%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP051^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n015.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:09 AM UTC 2026
% Result : ContradictoryAxioms 5.74s 2.04s
% Output : Refutation 5.74s
% Verified :
% SZS Type : Refutation
% Derivation depth : 15
% Number of leaves : 6
% Syntax : Number of formulae : 45 ( 38 unt; 0 typ; 0 def)
% Number of atoms : 158 ( 68 equ; 0 cnn)
% Maximal formula atoms : 3 ( 3 avg)
% Number of connectives : 273 ( 12 ~; 8 |; 10 &; 220 @)
% ( 0 <=>; 17 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 2 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 77 ( 73 usr; 9 con; 0-4 aty)
% ( 6 !!; 0 ??; 0 @@+; 0 @@-)
% Number of variables : 36 ( 20 ^; 16 !; 0 ?; 36 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
product_prod_nat_nat: $tType ).
thf(type_def_6,type,
a: $tType ).
thf(type_def_8,type,
b: $tType ).
thf(type_def_9,type,
nat: $tType ).
thf(type_def_10,type,
set_Pr1986765409at_nat: $tType ).
thf(type_def_11,type,
set_li664300135at_nat: $tType ).
thf(type_def_12,type,
set_nat: $tType ).
thf(type_def_13,type,
set_set_nat: $tType ).
thf(type_def_14,type,
set_se1612935105at_nat: $tType ).
thf(type_def_15,type,
list_P559422087at_nat: $tType ).
thf(func_def_0,type,
set_li664300135at_nat: $tType ).
thf(func_def_1,type,
set_se1612935105at_nat: $tType ).
thf(func_def_2,type,
list_P559422087at_nat: $tType ).
thf(func_def_3,type,
set_Pr1986765409at_nat: $tType ).
thf(func_def_4,type,
product_prod_nat_nat: $tType ).
thf(func_def_5,type,
set_set_nat: $tType ).
thf(func_def_6,type,
set_nat: $tType ).
thf(func_def_7,type,
nat: $tType ).
thf(func_def_8,type,
b: $tType ).
thf(func_def_9,type,
a: $tType ).
thf(func_def_10,type,
edmond1517640972ysis_a: ( product_prod_nat_nat > a ) > $o ).
thf(func_def_11,type,
edmond1517640973ysis_b: ( product_prod_nat_nat > b ) > $o ).
thf(func_def_12,type,
edmond1022345716sure_a: ( product_prod_nat_nat > a ) > nat > nat > nat ).
thf(func_def_13,type,
edmond1022345717sure_b: ( product_prod_nat_nat > b ) > nat > nat > nat ).
thf(func_def_14,type,
edmond475474835dges_a: ( product_prod_nat_nat > a ) > nat > nat > set_Pr1986765409at_nat ).
thf(func_def_15,type,
edmond475474836dges_b: ( product_prod_nat_nat > b ) > nat > nat > set_Pr1986765409at_nat ).
thf(func_def_16,type,
edmond771116670s_uE_a: ( product_prod_nat_nat > a ) > set_Pr1986765409at_nat ).
thf(func_def_17,type,
edmond771116671s_uE_b: ( product_prod_nat_nat > b ) > set_Pr1986765409at_nat ).
thf(func_def_18,type,
finite83082927at_nat: set_li664300135at_nat > nat ).
thf(func_def_19,type,
finite_card_nat: set_nat > nat ).
thf(func_def_20,type,
finite447719721at_nat: set_Pr1986765409at_nat > nat ).
thf(func_def_21,type,
finite_card_set_nat: set_set_nat > nat ).
thf(func_def_22,type,
finite1701894793at_nat: set_se1612935105at_nat > nat ).
thf(func_def_23,type,
finite1299096496at_nat: set_li664300135at_nat > $o ).
thf(func_def_24,type,
finite_finite_nat: set_nat > $o ).
thf(func_def_25,type,
finite772653738at_nat: set_Pr1986765409at_nat > $o ).
thf(func_def_26,type,
finite2012248349et_nat: set_set_nat > $o ).
thf(func_def_27,type,
finite1457549322at_nat: set_se1612935105at_nat > $o ).
thf(func_def_28,type,
finite_Graph_a: ( product_prod_nat_nat > a ) > $o ).
thf(func_def_29,type,
finite_Graph_b: ( product_prod_nat_nat > b ) > $o ).
thf(func_def_30,type,
e_a: ( product_prod_nat_nat > a ) > set_Pr1986765409at_nat ).
thf(func_def_31,type,
e_b: ( product_prod_nat_nat > b ) > set_Pr1986765409at_nat ).
thf(func_def_32,type,
v_a: ( product_prod_nat_nat > a ) > set_nat ).
thf(func_def_33,type,
v_b: ( product_prod_nat_nat > b ) > set_nat ).
thf(func_def_34,type,
adjacent_nodes_a: ( product_prod_nat_nat > a ) > nat > set_nat ).
thf(func_def_35,type,
adjacent_nodes_b: ( product_prod_nat_nat > b ) > nat > set_nat ).
thf(func_def_36,type,
connected_a: ( product_prod_nat_nat > a ) > nat > nat > $o ).
thf(func_def_37,type,
connected_b: ( product_prod_nat_nat > b ) > nat > nat > $o ).
thf(func_def_38,type,
dist_a: ( product_prod_nat_nat > a ) > nat > nat > nat > $o ).
thf(func_def_39,type,
dist_b: ( product_prod_nat_nat > b ) > nat > nat > nat > $o ).
thf(func_def_40,type,
incoming_a: ( product_prod_nat_nat > a ) > set_nat > set_Pr1986765409at_nat ).
thf(func_def_41,type,
incoming_b: ( product_prod_nat_nat > b ) > set_nat > set_Pr1986765409at_nat ).
thf(func_def_42,type,
isPath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
thf(func_def_43,type,
isPath_b: ( product_prod_nat_nat > b ) > nat > list_P559422087at_nat > nat > $o ).
thf(func_def_44,type,
isShortestPath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
thf(func_def_45,type,
isShortestPath_b: ( product_prod_nat_nat > b ) > nat > list_P559422087at_nat > nat > $o ).
thf(func_def_46,type,
isSimplePath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
thf(func_def_47,type,
isSimplePath_b: ( product_prod_nat_nat > b ) > nat > list_P559422087at_nat > nat > $o ).
thf(func_def_48,type,
min_dist_a: ( product_prod_nat_nat > a ) > nat > nat > nat ).
thf(func_def_49,type,
min_dist_b: ( product_prod_nat_nat > b ) > nat > nat > nat ).
thf(func_def_50,type,
outgoing_a: ( product_prod_nat_nat > a ) > set_nat > set_Pr1986765409at_nat ).
thf(func_def_51,type,
outgoing_b: ( product_prod_nat_nat > b ) > set_nat > set_Pr1986765409at_nat ).
thf(func_def_52,type,
reachableNodes_a: ( product_prod_nat_nat > a ) > nat > set_nat ).
thf(func_def_53,type,
reachableNodes_b: ( product_prod_nat_nat > b ) > nat > set_nat ).
thf(func_def_54,type,
zero_zero_nat: nat ).
thf(func_def_55,type,
set_Pr2131844118at_nat: list_P559422087at_nat > set_Pr1986765409at_nat ).
thf(func_def_56,type,
size_s1990949619at_nat: list_P559422087at_nat > nat ).
thf(func_def_57,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_58,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(func_def_59,type,
ord_le116442893at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o ).
thf(func_def_60,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_61,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_62,type,
ord_le841296385at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o ).
thf(func_def_63,type,
ord_le1613022364et_nat: set_set_nat > set_set_nat > $o ).
thf(func_def_64,type,
ord_le2096002913at_nat: set_se1612935105at_nat > set_se1612935105at_nat > $o ).
thf(func_def_65,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_66,type,
collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
thf(func_def_67,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(func_def_68,type,
collec1606769740at_nat: ( set_Pr1986765409at_nat > $o ) > set_se1612935105at_nat ).
thf(func_def_69,type,
member1608759472at_nat: list_P559422087at_nat > set_li664300135at_nat > $o ).
thf(func_def_70,type,
member_nat: nat > set_nat > $o ).
thf(func_def_71,type,
member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
thf(func_def_72,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(func_def_73,type,
member298845450at_nat: set_Pr1986765409at_nat > set_se1612935105at_nat > $o ).
thf(func_def_74,type,
c: product_prod_nat_nat > a ).
thf(func_def_75,type,
c2: product_prod_nat_nat > b ).
thf(func_def_76,type,
edges: set_Pr1986765409at_nat ).
thf(func_def_77,type,
p: list_P559422087at_nat ).
thf(func_def_78,type,
p2: list_P559422087at_nat ).
thf(func_def_79,type,
s: nat ).
thf(func_def_80,type,
t: nat ).
thf(func_def_104,type,
ph1:
!>[X0: $tType] : X0 ).
thf(func_def_105,type,
sK2: nat > list_P559422087at_nat ).
thf(f2264,plain,
$false,
inference(trivial_inequality_removal,[status(thm)],[f2263]) ).
thf(f2263,plain,
$false = $true,
inference(backward_demodulation,[status(thm)],[f1768,f2262]) ).
thf(f2262,plain,
( $false
= ( ord_less_eq_nat @ ( size_s1990949619at_nat @ p2 ) @ ( size_s1990949619at_nat @ p ) ) ),
inference(subsumption_resolution,[status(thm)],[f2261,f1743]) ).
thf(f1743,plain,
( ( size_s1990949619at_nat @ p )
!= ( size_s1990949619at_nat @ p2 ) ),
inference(equality_proxy_clausification,[status(thm)],[f1742]) ).
thf(f1742,plain,
( ( ( size_s1990949619at_nat @ p )
= ( size_s1990949619at_nat @ p2 ) )
= $false ),
inference(not_proxy_clausification,[status(thm)],[f1741]) ).
thf(f1741,plain,
( $true
= ( ( size_s1990949619at_nat @ p )
!= ( size_s1990949619at_nat @ p2 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1720]) ).
thf(f1720,plain,
( $true
= ( ( ( size_s1990949619at_nat @ p )
!= ( size_s1990949619at_nat @ p2 ) )
& ( ord_less_eq_nat @ ( size_s1990949619at_nat @ p ) @ ( size_s1990949619at_nat @ p2 ) ) ) ),
inference(beta_eta_normalization,[status(thm)],[f1484]) ).
thf(f1484,plain,
( ( ^ [Y0: nat,Y1: nat] :
( ( Y0 != Y1 )
& ( ord_less_eq_nat @ Y0 @ Y1 ) )
@ ( size_s1990949619at_nat @ p )
@ ( size_s1990949619at_nat @ p2 ) )
= $true ),
inference(definition_unfolding,[status(thm)],[f1233,f1185]) ).
thf(f1185,plain,
( ord_less_nat
= ( ^ [Y0: nat,Y1: nat] :
( ( Y0 != Y1 )
& ( ord_less_eq_nat @ Y0 @ Y1 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f825]) ).
thf(f825,plain,
( ord_less_nat
= ( ^ [Y0: nat,Y1: nat] :
( ( Y0 != Y1 )
& ( ord_less_eq_nat @ Y0 @ Y1 ) ) ) ),
inference(fool_elimination,[status(thm)],[f824]) ).
thf(f824,plain,
( ord_less_nat
= ( ^ [X0: nat,X1: nat] :
( ( ord_less_eq_nat @ X0 @ X1 )
& ( X0 != X1 ) ) ) ),
inference(rectify,[status(thm)],[f104]) ).
thf(f104,axiom,
( ord_less_nat
= ( ^ [X41: nat,X42: nat] :
( ( ord_less_eq_nat @ X41 @ X42 )
& ( X41 != X42 ) ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.Pn7hie6YQY/DTF2THF_17795.p',fact_103_nat__less__le) ).
thf(f1233,plain,
( ( ord_less_nat @ ( size_s1990949619at_nat @ p ) @ ( size_s1990949619at_nat @ p2 ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f905]) ).
thf(f905,plain,
( ( ord_less_nat @ ( size_s1990949619at_nat @ p ) @ ( size_s1990949619at_nat @ p2 ) )
= $true ),
inference(fool_elimination,[status(thm)],[f904]) ).
thf(f904,plain,
ord_less_nat @ ( size_s1990949619at_nat @ p ) @ ( size_s1990949619at_nat @ p2 ),
inference(rectify,[status(thm)],[f14]) ).
thf(f14,axiom,
ord_less_nat @ ( size_s1990949619at_nat @ p ) @ ( size_s1990949619at_nat @ p2 ),
file('/export/starexec/sandbox2/tmp/tmp.Pn7hie6YQY/DTF2THF_17795.p',fact_13__092_060open_062length_Ap_A_060_Alength_Ap_H_092_060close_062) ).
thf(f2261,plain,
( ( ( size_s1990949619at_nat @ p )
= ( size_s1990949619at_nat @ p2 ) )
| ( $false
= ( ord_less_eq_nat @ ( size_s1990949619at_nat @ p2 ) @ ( size_s1990949619at_nat @ p ) ) ) ),
inference(trivial_inequality_removal,[status(thm)],[f2230]) ).
thf(f2230,plain,
( ( $false
= ( ord_less_eq_nat @ ( size_s1990949619at_nat @ p2 ) @ ( size_s1990949619at_nat @ p ) ) )
| ( ( size_s1990949619at_nat @ p )
= ( size_s1990949619at_nat @ p2 ) )
| ( $false = $true ) ),
inference(superposition,[status(thm)],[f2222,f1740]) ).
thf(f1740,plain,
( $true
= ( ord_less_eq_nat @ ( size_s1990949619at_nat @ p ) @ ( size_s1990949619at_nat @ p2 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1720]) ).
thf(f2222,plain,
! [X2: nat,X1: nat] :
( ( $false
= ( ord_less_eq_nat @ X2 @ X1 ) )
| ( X1 = X2 )
| ( $false
= ( ord_less_eq_nat @ X1 @ X2 ) ) ),
inference(equality_proxy_clausification,[status(thm)],[f2221]) ).
thf(f2221,plain,
! [X2: nat,X1: nat] :
( ( $true
= ( X2 = X1 ) )
| ( $false
= ( ord_less_eq_nat @ X1 @ X2 ) )
| ( $false
= ( ord_less_eq_nat @ X2 @ X1 ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f2220]) ).
thf(f2220,plain,
! [X2: nat,X1: nat] :
( ( $false
= ( ord_less_eq_nat @ X1 @ X2 ) )
| ( $true
= ( ( ord_less_eq_nat @ X2 @ X1 )
=> ( X2 = X1 ) ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f2219]) ).
thf(f2219,plain,
! [X2: nat,X1: nat] :
( ( ( ord_less_eq_nat @ X1 @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ X1 )
=> ( X2 = X1 ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f2218]) ).
thf(f2218,plain,
! [X2: nat,X1: nat] :
( $true
= ( ^ [Y0: nat] :
( ( ord_less_eq_nat @ X1 @ Y0 )
=> ( ( ord_less_eq_nat @ Y0 @ X1 )
=> ( Y0 = X1 ) ) )
@ X2 ) ),
inference(pi_clausification,[status(thm)],[f2203]) ).
thf(f2203,plain,
! [X1: nat] :
( ( !! @ nat
@ ^ [Y0: nat] :
( ( ord_less_eq_nat @ X1 @ Y0 )
=> ( ( ord_less_eq_nat @ Y0 @ X1 )
=> ( Y0 = X1 ) ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f2202]) ).
thf(f2202,plain,
! [X1: nat] :
( $true
= ( ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq_nat @ Y0 @ Y1 )
=> ( ( ord_less_eq_nat @ Y1 @ Y0 )
=> ( Y1 = Y0 ) ) ) )
@ X1 ) ),
inference(pi_clausification,[status(thm)],[f1285]) ).
thf(f1285,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq_nat @ Y0 @ Y1 )
=> ( ( ord_less_eq_nat @ Y1 @ Y0 )
=> ( Y1 = Y0 ) ) ) ) ) ),
inference(cnf_transformation,[status(thm)],[f708]) ).
thf(f708,plain,
( $true
= ( !! @ nat
@ ^ [Y0: nat] :
( !! @ nat
@ ^ [Y1: nat] :
( ( ord_less_eq_nat @ Y0 @ Y1 )
=> ( ( ord_less_eq_nat @ Y1 @ Y0 )
=> ( Y1 = Y0 ) ) ) ) ) ),
inference(fool_elimination,[status(thm)],[f707]) ).
thf(f707,plain,
! [X0: nat,X1: nat] :
( ( ord_less_eq_nat @ X1 @ X0 )
=> ( ( ord_less_eq_nat @ X0 @ X1 )
=> ( X0 = X1 ) ) ),
inference(rectify,[status(thm)],[f97]) ).
thf(f97,axiom,
! [X25: nat,X37: nat] :
( ( ord_less_eq_nat @ X37 @ X25 )
=> ( ( ord_less_eq_nat @ X25 @ X37 )
=> ( X25 = X37 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.Pn7hie6YQY/DTF2THF_17795.p',fact_96_le__antisym) ).
thf(f1768,plain,
( $true
= ( ord_less_eq_nat @ ( size_s1990949619at_nat @ p2 ) @ ( size_s1990949619at_nat @ p ) ) ),
inference(binary_proxy_clausification,[status(thm)],[f1767]) ).
thf(f1767,plain,
( ( ( ( size_s1990949619at_nat @ p2 )
!= ( size_s1990949619at_nat @ p ) )
& ( ord_less_eq_nat @ ( size_s1990949619at_nat @ p2 ) @ ( size_s1990949619at_nat @ p ) ) )
= $true ),
inference(forward_demodulation,[status(thm)],[f1766,f1150]) ).
thf(f1150,plain,
( ( size_s1990949619at_nat @ p )
= ( min_dist_a @ c @ s @ t ) ),
inference(cnf_transformation,[status(thm)],[f376]) ).
thf(f376,plain,
( ( size_s1990949619at_nat @ p )
= ( min_dist_a @ c @ s @ t ) ),
inference(fool_elimination,[status(thm)],[f2]) ).
thf(f2,axiom,
( ( size_s1990949619at_nat @ p )
= ( min_dist_a @ c @ s @ t ) ),
file('/export/starexec/sandbox2/tmp/tmp.Pn7hie6YQY/DTF2THF_17795.p',fact_1_LENP) ).
thf(f1766,plain,
( ( ( ( size_s1990949619at_nat @ p2 )
!= ( min_dist_a @ c @ s @ t ) )
& ( ord_less_eq_nat @ ( size_s1990949619at_nat @ p2 ) @ ( min_dist_a @ c @ s @ t ) ) )
= $true ),
inference(forward_demodulation,[status(thm)],[f1612,f1329]) ).
thf(f1329,plain,
( ( size_s1990949619at_nat @ p2 )
= ( min_dist_b @ c2 @ s @ t ) ),
inference(cnf_transformation,[status(thm)],[f387]) ).
thf(f387,plain,
( ( size_s1990949619at_nat @ p2 )
= ( min_dist_b @ c2 @ s @ t ) ),
inference(fool_elimination,[status(thm)],[f3]) ).
thf(f3,axiom,
( ( size_s1990949619at_nat @ p2 )
= ( min_dist_b @ c2 @ s @ t ) ),
file('/export/starexec/sandbox2/tmp/tmp.Pn7hie6YQY/DTF2THF_17795.p',fact_2_LENP_H) ).
thf(f1612,plain,
( ( ( ( min_dist_b @ c2 @ s @ t )
!= ( min_dist_a @ c @ s @ t ) )
& ( ord_less_eq_nat @ ( min_dist_b @ c2 @ s @ t ) @ ( min_dist_a @ c @ s @ t ) ) )
= $true ),
inference(beta_eta_normalization,[status(thm)],[f1454]) ).
thf(f1454,plain,
( $true
= ( ^ [Y0: nat,Y1: nat] :
( ( Y0 != Y1 )
& ( ord_less_eq_nat @ Y0 @ Y1 ) )
@ ( min_dist_b @ c2 @ s @ t )
@ ( min_dist_a @ c @ s @ t ) ) ),
inference(definition_unfolding,[status(thm)],[f1174,f1185]) ).
thf(f1174,plain,
( ( ord_less_nat @ ( min_dist_b @ c2 @ s @ t ) @ ( min_dist_a @ c @ s @ t ) )
= $true ),
inference(cnf_transformation,[status(thm)],[f943]) ).
thf(f943,plain,
( ( ord_less_nat @ ( min_dist_b @ c2 @ s @ t ) @ ( min_dist_a @ c @ s @ t ) )
= $true ),
inference(fool_elimination,[status(thm)],[f942]) ).
thf(f942,plain,
ord_less_nat @ ( min_dist_b @ c2 @ s @ t ) @ ( min_dist_a @ c @ s @ t ),
inference(rectify,[status(thm)],[f7]) ).
thf(f7,axiom,
ord_less_nat @ ( min_dist_b @ c2 @ s @ t ) @ ( min_dist_a @ c @ s @ t ),
file('/export/starexec/sandbox2/tmp/tmp.Pn7hie6YQY/DTF2THF_17795.p',fact_6_SHORTER) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP051^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.14/0.33 % Computer : n015.cluster.edu
% 0.14/0.33 % Model : x86_64 x86_64
% 0.14/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.33 % Memory : 8042.1875MB
% 0.14/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.33 % CPULimit : 300
% 0.14/0.33 % WCLimit : 300
% 0.14/0.33 % DateTime : Tue Feb 24 08:24:46 EST 2026
% 0.14/0.34 % CPUTime :
% 0.14/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.46 thf(spec,logic,$$dhol).
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % File : ITP051^1 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.46 % Domain : Interactive Theorem Proving
% 0.19/0.46 % Problem : Sledgehammer EdmondsKarp_Termination_Abstract problem prob_318__7590952_1
% 0.19/0.46 % Version : Especial.
% 0.19/0.46 % English :
% 0.19/0.46
% 0.19/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.46 % Source : [Des21]
% 0.19/0.46 % Names : EdmondsKarp_Termination_Abstract/prob_318__7590952_1 [Des21]
% 0.19/0.46
% 0.19/0.46 % Status : ContradictoryAxioms
% 0.19/0.46 % Rating : 0.44 v9.1.0, 0.38 v9.0.0, 0.40 v8.2.0, 0.23 v8.1.0, 0.36 v7.5.0
% 0.19/0.46 % Syntax : Number of formulae : 430 ( 161 unt; 81 typ; 0 def)
% 0.19/0.46 % Number of atoms : 836 ( 202 equ; 0 cnn)
% 0.19/0.46 % Maximal formula atoms : 7 ( 2 avg)
% 0.19/0.46 % Number of connectives : 2328 ( 57 ~; 4 |; 50 &;1885 @)
% 0.19/0.46 % ( 0 <=>; 332 =>; 0 <=; 0 <~>)
% 0.19/0.46 % Maximal formula depth : 16 ( 6 avg)
% 0.19/0.46 % Number of types : 11 ( 10 usr)
% 0.19/0.46 % Number of type conns : 285 ( 285 >; 0 *; 0 +; 0 <<)
% 0.19/0.46 % Number of symbols : 73 ( 71 usr; 7 con; 0-4 aty)
% 0.19/0.46 % Number of variables : 694 ( 62 ^; 594 !; 38 ?; 694 :)
% 0.19/0.46 % SPC : TH0_CAX_EQU_NAR
% 0.19/0.46
% 0.19/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:31:25.892
% 0.19/0.46 %------------------------------------------------------------------------------
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% 0.19/0.46 % Relevant facts (348)
% 0.19/0.46 thf(fact_0_ek__analysis__axioms,axiom,
% 0.19/0.46 edmond1517640972ysis_a @ c ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis_axioms
% 0.19/0.46 thf(fact_1_LENP,axiom,
% 0.19/0.46 ( ( size_s1990949619at_nat @ p )
% 0.19/0.46 = ( min_dist_a @ c @ s @ t ) ) ).
% 0.19/0.46
% 0.19/0.46 % LENP
% 0.19/0.46 thf(fact_2_LENP_H,axiom,
% 0.19/0.46 ( ( size_s1990949619at_nat @ p2 )
% 0.19/0.46 = ( min_dist_b @ c2 @ s @ t ) ) ).
% 0.19/0.46
% 0.19/0.46 % LENP'
% 0.19/0.46 thf(fact_3_ek__analysis__defs_OekMeasure_Ocong,axiom,
% 0.19/0.46 edmond1022345717sure_b = edmond1022345717sure_b ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis_defs.ekMeasure.cong
% 0.19/0.46 thf(fact_4_ek__analysis__defs_OekMeasure_Ocong,axiom,
% 0.19/0.46 edmond1022345716sure_a = edmond1022345716sure_a ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis_defs.ekMeasure.cong
% 0.19/0.46 thf(fact_5_g_H_Oek__analysis__axioms,axiom,
% 0.19/0.46 edmond1517640973ysis_b @ c2 ).
% 0.19/0.46
% 0.19/0.46 % g'.ek_analysis_axioms
% 0.19/0.46 thf(fact_6_SHORTER,axiom,
% 0.19/0.46 ord_less_nat @ ( min_dist_b @ c2 @ s @ t ) @ ( min_dist_a @ c @ s @ t ) ).
% 0.19/0.46
% 0.19/0.46 % SHORTER
% 0.19/0.46 thf(fact_7_ek__analysis__defs_OuE_Ocong,axiom,
% 0.19/0.46 edmond771116671s_uE_b = edmond771116671s_uE_b ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis_defs.uE.cong
% 0.19/0.46 thf(fact_8_ek__analysis__defs_OuE_Ocong,axiom,
% 0.19/0.46 edmond771116670s_uE_a = edmond771116670s_uE_a ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis_defs.uE.cong
% 0.19/0.46 thf(fact_9_ek__analysis__defs_OspEdges_Ocong,axiom,
% 0.19/0.46 edmond475474835dges_a = edmond475474835dges_a ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis_defs.spEdges.cong
% 0.19/0.46 thf(fact_10_ek__analysis__defs_OspEdges_Ocong,axiom,
% 0.19/0.46 edmond475474836dges_b = edmond475474836dges_b ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis_defs.spEdges.cong
% 0.19/0.46 thf(fact_11_g_H_OFinite__Graph__axioms,axiom,
% 0.19/0.46 finite_Graph_b @ c2 ).
% 0.19/0.46
% 0.19/0.46 % g'.Finite_Graph_axioms
% 0.19/0.46 thf(fact_12_Finite__Graph__axioms,axiom,
% 0.19/0.46 finite_Graph_a @ c ).
% 0.19/0.46
% 0.19/0.46 % Finite_Graph_axioms
% 0.19/0.46 thf(fact_13__092_060open_062length_Ap_A_060_Alength_Ap_H_092_060close_062,axiom,
% 0.19/0.46 ord_less_nat @ ( size_s1990949619at_nat @ p ) @ ( size_s1990949619at_nat @ p2 ) ).
% 0.19/0.46
% 0.19/0.46 % \<open>length p < length p'\<close>
% 0.19/0.46 thf(fact_14_uE__eq,axiom,
% 0.19/0.46 ( ( edmond771116671s_uE_b @ c2 )
% 0.19/0.46 = ( edmond771116670s_uE_a @ c ) ) ).
% 0.19/0.46
% 0.19/0.46 % uE_eq
% 0.19/0.46 thf(fact_15_SP,axiom,
% 0.19/0.46 isShortestPath_a @ c @ s @ p @ t ).
% 0.19/0.46
% 0.19/0.46 % SP
% 0.19/0.46 thf(fact_16_CONN2,axiom,
% 0.19/0.46 connected_b @ c2 @ s @ t ).
% 0.19/0.46
% 0.19/0.46 % CONN2
% 0.19/0.46 thf(fact_17_SV,axiom,
% 0.19/0.46 member_nat @ s @ ( v_a @ c ) ).
% 0.19/0.46
% 0.19/0.46 % SV
% 0.19/0.46 thf(fact_18_P_H,axiom,
% 0.19/0.46 isPath_b @ c2 @ s @ p2 @ t ).
% 0.19/0.46
% 0.19/0.46 % P'
% 0.19/0.46 thf(fact_19_ek__analysis__def,axiom,
% 0.19/0.46 edmond1517640972ysis_a = finite_Graph_a ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis_def
% 0.19/0.46 thf(fact_20_ek__analysis__def,axiom,
% 0.19/0.46 edmond1517640973ysis_b = finite_Graph_b ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis_def
% 0.19/0.46 thf(fact_21_ek__analysis_Ointro,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > a] :
% 0.19/0.46 ( ( finite_Graph_a @ C )
% 0.19/0.46 => ( edmond1517640972ysis_a @ C ) ) ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis.intro
% 0.19/0.46 thf(fact_22_ek__analysis_Ointro,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > b] :
% 0.19/0.46 ( ( finite_Graph_b @ C )
% 0.19/0.46 => ( edmond1517640973ysis_b @ C ) ) ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis.intro
% 0.19/0.46 thf(fact_23_ek__analysis_Oaxioms,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > a] :
% 0.19/0.46 ( ( edmond1517640972ysis_a @ C )
% 0.19/0.46 => ( finite_Graph_a @ C ) ) ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis.axioms
% 0.19/0.46 thf(fact_24_ek__analysis_Oaxioms,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > b] :
% 0.19/0.46 ( ( edmond1517640973ysis_b @ C )
% 0.19/0.46 => ( finite_Graph_b @ C ) ) ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis.axioms
% 0.19/0.46 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,
% 0.19/0.46 ( ~ ( connected_b @ c2 @ s @ t )
% 0.19/0.46 => ( ord_less_nat @ ( edmond1022345717sure_b @ c2 @ s @ t ) @ ( edmond1022345716sure_a @ c @ s @ t ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % \<open>\<not> g'.connected s t \<Longrightarrow> g'.ekMeasure < ekMeasure\<close>
% 0.19/0.46 thf(fact_26__092_060open_062g_H_Odist_As_A_Ig_H_Omin__dist_As_At_J_At_092_060close_062,axiom,
% 0.19/0.46 dist_b @ c2 @ s @ ( min_dist_b @ c2 @ s @ t ) @ t ).
% 0.19/0.46
% 0.19/0.46 % \<open>g'.dist s (g'.min_dist s t) t\<close>
% 0.19/0.46 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,
% 0.19/0.46 ~ ! [P: list_P559422087at_nat] :
% 0.19/0.46 ( ( isPath_b @ c2 @ s @ P @ t )
% 0.19/0.46 => ( ( size_s1990949619at_nat @ P )
% 0.19/0.46 != ( min_dist_b @ c2 @ s @ t ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % \<open>\<And>thesis. (\<And>p'. \<lbrakk>g'.isPath s p' t; length p' = g'.min_dist s t\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.19/0.46 thf(fact_28_CONN,axiom,
% 0.19/0.46 connected_a @ c @ s @ t ).
% 0.19/0.46
% 0.19/0.46 % CONN
% 0.19/0.46 thf(fact_29_min__dist__less,axiom,
% 0.19/0.46 ! [Src: nat,V: nat,D: nat,D2: nat] :
% 0.19/0.46 ( ( connected_a @ c @ Src @ V )
% 0.19/0.46 => ( ( ( min_dist_a @ c @ Src @ V )
% 0.19/0.46 = D )
% 0.19/0.46 => ( ( ord_less_nat @ D2 @ D )
% 0.19/0.46 => ? [V2: nat] :
% 0.19/0.46 ( ( connected_a @ c @ Src @ V2 )
% 0.19/0.46 & ( ( min_dist_a @ c @ Src @ V2 )
% 0.19/0.46 = D2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % min_dist_less
% 0.19/0.46 thf(fact_30_g_H_Omin__dist__less,axiom,
% 0.19/0.46 ! [Src: nat,V: nat,D: nat,D2: nat] :
% 0.19/0.46 ( ( connected_b @ c2 @ Src @ V )
% 0.19/0.46 => ( ( ( min_dist_b @ c2 @ Src @ V )
% 0.19/0.46 = D )
% 0.19/0.46 => ( ( ord_less_nat @ D2 @ D )
% 0.19/0.46 => ? [V2: nat] :
% 0.19/0.46 ( ( connected_b @ c2 @ Src @ V2 )
% 0.19/0.46 & ( ( min_dist_b @ c2 @ Src @ V2 )
% 0.19/0.46 = D2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.min_dist_less
% 0.19/0.46 thf(fact_31_length__induct,axiom,
% 0.19/0.46 ! [P2: list_P559422087at_nat > $o,Xs: list_P559422087at_nat] :
% 0.19/0.46 ( ! [Xs2: list_P559422087at_nat] :
% 0.19/0.46 ( ! [Ys: list_P559422087at_nat] :
% 0.19/0.46 ( ( ord_less_nat @ ( size_s1990949619at_nat @ Ys ) @ ( size_s1990949619at_nat @ Xs2 ) )
% 0.19/0.46 => ( P2 @ Ys ) )
% 0.19/0.46 => ( P2 @ Xs2 ) )
% 0.19/0.46 => ( P2 @ Xs ) ) ).
% 0.19/0.46
% 0.19/0.46 % length_induct
% 0.19/0.46 thf(fact_32_finite__spEdges,axiom,
% 0.19/0.46 finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) ).
% 0.19/0.46
% 0.19/0.46 % finite_spEdges
% 0.19/0.46 thf(fact_33_g_H_Ofinite__spEdges,axiom,
% 0.19/0.46 finite772653738at_nat @ ( edmond475474836dges_b @ c2 @ s @ t ) ).
% 0.19/0.46
% 0.19/0.46 % g'.finite_spEdges
% 0.19/0.46 thf(fact_34_min__dist__z,axiom,
% 0.19/0.46 ! [V: nat] :
% 0.19/0.46 ( ( min_dist_a @ c @ V @ V )
% 0.19/0.46 = zero_zero_nat ) ).
% 0.19/0.46
% 0.19/0.46 % min_dist_z
% 0.19/0.46 thf(fact_35_g_H_Omin__dist__z,axiom,
% 0.19/0.46 ! [V: nat] :
% 0.19/0.46 ( ( min_dist_b @ c2 @ V @ V )
% 0.19/0.46 = zero_zero_nat ) ).
% 0.19/0.46
% 0.19/0.46 % g'.min_dist_z
% 0.19/0.46 thf(fact_36_g_H_Oconnected__def,axiom,
% 0.19/0.46 ! [U: nat,V: nat] :
% 0.19/0.46 ( ( connected_b @ c2 @ U @ V )
% 0.19/0.46 = ( ? [P3: list_P559422087at_nat] : ( isPath_b @ c2 @ U @ P3 @ V ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.connected_def
% 0.19/0.46 thf(fact_37_connected__inV__iff,axiom,
% 0.19/0.46 ! [U: nat,V: nat] :
% 0.19/0.46 ( ( connected_a @ c @ U @ V )
% 0.19/0.46 => ( ( member_nat @ V @ ( v_a @ c ) )
% 0.19/0.46 = ( member_nat @ U @ ( v_a @ c ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % connected_inV_iff
% 0.19/0.46 thf(fact_38_g_H_Oconnected__by__dist,axiom,
% 0.19/0.46 ! [V: nat,V3: nat] :
% 0.19/0.46 ( ( connected_b @ c2 @ V @ V3 )
% 0.19/0.46 = ( ? [D3: nat] : ( dist_b @ c2 @ V @ D3 @ V3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.connected_by_dist
% 0.19/0.46 thf(fact_39_obtain__shortest__path,axiom,
% 0.19/0.46 ! [U: nat,V: nat] :
% 0.19/0.46 ( ( connected_a @ c @ U @ V )
% 0.19/0.46 => ~ ! [P4: list_P559422087at_nat] :
% 0.19/0.46 ~ ( isShortestPath_a @ c @ U @ P4 @ V ) ) ).
% 0.19/0.46
% 0.19/0.46 % obtain_shortest_path
% 0.19/0.46 thf(fact_40_g_H_OisPath__distD,axiom,
% 0.19/0.46 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.46 ( ( isPath_b @ c2 @ U @ P5 @ V )
% 0.19/0.46 => ( dist_b @ c2 @ U @ ( size_s1990949619at_nat @ P5 ) @ V ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.isPath_distD
% 0.19/0.46 thf(fact_41_g_H_Odist__def,axiom,
% 0.19/0.46 ! [V: nat,D: nat,V3: nat] :
% 0.19/0.46 ( ( dist_b @ c2 @ V @ D @ V3 )
% 0.19/0.46 = ( ? [P3: list_P559422087at_nat] :
% 0.19/0.46 ( ( isPath_b @ c2 @ V @ P3 @ V3 )
% 0.19/0.46 & ( ( size_s1990949619at_nat @ P3 )
% 0.19/0.46 = D ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.dist_def
% 0.19/0.46 thf(fact_42_g_H_Omin__dist__is__dist,axiom,
% 0.19/0.46 ! [V: nat,V3: nat] :
% 0.19/0.46 ( ( connected_b @ c2 @ V @ V3 )
% 0.19/0.46 => ( dist_b @ c2 @ V @ ( min_dist_b @ c2 @ V @ V3 ) @ V3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.min_dist_is_dist
% 0.19/0.46 thf(fact_43_g_H_Oconnected__refl,axiom,
% 0.19/0.46 ! [V: nat] : ( connected_b @ c2 @ V @ V ) ).
% 0.19/0.46
% 0.19/0.46 % g'.connected_refl
% 0.19/0.46 thf(fact_44_connected__refl,axiom,
% 0.19/0.46 ! [V: nat] : ( connected_a @ c @ V @ V ) ).
% 0.19/0.46
% 0.19/0.46 % connected_refl
% 0.19/0.46 thf(fact_45_g_H_Odist__z__iff,axiom,
% 0.19/0.46 ! [V: nat,V3: nat] :
% 0.19/0.46 ( ( dist_b @ c2 @ V @ zero_zero_nat @ V3 )
% 0.19/0.46 = ( V3 = V ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.dist_z_iff
% 0.19/0.46 thf(fact_46_g_H_Odist__z,axiom,
% 0.19/0.46 ! [V: nat] : ( dist_b @ c2 @ V @ zero_zero_nat @ V ) ).
% 0.19/0.46
% 0.19/0.46 % g'.dist_z
% 0.19/0.46 thf(fact_47_g_H_Oconnected__distI,axiom,
% 0.19/0.46 ! [V: nat,D: nat,V3: nat] :
% 0.19/0.46 ( ( dist_b @ c2 @ V @ D @ V3 )
% 0.19/0.46 => ( connected_b @ c2 @ V @ V3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.connected_distI
% 0.19/0.46 thf(fact_48_g_H_Ofinite__uE,axiom,
% 0.19/0.46 finite772653738at_nat @ ( edmond771116671s_uE_b @ c2 ) ).
% 0.19/0.46
% 0.19/0.46 % g'.finite_uE
% 0.19/0.46 thf(fact_49_finite__uE,axiom,
% 0.19/0.46 finite772653738at_nat @ ( edmond771116670s_uE_a @ c ) ).
