TPTP Problem File: ITP052^1.p
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%------------------------------------------------------------------------------
% File : ITP052^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer EdmondsKarp_Termination_Abstract problem prob_430__7598256_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : EdmondsKarp_Termination_Abstract/prob_430__7598256_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 338 ( 114 unt; 58 typ; 0 def)
% Number of atoms : 729 ( 198 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 2470 ( 62 ~; 2 |; 35 &;2075 @)
% ( 0 <=>; 296 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 332 ( 332 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 52 usr; 7 con; 0-5 aty)
% Number of variables : 813 ( 62 ^; 725 !; 26 ?; 813 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:32:13.019
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P559422087at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1986765409at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_tf__capacity,type,
capacity: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (52)
thf(sy_c_Augmenting__Flow_ONFlow_Oaugment_001tf__capacity,type,
augmen887209518pacity: ( product_prod_nat_nat > capacity ) > ( product_prod_nat_nat > capacity ) > ( product_prod_nat_nat > capacity ) > product_prod_nat_nat > capacity ).
thf(sy_c_Augmenting__Path_ONPreflow_OaugmentingFlow_001tf__capacity,type,
augmen72931474pacity: ( product_prod_nat_nat > capacity ) > ( product_prod_nat_nat > capacity ) > list_P559422087at_nat > product_prod_nat_nat > capacity ).
thf(sy_c_Augmenting__Path_ONPreflow_OisAugmentingPath_001tf__capacity,type,
augmen1090971539pacity: ( product_prod_nat_nat > capacity ) > nat > nat > ( product_prod_nat_nat > capacity ) > list_P559422087at_nat > $o ).
thf(sy_c_Augmenting__Path_ONPreflow_OresCap_001tf__capacity,type,
augmen68920357pacity: ( product_prod_nat_nat > capacity ) > ( product_prod_nat_nat > capacity ) > list_P559422087at_nat > capacity ).
thf(sy_c_Graph_OFinite__Graph_001tf__capacity,type,
finite217307323pacity: ( product_prod_nat_nat > capacity ) > $o ).
thf(sy_c_Graph_OGraph_OisShortestPath_001tf__capacity,type,
isShor1936442771pacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).
thf(sy_c_Graph_OGraph_OisSimplePath_001tf__capacity,type,
isSimp1359852763pacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).
thf(sy_c_Groups_Oplus__class_Oplus_001tf__capacity,type,
plus_plus_capacity: capacity > capacity > capacity ).
thf(sy_c_Groups_Ozero__class_Ozero_001tf__capacity,type,
zero_zero_capacity: capacity ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
inf_in2138576462_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
inf_in586391887at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > set_Pr1986765409at_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_Pr2131844118at_nat: list_P559422087at_nat > set_Pr1986765409at_nat ).
thf(sy_c_Network_OFinite__Preflow_001tf__capacity,type,
finite1387368318pacity: ( product_prod_nat_nat > capacity ) > nat > nat > ( product_prod_nat_nat > capacity ) > $o ).
thf(sy_c_Network_OFlow_001tf__capacity,type,
flow_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > ( product_prod_nat_nat > capacity ) > $o ).
thf(sy_c_Network_OFlow_Oval_001tf__capacity,type,
val_capacity: ( product_prod_nat_nat > capacity ) > nat > ( product_prod_nat_nat > capacity ) > capacity ).
thf(sy_c_Network_ONCut_001tf__capacity,type,
nCut_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > set_nat > $o ).
thf(sy_c_Network_ONCut_Ocap_001tf__capacity,type,
cap_capacity: ( product_prod_nat_nat > capacity ) > set_nat > capacity ).
thf(sy_c_Network_ONFlow_001tf__capacity,type,
nFlow_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > ( product_prod_nat_nat > capacity ) > $o ).
thf(sy_c_Network_ONPreflow_001tf__capacity,type,
nPreflow_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > ( product_prod_nat_nat > capacity ) > $o ).
thf(sy_c_Network_ONetwork_001tf__capacity,type,
network_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > $o ).
thf(sy_c_Network_ONetwork_Oexcess_001tf__capacity,type,
excess_capacity: ( product_prod_nat_nat > capacity ) > ( product_prod_nat_nat > capacity ) > nat > capacity ).
thf(sy_c_Network_ONetwork_OisMaxFlow_001tf__capacity,type,
isMaxFlow_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > ( product_prod_nat_nat > capacity ) > $o ).
thf(sy_c_Network_ONetwork_Ois__max__flow__val_001tf__capacity,type,
is_max2144541363pacity: ( product_prod_nat_nat > capacity ) > nat > nat > capacity > $o ).
thf(sy_c_Network_OPreflow_001tf__capacity,type,
preflow_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > ( product_prod_nat_nat > capacity ) > $o ).
thf(sy_c_Network_OisMinCut_001tf__capacity,type,
isMinCut_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > set_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
bot_bo513358416_nat_o: product_prod_nat_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
bot_bo2130386637at_nat: set_Pr1986765409at_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le1015898640_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le116442893at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__capacity,type,
ord_less_capacity: capacity > capacity > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
ord_le1039616028_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
ord_le841296385at_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__capacity,type,
ord_less_eq_capacity: capacity > capacity > $o ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001t__Nat__Onat,type,
product_swap_nat_nat: product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Relation_Oconverse_001t__Nat__Onat_001t__Nat__Onat,type,
converse_nat_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat ).
thf(sy_c_Residual__Graph_ONetwork_Oflow__of__cf_001tf__capacity,type,
residu1549650759pacity: ( product_prod_nat_nat > capacity ) > ( product_prod_nat_nat > capacity ) > product_prod_nat_nat > capacity ).
thf(sy_c_Residual__Graph_ORGraph_001tf__capacity,type,
residu800046103pacity: ( product_prod_nat_nat > capacity ) > nat > nat > ( product_prod_nat_nat > capacity ) > $o ).
thf(sy_c_Residual__Graph_ORPreGraph_001tf__capacity,type,
residu164581374pacity: ( product_prod_nat_nat > capacity ) > nat > nat > ( product_prod_nat_nat > capacity ) > $o ).
thf(sy_c_Residual__Graph_OresidualGraph_001tf__capacity,type,
residu203630698pacity: ( product_prod_nat_nat > capacity ) > ( product_prod_nat_nat > capacity ) > product_prod_nat_nat > capacity ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
thf(sy_v_c,type,
c: product_prod_nat_nat > capacity ).
thf(sy_v_f,type,
f: product_prod_nat_nat > capacity ).
thf(sy_v_p,type,
p: list_P559422087at_nat ).
thf(sy_v_s,type,
s: nat ).
thf(sy_v_t,type,
t: nat ).
% Relevant facts (278)
thf(fact_0_t__not__s,axiom,
t != s ).
% t_not_s
thf(fact_1_cf_OisSPath__nt__parallel__pf,axiom,
! [S: nat,P: list_P559422087at_nat,T: nat] :
( ( isSimp1359852763pacity @ ( residu203630698pacity @ c @ f ) @ S @ P @ T )
=> ( ( inf_in586391887at_nat @ ( set_Pr2131844118at_nat @ P ) @ ( converse_nat_nat @ ( set_Pr2131844118at_nat @ P ) ) )
= bot_bo2130386637at_nat ) ) ).
% cf.isSPath_nt_parallel_pf
thf(fact_2_isSPath__nt__parallel__pf,axiom,
! [S: nat,P: list_P559422087at_nat,T: nat] :
( ( isSimp1359852763pacity @ c @ S @ P @ T )
=> ( ( inf_in586391887at_nat @ ( set_Pr2131844118at_nat @ P ) @ ( converse_nat_nat @ ( set_Pr2131844118at_nat @ P ) ) )
= bot_bo2130386637at_nat ) ) ).
% isSPath_nt_parallel_pf
thf(fact_3_NFlow__axioms,axiom,
nFlow_capacity @ c @ s @ t @ f ).
% NFlow_axioms
thf(fact_4_max__flow__val__unique,axiom,
! [Fv1: capacity,Fv2: capacity] :
( ( is_max2144541363pacity @ c @ s @ t @ Fv1 )
=> ( ( is_max2144541363pacity @ c @ s @ t @ Fv2 )
=> ( Fv1 = Fv2 ) ) ) ).
% max_flow_val_unique
thf(fact_5_isAugmentingPath__def,axiom,
! [P: list_P559422087at_nat] :
( ( augmen1090971539pacity @ c @ s @ t @ f @ P )
= ( isSimp1359852763pacity @ ( residu203630698pacity @ c @ f ) @ s @ P @ t ) ) ).
% isAugmentingPath_def
thf(fact_6_Finite__Preflow__axioms,axiom,
finite1387368318pacity @ c @ s @ t @ f ).
% Finite_Preflow_axioms
thf(fact_7_fofu__I__II,axiom,
( ( isMaxFlow_capacity @ c @ s @ t @ f )
=> ~ ? [X_1: list_P559422087at_nat] : ( augmen1090971539pacity @ c @ s @ t @ f @ X_1 ) ) ).
% fofu_I_II
thf(fact_8_noAugPath__iff__maxFlow,axiom,
( ( ~ ? [X: list_P559422087at_nat] : ( augmen1090971539pacity @ c @ s @ t @ f @ X ) )
= ( isMaxFlow_capacity @ c @ s @ t @ f ) ) ).
% noAugPath_iff_maxFlow
thf(fact_9_Preflow__axioms,axiom,
preflow_capacity @ c @ s @ t @ f ).
% Preflow_axioms
thf(fact_10_is__RGraph,axiom,
residu800046103pacity @ c @ s @ t @ ( residu203630698pacity @ c @ f ) ).
% is_RGraph
thf(fact_11_Flow__axioms,axiom,
flow_capacity @ c @ s @ t @ f ).
% Flow_axioms
thf(fact_12_NPreflow__axioms,axiom,
nPreflow_capacity @ c @ s @ t @ f ).
% NPreflow_axioms
thf(fact_13_converse__empty,axiom,
( ( converse_nat_nat @ bot_bo2130386637at_nat )
= bot_bo2130386637at_nat ) ).
