> ( ( isSimplePath_a @ C @ U @ P5 @ V )
=> ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_a @ C ) ) ) ) ) ) ).
% Finite_Graph.simplePath_length_less_V
thf(fact_271_Finite__Graph_OsimplePath__length__less__V,axiom,
! [C: product_prod_nat_nat > b,U: nat,P5: list_P559422087at_nat,V: nat] :
( ( finite_Graph_b @ C )
=> ( ( member_nat @ U @ ( v_b @ C ) )
=> ( ( isSimplePath_b @ C @ U @ P5 @ V )
=> ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_b @ C ) ) ) ) ) ) ).
% Finite_Graph.simplePath_length_less_V
thf(fact_272_psubset__card__mono,axiom,
! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
( ( finite772653738at_nat @ B2 )
=> ( ( ord_le116442893at_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite447719721at_nat @ A2 ) @ ( finite447719721at_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_273_psubset__card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_274_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_275_g_H_Oreachable__ss__V,axiom,
! [S: nat] :
( ( member_nat @ S @ ( v_b @ c2 ) )
=> ( ord_less_eq_set_nat @ ( reachableNodes_b @ c2 @ S ) @ ( v_b @ c2 ) ) ) ).
% g'.reachable_ss_V
thf(fact_276_spEdges__ss__E,axiom,
ord_le841296385at_nat @ ( edmond475474835dges_a @ c @ s @ t ) @ ( e_a @ c ) ).
% spEdges_ss_E
thf(fact_277_E__ss__uE,axiom,
ord_le841296385at_nat @ ( e_a @ c ) @ ( edmond771116670s_uE_a @ c ) ).
% E_ss_uE
thf(fact_278_Finite__Graph__EI,axiom,
( ( finite772653738at_nat @ ( e_a @ c ) )
=> ( finite_Graph_a @ c ) ) ).
% Finite_Graph_EI
thf(fact_279_Efin__imp__Vfin,axiom,
( ( finite772653738at_nat @ ( e_a @ c ) )
=> ( finite_finite_nat @ ( v_a @ c ) ) ) ).
% Efin_imp_Vfin
thf(fact_280_finite__E,axiom,
finite772653738at_nat @ ( e_a @ c ) ).
% finite_E
thf(fact_281_g_H_Oadjacent__nodes__finite,axiom,
! [U: nat] : ( finite_finite_nat @ ( adjacent_nodes_b @ c2 @ U ) ) ).
% g'.adjacent_nodes_finite
thf(fact_282_adjacent__nodes__finite,axiom,
! [U: nat] : ( finite_finite_nat @ ( adjacent_nodes_a @ c @ U ) ) ).
% adjacent_nodes_finite
thf(fact_283_Vfin__imp__Efin,axiom,
( ( finite_finite_nat @ ( v_a @ c ) )
=> ( finite772653738at_nat @ ( e_a @ c ) ) ) ).
% Vfin_imp_Efin
thf(fact_284_Graph_OreachableNodes_Ocong,axiom,
reachableNodes_a = reachableNodes_a ).
% Graph.reachableNodes.cong
thf(fact_285_Graph_OreachableNodes_Ocong,axiom,
reachableNodes_b = reachableNodes_b ).
% Graph.reachableNodes.cong
thf(fact_286_Graph_Oadjacent__nodes_Ocong,axiom,
adjacent_nodes_b = adjacent_nodes_b ).
% Graph.adjacent_nodes.cong
thf(fact_287_Graph_Oadjacent__nodes_Ocong,axiom,
adjacent_nodes_a = adjacent_nodes_a ).
% Graph.adjacent_nodes.cong
thf(fact_288_Graph_OE_Ocong,axiom,
e_a = e_a ).
% Graph.E.cong
thf(fact_289_Graph_OE_Ocong,axiom,
e_b = e_b ).