% 0.19/0.46
% 0.19/0.46 % finite_uE
% 0.19/0.46 thf(fact_50_g_H_Omin__dist__z__iff,axiom,
% 0.19/0.46 ! [V: nat,V3: nat] :
% 0.19/0.46 ( ( connected_b @ c2 @ V @ V3 )
% 0.19/0.46 => ( ( ( min_dist_b @ c2 @ V @ V3 )
% 0.19/0.46 = zero_zero_nat )
% 0.19/0.46 = ( V3 = V ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.min_dist_z_iff
% 0.19/0.46 thf(fact_51_mem__Collect__eq,axiom,
% 0.19/0.46 ! [A: set_Pr1986765409at_nat,P2: set_Pr1986765409at_nat > $o] :
% 0.19/0.46 ( ( member298845450at_nat @ A @ ( collec1606769740at_nat @ P2 ) )
% 0.19/0.46 = ( P2 @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % mem_Collect_eq
% 0.19/0.46 thf(fact_52_mem__Collect__eq,axiom,
% 0.19/0.46 ! [A: set_nat,P2: set_nat > $o] :
% 0.19/0.46 ( ( member_set_nat @ A @ ( collect_set_nat @ P2 ) )
% 0.19/0.46 = ( P2 @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % mem_Collect_eq
% 0.19/0.46 thf(fact_53_mem__Collect__eq,axiom,
% 0.19/0.46 ! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
% 0.19/0.46 ( ( member701585322at_nat @ A @ ( collec7649004at_nat @ P2 ) )
% 0.19/0.46 = ( P2 @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % mem_Collect_eq
% 0.19/0.46 thf(fact_54_mem__Collect__eq,axiom,
% 0.19/0.46 ! [A: nat,P2: nat > $o] :
% 0.19/0.46 ( ( member_nat @ A @ ( collect_nat @ P2 ) )
% 0.19/0.46 = ( P2 @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % mem_Collect_eq
% 0.19/0.46 thf(fact_55_Collect__mem__eq,axiom,
% 0.19/0.46 ! [A2: set_se1612935105at_nat] :
% 0.19/0.46 ( ( collec1606769740at_nat
% 0.19/0.46 @ ^ [X: set_Pr1986765409at_nat] : ( member298845450at_nat @ X @ A2 ) )
% 0.19/0.46 = A2 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mem_eq
% 0.19/0.46 thf(fact_56_Collect__mem__eq,axiom,
% 0.19/0.46 ! [A2: set_set_nat] :
% 0.19/0.46 ( ( collect_set_nat
% 0.19/0.46 @ ^ [X: set_nat] : ( member_set_nat @ X @ A2 ) )
% 0.19/0.46 = A2 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mem_eq
% 0.19/0.46 thf(fact_57_Collect__mem__eq,axiom,
% 0.19/0.46 ! [A2: set_Pr1986765409at_nat] :
% 0.19/0.46 ( ( collec7649004at_nat
% 0.19/0.46 @ ^ [X: product_prod_nat_nat] : ( member701585322at_nat @ X @ A2 ) )
% 0.19/0.46 = A2 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mem_eq
% 0.19/0.46 thf(fact_58_Collect__mem__eq,axiom,
% 0.19/0.46 ! [A2: set_nat] :
% 0.19/0.46 ( ( collect_nat
% 0.19/0.46 @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
% 0.19/0.46 = A2 ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_mem_eq
% 0.19/0.46 thf(fact_59_Collect__cong,axiom,
% 0.19/0.46 ! [P2: nat > $o,Q: nat > $o] :
% 0.19/0.46 ( ! [X2: nat] :
% 0.19/0.46 ( ( P2 @ X2 )
% 0.19/0.46 = ( Q @ X2 ) )
% 0.19/0.46 => ( ( collect_nat @ P2 )
% 0.19/0.46 = ( collect_nat @ Q ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_cong
% 0.19/0.46 thf(fact_60_Collect__cong,axiom,
% 0.19/0.46 ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.19/0.46 ( ! [X2: product_prod_nat_nat] :
% 0.19/0.46 ( ( P2 @ X2 )
% 0.19/0.46 = ( Q @ X2 ) )
% 0.19/0.46 => ( ( collec7649004at_nat @ P2 )
% 0.19/0.46 = ( collec7649004at_nat @ Q ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Collect_cong
% 0.19/0.46 thf(fact_61_min__dist__z__iff,axiom,
% 0.19/0.46 ! [V: nat,V3: nat] :
% 0.19/0.46 ( ( connected_a @ c @ V @ V3 )
% 0.19/0.46 => ( ( ( min_dist_a @ c @ V @ V3 )
% 0.19/0.46 = zero_zero_nat )
% 0.19/0.46 = ( V3 = V ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % min_dist_z_iff
% 0.19/0.46 thf(fact_62_ek__analysis_Ofinite__uE,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > b] :
% 0.19/0.46 ( ( edmond1517640973ysis_b @ C )
% 0.19/0.46 => ( finite772653738at_nat @ ( edmond771116671s_uE_b @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis.finite_uE
% 0.19/0.46 thf(fact_63_ek__analysis_Ofinite__uE,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > a] :
% 0.19/0.46 ( ( edmond1517640972ysis_a @ C )
% 0.19/0.46 => ( finite772653738at_nat @ ( edmond771116670s_uE_a @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis.finite_uE
% 0.19/0.46 thf(fact_64_ek__analysis_Ofinite__spEdges,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > a,S: nat,T: nat] :
% 0.19/0.46 ( ( edmond1517640972ysis_a @ C )
% 0.19/0.46 => ( finite772653738at_nat @ ( edmond475474835dges_a @ C @ S @ T ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis.finite_spEdges
% 0.19/0.46 thf(fact_65_ek__analysis_Ofinite__spEdges,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > b,S: nat,T: nat] :
% 0.19/0.46 ( ( edmond1517640973ysis_b @ C )
% 0.19/0.46 => ( finite772653738at_nat @ ( edmond475474836dges_b @ C @ S @ T ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ek_analysis.finite_spEdges
% 0.19/0.46 thf(fact_66_neq__if__length__neq,axiom,
% 0.19/0.46 ! [Xs: list_P559422087at_nat,Ys2: list_P559422087at_nat] :
% 0.19/0.46 ( ( ( size_s1990949619at_nat @ Xs )
% 0.19/0.46 != ( size_s1990949619at_nat @ Ys2 ) )
% 0.19/0.46 => ( Xs != Ys2 ) ) ).
% 0.19/0.46
% 0.19/0.46 % neq_if_length_neq
% 0.19/0.46 thf(fact_67_Ex__list__of__length,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ? [Xs2: list_P559422087at_nat] :
% 0.19/0.46 ( ( size_s1990949619at_nat @ Xs2 )
% 0.19/0.46 = N ) ).
% 0.19/0.46
% 0.19/0.46 % Ex_list_of_length
% 0.19/0.46 thf(fact_68_bot__nat__0_Onot__eq__extremum,axiom,
% 0.19/0.46 ! [A: nat] :
% 0.19/0.46 ( ( A != zero_zero_nat )
% 0.19/0.46 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % bot_nat_0.not_eq_extremum
% 0.19/0.46 thf(fact_69_less__nat__zero__code,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.19/0.46
% 0.19/0.46 % less_nat_zero_code
% 0.19/0.46 thf(fact_70_neq0__conv,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ( ( N != zero_zero_nat )
% 0.19/0.46 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % neq0_conv
% 0.19/0.46 thf(fact_71_not__gr__zero,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.19/0.46 = ( N = zero_zero_nat ) ) ).
% 0.19/0.46
% 0.19/0.46 % not_gr_zero
% 0.19/0.46 thf(fact_72_Graph_OisShortestPath__min__dist__def,axiom,
% 0.19/0.46 ( isShortestPath_a
% 0.19/0.46 = ( ^ [C2: product_prod_nat_nat > a,U2: nat,P3: list_P559422087at_nat,V4: nat] :
% 0.19/0.46 ( ( isPath_a @ C2 @ U2 @ P3 @ V4 )
% 0.19/0.46 & ( ( size_s1990949619at_nat @ P3 )
% 0.19/0.46 = ( min_dist_a @ C2 @ U2 @ V4 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.isShortestPath_min_dist_def
% 0.19/0.46 thf(fact_73_Graph_OisShortestPath__min__dist__def,axiom,
% 0.19/0.46 ( isShortestPath_b
% 0.19/0.46 = ( ^ [C2: product_prod_nat_nat > b,U2: nat,P3: list_P559422087at_nat,V4: nat] :
% 0.19/0.46 ( ( isPath_b @ C2 @ U2 @ P3 @ V4 )
% 0.19/0.46 & ( ( size_s1990949619at_nat @ P3 )
% 0.19/0.46 = ( min_dist_b @ C2 @ U2 @ V4 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.isShortestPath_min_dist_def
% 0.19/0.46 thf(fact_74_g_H_OisShortestPath__min__dist__def,axiom,
% 0.19/0.46 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.46 ( ( isShortestPath_b @ c2 @ U @ P5 @ V )
% 0.19/0.46 = ( ( isPath_b @ c2 @ U @ P5 @ V )
% 0.19/0.46 & ( ( size_s1990949619at_nat @ P5 )
% 0.19/0.46 = ( min_dist_b @ c2 @ U @ V ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.isShortestPath_min_dist_def
% 0.19/0.46 thf(fact_75_g_H_Omin__distI2,axiom,
% 0.19/0.46 ! [V: nat,V3: nat,Q: nat > $o] :
% 0.19/0.46 ( ( connected_b @ c2 @ V @ V3 )
% 0.19/0.46 => ( ! [D4: nat] :
% 0.19/0.46 ( ( dist_b @ c2 @ V @ D4 @ V3 )
% 0.19/0.46 => ( ! [D5: nat] :
% 0.19/0.46 ( ( dist_b @ c2 @ V @ D5 @ V3 )
% 0.19/0.46 => ( ord_less_eq_nat @ D4 @ D5 ) )
% 0.19/0.46 => ( Q @ D4 ) ) )
% 0.19/0.46 => ( Q @ ( min_dist_b @ c2 @ V @ V3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.min_distI2
% 0.19/0.46 thf(fact_76_isShortestPath__length__less__V,axiom,
% 0.19/0.46 ! [S: nat,P5: list_P559422087at_nat,T: nat] :
% 0.19/0.46 ( ( member_nat @ S @ ( v_a @ c ) )
% 0.19/0.46 => ( ( isShortestPath_a @ c @ S @ P5 @ T )
% 0.19/0.46 => ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_a @ c ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % isShortestPath_length_less_V
% 0.19/0.46 thf(fact_77_min__dist__less__V,axiom,
% 0.19/0.46 ! [S: nat,T: nat] :
% 0.19/0.46 ( ( member_nat @ S @ ( v_a @ c ) )
% 0.19/0.46 => ( ( connected_a @ c @ S @ T )
% 0.19/0.46 => ( ord_less_nat @ ( min_dist_a @ c @ S @ T ) @ ( finite_card_nat @ ( v_a @ c ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % min_dist_less_V
% 0.19/0.46 thf(fact_78_connected__def,axiom,
% 0.19/0.46 ! [U: nat,V: nat] :
% 0.19/0.46 ( ( connected_a @ c @ U @ V )
% 0.19/0.46 = ( ? [P3: list_P559422087at_nat] : ( isPath_a @ c @ U @ P3 @ V ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % connected_def
% 0.19/0.46 thf(fact_79_shortestPath__is__path,axiom,
% 0.19/0.46 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.46 ( ( isShortestPath_a @ c @ U @ P5 @ V )
% 0.19/0.46 => ( isPath_a @ c @ U @ P5 @ V ) ) ).
% 0.19/0.46
% 0.19/0.46 % shortestPath_is_path
% 0.19/0.46 thf(fact_80_g_H_OshortestPath__is__path,axiom,
% 0.19/0.46 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.46 ( ( isShortestPath_b @ c2 @ U @ P5 @ V )
% 0.19/0.46 => ( isPath_b @ c2 @ U @ P5 @ V ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.shortestPath_is_path
% 0.19/0.46 thf(fact_81_g_H_Oobtain__shortest__path,axiom,
% 0.19/0.46 ! [U: nat,V: nat] :
% 0.19/0.46 ( ( connected_b @ c2 @ U @ V )
% 0.19/0.46 => ~ ! [P4: list_P559422087at_nat] :
% 0.19/0.46 ~ ( isShortestPath_b @ c2 @ U @ P4 @ V ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.obtain_shortest_path
% 0.19/0.46 thf(fact_82_g_H_Omin__dist__le,axiom,
% 0.19/0.46 ! [Src: nat,V: nat,D2: nat] :
% 0.19/0.46 ( ( connected_b @ c2 @ Src @ V )
% 0.19/0.46 => ( ( ord_less_eq_nat @ D2 @ ( min_dist_b @ c2 @ Src @ V ) )
% 0.19/0.46 => ? [V2: nat] :
% 0.19/0.46 ( ( connected_b @ c2 @ Src @ V2 )
% 0.19/0.46 & ( ( min_dist_b @ c2 @ Src @ V2 )
% 0.19/0.46 = D2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.min_dist_le
% 0.19/0.46 thf(fact_83_g_H_Omin__distI__eq,axiom,
% 0.19/0.46 ! [V: nat,D: nat,V3: nat] :
% 0.19/0.46 ( ( dist_b @ c2 @ V @ D @ V3 )
% 0.19/0.46 => ( ! [D6: nat] :
% 0.19/0.46 ( ( dist_b @ c2 @ V @ D6 @ V3 )
% 0.19/0.46 => ( ord_less_eq_nat @ D @ D6 ) )
% 0.19/0.46 => ( ( min_dist_b @ c2 @ V @ V3 )
% 0.19/0.46 = D ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.min_distI_eq
% 0.19/0.46 thf(fact_84_g_H_Omin__dist__minD,axiom,
% 0.19/0.46 ! [V: nat,D: nat,V3: nat] :
% 0.19/0.46 ( ( dist_b @ c2 @ V @ D @ V3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( min_dist_b @ c2 @ V @ V3 ) @ D ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.min_dist_minD
% 0.19/0.46 thf(fact_85_min__dist__le,axiom,
% 0.19/0.46 ! [Src: nat,V: nat,D2: nat] :
% 0.19/0.46 ( ( connected_a @ c @ Src @ V )
% 0.19/0.46 => ( ( ord_less_eq_nat @ D2 @ ( min_dist_a @ c @ Src @ V ) )
% 0.19/0.46 => ? [V2: nat] :
% 0.19/0.46 ( ( connected_a @ c @ Src @ V2 )
% 0.19/0.46 & ( ( min_dist_a @ c @ Src @ V2 )
% 0.19/0.46 = D2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % min_dist_le
% 0.19/0.46 thf(fact_86_isShortestPath__def,axiom,
% 0.19/0.46 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.46 ( ( isShortestPath_a @ c @ U @ P5 @ V )
% 0.19/0.46 = ( ( isPath_a @ c @ U @ P5 @ V )
% 0.19/0.46 & ! [P6: list_P559422087at_nat] :
% 0.19/0.46 ( ( isPath_a @ c @ U @ P6 @ V )
% 0.19/0.46 => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P5 ) @ ( size_s1990949619at_nat @ P6 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % isShortestPath_def
% 0.19/0.46 thf(fact_87_isShortestPath__min__dist__def,axiom,
% 0.19/0.46 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.46 ( ( isShortestPath_a @ c @ U @ P5 @ V )
% 0.19/0.46 = ( ( isPath_a @ c @ U @ P5 @ V )
% 0.19/0.46 & ( ( size_s1990949619at_nat @ P5 )
% 0.19/0.46 = ( min_dist_a @ c @ U @ V ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % isShortestPath_min_dist_def
% 0.19/0.46 thf(fact_88_g_H_OisShortestPath__def,axiom,
% 0.19/0.46 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.46 ( ( isShortestPath_b @ c2 @ U @ P5 @ V )
% 0.19/0.46 = ( ( isPath_b @ c2 @ U @ P5 @ V )
% 0.19/0.46 & ! [P6: list_P559422087at_nat] :
% 0.19/0.46 ( ( isPath_b @ c2 @ U @ P6 @ V )
% 0.19/0.46 => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P5 ) @ ( size_s1990949619at_nat @ P6 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % g'.isShortestPath_def
% 0.19/0.46 thf(fact_89_le__zero__eq,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.19/0.46 = ( N = zero_zero_nat ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_zero_eq
% 0.19/0.46 thf(fact_90_le0,axiom,
% 0.19/0.46 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.19/0.46
% 0.19/0.46 % le0
% 0.19/0.46 thf(fact_91_bot__nat__0_Oextremum,axiom,
% 0.19/0.46 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.19/0.46
% 0.19/0.46 % bot_nat_0.extremum
% 0.19/0.46 thf(fact_92_zero__le,axiom,
% 0.19/0.46 ! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% 0.19/0.46
% 0.19/0.46 % zero_le
% 0.19/0.46 thf(fact_93_le__refl,axiom,
% 0.19/0.46 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.19/0.46
% 0.19/0.46 % le_refl
% 0.19/0.46 thf(fact_94_le__trans,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 => ( ( ord_less_eq_nat @ J @ K )
% 0.19/0.46 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_trans
% 0.19/0.46 thf(fact_95_eq__imp__le,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( M = N )
% 0.19/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_imp_le
% 0.19/0.46 thf(fact_96_le__antisym,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.19/0.46 => ( ( ord_less_eq_nat @ N @ M )
% 0.19/0.46 => ( M = N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_antisym
% 0.19/0.46 thf(fact_97_nat__le__linear,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.19/0.46 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_le_linear
% 0.19/0.46 thf(fact_98_Nat_Oex__has__greatest__nat,axiom,
% 0.19/0.46 ! [P2: nat > $o,K: nat,B: nat] :
% 0.19/0.46 ( ( P2 @ K )
% 0.19/0.46 => ( ! [Y: nat] :
% 0.19/0.46 ( ( P2 @ Y )
% 0.19/0.46 => ( ord_less_eq_nat @ Y @ B ) )
% 0.19/0.46 => ? [X2: nat] :
% 0.19/0.46 ( ( P2 @ X2 )
% 0.19/0.46 & ! [Y2: nat] :
% 0.19/0.46 ( ( P2 @ Y2 )
% 0.19/0.46 => ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.ex_has_greatest_nat
% 0.19/0.46 thf(fact_99_less__eq__nat_Osimps_I1_J,axiom,
% 0.19/0.46 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.19/0.46
% 0.19/0.46 % less_eq_nat.simps(1)
% 0.19/0.46 thf(fact_100_le__0__eq,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.19/0.46 = ( N = zero_zero_nat ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_0_eq
% 0.19/0.46 thf(fact_101_bot__nat__0_Oextremum__unique,axiom,
% 0.19/0.46 ! [A: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.19/0.46 = ( A = zero_zero_nat ) ) ).
% 0.19/0.46
% 0.19/0.46 % bot_nat_0.extremum_unique
% 0.19/0.46 thf(fact_102_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.19/0.46 ! [A: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.19/0.46 => ( A = zero_zero_nat ) ) ).
% 0.19/0.46
% 0.19/0.46 % bot_nat_0.extremum_uniqueI
% 0.19/0.46 thf(fact_103_nat__less__le,axiom,
% 0.19/0.46 ( ord_less_nat
% 0.19/0.46 = ( ^ [M2: nat,N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.19/0.46 & ( M2 != N2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_less_le
% 0.19/0.46 thf(fact_104_less__imp__le__nat,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_nat @ M @ N )
% 0.19/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_le_nat
% 0.19/0.46 thf(fact_105_le__eq__less__or__eq,axiom,
% 0.19/0.46 ( ord_less_eq_nat
% 0.19/0.46 = ( ^ [M2: nat,N2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ M2 @ N2 )
% 0.19/0.46 | ( M2 = N2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_eq_less_or_eq
% 0.19/0.46 thf(fact_106_less__or__eq__imp__le,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( ( ord_less_nat @ M @ N )
% 0.19/0.46 | ( M = N ) )
% 0.19/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_or_eq_imp_le
% 0.19/0.46 thf(fact_107_le__neq__implies__less,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.19/0.46 => ( ( M != N )
% 0.19/0.46 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_neq_implies_less
% 0.19/0.46 thf(fact_108_less__mono__imp__le__mono,axiom,
% 0.19/0.46 ! [F: nat > nat,I: nat,J: nat] :
% 0.19/0.46 ( ! [I2: nat,J2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ I2 @ J2 )
% 0.19/0.46 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.19/0.46 => ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_mono_imp_le_mono
% 0.19/0.46 thf(fact_109_ex__least__nat__le,axiom,
% 0.19/0.46 ! [P2: nat > $o,N: nat] :
% 0.19/0.46 ( ( P2 @ N )
% 0.19/0.46 => ( ~ ( P2 @ zero_zero_nat )
% 0.19/0.46 => ? [K2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ K2 @ N )
% 0.19/0.46 & ! [I3: nat] :
% 0.19/0.46 ( ( ord_less_nat @ I3 @ K2 )
% 0.19/0.46 => ~ ( P2 @ I3 ) )
% 0.19/0.46 & ( P2 @ K2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ex_least_nat_le
% 0.19/0.46 thf(fact_110_Graph_Omin__dist__le,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > b,Src: nat,V: nat,D2: nat] :
% 0.19/0.46 ( ( connected_b @ C @ Src @ V )
% 0.19/0.46 => ( ( ord_less_eq_nat @ D2 @ ( min_dist_b @ C @ Src @ V ) )
% 0.19/0.46 => ? [V2: nat] :
% 0.19/0.46 ( ( connected_b @ C @ Src @ V2 )
% 0.19/0.46 & ( ( min_dist_b @ C @ Src @ V2 )
% 0.19/0.46 = D2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.min_dist_le
% 0.19/0.46 thf(fact_111_Graph_Omin__dist__le,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > a,Src: nat,V: nat,D2: nat] :
% 0.19/0.46 ( ( connected_a @ C @ Src @ V )
% 0.19/0.46 => ( ( ord_less_eq_nat @ D2 @ ( min_dist_a @ C @ Src @ V ) )
% 0.19/0.46 => ? [V2: nat] :
% 0.19/0.46 ( ( connected_a @ C @ Src @ V2 )
% 0.19/0.46 & ( ( min_dist_a @ C @ Src @ V2 )
% 0.19/0.46 = D2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.min_dist_le
% 0.19/0.46 thf(fact_112_Graph_Omin__dist__minD,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > a,V: nat,D: nat,V3: nat] :
% 0.19/0.46 ( ( dist_a @ C @ V @ D @ V3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( min_dist_a @ C @ V @ V3 ) @ D ) ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.min_dist_minD
% 0.19/0.46 thf(fact_113_Graph_Omin__dist__minD,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > b,V: nat,D: nat,V3: nat] :
% 0.19/0.46 ( ( dist_b @ C @ V @ D @ V3 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( min_dist_b @ C @ V @ V3 ) @ D ) ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.min_dist_minD
% 0.19/0.46 thf(fact_114_Graph_Omin__distI__eq,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > a,V: nat,D: nat,V3: nat] :
% 0.19/0.46 ( ( dist_a @ C @ V @ D @ V3 )
% 0.19/0.46 => ( ! [D6: nat] :
% 0.19/0.46 ( ( dist_a @ C @ V @ D6 @ V3 )
% 0.19/0.46 => ( ord_less_eq_nat @ D @ D6 ) )
% 0.19/0.46 => ( ( min_dist_a @ C @ V @ V3 )
% 0.19/0.46 = D ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.min_distI_eq
% 0.19/0.46 thf(fact_115_Graph_Omin__distI__eq,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > b,V: nat,D: nat,V3: nat] :
% 0.19/0.46 ( ( dist_b @ C @ V @ D @ V3 )
% 0.19/0.46 => ( ! [D6: nat] :
% 0.19/0.46 ( ( dist_b @ C @ V @ D6 @ V3 )
% 0.19/0.46 => ( ord_less_eq_nat @ D @ D6 ) )
% 0.19/0.46 => ( ( min_dist_b @ C @ V @ V3 )
% 0.19/0.46 = D ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.min_distI_eq
% 0.19/0.46 thf(fact_116_Graph_Omin__distI2,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > b,V: nat,V3: nat,Q: nat > $o] :
% 0.19/0.46 ( ( connected_b @ C @ V @ V3 )
% 0.19/0.46 => ( ! [D4: nat] :
% 0.19/0.46 ( ( dist_b @ C @ V @ D4 @ V3 )
% 0.19/0.46 => ( ! [D5: nat] :
% 0.19/0.46 ( ( dist_b @ C @ V @ D5 @ V3 )
% 0.19/0.46 => ( ord_less_eq_nat @ D4 @ D5 ) )
% 0.19/0.46 => ( Q @ D4 ) ) )
% 0.19/0.46 => ( Q @ ( min_dist_b @ C @ V @ V3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.min_distI2
% 0.19/0.46 thf(fact_117_Graph_Omin__distI2,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > a,V: nat,V3: nat,Q: nat > $o] :
% 0.19/0.46 ( ( connected_a @ C @ V @ V3 )
% 0.19/0.46 => ( ! [D4: nat] :
% 0.19/0.46 ( ( dist_a @ C @ V @ D4 @ V3 )
% 0.19/0.46 => ( ! [D5: nat] :
% 0.19/0.46 ( ( dist_a @ C @ V @ D5 @ V3 )
% 0.19/0.46 => ( ord_less_eq_nat @ D4 @ D5 ) )
% 0.19/0.46 => ( Q @ D4 ) ) )
% 0.19/0.46 => ( Q @ ( min_dist_a @ C @ V @ V3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.min_distI2
% 0.19/0.46 thf(fact_118_Graph_OisShortestPath__def,axiom,
% 0.19/0.46 ( isShortestPath_a
% 0.19/0.46 = ( ^ [C2: product_prod_nat_nat > a,U2: nat,P3: list_P559422087at_nat,V4: nat] :
% 0.19/0.46 ( ( isPath_a @ C2 @ U2 @ P3 @ V4 )
% 0.19/0.46 & ! [P6: list_P559422087at_nat] :
% 0.19/0.46 ( ( isPath_a @ C2 @ U2 @ P6 @ V4 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P3 ) @ ( size_s1990949619at_nat @ P6 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.isShortestPath_def
% 0.19/0.46 thf(fact_119_Graph_OisShortestPath__def,axiom,
% 0.19/0.46 ( isShortestPath_b
% 0.19/0.46 = ( ^ [C2: product_prod_nat_nat > b,U2: nat,P3: list_P559422087at_nat,V4: nat] :
% 0.19/0.46 ( ( isPath_b @ C2 @ U2 @ P3 @ V4 )
% 0.19/0.46 & ! [P6: list_P559422087at_nat] :
% 0.19/0.46 ( ( isPath_b @ C2 @ U2 @ P6 @ V4 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P3 ) @ ( size_s1990949619at_nat @ P6 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.isShortestPath_def
% 0.19/0.46 thf(fact_120_zero__reorient,axiom,
% 0.19/0.46 ! [X3: nat] :
% 0.19/0.46 ( ( zero_zero_nat = X3 )
% 0.19/0.46 = ( X3 = zero_zero_nat ) ) ).
% 0.19/0.46
% 0.19/0.46 % zero_reorient
% 0.19/0.46 thf(fact_121_nat__neq__iff,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( M != N )
% 0.19/0.46 = ( ( ord_less_nat @ M @ N )
% 0.19/0.46 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_neq_iff
% 0.19/0.46 thf(fact_122_less__not__refl,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ~ ( ord_less_nat @ N @ N ) ).
% 0.19/0.46
% 0.19/0.46 % less_not_refl
% 0.19/0.46 thf(fact_123_less__not__refl2,axiom,
% 0.19/0.46 ! [N: nat,M: nat] :
% 0.19/0.46 ( ( ord_less_nat @ N @ M )
% 0.19/0.46 => ( M != N ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_not_refl2
% 0.19/0.46 thf(fact_124_less__not__refl3,axiom,
% 0.19/0.46 ! [S: nat,T: nat] :
% 0.19/0.46 ( ( ord_less_nat @ S @ T )
% 0.19/0.46 => ( S != T ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_not_refl3
% 0.19/0.46 thf(fact_125_less__irrefl__nat,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ~ ( ord_less_nat @ N @ N ) ).
% 0.19/0.46
% 0.19/0.46 % less_irrefl_nat
% 0.19/0.46 thf(fact_126_nat__less__induct,axiom,
% 0.19/0.46 ! [P2: nat > $o,N: nat] :
% 0.19/0.46 ( ! [N3: nat] :
% 0.19/0.46 ( ! [M3: nat] :
% 0.19/0.46 ( ( ord_less_nat @ M3 @ N3 )
% 0.19/0.46 => ( P2 @ M3 ) )
% 0.19/0.46 => ( P2 @ N3 ) )
% 0.19/0.46 => ( P2 @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_less_induct
% 0.19/0.46 thf(fact_127_infinite__descent,axiom,
% 0.19/0.46 ! [P2: nat > $o,N: nat] :
% 0.19/0.46 ( ! [N3: nat] :
% 0.19/0.46 ( ~ ( P2 @ N3 )
% 0.19/0.46 => ? [M3: nat] :
% 0.19/0.46 ( ( ord_less_nat @ M3 @ N3 )
% 0.19/0.46 & ~ ( P2 @ M3 ) ) )
% 0.19/0.46 => ( P2 @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % infinite_descent
% 0.19/0.46 thf(fact_128_linorder__neqE__nat,axiom,
% 0.19/0.46 ! [X3: nat,Y3: nat] :
% 0.19/0.46 ( ( X3 != Y3 )
% 0.19/0.46 => ( ~ ( ord_less_nat @ X3 @ Y3 )
% 0.19/0.46 => ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_neqE_nat
% 0.19/0.46 thf(fact_129_size__neq__size__imp__neq,axiom,
% 0.19/0.46 ! [X3: list_P559422087at_nat,Y3: list_P559422087at_nat] :
% 0.19/0.46 ( ( ( size_s1990949619at_nat @ X3 )
% 0.19/0.46 != ( size_s1990949619at_nat @ Y3 ) )
% 0.19/0.46 => ( X3 != Y3 ) ) ).
% 0.19/0.46
% 0.19/0.46 % size_neq_size_imp_neq
% 0.19/0.46 thf(fact_130_Graph_OisPath_Ocong,axiom,
% 0.19/0.46 isPath_b = isPath_b ).
% 0.19/0.46
% 0.19/0.46 % Graph.isPath.cong
% 0.19/0.46 thf(fact_131_Graph_OisPath_Ocong,axiom,
% 0.19/0.46 isPath_a = isPath_a ).
% 0.19/0.46
% 0.19/0.46 % Graph.isPath.cong
% 0.19/0.46 thf(fact_132_Graph_OV_Ocong,axiom,
% 0.19/0.46 v_a = v_a ).
% 0.19/0.46
% 0.19/0.46 % Graph.V.cong
% 0.19/0.46 thf(fact_133_Graph_OV_Ocong,axiom,
% 0.19/0.46 v_b = v_b ).
% 0.19/0.46
% 0.19/0.46 % Graph.V.cong
% 0.19/0.46 thf(fact_134_Graph_Omin__dist_Ocong,axiom,
% 0.19/0.46 min_dist_a = min_dist_a ).
% 0.19/0.46
% 0.19/0.46 % Graph.min_dist.cong
% 0.19/0.46 thf(fact_135_Graph_Omin__dist_Ocong,axiom,
% 0.19/0.46 min_dist_b = min_dist_b ).
% 0.19/0.46
% 0.19/0.46 % Graph.min_dist.cong
% 0.19/0.46 thf(fact_136_Graph_Oconnected_Ocong,axiom,
% 0.19/0.46 connected_b = connected_b ).
% 0.19/0.46
% 0.19/0.46 % Graph.connected.cong
% 0.19/0.46 thf(fact_137_Graph_Oconnected_Ocong,axiom,
% 0.19/0.46 connected_a = connected_a ).
% 0.19/0.46
% 0.19/0.46 % Graph.connected.cong
% 0.19/0.46 thf(fact_138_Graph_Oconnected__refl,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > b,V: nat] : ( connected_b @ C @ V @ V ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.connected_refl
% 0.19/0.46 thf(fact_139_Graph_Oconnected__refl,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > a,V: nat] : ( connected_a @ C @ V @ V ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.connected_refl
% 0.19/0.46 thf(fact_140_Graph_Odist_Ocong,axiom,
% 0.19/0.46 dist_b = dist_b ).
% 0.19/0.46
% 0.19/0.46 % Graph.dist.cong
% 0.19/0.46 thf(fact_141_Graph_Odist_Ocong,axiom,
% 0.19/0.46 dist_a = dist_a ).
% 0.19/0.46
% 0.19/0.46 % Graph.dist.cong
% 0.19/0.46 thf(fact_142_Graph_OisShortestPath_Ocong,axiom,
% 0.19/0.46 isShortestPath_a = isShortestPath_a ).
% 0.19/0.46
% 0.19/0.46 % Graph.isShortestPath.cong
% 0.19/0.46 thf(fact_143_Graph_OisShortestPath_Ocong,axiom,
% 0.19/0.46 isShortestPath_b = isShortestPath_b ).
% 0.19/0.46
% 0.19/0.46 % Graph.isShortestPath.cong
% 0.19/0.46 thf(fact_144_Finite__Graph_Omin__dist__less__V,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > b,S: nat,T: nat] :
% 0.19/0.46 ( ( finite_Graph_b @ C )
% 0.19/0.46 => ( ( member_nat @ S @ ( v_b @ C ) )
% 0.19/0.46 => ( ( connected_b @ C @ S @ T )
% 0.19/0.46 => ( ord_less_nat @ ( min_dist_b @ C @ S @ T ) @ ( finite_card_nat @ ( v_b @ C ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Finite_Graph.min_dist_less_V
% 0.19/0.46 thf(fact_145_Finite__Graph_Omin__dist__less__V,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > a,S: nat,T: nat] :
% 0.19/0.46 ( ( finite_Graph_a @ C )
% 0.19/0.46 => ( ( member_nat @ S @ ( v_a @ C ) )
% 0.19/0.46 => ( ( connected_a @ C @ S @ T )
% 0.19/0.46 => ( ord_less_nat @ ( min_dist_a @ C @ S @ T ) @ ( finite_card_nat @ ( v_a @ C ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Finite_Graph.min_dist_less_V
% 0.19/0.46 thf(fact_146_Finite__Graph_OisShortestPath__length__less__V,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > a,S: nat,P5: list_P559422087at_nat,T: nat] :
% 0.19/0.46 ( ( finite_Graph_a @ C )
% 0.19/0.46 => ( ( member_nat @ S @ ( v_a @ C ) )
% 0.19/0.46 => ( ( isShortestPath_a @ C @ S @ P5 @ T )
% 0.19/0.46 => ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_a @ C ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Finite_Graph.isShortestPath_length_less_V
% 0.19/0.46 thf(fact_147_Finite__Graph_OisShortestPath__length__less__V,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > b,S: nat,P5: list_P559422087at_nat,T: nat] :
% 0.19/0.46 ( ( finite_Graph_b @ C )
% 0.19/0.46 => ( ( member_nat @ S @ ( v_b @ C ) )
% 0.19/0.46 => ( ( isShortestPath_b @ C @ S @ P5 @ T )
% 0.19/0.46 => ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_b @ C ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Finite_Graph.isShortestPath_length_less_V
% 0.19/0.46 thf(fact_148_gr__zeroI,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ( ( N != zero_zero_nat )
% 0.19/0.46 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % gr_zeroI
% 0.19/0.46 thf(fact_149_not__less__zero,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.19/0.46
% 0.19/0.46 % not_less_zero
% 0.19/0.46 thf(fact_150_gr__implies__not__zero,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_nat @ M @ N )
% 0.19/0.46 => ( N != zero_zero_nat ) ) ).
% 0.19/0.46
% 0.19/0.46 % gr_implies_not_zero
% 0.19/0.46 thf(fact_151_zero__less__iff__neq__zero,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.19/0.46 = ( N != zero_zero_nat ) ) ).
% 0.19/0.46
% 0.19/0.46 % zero_less_iff_neq_zero
% 0.19/0.46 thf(fact_152_gr0I,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ( ( N != zero_zero_nat )
% 0.19/0.46 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % gr0I
% 0.19/0.46 thf(fact_153_not__gr0,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.19/0.46 = ( N = zero_zero_nat ) ) ).
% 0.19/0.46
% 0.19/0.46 % not_gr0
% 0.19/0.46 thf(fact_154_not__less0,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.19/0.46
% 0.19/0.46 % not_less0
% 0.19/0.46 thf(fact_155_less__zeroE,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.19/0.46
% 0.19/0.46 % less_zeroE
% 0.19/0.46 thf(fact_156_gr__implies__not0,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_nat @ M @ N )
% 0.19/0.46 => ( N != zero_zero_nat ) ) ).
% 0.19/0.46
% 0.19/0.46 % gr_implies_not0
% 0.19/0.46 thf(fact_157_infinite__descent0,axiom,
% 0.19/0.46 ! [P2: nat > $o,N: nat] :
% 0.19/0.46 ( ( P2 @ zero_zero_nat )
% 0.19/0.46 => ( ! [N3: nat] :
% 0.19/0.46 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.19/0.46 => ( ~ ( P2 @ N3 )
% 0.19/0.46 => ? [M3: nat] :
% 0.19/0.46 ( ( ord_less_nat @ M3 @ N3 )
% 0.19/0.46 & ~ ( P2 @ M3 ) ) ) )
% 0.19/0.46 => ( P2 @ N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % infinite_descent0
% 0.19/0.46 thf(fact_158_bot__nat__0_Oextremum__strict,axiom,
% 0.19/0.46 ! [A: nat] :
% 0.19/0.46 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.19/0.46
% 0.19/0.46 % bot_nat_0.extremum_strict
% 0.19/0.46 thf(fact_159_Graph_Omin__dist__z,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > a,V: nat] :
% 0.19/0.46 ( ( min_dist_a @ C @ V @ V )
% 0.19/0.46 = zero_zero_nat ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.min_dist_z
% 0.19/0.46 thf(fact_160_Graph_Omin__dist__z,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > b,V: nat] :
% 0.19/0.46 ( ( min_dist_b @ C @ V @ V )
% 0.19/0.46 = zero_zero_nat ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.min_dist_z
% 0.19/0.46 thf(fact_161_Graph_Odist__z,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > b,V: nat] : ( dist_b @ C @ V @ zero_zero_nat @ V ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.dist_z
% 0.19/0.46 thf(fact_162_Graph_Odist__z,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > a,V: nat] : ( dist_a @ C @ V @ zero_zero_nat @ V ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.dist_z
% 0.19/0.46 thf(fact_163_Graph_Odist__z__iff,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > b,V: nat,V3: nat] :
% 0.19/0.46 ( ( dist_b @ C @ V @ zero_zero_nat @ V3 )
% 0.19/0.46 = ( V3 = V ) ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.dist_z_iff
% 0.19/0.46 thf(fact_164_Graph_Odist__z__iff,axiom,
% 0.19/0.46 ! [C: product_prod_nat_nat > a,V: nat,V3: nat] :
% 0.19/0.46 ( ( dist_a @ C @ V @ zero_zero_nat @ V3 )
% 0.19/0.46 = ( V3 = V ) ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.dist_z_iff
% 0.19/0.46 thf(fact_165_Graph_Oconnected__def,axiom,
% 0.19/0.46 ( connected_b
% 0.19/0.46 = ( ^ [C2: product_prod_nat_nat > b,U2: nat,V4: nat] :
% 0.19/0.46 ? [P3: list_P559422087at_nat] : ( isPath_b @ C2 @ U2 @ P3 @ V4 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Graph.connected_def
% 0.19/0.46 thf(fact_166_Graph_Oconnected__def,axiom,
% 0.19/0.46 ( connected_a
% 0.19/0.46 = ( ^ [C2: product_prod_nat_nat > a,U2: nat,V4: nat] :
% 0.19/0.46 ? [P3: list_P559422087at_nat] : ( isPath_a @ C2 @ U2 @ P3 @ V4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.connected_def
% 0.19/0.47 thf(fact_167_Graph_Oconnected__inV__iff,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,U: nat,V: nat] :
% 0.19/0.47 ( ( connected_b @ C @ U @ V )
% 0.19/0.47 => ( ( member_nat @ V @ ( v_b @ C ) )
% 0.19/0.47 = ( member_nat @ U @ ( v_b @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.connected_inV_iff
% 0.19/0.47 thf(fact_168_Graph_Oconnected__inV__iff,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,U: nat,V: nat] :
% 0.19/0.47 ( ( connected_a @ C @ U @ V )
% 0.19/0.47 => ( ( member_nat @ V @ ( v_a @ C ) )
% 0.19/0.47 = ( member_nat @ U @ ( v_a @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.connected_inV_iff
% 0.19/0.47 thf(fact_169_Graph_OshortestPath__is__path,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.47 ( ( isShortestPath_a @ C @ U @ P5 @ V )
% 0.19/0.47 => ( isPath_a @ C @ U @ P5 @ V ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.shortestPath_is_path
% 0.19/0.47 thf(fact_170_Graph_OshortestPath__is__path,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.47 ( ( isShortestPath_b @ C @ U @ P5 @ V )
% 0.19/0.47 => ( isPath_b @ C @ U @ P5 @ V ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.shortestPath_is_path
% 0.19/0.47 thf(fact_171_Graph_Oconnected__distI,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,V: nat,D: nat,V3: nat] :
% 0.19/0.47 ( ( dist_b @ C @ V @ D @ V3 )
% 0.19/0.47 => ( connected_b @ C @ V @ V3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.connected_distI
% 0.19/0.47 thf(fact_172_Graph_Oconnected__distI,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,V: nat,D: nat,V3: nat] :
% 0.19/0.47 ( ( dist_a @ C @ V @ D @ V3 )
% 0.19/0.47 => ( connected_a @ C @ V @ V3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.connected_distI
% 0.19/0.47 thf(fact_173_Graph_Oconnected__by__dist,axiom,
% 0.19/0.47 ( connected_b
% 0.19/0.47 = ( ^ [C2: product_prod_nat_nat > b,V4: nat,V5: nat] :
% 0.19/0.47 ? [D3: nat] : ( dist_b @ C2 @ V4 @ D3 @ V5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.connected_by_dist
% 0.19/0.47 thf(fact_174_Graph_Oconnected__by__dist,axiom,
% 0.19/0.47 ( connected_a
% 0.19/0.47 = ( ^ [C2: product_prod_nat_nat > a,V4: nat,V5: nat] :
% 0.19/0.47 ? [D3: nat] : ( dist_a @ C2 @ V4 @ D3 @ V5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.connected_by_dist
% 0.19/0.47 thf(fact_175_Graph_Oobtain__shortest__path,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,U: nat,V: nat] :
% 0.19/0.47 ( ( connected_a @ C @ U @ V )
% 0.19/0.47 => ~ ! [P4: list_P559422087at_nat] :
% 0.19/0.47 ~ ( isShortestPath_a @ C @ U @ P4 @ V ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.obtain_shortest_path
% 0.19/0.47 thf(fact_176_Graph_Oobtain__shortest__path,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,U: nat,V: nat] :
% 0.19/0.47 ( ( connected_b @ C @ U @ V )
% 0.19/0.47 => ~ ! [P4: list_P559422087at_nat] :
% 0.19/0.47 ~ ( isShortestPath_b @ C @ U @ P4 @ V ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.obtain_shortest_path
% 0.19/0.47 thf(fact_177_Graph_Omin__dist__z__iff,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,V: nat,V3: nat] :
% 0.19/0.47 ( ( connected_b @ C @ V @ V3 )
% 0.19/0.47 => ( ( ( min_dist_b @ C @ V @ V3 )
% 0.19/0.47 = zero_zero_nat )
% 0.19/0.47 = ( V3 = V ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.min_dist_z_iff
% 0.19/0.47 thf(fact_178_Graph_Omin__dist__z__iff,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,V: nat,V3: nat] :
% 0.19/0.47 ( ( connected_a @ C @ V @ V3 )
% 0.19/0.47 => ( ( ( min_dist_a @ C @ V @ V3 )
% 0.19/0.47 = zero_zero_nat )
% 0.19/0.47 = ( V3 = V ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.min_dist_z_iff
% 0.19/0.47 thf(fact_179_Graph_Omin__dist__less,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,Src: nat,V: nat,D: nat,D2: nat] :
% 0.19/0.47 ( ( connected_b @ C @ Src @ V )
% 0.19/0.47 => ( ( ( min_dist_b @ C @ Src @ V )
% 0.19/0.47 = D )
% 0.19/0.47 => ( ( ord_less_nat @ D2 @ D )
% 0.19/0.47 => ? [V2: nat] :
% 0.19/0.47 ( ( connected_b @ C @ Src @ V2 )
% 0.19/0.47 & ( ( min_dist_b @ C @ Src @ V2 )
% 0.19/0.47 = D2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.min_dist_less
% 0.19/0.47 thf(fact_180_Graph_Omin__dist__less,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,Src: nat,V: nat,D: nat,D2: nat] :
% 0.19/0.47 ( ( connected_a @ C @ Src @ V )
% 0.19/0.47 => ( ( ( min_dist_a @ C @ Src @ V )
% 0.19/0.47 = D )
% 0.19/0.47 => ( ( ord_less_nat @ D2 @ D )
% 0.19/0.47 => ? [V2: nat] :
% 0.19/0.47 ( ( connected_a @ C @ Src @ V2 )
% 0.19/0.47 & ( ( min_dist_a @ C @ Src @ V2 )
% 0.19/0.47 = D2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.min_dist_less
% 0.19/0.47 thf(fact_181_Graph_Omin__dist__is__dist,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,V: nat,V3: nat] :
% 0.19/0.47 ( ( connected_b @ C @ V @ V3 )
% 0.19/0.47 => ( dist_b @ C @ V @ ( min_dist_b @ C @ V @ V3 ) @ V3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.min_dist_is_dist
% 0.19/0.47 thf(fact_182_Graph_Omin__dist__is__dist,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,V: nat,V3: nat] :
% 0.19/0.47 ( ( connected_a @ C @ V @ V3 )
% 0.19/0.47 => ( dist_a @ C @ V @ ( min_dist_a @ C @ V @ V3 ) @ V3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.min_dist_is_dist
% 0.19/0.47 thf(fact_183_Graph_OisPath__distD,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.47 ( ( isPath_b @ C @ U @ P5 @ V )
% 0.19/0.47 => ( dist_b @ C @ U @ ( size_s1990949619at_nat @ P5 ) @ V ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.isPath_distD
% 0.19/0.47 thf(fact_184_Graph_OisPath__distD,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.47 ( ( isPath_a @ C @ U @ P5 @ V )
% 0.19/0.47 => ( dist_a @ C @ U @ ( size_s1990949619at_nat @ P5 ) @ V ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.isPath_distD
% 0.19/0.47 thf(fact_185_Graph_Odist__def,axiom,
% 0.19/0.47 ( dist_b
% 0.19/0.47 = ( ^ [C2: product_prod_nat_nat > b,V4: nat,D3: nat,V5: nat] :
% 0.19/0.47 ? [P3: list_P559422087at_nat] :
% 0.19/0.47 ( ( isPath_b @ C2 @ V4 @ P3 @ V5 )
% 0.19/0.47 & ( ( size_s1990949619at_nat @ P3 )
% 0.19/0.47 = D3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.dist_def
% 0.19/0.47 thf(fact_186_Graph_Odist__def,axiom,
% 0.19/0.47 ( dist_a
% 0.19/0.47 = ( ^ [C2: product_prod_nat_nat > a,V4: nat,D3: nat,V5: nat] :
% 0.19/0.47 ? [P3: list_P559422087at_nat] :
% 0.19/0.47 ( ( isPath_a @ C2 @ V4 @ P3 @ V5 )
% 0.19/0.47 & ( ( size_s1990949619at_nat @ P3 )
% 0.19/0.47 = D3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.dist_def
% 0.19/0.47 thf(fact_187_card_Oinfinite,axiom,
% 0.19/0.47 ! [A2: set_li664300135at_nat] :
% 0.19/0.47 ( ~ ( finite1299096496at_nat @ A2 )
% 0.19/0.47 => ( ( finite83082927at_nat @ A2 )
% 0.19/0.47 = zero_zero_nat ) ) ).
