%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP052^2 : 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 : n002.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 08:55:10 AM UTC 2026
% Result : Theorem 2.84s 1.57s
% Output : Refutation 2.84s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP052^2 : 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
% 0.18/0.34 % Computer : n002.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Tue Feb 24 08:24:48 EST 2026
% 0.18/0.34 % CPUTime :
% 0.18/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.28/0.46 ---- Original DTF file ---
% 0.28/0.46 thf(spec,logic,$$dhol).
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % File : ITP052^2 : TPTP v9.2.1. Released v7.5.0.
% 0.28/0.46 % Domain : Interactive Theorem Proving
% 0.28/0.46 % Problem : Sledgehammer EdmondsKarp_Termination_Abstract problem prob_430__7598256_1
% 0.28/0.46 % Version : Especial.
% 0.28/0.46 % English :
% 0.28/0.46
% 0.28/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.28/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.28/0.46 % Source : [Des21]
% 0.28/0.46 % Names : EdmondsKarp_Termination_Abstract/prob_430__7598256_1 [Des21]
% 0.28/0.46
% 0.28/0.46 % Status : Theorem
% 0.28/0.46 % Rating : 0.33 v9.1.0, 0.00 v8.2.0, 0.33 v8.1.0, 0.00 v7.5.0
% 0.28/0.46 % Syntax : Number of formulae : 358 ( 75 unt; 58 typ; 0 def)
% 0.28/0.46 % Number of atoms : 830 ( 176 equ; 0 cnn)
% 0.28/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.28/0.46 % Number of connectives : 4724 ( 68 ~; 2 |; 36 &;4199 @)
% 0.28/0.46 % ( 0 <=>; 419 =>; 0 <=; 0 <~>)
% 0.28/0.46 % Maximal formula depth : 28 ( 9 avg)
% 0.28/0.46 % Number of types : 3 ( 2 usr)
% 0.28/0.46 % Number of type conns : 330 ( 330 >; 0 *; 0 +; 0 <<)
% 0.28/0.46 % Number of symbols : 57 ( 56 usr; 3 con; 0-6 aty)
% 0.28/0.46 % Number of variables : 1138 ( 56 ^;1005 !; 26 ?;1138 :)
% 0.28/0.46 % ( 51 !>; 0 ?*; 0 @-; 0 @+)
% 0.28/0.46 % SPC : TH1_THM_EQU_NAR
% 0.28/0.46
% 0.28/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:17:29.974
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 % Could-be-implicit typings (5)
% 0.28/0.46 thf(ty_t_Product__Type_Oprod,type,
% 0.28/0.46 product_prod: $tType > $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_List_Olist,type,
% 0.28/0.46 list: $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_tf_capacity,type,
% 0.28/0.46 capacity: $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Set_Oset,type,
% 0.28/0.46 set: $tType > $tType ).
% 0.28/0.46
% 0.28/0.46 thf(ty_t_Nat_Onat,type,
% 0.28/0.46 nat: $tType ).
% 0.28/0.46
% 0.28/0.46 % Explicit typings (53)
% 0.28/0.46 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.28/0.46 bounded_lattice:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_HOL_Otype,type,
% 0.28/0.46 type:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Obot,type,
% 0.28/0.46 bot:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.28/0.46 ord:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.28/0.46 order:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Lattices_Olattice,type,
% 0.28/0.46 lattice:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.28/0.46 linorder:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.28/0.46 preorder:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.28/0.46 order_bot:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.28/0.46 linordered_idom:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 0.28/0.46 semilattice_inf:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_cl_Lattices_Obounded__lattice__bot,type,
% 0.28/0.46 bounded_lattice_bot:
% 0.28/0.46 !>[A: $tType] : $o ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Augmenting__Flow_ONFlow_Oaugment,type,
% 0.28/0.46 augmenting_augment:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( ( product_prod @ nat @ nat ) > Capacity ) > ( ( product_prod @ nat @ nat ) > Capacity ) > ( product_prod @ nat @ nat ) > Capacity ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Augmenting__Path_ONPreflow_OaugmentingFlow,type,
% 0.28/0.46 augmen1604248148ngFlow:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( ( product_prod @ nat @ nat ) > Capacity ) > ( list @ ( product_prod @ nat @ nat ) ) > ( product_prod @ nat @ nat ) > Capacity ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Augmenting__Path_ONPreflow_OisAugmentingPath,type,
% 0.28/0.46 augmen2028041557ngPath:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > ( ( product_prod @ nat @ nat ) > Capacity ) > ( list @ ( product_prod @ nat @ nat ) ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Augmenting__Path_ONPreflow_OresCap,type,
% 0.28/0.46 augmenting_resCap:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( ( product_prod @ nat @ nat ) > Capacity ) > ( list @ ( product_prod @ nat @ nat ) ) > Capacity ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Graph_OFinite__Graph,type,
% 0.28/0.46 finite_Graph:
% 0.28/0.46 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Graph_OGraph_OisShortestPath,type,
% 0.28/0.46 isShortestPath:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Graph_OGraph_OisSimplePath,type,
% 0.28/0.46 isSimplePath:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.28/0.46 plus_plus:
% 0.28/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.28/0.46 zero_zero:
% 0.28/0.46 !>[A: $tType] : A ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.28/0.46 inf_inf:
% 0.28/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_List_Olist_Oset,type,
% 0.28/0.46 set2:
% 0.28/0.46 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Network_OFinite__Preflow,type,
% 0.28/0.46 finite_Preflow:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > ( ( product_prod @ nat @ nat ) > Capacity ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Network_OFlow,type,
% 0.28/0.46 flow:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > ( ( product_prod @ nat @ nat ) > Capacity ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Network_OFlow_Oval,type,
% 0.28/0.46 val:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( ( product_prod @ nat @ nat ) > Capacity ) > Capacity ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Network_ONCut,type,
% 0.28/0.46 nCut:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > ( set @ nat ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Network_ONCut_Ocap,type,
% 0.28/0.46 cap:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) > Capacity ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Network_ONFlow,type,
% 0.28/0.46 nFlow:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > ( ( product_prod @ nat @ nat ) > Capacity ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Network_ONPreflow,type,
% 0.28/0.46 nPreflow:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > ( ( product_prod @ nat @ nat ) > Capacity ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Network_ONetwork,type,
% 0.28/0.46 network:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Network_ONetwork_Oexcess,type,
% 0.28/0.46 excess:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( ( product_prod @ nat @ nat ) > Capacity ) > nat > Capacity ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Network_ONetwork_OisMaxFlow,type,
% 0.28/0.46 isMaxFlow:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > ( ( product_prod @ nat @ nat ) > Capacity ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Network_ONetwork_Ois__max__flow__val,type,
% 0.28/0.46 is_max_flow_val:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > Capacity > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Network_OPreflow,type,
% 0.28/0.46 preflow:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > ( ( product_prod @ nat @ nat ) > Capacity ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Network_OisMinCut,type,
% 0.28/0.46 isMinCut:
% 0.28/0.46 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > nat > nat > ( set @ nat ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.28/0.46 bot_bot:
% 0.28/0.46 !>[A: $tType] : A ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.28/0.46 ord_less:
% 0.28/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.28/0.46 ord_less_eq:
% 0.28/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Product__Type_OPair,type,
% 0.28/0.46 product_Pair:
% 0.28/0.46 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.28/0.46 product_swap:
% 0.28/0.46 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Relation_Oconverse,type,
% 0.28/0.46 converse:
% 0.28/0.46 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Residual__Graph_ONetwork_Oflow__of__cf,type,
% 0.28/0.46 residual_flow_of_cf:
% 0.28/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( ( product_prod @ nat @ nat ) > Capacity ) > ( product_prod @ nat @ nat ) > Capacity ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Residual__Graph_ORGraph,type,
% 0.28/0.46 residual_RGraph:
% 0.28/0.46 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > nat > nat > ( ( product_prod @ nat @ nat ) > A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Residual__Graph_ORPreGraph,type,
% 0.28/0.46 residual_RPreGraph:
% 0.28/0.46 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > nat > nat > ( ( product_prod @ nat @ nat ) > A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Residual__Graph_OresidualGraph,type,
% 0.28/0.46 residu1993642540lGraph:
% 0.28/0.46 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > ( ( product_prod @ nat @ nat ) > A ) > ( product_prod @ nat @ nat ) > A ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_Set_OCollect,type,
% 0.28/0.46 collect:
% 0.28/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_c_member,type,
% 0.28/0.46 member:
% 0.28/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_c,type,
% 0.28/0.46 c: ( product_prod @ nat @ nat ) > capacity ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_f,type,
% 0.28/0.46 f: ( product_prod @ nat @ nat ) > capacity ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_p,type,
% 0.28/0.46 p: list @ ( product_prod @ nat @ nat ) ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_s,type,
% 0.28/0.46 s: nat ).
% 0.28/0.46
% 0.28/0.46 thf(sy_v_t,type,
% 0.28/0.46 t: nat ).
% 0.28/0.46
% 0.28/0.46 % Relevant facts (256)
% 0.28/0.46 thf(fact_0_t__not__s,axiom,
% 0.28/0.46 t != s ).
% 0.28/0.46
% 0.28/0.46 % t_not_s
% 0.28/0.46 thf(fact_1_cf_OisSPath__nt__parallel__pf,axiom,
% 0.28/0.46 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat] :
% 0.28/0.46 ( ( isSimplePath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ S @ P @ T )
% 0.28/0.46 => ( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) @ ( converse @ nat @ nat @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) )
% 0.28/0.46 = ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % cf.isSPath_nt_parallel_pf
% 0.28/0.46 thf(fact_2_isSPath__nt__parallel__pf,axiom,
% 0.28/0.46 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat] :
% 0.28/0.46 ( ( isSimplePath @ capacity @ c @ S @ P @ T )
% 0.28/0.46 => ( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) @ ( converse @ nat @ nat @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) )
% 0.28/0.46 = ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % isSPath_nt_parallel_pf
% 0.28/0.46 thf(fact_3_NFlow__axioms,axiom,
% 0.28/0.46 nFlow @ capacity @ c @ s @ t @ f ).
% 0.28/0.46
% 0.28/0.46 % NFlow_axioms
% 0.28/0.46 thf(fact_4_max__flow__val__unique,axiom,
% 0.28/0.46 ! [Fv1: capacity,Fv2: capacity] :
% 0.28/0.46 ( ( is_max_flow_val @ capacity @ c @ s @ t @ Fv1 )
% 0.28/0.46 => ( ( is_max_flow_val @ capacity @ c @ s @ t @ Fv2 )
% 0.28/0.46 => ( Fv1 = Fv2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % max_flow_val_unique
% 0.28/0.46 thf(fact_5_isAugmentingPath__def,axiom,
% 0.28/0.46 ! [P: list @ ( product_prod @ nat @ nat )] :
% 0.28/0.46 ( ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ P )
% 0.28/0.46 = ( isSimplePath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ P @ t ) ) ).
% 0.28/0.46
% 0.28/0.46 % isAugmentingPath_def
% 0.28/0.46 thf(fact_6_Finite__Preflow__axioms,axiom,
% 0.28/0.46 finite_Preflow @ capacity @ c @ s @ t @ f ).
% 0.28/0.46
% 0.28/0.46 % Finite_Preflow_axioms
% 0.28/0.46 thf(fact_7_fofu__I__II,axiom,
% 0.28/0.46 ( ( isMaxFlow @ capacity @ c @ s @ t @ f )
% 0.28/0.46 => ~ ? [X_1: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ X_1 ) ) ).
% 0.28/0.46
% 0.28/0.46 % fofu_I_II
% 0.28/0.46 thf(fact_8_noAugPath__iff__maxFlow,axiom,
% 0.28/0.46 ( ( ~ ? [X: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ X ) )
% 0.28/0.46 = ( isMaxFlow @ capacity @ c @ s @ t @ f ) ) ).
% 0.28/0.46
% 0.28/0.46 % noAugPath_iff_maxFlow
% 0.28/0.46 thf(fact_9_Preflow__axioms,axiom,
% 0.28/0.46 preflow @ capacity @ c @ s @ t @ f ).
% 0.28/0.46
% 0.28/0.46 % Preflow_axioms
% 0.28/0.46 thf(fact_10_is__RGraph,axiom,
% 0.28/0.46 residual_RGraph @ capacity @ c @ s @ t @ ( residu1993642540lGraph @ capacity @ c @ f ) ).
% 0.28/0.46
% 0.28/0.46 % is_RGraph
% 0.28/0.46 thf(fact_11_Flow__axioms,axiom,
% 0.28/0.46 flow @ capacity @ c @ s @ t @ f ).
% 0.28/0.46
% 0.28/0.46 % Flow_axioms
% 0.28/0.46 thf(fact_12_NPreflow__axioms,axiom,
% 0.28/0.46 nPreflow @ capacity @ c @ s @ t @ f ).
% 0.28/0.46
% 0.28/0.46 % NPreflow_axioms
% 0.28/0.46 thf(fact_13_converse__empty,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( converse @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.28/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % converse_empty
% 0.28/0.46 thf(fact_14_inf__bot__left,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( bounded_lattice_bot @ A )
% 0.28/0.46 => ! [X2: A] :
% 0.28/0.46 ( ( inf_inf @ A @ ( bot_bot @ A ) @ X2 )
% 0.28/0.46 = ( bot_bot @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_bot_left
% 0.28/0.46 thf(fact_15_inf__bot__right,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( bounded_lattice_bot @ A )
% 0.28/0.46 => ! [X2: A] :
% 0.28/0.46 ( ( inf_inf @ A @ X2 @ ( bot_bot @ A ) )
% 0.28/0.46 = ( bot_bot @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_bot_right
% 0.28/0.46 thf(fact_16_is__RPreGraph,axiom,
% 0.28/0.46 residual_RPreGraph @ capacity @ c @ s @ t @ ( residu1993642540lGraph @ capacity @ c @ f ) ).
% 0.28/0.46
% 0.28/0.46 % is_RPreGraph
% 0.28/0.46 thf(fact_17_Network__axioms,axiom,
% 0.28/0.46 network @ capacity @ c @ s @ t ).
% 0.28/0.46
% 0.28/0.46 % Network_axioms
% 0.28/0.46 thf(fact_18_fo__rg__inv,axiom,
% 0.28/0.46 ( ( residual_flow_of_cf @ capacity @ c @ ( residu1993642540lGraph @ capacity @ c @ f ) )
% 0.28/0.46 = f ) ).
% 0.28/0.46
% 0.28/0.46 % fo_rg_inv
% 0.28/0.46 thf(fact_19_inf__right__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [X2: A,Y: A] :
% 0.28/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y ) @ Y )
% 0.28/0.46 = ( inf_inf @ A @ X2 @ Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_right_idem
% 0.28/0.46 thf(fact_20_inf_Oright__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
% 0.28/0.46 = ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.right_idem
% 0.28/0.46 thf(fact_21_inf__left__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [X2: A,Y: A] :
% 0.28/0.46 ( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ X2 @ Y ) )
% 0.28/0.46 = ( inf_inf @ A @ X2 @ Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_left_idem
% 0.28/0.46 thf(fact_22_inf_Oleft__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
% 0.28/0.46 = ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.left_idem
% 0.28/0.46 thf(fact_23_inf__idem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [X2: A] :
% 0.28/0.46 ( ( inf_inf @ A @ X2 @ X2 )
% 0.28/0.46 = X2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_idem
% 0.28/0.46 thf(fact_24_inf_Oidem,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( inf_inf @ A @ A2 @ A2 )
% 0.28/0.46 = A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.idem
% 0.28/0.46 thf(fact_25_inf__apply,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ B )
% 0.28/0.46 => ( ( inf_inf @ ( A > B ) )
% 0.28/0.46 = ( ^ [F: A > B,G: A > B,X3: A] : ( inf_inf @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_apply
% 0.28/0.46 thf(fact_26_converse__converse,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.28/0.46 ( ( converse @ B @ A @ ( converse @ A @ B @ R ) )
% 0.28/0.46 = R ) ).
% 0.28/0.46
% 0.28/0.46 % converse_converse
% 0.28/0.46 thf(fact_27_converse__inject,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A )] :
% 0.28/0.46 ( ( ( converse @ B @ A @ R )
% 0.28/0.46 = ( converse @ B @ A @ S ) )
% 0.28/0.46 = ( R = S ) ) ).
% 0.28/0.46
% 0.28/0.46 % converse_inject
% 0.28/0.46 thf(fact_28_augFlow__resFlow,axiom,
% 0.28/0.46 ! [P: list @ ( product_prod @ nat @ nat )] :
% 0.28/0.46 ( ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ P )
% 0.28/0.46 => ( flow @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ t @ ( augmen1604248148ngFlow @ capacity @ c @ f @ P ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % augFlow_resFlow
% 0.28/0.46 thf(fact_29_bot__empty__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( bot_bot @ ( A > $o ) )
% 0.28/0.46 = ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % bot_empty_eq
% 0.28/0.46 thf(fact_30_inf__left__commute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [X2: A,Y: A,Z: A] :
% 0.28/0.46 ( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y @ Z ) )
% 0.28/0.46 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X2 @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_left_commute
% 0.28/0.46 thf(fact_31_inf_Oleft__commute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [B2: A,A2: A,C: A] :
% 0.28/0.46 ( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A2 @ C ) )
% 0.28/0.46 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.left_commute
% 0.28/0.46 thf(fact_32_inf__commute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ( ( inf_inf @ A )
% 0.28/0.46 = ( ^ [X3: A,Y2: A] : ( inf_inf @ A @ Y2 @ X3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_commute
% 0.28/0.46 thf(fact_33_inf_Ocommute,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ( ( inf_inf @ A )
% 0.28/0.46 = ( ^ [A3: A,B3: A] : ( inf_inf @ A @ B3 @ A3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.commute
% 0.28/0.46 thf(fact_34_inf__assoc,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [X2: A,Y: A,Z: A] :
% 0.28/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y ) @ Z )
% 0.28/0.46 = ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_assoc
% 0.28/0.46 thf(fact_35_inf_Oassoc,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C )
% 0.28/0.46 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.assoc
% 0.28/0.46 thf(fact_36_boolean__algebra__cancel_Oinf2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [B4: A,K: A,B2: A,A2: A] :
% 0.28/0.46 ( ( B4
% 0.28/0.46 = ( inf_inf @ A @ K @ B2 ) )
% 0.28/0.46 => ( ( inf_inf @ A @ A2 @ B4 )
% 0.28/0.46 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % boolean_algebra_cancel.inf2
% 0.28/0.46 thf(fact_37_boolean__algebra__cancel_Oinf1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A4: A,K: A,A2: A,B2: A] :
% 0.28/0.46 ( ( A4
% 0.28/0.46 = ( inf_inf @ A @ K @ A2 ) )
% 0.28/0.46 => ( ( inf_inf @ A @ A4 @ B2 )
% 0.28/0.46 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % boolean_algebra_cancel.inf1
% 0.28/0.46 thf(fact_38_inf__fun__def,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ B )
% 0.28/0.46 => ( ( inf_inf @ ( A > B ) )
% 0.28/0.46 = ( ^ [F: A > B,G: A > B,X3: A] : ( inf_inf @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_fun_def
% 0.28/0.46 thf(fact_39_inf__sup__aci_I1_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A )
% 0.28/0.46 => ( ( inf_inf @ A )
% 0.28/0.46 = ( ^ [X3: A,Y2: A] : ( inf_inf @ A @ Y2 @ X3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_aci(1)
% 0.28/0.46 thf(fact_40_inf__sup__aci_I2_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A )
% 0.28/0.46 => ! [X2: A,Y: A,Z: A] :
% 0.28/0.46 ( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y ) @ Z )
% 0.28/0.46 = ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_aci(2)
% 0.28/0.46 thf(fact_41_inf__sup__aci_I3_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A )
% 0.28/0.46 => ! [X2: A,Y: A,Z: A] :
% 0.28/0.46 ( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y @ Z ) )
% 0.28/0.46 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X2 @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_aci(3)
% 0.28/0.46 thf(fact_42_inf__sup__aci_I4_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A )
% 0.28/0.46 => ! [X2: A,Y: A] :
% 0.28/0.46 ( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ X2 @ Y ) )
% 0.28/0.46 = ( inf_inf @ A @ X2 @ Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_aci(4)
% 0.28/0.46 thf(fact_43_converse__Int,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A )] :
% 0.28/0.46 ( ( converse @ B @ A @ ( inf_inf @ ( set @ ( product_prod @ B @ A ) ) @ R @ S ) )
% 0.28/0.46 = ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R ) @ ( converse @ B @ A @ S ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % converse_Int
% 0.28/0.46 thf(fact_44_augment__flow__presv,axiom,
% 0.28/0.46 ! [F2: ( product_prod @ nat @ nat ) > capacity] :
% 0.28/0.46 ( ( flow @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ t @ F2 )
% 0.28/0.46 => ( flow @ capacity @ c @ s @ t @ ( augmenting_augment @ capacity @ c @ f @ F2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % augment_flow_presv
% 0.28/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.28/0.46 ! [A: $tType,A2: A,P2: A > $o] :
% 0.28/0.46 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 0.28/0.46 = ( P2 @ A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % mem_Collect_eq
% 0.28/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.28/0.46 ! [A: $tType,A4: set @ A] :
% 0.28/0.46 ( ( collect @ A
% 0.28/0.46 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 0.28/0.46 = A4 ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_mem_eq
% 0.28/0.46 thf(fact_47_Collect__cong,axiom,
% 0.28/0.46 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.28/0.46 ( ! [X4: A] :
% 0.28/0.46 ( ( P2 @ X4 )
% 0.28/0.46 = ( Q @ X4 ) )
% 0.28/0.46 => ( ( collect @ A @ P2 )
% 0.28/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Collect_cong
% 0.28/0.46 thf(fact_48_ext,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,F3: A > B,G2: A > B] :
% 0.28/0.46 ( ! [X4: A] :
% 0.28/0.46 ( ( F3 @ X4 )
% 0.28/0.46 = ( G2 @ X4 ) )
% 0.28/0.46 => ( F3 = G2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % ext
% 0.28/0.46 thf(fact_49_cf_OisSPath__nt__parallel,axiom,
% 0.28/0.46 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat,E: product_prod @ nat @ nat] :
% 0.28/0.46 ( ( isSimplePath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ S @ P @ T )
% 0.28/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ E @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.28/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % cf.isSPath_nt_parallel
% 0.28/0.46 thf(fact_50_is__max__flow__val__def,axiom,
% 0.28/0.46 ! [Fv: capacity] :
% 0.28/0.46 ( ( is_max_flow_val @ capacity @ c @ s @ t @ Fv )
% 0.28/0.46 = ( ? [F: ( product_prod @ nat @ nat ) > capacity] :
% 0.28/0.46 ( ( isMaxFlow @ capacity @ c @ s @ t @ F )
% 0.28/0.46 & ( Fv
% 0.28/0.46 = ( val @ capacity @ c @ s @ F ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % is_max_flow_val_def
% 0.28/0.46 thf(fact_51_cf_OFinite__Graph__axioms,axiom,
% 0.28/0.46 finite_Graph @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) ).
% 0.28/0.46
% 0.28/0.46 % cf.Finite_Graph_axioms
% 0.28/0.46 thf(fact_52_ford__fulkerson_I1_J,axiom,
% 0.28/0.46 ( ( isMaxFlow @ capacity @ c @ s @ t @ f )
% 0.28/0.46 = ( ~ ? [X: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ X ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ford_fulkerson(1)
% 0.28/0.46 thf(fact_53_shortest__is__augmenting,axiom,
% 0.28/0.46 ! [P: list @ ( product_prod @ nat @ nat )] :
% 0.28/0.46 ( ( isShortestPath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ P @ t )
% 0.28/0.46 => ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ P ) ) ).
% 0.28/0.46
% 0.28/0.46 % shortest_is_augmenting
% 0.28/0.46 thf(fact_54_augmenting__path__imp__shortest,axiom,
% 0.28/0.46 ! [P: list @ ( product_prod @ nat @ nat )] :
% 0.28/0.46 ( ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ P )
% 0.28/0.46 => ? [P3: list @ ( product_prod @ nat @ nat )] : ( isShortestPath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ P3 @ t ) ) ).
% 0.28/0.46
% 0.28/0.46 % augmenting_path_imp_shortest
% 0.28/0.46 thf(fact_55_Graph_OisSPath__nt__parallel__pf,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat] :
% 0.28/0.46 ( ( isSimplePath @ Capacity @ C @ S @ P @ T )
% 0.28/0.46 => ( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) @ ( converse @ nat @ nat @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) )
% 0.28/0.46 = ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Graph.isSPath_nt_parallel_pf
% 0.28/0.46 thf(fact_56_RPreGraph_Omaxflow__imp__rgraph,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 0.28/0.46 ( ( residual_RPreGraph @ A @ C @ S @ T @ Cf )
% 0.28/0.46 => ( ( isMaxFlow @ A @ C @ S @ T @ ( residual_flow_of_cf @ A @ C @ Cf ) )
% 0.28/0.46 => ( residual_RGraph @ A @ C @ S @ T @ Cf ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % RPreGraph.maxflow_imp_rgraph
% 0.28/0.46 thf(fact_57_isSPath__nt__parallel,axiom,
% 0.28/0.46 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat,E: product_prod @ nat @ nat] :
% 0.28/0.46 ( ( isSimplePath @ capacity @ c @ S @ P @ T )
% 0.28/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ E @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.28/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % isSPath_nt_parallel
% 0.28/0.46 thf(fact_58_Finite__Graph__axioms,axiom,
% 0.28/0.46 finite_Graph @ capacity @ c ).
% 0.28/0.46
% 0.28/0.46 % Finite_Graph_axioms
% 0.28/0.46 thf(fact_59_shortestPath__is__simple,axiom,
% 0.28/0.46 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat] :
% 0.28/0.46 ( ( isShortestPath @ capacity @ c @ S @ P @ T )
% 0.28/0.46 => ( isSimplePath @ capacity @ c @ S @ P @ T ) ) ).
% 0.28/0.46
% 0.28/0.46 % shortestPath_is_simple
% 0.28/0.46 thf(fact_60_cf_OshortestPath__is__simple,axiom,
% 0.28/0.46 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat] :
% 0.28/0.46 ( ( isShortestPath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ S @ P @ T )
% 0.28/0.46 => ( isSimplePath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ S @ P @ T ) ) ).