% converse_empty
thf(fact_14_inf__bot__left,axiom,
! [X2: product_prod_nat_nat > $o] :
( ( inf_in2138576462_nat_o @ bot_bo513358416_nat_o @ X2 )
= bot_bo513358416_nat_o ) ).
% inf_bot_left
thf(fact_15_inf__bot__left,axiom,
! [X2: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ bot_bo2130386637at_nat @ X2 )
= bot_bo2130386637at_nat ) ).
% inf_bot_left
thf(fact_16_inf__bot__right,axiom,
! [X2: product_prod_nat_nat > $o] :
( ( inf_in2138576462_nat_o @ X2 @ bot_bo513358416_nat_o )
= bot_bo513358416_nat_o ) ).
% inf_bot_right
thf(fact_17_inf__bot__right,axiom,
! [X2: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ X2 @ bot_bo2130386637at_nat )
= bot_bo2130386637at_nat ) ).
% inf_bot_right
thf(fact_18_is__RPreGraph,axiom,
residu164581374pacity @ c @ s @ t @ ( residu203630698pacity @ c @ f ) ).
% is_RPreGraph
thf(fact_19_Network__axioms,axiom,
network_capacity @ c @ s @ t ).
% Network_axioms
thf(fact_20_fo__rg__inv,axiom,
( ( residu1549650759pacity @ c @ ( residu203630698pacity @ c @ f ) )
= f ) ).
% fo_rg_inv
thf(fact_21_inf__right__idem,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ ( inf_in586391887at_nat @ X2 @ Y ) @ Y )
= ( inf_in586391887at_nat @ X2 @ Y ) ) ).
% inf_right_idem
thf(fact_22_inf_Oright__idem,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ ( inf_in586391887at_nat @ A @ B ) @ B )
= ( inf_in586391887at_nat @ A @ B ) ) ).
% inf.right_idem
thf(fact_23_inf__left__idem,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ X2 @ ( inf_in586391887at_nat @ X2 @ Y ) )
= ( inf_in586391887at_nat @ X2 @ Y ) ) ).
% inf_left_idem
thf(fact_24_inf_Oleft__idem,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ A @ ( inf_in586391887at_nat @ A @ B ) )
= ( inf_in586391887at_nat @ A @ B ) ) ).
% inf.left_idem
thf(fact_25_inf__idem,axiom,
! [X2: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ X2 @ X2 )
= X2 ) ).
% inf_idem
thf(fact_26_inf_Oidem,axiom,
! [A: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_27_converse__converse,axiom,
! [R: set_Pr1986765409at_nat] :
( ( converse_nat_nat @ ( converse_nat_nat @ R ) )
= R ) ).
% converse_converse
thf(fact_28_converse__inject,axiom,
! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
( ( ( converse_nat_nat @ R )
= ( converse_nat_nat @ S ) )
= ( R = S ) ) ).
% converse_inject
thf(fact_29_augFlow__resFlow,axiom,
! [P: list_P559422087at_nat] :
( ( augmen1090971539pacity @ c @ s @ t @ f @ P )
=> ( flow_capacity @ ( residu203630698pacity @ c @ f ) @ s @ t @ ( augmen72931474pacity @ c @ f @ P ) ) ) ).
% augFlow_resFlow
thf(fact_30_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_31_bot__empty__eq,axiom,
( bot_bo513358416_nat_o
= ( ^ [X3: product_prod_nat_nat] : ( member701585322at_nat @ X3 @ bot_bo2130386637at_nat ) ) ) ).
% bot_empty_eq
thf(fact_32_inf__left__commute,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ X2 @ ( inf_in586391887at_nat @ Y @ Z ) )
= ( inf_in586391887at_nat @ Y @ ( inf_in586391887at_nat @ X2 @ Z ) ) ) ).
% inf_left_commute
thf(fact_33_inf_Oleft__commute,axiom,
! [B: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ B @ ( inf_in586391887at_nat @ A @ C ) )
= ( inf_in586391887at_nat @ A @ ( inf_in586391887at_nat @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_34_inf__commute,axiom,
( inf_in586391887at_nat
= ( ^ [X3: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] : ( inf_in586391887at_nat @ Y2 @ X3 ) ) ) ).
% inf_commute
thf(fact_35_inf_Ocommute,axiom,
( inf_in586391887at_nat
= ( ^ [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] : ( inf_in586391887at_nat @ B2 @ A2 ) ) ) ).
% inf.commute
thf(fact_36_inf__assoc,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ ( inf_in586391887at_nat @ X2 @ Y ) @ Z )
= ( inf_in586391887at_nat @ X2 @ ( inf_in586391887at_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_37_inf_Oassoc,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ ( inf_in586391887at_nat @ A @ B ) @ C )
= ( inf_in586391887at_nat @ A @ ( inf_in586391887at_nat @ B @ C ) ) ) ).
% inf.assoc
thf(fact_38_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_Pr1986765409at_nat,K: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( B3
= ( inf_in586391887at_nat @ K @ B ) )
=> ( ( inf_in586391887at_nat @ A @ B3 )
= ( inf_in586391887at_nat @ K @ ( inf_in586391887at_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_39_boolean__algebra__cancel_Oinf1,axiom,
! [A3: set_Pr1986765409at_nat,K: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( A3
= ( inf_in586391887at_nat @ K @ A ) )
=> ( ( inf_in586391887at_nat @ A3 @ B )
= ( inf_in586391887at_nat @ K @ ( inf_in586391887at_nat @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_40_inf__sup__aci_I1_J,axiom,
( inf_in586391887at_nat
= ( ^ [X3: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] : ( inf_in586391887at_nat @ Y2 @ X3 ) ) ) ).
% inf_sup_aci(1)
thf(fact_41_inf__sup__aci_I2_J,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ ( inf_in586391887at_nat @ X2 @ Y ) @ Z )
= ( inf_in586391887at_nat @ X2 @ ( inf_in586391887at_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_42_inf__sup__aci_I3_J,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ X2 @ ( inf_in586391887at_nat @ Y @ Z ) )
= ( inf_in586391887at_nat @ Y @ ( inf_in586391887at_nat @ X2 @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_43_inf__sup__aci_I4_J,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ X2 @ ( inf_in586391887at_nat @ X2 @ Y ) )
= ( inf_in586391887at_nat @ X2 @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_44_converse__Int,axiom,
! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
( ( converse_nat_nat @ ( inf_in586391887at_nat @ R @ S ) )
= ( inf_in586391887at_nat @ ( converse_nat_nat @ R ) @ ( converse_nat_nat @ S ) ) ) ).
% converse_Int
thf(fact_45_augment__flow__presv,axiom,
! [F: product_prod_nat_nat > capacity] :
( ( flow_capacity @ ( residu203630698pacity @ c @ f ) @ s @ t @ F )
=> ( flow_capacity @ c @ s @ t @ ( augmen887209518pacity @ c @ f @ F ) ) ) ).
% augment_flow_presv
thf(fact_46_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
( ( member701585322at_nat @ A @ ( collec7649004at_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_48_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A3: set_Pr1986765409at_nat] :
( ( collec7649004at_nat
@ ^ [X3: product_prod_nat_nat] : ( member701585322at_nat @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_50_Collect__cong,axiom,
! [P2: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X4: product_prod_nat_nat] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collec7649004at_nat @ P2 )
= ( collec7649004at_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_51_cf_OisSPath__nt__parallel,axiom,
! [S: nat,P: list_P559422087at_nat,T: nat,E: product_prod_nat_nat] :
( ( isSimp1359852763pacity @ ( residu203630698pacity @ c @ f ) @ S @ P @ T )
=> ( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
=> ~ ( member701585322at_nat @ ( product_swap_nat_nat @ E ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ).
% cf.isSPath_nt_parallel
thf(fact_52_is__max__flow__val__def,axiom,
! [Fv: capacity] :
( ( is_max2144541363pacity @ c @ s @ t @ Fv )
= ( ? [F2: product_prod_nat_nat > capacity] :
( ( isMaxFlow_capacity @ c @ s @ t @ F2 )
& ( Fv
= ( val_capacity @ c @ s @ F2 ) ) ) ) ) ).
% is_max_flow_val_def
thf(fact_53_cf_OFinite__Graph__axioms,axiom,
finite217307323pacity @ ( residu203630698pacity @ c @ f ) ).
% cf.Finite_Graph_axioms
thf(fact_54_ford__fulkerson_I1_J,axiom,
( ( isMaxFlow_capacity @ c @ s @ t @ f )
= ( ~ ? [X: list_P559422087at_nat] : ( augmen1090971539pacity @ c @ s @ t @ f @ X ) ) ) ).
% ford_fulkerson(1)
thf(fact_55_shortest__is__augmenting,axiom,
! [P: list_P559422087at_nat] :
( ( isShor1936442771pacity @ ( residu203630698pacity @ c @ f ) @ s @ P @ t )
=> ( augmen1090971539pacity @ c @ s @ t @ f @ P ) ) ).
% shortest_is_augmenting
thf(fact_56_augmenting__path__imp__shortest,axiom,
! [P: list_P559422087at_nat] :
( ( augmen1090971539pacity @ c @ s @ t @ f @ P )
=> ? [P3: list_P559422087at_nat] : ( isShor1936442771pacity @ ( residu203630698pacity @ c @ f ) @ s @ P3 @ t ) ) ).
% augmenting_path_imp_shortest
thf(fact_57_Graph_OisSPath__nt__parallel__pf,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat] :
( ( isSimp1359852763pacity @ C @ S @ P @ T )
=> ( ( inf_in586391887at_nat @ ( set_Pr2131844118at_nat @ P ) @ ( converse_nat_nat @ ( set_Pr2131844118at_nat @ P ) ) )
= bot_bo2130386637at_nat ) ) ).
% Graph.isSPath_nt_parallel_pf
thf(fact_58_RPreGraph_Omaxflow__imp__rgraph,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,Cf: product_prod_nat_nat > capacity] :
( ( residu164581374pacity @ C @ S @ T @ Cf )
=> ( ( isMaxFlow_capacity @ C @ S @ T @ ( residu1549650759pacity @ C @ Cf ) )
=> ( residu800046103pacity @ C @ S @ T @ Cf ) ) ) ).