% 0.19/0.47
% 0.19/0.47 % card.infinite
% 0.19/0.47 thf(fact_188_card_Oinfinite,axiom,
% 0.19/0.47 ! [A2: set_se1612935105at_nat] :
% 0.19/0.47 ( ~ ( finite1457549322at_nat @ A2 )
% 0.19/0.47 => ( ( finite1701894793at_nat @ A2 )
% 0.19/0.47 = zero_zero_nat ) ) ).
% 0.19/0.47
% 0.19/0.47 % card.infinite
% 0.19/0.47 thf(fact_189_card_Oinfinite,axiom,
% 0.19/0.47 ! [A2: set_set_nat] :
% 0.19/0.47 ( ~ ( finite2012248349et_nat @ A2 )
% 0.19/0.47 => ( ( finite_card_set_nat @ A2 )
% 0.19/0.47 = zero_zero_nat ) ) ).
% 0.19/0.47
% 0.19/0.47 % card.infinite
% 0.19/0.47 thf(fact_190_card_Oinfinite,axiom,
% 0.19/0.47 ! [A2: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ~ ( finite772653738at_nat @ A2 )
% 0.19/0.47 => ( ( finite447719721at_nat @ A2 )
% 0.19/0.47 = zero_zero_nat ) ) ).
% 0.19/0.47
% 0.19/0.47 % card.infinite
% 0.19/0.47 thf(fact_191_card_Oinfinite,axiom,
% 0.19/0.47 ! [A2: set_nat] :
% 0.19/0.47 ( ~ ( finite_finite_nat @ A2 )
% 0.19/0.47 => ( ( finite_card_nat @ A2 )
% 0.19/0.47 = zero_zero_nat ) ) ).
% 0.19/0.47
% 0.19/0.47 % card.infinite
% 0.19/0.47 thf(fact_192_card__spEdges__le,axiom,
% 0.19/0.47 ord_less_eq_nat @ ( finite447719721at_nat @ ( edmond475474835dges_a @ c @ s @ t ) ) @ ( finite447719721at_nat @ ( edmond771116670s_uE_a @ c ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_spEdges_le
% 0.19/0.47 thf(fact_193_g_H_Ocard__spEdges__le,axiom,
% 0.19/0.47 ord_less_eq_nat @ ( finite447719721at_nat @ ( edmond475474836dges_b @ c2 @ s @ t ) ) @ ( finite447719721at_nat @ ( edmond771116671s_uE_b @ c2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % g'.card_spEdges_le
% 0.19/0.47 thf(fact_194_simplePath__length__less__V,axiom,
% 0.19/0.47 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.47 ( ( member_nat @ U @ ( v_a @ c ) )
% 0.19/0.47 => ( ( isSimplePath_a @ c @ U @ P5 @ V )
% 0.19/0.47 => ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_a @ c ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % simplePath_length_less_V
% 0.19/0.47 thf(fact_195_card__ge__0__finite,axiom,
% 0.19/0.47 ! [A2: set_li664300135at_nat] :
% 0.19/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( finite83082927at_nat @ A2 ) )
% 0.19/0.47 => ( finite1299096496at_nat @ A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_ge_0_finite
% 0.19/0.47 thf(fact_196_card__ge__0__finite,axiom,
% 0.19/0.47 ! [A2: set_se1612935105at_nat] :
% 0.19/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( finite1701894793at_nat @ A2 ) )
% 0.19/0.47 => ( finite1457549322at_nat @ A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_ge_0_finite
% 0.19/0.47 thf(fact_197_card__ge__0__finite,axiom,
% 0.19/0.47 ! [A2: set_set_nat] :
% 0.19/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A2 ) )
% 0.19/0.47 => ( finite2012248349et_nat @ A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_ge_0_finite
% 0.19/0.47 thf(fact_198_card__ge__0__finite,axiom,
% 0.19/0.47 ! [A2: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( finite447719721at_nat @ A2 ) )
% 0.19/0.47 => ( finite772653738at_nat @ A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_ge_0_finite
% 0.19/0.47 thf(fact_199_card__ge__0__finite,axiom,
% 0.19/0.47 ! [A2: set_nat] :
% 0.19/0.47 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
% 0.19/0.47 => ( finite_finite_nat @ A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_ge_0_finite
% 0.19/0.47 thf(fact_200_min__distI2,axiom,
% 0.19/0.47 ! [V: nat,V3: nat,Q: nat > $o] :
% 0.19/0.47 ( ( connected_a @ c @ V @ V3 )
% 0.19/0.47 => ( ! [D4: nat] :
% 0.19/0.47 ( ( dist_a @ c @ V @ D4 @ V3 )
% 0.19/0.47 => ( ! [D5: nat] :
% 0.19/0.47 ( ( dist_a @ c @ V @ D5 @ V3 )
% 0.19/0.47 => ( ord_less_eq_nat @ D4 @ D5 ) )
% 0.19/0.47 => ( Q @ D4 ) ) )
% 0.19/0.47 => ( Q @ ( min_dist_a @ c @ V @ V3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % min_distI2
% 0.19/0.47 thf(fact_201_g_H_OisShortestPath__alt,axiom,
% 0.19/0.47 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.47 ( ( isShortestPath_b @ c2 @ U @ P5 @ V )
% 0.19/0.47 = ( ( isSimplePath_b @ c2 @ U @ P5 @ V )
% 0.19/0.47 & ( ( size_s1990949619at_nat @ P5 )
% 0.19/0.47 = ( min_dist_b @ c2 @ U @ V ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % g'.isShortestPath_alt
% 0.19/0.47 thf(fact_202_g_H_Oconnected__inV__iff,axiom,
% 0.19/0.47 ! [U: nat,V: nat] :
% 0.19/0.47 ( ( connected_b @ c2 @ U @ V )
% 0.19/0.47 => ( ( member_nat @ V @ ( v_b @ c2 ) )
% 0.19/0.47 = ( member_nat @ U @ ( v_b @ c2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % g'.connected_inV_iff
% 0.19/0.47 thf(fact_203_g_H_OisSPath__pathLE,axiom,
% 0.19/0.47 ! [S: nat,P5: list_P559422087at_nat,T: nat] :
% 0.19/0.47 ( ( isPath_b @ c2 @ S @ P5 @ T )
% 0.19/0.47 => ? [P: list_P559422087at_nat] : ( isSimplePath_b @ c2 @ S @ P @ T ) ) ).
% 0.19/0.47
% 0.19/0.47 % g'.isSPath_pathLE
% 0.19/0.47 thf(fact_204_isSPath__pathLE,axiom,
% 0.19/0.47 ! [S: nat,P5: list_P559422087at_nat,T: nat] :
% 0.19/0.47 ( ( isPath_a @ c @ S @ P5 @ T )
% 0.19/0.47 => ? [P: list_P559422087at_nat] : ( isSimplePath_a @ c @ S @ P @ T ) ) ).
% 0.19/0.47
% 0.19/0.47 % isSPath_pathLE
% 0.19/0.47 thf(fact_205_connected__by__dist,axiom,
% 0.19/0.47 ! [V: nat,V3: nat] :
% 0.19/0.47 ( ( connected_a @ c @ V @ V3 )
% 0.19/0.47 = ( ? [D3: nat] : ( dist_a @ c @ V @ D3 @ V3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % connected_by_dist
% 0.19/0.47 thf(fact_206_shortestPath__is__simple,axiom,
% 0.19/0.47 ! [S: nat,P5: list_P559422087at_nat,T: nat] :
% 0.19/0.47 ( ( isShortestPath_a @ c @ S @ P5 @ T )
% 0.19/0.47 => ( isSimplePath_a @ c @ S @ P5 @ T ) ) ).
% 0.19/0.47
% 0.19/0.47 % shortestPath_is_simple
% 0.19/0.47 thf(fact_207_g_H_OshortestPath__is__simple,axiom,
% 0.19/0.47 ! [S: nat,P5: list_P559422087at_nat,T: nat] :
% 0.19/0.47 ( ( isShortestPath_b @ c2 @ S @ P5 @ T )
% 0.19/0.47 => ( isSimplePath_b @ c2 @ S @ P5 @ T ) ) ).
% 0.19/0.47
% 0.19/0.47 % g'.shortestPath_is_simple
% 0.19/0.47 thf(fact_208_min__dist__minD,axiom,
% 0.19/0.47 ! [V: nat,D: nat,V3: nat] :
% 0.19/0.47 ( ( dist_a @ c @ V @ D @ V3 )
% 0.19/0.47 => ( ord_less_eq_nat @ ( min_dist_a @ c @ V @ V3 ) @ D ) ) ).
% 0.19/0.47
% 0.19/0.47 % min_dist_minD
% 0.19/0.47 thf(fact_209_min__distI__eq,axiom,
% 0.19/0.47 ! [V: nat,D: nat,V3: nat] :
% 0.19/0.47 ( ( dist_a @ c @ V @ D @ V3 )
% 0.19/0.47 => ( ! [D6: nat] :
% 0.19/0.47 ( ( dist_a @ c @ V @ D6 @ V3 )
% 0.19/0.47 => ( ord_less_eq_nat @ D @ D6 ) )
% 0.19/0.47 => ( ( min_dist_a @ c @ V @ V3 )
% 0.19/0.47 = D ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % min_distI_eq
% 0.19/0.47 thf(fact_210_isPath__distD,axiom,
% 0.19/0.47 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.47 ( ( isPath_a @ c @ U @ P5 @ V )
% 0.19/0.47 => ( dist_a @ c @ U @ ( size_s1990949619at_nat @ P5 ) @ V ) ) ).
% 0.19/0.47
% 0.19/0.47 % isPath_distD
% 0.19/0.47 thf(fact_211_dist__def,axiom,
% 0.19/0.47 ! [V: nat,D: nat,V3: nat] :
% 0.19/0.47 ( ( dist_a @ c @ V @ D @ V3 )
% 0.19/0.47 = ( ? [P3: list_P559422087at_nat] :
% 0.19/0.47 ( ( isPath_a @ c @ V @ P3 @ V3 )
% 0.19/0.47 & ( ( size_s1990949619at_nat @ P3 )
% 0.19/0.47 = D ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dist_def
% 0.19/0.47 thf(fact_212_min__dist__is__dist,axiom,
% 0.19/0.47 ! [V: nat,V3: nat] :
% 0.19/0.47 ( ( connected_a @ c @ V @ V3 )
% 0.19/0.47 => ( dist_a @ c @ V @ ( min_dist_a @ c @ V @ V3 ) @ V3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % min_dist_is_dist
% 0.19/0.47 thf(fact_213_g_H_OsimplePath__length__less__V,axiom,
% 0.19/0.47 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.47 ( ( member_nat @ U @ ( v_b @ c2 ) )
% 0.19/0.47 => ( ( isSimplePath_b @ c2 @ U @ P5 @ V )
% 0.19/0.47 => ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_b @ c2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % g'.simplePath_length_less_V
% 0.19/0.47 thf(fact_214_g_H_Omin__dist__less__V,axiom,
% 0.19/0.47 ! [S: nat,T: nat] :
% 0.19/0.47 ( ( member_nat @ S @ ( v_b @ c2 ) )
% 0.19/0.47 => ( ( connected_b @ c2 @ S @ T )
% 0.19/0.47 => ( ord_less_nat @ ( min_dist_b @ c2 @ S @ T ) @ ( finite_card_nat @ ( v_b @ c2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % g'.min_dist_less_V
% 0.19/0.47 thf(fact_215_g_H_OisShortestPath__length__less__V,axiom,
% 0.19/0.47 ! [S: nat,P5: list_P559422087at_nat,T: nat] :
% 0.19/0.47 ( ( member_nat @ S @ ( v_b @ c2 ) )
% 0.19/0.47 => ( ( isShortestPath_b @ c2 @ S @ P5 @ T )
% 0.19/0.47 => ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_b @ c2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % g'.isShortestPath_length_less_V
% 0.19/0.47 thf(fact_216_isShortestPath__alt,axiom,
% 0.19/0.47 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.47 ( ( isShortestPath_a @ c @ U @ P5 @ V )
% 0.19/0.47 = ( ( isSimplePath_a @ c @ U @ P5 @ V )
% 0.19/0.47 & ( ( size_s1990949619at_nat @ P5 )
% 0.19/0.47 = ( min_dist_a @ c @ U @ V ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % isShortestPath_alt
% 0.19/0.47 thf(fact_217_g_H_Ofinite__V,axiom,
% 0.19/0.47 finite_finite_nat @ ( v_b @ c2 ) ).
% 0.19/0.47
% 0.19/0.47 % g'.finite_V
% 0.19/0.47 thf(fact_218_finite__V,axiom,
% 0.19/0.47 finite_finite_nat @ ( v_a @ c ) ).
% 0.19/0.47
% 0.19/0.47 % finite_V
% 0.19/0.47 thf(fact_219_dist__z__iff,axiom,
% 0.19/0.47 ! [V: nat,V3: nat] :
% 0.19/0.47 ( ( dist_a @ c @ V @ zero_zero_nat @ V3 )
% 0.19/0.47 = ( V3 = V ) ) ).
% 0.19/0.47
% 0.19/0.47 % dist_z_iff
% 0.19/0.47 thf(fact_220_dist__z,axiom,
% 0.19/0.47 ! [V: nat] : ( dist_a @ c @ V @ zero_zero_nat @ V ) ).
% 0.19/0.47
% 0.19/0.47 % dist_z
% 0.19/0.47 thf(fact_221_connected__distI,axiom,
% 0.19/0.47 ! [V: nat,D: nat,V3: nat] :
% 0.19/0.47 ( ( dist_a @ c @ V @ D @ V3 )
% 0.19/0.47 => ( connected_a @ c @ V @ V3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % connected_distI
% 0.19/0.47 thf(fact_222_Veq,axiom,
% 0.19/0.47 ( ( v_b @ c2 )
% 0.19/0.47 = ( v_a @ c ) ) ).
% 0.19/0.47
% 0.19/0.47 % Veq
% 0.19/0.47 thf(fact_223_g_H_Oadjacent__nodes__ss__V,axiom,
% 0.19/0.47 ! [U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_b @ c2 @ U ) @ ( v_b @ c2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % g'.adjacent_nodes_ss_V
% 0.19/0.47 thf(fact_224_adjacent__nodes__ss__V,axiom,
% 0.19/0.47 ! [U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_a @ c @ U ) @ ( v_a @ c ) ) ).
% 0.19/0.47
% 0.19/0.47 % adjacent_nodes_ss_V
% 0.19/0.47 thf(fact_225_Graph_OisSimplePath_Ocong,axiom,
% 0.19/0.47 isSimplePath_a = isSimplePath_a ).
% 0.19/0.47
% 0.19/0.47 % Graph.isSimplePath.cong
% 0.19/0.47 thf(fact_226_Graph_OisSimplePath_Ocong,axiom,
% 0.19/0.47 isSimplePath_b = isSimplePath_b ).
% 0.19/0.47
% 0.19/0.47 % Graph.isSimplePath.cong
% 0.19/0.47 thf(fact_227_finite__subset,axiom,
% 0.19/0.47 ! [A2: set_se1612935105at_nat,B2: set_se1612935105at_nat] :
% 0.19/0.47 ( ( ord_le2096002913at_nat @ A2 @ B2 )
% 0.19/0.47 => ( ( finite1457549322at_nat @ B2 )
% 0.19/0.47 => ( finite1457549322at_nat @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_subset
% 0.19/0.47 thf(fact_228_finite__subset,axiom,
% 0.19/0.47 ! [A2: set_set_nat,B2: set_set_nat] :
% 0.19/0.47 ( ( ord_le1613022364et_nat @ A2 @ B2 )
% 0.19/0.47 => ( ( finite2012248349et_nat @ B2 )
% 0.19/0.47 => ( finite2012248349et_nat @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_subset
% 0.19/0.47 thf(fact_229_finite__subset,axiom,
% 0.19/0.47 ! [A2: set_nat,B2: set_nat] :
% 0.19/0.47 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.19/0.47 => ( ( finite_finite_nat @ B2 )
% 0.19/0.47 => ( finite_finite_nat @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_subset
% 0.19/0.47 thf(fact_230_finite__subset,axiom,
% 0.19/0.47 ! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( ord_le841296385at_nat @ A2 @ B2 )
% 0.19/0.47 => ( ( finite772653738at_nat @ B2 )
% 0.19/0.47 => ( finite772653738at_nat @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_subset
% 0.19/0.47 thf(fact_231_infinite__super,axiom,
% 0.19/0.47 ! [S2: set_nat,T2: set_nat] :
% 0.19/0.47 ( ( ord_less_eq_set_nat @ S2 @ T2 )
% 0.19/0.47 => ( ~ ( finite_finite_nat @ S2 )
% 0.19/0.47 => ~ ( finite_finite_nat @ T2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_super
% 0.19/0.47 thf(fact_232_infinite__super,axiom,
% 0.19/0.47 ! [S2: set_Pr1986765409at_nat,T2: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( ord_le841296385at_nat @ S2 @ T2 )
% 0.19/0.47 => ( ~ ( finite772653738at_nat @ S2 )
% 0.19/0.47 => ~ ( finite772653738at_nat @ T2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_super
% 0.19/0.47 thf(fact_233_rev__finite__subset,axiom,
% 0.19/0.47 ! [B2: set_nat,A2: set_nat] :
% 0.19/0.47 ( ( finite_finite_nat @ B2 )
% 0.19/0.47 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.19/0.47 => ( finite_finite_nat @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % rev_finite_subset
% 0.19/0.47 thf(fact_234_rev__finite__subset,axiom,
% 0.19/0.47 ! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( finite772653738at_nat @ B2 )
% 0.19/0.47 => ( ( ord_le841296385at_nat @ A2 @ B2 )
% 0.19/0.47 => ( finite772653738at_nat @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % rev_finite_subset
% 0.19/0.47 thf(fact_235_card__subset__eq,axiom,
% 0.19/0.47 ! [B2: set_nat,A2: set_nat] :
% 0.19/0.47 ( ( finite_finite_nat @ B2 )
% 0.19/0.47 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.19/0.47 => ( ( ( finite_card_nat @ A2 )
% 0.19/0.47 = ( finite_card_nat @ B2 ) )
% 0.19/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_subset_eq
% 0.19/0.47 thf(fact_236_card__subset__eq,axiom,
% 0.19/0.47 ! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( finite772653738at_nat @ B2 )
% 0.19/0.47 => ( ( ord_le841296385at_nat @ A2 @ B2 )
% 0.19/0.47 => ( ( ( finite447719721at_nat @ A2 )
% 0.19/0.47 = ( finite447719721at_nat @ B2 ) )
% 0.19/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_subset_eq
% 0.19/0.47 thf(fact_237_infinite__arbitrarily__large,axiom,
% 0.19/0.47 ! [A2: set_nat,N: nat] :
% 0.19/0.47 ( ~ ( finite_finite_nat @ A2 )
% 0.19/0.47 => ? [B3: set_nat] :
% 0.19/0.47 ( ( finite_finite_nat @ B3 )
% 0.19/0.47 & ( ( finite_card_nat @ B3 )
% 0.19/0.47 = N )
% 0.19/0.47 & ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_arbitrarily_large
% 0.19/0.47 thf(fact_238_infinite__arbitrarily__large,axiom,
% 0.19/0.47 ! [A2: set_Pr1986765409at_nat,N: nat] :
% 0.19/0.47 ( ~ ( finite772653738at_nat @ A2 )
% 0.19/0.47 => ? [B3: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( finite772653738at_nat @ B3 )
% 0.19/0.47 & ( ( finite447719721at_nat @ B3 )
% 0.19/0.47 = N )
% 0.19/0.47 & ( ord_le841296385at_nat @ B3 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_arbitrarily_large
% 0.19/0.47 thf(fact_239_Graph_OisSPath__pathLE,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,S: nat,P5: list_P559422087at_nat,T: nat] :
% 0.19/0.47 ( ( isPath_a @ C @ S @ P5 @ T )
% 0.19/0.47 => ? [P: list_P559422087at_nat] : ( isSimplePath_a @ C @ S @ P @ T ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.isSPath_pathLE
% 0.19/0.47 thf(fact_240_Graph_OisSPath__pathLE,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,S: nat,P5: list_P559422087at_nat,T: nat] :
% 0.19/0.47 ( ( isPath_b @ C @ S @ P5 @ T )
% 0.19/0.47 => ? [P: list_P559422087at_nat] : ( isSimplePath_b @ C @ S @ P @ T ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.isSPath_pathLE
% 0.19/0.47 thf(fact_241_Graph_OshortestPath__is__simple,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,S: nat,P5: list_P559422087at_nat,T: nat] :
% 0.19/0.47 ( ( isShortestPath_a @ C @ S @ P5 @ T )
% 0.19/0.47 => ( isSimplePath_a @ C @ S @ P5 @ T ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.shortestPath_is_simple
% 0.19/0.47 thf(fact_242_Graph_OshortestPath__is__simple,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,S: nat,P5: list_P559422087at_nat,T: nat] :
% 0.19/0.47 ( ( isShortestPath_b @ C @ S @ P5 @ T )
% 0.19/0.47 => ( isSimplePath_b @ C @ S @ P5 @ T ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.shortestPath_is_simple
% 0.19/0.47 thf(fact_243_finite__maxlen,axiom,
% 0.19/0.47 ! [M4: set_li664300135at_nat] :
% 0.19/0.47 ( ( finite1299096496at_nat @ M4 )
% 0.19/0.47 => ? [N3: nat] :
% 0.19/0.47 ! [X4: list_P559422087at_nat] :
% 0.19/0.47 ( ( member1608759472at_nat @ X4 @ M4 )
% 0.19/0.47 => ( ord_less_nat @ ( size_s1990949619at_nat @ X4 ) @ N3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_maxlen
% 0.19/0.47 thf(fact_244_card__psubset,axiom,
% 0.19/0.47 ! [B2: set_nat,A2: set_nat] :
% 0.19/0.47 ( ( finite_finite_nat @ B2 )
% 0.19/0.47 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.19/0.47 => ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
% 0.19/0.47 => ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_psubset
% 0.19/0.47 thf(fact_245_card__psubset,axiom,
% 0.19/0.47 ! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( finite772653738at_nat @ B2 )
% 0.19/0.47 => ( ( ord_le841296385at_nat @ A2 @ B2 )
% 0.19/0.47 => ( ( ord_less_nat @ ( finite447719721at_nat @ A2 ) @ ( finite447719721at_nat @ B2 ) )
% 0.19/0.47 => ( ord_le116442893at_nat @ A2 @ B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_psubset
% 0.19/0.47 thf(fact_246_card__mono,axiom,
% 0.19/0.47 ! [B2: set_nat,A2: set_nat] :
% 0.19/0.47 ( ( finite_finite_nat @ B2 )
% 0.19/0.47 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.19/0.47 => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_mono
% 0.19/0.47 thf(fact_247_card__mono,axiom,
% 0.19/0.47 ! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( finite772653738at_nat @ B2 )
% 0.19/0.47 => ( ( ord_le841296385at_nat @ A2 @ B2 )
% 0.19/0.47 => ( ord_less_eq_nat @ ( finite447719721at_nat @ A2 ) @ ( finite447719721at_nat @ B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_mono
% 0.19/0.47 thf(fact_248_card__seteq,axiom,
% 0.19/0.47 ! [B2: set_nat,A2: set_nat] :
% 0.19/0.47 ( ( finite_finite_nat @ B2 )
% 0.19/0.47 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.19/0.47 => ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
% 0.19/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_seteq
% 0.19/0.47 thf(fact_249_card__seteq,axiom,
% 0.19/0.47 ! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( finite772653738at_nat @ B2 )
% 0.19/0.47 => ( ( ord_le841296385at_nat @ A2 @ B2 )
% 0.19/0.47 => ( ( ord_less_eq_nat @ ( finite447719721at_nat @ B2 ) @ ( finite447719721at_nat @ A2 ) )
% 0.19/0.47 => ( A2 = B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % card_seteq
% 0.19/0.47 thf(fact_250_finite__if__finite__subsets__card__bdd,axiom,
% 0.19/0.47 ! [F2: set_nat,C3: nat] :
% 0.19/0.47 ( ! [G: set_nat] :
% 0.19/0.47 ( ( ord_less_eq_set_nat @ G @ F2 )
% 0.19/0.47 => ( ( finite_finite_nat @ G )
% 0.19/0.47 => ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C3 ) ) )
% 0.19/0.47 => ( ( finite_finite_nat @ F2 )
% 0.19/0.47 & ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_if_finite_subsets_card_bdd
% 0.19/0.47 thf(fact_251_finite__if__finite__subsets__card__bdd,axiom,
% 0.19/0.47 ! [F2: set_Pr1986765409at_nat,C3: nat] :
% 0.19/0.47 ( ! [G: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( ord_le841296385at_nat @ G @ F2 )
% 0.19/0.47 => ( ( finite772653738at_nat @ G )
% 0.19/0.47 => ( ord_less_eq_nat @ ( finite447719721at_nat @ G ) @ C3 ) ) )
% 0.19/0.47 => ( ( finite772653738at_nat @ F2 )
% 0.19/0.47 & ( ord_less_eq_nat @ ( finite447719721at_nat @ F2 ) @ C3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_if_finite_subsets_card_bdd
% 0.19/0.47 thf(fact_252_Finite__Graph__def,axiom,
% 0.19/0.47 ( finite_Graph_b
% 0.19/0.47 = ( ^ [C2: product_prod_nat_nat > b] : ( finite_finite_nat @ ( v_b @ C2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Finite_Graph_def
% 0.19/0.47 thf(fact_253_Finite__Graph__def,axiom,
% 0.19/0.47 ( finite_Graph_a
% 0.19/0.47 = ( ^ [C2: product_prod_nat_nat > a] : ( finite_finite_nat @ ( v_a @ C2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Finite_Graph_def
% 0.19/0.47 thf(fact_254_Finite__Graph_Ointro,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b] :
% 0.19/0.47 ( ( finite_finite_nat @ ( v_b @ C ) )
% 0.19/0.47 => ( finite_Graph_b @ C ) ) ).
% 0.19/0.47
% 0.19/0.47 % Finite_Graph.intro
% 0.19/0.47 thf(fact_255_Finite__Graph_Ointro,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a] :
% 0.19/0.47 ( ( finite_finite_nat @ ( v_a @ C ) )
% 0.19/0.47 => ( finite_Graph_a @ C ) ) ).
% 0.19/0.47
% 0.19/0.47 % Finite_Graph.intro
% 0.19/0.47 thf(fact_256_Finite__Graph_Ofinite__V,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b] :
% 0.19/0.47 ( ( finite_Graph_b @ C )
% 0.19/0.47 => ( finite_finite_nat @ ( v_b @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Finite_Graph.finite_V
% 0.19/0.47 thf(fact_257_Finite__Graph_Ofinite__V,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a] :
% 0.19/0.47 ( ( finite_Graph_a @ C )
% 0.19/0.47 => ( finite_finite_nat @ ( v_a @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Finite_Graph.finite_V
% 0.19/0.47 thf(fact_258_finite__psubset__induct,axiom,
% 0.19/0.47 ! [A2: set_Pr1986765409at_nat,P2: set_Pr1986765409at_nat > $o] :
% 0.19/0.47 ( ( finite772653738at_nat @ A2 )
% 0.19/0.47 => ( ! [A3: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( finite772653738at_nat @ A3 )
% 0.19/0.47 => ( ! [B4: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( ord_le116442893at_nat @ B4 @ A3 )
% 0.19/0.47 => ( P2 @ B4 ) )
% 0.19/0.47 => ( P2 @ A3 ) ) )
% 0.19/0.47 => ( P2 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_psubset_induct
% 0.19/0.47 thf(fact_259_finite__psubset__induct,axiom,
% 0.19/0.47 ! [A2: set_nat,P2: set_nat > $o] :
% 0.19/0.47 ( ( finite_finite_nat @ A2 )
% 0.19/0.47 => ( ! [A3: set_nat] :
% 0.19/0.47 ( ( finite_finite_nat @ A3 )
% 0.19/0.47 => ( ! [B4: set_nat] :
% 0.19/0.47 ( ( ord_less_set_nat @ B4 @ A3 )
% 0.19/0.47 => ( P2 @ B4 ) )
% 0.19/0.47 => ( P2 @ A3 ) ) )
% 0.19/0.47 => ( P2 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_psubset_induct
% 0.19/0.47 thf(fact_260_Graph_OisShortestPath__alt,axiom,
% 0.19/0.47 ( isShortestPath_a
% 0.19/0.47 = ( ^ [C2: product_prod_nat_nat > a,U2: nat,P3: list_P559422087at_nat,V4: nat] :
% 0.19/0.47 ( ( isSimplePath_a @ C2 @ U2 @ P3 @ V4 )
% 0.19/0.47 & ( ( size_s1990949619at_nat @ P3 )
% 0.19/0.47 = ( min_dist_a @ C2 @ U2 @ V4 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.isShortestPath_alt
% 0.19/0.47 thf(fact_261_Graph_OisShortestPath__alt,axiom,
% 0.19/0.47 ( isShortestPath_b
% 0.19/0.47 = ( ^ [C2: product_prod_nat_nat > b,U2: nat,P3: list_P559422087at_nat,V4: nat] :
% 0.19/0.47 ( ( isSimplePath_b @ C2 @ U2 @ P3 @ V4 )
% 0.19/0.47 & ( ( size_s1990949619at_nat @ P3 )
% 0.19/0.47 = ( min_dist_b @ C2 @ U2 @ V4 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.isShortestPath_alt
% 0.19/0.47 thf(fact_262_ek__analysis_Ocard__spEdges__le,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,S: nat,T: nat] :
% 0.19/0.47 ( ( edmond1517640972ysis_a @ C )
% 0.19/0.47 => ( ord_less_eq_nat @ ( finite447719721at_nat @ ( edmond475474835dges_a @ C @ S @ T ) ) @ ( finite447719721at_nat @ ( edmond771116670s_uE_a @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ek_analysis.card_spEdges_le
% 0.19/0.47 thf(fact_263_ek__analysis_Ocard__spEdges__le,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,S: nat,T: nat] :
% 0.19/0.47 ( ( edmond1517640973ysis_b @ C )
% 0.19/0.47 => ( ord_less_eq_nat @ ( finite447719721at_nat @ ( edmond475474836dges_b @ C @ S @ T ) ) @ ( finite447719721at_nat @ ( edmond771116671s_uE_b @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ek_analysis.card_spEdges_le
% 0.19/0.47 thf(fact_264_finite__has__minimal2,axiom,
% 0.19/0.47 ! [A2: set_nat,A: nat] :
% 0.19/0.47 ( ( finite_finite_nat @ A2 )
% 0.19/0.47 => ( ( member_nat @ A @ A2 )
% 0.19/0.47 => ? [X2: nat] :
% 0.19/0.47 ( ( member_nat @ X2 @ A2 )
% 0.19/0.47 & ( ord_less_eq_nat @ X2 @ A )
% 0.19/0.47 & ! [Xa: nat] :
% 0.19/0.47 ( ( member_nat @ Xa @ A2 )
% 0.19/0.47 => ( ( ord_less_eq_nat @ Xa @ X2 )
% 0.19/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_has_minimal2
% 0.19/0.47 thf(fact_265_finite__has__minimal2,axiom,
% 0.19/0.47 ! [A2: set_set_nat,A: set_nat] :
% 0.19/0.47 ( ( finite2012248349et_nat @ A2 )
% 0.19/0.47 => ( ( member_set_nat @ A @ A2 )
% 0.19/0.47 => ? [X2: set_nat] :
% 0.19/0.47 ( ( member_set_nat @ X2 @ A2 )
% 0.19/0.47 & ( ord_less_eq_set_nat @ X2 @ A )
% 0.19/0.47 & ! [Xa: set_nat] :
% 0.19/0.47 ( ( member_set_nat @ Xa @ A2 )
% 0.19/0.47 => ( ( ord_less_eq_set_nat @ Xa @ X2 )
% 0.19/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_has_minimal2
% 0.19/0.47 thf(fact_266_finite__has__minimal2,axiom,
% 0.19/0.47 ! [A2: set_se1612935105at_nat,A: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( finite1457549322at_nat @ A2 )
% 0.19/0.47 => ( ( member298845450at_nat @ A @ A2 )
% 0.19/0.47 => ? [X2: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( member298845450at_nat @ X2 @ A2 )
% 0.19/0.47 & ( ord_le841296385at_nat @ X2 @ A )
% 0.19/0.47 & ! [Xa: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( member298845450at_nat @ Xa @ A2 )
% 0.19/0.47 => ( ( ord_le841296385at_nat @ Xa @ X2 )
% 0.19/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_has_minimal2
% 0.19/0.47 thf(fact_267_finite__has__maximal2,axiom,
% 0.19/0.47 ! [A2: set_nat,A: nat] :
% 0.19/0.47 ( ( finite_finite_nat @ A2 )
% 0.19/0.47 => ( ( member_nat @ A @ A2 )
% 0.19/0.47 => ? [X2: nat] :
% 0.19/0.47 ( ( member_nat @ X2 @ A2 )
% 0.19/0.47 & ( ord_less_eq_nat @ A @ X2 )
% 0.19/0.47 & ! [Xa: nat] :
% 0.19/0.47 ( ( member_nat @ Xa @ A2 )
% 0.19/0.47 => ( ( ord_less_eq_nat @ X2 @ Xa )
% 0.19/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_has_maximal2
% 0.19/0.47 thf(fact_268_finite__has__maximal2,axiom,
% 0.19/0.47 ! [A2: set_set_nat,A: set_nat] :
% 0.19/0.47 ( ( finite2012248349et_nat @ A2 )
% 0.19/0.47 => ( ( member_set_nat @ A @ A2 )
% 0.19/0.47 => ? [X2: set_nat] :
% 0.19/0.47 ( ( member_set_nat @ X2 @ A2 )
% 0.19/0.47 & ( ord_less_eq_set_nat @ A @ X2 )
% 0.19/0.47 & ! [Xa: set_nat] :
% 0.19/0.47 ( ( member_set_nat @ Xa @ A2 )
% 0.19/0.47 => ( ( ord_less_eq_set_nat @ X2 @ Xa )
% 0.19/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_has_maximal2
% 0.19/0.47 thf(fact_269_finite__has__maximal2,axiom,
% 0.19/0.47 ! [A2: set_se1612935105at_nat,A: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( finite1457549322at_nat @ A2 )
% 0.19/0.47 => ( ( member298845450at_nat @ A @ A2 )
% 0.19/0.47 => ? [X2: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( member298845450at_nat @ X2 @ A2 )
% 0.19/0.47 & ( ord_le841296385at_nat @ A @ X2 )
% 0.19/0.47 & ! [Xa: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( member298845450at_nat @ Xa @ A2 )
% 0.19/0.47 => ( ( ord_le841296385at_nat @ X2 @ Xa )
% 0.19/0.47 => ( X2 = Xa ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_has_maximal2
% 0.19/0.47 thf(fact_270_Finite__Graph_OsimplePath__length__less__V,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.47 ( ( finite_Graph_a @ C )
% 0.19/0.47 => ( ( member_nat @ U @ ( v_a @ C ) )
% 0.19/0.47 => ( ( isSimplePath_a @ C @ U @ P5 @ V )
% 0.19/0.47 => ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_a @ C ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Finite_Graph.simplePath_length_less_V
% 0.19/0.47 thf(fact_271_Finite__Graph_OsimplePath__length__less__V,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,U: nat,P5: list_P559422087at_nat,V: nat] :
% 0.19/0.47 ( ( finite_Graph_b @ C )
% 0.19/0.47 => ( ( member_nat @ U @ ( v_b @ C ) )
% 0.19/0.47 => ( ( isSimplePath_b @ C @ U @ P5 @ V )
% 0.19/0.47 => ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_b @ C ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Finite_Graph.simplePath_length_less_V
% 0.19/0.47 thf(fact_272_psubset__card__mono,axiom,
% 0.19/0.47 ! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
% 0.19/0.47 ( ( finite772653738at_nat @ B2 )
% 0.19/0.47 => ( ( ord_le116442893at_nat @ A2 @ B2 )
% 0.19/0.47 => ( ord_less_nat @ ( finite447719721at_nat @ A2 ) @ ( finite447719721at_nat @ B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % psubset_card_mono
% 0.19/0.47 thf(fact_273_psubset__card__mono,axiom,
% 0.19/0.47 ! [B2: set_nat,A2: set_nat] :
% 0.19/0.47 ( ( finite_finite_nat @ B2 )
% 0.19/0.47 => ( ( ord_less_set_nat @ A2 @ B2 )
% 0.19/0.47 => ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % psubset_card_mono
% 0.19/0.47 thf(fact_274_reachable__ss__V,axiom,
% 0.19/0.47 ! [S: nat] :
% 0.19/0.47 ( ( member_nat @ S @ ( v_a @ c ) )
% 0.19/0.47 => ( ord_less_eq_set_nat @ ( reachableNodes_a @ c @ S ) @ ( v_a @ c ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % reachable_ss_V
% 0.19/0.47 thf(fact_275_g_H_Oreachable__ss__V,axiom,
% 0.19/0.47 ! [S: nat] :
% 0.19/0.47 ( ( member_nat @ S @ ( v_b @ c2 ) )
% 0.19/0.47 => ( ord_less_eq_set_nat @ ( reachableNodes_b @ c2 @ S ) @ ( v_b @ c2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % g'.reachable_ss_V
% 0.19/0.47 thf(fact_276_spEdges__ss__E,axiom,
% 0.19/0.47 ord_le841296385at_nat @ ( edmond475474835dges_a @ c @ s @ t ) @ ( e_a @ c ) ).