% Graph.E.cong
thf(fact_290_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_291_Graph_Oreachable__ss__V,axiom,
! [S: nat,C: product_prod_nat_nat > b] :
( ( member_nat @ S @ ( v_b @ C ) )
=> ( ord_less_eq_set_nat @ ( reachableNodes_b @ C @ S ) @ ( v_b @ C ) ) ) ).
% Graph.reachable_ss_V
thf(fact_292_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_293_Graph_OEfin__imp__Vfin,axiom,
! [C: product_prod_nat_nat > b] :
( ( finite772653738at_nat @ ( e_b @ C ) )
=> ( finite_finite_nat @ ( v_b @ C ) ) ) ).
% Graph.Efin_imp_Vfin
thf(fact_294_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_295_Graph_OVfin__imp__Efin,axiom,
! [C: product_prod_nat_nat > b] :
( ( finite_finite_nat @ ( v_b @ C ) )
=> ( finite772653738at_nat @ ( e_b @ C ) ) ) ).
% Graph.Vfin_imp_Efin
thf(fact_296_Graph_OFinite__Graph__EI,axiom,
! [C: product_prod_nat_nat > b] :
( ( finite772653738at_nat @ ( e_b @ C ) )
=> ( finite_Graph_b @ C ) ) ).
% Graph.Finite_Graph_EI
thf(fact_297_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_298_Finite__Graph_Ofinite__E,axiom,
! [C: product_prod_nat_nat > b] :
( ( finite_Graph_b @ C )
=> ( finite772653738at_nat @ ( e_b @ C ) ) ) ).
% Finite_Graph.finite_E
thf(fact_299_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_300_ek__analysis_OE__ss__uE,axiom,
! [C: product_prod_nat_nat > b] :
( ( edmond1517640973ysis_b @ C )
=> ( ord_le841296385at_nat @ ( e_b @ C ) @ ( edmond771116671s_uE_b @ C ) ) ) ).
% ek_analysis.E_ss_uE
thf(fact_301_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_302_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_303_ek__analysis_OspEdges__ss__E,axiom,
! [C: product_prod_nat_nat > b,S: nat,T: nat] :
( ( edmond1517640973ysis_b @ C )
=> ( ord_le841296385at_nat @ ( edmond475474836dges_b @ C @ S @ T ) @ ( e_b @ C ) ) ) ).
% ek_analysis.spEdges_ss_E
thf(fact_304_Graph_Oadjacent__nodes__ss__V,axiom,
! [C: product_prod_nat_nat > b,U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_b @ C @ U ) @ ( v_b @ C ) ) ).
% Graph.adjacent_nodes_ss_V
thf(fact_305_Graph_Oadjacent__nodes__ss__V,axiom,
! [C: product_prod_nat_nat > a,U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_a @ C @ U ) @ ( v_a @ C ) ) ).
% Graph.adjacent_nodes_ss_V
thf(fact_306_Finite__Graph_Oadjacent__nodes__finite,axiom,
! [C: product_prod_nat_nat > b,U: nat] :
( ( finite_Graph_b @ C )
=> ( finite_finite_nat @ ( adjacent_nodes_b @ C @ U ) ) ) ).
% Finite_Graph.adjacent_nodes_finite
thf(fact_307_Finite__Graph_Oadjacent__nodes__finite,axiom,
! [C: product_prod_nat_nat > a,U: nat] :
( ( finite_Graph_a @ C )
=> ( finite_finite_nat @ ( adjacent_nodes_a @ C @ U ) ) ) ).
% Finite_Graph.adjacent_nodes_finite
thf(fact_308_incoming_H__edges,axiom,
! [U3: set_nat] : ( ord_le841296385at_nat @ ( incoming_a @ c @ U3 ) @ ( e_a @ c ) ) ).
% incoming'_edges
thf(fact_309_outgoing_H__edges,axiom,
! [U3: set_nat] : ( ord_le841296385at_nat @ ( outgoing_a @ c @ U3 ) @ ( e_a @ c ) ) ).
% outgoing'_edges
thf(fact_310_g_H_OE__ss__uE,axiom,
ord_le841296385at_nat @ ( e_b @ c2 ) @ ( edmond771116671s_uE_b @ c2 ) ).
