%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP049^2 : 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 : n024.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 08:55:08 AM UTC 2026
% Result : Theorem 2.07s 1.18s
% Output : Refutation 2.07s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : ITP049^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.06 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.06/0.25 % Computer : n024.cluster.edu
% 0.06/0.25 % Model : x86_64 x86_64
% 0.06/0.25 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.06/0.25 % Memory : 8042.1875MB
% 0.06/0.25 % OS : Linux 3.10.0-693.el7.x86_64
% 0.06/0.25 % CPULimit : 300
% 0.06/0.25 % WCLimit : 300
% 0.06/0.25 % DateTime : Tue Feb 24 03:24:25 EST 2026
% 0.06/0.25 % CPUTime :
% 0.06/0.25 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.06/0.31 ---- Original DTF file ---
% 0.06/0.31 thf(spec,logic,$$dhol).
% 0.06/0.31 %------------------------------------------------------------------------------
% 0.06/0.31 % File : ITP049^2 : TPTP v9.2.1. Released v7.5.0.
% 0.06/0.31 % Domain : Interactive Theorem Proving
% 0.06/0.31 % Problem : Sledgehammer EdmondsKarp_Termination_Abstract problem prob_229__7588066_1
% 0.06/0.31 % Version : Especial.
% 0.06/0.31 % English :
% 0.06/0.31
% 0.06/0.31 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.06/0.31 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.06/0.31 % Source : [Des21]
% 0.06/0.31 % Names : EdmondsKarp_Termination_Abstract/prob_229__7588066_1 [Des21]
% 0.06/0.31
% 0.06/0.31 % Status : Theorem
% 0.06/0.31 % Rating : 0.00 v7.5.0
% 0.06/0.31 % Syntax : Number of formulae : 325 ( 74 unt; 44 typ; 0 def)
% 0.06/0.31 % Number of atoms : 789 ( 142 equ; 0 cnn)
% 0.06/0.31 % Maximal formula atoms : 13 ( 2 avg)
% 0.06/0.31 % Number of connectives : 5341 ( 81 ~; 7 |; 50 &;4788 @)
% 0.06/0.31 % ( 0 <=>; 415 =>; 0 <=; 0 <~>)
% 0.06/0.31 % Maximal formula depth : 28 ( 10 avg)
% 0.06/0.31 % Number of types : 3 ( 2 usr)
% 0.06/0.31 % Number of type conns : 236 ( 236 >; 0 *; 0 +; 0 <<)
% 0.06/0.31 % Number of symbols : 44 ( 42 usr; 3 con; 0-5 aty)
% 0.06/0.31 % Number of variables : 1195 ( 82 ^;1018 !; 50 ?;1195 :)
% 0.06/0.31 % ( 45 !>; 0 ?*; 0 @-; 0 @+)
% 0.06/0.31 % SPC : TH1_THM_EQU_NAR
% 0.06/0.31
% 0.06/0.31 % Comments : This file was generated by Sledgehammer 2021-02-23 16:16:10.366
% 0.06/0.31 %------------------------------------------------------------------------------
% 0.06/0.31 % Could-be-implicit typings (5)
% 0.06/0.31 thf(ty_t_Product__Type_Oprod,type,
% 0.06/0.31 product_prod: $tType > $tType > $tType ).
% 0.06/0.31
% 0.06/0.31 thf(ty_t_List_Olist,type,
% 0.06/0.31 list: $tType > $tType ).
% 0.06/0.31
% 0.06/0.31 thf(ty_t_Set_Oset,type,
% 0.06/0.31 set: $tType > $tType ).
% 0.06/0.31
% 0.06/0.31 thf(ty_t_Nat_Onat,type,
% 0.06/0.31 nat: $tType ).
% 0.06/0.31
% 0.06/0.31 thf(ty_tf_a,type,
% 0.06/0.31 a: $tType ).
% 0.06/0.31
% 0.06/0.31 % Explicit typings (39)
% 0.06/0.31 thf(sy_cl_HOL_Otype,type,
% 0.06/0.31 type:
% 0.06/0.31 !>[A: $tType] : $o ).
% 0.06/0.31
% 0.06/0.31 thf(sy_cl_Orderings_Oord,type,
% 0.06/0.31 ord:
% 0.06/0.31 !>[A: $tType] : $o ).
% 0.06/0.31
% 0.06/0.31 thf(sy_cl_Orderings_Oorder,type,
% 0.06/0.31 order:
% 0.06/0.31 !>[A: $tType] : $o ).
% 0.06/0.31
% 0.06/0.31 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.06/0.31 finite_finite:
% 0.06/0.31 !>[A: $tType] : $o ).
% 0.06/0.31
% 0.06/0.31 thf(sy_cl_Orderings_Olinorder,type,
% 0.06/0.31 linorder:
% 0.06/0.31 !>[A: $tType] : $o ).
% 0.06/0.31
% 0.06/0.31 thf(sy_cl_Orderings_Opreorder,type,
% 0.06/0.31 preorder:
% 0.06/0.31 !>[A: $tType] : $o ).
% 0.06/0.31
% 0.06/0.31 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.06/0.31 linordered_idom:
% 0.06/0.31 !>[A: $tType] : $o ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.06/0.31 bNF_Ca1785829860lChain:
% 0.06/0.31 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis,type,
% 0.06/0.31 edmond2129202899alysis:
% 0.06/0.31 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > $o ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis__defs_OspEdges,type,
% 0.06/0.31 edmond1803015688pEdges:
% 0.06/0.31 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > nat > nat > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Finite__Set_Ofinite,type,
% 0.06/0.31 finite_finite2:
% 0.06/0.31 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Graph_OFinite__Graph,type,
% 0.06/0.31 finite_Graph:
% 0.06/0.31 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > $o ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Graph_OGraph_OE,type,
% 0.06/0.31 e:
% 0.06/0.31 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Graph_OGraph_OV,type,
% 0.06/0.31 v:
% 0.06/0.31 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Graph_OGraph_Oadjacent__nodes,type,
% 0.06/0.31 adjacent_nodes:
% 0.06/0.31 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ nat ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Graph_OGraph_Oconnected,type,
% 0.06/0.31 connected:
% 0.06/0.31 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > $o ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Graph_OGraph_Oincoming,type,
% 0.06/0.31 incoming:
% 0.06/0.31 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Graph_OGraph_Oincoming_H,type,
% 0.06/0.31 incoming2:
% 0.06/0.31 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Graph_OGraph_OisPath,type,
% 0.06/0.31 isPath:
% 0.06/0.31 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Graph_OGraph_OisShortestPath,type,
% 0.06/0.31 isShortestPath:
% 0.06/0.31 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Graph_OGraph_OisSimplePath,type,
% 0.06/0.31 isSimplePath:
% 0.06/0.31 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Graph_OGraph_Ooutgoing,type,
% 0.06/0.31 outgoing:
% 0.06/0.31 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Graph_OGraph_Ooutgoing_H,type,
% 0.06/0.31 outgoing2:
% 0.06/0.31 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Graph_OGraph_OreachableNodes,type,
% 0.06/0.31 reachableNodes:
% 0.06/0.31 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ nat ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_List_Olist_Oset,type,
% 0.06/0.31 set2:
% 0.06/0.31 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.06/0.31 ord_less_eq:
% 0.06/0.31 !>[A: $tType] : ( A > A > $o ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Product__Type_OPair,type,
% 0.06/0.31 product_Pair:
% 0.06/0.31 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Product__Type_OSigma,type,
% 0.06/0.31 product_Sigma:
% 0.06/0.31 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.06/0.31 product_rec_prod:
% 0.06/0.31 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.06/0.31 product_swap:
% 0.06/0.31 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Relation_ODomain,type,
% 0.06/0.31 domain:
% 0.06/0.31 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Relation_OImage,type,
% 0.06/0.31 image:
% 0.06/0.31 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Relation_ORange,type,
% 0.06/0.31 range:
% 0.06/0.31 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Set_OCollect,type,
% 0.06/0.31 collect:
% 0.06/0.31 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_Transitive__Closure_Ortrancl,type,
% 0.06/0.31 transitive_rtrancl:
% 0.06/0.31 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_c_member,type,
% 0.06/0.31 member:
% 0.06/0.31 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.06/0.31
% 0.06/0.31 thf(sy_v_c,type,
% 0.06/0.31 c: ( product_prod @ nat @ nat ) > a ).
% 0.06/0.31
% 0.06/0.31 thf(sy_v_s,type,
% 0.06/0.31 s: nat ).
% 0.06/0.31
% 0.06/0.31 thf(sy_v_t,type,
% 0.06/0.31 t: nat ).
% 0.06/0.31
% 0.06/0.31 % Relevant facts (256)
% 0.06/0.31 thf(fact_0_ek__analysis__axioms,axiom,
% 0.06/0.31 edmond2129202899alysis @ a @ c ).
% 0.06/0.31
% 0.06/0.31 % ek_analysis_axioms
% 0.06/0.31 thf(fact_1__092_060open_062finite_AE_A_092_060Longrightarrow_062_Afinite_AspEdges_092_060close_062,axiom,
% 0.06/0.31 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
% 0.06/0.31 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % \<open>finite E \<Longrightarrow> finite spEdges\<close>
% 0.06/0.31 thf(fact_2_ek__analysis__defs_OspEdges_Ocong,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( linordered_idom @ A )
% 0.06/0.31 => ( ( edmond1803015688pEdges @ A )
% 0.06/0.31 = ( edmond1803015688pEdges @ A ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % ek_analysis_defs.spEdges.cong
% 0.06/0.31 thf(fact_3_finite__E,axiom,
% 0.06/0.31 finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) ).
% 0.06/0.31
% 0.06/0.31 % finite_E
% 0.06/0.31 thf(fact_4_spEdges__ss__E,axiom,
% 0.06/0.31 ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) @ ( e @ a @ c ) ).
% 0.06/0.31
% 0.06/0.31 % spEdges_ss_E
% 0.06/0.31 thf(fact_5_Finite__Graph__axioms,axiom,
% 0.06/0.31 finite_Graph @ a @ c ).
% 0.06/0.31
% 0.06/0.31 % Finite_Graph_axioms
% 0.06/0.31 thf(fact_6_finite__code,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( finite_finite @ A )
% 0.06/0.31 => ( ( finite_finite2 @ A )
% 0.06/0.31 = ( ^ [A2: set @ A] : $true ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_code
% 0.06/0.31 thf(fact_7_Finite__Graph__EI,axiom,
% 0.06/0.31 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
% 0.06/0.31 => ( finite_Graph @ a @ c ) ) ).
% 0.06/0.31
% 0.06/0.31 % Finite_Graph_EI
% 0.06/0.31 thf(fact_8_finite__set__choice,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
% 0.06/0.31 ( ( finite_finite2 @ A @ A3 )
% 0.06/0.31 => ( ! [X: A] :
% 0.06/0.31 ( ( member @ A @ X @ A3 )
% 0.06/0.31 => ? [X_1: B] : ( P @ X @ X_1 ) )
% 0.06/0.31 => ? [F: A > B] :
% 0.06/0.31 ! [X2: A] :
% 0.06/0.31 ( ( member @ A @ X2 @ A3 )
% 0.06/0.31 => ( P @ X2 @ ( F @ X2 ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_set_choice
% 0.06/0.31 thf(fact_9_finite,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( finite_finite @ A )
% 0.06/0.31 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite
% 0.06/0.31 thf(fact_10_connected__refl,axiom,
% 0.06/0.31 ! [V: nat] : ( connected @ a @ c @ V @ V ) ).
% 0.06/0.31
% 0.06/0.31 % connected_refl
% 0.06/0.31 thf(fact_11_spEdges__def,axiom,
% 0.06/0.31 ( ( edmond1803015688pEdges @ a @ c @ s @ t )
% 0.06/0.31 = ( collect @ ( product_prod @ nat @ nat )
% 0.06/0.31 @ ^ [E: product_prod @ nat @ nat] :
% 0.06/0.31 ? [P2: list @ ( product_prod @ nat @ nat )] :
% 0.06/0.31 ( ( member @ ( product_prod @ nat @ nat ) @ E @ ( set2 @ ( product_prod @ nat @ nat ) @ P2 ) )
% 0.06/0.31 & ( isShortestPath @ a @ c @ s @ P2 @ t ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % spEdges_def
% 0.06/0.31 thf(fact_12_finite__Range,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
% 0.06/0.31 => ( finite_finite2 @ B @ ( range @ A @ B @ R ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_Range
% 0.06/0.31 thf(fact_13_finite__Domain,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
% 0.06/0.31 => ( finite_finite2 @ A @ ( domain @ A @ B @ R ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_Domain
% 0.06/0.31 thf(fact_14_Efin__imp__Vfin,axiom,
% 0.06/0.31 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
% 0.06/0.31 => ( finite_finite2 @ nat @ ( v @ a @ c ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Efin_imp_Vfin
% 0.06/0.31 thf(fact_15_connected__inV__iff,axiom,
% 0.06/0.31 ! [U: nat,V: nat] :
% 0.06/0.31 ( ( connected @ a @ c @ U @ V )
% 0.06/0.31 => ( ( member @ nat @ V @ ( v @ a @ c ) )
% 0.06/0.31 = ( member @ nat @ U @ ( v @ a @ c ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % connected_inV_iff
% 0.06/0.31 thf(fact_16_obtain__shortest__path,axiom,
% 0.06/0.31 ! [U: nat,V: nat] :
% 0.06/0.31 ( ( connected @ a @ c @ U @ V )
% 0.06/0.31 => ~ ! [P3: list @ ( product_prod @ nat @ nat )] :
% 0.06/0.31 ~ ( isShortestPath @ a @ c @ U @ P3 @ V ) ) ).
% 0.06/0.31
% 0.06/0.31 % obtain_shortest_path
% 0.06/0.31 thf(fact_17_finite__Collect__conjI,axiom,
% 0.06/0.31 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.06/0.31 ( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.06/0.31 | ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
% 0.06/0.31 => ( finite_finite2 @ A
% 0.06/0.31 @ ( collect @ A
% 0.06/0.31 @ ^ [X3: A] :
% 0.06/0.31 ( ( P @ X3 )
% 0.06/0.31 & ( Q @ X3 ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_Collect_conjI
% 0.06/0.31 thf(fact_18_finite__Collect__disjI,axiom,
% 0.06/0.31 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.06/0.31 ( ( finite_finite2 @ A
% 0.06/0.31 @ ( collect @ A
% 0.06/0.31 @ ^ [X3: A] :
% 0.06/0.31 ( ( P @ X3 )
% 0.06/0.31 | ( Q @ X3 ) ) ) )
% 0.06/0.31 = ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.06/0.31 & ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_Collect_disjI
% 0.06/0.31 thf(fact_19_finite__Collect__bounded__ex,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,P: A > $o,Q: B > A > $o] :
% 0.06/0.31 ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.06/0.31 => ( ( finite_finite2 @ B
% 0.06/0.31 @ ( collect @ B
% 0.06/0.31 @ ^ [X3: B] :
% 0.06/0.31 ? [Y: A] :
% 0.06/0.31 ( ( P @ Y )
% 0.06/0.31 & ( Q @ X3 @ Y ) ) ) )
% 0.06/0.31 = ( ! [Y: A] :
% 0.06/0.31 ( ( P @ Y )
% 0.06/0.31 => ( finite_finite2 @ B
% 0.06/0.31 @ ( collect @ B
% 0.06/0.31 @ ^ [X3: B] : ( Q @ X3 @ Y ) ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_Collect_bounded_ex
% 0.06/0.31 thf(fact_20_finite__Collect__subsets,axiom,
% 0.06/0.31 ! [A: $tType,A3: set @ A] :
% 0.06/0.31 ( ( finite_finite2 @ A @ A3 )
% 0.06/0.31 => ( finite_finite2 @ ( set @ A )
% 0.06/0.31 @ ( collect @ ( set @ A )
% 0.06/0.31 @ ^ [B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_Collect_subsets
% 0.06/0.31 thf(fact_21_finite__V,axiom,
% 0.06/0.31 finite_finite2 @ nat @ ( v @ a @ c ) ).
% 0.06/0.31
% 0.06/0.31 % finite_V
% 0.06/0.31 thf(fact_22_Vfin__imp__Efin,axiom,
% 0.06/0.31 ( ( finite_finite2 @ nat @ ( v @ a @ c ) )
% 0.06/0.31 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Vfin_imp_Efin
% 0.06/0.31 thf(fact_23_Range__mono,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S )
% 0.06/0.31 => ( ord_less_eq @ ( set @ B ) @ ( range @ A @ B @ R ) @ ( range @ A @ B @ S ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Range_mono
% 0.06/0.31 thf(fact_24_Domain__mono,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S )
% 0.06/0.31 => ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R ) @ ( domain @ A @ B @ S ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Domain_mono
% 0.06/0.31 thf(fact_25_finite__image__set,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,P: A > $o,F2: A > B] :
% 0.06/0.31 ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.06/0.31 => ( finite_finite2 @ B
% 0.06/0.31 @ ( collect @ B
% 0.06/0.31 @ ^ [Uu: B] :
% 0.06/0.31 ? [X3: A] :
% 0.06/0.31 ( ( Uu
% 0.06/0.31 = ( F2 @ X3 ) )
% 0.06/0.31 & ( P @ X3 ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_image_set
% 0.06/0.31 thf(fact_26_finite__image__set2,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,C: $tType,P: A > $o,Q: B > $o,F2: A > B > C] :
% 0.06/0.31 ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.06/0.31 => ( ( finite_finite2 @ B @ ( collect @ B @ Q ) )
% 0.06/0.31 => ( finite_finite2 @ C
% 0.06/0.31 @ ( collect @ C
% 0.06/0.31 @ ^ [Uu: C] :
% 0.06/0.31 ? [X3: A,Y: B] :
% 0.06/0.31 ( ( Uu
% 0.06/0.31 = ( F2 @ X3 @ Y ) )
% 0.06/0.31 & ( P @ X3 )
% 0.06/0.31 & ( Q @ Y ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_image_set2
% 0.06/0.31 thf(fact_27_not__finite__existsD,axiom,
% 0.06/0.31 ! [A: $tType,P: A > $o] :
% 0.06/0.31 ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 0.06/0.31 => ? [X_12: A] : ( P @ X_12 ) ) ).
% 0.06/0.31
% 0.06/0.31 % not_finite_existsD
% 0.06/0.31 thf(fact_28_pigeonhole__infinite__rel,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,R2: A > B > $o] :
% 0.06/0.31 ( ~ ( finite_finite2 @ A @ A3 )
% 0.06/0.31 => ( ( finite_finite2 @ B @ B3 )
% 0.06/0.31 => ( ! [X: A] :
% 0.06/0.31 ( ( member @ A @ X @ A3 )
% 0.06/0.31 => ? [Xa: B] :
% 0.06/0.31 ( ( member @ B @ Xa @ B3 )
% 0.06/0.31 & ( R2 @ X @ Xa ) ) )
% 0.06/0.31 => ? [X: B] :
% 0.06/0.31 ( ( member @ B @ X @ B3 )
% 0.06/0.31 & ~ ( finite_finite2 @ A
% 0.06/0.31 @ ( collect @ A
% 0.06/0.31 @ ^ [A4: A] :
% 0.06/0.31 ( ( member @ A @ A4 @ A3 )
% 0.06/0.31 & ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % pigeonhole_infinite_rel
% 0.06/0.31 thf(fact_29_ek__analysis__def,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( linordered_idom @ A )
% 0.06/0.31 => ( ( edmond2129202899alysis @ A )
% 0.06/0.31 = ( finite_Graph @ A ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % ek_analysis_def
% 0.06/0.31 thf(fact_30_ek__analysis_Ointro,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( linordered_idom @ A )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > A] :
% 0.06/0.31 ( ( finite_Graph @ A @ C2 )
% 0.06/0.31 => ( edmond2129202899alysis @ A @ C2 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % ek_analysis.intro
% 0.06/0.31 thf(fact_31_ek__analysis_Oaxioms,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( linordered_idom @ A )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > A] :
% 0.06/0.31 ( ( edmond2129202899alysis @ A @ C2 )
% 0.06/0.31 => ( finite_Graph @ A @ C2 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % ek_analysis.axioms
% 0.06/0.31 thf(fact_32_ek__analysis__defs_OspEdges__def,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( linordered_idom @ A )
% 0.06/0.31 => ( ( edmond1803015688pEdges @ A )
% 0.06/0.31 = ( ^ [C3: ( product_prod @ nat @ nat ) > A,S2: nat,T2: nat] :
% 0.06/0.31 ( collect @ ( product_prod @ nat @ nat )
% 0.06/0.31 @ ^ [E: product_prod @ nat @ nat] :
% 0.06/0.31 ? [P2: list @ ( product_prod @ nat @ nat )] :
% 0.06/0.31 ( ( member @ ( product_prod @ nat @ nat ) @ E @ ( set2 @ ( product_prod @ nat @ nat ) @ P2 ) )
% 0.06/0.31 & ( isShortestPath @ A @ C3 @ S2 @ P2 @ T2 ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % ek_analysis_defs.spEdges_def
% 0.06/0.31 thf(fact_33_finite__has__minimal2,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( order @ A )
% 0.06/0.31 => ! [A3: set @ A,A5: A] :
% 0.06/0.31 ( ( finite_finite2 @ A @ A3 )
% 0.06/0.31 => ( ( member @ A @ A5 @ A3 )
% 0.06/0.31 => ? [X: A] :
% 0.06/0.31 ( ( member @ A @ X @ A3 )
% 0.06/0.31 & ( ord_less_eq @ A @ X @ A5 )
% 0.06/0.31 & ! [Xa: A] :
% 0.06/0.31 ( ( member @ A @ Xa @ A3 )
% 0.06/0.31 => ( ( ord_less_eq @ A @ Xa @ X )
% 0.06/0.31 => ( X = Xa ) ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_has_minimal2
% 0.06/0.31 thf(fact_34_finite__has__maximal2,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( order @ A )
% 0.06/0.31 => ! [A3: set @ A,A5: A] :
% 0.06/0.31 ( ( finite_finite2 @ A @ A3 )
% 0.06/0.31 => ( ( member @ A @ A5 @ A3 )
% 0.06/0.31 => ? [X: A] :
% 0.06/0.31 ( ( member @ A @ X @ A3 )
% 0.06/0.31 & ( ord_less_eq @ A @ A5 @ X )
% 0.06/0.31 & ! [Xa: A] :
% 0.06/0.31 ( ( member @ A @ Xa @ A3 )
% 0.06/0.31 => ( ( ord_less_eq @ A @ X @ Xa )
% 0.06/0.31 => ( X = Xa ) ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_has_maximal2
% 0.06/0.31 thf(fact_35_rev__finite__subset,axiom,
% 0.06/0.31 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.06/0.31 ( ( finite_finite2 @ A @ B3 )
% 0.06/0.31 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.06/0.31 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % rev_finite_subset
% 0.06/0.31 thf(fact_36_infinite__super,axiom,
% 0.06/0.31 ! [A: $tType,S3: set @ A,T3: set @ A] :
% 0.06/0.31 ( ( ord_less_eq @ ( set @ A ) @ S3 @ T3 )
% 0.06/0.31 => ( ~ ( finite_finite2 @ A @ S3 )
% 0.06/0.31 => ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % infinite_super
% 0.06/0.31 thf(fact_37_finite__subset,axiom,
% 0.06/0.31 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.06/0.31 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.06/0.31 => ( ( finite_finite2 @ A @ B3 )
% 0.06/0.31 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_subset
% 0.06/0.31 thf(fact_38_ek__analysis_OspEdges__ss__E,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( linordered_idom @ A )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > A,S: nat,T4: nat] :
% 0.06/0.31 ( ( edmond2129202899alysis @ A @ C2 )
% 0.06/0.31 => ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( edmond1803015688pEdges @ A @ C2 @ S @ T4 ) @ ( e @ A @ C2 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % ek_analysis.spEdges_ss_E
% 0.06/0.31 thf(fact_39_incoming_H__edges,axiom,
% 0.06/0.31 ! [U2: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming2 @ a @ c @ U2 ) @ ( e @ a @ c ) ) ).
% 0.06/0.31
% 0.06/0.31 % incoming'_edges
% 0.06/0.31 thf(fact_40_outgoing_H__edges,axiom,
% 0.06/0.31 ! [U2: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing2 @ a @ c @ U2 ) @ ( e @ a @ c ) ) ).
% 0.06/0.31
% 0.06/0.31 % outgoing'_edges
% 0.06/0.31 thf(fact_41_reachableNodes__def,axiom,
% 0.06/0.31 ! [U: nat] :
% 0.06/0.31 ( ( reachableNodes @ a @ c @ U )
% 0.06/0.31 = ( collect @ nat @ ( connected @ a @ c @ U ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % reachableNodes_def
% 0.06/0.31 thf(fact_42_List_Ofinite__set,axiom,
% 0.06/0.31 ! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% 0.06/0.31
% 0.06/0.31 % List.finite_set
% 0.06/0.31 thf(fact_43_Graph_OEfin__imp__Vfin,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity] :
% 0.06/0.31 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ Capacity @ C2 ) )
% 0.06/0.31 => ( finite_finite2 @ nat @ ( v @ Capacity @ C2 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.Efin_imp_Vfin
% 0.06/0.31 thf(fact_44_Graph_OVfin__imp__Efin,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity] :
% 0.06/0.31 ( ( finite_finite2 @ nat @ ( v @ Capacity @ C2 ) )
% 0.06/0.31 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ Capacity @ C2 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.Vfin_imp_Efin
% 0.06/0.31 thf(fact_45_mem__Collect__eq,axiom,
% 0.06/0.31 ! [A: $tType,A5: A,P: A > $o] :
% 0.06/0.31 ( ( member @ A @ A5 @ ( collect @ A @ P ) )
% 0.06/0.31 = ( P @ A5 ) ) ).
% 0.06/0.31
% 0.06/0.31 % mem_Collect_eq
% 0.06/0.31 thf(fact_46_Collect__mem__eq,axiom,
% 0.06/0.31 ! [A: $tType,A3: set @ A] :
% 0.06/0.31 ( ( collect @ A
% 0.06/0.31 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 0.06/0.31 = A3 ) ).
% 0.06/0.31
% 0.06/0.31 % Collect_mem_eq
% 0.06/0.31 thf(fact_47_Collect__cong,axiom,
% 0.06/0.31 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.06/0.31 ( ! [X: A] :
% 0.06/0.31 ( ( P @ X )
% 0.06/0.31 = ( Q @ X ) )
% 0.06/0.31 => ( ( collect @ A @ P )
% 0.06/0.31 = ( collect @ A @ Q ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Collect_cong
% 0.06/0.31 thf(fact_48_ext,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,F2: A > B,G: A > B] :
% 0.06/0.31 ( ! [X: A] :
% 0.06/0.31 ( ( F2 @ X )
% 0.06/0.31 = ( G @ X ) )
% 0.06/0.31 => ( F2 = G ) ) ).