% 0.19/0.47
% 0.19/0.47 % spEdges_ss_E
% 0.19/0.47 thf(fact_277_E__ss__uE,axiom,
% 0.19/0.47 ord_le841296385at_nat @ ( e_a @ c ) @ ( edmond771116670s_uE_a @ c ) ).
% 0.19/0.47
% 0.19/0.47 % E_ss_uE
% 0.19/0.47 thf(fact_278_Finite__Graph__EI,axiom,
% 0.19/0.47 ( ( finite772653738at_nat @ ( e_a @ c ) )
% 0.19/0.47 => ( finite_Graph_a @ c ) ) ).
% 0.19/0.47
% 0.19/0.47 % Finite_Graph_EI
% 0.19/0.47 thf(fact_279_Efin__imp__Vfin,axiom,
% 0.19/0.47 ( ( finite772653738at_nat @ ( e_a @ c ) )
% 0.19/0.47 => ( finite_finite_nat @ ( v_a @ c ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Efin_imp_Vfin
% 0.19/0.47 thf(fact_280_finite__E,axiom,
% 0.19/0.47 finite772653738at_nat @ ( e_a @ c ) ).
% 0.19/0.47
% 0.19/0.47 % finite_E
% 0.19/0.47 thf(fact_281_g_H_Oadjacent__nodes__finite,axiom,
% 0.19/0.47 ! [U: nat] : ( finite_finite_nat @ ( adjacent_nodes_b @ c2 @ U ) ) ).
% 0.19/0.47
% 0.19/0.47 % g'.adjacent_nodes_finite
% 0.19/0.47 thf(fact_282_adjacent__nodes__finite,axiom,
% 0.19/0.47 ! [U: nat] : ( finite_finite_nat @ ( adjacent_nodes_a @ c @ U ) ) ).
% 0.19/0.47
% 0.19/0.47 % adjacent_nodes_finite
% 0.19/0.47 thf(fact_283_Vfin__imp__Efin,axiom,
% 0.19/0.47 ( ( finite_finite_nat @ ( v_a @ c ) )
% 0.19/0.47 => ( finite772653738at_nat @ ( e_a @ c ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Vfin_imp_Efin
% 0.19/0.47 thf(fact_284_Graph_OreachableNodes_Ocong,axiom,
% 0.19/0.47 reachableNodes_a = reachableNodes_a ).
% 0.19/0.47
% 0.19/0.47 % Graph.reachableNodes.cong
% 0.19/0.47 thf(fact_285_Graph_OreachableNodes_Ocong,axiom,
% 0.19/0.47 reachableNodes_b = reachableNodes_b ).
% 0.19/0.47
% 0.19/0.47 % Graph.reachableNodes.cong
% 0.19/0.47 thf(fact_286_Graph_Oadjacent__nodes_Ocong,axiom,
% 0.19/0.47 adjacent_nodes_b = adjacent_nodes_b ).
% 0.19/0.47
% 0.19/0.47 % Graph.adjacent_nodes.cong
% 0.19/0.47 thf(fact_287_Graph_Oadjacent__nodes_Ocong,axiom,
% 0.19/0.47 adjacent_nodes_a = adjacent_nodes_a ).
% 0.19/0.47
% 0.19/0.47 % Graph.adjacent_nodes.cong
% 0.19/0.47 thf(fact_288_Graph_OE_Ocong,axiom,
% 0.19/0.47 e_a = e_a ).
% 0.19/0.47
% 0.19/0.47 % Graph.E.cong
% 0.19/0.47 thf(fact_289_Graph_OE_Ocong,axiom,
% 0.19/0.47 e_b = e_b ).
% 0.19/0.47
% 0.19/0.47 % Graph.E.cong
% 0.19/0.47 thf(fact_290_Graph_Oreachable__ss__V,axiom,
% 0.19/0.47 ! [S: nat,C: product_prod_nat_nat > a] :
% 0.19/0.47 ( ( member_nat @ S @ ( v_a @ C ) )
% 0.19/0.47 => ( ord_less_eq_set_nat @ ( reachableNodes_a @ C @ S ) @ ( v_a @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.reachable_ss_V
% 0.19/0.47 thf(fact_291_Graph_Oreachable__ss__V,axiom,
% 0.19/0.47 ! [S: nat,C: product_prod_nat_nat > b] :
% 0.19/0.47 ( ( member_nat @ S @ ( v_b @ C ) )
% 0.19/0.47 => ( ord_less_eq_set_nat @ ( reachableNodes_b @ C @ S ) @ ( v_b @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.reachable_ss_V
% 0.19/0.47 thf(fact_292_Graph_OEfin__imp__Vfin,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a] :
% 0.19/0.47 ( ( finite772653738at_nat @ ( e_a @ C ) )
% 0.19/0.47 => ( finite_finite_nat @ ( v_a @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.Efin_imp_Vfin
% 0.19/0.47 thf(fact_293_Graph_OEfin__imp__Vfin,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b] :
% 0.19/0.47 ( ( finite772653738at_nat @ ( e_b @ C ) )
% 0.19/0.47 => ( finite_finite_nat @ ( v_b @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.Efin_imp_Vfin
% 0.19/0.47 thf(fact_294_Graph_OVfin__imp__Efin,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a] :
% 0.19/0.47 ( ( finite_finite_nat @ ( v_a @ C ) )
% 0.19/0.47 => ( finite772653738at_nat @ ( e_a @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.Vfin_imp_Efin
% 0.19/0.47 thf(fact_295_Graph_OVfin__imp__Efin,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b] :
% 0.19/0.47 ( ( finite_finite_nat @ ( v_b @ C ) )
% 0.19/0.47 => ( finite772653738at_nat @ ( e_b @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.Vfin_imp_Efin
% 0.19/0.47 thf(fact_296_Graph_OFinite__Graph__EI,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b] :
% 0.19/0.47 ( ( finite772653738at_nat @ ( e_b @ C ) )
% 0.19/0.47 => ( finite_Graph_b @ C ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.Finite_Graph_EI
% 0.19/0.47 thf(fact_297_Graph_OFinite__Graph__EI,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a] :
% 0.19/0.47 ( ( finite772653738at_nat @ ( e_a @ C ) )
% 0.19/0.47 => ( finite_Graph_a @ C ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.Finite_Graph_EI
% 0.19/0.47 thf(fact_298_Finite__Graph_Ofinite__E,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b] :
% 0.19/0.47 ( ( finite_Graph_b @ C )
% 0.19/0.47 => ( finite772653738at_nat @ ( e_b @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Finite_Graph.finite_E
% 0.19/0.47 thf(fact_299_Finite__Graph_Ofinite__E,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a] :
% 0.19/0.47 ( ( finite_Graph_a @ C )
% 0.19/0.47 => ( finite772653738at_nat @ ( e_a @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Finite_Graph.finite_E
% 0.19/0.47 thf(fact_300_ek__analysis_OE__ss__uE,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b] :
% 0.19/0.47 ( ( edmond1517640973ysis_b @ C )
% 0.19/0.47 => ( ord_le841296385at_nat @ ( e_b @ C ) @ ( edmond771116671s_uE_b @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ek_analysis.E_ss_uE
% 0.19/0.47 thf(fact_301_ek__analysis_OE__ss__uE,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a] :
% 0.19/0.47 ( ( edmond1517640972ysis_a @ C )
% 0.19/0.47 => ( ord_le841296385at_nat @ ( e_a @ C ) @ ( edmond771116670s_uE_a @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ek_analysis.E_ss_uE
% 0.19/0.47 thf(fact_302_ek__analysis_OspEdges__ss__E,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,S: nat,T: nat] :
% 0.19/0.47 ( ( edmond1517640972ysis_a @ C )
% 0.19/0.47 => ( ord_le841296385at_nat @ ( edmond475474835dges_a @ C @ S @ T ) @ ( e_a @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ek_analysis.spEdges_ss_E
% 0.19/0.47 thf(fact_303_ek__analysis_OspEdges__ss__E,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,S: nat,T: nat] :
% 0.19/0.47 ( ( edmond1517640973ysis_b @ C )
% 0.19/0.47 => ( ord_le841296385at_nat @ ( edmond475474836dges_b @ C @ S @ T ) @ ( e_b @ C ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ek_analysis.spEdges_ss_E
% 0.19/0.47 thf(fact_304_Graph_Oadjacent__nodes__ss__V,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_b @ C @ U ) @ ( v_b @ C ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.adjacent_nodes_ss_V
% 0.19/0.47 thf(fact_305_Graph_Oadjacent__nodes__ss__V,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_a @ C @ U ) @ ( v_a @ C ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.adjacent_nodes_ss_V
% 0.19/0.47 thf(fact_306_Finite__Graph_Oadjacent__nodes__finite,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,U: nat] :
% 0.19/0.47 ( ( finite_Graph_b @ C )
% 0.19/0.47 => ( finite_finite_nat @ ( adjacent_nodes_b @ C @ U ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Finite_Graph.adjacent_nodes_finite
% 0.19/0.47 thf(fact_307_Finite__Graph_Oadjacent__nodes__finite,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,U: nat] :
% 0.19/0.47 ( ( finite_Graph_a @ C )
% 0.19/0.47 => ( finite_finite_nat @ ( adjacent_nodes_a @ C @ U ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Finite_Graph.adjacent_nodes_finite
% 0.19/0.47 thf(fact_308_incoming_H__edges,axiom,
% 0.19/0.47 ! [U3: set_nat] : ( ord_le841296385at_nat @ ( incoming_a @ c @ U3 ) @ ( e_a @ c ) ) ).
% 0.19/0.47
% 0.19/0.47 % incoming'_edges
% 0.19/0.47 thf(fact_309_outgoing_H__edges,axiom,
% 0.19/0.47 ! [U3: set_nat] : ( ord_le841296385at_nat @ ( outgoing_a @ c @ U3 ) @ ( e_a @ c ) ) ).
% 0.19/0.47
% 0.19/0.47 % outgoing'_edges
% 0.19/0.47 thf(fact_310_g_H_OE__ss__uE,axiom,
% 0.19/0.47 ord_le841296385at_nat @ ( e_b @ c2 ) @ ( edmond771116671s_uE_b @ c2 ) ).
% 0.19/0.47
% 0.19/0.47 % g'.E_ss_uE
% 0.19/0.47 thf(fact_311_g_H_OFinite__Graph__EI,axiom,
% 0.19/0.47 ( ( finite772653738at_nat @ ( e_b @ c2 ) )
% 0.19/0.47 => ( finite_Graph_b @ c2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % g'.Finite_Graph_EI
% 0.19/0.47 thf(fact_312_g_H_OEfin__imp__Vfin,axiom,
% 0.19/0.47 ( ( finite772653738at_nat @ ( e_b @ c2 ) )
% 0.19/0.47 => ( finite_finite_nat @ ( v_b @ c2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % g'.Efin_imp_Vfin
% 0.19/0.47 thf(fact_313_g_H_OspEdges__ss__E,axiom,
% 0.19/0.47 ord_le841296385at_nat @ ( edmond475474836dges_b @ c2 @ s @ t ) @ ( e_b @ c2 ) ).
% 0.19/0.47
% 0.19/0.47 % g'.spEdges_ss_E
% 0.19/0.47 thf(fact_314_g_H_Ofinite__E,axiom,
% 0.19/0.47 finite772653738at_nat @ ( e_b @ c2 ) ).
% 0.19/0.47
% 0.19/0.47 % g'.finite_E
% 0.19/0.47 thf(fact_315_g_H_OVfin__imp__Efin,axiom,
% 0.19/0.47 ( ( finite_finite_nat @ ( v_b @ c2 ) )
% 0.19/0.47 => ( finite772653738at_nat @ ( e_b @ c2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % g'.Vfin_imp_Efin
% 0.19/0.47 thf(fact_316_finite__outgoing_H,axiom,
% 0.19/0.47 ! [U3: set_nat] :
% 0.19/0.47 ( ( finite_finite_nat @ ( v_a @ c ) )
% 0.19/0.47 => ( finite772653738at_nat @ ( outgoing_a @ c @ U3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_outgoing'
% 0.19/0.47 thf(fact_317_finite__incoming_H,axiom,
% 0.19/0.47 ! [U3: set_nat] :
% 0.19/0.47 ( ( finite_finite_nat @ ( v_a @ c ) )
% 0.19/0.47 => ( finite772653738at_nat @ ( incoming_a @ c @ U3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_incoming'
% 0.19/0.47 thf(fact_318_Graph_Oincoming_H_Ocong,axiom,
% 0.19/0.47 incoming_a = incoming_a ).
% 0.19/0.47
% 0.19/0.47 % Graph.incoming'.cong
% 0.19/0.47 thf(fact_319_Graph_Oincoming_H_Ocong,axiom,
% 0.19/0.47 incoming_b = incoming_b ).
% 0.19/0.47
% 0.19/0.47 % Graph.incoming'.cong
% 0.19/0.47 thf(fact_320_Graph_Ooutgoing_H_Ocong,axiom,
% 0.19/0.47 outgoing_a = outgoing_a ).
% 0.19/0.47
% 0.19/0.47 % Graph.outgoing'.cong
% 0.19/0.47 thf(fact_321_Graph_Ooutgoing_H_Ocong,axiom,
% 0.19/0.47 outgoing_b = outgoing_b ).
% 0.19/0.47
% 0.19/0.47 % Graph.outgoing'.cong
% 0.19/0.47 thf(fact_322_Graph_Ooutgoing_H__edges,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,U3: set_nat] : ( ord_le841296385at_nat @ ( outgoing_a @ C @ U3 ) @ ( e_a @ C ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.outgoing'_edges
% 0.19/0.47 thf(fact_323_Graph_Ooutgoing_H__edges,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,U3: set_nat] : ( ord_le841296385at_nat @ ( outgoing_b @ C @ U3 ) @ ( e_b @ C ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.outgoing'_edges
% 0.19/0.47 thf(fact_324_Graph_Oincoming_H__edges,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,U3: set_nat] : ( ord_le841296385at_nat @ ( incoming_a @ C @ U3 ) @ ( e_a @ C ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.incoming'_edges
% 0.19/0.47 thf(fact_325_Graph_Oincoming_H__edges,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,U3: set_nat] : ( ord_le841296385at_nat @ ( incoming_b @ C @ U3 ) @ ( e_b @ C ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.incoming'_edges
% 0.19/0.47 thf(fact_326_Graph_Ofinite__outgoing_H,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,U3: set_nat] :
% 0.19/0.47 ( ( finite_finite_nat @ ( v_a @ C ) )
% 0.19/0.47 => ( finite772653738at_nat @ ( outgoing_a @ C @ U3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.finite_outgoing'
% 0.19/0.47 thf(fact_327_Graph_Ofinite__outgoing_H,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,U3: set_nat] :
% 0.19/0.47 ( ( finite_finite_nat @ ( v_b @ C ) )
% 0.19/0.47 => ( finite772653738at_nat @ ( outgoing_b @ C @ U3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.finite_outgoing'
% 0.19/0.47 thf(fact_328_Graph_Ofinite__incoming_H,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > a,U3: set_nat] :
% 0.19/0.47 ( ( finite_finite_nat @ ( v_a @ C ) )
% 0.19/0.47 => ( finite772653738at_nat @ ( incoming_a @ C @ U3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Graph.finite_incoming'
% 0.19/0.47 thf(fact_329_Graph_Ofinite__incoming_H,axiom,
% 0.19/0.47 ! [C: product_prod_nat_nat > b,U3: set_nat] :
% 0.19/0.47 ( ( finite_finite_nat @ ( v_b @ C ) )
% 0.31/0.47 => ( finite772653738at_nat @ ( incoming_b @ C @ U3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.finite_incoming'
% 0.31/0.47 thf(fact_330_g_H_Oincoming_H__edges,axiom,
% 0.31/0.47 ! [U3: set_nat] : ( ord_le841296385at_nat @ ( incoming_b @ c2 @ U3 ) @ ( e_b @ c2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.incoming'_edges
% 0.31/0.47 thf(fact_331_g_H_Ooutgoing_H__edges,axiom,
% 0.31/0.47 ! [U3: set_nat] : ( ord_le841296385at_nat @ ( outgoing_b @ c2 @ U3 ) @ ( e_b @ c2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.outgoing'_edges
% 0.31/0.47 thf(fact_332_g_H_Ofinite__outgoing_H,axiom,
% 0.31/0.47 ! [U3: set_nat] :
% 0.31/0.47 ( ( finite_finite_nat @ ( v_b @ c2 ) )
% 0.31/0.47 => ( finite772653738at_nat @ ( outgoing_b @ c2 @ U3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.finite_outgoing'
% 0.31/0.47 thf(fact_333_g_H_Ofinite__incoming_H,axiom,
% 0.31/0.47 ! [U3: set_nat] :
% 0.31/0.47 ( ( finite_finite_nat @ ( v_b @ c2 ) )
% 0.31/0.47 => ( finite772653738at_nat @ ( incoming_b @ c2 @ U3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.finite_incoming'
% 0.31/0.47 thf(fact_334__092_060open_062edges_A_092_060subseteq_062_AE_092_060close_062,axiom,
% 0.31/0.47 ord_le841296385at_nat @ edges @ ( e_a @ c ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>edges \<subseteq> E\<close>
% 0.31/0.47 thf(fact_335_psubsetI,axiom,
% 0.31/0.47 ! [A2: set_nat,B2: set_nat] :
% 0.31/0.47 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.31/0.47 => ( ( A2 != B2 )
% 0.31/0.47 => ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % psubsetI
% 0.31/0.47 thf(fact_336_psubsetI,axiom,
% 0.31/0.47 ! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 0.31/0.47 ( ( ord_le841296385at_nat @ A2 @ B2 )
% 0.31/0.47 => ( ( A2 != B2 )
% 0.31/0.47 => ( ord_le116442893at_nat @ A2 @ B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % psubsetI
% 0.31/0.47 thf(fact_337_subset__antisym,axiom,
% 0.31/0.47 ! [A2: set_nat,B2: set_nat] :
% 0.31/0.47 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.31/0.47 => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 0.31/0.47 => ( A2 = B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_antisym
% 0.31/0.47 thf(fact_338_subset__antisym,axiom,
% 0.31/0.47 ! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 0.31/0.47 ( ( ord_le841296385at_nat @ A2 @ B2 )
% 0.31/0.47 => ( ( ord_le841296385at_nat @ B2 @ A2 )
% 0.31/0.47 => ( A2 = B2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_antisym
% 0.31/0.47 thf(fact_339_subsetI,axiom,
% 0.31/0.47 ! [A2: set_nat,B2: set_nat] :
% 0.31/0.47 ( ! [X2: nat] :
% 0.31/0.47 ( ( member_nat @ X2 @ A2 )
% 0.31/0.47 => ( member_nat @ X2 @ B2 ) )
% 0.31/0.47 => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % subsetI
% 0.31/0.47 thf(fact_340_subsetI,axiom,
% 0.31/0.47 ! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 0.31/0.47 ( ! [X2: product_prod_nat_nat] :
% 0.31/0.47 ( ( member701585322at_nat @ X2 @ A2 )
% 0.31/0.47 => ( member701585322at_nat @ X2 @ B2 ) )
% 0.31/0.47 => ( ord_le841296385at_nat @ A2 @ B2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % subsetI
% 0.31/0.47 thf(fact_341_SP__EDGES,axiom,
% 0.31/0.47 ord_le841296385at_nat @ edges @ ( set_Pr2131844118at_nat @ p ) ).
% 0.31/0.47
% 0.31/0.47 % SP_EDGES
% 0.31/0.47 thf(fact_342_Collect__mono__iff,axiom,
% 0.31/0.47 ! [P2: nat > $o,Q: nat > $o] :
% 0.31/0.47 ( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
% 0.31/0.47 = ( ! [X: nat] :
% 0.31/0.47 ( ( P2 @ X )
% 0.31/0.47 => ( Q @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_mono_iff
% 0.31/0.47 thf(fact_343_Collect__mono__iff,axiom,
% 0.31/0.47 ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.31/0.47 ( ( ord_le841296385at_nat @ ( collec7649004at_nat @ P2 ) @ ( collec7649004at_nat @ Q ) )
% 0.31/0.47 = ( ! [X: product_prod_nat_nat] :
% 0.31/0.47 ( ( P2 @ X )
% 0.31/0.47 => ( Q @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_mono_iff
% 0.31/0.47 thf(fact_344_set__eq__subset,axiom,
% 0.31/0.47 ( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
% 0.31/0.47 = ( ^ [A4: set_nat,B5: set_nat] :
% 0.31/0.47 ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 0.31/0.47 & ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % set_eq_subset
% 0.31/0.47 thf(fact_345_set__eq__subset,axiom,
% 0.31/0.47 ( ( ^ [Y4: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] : ( Y4 = Z ) )
% 0.31/0.47 = ( ^ [A4: set_Pr1986765409at_nat,B5: set_Pr1986765409at_nat] :
% 0.31/0.47 ( ( ord_le841296385at_nat @ A4 @ B5 )
% 0.31/0.47 & ( ord_le841296385at_nat @ B5 @ A4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % set_eq_subset
% 0.31/0.47 thf(fact_346_subset__trans,axiom,
% 0.31/0.47 ! [A2: set_nat,B2: set_nat,C3: set_nat] :
% 0.31/0.47 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.31/0.47 => ( ( ord_less_eq_set_nat @ B2 @ C3 )
% 0.31/0.47 => ( ord_less_eq_set_nat @ A2 @ C3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_trans
% 0.31/0.47 thf(fact_347_subset__trans,axiom,
% 0.31/0.47 ! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,C3: set_Pr1986765409at_nat] :
% 0.31/0.47 ( ( ord_le841296385at_nat @ A2 @ B2 )
% 0.31/0.47 => ( ( ord_le841296385at_nat @ B2 @ C3 )
% 0.31/0.47 => ( ord_le841296385at_nat @ A2 @ C3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_trans
% 0.31/0.47
% 0.31/0.47 % Conjectures (1)
% 0.31/0.47 thf(conj_0,conjecture,
% 0.31/0.47 $false ).
% 0.31/0.47
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 ------------------------
% 2.72/1.52 ---- Embedded in THF ---
% 2.72/1.52 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.72/1.52 %%% Generated on Tue Feb 24 08:24:47 EST 2026
% 2.72/1.52 %%% using '$$dhol' embedding, version 1.3.0.
% 2.72/1.52 %%% Logic specification used:
% 2.72/1.52 %%% thf(spec, logic, $$dhol).
% 2.72/1.52
% 2.72/1.52 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.Pn7hie6YQY/DTF2DTF17795.p
% See solution above
% 2.72/1.52 ------------------------
% 2.72/1.56 ---- Cleaned THF ---
% 2.72/1.56 thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 2.72/1.56 set_li664300135at_nat: $tType ).
% 2.72/1.56
% 2.72/1.56 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 2.72/1.56 set_se1612935105at_nat: $tType ).
% 2.72/1.56
% 2.72/1.56 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.72/1.56 list_P559422087at_nat: $tType ).
% 2.72/1.56
% 2.72/1.56 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.72/1.56 set_Pr1986765409at_nat: $tType ).
% 2.72/1.56
% 2.72/1.56 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 2.72/1.56 product_prod_nat_nat: $tType ).
% 2.72/1.56
% 2.72/1.56 thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 2.72/1.56 set_set_nat: $tType ).
% 2.72/1.56
% 2.72/1.56 thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
% 2.72/1.56 set_nat: $tType ).
% 2.72/1.56
% 2.72/1.56 thf(ty_n_t__Nat__Onat,type,
% 2.72/1.56 nat: $tType ).
% 2.72/1.56
% 2.72/1.56 thf(ty_n_tf__b,type,
% 2.72/1.56 b: $tType ).
% 2.72/1.56
% 2.72/1.56 thf(ty_n_tf__a,type,
% 2.72/1.56 a: $tType ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis_001tf__a,type,
% 2.72/1.56 edmond1517640972ysis_a: ( product_prod_nat_nat > a ) > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis_001tf__b,type,
% 2.72/1.56 edmond1517640973ysis_b: ( product_prod_nat_nat > b ) > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis__defs_OekMeasure_001tf__a,type,
% 2.72/1.56 edmond1022345716sure_a: ( product_prod_nat_nat > a ) > nat > nat > nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis__defs_OekMeasure_001tf__b,type,
% 2.72/1.56 edmond1022345717sure_b: ( product_prod_nat_nat > b ) > nat > nat > nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis__defs_OspEdges_001tf__a,type,
% 2.72/1.56 edmond475474835dges_a: ( product_prod_nat_nat > a ) > nat > nat > set_Pr1986765409at_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis__defs_OspEdges_001tf__b,type,
% 2.72/1.56 edmond475474836dges_b: ( product_prod_nat_nat > b ) > nat > nat > set_Pr1986765409at_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis__defs_OuE_001tf__a,type,
% 2.72/1.56 edmond771116670s_uE_a: ( product_prod_nat_nat > a ) > set_Pr1986765409at_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__pndqeoznpl_Oek__analysis__defs_OuE_001tf__b,type,
% 2.72/1.56 edmond771116671s_uE_b: ( product_prod_nat_nat > b ) > set_Pr1986765409at_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.72/1.56 finite83082927at_nat: set_li664300135at_nat > nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 2.72/1.56 finite_card_nat: set_nat > nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 2.72/1.56 finite447719721at_nat: set_Pr1986765409at_nat > nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.72/1.56 finite_card_set_nat: set_set_nat > nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.72/1.56 finite1701894793at_nat: set_se1612935105at_nat > nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.72/1.56 finite1299096496at_nat: set_li664300135at_nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
% 2.72/1.56 finite_finite_nat: set_nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 2.72/1.56 finite772653738at_nat: set_Pr1986765409at_nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.72/1.56 finite2012248349et_nat: set_set_nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.72/1.56 finite1457549322at_nat: set_se1612935105at_nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OFinite__Graph_001tf__a,type,
% 2.72/1.56 finite_Graph_a: ( product_prod_nat_nat > a ) > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OFinite__Graph_001tf__b,type,
% 2.72/1.56 finite_Graph_b: ( product_prod_nat_nat > b ) > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_OE_001tf__a,type,
% 2.72/1.56 e_a: ( product_prod_nat_nat > a ) > set_Pr1986765409at_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_OE_001tf__b,type,
% 2.72/1.56 e_b: ( product_prod_nat_nat > b ) > set_Pr1986765409at_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_OV_001tf__a,type,
% 2.72/1.56 v_a: ( product_prod_nat_nat > a ) > set_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_OV_001tf__b,type,
% 2.72/1.56 v_b: ( product_prod_nat_nat > b ) > set_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_Oadjacent__nodes_001tf__a,type,
% 2.72/1.56 adjacent_nodes_a: ( product_prod_nat_nat > a ) > nat > set_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_Oadjacent__nodes_001tf__b,type,
% 2.72/1.56 adjacent_nodes_b: ( product_prod_nat_nat > b ) > nat > set_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_Oconnected_001tf__a,type,
% 2.72/1.56 connected_a: ( product_prod_nat_nat > a ) > nat > nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_Oconnected_001tf__b,type,
% 2.72/1.56 connected_b: ( product_prod_nat_nat > b ) > nat > nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_Odist_001tf__a,type,
% 2.72/1.56 dist_a: ( product_prod_nat_nat > a ) > nat > nat > nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_Odist_001tf__b,type,
% 2.72/1.56 dist_b: ( product_prod_nat_nat > b ) > nat > nat > nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_Oincoming_H_001tf__a,type,
% 2.72/1.56 incoming_a: ( product_prod_nat_nat > a ) > set_nat > set_Pr1986765409at_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_Oincoming_H_001tf__b,type,
% 2.72/1.56 incoming_b: ( product_prod_nat_nat > b ) > set_nat > set_Pr1986765409at_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_OisPath_001tf__a,type,
% 2.72/1.56 isPath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_OisPath_001tf__b,type,
% 2.72/1.56 isPath_b: ( product_prod_nat_nat > b ) > nat > list_P559422087at_nat > nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_OisShortestPath_001tf__a,type,
% 2.72/1.56 isShortestPath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_OisShortestPath_001tf__b,type,
% 2.72/1.56 isShortestPath_b: ( product_prod_nat_nat > b ) > nat > list_P559422087at_nat > nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_OisSimplePath_001tf__a,type,
% 2.72/1.56 isSimplePath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_nat > nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_OisSimplePath_001tf__b,type,
% 2.72/1.56 isSimplePath_b: ( product_prod_nat_nat > b ) > nat > list_P559422087at_nat > nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_Omin__dist_001tf__a,type,
% 2.72/1.56 min_dist_a: ( product_prod_nat_nat > a ) > nat > nat > nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_Omin__dist_001tf__b,type,
% 2.72/1.56 min_dist_b: ( product_prod_nat_nat > b ) > nat > nat > nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_Ooutgoing_H_001tf__a,type,
% 2.72/1.56 outgoing_a: ( product_prod_nat_nat > a ) > set_nat > set_Pr1986765409at_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_Ooutgoing_H_001tf__b,type,
% 2.72/1.56 outgoing_b: ( product_prod_nat_nat > b ) > set_nat > set_Pr1986765409at_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_OreachableNodes_001tf__a,type,
% 2.72/1.56 reachableNodes_a: ( product_prod_nat_nat > a ) > nat > set_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Graph_OGraph_OreachableNodes_001tf__b,type,
% 2.72/1.56 reachableNodes_b: ( product_prod_nat_nat > b ) > nat > set_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
% 2.72/1.56 zero_zero_nat: nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 2.72/1.56 set_Pr2131844118at_nat: list_P559422087at_nat > set_Pr1986765409at_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.72/1.56 size_s1990949619at_nat: list_P559422087at_nat > nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 2.72/1.56 ord_less_nat: nat > nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.72/1.56 ord_less_set_nat: set_nat > set_nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.72/1.56 ord_le116442893at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.72/1.56 ord_less_eq_nat: nat > nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.72/1.56 ord_less_eq_set_nat: set_nat > set_nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.72/1.56 ord_le841296385at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 2.72/1.56 ord_le1613022364et_nat: set_set_nat > set_set_nat > $o ).
% 2.72/1.56
% 2.72/1.56 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,
% 2.72/1.56 ord_le2096002913at_nat: set_se1612935105at_nat > set_se1612935105at_nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
% 2.72/1.56 collect_nat: ( nat > $o ) > set_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 2.72/1.56 collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.72/1.56 collect_set_nat: ( set_nat > $o ) > set_set_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_Set_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.72/1.56 collec1606769740at_nat: ( set_Pr1986765409at_nat > $o ) > set_se1612935105at_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.72/1.56 member1608759472at_nat: list_P559422087at_nat > set_li664300135at_nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_member_001t__Nat__Onat,type,
% 2.72/1.56 member_nat: nat > set_nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 2.72/1.56 member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 2.72/1.56 member_set_nat: set_nat > set_set_nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.72/1.56 member298845450at_nat: set_Pr1986765409at_nat > set_se1612935105at_nat > $o ).
% 2.72/1.56
% 2.72/1.56 thf(sy_v_c,type,
% 2.72/1.56 c: product_prod_nat_nat > a ).
% 2.72/1.56
% 2.72/1.56 thf(sy_v_c_H,type,
% 2.72/1.56 c2: product_prod_nat_nat > b ).
% 2.72/1.56
% 2.72/1.56 thf(sy_v_edges,type,
% 2.72/1.56 edges: set_Pr1986765409at_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_v_p,type,
% 2.72/1.56 p: list_P559422087at_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_v_p_H____,type,
% 2.72/1.56 p2: list_P559422087at_nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_v_s,type,
% 2.72/1.56 s: nat ).
% 2.72/1.56
% 2.72/1.56 thf(sy_v_t,type,
% 2.72/1.56 t: nat ).
% 2.72/1.56
% 2.72/1.56 thf(fact_0_ek__analysis__axioms,axiom,
% 2.72/1.56 edmond1517640972ysis_a @ c ).
% 2.72/1.56
% 2.72/1.56 thf(fact_1_LENP,axiom,
% 2.72/1.56 ( ( size_s1990949619at_nat @ p )
% 2.72/1.56 = ( min_dist_a @ c @ s @ t ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_2_LENP_H,axiom,
% 2.72/1.56 ( ( size_s1990949619at_nat @ p2 )
% 2.72/1.56 = ( min_dist_b @ c2 @ s @ t ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_3_ek__analysis__defs_OekMeasure_Ocong,axiom,
% 2.72/1.56 edmond1022345717sure_b = edmond1022345717sure_b ).
% 2.72/1.56
% 2.72/1.56 thf(fact_4_ek__analysis__defs_OekMeasure_Ocong,axiom,
% 2.72/1.56 edmond1022345716sure_a = edmond1022345716sure_a ).
% 2.72/1.56
% 2.72/1.56 thf(fact_5_g_H_Oek__analysis__axioms,axiom,
% 2.72/1.56 edmond1517640973ysis_b @ c2 ).
% 2.72/1.56
% 2.72/1.56 thf(fact_6_SHORTER,axiom,
% 2.72/1.56 ord_less_nat @ ( min_dist_b @ c2 @ s @ t ) @ ( min_dist_a @ c @ s @ t ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_7_ek__analysis__defs_OuE_Ocong,axiom,
% 2.72/1.56 edmond771116671s_uE_b = edmond771116671s_uE_b ).
% 2.72/1.56
% 2.72/1.56 thf(fact_8_ek__analysis__defs_OuE_Ocong,axiom,
% 2.72/1.56 edmond771116670s_uE_a = edmond771116670s_uE_a ).
% 2.72/1.56
% 2.72/1.56 thf(fact_9_ek__analysis__defs_OspEdges_Ocong,axiom,
% 2.72/1.56 edmond475474835dges_a = edmond475474835dges_a ).
% 2.72/1.56
% 2.72/1.56 thf(fact_10_ek__analysis__defs_OspEdges_Ocong,axiom,
% 2.72/1.56 edmond475474836dges_b = edmond475474836dges_b ).
% 2.72/1.56
% 2.72/1.56 thf(fact_11_g_H_OFinite__Graph__axioms,axiom,
% 2.72/1.56 finite_Graph_b @ c2 ).
% 2.72/1.56
% 2.72/1.56 thf(fact_12_Finite__Graph__axioms,axiom,
% 2.72/1.56 finite_Graph_a @ c ).
% 2.72/1.56
% 2.72/1.56 thf(fact_13__092_060open_062length_Ap_A_060_Alength_Ap_H_092_060close_062,axiom,
% 2.72/1.56 ord_less_nat @ ( size_s1990949619at_nat @ p ) @ ( size_s1990949619at_nat @ p2 ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_14_uE__eq,axiom,
% 2.72/1.56 ( ( edmond771116671s_uE_b @ c2 )
% 2.72/1.56 = ( edmond771116670s_uE_a @ c ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_15_SP,axiom,
% 2.72/1.56 isShortestPath_a @ c @ s @ p @ t ).
% 2.72/1.56
% 2.72/1.56 thf(fact_16_CONN2,axiom,
% 2.72/1.56 connected_b @ c2 @ s @ t ).
% 2.72/1.56
% 2.72/1.56 thf(fact_17_SV,axiom,
% 2.72/1.56 member_nat @ s @ ( v_a @ c ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_18_P_H,axiom,
% 2.72/1.56 isPath_b @ c2 @ s @ p2 @ t ).
% 2.72/1.56
% 2.72/1.56 thf(fact_19_ek__analysis__def,axiom,
% 2.72/1.56 edmond1517640972ysis_a = finite_Graph_a ).
% 2.72/1.56
% 2.72/1.56 thf(fact_20_ek__analysis__def,axiom,
% 2.72/1.56 edmond1517640973ysis_b = finite_Graph_b ).
% 2.72/1.56
% 2.72/1.56 thf(fact_21_ek__analysis_Ointro,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a] :
% 2.72/1.56 ( ( finite_Graph_a @ C )
% 2.72/1.56 => ( edmond1517640972ysis_a @ C ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_22_ek__analysis_Ointro,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b] :
% 2.72/1.56 ( ( finite_Graph_b @ C )
% 2.72/1.56 => ( edmond1517640973ysis_b @ C ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_23_ek__analysis_Oaxioms,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a] :
% 2.72/1.56 ( ( edmond1517640972ysis_a @ C )
% 2.72/1.56 => ( finite_Graph_a @ C ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_24_ek__analysis_Oaxioms,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b] :
% 2.72/1.56 ( ( edmond1517640973ysis_b @ C )
% 2.72/1.56 => ( finite_Graph_b @ C ) ) ).
% 2.72/1.56
% 2.72/1.56 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,
% 2.72/1.56 ( ~ ( connected_b @ c2 @ s @ t )
% 2.72/1.56 => ( ord_less_nat @ ( edmond1022345717sure_b @ c2 @ s @ t ) @ ( edmond1022345716sure_a @ c @ s @ t ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_26__092_060open_062g_H_Odist_As_A_Ig_H_Omin__dist_As_At_J_At_092_060close_062,axiom,
% 2.72/1.56 dist_b @ c2 @ s @ ( min_dist_b @ c2 @ s @ t ) @ t ).
% 2.72/1.56
% 2.72/1.56 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,
% 2.72/1.56 ~ ! [P: list_P559422087at_nat] :
% 2.72/1.56 ( ( isPath_b @ c2 @ s @ P @ t )
% 2.72/1.56 => ( ( size_s1990949619at_nat @ P )
% 2.72/1.56 != ( min_dist_b @ c2 @ s @ t ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_28_CONN,axiom,
% 2.72/1.56 connected_a @ c @ s @ t ).