% 0.28/0.46
% 0.28/0.46 % cf.shortestPath_is_simple
% 0.28/0.46 thf(fact_61_Graph_OisShortestPath_Ocong,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ( ( isShortestPath @ Capacity )
% 0.28/0.46 = ( isShortestPath @ Capacity ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Graph.isShortestPath.cong
% 0.28/0.46 thf(fact_62_Graph_OshortestPath__is__simple,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat] :
% 0.28/0.46 ( ( isShortestPath @ Capacity @ C @ S @ P @ T )
% 0.28/0.46 => ( isSimplePath @ Capacity @ C @ S @ P @ T ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Graph.shortestPath_is_simple
% 0.28/0.46 thf(fact_63_Graph_OisSPath__nt__parallel,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat,E: product_prod @ nat @ nat] :
% 0.28/0.46 ( ( isSimplePath @ Capacity @ C @ S @ P @ T )
% 0.28/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ E @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.28/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Graph.isSPath_nt_parallel
% 0.28/0.46 thf(fact_64_Graph_OisSimplePath_Ocong,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ( ( isSimplePath @ Capacity )
% 0.28/0.46 = ( isSimplePath @ Capacity ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Graph.isSimplePath.cong
% 0.28/0.46 thf(fact_65_RPreGraph_Othis__loc__rpg,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 0.28/0.46 ( ( residual_RPreGraph @ A @ C @ S @ T @ Cf )
% 0.28/0.46 => ( residual_RPreGraph @ A @ C @ S @ T @ Cf ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % RPreGraph.this_loc_rpg
% 0.28/0.46 thf(fact_66_NFlow_Oshortest__is__augmenting,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,P: list @ ( product_prod @ nat @ nat )] :
% 0.28/0.46 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ( isShortestPath @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ P @ T )
% 0.28/0.46 => ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ P ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow.shortest_is_augmenting
% 0.28/0.46 thf(fact_67_NFlow_Oaugmenting__path__imp__shortest,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,P: list @ ( product_prod @ nat @ nat )] :
% 0.28/0.46 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ P )
% 0.28/0.46 => ? [P3: list @ ( product_prod @ nat @ nat )] : ( isShortestPath @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ P3 @ T ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow.augmenting_path_imp_shortest
% 0.28/0.46 thf(fact_68_RGraph_Othis__loc,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 0.28/0.46 ( ( residual_RGraph @ A @ C @ S @ T @ Cf )
% 0.28/0.46 => ( residual_RGraph @ A @ C @ S @ T @ Cf ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % RGraph.this_loc
% 0.28/0.46 thf(fact_69_Network_Oflow__of__cf_Ocong,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ( ( residual_flow_of_cf @ Capacity )
% 0.28/0.46 = ( residual_flow_of_cf @ Capacity ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Network.flow_of_cf.cong
% 0.28/0.46 thf(fact_70_RPreGraph_Oaxioms_I1_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 0.28/0.46 ( ( residual_RPreGraph @ A @ C @ S @ T @ Cf )
% 0.28/0.46 => ( network @ A @ C @ S @ T ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % RPreGraph.axioms(1)
% 0.28/0.46 thf(fact_71_RGraph_Oaxioms_I1_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 0.28/0.46 ( ( residual_RGraph @ A @ C @ S @ T @ Cf )
% 0.28/0.46 => ( network @ A @ C @ S @ T ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % RGraph.axioms(1)
% 0.28/0.46 thf(fact_72_RGraph_Othis__loc__rpg,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 0.28/0.46 ( ( residual_RGraph @ A @ C @ S @ T @ Cf )
% 0.28/0.46 => ( residual_RPreGraph @ A @ C @ S @ T @ Cf ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % RGraph.this_loc_rpg
% 0.28/0.46 thf(fact_73_NPreflow_Ois__RPreGraph,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( residual_RPreGraph @ Capacity @ C @ S @ T @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NPreflow.is_RPreGraph
% 0.28/0.46 thf(fact_74_RPreGraph_OEX__RPG,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 0.28/0.46 ( ( residual_RPreGraph @ A @ C @ S @ T @ Cf )
% 0.28/0.46 => ? [F4: ( product_prod @ nat @ nat ) > A] :
% 0.28/0.46 ( ( nPreflow @ A @ C @ S @ T @ F4 )
% 0.28/0.46 & ( Cf
% 0.28/0.46 = ( residu1993642540lGraph @ A @ C @ F4 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % RPreGraph.EX_RPG
% 0.28/0.46 thf(fact_75_NFlow_Ois__RGraph,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( residual_RGraph @ Capacity @ C @ S @ T @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow.is_RGraph
% 0.28/0.46 thf(fact_76_RGraph_OEX__RG,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 0.28/0.46 ( ( residual_RGraph @ A @ C @ S @ T @ Cf )
% 0.28/0.46 => ? [F4: ( product_prod @ nat @ nat ) > A] :
% 0.28/0.46 ( ( nFlow @ A @ C @ S @ T @ F4 )
% 0.28/0.46 & ( Cf
% 0.28/0.46 = ( residu1993642540lGraph @ A @ C @ F4 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % RGraph.EX_RG
% 0.28/0.46 thf(fact_77_NPreflow_Ofo__rg__inv,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ( residual_flow_of_cf @ Capacity @ C @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) )
% 0.28/0.46 = F3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NPreflow.fo_rg_inv
% 0.28/0.46 thf(fact_78_RPreGraph_Org__fo__inv,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 0.28/0.46 ( ( residual_RPreGraph @ A @ C @ S @ T @ Cf )
% 0.28/0.46 => ( ( residu1993642540lGraph @ A @ C @ ( residual_flow_of_cf @ A @ C @ Cf ) )
% 0.28/0.46 = Cf ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % RPreGraph.rg_fo_inv
% 0.28/0.46 thf(fact_79_RPreGraph_Ois__NPreflow,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 0.28/0.46 ( ( residual_RPreGraph @ A @ C @ S @ T @ Cf )
% 0.28/0.46 => ( nPreflow @ A @ C @ S @ T @ ( residual_flow_of_cf @ A @ C @ Cf ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % RPreGraph.is_NPreflow
% 0.28/0.46 thf(fact_80_RGraph_Ois__NFlow,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 0.28/0.46 ( ( residual_RGraph @ A @ C @ S @ T @ Cf )
% 0.28/0.46 => ( nFlow @ A @ C @ S @ T @ ( residual_flow_of_cf @ A @ C @ Cf ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % RGraph.is_NFlow
% 0.28/0.46 thf(fact_81_augFlow__val,axiom,
% 0.28/0.46 ! [P: list @ ( product_prod @ nat @ nat )] :
% 0.28/0.46 ( ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ P )
% 0.28/0.46 => ( ( val @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ ( augmen1604248148ngFlow @ capacity @ c @ f @ P ) )
% 0.28/0.46 = ( augmenting_resCap @ capacity @ c @ f @ P ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % augFlow_val
% 0.28/0.46 thf(fact_82_augment__flow__value,axiom,
% 0.28/0.46 ! [F2: ( product_prod @ nat @ nat ) > capacity] :
% 0.28/0.46 ( ( flow @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ t @ F2 )
% 0.28/0.46 => ( ( val @ capacity @ c @ s @ ( augmenting_augment @ capacity @ c @ f @ F2 ) )
% 0.28/0.46 = ( plus_plus @ capacity @ ( val @ capacity @ c @ s @ f ) @ ( val @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ F2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % augment_flow_value
% 0.28/0.46 thf(fact_83_isMaxFlow__alt,axiom,
% 0.28/0.46 ! [F3: ( product_prod @ nat @ nat ) > capacity] :
% 0.28/0.46 ( ( isMaxFlow @ capacity @ c @ s @ t @ F3 )
% 0.28/0.46 = ( ( nFlow @ capacity @ c @ s @ t @ F3 )
% 0.28/0.46 & ! [F5: ( product_prod @ nat @ nat ) > capacity] :
% 0.28/0.46 ( ( nFlow @ capacity @ c @ s @ t @ F5 )
% 0.28/0.46 => ( ord_less_eq @ capacity @ ( val @ capacity @ c @ s @ F5 ) @ ( val @ capacity @ c @ s @ F3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % isMaxFlow_alt
% 0.28/0.46 thf(fact_84_isMaxFlow__def,axiom,
% 0.28/0.46 ! [F3: ( product_prod @ nat @ nat ) > capacity] :
% 0.28/0.46 ( ( isMaxFlow @ capacity @ c @ s @ t @ F3 )
% 0.28/0.46 = ( ( flow @ capacity @ c @ s @ t @ F3 )
% 0.28/0.46 & ! [F5: ( product_prod @ nat @ nat ) > capacity] :
% 0.28/0.46 ( ( flow @ capacity @ c @ s @ t @ F5 )
% 0.28/0.46 => ( ord_less_eq @ capacity @ ( val @ capacity @ c @ s @ F5 ) @ ( val @ capacity @ c @ s @ F3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % isMaxFlow_def
% 0.28/0.46 thf(fact_85_NPreflow_OaugFlow__resFlow,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,P: list @ ( product_prod @ nat @ nat )] :
% 0.28/0.46 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ P )
% 0.28/0.46 => ( flow @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ T @ ( augmen1604248148ngFlow @ Capacity @ C @ F3 @ P ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NPreflow.augFlow_resFlow
% 0.28/0.46 thf(fact_86_bot__apply,axiom,
% 0.28/0.46 ! [C2: $tType,D: $tType] :
% 0.28/0.46 ( ( bot @ C2 )
% 0.28/0.46 => ( ( bot_bot @ ( D > C2 ) )
% 0.28/0.46 = ( ^ [X3: D] : ( bot_bot @ C2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % bot_apply
% 0.28/0.46 thf(fact_87_maxFlow__minCut,axiom,
% 0.28/0.46 ! [F3: ( product_prod @ nat @ nat ) > capacity,K: set @ nat] :
% 0.28/0.46 ( ( isMaxFlow @ capacity @ c @ s @ t @ F3 )
% 0.28/0.46 => ( ( isMinCut @ capacity @ c @ s @ t @ K )
% 0.28/0.46 => ( ( val @ capacity @ c @ s @ F3 )
% 0.28/0.46 = ( cap @ capacity @ c @ K ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % maxFlow_minCut
% 0.28/0.46 thf(fact_88_order__refl,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( preorder @ A )
% 0.28/0.46 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % order_refl
% 0.28/0.46 thf(fact_89_inf_Obounded__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) )
% 0.28/0.46 = ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.46 & ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.bounded_iff
% 0.28/0.46 thf(fact_90_le__inf__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [X2: A,Y: A,Z: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ Y @ Z ) )
% 0.28/0.46 = ( ( ord_less_eq @ A @ X2 @ Y )
% 0.28/0.46 & ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_inf_iff
% 0.28/0.46 thf(fact_91_bot_Oextremum,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order_bot @ A )
% 0.28/0.46 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % bot.extremum
% 0.28/0.46 thf(fact_92_bot_Oextremum__unique,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order_bot @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.28/0.46 = ( A2
% 0.28/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % bot.extremum_unique
% 0.28/0.46 thf(fact_93_bot_Oextremum__uniqueI,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order_bot @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.28/0.46 => ( A2
% 0.28/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % bot.extremum_uniqueI
% 0.28/0.46 thf(fact_94_le__funD,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( ord @ B )
% 0.28/0.46 => ! [F3: A > B,G2: A > B,X2: A] :
% 0.28/0.46 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
% 0.28/0.46 => ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_funD
% 0.28/0.46 thf(fact_95_le__funE,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( ord @ B )
% 0.28/0.46 => ! [F3: A > B,G2: A > B,X2: A] :
% 0.28/0.46 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
% 0.28/0.46 => ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_funE
% 0.28/0.46 thf(fact_96_le__funI,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( ord @ B )
% 0.28/0.46 => ! [F3: A > B,G2: A > B] :
% 0.28/0.46 ( ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 0.28/0.46 => ( ord_less_eq @ ( A > B ) @ F3 @ G2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_funI
% 0.28/0.46 thf(fact_97_le__fun__def,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( ord @ B )
% 0.28/0.46 => ( ( ord_less_eq @ ( A > B ) )
% 0.28/0.46 = ( ^ [F: A > B,G: A > B] :
% 0.28/0.46 ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_fun_def
% 0.28/0.46 thf(fact_98_order__subst1,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType] :
% 0.28/0.46 ( ( ( order @ B )
% 0.28/0.46 & ( order @ A ) )
% 0.28/0.46 => ! [A2: A,F3: B > A,B2: B,C: B] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
% 0.28/0.46 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.28/0.46 => ( ! [X4: B,Y3: B] :
% 0.28/0.46 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 0.28/0.46 => ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
% 0.28/0.46 => ( ord_less_eq @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % order_subst1
% 0.28/0.46 thf(fact_99_order__subst2,axiom,
% 0.28/0.46 ! [A: $tType,C2: $tType] :
% 0.28/0.46 ( ( ( order @ C2 )
% 0.28/0.46 & ( order @ A ) )
% 0.28/0.46 => ! [A2: A,B2: A,F3: A > C2,C: C2] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.46 => ( ( ord_less_eq @ C2 @ ( F3 @ B2 ) @ C )
% 0.28/0.46 => ( ! [X4: A,Y3: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.28/0.46 => ( ord_less_eq @ C2 @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
% 0.28/0.46 => ( ord_less_eq @ C2 @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % order_subst2
% 0.28/0.46 thf(fact_100_ord__eq__le__subst,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType] :
% 0.28/0.46 ( ( ( ord @ B )
% 0.28/0.46 & ( ord @ A ) )
% 0.28/0.46 => ! [A2: A,F3: B > A,B2: B,C: B] :
% 0.28/0.46 ( ( A2
% 0.28/0.46 = ( F3 @ B2 ) )
% 0.28/0.46 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.28/0.46 => ( ! [X4: B,Y3: B] :
% 0.28/0.46 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 0.28/0.46 => ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
% 0.28/0.46 => ( ord_less_eq @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ord_eq_le_subst
% 0.28/0.46 thf(fact_101_ord__le__eq__subst,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType] :
% 0.28/0.46 ( ( ( ord @ B )
% 0.28/0.46 & ( ord @ A ) )
% 0.28/0.46 => ! [A2: A,B2: A,F3: A > B,C: B] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.46 => ( ( ( F3 @ B2 )
% 0.28/0.46 = C )
% 0.28/0.46 => ( ! [X4: A,Y3: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.28/0.46 => ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
% 0.28/0.46 => ( ord_less_eq @ B @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ord_le_eq_subst
% 0.28/0.46 thf(fact_102_eq__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A )
% 0.28/0.46 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.28/0.46 = ( ^ [X3: A,Y2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.28/0.46 & ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % eq_iff
% 0.28/0.46 thf(fact_103_antisym,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A )
% 0.28/0.46 => ! [X2: A,Y: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.28/0.46 => ( ( ord_less_eq @ A @ Y @ X2 )
% 0.28/0.46 => ( X2 = Y ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % antisym
% 0.28/0.46 thf(fact_104_linear,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A )
% 0.28/0.46 => ! [X2: A,Y: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.28/0.46 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % linear
% 0.28/0.46 thf(fact_105_eq__refl,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( preorder @ A )
% 0.28/0.46 => ! [X2: A,Y: A] :
% 0.28/0.46 ( ( X2 = Y )
% 0.28/0.46 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % eq_refl
% 0.28/0.46 thf(fact_106_le__cases,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A )
% 0.28/0.46 => ! [X2: A,Y: A] :
% 0.28/0.46 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 0.28/0.46 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_cases
% 0.28/0.46 thf(fact_107_order_Otrans,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.46 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.28/0.46 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % order.trans
% 0.28/0.46 thf(fact_108_le__cases3,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A )
% 0.28/0.46 => ! [X2: A,Y: A,Z: A] :
% 0.28/0.46 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 0.28/0.46 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 0.28/0.46 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 0.28/0.46 => ~ ( ord_less_eq @ A @ X2 @ Z ) )
% 0.28/0.46 => ( ( ( ord_less_eq @ A @ X2 @ Z )
% 0.28/0.46 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 0.28/0.46 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 0.28/0.46 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 0.28/0.46 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 0.28/0.46 => ~ ( ord_less_eq @ A @ Z @ X2 ) )
% 0.28/0.46 => ~ ( ( ord_less_eq @ A @ Z @ X2 )
% 0.28/0.46 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_cases3
% 0.28/0.46 thf(fact_109_antisym__conv,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A )
% 0.28/0.46 => ! [Y: A,X2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.28/0.46 => ( ( ord_less_eq @ A @ X2 @ Y )
% 0.28/0.46 = ( X2 = Y ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % antisym_conv
% 0.28/0.46 thf(fact_110_order__class_Oorder_Oeq__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A )
% 0.28/0.46 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.28/0.46 = ( ^ [A3: A,B3: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.28/0.46 & ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % order_class.order.eq_iff
% 0.28/0.46 thf(fact_111_ord__eq__le__trans,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ord @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( A2 = B2 )
% 0.28/0.46 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.28/0.46 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ord_eq_le_trans
% 0.28/0.46 thf(fact_112_ord__le__eq__trans,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( ord @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.46 => ( ( B2 = C )
% 0.28/0.46 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ord_le_eq_trans
% 0.28/0.46 thf(fact_113_order__class_Oorder_Oantisym,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.46 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.28/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % order_class.order.antisym
% 0.28/0.46 thf(fact_114_NPreflow_OresCap_Ocong,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ( ( augmenting_resCap @ Capacity )
% 0.28/0.46 = ( augmenting_resCap @ Capacity ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NPreflow.resCap.cong
% 0.28/0.46 thf(fact_115_order__trans,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( preorder @ A )
% 0.28/0.46 => ! [X2: A,Y: A,Z: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.28/0.46 => ( ( ord_less_eq @ A @ Y @ Z )
% 0.28/0.46 => ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % order_trans
% 0.28/0.46 thf(fact_116_dual__order_Orefl,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A )
% 0.28/0.46 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % dual_order.refl
% 0.28/0.46 thf(fact_117_linorder__wlog,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A )
% 0.28/0.46 => ! [P2: A > A > $o,A2: A,B2: A] :
% 0.28/0.46 ( ! [A5: A,B5: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.28/0.46 => ( P2 @ A5 @ B5 ) )
% 0.28/0.46 => ( ! [A5: A,B5: A] :
% 0.28/0.46 ( ( P2 @ B5 @ A5 )
% 0.28/0.46 => ( P2 @ A5 @ B5 ) )
% 0.28/0.46 => ( P2 @ A2 @ B2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % linorder_wlog
% 0.28/0.46 thf(fact_118_dual__order_Otrans,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A )
% 0.28/0.46 => ! [B2: A,A2: A,C: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.28/0.46 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.28/0.46 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % dual_order.trans
% 0.28/0.46 thf(fact_119_dual__order_Oeq__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A )
% 0.28/0.46 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 0.28/0.46 = ( ^ [A3: A,B3: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.28/0.46 & ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % dual_order.eq_iff
% 0.28/0.46 thf(fact_120_dual__order_Oantisym,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A )
% 0.28/0.46 => ! [B2: A,A2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.28/0.46 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % dual_order.antisym
% 0.28/0.46 thf(fact_121_inf_OcoboundedI2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [B2: A,C: A,A2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ B2 @ C )
% 0.28/0.46 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.coboundedI2
% 0.28/0.46 thf(fact_122_inf_OcoboundedI1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,C: A,B2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A2 @ C )
% 0.28/0.46 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.coboundedI1
% 0.28/0.46 thf(fact_123_inf_Oabsorb__iff2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ( ( ord_less_eq @ A )
% 0.28/0.46 = ( ^ [B3: A,A3: A] :
% 0.28/0.46 ( ( inf_inf @ A @ A3 @ B3 )
% 0.28/0.46 = B3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.absorb_iff2
% 0.28/0.46 thf(fact_124_inf_Oabsorb__iff1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ( ( ord_less_eq @ A )
% 0.28/0.46 = ( ^ [A3: A,B3: A] :
% 0.28/0.46 ( ( inf_inf @ A @ A3 @ B3 )
% 0.28/0.46 = A3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.absorb_iff1
% 0.28/0.46 thf(fact_125_inf_Ocobounded2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.cobounded2
% 0.28/0.46 thf(fact_126_inf_Ocobounded1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ A2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.cobounded1
% 0.28/0.46 thf(fact_127_inf_Oorder__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ( ( ord_less_eq @ A )
% 0.28/0.46 = ( ^ [A3: A,B3: A] :
% 0.28/0.46 ( A3
% 0.28/0.46 = ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.order_iff
% 0.28/0.46 thf(fact_128_inf__greatest,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [X2: A,Y: A,Z: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.28/0.46 => ( ( ord_less_eq @ A @ X2 @ Z )
% 0.28/0.46 => ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ Y @ Z ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_greatest
% 0.28/0.46 thf(fact_129_inf_OboundedI,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.46 => ( ( ord_less_eq @ A @ A2 @ C )
% 0.28/0.46 => ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.boundedI
% 0.28/0.46 thf(fact_130_inf_OboundedE,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) )
% 0.28/0.46 => ~ ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.46 => ~ ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.boundedE
% 0.28/0.46 thf(fact_131_inf__absorb2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [Y: A,X2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ Y @ X2 )
% 0.28/0.46 => ( ( inf_inf @ A @ X2 @ Y )
% 0.28/0.46 = Y ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_absorb2
% 0.28/0.46 thf(fact_132_inf__absorb1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [X2: A,Y: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.28/0.46 => ( ( inf_inf @ A @ X2 @ Y )
% 0.28/0.46 = X2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_absorb1
% 0.28/0.46 thf(fact_133_inf_Oabsorb2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [B2: A,A2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.28/0.46 => ( ( inf_inf @ A @ A2 @ B2 )
% 0.28/0.46 = B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.absorb2
% 0.28/0.46 thf(fact_134_inf_Oabsorb1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.46 => ( ( inf_inf @ A @ A2 @ B2 )
% 0.28/0.46 = A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.absorb1
% 0.28/0.46 thf(fact_135_le__iff__inf,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ( ( ord_less_eq @ A )
% 0.28/0.46 = ( ^ [X3: A,Y2: A] :
% 0.28/0.46 ( ( inf_inf @ A @ X3 @ Y2 )
% 0.28/0.46 = X3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_iff_inf
% 0.28/0.46 thf(fact_136_inf__unique,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [F3: A > A > A,X2: A,Y: A] :
% 0.28/0.46 ( ! [X4: A,Y3: A] : ( ord_less_eq @ A @ ( F3 @ X4 @ Y3 ) @ X4 )
% 0.28/0.46 => ( ! [X4: A,Y3: A] : ( ord_less_eq @ A @ ( F3 @ X4 @ Y3 ) @ Y3 )
% 0.28/0.46 => ( ! [X4: A,Y3: A,Z3: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.28/0.46 => ( ( ord_less_eq @ A @ X4 @ Z3 )
% 0.28/0.46 => ( ord_less_eq @ A @ X4 @ ( F3 @ Y3 @ Z3 ) ) ) )
% 0.28/0.46 => ( ( inf_inf @ A @ X2 @ Y )
% 0.28/0.46 = ( F3 @ X2 @ Y ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_unique
% 0.28/0.46 thf(fact_137_inf_OorderI,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( A2
% 0.28/0.46 = ( inf_inf @ A @ A2 @ B2 ) )
% 0.28/0.46 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.orderI
% 0.28/0.46 thf(fact_138_inf_OorderE,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.28/0.46 => ( A2
% 0.28/0.46 = ( inf_inf @ A @ A2 @ B2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.orderE
% 0.28/0.46 thf(fact_139_le__infI2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [B2: A,X2: A,A2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ B2 @ X2 )
% 0.28/0.46 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_infI2
% 0.28/0.46 thf(fact_140_le__infI1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,X2: A,B2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A2 @ X2 )
% 0.28/0.46 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_infI1
% 0.28/0.46 thf(fact_141_inf__mono,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,C: A,B2: A,D2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ A2 @ C )
% 0.28/0.46 => ( ( ord_less_eq @ A @ B2 @ D2 )
% 0.28/0.46 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ ( inf_inf @ A @ C @ D2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_mono
% 0.28/0.46 thf(fact_142_le__infI,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [X2: A,A2: A,B2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X2 @ A2 )
% 0.28/0.46 => ( ( ord_less_eq @ A @ X2 @ B2 )
% 0.28/0.46 => ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_infI
% 0.28/0.46 thf(fact_143_le__infE,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [X2: A,A2: A,B2: A] :
% 0.28/0.46 ( ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ A2 @ B2 ) )
% 0.28/0.46 => ~ ( ( ord_less_eq @ A @ X2 @ A2 )
% 0.28/0.46 => ~ ( ord_less_eq @ A @ X2 @ B2 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % le_infE
% 0.28/0.46 thf(fact_144_inf__le2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [X2: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y ) @ Y ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_le2
% 0.28/0.46 thf(fact_145_inf__le1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [X2: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y ) @ X2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_le1
% 0.28/0.46 thf(fact_146_inf__sup__ord_I1_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A )
% 0.28/0.46 => ! [X2: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y ) @ X2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_ord(1)
% 0.28/0.46 thf(fact_147_inf__sup__ord_I2_J,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( lattice @ A )
% 0.28/0.46 => ! [X2: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y ) @ Y ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf_sup_ord(2)
% 0.28/0.46 thf(fact_148_NPreflow_OisAugmentingPath_Ocong,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ( ( augmen2028041557ngPath @ Capacity )
% 0.28/0.46 = ( augmen2028041557ngPath @ Capacity ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NPreflow.isAugmentingPath.cong
% 0.28/0.46 thf(fact_149_NPreflow_OaugmentingFlow_Ocong,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ( ( augmen1604248148ngFlow @ Capacity )
% 0.28/0.46 = ( augmen1604248148ngFlow @ Capacity ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NPreflow.augmentingFlow.cong
% 0.28/0.46 thf(fact_150_NPreflow_OaugFlow__val,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,P: list @ ( product_prod @ nat @ nat )] :
% 0.28/0.46 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ P )
% 0.28/0.46 => ( ( val @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ ( augmen1604248148ngFlow @ Capacity @ C @ F3 @ P ) )
% 0.28/0.46 = ( augmenting_resCap @ Capacity @ C @ F3 @ P ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NPreflow.augFlow_val
% 0.28/0.46 thf(fact_151_bot__fun__def,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType] :
% 0.28/0.46 ( ( bot @ B )
% 0.28/0.46 => ( ( bot_bot @ ( A > B ) )
% 0.28/0.46 = ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % bot_fun_def
% 0.28/0.46 thf(fact_152_NPreflow_OisAugmentingPath__def,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,P: list @ ( product_prod @ nat @ nat )] :
% 0.28/0.46 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ P )
% 0.28/0.46 = ( isSimplePath @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ P @ T ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NPreflow.isAugmentingPath_def
% 0.28/0.46 thf(fact_153_ford__fulkerson_I2_J,axiom,
% 0.28/0.46 ( ( ~ ? [X: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ X ) )
% 0.28/0.46 = ( ? [K2: set @ nat] :
% 0.28/0.46 ( ( nCut @ capacity @ c @ s @ t @ K2 )
% 0.28/0.46 & ( ( val @ capacity @ c @ s @ f )
% 0.28/0.46 = ( cap @ capacity @ c @ K2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % ford_fulkerson(2)
% 0.28/0.46 thf(fact_154_fofu__III__I,axiom,
% 0.28/0.46 ( ? [K3: set @ nat] :
% 0.28/0.46 ( ( nCut @ capacity @ c @ s @ t @ K3 )
% 0.28/0.46 & ( ( val @ capacity @ c @ s @ f )
% 0.28/0.46 = ( cap @ capacity @ c @ K3 ) ) )
% 0.28/0.46 => ( isMaxFlow @ capacity @ c @ s @ t @ f ) ) ).
% 0.28/0.46
% 0.28/0.46 % fofu_III_I
% 0.28/0.46 thf(fact_155_fofu__II__III,axiom,
% 0.28/0.46 ( ~ ? [X_12: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ X_12 )
% 0.28/0.46 => ? [K4: set @ nat] :
% 0.28/0.46 ( ( nCut @ capacity @ c @ s @ t @ K4 )
% 0.28/0.46 & ( ( val @ capacity @ c @ s @ f )
% 0.28/0.46 = ( cap @ capacity @ c @ K4 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % fofu_II_III
% 0.28/0.46 thf(fact_156_NFlow_Oaugment__flow__value,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,F2: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ( flow @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ T @ F2 )
% 0.28/0.46 => ( ( val @ Capacity @ C @ S @ ( augmenting_augment @ Capacity @ C @ F3 @ F2 ) )
% 0.28/0.46 = ( plus_plus @ Capacity @ ( val @ Capacity @ C @ S @ F3 ) @ ( val @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ F2 ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow.augment_flow_value
% 0.28/0.46 thf(fact_157_Network_OmaxFlow__minCut,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,K: set @ nat] :
% 0.28/0.46 ( ( network @ Capacity @ C @ S @ T )
% 0.28/0.46 => ( ( isMaxFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ( isMinCut @ Capacity @ C @ S @ T @ K )
% 0.28/0.46 => ( ( val @ Capacity @ C @ S @ F3 )
% 0.28/0.46 = ( cap @ Capacity @ C @ K ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Network.maxFlow_minCut
% 0.28/0.46 thf(fact_158_Network_OisMaxFlow__alt,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( network @ Capacity @ C @ S @ T )
% 0.28/0.46 => ( ( isMaxFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 = ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 & ! [F5: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nFlow @ Capacity @ C @ S @ T @ F5 )
% 0.28/0.46 => ( ord_less_eq @ Capacity @ ( val @ Capacity @ C @ S @ F5 ) @ ( val @ Capacity @ C @ S @ F3 ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Network.isMaxFlow_alt
% 0.28/0.46 thf(fact_159_converse__mono,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A )] :
% 0.28/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R ) @ ( converse @ B @ A @ S ) )
% 0.28/0.46 = ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ R @ S ) ) ).
% 0.28/0.46
% 0.28/0.46 % converse_mono
% 0.28/0.46 thf(fact_160_NCut_Oaxioms_I1_J,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,K: set @ nat] :
% 0.28/0.46 ( ( nCut @ Capacity @ C @ S @ T @ K )
% 0.28/0.46 => ( network @ Capacity @ C @ S @ T ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NCut.axioms(1)
% 0.28/0.46 thf(fact_161_NCut_Ot__ni__cut,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,K: set @ nat] :
% 0.28/0.46 ( ( nCut @ Capacity @ C @ S @ T @ K )
% 0.28/0.46 => ~ ( member @ nat @ T @ K ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NCut.t_ni_cut
% 0.28/0.46 thf(fact_162_NCut_Os__in__cut,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,K: set @ nat] :
% 0.28/0.46 ( ( nCut @ Capacity @ C @ S @ T @ K )
% 0.28/0.46 => ( member @ nat @ S @ K ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NCut.s_in_cut
% 0.28/0.46 thf(fact_163_converse__subset__swap,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ A )] :
% 0.28/0.46 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ ( converse @ B @ A @ S ) )
% 0.28/0.46 = ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( converse @ A @ B @ R ) @ S ) ) ).
% 0.28/0.46
% 0.28/0.46 % converse_subset_swap
% 0.28/0.46 thf(fact_164_isMinCut__def,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ( ( isMinCut @ A )
% 0.28/0.46 = ( ^ [C3: ( product_prod @ nat @ nat ) > A,S2: nat,T2: nat,K2: set @ nat] :
% 0.28/0.46 ( ( nCut @ A @ C3 @ S2 @ T2 @ K2 )
% 0.28/0.46 & ! [K5: set @ nat] :
% 0.28/0.46 ( ( nCut @ A @ C3 @ S2 @ T2 @ K5 )
% 0.28/0.46 => ( ord_less_eq @ A @ ( cap @ A @ C3 @ K2 ) @ ( cap @ A @ C3 @ K5 ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % isMinCut_def
% 0.28/0.46 thf(fact_165_Network_Ot__not__s,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat] :
% 0.28/0.46 ( ( network @ Capacity @ C @ S @ T )
% 0.28/0.46 => ( T != S ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Network.t_not_s
% 0.28/0.46 thf(fact_166_Network_Os__not__t,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat] :
% 0.28/0.46 ( ( network @ Capacity @ C @ S @ T )
% 0.28/0.46 => ( S != T ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Network.s_not_t
% 0.28/0.46 thf(fact_167_Flow_Oval_Ocong,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ( ( val @ Capacity )
% 0.28/0.46 = ( val @ Capacity ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Flow.val.cong
% 0.28/0.46 thf(fact_168_Network_OisMaxFlow_Ocong,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ( ( isMaxFlow @ Capacity )
% 0.28/0.46 = ( isMaxFlow @ Capacity ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Network.isMaxFlow.cong
% 0.28/0.46 thf(fact_169_NFlow_Oaugment_Ocong,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ( ( augmenting_augment @ Capacity )
% 0.28/0.46 = ( augmenting_augment @ Capacity ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow.augment.cong
% 0.28/0.46 thf(fact_170_NCut_Ocap_Ocong,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ( ( cap @ Capacity )
% 0.28/0.46 = ( cap @ Capacity ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NCut.cap.cong
% 0.28/0.46 thf(fact_171_NFlow_Ofofu__II__III,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ~ ? [X_12: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ X_12 )
% 0.28/0.46 => ? [K4: set @ nat] :
% 0.28/0.46 ( ( nCut @ Capacity @ C @ S @ T @ K4 )
% 0.28/0.46 & ( ( val @ Capacity @ C @ S @ F3 )
% 0.28/0.46 = ( cap @ Capacity @ C @ K4 ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow.fofu_II_III
% 0.28/0.46 thf(fact_172_NFlow_Ofofu__III__I,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ? [K3: set @ nat] :
% 0.28/0.46 ( ( nCut @ Capacity @ C @ S @ T @ K3 )
% 0.28/0.46 & ( ( val @ Capacity @ C @ S @ F3 )
% 0.28/0.46 = ( cap @ Capacity @ C @ K3 ) ) )
% 0.28/0.46 => ( isMaxFlow @ Capacity @ C @ S @ T @ F3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow.fofu_III_I
% 0.28/0.46 thf(fact_173_Network_Ois__max__flow__val_Ocong,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ( ( is_max_flow_val @ Capacity )
% 0.28/0.46 = ( is_max_flow_val @ Capacity ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Network.is_max_flow_val.cong
% 0.28/0.46 thf(fact_174_NFlow_Oford__fulkerson_I2_J,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ( ~ ? [X: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ X ) )
% 0.28/0.46 = ( ? [K2: set @ nat] :
% 0.28/0.46 ( ( nCut @ Capacity @ C @ S @ T @ K2 )
% 0.28/0.46 & ( ( val @ Capacity @ C @ S @ F3 )
% 0.28/0.46 = ( cap @ Capacity @ C @ K2 ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow.ford_fulkerson(2)
% 0.28/0.46 thf(fact_175_NPreflow_Oaxioms_I1_J,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( network @ Capacity @ C @ S @ T ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NPreflow.axioms(1)
% 0.28/0.46 thf(fact_176_NFlow_Oaxioms_I1_J,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( nPreflow @ Capacity @ C @ S @ T @ F3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow.axioms(1)
% 0.28/0.46 thf(fact_177_NFlow_Oaxioms_I2_J,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( flow @ Capacity @ C @ S @ T @ F3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow.axioms(2)
% 0.28/0.46 thf(fact_178_NPreflow_Oaxioms_I2_J,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( preflow @ Capacity @ C @ S @ T @ F3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NPreflow.axioms(2)
% 0.28/0.46 thf(fact_179_Flow_Oaxioms_I1_J,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( flow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( preflow @ Capacity @ C @ S @ T @ F3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Flow.axioms(1)
% 0.28/0.46 thf(fact_180_Network_Omax__flow__val__unique,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,Fv1: Capacity,Fv2: Capacity] :
% 0.28/0.46 ( ( network @ Capacity @ C @ S @ T )
% 0.28/0.46 => ( ( is_max_flow_val @ Capacity @ C @ S @ T @ Fv1 )
% 0.28/0.46 => ( ( is_max_flow_val @ Capacity @ C @ S @ T @ Fv2 )
% 0.28/0.46 => ( Fv1 = Fv2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Network.max_flow_val_unique
% 0.28/0.46 thf(fact_181_Finite__Preflow_Oaxioms_I2_J,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( finite_Preflow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( finite_Graph @ Capacity @ C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Finite_Preflow.axioms(2)
% 0.28/0.46 thf(fact_182_Finite__Preflow_Oaxioms_I1_J,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( finite_Preflow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( preflow @ Capacity @ C @ S @ T @ F3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Finite_Preflow.axioms(1)
% 0.28/0.46 thf(fact_183_NFlow_Ointro,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ( flow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( nFlow @ Capacity @ C @ S @ T @ F3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow.intro
% 0.28/0.46 thf(fact_184_NFlow__def,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ( ( nFlow @ Capacity )
% 0.28/0.46 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,S2: nat,T2: nat,F: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nPreflow @ Capacity @ C3 @ S2 @ T2 @ F )
% 0.28/0.46 & ( flow @ Capacity @ C3 @ S2 @ T2 @ F ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow_def
% 0.28/0.46 thf(fact_185_NPreflow_Ointro,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( network @ Capacity @ C @ S @ T )
% 0.28/0.46 => ( ( preflow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( nPreflow @ Capacity @ C @ S @ T @ F3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NPreflow.intro
% 0.28/0.46 thf(fact_186_NPreflow__def,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ( ( nPreflow @ Capacity )
% 0.28/0.46 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,S2: nat,T2: nat,F: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( network @ Capacity @ C3 @ S2 @ T2 )
% 0.28/0.46 & ( preflow @ Capacity @ C3 @ S2 @ T2 @ F ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NPreflow_def
% 0.28/0.46 thf(fact_187_NFlow_Ofofu__I__II,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ( isMaxFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ~ ? [X_1: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ X_1 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow.fofu_I_II
% 0.28/0.46 thf(fact_188_NFlow_OnoAugPath__iff__maxFlow,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ( ~ ? [X: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ X ) )
% 0.28/0.46 = ( isMaxFlow @ Capacity @ C @ S @ T @ F3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow.noAugPath_iff_maxFlow
% 0.28/0.46 thf(fact_189_Finite__Preflow_Ointro,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( preflow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ( finite_Graph @ Capacity @ C )
% 0.28/0.46 => ( finite_Preflow @ Capacity @ C @ S @ T @ F3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Finite_Preflow.intro
% 0.28/0.46 thf(fact_190_Finite__Preflow__def,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ( ( finite_Preflow @ Capacity )
% 0.28/0.46 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,S2: nat,T2: nat,F: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( preflow @ Capacity @ C3 @ S2 @ T2 @ F )
% 0.28/0.46 & ( finite_Graph @ Capacity @ C3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Finite_Preflow_def
% 0.28/0.46 thf(fact_191_NFlow_Oaugment__flow__presv,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,F2: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ( flow @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ T @ F2 )
% 0.28/0.46 => ( flow @ Capacity @ C @ S @ T @ ( augmenting_augment @ Capacity @ C @ F3 @ F2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow.augment_flow_presv
% 0.28/0.46 thf(fact_192_NFlow_Oford__fulkerson_I1_J,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ( isMaxFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 = ( ~ ? [X: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ X ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow.ford_fulkerson(1)
% 0.28/0.46 thf(fact_193_Network_Ois__max__flow__val__def,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,Fv: Capacity] :
% 0.28/0.46 ( ( network @ Capacity @ C @ S @ T )
% 0.28/0.46 => ( ( is_max_flow_val @ Capacity @ C @ S @ T @ Fv )
% 0.28/0.46 = ( ? [F: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( isMaxFlow @ Capacity @ C @ S @ T @ F )
% 0.28/0.46 & ( Fv
% 0.28/0.46 = ( val @ Capacity @ C @ S @ F ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Network.is_max_flow_val_def
% 0.28/0.46 thf(fact_194_Network_OisMaxFlow__def,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( network @ Capacity @ C @ S @ T )
% 0.28/0.46 => ( ( isMaxFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 = ( ( flow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 & ! [F5: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( flow @ Capacity @ C @ S @ T @ F5 )
% 0.28/0.46 => ( ord_less_eq @ Capacity @ ( val @ Capacity @ C @ S @ F5 ) @ ( val @ Capacity @ C @ S @ F3 ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Network.isMaxFlow_def
% 0.28/0.46 thf(fact_195_cf_OisSPath__no__selfloop,axiom,
% 0.28/0.46 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat] :
% 0.28/0.46 ( ( isSimplePath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ U @ P @ V )
% 0.28/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % cf.isSPath_no_selfloop
% 0.28/0.46 thf(fact_196_cf_OisSPath__sg__incoming,axiom,
% 0.28/0.46 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,U2: nat] :
% 0.28/0.46 ( ( isSimplePath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ U @ P @ V )
% 0.28/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.28/0.46 => ( ( U1 != U2 )
% 0.28/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U2 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % cf.isSPath_sg_incoming
% 0.28/0.46 thf(fact_197_cf_OisSPath__sg__outgoing,axiom,
% 0.28/0.46 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,V2: nat] :
% 0.28/0.46 ( ( isSimplePath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ U @ P @ V )
% 0.28/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.28/0.46 => ( ( V1 != V2 )
% 0.28/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V2 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % cf.isSPath_sg_outgoing
% 0.28/0.46 thf(fact_198_isSPath__sg__outgoing,axiom,
% 0.28/0.46 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,V2: nat] :
% 0.28/0.46 ( ( isSimplePath @ capacity @ c @ U @ P @ V )
% 0.28/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.28/0.46 => ( ( V1 != V2 )
% 0.28/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V2 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % isSPath_sg_outgoing
% 0.28/0.46 thf(fact_199_isSPath__sg__incoming,axiom,
% 0.28/0.46 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,U2: nat] :
% 0.28/0.46 ( ( isSimplePath @ capacity @ c @ U @ P @ V )
% 0.28/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.28/0.46 => ( ( U1 != U2 )
% 0.28/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U2 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % isSPath_sg_incoming
% 0.28/0.46 thf(fact_200_isSPath__no__selfloop,axiom,
% 0.28/0.46 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat] :
% 0.28/0.46 ( ( isSimplePath @ capacity @ c @ U @ P @ V )
% 0.28/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % isSPath_no_selfloop
% 0.28/0.46 thf(fact_201_converse__iff,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,A2: A,B2: B,R: set @ ( product_prod @ B @ A )] :
% 0.28/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( converse @ B @ A @ R ) )
% 0.28/0.46 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A2 ) @ R ) ) ).
% 0.28/0.46
% 0.28/0.46 % converse_iff
% 0.28/0.46 thf(fact_202_subrelI,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.28/0.46 ( ! [X4: A,Y3: B] :
% 0.28/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R )
% 0.28/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ S ) )
% 0.28/0.46 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 0.28/0.46
% 0.28/0.46 % subrelI
% 0.28/0.46 thf(fact_203_converseD,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,A2: A,B2: B,R: set @ ( product_prod @ B @ A )] :
% 0.28/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( converse @ B @ A @ R ) )
% 0.28/0.46 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A2 ) @ R ) ) ).
% 0.28/0.46
% 0.28/0.46 % converseD
% 0.28/0.46 thf(fact_204_converseE,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R: set @ ( product_prod @ B @ A )] :
% 0.28/0.46 ( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R ) )
% 0.28/0.46 => ~ ! [X4: B,Y3: A] :
% 0.28/0.46 ( ( Yx
% 0.28/0.46 = ( product_Pair @ A @ B @ Y3 @ X4 ) )
% 0.28/0.46 => ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y3 ) @ R ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % converseE
% 0.28/0.46 thf(fact_205_converse_Ocases,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,A1: B,A22: A,R: set @ ( product_prod @ A @ B )] :
% 0.28/0.46 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R ) )
% 0.28/0.46 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A22 @ A1 ) @ R ) ) ).