% 0.06/0.31
% 0.06/0.31 % ext
% 0.06/0.31 thf(fact_49_isShortestPath__level__edge_I1_J,axiom,
% 0.06/0.31 ! [S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat,U: nat,V: nat] :
% 0.06/0.31 ( ( isShortestPath @ a @ c @ S @ P4 @ T4 )
% 0.06/0.31 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.31 => ( connected @ a @ c @ S @ U ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % isShortestPath_level_edge(1)
% 0.06/0.31 thf(fact_50_isShortestPath__level__edge_I2_J,axiom,
% 0.06/0.31 ! [S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat,U: nat,V: nat] :
% 0.06/0.31 ( ( isShortestPath @ a @ c @ S @ P4 @ T4 )
% 0.06/0.31 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.31 => ( connected @ a @ c @ U @ V ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % isShortestPath_level_edge(2)
% 0.06/0.31 thf(fact_51_isShortestPath__level__edge_I3_J,axiom,
% 0.06/0.31 ! [S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat,U: nat,V: nat] :
% 0.06/0.31 ( ( isShortestPath @ a @ c @ S @ P4 @ T4 )
% 0.06/0.31 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.31 => ( connected @ a @ c @ V @ T4 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % isShortestPath_level_edge(3)
% 0.06/0.31 thf(fact_52_incoming__edges,axiom,
% 0.06/0.31 ! [U: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming @ a @ c @ U ) @ ( e @ a @ c ) ) ).
% 0.06/0.31
% 0.06/0.31 % incoming_edges
% 0.06/0.31 thf(fact_53_reachable__ss__V,axiom,
% 0.06/0.31 ! [S: nat] :
% 0.06/0.31 ( ( member @ nat @ S @ ( v @ a @ c ) )
% 0.06/0.31 => ( ord_less_eq @ ( set @ nat ) @ ( reachableNodes @ a @ c @ S ) @ ( v @ a @ c ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % reachable_ss_V
% 0.06/0.31 thf(fact_54_connected__append__edge,axiom,
% 0.06/0.31 ! [U: nat,V: nat,W: nat] :
% 0.06/0.31 ( ( connected @ a @ c @ U @ V )
% 0.06/0.31 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V @ W ) @ ( e @ a @ c ) )
% 0.06/0.31 => ( connected @ a @ c @ U @ W ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % connected_append_edge
% 0.06/0.31 thf(fact_55_reachableNodes__append__edge,axiom,
% 0.06/0.31 ! [U: nat,S: nat,V: nat] :
% 0.06/0.31 ( ( member @ nat @ U @ ( reachableNodes @ a @ c @ S ) )
% 0.06/0.31 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( e @ a @ c ) )
% 0.06/0.31 => ( member @ nat @ V @ ( reachableNodes @ a @ c @ S ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % reachableNodes_append_edge
% 0.06/0.31 thf(fact_56_V__def,axiom,
% 0.06/0.31 ( ( v @ a @ c )
% 0.06/0.31 = ( collect @ nat
% 0.06/0.31 @ ^ [U3: nat] :
% 0.06/0.31 ? [V2: nat] :
% 0.06/0.31 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U3 @ V2 ) @ ( e @ a @ c ) )
% 0.06/0.31 | ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V2 @ U3 ) @ ( e @ a @ c ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % V_def
% 0.06/0.31 thf(fact_57_incoming__def,axiom,
% 0.06/0.31 ! [V: nat] :
% 0.06/0.31 ( ( incoming @ a @ c @ V )
% 0.06/0.31 = ( collect @ ( product_prod @ nat @ nat )
% 0.06/0.31 @ ^ [Uu: product_prod @ nat @ nat] :
% 0.06/0.31 ? [U3: nat] :
% 0.06/0.31 ( ( Uu
% 0.06/0.31 = ( product_Pair @ nat @ nat @ U3 @ V ) )
% 0.06/0.31 & ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U3 @ V ) @ ( e @ a @ c ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % incoming_def
% 0.06/0.31 thf(fact_58_outgoing_H__def,axiom,
% 0.06/0.31 ! [K: set @ nat] :
% 0.06/0.31 ( ( outgoing2 @ a @ c @ K )
% 0.06/0.31 = ( collect @ ( product_prod @ nat @ nat )
% 0.06/0.31 @ ^ [Uu: product_prod @ nat @ nat] :
% 0.06/0.31 ? [U3: nat,V2: nat] :
% 0.06/0.31 ( ( Uu
% 0.06/0.31 = ( product_Pair @ nat @ nat @ V2 @ U3 ) )
% 0.06/0.31 & ~ ( member @ nat @ U3 @ K )
% 0.06/0.31 & ( member @ nat @ V2 @ K )
% 0.06/0.31 & ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V2 @ U3 ) @ ( e @ a @ c ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % outgoing'_def
% 0.06/0.31 thf(fact_59_incoming_H__def,axiom,
% 0.06/0.31 ! [K: set @ nat] :
% 0.06/0.31 ( ( incoming2 @ a @ c @ K )
% 0.06/0.31 = ( collect @ ( product_prod @ nat @ nat )
% 0.06/0.31 @ ^ [Uu: product_prod @ nat @ nat] :
% 0.06/0.31 ? [U3: nat,V2: nat] :
% 0.06/0.31 ( ( Uu
% 0.06/0.31 = ( product_Pair @ nat @ nat @ U3 @ V2 ) )
% 0.06/0.31 & ~ ( member @ nat @ U3 @ K )
% 0.06/0.31 & ( member @ nat @ V2 @ K )
% 0.06/0.31 & ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U3 @ V2 ) @ ( e @ a @ c ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % incoming'_def
% 0.06/0.31 thf(fact_60_finite__Collect__le__nat,axiom,
% 0.06/0.31 ! [K: nat] :
% 0.06/0.31 ( finite_finite2 @ nat
% 0.06/0.31 @ ( collect @ nat
% 0.06/0.31 @ ^ [N: nat] : ( ord_less_eq @ nat @ N @ K ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_Collect_le_nat
% 0.06/0.31 thf(fact_61_finite__incoming,axiom,
% 0.06/0.31 ! [U: nat] :
% 0.06/0.31 ( ( finite_finite2 @ nat @ ( v @ a @ c ) )
% 0.06/0.31 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming @ a @ c @ U ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_incoming
% 0.06/0.31 thf(fact_62_finite__outgoing_H,axiom,
% 0.06/0.31 ! [U2: set @ nat] :
% 0.06/0.31 ( ( finite_finite2 @ nat @ ( v @ a @ c ) )
% 0.06/0.31 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing2 @ a @ c @ U2 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_outgoing'
% 0.06/0.31 thf(fact_63_finite__incoming_H,axiom,
% 0.06/0.31 ! [U2: set @ nat] :
% 0.06/0.31 ( ( finite_finite2 @ nat @ ( v @ a @ c ) )
% 0.06/0.31 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming2 @ a @ c @ U2 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_incoming'
% 0.06/0.31 thf(fact_64_adjacent__nodes__ss__V,axiom,
% 0.06/0.31 ! [U: nat] : ( ord_less_eq @ ( set @ nat ) @ ( adjacent_nodes @ a @ c @ U ) @ ( v @ a @ c ) ) ).
% 0.06/0.31
% 0.06/0.31 % adjacent_nodes_ss_V
% 0.06/0.31 thf(fact_65_Graph_Oincoming__def,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( incoming @ Capacity )
% 0.06/0.31 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,V2: nat] :
% 0.06/0.31 ( collect @ ( product_prod @ nat @ nat )
% 0.06/0.31 @ ^ [Uu: product_prod @ nat @ nat] :
% 0.06/0.31 ? [U3: nat] :
% 0.06/0.31 ( ( Uu
% 0.06/0.31 = ( product_Pair @ nat @ nat @ U3 @ V2 ) )
% 0.06/0.31 & ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U3 @ V2 ) @ ( e @ Capacity @ C3 ) ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.incoming_def
% 0.06/0.31 thf(fact_66_Graph_Oincoming_H__def,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( incoming2 @ Capacity )
% 0.06/0.31 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,K2: set @ nat] :
% 0.06/0.31 ( collect @ ( product_prod @ nat @ nat )
% 0.06/0.31 @ ^ [Uu: product_prod @ nat @ nat] :
% 0.06/0.31 ? [U3: nat,V2: nat] :
% 0.06/0.31 ( ( Uu
% 0.06/0.31 = ( product_Pair @ nat @ nat @ U3 @ V2 ) )
% 0.06/0.31 & ~ ( member @ nat @ U3 @ K2 )
% 0.06/0.31 & ( member @ nat @ V2 @ K2 )
% 0.06/0.31 & ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U3 @ V2 ) @ ( e @ Capacity @ C3 ) ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.incoming'_def
% 0.06/0.31 thf(fact_67_Graph_Ooutgoing_H__def,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( outgoing2 @ Capacity )
% 0.06/0.31 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,K2: set @ nat] :
% 0.06/0.31 ( collect @ ( product_prod @ nat @ nat )
% 0.06/0.31 @ ^ [Uu: product_prod @ nat @ nat] :
% 0.06/0.31 ? [U3: nat,V2: nat] :
% 0.06/0.31 ( ( Uu
% 0.06/0.31 = ( product_Pair @ nat @ nat @ V2 @ U3 ) )
% 0.06/0.31 & ~ ( member @ nat @ U3 @ K2 )
% 0.06/0.31 & ( member @ nat @ V2 @ K2 )
% 0.06/0.31 & ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V2 @ U3 ) @ ( e @ Capacity @ C3 ) ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.outgoing'_def
% 0.06/0.31 thf(fact_68_Graph_Oreachable__ss__V,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [S: nat,C2: ( product_prod @ nat @ nat ) > Capacity] :
% 0.06/0.31 ( ( member @ nat @ S @ ( v @ Capacity @ C2 ) )
% 0.06/0.31 => ( ord_less_eq @ ( set @ nat ) @ ( reachableNodes @ Capacity @ C2 @ S ) @ ( v @ Capacity @ C2 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.reachable_ss_V
% 0.06/0.31 thf(fact_69_Graph_OreachableNodes__append__edge,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [U: nat,C2: ( product_prod @ nat @ nat ) > Capacity,S: nat,V: nat] :
% 0.06/0.31 ( ( member @ nat @ U @ ( reachableNodes @ Capacity @ C2 @ S ) )
% 0.06/0.31 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( e @ Capacity @ C2 ) )
% 0.06/0.31 => ( member @ nat @ V @ ( reachableNodes @ Capacity @ C2 @ S ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.reachableNodes_append_edge
% 0.06/0.31 thf(fact_70_Graph_OreachableNodes_Ocong,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( reachableNodes @ Capacity )
% 0.06/0.31 = ( reachableNodes @ Capacity ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.reachableNodes.cong
% 0.06/0.31 thf(fact_71_Graph_Ooutgoing_H_Ocong,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( outgoing2 @ Capacity )
% 0.06/0.31 = ( outgoing2 @ Capacity ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.outgoing'.cong
% 0.06/0.31 thf(fact_72_Graph_Oincoming_H_Ocong,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( incoming2 @ Capacity )
% 0.06/0.31 = ( incoming2 @ Capacity ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.incoming'.cong
% 0.06/0.31 thf(fact_73_Graph_Oincoming_Ocong,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( incoming @ Capacity )
% 0.06/0.31 = ( incoming @ Capacity ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.incoming.cong
% 0.06/0.31 thf(fact_74_pred__equals__eq2,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( ( ( ^ [X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R2 ) )
% 0.06/0.31 = ( ^ [X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ S3 ) ) )
% 0.06/0.31 = ( R2 = S3 ) ) ).
% 0.06/0.31
% 0.06/0.31 % pred_equals_eq2
% 0.06/0.31 thf(fact_75_pred__subset__eq2,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( ( ord_less_eq @ ( A > B > $o )
% 0.06/0.31 @ ^ [X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R2 )
% 0.06/0.31 @ ^ [X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ S3 ) )
% 0.06/0.31 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ).
% 0.06/0.31
% 0.06/0.31 % pred_subset_eq2
% 0.06/0.31 thf(fact_76_Graph_OV__def,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( v @ Capacity )
% 0.06/0.31 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity] :
% 0.06/0.31 ( collect @ nat
% 0.06/0.31 @ ^ [U3: nat] :
% 0.06/0.31 ? [V2: nat] :
% 0.06/0.31 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U3 @ V2 ) @ ( e @ Capacity @ C3 ) )
% 0.06/0.31 | ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V2 @ U3 ) @ ( e @ Capacity @ C3 ) ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.V_def
% 0.06/0.31 thf(fact_77_subrelI,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( ! [X: A,Y2: B] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R )
% 0.06/0.31 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S ) )
% 0.06/0.31 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 0.06/0.31
% 0.06/0.31 % subrelI
% 0.06/0.31 thf(fact_78_Domain_Oinducts,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,X4: A,R: set @ ( product_prod @ A @ B ),P: A > $o] :
% 0.06/0.31 ( ( member @ A @ X4 @ ( domain @ A @ B @ R ) )
% 0.06/0.31 => ( ! [A6: A,B4: B] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B4 ) @ R )
% 0.06/0.31 => ( P @ A6 ) )
% 0.06/0.31 => ( P @ X4 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Domain.inducts
% 0.06/0.31 thf(fact_79_Domain_ODomainI,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A5: A,B5: B,R: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R )
% 0.06/0.31 => ( member @ A @ A5 @ ( domain @ A @ B @ R ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Domain.DomainI
% 0.06/0.31 thf(fact_80_Domain_Osimps,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A5: A,R: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( ( member @ A @ A5 @ ( domain @ A @ B @ R ) )
% 0.06/0.31 = ( ? [A4: A,B6: B] :
% 0.06/0.31 ( ( A5 = A4 )
% 0.06/0.31 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B6 ) @ R ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Domain.simps
% 0.06/0.31 thf(fact_81_Domain_Ocases,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A5: A,R: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( ( member @ A @ A5 @ ( domain @ A @ B @ R ) )
% 0.06/0.31 => ~ ! [B4: B] :
% 0.06/0.31 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ R ) ) ).
% 0.06/0.31
% 0.06/0.31 % Domain.cases
% 0.06/0.31 thf(fact_82_Domain__iff,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,A5: A,R: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( ( member @ A @ A5 @ ( domain @ A @ B @ R ) )
% 0.06/0.31 = ( ? [Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ Y ) @ R ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Domain_iff
% 0.06/0.31 thf(fact_83_DomainE,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A5: A,R: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( ( member @ A @ A5 @ ( domain @ A @ B @ R ) )
% 0.06/0.31 => ~ ! [B4: B] :
% 0.06/0.31 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ R ) ) ).
% 0.06/0.31
% 0.06/0.31 % DomainE
% 0.06/0.31 thf(fact_84_Range_Oinducts,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,X4: B,R: set @ ( product_prod @ A @ B ),P: B > $o] :
% 0.06/0.31 ( ( member @ B @ X4 @ ( range @ A @ B @ R ) )
% 0.06/0.31 => ( ! [A6: A,B4: B] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B4 ) @ R )
% 0.06/0.31 => ( P @ B4 ) )
% 0.06/0.31 => ( P @ X4 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Range.inducts
% 0.06/0.31 thf(fact_85_Range_Ointros,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A5: A,B5: B,R: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R )
% 0.06/0.31 => ( member @ B @ B5 @ ( range @ A @ B @ R ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Range.intros
% 0.06/0.31 thf(fact_86_Range_Osimps,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A5: B,R: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( ( member @ B @ A5 @ ( range @ A @ B @ R ) )
% 0.06/0.31 = ( ? [A4: A,B6: B] :
% 0.06/0.31 ( ( A5 = B6 )
% 0.06/0.31 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B6 ) @ R ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Range.simps
% 0.06/0.31 thf(fact_87_Range_Ocases,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A5: B,R: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( ( member @ B @ A5 @ ( range @ A @ B @ R ) )
% 0.06/0.31 => ~ ! [A6: A] :
% 0.06/0.31 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ A5 ) @ R ) ) ).
% 0.06/0.31
% 0.06/0.31 % Range.cases
% 0.06/0.31 thf(fact_88_Range__iff,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,A5: A,R: set @ ( product_prod @ B @ A )] :
% 0.06/0.31 ( ( member @ A @ A5 @ ( range @ B @ A @ R ) )
% 0.06/0.31 = ( ? [Y: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y @ A5 ) @ R ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Range_iff
% 0.06/0.31 thf(fact_89_RangeE,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,B5: A,R: set @ ( product_prod @ B @ A )] :
% 0.06/0.31 ( ( member @ A @ B5 @ ( range @ B @ A @ R ) )
% 0.06/0.31 => ~ ! [A6: B] :
% 0.06/0.31 ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A6 @ B5 ) @ R ) ) ).
% 0.06/0.31
% 0.06/0.31 % RangeE
% 0.06/0.31 thf(fact_90_Graph_OreachableNodes__def,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( reachableNodes @ Capacity )
% 0.06/0.31 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,U3: nat] : ( collect @ nat @ ( connected @ Capacity @ C3 @ U3 ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.reachableNodes_def
% 0.06/0.31 thf(fact_91_pred__subset__eq,axiom,
% 0.06/0.31 ! [A: $tType,R2: set @ A,S3: set @ A] :
% 0.06/0.31 ( ( ord_less_eq @ ( A > $o )
% 0.06/0.31 @ ^ [X3: A] : ( member @ A @ X3 @ R2 )
% 0.06/0.31 @ ^ [X3: A] : ( member @ A @ X3 @ S3 ) )
% 0.06/0.31 = ( ord_less_eq @ ( set @ A ) @ R2 @ S3 ) ) ).
% 0.06/0.31
% 0.06/0.31 % pred_subset_eq
% 0.06/0.31 thf(fact_92_Graph_Oconnected__append__edge,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat,W: nat] :
% 0.06/0.31 ( ( connected @ Capacity @ C2 @ U @ V )
% 0.06/0.31 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V @ W ) @ ( e @ Capacity @ C2 ) )
% 0.06/0.31 => ( connected @ Capacity @ C2 @ U @ W ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.connected_append_edge
% 0.06/0.31 thf(fact_93_Graph_Oincoming__edges,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming @ Capacity @ C2 @ U ) @ ( e @ Capacity @ C2 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.incoming_edges
% 0.06/0.31 thf(fact_94_Graph_OisShortestPath__level__edge_I1_J,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat,U: nat,V: nat] :
% 0.06/0.31 ( ( isShortestPath @ Capacity @ C2 @ S @ P4 @ T4 )
% 0.06/0.31 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.31 => ( connected @ Capacity @ C2 @ S @ U ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.isShortestPath_level_edge(1)
% 0.06/0.31 thf(fact_95_Graph_OisShortestPath__level__edge_I2_J,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat,U: nat,V: nat] :
% 0.06/0.31 ( ( isShortestPath @ Capacity @ C2 @ S @ P4 @ T4 )
% 0.06/0.31 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.31 => ( connected @ Capacity @ C2 @ U @ V ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.isShortestPath_level_edge(2)
% 0.06/0.31 thf(fact_96_Graph_OisShortestPath__level__edge_I3_J,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat,U: nat,V: nat] :
% 0.06/0.31 ( ( isShortestPath @ Capacity @ C2 @ S @ P4 @ T4 )
% 0.06/0.31 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.31 => ( connected @ Capacity @ C2 @ V @ T4 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.isShortestPath_level_edge(3)
% 0.06/0.31 thf(fact_97_Graph_Ooutgoing_H__edges,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U2: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing2 @ Capacity @ C2 @ U2 ) @ ( e @ Capacity @ C2 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.outgoing'_edges
% 0.06/0.31 thf(fact_98_Graph_Oincoming_H__edges,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U2: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming2 @ Capacity @ C2 @ U2 ) @ ( e @ Capacity @ C2 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.incoming'_edges
% 0.06/0.31 thf(fact_99_Domain__unfold,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType] :
% 0.06/0.31 ( ( domain @ A @ B )
% 0.06/0.31 = ( ^ [R3: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( collect @ A
% 0.06/0.31 @ ^ [X3: A] :
% 0.06/0.31 ? [Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R3 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Domain_unfold
% 0.06/0.31 thf(fact_100_Graph_Ofinite__incoming,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat] :
% 0.06/0.31 ( ( finite_finite2 @ nat @ ( v @ Capacity @ C2 ) )
% 0.06/0.31 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming @ Capacity @ C2 @ U ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.finite_incoming
% 0.06/0.31 thf(fact_101_Graph_OE_Ocong,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( e @ Capacity )
% 0.06/0.31 = ( e @ Capacity ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.E.cong
% 0.06/0.31 thf(fact_102_Graph_OV_Ocong,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( v @ Capacity )
% 0.06/0.31 = ( v @ Capacity ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.V.cong
% 0.06/0.31 thf(fact_103_Graph_Oconnected__refl,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,V: nat] : ( connected @ Capacity @ C2 @ V @ V ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.connected_refl
% 0.06/0.31 thf(fact_104_Graph_Oconnected_Ocong,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( connected @ Capacity )
% 0.06/0.31 = ( connected @ Capacity ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.connected.cong
% 0.06/0.31 thf(fact_105_Graph_Ofinite__outgoing_H,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U2: set @ nat] :
% 0.06/0.31 ( ( finite_finite2 @ nat @ ( v @ Capacity @ C2 ) )
% 0.06/0.31 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing2 @ Capacity @ C2 @ U2 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.finite_outgoing'
% 0.06/0.31 thf(fact_106_Graph_OisShortestPath_Ocong,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( isShortestPath @ Capacity )
% 0.06/0.31 = ( isShortestPath @ Capacity ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.isShortestPath.cong
% 0.06/0.31 thf(fact_107_Graph_Ofinite__incoming_H,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U2: set @ nat] :
% 0.06/0.31 ( ( finite_finite2 @ nat @ ( v @ Capacity @ C2 ) )
% 0.06/0.31 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming2 @ Capacity @ C2 @ U2 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.finite_incoming'
% 0.06/0.31 thf(fact_108_finite__list,axiom,
% 0.06/0.31 ! [A: $tType,A3: set @ A] :
% 0.06/0.31 ( ( finite_finite2 @ A @ A3 )
% 0.06/0.31 => ? [Xs2: list @ A] :
% 0.06/0.31 ( ( set2 @ A @ Xs2 )
% 0.06/0.31 = A3 ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_list
% 0.06/0.31 thf(fact_109_subset__code_I1_J,axiom,
% 0.06/0.31 ! [A: $tType,Xs: list @ A,B3: set @ A] :
% 0.06/0.31 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B3 )
% 0.06/0.31 = ( ! [X3: A] :
% 0.06/0.31 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.06/0.31 => ( member @ A @ X3 @ B3 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % subset_code(1)
% 0.06/0.31 thf(fact_110_Graph_Oconnected__inV__iff,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat] :
% 0.06/0.31 ( ( connected @ Capacity @ C2 @ U @ V )
% 0.06/0.31 => ( ( member @ nat @ V @ ( v @ Capacity @ C2 ) )
% 0.06/0.31 = ( member @ nat @ U @ ( v @ Capacity @ C2 ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.connected_inV_iff
% 0.06/0.31 thf(fact_111_Graph_Oobtain__shortest__path,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat] :
% 0.06/0.31 ( ( connected @ Capacity @ C2 @ U @ V )
% 0.06/0.31 => ~ ! [P3: list @ ( product_prod @ nat @ nat )] :
% 0.06/0.31 ~ ( isShortestPath @ Capacity @ C2 @ U @ P3 @ V ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.obtain_shortest_path
% 0.06/0.31 thf(fact_112_Finite__Graph_Ofinite__V,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( linordered_idom @ A )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > A] :
% 0.06/0.31 ( ( finite_Graph @ A @ C2 )
% 0.06/0.31 => ( finite_finite2 @ nat @ ( v @ A @ C2 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Finite_Graph.finite_V
% 0.06/0.31 thf(fact_113_Finite__Graph_Ointro,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( linordered_idom @ A )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > A] :
% 0.06/0.31 ( ( finite_finite2 @ nat @ ( v @ A @ C2 ) )
% 0.06/0.31 => ( finite_Graph @ A @ C2 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Finite_Graph.intro
% 0.06/0.31 thf(fact_114_Finite__Graph__def,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( linordered_idom @ A )
% 0.06/0.31 => ( ( finite_Graph @ A )
% 0.06/0.31 = ( ^ [C3: ( product_prod @ nat @ nat ) > A] : ( finite_finite2 @ nat @ ( v @ A @ C3 ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Finite_Graph_def
% 0.06/0.31 thf(fact_115_Finite__Graph_Ofinite__E,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( linordered_idom @ A )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > A] :
% 0.06/0.31 ( ( finite_Graph @ A @ C2 )
% 0.06/0.31 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ A @ C2 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Finite_Graph.finite_E
% 0.06/0.31 thf(fact_116_Graph_OFinite__Graph__EI,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity] :
% 0.06/0.31 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ Capacity @ C2 ) )
% 0.06/0.31 => ( finite_Graph @ Capacity @ C2 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.Finite_Graph_EI
% 0.06/0.31 thf(fact_117_outgoing__def,axiom,
% 0.06/0.31 ! [V: nat] :
% 0.06/0.31 ( ( outgoing @ a @ c @ V )
% 0.06/0.31 = ( collect @ ( product_prod @ nat @ nat )
% 0.06/0.31 @ ^ [Uu: product_prod @ nat @ nat] :
% 0.06/0.31 ? [U3: nat] :
% 0.06/0.31 ( ( Uu
% 0.06/0.31 = ( product_Pair @ nat @ nat @ V @ U3 ) )
% 0.06/0.31 & ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V @ U3 ) @ ( e @ a @ c ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % outgoing_def
% 0.06/0.31 thf(fact_118_outgoing__edges,axiom,
% 0.06/0.31 ! [U: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing @ a @ c @ U ) @ ( e @ a @ c ) ) ).
% 0.06/0.31
% 0.06/0.31 % outgoing_edges
% 0.06/0.31 thf(fact_119_E__ss__VxV,axiom,
% 0.06/0.31 ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( e @ a @ c )
% 0.06/0.31 @ ( product_Sigma @ nat @ nat @ ( v @ a @ c )
% 0.06/0.31 @ ^ [Uu: nat] : ( v @ a @ c ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % E_ss_VxV
% 0.06/0.31 thf(fact_120_connected__edgeRtc,axiom,
% 0.06/0.31 ! [U: nat,V: nat] :
% 0.06/0.31 ( ( connected @ a @ c @ U @ V )
% 0.06/0.31 = ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( transitive_rtrancl @ nat @ ( e @ a @ c ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % connected_edgeRtc
% 0.06/0.31 thf(fact_121_isSPath__no__selfloop,axiom,
% 0.06/0.31 ! [U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat] :
% 0.06/0.31 ( ( isSimplePath @ a @ c @ U @ P4 @ V )
% 0.06/0.31 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % isSPath_no_selfloop
% 0.06/0.31 thf(fact_122_isSPath__sg__incoming,axiom,
% 0.06/0.31 ! [U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,U22: nat] :
% 0.06/0.31 ( ( isSimplePath @ a @ c @ U @ P4 @ V )
% 0.06/0.31 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.31 => ( ( U1 != U22 )
% 0.06/0.31 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U22 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % isSPath_sg_incoming
% 0.06/0.31 thf(fact_123_isSPath__sg__outgoing,axiom,
% 0.06/0.31 ! [U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,V22: nat] :
% 0.06/0.31 ( ( isSimplePath @ a @ c @ U @ P4 @ V )
% 0.06/0.31 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.31 => ( ( V1 != V22 )
% 0.06/0.31 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V22 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % isSPath_sg_outgoing
% 0.06/0.31 thf(fact_124_reachableNodes__E__closed,axiom,
% 0.06/0.31 ! [S: nat] : ( ord_less_eq @ ( set @ nat ) @ ( image @ nat @ nat @ ( e @ a @ c ) @ ( reachableNodes @ a @ c @ S ) ) @ ( reachableNodes @ a @ c @ S ) ) ).