% 2.72/1.56
% 2.72/1.56 thf(fact_29_min__dist__less,axiom,
% 2.72/1.56 ! [Src: nat,V: nat,D: nat,D2: nat] :
% 2.72/1.56 ( ( connected_a @ c @ Src @ V )
% 2.72/1.56 => ( ( ( min_dist_a @ c @ Src @ V )
% 2.72/1.56 = D )
% 2.72/1.56 => ( ( ord_less_nat @ D2 @ D )
% 2.72/1.56 => ? [V2: nat] :
% 2.72/1.56 ( ( connected_a @ c @ Src @ V2 )
% 2.72/1.56 & ( ( min_dist_a @ c @ Src @ V2 )
% 2.72/1.56 = D2 ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_30_g_H_Omin__dist__less,axiom,
% 2.72/1.56 ! [Src: nat,V: nat,D: nat,D2: nat] :
% 2.72/1.56 ( ( connected_b @ c2 @ Src @ V )
% 2.72/1.56 => ( ( ( min_dist_b @ c2 @ Src @ V )
% 2.72/1.56 = D )
% 2.72/1.56 => ( ( ord_less_nat @ D2 @ D )
% 2.72/1.56 => ? [V2: nat] :
% 2.72/1.56 ( ( connected_b @ c2 @ Src @ V2 )
% 2.72/1.56 & ( ( min_dist_b @ c2 @ Src @ V2 )
% 2.72/1.56 = D2 ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_31_length__induct,axiom,
% 2.72/1.56 ! [P2: list_P559422087at_nat > $o,Xs: list_P559422087at_nat] :
% 2.72/1.56 ( ! [Xs2: list_P559422087at_nat] :
% 2.72/1.56 ( ! [Ys: list_P559422087at_nat] :
% 2.72/1.56 ( ( ord_less_nat @ ( size_s1990949619at_nat @ Ys ) @ ( size_s1990949619at_nat @ Xs2 ) )
% 2.72/1.56 => ( P2 @ Ys ) )
% 2.72/1.56 => ( P2 @ Xs2 ) )
% 2.72/1.56 => ( P2 @ Xs ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_32_finite__spEdges,axiom,
% 2.72/1.56 finite772653738at_nat @ ( edmond475474835dges_a @ c @ s @ t ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_33_g_H_Ofinite__spEdges,axiom,
% 2.72/1.56 finite772653738at_nat @ ( edmond475474836dges_b @ c2 @ s @ t ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_34_min__dist__z,axiom,
% 2.72/1.56 ! [V: nat] :
% 2.72/1.56 ( ( min_dist_a @ c @ V @ V )
% 2.72/1.56 = zero_zero_nat ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_35_g_H_Omin__dist__z,axiom,
% 2.72/1.56 ! [V: nat] :
% 2.72/1.56 ( ( min_dist_b @ c2 @ V @ V )
% 2.72/1.56 = zero_zero_nat ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_36_g_H_Oconnected__def,axiom,
% 2.72/1.56 ! [U: nat,V: nat] :
% 2.72/1.56 ( ( connected_b @ c2 @ U @ V )
% 2.72/1.56 = ( ? [P3: list_P559422087at_nat] : ( isPath_b @ c2 @ U @ P3 @ V ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_37_connected__inV__iff,axiom,
% 2.72/1.56 ! [U: nat,V: nat] :
% 2.72/1.56 ( ( connected_a @ c @ U @ V )
% 2.72/1.56 => ( ( member_nat @ V @ ( v_a @ c ) )
% 2.72/1.56 = ( member_nat @ U @ ( v_a @ c ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_38_g_H_Oconnected__by__dist,axiom,
% 2.72/1.56 ! [V: nat,V3: nat] :
% 2.72/1.56 ( ( connected_b @ c2 @ V @ V3 )
% 2.72/1.56 = ( ? [D3: nat] : ( dist_b @ c2 @ V @ D3 @ V3 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_39_obtain__shortest__path,axiom,
% 2.72/1.56 ! [U: nat,V: nat] :
% 2.72/1.56 ( ( connected_a @ c @ U @ V )
% 2.72/1.56 => ~ ! [P4: list_P559422087at_nat] :
% 2.72/1.56 ~ ( isShortestPath_a @ c @ U @ P4 @ V ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_40_g_H_OisPath__distD,axiom,
% 2.72/1.56 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( isPath_b @ c2 @ U @ P5 @ V )
% 2.72/1.56 => ( dist_b @ c2 @ U @ ( size_s1990949619at_nat @ P5 ) @ V ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_41_g_H_Odist__def,axiom,
% 2.72/1.56 ! [V: nat,D: nat,V3: nat] :
% 2.72/1.56 ( ( dist_b @ c2 @ V @ D @ V3 )
% 2.72/1.56 = ( ? [P3: list_P559422087at_nat] :
% 2.72/1.56 ( ( isPath_b @ c2 @ V @ P3 @ V3 )
% 2.72/1.56 & ( ( size_s1990949619at_nat @ P3 )
% 2.72/1.56 = D ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_42_g_H_Omin__dist__is__dist,axiom,
% 2.72/1.56 ! [V: nat,V3: nat] :
% 2.72/1.56 ( ( connected_b @ c2 @ V @ V3 )
% 2.72/1.56 => ( dist_b @ c2 @ V @ ( min_dist_b @ c2 @ V @ V3 ) @ V3 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_43_g_H_Oconnected__refl,axiom,
% 2.72/1.56 ! [V: nat] : ( connected_b @ c2 @ V @ V ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_44_connected__refl,axiom,
% 2.72/1.56 ! [V: nat] : ( connected_a @ c @ V @ V ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_45_g_H_Odist__z__iff,axiom,
% 2.72/1.56 ! [V: nat,V3: nat] :
% 2.72/1.56 ( ( dist_b @ c2 @ V @ zero_zero_nat @ V3 )
% 2.72/1.56 = ( V3 = V ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_46_g_H_Odist__z,axiom,
% 2.72/1.56 ! [V: nat] : ( dist_b @ c2 @ V @ zero_zero_nat @ V ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_47_g_H_Oconnected__distI,axiom,
% 2.72/1.56 ! [V: nat,D: nat,V3: nat] :
% 2.72/1.56 ( ( dist_b @ c2 @ V @ D @ V3 )
% 2.72/1.56 => ( connected_b @ c2 @ V @ V3 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_48_g_H_Ofinite__uE,axiom,
% 2.72/1.56 finite772653738at_nat @ ( edmond771116671s_uE_b @ c2 ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_49_finite__uE,axiom,
% 2.72/1.56 finite772653738at_nat @ ( edmond771116670s_uE_a @ c ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_50_g_H_Omin__dist__z__iff,axiom,
% 2.72/1.56 ! [V: nat,V3: nat] :
% 2.72/1.56 ( ( connected_b @ c2 @ V @ V3 )
% 2.72/1.56 => ( ( ( min_dist_b @ c2 @ V @ V3 )
% 2.72/1.56 = zero_zero_nat )
% 2.72/1.56 = ( V3 = V ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_51_mem__Collect__eq,axiom,
% 2.72/1.56 ! [A: set_Pr1986765409at_nat,P2: set_Pr1986765409at_nat > $o] :
% 2.72/1.56 ( ( member298845450at_nat @ A @ ( collec1606769740at_nat @ P2 ) )
% 2.72/1.56 = ( P2 @ A ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_52_mem__Collect__eq,axiom,
% 2.72/1.56 ! [A: set_nat,P2: set_nat > $o] :
% 2.72/1.56 ( ( member_set_nat @ A @ ( collect_set_nat @ P2 ) )
% 2.72/1.56 = ( P2 @ A ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_53_mem__Collect__eq,axiom,
% 2.72/1.56 ! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
% 2.72/1.56 ( ( member701585322at_nat @ A @ ( collec7649004at_nat @ P2 ) )
% 2.72/1.56 = ( P2 @ A ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_54_mem__Collect__eq,axiom,
% 2.72/1.56 ! [A: nat,P2: nat > $o] :
% 2.72/1.56 ( ( member_nat @ A @ ( collect_nat @ P2 ) )
% 2.72/1.56 = ( P2 @ A ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_55_Collect__mem__eq,axiom,
% 2.72/1.56 ! [A2: set_se1612935105at_nat] :
% 2.72/1.56 ( ( collec1606769740at_nat
% 2.72/1.56 @ ^ [X: set_Pr1986765409at_nat] : ( member298845450at_nat @ X @ A2 ) )
% 2.72/1.56 = A2 ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_56_Collect__mem__eq,axiom,
% 2.72/1.56 ! [A2: set_set_nat] :
% 2.72/1.56 ( ( collect_set_nat
% 2.72/1.56 @ ^ [X: set_nat] : ( member_set_nat @ X @ A2 ) )
% 2.72/1.56 = A2 ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_57_Collect__mem__eq,axiom,
% 2.72/1.56 ! [A2: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( collec7649004at_nat
% 2.72/1.56 @ ^ [X: product_prod_nat_nat] : ( member701585322at_nat @ X @ A2 ) )
% 2.72/1.56 = A2 ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_58_Collect__mem__eq,axiom,
% 2.72/1.56 ! [A2: set_nat] :
% 2.72/1.56 ( ( collect_nat
% 2.72/1.56 @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
% 2.72/1.56 = A2 ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_59_Collect__cong,axiom,
% 2.72/1.56 ! [P2: nat > $o,Q: nat > $o] :
% 2.72/1.56 ( ! [X2: nat] :
% 2.72/1.56 ( ( P2 @ X2 )
% 2.72/1.56 = ( Q @ X2 ) )
% 2.72/1.56 => ( ( collect_nat @ P2 )
% 2.72/1.56 = ( collect_nat @ Q ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_60_Collect__cong,axiom,
% 2.72/1.56 ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 2.72/1.56 ( ! [X2: product_prod_nat_nat] :
% 2.72/1.56 ( ( P2 @ X2 )
% 2.72/1.56 = ( Q @ X2 ) )
% 2.72/1.56 => ( ( collec7649004at_nat @ P2 )
% 2.72/1.56 = ( collec7649004at_nat @ Q ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_61_min__dist__z__iff,axiom,
% 2.72/1.56 ! [V: nat,V3: nat] :
% 2.72/1.56 ( ( connected_a @ c @ V @ V3 )
% 2.72/1.56 => ( ( ( min_dist_a @ c @ V @ V3 )
% 2.72/1.56 = zero_zero_nat )
% 2.72/1.56 = ( V3 = V ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_62_ek__analysis_Ofinite__uE,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b] :
% 2.72/1.56 ( ( edmond1517640973ysis_b @ C )
% 2.72/1.56 => ( finite772653738at_nat @ ( edmond771116671s_uE_b @ C ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_63_ek__analysis_Ofinite__uE,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a] :
% 2.72/1.56 ( ( edmond1517640972ysis_a @ C )
% 2.72/1.56 => ( finite772653738at_nat @ ( edmond771116670s_uE_a @ C ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_64_ek__analysis_Ofinite__spEdges,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,S: nat,T: nat] :
% 2.72/1.56 ( ( edmond1517640972ysis_a @ C )
% 2.72/1.56 => ( finite772653738at_nat @ ( edmond475474835dges_a @ C @ S @ T ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_65_ek__analysis_Ofinite__spEdges,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,S: nat,T: nat] :
% 2.72/1.56 ( ( edmond1517640973ysis_b @ C )
% 2.72/1.56 => ( finite772653738at_nat @ ( edmond475474836dges_b @ C @ S @ T ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_66_neq__if__length__neq,axiom,
% 2.72/1.56 ! [Xs: list_P559422087at_nat,Ys2: list_P559422087at_nat] :
% 2.72/1.56 ( ( ( size_s1990949619at_nat @ Xs )
% 2.72/1.56 != ( size_s1990949619at_nat @ Ys2 ) )
% 2.72/1.56 => ( Xs != Ys2 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_67_Ex__list__of__length,axiom,
% 2.72/1.56 ! [N: nat] :
% 2.72/1.56 ? [Xs2: list_P559422087at_nat] :
% 2.72/1.56 ( ( size_s1990949619at_nat @ Xs2 )
% 2.72/1.56 = N ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_68_bot__nat__0_Onot__eq__extremum,axiom,
% 2.72/1.56 ! [A: nat] :
% 2.72/1.56 ( ( A != zero_zero_nat )
% 2.72/1.56 = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_69_less__nat__zero__code,axiom,
% 2.72/1.56 ! [N: nat] :
% 2.72/1.56 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_70_neq0__conv,axiom,
% 2.72/1.56 ! [N: nat] :
% 2.72/1.56 ( ( N != zero_zero_nat )
% 2.72/1.56 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_71_not__gr__zero,axiom,
% 2.72/1.56 ! [N: nat] :
% 2.72/1.56 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.72/1.56 = ( N = zero_zero_nat ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_72_Graph_OisShortestPath__min__dist__def,axiom,
% 2.72/1.56 ( isShortestPath_a
% 2.72/1.56 = ( ^ [C2: product_prod_nat_nat > a,U2: nat,P3: list_P559422087at_nat,V4: nat] :
% 2.72/1.56 ( ( isPath_a @ C2 @ U2 @ P3 @ V4 )
% 2.72/1.56 & ( ( size_s1990949619at_nat @ P3 )
% 2.72/1.56 = ( min_dist_a @ C2 @ U2 @ V4 ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_73_Graph_OisShortestPath__min__dist__def,axiom,
% 2.72/1.56 ( isShortestPath_b
% 2.72/1.56 = ( ^ [C2: product_prod_nat_nat > b,U2: nat,P3: list_P559422087at_nat,V4: nat] :
% 2.72/1.56 ( ( isPath_b @ C2 @ U2 @ P3 @ V4 )
% 2.72/1.56 & ( ( size_s1990949619at_nat @ P3 )
% 2.72/1.56 = ( min_dist_b @ C2 @ U2 @ V4 ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_74_g_H_OisShortestPath__min__dist__def,axiom,
% 2.72/1.56 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( isShortestPath_b @ c2 @ U @ P5 @ V )
% 2.72/1.56 = ( ( isPath_b @ c2 @ U @ P5 @ V )
% 2.72/1.56 & ( ( size_s1990949619at_nat @ P5 )
% 2.72/1.56 = ( min_dist_b @ c2 @ U @ V ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_75_g_H_Omin__distI2,axiom,
% 2.72/1.56 ! [V: nat,V3: nat,Q: nat > $o] :
% 2.72/1.56 ( ( connected_b @ c2 @ V @ V3 )
% 2.72/1.56 => ( ! [D4: nat] :
% 2.72/1.56 ( ( dist_b @ c2 @ V @ D4 @ V3 )
% 2.72/1.56 => ( ! [D5: nat] :
% 2.72/1.56 ( ( dist_b @ c2 @ V @ D5 @ V3 )
% 2.72/1.56 => ( ord_less_eq_nat @ D4 @ D5 ) )
% 2.72/1.56 => ( Q @ D4 ) ) )
% 2.72/1.56 => ( Q @ ( min_dist_b @ c2 @ V @ V3 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_76_isShortestPath__length__less__V,axiom,
% 2.72/1.56 ! [S: nat,P5: list_P559422087at_nat,T: nat] :
% 2.72/1.56 ( ( member_nat @ S @ ( v_a @ c ) )
% 2.72/1.56 => ( ( isShortestPath_a @ c @ S @ P5 @ T )
% 2.72/1.56 => ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_a @ c ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_77_min__dist__less__V,axiom,
% 2.72/1.56 ! [S: nat,T: nat] :
% 2.72/1.56 ( ( member_nat @ S @ ( v_a @ c ) )
% 2.72/1.56 => ( ( connected_a @ c @ S @ T )
% 2.72/1.56 => ( ord_less_nat @ ( min_dist_a @ c @ S @ T ) @ ( finite_card_nat @ ( v_a @ c ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_78_connected__def,axiom,
% 2.72/1.56 ! [U: nat,V: nat] :
% 2.72/1.56 ( ( connected_a @ c @ U @ V )
% 2.72/1.56 = ( ? [P3: list_P559422087at_nat] : ( isPath_a @ c @ U @ P3 @ V ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_79_shortestPath__is__path,axiom,
% 2.72/1.56 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( isShortestPath_a @ c @ U @ P5 @ V )
% 2.72/1.56 => ( isPath_a @ c @ U @ P5 @ V ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_80_g_H_OshortestPath__is__path,axiom,
% 2.72/1.56 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( isShortestPath_b @ c2 @ U @ P5 @ V )
% 2.72/1.56 => ( isPath_b @ c2 @ U @ P5 @ V ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_81_g_H_Oobtain__shortest__path,axiom,
% 2.72/1.56 ! [U: nat,V: nat] :
% 2.72/1.56 ( ( connected_b @ c2 @ U @ V )
% 2.72/1.56 => ~ ! [P4: list_P559422087at_nat] :
% 2.72/1.56 ~ ( isShortestPath_b @ c2 @ U @ P4 @ V ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_82_g_H_Omin__dist__le,axiom,
% 2.72/1.56 ! [Src: nat,V: nat,D2: nat] :
% 2.72/1.56 ( ( connected_b @ c2 @ Src @ V )
% 2.72/1.56 => ( ( ord_less_eq_nat @ D2 @ ( min_dist_b @ c2 @ Src @ V ) )
% 2.72/1.56 => ? [V2: nat] :
% 2.72/1.56 ( ( connected_b @ c2 @ Src @ V2 )
% 2.72/1.56 & ( ( min_dist_b @ c2 @ Src @ V2 )
% 2.72/1.56 = D2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_83_g_H_Omin__distI__eq,axiom,
% 2.72/1.56 ! [V: nat,D: nat,V3: nat] :
% 2.72/1.56 ( ( dist_b @ c2 @ V @ D @ V3 )
% 2.72/1.56 => ( ! [D6: nat] :
% 2.72/1.56 ( ( dist_b @ c2 @ V @ D6 @ V3 )
% 2.72/1.56 => ( ord_less_eq_nat @ D @ D6 ) )
% 2.72/1.56 => ( ( min_dist_b @ c2 @ V @ V3 )
% 2.72/1.56 = D ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_84_g_H_Omin__dist__minD,axiom,
% 2.72/1.56 ! [V: nat,D: nat,V3: nat] :
% 2.72/1.56 ( ( dist_b @ c2 @ V @ D @ V3 )
% 2.72/1.56 => ( ord_less_eq_nat @ ( min_dist_b @ c2 @ V @ V3 ) @ D ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_85_min__dist__le,axiom,
% 2.72/1.56 ! [Src: nat,V: nat,D2: nat] :
% 2.72/1.56 ( ( connected_a @ c @ Src @ V )
% 2.72/1.56 => ( ( ord_less_eq_nat @ D2 @ ( min_dist_a @ c @ Src @ V ) )
% 2.72/1.56 => ? [V2: nat] :
% 2.72/1.56 ( ( connected_a @ c @ Src @ V2 )
% 2.72/1.56 & ( ( min_dist_a @ c @ Src @ V2 )
% 2.72/1.56 = D2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_86_isShortestPath__def,axiom,
% 2.72/1.56 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( isShortestPath_a @ c @ U @ P5 @ V )
% 2.72/1.56 = ( ( isPath_a @ c @ U @ P5 @ V )
% 2.72/1.56 & ! [P6: list_P559422087at_nat] :
% 2.72/1.56 ( ( isPath_a @ c @ U @ P6 @ V )
% 2.72/1.56 => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P5 ) @ ( size_s1990949619at_nat @ P6 ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_87_isShortestPath__min__dist__def,axiom,
% 2.72/1.56 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( isShortestPath_a @ c @ U @ P5 @ V )
% 2.72/1.56 = ( ( isPath_a @ c @ U @ P5 @ V )
% 2.72/1.56 & ( ( size_s1990949619at_nat @ P5 )
% 2.72/1.56 = ( min_dist_a @ c @ U @ V ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_88_g_H_OisShortestPath__def,axiom,
% 2.72/1.56 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( isShortestPath_b @ c2 @ U @ P5 @ V )
% 2.72/1.56 = ( ( isPath_b @ c2 @ U @ P5 @ V )
% 2.72/1.56 & ! [P6: list_P559422087at_nat] :
% 2.72/1.56 ( ( isPath_b @ c2 @ U @ P6 @ V )
% 2.72/1.56 => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P5 ) @ ( size_s1990949619at_nat @ P6 ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_89_le__zero__eq,axiom,
% 2.72/1.56 ! [N: nat] :
% 2.72/1.56 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.72/1.56 = ( N = zero_zero_nat ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_90_le0,axiom,
% 2.72/1.56 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_91_bot__nat__0_Oextremum,axiom,
% 2.72/1.56 ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_92_zero__le,axiom,
% 2.72/1.56 ! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_93_le__refl,axiom,
% 2.72/1.56 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_94_le__trans,axiom,
% 2.72/1.56 ! [I: nat,J: nat,K: nat] :
% 2.72/1.56 ( ( ord_less_eq_nat @ I @ J )
% 2.72/1.56 => ( ( ord_less_eq_nat @ J @ K )
% 2.72/1.56 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_95_eq__imp__le,axiom,
% 2.72/1.56 ! [M: nat,N: nat] :
% 2.72/1.56 ( ( M = N )
% 2.72/1.56 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_96_le__antisym,axiom,
% 2.72/1.56 ! [M: nat,N: nat] :
% 2.72/1.56 ( ( ord_less_eq_nat @ M @ N )
% 2.72/1.56 => ( ( ord_less_eq_nat @ N @ M )
% 2.72/1.56 => ( M = N ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_97_nat__le__linear,axiom,
% 2.72/1.56 ! [M: nat,N: nat] :
% 2.72/1.56 ( ( ord_less_eq_nat @ M @ N )
% 2.72/1.56 | ( ord_less_eq_nat @ N @ M ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_98_Nat_Oex__has__greatest__nat,axiom,
% 2.72/1.56 ! [P2: nat > $o,K: nat,B: nat] :
% 2.72/1.56 ( ( P2 @ K )
% 2.72/1.56 => ( ! [Y: nat] :
% 2.72/1.56 ( ( P2 @ Y )
% 2.72/1.56 => ( ord_less_eq_nat @ Y @ B ) )
% 2.72/1.56 => ? [X2: nat] :
% 2.72/1.56 ( ( P2 @ X2 )
% 2.72/1.56 & ! [Y2: nat] :
% 2.72/1.56 ( ( P2 @ Y2 )
% 2.72/1.56 => ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_99_less__eq__nat_Osimps_I1_J,axiom,
% 2.72/1.56 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_100_le__0__eq,axiom,
% 2.72/1.56 ! [N: nat] :
% 2.72/1.56 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 2.72/1.56 = ( N = zero_zero_nat ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_101_bot__nat__0_Oextremum__unique,axiom,
% 2.72/1.56 ! [A: nat] :
% 2.72/1.56 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.72/1.56 = ( A = zero_zero_nat ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_102_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.72/1.56 ! [A: nat] :
% 2.72/1.56 ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 2.72/1.56 => ( A = zero_zero_nat ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_103_nat__less__le,axiom,
% 2.72/1.56 ( ord_less_nat
% 2.72/1.56 = ( ^ [M2: nat,N2: nat] :
% 2.72/1.56 ( ( ord_less_eq_nat @ M2 @ N2 )
% 2.72/1.56 & ( M2 != N2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_104_less__imp__le__nat,axiom,
% 2.72/1.56 ! [M: nat,N: nat] :
% 2.72/1.56 ( ( ord_less_nat @ M @ N )
% 2.72/1.56 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_105_le__eq__less__or__eq,axiom,
% 2.72/1.56 ( ord_less_eq_nat
% 2.72/1.56 = ( ^ [M2: nat,N2: nat] :
% 2.72/1.56 ( ( ord_less_nat @ M2 @ N2 )
% 2.72/1.56 | ( M2 = N2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_106_less__or__eq__imp__le,axiom,
% 2.72/1.56 ! [M: nat,N: nat] :
% 2.72/1.56 ( ( ( ord_less_nat @ M @ N )
% 2.72/1.56 | ( M = N ) )
% 2.72/1.56 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_107_le__neq__implies__less,axiom,
% 2.72/1.56 ! [M: nat,N: nat] :
% 2.72/1.56 ( ( ord_less_eq_nat @ M @ N )
% 2.72/1.56 => ( ( M != N )
% 2.72/1.56 => ( ord_less_nat @ M @ N ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_108_less__mono__imp__le__mono,axiom,
% 2.72/1.56 ! [F: nat > nat,I: nat,J: nat] :
% 2.72/1.56 ( ! [I2: nat,J2: nat] :
% 2.72/1.56 ( ( ord_less_nat @ I2 @ J2 )
% 2.72/1.56 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 2.72/1.56 => ( ( ord_less_eq_nat @ I @ J )
% 2.72/1.56 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_109_ex__least__nat__le,axiom,
% 2.72/1.56 ! [P2: nat > $o,N: nat] :
% 2.72/1.56 ( ( P2 @ N )
% 2.72/1.56 => ( ~ ( P2 @ zero_zero_nat )
% 2.72/1.56 => ? [K2: nat] :
% 2.72/1.56 ( ( ord_less_eq_nat @ K2 @ N )
% 2.72/1.56 & ! [I3: nat] :
% 2.72/1.56 ( ( ord_less_nat @ I3 @ K2 )
% 2.72/1.56 => ~ ( P2 @ I3 ) )
% 2.72/1.56 & ( P2 @ K2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_110_Graph_Omin__dist__le,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,Src: nat,V: nat,D2: nat] :
% 2.72/1.56 ( ( connected_b @ C @ Src @ V )
% 2.72/1.56 => ( ( ord_less_eq_nat @ D2 @ ( min_dist_b @ C @ Src @ V ) )
% 2.72/1.56 => ? [V2: nat] :
% 2.72/1.56 ( ( connected_b @ C @ Src @ V2 )
% 2.72/1.56 & ( ( min_dist_b @ C @ Src @ V2 )
% 2.72/1.56 = D2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_111_Graph_Omin__dist__le,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,Src: nat,V: nat,D2: nat] :
% 2.72/1.56 ( ( connected_a @ C @ Src @ V )
% 2.72/1.56 => ( ( ord_less_eq_nat @ D2 @ ( min_dist_a @ C @ Src @ V ) )
% 2.72/1.56 => ? [V2: nat] :
% 2.72/1.56 ( ( connected_a @ C @ Src @ V2 )
% 2.72/1.56 & ( ( min_dist_a @ C @ Src @ V2 )
% 2.72/1.56 = D2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_112_Graph_Omin__dist__minD,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,V: nat,D: nat,V3: nat] :
% 2.72/1.56 ( ( dist_a @ C @ V @ D @ V3 )
% 2.72/1.56 => ( ord_less_eq_nat @ ( min_dist_a @ C @ V @ V3 ) @ D ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_113_Graph_Omin__dist__minD,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,V: nat,D: nat,V3: nat] :
% 2.72/1.56 ( ( dist_b @ C @ V @ D @ V3 )
% 2.72/1.56 => ( ord_less_eq_nat @ ( min_dist_b @ C @ V @ V3 ) @ D ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_114_Graph_Omin__distI__eq,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,V: nat,D: nat,V3: nat] :
% 2.72/1.56 ( ( dist_a @ C @ V @ D @ V3 )
% 2.72/1.56 => ( ! [D6: nat] :
% 2.72/1.56 ( ( dist_a @ C @ V @ D6 @ V3 )
% 2.72/1.56 => ( ord_less_eq_nat @ D @ D6 ) )
% 2.72/1.56 => ( ( min_dist_a @ C @ V @ V3 )
% 2.72/1.56 = D ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_115_Graph_Omin__distI__eq,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,V: nat,D: nat,V3: nat] :
% 2.72/1.56 ( ( dist_b @ C @ V @ D @ V3 )
% 2.72/1.56 => ( ! [D6: nat] :
% 2.72/1.56 ( ( dist_b @ C @ V @ D6 @ V3 )
% 2.72/1.56 => ( ord_less_eq_nat @ D @ D6 ) )
% 2.72/1.56 => ( ( min_dist_b @ C @ V @ V3 )
% 2.72/1.56 = D ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_116_Graph_Omin__distI2,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,V: nat,V3: nat,Q: nat > $o] :
% 2.72/1.56 ( ( connected_b @ C @ V @ V3 )
% 2.72/1.56 => ( ! [D4: nat] :
% 2.72/1.56 ( ( dist_b @ C @ V @ D4 @ V3 )
% 2.72/1.56 => ( ! [D5: nat] :
% 2.72/1.56 ( ( dist_b @ C @ V @ D5 @ V3 )
% 2.72/1.56 => ( ord_less_eq_nat @ D4 @ D5 ) )
% 2.72/1.56 => ( Q @ D4 ) ) )
% 2.72/1.56 => ( Q @ ( min_dist_b @ C @ V @ V3 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_117_Graph_Omin__distI2,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,V: nat,V3: nat,Q: nat > $o] :
% 2.72/1.56 ( ( connected_a @ C @ V @ V3 )
% 2.72/1.56 => ( ! [D4: nat] :
% 2.72/1.56 ( ( dist_a @ C @ V @ D4 @ V3 )
% 2.72/1.56 => ( ! [D5: nat] :
% 2.72/1.56 ( ( dist_a @ C @ V @ D5 @ V3 )
% 2.72/1.56 => ( ord_less_eq_nat @ D4 @ D5 ) )
% 2.72/1.56 => ( Q @ D4 ) ) )
% 2.72/1.56 => ( Q @ ( min_dist_a @ C @ V @ V3 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_118_Graph_OisShortestPath__def,axiom,
% 2.72/1.56 ( isShortestPath_a
% 2.72/1.56 = ( ^ [C2: product_prod_nat_nat > a,U2: nat,P3: list_P559422087at_nat,V4: nat] :
% 2.72/1.56 ( ( isPath_a @ C2 @ U2 @ P3 @ V4 )
% 2.72/1.56 & ! [P6: list_P559422087at_nat] :
% 2.72/1.56 ( ( isPath_a @ C2 @ U2 @ P6 @ V4 )
% 2.72/1.56 => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P3 ) @ ( size_s1990949619at_nat @ P6 ) ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_119_Graph_OisShortestPath__def,axiom,
% 2.72/1.56 ( isShortestPath_b
% 2.72/1.56 = ( ^ [C2: product_prod_nat_nat > b,U2: nat,P3: list_P559422087at_nat,V4: nat] :
% 2.72/1.56 ( ( isPath_b @ C2 @ U2 @ P3 @ V4 )
% 2.72/1.56 & ! [P6: list_P559422087at_nat] :
% 2.72/1.56 ( ( isPath_b @ C2 @ U2 @ P6 @ V4 )
% 2.72/1.56 => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P3 ) @ ( size_s1990949619at_nat @ P6 ) ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_120_zero__reorient,axiom,
% 2.72/1.56 ! [X3: nat] :
% 2.72/1.56 ( ( zero_zero_nat = X3 )
% 2.72/1.56 = ( X3 = zero_zero_nat ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_121_nat__neq__iff,axiom,
% 2.72/1.56 ! [M: nat,N: nat] :
% 2.72/1.56 ( ( M != N )
% 2.72/1.56 = ( ( ord_less_nat @ M @ N )
% 2.72/1.56 | ( ord_less_nat @ N @ M ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_122_less__not__refl,axiom,
% 2.72/1.56 ! [N: nat] :
% 2.72/1.56 ~ ( ord_less_nat @ N @ N ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_123_less__not__refl2,axiom,
% 2.72/1.56 ! [N: nat,M: nat] :
% 2.72/1.56 ( ( ord_less_nat @ N @ M )
% 2.72/1.56 => ( M != N ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_124_less__not__refl3,axiom,
% 2.72/1.56 ! [S: nat,T: nat] :
% 2.72/1.56 ( ( ord_less_nat @ S @ T )
% 2.72/1.56 => ( S != T ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_125_less__irrefl__nat,axiom,
% 2.72/1.56 ! [N: nat] :
% 2.72/1.56 ~ ( ord_less_nat @ N @ N ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_126_nat__less__induct,axiom,
% 2.72/1.56 ! [P2: nat > $o,N: nat] :
% 2.72/1.56 ( ! [N3: nat] :
% 2.72/1.56 ( ! [M3: nat] :
% 2.72/1.56 ( ( ord_less_nat @ M3 @ N3 )
% 2.72/1.56 => ( P2 @ M3 ) )
% 2.72/1.56 => ( P2 @ N3 ) )
% 2.72/1.56 => ( P2 @ N ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_127_infinite__descent,axiom,
% 2.72/1.56 ! [P2: nat > $o,N: nat] :
% 2.72/1.56 ( ! [N3: nat] :
% 2.72/1.56 ( ~ ( P2 @ N3 )
% 2.72/1.56 => ? [M3: nat] :
% 2.72/1.56 ( ( ord_less_nat @ M3 @ N3 )
% 2.72/1.56 & ~ ( P2 @ M3 ) ) )
% 2.72/1.56 => ( P2 @ N ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_128_linorder__neqE__nat,axiom,
% 2.72/1.56 ! [X3: nat,Y3: nat] :
% 2.72/1.56 ( ( X3 != Y3 )
% 2.72/1.56 => ( ~ ( ord_less_nat @ X3 @ Y3 )
% 2.72/1.56 => ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_129_size__neq__size__imp__neq,axiom,
% 2.72/1.56 ! [X3: list_P559422087at_nat,Y3: list_P559422087at_nat] :
% 2.72/1.56 ( ( ( size_s1990949619at_nat @ X3 )
% 2.72/1.56 != ( size_s1990949619at_nat @ Y3 ) )
% 2.72/1.56 => ( X3 != Y3 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_130_Graph_OisPath_Ocong,axiom,
% 2.72/1.56 isPath_b = isPath_b ).
% 2.72/1.56
% 2.72/1.56 thf(fact_131_Graph_OisPath_Ocong,axiom,
% 2.72/1.56 isPath_a = isPath_a ).
% 2.72/1.56
% 2.72/1.56 thf(fact_132_Graph_OV_Ocong,axiom,
% 2.72/1.56 v_a = v_a ).
% 2.72/1.56
% 2.72/1.56 thf(fact_133_Graph_OV_Ocong,axiom,
% 2.72/1.56 v_b = v_b ).
% 2.72/1.56
% 2.72/1.56 thf(fact_134_Graph_Omin__dist_Ocong,axiom,
% 2.72/1.56 min_dist_a = min_dist_a ).
% 2.72/1.56
% 2.72/1.56 thf(fact_135_Graph_Omin__dist_Ocong,axiom,
% 2.72/1.56 min_dist_b = min_dist_b ).
% 2.72/1.56
% 2.72/1.56 thf(fact_136_Graph_Oconnected_Ocong,axiom,
% 2.72/1.56 connected_b = connected_b ).
% 2.72/1.56
% 2.72/1.56 thf(fact_137_Graph_Oconnected_Ocong,axiom,
% 2.72/1.56 connected_a = connected_a ).
% 2.72/1.56
% 2.72/1.56 thf(fact_138_Graph_Oconnected__refl,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,V: nat] : ( connected_b @ C @ V @ V ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_139_Graph_Oconnected__refl,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,V: nat] : ( connected_a @ C @ V @ V ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_140_Graph_Odist_Ocong,axiom,
% 2.72/1.56 dist_b = dist_b ).
% 2.72/1.56
% 2.72/1.56 thf(fact_141_Graph_Odist_Ocong,axiom,
% 2.72/1.56 dist_a = dist_a ).
% 2.72/1.56
% 2.72/1.56 thf(fact_142_Graph_OisShortestPath_Ocong,axiom,
% 2.72/1.56 isShortestPath_a = isShortestPath_a ).
% 2.72/1.56
% 2.72/1.56 thf(fact_143_Graph_OisShortestPath_Ocong,axiom,
% 2.72/1.56 isShortestPath_b = isShortestPath_b ).