% RPreGraph.maxflow_imp_rgraph
thf(fact_59_isSPath__nt__parallel,axiom,
! [S: nat,P: list_P559422087at_nat,T: nat,E: product_prod_nat_nat] :
( ( isSimp1359852763pacity @ c @ S @ P @ T )
=> ( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
=> ~ ( member701585322at_nat @ ( product_swap_nat_nat @ E ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ).
% isSPath_nt_parallel
thf(fact_60_Finite__Graph__axioms,axiom,
finite217307323pacity @ c ).
% Finite_Graph_axioms
thf(fact_61_shortestPath__is__simple,axiom,
! [S: nat,P: list_P559422087at_nat,T: nat] :
( ( isShor1936442771pacity @ c @ S @ P @ T )
=> ( isSimp1359852763pacity @ c @ S @ P @ T ) ) ).
% shortestPath_is_simple
thf(fact_62_cf_OshortestPath__is__simple,axiom,
! [S: nat,P: list_P559422087at_nat,T: nat] :
( ( isShor1936442771pacity @ ( residu203630698pacity @ c @ f ) @ S @ P @ T )
=> ( isSimp1359852763pacity @ ( residu203630698pacity @ c @ f ) @ S @ P @ T ) ) ).
% cf.shortestPath_is_simple
thf(fact_63_Graph_OisShortestPath_Ocong,axiom,
isShor1936442771pacity = isShor1936442771pacity ).
% Graph.isShortestPath.cong
thf(fact_64_Graph_OshortestPath__is__simple,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat] :
( ( isShor1936442771pacity @ C @ S @ P @ T )
=> ( isSimp1359852763pacity @ C @ S @ P @ T ) ) ).
% Graph.shortestPath_is_simple
thf(fact_65_Graph_OisSPath__nt__parallel,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,E: product_prod_nat_nat] :
( ( isSimp1359852763pacity @ C @ S @ P @ T )
=> ( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
=> ~ ( member701585322at_nat @ ( product_swap_nat_nat @ E ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ).
% Graph.isSPath_nt_parallel
thf(fact_66_Graph_OisSimplePath_Ocong,axiom,
isSimp1359852763pacity = isSimp1359852763pacity ).
% Graph.isSimplePath.cong
thf(fact_67_RPreGraph_Othis__loc__rpg,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,Cf: product_prod_nat_nat > capacity] :
( ( residu164581374pacity @ C @ S @ T @ Cf )
=> ( residu164581374pacity @ C @ S @ T @ Cf ) ) ).
% RPreGraph.this_loc_rpg
thf(fact_68_NFlow_Oshortest__is__augmenting,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity,P: list_P559422087at_nat] :
( ( nFlow_capacity @ C @ S @ T @ F3 )
=> ( ( isShor1936442771pacity @ ( residu203630698pacity @ C @ F3 ) @ S @ P @ T )
=> ( augmen1090971539pacity @ C @ S @ T @ F3 @ P ) ) ) ).
% NFlow.shortest_is_augmenting
thf(fact_69_NFlow_Oaugmenting__path__imp__shortest,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity,P: list_P559422087at_nat] :
( ( nFlow_capacity @ C @ S @ T @ F3 )
=> ( ( augmen1090971539pacity @ C @ S @ T @ F3 @ P )
=> ? [P3: list_P559422087at_nat] : ( isShor1936442771pacity @ ( residu203630698pacity @ C @ F3 ) @ S @ P3 @ T ) ) ) ).
% NFlow.augmenting_path_imp_shortest
thf(fact_70_RGraph_Othis__loc,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,Cf: product_prod_nat_nat > capacity] :
( ( residu800046103pacity @ C @ S @ T @ Cf )
=> ( residu800046103pacity @ C @ S @ T @ Cf ) ) ).
% RGraph.this_loc
thf(fact_71_Network_Oflow__of__cf_Ocong,axiom,
residu1549650759pacity = residu1549650759pacity ).
% Network.flow_of_cf.cong
thf(fact_72_RPreGraph_Oaxioms_I1_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,Cf: product_prod_nat_nat > capacity] :
( ( residu164581374pacity @ C @ S @ T @ Cf )
=> ( network_capacity @ C @ S @ T ) ) ).
% RPreGraph.axioms(1)
thf(fact_73_RGraph_Oaxioms_I1_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,Cf: product_prod_nat_nat > capacity] :
( ( residu800046103pacity @ C @ S @ T @ Cf )
=> ( network_capacity @ C @ S @ T ) ) ).
% RGraph.axioms(1)
thf(fact_74_RGraph_Othis__loc__rpg,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,Cf: product_prod_nat_nat > capacity] :
( ( residu800046103pacity @ C @ S @ T @ Cf )
=> ( residu164581374pacity @ C @ S @ T @ Cf ) ) ).
% RGraph.this_loc_rpg
thf(fact_75_NPreflow_Ois__RPreGraph,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( nPreflow_capacity @ C @ S @ T @ F3 )
=> ( residu164581374pacity @ C @ S @ T @ ( residu203630698pacity @ C @ F3 ) ) ) ).
% NPreflow.is_RPreGraph
thf(fact_76_RPreGraph_OEX__RPG,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,Cf: product_prod_nat_nat > capacity] :
( ( residu164581374pacity @ C @ S @ T @ Cf )
=> ? [F4: product_prod_nat_nat > capacity] :
( ( nPreflow_capacity @ C @ S @ T @ F4 )
& ( Cf
= ( residu203630698pacity @ C @ F4 ) ) ) ) ).
% RPreGraph.EX_RPG
thf(fact_77_NFlow_Ois__RGraph,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( nFlow_capacity @ C @ S @ T @ F3 )
=> ( residu800046103pacity @ C @ S @ T @ ( residu203630698pacity @ C @ F3 ) ) ) ).
% NFlow.is_RGraph
thf(fact_78_RGraph_OEX__RG,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,Cf: product_prod_nat_nat > capacity] :
( ( residu800046103pacity @ C @ S @ T @ Cf )
=> ? [F4: product_prod_nat_nat > capacity] :
( ( nFlow_capacity @ C @ S @ T @ F4 )
& ( Cf
= ( residu203630698pacity @ C @ F4 ) ) ) ) ).
% RGraph.EX_RG
thf(fact_79_NPreflow_Ofo__rg__inv,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( nPreflow_capacity @ C @ S @ T @ F3 )
=> ( ( residu1549650759pacity @ C @ ( residu203630698pacity @ C @ F3 ) )
= F3 ) ) ).
% NPreflow.fo_rg_inv
thf(fact_80_RPreGraph_Org__fo__inv,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,Cf: product_prod_nat_nat > capacity] :
( ( residu164581374pacity @ C @ S @ T @ Cf )
=> ( ( residu203630698pacity @ C @ ( residu1549650759pacity @ C @ Cf ) )
= Cf ) ) ).
% RPreGraph.rg_fo_inv
thf(fact_81_RPreGraph_Ois__NPreflow,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,Cf: product_prod_nat_nat > capacity] :
( ( residu164581374pacity @ C @ S @ T @ Cf )
=> ( nPreflow_capacity @ C @ S @ T @ ( residu1549650759pacity @ C @ Cf ) ) ) ).
% RPreGraph.is_NPreflow
thf(fact_82_RGraph_Ois__NFlow,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,Cf: product_prod_nat_nat > capacity] :
( ( residu800046103pacity @ C @ S @ T @ Cf )
=> ( nFlow_capacity @ C @ S @ T @ ( residu1549650759pacity @ C @ Cf ) ) ) ).
% RGraph.is_NFlow
thf(fact_83_augFlow__val,axiom,
! [P: list_P559422087at_nat] :
( ( augmen1090971539pacity @ c @ s @ t @ f @ P )
=> ( ( val_capacity @ ( residu203630698pacity @ c @ f ) @ s @ ( augmen72931474pacity @ c @ f @ P ) )
= ( augmen68920357pacity @ c @ f @ P ) ) ) ).
% augFlow_val
thf(fact_84_augment__flow__value,axiom,
! [F: product_prod_nat_nat > capacity] :
( ( flow_capacity @ ( residu203630698pacity @ c @ f ) @ s @ t @ F )
=> ( ( val_capacity @ c @ s @ ( augmen887209518pacity @ c @ f @ F ) )
= ( plus_plus_capacity @ ( val_capacity @ c @ s @ f ) @ ( val_capacity @ ( residu203630698pacity @ c @ f ) @ s @ F ) ) ) ) ).
% augment_flow_value
thf(fact_85_isMaxFlow__alt,axiom,
! [F3: product_prod_nat_nat > capacity] :
( ( isMaxFlow_capacity @ c @ s @ t @ F3 )
= ( ( nFlow_capacity @ c @ s @ t @ F3 )
& ! [F5: product_prod_nat_nat > capacity] :
( ( nFlow_capacity @ c @ s @ t @ F5 )
=> ( ord_less_eq_capacity @ ( val_capacity @ c @ s @ F5 ) @ ( val_capacity @ c @ s @ F3 ) ) ) ) ) ).
% isMaxFlow_alt
thf(fact_86_isMaxFlow__def,axiom,
! [F3: product_prod_nat_nat > capacity] :
( ( isMaxFlow_capacity @ c @ s @ t @ F3 )
= ( ( flow_capacity @ c @ s @ t @ F3 )
& ! [F5: product_prod_nat_nat > capacity] :
( ( flow_capacity @ c @ s @ t @ F5 )
=> ( ord_less_eq_capacity @ ( val_capacity @ c @ s @ F5 ) @ ( val_capacity @ c @ s @ F3 ) ) ) ) ) ).
% isMaxFlow_def
thf(fact_87_NPreflow_OaugFlow__resFlow,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity,P: list_P559422087at_nat] :
( ( nPreflow_capacity @ C @ S @ T @ F3 )
=> ( ( augmen1090971539pacity @ C @ S @ T @ F3 @ P )
=> ( flow_capacity @ ( residu203630698pacity @ C @ F3 ) @ S @ T @ ( augmen72931474pacity @ C @ F3 @ P ) ) ) ) ).