% 0.06/0.31
% 0.06/0.31 % reachableNodes_E_closed
% 0.06/0.31 thf(fact_125_shortestPath__is__simple,axiom,
% 0.06/0.31 ! [S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat] :
% 0.06/0.31 ( ( isShortestPath @ a @ c @ S @ P4 @ T4 )
% 0.06/0.31 => ( isSimplePath @ a @ c @ S @ P4 @ T4 ) ) ).
% 0.06/0.31
% 0.06/0.31 % shortestPath_is_simple
% 0.06/0.31 thf(fact_126_ImageI,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A5: A,B5: B,R: set @ ( product_prod @ A @ B ),A3: set @ A] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R )
% 0.06/0.31 => ( ( member @ A @ A5 @ A3 )
% 0.06/0.31 => ( member @ B @ B5 @ ( image @ A @ B @ R @ A3 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % ImageI
% 0.06/0.31 thf(fact_127_finite__SigmaI,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] :
% 0.06/0.31 ( ( finite_finite2 @ A @ A3 )
% 0.06/0.31 => ( ! [A6: A] :
% 0.06/0.31 ( ( member @ A @ A6 @ A3 )
% 0.06/0.31 => ( finite_finite2 @ B @ ( B3 @ A6 ) ) )
% 0.06/0.31 => ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_SigmaI
% 0.06/0.31 thf(fact_128_adjacent__nodes__finite,axiom,
% 0.06/0.31 ! [U: nat] : ( finite_finite2 @ nat @ ( adjacent_nodes @ a @ c @ U ) ) ).
% 0.06/0.31
% 0.06/0.31 % adjacent_nodes_finite
% 0.06/0.31 thf(fact_129_finite__outgoing,axiom,
% 0.06/0.31 ! [U: nat] :
% 0.06/0.31 ( ( finite_finite2 @ nat @ ( v @ a @ c ) )
% 0.06/0.31 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing @ a @ c @ U ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_outgoing
% 0.06/0.31 thf(fact_130_Graph_Oadjacent__nodes_Ocong,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( adjacent_nodes @ Capacity )
% 0.06/0.31 = ( adjacent_nodes @ Capacity ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.adjacent_nodes.cong
% 0.06/0.31 thf(fact_131_Graph_OisSimplePath_Ocong,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( isSimplePath @ Capacity )
% 0.06/0.31 = ( isSimplePath @ Capacity ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.isSimplePath.cong
% 0.06/0.31 thf(fact_132_Graph_Ooutgoing_Ocong,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( outgoing @ Capacity )
% 0.06/0.31 = ( outgoing @ Capacity ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.outgoing.cong
% 0.06/0.31 thf(fact_133_rev__ImageI,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A5: A,A3: set @ A,B5: B,R: set @ ( product_prod @ A @ B )] :
% 0.06/0.31 ( ( member @ A @ A5 @ A3 )
% 0.06/0.31 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R )
% 0.06/0.31 => ( member @ B @ B5 @ ( image @ A @ B @ R @ A3 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % rev_ImageI
% 0.06/0.31 thf(fact_134_Image__iff,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,B5: A,R: set @ ( product_prod @ B @ A ),A3: set @ B] :
% 0.06/0.31 ( ( member @ A @ B5 @ ( image @ B @ A @ R @ A3 ) )
% 0.06/0.31 = ( ? [X3: B] :
% 0.06/0.31 ( ( member @ B @ X3 @ A3 )
% 0.06/0.31 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ B5 ) @ R ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Image_iff
% 0.06/0.31 thf(fact_135_ImageE,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,B5: A,R: set @ ( product_prod @ B @ A ),A3: set @ B] :
% 0.06/0.31 ( ( member @ A @ B5 @ ( image @ B @ A @ R @ A3 ) )
% 0.06/0.31 => ~ ! [X: B] :
% 0.06/0.31 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ B5 ) @ R )
% 0.06/0.31 => ~ ( member @ B @ X @ A3 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % ImageE
% 0.06/0.31 thf(fact_136_Image__subset,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),A3: set @ A,B3: set @ B,C4: set @ A] :
% 0.06/0.31 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R
% 0.06/0.31 @ ( product_Sigma @ A @ B @ A3
% 0.06/0.31 @ ^ [Uu: A] : B3 ) )
% 0.06/0.31 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R @ C4 ) @ B3 ) ) ).
% 0.06/0.31
% 0.06/0.31 % Image_subset
% 0.06/0.31 thf(fact_137_Graph_OshortestPath__is__simple,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat] :
% 0.06/0.31 ( ( isShortestPath @ Capacity @ C2 @ S @ P4 @ T4 )
% 0.06/0.31 => ( isSimplePath @ Capacity @ C2 @ S @ P4 @ T4 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.shortestPath_is_simple
% 0.06/0.31 thf(fact_138_Image__mono,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),R: set @ ( product_prod @ A @ B ),A7: set @ A,A3: set @ A] :
% 0.06/0.31 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ R )
% 0.06/0.31 => ( ( ord_less_eq @ ( set @ A ) @ A7 @ A3 )
% 0.06/0.31 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R4 @ A7 ) @ ( image @ A @ B @ R @ A3 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Image_mono
% 0.06/0.31 thf(fact_139_Relation_Ofinite__Image,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),A3: set @ A] :
% 0.06/0.31 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ R2 )
% 0.06/0.31 => ( finite_finite2 @ B @ ( image @ A @ B @ R2 @ A3 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Relation.finite_Image
% 0.06/0.31 thf(fact_140_Graph_OreachableNodes__E__closed,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,S: nat] : ( ord_less_eq @ ( set @ nat ) @ ( image @ nat @ nat @ ( e @ Capacity @ C2 ) @ ( reachableNodes @ Capacity @ C2 @ S ) ) @ ( reachableNodes @ Capacity @ C2 @ S ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.reachableNodes_E_closed
% 0.06/0.31 thf(fact_141_finite__cartesian__product,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B] :
% 0.06/0.31 ( ( finite_finite2 @ A @ A3 )
% 0.06/0.31 => ( ( finite_finite2 @ B @ B3 )
% 0.06/0.31 => ( finite_finite2 @ ( product_prod @ A @ B )
% 0.06/0.31 @ ( product_Sigma @ A @ B @ A3
% 0.06/0.31 @ ^ [Uu: A] : B3 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_cartesian_product
% 0.06/0.31 thf(fact_142_Graph_OisSPath__sg__outgoing,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,V22: nat] :
% 0.06/0.31 ( ( isSimplePath @ Capacity @ C2 @ U @ P4 @ V )
% 0.06/0.31 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.31 => ( ( V1 != V22 )
% 0.06/0.31 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V22 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.isSPath_sg_outgoing
% 0.06/0.31 thf(fact_143_Graph_OisSPath__sg__incoming,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,U22: nat] :
% 0.06/0.31 ( ( isSimplePath @ Capacity @ C2 @ U @ P4 @ V )
% 0.06/0.31 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.31 => ( ( U1 != U22 )
% 0.06/0.31 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U22 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.isSPath_sg_incoming
% 0.06/0.31 thf(fact_144_Graph_OisSPath__no__selfloop,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat] :
% 0.06/0.31 ( ( isSimplePath @ Capacity @ C2 @ U @ P4 @ V )
% 0.06/0.31 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.isSPath_no_selfloop
% 0.06/0.31 thf(fact_145_Graph_Oadjacent__nodes__ss__V,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat] : ( ord_less_eq @ ( set @ nat ) @ ( adjacent_nodes @ Capacity @ C2 @ U ) @ ( v @ Capacity @ C2 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.adjacent_nodes_ss_V
% 0.06/0.31 thf(fact_146_Graph_Ooutgoing__edges,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing @ Capacity @ C2 @ U ) @ ( e @ Capacity @ C2 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.outgoing_edges
% 0.06/0.31 thf(fact_147_Graph_Ooutgoing__def,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( outgoing @ Capacity )
% 0.06/0.31 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,V2: nat] :
% 0.06/0.31 ( collect @ ( product_prod @ nat @ nat )
% 0.06/0.31 @ ^ [Uu: product_prod @ nat @ nat] :
% 0.06/0.31 ? [U3: nat] :
% 0.06/0.31 ( ( Uu
% 0.06/0.31 = ( product_Pair @ nat @ nat @ V2 @ U3 ) )
% 0.06/0.31 & ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V2 @ U3 ) @ ( e @ Capacity @ C3 ) ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.outgoing_def
% 0.06/0.31 thf(fact_148_Graph_Oconnected__edgeRtc,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ( ( connected @ Capacity )
% 0.06/0.31 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,U3: nat,V2: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U3 @ V2 ) @ ( transitive_rtrancl @ nat @ ( e @ Capacity @ C3 ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.connected_edgeRtc
% 0.06/0.31 thf(fact_149_Graph_OE__ss__VxV,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity] :
% 0.06/0.31 ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( e @ Capacity @ C2 )
% 0.06/0.31 @ ( product_Sigma @ nat @ nat @ ( v @ Capacity @ C2 )
% 0.06/0.31 @ ^ [Uu: nat] : ( v @ Capacity @ C2 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.E_ss_VxV
% 0.06/0.31 thf(fact_150_Graph_Ofinite__outgoing,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat] :
% 0.06/0.31 ( ( finite_finite2 @ nat @ ( v @ Capacity @ C2 ) )
% 0.06/0.31 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing @ Capacity @ C2 @ U ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.finite_outgoing
% 0.06/0.31 thf(fact_151_Finite__Graph_Oadjacent__nodes__finite,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( linordered_idom @ A )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > A,U: nat] :
% 0.06/0.31 ( ( finite_Graph @ A @ C2 )
% 0.06/0.31 => ( finite_finite2 @ nat @ ( adjacent_nodes @ A @ C2 @ U ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Finite_Graph.adjacent_nodes_finite
% 0.06/0.31 thf(fact_152_isSPath__nt__parallel,axiom,
% 0.06/0.31 ! [S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat,E2: product_prod @ nat @ nat] :
% 0.06/0.31 ( ( isSimplePath @ a @ c @ S @ P4 @ T4 )
% 0.06/0.31 => ( ( member @ ( product_prod @ nat @ nat ) @ E2 @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.31 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ E2 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % isSPath_nt_parallel
% 0.06/0.31 thf(fact_153_mem__Sigma__iff,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A5: A,B5: B,A3: set @ A,B3: A > ( set @ B )] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 0.06/0.31 = ( ( member @ A @ A5 @ A3 )
% 0.06/0.31 & ( member @ B @ B5 @ ( B3 @ A5 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % mem_Sigma_iff
% 0.06/0.31 thf(fact_154_SigmaI,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A5: A,A3: set @ A,B5: B,B3: A > ( set @ B )] :
% 0.06/0.31 ( ( member @ A @ A5 @ A3 )
% 0.06/0.31 => ( ( member @ B @ B5 @ ( B3 @ A5 ) )
% 0.06/0.31 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % SigmaI
% 0.06/0.31 thf(fact_155_trancl__subset__Sigma__aux,axiom,
% 0.06/0.31 ! [A: $tType,A5: A,B5: A,R: set @ ( product_prod @ A @ A ),A3: set @ A] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.06/0.31 @ ( product_Sigma @ A @ A @ A3
% 0.06/0.31 @ ^ [Uu: A] : A3 ) )
% 0.06/0.31 => ( ( A5 = B5 )
% 0.06/0.31 | ( member @ A @ A5 @ A3 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % trancl_subset_Sigma_aux
% 0.06/0.31 thf(fact_156_old_Oprod_Oinject,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,A5: A,B5: B,A8: A,B7: B] :
% 0.06/0.31 ( ( ( product_Pair @ A @ B @ A5 @ B5 )
% 0.06/0.31 = ( product_Pair @ A @ B @ A8 @ B7 ) )
% 0.06/0.31 = ( ( A5 = A8 )
% 0.06/0.31 & ( B5 = B7 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % old.prod.inject
% 0.06/0.31 thf(fact_157_prod_Oinject,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 0.06/0.31 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.06/0.31 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.06/0.31 = ( ( X1 = Y1 )
% 0.06/0.31 & ( X22 = Y22 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % prod.inject
% 0.06/0.31 thf(fact_158_r__into__rtrancl,axiom,
% 0.06/0.31 ! [A: $tType,P4: product_prod @ A @ A,R: set @ ( product_prod @ A @ A )] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ P4 @ R )
% 0.06/0.31 => ( member @ ( product_prod @ A @ A ) @ P4 @ ( transitive_rtrancl @ A @ R ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % r_into_rtrancl
% 0.06/0.31 thf(fact_159_rtrancl__idemp,axiom,
% 0.06/0.31 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.06/0.31 ( ( transitive_rtrancl @ A @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 = ( transitive_rtrancl @ A @ R ) ) ).
% 0.06/0.31
% 0.06/0.31 % rtrancl_idemp
% 0.06/0.31 thf(fact_160_swap__swap,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,P4: product_prod @ A @ B] :
% 0.06/0.31 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P4 ) )
% 0.06/0.31 = P4 ) ).
% 0.06/0.31
% 0.06/0.31 % swap_swap
% 0.06/0.31 thf(fact_161_swap__simp,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,X4: B,Y3: A] :
% 0.06/0.31 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y3 ) )
% 0.06/0.31 = ( product_Pair @ A @ B @ Y3 @ X4 ) ) ).
% 0.06/0.31
% 0.06/0.31 % swap_simp
% 0.06/0.31 thf(fact_162_converse__rtrancl__induct2,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
% 0.06/0.31 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 0.06/0.31 => ( ( P @ Bx @ By )
% 0.06/0.31 => ( ! [A6: A,B4: B,Aa: A,Ba: B] :
% 0.06/0.31 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B4 ) @ ( product_Pair @ A @ B @ Aa @ Ba ) ) @ R )
% 0.06/0.31 => ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Aa @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 0.06/0.31 => ( ( P @ Aa @ Ba )
% 0.06/0.31 => ( P @ A6 @ B4 ) ) ) )
% 0.06/0.31 => ( P @ Ax @ Ay ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % converse_rtrancl_induct2
% 0.06/0.31 thf(fact_163_converse__rtranclE2,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,Xa2: A,Xb: B,Za: A,Zb: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )] :
% 0.06/0.31 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa2 @ Xb ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 0.06/0.31 => ( ( ( product_Pair @ A @ B @ Xa2 @ Xb )
% 0.06/0.31 != ( product_Pair @ A @ B @ Za @ Zb ) )
% 0.06/0.31 => ~ ! [A6: A,B4: B] :
% 0.06/0.31 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa2 @ Xb ) @ ( product_Pair @ A @ B @ A6 @ B4 ) ) @ R )
% 0.06/0.31 => ~ ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B4 ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % converse_rtranclE2
% 0.06/0.31 thf(fact_164_rtrancl__induct2,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
% 0.06/0.31 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 0.06/0.31 => ( ( P @ Ax @ Ay )
% 0.06/0.31 => ( ! [A6: A,B4: B,Aa: A,Ba: B] :
% 0.06/0.31 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A6 @ B4 ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 0.06/0.31 => ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B4 ) @ ( product_Pair @ A @ B @ Aa @ Ba ) ) @ R )
% 0.06/0.31 => ( ( P @ A6 @ B4 )
% 0.06/0.31 => ( P @ Aa @ Ba ) ) ) )
% 0.06/0.31 => ( P @ Bx @ By ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % rtrancl_induct2
% 0.06/0.31 thf(fact_165_old_Oprod_Oinducts,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.06/0.31 ( ! [A6: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B4 ) )
% 0.06/0.31 => ( P @ Prod ) ) ).
% 0.06/0.31
% 0.06/0.31 % old.prod.inducts
% 0.06/0.31 thf(fact_166_old_Oprod_Oexhaust,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 0.06/0.31 ~ ! [A6: A,B4: B] :
% 0.06/0.31 ( Y3
% 0.06/0.31 != ( product_Pair @ A @ B @ A6 @ B4 ) ) ).
% 0.06/0.31
% 0.06/0.31 % old.prod.exhaust
% 0.06/0.31 thf(fact_167_prod__induct7,axiom,
% 0.06/0.31 ! [G2: $tType,F3: $tType,E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
% 0.06/0.31 ( ! [A6: A,B4: B,C5: C,D2: D,E4: E3,F: F3,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) @ D2 @ ( product_Pair @ E3 @ ( product_prod @ F3 @ G2 ) @ E4 @ ( product_Pair @ F3 @ G2 @ F @ G3 ) ) ) ) ) ) )
% 0.06/0.31 => ( P @ X4 ) ) ).
% 0.06/0.31
% 0.06/0.31 % prod_induct7
% 0.06/0.31 thf(fact_168_prod__induct6,axiom,
% 0.06/0.31 ! [F3: $tType,E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) ) )] :
% 0.06/0.31 ( ! [A6: A,B4: B,C5: C,D2: D,E4: E3,F: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ F3 ) @ D2 @ ( product_Pair @ E3 @ F3 @ E4 @ F ) ) ) ) ) )
% 0.06/0.31 => ( P @ X4 ) ) ).
% 0.06/0.31
% 0.06/0.31 % prod_induct6
% 0.06/0.31 thf(fact_169_prod__induct5,axiom,
% 0.06/0.31 ! [E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) )] :
% 0.06/0.31 ( ! [A6: A,B4: B,C5: C,D2: D,E4: E3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E3 ) @ C5 @ ( product_Pair @ D @ E3 @ D2 @ E4 ) ) ) ) )
% 0.06/0.31 => ( P @ X4 ) ) ).
% 0.06/0.31
% 0.06/0.31 % prod_induct5
% 0.06/0.31 thf(fact_170_prod__induct4,axiom,
% 0.06/0.31 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.06/0.31 ( ! [A6: A,B4: B,C5: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C5 @ D2 ) ) ) )
% 0.06/0.31 => ( P @ X4 ) ) ).
% 0.06/0.31
% 0.06/0.31 % prod_induct4
% 0.06/0.31 thf(fact_171_prod__induct3,axiom,
% 0.06/0.31 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.06/0.31 ( ! [A6: A,B4: B,C5: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B4 @ C5 ) ) )
% 0.06/0.31 => ( P @ X4 ) ) ).
% 0.06/0.31
% 0.06/0.31 % prod_induct3
% 0.06/0.31 thf(fact_172_prod__cases7,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,F3: $tType,G2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
% 0.06/0.31 ~ ! [A6: A,B4: B,C5: C,D2: D,E4: E3,F: F3,G3: G2] :
% 0.06/0.31 ( Y3
% 0.06/0.31 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) @ D2 @ ( product_Pair @ E3 @ ( product_prod @ F3 @ G2 ) @ E4 @ ( product_Pair @ F3 @ G2 @ F @ G3 ) ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % prod_cases7
% 0.06/0.31 thf(fact_173_prod__cases6,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,F3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) ) )] :
% 0.06/0.31 ~ ! [A6: A,B4: B,C5: C,D2: D,E4: E3,F: F3] :
% 0.06/0.31 ( Y3
% 0.06/0.31 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ F3 ) @ D2 @ ( product_Pair @ E3 @ F3 @ E4 @ F ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % prod_cases6
% 0.06/0.31 thf(fact_174_prod__cases5,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) )] :
% 0.06/0.31 ~ ! [A6: A,B4: B,C5: C,D2: D,E4: E3] :
% 0.06/0.31 ( Y3
% 0.06/0.31 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E3 ) @ C5 @ ( product_Pair @ D @ E3 @ D2 @ E4 ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % prod_cases5
% 0.06/0.31 thf(fact_175_prod__cases4,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.06/0.31 ~ ! [A6: A,B4: B,C5: C,D2: D] :
% 0.06/0.31 ( Y3
% 0.06/0.31 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C5 @ D2 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % prod_cases4
% 0.06/0.31 thf(fact_176_prod__cases3,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.06/0.31 ~ ! [A6: A,B4: B,C5: C] :
% 0.06/0.31 ( Y3
% 0.06/0.31 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B4 @ C5 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % prod_cases3
% 0.06/0.31 thf(fact_177_Pair__inject,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,A5: A,B5: B,A8: A,B7: B] :
% 0.06/0.31 ( ( ( product_Pair @ A @ B @ A5 @ B5 )
% 0.06/0.31 = ( product_Pair @ A @ B @ A8 @ B7 ) )
% 0.06/0.31 => ~ ( ( A5 = A8 )
% 0.06/0.31 => ( B5 != B7 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Pair_inject
% 0.06/0.31 thf(fact_178_prod__cases,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] :
% 0.06/0.31 ( ! [A6: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B4 ) )
% 0.06/0.31 => ( P @ P4 ) ) ).
% 0.06/0.31
% 0.06/0.31 % prod_cases
% 0.06/0.31 thf(fact_179_surj__pair,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,P4: product_prod @ A @ B] :
% 0.06/0.31 ? [X: A,Y2: B] :
% 0.06/0.31 ( P4
% 0.06/0.31 = ( product_Pair @ A @ B @ X @ Y2 ) ) ).
% 0.06/0.31
% 0.06/0.31 % surj_pair
% 0.06/0.31 thf(fact_180_Graph_OisSPath__nt__parallel,axiom,
% 0.06/0.31 ! [Capacity: $tType] :
% 0.06/0.31 ( ( linordered_idom @ Capacity )
% 0.06/0.31 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat,E2: product_prod @ nat @ nat] :
% 0.06/0.31 ( ( isSimplePath @ Capacity @ C2 @ S @ P4 @ T4 )
% 0.06/0.31 => ( ( member @ ( product_prod @ nat @ nat ) @ E2 @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.31 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ E2 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Graph.isSPath_nt_parallel
% 0.06/0.31 thf(fact_181_Sigma__cong,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,C4: A > ( set @ B ),D3: A > ( set @ B )] :
% 0.06/0.31 ( ( A3 = B3 )
% 0.06/0.31 => ( ! [X: A] :
% 0.06/0.31 ( ( member @ A @ X @ B3 )
% 0.06/0.31 => ( ( C4 @ X )
% 0.06/0.31 = ( D3 @ X ) ) )
% 0.06/0.31 => ( ( product_Sigma @ A @ B @ A3 @ C4 )
% 0.06/0.31 = ( product_Sigma @ A @ B @ B3 @ D3 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Sigma_cong
% 0.06/0.31 thf(fact_182_Times__eq__cancel2,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,X4: A,C4: set @ A,A3: set @ B,B3: set @ B] :
% 0.06/0.31 ( ( member @ A @ X4 @ C4 )
% 0.06/0.31 => ( ( ( product_Sigma @ B @ A @ A3
% 0.06/0.31 @ ^ [Uu: B] : C4 )
% 0.06/0.31 = ( product_Sigma @ B @ A @ B3
% 0.06/0.31 @ ^ [Uu: B] : C4 ) )
% 0.06/0.31 = ( A3 = B3 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Times_eq_cancel2
% 0.06/0.31 thf(fact_183_SigmaE,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,C2: product_prod @ A @ B,A3: set @ A,B3: A > ( set @ B )] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 0.06/0.31 => ~ ! [X: A] :
% 0.06/0.31 ( ( member @ A @ X @ A3 )
% 0.06/0.31 => ! [Y2: B] :
% 0.06/0.31 ( ( member @ B @ Y2 @ ( B3 @ X ) )
% 0.06/0.31 => ( C2
% 0.06/0.31 != ( product_Pair @ A @ B @ X @ Y2 ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % SigmaE
% 0.06/0.31 thf(fact_184_SigmaD1,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A5: A,B5: B,A3: set @ A,B3: A > ( set @ B )] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 0.06/0.31 => ( member @ A @ A5 @ A3 ) ) ).
% 0.06/0.31
% 0.06/0.31 % SigmaD1
% 0.06/0.31 thf(fact_185_SigmaD2,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A5: A,B5: B,A3: set @ A,B3: A > ( set @ B )] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 0.06/0.31 => ( member @ B @ B5 @ ( B3 @ A5 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % SigmaD2
% 0.06/0.31 thf(fact_186_SigmaE2,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A5: A,B5: B,A3: set @ A,B3: A > ( set @ B )] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 0.06/0.31 => ~ ( ( member @ A @ A5 @ A3 )
% 0.06/0.31 => ~ ( member @ B @ B5 @ ( B3 @ A5 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % SigmaE2
% 0.06/0.31 thf(fact_187_rtranclE,axiom,
% 0.06/0.31 ! [A: $tType,A5: A,B5: A,R: set @ ( product_prod @ A @ A )] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 => ( ( A5 != B5 )
% 0.06/0.31 => ~ ! [Y2: A] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ Y2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ B5 ) @ R ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % rtranclE
% 0.06/0.31 thf(fact_188_rtrancl_Ocases,axiom,
% 0.06/0.31 ! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 => ( ( A22 != A1 )
% 0.06/0.31 => ~ ! [B4: A] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B4 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A22 ) @ R ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % rtrancl.cases
% 0.06/0.31 thf(fact_189_rtrancl_Osimps,axiom,
% 0.06/0.31 ! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 = ( ? [A4: A] :
% 0.06/0.31 ( ( A1 = A4 )
% 0.06/0.31 & ( A22 = A4 ) )
% 0.06/0.31 | ? [A4: A,B6: A,C3: A] :
% 0.06/0.31 ( ( A1 = A4 )
% 0.06/0.31 & ( A22 = C3 )
% 0.06/0.31 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B6 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ C3 ) @ R ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % rtrancl.simps
% 0.06/0.31 thf(fact_190_rtrancl__trans,axiom,
% 0.06/0.31 ! [A: $tType,X4: A,Y3: A,R: set @ ( product_prod @ A @ A ),Z: A] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % rtrancl_trans
% 0.06/0.31 thf(fact_191_rtrancl__induct,axiom,
% 0.06/0.31 ! [A: $tType,A5: A,B5: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 => ( ( P @ A5 )
% 0.06/0.31 => ( ! [Y2: A,Z2: A] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ Y2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ R )
% 0.06/0.31 => ( ( P @ Y2 )
% 0.06/0.31 => ( P @ Z2 ) ) ) )
% 0.06/0.31 => ( P @ B5 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % rtrancl_induct
% 0.06/0.31 thf(fact_192_rtrancl_Oinducts,axiom,
% 0.06/0.31 ! [A: $tType,X1: A,X22: A,R: set @ ( product_prod @ A @ A ),P: A > A > $o] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X1 @ X22 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 => ( ! [A6: A] : ( P @ A6 @ A6 )
% 0.06/0.31 => ( ! [A6: A,B4: A,C5: A] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B4 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 => ( ( P @ A6 @ B4 )
% 0.06/0.31 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C5 ) @ R )
% 0.06/0.31 => ( P @ A6 @ C5 ) ) ) )
% 0.06/0.31 => ( P @ X1 @ X22 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % rtrancl.inducts
% 0.06/0.31 thf(fact_193_converse__rtranclE,axiom,
% 0.06/0.31 ! [A: $tType,X4: A,Z: A,R: set @ ( product_prod @ A @ A )] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 => ( ( X4 != Z )
% 0.06/0.31 => ~ ! [Y2: A] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y2 ) @ R )
% 0.06/0.31 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % converse_rtranclE
% 0.06/0.31 thf(fact_194_rtrancl_Ortrancl__refl,axiom,
% 0.06/0.31 ! [A: $tType,A5: A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ ( transitive_rtrancl @ A @ R ) ) ).