% 2.72/1.56
% 2.72/1.56 thf(fact_144_Finite__Graph_Omin__dist__less__V,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,S: nat,T: nat] :
% 2.72/1.56 ( ( finite_Graph_b @ C )
% 2.72/1.56 => ( ( member_nat @ S @ ( v_b @ C ) )
% 2.72/1.56 => ( ( connected_b @ C @ S @ T )
% 2.72/1.56 => ( ord_less_nat @ ( min_dist_b @ C @ S @ T ) @ ( finite_card_nat @ ( v_b @ C ) ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_145_Finite__Graph_Omin__dist__less__V,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,S: nat,T: nat] :
% 2.72/1.56 ( ( finite_Graph_a @ C )
% 2.72/1.56 => ( ( member_nat @ S @ ( v_a @ C ) )
% 2.72/1.56 => ( ( connected_a @ C @ S @ T )
% 2.72/1.56 => ( ord_less_nat @ ( min_dist_a @ C @ S @ T ) @ ( finite_card_nat @ ( v_a @ C ) ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_146_Finite__Graph_OisShortestPath__length__less__V,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,S: nat,P5: list_P559422087at_nat,T: nat] :
% 2.72/1.56 ( ( finite_Graph_a @ C )
% 2.72/1.56 => ( ( member_nat @ S @ ( v_a @ C ) )
% 2.72/1.56 => ( ( isShortestPath_a @ C @ S @ P5 @ T )
% 2.72/1.56 => ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_a @ C ) ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_147_Finite__Graph_OisShortestPath__length__less__V,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,S: nat,P5: list_P559422087at_nat,T: nat] :
% 2.72/1.56 ( ( finite_Graph_b @ C )
% 2.72/1.56 => ( ( member_nat @ S @ ( v_b @ C ) )
% 2.72/1.56 => ( ( isShortestPath_b @ C @ S @ P5 @ T )
% 2.72/1.56 => ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_b @ C ) ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_148_gr__zeroI,axiom,
% 2.72/1.56 ! [N: nat] :
% 2.72/1.56 ( ( N != zero_zero_nat )
% 2.72/1.56 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_149_not__less__zero,axiom,
% 2.72/1.56 ! [N: nat] :
% 2.72/1.56 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_150_gr__implies__not__zero,axiom,
% 2.72/1.56 ! [M: nat,N: nat] :
% 2.72/1.56 ( ( ord_less_nat @ M @ N )
% 2.72/1.56 => ( N != zero_zero_nat ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_151_zero__less__iff__neq__zero,axiom,
% 2.72/1.56 ! [N: nat] :
% 2.72/1.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 2.72/1.56 = ( N != zero_zero_nat ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_152_gr0I,axiom,
% 2.72/1.56 ! [N: nat] :
% 2.72/1.56 ( ( N != zero_zero_nat )
% 2.72/1.56 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_153_not__gr0,axiom,
% 2.72/1.56 ! [N: nat] :
% 2.72/1.56 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 2.72/1.56 = ( N = zero_zero_nat ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_154_not__less0,axiom,
% 2.72/1.56 ! [N: nat] :
% 2.72/1.56 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_155_less__zeroE,axiom,
% 2.72/1.56 ! [N: nat] :
% 2.72/1.56 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_156_gr__implies__not0,axiom,
% 2.72/1.56 ! [M: nat,N: nat] :
% 2.72/1.56 ( ( ord_less_nat @ M @ N )
% 2.72/1.56 => ( N != zero_zero_nat ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_157_infinite__descent0,axiom,
% 2.72/1.56 ! [P2: nat > $o,N: nat] :
% 2.72/1.56 ( ( P2 @ zero_zero_nat )
% 2.72/1.56 => ( ! [N3: nat] :
% 2.72/1.56 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 2.72/1.56 => ( ~ ( P2 @ N3 )
% 2.72/1.56 => ? [M3: nat] :
% 2.72/1.56 ( ( ord_less_nat @ M3 @ N3 )
% 2.72/1.56 & ~ ( P2 @ M3 ) ) ) )
% 2.72/1.56 => ( P2 @ N ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_158_bot__nat__0_Oextremum__strict,axiom,
% 2.72/1.56 ! [A: nat] :
% 2.72/1.56 ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_159_Graph_Omin__dist__z,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,V: nat] :
% 2.72/1.56 ( ( min_dist_a @ C @ V @ V )
% 2.72/1.56 = zero_zero_nat ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_160_Graph_Omin__dist__z,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,V: nat] :
% 2.72/1.56 ( ( min_dist_b @ C @ V @ V )
% 2.72/1.56 = zero_zero_nat ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_161_Graph_Odist__z,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,V: nat] : ( dist_b @ C @ V @ zero_zero_nat @ V ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_162_Graph_Odist__z,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,V: nat] : ( dist_a @ C @ V @ zero_zero_nat @ V ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_163_Graph_Odist__z__iff,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,V: nat,V3: nat] :
% 2.72/1.56 ( ( dist_b @ C @ V @ zero_zero_nat @ V3 )
% 2.72/1.56 = ( V3 = V ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_164_Graph_Odist__z__iff,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,V: nat,V3: nat] :
% 2.72/1.56 ( ( dist_a @ C @ V @ zero_zero_nat @ V3 )
% 2.72/1.56 = ( V3 = V ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_165_Graph_Oconnected__def,axiom,
% 2.72/1.56 ( connected_b
% 2.72/1.56 = ( ^ [C2: product_prod_nat_nat > b,U2: nat,V4: nat] :
% 2.72/1.56 ? [P3: list_P559422087at_nat] : ( isPath_b @ C2 @ U2 @ P3 @ V4 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_166_Graph_Oconnected__def,axiom,
% 2.72/1.56 ( connected_a
% 2.72/1.56 = ( ^ [C2: product_prod_nat_nat > a,U2: nat,V4: nat] :
% 2.72/1.56 ? [P3: list_P559422087at_nat] : ( isPath_a @ C2 @ U2 @ P3 @ V4 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_167_Graph_Oconnected__inV__iff,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,U: nat,V: nat] :
% 2.72/1.56 ( ( connected_b @ C @ U @ V )
% 2.72/1.56 => ( ( member_nat @ V @ ( v_b @ C ) )
% 2.72/1.56 = ( member_nat @ U @ ( v_b @ C ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_168_Graph_Oconnected__inV__iff,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,U: nat,V: nat] :
% 2.72/1.56 ( ( connected_a @ C @ U @ V )
% 2.72/1.56 => ( ( member_nat @ V @ ( v_a @ C ) )
% 2.72/1.56 = ( member_nat @ U @ ( v_a @ C ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_169_Graph_OshortestPath__is__path,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( isShortestPath_a @ C @ U @ P5 @ V )
% 2.72/1.56 => ( isPath_a @ C @ U @ P5 @ V ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_170_Graph_OshortestPath__is__path,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( isShortestPath_b @ C @ U @ P5 @ V )
% 2.72/1.56 => ( isPath_b @ C @ U @ P5 @ V ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_171_Graph_Oconnected__distI,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,V: nat,D: nat,V3: nat] :
% 2.72/1.56 ( ( dist_b @ C @ V @ D @ V3 )
% 2.72/1.56 => ( connected_b @ C @ V @ V3 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_172_Graph_Oconnected__distI,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,V: nat,D: nat,V3: nat] :
% 2.72/1.56 ( ( dist_a @ C @ V @ D @ V3 )
% 2.72/1.56 => ( connected_a @ C @ V @ V3 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_173_Graph_Oconnected__by__dist,axiom,
% 2.72/1.56 ( connected_b
% 2.72/1.56 = ( ^ [C2: product_prod_nat_nat > b,V4: nat,V5: nat] :
% 2.72/1.56 ? [D3: nat] : ( dist_b @ C2 @ V4 @ D3 @ V5 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_174_Graph_Oconnected__by__dist,axiom,
% 2.72/1.56 ( connected_a
% 2.72/1.56 = ( ^ [C2: product_prod_nat_nat > a,V4: nat,V5: nat] :
% 2.72/1.56 ? [D3: nat] : ( dist_a @ C2 @ V4 @ D3 @ V5 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_175_Graph_Oobtain__shortest__path,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,U: nat,V: nat] :
% 2.72/1.56 ( ( connected_a @ C @ U @ V )
% 2.72/1.56 => ~ ! [P4: list_P559422087at_nat] :
% 2.72/1.56 ~ ( isShortestPath_a @ C @ U @ P4 @ V ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_176_Graph_Oobtain__shortest__path,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,U: nat,V: nat] :
% 2.72/1.56 ( ( connected_b @ C @ U @ V )
% 2.72/1.56 => ~ ! [P4: list_P559422087at_nat] :
% 2.72/1.56 ~ ( isShortestPath_b @ C @ U @ P4 @ V ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_177_Graph_Omin__dist__z__iff,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,V: nat,V3: nat] :
% 2.72/1.56 ( ( connected_b @ C @ V @ V3 )
% 2.72/1.56 => ( ( ( min_dist_b @ C @ V @ V3 )
% 2.72/1.56 = zero_zero_nat )
% 2.72/1.56 = ( V3 = V ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_178_Graph_Omin__dist__z__iff,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,V: nat,V3: nat] :
% 2.72/1.56 ( ( connected_a @ C @ V @ V3 )
% 2.72/1.56 => ( ( ( min_dist_a @ C @ V @ V3 )
% 2.72/1.56 = zero_zero_nat )
% 2.72/1.56 = ( V3 = V ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_179_Graph_Omin__dist__less,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,Src: nat,V: nat,D: nat,D2: nat] :
% 2.72/1.56 ( ( connected_b @ C @ Src @ V )
% 2.72/1.56 => ( ( ( min_dist_b @ C @ Src @ V )
% 2.72/1.56 = D )
% 2.72/1.56 => ( ( ord_less_nat @ D2 @ D )
% 2.72/1.56 => ? [V2: nat] :
% 2.72/1.56 ( ( connected_b @ C @ Src @ V2 )
% 2.72/1.56 & ( ( min_dist_b @ C @ Src @ V2 )
% 2.72/1.56 = D2 ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_180_Graph_Omin__dist__less,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,Src: nat,V: nat,D: nat,D2: nat] :
% 2.72/1.56 ( ( connected_a @ C @ Src @ V )
% 2.72/1.56 => ( ( ( min_dist_a @ C @ Src @ V )
% 2.72/1.56 = D )
% 2.72/1.56 => ( ( ord_less_nat @ D2 @ D )
% 2.72/1.56 => ? [V2: nat] :
% 2.72/1.56 ( ( connected_a @ C @ Src @ V2 )
% 2.72/1.56 & ( ( min_dist_a @ C @ Src @ V2 )
% 2.72/1.56 = D2 ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_181_Graph_Omin__dist__is__dist,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,V: nat,V3: nat] :
% 2.72/1.56 ( ( connected_b @ C @ V @ V3 )
% 2.72/1.56 => ( dist_b @ C @ V @ ( min_dist_b @ C @ V @ V3 ) @ V3 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_182_Graph_Omin__dist__is__dist,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,V: nat,V3: nat] :
% 2.72/1.56 ( ( connected_a @ C @ V @ V3 )
% 2.72/1.56 => ( dist_a @ C @ V @ ( min_dist_a @ C @ V @ V3 ) @ V3 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_183_Graph_OisPath__distD,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( isPath_b @ C @ U @ P5 @ V )
% 2.72/1.56 => ( dist_b @ C @ U @ ( size_s1990949619at_nat @ P5 ) @ V ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_184_Graph_OisPath__distD,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( isPath_a @ C @ U @ P5 @ V )
% 2.72/1.56 => ( dist_a @ C @ U @ ( size_s1990949619at_nat @ P5 ) @ V ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_185_Graph_Odist__def,axiom,
% 2.72/1.56 ( dist_b
% 2.72/1.56 = ( ^ [C2: product_prod_nat_nat > b,V4: nat,D3: nat,V5: nat] :
% 2.72/1.56 ? [P3: list_P559422087at_nat] :
% 2.72/1.56 ( ( isPath_b @ C2 @ V4 @ P3 @ V5 )
% 2.72/1.56 & ( ( size_s1990949619at_nat @ P3 )
% 2.72/1.56 = D3 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_186_Graph_Odist__def,axiom,
% 2.72/1.56 ( dist_a
% 2.72/1.56 = ( ^ [C2: product_prod_nat_nat > a,V4: nat,D3: nat,V5: nat] :
% 2.72/1.56 ? [P3: list_P559422087at_nat] :
% 2.72/1.56 ( ( isPath_a @ C2 @ V4 @ P3 @ V5 )
% 2.72/1.56 & ( ( size_s1990949619at_nat @ P3 )
% 2.72/1.56 = D3 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_187_card_Oinfinite,axiom,
% 2.72/1.56 ! [A2: set_li664300135at_nat] :
% 2.72/1.56 ( ~ ( finite1299096496at_nat @ A2 )
% 2.72/1.56 => ( ( finite83082927at_nat @ A2 )
% 2.72/1.56 = zero_zero_nat ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_188_card_Oinfinite,axiom,
% 2.72/1.56 ! [A2: set_se1612935105at_nat] :
% 2.72/1.56 ( ~ ( finite1457549322at_nat @ A2 )
% 2.72/1.56 => ( ( finite1701894793at_nat @ A2 )
% 2.72/1.56 = zero_zero_nat ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_189_card_Oinfinite,axiom,
% 2.72/1.56 ! [A2: set_set_nat] :
% 2.72/1.56 ( ~ ( finite2012248349et_nat @ A2 )
% 2.72/1.56 => ( ( finite_card_set_nat @ A2 )
% 2.72/1.56 = zero_zero_nat ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_190_card_Oinfinite,axiom,
% 2.72/1.56 ! [A2: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ~ ( finite772653738at_nat @ A2 )
% 2.72/1.56 => ( ( finite447719721at_nat @ A2 )
% 2.72/1.56 = zero_zero_nat ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_191_card_Oinfinite,axiom,
% 2.72/1.56 ! [A2: set_nat] :
% 2.72/1.56 ( ~ ( finite_finite_nat @ A2 )
% 2.72/1.56 => ( ( finite_card_nat @ A2 )
% 2.72/1.56 = zero_zero_nat ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_192_card__spEdges__le,axiom,
% 2.72/1.56 ord_less_eq_nat @ ( finite447719721at_nat @ ( edmond475474835dges_a @ c @ s @ t ) ) @ ( finite447719721at_nat @ ( edmond771116670s_uE_a @ c ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_193_g_H_Ocard__spEdges__le,axiom,
% 2.72/1.56 ord_less_eq_nat @ ( finite447719721at_nat @ ( edmond475474836dges_b @ c2 @ s @ t ) ) @ ( finite447719721at_nat @ ( edmond771116671s_uE_b @ c2 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_194_simplePath__length__less__V,axiom,
% 2.72/1.56 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( member_nat @ U @ ( v_a @ c ) )
% 2.72/1.56 => ( ( isSimplePath_a @ c @ U @ P5 @ V )
% 2.72/1.56 => ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_a @ c ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_195_card__ge__0__finite,axiom,
% 2.72/1.56 ! [A2: set_li664300135at_nat] :
% 2.72/1.56 ( ( ord_less_nat @ zero_zero_nat @ ( finite83082927at_nat @ A2 ) )
% 2.72/1.56 => ( finite1299096496at_nat @ A2 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_196_card__ge__0__finite,axiom,
% 2.72/1.56 ! [A2: set_se1612935105at_nat] :
% 2.72/1.56 ( ( ord_less_nat @ zero_zero_nat @ ( finite1701894793at_nat @ A2 ) )
% 2.72/1.56 => ( finite1457549322at_nat @ A2 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_197_card__ge__0__finite,axiom,
% 2.72/1.56 ! [A2: set_set_nat] :
% 2.72/1.56 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A2 ) )
% 2.72/1.56 => ( finite2012248349et_nat @ A2 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_198_card__ge__0__finite,axiom,
% 2.72/1.56 ! [A2: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( ord_less_nat @ zero_zero_nat @ ( finite447719721at_nat @ A2 ) )
% 2.72/1.56 => ( finite772653738at_nat @ A2 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_199_card__ge__0__finite,axiom,
% 2.72/1.56 ! [A2: set_nat] :
% 2.72/1.56 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
% 2.72/1.56 => ( finite_finite_nat @ A2 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_200_min__distI2,axiom,
% 2.72/1.56 ! [V: nat,V3: nat,Q: nat > $o] :
% 2.72/1.56 ( ( connected_a @ c @ V @ V3 )
% 2.72/1.56 => ( ! [D4: nat] :
% 2.72/1.56 ( ( dist_a @ c @ V @ D4 @ V3 )
% 2.72/1.56 => ( ! [D5: nat] :
% 2.72/1.56 ( ( dist_a @ c @ V @ D5 @ V3 )
% 2.72/1.56 => ( ord_less_eq_nat @ D4 @ D5 ) )
% 2.72/1.56 => ( Q @ D4 ) ) )
% 2.72/1.56 => ( Q @ ( min_dist_a @ c @ V @ V3 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_201_g_H_OisShortestPath__alt,axiom,
% 2.72/1.56 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( isShortestPath_b @ c2 @ U @ P5 @ V )
% 2.72/1.56 = ( ( isSimplePath_b @ c2 @ U @ P5 @ V )
% 2.72/1.56 & ( ( size_s1990949619at_nat @ P5 )
% 2.72/1.56 = ( min_dist_b @ c2 @ U @ V ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_202_g_H_Oconnected__inV__iff,axiom,
% 2.72/1.56 ! [U: nat,V: nat] :
% 2.72/1.56 ( ( connected_b @ c2 @ U @ V )
% 2.72/1.56 => ( ( member_nat @ V @ ( v_b @ c2 ) )
% 2.72/1.56 = ( member_nat @ U @ ( v_b @ c2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_203_g_H_OisSPath__pathLE,axiom,
% 2.72/1.56 ! [S: nat,P5: list_P559422087at_nat,T: nat] :
% 2.72/1.56 ( ( isPath_b @ c2 @ S @ P5 @ T )
% 2.72/1.56 => ? [P: list_P559422087at_nat] : ( isSimplePath_b @ c2 @ S @ P @ T ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_204_isSPath__pathLE,axiom,
% 2.72/1.56 ! [S: nat,P5: list_P559422087at_nat,T: nat] :
% 2.72/1.56 ( ( isPath_a @ c @ S @ P5 @ T )
% 2.72/1.56 => ? [P: list_P559422087at_nat] : ( isSimplePath_a @ c @ S @ P @ T ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_205_connected__by__dist,axiom,
% 2.72/1.56 ! [V: nat,V3: nat] :
% 2.72/1.56 ( ( connected_a @ c @ V @ V3 )
% 2.72/1.56 = ( ? [D3: nat] : ( dist_a @ c @ V @ D3 @ V3 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_206_shortestPath__is__simple,axiom,
% 2.72/1.56 ! [S: nat,P5: list_P559422087at_nat,T: nat] :
% 2.72/1.56 ( ( isShortestPath_a @ c @ S @ P5 @ T )
% 2.72/1.56 => ( isSimplePath_a @ c @ S @ P5 @ T ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_207_g_H_OshortestPath__is__simple,axiom,
% 2.72/1.56 ! [S: nat,P5: list_P559422087at_nat,T: nat] :
% 2.72/1.56 ( ( isShortestPath_b @ c2 @ S @ P5 @ T )
% 2.72/1.56 => ( isSimplePath_b @ c2 @ S @ P5 @ T ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_208_min__dist__minD,axiom,
% 2.72/1.56 ! [V: nat,D: nat,V3: nat] :
% 2.72/1.56 ( ( dist_a @ c @ V @ D @ V3 )
% 2.72/1.56 => ( ord_less_eq_nat @ ( min_dist_a @ c @ V @ V3 ) @ D ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_209_min__distI__eq,axiom,
% 2.72/1.56 ! [V: nat,D: nat,V3: nat] :
% 2.72/1.56 ( ( dist_a @ c @ V @ D @ V3 )
% 2.72/1.56 => ( ! [D6: nat] :
% 2.72/1.56 ( ( dist_a @ c @ V @ D6 @ V3 )
% 2.72/1.56 => ( ord_less_eq_nat @ D @ D6 ) )
% 2.72/1.56 => ( ( min_dist_a @ c @ V @ V3 )
% 2.72/1.56 = D ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_210_isPath__distD,axiom,
% 2.72/1.56 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( isPath_a @ c @ U @ P5 @ V )
% 2.72/1.56 => ( dist_a @ c @ U @ ( size_s1990949619at_nat @ P5 ) @ V ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_211_dist__def,axiom,
% 2.72/1.56 ! [V: nat,D: nat,V3: nat] :
% 2.72/1.56 ( ( dist_a @ c @ V @ D @ V3 )
% 2.72/1.56 = ( ? [P3: list_P559422087at_nat] :
% 2.72/1.56 ( ( isPath_a @ c @ V @ P3 @ V3 )
% 2.72/1.56 & ( ( size_s1990949619at_nat @ P3 )
% 2.72/1.56 = D ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_212_min__dist__is__dist,axiom,
% 2.72/1.56 ! [V: nat,V3: nat] :
% 2.72/1.56 ( ( connected_a @ c @ V @ V3 )
% 2.72/1.56 => ( dist_a @ c @ V @ ( min_dist_a @ c @ V @ V3 ) @ V3 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_213_g_H_OsimplePath__length__less__V,axiom,
% 2.72/1.56 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( member_nat @ U @ ( v_b @ c2 ) )
% 2.72/1.56 => ( ( isSimplePath_b @ c2 @ U @ P5 @ V )
% 2.72/1.56 => ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_b @ c2 ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_214_g_H_Omin__dist__less__V,axiom,
% 2.72/1.56 ! [S: nat,T: nat] :
% 2.72/1.56 ( ( member_nat @ S @ ( v_b @ c2 ) )
% 2.72/1.56 => ( ( connected_b @ c2 @ S @ T )
% 2.72/1.56 => ( ord_less_nat @ ( min_dist_b @ c2 @ S @ T ) @ ( finite_card_nat @ ( v_b @ c2 ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_215_g_H_OisShortestPath__length__less__V,axiom,
% 2.72/1.56 ! [S: nat,P5: list_P559422087at_nat,T: nat] :
% 2.72/1.56 ( ( member_nat @ S @ ( v_b @ c2 ) )
% 2.72/1.56 => ( ( isShortestPath_b @ c2 @ S @ P5 @ T )
% 2.72/1.56 => ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_b @ c2 ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_216_isShortestPath__alt,axiom,
% 2.72/1.56 ! [U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( isShortestPath_a @ c @ U @ P5 @ V )
% 2.72/1.56 = ( ( isSimplePath_a @ c @ U @ P5 @ V )
% 2.72/1.56 & ( ( size_s1990949619at_nat @ P5 )
% 2.72/1.56 = ( min_dist_a @ c @ U @ V ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_217_g_H_Ofinite__V,axiom,
% 2.72/1.56 finite_finite_nat @ ( v_b @ c2 ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_218_finite__V,axiom,
% 2.72/1.56 finite_finite_nat @ ( v_a @ c ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_219_dist__z__iff,axiom,
% 2.72/1.56 ! [V: nat,V3: nat] :
% 2.72/1.56 ( ( dist_a @ c @ V @ zero_zero_nat @ V3 )
% 2.72/1.56 = ( V3 = V ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_220_dist__z,axiom,
% 2.72/1.56 ! [V: nat] : ( dist_a @ c @ V @ zero_zero_nat @ V ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_221_connected__distI,axiom,
% 2.72/1.56 ! [V: nat,D: nat,V3: nat] :
% 2.72/1.56 ( ( dist_a @ c @ V @ D @ V3 )
% 2.72/1.56 => ( connected_a @ c @ V @ V3 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_222_Veq,axiom,
% 2.72/1.56 ( ( v_b @ c2 )
% 2.72/1.56 = ( v_a @ c ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_223_g_H_Oadjacent__nodes__ss__V,axiom,
% 2.72/1.56 ! [U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_b @ c2 @ U ) @ ( v_b @ c2 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_224_adjacent__nodes__ss__V,axiom,
% 2.72/1.56 ! [U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_a @ c @ U ) @ ( v_a @ c ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_225_Graph_OisSimplePath_Ocong,axiom,
% 2.72/1.56 isSimplePath_a = isSimplePath_a ).
% 2.72/1.56
% 2.72/1.56 thf(fact_226_Graph_OisSimplePath_Ocong,axiom,
% 2.72/1.56 isSimplePath_b = isSimplePath_b ).
% 2.72/1.56
% 2.72/1.56 thf(fact_227_finite__subset,axiom,
% 2.72/1.56 ! [A2: set_se1612935105at_nat,B2: set_se1612935105at_nat] :
% 2.72/1.56 ( ( ord_le2096002913at_nat @ A2 @ B2 )
% 2.72/1.56 => ( ( finite1457549322at_nat @ B2 )
% 2.72/1.56 => ( finite1457549322at_nat @ A2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_228_finite__subset,axiom,
% 2.72/1.56 ! [A2: set_set_nat,B2: set_set_nat] :
% 2.72/1.56 ( ( ord_le1613022364et_nat @ A2 @ B2 )
% 2.72/1.56 => ( ( finite2012248349et_nat @ B2 )
% 2.72/1.56 => ( finite2012248349et_nat @ A2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_229_finite__subset,axiom,
% 2.72/1.56 ! [A2: set_nat,B2: set_nat] :
% 2.72/1.56 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.72/1.56 => ( ( finite_finite_nat @ B2 )
% 2.72/1.56 => ( finite_finite_nat @ A2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_230_finite__subset,axiom,
% 2.72/1.56 ! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( ord_le841296385at_nat @ A2 @ B2 )
% 2.72/1.56 => ( ( finite772653738at_nat @ B2 )
% 2.72/1.56 => ( finite772653738at_nat @ A2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_231_infinite__super,axiom,
% 2.72/1.56 ! [S2: set_nat,T2: set_nat] :
% 2.72/1.56 ( ( ord_less_eq_set_nat @ S2 @ T2 )
% 2.72/1.56 => ( ~ ( finite_finite_nat @ S2 )
% 2.72/1.56 => ~ ( finite_finite_nat @ T2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_232_infinite__super,axiom,
% 2.72/1.56 ! [S2: set_Pr1986765409at_nat,T2: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( ord_le841296385at_nat @ S2 @ T2 )
% 2.72/1.56 => ( ~ ( finite772653738at_nat @ S2 )
% 2.72/1.56 => ~ ( finite772653738at_nat @ T2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_233_rev__finite__subset,axiom,
% 2.72/1.56 ! [B2: set_nat,A2: set_nat] :
% 2.72/1.56 ( ( finite_finite_nat @ B2 )
% 2.72/1.56 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.72/1.56 => ( finite_finite_nat @ A2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_234_rev__finite__subset,axiom,
% 2.72/1.56 ! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( finite772653738at_nat @ B2 )
% 2.72/1.56 => ( ( ord_le841296385at_nat @ A2 @ B2 )
% 2.72/1.56 => ( finite772653738at_nat @ A2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_235_card__subset__eq,axiom,
% 2.72/1.56 ! [B2: set_nat,A2: set_nat] :
% 2.72/1.56 ( ( finite_finite_nat @ B2 )
% 2.72/1.56 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.72/1.56 => ( ( ( finite_card_nat @ A2 )
% 2.72/1.56 = ( finite_card_nat @ B2 ) )
% 2.72/1.56 => ( A2 = B2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_236_card__subset__eq,axiom,
% 2.72/1.56 ! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( finite772653738at_nat @ B2 )
% 2.72/1.56 => ( ( ord_le841296385at_nat @ A2 @ B2 )
% 2.72/1.56 => ( ( ( finite447719721at_nat @ A2 )
% 2.72/1.56 = ( finite447719721at_nat @ B2 ) )
% 2.72/1.56 => ( A2 = B2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_237_infinite__arbitrarily__large,axiom,
% 2.72/1.56 ! [A2: set_nat,N: nat] :
% 2.72/1.56 ( ~ ( finite_finite_nat @ A2 )
% 2.72/1.56 => ? [B3: set_nat] :
% 2.72/1.56 ( ( finite_finite_nat @ B3 )
% 2.72/1.56 & ( ( finite_card_nat @ B3 )
% 2.72/1.56 = N )
% 2.72/1.56 & ( ord_less_eq_set_nat @ B3 @ A2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_238_infinite__arbitrarily__large,axiom,
% 2.72/1.56 ! [A2: set_Pr1986765409at_nat,N: nat] :
% 2.72/1.56 ( ~ ( finite772653738at_nat @ A2 )
% 2.72/1.56 => ? [B3: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( finite772653738at_nat @ B3 )
% 2.72/1.56 & ( ( finite447719721at_nat @ B3 )
% 2.72/1.56 = N )
% 2.72/1.56 & ( ord_le841296385at_nat @ B3 @ A2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_239_Graph_OisSPath__pathLE,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,S: nat,P5: list_P559422087at_nat,T: nat] :
% 2.72/1.56 ( ( isPath_a @ C @ S @ P5 @ T )
% 2.72/1.56 => ? [P: list_P559422087at_nat] : ( isSimplePath_a @ C @ S @ P @ T ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_240_Graph_OisSPath__pathLE,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,S: nat,P5: list_P559422087at_nat,T: nat] :
% 2.72/1.56 ( ( isPath_b @ C @ S @ P5 @ T )
% 2.72/1.56 => ? [P: list_P559422087at_nat] : ( isSimplePath_b @ C @ S @ P @ T ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_241_Graph_OshortestPath__is__simple,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,S: nat,P5: list_P559422087at_nat,T: nat] :
% 2.72/1.56 ( ( isShortestPath_a @ C @ S @ P5 @ T )
% 2.72/1.56 => ( isSimplePath_a @ C @ S @ P5 @ T ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_242_Graph_OshortestPath__is__simple,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,S: nat,P5: list_P559422087at_nat,T: nat] :
% 2.72/1.56 ( ( isShortestPath_b @ C @ S @ P5 @ T )
% 2.72/1.56 => ( isSimplePath_b @ C @ S @ P5 @ T ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_243_finite__maxlen,axiom,
% 2.72/1.56 ! [M4: set_li664300135at_nat] :
% 2.72/1.56 ( ( finite1299096496at_nat @ M4 )
% 2.72/1.56 => ? [N3: nat] :
% 2.72/1.56 ! [X4: list_P559422087at_nat] :
% 2.72/1.56 ( ( member1608759472at_nat @ X4 @ M4 )
% 2.72/1.56 => ( ord_less_nat @ ( size_s1990949619at_nat @ X4 ) @ N3 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_244_card__psubset,axiom,
% 2.72/1.56 ! [B2: set_nat,A2: set_nat] :
% 2.72/1.56 ( ( finite_finite_nat @ B2 )
% 2.72/1.56 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.72/1.56 => ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
% 2.72/1.56 => ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_245_card__psubset,axiom,
% 2.72/1.56 ! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( finite772653738at_nat @ B2 )
% 2.72/1.56 => ( ( ord_le841296385at_nat @ A2 @ B2 )
% 2.72/1.56 => ( ( ord_less_nat @ ( finite447719721at_nat @ A2 ) @ ( finite447719721at_nat @ B2 ) )
% 2.72/1.56 => ( ord_le116442893at_nat @ A2 @ B2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_246_card__mono,axiom,
% 2.72/1.56 ! [B2: set_nat,A2: set_nat] :
% 2.72/1.56 ( ( finite_finite_nat @ B2 )
% 2.72/1.56 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.72/1.56 => ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_247_card__mono,axiom,
% 2.72/1.56 ! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( finite772653738at_nat @ B2 )
% 2.72/1.56 => ( ( ord_le841296385at_nat @ A2 @ B2 )
% 2.72/1.56 => ( ord_less_eq_nat @ ( finite447719721at_nat @ A2 ) @ ( finite447719721at_nat @ B2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_248_card__seteq,axiom,
% 2.72/1.56 ! [B2: set_nat,A2: set_nat] :
% 2.72/1.56 ( ( finite_finite_nat @ B2 )
% 2.72/1.56 => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.72/1.56 => ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
% 2.72/1.56 => ( A2 = B2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_249_card__seteq,axiom,
% 2.72/1.56 ! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( finite772653738at_nat @ B2 )
% 2.72/1.56 => ( ( ord_le841296385at_nat @ A2 @ B2 )
% 2.72/1.56 => ( ( ord_less_eq_nat @ ( finite447719721at_nat @ B2 ) @ ( finite447719721at_nat @ A2 ) )
% 2.72/1.56 => ( A2 = B2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_250_finite__if__finite__subsets__card__bdd,axiom,
% 2.72/1.56 ! [F2: set_nat,C3: nat] :
% 2.72/1.56 ( ! [G: set_nat] :
% 2.72/1.56 ( ( ord_less_eq_set_nat @ G @ F2 )
% 2.72/1.56 => ( ( finite_finite_nat @ G )
% 2.72/1.56 => ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C3 ) ) )
% 2.72/1.56 => ( ( finite_finite_nat @ F2 )
% 2.72/1.56 & ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C3 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_251_finite__if__finite__subsets__card__bdd,axiom,
% 2.72/1.56 ! [F2: set_Pr1986765409at_nat,C3: nat] :
% 2.72/1.56 ( ! [G: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( ord_le841296385at_nat @ G @ F2 )
% 2.72/1.56 => ( ( finite772653738at_nat @ G )
% 2.72/1.56 => ( ord_less_eq_nat @ ( finite447719721at_nat @ G ) @ C3 ) ) )
% 2.72/1.56 => ( ( finite772653738at_nat @ F2 )
% 2.72/1.56 & ( ord_less_eq_nat @ ( finite447719721at_nat @ F2 ) @ C3 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_252_Finite__Graph__def,axiom,
% 2.72/1.56 ( finite_Graph_b
% 2.72/1.56 = ( ^ [C2: product_prod_nat_nat > b] : ( finite_finite_nat @ ( v_b @ C2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_253_Finite__Graph__def,axiom,
% 2.72/1.56 ( finite_Graph_a
% 2.72/1.56 = ( ^ [C2: product_prod_nat_nat > a] : ( finite_finite_nat @ ( v_a @ C2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_254_Finite__Graph_Ointro,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b] :
% 2.72/1.56 ( ( finite_finite_nat @ ( v_b @ C ) )
% 2.72/1.56 => ( finite_Graph_b @ C ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_255_Finite__Graph_Ointro,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a] :
% 2.72/1.56 ( ( finite_finite_nat @ ( v_a @ C ) )
% 2.72/1.56 => ( finite_Graph_a @ C ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_256_Finite__Graph_Ofinite__V,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b] :
% 2.72/1.56 ( ( finite_Graph_b @ C )
% 2.72/1.56 => ( finite_finite_nat @ ( v_b @ C ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_257_Finite__Graph_Ofinite__V,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a] :
% 2.72/1.56 ( ( finite_Graph_a @ C )
% 2.72/1.56 => ( finite_finite_nat @ ( v_a @ C ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_258_finite__psubset__induct,axiom,
% 2.72/1.56 ! [A2: set_Pr1986765409at_nat,P2: set_Pr1986765409at_nat > $o] :
% 2.72/1.56 ( ( finite772653738at_nat @ A2 )
% 2.72/1.56 => ( ! [A3: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( finite772653738at_nat @ A3 )
% 2.72/1.56 => ( ! [B4: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( ord_le116442893at_nat @ B4 @ A3 )
% 2.72/1.56 => ( P2 @ B4 ) )
% 2.72/1.56 => ( P2 @ A3 ) ) )
% 2.72/1.56 => ( P2 @ A2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_259_finite__psubset__induct,axiom,
% 2.72/1.56 ! [A2: set_nat,P2: set_nat > $o] :
% 2.72/1.56 ( ( finite_finite_nat @ A2 )
% 2.72/1.56 => ( ! [A3: set_nat] :
% 2.72/1.56 ( ( finite_finite_nat @ A3 )
% 2.72/1.56 => ( ! [B4: set_nat] :
% 2.72/1.56 ( ( ord_less_set_nat @ B4 @ A3 )
% 2.72/1.56 => ( P2 @ B4 ) )
% 2.72/1.56 => ( P2 @ A3 ) ) )
% 2.72/1.56 => ( P2 @ A2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_260_Graph_OisShortestPath__alt,axiom,
% 2.72/1.56 ( isShortestPath_a
% 2.72/1.56 = ( ^ [C2: product_prod_nat_nat > a,U2: nat,P3: list_P559422087at_nat,V4: nat] :
% 2.72/1.56 ( ( isSimplePath_a @ C2 @ U2 @ P3 @ V4 )
% 2.72/1.56 & ( ( size_s1990949619at_nat @ P3 )
% 2.72/1.56 = ( min_dist_a @ C2 @ U2 @ V4 ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_261_Graph_OisShortestPath__alt,axiom,
% 2.72/1.56 ( isShortestPath_b
% 2.72/1.56 = ( ^ [C2: product_prod_nat_nat > b,U2: nat,P3: list_P559422087at_nat,V4: nat] :
% 2.72/1.56 ( ( isSimplePath_b @ C2 @ U2 @ P3 @ V4 )
% 2.72/1.56 & ( ( size_s1990949619at_nat @ P3 )
% 2.72/1.56 = ( min_dist_b @ C2 @ U2 @ V4 ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_262_ek__analysis_Ocard__spEdges__le,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,S: nat,T: nat] :
% 2.72/1.56 ( ( edmond1517640972ysis_a @ C )
% 2.72/1.56 => ( ord_less_eq_nat @ ( finite447719721at_nat @ ( edmond475474835dges_a @ C @ S @ T ) ) @ ( finite447719721at_nat @ ( edmond771116670s_uE_a @ C ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_263_ek__analysis_Ocard__spEdges__le,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,S: nat,T: nat] :
% 2.72/1.56 ( ( edmond1517640973ysis_b @ C )
% 2.72/1.56 => ( ord_less_eq_nat @ ( finite447719721at_nat @ ( edmond475474836dges_b @ C @ S @ T ) ) @ ( finite447719721at_nat @ ( edmond771116671s_uE_b @ C ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_264_finite__has__minimal2,axiom,
% 2.72/1.56 ! [A2: set_nat,A: nat] :
% 2.72/1.56 ( ( finite_finite_nat @ A2 )
% 2.72/1.56 => ( ( member_nat @ A @ A2 )
% 2.72/1.56 => ? [X2: nat] :
% 2.72/1.56 ( ( member_nat @ X2 @ A2 )
% 2.72/1.56 & ( ord_less_eq_nat @ X2 @ A )
% 2.72/1.56 & ! [Xa: nat] :
% 2.72/1.56 ( ( member_nat @ Xa @ A2 )
% 2.72/1.56 => ( ( ord_less_eq_nat @ Xa @ X2 )
% 2.72/1.56 => ( X2 = Xa ) ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_265_finite__has__minimal2,axiom,
% 2.72/1.56 ! [A2: set_set_nat,A: set_nat] :
% 2.72/1.56 ( ( finite2012248349et_nat @ A2 )
% 2.72/1.56 => ( ( member_set_nat @ A @ A2 )
% 2.72/1.56 => ? [X2: set_nat] :
% 2.72/1.56 ( ( member_set_nat @ X2 @ A2 )
% 2.72/1.56 & ( ord_less_eq_set_nat @ X2 @ A )
% 2.72/1.56 & ! [Xa: set_nat] :
% 2.72/1.56 ( ( member_set_nat @ Xa @ A2 )
% 2.72/1.56 => ( ( ord_less_eq_set_nat @ Xa @ X2 )
% 2.72/1.56 => ( X2 = Xa ) ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_266_finite__has__minimal2,axiom,
% 2.72/1.56 ! [A2: set_se1612935105at_nat,A: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( finite1457549322at_nat @ A2 )
% 2.72/1.56 => ( ( member298845450at_nat @ A @ A2 )
% 2.72/1.56 => ? [X2: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( member298845450at_nat @ X2 @ A2 )
% 2.72/1.56 & ( ord_le841296385at_nat @ X2 @ A )
% 2.72/1.56 & ! [Xa: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( member298845450at_nat @ Xa @ A2 )
% 2.72/1.56 => ( ( ord_le841296385at_nat @ Xa @ X2 )
% 2.72/1.56 => ( X2 = Xa ) ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_267_finite__has__maximal2,axiom,
% 2.72/1.56 ! [A2: set_nat,A: nat] :
% 2.72/1.56 ( ( finite_finite_nat @ A2 )
% 2.72/1.56 => ( ( member_nat @ A @ A2 )
% 2.72/1.56 => ? [X2: nat] :
% 2.72/1.56 ( ( member_nat @ X2 @ A2 )
% 2.72/1.56 & ( ord_less_eq_nat @ A @ X2 )
% 2.72/1.56 & ! [Xa: nat] :
% 2.72/1.56 ( ( member_nat @ Xa @ A2 )
% 2.72/1.56 => ( ( ord_less_eq_nat @ X2 @ Xa )
% 2.72/1.56 => ( X2 = Xa ) ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_268_finite__has__maximal2,axiom,
% 2.72/1.56 ! [A2: set_set_nat,A: set_nat] :
% 2.72/1.56 ( ( finite2012248349et_nat @ A2 )
% 2.72/1.56 => ( ( member_set_nat @ A @ A2 )
% 2.72/1.56 => ? [X2: set_nat] :
% 2.72/1.56 ( ( member_set_nat @ X2 @ A2 )
% 2.72/1.56 & ( ord_less_eq_set_nat @ A @ X2 )
% 2.72/1.56 & ! [Xa: set_nat] :
% 2.72/1.56 ( ( member_set_nat @ Xa @ A2 )
% 2.72/1.56 => ( ( ord_less_eq_set_nat @ X2 @ Xa )
% 2.72/1.56 => ( X2 = Xa ) ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_269_finite__has__maximal2,axiom,
% 2.72/1.56 ! [A2: set_se1612935105at_nat,A: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( finite1457549322at_nat @ A2 )
% 2.72/1.56 => ( ( member298845450at_nat @ A @ A2 )
% 2.72/1.56 => ? [X2: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( member298845450at_nat @ X2 @ A2 )
% 2.72/1.56 & ( ord_le841296385at_nat @ A @ X2 )
% 2.72/1.56 & ! [Xa: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( member298845450at_nat @ Xa @ A2 )
% 2.72/1.56 => ( ( ord_le841296385at_nat @ X2 @ Xa )
% 2.72/1.56 => ( X2 = Xa ) ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_270_Finite__Graph_OsimplePath__length__less__V,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( finite_Graph_a @ C )
% 2.72/1.56 => ( ( member_nat @ U @ ( v_a @ C ) )
% 2.72/1.56 => ( ( isSimplePath_a @ C @ U @ P5 @ V )
% 2.72/1.56 => ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_a @ C ) ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_271_Finite__Graph_OsimplePath__length__less__V,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,U: nat,P5: list_P559422087at_nat,V: nat] :
% 2.72/1.56 ( ( finite_Graph_b @ C )
% 2.72/1.56 => ( ( member_nat @ U @ ( v_b @ C ) )
% 2.72/1.56 => ( ( isSimplePath_b @ C @ U @ P5 @ V )
% 2.72/1.56 => ( ord_less_nat @ ( size_s1990949619at_nat @ P5 ) @ ( finite_card_nat @ ( v_b @ C ) ) ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_272_psubset__card__mono,axiom,
% 2.72/1.56 ! [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( finite772653738at_nat @ B2 )
% 2.72/1.56 => ( ( ord_le116442893at_nat @ A2 @ B2 )
% 2.72/1.56 => ( ord_less_nat @ ( finite447719721at_nat @ A2 ) @ ( finite447719721at_nat @ B2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_273_psubset__card__mono,axiom,
% 2.72/1.56 ! [B2: set_nat,A2: set_nat] :
% 2.72/1.56 ( ( finite_finite_nat @ B2 )
% 2.72/1.56 => ( ( ord_less_set_nat @ A2 @ B2 )
% 2.72/1.56 => ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_274_reachable__ss__V,axiom,
% 2.72/1.56 ! [S: nat] :
% 2.72/1.56 ( ( member_nat @ S @ ( v_a @ c ) )
% 2.72/1.56 => ( ord_less_eq_set_nat @ ( reachableNodes_a @ c @ S ) @ ( v_a @ c ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_275_g_H_Oreachable__ss__V,axiom,
% 2.72/1.56 ! [S: nat] :
% 2.72/1.56 ( ( member_nat @ S @ ( v_b @ c2 ) )
% 2.72/1.56 => ( ord_less_eq_set_nat @ ( reachableNodes_b @ c2 @ S ) @ ( v_b @ c2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_276_spEdges__ss__E,axiom,
% 2.72/1.56 ord_le841296385at_nat @ ( edmond475474835dges_a @ c @ s @ t ) @ ( e_a @ c ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_277_E__ss__uE,axiom,
% 2.72/1.56 ord_le841296385at_nat @ ( e_a @ c ) @ ( edmond771116670s_uE_a @ c ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_278_Finite__Graph__EI,axiom,
% 2.72/1.56 ( ( finite772653738at_nat @ ( e_a @ c ) )
% 2.72/1.56 => ( finite_Graph_a @ c ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_279_Efin__imp__Vfin,axiom,
% 2.72/1.56 ( ( finite772653738at_nat @ ( e_a @ c ) )
% 2.72/1.56 => ( finite_finite_nat @ ( v_a @ c ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_280_finite__E,axiom,
% 2.72/1.56 finite772653738at_nat @ ( e_a @ c ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_281_g_H_Oadjacent__nodes__finite,axiom,
% 2.72/1.56 ! [U: nat] : ( finite_finite_nat @ ( adjacent_nodes_b @ c2 @ U ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_282_adjacent__nodes__finite,axiom,
% 2.72/1.56 ! [U: nat] : ( finite_finite_nat @ ( adjacent_nodes_a @ c @ U ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_283_Vfin__imp__Efin,axiom,
% 2.72/1.56 ( ( finite_finite_nat @ ( v_a @ c ) )
% 2.72/1.56 => ( finite772653738at_nat @ ( e_a @ c ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_284_Graph_OreachableNodes_Ocong,axiom,
% 2.72/1.56 reachableNodes_a = reachableNodes_a ).
% 2.72/1.56
% 2.72/1.56 thf(fact_285_Graph_OreachableNodes_Ocong,axiom,
% 2.72/1.56 reachableNodes_b = reachableNodes_b ).
% 2.72/1.56
% 2.72/1.56 thf(fact_286_Graph_Oadjacent__nodes_Ocong,axiom,
% 2.72/1.56 adjacent_nodes_b = adjacent_nodes_b ).
% 2.72/1.56
% 2.72/1.56 thf(fact_287_Graph_Oadjacent__nodes_Ocong,axiom,
% 2.72/1.56 adjacent_nodes_a = adjacent_nodes_a ).
% 2.72/1.56
% 2.72/1.56 thf(fact_288_Graph_OE_Ocong,axiom,
% 2.72/1.56 e_a = e_a ).
% 2.72/1.56
% 2.72/1.56 thf(fact_289_Graph_OE_Ocong,axiom,
% 2.72/1.56 e_b = e_b ).