% g'.E_ss_uE
thf(fact_311_g_H_OFinite__Graph__EI,axiom,
( ( finite772653738at_nat @ ( e_b @ c2 ) )
=> ( finite_Graph_b @ c2 ) ) ).
% g'.Finite_Graph_EI
thf(fact_312_g_H_OEfin__imp__Vfin,axiom,
( ( finite772653738at_nat @ ( e_b @ c2 ) )
=> ( finite_finite_nat @ ( v_b @ c2 ) ) ) ).
% g'.Efin_imp_Vfin
thf(fact_313_g_H_OspEdges__ss__E,axiom,
ord_le841296385at_nat @ ( edmond475474836dges_b @ c2 @ s @ t ) @ ( e_b @ c2 ) ).
% g'.spEdges_ss_E
thf(fact_314_g_H_Ofinite__E,axiom,
finite772653738at_nat @ ( e_b @ c2 ) ).
% g'.finite_E
thf(fact_315_g_H_OVfin__imp__Efin,axiom,
( ( finite_finite_nat @ ( v_b @ c2 ) )
=> ( finite772653738at_nat @ ( e_b @ c2 ) ) ) ).
% g'.Vfin_imp_Efin
thf(fact_316_finite__outgoing_H,axiom,
! [U3: set_nat] :
( ( finite_finite_nat @ ( v_a @ c ) )
=> ( finite772653738at_nat @ ( outgoing_a @ c @ U3 ) ) ) ).
% finite_outgoing'
thf(fact_317_finite__incoming_H,axiom,
! [U3: set_nat] :
( ( finite_finite_nat @ ( v_a @ c ) )
=> ( finite772653738at_nat @ ( incoming_a @ c @ U3 ) ) ) ).
% finite_incoming'
thf(fact_318_Graph_Oincoming_H_Ocong,axiom,
incoming_a = incoming_a ).
% Graph.incoming'.cong
thf(fact_319_Graph_Oincoming_H_Ocong,axiom,
incoming_b = incoming_b ).
% Graph.incoming'.cong
thf(fact_320_Graph_Ooutgoing_H_Ocong,axiom,
outgoing_a = outgoing_a ).
% Graph.outgoing'.cong
thf(fact_321_Graph_Ooutgoing_H_Ocong,axiom,
outgoing_b = outgoing_b ).
% Graph.outgoing'.cong
thf(fact_322_Graph_Ooutgoing_H__edges,axiom,
! [C: product_prod_nat_nat > a,U3: set_nat] : ( ord_le841296385at_nat @ ( outgoing_a @ C @ U3 ) @ ( e_a @ C ) ) ).
% Graph.outgoing'_edges
thf(fact_323_Graph_Ooutgoing_H__edges,axiom,
! [C: product_prod_nat_nat > b,U3: set_nat] : ( ord_le841296385at_nat @ ( outgoing_b @ C @ U3 ) @ ( e_b @ C ) ) ).
% Graph.outgoing'_edges
thf(fact_324_Graph_Oincoming_H__edges,axiom,
! [C: product_prod_nat_nat > a,U3: set_nat] : ( ord_le841296385at_nat @ ( incoming_a @ C @ U3 ) @ ( e_a @ C ) ) ).
% Graph.incoming'_edges
thf(fact_325_Graph_Oincoming_H__edges,axiom,
! [C: product_prod_nat_nat > b,U3: set_nat] : ( ord_le841296385at_nat @ ( incoming_b @ C @ U3 ) @ ( e_b @ C ) ) ).
% Graph.incoming'_edges
thf(fact_326_Graph_Ofinite__outgoing_H,axiom,
! [C: product_prod_nat_nat > a,U3: set_nat] :
( ( finite_finite_nat @ ( v_a @ C ) )
=> ( finite772653738at_nat @ ( outgoing_a @ C @ U3 ) ) ) ).