% 0.28/0.46
% 0.28/0.46 % converse.cases
% 0.28/0.46 thf(fact_206_converse_Osimps,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,A1: B,A22: A,R: set @ ( product_prod @ A @ B )] :
% 0.28/0.46 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R ) )
% 0.28/0.46 = ( ? [A3: A,B3: B] :
% 0.28/0.46 ( ( A1 = B3 )
% 0.28/0.46 & ( A22 = A3 )
% 0.28/0.46 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % converse.simps
% 0.28/0.46 thf(fact_207_converse_Ointros,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,A2: A,B2: B,R: set @ ( product_prod @ A @ B )] :
% 0.28/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R )
% 0.28/0.46 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A2 ) @ ( converse @ A @ B @ R ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % converse.intros
% 0.28/0.46 thf(fact_208_converse_Oinducts,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,X1: B,X22: A,R: set @ ( product_prod @ A @ B ),P2: B > A > $o] :
% 0.28/0.46 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X1 @ X22 ) @ ( converse @ A @ B @ R ) )
% 0.28/0.46 => ( ! [A5: A,B5: B] :
% 0.28/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R )
% 0.28/0.46 => ( P2 @ B5 @ A5 ) )
% 0.28/0.46 => ( P2 @ X1 @ X22 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % converse.inducts
% 0.28/0.46 thf(fact_209_Graph_OisSPath__sg__outgoing,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,V2: nat] :
% 0.28/0.46 ( ( isSimplePath @ Capacity @ C @ U @ P @ V )
% 0.28/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.28/0.46 => ( ( V1 != V2 )
% 0.28/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V2 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Graph.isSPath_sg_outgoing
% 0.28/0.46 thf(fact_210_Graph_OisSPath__sg__incoming,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,U2: nat] :
% 0.28/0.46 ( ( isSimplePath @ Capacity @ C @ U @ P @ V )
% 0.28/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.28/0.46 => ( ( U1 != U2 )
% 0.28/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U2 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Graph.isSPath_sg_incoming
% 0.28/0.46 thf(fact_211_Graph_OisSPath__no__selfloop,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat] :
% 0.28/0.46 ( ( isSimplePath @ Capacity @ C @ U @ P @ V )
% 0.28/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Graph.isSPath_no_selfloop
% 0.28/0.46 thf(fact_212_augment__flow__presv__cap,axiom,
% 0.28/0.46 ! [F2: ( product_prod @ nat @ nat ) > capacity,U: nat,V: nat] :
% 0.28/0.46 ( ( flow @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ t @ F2 )
% 0.28/0.46 => ( ( ord_less_eq @ capacity @ ( zero_zero @ capacity ) @ ( augmenting_augment @ capacity @ c @ f @ F2 @ ( product_Pair @ nat @ nat @ U @ V ) ) )
% 0.28/0.46 & ( ord_less_eq @ capacity @ ( augmenting_augment @ capacity @ c @ f @ F2 @ ( product_Pair @ nat @ nat @ U @ V ) ) @ ( c @ ( product_Pair @ nat @ nat @ U @ V ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % augment_flow_presv_cap
% 0.28/0.46 thf(fact_213_swap__simp,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,X2: B,Y: A] :
% 0.28/0.46 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
% 0.28/0.46 = ( product_Pair @ A @ B @ Y @ X2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % swap_simp
% 0.28/0.46 thf(fact_214_f__non__negative,axiom,
% 0.28/0.46 ! [E: product_prod @ nat @ nat] : ( ord_less_eq @ capacity @ ( zero_zero @ capacity ) @ ( f @ E ) ) ).
% 0.28/0.46
% 0.28/0.46 % f_non_negative
% 0.28/0.46 thf(fact_215_old_Oprod_Oinject,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B6: B] :
% 0.28/0.46 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.28/0.46 = ( product_Pair @ A @ B @ A6 @ B6 ) )
% 0.28/0.46 = ( ( A2 = A6 )
% 0.28/0.46 & ( B2 = B6 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % old.prod.inject
% 0.28/0.46 thf(fact_216_prod_Oinject,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 0.28/0.46 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.28/0.46 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.28/0.46 = ( ( X1 = Y1 )
% 0.28/0.46 & ( X22 = Y22 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % prod.inject
% 0.28/0.46 thf(fact_217_cap__non__negative,axiom,
% 0.28/0.46 ! [U3: nat,V3: nat] : ( ord_less_eq @ capacity @ ( zero_zero @ capacity ) @ ( c @ ( product_Pair @ nat @ nat @ U3 @ V3 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % cap_non_negative
% 0.28/0.46 thf(fact_218_capacity__const,axiom,
% 0.28/0.46 ! [E2: product_prod @ nat @ nat] :
% 0.28/0.46 ( ( ord_less_eq @ capacity @ ( zero_zero @ capacity ) @ ( f @ E2 ) )
% 0.28/0.46 & ( ord_less_eq @ capacity @ ( f @ E2 ) @ ( c @ E2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % capacity_const
% 0.28/0.46 thf(fact_219_swap__swap,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 0.28/0.46 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
% 0.28/0.46 = P ) ).
% 0.28/0.46
% 0.28/0.46 % swap_swap
% 0.28/0.46 thf(fact_220_resE__nonNegative,axiom,
% 0.28/0.46 ! [E: product_prod @ nat @ nat] : ( ord_less_eq @ capacity @ ( zero_zero @ capacity ) @ ( residu1993642540lGraph @ capacity @ c @ f @ E ) ) ).
% 0.28/0.46
% 0.28/0.46 % resE_nonNegative
% 0.28/0.46 thf(fact_221_RPreGraph_OresE__nonNegative,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A,E: product_prod @ nat @ nat] :
% 0.28/0.46 ( ( residual_RPreGraph @ A @ C @ S @ T @ Cf )
% 0.28/0.46 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( Cf @ E ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % RPreGraph.resE_nonNegative
% 0.28/0.46 thf(fact_222_Preflow_Ocapacity__const,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.28/0.46 ( ( preflow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ! [E2: product_prod @ nat @ nat] :
% 0.28/0.46 ( ( ord_less_eq @ Capacity @ ( zero_zero @ Capacity ) @ ( F3 @ E2 ) )
% 0.28/0.46 & ( ord_less_eq @ Capacity @ ( F3 @ E2 ) @ ( C @ E2 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Preflow.capacity_const
% 0.28/0.46 thf(fact_223_Preflow_Of__non__negative,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,E: product_prod @ nat @ nat] :
% 0.28/0.46 ( ( preflow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ord_less_eq @ Capacity @ ( zero_zero @ Capacity ) @ ( F3 @ E ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Preflow.f_non_negative
% 0.28/0.46 thf(fact_224_old_Oprod_Oinducts,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.28/0.46 ( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.28/0.46 => ( P2 @ Prod ) ) ).
% 0.28/0.46
% 0.28/0.46 % old.prod.inducts
% 0.28/0.46 thf(fact_225_old_Oprod_Oexhaust,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.28/0.46 ~ ! [A5: A,B5: B] :
% 0.28/0.46 ( Y
% 0.28/0.46 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% 0.28/0.46
% 0.28/0.46 % old.prod.exhaust
% 0.28/0.46 thf(fact_226_prod__induct7,axiom,
% 0.28/0.46 ! [G3: $tType,F6: $tType,E3: $tType,D: $tType,C2: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) ) )] :
% 0.28/0.46 ( ! [A5: A,B5: B,C4: C2,D3: D,E4: E3,F4: F6,G4: G3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) @ D3 @ ( product_Pair @ E3 @ ( product_prod @ F6 @ G3 ) @ E4 @ ( product_Pair @ F6 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
% 0.28/0.46 => ( P2 @ X2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % prod_induct7
% 0.28/0.46 thf(fact_227_prod__induct6,axiom,
% 0.28/0.46 ! [F6: $tType,E3: $tType,D: $tType,C2: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) ) )] :
% 0.28/0.46 ( ! [A5: A,B5: B,C4: C2,D3: D,E4: E3,F4: F6] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) ) @ B5 @ ( product_Pair @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E3 @ F6 ) @ D3 @ ( product_Pair @ E3 @ F6 @ E4 @ F4 ) ) ) ) ) )
% 0.28/0.46 => ( P2 @ X2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % prod_induct6
% 0.28/0.46 thf(fact_228_prod__induct5,axiom,
% 0.28/0.46 ! [E3: $tType,D: $tType,C2: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ E3 ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ E3 ) ) )] :
% 0.28/0.46 ( ! [A5: A,B5: B,C4: C2,D3: D,E4: E3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ E3 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D @ E3 ) ) @ B5 @ ( product_Pair @ C2 @ ( product_prod @ D @ E3 ) @ C4 @ ( product_Pair @ D @ E3 @ D3 @ E4 ) ) ) ) )
% 0.28/0.46 => ( P2 @ X2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % prod_induct5
% 0.28/0.46 thf(fact_229_prod__induct4,axiom,
% 0.28/0.46 ! [D: $tType,C2: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ D ) )] :
% 0.28/0.46 ( ! [A5: A,B5: B,C4: C2,D3: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C2 @ D ) @ B5 @ ( product_Pair @ C2 @ D @ C4 @ D3 ) ) ) )
% 0.28/0.46 => ( P2 @ X2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % prod_induct4
% 0.28/0.46 thf(fact_230_prod__induct3,axiom,
% 0.28/0.46 ! [C2: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C2 ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C2 )] :
% 0.28/0.46 ( ! [A5: A,B5: B,C4: C2] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C2 ) @ A5 @ ( product_Pair @ B @ C2 @ B5 @ C4 ) ) )
% 0.28/0.46 => ( P2 @ X2 ) ) ).
% 0.28/0.46
% 0.28/0.46 % prod_induct3
% 0.28/0.46 thf(fact_231_prod__cases7,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,C2: $tType,D: $tType,E3: $tType,F6: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) ) )] :
% 0.28/0.46 ~ ! [A5: A,B5: B,C4: C2,D3: D,E4: E3,F4: F6,G4: G3] :
% 0.28/0.46 ( Y
% 0.28/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) @ D3 @ ( product_Pair @ E3 @ ( product_prod @ F6 @ G3 ) @ E4 @ ( product_Pair @ F6 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % prod_cases7
% 0.28/0.46 thf(fact_232_prod__cases6,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,C2: $tType,D: $tType,E3: $tType,F6: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) ) )] :
% 0.28/0.46 ~ ! [A5: A,B5: B,C4: C2,D3: D,E4: E3,F4: F6] :
% 0.28/0.46 ( Y
% 0.28/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) ) @ B5 @ ( product_Pair @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E3 @ F6 ) @ D3 @ ( product_Pair @ E3 @ F6 @ E4 @ F4 ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % prod_cases6
% 0.28/0.46 thf(fact_233_prod__cases5,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,C2: $tType,D: $tType,E3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ E3 ) ) )] :
% 0.28/0.46 ~ ! [A5: A,B5: B,C4: C2,D3: D,E4: E3] :
% 0.28/0.46 ( Y
% 0.28/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ E3 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D @ E3 ) ) @ B5 @ ( product_Pair @ C2 @ ( product_prod @ D @ E3 ) @ C4 @ ( product_Pair @ D @ E3 @ D3 @ E4 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % prod_cases5
% 0.28/0.46 thf(fact_234_prod__cases4,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,C2: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ D ) )] :
% 0.28/0.46 ~ ! [A5: A,B5: B,C4: C2,D3: D] :
% 0.28/0.46 ( Y
% 0.28/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C2 @ D ) @ B5 @ ( product_Pair @ C2 @ D @ C4 @ D3 ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % prod_cases4
% 0.28/0.46 thf(fact_235_prod__cases3,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,C2: $tType,Y: product_prod @ A @ ( product_prod @ B @ C2 )] :
% 0.28/0.46 ~ ! [A5: A,B5: B,C4: C2] :
% 0.28/0.46 ( Y
% 0.28/0.46 != ( product_Pair @ A @ ( product_prod @ B @ C2 ) @ A5 @ ( product_Pair @ B @ C2 @ B5 @ C4 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % prod_cases3
% 0.28/0.46 thf(fact_236_Pair__inject,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B6: B] :
% 0.28/0.46 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.28/0.46 = ( product_Pair @ A @ B @ A6 @ B6 ) )
% 0.28/0.46 => ~ ( ( A2 = A6 )
% 0.28/0.46 => ( B2 != B6 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Pair_inject
% 0.28/0.46 thf(fact_237_prod__cases,axiom,
% 0.28/0.46 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 0.28/0.46 ( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.28/0.46 => ( P2 @ P ) ) ).
% 0.28/0.46
% 0.28/0.46 % prod_cases
% 0.28/0.46 thf(fact_238_surj__pair,axiom,
% 0.28/0.46 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 0.28/0.46 ? [X4: A,Y3: B] :
% 0.28/0.46 ( P
% 0.28/0.46 = ( product_Pair @ A @ B @ X4 @ Y3 ) ) ).
% 0.28/0.46
% 0.28/0.46 % surj_pair
% 0.28/0.46 thf(fact_239_Network_Ocap__non__negative,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat] :
% 0.28/0.46 ( ( network @ Capacity @ C @ S @ T )
% 0.28/0.46 => ! [U3: nat,V3: nat] : ( ord_less_eq @ Capacity @ ( zero_zero @ Capacity ) @ ( C @ ( product_Pair @ nat @ nat @ U3 @ V3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Network.cap_non_negative
% 0.28/0.46 thf(fact_240_NPreflow_OresE__nonNegative,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,E: product_prod @ nat @ nat] :
% 0.28/0.46 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ord_less_eq @ Capacity @ ( zero_zero @ Capacity ) @ ( residu1993642540lGraph @ Capacity @ C @ F3 @ E ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NPreflow.resE_nonNegative
% 0.28/0.46 thf(fact_241_NFlow_Oaugment__flow__presv__cap,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,F2: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat] :
% 0.28/0.46 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 0.28/0.46 => ( ( flow @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ T @ F2 )
% 0.28/0.46 => ( ( ord_less_eq @ Capacity @ ( zero_zero @ Capacity ) @ ( augmenting_augment @ Capacity @ C @ F3 @ F2 @ ( product_Pair @ nat @ nat @ U @ V ) ) )
% 0.28/0.46 & ( ord_less_eq @ Capacity @ ( augmenting_augment @ Capacity @ C @ F3 @ F2 @ ( product_Pair @ nat @ nat @ U @ V ) ) @ ( C @ ( product_Pair @ nat @ nat @ U @ V ) ) ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % NFlow.augment_flow_presv_cap
% 0.28/0.46 thf(fact_242_excess__s__non__pos,axiom,
% 0.28/0.46 ord_less_eq @ capacity @ ( excess @ capacity @ c @ f @ s ) @ ( zero_zero @ capacity ) ).
% 0.28/0.46
% 0.28/0.46 % excess_s_non_pos
% 0.28/0.46 thf(fact_243_resCap__gzero,axiom,
% 0.28/0.46 ! [P: list @ ( product_prod @ nat @ nat )] :
% 0.28/0.46 ( ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ P )
% 0.28/0.46 => ( ord_less @ capacity @ ( zero_zero @ capacity ) @ ( augmenting_resCap @ capacity @ c @ f @ P ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % resCap_gzero
% 0.28/0.46 thf(fact_244_Network_Oexcess_Ocong,axiom,
% 0.28/0.46 ! [Capacity: $tType] :
% 0.28/0.46 ( ( linordered_idom @ Capacity )
% 0.28/0.46 => ( ( excess @ Capacity )
% 0.28/0.46 = ( excess @ Capacity ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % Network.excess.cong
% 0.28/0.46 thf(fact_245_bot_Onot__eq__extremum,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order_bot @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ( ( A2
% 0.28/0.46 != ( bot_bot @ A ) )
% 0.28/0.46 = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % bot.not_eq_extremum
% 0.28/0.46 thf(fact_246_bot_Oextremum__strict,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order_bot @ A )
% 0.28/0.46 => ! [A2: A] :
% 0.28/0.46 ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % bot.extremum_strict
% 0.28/0.46 thf(fact_247_less__infI1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,X2: A,B2: A] :
% 0.28/0.46 ( ( ord_less @ A @ A2 @ X2 )
% 0.28/0.46 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_infI1
% 0.28/0.46 thf(fact_248_less__infI2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [B2: A,X2: A,A2: A] :
% 0.28/0.46 ( ( ord_less @ A @ B2 @ X2 )
% 0.28/0.46 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % less_infI2
% 0.28/0.46 thf(fact_249_inf_Ostrict__boundedE,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,B2: A,C: A] :
% 0.28/0.46 ( ( ord_less @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) )
% 0.28/0.46 => ~ ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.46 => ~ ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.strict_boundedE
% 0.28/0.46 thf(fact_250_inf_Ostrict__order__iff,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ( ( ord_less @ A )
% 0.28/0.46 = ( ^ [A3: A,B3: A] :
% 0.28/0.46 ( ( A3
% 0.28/0.46 = ( inf_inf @ A @ A3 @ B3 ) )
% 0.28/0.46 & ( A3 != B3 ) ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.strict_order_iff
% 0.28/0.46 thf(fact_251_inf_Ostrict__coboundedI1,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [A2: A,C: A,B2: A] :
% 0.28/0.46 ( ( ord_less @ A @ A2 @ C )
% 0.28/0.46 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.strict_coboundedI1
% 0.28/0.46 thf(fact_252_inf_Ostrict__coboundedI2,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A )
% 0.28/0.46 => ! [B2: A,C: A,A2: A] :
% 0.28/0.46 ( ( ord_less @ A @ B2 @ C )
% 0.28/0.46 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % inf.strict_coboundedI2
% 0.28/0.46 thf(fact_253_dual__order_Ostrict__implies__not__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A )
% 0.28/0.46 => ! [B2: A,A2: A] :
% 0.28/0.46 ( ( ord_less @ A @ B2 @ A2 )
% 0.28/0.46 => ( A2 != B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % dual_order.strict_implies_not_eq
% 0.28/0.46 thf(fact_254_order_Ostrict__implies__not__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( order @ A )
% 0.28/0.46 => ! [A2: A,B2: A] :
% 0.28/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.28/0.46 => ( A2 != B2 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % order.strict_implies_not_eq
% 0.28/0.46 thf(fact_255_not__less__iff__gr__or__eq,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linorder @ A )
% 0.28/0.46 => ! [X2: A,Y: A] :
% 0.28/0.46 ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 0.28/0.46 = ( ( ord_less @ A @ Y @ X2 )
% 0.28/0.46 | ( X2 = Y ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 % not_less_iff_gr_or_eq
% 0.28/0.46
% 0.28/0.46 % Subclasses (5)
% 0.28/0.46 thf(subcl_Rings_Olinordered__idom___HOL_Otype,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ( type @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(subcl_Rings_Olinordered__idom___Orderings_Oord,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ( ord @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(subcl_Rings_Olinordered__idom___Orderings_Oorder,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ( order @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(subcl_Rings_Olinordered__idom___Orderings_Olinorder,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ( linorder @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(subcl_Rings_Olinordered__idom___Orderings_Opreorder,axiom,
% 0.28/0.46 ! [A: $tType] :
% 0.28/0.46 ( ( linordered_idom @ A )
% 0.28/0.46 => ( preorder @ A ) ) ).
% 0.28/0.46
% 0.28/0.46 % Type constructors (36)
% 0.28/0.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 0.28/0.46 bounded_lattice @ $o ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 0.28/0.46 ! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( bounded_lattice @ A8 )
% 0.28/0.46 => ( bounded_lattice @ ( A7 > A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( bounded_lattice @ A8 )
% 0.28/0.46 => ( bounded_lattice_bot @ ( A7 > A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( semilattice_inf @ A8 )
% 0.28/0.46 => ( semilattice_inf @ ( A7 > A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( order_bot @ A8 )
% 0.28/0.46 => ( order_bot @ ( A7 > A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( preorder @ A8 )
% 0.28/0.46 => ( preorder @ ( A7 > A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( lattice @ A8 )
% 0.28/0.46 => ( lattice @ ( A7 > A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( order @ A8 )
% 0.28/0.46 => ( order @ ( A7 > A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Orderings_Oord,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( ord @ A8 )
% 0.28/0.46 => ( ord @ ( A7 > A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_fun___Orderings_Obot,axiom,
% 0.28/0.46 ! [A7: $tType,A8: $tType] :
% 0.28/0.46 ( ( bot @ A8 )
% 0.28/0.46 => ( bot @ ( A7 > A8 ) ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_3,axiom,
% 0.28/0.46 semilattice_inf @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Orderings_Oorder__bot_4,axiom,
% 0.28/0.46 order_bot @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Orderings_Opreorder_5,axiom,
% 0.28/0.46 preorder @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.28/0.46 linorder @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Lattices_Olattice_6,axiom,
% 0.28/0.46 lattice @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Orderings_Oorder_7,axiom,
% 0.28/0.46 order @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Orderings_Oord_8,axiom,
% 0.28/0.46 ord @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Nat_Onat___Orderings_Obot_9,axiom,
% 0.28/0.46 bot @ nat ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_10,axiom,
% 0.28/0.46 ! [A7: $tType] : ( bounded_lattice_bot @ ( set @ A7 ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_11,axiom,
% 0.28/0.46 ! [A7: $tType] : ( semilattice_inf @ ( set @ A7 ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Orderings_Oorder__bot_12,axiom,
% 0.28/0.46 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Orderings_Opreorder_13,axiom,
% 0.28/0.46 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Lattices_Olattice_14,axiom,
% 0.28/0.46 ! [A7: $tType] : ( lattice @ ( set @ A7 ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Orderings_Oorder_15,axiom,
% 0.28/0.46 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Orderings_Oord_16,axiom,
% 0.28/0.46 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_Set_Oset___Orderings_Obot_17,axiom,
% 0.28/0.46 ! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_18,axiom,
% 0.28/0.46 bounded_lattice_bot @ $o ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_19,axiom,
% 0.28/0.46 semilattice_inf @ $o ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Orderings_Oorder__bot_20,axiom,
% 0.28/0.46 order_bot @ $o ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Orderings_Opreorder_21,axiom,
% 0.28/0.46 preorder @ $o ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder_22,axiom,
% 0.28/0.46 linorder @ $o ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Lattices_Olattice_23,axiom,
% 0.28/0.46 lattice @ $o ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Orderings_Oorder_24,axiom,
% 0.28/0.46 order @ $o ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Orderings_Oord_25,axiom,
% 0.28/0.46 ord @ $o ).
% 0.28/0.46
% 0.28/0.46 thf(tcon_HOL_Obool___Orderings_Obot_26,axiom,
% 0.28/0.46 bot @ $o ).
% 0.28/0.46
% 0.28/0.46 % Free types (1)
% 0.28/0.46 thf(tfree_0,hypothesis,
% 0.28/0.46 linordered_idom @ capacity ).
% 0.28/0.46
% 0.28/0.46 % Conjectures (2)
% 0.28/0.46 thf(conj_0,hypothesis,
% 0.28/0.46 augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ p ).
% 0.28/0.46
% 0.28/0.46 thf(conj_1,conjecture,
% 0.28/0.46 ( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set2 @ ( product_prod @ nat @ nat ) @ p ) @ ( converse @ nat @ nat @ ( set2 @ ( product_prod @ nat @ nat ) @ p ) ) )
% 0.28/0.46 = ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ).
% 0.28/0.46
% 0.28/0.46 %------------------------------------------------------------------------------
% 0.28/0.46 ------------------------
% 2.53/1.49 ---- Embedded in THF ---
% 2.53/1.49 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.53/1.49 %%% Generated on Tue Feb 24 08:24:49 EST 2026
% 2.53/1.49 %%% using '$$dhol' embedding, version 1.3.0.
% 2.53/1.49 %%% Logic specification used:
% 2.53/1.49 %%% thf(spec, logic, $$dhol).
% 2.53/1.49
% 2.53/1.49 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.7uUisdwD1b/DTF2DTF5283.p
% 2.53/1.49 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.53/1.49 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.53/1.49 thf(ty_tf_capacity, type, capacity: $tType).
% 2.53/1.49 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.53/1.49 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.53/1.49 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: ($o))).
% 2.53/1.49 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.53/1.49 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: ($o))).
% 2.53/1.49 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.53/1.49 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.53/1.49 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: ($o))).
% 2.53/1.49 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.53/1.49 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.53/1.49 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 2.53/1.49 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 2.53/1.49 thf(sy_cl_Lattices_Osemilattice__inf, type, semilattice_inf: (!> [A:$tType]: ($o))).
% 2.53/1.49 thf(sy_cl_Lattices_Obounded__lattice__bot, type, bounded_lattice_bot: (!> [A:$tType]: ($o))).
% 2.53/1.49 thf(sy_c_Augmenting__Flow_ONFlow_Oaugment, type, augmenting_augment: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > ((((product_prod @ nat) @ nat) > Capacity) > ((((product_prod @ nat) @ nat) > Capacity) > (((product_prod @ nat) @ nat) > Capacity))))))).
% 2.53/1.49 thf(sy_c_Augmenting__Path_ONPreflow_OaugmentingFlow, type, augmen1604248148ngFlow: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > ((((product_prod @ nat) @ nat) > Capacity) > ((list @ ((product_prod @ nat) @ nat)) > (((product_prod @ nat) @ nat) > Capacity))))))).
% 2.53/1.49 thf(sy_c_Augmenting__Path_ONPreflow_OisAugmentingPath, type, augmen2028041557ngPath: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > ((((product_prod @ nat) @ nat) > Capacity) > ((list @ ((product_prod @ nat) @ nat)) > $o)))))))).
% 2.53/1.49 thf(sy_c_Augmenting__Path_ONPreflow_OresCap, type, augmenting_resCap: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > ((((product_prod @ nat) @ nat) > Capacity) > ((list @ ((product_prod @ nat) @ nat)) > Capacity)))))).
% 2.53/1.49 thf(sy_c_Graph_OFinite__Graph, type, finite_Graph: (!> [A:$tType]: (((((product_prod @ nat) @ nat) > A) > $o)))).
% 2.53/1.49 thf(sy_c_Graph_OGraph_OisShortestPath, type, isShortestPath: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > ((list @ ((product_prod @ nat) @ nat)) > (nat > $o))))))).
% 2.53/1.49 thf(sy_c_Graph_OGraph_OisSimplePath, type, isSimplePath: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > ((list @ ((product_prod @ nat) @ nat)) > (nat > $o))))))).
% 2.53/1.49 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.53/1.49 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.53/1.49 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 2.53/1.49 thf(sy_c_List_Olist_Oset, type, set2: (!> [A:$tType]: (((list @ A) > (set @ A))))).
% 2.53/1.49 thf(sy_c_Network_OFinite__Preflow, type, finite_Preflow: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > ((((product_prod @ nat) @ nat) > Capacity) > $o))))))).
% 2.53/1.49 thf(sy_c_Network_OFlow, type, flow: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > ((((product_prod @ nat) @ nat) > Capacity) > $o))))))).
% 2.53/1.49 thf(sy_c_Network_OFlow_Oval, type, val: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > ((((product_prod @ nat) @ nat) > Capacity) > Capacity)))))).
% 2.53/1.49 thf(sy_c_Network_ONCut, type, nCut: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > ((set @ nat) > $o))))))).
% 2.53/1.49 thf(sy_c_Network_ONCut_Ocap, type, cap: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > ((set @ nat) > Capacity))))).
% 2.53/1.49 thf(sy_c_Network_ONFlow, type, nFlow: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > ((((product_prod @ nat) @ nat) > Capacity) > $o))))))).
% 2.53/1.49 thf(sy_c_Network_ONPreflow, type, nPreflow: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > ((((product_prod @ nat) @ nat) > Capacity) > $o))))))).
% 2.53/1.49 thf(sy_c_Network_ONetwork, type, network: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > $o)))))).
% 2.53/1.49 thf(sy_c_Network_ONetwork_Oexcess, type, excess: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > ((((product_prod @ nat) @ nat) > Capacity) > (nat > Capacity)))))).
% 2.53/1.49 thf(sy_c_Network_ONetwork_OisMaxFlow, type, isMaxFlow: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > ((((product_prod @ nat) @ nat) > Capacity) > $o))))))).
% 2.53/1.49 thf(sy_c_Network_ONetwork_Ois__max__flow__val, type, is_max_flow_val: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > (Capacity > $o))))))).
% 2.53/1.49 thf(sy_c_Network_OPreflow, type, preflow: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > ((((product_prod @ nat) @ nat) > Capacity) > $o))))))).
% 2.53/1.49 thf(sy_c_Network_OisMinCut, type, isMinCut: (!> [A:$tType]: (((((product_prod @ nat) @ nat) > A) > (nat > (nat > ((set @ nat) > $o))))))).
% 2.53/1.49 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.53/1.49 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.53/1.49 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.53/1.49 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.53/1.49 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 2.53/1.49 thf(sy_c_Relation_Oconverse, type, converse: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ B) @ A)))))).
% 2.53/1.49 thf(sy_c_Residual__Graph_ONetwork_Oflow__of__cf, type, residual_flow_of_cf: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > ((((product_prod @ nat) @ nat) > Capacity) > (((product_prod @ nat) @ nat) > Capacity)))))).
% 2.53/1.49 thf(sy_c_Residual__Graph_ORGraph, type, residual_RGraph: (!> [A:$tType]: (((((product_prod @ nat) @ nat) > A) > (nat > (nat > ((((product_prod @ nat) @ nat) > A) > $o))))))).
% 2.53/1.49 thf(sy_c_Residual__Graph_ORPreGraph, type, residual_RPreGraph: (!> [A:$tType]: (((((product_prod @ nat) @ nat) > A) > (nat > (nat > ((((product_prod @ nat) @ nat) > A) > $o))))))).
% 2.53/1.49 thf(sy_c_Residual__Graph_OresidualGraph, type, residu1993642540lGraph: (!> [A:$tType]: (((((product_prod @ nat) @ nat) > A) > ((((product_prod @ nat) @ nat) > A) > (((product_prod @ nat) @ nat) > A)))))).
% 2.53/1.49 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.53/1.49 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.53/1.49 thf(sy_v_c, type, c: (((product_prod @ nat) @ nat) > capacity)).
% 2.53/1.49 thf(sy_v_f, type, f: (((product_prod @ nat) @ nat) > capacity)).
% 2.53/1.49 thf(sy_v_p, type, p: (list @ ((product_prod @ nat) @ nat))).
% 2.53/1.49 thf(sy_v_s, type, s: nat).
% 2.53/1.49 thf(sy_v_t, type, t: nat).
% 2.53/1.49 thf(fact_0_t__not__s, axiom, (t != s)).
% 2.53/1.49 thf(fact_1_cf_OisSPath__nt__parallel__pf, axiom, (! [S:nat,P:(list @ ((product_prod @ nat) @ nat)),T:nat]: (((((((isSimplePath @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) @ S) @ P) @ T) => ((((inf_inf @ (set @ ((product_prod @ nat) @ nat))) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) @ (((converse @ nat) @ nat) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P))) = (bot_bot @ (set @ ((product_prod @ nat) @ nat)))))))).
% 2.53/1.49 thf(fact_2_isSPath__nt__parallel__pf, axiom, (! [S:nat,P:(list @ ((product_prod @ nat) @ nat)),T:nat]: (((((((isSimplePath @ capacity) @ c) @ S) @ P) @ T) => ((((inf_inf @ (set @ ((product_prod @ nat) @ nat))) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) @ (((converse @ nat) @ nat) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P))) = (bot_bot @ (set @ ((product_prod @ nat) @ nat)))))))).
% 2.53/1.49 thf(fact_3_NFlow__axioms, axiom, (((((nFlow @ capacity) @ c) @ s) @ t) @ f)).
% 2.53/1.49 thf(fact_4_max__flow__val__unique, axiom, (! [Fv1:capacity,Fv2:capacity]: (((((((is_max_flow_val @ capacity) @ c) @ s) @ t) @ Fv1) => ((((((is_max_flow_val @ capacity) @ c) @ s) @ t) @ Fv2) => (Fv1 = Fv2)))))).
% 2.53/1.49 thf(fact_5_isAugmentingPath__def, axiom, (! [P:(list @ ((product_prod @ nat) @ nat))]: ((((((((augmen2028041557ngPath @ capacity) @ c) @ s) @ t) @ f) @ P) = (((((isSimplePath @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) @ s) @ P) @ t))))).
% 2.53/1.49 thf(fact_6_Finite__Preflow__axioms, axiom, (((((finite_Preflow @ capacity) @ c) @ s) @ t) @ f)).
% 2.53/1.49 thf(fact_7_fofu__I__II, axiom, ((((((isMaxFlow @ capacity) @ c) @ s) @ t) @ f) => (~ ((? [X_1:(list @ ((product_prod @ nat) @ nat))]: (((((((augmen2028041557ngPath @ capacity) @ c) @ s) @ t) @ f) @ X_1))))))).
% 2.53/1.49 thf(fact_8_noAugPath__iff__maxFlow, axiom, ((~ ((? [X:(list @ ((product_prod @ nat) @ nat))]: (((((((augmen2028041557ngPath @ capacity) @ c) @ s) @ t) @ f) @ X))))) = (((((isMaxFlow @ capacity) @ c) @ s) @ t) @ f))).
% 2.53/1.49 thf(fact_9_Preflow__axioms, axiom, (((((preflow @ capacity) @ c) @ s) @ t) @ f)).
% 2.53/1.49 thf(fact_10_is__RGraph, axiom, (((((residual_RGraph @ capacity) @ c) @ s) @ t) @ (((residu1993642540lGraph @ capacity) @ c) @ f))).
% 2.53/1.49 thf(fact_11_Flow__axioms, axiom, (((((flow @ capacity) @ c) @ s) @ t) @ f)).
% 2.53/1.49 thf(fact_12_NPreflow__axioms, axiom, (((((nPreflow @ capacity) @ c) @ s) @ t) @ f)).
% 2.53/1.49 thf(fact_13_converse__empty, axiom, (! [B:$tType,A:$tType]: (((((converse @ B) @ A) @ (bot_bot @ (set @ ((product_prod @ B) @ A)))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 2.53/1.49 thf(fact_14_inf__bot__left, axiom, (! [A:$tType]: (((bounded_lattice_bot @ A) => (! [X2:A]: (((((inf_inf @ A) @ (bot_bot @ A)) @ X2) = (bot_bot @ A)))))))).
% 2.53/1.49 thf(fact_15_inf__bot__right, axiom, (! [A:$tType]: (((bounded_lattice_bot @ A) => (! [X2:A]: (((((inf_inf @ A) @ X2) @ (bot_bot @ A)) = (bot_bot @ A)))))))).
% 2.53/1.49 thf(fact_16_is__RPreGraph, axiom, (((((residual_RPreGraph @ capacity) @ c) @ s) @ t) @ (((residu1993642540lGraph @ capacity) @ c) @ f))).
% 2.53/1.49 thf(fact_17_Network__axioms, axiom, ((((network @ capacity) @ c) @ s) @ t)).
% 2.53/1.49 thf(fact_18_fo__rg__inv, axiom, ((((residual_flow_of_cf @ capacity) @ c) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) = f)).
% 2.53/1.49 thf(fact_19_inf__right__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A,Y:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X2) @ Y)) @ Y) = (((inf_inf @ A) @ X2) @ Y)))))))).
% 2.53/1.49 thf(fact_20_inf_Oright__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ B2) = (((inf_inf @ A) @ A2) @ B2)))))))).
% 2.53/1.49 thf(fact_21_inf__left__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A,Y:A]: (((((inf_inf @ A) @ X2) @ (((inf_inf @ A) @ X2) @ Y)) = (((inf_inf @ A) @ X2) @ Y)))))))).
% 2.53/1.49 thf(fact_22_inf_Oleft__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A]: (((((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ A2) @ B2)) = (((inf_inf @ A) @ A2) @ B2)))))))).
% 2.53/1.49 thf(fact_23_inf__idem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A]: (((((inf_inf @ A) @ X2) @ X2) = X2))))))).
% 2.53/1.49 thf(fact_24_inf_Oidem, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A]: (((((inf_inf @ A) @ A2) @ A2) = A2))))))).