% 2.72/1.56
% 2.72/1.56 thf(fact_290_Graph_Oreachable__ss__V,axiom,
% 2.72/1.56 ! [S: nat,C: product_prod_nat_nat > a] :
% 2.72/1.56 ( ( member_nat @ S @ ( v_a @ C ) )
% 2.72/1.56 => ( ord_less_eq_set_nat @ ( reachableNodes_a @ C @ S ) @ ( v_a @ C ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_291_Graph_Oreachable__ss__V,axiom,
% 2.72/1.56 ! [S: nat,C: product_prod_nat_nat > b] :
% 2.72/1.56 ( ( member_nat @ S @ ( v_b @ C ) )
% 2.72/1.56 => ( ord_less_eq_set_nat @ ( reachableNodes_b @ C @ S ) @ ( v_b @ C ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_292_Graph_OEfin__imp__Vfin,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a] :
% 2.72/1.56 ( ( finite772653738at_nat @ ( e_a @ C ) )
% 2.72/1.56 => ( finite_finite_nat @ ( v_a @ C ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_293_Graph_OEfin__imp__Vfin,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b] :
% 2.72/1.56 ( ( finite772653738at_nat @ ( e_b @ C ) )
% 2.72/1.56 => ( finite_finite_nat @ ( v_b @ C ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_294_Graph_OVfin__imp__Efin,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a] :
% 2.72/1.56 ( ( finite_finite_nat @ ( v_a @ C ) )
% 2.72/1.56 => ( finite772653738at_nat @ ( e_a @ C ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_295_Graph_OVfin__imp__Efin,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b] :
% 2.72/1.56 ( ( finite_finite_nat @ ( v_b @ C ) )
% 2.72/1.56 => ( finite772653738at_nat @ ( e_b @ C ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_296_Graph_OFinite__Graph__EI,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b] :
% 2.72/1.56 ( ( finite772653738at_nat @ ( e_b @ C ) )
% 2.72/1.56 => ( finite_Graph_b @ C ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_297_Graph_OFinite__Graph__EI,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a] :
% 2.72/1.56 ( ( finite772653738at_nat @ ( e_a @ C ) )
% 2.72/1.56 => ( finite_Graph_a @ C ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_298_Finite__Graph_Ofinite__E,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b] :
% 2.72/1.56 ( ( finite_Graph_b @ C )
% 2.72/1.56 => ( finite772653738at_nat @ ( e_b @ C ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_299_Finite__Graph_Ofinite__E,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a] :
% 2.72/1.56 ( ( finite_Graph_a @ C )
% 2.72/1.56 => ( finite772653738at_nat @ ( e_a @ C ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_300_ek__analysis_OE__ss__uE,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b] :
% 2.72/1.56 ( ( edmond1517640973ysis_b @ C )
% 2.72/1.56 => ( ord_le841296385at_nat @ ( e_b @ C ) @ ( edmond771116671s_uE_b @ C ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_301_ek__analysis_OE__ss__uE,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a] :
% 2.72/1.56 ( ( edmond1517640972ysis_a @ C )
% 2.72/1.56 => ( ord_le841296385at_nat @ ( e_a @ C ) @ ( edmond771116670s_uE_a @ C ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_302_ek__analysis_OspEdges__ss__E,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,S: nat,T: nat] :
% 2.72/1.56 ( ( edmond1517640972ysis_a @ C )
% 2.72/1.56 => ( ord_le841296385at_nat @ ( edmond475474835dges_a @ C @ S @ T ) @ ( e_a @ C ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_303_ek__analysis_OspEdges__ss__E,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,S: nat,T: nat] :
% 2.72/1.56 ( ( edmond1517640973ysis_b @ C )
% 2.72/1.56 => ( ord_le841296385at_nat @ ( edmond475474836dges_b @ C @ S @ T ) @ ( e_b @ C ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_304_Graph_Oadjacent__nodes__ss__V,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_b @ C @ U ) @ ( v_b @ C ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_305_Graph_Oadjacent__nodes__ss__V,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,U: nat] : ( ord_less_eq_set_nat @ ( adjacent_nodes_a @ C @ U ) @ ( v_a @ C ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_306_Finite__Graph_Oadjacent__nodes__finite,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,U: nat] :
% 2.72/1.56 ( ( finite_Graph_b @ C )
% 2.72/1.56 => ( finite_finite_nat @ ( adjacent_nodes_b @ C @ U ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_307_Finite__Graph_Oadjacent__nodes__finite,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,U: nat] :
% 2.72/1.56 ( ( finite_Graph_a @ C )
% 2.72/1.56 => ( finite_finite_nat @ ( adjacent_nodes_a @ C @ U ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_308_incoming_H__edges,axiom,
% 2.72/1.56 ! [U3: set_nat] : ( ord_le841296385at_nat @ ( incoming_a @ c @ U3 ) @ ( e_a @ c ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_309_outgoing_H__edges,axiom,
% 2.72/1.56 ! [U3: set_nat] : ( ord_le841296385at_nat @ ( outgoing_a @ c @ U3 ) @ ( e_a @ c ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_310_g_H_OE__ss__uE,axiom,
% 2.72/1.56 ord_le841296385at_nat @ ( e_b @ c2 ) @ ( edmond771116671s_uE_b @ c2 ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_311_g_H_OFinite__Graph__EI,axiom,
% 2.72/1.56 ( ( finite772653738at_nat @ ( e_b @ c2 ) )
% 2.72/1.56 => ( finite_Graph_b @ c2 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_312_g_H_OEfin__imp__Vfin,axiom,
% 2.72/1.56 ( ( finite772653738at_nat @ ( e_b @ c2 ) )
% 2.72/1.56 => ( finite_finite_nat @ ( v_b @ c2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_313_g_H_OspEdges__ss__E,axiom,
% 2.72/1.56 ord_le841296385at_nat @ ( edmond475474836dges_b @ c2 @ s @ t ) @ ( e_b @ c2 ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_314_g_H_Ofinite__E,axiom,
% 2.72/1.56 finite772653738at_nat @ ( e_b @ c2 ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_315_g_H_OVfin__imp__Efin,axiom,
% 2.72/1.56 ( ( finite_finite_nat @ ( v_b @ c2 ) )
% 2.72/1.56 => ( finite772653738at_nat @ ( e_b @ c2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_316_finite__outgoing_H,axiom,
% 2.72/1.56 ! [U3: set_nat] :
% 2.72/1.56 ( ( finite_finite_nat @ ( v_a @ c ) )
% 2.72/1.56 => ( finite772653738at_nat @ ( outgoing_a @ c @ U3 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_317_finite__incoming_H,axiom,
% 2.72/1.56 ! [U3: set_nat] :
% 2.72/1.56 ( ( finite_finite_nat @ ( v_a @ c ) )
% 2.72/1.56 => ( finite772653738at_nat @ ( incoming_a @ c @ U3 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_318_Graph_Oincoming_H_Ocong,axiom,
% 2.72/1.56 incoming_a = incoming_a ).
% 2.72/1.56
% 2.72/1.56 thf(fact_319_Graph_Oincoming_H_Ocong,axiom,
% 2.72/1.56 incoming_b = incoming_b ).
% 2.72/1.56
% 2.72/1.56 thf(fact_320_Graph_Ooutgoing_H_Ocong,axiom,
% 2.72/1.56 outgoing_a = outgoing_a ).
% 2.72/1.56
% 2.72/1.56 thf(fact_321_Graph_Ooutgoing_H_Ocong,axiom,
% 2.72/1.56 outgoing_b = outgoing_b ).
% 2.72/1.56
% 2.72/1.56 thf(fact_322_Graph_Ooutgoing_H__edges,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,U3: set_nat] : ( ord_le841296385at_nat @ ( outgoing_a @ C @ U3 ) @ ( e_a @ C ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_323_Graph_Ooutgoing_H__edges,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,U3: set_nat] : ( ord_le841296385at_nat @ ( outgoing_b @ C @ U3 ) @ ( e_b @ C ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_324_Graph_Oincoming_H__edges,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,U3: set_nat] : ( ord_le841296385at_nat @ ( incoming_a @ C @ U3 ) @ ( e_a @ C ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_325_Graph_Oincoming_H__edges,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,U3: set_nat] : ( ord_le841296385at_nat @ ( incoming_b @ C @ U3 ) @ ( e_b @ C ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_326_Graph_Ofinite__outgoing_H,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,U3: set_nat] :
% 2.72/1.56 ( ( finite_finite_nat @ ( v_a @ C ) )
% 2.72/1.56 => ( finite772653738at_nat @ ( outgoing_a @ C @ U3 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_327_Graph_Ofinite__outgoing_H,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,U3: set_nat] :
% 2.72/1.56 ( ( finite_finite_nat @ ( v_b @ C ) )
% 2.72/1.56 => ( finite772653738at_nat @ ( outgoing_b @ C @ U3 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_328_Graph_Ofinite__incoming_H,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > a,U3: set_nat] :
% 2.72/1.56 ( ( finite_finite_nat @ ( v_a @ C ) )
% 2.72/1.56 => ( finite772653738at_nat @ ( incoming_a @ C @ U3 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_329_Graph_Ofinite__incoming_H,axiom,
% 2.72/1.56 ! [C: product_prod_nat_nat > b,U3: set_nat] :
% 2.72/1.56 ( ( finite_finite_nat @ ( v_b @ C ) )
% 2.72/1.56 => ( finite772653738at_nat @ ( incoming_b @ C @ U3 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_330_g_H_Oincoming_H__edges,axiom,
% 2.72/1.56 ! [U3: set_nat] : ( ord_le841296385at_nat @ ( incoming_b @ c2 @ U3 ) @ ( e_b @ c2 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_331_g_H_Ooutgoing_H__edges,axiom,
% 2.72/1.56 ! [U3: set_nat] : ( ord_le841296385at_nat @ ( outgoing_b @ c2 @ U3 ) @ ( e_b @ c2 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_332_g_H_Ofinite__outgoing_H,axiom,
% 2.72/1.56 ! [U3: set_nat] :
% 2.72/1.56 ( ( finite_finite_nat @ ( v_b @ c2 ) )
% 2.72/1.56 => ( finite772653738at_nat @ ( outgoing_b @ c2 @ U3 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_333_g_H_Ofinite__incoming_H,axiom,
% 2.72/1.56 ! [U3: set_nat] :
% 2.72/1.56 ( ( finite_finite_nat @ ( v_b @ c2 ) )
% 2.72/1.56 => ( finite772653738at_nat @ ( incoming_b @ c2 @ U3 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_334__092_060open_062edges_A_092_060subseteq_062_AE_092_060close_062,axiom,
% 2.72/1.56 ord_le841296385at_nat @ edges @ ( e_a @ c ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_335_psubsetI,axiom,
% 2.72/1.56 ! [A2: set_nat,B2: set_nat] :
% 2.72/1.56 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.72/1.56 => ( ( A2 != B2 )
% 2.72/1.56 => ( ord_less_set_nat @ A2 @ B2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_336_psubsetI,axiom,
% 2.72/1.56 ! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( ord_le841296385at_nat @ A2 @ B2 )
% 2.72/1.56 => ( ( A2 != B2 )
% 2.72/1.56 => ( ord_le116442893at_nat @ A2 @ B2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_337_subset__antisym,axiom,
% 2.72/1.56 ! [A2: set_nat,B2: set_nat] :
% 2.72/1.56 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.72/1.56 => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 2.72/1.56 => ( A2 = B2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_338_subset__antisym,axiom,
% 2.72/1.56 ! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( ord_le841296385at_nat @ A2 @ B2 )
% 2.72/1.56 => ( ( ord_le841296385at_nat @ B2 @ A2 )
% 2.72/1.56 => ( A2 = B2 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_339_subsetI,axiom,
% 2.72/1.56 ! [A2: set_nat,B2: set_nat] :
% 2.72/1.56 ( ! [X2: nat] :
% 2.72/1.56 ( ( member_nat @ X2 @ A2 )
% 2.72/1.56 => ( member_nat @ X2 @ B2 ) )
% 2.72/1.56 => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_340_subsetI,axiom,
% 2.72/1.56 ! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ! [X2: product_prod_nat_nat] :
% 2.72/1.56 ( ( member701585322at_nat @ X2 @ A2 )
% 2.72/1.56 => ( member701585322at_nat @ X2 @ B2 ) )
% 2.72/1.56 => ( ord_le841296385at_nat @ A2 @ B2 ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_341_SP__EDGES,axiom,
% 2.72/1.56 ord_le841296385at_nat @ edges @ ( set_Pr2131844118at_nat @ p ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_342_Collect__mono__iff,axiom,
% 2.72/1.56 ! [P2: nat > $o,Q: nat > $o] :
% 2.72/1.56 ( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
% 2.72/1.56 = ( ! [X: nat] :
% 2.72/1.56 ( ( P2 @ X )
% 2.72/1.56 => ( Q @ X ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_343_Collect__mono__iff,axiom,
% 2.72/1.56 ! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 2.72/1.56 ( ( ord_le841296385at_nat @ ( collec7649004at_nat @ P2 ) @ ( collec7649004at_nat @ Q ) )
% 2.72/1.56 = ( ! [X: product_prod_nat_nat] :
% 2.72/1.56 ( ( P2 @ X )
% 2.72/1.56 => ( Q @ X ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_344_set__eq__subset,axiom,
% 2.72/1.56 ( ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z ) )
% 2.72/1.56 = ( ^ [A4: set_nat,B5: set_nat] :
% 2.72/1.56 ( ( ord_less_eq_set_nat @ A4 @ B5 )
% 2.72/1.56 & ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_345_set__eq__subset,axiom,
% 2.72/1.56 ( ( ^ [Y4: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] : ( Y4 = Z ) )
% 2.72/1.56 = ( ^ [A4: set_Pr1986765409at_nat,B5: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( ord_le841296385at_nat @ A4 @ B5 )
% 2.72/1.56 & ( ord_le841296385at_nat @ B5 @ A4 ) ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_346_subset__trans,axiom,
% 2.72/1.56 ! [A2: set_nat,B2: set_nat,C3: set_nat] :
% 2.72/1.56 ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 2.72/1.56 => ( ( ord_less_eq_set_nat @ B2 @ C3 )
% 2.72/1.56 => ( ord_less_eq_set_nat @ A2 @ C3 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(fact_347_subset__trans,axiom,
% 2.72/1.56 ! [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat,C3: set_Pr1986765409at_nat] :
% 2.72/1.56 ( ( ord_le841296385at_nat @ A2 @ B2 )
% 2.72/1.56 => ( ( ord_le841296385at_nat @ B2 @ C3 )
% 2.72/1.56 => ( ord_le841296385at_nat @ A2 @ C3 ) ) ) ).
% 2.72/1.56
% 2.72/1.56 thf(conj_0,conjecture,
% 2.72/1.56 $false ).
% 2.72/1.56 ------------------------
% 2.72/1.58 % (17947)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/2Mi)
% 2.72/1.58 % (17945)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/4Mi)
% 2.72/1.58 % (17949)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/275Mi)
% 2.72/1.58 % (17950)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/18Mi)
% 2.72/1.58 % (17947)Instruction limit reached!
% 2.72/1.58 % (17947)------------------------------
% 2.72/1.58 % (17947)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.58 % (17947)Termination reason: Unknown
% 2.72/1.58 % (17947)Termination phase: shuffling
% 2.72/1.58
% 2.72/1.58 % (17947)Memory used [KB]: 1407
% 2.72/1.58 % (17947)Time elapsed: 0.002 s
% 2.72/1.58 % (17947)Instructions burned: 2 (million)
% 2.72/1.58 % (17947)------------------------------
% 2.72/1.58 % (17947)------------------------------
% 2.72/1.58 % (17945)Instruction limit reached!
% 2.72/1.58 % (17945)------------------------------
% 2.72/1.58 % (17945)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.58 % (17945)Termination reason: Unknown
% 2.72/1.58 % (17946)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/27Mi)
% 2.72/1.58 % (17945)Termination phase: shuffling
% 2.72/1.58
% 2.72/1.58 % (17945)Memory used [KB]: 1407
% 2.72/1.58 % (17945)Time elapsed: 0.003 s
% 2.72/1.58 % (17945)Instructions burned: 6 (million)
% 2.72/1.58 % (17945)------------------------------
% 2.72/1.58 % (17945)------------------------------
% 2.72/1.59 % (17950)Instruction limit reached!
% 2.72/1.59 % (17950)------------------------------
% 2.72/1.59 % (17950)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.59 % (17950)Termination reason: Unknown
% 2.72/1.59 % (17950)Termination phase: shuffling
% 2.72/1.59
% 2.72/1.59 % (17950)Memory used [KB]: 1663
% 2.72/1.59 % (17950)Time elapsed: 0.007 s
% 2.72/1.59 % (17950)Instructions burned: 19 (million)
% 2.72/1.59 % (17950)------------------------------
% 2.72/1.59 % (17950)------------------------------
% 2.72/1.59 % (17944)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/183Mi)
% 2.72/1.59 % (17948)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/2Mi)
% 2.72/1.59 % (17948)Instruction limit reached!
% 2.72/1.59 % (17948)------------------------------
% 2.72/1.59 % (17948)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.59 % (17948)Termination reason: Unknown
% 2.72/1.59 % (17948)Termination phase: shuffling
% 2.72/1.59
% 2.72/1.59 % (17948)Memory used [KB]: 1407
% 2.72/1.59 % (17948)Time elapsed: 0.003 s
% 2.72/1.59 % (17948)Instructions burned: 3 (million)
% 2.72/1.59 % (17948)------------------------------
% 2.72/1.59 % (17948)------------------------------
% 2.72/1.59 % (17951)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/3Mi)
% 2.72/1.59 % (17952)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/37Mi)
% 2.72/1.59 % (17951)Instruction limit reached!
% 2.72/1.59 % (17951)------------------------------
% 2.72/1.59 % (17951)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.59 % (17951)Termination reason: Unknown
% 2.72/1.59 % (17951)Termination phase: shuffling
% 2.72/1.59
% 2.72/1.59 % (17951)Memory used [KB]: 1407
% 2.72/1.59 % (17951)Time elapsed: 0.002 s
% 2.72/1.59 % (17951)Instructions burned: 3 (million)
% 2.72/1.59 % (17951)------------------------------
% 2.72/1.59 % (17951)------------------------------
% 2.72/1.59 % (17949)Refutation not found, incomplete strategy
% 2.72/1.59 % (17949)------------------------------
% 2.72/1.59 % (17949)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.72/1.59 % (17949)Termination reason: Refutation not found, incomplete strategy
% 2.72/1.59
% 2.72/1.59
% 2.72/1.59 % (17949)Memory used [KB]: 6268
% 2.72/1.59 % (17949)Time elapsed: 0.013 s
% 2.72/1.59 % (17949)Instructions burned: 40 (million)
% 2.72/1.59 % (17949)------------------------------
% 2.72/1.59 % (17949)------------------------------
% 2.72/1.59 % (17953)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/15Mi)
% 2.80/1.60 % (17946)Instruction limit reached!
% 2.80/1.60 % (17946)------------------------------
% 2.80/1.60 % (17946)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.60 % (17946)Termination reason: Unknown
% 2.80/1.60 % (17946)Termination phase: Property scanning
% 2.80/1.60
% 2.80/1.60 % (17946)Memory used [KB]: 1791
% 2.80/1.60 % (17946)Time elapsed: 0.018 s
% 2.80/1.60 % (17946)Instructions burned: 27 (million)
% 2.80/1.60 % (17946)------------------------------
% 2.80/1.60 % (17946)------------------------------
% 2.80/1.60 % (17953)Instruction limit reached!
% 2.80/1.60 % (17953)------------------------------
% 2.80/1.60 % (17953)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.60 % (17953)Termination reason: Unknown
% 2.80/1.60 % (17953)Termination phase: shuffling
% 2.80/1.60
% 2.80/1.60 % (17953)Memory used [KB]: 1663
% 2.80/1.60 % (17953)Time elapsed: 0.006 s
% 2.80/1.60 % (17953)Instructions burned: 16 (million)
% 2.80/1.60 % (17953)------------------------------
% 2.80/1.60 % (17953)------------------------------
% 2.80/1.60 % (17955)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/1041Mi)
% 2.80/1.60 % (17952)Instruction limit reached!
% 2.80/1.60 % (17952)------------------------------
% 2.80/1.60 % (17952)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.60 % (17952)Termination reason: Unknown
% 2.80/1.60 % (17952)Termination phase: shuffling
% 2.80/1.60
% 2.80/1.60 % (17952)Memory used [KB]: 2174
% 2.80/1.60 % (17952)Time elapsed: 0.011 s
% 2.80/1.60 % (17952)Instructions burned: 38 (million)
% 2.80/1.60 % (17952)------------------------------
% 2.80/1.60 % (17952)------------------------------
% 2.80/1.60 % (17956)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/7Mi)
% 2.80/1.60 % (17956)Instruction limit reached!
% 2.80/1.60 % (17956)------------------------------
% 2.80/1.60 % (17956)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.60 % (17956)Termination reason: Unknown
% 2.80/1.60 % (17956)Termination phase: shuffling
% 2.80/1.60
% 2.80/1.60 % (17956)Memory used [KB]: 1407
% 2.80/1.60 % (17956)Time elapsed: 0.003 s
% 2.80/1.60 % (17956)Instructions burned: 7 (million)
% 2.80/1.60 % (17956)------------------------------
% 2.80/1.60 % (17956)------------------------------
% 2.80/1.61 % (17954)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/3Mi)
% 2.80/1.61 % (17954)Instruction limit reached!
% 2.80/1.61 % (17954)------------------------------
% 2.80/1.61 % (17954)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.61 % (17954)Termination reason: Unknown
% 2.80/1.61 % (17954)Termination phase: shuffling
% 2.80/1.61
% 2.80/1.61 % (17954)Memory used [KB]: 1407
% 2.80/1.61 % (17954)Time elapsed: 0.004 s
% 2.80/1.61 % (17954)Instructions burned: 3 (million)
% 2.80/1.61 % (17954)------------------------------
% 2.80/1.61 % (17954)------------------------------
% 2.80/1.61 % (17958)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/3Mi)
% 2.80/1.61 % (17959)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/3Mi)
% 2.80/1.61 % (17958)Instruction limit reached!
% 2.80/1.61 % (17958)------------------------------
% 2.80/1.61 % (17958)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.61 % (17958)Termination reason: Unknown
% 2.80/1.61 % (17958)Termination phase: shuffling
% 2.80/1.61
% 2.80/1.61 % (17958)Memory used [KB]: 1407
% 2.80/1.61 % (17958)Time elapsed: 0.002 s
% 2.80/1.61 % (17958)Instructions burned: 4 (million)
% 2.80/1.61 % (17958)------------------------------
% 2.80/1.61 % (17958)------------------------------
% 2.80/1.61 % (17959)Instruction limit reached!
% 2.80/1.61 % (17959)------------------------------
% 2.80/1.61 % (17959)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.61 % (17959)Termination reason: Unknown
% 2.80/1.61 % (17959)Termination phase: shuffling
% 2.80/1.61
% 2.80/1.61 % (17959)Memory used [KB]: 1407
% 2.80/1.61 % (17959)Time elapsed: 0.002 s
% 2.80/1.61 % (17959)Instructions burned: 3 (million)
% 2.80/1.61 % (17959)------------------------------
% 2.80/1.61 % (17959)------------------------------
% 2.80/1.61 % (17960)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/7Mi)
% 2.80/1.62 % (17960)Instruction limit reached!
% 2.80/1.62 % (17960)------------------------------
% 2.80/1.62 % (17960)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.62 % (17960)Termination reason: Unknown
% 2.80/1.62 % (17960)Termination phase: shuffling
% 2.80/1.62
% 2.80/1.62 % (17960)Memory used [KB]: 1407
% 2.80/1.62 % (17960)Time elapsed: 0.003 s
% 2.80/1.62 % (17960)Instructions burned: 7 (million)
% 2.80/1.62 % (17960)------------------------------
% 2.80/1.62 % (17960)------------------------------
% 2.80/1.62 % (17957)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/16Mi)
% 2.80/1.62 % (17964)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/4Mi)
% 2.80/1.62 % (17965)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/18Mi)
% 2.80/1.62 % (17964)Instruction limit reached!
% 2.80/1.62 % (17964)------------------------------
% 2.80/1.62 % (17964)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.62 % (17964)Termination reason: Unknown
% 2.80/1.62 % (17964)Termination phase: shuffling
% 2.80/1.62
% 2.80/1.62 % (17964)Memory used [KB]: 1407
% 2.80/1.62 % (17964)Time elapsed: 0.003 s
% 2.80/1.62 % (17964)Instructions burned: 6 (million)
% 2.80/1.62 % (17964)------------------------------
% 2.80/1.62 % (17964)------------------------------
% 2.80/1.62 % (17966)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/710Mi)
% 2.80/1.63 % (17965)Instruction limit reached!
% 2.80/1.63 % (17965)------------------------------
% 2.80/1.63 % (17965)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.63 % (17965)Termination reason: Unknown
% 2.80/1.63 % (17965)Termination phase: shuffling
% 2.80/1.63
% 2.80/1.63 % (17965)Memory used [KB]: 1663
% 2.80/1.63 % (17965)Time elapsed: 0.007 s
% 2.80/1.63 % (17965)Instructions burned: 18 (million)
% 2.80/1.63 % (17965)------------------------------
% 2.80/1.63 % (17965)------------------------------
% 2.80/1.63 % (17962)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/3Mi)
% 2.80/1.63 % (17962)Instruction limit reached!
% 2.80/1.63 % (17962)------------------------------
% 2.80/1.63 % (17962)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.63 % (17962)Termination reason: Unknown
% 2.80/1.63 % (17962)Termination phase: shuffling
% 2.80/1.63
% 2.80/1.63 % (17962)Memory used [KB]: 1407
% 2.80/1.63 % (17962)Time elapsed: 0.004 s
% 2.80/1.63 % (17962)Instructions burned: 3 (million)
% 2.80/1.63 % (17962)------------------------------
% 2.80/1.63 % (17962)------------------------------
% 2.80/1.63 % (17957)Instruction limit reached!
% 2.80/1.63 % (17957)------------------------------
% 2.80/1.63 % (17957)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.63 % (17957)Termination reason: Unknown
% 2.80/1.63 % (17957)Termination phase: shuffling
% 2.80/1.63
% 2.80/1.63 % (17957)Memory used [KB]: 1663
% 2.80/1.63 % (17957)Time elapsed: 0.014 s
% 2.80/1.63 % (17957)Instructions burned: 17 (million)
% 2.80/1.63 % (17957)------------------------------
% 2.80/1.63 % (17957)------------------------------
% 2.80/1.63 % (17967)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/6Mi)
% 2.80/1.63 % (17967)Instruction limit reached!
% 2.80/1.63 % (17967)------------------------------
% 2.80/1.63 % (17967)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.63 % (17967)Termination reason: Unknown
% 2.80/1.63 % (17967)Termination phase: shuffling
% 2.80/1.63
% 2.80/1.63 % (17967)Memory used [KB]: 1407
% 2.80/1.63 % (17967)Time elapsed: 0.003 s
% 2.80/1.63 % (17967)Instructions burned: 6 (million)
% 2.80/1.63 % (17967)------------------------------
% 2.80/1.63 % (17967)------------------------------
% 2.80/1.64 % (17968)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/902Mi)
% 2.80/1.64 % (17966)Refutation not found, incomplete strategy
% 2.80/1.64 % (17966)------------------------------
% 2.80/1.64 % (17966)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.64 % (17966)Termination reason: Refutation not found, incomplete strategy
% 2.80/1.64
% 2.80/1.64
% 2.80/1.64 % (17966)Memory used [KB]: 6268
% 2.80/1.64 % (17966)Time elapsed: 0.013 s
% 2.80/1.64 % (17966)Instructions burned: 40 (million)
% 2.80/1.64 % (17966)------------------------------
% 2.80/1.64 % (17966)------------------------------
% 2.80/1.64 % (17970)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/5Mi)
% 2.80/1.64 % (17970)Instruction limit reached!
% 2.80/1.64 % (17970)------------------------------
% 2.80/1.64 % (17970)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.64 % (17970)Termination reason: Unknown
% 2.80/1.64 % (17970)Termination phase: shuffling
% 2.80/1.64
% 2.80/1.64 % (17970)Memory used [KB]: 1535
% 2.80/1.64 % (17970)Time elapsed: 0.003 s
% 2.80/1.64 % (17970)Instructions burned: 7 (million)
% 2.80/1.64 % (17970)------------------------------
% 2.80/1.64 % (17970)------------------------------
% 2.80/1.64 % (17971)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/6Mi)
% 2.80/1.64 % (17971)Instruction limit reached!
% 2.80/1.64 % (17971)------------------------------
% 2.80/1.64 % (17971)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.64 % (17971)Termination reason: Unknown
% 2.80/1.64 % (17971)Termination phase: shuffling
% 2.80/1.64
% 2.80/1.64 % (17971)Memory used [KB]: 1535
% 2.80/1.64 % (17971)Time elapsed: 0.003 s
% 2.80/1.64 % (17971)Instructions burned: 8 (million)
% 2.80/1.64 % (17971)------------------------------
% 2.80/1.64 % (17971)------------------------------
% 2.80/1.65 % (17969)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/21Mi)
% 2.80/1.65 % (17972)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/377Mi)
% 2.80/1.65 % (17973)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/779Mi)
% 2.80/1.65 % (17974)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/19Mi)
% 2.80/1.66 % (17969)Instruction limit reached!
% 2.80/1.66 % (17969)------------------------------
% 2.80/1.66 % (17969)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.66 % (17969)Termination reason: Unknown
% 2.80/1.66 % (17969)Termination phase: shuffling
% 2.80/1.66
% 2.80/1.66 % (17969)Memory used [KB]: 1791
% 2.80/1.66 % (17969)Time elapsed: 0.012 s
% 2.80/1.66 % (17969)Instructions burned: 22 (million)
% 2.80/1.66 % (17969)------------------------------
% 2.80/1.66 % (17969)------------------------------
% 2.80/1.66 % (17974)Instruction limit reached!
% 2.80/1.66 % (17974)------------------------------
% 2.80/1.66 % (17974)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.66 % (17974)Termination reason: Unknown
% 2.80/1.66 % (17974)Termination phase: shuffling
% 2.80/1.66
% 2.80/1.66 % (17974)Memory used [KB]: 1791
% 2.80/1.66 % (17974)Time elapsed: 0.007 s
% 2.80/1.66 % (17974)Instructions burned: 22 (million)
% 2.80/1.66 % (17974)------------------------------
% 2.80/1.66 % (17974)------------------------------
% 2.80/1.67 % (17983)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/17Mi)
% 2.80/1.67 % (17944)Instruction limit reached!
% 2.80/1.67 % (17944)------------------------------
% 2.80/1.67 % (17944)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.67 % (17944)Termination reason: Unknown
% 2.80/1.67 % (17944)Termination phase: Saturation
% 2.80/1.67
% 2.80/1.67 % (17944)Memory used [KB]: 7547
% 2.80/1.67 % (17944)Time elapsed: 0.085 s
% 2.80/1.67 % (17944)Instructions burned: 184 (million)
% 2.80/1.67 % (17944)------------------------------
% 2.80/1.67 % (17944)------------------------------
% 2.80/1.68 % (17983)Instruction limit reached!
% 2.80/1.68 % (17983)------------------------------
% 2.80/1.68 % (17983)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.68 % (17983)Termination reason: Unknown
% 2.80/1.68 % (17983)Termination phase: shuffling
% 2.80/1.68
% 2.80/1.68 % (17983)Memory used [KB]: 1791
% 2.80/1.68 % (17983)Time elapsed: 0.007 s
% 2.80/1.68 % (17983)Instructions burned: 19 (million)
% 2.80/1.68 % (17983)------------------------------
% 2.80/1.68 % (17983)------------------------------
% 2.80/1.68 % (17968)Refutation not found, incomplete strategy
% 2.80/1.68 % (17968)------------------------------
% 2.80/1.68 % (17968)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.68 % (17968)Termination reason: Refutation not found, incomplete strategy
% 2.80/1.68
% 2.80/1.68
% 2.80/1.68 % (17968)Memory used [KB]: 7291
% 2.80/1.68 % (17968)Time elapsed: 0.041 s
% 2.80/1.68 % (17968)Instructions burned: 109 (million)
% 2.80/1.68 % (17968)------------------------------
% 2.80/1.68 % (17968)------------------------------
% 2.80/1.68 % (17982)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/879Mi)
% 2.80/1.68 % (17984)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/3Mi)
% 2.80/1.69 % (17984)Instruction limit reached!
% 2.80/1.69 % (17984)------------------------------
% 2.80/1.69 % (17984)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.69 % (17984)Termination reason: Unknown
% 2.80/1.69 % (17984)Termination phase: shuffling
% 2.80/1.69
% 2.80/1.69 % (17984)Memory used [KB]: 1407
% 2.80/1.69 % (17984)Time elapsed: 0.002 s
% 2.80/1.69 % (17984)Instructions burned: 3 (million)
% 2.80/1.69 % (17984)------------------------------
% 2.80/1.69 % (17984)------------------------------
% 2.80/1.69 % (17988)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/30Mi)
% 2.80/1.69 % (17992)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_17795 for (2998ds/100Mi)
% 2.80/1.70 % (17982)Refutation not found, incomplete strategy
% 2.80/1.70 % (17982)------------------------------
% 2.80/1.70 % (17982)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.70 % (17982)Termination reason: Refutation not found, incomplete strategy
% 2.80/1.70
% 2.80/1.70
% 2.80/1.70 % (17982)Memory used [KB]: 6268
% 2.80/1.70 % (17982)Time elapsed: 0.022 s
% 2.80/1.70 % (17982)Instructions burned: 47 (million)
% 2.80/1.70 % (17982)------------------------------
% 2.80/1.70 % (17982)------------------------------
% 2.80/1.70 % (17989)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_17795 for (2999ds/127Mi)
% 2.80/1.71 % (17988)Instruction limit reached!
% 2.80/1.71 % (17988)------------------------------
% 2.80/1.71 % (17988)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.71 % (17988)Termination reason: Unknown
% 2.80/1.71 % (17988)Termination phase: Property scanning
% 2.80/1.71
% 2.80/1.71 % (17988)Memory used [KB]: 1791
% 2.80/1.71 % (17988)Time elapsed: 0.018 s
% 2.80/1.71 % (17988)Instructions burned: 31 (million)
% 2.80/1.71 % (17988)------------------------------
% 2.80/1.71 % (17988)------------------------------
% 2.80/1.71 % (17995)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_17795 for (2998ds/3Mi)
% 2.80/1.72 % (17995)Instruction limit reached!
% 2.80/1.72 % (17995)------------------------------
% 2.80/1.72 % (17995)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.72 % (17995)Termination reason: Unknown
% 2.80/1.72 % (17995)Termination phase: shuffling
% 2.80/1.72
% 2.80/1.72 % (17995)Memory used [KB]: 1407
% 2.80/1.72 % (17995)Time elapsed: 0.004 s
% 2.80/1.72 % (17995)Instructions burned: 5 (million)
% 2.80/1.72 % (17995)------------------------------
% 2.80/1.72 % (17995)------------------------------
% 2.80/1.73 % (17999)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_17795 for (2998ds/20Mi)
% 2.80/1.74 % (18001)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_17795 for (2998ds/86Mi)
% 2.80/1.74 % (17999)Instruction limit reached!
% 2.80/1.74 % (17999)------------------------------
% 2.80/1.74 % (17999)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.74 % (17999)Termination reason: Unknown
% 2.80/1.74 % (17999)Termination phase: shuffling
% 2.80/1.74
% 2.80/1.74 % (17999)Memory used [KB]: 1791
% 2.80/1.74 % (17999)Time elapsed: 0.012 s
% 2.80/1.74 % (17999)Instructions burned: 22 (million)
% 2.80/1.74 % (17999)------------------------------
% 2.80/1.74 % (17999)------------------------------
% 2.80/1.74 % (17992)Instruction limit reached!
% 2.80/1.74 % (17992)------------------------------
% 2.80/1.74 % (17992)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.74 % (17992)Termination reason: Unknown
% 2.80/1.74 % (17992)Termination phase: Property scanning
% 2.80/1.74
% 2.80/1.74 % (17992)Memory used [KB]: 2430
% 2.80/1.74 % (17992)Time elapsed: 0.045 s
% 2.80/1.74 % (17992)Instructions burned: 103 (million)
% 2.80/1.74 % (17992)------------------------------
% 2.80/1.74 % (17992)------------------------------
% 2.80/1.75 % (18008)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_17795 for (2998ds/39Mi)
% 2.80/1.75 % (17989)Instruction limit reached!
% 2.80/1.75 % (17989)------------------------------
% 2.80/1.75 % (17989)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.75 % (17989)Termination reason: Unknown
% 2.80/1.75 % (17989)Termination phase: Saturation
% 2.80/1.75
% 2.80/1.75 % (17989)Memory used [KB]: 7036
% 2.80/1.75 % (17989)Time elapsed: 0.056 s
% 2.80/1.75 % (17989)Instructions burned: 127 (million)
% 2.80/1.75 % (17989)------------------------------
% 2.80/1.75 % (17989)------------------------------
% 2.80/1.75 % (18005)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_17795 for (2998ds/107Mi)
% 2.80/1.76 % (18008)Instruction limit reached!
% 2.80/1.76 % (18008)------------------------------
% 2.80/1.76 % (18008)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.76 % (18008)Termination reason: Unknown
% 2.80/1.76 % (18008)Termination phase: Property scanning
% 2.80/1.76
% 2.80/1.76 % (18008)Memory used [KB]: 2174
% 2.80/1.76 % (18008)Time elapsed: 0.012 s
% 2.80/1.76 % (18008)Instructions burned: 40 (million)
% 2.80/1.76 % (18008)------------------------------
% 2.80/1.76 % (18008)------------------------------
% 2.80/1.77 % (18014)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_17795 for (2998ds/46Mi)
% 2.80/1.77 % (18011)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_17795 for (2998ds/448Mi)
% 2.80/1.77 % (18005)Refutation not found, incomplete strategy
% 2.80/1.77 % (18005)------------------------------
% 2.80/1.77 % (18005)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.77 % (18005)Termination reason: Refutation not found, incomplete strategy
% 2.80/1.77
% 2.80/1.77
% 2.80/1.77 % (18005)Memory used [KB]: 6268
% 2.80/1.77 % (18005)Time elapsed: 0.023 s
% 2.80/1.77 % (18005)Instructions burned: 45 (million)
% 2.80/1.77 % (18005)------------------------------
% 2.80/1.77 % (18005)------------------------------
% 2.80/1.78 % (18001)Instruction limit reached!
% 2.80/1.78 % (18001)------------------------------
% 2.80/1.78 % (18001)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.78 % (18001)Termination reason: Unknown
% 2.80/1.78 % (18001)Termination phase: Property scanning
% 2.80/1.78
% 2.80/1.78 % (18001)Memory used [KB]: 2174
% 2.80/1.78 % (18001)Time elapsed: 0.044 s
% 2.80/1.78 % (18001)Instructions burned: 88 (million)
% 2.80/1.78 % (18001)------------------------------
% 2.80/1.78 % (18001)------------------------------
% 2.80/1.79 % (18017)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_17795 for (2997ds/98Mi)
% 2.80/1.79 % (18014)Refutation not found, incomplete strategy
% 2.80/1.79 % (18014)------------------------------
% 2.80/1.79 % (18014)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.79 % (18014)Termination reason: Refutation not found, incomplete strategy
% 2.80/1.79
% 2.80/1.79
% 2.80/1.79 % (18014)Memory used [KB]: 6268
% 2.80/1.79 % (18014)Time elapsed: 0.018 s
% 2.80/1.79 % (18014)Instructions burned: 45 (million)
% 2.80/1.79 % (18014)------------------------------
% 2.80/1.79 % (18014)------------------------------
% 2.80/1.79 % (18018)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_17795 for (2997ds/507Mi)
% 2.80/1.80 % (18019)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_17795 for (2997ds/149Mi)
% 2.80/1.81 % (17972)Instruction limit reached!
% 2.80/1.81 % (17972)------------------------------
% 2.80/1.81 % (17972)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.81 % (17972)Termination reason: Unknown
% 2.80/1.81 % (17972)Termination phase: Saturation
% 2.80/1.81
% 2.80/1.81 % (17972)Memory used [KB]: 8443
% 2.80/1.81 % (17972)Time elapsed: 0.165 s
% 2.80/1.81 % (17972)Instructions burned: 377 (million)
% 2.80/1.81 % (17972)------------------------------
% 2.80/1.81 % (17972)------------------------------
% 2.80/1.82 % (18022)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_17795 for (2997ds/39Mi)
% 2.80/1.83 % (18017)Instruction limit reached!