% NPreflow.augFlow_resFlow
thf(fact_88_bot__apply,axiom,
( bot_bo513358416_nat_o
= ( ^ [X3: product_prod_nat_nat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_89_maxFlow__minCut,axiom,
! [F3: product_prod_nat_nat > capacity,K: set_nat] :
( ( isMaxFlow_capacity @ c @ s @ t @ F3 )
=> ( ( isMinCut_capacity @ c @ s @ t @ K )
=> ( ( val_capacity @ c @ s @ F3 )
= ( cap_capacity @ c @ K ) ) ) ) ).
% maxFlow_minCut
thf(fact_90_order__refl,axiom,
! [X2: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_91_order__refl,axiom,
! [X2: capacity] : ( ord_less_eq_capacity @ X2 @ X2 ) ).
% order_refl
thf(fact_92_inf_Obounded__iff,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ ( inf_in586391887at_nat @ B @ C ) )
= ( ( ord_le841296385at_nat @ A @ B )
& ( ord_le841296385at_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_93_le__inf__iff,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ ( inf_in586391887at_nat @ Y @ Z ) )
= ( ( ord_le841296385at_nat @ X2 @ Y )
& ( ord_le841296385at_nat @ X2 @ Z ) ) ) ).
% le_inf_iff
thf(fact_94_bot_Oextremum,axiom,
! [A: product_prod_nat_nat > $o] : ( ord_le1039616028_nat_o @ bot_bo513358416_nat_o @ A ) ).
% bot.extremum
thf(fact_95_bot_Oextremum,axiom,
! [A: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ bot_bo2130386637at_nat @ A ) ).
% bot.extremum
thf(fact_96_bot_Oextremum__unique,axiom,
! [A: product_prod_nat_nat > $o] :
( ( ord_le1039616028_nat_o @ A @ bot_bo513358416_nat_o )
= ( A = bot_bo513358416_nat_o ) ) ).
% bot.extremum_unique
thf(fact_97_bot_Oextremum__unique,axiom,
! [A: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ bot_bo2130386637at_nat )
= ( A = bot_bo2130386637at_nat ) ) ).
% bot.extremum_unique
thf(fact_98_bot_Oextremum__uniqueI,axiom,
! [A: product_prod_nat_nat > $o] :
( ( ord_le1039616028_nat_o @ A @ bot_bo513358416_nat_o )
=> ( A = bot_bo513358416_nat_o ) ) ).
% bot.extremum_uniqueI
thf(fact_99_bot_Oextremum__uniqueI,axiom,
! [A: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ bot_bo2130386637at_nat )
=> ( A = bot_bo2130386637at_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_100_order__subst1,axiom,
! [A: capacity,F3: set_Pr1986765409at_nat > capacity,B: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_less_eq_capacity @ A @ ( F3 @ B ) )
=> ( ( ord_le841296385at_nat @ B @ C )
=> ( ! [X4: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X4 @ Y3 )
=> ( ord_less_eq_capacity @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_capacity @ A @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_101_order__subst1,axiom,
! [A: set_Pr1986765409at_nat,F3: capacity > set_Pr1986765409at_nat,B: capacity,C: capacity] :
( ( ord_le841296385at_nat @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_capacity @ B @ C )
=> ( ! [X4: capacity,Y3: capacity] :
( ( ord_less_eq_capacity @ X4 @ Y3 )
=> ( ord_le841296385at_nat @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le841296385at_nat @ A @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_102_order__subst1,axiom,
! [A: set_Pr1986765409at_nat,F3: set_Pr1986765409at_nat > set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ ( F3 @ B ) )
=> ( ( ord_le841296385at_nat @ B @ C )
=> ( ! [X4: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X4 @ Y3 )
=> ( ord_le841296385at_nat @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le841296385at_nat @ A @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_103_order__subst1,axiom,
! [A: capacity,F3: capacity > capacity,B: capacity,C: capacity] :
( ( ord_less_eq_capacity @ A @ ( F3 @ B ) )
=> ( ( ord_less_eq_capacity @ B @ C )
=> ( ! [X4: capacity,Y3: capacity] :
( ( ord_less_eq_capacity @ X4 @ Y3 )
=> ( ord_less_eq_capacity @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_capacity @ A @ ( F3 @ C ) ) ) ) ) ).
% order_subst1
thf(fact_104_order__subst2,axiom,
! [A: capacity,B: capacity,F3: capacity > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_less_eq_capacity @ A @ B )
=> ( ( ord_le841296385at_nat @ ( F3 @ B ) @ C )
=> ( ! [X4: capacity,Y3: capacity] :
( ( ord_less_eq_capacity @ X4 @ Y3 )
=> ( ord_le841296385at_nat @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le841296385at_nat @ ( F3 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_105_order__subst2,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,F3: set_Pr1986765409at_nat > capacity,C: capacity] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ( ord_less_eq_capacity @ ( F3 @ B ) @ C )
=> ( ! [X4: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X4 @ Y3 )
=> ( ord_less_eq_capacity @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_capacity @ ( F3 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_106_order__subst2,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,F3: set_Pr1986765409at_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ( ord_le841296385at_nat @ ( F3 @ B ) @ C )
=> ( ! [X4: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X4 @ Y3 )
=> ( ord_le841296385at_nat @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le841296385at_nat @ ( F3 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_107_order__subst2,axiom,
! [A: capacity,B: capacity,F3: capacity > capacity,C: capacity] :
( ( ord_less_eq_capacity @ A @ B )
=> ( ( ord_less_eq_capacity @ ( F3 @ B ) @ C )
=> ( ! [X4: capacity,Y3: capacity] :
( ( ord_less_eq_capacity @ X4 @ Y3 )
=> ( ord_less_eq_capacity @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_capacity @ ( F3 @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_108_ord__eq__le__subst,axiom,
! [A: set_Pr1986765409at_nat,F3: capacity > set_Pr1986765409at_nat,B: capacity,C: capacity] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_capacity @ B @ C )
=> ( ! [X4: capacity,Y3: capacity] :
( ( ord_less_eq_capacity @ X4 @ Y3 )
=> ( ord_le841296385at_nat @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le841296385at_nat @ A @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_109_ord__eq__le__subst,axiom,
! [A: capacity,F3: set_Pr1986765409at_nat > capacity,B: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_le841296385at_nat @ B @ C )
=> ( ! [X4: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X4 @ Y3 )
=> ( ord_less_eq_capacity @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_capacity @ A @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_110_ord__eq__le__subst,axiom,
! [A: set_Pr1986765409at_nat,F3: set_Pr1986765409at_nat > set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_le841296385at_nat @ B @ C )
=> ( ! [X4: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X4 @ Y3 )
=> ( ord_le841296385at_nat @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le841296385at_nat @ A @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_111_ord__eq__le__subst,axiom,
! [A: capacity,F3: capacity > capacity,B: capacity,C: capacity] :
( ( A
= ( F3 @ B ) )
=> ( ( ord_less_eq_capacity @ B @ C )
=> ( ! [X4: capacity,Y3: capacity] :
( ( ord_less_eq_capacity @ X4 @ Y3 )
=> ( ord_less_eq_capacity @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_capacity @ A @ ( F3 @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_112_ord__le__eq__subst,axiom,
! [A: capacity,B: capacity,F3: capacity > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_less_eq_capacity @ A @ B )
=> ( ( ( F3 @ B )
= C )
=> ( ! [X4: capacity,Y3: capacity] :
( ( ord_less_eq_capacity @ X4 @ Y3 )
=> ( ord_le841296385at_nat @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le841296385at_nat @ ( F3 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_113_ord__le__eq__subst,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,F3: set_Pr1986765409at_nat > capacity,C: capacity] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ( ( F3 @ B )
= C )
=> ( ! [X4: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X4 @ Y3 )
=> ( ord_less_eq_capacity @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_capacity @ ( F3 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_114_ord__le__eq__subst,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,F3: set_Pr1986765409at_nat > set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ( ( F3 @ B )
= C )
=> ( ! [X4: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X4 @ Y3 )
=> ( ord_le841296385at_nat @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_le841296385at_nat @ ( F3 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_115_ord__le__eq__subst,axiom,
! [A: capacity,B: capacity,F3: capacity > capacity,C: capacity] :
( ( ord_less_eq_capacity @ A @ B )
=> ( ( ( F3 @ B )
= C )
=> ( ! [X4: capacity,Y3: capacity] :
( ( ord_less_eq_capacity @ X4 @ Y3 )
=> ( ord_less_eq_capacity @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq_capacity @ ( F3 @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_116_eq__iff,axiom,
( ( ^ [Y4: set_Pr1986765409at_nat,Z2: set_Pr1986765409at_nat] : Y4 = Z2 )
= ( ^ [X3: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X3 @ Y2 )
& ( ord_le841296385at_nat @ Y2 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_117_eq__iff,axiom,
( ( ^ [Y4: capacity,Z2: capacity] : Y4 = Z2 )
= ( ^ [X3: capacity,Y2: capacity] :
( ( ord_less_eq_capacity @ X3 @ Y2 )
& ( ord_less_eq_capacity @ Y2 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_118_antisym,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y )
=> ( ( ord_le841296385at_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% antisym
thf(fact_119_antisym,axiom,
! [X2: capacity,Y: capacity] :
( ( ord_less_eq_capacity @ X2 @ Y )
=> ( ( ord_less_eq_capacity @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% antisym
thf(fact_120_linear,axiom,
! [X2: capacity,Y: capacity] :
( ( ord_less_eq_capacity @ X2 @ Y )
| ( ord_less_eq_capacity @ Y @ X2 ) ) ).
% linear
thf(fact_121_eq__refl,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat] :
( ( X2 = Y )
=> ( ord_le841296385at_nat @ X2 @ Y ) ) ).
% eq_refl
thf(fact_122_eq__refl,axiom,
! [X2: capacity,Y: capacity] :
( ( X2 = Y )
=> ( ord_less_eq_capacity @ X2 @ Y ) ) ).
% eq_refl
thf(fact_123_le__cases,axiom,
! [X2: capacity,Y: capacity] :
( ~ ( ord_less_eq_capacity @ X2 @ Y )
=> ( ord_less_eq_capacity @ Y @ X2 ) ) ).