% 2.53/1.49 thf(fact_25_inf__apply, axiom, (! [B:$tType,A:$tType]: (((semilattice_inf @ B) => ((inf_inf @ (A > B)) = (^ [F:(A > B),G:(A > B),X3:A]: ((((inf_inf @ B) @ (F @ X3)) @ (G @ X3))))))))).
% 2.53/1.49 thf(fact_26_converse__converse, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B))]: (((((converse @ B) @ A) @ (((converse @ A) @ B) @ R)) = R)))).
% 2.53/1.49 thf(fact_27_converse__inject, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ B) @ A)),S:(set @ ((product_prod @ B) @ A))]: ((((((converse @ B) @ A) @ R) = (((converse @ B) @ A) @ S)) = (R = S))))).
% 2.53/1.49 thf(fact_28_augFlow__resFlow, axiom, (! [P:(list @ ((product_prod @ nat) @ nat))]: ((((((((augmen2028041557ngPath @ capacity) @ c) @ s) @ t) @ f) @ P) => (((((flow @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) @ s) @ t) @ ((((augmen1604248148ngFlow @ capacity) @ c) @ f) @ P)))))).
% 2.53/1.49 thf(fact_29_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X3:A]: ((((member @ A) @ X3) @ (bot_bot @ (set @ A))))))))).
% 2.53/1.49 thf(fact_30_inf__left__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A,Y:A,Z:A]: (((((inf_inf @ A) @ X2) @ (((inf_inf @ A) @ Y) @ Z)) = (((inf_inf @ A) @ Y) @ (((inf_inf @ A) @ X2) @ Z))))))))).
% 2.53/1.49 thf(fact_31_inf_Oleft__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,A2:A,C:A]: (((((inf_inf @ A) @ B2) @ (((inf_inf @ A) @ A2) @ C)) = (((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C))))))))).
% 2.53/1.49 thf(fact_32_inf__commute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((inf_inf @ A) = (^ [X3:A,Y2:A]: ((((inf_inf @ A) @ Y2) @ X3)))))))).
% 2.53/1.49 thf(fact_33_inf_Ocommute, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((inf_inf @ A) = (^ [A3:A,B3:A]: ((((inf_inf @ A) @ B3) @ A3)))))))).
% 2.53/1.49 thf(fact_34_inf__assoc, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A,Y:A,Z:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X2) @ Y)) @ Z) = (((inf_inf @ A) @ X2) @ (((inf_inf @ A) @ Y) @ Z))))))))).
% 2.53/1.49 thf(fact_35_inf_Oassoc, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A,C:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ C) = (((inf_inf @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C))))))))).
% 2.53/1.49 thf(fact_36_boolean__algebra__cancel_Oinf2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B4:A,K:A,B2:A,A2:A]: (((B4 = (((inf_inf @ A) @ K) @ B2)) => ((((inf_inf @ A) @ A2) @ B4) = (((inf_inf @ A) @ K) @ (((inf_inf @ A) @ A2) @ B2)))))))))).
% 2.53/1.49 thf(fact_37_boolean__algebra__cancel_Oinf1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A4:A,K:A,A2:A,B2:A]: (((A4 = (((inf_inf @ A) @ K) @ A2)) => ((((inf_inf @ A) @ A4) @ B2) = (((inf_inf @ A) @ K) @ (((inf_inf @ A) @ A2) @ B2)))))))))).
% 2.53/1.49 thf(fact_38_inf__fun__def, axiom, (! [B:$tType,A:$tType]: (((semilattice_inf @ B) => ((inf_inf @ (A > B)) = (^ [F:(A > B),G:(A > B),X3:A]: ((((inf_inf @ B) @ (F @ X3)) @ (G @ X3))))))))).
% 2.53/1.49 thf(fact_39_inf__sup__aci_I1_J, axiom, (! [A:$tType]: (((lattice @ A) => ((inf_inf @ A) = (^ [X3:A,Y2:A]: ((((inf_inf @ A) @ Y2) @ X3)))))))).
% 2.53/1.49 thf(fact_40_inf__sup__aci_I2_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X2:A,Y:A,Z:A]: (((((inf_inf @ A) @ (((inf_inf @ A) @ X2) @ Y)) @ Z) = (((inf_inf @ A) @ X2) @ (((inf_inf @ A) @ Y) @ Z))))))))).
% 2.53/1.49 thf(fact_41_inf__sup__aci_I3_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X2:A,Y:A,Z:A]: (((((inf_inf @ A) @ X2) @ (((inf_inf @ A) @ Y) @ Z)) = (((inf_inf @ A) @ Y) @ (((inf_inf @ A) @ X2) @ Z))))))))).
% 2.53/1.49 thf(fact_42_inf__sup__aci_I4_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X2:A,Y:A]: (((((inf_inf @ A) @ X2) @ (((inf_inf @ A) @ X2) @ Y)) = (((inf_inf @ A) @ X2) @ Y)))))))).
% 2.53/1.49 thf(fact_43_converse__Int, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ B) @ A)),S:(set @ ((product_prod @ B) @ A))]: (((((converse @ B) @ A) @ (((inf_inf @ (set @ ((product_prod @ B) @ A))) @ R) @ S)) = (((inf_inf @ (set @ ((product_prod @ A) @ B))) @ (((converse @ B) @ A) @ R)) @ (((converse @ B) @ A) @ S)))))).
% 2.53/1.49 thf(fact_44_augment__flow__presv, axiom, (! [F2:(((product_prod @ nat) @ nat) > capacity)]: (((((((flow @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) @ s) @ t) @ F2) => (((((flow @ capacity) @ c) @ s) @ t) @ ((((augmenting_augment @ capacity) @ c) @ f) @ F2)))))).
% 2.53/1.49 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P2:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P2)) = (P2 @ A2))))).
% 2.53/1.49 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A4)))) = A4)))).
% 2.53/1.49 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X4:A]: (((P2 @ X4) = (Q @ X4)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q)))))).
% 2.53/1.49 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F3:(A > B),G2:(A > B)]: (((! [X4:A]: (((F3 @ X4) = (G2 @ X4)))) => (F3 = G2))))).
% 2.53/1.49 thf(fact_49_cf_OisSPath__nt__parallel, axiom, (! [S:nat,P:(list @ ((product_prod @ nat) @ nat)),T:nat,E:((product_prod @ nat) @ nat)]: (((((((isSimplePath @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) @ S) @ P) @ T) => ((((member @ ((product_prod @ nat) @ nat)) @ E) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ (((product_swap @ nat) @ nat) @ E)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P))))))))).
% 2.53/1.49 thf(fact_50_is__max__flow__val__def, axiom, (! [Fv:capacity]: (((((((is_max_flow_val @ capacity) @ c) @ s) @ t) @ Fv) = (? [F:(((product_prod @ nat) @ nat) > capacity)]: (((((((isMaxFlow @ capacity) @ c) @ s) @ t) @ F) & (Fv = ((((val @ capacity) @ c) @ s) @ F))))))))).
% 2.53/1.49 thf(fact_51_cf_OFinite__Graph__axioms, axiom, ((finite_Graph @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f))).
% 2.53/1.49 thf(fact_52_ford__fulkerson_I1_J, axiom, ((((((isMaxFlow @ capacity) @ c) @ s) @ t) @ f) = (~ ((? [X:(list @ ((product_prod @ nat) @ nat))]: (((((((augmen2028041557ngPath @ capacity) @ c) @ s) @ t) @ f) @ X))))))).
% 2.53/1.49 thf(fact_53_shortest__is__augmenting, axiom, (! [P:(list @ ((product_prod @ nat) @ nat))]: (((((((isShortestPath @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) @ s) @ P) @ t) => ((((((augmen2028041557ngPath @ capacity) @ c) @ s) @ t) @ f) @ P))))).
% 2.53/1.49 thf(fact_54_augmenting__path__imp__shortest, axiom, (! [P:(list @ ((product_prod @ nat) @ nat))]: ((((((((augmen2028041557ngPath @ capacity) @ c) @ s) @ t) @ f) @ P) => (? [P3:(list @ ((product_prod @ nat) @ nat))]: ((((((isShortestPath @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) @ s) @ P3) @ t))))))).
% 2.53/1.49 thf(fact_55_Graph_OisSPath__nt__parallel__pf, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,P:(list @ ((product_prod @ nat) @ nat)),T:nat]: (((((((isSimplePath @ Capacity) @ C) @ S) @ P) @ T) => ((((inf_inf @ (set @ ((product_prod @ nat) @ nat))) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) @ (((converse @ nat) @ nat) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P))) = (bot_bot @ (set @ ((product_prod @ nat) @ nat))))))))))).
% 2.53/1.49 thf(fact_56_RPreGraph_Omaxflow__imp__rgraph, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,T:nat,Cf:(((product_prod @ nat) @ nat) > A)]: (((((((residual_RPreGraph @ A) @ C) @ S) @ T) @ Cf) => ((((((isMaxFlow @ A) @ C) @ S) @ T) @ (((residual_flow_of_cf @ A) @ C) @ Cf)) => (((((residual_RGraph @ A) @ C) @ S) @ T) @ Cf))))))))).
% 2.53/1.49 thf(fact_57_isSPath__nt__parallel, axiom, (! [S:nat,P:(list @ ((product_prod @ nat) @ nat)),T:nat,E:((product_prod @ nat) @ nat)]: (((((((isSimplePath @ capacity) @ c) @ S) @ P) @ T) => ((((member @ ((product_prod @ nat) @ nat)) @ E) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ (((product_swap @ nat) @ nat) @ E)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P))))))))).
% 2.53/1.49 thf(fact_58_Finite__Graph__axioms, axiom, ((finite_Graph @ capacity) @ c)).
% 2.53/1.49 thf(fact_59_shortestPath__is__simple, axiom, (! [S:nat,P:(list @ ((product_prod @ nat) @ nat)),T:nat]: (((((((isShortestPath @ capacity) @ c) @ S) @ P) @ T) => (((((isSimplePath @ capacity) @ c) @ S) @ P) @ T))))).
% 2.53/1.49 thf(fact_60_cf_OshortestPath__is__simple, axiom, (! [S:nat,P:(list @ ((product_prod @ nat) @ nat)),T:nat]: (((((((isShortestPath @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) @ S) @ P) @ T) => (((((isSimplePath @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) @ S) @ P) @ T))))).
% 2.53/1.49 thf(fact_61_Graph_OisShortestPath_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isShortestPath @ Capacity) = (isShortestPath @ Capacity)))))).
% 2.53/1.49 thf(fact_62_Graph_OshortestPath__is__simple, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,P:(list @ ((product_prod @ nat) @ nat)),T:nat]: (((((((isShortestPath @ Capacity) @ C) @ S) @ P) @ T) => (((((isSimplePath @ Capacity) @ C) @ S) @ P) @ T)))))))).
% 2.53/1.49 thf(fact_63_Graph_OisSPath__nt__parallel, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,P:(list @ ((product_prod @ nat) @ nat)),T:nat,E:((product_prod @ nat) @ nat)]: (((((((isSimplePath @ Capacity) @ C) @ S) @ P) @ T) => ((((member @ ((product_prod @ nat) @ nat)) @ E) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ (((product_swap @ nat) @ nat) @ E)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)))))))))))).
% 2.53/1.49 thf(fact_64_Graph_OisSimplePath_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isSimplePath @ Capacity) = (isSimplePath @ Capacity)))))).
% 2.53/1.49 thf(fact_65_RPreGraph_Othis__loc__rpg, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,T:nat,Cf:(((product_prod @ nat) @ nat) > A)]: (((((((residual_RPreGraph @ A) @ C) @ S) @ T) @ Cf) => (((((residual_RPreGraph @ A) @ C) @ S) @ T) @ Cf)))))))).
% 2.53/1.49 thf(fact_66_NFlow_Oshortest__is__augmenting, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity),P:(list @ ((product_prod @ nat) @ nat))]: (((((((nFlow @ Capacity) @ C) @ S) @ T) @ F3) => ((((((isShortestPath @ Capacity) @ (((residu1993642540lGraph @ Capacity) @ C) @ F3)) @ S) @ P) @ T) => ((((((augmen2028041557ngPath @ Capacity) @ C) @ S) @ T) @ F3) @ P))))))))).
% 2.53/1.49 thf(fact_67_NFlow_Oaugmenting__path__imp__shortest, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity),P:(list @ ((product_prod @ nat) @ nat))]: (((((((nFlow @ Capacity) @ C) @ S) @ T) @ F3) => (((((((augmen2028041557ngPath @ Capacity) @ C) @ S) @ T) @ F3) @ P) => (? [P3:(list @ ((product_prod @ nat) @ nat))]: ((((((isShortestPath @ Capacity) @ (((residu1993642540lGraph @ Capacity) @ C) @ F3)) @ S) @ P3) @ T))))))))))).
% 2.53/1.49 thf(fact_68_RGraph_Othis__loc, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,T:nat,Cf:(((product_prod @ nat) @ nat) > A)]: (((((((residual_RGraph @ A) @ C) @ S) @ T) @ Cf) => (((((residual_RGraph @ A) @ C) @ S) @ T) @ Cf)))))))).
% 2.53/1.49 thf(fact_69_Network_Oflow__of__cf_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((residual_flow_of_cf @ Capacity) = (residual_flow_of_cf @ Capacity)))))).
% 2.53/1.49 thf(fact_70_RPreGraph_Oaxioms_I1_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,T:nat,Cf:(((product_prod @ nat) @ nat) > A)]: (((((((residual_RPreGraph @ A) @ C) @ S) @ T) @ Cf) => ((((network @ A) @ C) @ S) @ T)))))))).
% 2.53/1.49 thf(fact_71_RGraph_Oaxioms_I1_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,T:nat,Cf:(((product_prod @ nat) @ nat) > A)]: (((((((residual_RGraph @ A) @ C) @ S) @ T) @ Cf) => ((((network @ A) @ C) @ S) @ T)))))))).
% 2.53/1.49 thf(fact_72_RGraph_Othis__loc__rpg, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,T:nat,Cf:(((product_prod @ nat) @ nat) > A)]: (((((((residual_RGraph @ A) @ C) @ S) @ T) @ Cf) => (((((residual_RPreGraph @ A) @ C) @ S) @ T) @ Cf)))))))).
% 2.53/1.49 thf(fact_73_NPreflow_Ois__RPreGraph, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity)]: (((((((nPreflow @ Capacity) @ C) @ S) @ T) @ F3) => (((((residual_RPreGraph @ Capacity) @ C) @ S) @ T) @ (((residu1993642540lGraph @ Capacity) @ C) @ F3))))))))).
% 2.53/1.49 thf(fact_74_RPreGraph_OEX__RPG, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,T:nat,Cf:(((product_prod @ nat) @ nat) > A)]: (((((((residual_RPreGraph @ A) @ C) @ S) @ T) @ Cf) => (? [F4:(((product_prod @ nat) @ nat) > A)]: (((((((nPreflow @ A) @ C) @ S) @ T) @ F4) & (Cf = (((residu1993642540lGraph @ A) @ C) @ F4)))))))))))).
% 2.53/1.49 thf(fact_75_NFlow_Ois__RGraph, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity)]: (((((((nFlow @ Capacity) @ C) @ S) @ T) @ F3) => (((((residual_RGraph @ Capacity) @ C) @ S) @ T) @ (((residu1993642540lGraph @ Capacity) @ C) @ F3))))))))).
% 2.53/1.49 thf(fact_76_RGraph_OEX__RG, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,T:nat,Cf:(((product_prod @ nat) @ nat) > A)]: (((((((residual_RGraph @ A) @ C) @ S) @ T) @ Cf) => (? [F4:(((product_prod @ nat) @ nat) > A)]: (((((((nFlow @ A) @ C) @ S) @ T) @ F4) & (Cf = (((residu1993642540lGraph @ A) @ C) @ F4)))))))))))).
% 2.53/1.49 thf(fact_77_NPreflow_Ofo__rg__inv, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity)]: (((((((nPreflow @ Capacity) @ C) @ S) @ T) @ F3) => ((((residual_flow_of_cf @ Capacity) @ C) @ (((residu1993642540lGraph @ Capacity) @ C) @ F3)) = F3)))))))).
% 2.53/1.49 thf(fact_78_RPreGraph_Org__fo__inv, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,T:nat,Cf:(((product_prod @ nat) @ nat) > A)]: (((((((residual_RPreGraph @ A) @ C) @ S) @ T) @ Cf) => ((((residu1993642540lGraph @ A) @ C) @ (((residual_flow_of_cf @ A) @ C) @ Cf)) = Cf)))))))).
% 2.53/1.49 thf(fact_79_RPreGraph_Ois__NPreflow, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,T:nat,Cf:(((product_prod @ nat) @ nat) > A)]: (((((((residual_RPreGraph @ A) @ C) @ S) @ T) @ Cf) => (((((nPreflow @ A) @ C) @ S) @ T) @ (((residual_flow_of_cf @ A) @ C) @ Cf))))))))).
% 2.53/1.49 thf(fact_80_RGraph_Ois__NFlow, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,T:nat,Cf:(((product_prod @ nat) @ nat) > A)]: (((((((residual_RGraph @ A) @ C) @ S) @ T) @ Cf) => (((((nFlow @ A) @ C) @ S) @ T) @ (((residual_flow_of_cf @ A) @ C) @ Cf))))))))).
% 2.53/1.49 thf(fact_81_augFlow__val, axiom, (! [P:(list @ ((product_prod @ nat) @ nat))]: ((((((((augmen2028041557ngPath @ capacity) @ c) @ s) @ t) @ f) @ P) => (((((val @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) @ s) @ ((((augmen1604248148ngFlow @ capacity) @ c) @ f) @ P)) = ((((augmenting_resCap @ capacity) @ c) @ f) @ P)))))).
% 2.53/1.49 thf(fact_82_augment__flow__value, axiom, (! [F2:(((product_prod @ nat) @ nat) > capacity)]: (((((((flow @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) @ s) @ t) @ F2) => (((((val @ capacity) @ c) @ s) @ ((((augmenting_augment @ capacity) @ c) @ f) @ F2)) = (((plus_plus @ capacity) @ ((((val @ capacity) @ c) @ s) @ f)) @ ((((val @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) @ s) @ F2))))))).
% 2.53/1.49 thf(fact_83_isMaxFlow__alt, axiom, (! [F3:(((product_prod @ nat) @ nat) > capacity)]: (((((((isMaxFlow @ capacity) @ c) @ s) @ t) @ F3) = ((((((nFlow @ capacity) @ c) @ s) @ t) @ F3) & (! [F5:(((product_prod @ nat) @ nat) > capacity)]: (((((((nFlow @ capacity) @ c) @ s) @ t) @ F5) => (((ord_less_eq @ capacity) @ ((((val @ capacity) @ c) @ s) @ F5)) @ ((((val @ capacity) @ c) @ s) @ F3)))))))))).
% 2.53/1.49 thf(fact_84_isMaxFlow__def, axiom, (! [F3:(((product_prod @ nat) @ nat) > capacity)]: (((((((isMaxFlow @ capacity) @ c) @ s) @ t) @ F3) = ((((((flow @ capacity) @ c) @ s) @ t) @ F3) & (! [F5:(((product_prod @ nat) @ nat) > capacity)]: (((((((flow @ capacity) @ c) @ s) @ t) @ F5) => (((ord_less_eq @ capacity) @ ((((val @ capacity) @ c) @ s) @ F5)) @ ((((val @ capacity) @ c) @ s) @ F3)))))))))).
% 2.53/1.49 thf(fact_85_NPreflow_OaugFlow__resFlow, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity),P:(list @ ((product_prod @ nat) @ nat))]: (((((((nPreflow @ Capacity) @ C) @ S) @ T) @ F3) => (((((((augmen2028041557ngPath @ Capacity) @ C) @ S) @ T) @ F3) @ P) => (((((flow @ Capacity) @ (((residu1993642540lGraph @ Capacity) @ C) @ F3)) @ S) @ T) @ ((((augmen1604248148ngFlow @ Capacity) @ C) @ F3) @ P)))))))))).
% 2.53/1.49 thf(fact_86_bot__apply, axiom, (! [C2:$tType,D:$tType]: (((bot @ C2) => ((bot_bot @ (D > C2)) = (^ [X3:D]: ((bot_bot @ C2)))))))).
% 2.53/1.49 thf(fact_87_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))))))).
% 2.53/1.49 thf(fact_88_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A]: ((((ord_less_eq @ A) @ X2) @ X2))))))).
% 2.53/1.49 thf(fact_89_inf_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C)) = ((((ord_less_eq @ A) @ A2) @ B2) & (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.53/1.49 thf(fact_90_le__inf__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X2) @ (((inf_inf @ A) @ Y) @ Z)) = ((((ord_less_eq @ A) @ X2) @ Y) & (((ord_less_eq @ A) @ X2) @ Z))))))))).
% 2.53/1.49 thf(fact_91_bot_Oextremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A2))))))).
% 2.53/1.49 thf(fact_92_bot_Oextremum__unique, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) = (A2 = (bot_bot @ A))))))))).
% 2.53/1.49 thf(fact_93_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) => (A2 = (bot_bot @ A))))))))).
% 2.53/1.49 thf(fact_94_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G2:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G2) => (((ord_less_eq @ B) @ (F3 @ X2)) @ (G2 @ X2))))))))).
% 2.53/1.49 thf(fact_95_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G2:(A > B),X2:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G2) => (((ord_less_eq @ B) @ (F3 @ X2)) @ (G2 @ X2))))))))).
% 2.53/1.49 thf(fact_96_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F3:(A > B),G2:(A > B)]: (((! [X4:A]: ((((ord_less_eq @ B) @ (F3 @ X4)) @ (G2 @ X4)))) => (((ord_less_eq @ (A > B)) @ F3) @ G2)))))))).
% 2.53/1.49 thf(fact_97_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F:(A > B),G:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3))))))))))).
% 2.53/1.49 thf(fact_98_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F3:(B > A),B2:B,C:B]: (((((ord_less_eq @ A) @ A2) @ (F3 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X4:B,Y3:B]: (((((ord_less_eq @ B) @ X4) @ Y3) => (((ord_less_eq @ A) @ (F3 @ X4)) @ (F3 @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C))))))))))).
% 2.53/1.49 thf(fact_99_order__subst2, axiom, (! [A:$tType,C2:$tType]: ((((order @ C2) & (order @ A)) => (! [A2:A,B2:A,F3:(A > C2),C:C2]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C2) @ (F3 @ B2)) @ C) => ((! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => (((ord_less_eq @ C2) @ (F3 @ X4)) @ (F3 @ Y3))))) => (((ord_less_eq @ C2) @ (F3 @ A2)) @ C)))))))))).
% 2.53/1.49 thf(fact_100_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F3:(B > A),B2:B,C:B]: (((A2 = (F3 @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X4:B,Y3:B]: (((((ord_less_eq @ B) @ X4) @ Y3) => (((ord_less_eq @ A) @ (F3 @ X4)) @ (F3 @ Y3))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C))))))))))).
% 2.53/1.49 thf(fact_101_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F3:(A > B),C:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F3 @ B2) = C) => ((! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => (((ord_less_eq @ B) @ (F3 @ X4)) @ (F3 @ Y3))))) => (((ord_less_eq @ B) @ (F3 @ A2)) @ C)))))))))).
% 2.53/1.49 thf(fact_102_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) & (((ord_less_eq @ A) @ Y2) @ X3))))))))).
% 2.53/1.49 thf(fact_103_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ X2) => (X2 = Y))))))))).
% 2.53/1.49 thf(fact_104_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) | (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.53/1.49 thf(fact_105_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A]: (((X2 = Y) => (((ord_less_eq @ A) @ X2) @ Y)))))))).
% 2.53/1.49 thf(fact_106_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less_eq @ A) @ X2) @ Y))) => (((ord_less_eq @ A) @ Y) @ X2)))))))).
% 2.53/1.49 thf(fact_107_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.53/1.49 thf(fact_108_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A,Z:A]: ((((((ord_less_eq @ A) @ X2) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ Z)))) => (((((ord_less_eq @ A) @ Y) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Z)))) => (((((ord_less_eq @ A) @ X2) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y)))) => (((((ord_less_eq @ A) @ Z) @ Y) => (~ ((((ord_less_eq @ A) @ Y) @ X2)))) => (((((ord_less_eq @ A) @ Y) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X2)))) => (~ (((((ord_less_eq @ A) @ Z) @ X2) => (~ ((((ord_less_eq @ A) @ X2) @ Y))))))))))))))))).
% 2.53/1.49 thf(fact_109_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((ord_less_eq @ A) @ X2) @ Y) = (X2 = Y))))))))).
% 2.53/1.49 thf(fact_110_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) & (((ord_less_eq @ A) @ B3) @ A3))))))))).
% 2.53/1.49 thf(fact_111_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.53/1.49 thf(fact_112_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.53/1.49 thf(fact_113_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.53/1.49 thf(fact_114_NPreflow_OresCap_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((augmenting_resCap @ Capacity) = (augmenting_resCap @ Capacity)))))).
% 2.53/1.49 thf(fact_115_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ Y) @ Z) => (((ord_less_eq @ A) @ X2) @ Z))))))))).
% 2.53/1.49 thf(fact_116_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.53/1.49 thf(fact_117_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P2:(A > (A > $o)),A2:A,B2:A]: (((! [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((P2 @ A5) @ B5)))) => ((! [A5:A,B5:A]: ((((P2 @ B5) @ A5) => ((P2 @ A5) @ B5)))) => ((P2 @ A2) @ B2))))))))).
% 2.53/1.49 thf(fact_118_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 2.53/1.49 thf(fact_119_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z2:A]: ((Y4 = Z2))) = (^ [A3:A,B3:A]: (((((ord_less_eq @ A) @ B3) @ A3) & (((ord_less_eq @ A) @ A3) @ B3))))))))).
% 2.53/1.49 thf(fact_120_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.53/1.49 thf(fact_121_inf_OcoboundedI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,C:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ C)))))))).
% 2.53/1.49 thf(fact_122_inf_OcoboundedI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,C:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ C) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ C)))))))).
% 2.53/1.49 thf(fact_123_inf_Oabsorb__iff2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [B3:A,A3:A]: (((((inf_inf @ A) @ A3) @ B3) = B3)))))))).
% 2.53/1.49 thf(fact_124_inf_Oabsorb__iff1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [A3:A,B3:A]: (((((inf_inf @ A) @ A3) @ B3) = A3)))))))).
% 2.53/1.49 thf(fact_125_inf_Ocobounded2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ B2))))))).
% 2.53/1.49 thf(fact_126_inf_Ocobounded1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ A2))))))).
% 2.53/1.49 thf(fact_127_inf_Oorder__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [A3:A,B3:A]: ((A3 = (((inf_inf @ A) @ A3) @ B3))))))))).
% 2.53/1.49 thf(fact_128_inf__greatest, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A,Y:A,Z:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((ord_less_eq @ A) @ X2) @ Z) => (((ord_less_eq @ A) @ X2) @ (((inf_inf @ A) @ Y) @ Z)))))))))).
% 2.53/1.49 thf(fact_129_inf_OboundedI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ A2) @ C) => (((ord_less_eq @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C)))))))))).
% 2.53/1.49 thf(fact_130_inf_OboundedE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C)) => (~ (((((ord_less_eq @ A) @ A2) @ B2) => (~ ((((ord_less_eq @ A) @ A2) @ C))))))))))))).
% 2.53/1.49 thf(fact_131_inf__absorb2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [Y:A,X2:A]: (((((ord_less_eq @ A) @ Y) @ X2) => ((((inf_inf @ A) @ X2) @ Y) = Y)))))))).
% 2.53/1.49 thf(fact_132_inf__absorb1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) => ((((inf_inf @ A) @ X2) @ Y) = X2)))))))).
% 2.53/1.49 thf(fact_133_inf_Oabsorb2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((inf_inf @ A) @ A2) @ B2) = B2)))))))).
% 2.53/1.49 thf(fact_134_inf_Oabsorb1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((inf_inf @ A) @ A2) @ B2) = A2)))))))).
% 2.53/1.49 thf(fact_135_le__iff__inf, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less_eq @ A) = (^ [X3:A,Y2:A]: (((((inf_inf @ A) @ X3) @ Y2) = X3)))))))).
% 2.53/1.49 thf(fact_136_inf__unique, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [F3:(A > (A > A)),X2:A,Y:A]: (((! [X4:A,Y3:A]: ((((ord_less_eq @ A) @ ((F3 @ X4) @ Y3)) @ X4))) => ((! [X4:A,Y3:A]: ((((ord_less_eq @ A) @ ((F3 @ X4) @ Y3)) @ Y3))) => ((! [X4:A,Y3:A,Z3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => ((((ord_less_eq @ A) @ X4) @ Z3) => (((ord_less_eq @ A) @ X4) @ ((F3 @ Y3) @ Z3)))))) => ((((inf_inf @ A) @ X2) @ Y) = ((F3 @ X2) @ Y))))))))))).
% 2.53/1.49 thf(fact_137_inf_OorderI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A]: (((A2 = (((inf_inf @ A) @ A2) @ B2)) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.53/1.49 thf(fact_138_inf_OorderE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (A2 = (((inf_inf @ A) @ A2) @ B2))))))))).
% 2.53/1.49 thf(fact_139_le__infI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,X2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ X2) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ X2)))))))).
% 2.53/1.49 thf(fact_140_le__infI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,X2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ X2) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ X2)))))))).
% 2.53/1.49 thf(fact_141_inf__mono, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,C:A,B2:A,D2:A]: (((((ord_less_eq @ A) @ A2) @ C) => ((((ord_less_eq @ A) @ B2) @ D2) => (((ord_less_eq @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ (((inf_inf @ A) @ C) @ D2)))))))))).
% 2.53/1.49 thf(fact_142_le__infI, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ X2) @ A2) => ((((ord_less_eq @ A) @ X2) @ B2) => (((ord_less_eq @ A) @ X2) @ (((inf_inf @ A) @ A2) @ B2)))))))))).
% 2.53/1.49 thf(fact_143_le__infE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ X2) @ (((inf_inf @ A) @ A2) @ B2)) => (~ (((((ord_less_eq @ A) @ X2) @ A2) => (~ ((((ord_less_eq @ A) @ X2) @ B2))))))))))))).
% 2.53/1.49 thf(fact_144_inf__le2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A,Y:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X2) @ Y)) @ Y))))))).
% 2.53/1.49 thf(fact_145_inf__le1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [X2:A,Y:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X2) @ Y)) @ X2))))))).
% 2.53/1.49 thf(fact_146_inf__sup__ord_I1_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X2:A,Y:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X2) @ Y)) @ X2))))))).
% 2.53/1.49 thf(fact_147_inf__sup__ord_I2_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X2:A,Y:A]: ((((ord_less_eq @ A) @ (((inf_inf @ A) @ X2) @ Y)) @ Y))))))).
% 2.53/1.49 thf(fact_148_NPreflow_OisAugmentingPath_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((augmen2028041557ngPath @ Capacity) = (augmen2028041557ngPath @ Capacity)))))).
% 2.53/1.49 thf(fact_149_NPreflow_OaugmentingFlow_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((augmen1604248148ngFlow @ Capacity) = (augmen1604248148ngFlow @ Capacity)))))).
% 2.53/1.49 thf(fact_150_NPreflow_OaugFlow__val, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity),P:(list @ ((product_prod @ nat) @ nat))]: (((((((nPreflow @ Capacity) @ C) @ S) @ T) @ F3) => (((((((augmen2028041557ngPath @ Capacity) @ C) @ S) @ T) @ F3) @ P) => (((((val @ Capacity) @ (((residu1993642540lGraph @ Capacity) @ C) @ F3)) @ S) @ ((((augmen1604248148ngFlow @ Capacity) @ C) @ F3) @ P)) = ((((augmenting_resCap @ Capacity) @ C) @ F3) @ P)))))))))).
% 2.53/1.49 thf(fact_151_bot__fun__def, axiom, (! [B:$tType,A:$tType]: (((bot @ B) => ((bot_bot @ (A > B)) = (^ [X3:A]: ((bot_bot @ B)))))))).
% 2.53/1.49 thf(fact_152_NPreflow_OisAugmentingPath__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity),P:(list @ ((product_prod @ nat) @ nat))]: (((((((nPreflow @ Capacity) @ C) @ S) @ T) @ F3) => (((((((augmen2028041557ngPath @ Capacity) @ C) @ S) @ T) @ F3) @ P) = (((((isSimplePath @ Capacity) @ (((residu1993642540lGraph @ Capacity) @ C) @ F3)) @ S) @ P) @ T))))))))).
% 2.53/1.49 thf(fact_153_ford__fulkerson_I2_J, axiom, ((~ ((? [X:(list @ ((product_prod @ nat) @ nat))]: (((((((augmen2028041557ngPath @ capacity) @ c) @ s) @ t) @ f) @ X))))) = (? [K2:(set @ nat)]: (((((((nCut @ capacity) @ c) @ s) @ t) @ K2) & (((((val @ capacity) @ c) @ s) @ f) = (((cap @ capacity) @ c) @ K2))))))).
% 2.53/1.49 thf(fact_154_fofu__III__I, axiom, ((? [K3:(set @ nat)]: (((((((nCut @ capacity) @ c) @ s) @ t) @ K3) & (((((val @ capacity) @ c) @ s) @ f) = (((cap @ capacity) @ c) @ K3))))) => (((((isMaxFlow @ capacity) @ c) @ s) @ t) @ f))).
% 2.53/1.49 thf(fact_155_fofu__II__III, axiom, ((~ ((? [X_12:(list @ ((product_prod @ nat) @ nat))]: (((((((augmen2028041557ngPath @ capacity) @ c) @ s) @ t) @ f) @ X_12))))) => (? [K4:(set @ nat)]: (((((((nCut @ capacity) @ c) @ s) @ t) @ K4) & (((((val @ capacity) @ c) @ s) @ f) = (((cap @ capacity) @ c) @ K4))))))).
% 2.53/1.49 thf(fact_156_NFlow_Oaugment__flow__value, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity),F2:(((product_prod @ nat) @ nat) > Capacity)]: (((((((nFlow @ Capacity) @ C) @ S) @ T) @ F3) => ((((((flow @ Capacity) @ (((residu1993642540lGraph @ Capacity) @ C) @ F3)) @ S) @ T) @ F2) => (((((val @ Capacity) @ C) @ S) @ ((((augmenting_augment @ Capacity) @ C) @ F3) @ F2)) = (((plus_plus @ Capacity) @ ((((val @ Capacity) @ C) @ S) @ F3)) @ ((((val @ Capacity) @ (((residu1993642540lGraph @ Capacity) @ C) @ F3)) @ S) @ F2))))))))))).
% 2.53/1.49 thf(fact_157_Network_OmaxFlow__minCut, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity),K:(set @ nat)]: ((((((network @ Capacity) @ C) @ S) @ T) => ((((((isMaxFlow @ Capacity) @ C) @ S) @ T) @ F3) => ((((((isMinCut @ Capacity) @ C) @ S) @ T) @ K) => (((((val @ Capacity) @ C) @ S) @ F3) = (((cap @ Capacity) @ C) @ K))))))))))).
% 2.53/1.49 thf(fact_158_Network_OisMaxFlow__alt, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [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)))))))))))))).
% 2.53/1.49 thf(fact_159_converse__mono, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ B) @ A)),S:(set @ ((product_prod @ B) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ (((converse @ B) @ A) @ R)) @ (((converse @ B) @ A) @ S)) = (((ord_less_eq @ (set @ ((product_prod @ B) @ A))) @ R) @ S))))).