% 0.06/0.31
% 0.06/0.31 % rtrancl.rtrancl_refl
% 0.06/0.31 thf(fact_195_converse__rtrancl__induct,axiom,
% 0.06/0.31 ! [A: $tType,A5: A,B5: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 => ( ( P @ B5 )
% 0.06/0.31 => ( ! [Y2: A,Z2: A] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ R )
% 0.06/0.31 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z2 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 => ( ( P @ Z2 )
% 0.06/0.31 => ( P @ Y2 ) ) ) )
% 0.06/0.31 => ( P @ A5 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % converse_rtrancl_induct
% 0.06/0.31 thf(fact_196_rtrancl_Ortrancl__into__rtrancl,axiom,
% 0.06/0.31 ! [A: $tType,A5: A,B5: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C2 ) @ R )
% 0.06/0.31 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ C2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % rtrancl.rtrancl_into_rtrancl
% 0.06/0.31 thf(fact_197_converse__rtrancl__into__rtrancl,axiom,
% 0.06/0.31 ! [A: $tType,A5: A,B5: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 0.06/0.31 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ R )
% 0.06/0.31 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.06/0.31 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ C2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % converse_rtrancl_into_rtrancl
% 0.06/0.31 thf(fact_198_rtrancl__mono,axiom,
% 0.06/0.31 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
% 0.06/0.31 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
% 0.06/0.31 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R ) @ ( transitive_rtrancl @ A @ S ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % rtrancl_mono
% 0.06/0.31 thf(fact_199_rtrancl__subset,axiom,
% 0.06/0.31 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 0.06/0.31 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 )
% 0.06/0.31 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S3 @ ( transitive_rtrancl @ A @ R2 ) )
% 0.06/0.31 => ( ( transitive_rtrancl @ A @ S3 )
% 0.06/0.31 = ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % rtrancl_subset
% 0.06/0.31 thf(fact_200_rtrancl__subset__rtrancl,axiom,
% 0.06/0.31 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
% 0.06/0.31 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( transitive_rtrancl @ A @ S ) )
% 0.06/0.31 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R ) @ ( transitive_rtrancl @ A @ S ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % rtrancl_subset_rtrancl
% 0.06/0.31 thf(fact_201_Sigma__mono,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A3: set @ A,C4: set @ A,B3: A > ( set @ B ),D3: A > ( set @ B )] :
% 0.06/0.31 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C4 )
% 0.06/0.31 => ( ! [X: A] :
% 0.06/0.31 ( ( member @ A @ X @ A3 )
% 0.06/0.31 => ( ord_less_eq @ ( set @ B ) @ ( B3 @ X ) @ ( D3 @ X ) ) )
% 0.06/0.31 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ C4 @ D3 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Sigma_mono
% 0.06/0.31 thf(fact_202_Image__closed__trancl,axiom,
% 0.06/0.31 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X5: set @ A] :
% 0.06/0.31 ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ X5 ) @ X5 )
% 0.06/0.31 => ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ X5 )
% 0.06/0.31 = X5 ) ) ).
% 0.06/0.31
% 0.06/0.31 % Image_closed_trancl
% 0.06/0.31 thf(fact_203_Not__Domain__rtrancl,axiom,
% 0.06/0.31 ! [A: $tType,X4: A,R2: set @ ( product_prod @ A @ A ),Y3: A] :
% 0.06/0.31 ( ~ ( member @ A @ X4 @ ( domain @ A @ A @ R2 ) )
% 0.06/0.31 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ ( transitive_rtrancl @ A @ R2 ) )
% 0.06/0.31 = ( X4 = Y3 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Not_Domain_rtrancl
% 0.06/0.31 thf(fact_204_Times__subset__cancel2,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType,X4: A,C4: set @ A,A3: set @ B,B3: set @ B] :
% 0.06/0.31 ( ( member @ A @ X4 @ C4 )
% 0.06/0.31 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
% 0.06/0.31 @ ( product_Sigma @ B @ A @ A3
% 0.06/0.31 @ ^ [Uu: B] : C4 )
% 0.06/0.31 @ ( product_Sigma @ B @ A @ B3
% 0.06/0.31 @ ^ [Uu: B] : C4 ) )
% 0.06/0.31 = ( ord_less_eq @ ( set @ B ) @ A3 @ B3 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % Times_subset_cancel2
% 0.06/0.31 thf(fact_205_finite__rtrancl__Image,axiom,
% 0.06/0.31 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A] :
% 0.06/0.31 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
% 0.06/0.31 => ( ( finite_finite2 @ A @ A3 )
% 0.06/0.31 => ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ A3 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % finite_rtrancl_Image
% 0.06/0.31 thf(fact_206_infinite__cartesian__product,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B] :
% 0.06/0.31 ( ~ ( finite_finite2 @ A @ A3 )
% 0.06/0.31 => ( ~ ( finite_finite2 @ B @ B3 )
% 0.06/0.31 => ~ ( finite_finite2 @ ( product_prod @ A @ B )
% 0.06/0.31 @ ( product_Sigma @ A @ B @ A3
% 0.06/0.31 @ ^ [Uu: A] : B3 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % infinite_cartesian_product
% 0.06/0.31 thf(fact_207_old_Oprod_Orec,axiom,
% 0.06/0.31 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A5: A,B5: B] :
% 0.06/0.31 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.06/0.31 = ( F1 @ A5 @ B5 ) ) ).
% 0.06/0.31
% 0.06/0.31 % old.prod.rec
% 0.06/0.31 thf(fact_208_predicate1I,axiom,
% 0.06/0.31 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.06/0.31 ( ! [X: A] :
% 0.06/0.31 ( ( P @ X )
% 0.06/0.31 => ( Q @ X ) )
% 0.06/0.31 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 0.06/0.31
% 0.06/0.31 % predicate1I
% 0.06/0.31 thf(fact_209_order__refl,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( preorder @ A )
% 0.06/0.31 => ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% 0.06/0.31
% 0.06/0.31 % order_refl
% 0.06/0.31 thf(fact_210_predicate2I,axiom,
% 0.06/0.31 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 0.06/0.31 ( ! [X: A,Y2: B] :
% 0.06/0.31 ( ( P @ X @ Y2 )
% 0.06/0.31 => ( Q @ X @ Y2 ) )
% 0.06/0.31 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 0.06/0.31
% 0.06/0.31 % predicate2I
% 0.06/0.31 thf(fact_211_order__subst1,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType] :
% 0.06/0.31 ( ( ( order @ B )
% 0.06/0.31 & ( order @ A ) )
% 0.06/0.31 => ! [A5: A,F2: B > A,B5: B,C2: B] :
% 0.06/0.31 ( ( ord_less_eq @ A @ A5 @ ( F2 @ B5 ) )
% 0.06/0.31 => ( ( ord_less_eq @ B @ B5 @ C2 )
% 0.06/0.31 => ( ! [X: B,Y2: B] :
% 0.06/0.31 ( ( ord_less_eq @ B @ X @ Y2 )
% 0.06/0.31 => ( ord_less_eq @ A @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
% 0.06/0.31 => ( ord_less_eq @ A @ A5 @ ( F2 @ C2 ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % order_subst1
% 0.06/0.31 thf(fact_212_order__subst2,axiom,
% 0.06/0.31 ! [A: $tType,C: $tType] :
% 0.06/0.31 ( ( ( order @ C )
% 0.06/0.31 & ( order @ A ) )
% 0.06/0.31 => ! [A5: A,B5: A,F2: A > C,C2: C] :
% 0.06/0.31 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.06/0.31 => ( ( ord_less_eq @ C @ ( F2 @ B5 ) @ C2 )
% 0.06/0.31 => ( ! [X: A,Y2: A] :
% 0.06/0.31 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.06/0.31 => ( ord_less_eq @ C @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
% 0.06/0.31 => ( ord_less_eq @ C @ ( F2 @ A5 ) @ C2 ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % order_subst2
% 0.06/0.31 thf(fact_213_ord__eq__le__subst,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType] :
% 0.06/0.31 ( ( ( ord @ B )
% 0.06/0.31 & ( ord @ A ) )
% 0.06/0.31 => ! [A5: A,F2: B > A,B5: B,C2: B] :
% 0.06/0.31 ( ( A5
% 0.06/0.31 = ( F2 @ B5 ) )
% 0.06/0.31 => ( ( ord_less_eq @ B @ B5 @ C2 )
% 0.06/0.31 => ( ! [X: B,Y2: B] :
% 0.06/0.31 ( ( ord_less_eq @ B @ X @ Y2 )
% 0.06/0.31 => ( ord_less_eq @ A @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
% 0.06/0.31 => ( ord_less_eq @ A @ A5 @ ( F2 @ C2 ) ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % ord_eq_le_subst
% 0.06/0.31 thf(fact_214_ord__le__eq__subst,axiom,
% 0.06/0.31 ! [A: $tType,B: $tType] :
% 0.06/0.31 ( ( ( ord @ B )
% 0.06/0.31 & ( ord @ A ) )
% 0.06/0.31 => ! [A5: A,B5: A,F2: A > B,C2: B] :
% 0.06/0.31 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.06/0.31 => ( ( ( F2 @ B5 )
% 0.06/0.31 = C2 )
% 0.06/0.31 => ( ! [X: A,Y2: A] :
% 0.06/0.31 ( ( ord_less_eq @ A @ X @ Y2 )
% 0.06/0.31 => ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
% 0.06/0.31 => ( ord_less_eq @ B @ ( F2 @ A5 ) @ C2 ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % ord_le_eq_subst
% 0.06/0.31 thf(fact_215_eq__iff,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( order @ A )
% 0.06/0.31 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 0.06/0.31 = ( ^ [X3: A,Y: A] :
% 0.06/0.31 ( ( ord_less_eq @ A @ X3 @ Y )
% 0.06/0.31 & ( ord_less_eq @ A @ Y @ X3 ) ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % eq_iff
% 0.06/0.31 thf(fact_216_antisym,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( order @ A )
% 0.06/0.31 => ! [X4: A,Y3: A] :
% 0.06/0.31 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.06/0.31 => ( ( ord_less_eq @ A @ Y3 @ X4 )
% 0.06/0.31 => ( X4 = Y3 ) ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % antisym
% 0.06/0.31 thf(fact_217_linear,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( linorder @ A )
% 0.06/0.31 => ! [X4: A,Y3: A] :
% 0.06/0.31 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.06/0.31 | ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 0.06/0.31
% 0.06/0.31 % linear
% 0.06/0.31 thf(fact_218_eq__refl,axiom,
% 0.06/0.31 ! [A: $tType] :
% 0.06/0.31 ( ( preorder @ A )
% 0.06/0.31 => ! [X4: A,Y3: A] :
% 0.06/0.32 ( ( X4 = Y3 )
% 0.06/0.32 => ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % eq_refl
% 0.06/0.32 thf(fact_219_le__cases,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( linorder @ A )
% 0.06/0.32 => ! [X4: A,Y3: A] :
% 0.06/0.32 ( ~ ( ord_less_eq @ A @ X4 @ Y3 )
% 0.06/0.32 => ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % le_cases
% 0.06/0.32 thf(fact_220_order_Otrans,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( order @ A )
% 0.06/0.32 => ! [A5: A,B5: A,C2: A] :
% 0.06/0.32 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.06/0.32 => ( ( ord_less_eq @ A @ B5 @ C2 )
% 0.06/0.32 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % order.trans
% 0.06/0.32 thf(fact_221_le__cases3,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( linorder @ A )
% 0.06/0.32 => ! [X4: A,Y3: A,Z: A] :
% 0.06/0.32 ( ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.06/0.32 => ~ ( ord_less_eq @ A @ Y3 @ Z ) )
% 0.06/0.32 => ( ( ( ord_less_eq @ A @ Y3 @ X4 )
% 0.06/0.32 => ~ ( ord_less_eq @ A @ X4 @ Z ) )
% 0.06/0.32 => ( ( ( ord_less_eq @ A @ X4 @ Z )
% 0.06/0.32 => ~ ( ord_less_eq @ A @ Z @ Y3 ) )
% 0.06/0.32 => ( ( ( ord_less_eq @ A @ Z @ Y3 )
% 0.06/0.32 => ~ ( ord_less_eq @ A @ Y3 @ X4 ) )
% 0.06/0.32 => ( ( ( ord_less_eq @ A @ Y3 @ Z )
% 0.06/0.32 => ~ ( ord_less_eq @ A @ Z @ X4 ) )
% 0.06/0.32 => ~ ( ( ord_less_eq @ A @ Z @ X4 )
% 0.06/0.32 => ~ ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % le_cases3
% 0.06/0.32 thf(fact_222_antisym__conv,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( order @ A )
% 0.06/0.32 => ! [Y3: A,X4: A] :
% 0.06/0.32 ( ( ord_less_eq @ A @ Y3 @ X4 )
% 0.06/0.32 => ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.06/0.32 = ( X4 = Y3 ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % antisym_conv
% 0.06/0.32 thf(fact_223_order__class_Oorder_Oeq__iff,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( order @ A )
% 0.06/0.32 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 0.06/0.32 = ( ^ [A4: A,B6: A] :
% 0.06/0.32 ( ( ord_less_eq @ A @ A4 @ B6 )
% 0.06/0.32 & ( ord_less_eq @ A @ B6 @ A4 ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % order_class.order.eq_iff
% 0.06/0.32 thf(fact_224_ord__eq__le__trans,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( ord @ A )
% 0.06/0.32 => ! [A5: A,B5: A,C2: A] :
% 0.06/0.32 ( ( A5 = B5 )
% 0.06/0.32 => ( ( ord_less_eq @ A @ B5 @ C2 )
% 0.06/0.32 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % ord_eq_le_trans
% 0.06/0.32 thf(fact_225_ord__le__eq__trans,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( ord @ A )
% 0.06/0.32 => ! [A5: A,B5: A,C2: A] :
% 0.06/0.32 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.06/0.32 => ( ( B5 = C2 )
% 0.06/0.32 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % ord_le_eq_trans
% 0.06/0.32 thf(fact_226_order__class_Oorder_Oantisym,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( order @ A )
% 0.06/0.32 => ! [A5: A,B5: A] :
% 0.06/0.32 ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.06/0.32 => ( ( ord_less_eq @ A @ B5 @ A5 )
% 0.06/0.32 => ( A5 = B5 ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % order_class.order.antisym
% 0.06/0.32 thf(fact_227_order__trans,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( preorder @ A )
% 0.06/0.32 => ! [X4: A,Y3: A,Z: A] :
% 0.06/0.32 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 0.06/0.32 => ( ( ord_less_eq @ A @ Y3 @ Z )
% 0.06/0.32 => ( ord_less_eq @ A @ X4 @ Z ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % order_trans
% 0.06/0.32 thf(fact_228_dual__order_Orefl,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( order @ A )
% 0.06/0.32 => ! [A5: A] : ( ord_less_eq @ A @ A5 @ A5 ) ) ).
% 0.06/0.32
% 0.06/0.32 % dual_order.refl
% 0.06/0.32 thf(fact_229_linorder__wlog,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( linorder @ A )
% 0.06/0.32 => ! [P: A > A > $o,A5: A,B5: A] :
% 0.06/0.32 ( ! [A6: A,B4: A] :
% 0.06/0.32 ( ( ord_less_eq @ A @ A6 @ B4 )
% 0.06/0.32 => ( P @ A6 @ B4 ) )
% 0.06/0.32 => ( ! [A6: A,B4: A] :
% 0.06/0.32 ( ( P @ B4 @ A6 )
% 0.06/0.32 => ( P @ A6 @ B4 ) )
% 0.06/0.32 => ( P @ A5 @ B5 ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % linorder_wlog
% 0.06/0.32 thf(fact_230_dual__order_Otrans,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( order @ A )
% 0.06/0.32 => ! [B5: A,A5: A,C2: A] :
% 0.06/0.32 ( ( ord_less_eq @ A @ B5 @ A5 )
% 0.06/0.32 => ( ( ord_less_eq @ A @ C2 @ B5 )
% 0.06/0.32 => ( ord_less_eq @ A @ C2 @ A5 ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % dual_order.trans
% 0.06/0.32 thf(fact_231_dual__order_Oeq__iff,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( order @ A )
% 0.06/0.32 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 0.06/0.32 = ( ^ [A4: A,B6: A] :
% 0.06/0.32 ( ( ord_less_eq @ A @ B6 @ A4 )
% 0.06/0.32 & ( ord_less_eq @ A @ A4 @ B6 ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % dual_order.eq_iff
% 0.06/0.32 thf(fact_232_dual__order_Oantisym,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( order @ A )
% 0.06/0.32 => ! [B5: A,A5: A] :
% 0.06/0.32 ( ( ord_less_eq @ A @ B5 @ A5 )
% 0.06/0.32 => ( ( ord_less_eq @ A @ A5 @ B5 )
% 0.06/0.32 => ( A5 = B5 ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % dual_order.antisym
% 0.06/0.32 thf(fact_233_predicate2D,axiom,
% 0.06/0.32 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X4: A,Y3: B] :
% 0.06/0.32 ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.06/0.32 => ( ( P @ X4 @ Y3 )
% 0.06/0.32 => ( Q @ X4 @ Y3 ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % predicate2D
% 0.06/0.32 thf(fact_234_rev__predicate2D,axiom,
% 0.06/0.32 ! [A: $tType,B: $tType,P: A > B > $o,X4: A,Y3: B,Q: A > B > $o] :
% 0.06/0.32 ( ( P @ X4 @ Y3 )
% 0.06/0.32 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 0.06/0.32 => ( Q @ X4 @ Y3 ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % rev_predicate2D
% 0.06/0.32 thf(fact_235_predicate1D,axiom,
% 0.06/0.32 ! [A: $tType,P: A > $o,Q: A > $o,X4: A] :
% 0.06/0.32 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.06/0.32 => ( ( P @ X4 )
% 0.06/0.32 => ( Q @ X4 ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % predicate1D
% 0.06/0.32 thf(fact_236_rev__predicate1D,axiom,
% 0.06/0.32 ! [A: $tType,P: A > $o,X4: A,Q: A > $o] :
% 0.06/0.32 ( ( P @ X4 )
% 0.06/0.32 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 0.06/0.32 => ( Q @ X4 ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % rev_predicate1D
% 0.06/0.32 thf(fact_237_le__funD,axiom,
% 0.06/0.32 ! [B: $tType,A: $tType] :
% 0.06/0.32 ( ( ord @ B )
% 0.06/0.32 => ! [F2: A > B,G: A > B,X4: A] :
% 0.06/0.32 ( ( ord_less_eq @ ( A > B ) @ F2 @ G )
% 0.06/0.32 => ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % le_funD
% 0.06/0.32 thf(fact_238_le__funE,axiom,
% 0.06/0.32 ! [B: $tType,A: $tType] :
% 0.06/0.32 ( ( ord @ B )
% 0.06/0.32 => ! [F2: A > B,G: A > B,X4: A] :
% 0.06/0.32 ( ( ord_less_eq @ ( A > B ) @ F2 @ G )
% 0.06/0.32 => ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % le_funE
% 0.06/0.32 thf(fact_239_le__funI,axiom,
% 0.06/0.32 ! [B: $tType,A: $tType] :
% 0.06/0.32 ( ( ord @ B )
% 0.06/0.32 => ! [F2: A > B,G: A > B] :
% 0.06/0.32 ( ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) )
% 0.06/0.32 => ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % le_funI
% 0.06/0.32 thf(fact_240_le__fun__def,axiom,
% 0.06/0.32 ! [B: $tType,A: $tType] :
% 0.06/0.32 ( ( ord @ B )
% 0.06/0.32 => ( ( ord_less_eq @ ( A > B ) )
% 0.06/0.32 = ( ^ [F4: A > B,G4: A > B] :
% 0.06/0.32 ! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % le_fun_def
% 0.06/0.32 thf(fact_241_relChain__def,axiom,
% 0.06/0.32 ! [B: $tType,A: $tType] :
% 0.06/0.32 ( ( ord @ B )
% 0.06/0.32 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 0.06/0.32 = ( ^ [R3: set @ ( product_prod @ A @ A ),As: A > B] :
% 0.06/0.32 ! [I: A,J: A] :
% 0.06/0.32 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R3 )
% 0.06/0.32 => ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % relChain_def
% 0.06/0.32 thf(fact_242_rtc__isPath,axiom,
% 0.06/0.32 ! [U: nat,V: nat] :
% 0.06/0.32 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( transitive_rtrancl @ nat @ ( e @ a @ c ) ) )
% 0.06/0.32 => ? [P3: list @ ( product_prod @ nat @ nat )] : ( isPath @ a @ c @ U @ P3 @ V ) ) ).
% 0.06/0.32
% 0.06/0.32 % rtc_isPath
% 0.06/0.32 thf(fact_243_isSPath__pathLE,axiom,
% 0.06/0.32 ! [S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat] :
% 0.06/0.32 ( ( isPath @ a @ c @ S @ P4 @ T4 )
% 0.06/0.32 => ? [P5: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ a @ c @ S @ P5 @ T4 ) ) ).
% 0.06/0.32
% 0.06/0.32 % isSPath_pathLE
% 0.06/0.32 thf(fact_244_connected__def,axiom,
% 0.06/0.32 ! [U: nat,V: nat] :
% 0.06/0.32 ( ( connected @ a @ c @ U @ V )
% 0.06/0.32 = ( ? [P2: list @ ( product_prod @ nat @ nat )] : ( isPath @ a @ c @ U @ P2 @ V ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % connected_def
% 0.06/0.32 thf(fact_245_shortestPath__is__path,axiom,
% 0.06/0.32 ! [U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.06/0.32 ( ( isShortestPath @ a @ c @ U @ P4 @ V )
% 0.06/0.32 => ( isPath @ a @ c @ U @ P4 @ V ) ) ).
% 0.06/0.32
% 0.06/0.32 % shortestPath_is_path
% 0.06/0.32 thf(fact_246_isPath__ex__edge1,axiom,
% 0.06/0.32 ! [U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
% 0.06/0.32 ( ( isPath @ a @ c @ U @ P4 @ V )
% 0.06/0.32 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.32 => ( ( U1 != U )
% 0.06/0.32 => ? [U23: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U23 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % isPath_ex_edge1
% 0.06/0.32 thf(fact_247_isPath__ex__edge2,axiom,
% 0.06/0.32 ! [U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
% 0.06/0.32 ( ( isPath @ a @ c @ U @ P4 @ V )
% 0.06/0.32 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.32 => ( ( V1 != V )
% 0.06/0.32 => ? [V23: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V1 @ V23 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % isPath_ex_edge2
% 0.06/0.32 thf(fact_248_isPath__edgeset,axiom,
% 0.06/0.32 ! [U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,E2: product_prod @ nat @ nat] :
% 0.06/0.32 ( ( isPath @ a @ c @ U @ P4 @ V )
% 0.06/0.32 => ( ( member @ ( product_prod @ nat @ nat ) @ E2 @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.32 => ( member @ ( product_prod @ nat @ nat ) @ E2 @ ( e @ a @ c ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % isPath_edgeset
% 0.06/0.32 thf(fact_249_isPath__rtc,axiom,
% 0.06/0.32 ! [U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.06/0.32 ( ( isPath @ a @ c @ U @ P4 @ V )
% 0.06/0.32 => ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( transitive_rtrancl @ nat @ ( e @ a @ c ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % isPath_rtc
% 0.06/0.32 thf(fact_250_Graph_OisPath__ex__edge2,axiom,
% 0.06/0.32 ! [Capacity: $tType] :
% 0.06/0.32 ( ( linordered_idom @ Capacity )
% 0.06/0.32 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
% 0.06/0.32 ( ( isPath @ Capacity @ C2 @ U @ P4 @ V )
% 0.06/0.32 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.32 => ( ( V1 != V )
% 0.06/0.32 => ? [V23: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V1 @ V23 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % Graph.isPath_ex_edge2
% 0.06/0.32 thf(fact_251_Graph_OisPath__ex__edge1,axiom,
% 0.06/0.32 ! [Capacity: $tType] :
% 0.06/0.32 ( ( linordered_idom @ Capacity )
% 0.06/0.32 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
% 0.06/0.32 ( ( isPath @ Capacity @ C2 @ U @ P4 @ V )
% 0.06/0.32 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.32 => ( ( U1 != U )
% 0.06/0.32 => ? [U23: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U23 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % Graph.isPath_ex_edge1
% 0.06/0.32 thf(fact_252_Graph_OisPath__edgeset,axiom,
% 0.06/0.32 ! [Capacity: $tType] :
% 0.06/0.32 ( ( linordered_idom @ Capacity )
% 0.06/0.32 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,E2: product_prod @ nat @ nat] :
% 0.06/0.32 ( ( isPath @ Capacity @ C2 @ U @ P4 @ V )
% 0.06/0.32 => ( ( member @ ( product_prod @ nat @ nat ) @ E2 @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 0.06/0.32 => ( member @ ( product_prod @ nat @ nat ) @ E2 @ ( e @ Capacity @ C2 ) ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % Graph.isPath_edgeset
% 0.06/0.32 thf(fact_253_Graph_Oconnected__def,axiom,
% 0.06/0.32 ! [Capacity: $tType] :
% 0.06/0.32 ( ( linordered_idom @ Capacity )
% 0.06/0.32 => ( ( connected @ Capacity )
% 0.06/0.32 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,U3: nat,V2: nat] :
% 0.06/0.32 ? [P2: list @ ( product_prod @ nat @ nat )] : ( isPath @ Capacity @ C3 @ U3 @ P2 @ V2 ) ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % Graph.connected_def
% 0.06/0.32 thf(fact_254_Graph_OshortestPath__is__path,axiom,
% 0.06/0.32 ! [Capacity: $tType] :
% 0.06/0.32 ( ( linordered_idom @ Capacity )
% 0.06/0.32 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.06/0.32 ( ( isShortestPath @ Capacity @ C2 @ U @ P4 @ V )
% 0.06/0.32 => ( isPath @ Capacity @ C2 @ U @ P4 @ V ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % Graph.shortestPath_is_path
% 0.06/0.32 thf(fact_255_Graph_OisPath_Ocong,axiom,
% 0.06/0.32 ! [Capacity: $tType] :
% 0.06/0.32 ( ( linordered_idom @ Capacity )
% 0.06/0.32 => ( ( isPath @ Capacity )
% 0.06/0.32 = ( isPath @ Capacity ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % Graph.isPath.cong
% 0.06/0.32
% 0.06/0.32 % Subclasses (5)
% 0.06/0.32 thf(subcl_Rings_Olinordered__idom___HOL_Otype,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( linordered_idom @ A )
% 0.06/0.32 => ( type @ A ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(subcl_Rings_Olinordered__idom___Orderings_Oord,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( linordered_idom @ A )
% 0.06/0.32 => ( ord @ A ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(subcl_Rings_Olinordered__idom___Orderings_Oorder,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( linordered_idom @ A )
% 0.06/0.32 => ( order @ A ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(subcl_Rings_Olinordered__idom___Orderings_Olinorder,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( linordered_idom @ A )
% 0.06/0.32 => ( linorder @ A ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(subcl_Rings_Olinordered__idom___Orderings_Opreorder,axiom,
% 0.06/0.32 ! [A: $tType] :
% 0.06/0.32 ( ( linordered_idom @ A )
% 0.06/0.32 => ( preorder @ A ) ) ).