% 2.80/1.83 % (18017)------------------------------
% 2.80/1.83 % (18017)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.83 % (18017)Termination reason: Unknown
% 2.80/1.83 % (18017)Termination phase: Function definition elimination
% 2.80/1.83
% 2.80/1.83 % (18017)Memory used [KB]: 2302
% 2.80/1.83 % (18017)Time elapsed: 0.046 s
% 2.80/1.83 % (18017)Instructions burned: 98 (million)
% 2.80/1.83 % (18017)------------------------------
% 2.80/1.83 % (18017)------------------------------
% 2.80/1.83 % (18022)Instruction limit reached!
% 2.80/1.83 % (18022)------------------------------
% 2.80/1.83 % (18022)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.83 % (18022)Termination reason: Unknown
% 2.80/1.83 % (18022)Termination phase: Unused predicate definition removal
% 2.80/1.83
% 2.80/1.83 % (18022)Memory used [KB]: 1791
% 2.80/1.83 % (18022)Time elapsed: 0.013 s
% 2.80/1.83 % (18022)Instructions burned: 40 (million)
% 2.80/1.83 % (18022)------------------------------
% 2.80/1.83 % (18022)------------------------------
% 2.80/1.84 % (18026)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_17795 for (2997ds/1087Mi)
% 2.80/1.85 % (18024)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_17795 for (2997ds/6Mi)
% 2.80/1.85 % (18024)Instruction limit reached!
% 2.80/1.85 % (18024)------------------------------
% 2.80/1.85 % (18024)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.85 % (18024)Termination reason: Unknown
% 2.80/1.85 % (18024)Termination phase: shuffling
% 2.80/1.85
% 2.80/1.85 % (18024)Memory used [KB]: 1535
% 2.80/1.85 % (18024)Time elapsed: 0.005 s
% 2.80/1.85 % (18024)Instructions burned: 7 (million)
% 2.80/1.85 % (18024)------------------------------
% 2.80/1.85 % (18024)------------------------------
% 2.80/1.85 % (18019)Instruction limit reached!
% 2.80/1.85 % (18019)------------------------------
% 2.80/1.85 % (18019)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.85 % (18019)Termination reason: Unknown
% 2.80/1.85 % (18019)Termination phase: Saturation
% 2.80/1.85
% 2.80/1.85 % (18019)Memory used [KB]: 7164
% 2.80/1.85 % (18019)Time elapsed: 0.058 s
% 2.80/1.85 % (18019)Instructions burned: 149 (million)
% 2.80/1.85 % (18019)------------------------------
% 2.80/1.85 % (18019)------------------------------
% 2.80/1.86 % (18026)Refutation not found, incomplete strategy
% 2.80/1.86 % (18026)------------------------------
% 2.80/1.86 % (18026)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.86 % (18026)Termination reason: Refutation not found, incomplete strategy
% 2.80/1.86
% 2.80/1.86
% 2.80/1.86 % (18026)Memory used [KB]: 6268
% 2.80/1.86 % (18026)Time elapsed: 0.014 s
% 2.80/1.86 % (18026)Instructions burned: 42 (million)
% 2.80/1.86 % (18026)------------------------------
% 2.80/1.86 % (18026)------------------------------
% 2.80/1.86 % (18018)Refutation not found, incomplete strategy
% 2.80/1.86 % (18018)------------------------------
% 2.80/1.86 % (18018)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.87 % (18018)Termination reason: Refutation not found, incomplete strategy
% 2.80/1.87
% 2.80/1.87
% 2.80/1.87 % (18018)Memory used [KB]: 7675
% 2.80/1.87 % (18018)Time elapsed: 0.072 s
% 2.80/1.87 % (18018)Instructions burned: 153 (million)
% 2.80/1.87 % (18018)------------------------------
% 2.80/1.87 % (18018)------------------------------
% 2.80/1.87 % (18027)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_17795 for (2997ds/47Mi)
% 2.80/1.87 % (18029)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_17795 for (2997ds/432Mi)
% 2.80/1.87 % (18028)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_17795 for (2997ds/44Mi)
% 2.80/1.88 % (18031)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_17795 for (2997ds/62Mi)
% 2.80/1.89 % (18028)Instruction limit reached!
% 2.80/1.89 % (18028)------------------------------
% 2.80/1.89 % (18028)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.89 % (18028)Termination reason: Unknown
% 2.80/1.89 % (18028)Termination phase: Property scanning
% 2.80/1.89
% 2.80/1.89 % (18028)Memory used [KB]: 2174
% 2.80/1.89 % (18028)Time elapsed: 0.021 s
% 2.80/1.89 % (18028)Instructions burned: 44 (million)
% 2.80/1.89 % (18028)------------------------------
% 2.80/1.89 % (18028)------------------------------
% 2.80/1.89 % (18027)Instruction limit reached!
% 2.80/1.89 % (18027)------------------------------
% 2.80/1.89 % (18027)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.89 % (18027)Termination reason: Unknown
% 2.80/1.89 % (18027)Termination phase: Preprocessing 3
% 2.80/1.89
% 2.80/1.89 % (18027)Memory used [KB]: 2046
% 2.80/1.89 % (18027)Time elapsed: 0.026 s
% 2.80/1.89 % (18027)Instructions burned: 47 (million)
% 2.80/1.89 % (18027)------------------------------
% 2.80/1.89 % (18027)------------------------------
% 2.80/1.89 % (18031)Instruction limit reached!
% 2.80/1.89 % (18031)------------------------------
% 2.80/1.89 % (18031)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.89 % (18031)Termination reason: Unknown
% 2.80/1.89 % (18031)Termination phase: Property scanning
% 2.80/1.89
% 2.80/1.89 % (18031)Memory used [KB]: 2174
% 2.80/1.89 % (18031)Time elapsed: 0.018 s
% 2.80/1.89 % (18031)Instructions burned: 64 (million)
% 2.80/1.89 % (18031)------------------------------
% 2.80/1.89 % (18031)------------------------------
% 2.80/1.90 % (18033)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_17795 for (2996ds/96Mi)
% 2.80/1.91 % (18032)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_17795 for (2996ds/31Mi)
% 2.80/1.91 % (18034)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_17795 for (2996ds/453Mi)
% 2.80/1.92 % (18032)Instruction limit reached!
% 2.80/1.92 % (18032)------------------------------
% 2.80/1.92 % (18032)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.92 % (18032)Termination reason: Unknown
% 2.80/1.92 % (18032)Termination phase: Property scanning
% 2.80/1.92
% 2.80/1.92 % (18032)Memory used [KB]: 1791
% 2.80/1.92 % (18032)Time elapsed: 0.016 s
% 2.80/1.92 % (18032)Instructions burned: 31 (million)
% 2.80/1.92 % (18032)------------------------------
% 2.80/1.92 % (18032)------------------------------
% 2.80/1.94 % (18036)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_17795 for (2996ds/71Mi)
% 2.80/1.96 % (18033)Instruction limit reached!
% 2.80/1.96 % (18033)------------------------------
% 2.80/1.96 % (18033)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.96 % (18033)Termination reason: Unknown
% 2.80/1.96 % (18033)Termination phase: Property scanning
% 2.80/1.96
% 2.80/1.96 % (18033)Memory used [KB]: 2174
% 2.80/1.96 % (18033)Time elapsed: 0.044 s
% 2.80/1.96 % (18033)Instructions burned: 96 (million)
% 2.80/1.96 % (18033)------------------------------
% 2.80/1.96 % (18033)------------------------------
% 2.80/1.97 % (18011)Instruction limit reached!
% 2.80/1.97 % (18011)------------------------------
% 2.80/1.97 % (18011)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.97 % (18011)Termination reason: Unknown
% 2.80/1.97 % (18011)Termination phase: Saturation
% 2.80/1.97
% 2.80/1.97 % (18011)Memory used [KB]: 8699
% 2.80/1.97 % (18011)Time elapsed: 0.199 s
% 2.80/1.97 % (18011)Instructions burned: 449 (million)
% 2.80/1.97 % (18011)------------------------------
% 2.80/1.97 % (18011)------------------------------
% 2.80/1.97 % (18036)Instruction limit reached!
% 2.80/1.97 % (18036)------------------------------
% 2.80/1.97 % (18036)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.97 % (18036)Termination reason: Unknown
% 2.80/1.97 % (18036)Termination phase: Property scanning
% 2.80/1.97
% 2.80/1.97 % (18036)Memory used [KB]: 2174
% 2.80/1.97 % (18036)Time elapsed: 0.037 s
% 2.80/1.97 % (18036)Instructions burned: 71 (million)
% 2.80/1.97 % (18036)------------------------------
% 2.80/1.97 % (18036)------------------------------
% 2.80/1.98 % (18039)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_17795 for (2996ds/83Mi)
% 2.80/1.98 % (18040)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_17795 for (2996ds/11Mi)
% 2.80/1.99 % (18040)Instruction limit reached!
% 2.80/1.99 % (18040)------------------------------
% 2.80/1.99 % (18040)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.99 % (18040)Termination reason: Unknown
% 2.80/1.99 % (18040)Termination phase: shuffling
% 2.80/1.99
% 2.80/1.99 % (18040)Memory used [KB]: 1663
% 2.80/1.99 % (18040)Time elapsed: 0.007 s
% 2.80/1.99 % (18040)Instructions burned: 12 (million)
% 2.80/1.99 % (18040)------------------------------
% 2.80/1.99 % (18040)------------------------------
% 2.80/1.99 % (18039)Refutation not found, incomplete strategy
% 2.80/1.99 % (18039)------------------------------
% 2.80/1.99 % (18039)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/1.99 % (18039)Termination reason: Refutation not found, incomplete strategy
% 2.80/1.99
% 2.80/1.99
% 2.80/1.99 % (18039)Memory used [KB]: 6268
% 2.80/1.99 % (18039)Time elapsed: 0.016 s
% 2.80/1.99 % (18039)Instructions burned: 45 (million)
% 2.80/1.99 % (18039)------------------------------
% 2.80/1.99 % (18039)------------------------------
% 2.80/2.00 % (17973)Instruction limit reached!
% 2.80/2.00 % (17973)------------------------------
% 2.80/2.00 % (17973)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/2.00 % (17973)Termination reason: Unknown
% 2.80/2.00 % (17973)Termination phase: Saturation
% 2.80/2.00
% 2.80/2.00 % (17973)Memory used [KB]: 9338
% 2.80/2.00 % (17973)Time elapsed: 0.350 s
% 2.80/2.00 % (17973)Instructions burned: 779 (million)
% 2.80/2.00 % (17973)------------------------------
% 2.80/2.00 % (17973)------------------------------
% 2.80/2.00 % (18041)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_17795 for (2995ds/41Mi)
% 2.80/2.00 % (18044)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_17795 for (2995ds/6Mi)
% 2.80/2.00 % (18046)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_17795 for (2995ds/150Mi)
% 2.80/2.00 % (18044)Instruction limit reached!
% 2.80/2.00 % (18044)------------------------------
% 2.80/2.00 % (18044)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/2.00 % (18044)Termination reason: Unknown
% 2.80/2.00 % (18044)Termination phase: shuffling
% 2.80/2.00
% 2.80/2.00 % (18044)Memory used [KB]: 1407
% 2.80/2.00 % (18044)Time elapsed: 0.003 s
% 2.80/2.00 % (18044)Instructions burned: 6 (million)
% 2.80/2.00 % (18044)------------------------------
% 2.80/2.00 % (18044)------------------------------
% 2.80/2.01 % (18041)Instruction limit reached!
% 2.80/2.01 % (18041)------------------------------
% 2.80/2.01 % (18041)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.80/2.01 % (18041)Termination reason: Unknown
% 2.80/2.01 % (18041)Termination phase: Property scanning
% 2.80/2.01
% 2.80/2.01 % (18041)Memory used [KB]: 2174
% 2.80/2.01 % (18041)Time elapsed: 0.013 s
% 2.80/2.01 % (18047)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_17795 for (2995ds/343Mi)
% 2.80/2.01 % (18041)Instructions burned: 42 (million)
% 2.80/2.01 % (18041)------------------------------
% 2.80/2.01 % (18041)------------------------------
% 5.74/2.01 % (18048)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_17795 for (2995ds/109Mi)
% 5.74/2.02 % (18049)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_17795 for (2995ds/253Mi)
% 5.74/2.03 % (18047)Refutation not found, incomplete strategy
% 5.74/2.03 % (18047)------------------------------
% 5.74/2.03 % (18047)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.74/2.03 % (18047)Termination reason: Refutation not found, incomplete strategy
% 5.74/2.03
% 5.74/2.03
% 5.74/2.03 % (18047)Memory used [KB]: 6268
% 5.74/2.03 % (18047)Time elapsed: 0.016 s
% 5.74/2.03 % (18047)Instructions burned: 49 (million)
% 5.74/2.03 % (18047)------------------------------
% 5.74/2.03 % (18047)------------------------------
% 5.74/2.03 % (18048)Refutation not found, incomplete strategy
% 5.74/2.03 % (18048)------------------------------
% 5.74/2.03 % (18048)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.74/2.03 % (18048)Termination reason: Refutation not found, incomplete strategy
% 5.74/2.03
% 5.74/2.03
% 5.74/2.03 % (18048)Memory used [KB]: 6268
% 5.74/2.03 % (18048)Time elapsed: 0.014 s
% 5.74/2.03 % (18048)Instructions burned: 43 (million)
% 5.74/2.03 % (18048)------------------------------
% 5.74/2.03 % (18048)------------------------------
% 5.74/2.03 % (18049)Refutation not found, incomplete strategy
% 5.74/2.03 % (18049)------------------------------
% 5.74/2.03 % (18049)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.74/2.03 % (18049)Termination reason: Refutation not found, incomplete strategy
% 5.74/2.03
% 5.74/2.03
% 5.74/2.03 % (18049)Memory used [KB]: 6268
% 5.74/2.03 % (18049)Time elapsed: 0.013 s
% 5.74/2.03 % (18049)Instructions burned: 40 (million)
% 5.74/2.03 % (18049)------------------------------
% 5.74/2.03 % (18049)------------------------------
% 5.74/2.03 % (18034)First to succeed.
% 5.74/2.03 % (17955)Instruction limit reached!
% 5.74/2.03 % (17955)------------------------------
% 5.74/2.03 % (17955)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.74/2.03 % (17955)Termination reason: Unknown
% 5.74/2.03 % (17955)Termination phase: Saturation
% 5.74/2.03
% 5.74/2.03 % (17955)Memory used [KB]: 10362
% 5.74/2.03 % (17955)Time elapsed: 0.435 s
% 5.74/2.03 % (17955)Instructions burned: 1043 (million)
% 5.74/2.03 % (17955)------------------------------
% 5.74/2.03 % (17955)------------------------------
% 5.74/2.04 % (18050)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_17795 for (2995ds/140Mi)
% 5.74/2.04 % (18034)Refutation found. Thanks to Tanya!
% 5.74/2.04 % SZS status ContradictoryAxioms for DTF2THF_17795
% 5.74/2.04 % SZS output start Proof for DTF2THF_17795
% 5.74/2.04 thf(type_def_5, type, product_prod_nat_nat: $tType).
% 5.74/2.04 thf(type_def_6, type, a: $tType).
% 5.74/2.04 thf(type_def_8, type, b: $tType).
% 5.74/2.04 thf(type_def_9, type, nat: $tType).
% 5.74/2.04 thf(type_def_10, type, set_Pr1986765409at_nat: $tType).
% 5.74/2.04 thf(type_def_11, type, set_li664300135at_nat: $tType).
% 5.74/2.04 thf(type_def_12, type, set_nat: $tType).
% 5.74/2.04 thf(type_def_13, type, set_set_nat: $tType).
% 5.74/2.04 thf(type_def_14, type, set_se1612935105at_nat: $tType).
% 5.74/2.04 thf(type_def_15, type, list_P559422087at_nat: $tType).
% 5.74/2.04 thf(func_def_0, type, set_li664300135at_nat: $tType).
% 5.74/2.04 thf(func_def_1, type, set_se1612935105at_nat: $tType).
% 5.74/2.04 thf(func_def_2, type, list_P559422087at_nat: $tType).
% 5.74/2.04 thf(func_def_3, type, set_Pr1986765409at_nat: $tType).
% 5.74/2.04 thf(func_def_4, type, product_prod_nat_nat: $tType).
% 5.74/2.04 thf(func_def_5, type, set_set_nat: $tType).
% 5.74/2.04 thf(func_def_6, type, set_nat: $tType).
% 5.74/2.04 thf(func_def_7, type, nat: $tType).
% 5.74/2.04 thf(func_def_8, type, b: $tType).
% 5.74/2.04 thf(func_def_9, type, a: $tType).
% 5.74/2.04 thf(func_def_10, type, edmond1517640972ysis_a: (product_prod_nat_nat > a) > $o).
% 5.74/2.04 thf(func_def_11, type, edmond1517640973ysis_b: (product_prod_nat_nat > b) > $o).
% 5.74/2.04 thf(func_def_12, type, edmond1022345716sure_a: (product_prod_nat_nat > a) > nat > nat > nat).
% 5.74/2.04 thf(func_def_13, type, edmond1022345717sure_b: (product_prod_nat_nat > b) > nat > nat > nat).
% 5.74/2.04 thf(func_def_14, type, edmond475474835dges_a: (product_prod_nat_nat > a) > nat > nat > set_Pr1986765409at_nat).
% 5.74/2.04 thf(func_def_15, type, edmond475474836dges_b: (product_prod_nat_nat > b) > nat > nat > set_Pr1986765409at_nat).
% 5.74/2.04 thf(func_def_16, type, edmond771116670s_uE_a: (product_prod_nat_nat > a) > set_Pr1986765409at_nat).
% 5.74/2.04 thf(func_def_17, type, edmond771116671s_uE_b: (product_prod_nat_nat > b) > set_Pr1986765409at_nat).
% 5.74/2.04 thf(func_def_18, type, finite83082927at_nat: set_li664300135at_nat > nat).
% 5.74/2.04 thf(func_def_19, type, finite_card_nat: set_nat > nat).
% 5.74/2.04 thf(func_def_20, type, finite447719721at_nat: set_Pr1986765409at_nat > nat).
% 5.74/2.04 thf(func_def_21, type, finite_card_set_nat: set_set_nat > nat).
% 5.74/2.04 thf(func_def_22, type, finite1701894793at_nat: set_se1612935105at_nat > nat).
% 5.74/2.04 thf(func_def_23, type, finite1299096496at_nat: set_li664300135at_nat > $o).
% 5.74/2.04 thf(func_def_24, type, finite_finite_nat: set_nat > $o).
% 5.74/2.04 thf(func_def_25, type, finite772653738at_nat: set_Pr1986765409at_nat > $o).
% 5.74/2.04 thf(func_def_26, type, finite2012248349et_nat: set_set_nat > $o).
% 5.74/2.04 thf(func_def_27, type, finite1457549322at_nat: set_se1612935105at_nat > $o).
% 5.74/2.04 thf(func_def_28, type, finite_Graph_a: (product_prod_nat_nat > a) > $o).
% 5.74/2.04 thf(func_def_29, type, finite_Graph_b: (product_prod_nat_nat > b) > $o).
% 5.74/2.04 thf(func_def_30, type, e_a: (product_prod_nat_nat > a) > set_Pr1986765409at_nat).
% 5.74/2.04 thf(func_def_31, type, e_b: (product_prod_nat_nat > b) > set_Pr1986765409at_nat).
% 5.74/2.04 thf(func_def_32, type, v_a: (product_prod_nat_nat > a) > set_nat).
% 5.74/2.04 thf(func_def_33, type, v_b: (product_prod_nat_nat > b) > set_nat).
% 5.74/2.04 thf(func_def_34, type, adjacent_nodes_a: (product_prod_nat_nat > a) > nat > set_nat).
% 5.74/2.04 thf(func_def_35, type, adjacent_nodes_b: (product_prod_nat_nat > b) > nat > set_nat).
% 5.74/2.04 thf(func_def_36, type, connected_a: (product_prod_nat_nat > a) > nat > nat > $o).
% 5.74/2.04 thf(func_def_37, type, connected_b: (product_prod_nat_nat > b) > nat > nat > $o).
% 5.74/2.04 thf(func_def_38, type, dist_a: (product_prod_nat_nat > a) > nat > nat > nat > $o).
% 5.74/2.04 thf(func_def_39, type, dist_b: (product_prod_nat_nat > b) > nat > nat > nat > $o).
% 5.74/2.04 thf(func_def_40, type, incoming_a: (product_prod_nat_nat > a) > set_nat > set_Pr1986765409at_nat).
% 5.74/2.04 thf(func_def_41, type, incoming_b: (product_prod_nat_nat > b) > set_nat > set_Pr1986765409at_nat).
% 5.74/2.04 thf(func_def_42, type, isPath_a: (product_prod_nat_nat > a) > nat > list_P559422087at_nat > nat > $o).
% 5.74/2.04 thf(func_def_43, type, isPath_b: (product_prod_nat_nat > b) > nat > list_P559422087at_nat > nat > $o).
% 5.74/2.04 thf(func_def_44, type, isShortestPath_a: (product_prod_nat_nat > a) > nat > list_P559422087at_nat > nat > $o).
% 5.74/2.04 thf(func_def_45, type, isShortestPath_b: (product_prod_nat_nat > b) > nat > list_P559422087at_nat > nat > $o).
% 5.74/2.04 thf(func_def_46, type, isSimplePath_a: (product_prod_nat_nat > a) > nat > list_P559422087at_nat > nat > $o).
% 5.74/2.04 thf(func_def_47, type, isSimplePath_b: (product_prod_nat_nat > b) > nat > list_P559422087at_nat > nat > $o).
% 5.74/2.04 thf(func_def_48, type, min_dist_a: (product_prod_nat_nat > a) > nat > nat > nat).
% 5.74/2.04 thf(func_def_49, type, min_dist_b: (product_prod_nat_nat > b) > nat > nat > nat).
% 5.74/2.04 thf(func_def_50, type, outgoing_a: (product_prod_nat_nat > a) > set_nat > set_Pr1986765409at_nat).
% 5.74/2.04 thf(func_def_51, type, outgoing_b: (product_prod_nat_nat > b) > set_nat > set_Pr1986765409at_nat).
% 5.74/2.04 thf(func_def_52, type, reachableNodes_a: (product_prod_nat_nat > a) > nat > set_nat).
% 5.74/2.04 thf(func_def_53, type, reachableNodes_b: (product_prod_nat_nat > b) > nat > set_nat).
% 5.74/2.04 thf(func_def_54, type, zero_zero_nat: nat).
% 5.74/2.04 thf(func_def_55, type, set_Pr2131844118at_nat: list_P559422087at_nat > set_Pr1986765409at_nat).
% 5.74/2.04 thf(func_def_56, type, size_s1990949619at_nat: list_P559422087at_nat > nat).
% 5.74/2.04 thf(func_def_57, type, ord_less_nat: nat > nat > $o).
% 5.74/2.04 thf(func_def_58, type, ord_less_set_nat: set_nat > set_nat > $o).
% 5.74/2.04 thf(func_def_59, type, ord_le116442893at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o).
% 5.74/2.04 thf(func_def_60, type, ord_less_eq_nat: nat > nat > $o).
% 5.74/2.04 thf(func_def_61, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 5.74/2.04 thf(func_def_62, type, ord_le841296385at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o).
% 5.74/2.04 thf(func_def_63, type, ord_le1613022364et_nat: set_set_nat > set_set_nat > $o).
% 5.74/2.04 thf(func_def_64, type, ord_le2096002913at_nat: set_se1612935105at_nat > set_se1612935105at_nat > $o).
% 5.74/2.04 thf(func_def_65, type, collect_nat: (nat > $o) > set_nat).
% 5.74/2.04 thf(func_def_66, type, collec7649004at_nat: (product_prod_nat_nat > $o) > set_Pr1986765409at_nat).
% 5.74/2.04 thf(func_def_67, type, collect_set_nat: (set_nat > $o) > set_set_nat).
% 5.74/2.04 thf(func_def_68, type, collec1606769740at_nat: (set_Pr1986765409at_nat > $o) > set_se1612935105at_nat).
% 5.74/2.04 thf(func_def_69, type, member1608759472at_nat: list_P559422087at_nat > set_li664300135at_nat > $o).
% 5.74/2.04 thf(func_def_70, type, member_nat: nat > set_nat > $o).
% 5.74/2.04 thf(func_def_71, type, member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o).
% 5.74/2.04 thf(func_def_72, type, member_set_nat: set_nat > set_set_nat > $o).
% 5.74/2.04 thf(func_def_73, type, member298845450at_nat: set_Pr1986765409at_nat > set_se1612935105at_nat > $o).
% 5.74/2.04 thf(func_def_74, type, c: product_prod_nat_nat > a).
% 5.74/2.04 thf(func_def_75, type, c2: product_prod_nat_nat > b).
% 5.74/2.04 thf(func_def_76, type, edges: set_Pr1986765409at_nat).
% 5.74/2.04 thf(func_def_77, type, p: list_P559422087at_nat).
% 5.74/2.04 thf(func_def_78, type, p2: list_P559422087at_nat).
% 5.74/2.04 thf(func_def_79, type, s: nat).
% 5.74/2.04 thf(func_def_80, type, t: nat).
% 5.74/2.04 thf(func_def_104, type, ph1: !>[X0: $tType]:(X0)).
% 5.74/2.04 thf(func_def_105, type, sK2: nat > list_P559422087at_nat).
% 5.74/2.04 thf(f2264,plain,(
% 5.74/2.04 $false),
% 5.74/2.04 inference(trivial_inequality_removal,[status(thm)],[f2263])).
% 5.74/2.04 thf(f2263,plain,(
% 5.74/2.04 ($false = $true)),
% 5.74/2.04 inference(backward_demodulation,[status(thm)],[f1768,f2262])).
% 5.74/2.04 thf(f2262,plain,(
% 5.74/2.04 ($false = (ord_less_eq_nat @ (size_s1990949619at_nat @ p2) @ (size_s1990949619at_nat @ p)))),
% 5.74/2.04 inference(subsumption_resolution,[status(thm)],[f2261,f1743])).
% 5.74/2.04 thf(f1743,plain,(
% 5.74/2.04 ((size_s1990949619at_nat @ p) != (size_s1990949619at_nat @ p2))),
% 5.74/2.04 inference(equality_proxy_clausification,[status(thm)],[f1742])).
% 5.74/2.04 thf(f1742,plain,(
% 5.74/2.04 (((size_s1990949619at_nat @ p) = (size_s1990949619at_nat @ p2)) = $false)),
% 5.74/2.04 inference(not_proxy_clausification,[status(thm)],[f1741])).
% 5.74/2.04 thf(f1741,plain,(
% 5.74/2.04 ($true = (~ ((size_s1990949619at_nat @ p) = (size_s1990949619at_nat @ p2))))),
% 5.74/2.04 inference(binary_proxy_clausification,[status(thm)],[f1720])).
% 5.74/2.04 thf(f1720,plain,(
% 5.74/2.04 ($true = ((~ ((size_s1990949619at_nat @ p) = (size_s1990949619at_nat @ p2))) & (ord_less_eq_nat @ (size_s1990949619at_nat @ p) @ (size_s1990949619at_nat @ p2))))),
% 5.74/2.04 inference(beta_eta_normalization,[status(thm)],[f1484])).
% 5.74/2.04 thf(f1484,plain,(
% 5.74/2.04 (((^[Y0 : nat]: ((^[Y1 : nat]: ((~ (Y0 = Y1)) & (ord_less_eq_nat @ Y0 @ Y1))))) @ (size_s1990949619at_nat @ p) @ (size_s1990949619at_nat @ p2)) = $true)),
% 5.74/2.04 inference(definition_unfolding,[status(thm)],[f1233,f1185])).
% 5.74/2.04 thf(f1185,plain,(
% 5.74/2.04 (ord_less_nat = (^[Y0 : nat]: ((^[Y1 : nat]: ((~ (Y0 = Y1)) & (ord_less_eq_nat @ Y0 @ Y1))))))),
% 5.74/2.04 inference(cnf_transformation,[status(thm)],[f825])).
% 5.74/2.04 thf(f825,plain,(
% 5.74/2.04 (ord_less_nat = (^[Y0 : nat]: ((^[Y1 : nat]: ((~ (Y0 = Y1)) & (ord_less_eq_nat @ Y0 @ Y1))))))),
% 5.74/2.04 inference(fool_elimination,[status(thm)],[f824])).
% 5.74/2.04 thf(f824,plain,(
% 5.74/2.04 (ord_less_nat = (^[X0 : nat, X1 : nat] : ((ord_less_eq_nat @ X0 @ X1) & (X0 != X1))))),
% 5.74/2.04 inference(rectify,[status(thm)],[f104])).
% 5.74/2.04 thf(f104,axiom,(
% 5.74/2.04 (ord_less_nat = (^[X41 : nat, X42 : nat] : ((ord_less_eq_nat @ X41 @ X42) & (X41 != X42))))),
% 5.74/2.04 file('/export/starexec/sandbox2/tmp/tmp.Pn7hie6YQY/DTF2THF_17795.p',fact_103_nat__less__le)).
% 5.74/2.04 thf(f1233,plain,(
% 5.74/2.04 ((ord_less_nat @ (size_s1990949619at_nat @ p) @ (size_s1990949619at_nat @ p2)) = $true)),
% 5.74/2.04 inference(cnf_transformation,[status(thm)],[f905])).
% 5.74/2.04 thf(f905,plain,(
% 5.74/2.04 ((ord_less_nat @ (size_s1990949619at_nat @ p) @ (size_s1990949619at_nat @ p2)) = $true)),
% 5.74/2.04 inference(fool_elimination,[status(thm)],[f904])).
% 5.74/2.04 thf(f904,plain,(
% 5.74/2.04 (ord_less_nat @ (size_s1990949619at_nat @ p) @ (size_s1990949619at_nat @ p2))),
% 5.74/2.04 inference(rectify,[status(thm)],[f14])).
% 5.74/2.04 thf(f14,axiom,(
% 5.74/2.04 (ord_less_nat @ (size_s1990949619at_nat @ p) @ (size_s1990949619at_nat @ p2))),
% 5.74/2.04 file('/export/starexec/sandbox2/tmp/tmp.Pn7hie6YQY/DTF2THF_17795.p',fact_13__092_060open_062length_Ap_A_060_Alength_Ap_H_092_060close_062)).
% 5.74/2.04 thf(f2261,plain,(
% 5.74/2.04 ((size_s1990949619at_nat @ p) = (size_s1990949619at_nat @ p2)) | ($false = (ord_less_eq_nat @ (size_s1990949619at_nat @ p2) @ (size_s1990949619at_nat @ p)))),
% 5.74/2.04 inference(trivial_inequality_removal,[status(thm)],[f2230])).
% 5.74/2.04 thf(f2230,plain,(
% 5.74/2.04 ($false = (ord_less_eq_nat @ (size_s1990949619at_nat @ p2) @ (size_s1990949619at_nat @ p))) | ((size_s1990949619at_nat @ p) = (size_s1990949619at_nat @ p2)) | ($false = $true)),
% 5.74/2.04 inference(superposition,[status(thm)],[f2222,f1740])).
% 5.74/2.04 thf(f1740,plain,(
% 5.74/2.04 ($true = (ord_less_eq_nat @ (size_s1990949619at_nat @ p) @ (size_s1990949619at_nat @ p2)))),
% 5.74/2.04 inference(binary_proxy_clausification,[status(thm)],[f1720])).
% 5.74/2.04 thf(f2222,plain,(
% 5.74/2.04 ( ! [X2 : nat,X1 : nat] : (($false = (ord_less_eq_nat @ X2 @ X1)) | (X1 = X2) | ($false = (ord_less_eq_nat @ X1 @ X2))) )),
% 5.74/2.04 inference(equality_proxy_clausification,[status(thm)],[f2221])).
% 5.74/2.04 thf(f2221,plain,(
% 5.74/2.04 ( ! [X2 : nat,X1 : nat] : (($true = (X2 = X1)) | ($false = (ord_less_eq_nat @ X1 @ X2)) | ($false = (ord_less_eq_nat @ X2 @ X1))) )),
% 5.74/2.04 inference(binary_proxy_clausification,[status(thm)],[f2220])).
% 5.74/2.04 thf(f2220,plain,(
% 5.74/2.04 ( ! [X2 : nat,X1 : nat] : (($false = (ord_less_eq_nat @ X1 @ X2)) | ($true = ((ord_less_eq_nat @ X2 @ X1) => (X2 = X1)))) )),
% 5.74/2.04 inference(binary_proxy_clausification,[status(thm)],[f2219])).
% 5.74/2.04 thf(f2219,plain,(
% 5.74/2.04 ( ! [X2 : nat,X1 : nat] : ((((ord_less_eq_nat @ X1 @ X2) => ((ord_less_eq_nat @ X2 @ X1) => (X2 = X1))) = $true)) )),
% 5.74/2.04 inference(beta_eta_normalization,[status(thm)],[f2218])).
% 5.74/2.04 thf(f2218,plain,(
% 5.74/2.04 ( ! [X2 : nat,X1 : nat] : (($true = ((^[Y0 : nat]: ((ord_less_eq_nat @ X1 @ Y0) => ((ord_less_eq_nat @ Y0 @ X1) => (Y0 = X1)))) @ X2))) )),
% 5.74/2.04 inference(pi_clausification,[status(thm)],[f2203])).
% 5.74/2.04 thf(f2203,plain,(
% 5.74/2.04 ( ! [X1 : nat] : (((!! @ nat @ (^[Y0 : nat]: ((ord_less_eq_nat @ X1 @ Y0) => ((ord_less_eq_nat @ Y0 @ X1) => (Y0 = X1))))) = $true)) )),
% 5.74/2.04 inference(beta_eta_normalization,[status(thm)],[f2202])).
% 5.74/2.04 thf(f2202,plain,(
% 5.74/2.04 ( ! [X1 : nat] : (($true = ((^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less_eq_nat @ Y0 @ Y1) => ((ord_less_eq_nat @ Y1 @ Y0) => (Y1 = Y0)))))) @ X1))) )),
% 5.74/2.04 inference(pi_clausification,[status(thm)],[f1285])).
% 5.74/2.04 thf(f1285,plain,(
% 5.74/2.04 ($true = (!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less_eq_nat @ Y0 @ Y1) => ((ord_less_eq_nat @ Y1 @ Y0) => (Y1 = Y0))))))))),
% 5.74/2.04 inference(cnf_transformation,[status(thm)],[f708])).
% 5.74/2.04 thf(f708,plain,(
% 5.74/2.04 ($true = (!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less_eq_nat @ Y0 @ Y1) => ((ord_less_eq_nat @ Y1 @ Y0) => (Y1 = Y0))))))))),
% 5.74/2.04 inference(fool_elimination,[status(thm)],[f707])).
% 5.74/2.04 thf(f707,plain,(
% 5.74/2.04 ! [X0 : nat,X1 : nat] : ((ord_less_eq_nat @ X1 @ X0) => ((ord_less_eq_nat @ X0 @ X1) => (X0 = X1)))),
% 5.74/2.04 inference(rectify,[status(thm)],[f97])).
% 5.74/2.04 thf(f97,axiom,(
% 5.74/2.04 ! [X25 : nat,X37 : nat] : ((ord_less_eq_nat @ X37 @ X25) => ((ord_less_eq_nat @ X25 @ X37) => (X25 = X37)))),
% 5.74/2.04 file('/export/starexec/sandbox2/tmp/tmp.Pn7hie6YQY/DTF2THF_17795.p',fact_96_le__antisym)).
% 5.74/2.04 thf(f1768,plain,(
% 5.74/2.04 ($true = (ord_less_eq_nat @ (size_s1990949619at_nat @ p2) @ (size_s1990949619at_nat @ p)))),
% 5.74/2.04 inference(binary_proxy_clausification,[status(thm)],[f1767])).
% 5.74/2.04 thf(f1767,plain,(
% 5.74/2.04 (((~ ((size_s1990949619at_nat @ p2) = (size_s1990949619at_nat @ p))) & (ord_less_eq_nat @ (size_s1990949619at_nat @ p2) @ (size_s1990949619at_nat @ p))) = $true)),
% 5.74/2.04 inference(forward_demodulation,[status(thm)],[f1766,f1150])).
% 5.74/2.04 thf(f1150,plain,(
% 5.74/2.04 ((size_s1990949619at_nat @ p) = (min_dist_a @ c @ s @ t))),
% 5.74/2.04 inference(cnf_transformation,[status(thm)],[f376])).
% 5.74/2.04 thf(f376,plain,(
% 5.74/2.04 ((size_s1990949619at_nat @ p) = (min_dist_a @ c @ s @ t))),
% 5.74/2.04 inference(fool_elimination,[status(thm)],[f2])).
% 5.74/2.04 thf(f2,axiom,(
% 5.74/2.04 ((size_s1990949619at_nat @ p) = (min_dist_a @ c @ s @ t))),
% 5.74/2.04 file('/export/starexec/sandbox2/tmp/tmp.Pn7hie6YQY/DTF2THF_17795.p',fact_1_LENP)).
% 5.74/2.04 thf(f1766,plain,(
% 5.74/2.04 (((~ ((size_s1990949619at_nat @ p2) = (min_dist_a @ c @ s @ t))) & (ord_less_eq_nat @ (size_s1990949619at_nat @ p2) @ (min_dist_a @ c @ s @ t))) = $true)),
% 5.74/2.04 inference(forward_demodulation,[status(thm)],[f1612,f1329])).
% 5.74/2.04 thf(f1329,plain,(
% 5.74/2.04 ((size_s1990949619at_nat @ p2) = (min_dist_b @ c2 @ s @ t))),
% 5.74/2.04 inference(cnf_transformation,[status(thm)],[f387])).
% 5.74/2.04 thf(f387,plain,(
% 5.74/2.04 ((size_s1990949619at_nat @ p2) = (min_dist_b @ c2 @ s @ t))),
% 5.74/2.04 inference(fool_elimination,[status(thm)],[f3])).
% 5.74/2.04 thf(f3,axiom,(
% 5.74/2.04 ((size_s1990949619at_nat @ p2) = (min_dist_b @ c2 @ s @ t))),
% 5.74/2.04 file('/export/starexec/sandbox2/tmp/tmp.Pn7hie6YQY/DTF2THF_17795.p',fact_2_LENP_H)).
% 5.74/2.04 thf(f1612,plain,(
% 5.74/2.04 (((~ ((min_dist_b @ c2 @ s @ t) = (min_dist_a @ c @ s @ t))) & (ord_less_eq_nat @ (min_dist_b @ c2 @ s @ t) @ (min_dist_a @ c @ s @ t))) = $true)),
% 5.74/2.04 inference(beta_eta_normalization,[status(thm)],[f1454])).
% 5.74/2.04 thf(f1454,plain,(
% 5.74/2.04 ($true = ((^[Y0 : nat]: ((^[Y1 : nat]: ((~ (Y0 = Y1)) & (ord_less_eq_nat @ Y0 @ Y1))))) @ (min_dist_b @ c2 @ s @ t) @ (min_dist_a @ c @ s @ t)))),
% 5.74/2.04 inference(definition_unfolding,[status(thm)],[f1174,f1185])).
% 5.74/2.04 thf(f1174,plain,(
% 5.74/2.04 ((ord_less_nat @ (min_dist_b @ c2 @ s @ t) @ (min_dist_a @ c @ s @ t)) = $true)),
% 5.74/2.04 inference(cnf_transformation,[status(thm)],[f943])).
% 5.74/2.04 thf(f943,plain,(
% 5.74/2.04 ((ord_less_nat @ (min_dist_b @ c2 @ s @ t) @ (min_dist_a @ c @ s @ t)) = $true)),
% 5.74/2.04 inference(fool_elimination,[status(thm)],[f942])).
% 5.74/2.04 thf(f942,plain,(
% 5.74/2.04 (ord_less_nat @ (min_dist_b @ c2 @ s @ t) @ (min_dist_a @ c @ s @ t))),
% 5.74/2.04 inference(rectify,[status(thm)],[f7])).
% 5.74/2.04 thf(f7,axiom,(
% 5.74/2.04 (ord_less_nat @ (min_dist_b @ c2 @ s @ t) @ (min_dist_a @ c @ s @ t))),
% 5.74/2.04 file('/export/starexec/sandbox2/tmp/tmp.Pn7hie6YQY/DTF2THF_17795.p',fact_6_SHORTER)).
% 5.74/2.04 % SZS output end Proof for DTF2THF_17795
% 5.74/2.04 % (18034)------------------------------
% 5.74/2.04 % (18034)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.74/2.04 % (18034)Termination reason: Refutation
% 5.74/2.04
% 5.74/2.04 % (18034)Memory used [KB]: 7931
% 5.74/2.04 % (18034)Time elapsed: 0.144 s
% 5.74/2.04 % (18034)Instructions burned: 338 (million)
% 5.74/2.04 % (18034)------------------------------
% 5.74/2.04 % (18034)------------------------------
% 5.74/2.04 % (17943)Success in time 0.46 s
%------------------------------------------------------------------------------