% Graph.finite_outgoing'
thf(fact_327_Graph_Ofinite__outgoing_H,axiom,
! [C: product_prod_nat_nat > b,U3: set_nat] :
( ( finite_finite_nat @ ( v_b @ C ) )
=> ( finite772653738at_nat @ ( outgoing_b @ C @ U3 ) ) ) ).
% Graph.finite_outgoing'
thf(fact_328_Graph_Ofinite__incoming_H,axiom,
! [C: product_prod_nat_nat > a,U3: set_nat] :
( ( finite_finite_nat @ ( v_a @ C ) )
=> ( finite772653738at_nat @ ( incoming_a @ C @ U3 ) ) ) ).
% Graph.finite_incoming'
thf(fact_329_Graph_Ofinite__incoming_H,axiom,
! [C: product_prod_nat_nat > b,U3: set_nat] :
( ( finite_finite_nat @ ( v_b @ C ) )
=> ( finite772653738at_nat @ ( incoming_b @ C @ U3 ) ) ) ).
% Graph.finite_incoming'
thf(fact_330_g_H_Oincoming_H__edges,axiom,
! [U3: set_nat] : ( ord_le841296385at_nat @ ( incoming_b @ c2 @ U3 ) @ ( e_b @ c2 ) ) ).
% g'.incoming'_edges
thf(fact_331_g_H_Ooutgoing_H__edges,axiom,
! [U3: set_nat] : ( ord_le841296385at_nat @ ( outgoing_b @ c2 @ U3 ) @ ( e_b @ c2 ) ) ).
% g'.outgoing'_edges
thf(fact_332_g_H_Ofinite__outgoing_H,axiom,
! [U3: set_nat] :
( ( finite_finite_nat @ ( v_b @ c2 ) )
=> ( finite772653738at_nat @ ( outgoing_b @ c2 @ U3 ) ) ) ).
% g'.finite_outgoing'
thf(fact_333_g_H_Ofinite__incoming_H,axiom,
! [U3: set_nat] :
( ( finite_finite_nat @ ( v_b @ c2 ) )
=> ( finite772653738at_nat @ ( incoming_b @ c2 @ U3 ) ) ) ).
% g'.finite_incoming'
thf(fact_334__092_060open_062edges_A_092_060subseteq_062_AE_092_060close_062,axiom,
ord_le841296385at_nat @ edges @ ( e_a @ c ) ).
% \<open>edges \<subseteq> E\<close>
thf(fact_335_psubsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_336_psubsetI,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_le116442893at_nat @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_337_subset__antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_338_subset__antisym,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( ord_le841296385at_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_339_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_340_subsetI,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ! [X2: product_prod_nat_nat] :
( ( member701585322at_nat @ X2 @ A2 )
=> ( member701585322at_nat @ X2 @ B2 ) )
=> ( ord_le841296385at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_341_SP__EDGES,axiom,
ord_le841296385at_nat @ edges @ ( set_Pr2131844118at_nat @ p ) ).
% SP_EDGES
thf(fact_342_Collect__mono__iff,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_343_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 ) )
= ( ! [X: product_prod_nat_nat] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_344_set__eq__subset,axiom,
( ( ^ [Y4: set_nat,Z: set_nat] : Y4 = Z )
= ( ^ [A4: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_345_set__eq__subset,axiom,
( ( ^ [Y4: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] : Y4 = Z )
= ( ^ [A4: set_Pr1986765409at_nat,B5: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A4 @ B5 )
& ( ord_le841296385at_nat @ B5 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_346_subset__trans,axiom,
! [A2: set_nat,B2: set_nat,C3: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C3 )
=> ( ord_less_eq_set_nat @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_347_subset__trans,axiom,
! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,C3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
=> ( ( ord_le841296385at_nat @ B2 @ C3 )
=> ( ord_le841296385at_nat @ A2 @ C3 ) ) ) ).
% subset_trans
% Conjectures (1)
thf(conj_0,conjecture,
$false ).
%------------------------------------------------------------------------------