% 0.06/0.32
% 0.06/0.32 % Type constructors (18)
% 0.06/0.32 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.06/0.32 ! [A9: $tType,A10: $tType] :
% 0.06/0.32 ( ( preorder @ A10 )
% 0.06/0.32 => ( preorder @ ( A9 > A10 ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.06/0.32 ! [A9: $tType,A10: $tType] :
% 0.06/0.32 ( ( ( finite_finite @ A9 )
% 0.06/0.32 & ( finite_finite @ A10 ) )
% 0.06/0.32 => ( finite_finite @ ( A9 > A10 ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.06/0.32 ! [A9: $tType,A10: $tType] :
% 0.06/0.32 ( ( order @ A10 )
% 0.06/0.32 => ( order @ ( A9 > A10 ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_fun___Orderings_Oord,axiom,
% 0.06/0.32 ! [A9: $tType,A10: $tType] :
% 0.06/0.32 ( ( ord @ A10 )
% 0.06/0.32 => ( ord @ ( A9 > A10 ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.06/0.32 preorder @ nat ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.06/0.32 linorder @ nat ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.06/0.32 order @ nat ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.06/0.32 ord @ nat ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.06/0.32 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
% 0.06/0.32 ! [A9: $tType] :
% 0.06/0.32 ( ( finite_finite @ A9 )
% 0.06/0.32 => ( finite_finite @ ( set @ A9 ) ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 0.06/0.32 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 0.06/0.32 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
% 0.06/0.32 preorder @ $o ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_HOL_Obool___Orderings_Olinorder_9,axiom,
% 0.06/0.32 linorder @ $o ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_HOL_Obool___Finite__Set_Ofinite_10,axiom,
% 0.06/0.32 finite_finite @ $o ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
% 0.06/0.32 order @ $o ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
% 0.06/0.32 ord @ $o ).
% 0.06/0.32
% 0.06/0.32 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_13,axiom,
% 0.06/0.32 ! [A9: $tType,A10: $tType] :
% 0.06/0.32 ( ( ( finite_finite @ A9 )
% 0.06/0.32 & ( finite_finite @ A10 ) )
% 0.06/0.32 => ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
% 0.06/0.32
% 0.06/0.32 % Free types (1)
% 0.06/0.32 thf(tfree_0,hypothesis,
% 0.06/0.32 linordered_idom @ a ).
% 0.06/0.32
% 0.06/0.32 % Conjectures (1)
% 0.06/0.32 thf(conj_0,conjecture,
% 0.06/0.32 finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) ).
% 0.06/0.32
% 0.06/0.32 %------------------------------------------------------------------------------
% 0.06/0.32 ------------------------
% 1.80/1.10 ---- Embedded in THF ---
% 1.80/1.10 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 1.80/1.10 %%% Generated on Tue Feb 24 03:24:26 EST 2026
% 1.80/1.10 %%% using '$$dhol' embedding, version 1.3.0.
% 1.80/1.10 %%% Logic specification used:
% 1.80/1.10 %%% thf(spec, logic, $$dhol).
% 1.80/1.10
% 1.80/1.10 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.vhIah8fDUX/DTF2DTF23290.p
% 1.80/1.10 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 1.80/1.10 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 1.80/1.10 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 1.80/1.10 thf(ty_t_Nat_Onat, type, nat: $tType).
% 1.80/1.10 thf(ty_tf_a, type, a: $tType).
% 1.80/1.10 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 1.80/1.10 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 1.80/1.10 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 1.80/1.10 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 1.80/1.10 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 1.80/1.10 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 1.80/1.10 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 1.80/1.10 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca1785829860lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 1.80/1.10 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis, type, edmond2129202899alysis: (!> [A:$tType]: (((((product_prod @ nat) @ nat) > A) > $o)))).
% 1.80/1.10 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis__defs_OspEdges, type, edmond1803015688pEdges: (!> [A:$tType]: (((((product_prod @ nat) @ nat) > A) > (nat > (nat > (set @ ((product_prod @ nat) @ nat)))))))).
% 1.80/1.10 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 1.80/1.10 thf(sy_c_Graph_OFinite__Graph, type, finite_Graph: (!> [A:$tType]: (((((product_prod @ nat) @ nat) > A) > $o)))).
% 1.80/1.10 thf(sy_c_Graph_OGraph_OE, type, e: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (set @ ((product_prod @ nat) @ nat)))))).
% 1.80/1.10 thf(sy_c_Graph_OGraph_OV, type, v: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (set @ nat))))).
% 1.80/1.10 thf(sy_c_Graph_OGraph_Oadjacent__nodes, type, adjacent_nodes: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (set @ nat)))))).
% 1.80/1.10 thf(sy_c_Graph_OGraph_Oconnected, type, connected: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > $o)))))).
% 1.80/1.10 thf(sy_c_Graph_OGraph_Oincoming, type, incoming: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (set @ ((product_prod @ nat) @ nat))))))).
% 1.80/1.10 thf(sy_c_Graph_OGraph_Oincoming_H, type, incoming2: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > ((set @ nat) > (set @ ((product_prod @ nat) @ nat))))))).
% 1.80/1.10 thf(sy_c_Graph_OGraph_OisPath, type, isPath: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > ((list @ ((product_prod @ nat) @ nat)) > (nat > $o))))))).
% 1.80/1.10 thf(sy_c_Graph_OGraph_OisShortestPath, type, isShortestPath: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > ((list @ ((product_prod @ nat) @ nat)) > (nat > $o))))))).
% 1.80/1.10 thf(sy_c_Graph_OGraph_OisSimplePath, type, isSimplePath: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > ((list @ ((product_prod @ nat) @ nat)) > (nat > $o))))))).
% 1.80/1.10 thf(sy_c_Graph_OGraph_Ooutgoing, type, outgoing: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (set @ ((product_prod @ nat) @ nat))))))).
% 1.80/1.10 thf(sy_c_Graph_OGraph_Ooutgoing_H, type, outgoing2: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > ((set @ nat) > (set @ ((product_prod @ nat) @ nat))))))).
% 1.80/1.10 thf(sy_c_Graph_OGraph_OreachableNodes, type, reachableNodes: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (set @ nat)))))).
% 1.80/1.10 thf(sy_c_List_Olist_Oset, type, set2: (!> [A:$tType]: (((list @ A) > (set @ A))))).
% 1.80/1.10 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 1.80/1.10 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 1.80/1.10 thf(sy_c_Product__Type_OSigma, type, product_Sigma: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ ((product_prod @ A) @ B))))))).
% 1.80/1.10 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 1.80/1.10 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 1.80/1.10 thf(sy_c_Relation_ODomain, type, domain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ A))))).
% 1.80/1.10 thf(sy_c_Relation_OImage, type, image: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > ((set @ A) > (set @ B)))))).
% 1.80/1.10 thf(sy_c_Relation_ORange, type, range: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ B))))).
% 1.80/1.10 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 1.80/1.10 thf(sy_c_Transitive__Closure_Ortrancl, type, transitive_rtrancl: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ A) @ A)))))).
% 1.80/1.10 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 1.80/1.10 thf(sy_v_c, type, c: (((product_prod @ nat) @ nat) > a)).
% 1.80/1.10 thf(sy_v_s, type, s: nat).
% 1.80/1.10 thf(sy_v_t, type, t: nat).
% 1.80/1.10 thf(fact_0_ek__analysis__axioms, axiom, ((edmond2129202899alysis @ a) @ c)).
% 1.80/1.10 thf(fact_1__092_060open_062finite_AE_A_092_060Longrightarrow_062_Afinite_AspEdges_092_060close_062, axiom, (((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ a) @ c)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((((edmond1803015688pEdges @ a) @ c) @ s) @ t)))).
% 1.80/1.10 thf(fact_2_ek__analysis__defs_OspEdges_Ocong, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((edmond1803015688pEdges @ A) = (edmond1803015688pEdges @ A)))))).
% 1.80/1.10 thf(fact_3_finite__E, axiom, ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ a) @ c))).
% 1.80/1.10 thf(fact_4_spEdges__ss__E, axiom, (((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ ((((edmond1803015688pEdges @ a) @ c) @ s) @ t)) @ ((e @ a) @ c))).
% 1.80/1.10 thf(fact_5_Finite__Graph__axioms, axiom, ((finite_Graph @ a) @ c)).
% 1.80/1.10 thf(fact_6_finite__code, axiom, (! [A:$tType]: (((finite_finite @ A) => ((finite_finite2 @ A) = (^ [A2:(set @ A)]: ($true))))))).
% 1.80/1.10 thf(fact_7_Finite__Graph__EI, axiom, (((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ a) @ c)) => ((finite_Graph @ a) @ c))).
% 1.80/1.10 thf(fact_8_finite__set__choice, axiom, (! [B:$tType,A:$tType,A3:(set @ A),P:(A > (B > $o))]: ((((finite_finite2 @ A) @ A3) => ((! [X:A]: (((((member @ A) @ X) @ A3) => (? [X_1:B]: (((P @ X) @ X_1)))))) => (? [F:(A > B)]: ((! [X2:A]: (((((member @ A) @ X2) @ A3) => ((P @ X2) @ (F @ X2)))))))))))).
% 1.80/1.10 thf(fact_9_finite, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A3:(set @ A)]: (((finite_finite2 @ A) @ A3))))))).
% 1.80/1.10 thf(fact_10_connected__refl, axiom, (! [V:nat]: (((((connected @ a) @ c) @ V) @ V)))).
% 1.80/1.10 thf(fact_11_spEdges__def, axiom, (((((edmond1803015688pEdges @ a) @ c) @ s) @ t) = ((collect @ ((product_prod @ nat) @ nat)) @ (^ [E:((product_prod @ nat) @ nat)]: ((? [P2:(list @ ((product_prod @ nat) @ nat))]: (((((member @ ((product_prod @ nat) @ nat)) @ E) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P2)) & (((((isShortestPath @ a) @ c) @ s) @ P2) @ t))))))))).
% 1.80/1.10 thf(fact_12_finite__Range, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B))]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ R) => ((finite_finite2 @ B) @ (((range @ A) @ B) @ R)))))).
% 1.80/1.10 thf(fact_13_finite__Domain, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B))]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ R) => ((finite_finite2 @ A) @ (((domain @ A) @ B) @ R)))))).
% 1.80/1.10 thf(fact_14_Efin__imp__Vfin, axiom, (((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ a) @ c)) => ((finite_finite2 @ nat) @ ((v @ a) @ c)))).
% 1.80/1.10 thf(fact_15_connected__inV__iff, axiom, (! [U:nat,V:nat]: ((((((connected @ a) @ c) @ U) @ V) => ((((member @ nat) @ V) @ ((v @ a) @ c)) = (((member @ nat) @ U) @ ((v @ a) @ c))))))).
% 1.80/1.10 thf(fact_16_obtain__shortest__path, axiom, (! [U:nat,V:nat]: ((((((connected @ a) @ c) @ U) @ V) => (~ ((! [P3:(list @ ((product_prod @ nat) @ nat))]: ((~ ((((((isShortestPath @ a) @ c) @ U) @ P3) @ V))))))))))).
% 1.80/1.10 thf(fact_17_finite__Collect__conjI, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((finite_finite2 @ A) @ ((collect @ A) @ P)) | ((finite_finite2 @ A) @ ((collect @ A) @ Q))) => ((finite_finite2 @ A) @ ((collect @ A) @ (^ [X3:A]: (((P @ X3) & (Q @ X3)))))))))).
% 1.80/1.10 thf(fact_18_finite__Collect__disjI, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: ((((finite_finite2 @ A) @ ((collect @ A) @ (^ [X3:A]: (((P @ X3) | (Q @ X3)))))) = (((finite_finite2 @ A) @ ((collect @ A) @ P)) & ((finite_finite2 @ A) @ ((collect @ A) @ Q))))))).
% 1.80/1.10 thf(fact_19_finite__Collect__bounded__ex, axiom, (! [B:$tType,A:$tType,P:(A > $o),Q:(B > (A > $o))]: ((((finite_finite2 @ A) @ ((collect @ A) @ P)) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [X3:B]: ((? [Y:A]: (((P @ Y) & ((Q @ X3) @ Y)))))))) = (! [Y:A]: (((P @ Y) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [X3:B]: (((Q @ X3) @ Y))))))))))))).
% 1.80/1.10 thf(fact_20_finite__Collect__subsets, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((finite_finite2 @ (set @ A)) @ ((collect @ (set @ A)) @ (^ [B2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B2) @ A3))))))))).
% 1.80/1.10 thf(fact_21_finite__V, axiom, ((finite_finite2 @ nat) @ ((v @ a) @ c))).
% 1.80/1.10 thf(fact_22_Vfin__imp__Efin, axiom, (((finite_finite2 @ nat) @ ((v @ a) @ c)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ a) @ c)))).
% 1.80/1.10 thf(fact_23_Range__mono, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S) => (((ord_less_eq @ (set @ B)) @ (((range @ A) @ B) @ R)) @ (((range @ A) @ B) @ S)))))).
% 1.80/1.10 thf(fact_24_Domain__mono, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S) => (((ord_less_eq @ (set @ A)) @ (((domain @ A) @ B) @ R)) @ (((domain @ A) @ B) @ S)))))).
% 1.80/1.10 thf(fact_25_finite__image__set, axiom, (! [A:$tType,B:$tType,P:(A > $o),F2:(A > B)]: ((((finite_finite2 @ A) @ ((collect @ A) @ P)) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [Uu:B]: ((? [X3:A]: (((Uu = (F2 @ X3)) & (P @ X3)))))))))))).
% 1.80/1.10 thf(fact_26_finite__image__set2, axiom, (! [A:$tType,B:$tType,C:$tType,P:(A > $o),Q:(B > $o),F2:(A > (B > C))]: ((((finite_finite2 @ A) @ ((collect @ A) @ P)) => (((finite_finite2 @ B) @ ((collect @ B) @ Q)) => ((finite_finite2 @ C) @ ((collect @ C) @ (^ [Uu:C]: ((? [X3:A,Y:B]: (((Uu = ((F2 @ X3) @ Y)) & ((P @ X3) & (Q @ Y)))))))))))))).
% 1.80/1.10 thf(fact_27_not__finite__existsD, axiom, (! [A:$tType,P:(A > $o)]: (((~ (((finite_finite2 @ A) @ ((collect @ A) @ P)))) => (? [X_12:A]: ((P @ X_12))))))).
% 1.80/1.10 thf(fact_28_pigeonhole__infinite__rel, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B),R2:(A > (B > $o))]: (((~ (((finite_finite2 @ A) @ A3))) => (((finite_finite2 @ B) @ B3) => ((! [X:A]: (((((member @ A) @ X) @ A3) => (? [Xa:B]: (((((member @ B) @ Xa) @ B3) & ((R2 @ X) @ Xa))))))) => (? [X:B]: (((((member @ B) @ X) @ B3) & (~ (((finite_finite2 @ A) @ ((collect @ A) @ (^ [A4:A]: (((((member @ A) @ A4) @ A3) & ((R2 @ A4) @ X))))))))))))))))).
% 1.80/1.10 thf(fact_29_ek__analysis__def, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((edmond2129202899alysis @ A) = (finite_Graph @ A)))))).
% 1.80/1.10 thf(fact_30_ek__analysis_Ointro, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:(((product_prod @ nat) @ nat) > A)]: ((((finite_Graph @ A) @ C2) => ((edmond2129202899alysis @ A) @ C2)))))))).
% 1.80/1.10 thf(fact_31_ek__analysis_Oaxioms, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:(((product_prod @ nat) @ nat) > A)]: ((((edmond2129202899alysis @ A) @ C2) => ((finite_Graph @ A) @ C2)))))))).
% 1.80/1.10 thf(fact_32_ek__analysis__defs_OspEdges__def, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((edmond1803015688pEdges @ A) = (^ [C3:(((product_prod @ nat) @ nat) > A),S2:nat,T2:nat]: (((collect @ ((product_prod @ nat) @ nat)) @ (^ [E:((product_prod @ nat) @ nat)]: ((? [P2:(list @ ((product_prod @ nat) @ nat))]: (((((member @ ((product_prod @ nat) @ nat)) @ E) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P2)) & (((((isShortestPath @ A) @ C3) @ S2) @ P2) @ T2)))))))))))))).
% 1.80/1.10 thf(fact_33_finite__has__minimal2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A),A5:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A5) @ A3) => (? [X:A]: (((((member @ A) @ X) @ A3) & ((((ord_less_eq @ A) @ X) @ A5) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ Xa) @ X) => (X = Xa))))))))))))))))).
% 1.80/1.10 thf(fact_34_finite__has__maximal2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A),A5:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A5) @ A3) => (? [X:A]: (((((member @ A) @ X) @ A3) & ((((ord_less_eq @ A) @ A5) @ X) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ X) @ Xa) => (X = Xa))))))))))))))))).
% 1.80/1.10 thf(fact_35_rev__finite__subset, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((finite_finite2 @ A) @ A3)))))).
% 1.80/1.10 thf(fact_36_infinite__super, axiom, (! [A:$tType,S3:(set @ A),T3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S3) @ T3) => ((~ (((finite_finite2 @ A) @ S3))) => (~ (((finite_finite2 @ A) @ T3)))))))).
% 1.80/1.10 thf(fact_37_finite__subset, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((finite_finite2 @ A) @ B3) => ((finite_finite2 @ A) @ A3)))))).
% 1.80/1.10 thf(fact_38_ek__analysis_OspEdges__ss__E, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:(((product_prod @ nat) @ nat) > A),S:nat,T4:nat]: ((((edmond2129202899alysis @ A) @ C2) => (((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ ((((edmond1803015688pEdges @ A) @ C2) @ S) @ T4)) @ ((e @ A) @ C2))))))))).
% 1.80/1.10 thf(fact_39_incoming_H__edges, axiom, (! [U2:(set @ nat)]: ((((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ (((incoming2 @ a) @ c) @ U2)) @ ((e @ a) @ c))))).
% 1.80/1.10 thf(fact_40_outgoing_H__edges, axiom, (! [U2:(set @ nat)]: ((((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ (((outgoing2 @ a) @ c) @ U2)) @ ((e @ a) @ c))))).
% 1.80/1.10 thf(fact_41_reachableNodes__def, axiom, (! [U:nat]: (((((reachableNodes @ a) @ c) @ U) = ((collect @ nat) @ (((connected @ a) @ c) @ U)))))).
% 1.80/1.10 thf(fact_42_List_Ofinite__set, axiom, (! [A:$tType,Xs:(list @ A)]: (((finite_finite2 @ A) @ ((set2 @ A) @ Xs))))).
% 1.80/1.10 thf(fact_43_Graph_OEfin__imp__Vfin, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity)]: ((((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ Capacity) @ C2)) => ((finite_finite2 @ nat) @ ((v @ Capacity) @ C2))))))))).
% 1.80/1.10 thf(fact_44_Graph_OVfin__imp__Efin, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity)]: ((((finite_finite2 @ nat) @ ((v @ Capacity) @ C2)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ Capacity) @ C2))))))))).
% 1.80/1.10 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A5:A,P:(A > $o)]: (((((member @ A) @ A5) @ ((collect @ A) @ P)) = (P @ A5))))).
% 1.80/1.10 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A3)))) = A3)))).
% 1.80/1.10 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P @ X) = (Q @ X)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 1.80/1.10 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F2:(A > B),G:(A > B)]: (((! [X:A]: (((F2 @ X) = (G @ X)))) => (F2 = G))))).
% 1.80/1.10 thf(fact_49_isShortestPath__level__edge_I1_J, axiom, (! [S:nat,P4:(list @ ((product_prod @ nat) @ nat)),T4:nat,U:nat,V:nat]: (((((((isShortestPath @ a) @ c) @ S) @ P4) @ T4) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => ((((connected @ a) @ c) @ S) @ U)))))).
% 1.80/1.10 thf(fact_50_isShortestPath__level__edge_I2_J, axiom, (! [S:nat,P4:(list @ ((product_prod @ nat) @ nat)),T4:nat,U:nat,V:nat]: (((((((isShortestPath @ a) @ c) @ S) @ P4) @ T4) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => ((((connected @ a) @ c) @ U) @ V)))))).
% 1.80/1.10 thf(fact_51_isShortestPath__level__edge_I3_J, axiom, (! [S:nat,P4:(list @ ((product_prod @ nat) @ nat)),T4:nat,U:nat,V:nat]: (((((((isShortestPath @ a) @ c) @ S) @ P4) @ T4) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => ((((connected @ a) @ c) @ V) @ T4)))))).
% 1.80/1.10 thf(fact_52_incoming__edges, axiom, (! [U:nat]: ((((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ (((incoming @ a) @ c) @ U)) @ ((e @ a) @ c))))).
% 1.80/1.10 thf(fact_53_reachable__ss__V, axiom, (! [S:nat]: (((((member @ nat) @ S) @ ((v @ a) @ c)) => (((ord_less_eq @ (set @ nat)) @ (((reachableNodes @ a) @ c) @ S)) @ ((v @ a) @ c)))))).
% 1.80/1.10 thf(fact_54_connected__append__edge, axiom, (! [U:nat,V:nat,W:nat]: ((((((connected @ a) @ c) @ U) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ V) @ W)) @ ((e @ a) @ c)) => ((((connected @ a) @ c) @ U) @ W)))))).
% 1.80/1.10 thf(fact_55_reachableNodes__append__edge, axiom, (! [U:nat,S:nat,V:nat]: (((((member @ nat) @ U) @ (((reachableNodes @ a) @ c) @ S)) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((e @ a) @ c)) => (((member @ nat) @ V) @ (((reachableNodes @ a) @ c) @ S))))))).
% 1.80/1.10 thf(fact_56_V__def, axiom, (((v @ a) @ c) = ((collect @ nat) @ (^ [U3:nat]: ((? [V2:nat]: (((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U3) @ V2)) @ ((e @ a) @ c)) | (((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ V2) @ U3)) @ ((e @ a) @ c)))))))))).
% 1.80/1.10 thf(fact_57_incoming__def, axiom, (! [V:nat]: (((((incoming @ a) @ c) @ V) = ((collect @ ((product_prod @ nat) @ nat)) @ (^ [Uu:((product_prod @ nat) @ nat)]: ((? [U3:nat]: (((Uu = ((((product_Pair @ nat) @ nat) @ U3) @ V)) & (((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U3) @ V)) @ ((e @ a) @ c)))))))))))).
% 1.80/1.10 thf(fact_58_outgoing_H__def, axiom, (! [K:(set @ nat)]: (((((outgoing2 @ a) @ c) @ K) = ((collect @ ((product_prod @ nat) @ nat)) @ (^ [Uu:((product_prod @ nat) @ nat)]: ((? [U3:nat,V2:nat]: (((Uu = ((((product_Pair @ nat) @ nat) @ V2) @ U3)) & ((~ ((((member @ nat) @ U3) @ K))) & ((((member @ nat) @ V2) @ K) & (((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ V2) @ U3)) @ ((e @ a) @ c)))))))))))))).
% 1.80/1.10 thf(fact_59_incoming_H__def, axiom, (! [K:(set @ nat)]: (((((incoming2 @ a) @ c) @ K) = ((collect @ ((product_prod @ nat) @ nat)) @ (^ [Uu:((product_prod @ nat) @ nat)]: ((? [U3:nat,V2:nat]: (((Uu = ((((product_Pair @ nat) @ nat) @ U3) @ V2)) & ((~ ((((member @ nat) @ U3) @ K))) & ((((member @ nat) @ V2) @ K) & (((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U3) @ V2)) @ ((e @ a) @ c)))))))))))))).
% 1.80/1.10 thf(fact_60_finite__Collect__le__nat, axiom, (! [K:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N:nat]: ((((ord_less_eq @ nat) @ N) @ K)))))))).
% 1.80/1.10 thf(fact_61_finite__incoming, axiom, (! [U:nat]: ((((finite_finite2 @ nat) @ ((v @ a) @ c)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ (((incoming @ a) @ c) @ U)))))).
% 1.80/1.10 thf(fact_62_finite__outgoing_H, axiom, (! [U2:(set @ nat)]: ((((finite_finite2 @ nat) @ ((v @ a) @ c)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ (((outgoing2 @ a) @ c) @ U2)))))).
% 1.80/1.10 thf(fact_63_finite__incoming_H, axiom, (! [U2:(set @ nat)]: ((((finite_finite2 @ nat) @ ((v @ a) @ c)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ (((incoming2 @ a) @ c) @ U2)))))).
% 1.80/1.10 thf(fact_64_adjacent__nodes__ss__V, axiom, (! [U:nat]: ((((ord_less_eq @ (set @ nat)) @ (((adjacent_nodes @ a) @ c) @ U)) @ ((v @ a) @ c))))).
% 1.80/1.10 thf(fact_65_Graph_Oincoming__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((incoming @ Capacity) = (^ [C3:(((product_prod @ nat) @ nat) > Capacity),V2:nat]: (((collect @ ((product_prod @ nat) @ nat)) @ (^ [Uu:((product_prod @ nat) @ nat)]: ((? [U3:nat]: (((Uu = ((((product_Pair @ nat) @ nat) @ U3) @ V2)) & (((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U3) @ V2)) @ ((e @ Capacity) @ C3))))))))))))))).
% 1.80/1.10 thf(fact_66_Graph_Oincoming_H__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((incoming2 @ Capacity) = (^ [C3:(((product_prod @ nat) @ nat) > Capacity),K2:(set @ nat)]: (((collect @ ((product_prod @ nat) @ nat)) @ (^ [Uu:((product_prod @ nat) @ nat)]: ((? [U3:nat,V2:nat]: (((Uu = ((((product_Pair @ nat) @ nat) @ U3) @ V2)) & ((~ ((((member @ nat) @ U3) @ K2))) & ((((member @ nat) @ V2) @ K2) & (((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U3) @ V2)) @ ((e @ Capacity) @ C3))))))))))))))))).
% 1.80/1.10 thf(fact_67_Graph_Ooutgoing_H__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((outgoing2 @ Capacity) = (^ [C3:(((product_prod @ nat) @ nat) > Capacity),K2:(set @ nat)]: (((collect @ ((product_prod @ nat) @ nat)) @ (^ [Uu:((product_prod @ nat) @ nat)]: ((? [U3:nat,V2:nat]: (((Uu = ((((product_Pair @ nat) @ nat) @ V2) @ U3)) & ((~ ((((member @ nat) @ U3) @ K2))) & ((((member @ nat) @ V2) @ K2) & (((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ V2) @ U3)) @ ((e @ Capacity) @ C3))))))))))))))))).
% 1.80/1.10 thf(fact_68_Graph_Oreachable__ss__V, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [S:nat,C2:(((product_prod @ nat) @ nat) > Capacity)]: (((((member @ nat) @ S) @ ((v @ Capacity) @ C2)) => (((ord_less_eq @ (set @ nat)) @ (((reachableNodes @ Capacity) @ C2) @ S)) @ ((v @ Capacity) @ C2))))))))).
% 1.80/1.10 thf(fact_69_Graph_OreachableNodes__append__edge, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [U:nat,C2:(((product_prod @ nat) @ nat) > Capacity),S:nat,V:nat]: (((((member @ nat) @ U) @ (((reachableNodes @ Capacity) @ C2) @ S)) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((e @ Capacity) @ C2)) => (((member @ nat) @ V) @ (((reachableNodes @ Capacity) @ C2) @ S)))))))))).
% 1.80/1.10 thf(fact_70_Graph_OreachableNodes_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((reachableNodes @ Capacity) = (reachableNodes @ Capacity)))))).
% 1.80/1.10 thf(fact_71_Graph_Ooutgoing_H_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((outgoing2 @ Capacity) = (outgoing2 @ Capacity)))))).
% 1.80/1.10 thf(fact_72_Graph_Oincoming_H_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((incoming2 @ Capacity) = (incoming2 @ Capacity)))))).
% 1.80/1.10 thf(fact_73_Graph_Oincoming_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((incoming @ Capacity) = (incoming @ Capacity)))))).
% 1.80/1.10 thf(fact_74_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: ((((^ [X3:A,Y:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y)) @ R2))) = (^ [X3:A,Y:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y)) @ S3)))) = (R2 = S3))))).
% 1.80/1.10 thf(fact_75_pred__subset__eq2, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),S3:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (A > (B > $o))) @ (^ [X3:A,Y:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y)) @ R2)))) @ (^ [X3:A,Y:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y)) @ S3)))) = (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R2) @ S3))))).