% le_cases
thf(fact_124_order_Otrans,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ( ord_le841296385at_nat @ B @ C )
=> ( ord_le841296385at_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_125_order_Otrans,axiom,
! [A: capacity,B: capacity,C: capacity] :
( ( ord_less_eq_capacity @ A @ B )
=> ( ( ord_less_eq_capacity @ B @ C )
=> ( ord_less_eq_capacity @ A @ C ) ) ) ).
% order.trans
thf(fact_126_le__cases3,axiom,
! [X2: capacity,Y: capacity,Z: capacity] :
( ( ( ord_less_eq_capacity @ X2 @ Y )
=> ~ ( ord_less_eq_capacity @ Y @ Z ) )
=> ( ( ( ord_less_eq_capacity @ Y @ X2 )
=> ~ ( ord_less_eq_capacity @ X2 @ Z ) )
=> ( ( ( ord_less_eq_capacity @ X2 @ Z )
=> ~ ( ord_less_eq_capacity @ Z @ Y ) )
=> ( ( ( ord_less_eq_capacity @ Z @ Y )
=> ~ ( ord_less_eq_capacity @ Y @ X2 ) )
=> ( ( ( ord_less_eq_capacity @ Y @ Z )
=> ~ ( ord_less_eq_capacity @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_capacity @ Z @ X2 )
=> ~ ( ord_less_eq_capacity @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_127_antisym__conv,axiom,
! [Y: set_Pr1986765409at_nat,X2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ Y @ X2 )
=> ( ( ord_le841296385at_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_128_antisym__conv,axiom,
! [Y: capacity,X2: capacity] :
( ( ord_less_eq_capacity @ Y @ X2 )
=> ( ( ord_less_eq_capacity @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv
thf(fact_129_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr1986765409at_nat,Z2: set_Pr1986765409at_nat] : Y4 = Z2 )
= ( ^ [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A2 @ B2 )
& ( ord_le841296385at_nat @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_130_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: capacity,Z2: capacity] : Y4 = Z2 )
= ( ^ [A2: capacity,B2: capacity] :
( ( ord_less_eq_capacity @ A2 @ B2 )
& ( ord_less_eq_capacity @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_131_ord__eq__le__trans,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( A = B )
=> ( ( ord_le841296385at_nat @ B @ C )
=> ( ord_le841296385at_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_132_ord__eq__le__trans,axiom,
! [A: capacity,B: capacity,C: capacity] :
( ( A = B )
=> ( ( ord_less_eq_capacity @ B @ C )
=> ( ord_less_eq_capacity @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_133_ord__le__eq__trans,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ( B = C )
=> ( ord_le841296385at_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_134_ord__le__eq__trans,axiom,
! [A: capacity,B: capacity,C: capacity] :
( ( ord_less_eq_capacity @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_capacity @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_135_order__class_Oorder_Oantisym,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ( ord_le841296385at_nat @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_136_order__class_Oorder_Oantisym,axiom,
! [A: capacity,B: capacity] :
( ( ord_less_eq_capacity @ A @ B )
=> ( ( ord_less_eq_capacity @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_137_NPreflow_OresCap_Ocong,axiom,
augmen68920357pacity = augmen68920357pacity ).
% NPreflow.resCap.cong
thf(fact_138_order__trans,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y )
=> ( ( ord_le841296385at_nat @ Y @ Z )
=> ( ord_le841296385at_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_139_order__trans,axiom,
! [X2: capacity,Y: capacity,Z: capacity] :
( ( ord_less_eq_capacity @ X2 @ Y )
=> ( ( ord_less_eq_capacity @ Y @ Z )
=> ( ord_less_eq_capacity @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_140_dual__order_Orefl,axiom,
! [A: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ A @ A ) ).
% dual_order.refl
thf(fact_141_dual__order_Orefl,axiom,
! [A: capacity] : ( ord_less_eq_capacity @ A @ A ) ).
% dual_order.refl
thf(fact_142_linorder__wlog,axiom,
! [P2: capacity > capacity > $o,A: capacity,B: capacity] :
( ! [A4: capacity,B4: capacity] :
( ( ord_less_eq_capacity @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: capacity,B4: capacity] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_143_dual__order_Otrans,axiom,
! [B: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ B @ A )
=> ( ( ord_le841296385at_nat @ C @ B )
=> ( ord_le841296385at_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_144_dual__order_Otrans,axiom,
! [B: capacity,A: capacity,C: capacity] :
( ( ord_less_eq_capacity @ B @ A )
=> ( ( ord_less_eq_capacity @ C @ B )
=> ( ord_less_eq_capacity @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_145_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: set_Pr1986765409at_nat,Z2: set_Pr1986765409at_nat] : Y4 = Z2 )
= ( ^ [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ B2 @ A2 )
& ( ord_le841296385at_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_146_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: capacity,Z2: capacity] : Y4 = Z2 )
= ( ^ [A2: capacity,B2: capacity] :
( ( ord_less_eq_capacity @ B2 @ A2 )
& ( ord_less_eq_capacity @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_147_dual__order_Oantisym,axiom,
! [B: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ B @ A )
=> ( ( ord_le841296385at_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_148_dual__order_Oantisym,axiom,
! [B: capacity,A: capacity] :
( ( ord_less_eq_capacity @ B @ A )
=> ( ( ord_less_eq_capacity @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_149_inf_OcoboundedI2,axiom,
! [B: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ B @ C )
=> ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_150_inf_OcoboundedI1,axiom,
! [A: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ C )
=> ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_151_inf_Oabsorb__iff2,axiom,
( ord_le841296385at_nat
= ( ^ [B2: set_Pr1986765409at_nat,A2: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_152_inf_Oabsorb__iff1,axiom,
( ord_le841296385at_nat
= ( ^ [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb_iff1
thf(fact_153_inf_Ocobounded2,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_154_inf_Ocobounded1,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_155_inf_Oorder__iff,axiom,
( ord_le841296385at_nat
= ( ^ [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( A2
= ( inf_in586391887at_nat @ A2 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_156_inf__greatest,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat,Z: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y )
=> ( ( ord_le841296385at_nat @ X2 @ Z )
=> ( ord_le841296385at_nat @ X2 @ ( inf_in586391887at_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_157_inf_OboundedI,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ( ord_le841296385at_nat @ A @ C )
=> ( ord_le841296385at_nat @ A @ ( inf_in586391887at_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_158_inf_OboundedE,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ ( inf_in586391887at_nat @ B @ C ) )
=> ~ ( ( ord_le841296385at_nat @ A @ B )
=> ~ ( ord_le841296385at_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_159_inf__absorb2,axiom,
! [Y: set_Pr1986765409at_nat,X2: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ Y @ X2 )
=> ( ( inf_in586391887at_nat @ X2 @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_160_inf__absorb1,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ Y )
=> ( ( inf_in586391887at_nat @ X2 @ Y )
= X2 ) ) ).
% inf_absorb1
thf(fact_161_inf_Oabsorb2,axiom,
! [B: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ B @ A )
=> ( ( inf_in586391887at_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_162_inf_Oabsorb1,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ B )
=> ( ( inf_in586391887at_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_163_le__iff__inf,axiom,
( ord_le841296385at_nat
= ( ^ [X3: set_Pr1986765409at_nat,Y2: set_Pr1986765409at_nat] :
( ( inf_in586391887at_nat @ X3 @ Y2 )
= X3 ) ) ) ).
% le_iff_inf
thf(fact_164_inf__unique,axiom,
! [F3: set_Pr1986765409at_nat > set_Pr1986765409at_nat > set_Pr1986765409at_nat,X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat] :
( ! [X4: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( F3 @ X4 @ Y3 ) @ X4 )
=> ( ! [X4: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( F3 @ X4 @ Y3 ) @ Y3 )
=> ( ! [X4: set_Pr1986765409at_nat,Y3: set_Pr1986765409at_nat,Z3: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X4 @ Y3 )
=> ( ( ord_le841296385at_nat @ X4 @ Z3 )
=> ( ord_le841296385at_nat @ X4 @ ( F3 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_in586391887at_nat @ X2 @ Y )
= ( F3 @ X2 @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_165_inf_OorderI,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( A
= ( inf_in586391887at_nat @ A @ B ) )
=> ( ord_le841296385at_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_166_inf_OorderE,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ B )
=> ( A
= ( inf_in586391887at_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_167_le__infI2,axiom,
! [B: set_Pr1986765409at_nat,X2: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ B @ X2 )
=> ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A @ B ) @ X2 ) ) ).
% le_infI2
thf(fact_168_le__infI1,axiom,
! [A: set_Pr1986765409at_nat,X2: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ X2 )
=> ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A @ B ) @ X2 ) ) ).
% le_infI1
thf(fact_169_inf__mono,axiom,
! [A: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,D: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ A @ C )
=> ( ( ord_le841296385at_nat @ B @ D )
=> ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ A @ B ) @ ( inf_in586391887at_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_170_le__infI,axiom,
! [X2: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ A )
=> ( ( ord_le841296385at_nat @ X2 @ B )
=> ( ord_le841296385at_nat @ X2 @ ( inf_in586391887at_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_171_le__infE,axiom,
! [X2: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ X2 @ ( inf_in586391887at_nat @ A @ B ) )
=> ~ ( ( ord_le841296385at_nat @ X2 @ A )
=> ~ ( ord_le841296385at_nat @ X2 @ B ) ) ) ).
% le_infE
thf(fact_172_inf__le2,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ X2 @ Y ) @ Y ) ).
% inf_le2
thf(fact_173_inf__le1,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ X2 @ Y ) @ X2 ) ).
% inf_le1
thf(fact_174_inf__sup__ord_I1_J,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ X2 @ Y ) @ X2 ) ).
% inf_sup_ord(1)
thf(fact_175_inf__sup__ord_I2_J,axiom,
! [X2: set_Pr1986765409at_nat,Y: set_Pr1986765409at_nat] : ( ord_le841296385at_nat @ ( inf_in586391887at_nat @ X2 @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_176_NPreflow_OisAugmentingPath_Ocong,axiom,
augmen1090971539pacity = augmen1090971539pacity ).