% 2.53/1.49 thf(fact_160_NCut_Oaxioms_I1_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,K:(set @ nat)]: (((((((nCut @ Capacity) @ C) @ S) @ T) @ K) => ((((network @ Capacity) @ C) @ S) @ T)))))))).
% 2.53/1.49 thf(fact_161_NCut_Ot__ni__cut, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,K:(set @ nat)]: (((((((nCut @ Capacity) @ C) @ S) @ T) @ K) => (~ ((((member @ nat) @ T) @ K)))))))))).
% 2.53/1.49 thf(fact_162_NCut_Os__in__cut, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,K:(set @ nat)]: (((((((nCut @ Capacity) @ C) @ S) @ T) @ K) => (((member @ nat) @ S) @ K)))))))).
% 2.53/1.49 thf(fact_163_converse__subset__swap, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ B) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ (((converse @ B) @ A) @ S)) = (((ord_less_eq @ (set @ ((product_prod @ B) @ A))) @ (((converse @ A) @ B) @ R)) @ S))))).
% 2.53/1.49 thf(fact_164_isMinCut__def, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((isMinCut @ A) = (^ [C3:(((product_prod @ nat) @ nat) > A),S2:nat,T2:nat,K2:(set @ nat)]: (((((((nCut @ A) @ C3) @ S2) @ T2) @ K2) & (! [K5:(set @ nat)]: (((((((nCut @ A) @ C3) @ S2) @ T2) @ K5) => (((ord_less_eq @ A) @ (((cap @ A) @ C3) @ K2)) @ (((cap @ A) @ C3) @ K5))))))))))))).
% 2.53/1.49 thf(fact_165_Network_Ot__not__s, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat]: ((((((network @ Capacity) @ C) @ S) @ T) => (T != S)))))))).
% 2.53/1.49 thf(fact_166_Network_Os__not__t, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat]: ((((((network @ Capacity) @ C) @ S) @ T) => (S != T)))))))).
% 2.53/1.49 thf(fact_167_Flow_Oval_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((val @ Capacity) = (val @ Capacity)))))).
% 2.53/1.49 thf(fact_168_Network_OisMaxFlow_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isMaxFlow @ Capacity) = (isMaxFlow @ Capacity)))))).
% 2.53/1.49 thf(fact_169_NFlow_Oaugment_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((augmenting_augment @ Capacity) = (augmenting_augment @ Capacity)))))).
% 2.53/1.49 thf(fact_170_NCut_Ocap_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((cap @ Capacity) = (cap @ Capacity)))))).
% 2.53/1.49 thf(fact_171_NFlow_Ofofu__II__III, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity)]: (((((((nFlow @ Capacity) @ C) @ S) @ T) @ F3) => ((~ ((? [X_12:(list @ ((product_prod @ nat) @ nat))]: (((((((augmen2028041557ngPath @ Capacity) @ C) @ S) @ T) @ F3) @ X_12))))) => (? [K4:(set @ nat)]: (((((((nCut @ Capacity) @ C) @ S) @ T) @ K4) & (((((val @ Capacity) @ C) @ S) @ F3) = (((cap @ Capacity) @ C) @ K4))))))))))))).
% 2.53/1.49 thf(fact_172_NFlow_Ofofu__III__I, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [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))))))))).
% 2.53/1.49 thf(fact_173_Network_Ois__max__flow__val_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((is_max_flow_val @ Capacity) = (is_max_flow_val @ Capacity)))))).
% 2.53/1.49 thf(fact_174_NFlow_Oford__fulkerson_I2_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity)]: (((((((nFlow @ Capacity) @ C) @ S) @ T) @ F3) => ((~ ((? [X:(list @ ((product_prod @ nat) @ nat))]: (((((((augmen2028041557ngPath @ Capacity) @ C) @ S) @ T) @ F3) @ X))))) = (? [K2:(set @ nat)]: (((((((nCut @ Capacity) @ C) @ S) @ T) @ K2) & (((((val @ Capacity) @ C) @ S) @ F3) = (((cap @ Capacity) @ C) @ K2))))))))))))).
% 2.53/1.49 thf(fact_175_NPreflow_Oaxioms_I1_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity)]: (((((((nPreflow @ Capacity) @ C) @ S) @ T) @ F3) => ((((network @ Capacity) @ C) @ S) @ T)))))))).
% 2.53/1.49 thf(fact_176_NFlow_Oaxioms_I1_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [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)))))))).
% 2.53/1.49 thf(fact_177_NFlow_Oaxioms_I2_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity)]: (((((((nFlow @ Capacity) @ C) @ S) @ T) @ F3) => (((((flow @ Capacity) @ C) @ S) @ T) @ F3)))))))).
% 2.53/1.49 thf(fact_178_NPreflow_Oaxioms_I2_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [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)))))))).
% 2.53/1.49 thf(fact_179_Flow_Oaxioms_I1_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity)]: (((((((flow @ Capacity) @ C) @ S) @ T) @ F3) => (((((preflow @ Capacity) @ C) @ S) @ T) @ F3)))))))).
% 2.53/1.49 thf(fact_180_Network_Omax__flow__val__unique, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,Fv1:Capacity,Fv2:Capacity]: ((((((network @ Capacity) @ C) @ S) @ T) => ((((((is_max_flow_val @ Capacity) @ C) @ S) @ T) @ Fv1) => ((((((is_max_flow_val @ Capacity) @ C) @ S) @ T) @ Fv2) => (Fv1 = Fv2)))))))))).
% 2.53/1.49 thf(fact_181_Finite__Preflow_Oaxioms_I2_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity)]: (((((((finite_Preflow @ Capacity) @ C) @ S) @ T) @ F3) => ((finite_Graph @ Capacity) @ C)))))))).
% 2.53/1.49 thf(fact_182_Finite__Preflow_Oaxioms_I1_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity)]: (((((((finite_Preflow @ Capacity) @ C) @ S) @ T) @ F3) => (((((preflow @ Capacity) @ C) @ S) @ T) @ F3)))))))).
% 2.53/1.49 thf(fact_183_NFlow_Ointro, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity)]: (((((((nPreflow @ Capacity) @ C) @ S) @ T) @ F3) => ((((((flow @ Capacity) @ C) @ S) @ T) @ F3) => (((((nFlow @ Capacity) @ C) @ S) @ T) @ F3))))))))).
% 2.53/1.49 thf(fact_184_NFlow__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((nFlow @ Capacity) = (^ [C3:(((product_prod @ nat) @ nat) > Capacity),S2:nat,T2:nat,F:(((product_prod @ nat) @ nat) > Capacity)]: (((((((nPreflow @ Capacity) @ C3) @ S2) @ T2) @ F) & (((((flow @ Capacity) @ C3) @ S2) @ T2) @ F))))))))).
% 2.53/1.49 thf(fact_185_NPreflow_Ointro, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity)]: ((((((network @ Capacity) @ C) @ S) @ T) => ((((((preflow @ Capacity) @ C) @ S) @ T) @ F3) => (((((nPreflow @ Capacity) @ C) @ S) @ T) @ F3))))))))).
% 2.53/1.49 thf(fact_186_NPreflow__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((nPreflow @ Capacity) = (^ [C3:(((product_prod @ nat) @ nat) > Capacity),S2:nat,T2:nat,F:(((product_prod @ nat) @ nat) > Capacity)]: ((((((network @ Capacity) @ C3) @ S2) @ T2) & (((((preflow @ Capacity) @ C3) @ S2) @ T2) @ F))))))))).
% 2.53/1.49 thf(fact_187_NFlow_Ofofu__I__II, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity)]: (((((((nFlow @ Capacity) @ C) @ S) @ T) @ F3) => ((((((isMaxFlow @ Capacity) @ C) @ S) @ T) @ F3) => (~ ((? [X_1:(list @ ((product_prod @ nat) @ nat))]: (((((((augmen2028041557ngPath @ Capacity) @ C) @ S) @ T) @ F3) @ X_1))))))))))))).
% 2.53/1.49 thf(fact_188_NFlow_OnoAugPath__iff__maxFlow, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity)]: (((((((nFlow @ Capacity) @ C) @ S) @ T) @ F3) => ((~ ((? [X:(list @ ((product_prod @ nat) @ nat))]: (((((((augmen2028041557ngPath @ Capacity) @ C) @ S) @ T) @ F3) @ X))))) = (((((isMaxFlow @ Capacity) @ C) @ S) @ T) @ F3))))))))).
% 2.53/1.49 thf(fact_189_Finite__Preflow_Ointro, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity)]: (((((((preflow @ Capacity) @ C) @ S) @ T) @ F3) => (((finite_Graph @ Capacity) @ C) => (((((finite_Preflow @ Capacity) @ C) @ S) @ T) @ F3))))))))).
% 2.53/1.49 thf(fact_190_Finite__Preflow__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((finite_Preflow @ Capacity) = (^ [C3:(((product_prod @ nat) @ nat) > Capacity),S2:nat,T2:nat,F:(((product_prod @ nat) @ nat) > Capacity)]: (((((((preflow @ Capacity) @ C3) @ S2) @ T2) @ F) & ((finite_Graph @ Capacity) @ C3))))))))).
% 2.53/1.49 thf(fact_191_NFlow_Oaugment__flow__presv, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity),F2:(((product_prod @ nat) @ nat) > Capacity)]: (((((((nFlow @ Capacity) @ C) @ S) @ T) @ F3) => ((((((flow @ Capacity) @ (((residu1993642540lGraph @ Capacity) @ C) @ F3)) @ S) @ T) @ F2) => (((((flow @ Capacity) @ C) @ S) @ T) @ ((((augmenting_augment @ Capacity) @ C) @ F3) @ F2)))))))))).
% 2.53/1.49 thf(fact_192_NFlow_Oford__fulkerson_I1_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [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 @ ((product_prod @ nat) @ nat))]: (((((((augmen2028041557ngPath @ Capacity) @ C) @ S) @ T) @ F3) @ X))))))))))))).
% 2.53/1.49 thf(fact_193_Network_Ois__max__flow__val__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,Fv:Capacity]: ((((((network @ Capacity) @ C) @ S) @ T) => ((((((is_max_flow_val @ Capacity) @ C) @ S) @ T) @ Fv) = (? [F:(((product_prod @ nat) @ nat) > Capacity)]: (((((((isMaxFlow @ Capacity) @ C) @ S) @ T) @ F) & (Fv = ((((val @ Capacity) @ C) @ S) @ F))))))))))))).
% 2.53/1.49 thf(fact_194_Network_OisMaxFlow__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [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)))))))))))))).
% 2.53/1.49 thf(fact_195_cf_OisSPath__no__selfloop, axiom, (! [U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat]: (((((((isSimplePath @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) @ U) @ P) @ V) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ U1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)))))))).
% 2.53/1.49 thf(fact_196_cf_OisSPath__sg__incoming, axiom, (! [U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat,U2:nat]: (((((((isSimplePath @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) @ U) @ P) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((U1 != U2) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U2) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)))))))))).
% 2.53/1.49 thf(fact_197_cf_OisSPath__sg__outgoing, axiom, (! [U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat,V2:nat]: (((((((isSimplePath @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) @ U) @ P) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((V1 != V2) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V2)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)))))))))).
% 2.53/1.49 thf(fact_198_isSPath__sg__outgoing, axiom, (! [U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat,V2:nat]: (((((((isSimplePath @ capacity) @ c) @ U) @ P) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((V1 != V2) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V2)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)))))))))).
% 2.53/1.49 thf(fact_199_isSPath__sg__incoming, axiom, (! [U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat,U2:nat]: (((((((isSimplePath @ capacity) @ c) @ U) @ P) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((U1 != U2) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U2) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)))))))))).
% 2.53/1.49 thf(fact_200_isSPath__no__selfloop, axiom, (! [U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat]: (((((((isSimplePath @ capacity) @ c) @ U) @ P) @ V) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ U1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)))))))).
% 2.53/1.49 thf(fact_201_converse__iff, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,R:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ (((converse @ B) @ A) @ R)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ B2) @ A2)) @ R))))).
% 2.53/1.49 thf(fact_202_subrelI, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((! [X4:A,Y3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y3)) @ R) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X4) @ Y3)) @ S)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S))))).
% 2.53/1.49 thf(fact_203_converseD, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,R:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ (((converse @ B) @ A) @ R)) => (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ B2) @ A2)) @ R))))).
% 2.53/1.49 thf(fact_204_converseE, axiom, (! [A:$tType,B:$tType,Yx:((product_prod @ A) @ B),R:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ Yx) @ (((converse @ B) @ A) @ R)) => (~ ((! [X4:B,Y3:A]: (((Yx = ((((product_Pair @ A) @ B) @ Y3) @ X4)) => (~ ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X4) @ Y3)) @ R)))))))))))).
% 2.53/1.49 thf(fact_205_converse_Ocases, axiom, (! [B:$tType,A:$tType,A1:B,A22:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A1) @ A22)) @ (((converse @ A) @ B) @ R)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A22) @ A1)) @ R))))).
% 2.53/1.49 thf(fact_206_converse_Osimps, axiom, (! [B:$tType,A:$tType,A1:B,A22:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A1) @ A22)) @ (((converse @ A) @ B) @ R)) = (? [A3:A,B3:B]: (((A1 = B3) & ((A22 = A3) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R))))))))).
% 2.53/1.49 thf(fact_207_converse_Ointros, axiom, (! [B:$tType,A:$tType,A2:A,B2:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) @ R) => (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ B2) @ A2)) @ (((converse @ A) @ B) @ R)))))).
% 2.53/1.49 thf(fact_208_converse_Oinducts, axiom, (! [B:$tType,A:$tType,X1:B,X22:A,R:(set @ ((product_prod @ A) @ B)),P2:(B > (A > $o))]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X1) @ X22)) @ (((converse @ A) @ B) @ R)) => ((! [A5:A,B5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ R) => ((P2 @ B5) @ A5)))) => ((P2 @ X1) @ X22)))))).
% 2.53/1.49 thf(fact_209_Graph_OisSPath__sg__outgoing, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat,V2:nat]: (((((((isSimplePath @ Capacity) @ C) @ U) @ P) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((V1 != V2) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V2)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P))))))))))))).
% 2.53/1.49 thf(fact_210_Graph_OisSPath__sg__incoming, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat,U2:nat]: (((((((isSimplePath @ Capacity) @ C) @ U) @ P) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((U1 != U2) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U2) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P))))))))))))).
% 2.53/1.49 thf(fact_211_Graph_OisSPath__no__selfloop, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat]: (((((((isSimplePath @ Capacity) @ C) @ U) @ P) @ V) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ U1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P))))))))))).
% 2.53/1.49 thf(fact_212_augment__flow__presv__cap, axiom, (! [F2:(((product_prod @ nat) @ nat) > capacity),U:nat,V:nat]: (((((((flow @ capacity) @ (((residu1993642540lGraph @ capacity) @ c) @ f)) @ s) @ t) @ F2) => ((((ord_less_eq @ capacity) @ (zero_zero @ capacity)) @ (((((augmenting_augment @ capacity) @ c) @ f) @ F2) @ ((((product_Pair @ nat) @ nat) @ U) @ V))) & (((ord_less_eq @ capacity) @ (((((augmenting_augment @ capacity) @ c) @ f) @ F2) @ ((((product_Pair @ nat) @ nat) @ U) @ V))) @ (c @ ((((product_Pair @ nat) @ nat) @ U) @ V)))))))).
% 2.53/1.49 thf(fact_213_swap__simp, axiom, (! [A:$tType,B:$tType,X2:B,Y:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X2) @ Y)) = ((((product_Pair @ A) @ B) @ Y) @ X2))))).
% 2.53/1.49 thf(fact_214_f__non__negative, axiom, (! [E:((product_prod @ nat) @ nat)]: ((((ord_less_eq @ capacity) @ (zero_zero @ capacity)) @ (f @ E))))).
% 2.53/1.49 thf(fact_215_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A6:A,B6:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A6) @ B6)) = ((A2 = A6) & (B2 = B6)))))).
% 2.53/1.49 thf(fact_216_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y22:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y22)) = ((X1 = Y1) & (X22 = Y22)))))).
% 2.53/1.49 thf(fact_217_cap__non__negative, axiom, (! [U3:nat,V3:nat]: ((((ord_less_eq @ capacity) @ (zero_zero @ capacity)) @ (c @ ((((product_Pair @ nat) @ nat) @ U3) @ V3)))))).
% 2.53/1.49 thf(fact_218_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)))))).
% 2.53/1.49 thf(fact_219_swap__swap, axiom, (! [B:$tType,A:$tType,P:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P)) = P)))).
% 2.53/1.49 thf(fact_220_resE__nonNegative, axiom, (! [E:((product_prod @ nat) @ nat)]: ((((ord_less_eq @ capacity) @ (zero_zero @ capacity)) @ ((((residu1993642540lGraph @ capacity) @ c) @ f) @ E))))).
% 2.53/1.49 thf(fact_221_RPreGraph_OresE__nonNegative, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,T:nat,Cf:(((product_prod @ nat) @ nat) > A),E:((product_prod @ nat) @ nat)]: (((((((residual_RPreGraph @ A) @ C) @ S) @ T) @ Cf) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (Cf @ E))))))))).
% 2.53/1.49 thf(fact_222_Preflow_Ocapacity__const, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [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)))))))))))).
% 2.53/1.49 thf(fact_223_Preflow_Of__non__negative, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity),E:((product_prod @ nat) @ nat)]: (((((((preflow @ Capacity) @ C) @ S) @ T) @ F3) => (((ord_less_eq @ Capacity) @ (zero_zero @ Capacity)) @ (F3 @ E))))))))).
% 2.53/1.49 thf(fact_224_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P2 @ Prod))))).
% 2.53/1.49 thf(fact_225_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B5:B]: ((Y != ((((product_Pair @ A) @ B) @ A5) @ B5))))))))).
% 2.53/1.49 thf(fact_226_prod__induct7, axiom, (! [G3:$tType,F6:$tType,E3:$tType,D:$tType,C2:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F6) @ G3)))))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F6) @ G3))))))]: (((! [A5:A,B5:B,C4:C2,D3:D,E4:E3,F4:F6,G4:G3]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F6) @ G3)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F6) @ G3))))) @ B5) @ ((((product_Pair @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F6) @ G3)))) @ C4) @ ((((product_Pair @ D) @ ((product_prod @ E3) @ ((product_prod @ F6) @ G3))) @ D3) @ ((((product_Pair @ E3) @ ((product_prod @ F6) @ G3)) @ E4) @ ((((product_Pair @ F6) @ G3) @ F4) @ G4))))))))) => (P2 @ X2))))).
% 2.53/1.49 thf(fact_227_prod__induct6, axiom, (! [F6:$tType,E3:$tType,D:$tType,C2:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ F6))))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ F6)))))]: (((! [A5:A,B5:B,C4:C2,D3:D,E4:E3,F4:F6]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ F6))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ F6)))) @ B5) @ ((((product_Pair @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ F6))) @ C4) @ ((((product_Pair @ D) @ ((product_prod @ E3) @ F6)) @ D3) @ ((((product_Pair @ E3) @ F6) @ E4) @ F4)))))))) => (P2 @ X2))))).
% 2.53/1.49 thf(fact_228_prod__induct5, axiom, (! [E3:$tType,D:$tType,C2:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ E3)))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ E3))))]: (((! [A5:A,B5:B,C4:C2,D3:D,E4:E3]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ E3)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ E3))) @ B5) @ ((((product_Pair @ C2) @ ((product_prod @ D) @ E3)) @ C4) @ ((((product_Pair @ D) @ E3) @ D3) @ E4))))))) => (P2 @ X2))))).
% 2.53/1.49 thf(fact_229_prod__induct4, axiom, (! [D:$tType,C2:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ D))) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ D)))]: (((! [A5:A,B5:B,C4:C2,D3:D]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C2) @ D)) @ B5) @ ((((product_Pair @ C2) @ D) @ C4) @ D3)))))) => (P2 @ X2))))).
% 2.53/1.49 thf(fact_230_prod__induct3, axiom, (! [C2:$tType,B:$tType,A:$tType,P2:(((product_prod @ A) @ ((product_prod @ B) @ C2)) > $o),X2:((product_prod @ A) @ ((product_prod @ B) @ C2))]: (((! [A5:A,B5:B,C4:C2]: ((P2 @ ((((product_Pair @ A) @ ((product_prod @ B) @ C2)) @ A5) @ ((((product_Pair @ B) @ C2) @ B5) @ C4))))) => (P2 @ X2))))).
% 2.53/1.49 thf(fact_231_prod__cases7, axiom, (! [A:$tType,B:$tType,C2:$tType,D:$tType,E3:$tType,F6:$tType,G3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F6) @ G3))))))]: ((~ ((! [A5:A,B5:B,C4:C2,D3:D,E4:E3,F4:F6,G4:G3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F6) @ G3)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F6) @ G3))))) @ B5) @ ((((product_Pair @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F6) @ G3)))) @ C4) @ ((((product_Pair @ D) @ ((product_prod @ E3) @ ((product_prod @ F6) @ G3))) @ D3) @ ((((product_Pair @ E3) @ ((product_prod @ F6) @ G3)) @ E4) @ ((((product_Pair @ F6) @ G3) @ F4) @ G4)))))))))))))).
% 2.53/1.49 thf(fact_232_prod__cases6, axiom, (! [A:$tType,B:$tType,C2:$tType,D:$tType,E3:$tType,F6:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ F6)))))]: ((~ ((! [A5:A,B5:B,C4:C2,D3:D,E4:E3,F4:F6]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ F6))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ F6)))) @ B5) @ ((((product_Pair @ C2) @ ((product_prod @ D) @ ((product_prod @ E3) @ F6))) @ C4) @ ((((product_Pair @ D) @ ((product_prod @ E3) @ F6)) @ D3) @ ((((product_Pair @ E3) @ F6) @ E4) @ F4))))))))))))).
% 2.53/1.49 thf(fact_233_prod__cases5, axiom, (! [A:$tType,B:$tType,C2:$tType,D:$tType,E3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ E3))))]: ((~ ((! [A5:A,B5:B,C4:C2,D3:D,E4:E3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ E3)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C2) @ ((product_prod @ D) @ E3))) @ B5) @ ((((product_Pair @ C2) @ ((product_prod @ D) @ E3)) @ C4) @ ((((product_Pair @ D) @ E3) @ D3) @ E4)))))))))))).
% 2.53/1.49 thf(fact_234_prod__cases4, axiom, (! [A:$tType,B:$tType,C2:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ D)))]: ((~ ((! [A5:A,B5:B,C4:C2,D3:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C2) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C2) @ D)) @ B5) @ ((((product_Pair @ C2) @ D) @ C4) @ D3))))))))))).
% 2.53/1.49 thf(fact_235_prod__cases3, axiom, (! [A:$tType,B:$tType,C2:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C2))]: ((~ ((! [A5:A,B5:B,C4:C2]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C2)) @ A5) @ ((((product_Pair @ B) @ C2) @ B5) @ C4)))))))))).
% 2.53/1.49 thf(fact_236_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A6:A,B6:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A6) @ B6)) => (~ (((A2 = A6) => (B2 != B6)))))))).
% 2.53/1.49 thf(fact_237_prod__cases, axiom, (! [B:$tType,A:$tType,P2:(((product_prod @ A) @ B) > $o),P:((product_prod @ A) @ B)]: (((! [A5:A,B5:B]: ((P2 @ ((((product_Pair @ A) @ B) @ A5) @ B5)))) => (P2 @ P))))).
% 2.53/1.49 thf(fact_238_surj__pair, axiom, (! [A:$tType,B:$tType,P:((product_prod @ A) @ B)]: ((? [X4:A,Y3:B]: ((P = ((((product_Pair @ A) @ B) @ X4) @ Y3))))))).
% 2.53/1.49 thf(fact_239_Network_Ocap__non__negative, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [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)))))))))))).
% 2.53/1.49 thf(fact_240_NPreflow_OresE__nonNegative, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity),E:((product_prod @ nat) @ nat)]: (((((((nPreflow @ Capacity) @ C) @ S) @ T) @ F3) => (((ord_less_eq @ Capacity) @ (zero_zero @ Capacity)) @ ((((residu1993642540lGraph @ Capacity) @ C) @ F3) @ E))))))))).
% 2.53/1.49 thf(fact_241_NFlow_Oaugment__flow__presv__cap, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T:nat,F3:(((product_prod @ nat) @ nat) > Capacity),F2:(((product_prod @ nat) @ nat) > Capacity),U:nat,V:nat]: (((((((nFlow @ Capacity) @ C) @ S) @ T) @ F3) => ((((((flow @ Capacity) @ (((residu1993642540lGraph @ Capacity) @ C) @ F3)) @ S) @ T) @ F2) => ((((ord_less_eq @ Capacity) @ (zero_zero @ Capacity)) @ (((((augmenting_augment @ Capacity) @ C) @ F3) @ F2) @ ((((product_Pair @ nat) @ nat) @ U) @ V))) & (((ord_less_eq @ Capacity) @ (((((augmenting_augment @ Capacity) @ C) @ F3) @ F2) @ ((((product_Pair @ nat) @ nat) @ U) @ V))) @ (C @ ((((product_Pair @ nat) @ nat) @ U) @ V)))))))))))).
% 2.53/1.49 thf(fact_242_excess__s__non__pos, axiom, (((ord_less_eq @ capacity) @ ((((excess @ capacity) @ c) @ f) @ s)) @ (zero_zero @ capacity))).
% 2.53/1.49 thf(fact_243_resCap__gzero, axiom, (! [P:(list @ ((product_prod @ nat) @ nat))]: ((((((((augmen2028041557ngPath @ capacity) @ c) @ s) @ t) @ f) @ P) => (((ord_less @ capacity) @ (zero_zero @ capacity)) @ ((((augmenting_resCap @ capacity) @ c) @ f) @ P)))))).
% 2.53/1.49 thf(fact_244_Network_Oexcess_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((excess @ Capacity) = (excess @ Capacity)))))).
% 2.53/1.49 thf(fact_245_bot_Onot__eq__extremum, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: (((A2 != (bot_bot @ A)) = (((ord_less @ A) @ (bot_bot @ A)) @ A2)))))))).
% 2.53/1.49 thf(fact_246_bot_Oextremum__strict, axiom, (! [A:$tType]: (((order_bot @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ (bot_bot @ A)))))))))).
% 2.53/1.49 thf(fact_247_less__infI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,X2:A,B2:A]: (((((ord_less @ A) @ A2) @ X2) => (((ord_less @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ X2)))))))).
% 2.53/1.49 thf(fact_248_less__infI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,X2:A,A2:A]: (((((ord_less @ A) @ B2) @ X2) => (((ord_less @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ X2)))))))).
% 2.53/1.49 thf(fact_249_inf_Ostrict__boundedE, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ (((inf_inf @ A) @ B2) @ C)) => (~ (((((ord_less @ A) @ A2) @ B2) => (~ ((((ord_less @ A) @ A2) @ C))))))))))))).
% 2.53/1.49 thf(fact_250_inf_Ostrict__order__iff, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((ord_less @ A) = (^ [A3:A,B3:A]: (((A3 = (((inf_inf @ A) @ A3) @ B3)) & (A3 != B3))))))))).
% 2.53/1.49 thf(fact_251_inf_Ostrict__coboundedI1, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A2:A,C:A,B2:A]: (((((ord_less @ A) @ A2) @ C) => (((ord_less @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ C)))))))).
% 2.53/1.49 thf(fact_252_inf_Ostrict__coboundedI2, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [B2:A,C:A,A2:A]: (((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ (((inf_inf @ A) @ A2) @ B2)) @ C)))))))).
% 2.53/1.49 thf(fact_253_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (A2 != B2)))))))).
% 2.53/1.49 thf(fact_254_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (A2 != B2)))))))).
% 2.53/1.49 thf(fact_255_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X2:A,Y:A]: (((~ ((((ord_less @ A) @ X2) @ Y))) = ((((ord_less @ A) @ Y) @ X2) | (X2 = Y))))))))).
% 2.53/1.49 thf(subcl_Rings_Olinordered__idom___HOL_Otype, axiom, (! [A:$tType]: (((linordered_idom @ A) => (type @ A))))).
% 2.53/1.49 thf(subcl_Rings_Olinordered__idom___Orderings_Oord, axiom, (! [A:$tType]: (((linordered_idom @ A) => (ord @ A))))).
% 2.53/1.49 thf(subcl_Rings_Olinordered__idom___Orderings_Oorder, axiom, (! [A:$tType]: (((linordered_idom @ A) => (order @ A))))).
% 2.53/1.49 thf(subcl_Rings_Olinordered__idom___Orderings_Olinorder, axiom, (! [A:$tType]: (((linordered_idom @ A) => (linorder @ A))))).
% 2.53/1.49 thf(subcl_Rings_Olinordered__idom___Orderings_Opreorder, axiom, (! [A:$tType]: (((linordered_idom @ A) => (preorder @ A))))).
% 2.53/1.49 thf(tcon_HOL_Obool___Lattices_Obounded__lattice, axiom, (bounded_lattice @ $o)).
% 2.53/1.49 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1, axiom, (! [A7:$tType]: ((bounded_lattice @ (set @ A7))))).
% 2.53/1.49 thf(tcon_fun___Lattices_Obounded__lattice_2, axiom, (! [A7:$tType,A8:$tType]: (((bounded_lattice @ A8) => (bounded_lattice @ (A7 > A8)))))).
% 2.53/1.49 thf(tcon_fun___Lattices_Obounded__lattice__bot, axiom, (! [A7:$tType,A8:$tType]: (((bounded_lattice @ A8) => (bounded_lattice_bot @ (A7 > A8)))))).
% 2.53/1.49 thf(tcon_fun___Lattices_Osemilattice__inf, axiom, (! [A7:$tType,A8:$tType]: (((semilattice_inf @ A8) => (semilattice_inf @ (A7 > A8)))))).
% 2.53/1.49 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A7:$tType,A8:$tType]: (((order_bot @ A8) => (order_bot @ (A7 > A8)))))).
% 2.53/1.49 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: (((preorder @ A8) => (preorder @ (A7 > A8)))))).
% 2.53/1.49 thf(tcon_fun___Lattices_Olattice, axiom, (! [A7:$tType,A8:$tType]: (((lattice @ A8) => (lattice @ (A7 > A8)))))).
% 2.53/1.49 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: (((order @ A8) => (order @ (A7 > A8)))))).
% 2.53/1.49 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: (((ord @ A8) => (ord @ (A7 > A8)))))).
% 2.53/1.49 thf(tcon_fun___Orderings_Obot, axiom, (! [A7:$tType,A8:$tType]: (((bot @ A8) => (bot @ (A7 > A8)))))).
% 2.53/1.49 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_3, axiom, (semilattice_inf @ nat)).
% 2.53/1.49 thf(tcon_Nat_Onat___Orderings_Oorder__bot_4, axiom, (order_bot @ nat)).
% 2.53/1.49 thf(tcon_Nat_Onat___Orderings_Opreorder_5, axiom, (preorder @ nat)).
% 2.53/1.49 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 2.53/1.49 thf(tcon_Nat_Onat___Lattices_Olattice_6, axiom, (lattice @ nat)).
% 2.53/1.49 thf(tcon_Nat_Onat___Orderings_Oorder_7, axiom, (order @ nat)).
% 2.53/1.49 thf(tcon_Nat_Onat___Orderings_Oord_8, axiom, (ord @ nat)).
% 2.53/1.49 thf(tcon_Nat_Onat___Orderings_Obot_9, axiom, (bot @ nat)).
% 2.53/1.49 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_10, axiom, (! [A7:$tType]: ((bounded_lattice_bot @ (set @ A7))))).
% 2.53/1.49 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_11, axiom, (! [A7:$tType]: ((semilattice_inf @ (set @ A7))))).
% 2.53/1.49 thf(tcon_Set_Oset___Orderings_Oorder__bot_12, axiom, (! [A7:$tType]: ((order_bot @ (set @ A7))))).
% 2.53/1.49 thf(tcon_Set_Oset___Orderings_Opreorder_13, axiom, (! [A7:$tType]: ((preorder @ (set @ A7))))).
% 2.53/1.49 thf(tcon_Set_Oset___Lattices_Olattice_14, axiom, (! [A7:$tType]: ((lattice @ (set @ A7))))).
% 2.53/1.49 thf(tcon_Set_Oset___Orderings_Oorder_15, axiom, (! [A7:$tType]: ((order @ (set @ A7))))).
% 2.53/1.49 thf(tcon_Set_Oset___Orderings_Oord_16, axiom, (! [A7:$tType]: ((ord @ (set @ A7))))).
% 2.53/1.49 thf(tcon_Set_Oset___Orderings_Obot_17, axiom, (! [A7:$tType]: ((bot @ (set @ A7))))).
% 2.53/1.49 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_18, axiom, (bounded_lattice_bot @ $o)).
% 2.53/1.49 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_19, axiom, (semilattice_inf @ $o)).
% 2.53/1.49 thf(tcon_HOL_Obool___Orderings_Oorder__bot_20, axiom, (order_bot @ $o)).
% 2.53/1.49 thf(tcon_HOL_Obool___Orderings_Opreorder_21, axiom, (preorder @ $o)).
% 2.53/1.49 thf(tcon_HOL_Obool___Orderings_Olinorder_22, axiom, (linorder @ $o)).
% 2.53/1.49 thf(tcon_HOL_Obool___Lattices_Olattice_23, axiom, (lattice @ $o)).
% 2.53/1.49 thf(tcon_HOL_Obool___Orderings_Oorder_24, axiom, (order @ $o)).
% 2.53/1.49 thf(tcon_HOL_Obool___Orderings_Oord_25, axiom, (ord @ $o)).
% 2.53/1.49 thf(tcon_HOL_Obool___Orderings_Obot_26, axiom, (bot @ $o)).
% 2.53/1.49 thf(tfree_0, hypothesis, (linordered_idom @ capacity)).
% 2.53/1.49 thf(conj_0, hypothesis, ((((((augmen2028041557ngPath @ capacity) @ c) @ s) @ t) @ f) @ p)).
% 2.53/1.49 thf(conj_1, conjecture, ((((inf_inf @ (set @ ((product_prod @ nat) @ nat))) @ ((set2 @ ((product_prod @ nat) @ nat)) @ p)) @ (((converse @ nat) @ nat) @ ((set2 @ ((product_prod @ nat) @ nat)) @ p))) = (bot_bot @ (set @ ((product_prod @ nat) @ nat))))).
% 2.53/1.49 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.7uUisdwD1b/DTF2DTF5283.p
% 2.53/1.49 ------------------------
% 2.84/1.52 ---- Cleaned THF ---
% 2.84/1.52 thf(ty_t_Product__Type_Oprod,type,
% 2.84/1.52 product_prod: $tType > $tType > $tType ).
% 2.84/1.52
% 2.84/1.52 thf(ty_t_List_Olist,type,
% 2.84/1.52 list: $tType > $tType ).
% 2.84/1.52
% 2.84/1.52 thf(ty_tf_capacity,type,
% 2.84/1.52 capacity: $tType ).
% 2.84/1.52
% 2.84/1.52 thf(ty_t_Set_Oset,type,
% 2.84/1.52 set: $tType > $tType ).
% 2.84/1.52
% 2.84/1.52 thf(ty_t_Nat_Onat,type,
% 2.84/1.52 nat: $tType ).