% 1.80/1.10 thf(fact_76_Graph_OV__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((v @ Capacity) = (^ [C3:(((product_prod @ nat) @ nat) > Capacity)]: (((collect @ nat) @ (^ [U3:nat]: ((? [V2:nat]: (((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U3) @ V2)) @ ((e @ Capacity) @ C3)) | (((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ V2) @ U3)) @ ((e @ Capacity) @ C3))))))))))))))).
% 1.80/1.10 thf(fact_77_subrelI, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((! [X:A,Y2:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ R) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y2)) @ S)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S))))).
% 1.80/1.10 thf(fact_78_Domain_Oinducts, axiom, (! [B:$tType,A:$tType,X4:A,R:(set @ ((product_prod @ A) @ B)),P:(A > $o)]: (((((member @ A) @ X4) @ (((domain @ A) @ B) @ R)) => ((! [A6:A,B4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A6) @ B4)) @ R) => (P @ A6)))) => (P @ X4)))))).
% 1.80/1.10 thf(fact_79_Domain_ODomainI, axiom, (! [B:$tType,A:$tType,A5:A,B5:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ R) => (((member @ A) @ A5) @ (((domain @ A) @ B) @ R)))))).
% 1.80/1.10 thf(fact_80_Domain_Osimps, axiom, (! [B:$tType,A:$tType,A5:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A5) @ (((domain @ A) @ B) @ R)) = (? [A4:A,B6:B]: (((A5 = A4) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B6)) @ R)))))))).
% 1.80/1.10 thf(fact_81_Domain_Ocases, axiom, (! [B:$tType,A:$tType,A5:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A5) @ (((domain @ A) @ B) @ R)) => (~ ((! [B4:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B4)) @ R))))))))))).
% 1.80/1.10 thf(fact_82_Domain__iff, axiom, (! [A:$tType,B:$tType,A5:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A5) @ (((domain @ A) @ B) @ R)) = (? [Y:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ Y)) @ R))))))).
% 1.80/1.10 thf(fact_83_DomainE, axiom, (! [B:$tType,A:$tType,A5:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A5) @ (((domain @ A) @ B) @ R)) => (~ ((! [B4:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B4)) @ R))))))))))).
% 1.80/1.10 thf(fact_84_Range_Oinducts, axiom, (! [A:$tType,B:$tType,X4:B,R:(set @ ((product_prod @ A) @ B)),P:(B > $o)]: (((((member @ B) @ X4) @ (((range @ A) @ B) @ R)) => ((! [A6:A,B4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A6) @ B4)) @ R) => (P @ B4)))) => (P @ X4)))))).
% 1.80/1.10 thf(fact_85_Range_Ointros, axiom, (! [B:$tType,A:$tType,A5:A,B5:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ R) => (((member @ B) @ B5) @ (((range @ A) @ B) @ R)))))).
% 1.80/1.10 thf(fact_86_Range_Osimps, axiom, (! [B:$tType,A:$tType,A5:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ B) @ A5) @ (((range @ A) @ B) @ R)) = (? [A4:A,B6:B]: (((A5 = B6) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B6)) @ R)))))))).
% 1.80/1.10 thf(fact_87_Range_Ocases, axiom, (! [B:$tType,A:$tType,A5:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ B) @ A5) @ (((range @ A) @ B) @ R)) => (~ ((! [A6:A]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A6) @ A5)) @ R))))))))))).
% 1.80/1.10 thf(fact_88_Range__iff, axiom, (! [A:$tType,B:$tType,A5:A,R:(set @ ((product_prod @ B) @ A))]: (((((member @ A) @ A5) @ (((range @ B) @ A) @ R)) = (? [Y:B]: ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ Y) @ A5)) @ R))))))).
% 1.80/1.10 thf(fact_89_RangeE, axiom, (! [A:$tType,B:$tType,B5:A,R:(set @ ((product_prod @ B) @ A))]: (((((member @ A) @ B5) @ (((range @ B) @ A) @ R)) => (~ ((! [A6:B]: ((~ ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A6) @ B5)) @ R))))))))))).
% 1.80/1.10 thf(fact_90_Graph_OreachableNodes__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((reachableNodes @ Capacity) = (^ [C3:(((product_prod @ nat) @ nat) > Capacity),U3:nat]: (((collect @ nat) @ (((connected @ Capacity) @ C3) @ U3))))))))).
% 1.80/1.10 thf(fact_91_pred__subset__eq, axiom, (! [A:$tType,R2:(set @ A),S3:(set @ A)]: (((((ord_less_eq @ (A > $o)) @ (^ [X3:A]: ((((member @ A) @ X3) @ R2)))) @ (^ [X3:A]: ((((member @ A) @ X3) @ S3)))) = (((ord_less_eq @ (set @ A)) @ R2) @ S3))))).
% 1.80/1.10 thf(fact_92_Graph_Oconnected__append__edge, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U:nat,V:nat,W:nat]: ((((((connected @ Capacity) @ C2) @ U) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ V) @ W)) @ ((e @ Capacity) @ C2)) => ((((connected @ Capacity) @ C2) @ U) @ W))))))))).
% 1.80/1.10 thf(fact_93_Graph_Oincoming__edges, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U:nat]: ((((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ (((incoming @ Capacity) @ C2) @ U)) @ ((e @ Capacity) @ C2)))))))).
% 1.80/1.10 thf(fact_94_Graph_OisShortestPath__level__edge_I1_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),S:nat,P4:(list @ ((product_prod @ nat) @ nat)),T4:nat,U:nat,V:nat]: (((((((isShortestPath @ Capacity) @ C2) @ S) @ P4) @ T4) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => ((((connected @ Capacity) @ C2) @ S) @ U))))))))).
% 1.80/1.10 thf(fact_95_Graph_OisShortestPath__level__edge_I2_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),S:nat,P4:(list @ ((product_prod @ nat) @ nat)),T4:nat,U:nat,V:nat]: (((((((isShortestPath @ Capacity) @ C2) @ S) @ P4) @ T4) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => ((((connected @ Capacity) @ C2) @ U) @ V))))))))).
% 1.80/1.10 thf(fact_96_Graph_OisShortestPath__level__edge_I3_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),S:nat,P4:(list @ ((product_prod @ nat) @ nat)),T4:nat,U:nat,V:nat]: (((((((isShortestPath @ Capacity) @ C2) @ S) @ P4) @ T4) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => ((((connected @ Capacity) @ C2) @ V) @ T4))))))))).
% 1.80/1.10 thf(fact_97_Graph_Ooutgoing_H__edges, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U2:(set @ nat)]: ((((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ (((outgoing2 @ Capacity) @ C2) @ U2)) @ ((e @ Capacity) @ C2)))))))).
% 1.80/1.10 thf(fact_98_Graph_Oincoming_H__edges, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U2:(set @ nat)]: ((((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ (((incoming2 @ Capacity) @ C2) @ U2)) @ ((e @ Capacity) @ C2)))))))).
% 1.80/1.10 thf(fact_99_Domain__unfold, axiom, (! [B:$tType,A:$tType]: ((((domain @ A) @ B) = (^ [R3:(set @ ((product_prod @ A) @ B))]: (((collect @ A) @ (^ [X3:A]: ((? [Y:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y)) @ R3)))))))))))).
% 1.80/1.10 thf(fact_100_Graph_Ofinite__incoming, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U:nat]: ((((finite_finite2 @ nat) @ ((v @ Capacity) @ C2)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ (((incoming @ Capacity) @ C2) @ U))))))))).
% 1.80/1.10 thf(fact_101_Graph_OE_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((e @ Capacity) = (e @ Capacity)))))).
% 1.80/1.10 thf(fact_102_Graph_OV_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((v @ Capacity) = (v @ Capacity)))))).
% 1.80/1.10 thf(fact_103_Graph_Oconnected__refl, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),V:nat]: (((((connected @ Capacity) @ C2) @ V) @ V))))))).
% 1.80/1.10 thf(fact_104_Graph_Oconnected_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((connected @ Capacity) = (connected @ Capacity)))))).
% 1.80/1.10 thf(fact_105_Graph_Ofinite__outgoing_H, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U2:(set @ nat)]: ((((finite_finite2 @ nat) @ ((v @ Capacity) @ C2)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ (((outgoing2 @ Capacity) @ C2) @ U2))))))))).
% 1.80/1.10 thf(fact_106_Graph_OisShortestPath_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isShortestPath @ Capacity) = (isShortestPath @ Capacity)))))).
% 1.80/1.10 thf(fact_107_Graph_Ofinite__incoming_H, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U2:(set @ nat)]: ((((finite_finite2 @ nat) @ ((v @ Capacity) @ C2)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ (((incoming2 @ Capacity) @ C2) @ U2))))))))).
% 1.80/1.10 thf(fact_108_finite__list, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (? [Xs2:(list @ A)]: ((((set2 @ A) @ Xs2) = A3))))))).
% 1.80/1.10 thf(fact_109_subset__code_I1_J, axiom, (! [A:$tType,Xs:(list @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ B3) = (! [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) => (((member @ A) @ X3) @ B3)))))))).
% 1.80/1.10 thf(fact_110_Graph_Oconnected__inV__iff, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U:nat,V:nat]: ((((((connected @ Capacity) @ C2) @ U) @ V) => ((((member @ nat) @ V) @ ((v @ Capacity) @ C2)) = (((member @ nat) @ U) @ ((v @ Capacity) @ C2)))))))))).
% 1.80/1.10 thf(fact_111_Graph_Oobtain__shortest__path, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U:nat,V:nat]: ((((((connected @ Capacity) @ C2) @ U) @ V) => (~ ((! [P3:(list @ ((product_prod @ nat) @ nat))]: ((~ ((((((isShortestPath @ Capacity) @ C2) @ U) @ P3) @ V)))))))))))))).
% 1.80/1.10 thf(fact_112_Finite__Graph_Ofinite__V, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:(((product_prod @ nat) @ nat) > A)]: ((((finite_Graph @ A) @ C2) => ((finite_finite2 @ nat) @ ((v @ A) @ C2))))))))).
% 1.80/1.10 thf(fact_113_Finite__Graph_Ointro, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:(((product_prod @ nat) @ nat) > A)]: ((((finite_finite2 @ nat) @ ((v @ A) @ C2)) => ((finite_Graph @ A) @ C2)))))))).
% 1.80/1.10 thf(fact_114_Finite__Graph__def, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((finite_Graph @ A) = (^ [C3:(((product_prod @ nat) @ nat) > A)]: (((finite_finite2 @ nat) @ ((v @ A) @ C3))))))))).
% 1.80/1.10 thf(fact_115_Finite__Graph_Ofinite__E, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:(((product_prod @ nat) @ nat) > A)]: ((((finite_Graph @ A) @ C2) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ A) @ C2))))))))).
% 1.80/1.10 thf(fact_116_Graph_OFinite__Graph__EI, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity)]: ((((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ Capacity) @ C2)) => ((finite_Graph @ Capacity) @ C2)))))))).
% 1.80/1.10 thf(fact_117_outgoing__def, axiom, (! [V:nat]: (((((outgoing @ a) @ c) @ V) = ((collect @ ((product_prod @ nat) @ nat)) @ (^ [Uu:((product_prod @ nat) @ nat)]: ((? [U3:nat]: (((Uu = ((((product_Pair @ nat) @ nat) @ V) @ U3)) & (((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ V) @ U3)) @ ((e @ a) @ c)))))))))))).
% 1.80/1.10 thf(fact_118_outgoing__edges, axiom, (! [U:nat]: ((((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ (((outgoing @ a) @ c) @ U)) @ ((e @ a) @ c))))).
% 1.80/1.10 thf(fact_119_E__ss__VxV, axiom, (((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ ((e @ a) @ c)) @ ((((product_Sigma @ nat) @ nat) @ ((v @ a) @ c)) @ (^ [Uu:nat]: (((v @ a) @ c)))))).
% 1.80/1.10 thf(fact_120_connected__edgeRtc, axiom, (! [U:nat,V:nat]: ((((((connected @ a) @ c) @ U) @ V) = (((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((transitive_rtrancl @ nat) @ ((e @ a) @ c))))))).
% 1.80/1.10 thf(fact_121_isSPath__no__selfloop, axiom, (! [U:nat,P4:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat]: (((((((isSimplePath @ a) @ c) @ U) @ P4) @ V) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ U1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)))))))).
% 1.80/1.10 thf(fact_122_isSPath__sg__incoming, axiom, (! [U:nat,P4:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat,U22:nat]: (((((((isSimplePath @ a) @ c) @ U) @ P4) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => ((U1 != U22) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U22) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)))))))))).
% 1.80/1.10 thf(fact_123_isSPath__sg__outgoing, axiom, (! [U:nat,P4:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat,V22:nat]: (((((((isSimplePath @ a) @ c) @ U) @ P4) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => ((V1 != V22) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V22)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)))))))))).
% 1.80/1.10 thf(fact_124_reachableNodes__E__closed, axiom, (! [S:nat]: ((((ord_less_eq @ (set @ nat)) @ ((((image @ nat) @ nat) @ ((e @ a) @ c)) @ (((reachableNodes @ a) @ c) @ S))) @ (((reachableNodes @ a) @ c) @ S))))).
% 1.80/1.10 thf(fact_125_shortestPath__is__simple, axiom, (! [S:nat,P4:(list @ ((product_prod @ nat) @ nat)),T4:nat]: (((((((isShortestPath @ a) @ c) @ S) @ P4) @ T4) => (((((isSimplePath @ a) @ c) @ S) @ P4) @ T4))))).
% 1.80/1.10 thf(fact_126_ImageI, axiom, (! [B:$tType,A:$tType,A5:A,B5:B,R:(set @ ((product_prod @ A) @ B)),A3:(set @ A)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ R) => ((((member @ A) @ A5) @ A3) => (((member @ B) @ B5) @ ((((image @ A) @ B) @ R) @ A3))))))).
% 1.80/1.10 thf(fact_127_finite__SigmaI, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(A > (set @ B))]: ((((finite_finite2 @ A) @ A3) => ((! [A6:A]: (((((member @ A) @ A6) @ A3) => ((finite_finite2 @ B) @ (B3 @ A6))))) => ((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3))))))).
% 1.80/1.10 thf(fact_128_adjacent__nodes__finite, axiom, (! [U:nat]: (((finite_finite2 @ nat) @ (((adjacent_nodes @ a) @ c) @ U))))).
% 1.80/1.10 thf(fact_129_finite__outgoing, axiom, (! [U:nat]: ((((finite_finite2 @ nat) @ ((v @ a) @ c)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ (((outgoing @ a) @ c) @ U)))))).
% 1.80/1.10 thf(fact_130_Graph_Oadjacent__nodes_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((adjacent_nodes @ Capacity) = (adjacent_nodes @ Capacity)))))).
% 1.80/1.10 thf(fact_131_Graph_OisSimplePath_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isSimplePath @ Capacity) = (isSimplePath @ Capacity)))))).
% 1.80/1.10 thf(fact_132_Graph_Ooutgoing_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((outgoing @ Capacity) = (outgoing @ Capacity)))))).
% 1.80/1.10 thf(fact_133_rev__ImageI, axiom, (! [B:$tType,A:$tType,A5:A,A3:(set @ A),B5:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A5) @ A3) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ R) => (((member @ B) @ B5) @ ((((image @ A) @ B) @ R) @ A3))))))).
% 1.80/1.10 thf(fact_134_Image__iff, axiom, (! [A:$tType,B:$tType,B5:A,R:(set @ ((product_prod @ B) @ A)),A3:(set @ B)]: (((((member @ A) @ B5) @ ((((image @ B) @ A) @ R) @ A3)) = (? [X3:B]: (((((member @ B) @ X3) @ A3) & (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X3) @ B5)) @ R)))))))).
% 1.80/1.10 thf(fact_135_ImageE, axiom, (! [A:$tType,B:$tType,B5:A,R:(set @ ((product_prod @ B) @ A)),A3:(set @ B)]: (((((member @ A) @ B5) @ ((((image @ B) @ A) @ R) @ A3)) => (~ ((! [X:B]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ B5)) @ R) => (~ ((((member @ B) @ X) @ A3)))))))))))).
% 1.80/1.10 thf(fact_136_Image__subset, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),A3:(set @ A),B3:(set @ B),C4:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B3)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ R) @ C4)) @ B3))))).
% 1.80/1.10 thf(fact_137_Graph_OshortestPath__is__simple, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),S:nat,P4:(list @ ((product_prod @ nat) @ nat)),T4:nat]: (((((((isShortestPath @ Capacity) @ C2) @ S) @ P4) @ T4) => (((((isSimplePath @ Capacity) @ C2) @ S) @ P4) @ T4)))))))).
% 1.80/1.10 thf(fact_138_Image__mono, axiom, (! [B:$tType,A:$tType,R4:(set @ ((product_prod @ A) @ B)),R:(set @ ((product_prod @ A) @ B)),A7:(set @ A),A3:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R4) @ R) => ((((ord_less_eq @ (set @ A)) @ A7) @ A3) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ R4) @ A7)) @ ((((image @ A) @ B) @ R) @ A3))))))).
% 1.80/1.10 thf(fact_139_Relation_Ofinite__Image, axiom, (! [B:$tType,A:$tType,R2:(set @ ((product_prod @ A) @ B)),A3:(set @ A)]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ R2) => ((finite_finite2 @ B) @ ((((image @ A) @ B) @ R2) @ A3)))))).
% 1.80/1.10 thf(fact_140_Graph_OreachableNodes__E__closed, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),S:nat]: ((((ord_less_eq @ (set @ nat)) @ ((((image @ nat) @ nat) @ ((e @ Capacity) @ C2)) @ (((reachableNodes @ Capacity) @ C2) @ S))) @ (((reachableNodes @ Capacity) @ C2) @ S)))))))).
% 1.80/1.10 thf(fact_141_finite__cartesian__product, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ B) @ B3) => ((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B3))))))))).
% 1.80/1.10 thf(fact_142_Graph_OisSPath__sg__outgoing, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U:nat,P4:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat,V22:nat]: (((((((isSimplePath @ Capacity) @ C2) @ U) @ P4) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => ((V1 != V22) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V22)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4))))))))))))).
% 1.80/1.10 thf(fact_143_Graph_OisSPath__sg__incoming, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U:nat,P4:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat,U22:nat]: (((((((isSimplePath @ Capacity) @ C2) @ U) @ P4) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => ((U1 != U22) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U22) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4))))))))))))).
% 1.80/1.10 thf(fact_144_Graph_OisSPath__no__selfloop, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U:nat,P4:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat]: (((((((isSimplePath @ Capacity) @ C2) @ U) @ P4) @ V) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ U1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4))))))))))).
% 1.80/1.10 thf(fact_145_Graph_Oadjacent__nodes__ss__V, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U:nat]: ((((ord_less_eq @ (set @ nat)) @ (((adjacent_nodes @ Capacity) @ C2) @ U)) @ ((v @ Capacity) @ C2)))))))).
% 1.80/1.10 thf(fact_146_Graph_Ooutgoing__edges, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U:nat]: ((((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ (((outgoing @ Capacity) @ C2) @ U)) @ ((e @ Capacity) @ C2)))))))).
% 1.80/1.10 thf(fact_147_Graph_Ooutgoing__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((outgoing @ Capacity) = (^ [C3:(((product_prod @ nat) @ nat) > Capacity),V2:nat]: (((collect @ ((product_prod @ nat) @ nat)) @ (^ [Uu:((product_prod @ nat) @ nat)]: ((? [U3:nat]: (((Uu = ((((product_Pair @ nat) @ nat) @ V2) @ U3)) & (((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ V2) @ U3)) @ ((e @ Capacity) @ C3))))))))))))))).
% 1.80/1.10 thf(fact_148_Graph_Oconnected__edgeRtc, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((connected @ Capacity) = (^ [C3:(((product_prod @ nat) @ nat) > Capacity),U3:nat,V2:nat]: ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U3) @ V2)) @ ((transitive_rtrancl @ nat) @ ((e @ Capacity) @ C3)))))))))).
% 1.80/1.10 thf(fact_149_Graph_OE__ss__VxV, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity)]: ((((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ ((e @ Capacity) @ C2)) @ ((((product_Sigma @ nat) @ nat) @ ((v @ Capacity) @ C2)) @ (^ [Uu:nat]: (((v @ Capacity) @ C2))))))))))).
% 1.80/1.10 thf(fact_150_Graph_Ofinite__outgoing, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U:nat]: ((((finite_finite2 @ nat) @ ((v @ Capacity) @ C2)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ (((outgoing @ Capacity) @ C2) @ U))))))))).
% 1.80/1.10 thf(fact_151_Finite__Graph_Oadjacent__nodes__finite, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:(((product_prod @ nat) @ nat) > A),U:nat]: ((((finite_Graph @ A) @ C2) => ((finite_finite2 @ nat) @ (((adjacent_nodes @ A) @ C2) @ U))))))))).
% 1.80/1.10 thf(fact_152_isSPath__nt__parallel, axiom, (! [S:nat,P4:(list @ ((product_prod @ nat) @ nat)),T4:nat,E2:((product_prod @ nat) @ nat)]: (((((((isSimplePath @ a) @ c) @ S) @ P4) @ T4) => ((((member @ ((product_prod @ nat) @ nat)) @ E2) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ (((product_swap @ nat) @ nat) @ E2)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4))))))))).
% 1.80/1.10 thf(fact_153_mem__Sigma__iff, axiom, (! [B:$tType,A:$tType,A5:A,B5:B,A3:(set @ A),B3:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) = ((((member @ A) @ A5) @ A3) & (((member @ B) @ B5) @ (B3 @ A5))))))).
% 1.80/1.10 thf(fact_154_SigmaI, axiom, (! [B:$tType,A:$tType,A5:A,A3:(set @ A),B5:B,B3:(A > (set @ B))]: (((((member @ A) @ A5) @ A3) => ((((member @ B) @ B5) @ (B3 @ A5)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3))))))).
% 1.80/1.10 thf(fact_155_trancl__subset__Sigma__aux, axiom, (! [A:$tType,A5:A,B5:A,R:(set @ ((product_prod @ A) @ A)),A3:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ B5)) @ ((transitive_rtrancl @ A) @ R)) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ A3) @ (^ [Uu:A]: (A3)))) => ((A5 = B5) | (((member @ A) @ A5) @ A3))))))).
% 1.80/1.10 thf(fact_156_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A5:A,B5:B,A8:A,B7:B]: (((((((product_Pair @ A) @ B) @ A5) @ B5) = ((((product_Pair @ A) @ B) @ A8) @ B7)) = ((A5 = A8) & (B5 = B7)))))).
% 1.80/1.10 thf(fact_157_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y22:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y22)) = ((X1 = Y1) & (X22 = Y22)))))).
% 1.80/1.10 thf(fact_158_r__into__rtrancl, axiom, (! [A:$tType,P4:((product_prod @ A) @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ P4) @ R) => (((member @ ((product_prod @ A) @ A)) @ P4) @ ((transitive_rtrancl @ A) @ R)))))).
% 1.80/1.10 thf(fact_159_rtrancl__idemp, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((transitive_rtrancl @ A) @ ((transitive_rtrancl @ A) @ R)) = ((transitive_rtrancl @ A) @ R))))).
% 1.80/1.10 thf(fact_160_swap__swap, axiom, (! [B:$tType,A:$tType,P4:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P4)) = P4)))).
% 1.80/1.10 thf(fact_161_swap__simp, axiom, (! [A:$tType,B:$tType,X4:B,Y3:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X4) @ Y3)) = ((((product_Pair @ A) @ B) @ Y3) @ X4))))).
% 1.80/1.10 thf(fact_162_converse__rtrancl__induct2, axiom, (! [A:$tType,B:$tType,Ax:A,Ay:B,Bx:A,By:B,R:(set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))),P:(A > (B > $o))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ Bx) @ By))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R)) => (((P @ Bx) @ By) => ((! [A6:A,B4:B,Aa:A,Ba:B]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A6) @ B4)) @ ((((product_Pair @ A) @ B) @ Aa) @ Ba))) @ R) => ((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Aa) @ Ba)) @ ((((product_Pair @ A) @ B) @ Bx) @ By))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R)) => (((P @ Aa) @ Ba) => ((P @ A6) @ B4)))))) => ((P @ Ax) @ Ay))))))).
% 1.80/1.10 thf(fact_163_converse__rtranclE2, axiom, (! [B:$tType,A:$tType,Xa2:A,Xb:B,Za:A,Zb:B,R:(set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Xa2) @ Xb)) @ ((((product_Pair @ A) @ B) @ Za) @ Zb))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R)) => ((((((product_Pair @ A) @ B) @ Xa2) @ Xb) != ((((product_Pair @ A) @ B) @ Za) @ Zb)) => (~ ((! [A6:A,B4:B]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Xa2) @ Xb)) @ ((((product_Pair @ A) @ B) @ A6) @ B4))) @ R) => (~ ((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A6) @ B4)) @ ((((product_Pair @ A) @ B) @ Za) @ Zb))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R)))))))))))))).
% 1.80/1.10 thf(fact_164_rtrancl__induct2, axiom, (! [A:$tType,B:$tType,Ax:A,Ay:B,Bx:A,By:B,R:(set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))),P:(A > (B > $o))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ Bx) @ By))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R)) => (((P @ Ax) @ Ay) => ((! [A6:A,B4:B,Aa:A,Ba:B]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ A6) @ B4))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R)) => ((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A6) @ B4)) @ ((((product_Pair @ A) @ B) @ Aa) @ Ba))) @ R) => (((P @ A6) @ B4) => ((P @ Aa) @ Ba)))))) => ((P @ Bx) @ By))))))).
% 1.80/1.10 thf(fact_165_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A6:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A6) @ B4)))) => (P @ Prod))))).
% 1.80/1.10 thf(fact_166_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y3:((product_prod @ A) @ B)]: ((~ ((! [A6:A,B4:B]: ((Y3 != ((((product_Pair @ A) @ B) @ A6) @ B4))))))))).
% 1.80/1.10 thf(fact_167_prod__induct7, axiom, (! [G2:$tType,F3:$tType,E3:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F3) @ G2)))))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F3) @ G2))))))]: (((! [A6:A,B4:B,C5:C,D2:D,E4:E3,F:F3,G3:G2]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F3) @ G2)))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F3) @ G2))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F3) @ G2)))) @ C5) @ ((((product_Pair @ D) @ ((product_prod @ E3) @ ((product_prod @ F3) @ G2))) @ D2) @ ((((product_Pair @ E3) @ ((product_prod @ F3) @ G2)) @ E4) @ ((((product_Pair @ F3) @ G2) @ F) @ G3))))))))) => (P @ X4))))).
% 1.80/1.10 thf(fact_168_prod__induct6, axiom, (! [F3:$tType,E3:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F3))))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F3)))))]: (((! [A6:A,B4:B,C5:C,D2:D,E4:E3,F:F3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F3))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F3)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F3))) @ C5) @ ((((product_Pair @ D) @ ((product_prod @ E3) @ F3)) @ D2) @ ((((product_Pair @ E3) @ F3) @ E4) @ F)))))))) => (P @ X4))))).
% 1.80/1.10 thf(fact_169_prod__induct5, axiom, (! [E3:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E3)))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E3))))]: (((! [A6:A,B4:B,C5:C,D2:D,E4:E3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E3)))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E3))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E3)) @ C5) @ ((((product_Pair @ D) @ E3) @ D2) @ E4))))))) => (P @ X4))))).
% 1.80/1.10 thf(fact_170_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A6:A,B4:B,C5:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C5) @ D2)))))) => (P @ X4))))).