% NPreflow.isAugmentingPath.cong
thf(fact_177_NPreflow_OaugmentingFlow_Ocong,axiom,
augmen72931474pacity = augmen72931474pacity ).
% NPreflow.augmentingFlow.cong
thf(fact_178_NPreflow_OaugFlow__val,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity,P: list_P559422087at_nat] :
( ( nPreflow_capacity @ C @ S @ T @ F3 )
=> ( ( augmen1090971539pacity @ C @ S @ T @ F3 @ P )
=> ( ( val_capacity @ ( residu203630698pacity @ C @ F3 ) @ S @ ( augmen72931474pacity @ C @ F3 @ P ) )
= ( augmen68920357pacity @ C @ F3 @ P ) ) ) ) ).
% NPreflow.augFlow_val
thf(fact_179_bot__fun__def,axiom,
( bot_bo513358416_nat_o
= ( ^ [X3: product_prod_nat_nat] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_180_NPreflow_OisAugmentingPath__def,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity,P: list_P559422087at_nat] :
( ( nPreflow_capacity @ C @ S @ T @ F3 )
=> ( ( augmen1090971539pacity @ C @ S @ T @ F3 @ P )
= ( isSimp1359852763pacity @ ( residu203630698pacity @ C @ F3 ) @ S @ P @ T ) ) ) ).
% NPreflow.isAugmentingPath_def
thf(fact_181_ford__fulkerson_I2_J,axiom,
( ( ~ ? [X: list_P559422087at_nat] : ( augmen1090971539pacity @ c @ s @ t @ f @ X ) )
= ( ? [K2: set_nat] :
( ( nCut_capacity @ c @ s @ t @ K2 )
& ( ( val_capacity @ c @ s @ f )
= ( cap_capacity @ c @ K2 ) ) ) ) ) ).
% ford_fulkerson(2)
thf(fact_182_fofu__III__I,axiom,
( ? [K3: set_nat] :
( ( nCut_capacity @ c @ s @ t @ K3 )
& ( ( val_capacity @ c @ s @ f )
= ( cap_capacity @ c @ K3 ) ) )
=> ( isMaxFlow_capacity @ c @ s @ t @ f ) ) ).
% fofu_III_I
thf(fact_183_fofu__II__III,axiom,
( ~ ? [X_12: list_P559422087at_nat] : ( augmen1090971539pacity @ c @ s @ t @ f @ X_12 )
=> ? [K4: set_nat] :
( ( nCut_capacity @ c @ s @ t @ K4 )
& ( ( val_capacity @ c @ s @ f )
= ( cap_capacity @ c @ K4 ) ) ) ) ).
% fofu_II_III
thf(fact_184_NFlow_Oaugment__flow__value,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity,F: product_prod_nat_nat > capacity] :
( ( nFlow_capacity @ C @ S @ T @ F3 )
=> ( ( flow_capacity @ ( residu203630698pacity @ C @ F3 ) @ S @ T @ F )
=> ( ( val_capacity @ C @ S @ ( augmen887209518pacity @ C @ F3 @ F ) )
= ( plus_plus_capacity @ ( val_capacity @ C @ S @ F3 ) @ ( val_capacity @ ( residu203630698pacity @ C @ F3 ) @ S @ F ) ) ) ) ) ).
% NFlow.augment_flow_value
thf(fact_185_Network_OmaxFlow__minCut,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity,K: set_nat] :
( ( network_capacity @ C @ S @ T )
=> ( ( isMaxFlow_capacity @ C @ S @ T @ F3 )
=> ( ( isMinCut_capacity @ C @ S @ T @ K )
=> ( ( val_capacity @ C @ S @ F3 )
= ( cap_capacity @ C @ K ) ) ) ) ) ).
% Network.maxFlow_minCut
thf(fact_186_Network_OisMaxFlow__alt,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( network_capacity @ C @ S @ T )
=> ( ( isMaxFlow_capacity @ C @ S @ T @ F3 )
= ( ( nFlow_capacity @ C @ S @ T @ F3 )
& ! [F5: product_prod_nat_nat > capacity] :
( ( nFlow_capacity @ C @ S @ T @ F5 )
=> ( ord_less_eq_capacity @ ( val_capacity @ C @ S @ F5 ) @ ( val_capacity @ C @ S @ F3 ) ) ) ) ) ) ).
% Network.isMaxFlow_alt
thf(fact_187_converse__mono,axiom,
! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ ( converse_nat_nat @ R ) @ ( converse_nat_nat @ S ) )
= ( ord_le841296385at_nat @ R @ S ) ) ).
% converse_mono
thf(fact_188_NCut_Oaxioms_I1_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,K: set_nat] :
( ( nCut_capacity @ C @ S @ T @ K )
=> ( network_capacity @ C @ S @ T ) ) ).
% NCut.axioms(1)
thf(fact_189_NCut_Ot__ni__cut,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,K: set_nat] :
( ( nCut_capacity @ C @ S @ T @ K )
=> ~ ( member_nat @ T @ K ) ) ).
% NCut.t_ni_cut
thf(fact_190_NCut_Os__in__cut,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,K: set_nat] :
( ( nCut_capacity @ C @ S @ T @ K )
=> ( member_nat @ S @ K ) ) ).
% NCut.s_in_cut
thf(fact_191_converse__subset__swap,axiom,
! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
( ( ord_le841296385at_nat @ R @ ( converse_nat_nat @ S ) )
= ( ord_le841296385at_nat @ ( converse_nat_nat @ R ) @ S ) ) ).
% converse_subset_swap
thf(fact_192_isMinCut__def,axiom,
( isMinCut_capacity
= ( ^ [C2: product_prod_nat_nat > capacity,S2: nat,T2: nat,K2: set_nat] :
( ( nCut_capacity @ C2 @ S2 @ T2 @ K2 )
& ! [K5: set_nat] :
( ( nCut_capacity @ C2 @ S2 @ T2 @ K5 )
=> ( ord_less_eq_capacity @ ( cap_capacity @ C2 @ K2 ) @ ( cap_capacity @ C2 @ K5 ) ) ) ) ) ) ).
% isMinCut_def
thf(fact_193_Network_Ot__not__s,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat] :
( ( network_capacity @ C @ S @ T )
=> ( T != S ) ) ).
% Network.t_not_s
thf(fact_194_Network_Os__not__t,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat] :
( ( network_capacity @ C @ S @ T )
=> ( S != T ) ) ).
% Network.s_not_t
thf(fact_195_Flow_Oval_Ocong,axiom,
val_capacity = val_capacity ).
% Flow.val.cong
thf(fact_196_Network_OisMaxFlow_Ocong,axiom,
isMaxFlow_capacity = isMaxFlow_capacity ).
% Network.isMaxFlow.cong
thf(fact_197_NFlow_Oaugment_Ocong,axiom,
augmen887209518pacity = augmen887209518pacity ).
% NFlow.augment.cong
thf(fact_198_NCut_Ocap_Ocong,axiom,
cap_capacity = cap_capacity ).
% NCut.cap.cong
thf(fact_199_NFlow_Ofofu__II__III,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( nFlow_capacity @ C @ S @ T @ F3 )
=> ( ~ ? [X_12: list_P559422087at_nat] : ( augmen1090971539pacity @ C @ S @ T @ F3 @ X_12 )
=> ? [K4: set_nat] :
( ( nCut_capacity @ C @ S @ T @ K4 )
& ( ( val_capacity @ C @ S @ F3 )
= ( cap_capacity @ C @ K4 ) ) ) ) ) ).
% NFlow.fofu_II_III
thf(fact_200_NFlow_Ofofu__III__I,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( nFlow_capacity @ C @ S @ T @ F3 )
=> ( ? [K3: set_nat] :
( ( nCut_capacity @ C @ S @ T @ K3 )
& ( ( val_capacity @ C @ S @ F3 )
= ( cap_capacity @ C @ K3 ) ) )
=> ( isMaxFlow_capacity @ C @ S @ T @ F3 ) ) ) ).
% NFlow.fofu_III_I
thf(fact_201_Network_Ois__max__flow__val_Ocong,axiom,
is_max2144541363pacity = is_max2144541363pacity ).
% Network.is_max_flow_val.cong
thf(fact_202_NFlow_Oford__fulkerson_I2_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( nFlow_capacity @ C @ S @ T @ F3 )
=> ( ( ~ ? [X: list_P559422087at_nat] : ( augmen1090971539pacity @ C @ S @ T @ F3 @ X ) )
= ( ? [K2: set_nat] :
( ( nCut_capacity @ C @ S @ T @ K2 )
& ( ( val_capacity @ C @ S @ F3 )
= ( cap_capacity @ C @ K2 ) ) ) ) ) ) ).
% NFlow.ford_fulkerson(2)
thf(fact_203_NPreflow_Oaxioms_I1_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( nPreflow_capacity @ C @ S @ T @ F3 )
=> ( network_capacity @ C @ S @ T ) ) ).
% NPreflow.axioms(1)
thf(fact_204_NFlow_Oaxioms_I1_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( nFlow_capacity @ C @ S @ T @ F3 )
=> ( nPreflow_capacity @ C @ S @ T @ F3 ) ) ).
% NFlow.axioms(1)
thf(fact_205_NFlow_Oaxioms_I2_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( nFlow_capacity @ C @ S @ T @ F3 )
=> ( flow_capacity @ C @ S @ T @ F3 ) ) ).
% NFlow.axioms(2)
thf(fact_206_NPreflow_Oaxioms_I2_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( nPreflow_capacity @ C @ S @ T @ F3 )
=> ( preflow_capacity @ C @ S @ T @ F3 ) ) ).
% NPreflow.axioms(2)
thf(fact_207_Flow_Oaxioms_I1_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( flow_capacity @ C @ S @ T @ F3 )
=> ( preflow_capacity @ C @ S @ T @ F3 ) ) ).