% 2.84/1.52
% 2.84/1.52 thf(sy_cl_Lattices_Obounded__lattice,type,
% 2.84/1.52 bounded_lattice:
% 2.84/1.52 !>[A: $tType] : $o ).
% 2.84/1.52
% 2.84/1.52 thf(sy_cl_HOL_Otype,type,
% 2.84/1.52 type:
% 2.84/1.52 !>[A: $tType] : $o ).
% 2.84/1.52
% 2.84/1.52 thf(sy_cl_Orderings_Obot,type,
% 2.84/1.52 bot:
% 2.84/1.52 !>[A: $tType] : $o ).
% 2.84/1.52
% 2.84/1.52 thf(sy_cl_Orderings_Oord,type,
% 2.84/1.52 ord:
% 2.84/1.52 !>[A: $tType] : $o ).
% 2.84/1.52
% 2.84/1.52 thf(sy_cl_Orderings_Oorder,type,
% 2.84/1.52 order:
% 2.84/1.52 !>[A: $tType] : $o ).
% 2.84/1.52
% 2.84/1.52 thf(sy_cl_Lattices_Olattice,type,
% 2.84/1.52 lattice:
% 2.84/1.52 !>[A: $tType] : $o ).
% 2.84/1.52
% 2.84/1.52 thf(sy_cl_Orderings_Olinorder,type,
% 2.84/1.52 linorder:
% 2.84/1.52 !>[A: $tType] : $o ).
% 2.84/1.52
% 2.84/1.52 thf(sy_cl_Orderings_Opreorder,type,
% 2.84/1.52 preorder:
% 2.84/1.52 !>[A: $tType] : $o ).
% 2.84/1.52
% 2.84/1.52 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.84/1.52 order_bot:
% 2.84/1.52 !>[A: $tType] : $o ).
% 2.84/1.52
% 2.84/1.52 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.84/1.52 linordered_idom:
% 2.84/1.52 !>[A: $tType] : $o ).
% 2.84/1.52
% 2.84/1.52 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 2.84/1.52 semilattice_inf:
% 2.84/1.52 !>[A: $tType] : $o ).
% 2.84/1.52
% 2.84/1.52 thf(sy_cl_Lattices_Obounded__lattice__bot,type,
% 2.84/1.52 bounded_lattice_bot:
% 2.84/1.52 !>[A: $tType] : $o ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Augmenting__Flow_ONFlow_Oaugment,type,
% 2.84/1.52 augmenting_augment:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( ( product_prod @ nat @ nat ) > Capacity ) > ( ( product_prod @ nat @ nat ) > Capacity ) > ( product_prod @ nat @ nat ) > Capacity ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Augmenting__Path_ONPreflow_OaugmentingFlow,type,
% 2.84/1.52 augmen1604248148ngFlow:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( ( product_prod @ nat @ nat ) > Capacity ) > ( list @ ( product_prod @ nat @ nat ) ) > ( product_prod @ nat @ nat ) > Capacity ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Augmenting__Path_ONPreflow_OisAugmentingPath,type,
% 2.84/1.52 augmen2028041557ngPath:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > ( ( product_prod @ nat @ nat ) > Capacity ) > ( list @ ( product_prod @ nat @ nat ) ) > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Augmenting__Path_ONPreflow_OresCap,type,
% 2.84/1.52 augmenting_resCap:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( ( product_prod @ nat @ nat ) > Capacity ) > ( list @ ( product_prod @ nat @ nat ) ) > Capacity ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Graph_OFinite__Graph,type,
% 2.84/1.52 finite_Graph:
% 2.84/1.52 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Graph_OGraph_OisShortestPath,type,
% 2.84/1.52 isShortestPath:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Graph_OGraph_OisSimplePath,type,
% 2.84/1.52 isSimplePath:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.84/1.52 plus_plus:
% 2.84/1.52 !>[A: $tType] : ( A > A > A ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.84/1.52 zero_zero:
% 2.84/1.52 !>[A: $tType] : A ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 2.84/1.52 inf_inf:
% 2.84/1.52 !>[A: $tType] : ( A > A > A ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_List_Olist_Oset,type,
% 2.84/1.52 set2:
% 2.84/1.52 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Network_OFinite__Preflow,type,
% 2.84/1.52 finite_Preflow:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > ( ( product_prod @ nat @ nat ) > Capacity ) > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Network_OFlow,type,
% 2.84/1.52 flow:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > ( ( product_prod @ nat @ nat ) > Capacity ) > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Network_OFlow_Oval,type,
% 2.84/1.52 val:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( ( product_prod @ nat @ nat ) > Capacity ) > Capacity ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Network_ONCut,type,
% 2.84/1.52 nCut:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > ( set @ nat ) > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Network_ONCut_Ocap,type,
% 2.84/1.52 cap:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) > Capacity ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Network_ONFlow,type,
% 2.84/1.52 nFlow:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > ( ( product_prod @ nat @ nat ) > Capacity ) > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Network_ONPreflow,type,
% 2.84/1.52 nPreflow:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > ( ( product_prod @ nat @ nat ) > Capacity ) > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Network_ONetwork,type,
% 2.84/1.52 network:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Network_ONetwork_Oexcess,type,
% 2.84/1.52 excess:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( ( product_prod @ nat @ nat ) > Capacity ) > nat > Capacity ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Network_ONetwork_OisMaxFlow,type,
% 2.84/1.52 isMaxFlow:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > ( ( product_prod @ nat @ nat ) > Capacity ) > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Network_ONetwork_Ois__max__flow__val,type,
% 2.84/1.52 is_max_flow_val:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > Capacity > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Network_OPreflow,type,
% 2.84/1.52 preflow:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > ( ( product_prod @ nat @ nat ) > Capacity ) > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Network_OisMinCut,type,
% 2.84/1.52 isMinCut:
% 2.84/1.52 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > nat > nat > ( set @ nat ) > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.84/1.52 bot_bot:
% 2.84/1.52 !>[A: $tType] : A ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.84/1.52 ord_less:
% 2.84/1.52 !>[A: $tType] : ( A > A > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.84/1.52 ord_less_eq:
% 2.84/1.52 !>[A: $tType] : ( A > A > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Product__Type_OPair,type,
% 2.84/1.52 product_Pair:
% 2.84/1.52 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 2.84/1.52 product_swap:
% 2.84/1.52 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Relation_Oconverse,type,
% 2.84/1.52 converse:
% 2.84/1.52 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ A ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Residual__Graph_ONetwork_Oflow__of__cf,type,
% 2.84/1.52 residual_flow_of_cf:
% 2.84/1.52 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( ( product_prod @ nat @ nat ) > Capacity ) > ( product_prod @ nat @ nat ) > Capacity ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Residual__Graph_ORGraph,type,
% 2.84/1.52 residual_RGraph:
% 2.84/1.52 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > nat > nat > ( ( product_prod @ nat @ nat ) > A ) > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Residual__Graph_ORPreGraph,type,
% 2.84/1.52 residual_RPreGraph:
% 2.84/1.52 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > nat > nat > ( ( product_prod @ nat @ nat ) > A ) > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Residual__Graph_OresidualGraph,type,
% 2.84/1.52 residu1993642540lGraph:
% 2.84/1.52 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > ( ( product_prod @ nat @ nat ) > A ) > ( product_prod @ nat @ nat ) > A ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_Set_OCollect,type,
% 2.84/1.52 collect:
% 2.84/1.52 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_c_member,type,
% 2.84/1.52 member:
% 2.84/1.52 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_v_c,type,
% 2.84/1.52 c: ( product_prod @ nat @ nat ) > capacity ).
% 2.84/1.52
% 2.84/1.52 thf(sy_v_f,type,
% 2.84/1.52 f: ( product_prod @ nat @ nat ) > capacity ).
% 2.84/1.52
% 2.84/1.52 thf(sy_v_p,type,
% 2.84/1.52 p: list @ ( product_prod @ nat @ nat ) ).
% 2.84/1.52
% 2.84/1.52 thf(sy_v_s,type,
% 2.84/1.52 s: nat ).
% 2.84/1.52
% 2.84/1.52 thf(sy_v_t,type,
% 2.84/1.52 t: nat ).
% 2.84/1.52
% 2.84/1.52 thf(fact_0_t__not__s,axiom,
% 2.84/1.52 t != s ).
% 2.84/1.52
% 2.84/1.52 thf(fact_1_cf_OisSPath__nt__parallel__pf,axiom,
% 2.84/1.52 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat] :
% 2.84/1.52 ( ( isSimplePath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ S @ P @ T )
% 2.84/1.52 => ( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) @ ( converse @ nat @ nat @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) )
% 2.84/1.52 = ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_2_isSPath__nt__parallel__pf,axiom,
% 2.84/1.52 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat] :
% 2.84/1.52 ( ( isSimplePath @ capacity @ c @ S @ P @ T )
% 2.84/1.52 => ( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) @ ( converse @ nat @ nat @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) )
% 2.84/1.52 = ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_3_NFlow__axioms,axiom,
% 2.84/1.52 nFlow @ capacity @ c @ s @ t @ f ).
% 2.84/1.52
% 2.84/1.52 thf(fact_4_max__flow__val__unique,axiom,
% 2.84/1.52 ! [Fv1: capacity,Fv2: capacity] :
% 2.84/1.52 ( ( is_max_flow_val @ capacity @ c @ s @ t @ Fv1 )
% 2.84/1.52 => ( ( is_max_flow_val @ capacity @ c @ s @ t @ Fv2 )
% 2.84/1.52 => ( Fv1 = Fv2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_5_isAugmentingPath__def,axiom,
% 2.84/1.52 ! [P: list @ ( product_prod @ nat @ nat )] :
% 2.84/1.52 ( ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ P )
% 2.84/1.52 = ( isSimplePath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ P @ t ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_6_Finite__Preflow__axioms,axiom,
% 2.84/1.52 finite_Preflow @ capacity @ c @ s @ t @ f ).
% 2.84/1.52
% 2.84/1.52 thf(fact_7_fofu__I__II,axiom,
% 2.84/1.52 ( ( isMaxFlow @ capacity @ c @ s @ t @ f )
% 2.84/1.52 => ~ ? [X_1: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ X_1 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_8_noAugPath__iff__maxFlow,axiom,
% 2.84/1.52 ( ( ~ ? [X: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ X ) )
% 2.84/1.52 = ( isMaxFlow @ capacity @ c @ s @ t @ f ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_9_Preflow__axioms,axiom,
% 2.84/1.52 preflow @ capacity @ c @ s @ t @ f ).
% 2.84/1.52
% 2.84/1.52 thf(fact_10_is__RGraph,axiom,
% 2.84/1.52 residual_RGraph @ capacity @ c @ s @ t @ ( residu1993642540lGraph @ capacity @ c @ f ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_11_Flow__axioms,axiom,
% 2.84/1.52 flow @ capacity @ c @ s @ t @ f ).
% 2.84/1.52
% 2.84/1.52 thf(fact_12_NPreflow__axioms,axiom,
% 2.84/1.52 nPreflow @ capacity @ c @ s @ t @ f ).
% 2.84/1.52
% 2.84/1.52 thf(fact_13_converse__empty,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType] :
% 2.84/1.52 ( ( converse @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
% 2.84/1.52 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_14_inf__bot__left,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( bounded_lattice_bot @ A )
% 2.84/1.52 => ! [X2: A] :
% 2.84/1.52 ( ( inf_inf @ A @ ( bot_bot @ A ) @ X2 )
% 2.84/1.52 = ( bot_bot @ A ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_15_inf__bot__right,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( bounded_lattice_bot @ A )
% 2.84/1.52 => ! [X2: A] :
% 2.84/1.52 ( ( inf_inf @ A @ X2 @ ( bot_bot @ A ) )
% 2.84/1.52 = ( bot_bot @ A ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_16_is__RPreGraph,axiom,
% 2.84/1.52 residual_RPreGraph @ capacity @ c @ s @ t @ ( residu1993642540lGraph @ capacity @ c @ f ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_17_Network__axioms,axiom,
% 2.84/1.52 network @ capacity @ c @ s @ t ).
% 2.84/1.52
% 2.84/1.52 thf(fact_18_fo__rg__inv,axiom,
% 2.84/1.52 ( ( residual_flow_of_cf @ capacity @ c @ ( residu1993642540lGraph @ capacity @ c @ f ) )
% 2.84/1.52 = f ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_19_inf__right__idem,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [X2: A,Y: A] :
% 2.84/1.52 ( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y ) @ Y )
% 2.84/1.52 = ( inf_inf @ A @ X2 @ Y ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_20_inf_Oright__idem,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,B2: A] :
% 2.84/1.52 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
% 2.84/1.52 = ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_21_inf__left__idem,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [X2: A,Y: A] :
% 2.84/1.52 ( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ X2 @ Y ) )
% 2.84/1.52 = ( inf_inf @ A @ X2 @ Y ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_22_inf_Oleft__idem,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,B2: A] :
% 2.84/1.52 ( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
% 2.84/1.52 = ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_23_inf__idem,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [X2: A] :
% 2.84/1.52 ( ( inf_inf @ A @ X2 @ X2 )
% 2.84/1.52 = X2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_24_inf_Oidem,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A] :
% 2.84/1.52 ( ( inf_inf @ A @ A2 @ A2 )
% 2.84/1.52 = A2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_25_inf__apply,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ B )
% 2.84/1.52 => ( ( inf_inf @ ( A > B ) )
% 2.84/1.52 = ( ^ [F: A > B,G: A > B,X3: A] : ( inf_inf @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_26_converse__converse,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 2.84/1.52 ( ( converse @ B @ A @ ( converse @ A @ B @ R ) )
% 2.84/1.52 = R ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_27_converse__inject,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A )] :
% 2.84/1.52 ( ( ( converse @ B @ A @ R )
% 2.84/1.52 = ( converse @ B @ A @ S ) )
% 2.84/1.52 = ( R = S ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_28_augFlow__resFlow,axiom,
% 2.84/1.52 ! [P: list @ ( product_prod @ nat @ nat )] :
% 2.84/1.52 ( ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ P )
% 2.84/1.52 => ( flow @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ t @ ( augmen1604248148ngFlow @ capacity @ c @ f @ P ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_29_bot__empty__eq,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( bot_bot @ ( A > $o ) )
% 2.84/1.52 = ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_30_inf__left__commute,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [X2: A,Y: A,Z: A] :
% 2.84/1.52 ( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y @ Z ) )
% 2.84/1.52 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X2 @ Z ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_31_inf_Oleft__commute,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [B2: A,A2: A,C: A] :
% 2.84/1.52 ( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A2 @ C ) )
% 2.84/1.52 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_32_inf__commute,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ( ( inf_inf @ A )
% 2.84/1.52 = ( ^ [X3: A,Y2: A] : ( inf_inf @ A @ Y2 @ X3 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_33_inf_Ocommute,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ( ( inf_inf @ A )
% 2.84/1.52 = ( ^ [A3: A,B3: A] : ( inf_inf @ A @ B3 @ A3 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_34_inf__assoc,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [X2: A,Y: A,Z: A] :
% 2.84/1.52 ( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y ) @ Z )
% 2.84/1.52 = ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_35_inf_Oassoc,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,B2: A,C: A] :
% 2.84/1.52 ( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C )
% 2.84/1.52 = ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_36_boolean__algebra__cancel_Oinf2,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [B4: A,K: A,B2: A,A2: A] :
% 2.84/1.52 ( ( B4
% 2.84/1.52 = ( inf_inf @ A @ K @ B2 ) )
% 2.84/1.52 => ( ( inf_inf @ A @ A2 @ B4 )
% 2.84/1.52 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_37_boolean__algebra__cancel_Oinf1,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A4: A,K: A,A2: A,B2: A] :
% 2.84/1.52 ( ( A4
% 2.84/1.52 = ( inf_inf @ A @ K @ A2 ) )
% 2.84/1.52 => ( ( inf_inf @ A @ A4 @ B2 )
% 2.84/1.52 = ( inf_inf @ A @ K @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_38_inf__fun__def,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ B )
% 2.84/1.52 => ( ( inf_inf @ ( A > B ) )
% 2.84/1.52 = ( ^ [F: A > B,G: A > B,X3: A] : ( inf_inf @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_39_inf__sup__aci_I1_J,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( lattice @ A )
% 2.84/1.52 => ( ( inf_inf @ A )
% 2.84/1.52 = ( ^ [X3: A,Y2: A] : ( inf_inf @ A @ Y2 @ X3 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_40_inf__sup__aci_I2_J,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( lattice @ A )
% 2.84/1.52 => ! [X2: A,Y: A,Z: A] :
% 2.84/1.52 ( ( inf_inf @ A @ ( inf_inf @ A @ X2 @ Y ) @ Z )
% 2.84/1.52 = ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_41_inf__sup__aci_I3_J,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( lattice @ A )
% 2.84/1.52 => ! [X2: A,Y: A,Z: A] :
% 2.84/1.52 ( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ Y @ Z ) )
% 2.84/1.52 = ( inf_inf @ A @ Y @ ( inf_inf @ A @ X2 @ Z ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_42_inf__sup__aci_I4_J,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( lattice @ A )
% 2.84/1.52 => ! [X2: A,Y: A] :
% 2.84/1.52 ( ( inf_inf @ A @ X2 @ ( inf_inf @ A @ X2 @ Y ) )
% 2.84/1.52 = ( inf_inf @ A @ X2 @ Y ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_43_converse__Int,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A )] :
% 2.84/1.52 ( ( converse @ B @ A @ ( inf_inf @ ( set @ ( product_prod @ B @ A ) ) @ R @ S ) )
% 2.84/1.52 = ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R ) @ ( converse @ B @ A @ S ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_44_augment__flow__presv,axiom,
% 2.84/1.52 ! [F2: ( product_prod @ nat @ nat ) > capacity] :
% 2.84/1.52 ( ( flow @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ t @ F2 )
% 2.84/1.52 => ( flow @ capacity @ c @ s @ t @ ( augmenting_augment @ capacity @ c @ f @ F2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_45_mem__Collect__eq,axiom,
% 2.84/1.52 ! [A: $tType,A2: A,P2: A > $o] :
% 2.84/1.52 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 2.84/1.52 = ( P2 @ A2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_46_Collect__mem__eq,axiom,
% 2.84/1.52 ! [A: $tType,A4: set @ A] :
% 2.84/1.52 ( ( collect @ A
% 2.84/1.52 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 2.84/1.52 = A4 ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_47_Collect__cong,axiom,
% 2.84/1.52 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.84/1.52 ( ! [X4: A] :
% 2.84/1.52 ( ( P2 @ X4 )
% 2.84/1.52 = ( Q @ X4 ) )
% 2.84/1.52 => ( ( collect @ A @ P2 )
% 2.84/1.52 = ( collect @ A @ Q ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_48_ext,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType,F3: A > B,G2: A > B] :
% 2.84/1.52 ( ! [X4: A] :
% 2.84/1.52 ( ( F3 @ X4 )
% 2.84/1.52 = ( G2 @ X4 ) )
% 2.84/1.52 => ( F3 = G2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_49_cf_OisSPath__nt__parallel,axiom,
% 2.84/1.52 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat,E: product_prod @ nat @ nat] :
% 2.84/1.52 ( ( isSimplePath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ S @ P @ T )
% 2.84/1.52 => ( ( member @ ( product_prod @ nat @ nat ) @ E @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.84/1.52 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_50_is__max__flow__val__def,axiom,
% 2.84/1.52 ! [Fv: capacity] :
% 2.84/1.52 ( ( is_max_flow_val @ capacity @ c @ s @ t @ Fv )
% 2.84/1.52 = ( ? [F: ( product_prod @ nat @ nat ) > capacity] :
% 2.84/1.52 ( ( isMaxFlow @ capacity @ c @ s @ t @ F )
% 2.84/1.52 & ( Fv
% 2.84/1.52 = ( val @ capacity @ c @ s @ F ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_51_cf_OFinite__Graph__axioms,axiom,
% 2.84/1.52 finite_Graph @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_52_ford__fulkerson_I1_J,axiom,
% 2.84/1.52 ( ( isMaxFlow @ capacity @ c @ s @ t @ f )
% 2.84/1.52 = ( ~ ? [X: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ X ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_53_shortest__is__augmenting,axiom,
% 2.84/1.52 ! [P: list @ ( product_prod @ nat @ nat )] :
% 2.84/1.52 ( ( isShortestPath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ P @ t )
% 2.84/1.52 => ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ P ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_54_augmenting__path__imp__shortest,axiom,
% 2.84/1.52 ! [P: list @ ( product_prod @ nat @ nat )] :
% 2.84/1.52 ( ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ P )
% 2.84/1.52 => ? [P3: list @ ( product_prod @ nat @ nat )] : ( isShortestPath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ P3 @ t ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_55_Graph_OisSPath__nt__parallel__pf,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat] :
% 2.84/1.52 ( ( isSimplePath @ Capacity @ C @ S @ P @ T )
% 2.84/1.52 => ( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) @ ( converse @ nat @ nat @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) )
% 2.84/1.52 = ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_56_RPreGraph_Omaxflow__imp__rgraph,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 2.84/1.52 ( ( residual_RPreGraph @ A @ C @ S @ T @ Cf )
% 2.84/1.52 => ( ( isMaxFlow @ A @ C @ S @ T @ ( residual_flow_of_cf @ A @ C @ Cf ) )
% 2.84/1.52 => ( residual_RGraph @ A @ C @ S @ T @ Cf ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_57_isSPath__nt__parallel,axiom,
% 2.84/1.52 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat,E: product_prod @ nat @ nat] :
% 2.84/1.52 ( ( isSimplePath @ capacity @ c @ S @ P @ T )
% 2.84/1.52 => ( ( member @ ( product_prod @ nat @ nat ) @ E @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.84/1.52 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_58_Finite__Graph__axioms,axiom,
% 2.84/1.52 finite_Graph @ capacity @ c ).
% 2.84/1.52
% 2.84/1.52 thf(fact_59_shortestPath__is__simple,axiom,
% 2.84/1.52 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat] :
% 2.84/1.52 ( ( isShortestPath @ capacity @ c @ S @ P @ T )
% 2.84/1.52 => ( isSimplePath @ capacity @ c @ S @ P @ T ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_60_cf_OshortestPath__is__simple,axiom,
% 2.84/1.52 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat] :
% 2.84/1.52 ( ( isShortestPath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ S @ P @ T )
% 2.84/1.52 => ( isSimplePath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ S @ P @ T ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_61_Graph_OisShortestPath_Ocong,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ( ( isShortestPath @ Capacity )
% 2.84/1.52 = ( isShortestPath @ Capacity ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_62_Graph_OshortestPath__is__simple,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat] :
% 2.84/1.52 ( ( isShortestPath @ Capacity @ C @ S @ P @ T )
% 2.84/1.52 => ( isSimplePath @ Capacity @ C @ S @ P @ T ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_63_Graph_OisSPath__nt__parallel,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T: nat,E: product_prod @ nat @ nat] :
% 2.84/1.52 ( ( isSimplePath @ Capacity @ C @ S @ P @ T )
% 2.84/1.52 => ( ( member @ ( product_prod @ nat @ nat ) @ E @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.84/1.52 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_64_Graph_OisSimplePath_Ocong,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ( ( isSimplePath @ Capacity )
% 2.84/1.52 = ( isSimplePath @ Capacity ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_65_RPreGraph_Othis__loc__rpg,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 2.84/1.52 ( ( residual_RPreGraph @ A @ C @ S @ T @ Cf )
% 2.84/1.52 => ( residual_RPreGraph @ A @ C @ S @ T @ Cf ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_66_NFlow_Oshortest__is__augmenting,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,P: list @ ( product_prod @ nat @ nat )] :
% 2.84/1.52 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ( isShortestPath @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ P @ T )
% 2.84/1.52 => ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ P ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_67_NFlow_Oaugmenting__path__imp__shortest,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,P: list @ ( product_prod @ nat @ nat )] :
% 2.84/1.52 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ P )
% 2.84/1.52 => ? [P3: list @ ( product_prod @ nat @ nat )] : ( isShortestPath @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ P3 @ T ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_68_RGraph_Othis__loc,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 2.84/1.52 ( ( residual_RGraph @ A @ C @ S @ T @ Cf )
% 2.84/1.52 => ( residual_RGraph @ A @ C @ S @ T @ Cf ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_69_Network_Oflow__of__cf_Ocong,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ( ( residual_flow_of_cf @ Capacity )
% 2.84/1.52 = ( residual_flow_of_cf @ Capacity ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_70_RPreGraph_Oaxioms_I1_J,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 2.84/1.52 ( ( residual_RPreGraph @ A @ C @ S @ T @ Cf )
% 2.84/1.52 => ( network @ A @ C @ S @ T ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_71_RGraph_Oaxioms_I1_J,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 2.84/1.52 ( ( residual_RGraph @ A @ C @ S @ T @ Cf )
% 2.84/1.52 => ( network @ A @ C @ S @ T ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_72_RGraph_Othis__loc__rpg,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 2.84/1.52 ( ( residual_RGraph @ A @ C @ S @ T @ Cf )
% 2.84/1.52 => ( residual_RPreGraph @ A @ C @ S @ T @ Cf ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_73_NPreflow_Ois__RPreGraph,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( residual_RPreGraph @ Capacity @ C @ S @ T @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_74_RPreGraph_OEX__RPG,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 2.84/1.52 ( ( residual_RPreGraph @ A @ C @ S @ T @ Cf )
% 2.84/1.52 => ? [F4: ( product_prod @ nat @ nat ) > A] :
% 2.84/1.52 ( ( nPreflow @ A @ C @ S @ T @ F4 )
% 2.84/1.52 & ( Cf
% 2.84/1.52 = ( residu1993642540lGraph @ A @ C @ F4 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_75_NFlow_Ois__RGraph,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( residual_RGraph @ Capacity @ C @ S @ T @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_76_RGraph_OEX__RG,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 2.84/1.52 ( ( residual_RGraph @ A @ C @ S @ T @ Cf )
% 2.84/1.52 => ? [F4: ( product_prod @ nat @ nat ) > A] :
% 2.84/1.52 ( ( nFlow @ A @ C @ S @ T @ F4 )
% 2.84/1.52 & ( Cf
% 2.84/1.52 = ( residu1993642540lGraph @ A @ C @ F4 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_77_NPreflow_Ofo__rg__inv,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ( residual_flow_of_cf @ Capacity @ C @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) )
% 2.84/1.52 = F3 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_78_RPreGraph_Org__fo__inv,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 2.84/1.52 ( ( residual_RPreGraph @ A @ C @ S @ T @ Cf )
% 2.84/1.52 => ( ( residu1993642540lGraph @ A @ C @ ( residual_flow_of_cf @ A @ C @ Cf ) )
% 2.84/1.52 = Cf ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_79_RPreGraph_Ois__NPreflow,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 2.84/1.52 ( ( residual_RPreGraph @ A @ C @ S @ T @ Cf )
% 2.84/1.52 => ( nPreflow @ A @ C @ S @ T @ ( residual_flow_of_cf @ A @ C @ Cf ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_80_RGraph_Ois__NFlow,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A] :
% 2.84/1.52 ( ( residual_RGraph @ A @ C @ S @ T @ Cf )
% 2.84/1.52 => ( nFlow @ A @ C @ S @ T @ ( residual_flow_of_cf @ A @ C @ Cf ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_81_augFlow__val,axiom,
% 2.84/1.52 ! [P: list @ ( product_prod @ nat @ nat )] :
% 2.84/1.52 ( ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ P )
% 2.84/1.52 => ( ( val @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ ( augmen1604248148ngFlow @ capacity @ c @ f @ P ) )
% 2.84/1.52 = ( augmenting_resCap @ capacity @ c @ f @ P ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_82_augment__flow__value,axiom,
% 2.84/1.52 ! [F2: ( product_prod @ nat @ nat ) > capacity] :
% 2.84/1.52 ( ( flow @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ t @ F2 )
% 2.84/1.52 => ( ( val @ capacity @ c @ s @ ( augmenting_augment @ capacity @ c @ f @ F2 ) )
% 2.84/1.52 = ( plus_plus @ capacity @ ( val @ capacity @ c @ s @ f ) @ ( val @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ F2 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_83_isMaxFlow__alt,axiom,
% 2.84/1.52 ! [F3: ( product_prod @ nat @ nat ) > capacity] :
% 2.84/1.52 ( ( isMaxFlow @ capacity @ c @ s @ t @ F3 )
% 2.84/1.52 = ( ( nFlow @ capacity @ c @ s @ t @ F3 )
% 2.84/1.52 & ! [F5: ( product_prod @ nat @ nat ) > capacity] :
% 2.84/1.52 ( ( nFlow @ capacity @ c @ s @ t @ F5 )
% 2.84/1.52 => ( ord_less_eq @ capacity @ ( val @ capacity @ c @ s @ F5 ) @ ( val @ capacity @ c @ s @ F3 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_84_isMaxFlow__def,axiom,
% 2.84/1.52 ! [F3: ( product_prod @ nat @ nat ) > capacity] :
% 2.84/1.52 ( ( isMaxFlow @ capacity @ c @ s @ t @ F3 )
% 2.84/1.52 = ( ( flow @ capacity @ c @ s @ t @ F3 )
% 2.84/1.52 & ! [F5: ( product_prod @ nat @ nat ) > capacity] :
% 2.84/1.52 ( ( flow @ capacity @ c @ s @ t @ F5 )
% 2.84/1.52 => ( ord_less_eq @ capacity @ ( val @ capacity @ c @ s @ F5 ) @ ( val @ capacity @ c @ s @ F3 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_85_NPreflow_OaugFlow__resFlow,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,P: list @ ( product_prod @ nat @ nat )] :
% 2.84/1.52 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ P )
% 2.84/1.52 => ( flow @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ T @ ( augmen1604248148ngFlow @ Capacity @ C @ F3 @ P ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_86_bot__apply,axiom,
% 2.84/1.52 ! [C2: $tType,D: $tType] :
% 2.84/1.52 ( ( bot @ C2 )
% 2.84/1.52 => ( ( bot_bot @ ( D > C2 ) )
% 2.84/1.52 = ( ^ [X3: D] : ( bot_bot @ C2 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_87_maxFlow__minCut,axiom,
% 2.84/1.52 ! [F3: ( product_prod @ nat @ nat ) > capacity,K: set @ nat] :
% 2.84/1.52 ( ( isMaxFlow @ capacity @ c @ s @ t @ F3 )
% 2.84/1.52 => ( ( isMinCut @ capacity @ c @ s @ t @ K )
% 2.84/1.52 => ( ( val @ capacity @ c @ s @ F3 )
% 2.84/1.52 = ( cap @ capacity @ c @ K ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_88_order__refl,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( preorder @ A )
% 2.84/1.52 => ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_89_inf_Obounded__iff,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,B2: A,C: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) )
% 2.84/1.52 = ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.84/1.52 & ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_90_le__inf__iff,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [X2: A,Y: A,Z: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ Y @ Z ) )
% 2.84/1.52 = ( ( ord_less_eq @ A @ X2 @ Y )
% 2.84/1.52 & ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_91_bot_Oextremum,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order_bot @ A )
% 2.84/1.52 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_92_bot_Oextremum__unique,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order_bot @ A )
% 2.84/1.52 => ! [A2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 2.84/1.52 = ( A2
% 2.84/1.52 = ( bot_bot @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_93_bot_Oextremum__uniqueI,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order_bot @ A )
% 2.84/1.52 => ! [A2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 2.84/1.52 => ( A2
% 2.84/1.52 = ( bot_bot @ A ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_94_le__funD,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType] :
% 2.84/1.52 ( ( ord @ B )
% 2.84/1.52 => ! [F3: A > B,G2: A > B,X2: A] :
% 2.84/1.52 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
% 2.84/1.52 => ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_95_le__funE,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType] :
% 2.84/1.52 ( ( ord @ B )
% 2.84/1.52 => ! [F3: A > B,G2: A > B,X2: A] :
% 2.84/1.52 ( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
% 2.84/1.52 => ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_96_le__funI,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType] :
% 2.84/1.52 ( ( ord @ B )
% 2.84/1.52 => ! [F3: A > B,G2: A > B] :
% 2.84/1.52 ( ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) )
% 2.84/1.52 => ( ord_less_eq @ ( A > B ) @ F3 @ G2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_97_le__fun__def,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType] :
% 2.84/1.52 ( ( ord @ B )
% 2.84/1.52 => ( ( ord_less_eq @ ( A > B ) )
% 2.84/1.52 = ( ^ [F: A > B,G: A > B] :
% 2.84/1.52 ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_98_order__subst1,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType] :
% 2.84/1.52 ( ( ( order @ B )
% 2.84/1.52 & ( order @ A ) )
% 2.84/1.52 => ! [A2: A,F3: B > A,B2: B,C: B] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
% 2.84/1.52 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.84/1.52 => ( ! [X4: B,Y3: B] :
% 2.84/1.52 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 2.84/1.52 => ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
% 2.84/1.52 => ( ord_less_eq @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_99_order__subst2,axiom,
% 2.84/1.52 ! [A: $tType,C2: $tType] :
% 2.84/1.52 ( ( ( order @ C2 )
% 2.84/1.52 & ( order @ A ) )
% 2.84/1.52 => ! [A2: A,B2: A,F3: A > C2,C: C2] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.84/1.52 => ( ( ord_less_eq @ C2 @ ( F3 @ B2 ) @ C )
% 2.84/1.52 => ( ! [X4: A,Y3: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.84/1.52 => ( ord_less_eq @ C2 @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
% 2.84/1.52 => ( ord_less_eq @ C2 @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_100_ord__eq__le__subst,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType] :
% 2.84/1.52 ( ( ( ord @ B )
% 2.84/1.52 & ( ord @ A ) )
% 2.84/1.52 => ! [A2: A,F3: B > A,B2: B,C: B] :
% 2.84/1.52 ( ( A2
% 2.84/1.52 = ( F3 @ B2 ) )
% 2.84/1.52 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.84/1.52 => ( ! [X4: B,Y3: B] :
% 2.84/1.52 ( ( ord_less_eq @ B @ X4 @ Y3 )
% 2.84/1.52 => ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
% 2.84/1.52 => ( ord_less_eq @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_101_ord__le__eq__subst,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType] :
% 2.84/1.52 ( ( ( ord @ B )
% 2.84/1.52 & ( ord @ A ) )
% 2.84/1.52 => ! [A2: A,B2: A,F3: A > B,C: B] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.84/1.52 => ( ( ( F3 @ B2 )
% 2.84/1.52 = C )
% 2.84/1.52 => ( ! [X4: A,Y3: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.84/1.52 => ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
% 2.84/1.52 => ( ord_less_eq @ B @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_102_eq__iff,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order @ A )
% 2.84/1.52 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.84/1.52 = ( ^ [X3: A,Y2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.84/1.52 & ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_103_antisym,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order @ A )
% 2.84/1.52 => ! [X2: A,Y: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.84/1.52 => ( ( ord_less_eq @ A @ Y @ X2 )
% 2.84/1.52 => ( X2 = Y ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_104_linear,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linorder @ A )
% 2.84/1.52 => ! [X2: A,Y: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.84/1.52 | ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_105_eq__refl,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( preorder @ A )
% 2.84/1.52 => ! [X2: A,Y: A] :
% 2.84/1.52 ( ( X2 = Y )
% 2.84/1.52 => ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_106_le__cases,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linorder @ A )
% 2.84/1.52 => ! [X2: A,Y: A] :
% 2.84/1.52 ( ~ ( ord_less_eq @ A @ X2 @ Y )
% 2.84/1.52 => ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_107_order_Otrans,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order @ A )
% 2.84/1.52 => ! [A2: A,B2: A,C: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.84/1.52 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.84/1.52 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_108_le__cases3,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linorder @ A )
% 2.84/1.52 => ! [X2: A,Y: A,Z: A] :
% 2.84/1.52 ( ( ( ord_less_eq @ A @ X2 @ Y )
% 2.84/1.52 => ~ ( ord_less_eq @ A @ Y @ Z ) )
% 2.84/1.52 => ( ( ( ord_less_eq @ A @ Y @ X2 )
% 2.84/1.52 => ~ ( ord_less_eq @ A @ X2 @ Z ) )
% 2.84/1.52 => ( ( ( ord_less_eq @ A @ X2 @ Z )
% 2.84/1.52 => ~ ( ord_less_eq @ A @ Z @ Y ) )
% 2.84/1.52 => ( ( ( ord_less_eq @ A @ Z @ Y )
% 2.84/1.52 => ~ ( ord_less_eq @ A @ Y @ X2 ) )
% 2.84/1.52 => ( ( ( ord_less_eq @ A @ Y @ Z )
% 2.84/1.52 => ~ ( ord_less_eq @ A @ Z @ X2 ) )
% 2.84/1.52 => ~ ( ( ord_less_eq @ A @ Z @ X2 )
% 2.84/1.52 => ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_109_antisym__conv,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order @ A )
% 2.84/1.52 => ! [Y: A,X2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ Y @ X2 )
% 2.84/1.52 => ( ( ord_less_eq @ A @ X2 @ Y )
% 2.84/1.52 = ( X2 = Y ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_110_order__class_Oorder_Oeq__iff,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order @ A )
% 2.84/1.52 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.84/1.52 = ( ^ [A3: A,B3: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.84/1.52 & ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_111_ord__eq__le__trans,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( ord @ A )
% 2.84/1.52 => ! [A2: A,B2: A,C: A] :
% 2.84/1.52 ( ( A2 = B2 )
% 2.84/1.52 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.84/1.52 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_112_ord__le__eq__trans,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( ord @ A )
% 2.84/1.52 => ! [A2: A,B2: A,C: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.84/1.52 => ( ( B2 = C )
% 2.84/1.52 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_113_order__class_Oorder_Oantisym,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order @ A )
% 2.84/1.52 => ! [A2: A,B2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.84/1.52 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.84/1.52 => ( A2 = B2 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_114_NPreflow_OresCap_Ocong,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ( ( augmenting_resCap @ Capacity )
% 2.84/1.52 = ( augmenting_resCap @ Capacity ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_115_order__trans,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( preorder @ A )
% 2.84/1.52 => ! [X2: A,Y: A,Z: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.84/1.52 => ( ( ord_less_eq @ A @ Y @ Z )
% 2.84/1.52 => ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_116_dual__order_Orefl,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order @ A )
% 2.84/1.52 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_117_linorder__wlog,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linorder @ A )
% 2.84/1.52 => ! [P2: A > A > $o,A2: A,B2: A] :
% 2.84/1.52 ( ! [A5: A,B5: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.84/1.52 => ( P2 @ A5 @ B5 ) )
% 2.84/1.52 => ( ! [A5: A,B5: A] :
% 2.84/1.52 ( ( P2 @ B5 @ A5 )
% 2.84/1.52 => ( P2 @ A5 @ B5 ) )
% 2.84/1.52 => ( P2 @ A2 @ B2 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_118_dual__order_Otrans,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order @ A )
% 2.84/1.52 => ! [B2: A,A2: A,C: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.84/1.52 => ( ( ord_less_eq @ A @ C @ B2 )
% 2.84/1.52 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_119_dual__order_Oeq__iff,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order @ A )
% 2.84/1.52 => ( ( ^ [Y4: A,Z2: A] : ( Y4 = Z2 ) )
% 2.84/1.52 = ( ^ [A3: A,B3: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.84/1.52 & ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_120_dual__order_Oantisym,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order @ A )
% 2.84/1.52 => ! [B2: A,A2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.84/1.52 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.84/1.52 => ( A2 = B2 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_121_inf_OcoboundedI2,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [B2: A,C: A,A2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ B2 @ C )
% 2.84/1.52 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_122_inf_OcoboundedI1,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,C: A,B2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A2 @ C )
% 2.84/1.52 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_123_inf_Oabsorb__iff2,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ( ( ord_less_eq @ A )
% 2.84/1.52 = ( ^ [B3: A,A3: A] :
% 2.84/1.52 ( ( inf_inf @ A @ A3 @ B3 )
% 2.84/1.52 = B3 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_124_inf_Oabsorb__iff1,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ( ( ord_less_eq @ A )
% 2.84/1.52 = ( ^ [A3: A,B3: A] :
% 2.84/1.52 ( ( inf_inf @ A @ A3 @ B3 )
% 2.84/1.52 = A3 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_125_inf_Ocobounded2,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_126_inf_Ocobounded1,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ A2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_127_inf_Oorder__iff,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ( ( ord_less_eq @ A )
% 2.84/1.52 = ( ^ [A3: A,B3: A] :
% 2.84/1.52 ( A3
% 2.84/1.52 = ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_128_inf__greatest,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [X2: A,Y: A,Z: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.84/1.52 => ( ( ord_less_eq @ A @ X2 @ Z )
% 2.84/1.52 => ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ Y @ Z ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_129_inf_OboundedI,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,B2: A,C: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.84/1.52 => ( ( ord_less_eq @ A @ A2 @ C )
% 2.84/1.52 => ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_130_inf_OboundedE,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,B2: A,C: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) )
% 2.84/1.52 => ~ ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.84/1.52 => ~ ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_131_inf__absorb2,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [Y: A,X2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ Y @ X2 )