% 1.80/1.10 thf(fact_171_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X4:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A6:A,B4:B,C5:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A6) @ ((((product_Pair @ B) @ C) @ B4) @ C5))))) => (P @ X4))))).
% 1.80/1.10 thf(fact_172_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E3:$tType,F3:$tType,G2:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F3) @ G2))))))]: ((~ ((! [A6:A,B4:B,C5:C,D2:D,E4:E3,F:F3,G3:G2]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F3) @ G2)))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F3) @ G2))))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ ((product_prod @ F3) @ G2)))) @ C5) @ ((((product_Pair @ D) @ ((product_prod @ E3) @ ((product_prod @ F3) @ G2))) @ D2) @ ((((product_Pair @ E3) @ ((product_prod @ F3) @ G2)) @ E4) @ ((((product_Pair @ F3) @ G2) @ F) @ G3)))))))))))))).
% 1.80/1.10 thf(fact_173_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E3:$tType,F3:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F3)))))]: ((~ ((! [A6:A,B4:B,C5:C,D2:D,E4:E3,F:F3]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F3))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F3)))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E3) @ F3))) @ C5) @ ((((product_Pair @ D) @ ((product_prod @ E3) @ F3)) @ D2) @ ((((product_Pair @ E3) @ F3) @ E4) @ F))))))))))))).
% 1.80/1.10 thf(fact_174_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E3:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E3))))]: ((~ ((! [A6:A,B4:B,C5:C,D2:D,E4:E3]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E3)))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E3))) @ B4) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E3)) @ C5) @ ((((product_Pair @ D) @ E3) @ D2) @ E4)))))))))))).
% 1.80/1.10 thf(fact_175_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A6:A,B4:B,C5:C,D2:D]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B4) @ ((((product_Pair @ C) @ D) @ C5) @ D2))))))))))).
% 1.80/1.10 thf(fact_176_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A6:A,B4:B,C5:C]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A6) @ ((((product_Pair @ B) @ C) @ B4) @ C5)))))))))).
% 1.80/1.10 thf(fact_177_Pair__inject, axiom, (! [A:$tType,B:$tType,A5:A,B5:B,A8:A,B7:B]: (((((((product_Pair @ A) @ B) @ A5) @ B5) = ((((product_Pair @ A) @ B) @ A8) @ B7)) => (~ (((A5 = A8) => (B5 != B7)))))))).
% 1.80/1.10 thf(fact_178_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P4:((product_prod @ A) @ B)]: (((! [A6:A,B4:B]: ((P @ ((((product_Pair @ A) @ B) @ A6) @ B4)))) => (P @ P4))))).
% 1.80/1.10 thf(fact_179_surj__pair, axiom, (! [A:$tType,B:$tType,P4:((product_prod @ A) @ B)]: ((? [X:A,Y2:B]: ((P4 = ((((product_Pair @ A) @ B) @ X) @ Y2))))))).
% 1.80/1.10 thf(fact_180_Graph_OisSPath__nt__parallel, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),S:nat,P4:(list @ ((product_prod @ nat) @ nat)),T4:nat,E2:((product_prod @ nat) @ nat)]: (((((((isSimplePath @ Capacity) @ C2) @ S) @ P4) @ T4) => ((((member @ ((product_prod @ nat) @ nat)) @ E2) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ (((product_swap @ nat) @ nat) @ E2)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)))))))))))).
% 1.80/1.10 thf(fact_181_Sigma__cong, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ A),C4:(A > (set @ B)),D3:(A > (set @ B))]: (((A3 = B3) => ((! [X:A]: (((((member @ A) @ X) @ B3) => ((C4 @ X) = (D3 @ X))))) => (((((product_Sigma @ A) @ B) @ A3) @ C4) = ((((product_Sigma @ A) @ B) @ B3) @ D3))))))).
% 1.80/1.10 thf(fact_182_Times__eq__cancel2, axiom, (! [A:$tType,B:$tType,X4:A,C4:(set @ A),A3:(set @ B),B3:(set @ B)]: (((((member @ A) @ X4) @ C4) => ((((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu:B]: (C4))) = ((((product_Sigma @ B) @ A) @ B3) @ (^ [Uu:B]: (C4)))) = (A3 = B3)))))).
% 1.80/1.10 thf(fact_183_SigmaE, axiom, (! [A:$tType,B:$tType,C2:((product_prod @ A) @ B),A3:(set @ A),B3:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ C2) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) => (~ ((! [X:A]: (((((member @ A) @ X) @ A3) => (! [Y2:B]: (((((member @ B) @ Y2) @ (B3 @ X)) => (C2 != ((((product_Pair @ A) @ B) @ X) @ Y2)))))))))))))).
% 1.80/1.10 thf(fact_184_SigmaD1, axiom, (! [B:$tType,A:$tType,A5:A,B5:B,A3:(set @ A),B3:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) => (((member @ A) @ A5) @ A3))))).
% 1.80/1.10 thf(fact_185_SigmaD2, axiom, (! [B:$tType,A:$tType,A5:A,B5:B,A3:(set @ A),B3:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) => (((member @ B) @ B5) @ (B3 @ A5)))))).
% 1.80/1.10 thf(fact_186_SigmaE2, axiom, (! [B:$tType,A:$tType,A5:A,B5:B,A3:(set @ A),B3:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) => (~ (((((member @ A) @ A5) @ A3) => (~ ((((member @ B) @ B5) @ (B3 @ A5))))))))))).
% 1.80/1.10 thf(fact_187_rtranclE, axiom, (! [A:$tType,A5:A,B5:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ B5)) @ ((transitive_rtrancl @ A) @ R)) => ((A5 != B5) => (~ ((! [Y2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ Y2)) @ ((transitive_rtrancl @ A) @ R)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ B5)) @ R))))))))))))).
% 1.80/1.10 thf(fact_188_rtrancl_Ocases, axiom, (! [A:$tType,A1:A,A22:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A1) @ A22)) @ ((transitive_rtrancl @ A) @ R)) => ((A22 != A1) => (~ ((! [B4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A1) @ B4)) @ ((transitive_rtrancl @ A) @ R)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B4) @ A22)) @ R))))))))))))).
% 1.80/1.10 thf(fact_189_rtrancl_Osimps, axiom, (! [A:$tType,A1:A,A22:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A1) @ A22)) @ ((transitive_rtrancl @ A) @ R)) = ((? [A4:A]: (((A1 = A4) & (A22 = A4)))) | (? [A4:A,B6:A,C3:A]: (((A1 = A4) & ((A22 = C3) & ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A4) @ B6)) @ ((transitive_rtrancl @ A) @ R)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B6) @ C3)) @ R))))))))))).
% 1.80/1.10 thf(fact_190_rtrancl__trans, axiom, (! [A:$tType,X4:A,Y3:A,R:(set @ ((product_prod @ A) @ A)),Z:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y3)) @ ((transitive_rtrancl @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z)) @ ((transitive_rtrancl @ A) @ R)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Z)) @ ((transitive_rtrancl @ A) @ R))))))).
% 1.80/1.10 thf(fact_191_rtrancl__induct, axiom, (! [A:$tType,A5:A,B5:A,R:(set @ ((product_prod @ A) @ A)),P:(A > $o)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ B5)) @ ((transitive_rtrancl @ A) @ R)) => ((P @ A5) => ((! [Y2:A,Z2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ Y2)) @ ((transitive_rtrancl @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ Z2)) @ R) => ((P @ Y2) => (P @ Z2)))))) => (P @ B5))))))).
% 1.80/1.10 thf(fact_192_rtrancl_Oinducts, axiom, (! [A:$tType,X1:A,X22:A,R:(set @ ((product_prod @ A) @ A)),P:(A > (A > $o))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X1) @ X22)) @ ((transitive_rtrancl @ A) @ R)) => ((! [A6:A]: (((P @ A6) @ A6))) => ((! [A6:A,B4:A,C5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A6) @ B4)) @ ((transitive_rtrancl @ A) @ R)) => (((P @ A6) @ B4) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B4) @ C5)) @ R) => ((P @ A6) @ C5)))))) => ((P @ X1) @ X22))))))).
% 1.80/1.10 thf(fact_193_converse__rtranclE, axiom, (! [A:$tType,X4:A,Z:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Z)) @ ((transitive_rtrancl @ A) @ R)) => ((X4 != Z) => (~ ((! [Y2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y2)) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ Z)) @ ((transitive_rtrancl @ A) @ R)))))))))))))).
% 1.80/1.10 thf(fact_194_rtrancl_Ortrancl__refl, axiom, (! [A:$tType,A5:A,R:(set @ ((product_prod @ A) @ A))]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ A5)) @ ((transitive_rtrancl @ A) @ R))))).
% 1.80/1.10 thf(fact_195_converse__rtrancl__induct, axiom, (! [A:$tType,A5:A,B5:A,R:(set @ ((product_prod @ A) @ A)),P:(A > $o)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ B5)) @ ((transitive_rtrancl @ A) @ R)) => ((P @ B5) => ((! [Y2:A,Z2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y2) @ Z2)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z2) @ B5)) @ ((transitive_rtrancl @ A) @ R)) => ((P @ Z2) => (P @ Y2)))))) => (P @ A5))))))).
% 1.80/1.10 thf(fact_196_rtrancl_Ortrancl__into__rtrancl, axiom, (! [A:$tType,A5:A,B5:A,R:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ B5)) @ ((transitive_rtrancl @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ C2)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ C2)) @ ((transitive_rtrancl @ A) @ R))))))).
% 1.80/1.10 thf(fact_197_converse__rtrancl__into__rtrancl, axiom, (! [A:$tType,A5:A,B5:A,R:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ B5)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ C2)) @ ((transitive_rtrancl @ A) @ R)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ C2)) @ ((transitive_rtrancl @ A) @ R))))))).
% 1.80/1.10 thf(fact_198_rtrancl__mono, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R) @ S) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ ((transitive_rtrancl @ A) @ R)) @ ((transitive_rtrancl @ A) @ S)))))).
% 1.80/1.10 thf(fact_199_rtrancl__subset, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),S3:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R2) @ S3) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ S3) @ ((transitive_rtrancl @ A) @ R2)) => (((transitive_rtrancl @ A) @ S3) = ((transitive_rtrancl @ A) @ R2))))))).
% 1.80/1.10 thf(fact_200_rtrancl__subset__rtrancl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R) @ ((transitive_rtrancl @ A) @ S)) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ ((transitive_rtrancl @ A) @ R)) @ ((transitive_rtrancl @ A) @ S)))))).
% 1.80/1.10 thf(fact_201_Sigma__mono, axiom, (! [B:$tType,A:$tType,A3:(set @ A),C4:(set @ A),B3:(A > (set @ B)),D3:(A > (set @ B))]: (((((ord_less_eq @ (set @ A)) @ A3) @ C4) => ((! [X:A]: (((((member @ A) @ X) @ A3) => (((ord_less_eq @ (set @ B)) @ (B3 @ X)) @ (D3 @ X))))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A3) @ B3)) @ ((((product_Sigma @ A) @ B) @ C4) @ D3))))))).
% 1.80/1.10 thf(fact_202_Image__closed__trancl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),X5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ R) @ X5)) @ X5) => (((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R)) @ X5) = X5))))).
% 1.80/1.10 thf(fact_203_Not__Domain__rtrancl, axiom, (! [A:$tType,X4:A,R2:(set @ ((product_prod @ A) @ A)),Y3:A]: (((~ ((((member @ A) @ X4) @ (((domain @ A) @ A) @ R2)))) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X4) @ Y3)) @ ((transitive_rtrancl @ A) @ R2)) = (X4 = Y3)))))).
% 1.80/1.10 thf(fact_204_Times__subset__cancel2, axiom, (! [A:$tType,B:$tType,X4:A,C4:(set @ A),A3:(set @ B),B3:(set @ B)]: (((((member @ A) @ X4) @ C4) => ((((ord_less_eq @ (set @ ((product_prod @ B) @ A))) @ ((((product_Sigma @ B) @ A) @ A3) @ (^ [Uu:B]: (C4)))) @ ((((product_Sigma @ B) @ A) @ B3) @ (^ [Uu:B]: (C4)))) = (((ord_less_eq @ (set @ B)) @ A3) @ B3)))))).
% 1.80/1.10 thf(fact_205_finite__rtrancl__Image, axiom, (! [A:$tType,R2:(set @ ((product_prod @ A) @ A)),A3:(set @ A)]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R2) => (((finite_finite2 @ A) @ A3) => ((finite_finite2 @ A) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R2)) @ A3))))))).
% 1.80/1.10 thf(fact_206_infinite__cartesian__product, axiom, (! [B:$tType,A:$tType,A3:(set @ A),B3:(set @ B)]: (((~ (((finite_finite2 @ A) @ A3))) => ((~ (((finite_finite2 @ B) @ B3))) => (~ (((finite_finite2 @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A3) @ (^ [Uu:A]: (B3))))))))))).
% 1.80/1.10 thf(fact_207_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A5:A,B5:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A5) @ B5)) = ((F1 @ A5) @ B5))))).
% 1.80/1.10 thf(fact_208_predicate1I, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P @ X) => (Q @ X)))) => (((ord_less_eq @ (A > $o)) @ P) @ Q))))).
% 1.80/1.10 thf(fact_209_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A]: ((((ord_less_eq @ A) @ X4) @ X4))))))).
% 1.80/1.10 thf(fact_210_predicate2I, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Q:(A > (B > $o))]: (((! [X:A,Y2:B]: ((((P @ X) @ Y2) => ((Q @ X) @ Y2)))) => (((ord_less_eq @ (A > (B > $o))) @ P) @ Q))))).
% 1.80/1.10 thf(fact_211_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A5:A,F2:(B > A),B5:B,C2:B]: (((((ord_less_eq @ A) @ A5) @ (F2 @ B5)) => ((((ord_less_eq @ B) @ B5) @ C2) => ((! [X:B,Y2:B]: (((((ord_less_eq @ B) @ X) @ Y2) => (((ord_less_eq @ A) @ (F2 @ X)) @ (F2 @ Y2))))) => (((ord_less_eq @ A) @ A5) @ (F2 @ C2))))))))))).
% 1.80/1.10 thf(fact_212_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A5:A,B5:A,F2:(A > C),C2:C]: (((((ord_less_eq @ A) @ A5) @ B5) => ((((ord_less_eq @ C) @ (F2 @ B5)) @ C2) => ((! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => (((ord_less_eq @ C) @ (F2 @ X)) @ (F2 @ Y2))))) => (((ord_less_eq @ C) @ (F2 @ A5)) @ C2)))))))))).
% 1.80/1.10 thf(fact_213_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A5:A,F2:(B > A),B5:B,C2:B]: (((A5 = (F2 @ B5)) => ((((ord_less_eq @ B) @ B5) @ C2) => ((! [X:B,Y2:B]: (((((ord_less_eq @ B) @ X) @ Y2) => (((ord_less_eq @ A) @ (F2 @ X)) @ (F2 @ Y2))))) => (((ord_less_eq @ A) @ A5) @ (F2 @ C2))))))))))).
% 1.80/1.10 thf(fact_214_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A5:A,B5:A,F2:(A > B),C2:B]: (((((ord_less_eq @ A) @ A5) @ B5) => (((F2 @ B5) = C2) => ((! [X:A,Y2:A]: (((((ord_less_eq @ A) @ X) @ Y2) => (((ord_less_eq @ B) @ (F2 @ X)) @ (F2 @ Y2))))) => (((ord_less_eq @ B) @ (F2 @ A5)) @ C2)))))))))).
% 1.80/1.10 thf(fact_215_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z3:A]: ((Y4 = Z3))) = (^ [X3:A,Y:A]: (((((ord_less_eq @ A) @ X3) @ Y) & (((ord_less_eq @ A) @ Y) @ X3))))))))).
% 1.80/1.10 thf(fact_216_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ X4) => (X4 = Y3))))))))).
% 1.80/1.10 thf(fact_217_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A]: (((((ord_less_eq @ A) @ X4) @ Y3) | (((ord_less_eq @ A) @ Y3) @ X4)))))))).
% 1.80/1.10 thf(fact_218_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y3:A]: (((X4 = Y3) => (((ord_less_eq @ A) @ X4) @ Y3)))))))).
% 1.80/1.10 thf(fact_219_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X4) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X4)))))))).
% 1.80/1.10 thf(fact_220_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B5:A,C2:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((((ord_less_eq @ A) @ B5) @ C2) => (((ord_less_eq @ A) @ A5) @ C2))))))))).
% 1.80/1.10 thf(fact_221_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X4:A,Y3:A,Z:A]: ((((((ord_less_eq @ A) @ X4) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ Z)))) => (((((ord_less_eq @ A) @ Y3) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ Z)))) => (((((ord_less_eq @ A) @ X4) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y3)))) => (((((ord_less_eq @ A) @ Z) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ X4)))) => (((((ord_less_eq @ A) @ Y3) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X4)))) => (~ (((((ord_less_eq @ A) @ Z) @ X4) => (~ ((((ord_less_eq @ A) @ X4) @ Y3))))))))))))))))).
% 1.80/1.10 thf(fact_222_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y3:A,X4:A]: (((((ord_less_eq @ A) @ Y3) @ X4) => ((((ord_less_eq @ A) @ X4) @ Y3) = (X4 = Y3))))))))).
% 1.80/1.10 thf(fact_223_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z3:A]: ((Y4 = Z3))) = (^ [A4:A,B6:A]: (((((ord_less_eq @ A) @ A4) @ B6) & (((ord_less_eq @ A) @ B6) @ A4))))))))).
% 1.80/1.10 thf(fact_224_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A5:A,B5:A,C2:A]: (((A5 = B5) => ((((ord_less_eq @ A) @ B5) @ C2) => (((ord_less_eq @ A) @ A5) @ C2))))))))).
% 1.80/1.10 thf(fact_225_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A5:A,B5:A,C2:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((B5 = C2) => (((ord_less_eq @ A) @ A5) @ C2))))))))).
% 1.80/1.10 thf(fact_226_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A,B5:A]: (((((ord_less_eq @ A) @ A5) @ B5) => ((((ord_less_eq @ A) @ B5) @ A5) => (A5 = B5))))))))).
% 1.80/1.10 thf(fact_227_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X4:A,Y3:A,Z:A]: (((((ord_less_eq @ A) @ X4) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z) => (((ord_less_eq @ A) @ X4) @ Z))))))))).
% 1.80/1.10 thf(fact_228_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A5:A]: ((((ord_less_eq @ A) @ A5) @ A5))))))).
% 1.80/1.10 thf(fact_229_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A5:A,B5:A]: (((! [A6:A,B4:A]: (((((ord_less_eq @ A) @ A6) @ B4) => ((P @ A6) @ B4)))) => ((! [A6:A,B4:A]: ((((P @ B4) @ A6) => ((P @ A6) @ B4)))) => ((P @ A5) @ B5))))))))).
% 1.80/1.10 thf(fact_230_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B5:A,A5:A,C2:A]: (((((ord_less_eq @ A) @ B5) @ A5) => ((((ord_less_eq @ A) @ C2) @ B5) => (((ord_less_eq @ A) @ C2) @ A5))))))))).
% 1.80/1.10 thf(fact_231_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z3:A]: ((Y4 = Z3))) = (^ [A4:A,B6:A]: (((((ord_less_eq @ A) @ B6) @ A4) & (((ord_less_eq @ A) @ A4) @ B6))))))))).
% 1.80/1.10 thf(fact_232_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B5:A,A5:A]: (((((ord_less_eq @ A) @ B5) @ A5) => ((((ord_less_eq @ A) @ A5) @ B5) => (A5 = B5))))))))).
% 1.80/1.10 thf(fact_233_predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Q:(A > (B > $o)),X4:A,Y3:B]: (((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => (((P @ X4) @ Y3) => ((Q @ X4) @ Y3)))))).
% 1.80/1.10 thf(fact_234_rev__predicate2D, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),X4:A,Y3:B,Q:(A > (B > $o))]: ((((P @ X4) @ Y3) => ((((ord_less_eq @ (A > (B > $o))) @ P) @ Q) => ((Q @ X4) @ Y3)))))).
% 1.80/1.10 thf(fact_235_predicate1D, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o),X4:A]: (((((ord_less_eq @ (A > $o)) @ P) @ Q) => ((P @ X4) => (Q @ X4)))))).
% 1.80/1.10 thf(fact_236_rev__predicate1D, axiom, (! [A:$tType,P:(A > $o),X4:A,Q:(A > $o)]: (((P @ X4) => ((((ord_less_eq @ (A > $o)) @ P) @ Q) => (Q @ X4)))))).
% 1.80/1.10 thf(fact_237_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G:(A > B),X4:A]: (((((ord_less_eq @ (A > B)) @ F2) @ G) => (((ord_less_eq @ B) @ (F2 @ X4)) @ (G @ X4))))))))).
% 1.80/1.10 thf(fact_238_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G:(A > B),X4:A]: (((((ord_less_eq @ (A > B)) @ F2) @ G) => (((ord_less_eq @ B) @ (F2 @ X4)) @ (G @ X4))))))))).
% 1.80/1.10 thf(fact_239_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F2:(A > B),G:(A > B)]: (((! [X:A]: ((((ord_less_eq @ B) @ (F2 @ X)) @ (G @ X)))) => (((ord_less_eq @ (A > B)) @ F2) @ G)))))))).
% 1.80/1.10 thf(fact_240_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F4:(A > B),G4:(A > B)]: ((! [X3:A]: ((((ord_less_eq @ B) @ (F4 @ X3)) @ (G4 @ X3))))))))))).
% 1.80/1.10 thf(fact_241_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca1785829860lChain @ A) @ B) = (^ [R3:(set @ ((product_prod @ A) @ A)),As:(A > B)]: ((! [I:A,J:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R3) => (((ord_less_eq @ B) @ (As @ I)) @ (As @ J)))))))))))).
% 1.80/1.10 thf(fact_242_rtc__isPath, axiom, (! [U:nat,V:nat]: (((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((transitive_rtrancl @ nat) @ ((e @ a) @ c))) => (? [P3:(list @ ((product_prod @ nat) @ nat))]: ((((((isPath @ a) @ c) @ U) @ P3) @ V))))))).
% 1.80/1.10 thf(fact_243_isSPath__pathLE, axiom, (! [S:nat,P4:(list @ ((product_prod @ nat) @ nat)),T4:nat]: (((((((isPath @ a) @ c) @ S) @ P4) @ T4) => (? [P5:(list @ ((product_prod @ nat) @ nat))]: ((((((isSimplePath @ a) @ c) @ S) @ P5) @ T4))))))).
% 1.80/1.10 thf(fact_244_connected__def, axiom, (! [U:nat,V:nat]: ((((((connected @ a) @ c) @ U) @ V) = (? [P2:(list @ ((product_prod @ nat) @ nat))]: ((((((isPath @ a) @ c) @ U) @ P2) @ V))))))).
% 1.80/1.10 thf(fact_245_shortestPath__is__path, axiom, (! [U:nat,P4:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isShortestPath @ a) @ c) @ U) @ P4) @ V) => (((((isPath @ a) @ c) @ U) @ P4) @ V))))).
% 1.80/1.10 thf(fact_246_isPath__ex__edge1, axiom, (! [U:nat,P4:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat]: (((((((isPath @ a) @ c) @ U) @ P4) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => ((U1 != U) => (? [U23:nat]: ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U23) @ U1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)))))))))).
% 1.80/1.10 thf(fact_247_isPath__ex__edge2, axiom, (! [U:nat,P4:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat]: (((((((isPath @ a) @ c) @ U) @ P4) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => ((V1 != V) => (? [V23:nat]: ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ V1) @ V23)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)))))))))).
% 1.80/1.10 thf(fact_248_isPath__edgeset, axiom, (! [U:nat,P4:(list @ ((product_prod @ nat) @ nat)),V:nat,E2:((product_prod @ nat) @ nat)]: (((((((isPath @ a) @ c) @ U) @ P4) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ E2) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => (((member @ ((product_prod @ nat) @ nat)) @ E2) @ ((e @ a) @ c))))))).
% 1.80/1.10 thf(fact_249_isPath__rtc, axiom, (! [U:nat,P4:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isPath @ a) @ c) @ U) @ P4) @ V) => (((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((transitive_rtrancl @ nat) @ ((e @ a) @ c))))))).
% 1.80/1.10 thf(fact_250_Graph_OisPath__ex__edge2, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U:nat,P4:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat]: (((((((isPath @ Capacity) @ C2) @ U) @ P4) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => ((V1 != V) => (? [V23:nat]: ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ V1) @ V23)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4))))))))))))).
% 1.80/1.10 thf(fact_251_Graph_OisPath__ex__edge1, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U:nat,P4:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat]: (((((((isPath @ Capacity) @ C2) @ U) @ P4) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => ((U1 != U) => (? [U23:nat]: ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U23) @ U1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4))))))))))))).
% 1.80/1.10 thf(fact_252_Graph_OisPath__edgeset, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U:nat,P4:(list @ ((product_prod @ nat) @ nat)),V:nat,E2:((product_prod @ nat) @ nat)]: (((((((isPath @ Capacity) @ C2) @ U) @ P4) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ E2) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P4)) => (((member @ ((product_prod @ nat) @ nat)) @ E2) @ ((e @ Capacity) @ C2)))))))))).
% 1.80/1.10 thf(fact_253_Graph_Oconnected__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((connected @ Capacity) = (^ [C3:(((product_prod @ nat) @ nat) > Capacity),U3:nat,V2:nat]: ((? [P2:(list @ ((product_prod @ nat) @ nat))]: ((((((isPath @ Capacity) @ C3) @ U3) @ P2) @ V2)))))))))).
% 1.80/1.10 thf(fact_254_Graph_OshortestPath__is__path, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C2:(((product_prod @ nat) @ nat) > Capacity),U:nat,P4:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isShortestPath @ Capacity) @ C2) @ U) @ P4) @ V) => (((((isPath @ Capacity) @ C2) @ U) @ P4) @ V)))))))).
% 1.80/1.10 thf(fact_255_Graph_OisPath_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isPath @ Capacity) = (isPath @ Capacity)))))).
% 1.80/1.10 thf(subcl_Rings_Olinordered__idom___HOL_Otype, axiom, (! [A:$tType]: (((linordered_idom @ A) => (type @ A))))).
% 1.80/1.10 thf(subcl_Rings_Olinordered__idom___Orderings_Oord, axiom, (! [A:$tType]: (((linordered_idom @ A) => (ord @ A))))).
% 1.80/1.10 thf(subcl_Rings_Olinordered__idom___Orderings_Oorder, axiom, (! [A:$tType]: (((linordered_idom @ A) => (order @ A))))).
% 1.80/1.10 thf(subcl_Rings_Olinordered__idom___Orderings_Olinorder, axiom, (! [A:$tType]: (((linordered_idom @ A) => (linorder @ A))))).
% 1.80/1.10 thf(subcl_Rings_Olinordered__idom___Orderings_Opreorder, axiom, (! [A:$tType]: (((linordered_idom @ A) => (preorder @ A))))).
% 1.80/1.10 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A9:$tType,A10:$tType]: (((preorder @ A10) => (preorder @ (A9 > A10)))))).
% 1.80/1.10 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ (A9 > A10)))))).
% 1.80/1.10 thf(tcon_fun___Orderings_Oorder, axiom, (! [A9:$tType,A10:$tType]: (((order @ A10) => (order @ (A9 > A10)))))).
% 1.80/1.10 thf(tcon_fun___Orderings_Oord, axiom, (! [A9:$tType,A10:$tType]: (((ord @ A10) => (ord @ (A9 > A10)))))).