% Flow.axioms(1)
thf(fact_208_Network_Omax__flow__val__unique,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,Fv1: capacity,Fv2: capacity] :
( ( network_capacity @ C @ S @ T )
=> ( ( is_max2144541363pacity @ C @ S @ T @ Fv1 )
=> ( ( is_max2144541363pacity @ C @ S @ T @ Fv2 )
=> ( Fv1 = Fv2 ) ) ) ) ).
% Network.max_flow_val_unique
thf(fact_209_Finite__Preflow_Oaxioms_I2_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( finite1387368318pacity @ C @ S @ T @ F3 )
=> ( finite217307323pacity @ C ) ) ).
% Finite_Preflow.axioms(2)
thf(fact_210_Finite__Preflow_Oaxioms_I1_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( finite1387368318pacity @ C @ S @ T @ F3 )
=> ( preflow_capacity @ C @ S @ T @ F3 ) ) ).
% Finite_Preflow.axioms(1)
thf(fact_211_NFlow_Ointro,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( nPreflow_capacity @ C @ S @ T @ F3 )
=> ( ( flow_capacity @ C @ S @ T @ F3 )
=> ( nFlow_capacity @ C @ S @ T @ F3 ) ) ) ).
% NFlow.intro
thf(fact_212_NFlow__def,axiom,
( nFlow_capacity
= ( ^ [C2: product_prod_nat_nat > capacity,S2: nat,T2: nat,F2: product_prod_nat_nat > capacity] :
( ( nPreflow_capacity @ C2 @ S2 @ T2 @ F2 )
& ( flow_capacity @ C2 @ S2 @ T2 @ F2 ) ) ) ) ).
% NFlow_def
thf(fact_213_NPreflow_Ointro,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( network_capacity @ C @ S @ T )
=> ( ( preflow_capacity @ C @ S @ T @ F3 )
=> ( nPreflow_capacity @ C @ S @ T @ F3 ) ) ) ).
% NPreflow.intro
thf(fact_214_NPreflow__def,axiom,
( nPreflow_capacity
= ( ^ [C2: product_prod_nat_nat > capacity,S2: nat,T2: nat,F2: product_prod_nat_nat > capacity] :
( ( network_capacity @ C2 @ S2 @ T2 )
& ( preflow_capacity @ C2 @ S2 @ T2 @ F2 ) ) ) ) ).
% NPreflow_def
thf(fact_215_NFlow_Ofofu__I__II,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( nFlow_capacity @ C @ S @ T @ F3 )
=> ( ( isMaxFlow_capacity @ C @ S @ T @ F3 )
=> ~ ? [X_1: list_P559422087at_nat] : ( augmen1090971539pacity @ C @ S @ T @ F3 @ X_1 ) ) ) ).
% NFlow.fofu_I_II
thf(fact_216_NFlow_OnoAugPath__iff__maxFlow,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( nFlow_capacity @ C @ S @ T @ F3 )
=> ( ( ~ ? [X: list_P559422087at_nat] : ( augmen1090971539pacity @ C @ S @ T @ F3 @ X ) )
= ( isMaxFlow_capacity @ C @ S @ T @ F3 ) ) ) ).
% NFlow.noAugPath_iff_maxFlow
thf(fact_217_Finite__Preflow_Ointro,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( preflow_capacity @ C @ S @ T @ F3 )
=> ( ( finite217307323pacity @ C )
=> ( finite1387368318pacity @ C @ S @ T @ F3 ) ) ) ).
% Finite_Preflow.intro
thf(fact_218_Finite__Preflow__def,axiom,
( finite1387368318pacity
= ( ^ [C2: product_prod_nat_nat > capacity,S2: nat,T2: nat,F2: product_prod_nat_nat > capacity] :
( ( preflow_capacity @ C2 @ S2 @ T2 @ F2 )
& ( finite217307323pacity @ C2 ) ) ) ) ).
% Finite_Preflow_def
thf(fact_219_NFlow_Oaugment__flow__presv,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity,F: product_prod_nat_nat > capacity] :
( ( nFlow_capacity @ C @ S @ T @ F3 )
=> ( ( flow_capacity @ ( residu203630698pacity @ C @ F3 ) @ S @ T @ F )
=> ( flow_capacity @ C @ S @ T @ ( augmen887209518pacity @ C @ F3 @ F ) ) ) ) ).
% NFlow.augment_flow_presv
thf(fact_220_NFlow_Oford__fulkerson_I1_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( nFlow_capacity @ C @ S @ T @ F3 )
=> ( ( isMaxFlow_capacity @ C @ S @ T @ F3 )
= ( ~ ? [X: list_P559422087at_nat] : ( augmen1090971539pacity @ C @ S @ T @ F3 @ X ) ) ) ) ).
% NFlow.ford_fulkerson(1)
thf(fact_221_Network_Ois__max__flow__val__def,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,Fv: capacity] :
( ( network_capacity @ C @ S @ T )
=> ( ( is_max2144541363pacity @ C @ S @ T @ Fv )
= ( ? [F2: product_prod_nat_nat > capacity] :
( ( isMaxFlow_capacity @ C @ S @ T @ F2 )
& ( Fv
= ( val_capacity @ C @ S @ F2 ) ) ) ) ) ) ).
% Network.is_max_flow_val_def
thf(fact_222_Network_OisMaxFlow__def,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( network_capacity @ C @ S @ T )
=> ( ( isMaxFlow_capacity @ C @ S @ T @ F3 )
= ( ( flow_capacity @ C @ S @ T @ F3 )
& ! [F5: product_prod_nat_nat > capacity] :
( ( flow_capacity @ C @ S @ T @ F5 )
=> ( ord_less_eq_capacity @ ( val_capacity @ C @ S @ F5 ) @ ( val_capacity @ C @ S @ F3 ) ) ) ) ) ) ).
% Network.isMaxFlow_def
thf(fact_223_cf_OisSPath__no__selfloop,axiom,
! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat] :
( ( isSimp1359852763pacity @ ( residu203630698pacity @ c @ f ) @ U @ P @ V )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ).
% cf.isSPath_no_selfloop
thf(fact_224_cf_OisSPath__sg__incoming,axiom,
! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,U2: nat] :
( ( isSimp1359852763pacity @ ( residu203630698pacity @ c @ f ) @ U @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( U1 != U2 )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U2 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% cf.isSPath_sg_incoming
thf(fact_225_cf_OisSPath__sg__outgoing,axiom,
! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,V2: nat] :
( ( isSimp1359852763pacity @ ( residu203630698pacity @ c @ f ) @ U @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( V1 != V2 )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V2 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% cf.isSPath_sg_outgoing
thf(fact_226_isSPath__sg__outgoing,axiom,
! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,V2: nat] :
( ( isSimp1359852763pacity @ c @ U @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( V1 != V2 )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V2 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% isSPath_sg_outgoing
thf(fact_227_isSPath__sg__incoming,axiom,
! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,U2: nat] :
( ( isSimp1359852763pacity @ c @ U @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( U1 != U2 )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U2 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% isSPath_sg_incoming
thf(fact_228_isSPath__no__selfloop,axiom,
! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat] :
( ( isSimp1359852763pacity @ c @ U @ P @ V )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ).
% isSPath_no_selfloop
thf(fact_229_converse__iff,axiom,
! [A: nat,B: nat,R: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( converse_nat_nat @ R ) )
= ( member701585322at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R ) ) ).
% converse_iff
thf(fact_230_subrelI,axiom,
! [R: set_Pr1986765409at_nat,S: set_Pr1986765409at_nat] :
( ! [X4: nat,Y3: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ S ) )
=> ( ord_le841296385at_nat @ R @ S ) ) ).
% subrelI
thf(fact_231_converseD,axiom,
! [A: nat,B: nat,R: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( converse_nat_nat @ R ) )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R ) ) ).
% converseD
thf(fact_232_converseE,axiom,
! [Yx: product_prod_nat_nat,R: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ Yx @ ( converse_nat_nat @ R ) )
=> ~ ! [X4: nat,Y3: nat] :
( ( Yx
= ( product_Pair_nat_nat @ Y3 @ X4 ) )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R ) ) ) ).
% converseE
thf(fact_233_converse_Ocases,axiom,
! [A1: nat,A22: nat,R: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( converse_nat_nat @ R ) )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ A22 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_234_converse_Osimps,axiom,
! [A1: nat,A22: nat,R: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A1 @ A22 ) @ ( converse_nat_nat @ R ) )
= ( ? [A2: nat,B2: nat] :
( ( A1 = B2 )
& ( A22 = A2 )
& ( member701585322at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_235_converse_Ointros,axiom,
! [A: nat,B: nat,R: set_Pr1986765409at_nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
=> ( member701585322at_nat @ ( product_Pair_nat_nat @ B @ A ) @ ( converse_nat_nat @ R ) ) ) ).
% converse.intros
thf(fact_236_converse_Oinducts,axiom,
! [X1: nat,X22: nat,R: set_Pr1986765409at_nat,P2: nat > nat > $o] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) @ ( converse_nat_nat @ R ) )
=> ( ! [A4: nat,B4: nat] :
( ( member701585322at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ R )
=> ( P2 @ B4 @ A4 ) )
=> ( P2 @ X1 @ X22 ) ) ) ).
% converse.inducts
thf(fact_237_Graph_OisSPath__sg__outgoing,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,V2: nat] :
( ( isSimp1359852763pacity @ C @ U @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( V1 != V2 )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V2 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% Graph.isSPath_sg_outgoing
thf(fact_238_Graph_OisSPath__sg__incoming,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,U2: nat] :
( ( isSimp1359852763pacity @ C @ U @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( U1 != U2 )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U2 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% Graph.isSPath_sg_incoming
thf(fact_239_Graph_OisSPath__no__selfloop,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat] :
( ( isSimp1359852763pacity @ C @ U @ P @ V )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ).
% Graph.isSPath_no_selfloop
thf(fact_240_augment__flow__presv__cap,axiom,
! [F: product_prod_nat_nat > capacity,U: nat,V: nat] :
( ( flow_capacity @ ( residu203630698pacity @ c @ f ) @ s @ t @ F )
=> ( ( ord_less_eq_capacity @ zero_zero_capacity @ ( augmen887209518pacity @ c @ f @ F @ ( product_Pair_nat_nat @ U @ V ) ) )
& ( ord_less_eq_capacity @ ( augmen887209518pacity @ c @ f @ F @ ( product_Pair_nat_nat @ U @ V ) ) @ ( c @ ( product_Pair_nat_nat @ U @ V ) ) ) ) ) ).