% 2.84/1.52 => ( ( inf_inf @ A @ X2 @ Y )
% 2.84/1.52 = Y ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_132_inf__absorb1,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [X2: A,Y: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.84/1.52 => ( ( inf_inf @ A @ X2 @ Y )
% 2.84/1.52 = X2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_133_inf_Oabsorb2,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [B2: A,A2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.84/1.52 => ( ( inf_inf @ A @ A2 @ B2 )
% 2.84/1.52 = B2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_134_inf_Oabsorb1,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,B2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.84/1.52 => ( ( inf_inf @ A @ A2 @ B2 )
% 2.84/1.52 = A2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_135_le__iff__inf,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ( ( ord_less_eq @ A )
% 2.84/1.52 = ( ^ [X3: A,Y2: A] :
% 2.84/1.52 ( ( inf_inf @ A @ X3 @ Y2 )
% 2.84/1.52 = X3 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_136_inf__unique,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [F3: A > A > A,X2: A,Y: A] :
% 2.84/1.52 ( ! [X4: A,Y3: A] : ( ord_less_eq @ A @ ( F3 @ X4 @ Y3 ) @ X4 )
% 2.84/1.52 => ( ! [X4: A,Y3: A] : ( ord_less_eq @ A @ ( F3 @ X4 @ Y3 ) @ Y3 )
% 2.84/1.52 => ( ! [X4: A,Y3: A,Z3: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.84/1.52 => ( ( ord_less_eq @ A @ X4 @ Z3 )
% 2.84/1.52 => ( ord_less_eq @ A @ X4 @ ( F3 @ Y3 @ Z3 ) ) ) )
% 2.84/1.52 => ( ( inf_inf @ A @ X2 @ Y )
% 2.84/1.52 = ( F3 @ X2 @ Y ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_137_inf_OorderI,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,B2: A] :
% 2.84/1.52 ( ( A2
% 2.84/1.52 = ( inf_inf @ A @ A2 @ B2 ) )
% 2.84/1.52 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_138_inf_OorderE,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,B2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.84/1.52 => ( A2
% 2.84/1.52 = ( inf_inf @ A @ A2 @ B2 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_139_le__infI2,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [B2: A,X2: A,A2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ B2 @ X2 )
% 2.84/1.52 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_140_le__infI1,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,X2: A,B2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A2 @ X2 )
% 2.84/1.52 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_141_inf__mono,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,C: A,B2: A,D2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ A2 @ C )
% 2.84/1.52 => ( ( ord_less_eq @ A @ B2 @ D2 )
% 2.84/1.52 => ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ ( inf_inf @ A @ C @ D2 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_142_le__infI,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [X2: A,A2: A,B2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ X2 @ A2 )
% 2.84/1.52 => ( ( ord_less_eq @ A @ X2 @ B2 )
% 2.84/1.52 => ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_143_le__infE,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [X2: A,A2: A,B2: A] :
% 2.84/1.52 ( ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ A2 @ B2 ) )
% 2.84/1.52 => ~ ( ( ord_less_eq @ A @ X2 @ A2 )
% 2.84/1.52 => ~ ( ord_less_eq @ A @ X2 @ B2 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_144_inf__le2,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [X2: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y ) @ Y ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_145_inf__le1,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [X2: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y ) @ X2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_146_inf__sup__ord_I1_J,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( lattice @ A )
% 2.84/1.52 => ! [X2: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y ) @ X2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_147_inf__sup__ord_I2_J,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( lattice @ A )
% 2.84/1.52 => ! [X2: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y ) @ Y ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_148_NPreflow_OisAugmentingPath_Ocong,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ( ( augmen2028041557ngPath @ Capacity )
% 2.84/1.52 = ( augmen2028041557ngPath @ Capacity ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_149_NPreflow_OaugmentingFlow_Ocong,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ( ( augmen1604248148ngFlow @ Capacity )
% 2.84/1.52 = ( augmen1604248148ngFlow @ Capacity ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_150_NPreflow_OaugFlow__val,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,P: list @ ( product_prod @ nat @ nat )] :
% 2.84/1.52 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ P )
% 2.84/1.52 => ( ( val @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ ( augmen1604248148ngFlow @ Capacity @ C @ F3 @ P ) )
% 2.84/1.52 = ( augmenting_resCap @ Capacity @ C @ F3 @ P ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_151_bot__fun__def,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType] :
% 2.84/1.52 ( ( bot @ B )
% 2.84/1.52 => ( ( bot_bot @ ( A > B ) )
% 2.84/1.52 = ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_152_NPreflow_OisAugmentingPath__def,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,P: list @ ( product_prod @ nat @ nat )] :
% 2.84/1.52 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ P )
% 2.84/1.52 = ( isSimplePath @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ P @ T ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_153_ford__fulkerson_I2_J,axiom,
% 2.84/1.52 ( ( ~ ? [X: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ X ) )
% 2.84/1.52 = ( ? [K2: set @ nat] :
% 2.84/1.52 ( ( nCut @ capacity @ c @ s @ t @ K2 )
% 2.84/1.52 & ( ( val @ capacity @ c @ s @ f )
% 2.84/1.52 = ( cap @ capacity @ c @ K2 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_154_fofu__III__I,axiom,
% 2.84/1.52 ( ? [K3: set @ nat] :
% 2.84/1.52 ( ( nCut @ capacity @ c @ s @ t @ K3 )
% 2.84/1.52 & ( ( val @ capacity @ c @ s @ f )
% 2.84/1.52 = ( cap @ capacity @ c @ K3 ) ) )
% 2.84/1.52 => ( isMaxFlow @ capacity @ c @ s @ t @ f ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_155_fofu__II__III,axiom,
% 2.84/1.52 ( ~ ? [X_12: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ X_12 )
% 2.84/1.52 => ? [K4: set @ nat] :
% 2.84/1.52 ( ( nCut @ capacity @ c @ s @ t @ K4 )
% 2.84/1.52 & ( ( val @ capacity @ c @ s @ f )
% 2.84/1.52 = ( cap @ capacity @ c @ K4 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_156_NFlow_Oaugment__flow__value,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,F2: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ( flow @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ T @ F2 )
% 2.84/1.52 => ( ( val @ Capacity @ C @ S @ ( augmenting_augment @ Capacity @ C @ F3 @ F2 ) )
% 2.84/1.52 = ( plus_plus @ Capacity @ ( val @ Capacity @ C @ S @ F3 ) @ ( val @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ F2 ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_157_Network_OmaxFlow__minCut,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,K: set @ nat] :
% 2.84/1.52 ( ( network @ Capacity @ C @ S @ T )
% 2.84/1.52 => ( ( isMaxFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ( isMinCut @ Capacity @ C @ S @ T @ K )
% 2.84/1.52 => ( ( val @ Capacity @ C @ S @ F3 )
% 2.84/1.52 = ( cap @ Capacity @ C @ K ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_158_Network_OisMaxFlow__alt,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( network @ Capacity @ C @ S @ T )
% 2.84/1.52 => ( ( isMaxFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 = ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 & ! [F5: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nFlow @ Capacity @ C @ S @ T @ F5 )
% 2.84/1.52 => ( ord_less_eq @ Capacity @ ( val @ Capacity @ C @ S @ F5 ) @ ( val @ Capacity @ C @ S @ F3 ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_159_converse__mono,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ A ),S: set @ ( product_prod @ B @ A )] :
% 2.84/1.52 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( converse @ B @ A @ R ) @ ( converse @ B @ A @ S ) )
% 2.84/1.52 = ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ R @ S ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_160_NCut_Oaxioms_I1_J,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,K: set @ nat] :
% 2.84/1.52 ( ( nCut @ Capacity @ C @ S @ T @ K )
% 2.84/1.52 => ( network @ Capacity @ C @ S @ T ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_161_NCut_Ot__ni__cut,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,K: set @ nat] :
% 2.84/1.52 ( ( nCut @ Capacity @ C @ S @ T @ K )
% 2.84/1.52 => ~ ( member @ nat @ T @ K ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_162_NCut_Os__in__cut,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,K: set @ nat] :
% 2.84/1.52 ( ( nCut @ Capacity @ C @ S @ T @ K )
% 2.84/1.52 => ( member @ nat @ S @ K ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_163_converse__subset__swap,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ A )] :
% 2.84/1.52 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ ( converse @ B @ A @ S ) )
% 2.84/1.52 = ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( converse @ A @ B @ R ) @ S ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_164_isMinCut__def,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ( ( isMinCut @ A )
% 2.84/1.52 = ( ^ [C3: ( product_prod @ nat @ nat ) > A,S2: nat,T2: nat,K2: set @ nat] :
% 2.84/1.52 ( ( nCut @ A @ C3 @ S2 @ T2 @ K2 )
% 2.84/1.52 & ! [K5: set @ nat] :
% 2.84/1.52 ( ( nCut @ A @ C3 @ S2 @ T2 @ K5 )
% 2.84/1.52 => ( ord_less_eq @ A @ ( cap @ A @ C3 @ K2 ) @ ( cap @ A @ C3 @ K5 ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_165_Network_Ot__not__s,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat] :
% 2.84/1.52 ( ( network @ Capacity @ C @ S @ T )
% 2.84/1.52 => ( T != S ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_166_Network_Os__not__t,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat] :
% 2.84/1.52 ( ( network @ Capacity @ C @ S @ T )
% 2.84/1.52 => ( S != T ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_167_Flow_Oval_Ocong,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ( ( val @ Capacity )
% 2.84/1.52 = ( val @ Capacity ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_168_Network_OisMaxFlow_Ocong,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ( ( isMaxFlow @ Capacity )
% 2.84/1.52 = ( isMaxFlow @ Capacity ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_169_NFlow_Oaugment_Ocong,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ( ( augmenting_augment @ Capacity )
% 2.84/1.52 = ( augmenting_augment @ Capacity ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_170_NCut_Ocap_Ocong,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ( ( cap @ Capacity )
% 2.84/1.52 = ( cap @ Capacity ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_171_NFlow_Ofofu__II__III,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ~ ? [X_12: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ X_12 )
% 2.84/1.52 => ? [K4: set @ nat] :
% 2.84/1.52 ( ( nCut @ Capacity @ C @ S @ T @ K4 )
% 2.84/1.52 & ( ( val @ Capacity @ C @ S @ F3 )
% 2.84/1.52 = ( cap @ Capacity @ C @ K4 ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_172_NFlow_Ofofu__III__I,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ? [K3: set @ nat] :
% 2.84/1.52 ( ( nCut @ Capacity @ C @ S @ T @ K3 )
% 2.84/1.52 & ( ( val @ Capacity @ C @ S @ F3 )
% 2.84/1.52 = ( cap @ Capacity @ C @ K3 ) ) )
% 2.84/1.52 => ( isMaxFlow @ Capacity @ C @ S @ T @ F3 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_173_Network_Ois__max__flow__val_Ocong,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ( ( is_max_flow_val @ Capacity )
% 2.84/1.52 = ( is_max_flow_val @ Capacity ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_174_NFlow_Oford__fulkerson_I2_J,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ( ~ ? [X: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ X ) )
% 2.84/1.52 = ( ? [K2: set @ nat] :
% 2.84/1.52 ( ( nCut @ Capacity @ C @ S @ T @ K2 )
% 2.84/1.52 & ( ( val @ Capacity @ C @ S @ F3 )
% 2.84/1.52 = ( cap @ Capacity @ C @ K2 ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_175_NPreflow_Oaxioms_I1_J,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( network @ Capacity @ C @ S @ T ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_176_NFlow_Oaxioms_I1_J,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( nPreflow @ Capacity @ C @ S @ T @ F3 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_177_NFlow_Oaxioms_I2_J,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( flow @ Capacity @ C @ S @ T @ F3 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_178_NPreflow_Oaxioms_I2_J,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( preflow @ Capacity @ C @ S @ T @ F3 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_179_Flow_Oaxioms_I1_J,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( flow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( preflow @ Capacity @ C @ S @ T @ F3 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_180_Network_Omax__flow__val__unique,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,Fv1: Capacity,Fv2: Capacity] :
% 2.84/1.52 ( ( network @ Capacity @ C @ S @ T )
% 2.84/1.52 => ( ( is_max_flow_val @ Capacity @ C @ S @ T @ Fv1 )
% 2.84/1.52 => ( ( is_max_flow_val @ Capacity @ C @ S @ T @ Fv2 )
% 2.84/1.52 => ( Fv1 = Fv2 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_181_Finite__Preflow_Oaxioms_I2_J,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( finite_Preflow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( finite_Graph @ Capacity @ C ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_182_Finite__Preflow_Oaxioms_I1_J,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( finite_Preflow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( preflow @ Capacity @ C @ S @ T @ F3 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_183_NFlow_Ointro,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ( flow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( nFlow @ Capacity @ C @ S @ T @ F3 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_184_NFlow__def,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ( ( nFlow @ Capacity )
% 2.84/1.52 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,S2: nat,T2: nat,F: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nPreflow @ Capacity @ C3 @ S2 @ T2 @ F )
% 2.84/1.52 & ( flow @ Capacity @ C3 @ S2 @ T2 @ F ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_185_NPreflow_Ointro,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( network @ Capacity @ C @ S @ T )
% 2.84/1.52 => ( ( preflow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( nPreflow @ Capacity @ C @ S @ T @ F3 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_186_NPreflow__def,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ( ( nPreflow @ Capacity )
% 2.84/1.52 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,S2: nat,T2: nat,F: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( network @ Capacity @ C3 @ S2 @ T2 )
% 2.84/1.52 & ( preflow @ Capacity @ C3 @ S2 @ T2 @ F ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_187_NFlow_Ofofu__I__II,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ( isMaxFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ~ ? [X_1: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ X_1 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_188_NFlow_OnoAugPath__iff__maxFlow,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ( ~ ? [X: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ X ) )
% 2.84/1.52 = ( isMaxFlow @ Capacity @ C @ S @ T @ F3 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_189_Finite__Preflow_Ointro,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( preflow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ( finite_Graph @ Capacity @ C )
% 2.84/1.52 => ( finite_Preflow @ Capacity @ C @ S @ T @ F3 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_190_Finite__Preflow__def,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ( ( finite_Preflow @ Capacity )
% 2.84/1.52 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,S2: nat,T2: nat,F: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( preflow @ Capacity @ C3 @ S2 @ T2 @ F )
% 2.84/1.52 & ( finite_Graph @ Capacity @ C3 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_191_NFlow_Oaugment__flow__presv,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,F2: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ( flow @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ T @ F2 )
% 2.84/1.52 => ( flow @ Capacity @ C @ S @ T @ ( augmenting_augment @ Capacity @ C @ F3 @ F2 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_192_NFlow_Oford__fulkerson_I1_J,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ( isMaxFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 = ( ~ ? [X: list @ ( product_prod @ nat @ nat )] : ( augmen2028041557ngPath @ Capacity @ C @ S @ T @ F3 @ X ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_193_Network_Ois__max__flow__val__def,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,Fv: Capacity] :
% 2.84/1.52 ( ( network @ Capacity @ C @ S @ T )
% 2.84/1.52 => ( ( is_max_flow_val @ Capacity @ C @ S @ T @ Fv )
% 2.84/1.52 = ( ? [F: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( isMaxFlow @ Capacity @ C @ S @ T @ F )
% 2.84/1.52 & ( Fv
% 2.84/1.52 = ( val @ Capacity @ C @ S @ F ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_194_Network_OisMaxFlow__def,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( network @ Capacity @ C @ S @ T )
% 2.84/1.52 => ( ( isMaxFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 = ( ( flow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 & ! [F5: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( flow @ Capacity @ C @ S @ T @ F5 )
% 2.84/1.52 => ( ord_less_eq @ Capacity @ ( val @ Capacity @ C @ S @ F5 ) @ ( val @ Capacity @ C @ S @ F3 ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_195_cf_OisSPath__no__selfloop,axiom,
% 2.84/1.52 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat] :
% 2.84/1.52 ( ( isSimplePath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ U @ P @ V )
% 2.84/1.52 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_196_cf_OisSPath__sg__incoming,axiom,
% 2.84/1.52 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,U2: nat] :
% 2.84/1.52 ( ( isSimplePath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ U @ P @ V )
% 2.84/1.52 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.84/1.52 => ( ( U1 != U2 )
% 2.84/1.52 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U2 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_197_cf_OisSPath__sg__outgoing,axiom,
% 2.84/1.52 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,V2: nat] :
% 2.84/1.52 ( ( isSimplePath @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ U @ P @ V )
% 2.84/1.52 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.84/1.52 => ( ( V1 != V2 )
% 2.84/1.52 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V2 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_198_isSPath__sg__outgoing,axiom,
% 2.84/1.52 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,V2: nat] :
% 2.84/1.52 ( ( isSimplePath @ capacity @ c @ U @ P @ V )
% 2.84/1.52 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.84/1.52 => ( ( V1 != V2 )
% 2.84/1.52 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V2 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_199_isSPath__sg__incoming,axiom,
% 2.84/1.52 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,U2: nat] :
% 2.84/1.52 ( ( isSimplePath @ capacity @ c @ U @ P @ V )
% 2.84/1.52 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.84/1.52 => ( ( U1 != U2 )
% 2.84/1.52 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U2 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_200_isSPath__no__selfloop,axiom,
% 2.84/1.52 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat] :
% 2.84/1.52 ( ( isSimplePath @ capacity @ c @ U @ P @ V )
% 2.84/1.52 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_201_converse__iff,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,A2: A,B2: B,R: set @ ( product_prod @ B @ A )] :
% 2.84/1.52 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( converse @ B @ A @ R ) )
% 2.84/1.52 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A2 ) @ R ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_202_subrelI,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.84/1.52 ( ! [X4: A,Y3: B] :
% 2.84/1.52 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R )
% 2.84/1.52 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ S ) )
% 2.84/1.52 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_203_converseD,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,A2: A,B2: B,R: set @ ( product_prod @ B @ A )] :
% 2.84/1.52 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( converse @ B @ A @ R ) )
% 2.84/1.52 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A2 ) @ R ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_204_converseE,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R: set @ ( product_prod @ B @ A )] :
% 2.84/1.52 ( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R ) )
% 2.84/1.52 => ~ ! [X4: B,Y3: A] :
% 2.84/1.52 ( ( Yx
% 2.84/1.52 = ( product_Pair @ A @ B @ Y3 @ X4 ) )
% 2.84/1.52 => ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y3 ) @ R ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_205_converse_Ocases,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType,A1: B,A22: A,R: set @ ( product_prod @ A @ B )] :
% 2.84/1.52 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R ) )
% 2.84/1.52 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A22 @ A1 ) @ R ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_206_converse_Osimps,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType,A1: B,A22: A,R: set @ ( product_prod @ A @ B )] :
% 2.84/1.52 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R ) )
% 2.84/1.52 = ( ? [A3: A,B3: B] :
% 2.84/1.52 ( ( A1 = B3 )
% 2.84/1.52 & ( A22 = A3 )
% 2.84/1.52 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_207_converse_Ointros,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType,A2: A,B2: B,R: set @ ( product_prod @ A @ B )] :
% 2.84/1.52 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R )
% 2.84/1.52 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A2 ) @ ( converse @ A @ B @ R ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_208_converse_Oinducts,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType,X1: B,X22: A,R: set @ ( product_prod @ A @ B ),P2: B > A > $o] :
% 2.84/1.52 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X1 @ X22 ) @ ( converse @ A @ B @ R ) )
% 2.84/1.52 => ( ! [A5: A,B5: B] :
% 2.84/1.52 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R )
% 2.84/1.52 => ( P2 @ B5 @ A5 ) )
% 2.84/1.52 => ( P2 @ X1 @ X22 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_209_Graph_OisSPath__sg__outgoing,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,V2: nat] :
% 2.84/1.52 ( ( isSimplePath @ Capacity @ C @ U @ P @ V )
% 2.84/1.52 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.84/1.52 => ( ( V1 != V2 )
% 2.84/1.52 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V2 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_210_Graph_OisSPath__sg__incoming,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,U2: nat] :
% 2.84/1.52 ( ( isSimplePath @ Capacity @ C @ U @ P @ V )
% 2.84/1.52 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.84/1.52 => ( ( U1 != U2 )
% 2.84/1.52 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U2 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_211_Graph_OisSPath__no__selfloop,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat] :
% 2.84/1.52 ( ( isSimplePath @ Capacity @ C @ U @ P @ V )
% 2.84/1.52 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_212_augment__flow__presv__cap,axiom,
% 2.84/1.52 ! [F2: ( product_prod @ nat @ nat ) > capacity,U: nat,V: nat] :
% 2.84/1.52 ( ( flow @ capacity @ ( residu1993642540lGraph @ capacity @ c @ f ) @ s @ t @ F2 )
% 2.84/1.52 => ( ( ord_less_eq @ capacity @ ( zero_zero @ capacity ) @ ( augmenting_augment @ capacity @ c @ f @ F2 @ ( product_Pair @ nat @ nat @ U @ V ) ) )
% 2.84/1.52 & ( ord_less_eq @ capacity @ ( augmenting_augment @ capacity @ c @ f @ F2 @ ( product_Pair @ nat @ nat @ U @ V ) ) @ ( c @ ( product_Pair @ nat @ nat @ U @ V ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_213_swap__simp,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,X2: B,Y: A] :
% 2.84/1.52 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X2 @ Y ) )
% 2.84/1.52 = ( product_Pair @ A @ B @ Y @ X2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_214_f__non__negative,axiom,
% 2.84/1.52 ! [E: product_prod @ nat @ nat] : ( ord_less_eq @ capacity @ ( zero_zero @ capacity ) @ ( f @ E ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_215_old_Oprod_Oinject,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B6: B] :
% 2.84/1.52 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.84/1.52 = ( product_Pair @ A @ B @ A6 @ B6 ) )
% 2.84/1.52 = ( ( A2 = A6 )
% 2.84/1.52 & ( B2 = B6 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_216_prod_Oinject,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 2.84/1.52 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 2.84/1.52 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 2.84/1.52 = ( ( X1 = Y1 )
% 2.84/1.52 & ( X22 = Y22 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_217_cap__non__negative,axiom,
% 2.84/1.52 ! [U3: nat,V3: nat] : ( ord_less_eq @ capacity @ ( zero_zero @ capacity ) @ ( c @ ( product_Pair @ nat @ nat @ U3 @ V3 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_218_capacity__const,axiom,
% 2.84/1.52 ! [E2: product_prod @ nat @ nat] :
% 2.84/1.52 ( ( ord_less_eq @ capacity @ ( zero_zero @ capacity ) @ ( f @ E2 ) )
% 2.84/1.52 & ( ord_less_eq @ capacity @ ( f @ E2 ) @ ( c @ E2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_219_swap__swap,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 2.84/1.52 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
% 2.84/1.52 = P ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_220_resE__nonNegative,axiom,
% 2.84/1.52 ! [E: product_prod @ nat @ nat] : ( ord_less_eq @ capacity @ ( zero_zero @ capacity ) @ ( residu1993642540lGraph @ capacity @ c @ f @ E ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_221_RPreGraph_OresE__nonNegative,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat,Cf: ( product_prod @ nat @ nat ) > A,E: product_prod @ nat @ nat] :
% 2.84/1.52 ( ( residual_RPreGraph @ A @ C @ S @ T @ Cf )
% 2.84/1.52 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( Cf @ E ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_222_Preflow_Ocapacity__const,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.84/1.52 ( ( preflow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ! [E2: product_prod @ nat @ nat] :
% 2.84/1.52 ( ( ord_less_eq @ Capacity @ ( zero_zero @ Capacity ) @ ( F3 @ E2 ) )
% 2.84/1.52 & ( ord_less_eq @ Capacity @ ( F3 @ E2 ) @ ( C @ E2 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_223_Preflow_Of__non__negative,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,E: product_prod @ nat @ nat] :
% 2.84/1.52 ( ( preflow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ord_less_eq @ Capacity @ ( zero_zero @ Capacity ) @ ( F3 @ E ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_224_old_Oprod_Oinducts,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.84/1.52 ( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.84/1.52 => ( P2 @ Prod ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_225_old_Oprod_Oexhaust,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 2.84/1.52 ~ ! [A5: A,B5: B] :
% 2.84/1.52 ( Y
% 2.84/1.52 != ( product_Pair @ A @ B @ A5 @ B5 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_226_prod__induct7,axiom,
% 2.84/1.52 ! [G3: $tType,F6: $tType,E3: $tType,D: $tType,C2: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) ) )] :
% 2.84/1.52 ( ! [A5: A,B5: B,C4: C2,D3: D,E4: E3,F4: F6,G4: G3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) @ D3 @ ( product_Pair @ E3 @ ( product_prod @ F6 @ G3 ) @ E4 @ ( product_Pair @ F6 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
% 2.84/1.52 => ( P2 @ X2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_227_prod__induct6,axiom,
% 2.84/1.52 ! [F6: $tType,E3: $tType,D: $tType,C2: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) ) )] :
% 2.84/1.52 ( ! [A5: A,B5: B,C4: C2,D3: D,E4: E3,F4: F6] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) ) @ B5 @ ( product_Pair @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E3 @ F6 ) @ D3 @ ( product_Pair @ E3 @ F6 @ E4 @ F4 ) ) ) ) ) )
% 2.84/1.52 => ( P2 @ X2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_228_prod__induct5,axiom,
% 2.84/1.52 ! [E3: $tType,D: $tType,C2: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ E3 ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ E3 ) ) )] :
% 2.84/1.52 ( ! [A5: A,B5: B,C4: C2,D3: D,E4: E3] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ E3 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D @ E3 ) ) @ B5 @ ( product_Pair @ C2 @ ( product_prod @ D @ E3 ) @ C4 @ ( product_Pair @ D @ E3 @ D3 @ E4 ) ) ) ) )
% 2.84/1.52 => ( P2 @ X2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_229_prod__induct4,axiom,
% 2.84/1.52 ! [D: $tType,C2: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ D ) )] :
% 2.84/1.52 ( ! [A5: A,B5: B,C4: C2,D3: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C2 @ D ) @ B5 @ ( product_Pair @ C2 @ D @ C4 @ D3 ) ) ) )
% 2.84/1.52 => ( P2 @ X2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_230_prod__induct3,axiom,
% 2.84/1.52 ! [C2: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C2 ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C2 )] :
% 2.84/1.52 ( ! [A5: A,B5: B,C4: C2] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C2 ) @ A5 @ ( product_Pair @ B @ C2 @ B5 @ C4 ) ) )
% 2.84/1.52 => ( P2 @ X2 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_231_prod__cases7,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,C2: $tType,D: $tType,E3: $tType,F6: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) ) )] :
% 2.84/1.52 ~ ! [A5: A,B5: B,C4: C2,D3: D,E4: E3,F4: F6,G4: G3] :
% 2.84/1.52 ( Y
% 2.84/1.52 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) ) @ B5 @ ( product_Pair @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E3 @ ( product_prod @ F6 @ G3 ) ) @ D3 @ ( product_Pair @ E3 @ ( product_prod @ F6 @ G3 ) @ E4 @ ( product_Pair @ F6 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_232_prod__cases6,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,C2: $tType,D: $tType,E3: $tType,F6: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) ) )] :
% 2.84/1.52 ~ ! [A5: A,B5: B,C4: C2,D3: D,E4: E3,F4: F6] :
% 2.84/1.52 ( Y
% 2.84/1.52 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) ) @ B5 @ ( product_Pair @ C2 @ ( product_prod @ D @ ( product_prod @ E3 @ F6 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E3 @ F6 ) @ D3 @ ( product_Pair @ E3 @ F6 @ E4 @ F4 ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_233_prod__cases5,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,C2: $tType,D: $tType,E3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ E3 ) ) )] :
% 2.84/1.52 ~ ! [A5: A,B5: B,C4: C2,D3: D,E4: E3] :
% 2.84/1.52 ( Y
% 2.84/1.52 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D @ E3 ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D @ E3 ) ) @ B5 @ ( product_Pair @ C2 @ ( product_prod @ D @ E3 ) @ C4 @ ( product_Pair @ D @ E3 @ D3 @ E4 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_234_prod__cases4,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,C2: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ D ) )] :
% 2.84/1.52 ~ ! [A5: A,B5: B,C4: C2,D3: D] :
% 2.84/1.52 ( Y
% 2.84/1.52 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C2 @ D ) @ B5 @ ( product_Pair @ C2 @ D @ C4 @ D3 ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_235_prod__cases3,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,C2: $tType,Y: product_prod @ A @ ( product_prod @ B @ C2 )] :
% 2.84/1.52 ~ ! [A5: A,B5: B,C4: C2] :
% 2.84/1.52 ( Y
% 2.84/1.52 != ( product_Pair @ A @ ( product_prod @ B @ C2 ) @ A5 @ ( product_Pair @ B @ C2 @ B5 @ C4 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_236_Pair__inject,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B6: B] :
% 2.84/1.52 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.84/1.52 = ( product_Pair @ A @ B @ A6 @ B6 ) )
% 2.84/1.52 => ~ ( ( A2 = A6 )
% 2.84/1.52 => ( B2 != B6 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_237_prod__cases,axiom,
% 2.84/1.52 ! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 2.84/1.52 ( ! [A5: A,B5: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.84/1.52 => ( P2 @ P ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_238_surj__pair,axiom,
% 2.84/1.52 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 2.84/1.52 ? [X4: A,Y3: B] :
% 2.84/1.52 ( P
% 2.84/1.52 = ( product_Pair @ A @ B @ X4 @ Y3 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_239_Network_Ocap__non__negative,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat] :
% 2.84/1.52 ( ( network @ Capacity @ C @ S @ T )
% 2.84/1.52 => ! [U3: nat,V3: nat] : ( ord_less_eq @ Capacity @ ( zero_zero @ Capacity ) @ ( C @ ( product_Pair @ nat @ nat @ U3 @ V3 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_240_NPreflow_OresE__nonNegative,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,E: product_prod @ nat @ nat] :
% 2.84/1.52 ( ( nPreflow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ord_less_eq @ Capacity @ ( zero_zero @ Capacity ) @ ( residu1993642540lGraph @ Capacity @ C @ F3 @ E ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_241_NFlow_Oaugment__flow__presv__cap,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T: nat,F3: ( product_prod @ nat @ nat ) > Capacity,F2: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat] :
% 2.84/1.52 ( ( nFlow @ Capacity @ C @ S @ T @ F3 )
% 2.84/1.52 => ( ( flow @ Capacity @ ( residu1993642540lGraph @ Capacity @ C @ F3 ) @ S @ T @ F2 )
% 2.84/1.52 => ( ( ord_less_eq @ Capacity @ ( zero_zero @ Capacity ) @ ( augmenting_augment @ Capacity @ C @ F3 @ F2 @ ( product_Pair @ nat @ nat @ U @ V ) ) )
% 2.84/1.52 & ( ord_less_eq @ Capacity @ ( augmenting_augment @ Capacity @ C @ F3 @ F2 @ ( product_Pair @ nat @ nat @ U @ V ) ) @ ( C @ ( product_Pair @ nat @ nat @ U @ V ) ) ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_242_excess__s__non__pos,axiom,
% 2.84/1.52 ord_less_eq @ capacity @ ( excess @ capacity @ c @ f @ s ) @ ( zero_zero @ capacity ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_243_resCap__gzero,axiom,
% 2.84/1.52 ! [P: list @ ( product_prod @ nat @ nat )] :
% 2.84/1.52 ( ( augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ P )
% 2.84/1.52 => ( ord_less @ capacity @ ( zero_zero @ capacity ) @ ( augmenting_resCap @ capacity @ c @ f @ P ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_244_Network_Oexcess_Ocong,axiom,
% 2.84/1.52 ! [Capacity: $tType] :
% 2.84/1.52 ( ( linordered_idom @ Capacity )
% 2.84/1.52 => ( ( excess @ Capacity )
% 2.84/1.52 = ( excess @ Capacity ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_245_bot_Onot__eq__extremum,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order_bot @ A )
% 2.84/1.52 => ! [A2: A] :
% 2.84/1.52 ( ( A2
% 2.84/1.52 != ( bot_bot @ A ) )
% 2.84/1.52 = ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_246_bot_Oextremum__strict,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order_bot @ A )
% 2.84/1.52 => ! [A2: A] :
% 2.84/1.52 ~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_247_less__infI1,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,X2: A,B2: A] :
% 2.84/1.52 ( ( ord_less @ A @ A2 @ X2 )
% 2.84/1.52 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_248_less__infI2,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [B2: A,X2: A,A2: A] :
% 2.84/1.52 ( ( ord_less @ A @ B2 @ X2 )
% 2.84/1.52 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_249_inf_Ostrict__boundedE,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,B2: A,C: A] :
% 2.84/1.52 ( ( ord_less @ A @ A2 @ ( inf_inf @ A @ B2 @ C ) )
% 2.84/1.52 => ~ ( ( ord_less @ A @ A2 @ B2 )
% 2.84/1.52 => ~ ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_250_inf_Ostrict__order__iff,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ( ( ord_less @ A )
% 2.84/1.52 = ( ^ [A3: A,B3: A] :
% 2.84/1.52 ( ( A3
% 2.84/1.52 = ( inf_inf @ A @ A3 @ B3 ) )
% 2.84/1.52 & ( A3 != B3 ) ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_251_inf_Ostrict__coboundedI1,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [A2: A,C: A,B2: A] :
% 2.84/1.52 ( ( ord_less @ A @ A2 @ C )
% 2.84/1.52 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_252_inf_Ostrict__coboundedI2,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A )
% 2.84/1.52 => ! [B2: A,C: A,A2: A] :
% 2.84/1.52 ( ( ord_less @ A @ B2 @ C )
% 2.84/1.52 => ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_253_dual__order_Ostrict__implies__not__eq,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order @ A )
% 2.84/1.52 => ! [B2: A,A2: A] :
% 2.84/1.52 ( ( ord_less @ A @ B2 @ A2 )
% 2.84/1.52 => ( A2 != B2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_254_order_Ostrict__implies__not__eq,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( order @ A )
% 2.84/1.52 => ! [A2: A,B2: A] :
% 2.84/1.52 ( ( ord_less @ A @ A2 @ B2 )
% 2.84/1.52 => ( A2 != B2 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(fact_255_not__less__iff__gr__or__eq,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linorder @ A )
% 2.84/1.52 => ! [X2: A,Y: A] :
% 2.84/1.52 ( ( ~ ( ord_less @ A @ X2 @ Y ) )
% 2.84/1.52 = ( ( ord_less @ A @ Y @ X2 )
% 2.84/1.52 | ( X2 = Y ) ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(subcl_Rings_Olinordered__idom___HOL_Otype,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ( type @ A ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(subcl_Rings_Olinordered__idom___Orderings_Oord,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ( ord @ A ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(subcl_Rings_Olinordered__idom___Orderings_Oorder,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ( order @ A ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(subcl_Rings_Olinordered__idom___Orderings_Olinorder,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ( linorder @ A ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(subcl_Rings_Olinordered__idom___Orderings_Opreorder,axiom,
% 2.84/1.52 ! [A: $tType] :
% 2.84/1.52 ( ( linordered_idom @ A )
% 2.84/1.52 => ( preorder @ A ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 2.84/1.52 bounded_lattice @ $o ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 2.84/1.52 ! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 2.84/1.52 ! [A7: $tType,A8: $tType] :
% 2.84/1.52 ( ( bounded_lattice @ A8 )
% 2.84/1.52 => ( bounded_lattice @ ( A7 > A8 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
% 2.84/1.52 ! [A7: $tType,A8: $tType] :
% 2.84/1.52 ( ( bounded_lattice @ A8 )
% 2.84/1.52 => ( bounded_lattice_bot @ ( A7 > A8 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 2.84/1.52 ! [A7: $tType,A8: $tType] :
% 2.84/1.52 ( ( semilattice_inf @ A8 )
% 2.84/1.52 => ( semilattice_inf @ ( A7 > A8 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.84/1.52 ! [A7: $tType,A8: $tType] :
% 2.84/1.52 ( ( order_bot @ A8 )
% 2.84/1.52 => ( order_bot @ ( A7 > A8 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.84/1.52 ! [A7: $tType,A8: $tType] :
% 2.84/1.52 ( ( preorder @ A8 )
% 2.84/1.52 => ( preorder @ ( A7 > A8 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_fun___Lattices_Olattice,axiom,
% 2.84/1.52 ! [A7: $tType,A8: $tType] :
% 2.84/1.52 ( ( lattice @ A8 )
% 2.84/1.52 => ( lattice @ ( A7 > A8 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.84/1.52 ! [A7: $tType,A8: $tType] :
% 2.84/1.52 ( ( order @ A8 )
% 2.84/1.52 => ( order @ ( A7 > A8 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_fun___Orderings_Oord,axiom,
% 2.84/1.52 ! [A7: $tType,A8: $tType] :
% 2.84/1.52 ( ( ord @ A8 )
% 2.84/1.52 => ( ord @ ( A7 > A8 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_fun___Orderings_Obot,axiom,
% 2.84/1.52 ! [A7: $tType,A8: $tType] :
% 2.84/1.52 ( ( bot @ A8 )
% 2.84/1.52 => ( bot @ ( A7 > A8 ) ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_3,axiom,
% 2.84/1.52 semilattice_inf @ nat ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Nat_Onat___Orderings_Oorder__bot_4,axiom,
% 2.84/1.52 order_bot @ nat ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Nat_Onat___Orderings_Opreorder_5,axiom,
% 2.84/1.52 preorder @ nat ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.84/1.52 linorder @ nat ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Nat_Onat___Lattices_Olattice_6,axiom,
% 2.84/1.52 lattice @ nat ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Nat_Onat___Orderings_Oorder_7,axiom,
% 2.84/1.52 order @ nat ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Nat_Onat___Orderings_Oord_8,axiom,
% 2.84/1.52 ord @ nat ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Nat_Onat___Orderings_Obot_9,axiom,
% 2.84/1.52 bot @ nat ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_10,axiom,
% 2.84/1.52 ! [A7: $tType] : ( bounded_lattice_bot @ ( set @ A7 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_11,axiom,
% 2.84/1.52 ! [A7: $tType] : ( semilattice_inf @ ( set @ A7 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Set_Oset___Orderings_Oorder__bot_12,axiom,
% 2.84/1.52 ! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Set_Oset___Orderings_Opreorder_13,axiom,
% 2.84/1.52 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Set_Oset___Lattices_Olattice_14,axiom,
% 2.84/1.52 ! [A7: $tType] : ( lattice @ ( set @ A7 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Set_Oset___Orderings_Oorder_15,axiom,
% 2.84/1.52 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Set_Oset___Orderings_Oord_16,axiom,
% 2.84/1.52 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_Set_Oset___Orderings_Obot_17,axiom,
% 2.84/1.52 ! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_18,axiom,
% 2.84/1.52 bounded_lattice_bot @ $o ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_19,axiom,
% 2.84/1.52 semilattice_inf @ $o ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_HOL_Obool___Orderings_Oorder__bot_20,axiom,
% 2.84/1.52 order_bot @ $o ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_HOL_Obool___Orderings_Opreorder_21,axiom,
% 2.84/1.52 preorder @ $o ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_HOL_Obool___Orderings_Olinorder_22,axiom,
% 2.84/1.52 linorder @ $o ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_HOL_Obool___Lattices_Olattice_23,axiom,
% 2.84/1.52 lattice @ $o ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_HOL_Obool___Orderings_Oorder_24,axiom,
% 2.84/1.52 order @ $o ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_HOL_Obool___Orderings_Oord_25,axiom,
% 2.84/1.52 ord @ $o ).