% augment_flow_presv_cap
thf(fact_241_swap__simp,axiom,
! [X2: nat,Y: nat] :
( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X2 @ Y ) )
= ( product_Pair_nat_nat @ Y @ X2 ) ) ).
% swap_simp
thf(fact_242_f__non__negative,axiom,
! [E: product_prod_nat_nat] : ( ord_less_eq_capacity @ zero_zero_capacity @ ( f @ E ) ) ).
% f_non_negative
thf(fact_243_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A5: nat,B5: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A5 @ B5 ) )
= ( ( A = A5 )
& ( B = B5 ) ) ) ).
% old.prod.inject
thf(fact_244_prod_Oinject,axiom,
! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X22 )
= ( product_Pair_nat_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.inject
thf(fact_245_cap__non__negative,axiom,
! [U3: nat,V3: nat] : ( ord_less_eq_capacity @ zero_zero_capacity @ ( c @ ( product_Pair_nat_nat @ U3 @ V3 ) ) ) ).
% cap_non_negative
thf(fact_246_capacity__const,axiom,
! [E2: product_prod_nat_nat] :
( ( ord_less_eq_capacity @ zero_zero_capacity @ ( f @ E2 ) )
& ( ord_less_eq_capacity @ ( f @ E2 ) @ ( c @ E2 ) ) ) ).
% capacity_const
thf(fact_247_swap__swap,axiom,
! [P: product_prod_nat_nat] :
( ( product_swap_nat_nat @ ( product_swap_nat_nat @ P ) )
= P ) ).
% swap_swap
thf(fact_248_resE__nonNegative,axiom,
! [E: product_prod_nat_nat] : ( ord_less_eq_capacity @ zero_zero_capacity @ ( residu203630698pacity @ c @ f @ E ) ) ).
% resE_nonNegative
thf(fact_249_RPreGraph_OresE__nonNegative,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,Cf: product_prod_nat_nat > capacity,E: product_prod_nat_nat] :
( ( residu164581374pacity @ C @ S @ T @ Cf )
=> ( ord_less_eq_capacity @ zero_zero_capacity @ ( Cf @ E ) ) ) ).
% RPreGraph.resE_nonNegative
thf(fact_250_Preflow_Ocapacity__const,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity] :
( ( preflow_capacity @ C @ S @ T @ F3 )
=> ! [E2: product_prod_nat_nat] :
( ( ord_less_eq_capacity @ zero_zero_capacity @ ( F3 @ E2 ) )
& ( ord_less_eq_capacity @ ( F3 @ E2 ) @ ( C @ E2 ) ) ) ) ).
% Preflow.capacity_const
thf(fact_251_Preflow_Of__non__negative,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity,E: product_prod_nat_nat] :
( ( preflow_capacity @ C @ S @ T @ F3 )
=> ( ord_less_eq_capacity @ zero_zero_capacity @ ( F3 @ E ) ) ) ).
% Preflow.f_non_negative
thf(fact_252_old_Oprod_Oinducts,axiom,
! [P2: product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ! [A4: nat,B4: nat] : ( P2 @ ( product_Pair_nat_nat @ A4 @ B4 ) )
=> ( P2 @ Prod ) ) ).
% old.prod.inducts
thf(fact_253_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A4: nat,B4: nat] :
( Y
!= ( product_Pair_nat_nat @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_254_Pair__inject,axiom,
! [A: nat,B: nat,A5: nat,B5: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A5 @ B5 ) )
=> ~ ( ( A = A5 )
=> ( B != B5 ) ) ) ).
% Pair_inject
thf(fact_255_prod__cases,axiom,
! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
( ! [A4: nat,B4: nat] : ( P2 @ ( product_Pair_nat_nat @ A4 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_256_surj__pair,axiom,
! [P: product_prod_nat_nat] :
? [X4: nat,Y3: nat] :
( P
= ( product_Pair_nat_nat @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_257_Network_Ocap__non__negative,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat] :
( ( network_capacity @ C @ S @ T )
=> ! [U3: nat,V3: nat] : ( ord_less_eq_capacity @ zero_zero_capacity @ ( C @ ( product_Pair_nat_nat @ U3 @ V3 ) ) ) ) ).
% Network.cap_non_negative
thf(fact_258_NPreflow_OresE__nonNegative,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity,E: product_prod_nat_nat] :
( ( nPreflow_capacity @ C @ S @ T @ F3 )
=> ( ord_less_eq_capacity @ zero_zero_capacity @ ( residu203630698pacity @ C @ F3 @ E ) ) ) ).
% NPreflow.resE_nonNegative
thf(fact_259_NFlow_Oaugment__flow__presv__cap,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat,F3: product_prod_nat_nat > capacity,F: product_prod_nat_nat > capacity,U: nat,V: nat] :
( ( nFlow_capacity @ C @ S @ T @ F3 )
=> ( ( flow_capacity @ ( residu203630698pacity @ C @ F3 ) @ S @ T @ F )
=> ( ( ord_less_eq_capacity @ zero_zero_capacity @ ( augmen887209518pacity @ C @ F3 @ F @ ( product_Pair_nat_nat @ U @ V ) ) )
& ( ord_less_eq_capacity @ ( augmen887209518pacity @ C @ F3 @ F @ ( product_Pair_nat_nat @ U @ V ) ) @ ( C @ ( product_Pair_nat_nat @ U @ V ) ) ) ) ) ) ).
% NFlow.augment_flow_presv_cap
thf(fact_260_excess__s__non__pos,axiom,
ord_less_eq_capacity @ ( excess_capacity @ c @ f @ s ) @ zero_zero_capacity ).
% excess_s_non_pos
thf(fact_261_resCap__gzero,axiom,
! [P: list_P559422087at_nat] :
( ( augmen1090971539pacity @ c @ s @ t @ f @ P )
=> ( ord_less_capacity @ zero_zero_capacity @ ( augmen68920357pacity @ c @ f @ P ) ) ) ).
% resCap_gzero
thf(fact_262_Network_Oexcess_Ocong,axiom,
excess_capacity = excess_capacity ).
% Network.excess.cong
thf(fact_263_bot_Onot__eq__extremum,axiom,
! [A: product_prod_nat_nat > $o] :
( ( A != bot_bo513358416_nat_o )
= ( ord_le1015898640_nat_o @ bot_bo513358416_nat_o @ A ) ) ).
% bot.not_eq_extremum
thf(fact_264_bot_Onot__eq__extremum,axiom,
! [A: set_Pr1986765409at_nat] :
( ( A != bot_bo2130386637at_nat )
= ( ord_le116442893at_nat @ bot_bo2130386637at_nat @ A ) ) ).
% bot.not_eq_extremum
thf(fact_265_bot_Oextremum__strict,axiom,
! [A: product_prod_nat_nat > $o] :
~ ( ord_le1015898640_nat_o @ A @ bot_bo513358416_nat_o ) ).
% bot.extremum_strict
thf(fact_266_bot_Oextremum__strict,axiom,
! [A: set_Pr1986765409at_nat] :
~ ( ord_le116442893at_nat @ A @ bot_bo2130386637at_nat ) ).
% bot.extremum_strict
thf(fact_267_less__infI1,axiom,
! [A: set_Pr1986765409at_nat,X2: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le116442893at_nat @ A @ X2 )
=> ( ord_le116442893at_nat @ ( inf_in586391887at_nat @ A @ B ) @ X2 ) ) ).
% less_infI1
thf(fact_268_less__infI2,axiom,
! [B: set_Pr1986765409at_nat,X2: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( ord_le116442893at_nat @ B @ X2 )
=> ( ord_le116442893at_nat @ ( inf_in586391887at_nat @ A @ B ) @ X2 ) ) ).
% less_infI2
thf(fact_269_inf_Ostrict__boundedE,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat] :
( ( ord_le116442893at_nat @ A @ ( inf_in586391887at_nat @ B @ C ) )
=> ~ ( ( ord_le116442893at_nat @ A @ B )
=> ~ ( ord_le116442893at_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_270_inf_Ostrict__order__iff,axiom,
( ord_le116442893at_nat
= ( ^ [A2: set_Pr1986765409at_nat,B2: set_Pr1986765409at_nat] :
( ( A2
= ( inf_in586391887at_nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_271_inf_Ostrict__coboundedI1,axiom,
! [A: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le116442893at_nat @ A @ C )
=> ( ord_le116442893at_nat @ ( inf_in586391887at_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_272_inf_Ostrict__coboundedI2,axiom,
! [B: set_Pr1986765409at_nat,C: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( ord_le116442893at_nat @ B @ C )
=> ( ord_le116442893at_nat @ ( inf_in586391887at_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_273_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_Pr1986765409at_nat,A: set_Pr1986765409at_nat] :
( ( ord_le116442893at_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_274_dual__order_Ostrict__implies__not__eq,axiom,
! [B: capacity,A: capacity] :
( ( ord_less_capacity @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_275_order_Ostrict__implies__not__eq,axiom,
! [A: set_Pr1986765409at_nat,B: set_Pr1986765409at_nat] :
( ( ord_le116442893at_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_276_order_Ostrict__implies__not__eq,axiom,
! [A: capacity,B: capacity] :
( ( ord_less_capacity @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_277_not__less__iff__gr__or__eq,axiom,
! [X2: capacity,Y: capacity] :
( ( ~ ( ord_less_capacity @ X2 @ Y ) )
= ( ( ord_less_capacity @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
% Conjectures (2)
thf(conj_0,hypothesis,
augmen1090971539pacity @ c @ s @ t @ f @ p ).
thf(conj_1,conjecture,
( ( inf_in586391887at_nat @ ( set_Pr2131844118at_nat @ p ) @ ( converse_nat_nat @ ( set_Pr2131844118at_nat @ p ) ) )
= bot_bo2130386637at_nat ) ).
%------------------------------------------------------------------------------