% 2.84/1.52
% 2.84/1.52 thf(tcon_HOL_Obool___Orderings_Obot_26,axiom,
% 2.84/1.52 bot @ $o ).
% 2.84/1.52
% 2.84/1.52 thf(tfree_0,hypothesis,
% 2.84/1.52 linordered_idom @ capacity ).
% 2.84/1.52
% 2.84/1.52 thf(conj_0,hypothesis,
% 2.84/1.52 augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ p ).
% 2.84/1.52
% 2.84/1.52 thf(conj_1,conjecture,
% 2.84/1.52 ( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set2 @ ( product_prod @ nat @ nat ) @ p ) @ ( converse @ nat @ nat @ ( set2 @ ( product_prod @ nat @ nat ) @ p ) ) )
% 2.84/1.52 = ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ).
% 2.84/1.52 ------------------------
% 2.84/1.54 % (5511)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_5283 for (2999ds/18Mi)
% 2.84/1.54 % (5505)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_5283 for (2999ds/183Mi)
% 2.84/1.54 % (5510)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_5283 for (2999ds/275Mi)
% 2.84/1.55 % (5511)Instruction limit reached!
% 2.84/1.55 % (5511)------------------------------
% 2.84/1.55 % (5511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.55 % (5511)Termination reason: Unknown
% 2.84/1.55 % (5511)Termination phase: shuffling
% 2.84/1.55
% 2.84/1.55 % (5511)Memory used [KB]: 1663
% 2.84/1.55 % (5511)Time elapsed: 0.008 s
% 2.84/1.55 % (5511)Instructions burned: 19 (million)
% 2.84/1.55 % (5511)------------------------------
% 2.84/1.55 % (5511)------------------------------
% 2.84/1.55 % (5508)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_5283 for (2999ds/2Mi)
% 2.84/1.55 % (5507)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_5283 for (2999ds/27Mi)
% 2.84/1.55 % (5509)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_5283 for (2999ds/2Mi)
% 2.84/1.55 % (5508)Instruction limit reached!
% 2.84/1.55 % (5508)------------------------------
% 2.84/1.55 % (5508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.55 % (5508)Termination reason: Unknown
% 2.84/1.55 % (5508)Termination phase: shuffling
% 2.84/1.55
% 2.84/1.55 % (5508)Memory used [KB]: 1407
% 2.84/1.55 % (5508)Time elapsed: 0.003 s
% 2.84/1.55 % (5508)Instructions burned: 2 (million)
% 2.84/1.55 % (5508)------------------------------
% 2.84/1.55 % (5508)------------------------------
% 2.84/1.55 % (5509)Instruction limit reached!
% 2.84/1.55 % (5509)------------------------------
% 2.84/1.55 % (5509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.55 % (5509)Termination reason: Unknown
% 2.84/1.55 % (5509)Termination phase: shuffling
% 2.84/1.55
% 2.84/1.55 % (5509)Memory used [KB]: 1407
% 2.84/1.55 % (5509)Time elapsed: 0.003 s
% 2.84/1.55 % (5509)Instructions burned: 2 (million)
% 2.84/1.55 % (5509)------------------------------
% 2.84/1.55 % (5509)------------------------------
% 2.84/1.55 % (5506)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_5283 for (2999ds/4Mi)
% 2.84/1.55 % (5506)Instruction limit reached!
% 2.84/1.55 % (5506)------------------------------
% 2.84/1.55 % (5506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.55 % (5506)Termination reason: Unknown
% 2.84/1.55 % (5506)Termination phase: shuffling
% 2.84/1.55
% 2.84/1.55 % (5506)Memory used [KB]: 1407
% 2.84/1.55 % (5506)Time elapsed: 0.004 s
% 2.84/1.55 % (5506)Instructions burned: 4 (million)
% 2.84/1.55 % (5506)------------------------------
% 2.84/1.55 % (5506)------------------------------
% 2.84/1.56 % (5512)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_5283 for (2999ds/3Mi)
% 2.84/1.56 % (5512)Instruction limit reached!
% 2.84/1.56 % (5512)------------------------------
% 2.84/1.56 % (5512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.56 % (5512)Termination reason: Unknown
% 2.84/1.56 % (5512)Termination phase: shuffling
% 2.84/1.56
% 2.84/1.56 % (5512)Memory used [KB]: 1407
% 2.84/1.56 % (5512)Time elapsed: 0.003 s
% 2.84/1.56 % (5512)Instructions burned: 5 (million)
% 2.84/1.56 % (5512)------------------------------
% 2.84/1.56 % (5512)------------------------------
% 2.84/1.56 % (5513)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_5283 for (2999ds/37Mi)
% 2.84/1.56 % (5507)Instruction limit reached!
% 2.84/1.56 % (5507)------------------------------
% 2.84/1.56 % (5507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.56 % (5507)Termination reason: Unknown
% 2.84/1.56 % (5507)Termination phase: shuffling
% 2.84/1.56
% 2.84/1.56 % (5507)Memory used [KB]: 1791
% 2.84/1.56 % (5507)Time elapsed: 0.018 s
% 2.84/1.56 % (5507)Instructions burned: 27 (million)
% 2.84/1.56 % (5507)------------------------------
% 2.84/1.56 % (5507)------------------------------
% 2.84/1.57 % (5514)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_5283 for (2999ds/15Mi)
% 2.84/1.57 % (5516)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_5283 for (2999ds/1041Mi)
% 2.84/1.57 % (5510)First to succeed.
% 2.84/1.57 % (5515)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_5283 for (2999ds/3Mi)
% 2.84/1.57 % (5510)Refutation found. Thanks to Tanya!
% 2.84/1.57 % SZS status Theorem for DTF2THF_5283
% 2.84/1.57 % SZS output start Proof for DTF2THF_5283
% 2.84/1.57 thf(type_def_5, type, nat: $tType).
% 2.84/1.57 thf(type_def_6, type, product_prod: ($tType * $tType) > $tType).
% 2.84/1.57 thf(type_def_8, type, list: ($tType) > $tType).
% 2.84/1.57 thf(type_def_9, type, set: ($tType) > $tType).
% 2.84/1.57 thf(type_def_10, type, capacity: $tType).
% 2.84/1.57 thf(func_def_0, type, product_prod: ($tType * $tType) > $tType).
% 2.84/1.57 thf(func_def_1, type, list: ($tType) > $tType).
% 2.84/1.57 thf(func_def_2, type, capacity: $tType).
% 2.84/1.57 thf(func_def_3, type, set: ($tType) > $tType).
% 2.84/1.57 thf(func_def_4, type, nat: $tType).
% 2.84/1.57 thf(func_def_5, type, bounded_lattice: !>[X0: $tType]:($o)).
% 2.84/1.57 thf(func_def_6, type, type: !>[X0: $tType]:($o)).
% 2.84/1.57 thf(func_def_7, type, bot: !>[X0: $tType]:($o)).
% 2.84/1.57 thf(func_def_8, type, ord: !>[X0: $tType]:($o)).
% 2.84/1.57 thf(func_def_9, type, order: !>[X0: $tType]:($o)).
% 2.84/1.57 thf(func_def_10, type, lattice: !>[X0: $tType]:($o)).
% 2.84/1.57 thf(func_def_11, type, linorder: !>[X0: $tType]:($o)).
% 2.84/1.57 thf(func_def_12, type, preorder: !>[X0: $tType]:($o)).
% 2.84/1.57 thf(func_def_13, type, order_bot: !>[X0: $tType]:($o)).
% 2.84/1.57 thf(func_def_14, type, linordered_idom: !>[X0: $tType]:($o)).
% 2.84/1.57 thf(func_def_15, type, semilattice_inf: !>[X0: $tType]:($o)).
% 2.84/1.57 thf(func_def_16, type, bounded_lattice_bot: !>[X0: $tType]:($o)).
% 2.84/1.57 thf(func_def_17, type, augmenting_augment: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > (product_prod @ nat @ nat > X0) > (product_prod @ nat @ nat > X0) > product_prod @ nat @ nat > X0)).
% 2.84/1.57 thf(func_def_18, type, augmen1604248148ngFlow: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > (product_prod @ nat @ nat > X0) > list @ product_prod @ nat @ nat > product_prod @ nat @ nat > X0)).
% 2.84/1.57 thf(func_def_19, type, augmen2028041557ngPath: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > (product_prod @ nat @ nat > X0) > list @ product_prod @ nat @ nat > $o)).
% 2.84/1.57 thf(func_def_20, type, augmenting_resCap: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > (product_prod @ nat @ nat > X0) > list @ product_prod @ nat @ nat > X0)).
% 2.84/1.57 thf(func_def_21, type, finite_Graph: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > $o)).
% 2.84/1.57 thf(func_def_22, type, isShortestPath: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > list @ product_prod @ nat @ nat > nat > $o)).
% 2.84/1.57 thf(func_def_23, type, isSimplePath: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > list @ product_prod @ nat @ nat > nat > $o)).
% 2.84/1.57 thf(func_def_24, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.84/1.57 thf(func_def_25, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.84/1.57 thf(func_def_26, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.84/1.57 thf(func_def_27, type, set2: !>[X0: $tType]:(list @ X0 > set @ X0)).
% 2.84/1.57 thf(func_def_28, type, finite_Preflow: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > (product_prod @ nat @ nat > X0) > $o)).
% 2.84/1.57 thf(func_def_29, type, flow: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > (product_prod @ nat @ nat > X0) > $o)).
% 2.84/1.57 thf(func_def_30, type, val: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > (product_prod @ nat @ nat > X0) > X0)).
% 2.84/1.57 thf(func_def_31, type, nCut: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > set @ nat > $o)).
% 2.84/1.57 thf(func_def_32, type, cap: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > set @ nat > X0)).
% 2.84/1.57 thf(func_def_33, type, nFlow: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > (product_prod @ nat @ nat > X0) > $o)).
% 2.84/1.57 thf(func_def_34, type, nPreflow: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > (product_prod @ nat @ nat > X0) > $o)).
% 2.84/1.57 thf(func_def_35, type, network: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > $o)).
% 2.84/1.57 thf(func_def_36, type, excess: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > (product_prod @ nat @ nat > X0) > nat > X0)).
% 2.84/1.57 thf(func_def_37, type, isMaxFlow: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > (product_prod @ nat @ nat > X0) > $o)).
% 2.84/1.57 thf(func_def_38, type, is_max_flow_val: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > X0 > $o)).
% 2.84/1.57 thf(func_def_39, type, preflow: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > (product_prod @ nat @ nat > X0) > $o)).
% 2.84/1.57 thf(func_def_40, type, isMinCut: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > set @ nat > $o)).
% 2.84/1.57 thf(func_def_41, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.84/1.57 thf(func_def_42, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.84/1.57 thf(func_def_43, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.84/1.57 thf(func_def_44, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.84/1.57 thf(func_def_45, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 2.84/1.57 thf(func_def_46, type, converse: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ product_prod @ X1 @ X0)).
% 2.84/1.57 thf(func_def_47, type, residual_flow_of_cf: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > (product_prod @ nat @ nat > X0) > product_prod @ nat @ nat > X0)).
% 2.84/1.57 thf(func_def_48, type, residual_RGraph: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > (product_prod @ nat @ nat > X0) > $o)).
% 2.84/1.57 thf(func_def_49, type, residual_RPreGraph: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > (product_prod @ nat @ nat > X0) > $o)).
% 2.84/1.57 thf(func_def_50, type, residu1993642540lGraph: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > (product_prod @ nat @ nat > X0) > product_prod @ nat @ nat > X0)).
% 2.84/1.57 thf(func_def_51, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.84/1.57 thf(func_def_52, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.84/1.57 thf(func_def_53, type, c: product_prod @ nat @ nat > capacity).
% 2.84/1.57 thf(func_def_54, type, f: product_prod @ nat @ nat > capacity).
% 2.84/1.57 thf(func_def_55, type, p: list @ product_prod @ nat @ nat).
% 2.84/1.57 thf(func_def_56, type, s: nat).
% 2.84/1.57 thf(func_def_57, type, t: nat).
% 2.84/1.57 thf(func_def_72, type, sK0: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.84/1.57 thf(func_def_73, type, sK1: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.84/1.57 thf(func_def_74, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.84/1.57 thf(func_def_75, type, sK3: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.84/1.57 thf(func_def_76, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.84/1.57 thf(func_def_77, type, sK5: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 2.84/1.57 thf(func_def_78, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.84/1.57 thf(func_def_79, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.84/1.57 thf(func_def_80, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.84/1.57 thf(func_def_81, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.84/1.57 thf(func_def_82, type, sK10: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.84/1.57 thf(func_def_83, type, sK11: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.84/1.57 thf(func_def_84, type, sK12: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 2.84/1.57 thf(func_def_85, type, sK13: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 2.84/1.57 thf(func_def_86, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 2.84/1.57 thf(func_def_87, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 2.84/1.57 thf(func_def_88, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 2.84/1.57 thf(func_def_89, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 2.84/1.57 thf(func_def_90, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 2.84/1.57 thf(func_def_91, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 2.84/1.57 thf(func_def_92, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 2.84/1.57 thf(func_def_93, type, sK21: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 2.84/1.57 thf(func_def_94, type, sK22: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 2.84/1.57 thf(func_def_95, type, sK23: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 2.84/1.57 thf(func_def_96, type, sK24: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 2.84/1.57 thf(func_def_97, type, sK25: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > set @ product_prod @ X1 @ X0 > X1)).
% 2.84/1.57 thf(func_def_98, type, sK26: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > set @ product_prod @ X1 @ X0 > X0)).
% 2.84/1.57 thf(func_def_99, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.84/1.57 thf(func_def_100, type, sK28: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.84/1.57 thf(func_def_101, type, sK29: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.84/1.57 thf(func_def_102, type, sK30: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.84/1.57 thf(func_def_103, type, sK31: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.84/1.57 thf(func_def_104, type, sK32: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.84/1.57 thf(func_def_105, type, sK33: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 2.84/1.57 thf(func_def_106, type, sK34: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 2.84/1.57 thf(func_def_107, type, sK35: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 2.84/1.57 thf(func_def_108, type, sK36: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 2.84/1.57 thf(func_def_109, type, sK37: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 2.84/1.57 thf(func_def_110, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 2.84/1.57 thf(func_def_111, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 2.84/1.57 thf(func_def_112, type, sK40: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.84/1.57 thf(func_def_113, type, sK41: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.84/1.57 thf(func_def_114, type, sK42: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.84/1.57 thf(func_def_115, type, sK43: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.84/1.57 thf(func_def_116, type, sK44: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.84/1.57 thf(func_def_117, type, sK45: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.84/1.57 thf(func_def_118, type, sK46: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.84/1.57 thf(func_def_119, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.84/1.57 thf(func_def_120, type, sK48: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.84/1.57 thf(func_def_121, type, sK49: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.84/1.57 thf(func_def_122, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.84/1.57 thf(func_def_123, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.84/1.57 thf(func_def_124, type, sK52: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X1)).
% 2.84/1.57 thf(func_def_125, type, sK53: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X0)).
% 2.84/1.57 thf(func_def_126, type, sK54: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 2.84/1.57 thf(func_def_127, type, sK55: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 2.84/1.57 thf(func_def_128, type, sK56: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 2.84/1.57 thf(func_def_129, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 2.84/1.57 thf(func_def_130, type, sK58: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 2.84/1.57 thf(func_def_131, type, sK59: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.84/1.57 thf(func_def_132, type, sK60: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.84/1.57 thf(func_def_133, type, sK61: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.84/1.57 thf(func_def_134, type, sK62: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.84/1.57 thf(func_def_135, type, sK63: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > product_prod @ X0 @ X1 > X1)).
% 2.84/1.57 thf(func_def_136, type, sK64: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > product_prod @ X0 @ X1 > X0)).
% 2.84/1.57 thf(func_def_137, type, sK65: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > X1 > X0 > X1)).
% 2.84/1.57 thf(func_def_138, type, sK66: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > X1 > X0 > X0)).
% 2.84/1.57 thf(func_def_140, type, ph68: !>[X0: $tType]:(X0)).
% 2.84/1.57 thf(f1058,plain,(
% 2.84/1.57 $false),
% 2.84/1.57 inference(subsumption_resolution,[status(thm)],[f1057,f990])).
% 2.84/1.57 thf(f990,plain,(
% 2.84/1.57 (linordered_idom @ capacity = $true)),
% 2.84/1.57 inference(cnf_transformation,[status(thm)],[f416])).
% 2.84/1.57 thf(f416,plain,(
% 2.84/1.57 (linordered_idom @ capacity = $true)),
% 2.84/1.57 inference(fool_elimination,[status(thm)],[f415])).
% 2.84/1.57 thf(f415,plain,(
% 2.84/1.57 linordered_idom @ capacity),
% 2.84/1.57 inference(rectify,[status(thm)],[f298])).
% 2.84/1.57 thf(f298,axiom,(
% 2.84/1.57 linordered_idom @ capacity),
% 2.84/1.57 file('/export/starexec/sandbox2/tmp/tmp.7uUisdwD1b/DTF2THF_5283.p',tfree_0)).
% 2.84/1.57 thf(f1057,plain,(
% 2.84/1.57 (linordered_idom @ capacity != $true)),
% 2.84/1.57 inference(trivial_inequality_removal,[status(thm)],[f1056])).
% 2.84/1.57 thf(f1056,plain,(
% 2.84/1.57 ($true != $true) | (linordered_idom @ capacity != $true)),
% 2.84/1.57 inference(superposition,[status(thm)],[f1054,f1043])).
% 2.84/1.57 thf(f1043,plain,(
% 2.84/1.57 ((isSimplePath @ capacity @ (residu1993642540lGraph @ capacity @ c @ f) @ s @ p @ t) = $true)),
% 2.84/1.57 inference(superposition,[status(thm)],[f974,f991])).
% 2.84/1.57 thf(f991,plain,(
% 2.84/1.57 ((augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ p) = $true)),
% 2.84/1.57 inference(cnf_transformation,[status(thm)],[f732])).
% 2.84/1.57 thf(f732,plain,(
% 2.84/1.57 ((augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ p) = $true)),
% 2.84/1.57 inference(fool_elimination,[status(thm)],[f731])).
% 2.84/1.57 thf(f731,plain,(
% 2.84/1.57 (augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ p)),
% 2.84/1.57 inference(rectify,[status(thm)],[f299])).
% 2.84/1.57 thf(f299,axiom,(
% 2.84/1.57 (augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ p)),
% 2.84/1.57 file('/export/starexec/sandbox2/tmp/tmp.7uUisdwD1b/DTF2THF_5283.p',conj_0)).
% 2.84/1.57 thf(f974,plain,(
% 2.84/1.57 ( ! [X0 : list @ product_prod @ nat @ nat] : (((augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ X0) = (isSimplePath @ capacity @ (residu1993642540lGraph @ capacity @ c @ f) @ s @ X0 @ t))) )),
% 2.84/1.57 inference(cnf_transformation,[status(thm)],[f875])).
% 2.84/1.57 thf(f875,plain,(
% 2.84/1.57 ! [X0 : list @ product_prod @ nat @ nat] : ((augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ X0) = (isSimplePath @ capacity @ (residu1993642540lGraph @ capacity @ c @ f) @ s @ X0 @ t))),
% 2.84/1.57 inference(rectify,[status(thm)],[f6])).
% 2.84/1.57 thf(f6,axiom,(
% 2.84/1.57 ! [X4 : list @ product_prod @ nat @ nat] : ((augmen2028041557ngPath @ capacity @ c @ s @ t @ f @ X4) = (isSimplePath @ capacity @ (residu1993642540lGraph @ capacity @ c @ f) @ s @ X4 @ t))),
% 2.84/1.57 file('/export/starexec/sandbox2/tmp/tmp.7uUisdwD1b/DTF2THF_5283.p',fact_5_isAugmentingPath__def)).
% 2.84/1.57 thf(f1054,plain,(
% 2.84/1.57 ( ! [X0 : $tType,X2 : nat,X3 : nat,X1 : product_prod @ nat @ nat > X0] : (($true != (isSimplePath @ X0 @ X1 @ X2 @ p @ X3)) | (linordered_idom @ X0 != $true)) )),
% 2.84/1.57 inference(trivial_inequality_removal,[status(thm)],[f1052])).
% 2.84/1.57 thf(f1052,plain,(
% 2.84/1.57 ( ! [X0 : $tType,X2 : nat,X3 : nat,X1 : product_prod @ nat @ nat > X0] : ((linordered_idom @ X0 != $true) | ($true != (isSimplePath @ X0 @ X1 @ X2 @ p @ X3)) | (bot_bot @ set @ product_prod @ nat @ nat != bot_bot @ set @ product_prod @ nat @ nat)) )),
% 2.84/1.57 inference(superposition,[status(thm)],[f1019,f1017])).
% 2.84/1.57 thf(f1017,plain,(
% 2.84/1.57 ( ! [X0 : $tType,X2 : nat,X3 : list @ product_prod @ nat @ nat,X1 : product_prod @ nat @ nat > X0,X4 : nat] : ((bot_bot @ set @ product_prod @ nat @ nat = (inf_inf @ set @ product_prod @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ X3) @ (converse @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ X3)))) | (linordered_idom @ X0 != $true) | ($true != (isSimplePath @ X0 @ X1 @ X2 @ X3 @ X4))) )),
% 2.84/1.57 inference(cnf_transformation,[status(thm)],[f924])).
% 2.84/1.57 thf(f924,plain,(
% 2.84/1.57 ! [X0 : $tType] : ((linordered_idom @ X0 != $true) | ! [X1 : product_prod @ nat @ nat > X0,X2 : nat,X3 : list @ product_prod @ nat @ nat,X4 : nat] : (($true != (isSimplePath @ X0 @ X1 @ X2 @ X3 @ X4)) | (bot_bot @ set @ product_prod @ nat @ nat = (inf_inf @ set @ product_prod @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ X3) @ (converse @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ X3))))))),
% 2.84/1.57 inference(ennf_transformation,[status(thm)],[f372])).
% 2.84/1.57 thf(f372,plain,(
% 2.84/1.57 ! [X0 : $tType] : ((linordered_idom @ X0 = $true) => ! [X1 : product_prod @ nat @ nat > X0,X2 : nat,X3 : list @ product_prod @ nat @ nat,X4 : nat] : (($true = (isSimplePath @ X0 @ X1 @ X2 @ X3 @ X4)) => (bot_bot @ set @ product_prod @ nat @ nat = (inf_inf @ set @ product_prod @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ X3) @ (converse @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ X3))))))),
% 2.84/1.57 inference(fool_elimination,[status(thm)],[f371])).
% 2.84/1.57 thf(f371,plain,(
% 2.84/1.57 ! [X0 : $tType] : (linordered_idom @ X0 => ! [X1 : product_prod @ nat @ nat > X0,X2 : nat,X3 : list @ product_prod @ nat @ nat,X4 : nat] : ((isSimplePath @ X0 @ X1 @ X2 @ X3 @ X4) => (bot_bot @ set @ product_prod @ nat @ nat = (inf_inf @ set @ product_prod @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ X3) @ (converse @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ X3))))))),
% 2.84/1.57 inference(rectify,[status(thm)],[f56])).
% 2.84/1.57 thf(f56,axiom,(
% 2.84/1.57 ! [X1 : $tType] : (linordered_idom @ X1 => ! [X19 : product_prod @ nat @ nat > X1,X3 : nat,X4 : list @ product_prod @ nat @ nat,X5 : nat] : ((isSimplePath @ X1 @ X19 @ X3 @ X4 @ X5) => ((inf_inf @ set @ product_prod @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ X4) @ (converse @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ X4))) = bot_bot @ set @ product_prod @ nat @ nat)))),
% 2.84/1.57 file('/export/starexec/sandbox2/tmp/tmp.7uUisdwD1b/DTF2THF_5283.p',fact_55_Graph_OisSPath__nt__parallel__pf)).
% 2.84/1.57 thf(f1019,plain,(
% 2.84/1.57 (bot_bot @ set @ product_prod @ nat @ nat != (inf_inf @ set @ product_prod @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ p) @ (converse @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ p))))),
% 2.84/1.57 inference(cnf_transformation,[status(thm)],[f857])).
% 2.84/1.57 thf(f857,plain,(
% 2.84/1.57 (bot_bot @ set @ product_prod @ nat @ nat != (inf_inf @ set @ product_prod @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ p) @ (converse @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ p))))),
% 2.84/1.57 inference(flattening,[status(thm)],[f301])).
% 2.84/1.57 thf(f301,negated_conjecture,(
% 2.84/1.57 ~(bot_bot @ set @ product_prod @ nat @ nat = (inf_inf @ set @ product_prod @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ p) @ (converse @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ p))))),
% 2.84/1.57 inference(negated_conjecture,[status(cth)],[f300])).
% 2.84/1.57 thf(f300,conjecture,(
% 2.84/1.57 (bot_bot @ set @ product_prod @ nat @ nat = (inf_inf @ set @ product_prod @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ p) @ (converse @ nat @ nat @ (set2 @ product_prod @ nat @ nat @ p))))),
% 2.84/1.57 file('/export/starexec/sandbox2/tmp/tmp.7uUisdwD1b/DTF2THF_5283.p',conj_1)).
% 2.84/1.57 % SZS output end Proof for DTF2THF_5283
% 2.84/1.57 % (5510)------------------------------
% 2.84/1.57 % (5510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.84/1.57 % (5510)Termination reason: Refutation
% 2.84/1.57
% 2.84/1.57 % (5510)Memory used [KB]: 6780
% 2.84/1.57 % (5510)Time elapsed: 0.032 s
% 2.84/1.57 % (5510)Instructions burned: 87 (million)
% 2.84/1.57 % (5510)------------------------------
% 2.84/1.57 % (5510)------------------------------
% 2.84/1.57 % (5504)Success in time 0.048 s
%------------------------------------------------------------------------------