% 1.80/1.10 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, (preorder @ nat)).
% 1.80/1.10 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 1.80/1.10 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, (order @ nat)).
% 1.80/1.10 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, (ord @ nat)).
% 1.80/1.10 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A9:$tType]: ((preorder @ (set @ A9))))).
% 1.80/1.10 thf(tcon_Set_Oset___Finite__Set_Ofinite_5, axiom, (! [A9:$tType]: (((finite_finite @ A9) => (finite_finite @ (set @ A9)))))).
% 1.80/1.10 thf(tcon_Set_Oset___Orderings_Oorder_6, axiom, (! [A9:$tType]: ((order @ (set @ A9))))).
% 1.80/1.10 thf(tcon_Set_Oset___Orderings_Oord_7, axiom, (! [A9:$tType]: ((ord @ (set @ A9))))).
% 1.80/1.10 thf(tcon_HOL_Obool___Orderings_Opreorder_8, axiom, (preorder @ $o)).
% 1.80/1.10 thf(tcon_HOL_Obool___Orderings_Olinorder_9, axiom, (linorder @ $o)).
% 1.80/1.10 thf(tcon_HOL_Obool___Finite__Set_Ofinite_10, axiom, (finite_finite @ $o)).
% 1.80/1.10 thf(tcon_HOL_Obool___Orderings_Oorder_11, axiom, (order @ $o)).
% 1.80/1.10 thf(tcon_HOL_Obool___Orderings_Oord_12, axiom, (ord @ $o)).
% 1.80/1.10 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_13, axiom, (! [A9:$tType,A10:$tType]: ((((finite_finite @ A9) & (finite_finite @ A10)) => (finite_finite @ ((product_prod @ A9) @ A10)))))).
% 1.80/1.10 thf(tfree_0, hypothesis, (linordered_idom @ a)).
% 1.80/1.10 thf(conj_0, conjecture, ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((((edmond1803015688pEdges @ a) @ c) @ s) @ t))).
% 1.80/1.10 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.vhIah8fDUX/DTF2DTF23290.p
% 1.80/1.10 ------------------------
% 2.07/1.13 ---- Cleaned THF ---
% 2.07/1.13 thf(ty_t_Product__Type_Oprod,type,
% 2.07/1.13 product_prod: $tType > $tType > $tType ).
% 2.07/1.13
% 2.07/1.13 thf(ty_t_List_Olist,type,
% 2.07/1.13 list: $tType > $tType ).
% 2.07/1.13
% 2.07/1.13 thf(ty_t_Set_Oset,type,
% 2.07/1.13 set: $tType > $tType ).
% 2.07/1.13
% 2.07/1.13 thf(ty_t_Nat_Onat,type,
% 2.07/1.13 nat: $tType ).
% 2.07/1.13
% 2.07/1.13 thf(ty_tf_a,type,
% 2.07/1.13 a: $tType ).
% 2.07/1.13
% 2.07/1.13 thf(sy_cl_HOL_Otype,type,
% 2.07/1.13 type:
% 2.07/1.13 !>[A: $tType] : $o ).
% 2.07/1.13
% 2.07/1.13 thf(sy_cl_Orderings_Oord,type,
% 2.07/1.13 ord:
% 2.07/1.13 !>[A: $tType] : $o ).
% 2.07/1.13
% 2.07/1.13 thf(sy_cl_Orderings_Oorder,type,
% 2.07/1.13 order:
% 2.07/1.13 !>[A: $tType] : $o ).
% 2.07/1.13
% 2.07/1.13 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.07/1.13 finite_finite:
% 2.07/1.13 !>[A: $tType] : $o ).
% 2.07/1.13
% 2.07/1.13 thf(sy_cl_Orderings_Olinorder,type,
% 2.07/1.13 linorder:
% 2.07/1.13 !>[A: $tType] : $o ).
% 2.07/1.13
% 2.07/1.13 thf(sy_cl_Orderings_Opreorder,type,
% 2.07/1.13 preorder:
% 2.07/1.13 !>[A: $tType] : $o ).
% 2.07/1.13
% 2.07/1.13 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.07/1.13 linordered_idom:
% 2.07/1.13 !>[A: $tType] : $o ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 2.07/1.13 bNF_Ca1785829860lChain:
% 2.07/1.13 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis,type,
% 2.07/1.13 edmond2129202899alysis:
% 2.07/1.13 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > $o ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis__defs_OspEdges,type,
% 2.07/1.13 edmond1803015688pEdges:
% 2.07/1.13 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > nat > nat > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Finite__Set_Ofinite,type,
% 2.07/1.13 finite_finite2:
% 2.07/1.13 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Graph_OFinite__Graph,type,
% 2.07/1.13 finite_Graph:
% 2.07/1.13 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > $o ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Graph_OGraph_OE,type,
% 2.07/1.13 e:
% 2.07/1.13 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Graph_OGraph_OV,type,
% 2.07/1.13 v:
% 2.07/1.13 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Graph_OGraph_Oadjacent__nodes,type,
% 2.07/1.13 adjacent_nodes:
% 2.07/1.13 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ nat ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Graph_OGraph_Oconnected,type,
% 2.07/1.13 connected:
% 2.07/1.13 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > $o ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Graph_OGraph_Oincoming,type,
% 2.07/1.13 incoming:
% 2.07/1.13 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Graph_OGraph_Oincoming_H,type,
% 2.07/1.13 incoming2:
% 2.07/1.13 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Graph_OGraph_OisPath,type,
% 2.07/1.13 isPath:
% 2.07/1.13 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Graph_OGraph_OisShortestPath,type,
% 2.07/1.13 isShortestPath:
% 2.07/1.13 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Graph_OGraph_OisSimplePath,type,
% 2.07/1.13 isSimplePath:
% 2.07/1.13 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Graph_OGraph_Ooutgoing,type,
% 2.07/1.13 outgoing:
% 2.07/1.13 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Graph_OGraph_Ooutgoing_H,type,
% 2.07/1.13 outgoing2:
% 2.07/1.13 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Graph_OGraph_OreachableNodes,type,
% 2.07/1.13 reachableNodes:
% 2.07/1.13 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ nat ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_List_Olist_Oset,type,
% 2.07/1.13 set2:
% 2.07/1.13 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.07/1.13 ord_less_eq:
% 2.07/1.13 !>[A: $tType] : ( A > A > $o ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Product__Type_OPair,type,
% 2.07/1.13 product_Pair:
% 2.07/1.13 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Product__Type_OSigma,type,
% 2.07/1.13 product_Sigma:
% 2.07/1.13 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.07/1.13 product_rec_prod:
% 2.07/1.13 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 2.07/1.13 product_swap:
% 2.07/1.13 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Relation_ODomain,type,
% 2.07/1.13 domain:
% 2.07/1.13 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Relation_OImage,type,
% 2.07/1.13 image:
% 2.07/1.13 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Relation_ORange,type,
% 2.07/1.13 range:
% 2.07/1.13 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Set_OCollect,type,
% 2.07/1.13 collect:
% 2.07/1.13 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_Transitive__Closure_Ortrancl,type,
% 2.07/1.13 transitive_rtrancl:
% 2.07/1.13 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_c_member,type,
% 2.07/1.13 member:
% 2.07/1.13 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.07/1.13
% 2.07/1.13 thf(sy_v_c,type,
% 2.07/1.13 c: ( product_prod @ nat @ nat ) > a ).
% 2.07/1.13
% 2.07/1.13 thf(sy_v_s,type,
% 2.07/1.13 s: nat ).
% 2.07/1.13
% 2.07/1.13 thf(sy_v_t,type,
% 2.07/1.13 t: nat ).
% 2.07/1.13
% 2.07/1.13 thf(fact_0_ek__analysis__axioms,axiom,
% 2.07/1.13 edmond2129202899alysis @ a @ c ).
% 2.07/1.13
% 2.07/1.13 thf(fact_1__092_060open_062finite_AE_A_092_060Longrightarrow_062_Afinite_AspEdges_092_060close_062,axiom,
% 2.07/1.13 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
% 2.07/1.13 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_2_ek__analysis__defs_OspEdges_Ocong,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linordered_idom @ A )
% 2.07/1.13 => ( ( edmond1803015688pEdges @ A )
% 2.07/1.13 = ( edmond1803015688pEdges @ A ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_3_finite__E,axiom,
% 2.07/1.13 finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_4_spEdges__ss__E,axiom,
% 2.07/1.13 ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) @ ( e @ a @ c ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_5_Finite__Graph__axioms,axiom,
% 2.07/1.13 finite_Graph @ a @ c ).
% 2.07/1.13
% 2.07/1.13 thf(fact_6_finite__code,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( finite_finite @ A )
% 2.07/1.13 => ( ( finite_finite2 @ A )
% 2.07/1.13 = ( ^ [A2: set @ A] : $true ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_7_Finite__Graph__EI,axiom,
% 2.07/1.13 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
% 2.07/1.13 => ( finite_Graph @ a @ c ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_8_finite__set__choice,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
% 2.07/1.13 ( ( finite_finite2 @ A @ A3 )
% 2.07/1.13 => ( ! [X: A] :
% 2.07/1.13 ( ( member @ A @ X @ A3 )
% 2.07/1.13 => ? [X_1: B] : ( P @ X @ X_1 ) )
% 2.07/1.13 => ? [F: A > B] :
% 2.07/1.13 ! [X2: A] :
% 2.07/1.13 ( ( member @ A @ X2 @ A3 )
% 2.07/1.13 => ( P @ X2 @ ( F @ X2 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_9_finite,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( finite_finite @ A )
% 2.07/1.13 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_10_connected__refl,axiom,
% 2.07/1.13 ! [V: nat] : ( connected @ a @ c @ V @ V ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_11_spEdges__def,axiom,
% 2.07/1.13 ( ( edmond1803015688pEdges @ a @ c @ s @ t )
% 2.07/1.13 = ( collect @ ( product_prod @ nat @ nat )
% 2.07/1.13 @ ^ [E: product_prod @ nat @ nat] :
% 2.07/1.13 ? [P2: list @ ( product_prod @ nat @ nat )] :
% 2.07/1.13 ( ( member @ ( product_prod @ nat @ nat ) @ E @ ( set2 @ ( product_prod @ nat @ nat ) @ P2 ) )
% 2.07/1.13 & ( isShortestPath @ a @ c @ s @ P2 @ t ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_12_finite__Range,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
% 2.07/1.13 => ( finite_finite2 @ B @ ( range @ A @ B @ R ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_13_finite__Domain,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
% 2.07/1.13 => ( finite_finite2 @ A @ ( domain @ A @ B @ R ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_14_Efin__imp__Vfin,axiom,
% 2.07/1.13 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
% 2.07/1.13 => ( finite_finite2 @ nat @ ( v @ a @ c ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_15_connected__inV__iff,axiom,
% 2.07/1.13 ! [U: nat,V: nat] :
% 2.07/1.13 ( ( connected @ a @ c @ U @ V )
% 2.07/1.13 => ( ( member @ nat @ V @ ( v @ a @ c ) )
% 2.07/1.13 = ( member @ nat @ U @ ( v @ a @ c ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_16_obtain__shortest__path,axiom,
% 2.07/1.13 ! [U: nat,V: nat] :
% 2.07/1.13 ( ( connected @ a @ c @ U @ V )
% 2.07/1.13 => ~ ! [P3: list @ ( product_prod @ nat @ nat )] :
% 2.07/1.13 ~ ( isShortestPath @ a @ c @ U @ P3 @ V ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_17_finite__Collect__conjI,axiom,
% 2.07/1.13 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.07/1.13 ( ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.07/1.13 | ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
% 2.07/1.13 => ( finite_finite2 @ A
% 2.07/1.13 @ ( collect @ A
% 2.07/1.13 @ ^ [X3: A] :
% 2.07/1.13 ( ( P @ X3 )
% 2.07/1.13 & ( Q @ X3 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_18_finite__Collect__disjI,axiom,
% 2.07/1.13 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.07/1.13 ( ( finite_finite2 @ A
% 2.07/1.13 @ ( collect @ A
% 2.07/1.13 @ ^ [X3: A] :
% 2.07/1.13 ( ( P @ X3 )
% 2.07/1.13 | ( Q @ X3 ) ) ) )
% 2.07/1.13 = ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.07/1.13 & ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_19_finite__Collect__bounded__ex,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,P: A > $o,Q: B > A > $o] :
% 2.07/1.13 ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.07/1.13 => ( ( finite_finite2 @ B
% 2.07/1.13 @ ( collect @ B
% 2.07/1.13 @ ^ [X3: B] :
% 2.07/1.13 ? [Y: A] :
% 2.07/1.13 ( ( P @ Y )
% 2.07/1.13 & ( Q @ X3 @ Y ) ) ) )
% 2.07/1.13 = ( ! [Y: A] :
% 2.07/1.13 ( ( P @ Y )
% 2.07/1.13 => ( finite_finite2 @ B
% 2.07/1.13 @ ( collect @ B
% 2.07/1.13 @ ^ [X3: B] : ( Q @ X3 @ Y ) ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_20_finite__Collect__subsets,axiom,
% 2.07/1.13 ! [A: $tType,A3: set @ A] :
% 2.07/1.13 ( ( finite_finite2 @ A @ A3 )
% 2.07/1.13 => ( finite_finite2 @ ( set @ A )
% 2.07/1.13 @ ( collect @ ( set @ A )
% 2.07/1.13 @ ^ [B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_21_finite__V,axiom,
% 2.07/1.13 finite_finite2 @ nat @ ( v @ a @ c ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_22_Vfin__imp__Efin,axiom,
% 2.07/1.13 ( ( finite_finite2 @ nat @ ( v @ a @ c ) )
% 2.07/1.13 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_23_Range__mono,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S )
% 2.07/1.13 => ( ord_less_eq @ ( set @ B ) @ ( range @ A @ B @ R ) @ ( range @ A @ B @ S ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_24_Domain__mono,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S )
% 2.07/1.13 => ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R ) @ ( domain @ A @ B @ S ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_25_finite__image__set,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,P: A > $o,F2: A > B] :
% 2.07/1.13 ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.07/1.13 => ( finite_finite2 @ B
% 2.07/1.13 @ ( collect @ B
% 2.07/1.13 @ ^ [Uu: B] :
% 2.07/1.13 ? [X3: A] :
% 2.07/1.13 ( ( Uu
% 2.07/1.13 = ( F2 @ X3 ) )
% 2.07/1.13 & ( P @ X3 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_26_finite__image__set2,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,C: $tType,P: A > $o,Q: B > $o,F2: A > B > C] :
% 2.07/1.13 ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.07/1.13 => ( ( finite_finite2 @ B @ ( collect @ B @ Q ) )
% 2.07/1.13 => ( finite_finite2 @ C
% 2.07/1.13 @ ( collect @ C
% 2.07/1.13 @ ^ [Uu: C] :
% 2.07/1.13 ? [X3: A,Y: B] :
% 2.07/1.13 ( ( Uu
% 2.07/1.13 = ( F2 @ X3 @ Y ) )
% 2.07/1.13 & ( P @ X3 )
% 2.07/1.13 & ( Q @ Y ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_27_not__finite__existsD,axiom,
% 2.07/1.13 ! [A: $tType,P: A > $o] :
% 2.07/1.13 ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
% 2.07/1.13 => ? [X_12: A] : ( P @ X_12 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_28_pigeonhole__infinite__rel,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B,R2: A > B > $o] :
% 2.07/1.13 ( ~ ( finite_finite2 @ A @ A3 )
% 2.07/1.13 => ( ( finite_finite2 @ B @ B3 )
% 2.07/1.13 => ( ! [X: A] :
% 2.07/1.13 ( ( member @ A @ X @ A3 )
% 2.07/1.13 => ? [Xa: B] :
% 2.07/1.13 ( ( member @ B @ Xa @ B3 )
% 2.07/1.13 & ( R2 @ X @ Xa ) ) )
% 2.07/1.13 => ? [X: B] :
% 2.07/1.13 ( ( member @ B @ X @ B3 )
% 2.07/1.13 & ~ ( finite_finite2 @ A
% 2.07/1.13 @ ( collect @ A
% 2.07/1.13 @ ^ [A4: A] :
% 2.07/1.13 ( ( member @ A @ A4 @ A3 )
% 2.07/1.13 & ( R2 @ A4 @ X ) ) ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_29_ek__analysis__def,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linordered_idom @ A )
% 2.07/1.13 => ( ( edmond2129202899alysis @ A )
% 2.07/1.13 = ( finite_Graph @ A ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_30_ek__analysis_Ointro,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linordered_idom @ A )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > A] :
% 2.07/1.13 ( ( finite_Graph @ A @ C2 )
% 2.07/1.13 => ( edmond2129202899alysis @ A @ C2 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_31_ek__analysis_Oaxioms,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linordered_idom @ A )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > A] :
% 2.07/1.13 ( ( edmond2129202899alysis @ A @ C2 )
% 2.07/1.13 => ( finite_Graph @ A @ C2 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_32_ek__analysis__defs_OspEdges__def,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linordered_idom @ A )
% 2.07/1.13 => ( ( edmond1803015688pEdges @ A )
% 2.07/1.13 = ( ^ [C3: ( product_prod @ nat @ nat ) > A,S2: nat,T2: nat] :
% 2.07/1.13 ( collect @ ( product_prod @ nat @ nat )
% 2.07/1.13 @ ^ [E: product_prod @ nat @ nat] :
% 2.07/1.13 ? [P2: list @ ( product_prod @ nat @ nat )] :
% 2.07/1.13 ( ( member @ ( product_prod @ nat @ nat ) @ E @ ( set2 @ ( product_prod @ nat @ nat ) @ P2 ) )
% 2.07/1.13 & ( isShortestPath @ A @ C3 @ S2 @ P2 @ T2 ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_33_finite__has__minimal2,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( order @ A )
% 2.07/1.13 => ! [A3: set @ A,A5: A] :
% 2.07/1.13 ( ( finite_finite2 @ A @ A3 )
% 2.07/1.13 => ( ( member @ A @ A5 @ A3 )
% 2.07/1.13 => ? [X: A] :
% 2.07/1.13 ( ( member @ A @ X @ A3 )
% 2.07/1.13 & ( ord_less_eq @ A @ X @ A5 )
% 2.07/1.13 & ! [Xa: A] :
% 2.07/1.13 ( ( member @ A @ Xa @ A3 )
% 2.07/1.13 => ( ( ord_less_eq @ A @ Xa @ X )
% 2.07/1.13 => ( X = Xa ) ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_34_finite__has__maximal2,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( order @ A )
% 2.07/1.13 => ! [A3: set @ A,A5: A] :
% 2.07/1.13 ( ( finite_finite2 @ A @ A3 )
% 2.07/1.13 => ( ( member @ A @ A5 @ A3 )
% 2.07/1.13 => ? [X: A] :
% 2.07/1.13 ( ( member @ A @ X @ A3 )
% 2.07/1.13 & ( ord_less_eq @ A @ A5 @ X )
% 2.07/1.13 & ! [Xa: A] :
% 2.07/1.13 ( ( member @ A @ Xa @ A3 )
% 2.07/1.13 => ( ( ord_less_eq @ A @ X @ Xa )
% 2.07/1.13 => ( X = Xa ) ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_35_rev__finite__subset,axiom,
% 2.07/1.13 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 2.07/1.13 ( ( finite_finite2 @ A @ B3 )
% 2.07/1.13 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.07/1.13 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_36_infinite__super,axiom,
% 2.07/1.13 ! [A: $tType,S3: set @ A,T3: set @ A] :
% 2.07/1.13 ( ( ord_less_eq @ ( set @ A ) @ S3 @ T3 )
% 2.07/1.13 => ( ~ ( finite_finite2 @ A @ S3 )
% 2.07/1.13 => ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_37_finite__subset,axiom,
% 2.07/1.13 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.07/1.13 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.07/1.13 => ( ( finite_finite2 @ A @ B3 )
% 2.07/1.13 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_38_ek__analysis_OspEdges__ss__E,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linordered_idom @ A )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > A,S: nat,T4: nat] :
% 2.07/1.13 ( ( edmond2129202899alysis @ A @ C2 )
% 2.07/1.13 => ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( edmond1803015688pEdges @ A @ C2 @ S @ T4 ) @ ( e @ A @ C2 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_39_incoming_H__edges,axiom,
% 2.07/1.13 ! [U2: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming2 @ a @ c @ U2 ) @ ( e @ a @ c ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_40_outgoing_H__edges,axiom,
% 2.07/1.13 ! [U2: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing2 @ a @ c @ U2 ) @ ( e @ a @ c ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_41_reachableNodes__def,axiom,
% 2.07/1.13 ! [U: nat] :
% 2.07/1.13 ( ( reachableNodes @ a @ c @ U )
% 2.07/1.13 = ( collect @ nat @ ( connected @ a @ c @ U ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_42_List_Ofinite__set,axiom,
% 2.07/1.13 ! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_43_Graph_OEfin__imp__Vfin,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity] :
% 2.07/1.13 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ Capacity @ C2 ) )
% 2.07/1.13 => ( finite_finite2 @ nat @ ( v @ Capacity @ C2 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_44_Graph_OVfin__imp__Efin,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity] :
% 2.07/1.13 ( ( finite_finite2 @ nat @ ( v @ Capacity @ C2 ) )
% 2.07/1.13 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ Capacity @ C2 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_45_mem__Collect__eq,axiom,
% 2.07/1.13 ! [A: $tType,A5: A,P: A > $o] :
% 2.07/1.13 ( ( member @ A @ A5 @ ( collect @ A @ P ) )
% 2.07/1.13 = ( P @ A5 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_46_Collect__mem__eq,axiom,
% 2.07/1.13 ! [A: $tType,A3: set @ A] :
% 2.07/1.13 ( ( collect @ A
% 2.07/1.13 @ ^ [X3: A] : ( member @ A @ X3 @ A3 ) )
% 2.07/1.13 = A3 ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_47_Collect__cong,axiom,
% 2.07/1.13 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.07/1.13 ( ! [X: A] :
% 2.07/1.13 ( ( P @ X )
% 2.07/1.13 = ( Q @ X ) )
% 2.07/1.13 => ( ( collect @ A @ P )
% 2.07/1.13 = ( collect @ A @ Q ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_48_ext,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,F2: A > B,G: A > B] :
% 2.07/1.13 ( ! [X: A] :
% 2.07/1.13 ( ( F2 @ X )
% 2.07/1.13 = ( G @ X ) )
% 2.07/1.13 => ( F2 = G ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_49_isShortestPath__level__edge_I1_J,axiom,
% 2.07/1.13 ! [S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat,U: nat,V: nat] :
% 2.07/1.13 ( ( isShortestPath @ a @ c @ S @ P4 @ T4 )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ( connected @ a @ c @ S @ U ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_50_isShortestPath__level__edge_I2_J,axiom,
% 2.07/1.13 ! [S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat,U: nat,V: nat] :
% 2.07/1.13 ( ( isShortestPath @ a @ c @ S @ P4 @ T4 )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ( connected @ a @ c @ U @ V ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_51_isShortestPath__level__edge_I3_J,axiom,
% 2.07/1.13 ! [S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat,U: nat,V: nat] :
% 2.07/1.13 ( ( isShortestPath @ a @ c @ S @ P4 @ T4 )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ( connected @ a @ c @ V @ T4 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_52_incoming__edges,axiom,
% 2.07/1.13 ! [U: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming @ a @ c @ U ) @ ( e @ a @ c ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_53_reachable__ss__V,axiom,
% 2.07/1.13 ! [S: nat] :
% 2.07/1.13 ( ( member @ nat @ S @ ( v @ a @ c ) )
% 2.07/1.13 => ( ord_less_eq @ ( set @ nat ) @ ( reachableNodes @ a @ c @ S ) @ ( v @ a @ c ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_54_connected__append__edge,axiom,
% 2.07/1.13 ! [U: nat,V: nat,W: nat] :
% 2.07/1.13 ( ( connected @ a @ c @ U @ V )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V @ W ) @ ( e @ a @ c ) )
% 2.07/1.13 => ( connected @ a @ c @ U @ W ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_55_reachableNodes__append__edge,axiom,
% 2.07/1.13 ! [U: nat,S: nat,V: nat] :
% 2.07/1.13 ( ( member @ nat @ U @ ( reachableNodes @ a @ c @ S ) )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( e @ a @ c ) )
% 2.07/1.13 => ( member @ nat @ V @ ( reachableNodes @ a @ c @ S ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_56_V__def,axiom,
% 2.07/1.13 ( ( v @ a @ c )
% 2.07/1.13 = ( collect @ nat
% 2.07/1.13 @ ^ [U3: nat] :
% 2.07/1.13 ? [V2: nat] :
% 2.07/1.13 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U3 @ V2 ) @ ( e @ a @ c ) )
% 2.07/1.13 | ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V2 @ U3 ) @ ( e @ a @ c ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_57_incoming__def,axiom,
% 2.07/1.13 ! [V: nat] :
% 2.07/1.13 ( ( incoming @ a @ c @ V )
% 2.07/1.13 = ( collect @ ( product_prod @ nat @ nat )
% 2.07/1.13 @ ^ [Uu: product_prod @ nat @ nat] :
% 2.07/1.13 ? [U3: nat] :
% 2.07/1.13 ( ( Uu
% 2.07/1.13 = ( product_Pair @ nat @ nat @ U3 @ V ) )
% 2.07/1.13 & ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U3 @ V ) @ ( e @ a @ c ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_58_outgoing_H__def,axiom,
% 2.07/1.13 ! [K: set @ nat] :
% 2.07/1.13 ( ( outgoing2 @ a @ c @ K )
% 2.07/1.13 = ( collect @ ( product_prod @ nat @ nat )
% 2.07/1.13 @ ^ [Uu: product_prod @ nat @ nat] :
% 2.07/1.13 ? [U3: nat,V2: nat] :
% 2.07/1.13 ( ( Uu
% 2.07/1.13 = ( product_Pair @ nat @ nat @ V2 @ U3 ) )
% 2.07/1.13 & ~ ( member @ nat @ U3 @ K )
% 2.07/1.13 & ( member @ nat @ V2 @ K )
% 2.07/1.13 & ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V2 @ U3 ) @ ( e @ a @ c ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_59_incoming_H__def,axiom,
% 2.07/1.13 ! [K: set @ nat] :
% 2.07/1.13 ( ( incoming2 @ a @ c @ K )
% 2.07/1.13 = ( collect @ ( product_prod @ nat @ nat )
% 2.07/1.13 @ ^ [Uu: product_prod @ nat @ nat] :
% 2.07/1.13 ? [U3: nat,V2: nat] :
% 2.07/1.13 ( ( Uu
% 2.07/1.13 = ( product_Pair @ nat @ nat @ U3 @ V2 ) )
% 2.07/1.13 & ~ ( member @ nat @ U3 @ K )
% 2.07/1.13 & ( member @ nat @ V2 @ K )
% 2.07/1.13 & ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U3 @ V2 ) @ ( e @ a @ c ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_60_finite__Collect__le__nat,axiom,
% 2.07/1.13 ! [K: nat] :
% 2.07/1.13 ( finite_finite2 @ nat
% 2.07/1.13 @ ( collect @ nat
% 2.07/1.13 @ ^ [N: nat] : ( ord_less_eq @ nat @ N @ K ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_61_finite__incoming,axiom,
% 2.07/1.13 ! [U: nat] :
% 2.07/1.13 ( ( finite_finite2 @ nat @ ( v @ a @ c ) )
% 2.07/1.13 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming @ a @ c @ U ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_62_finite__outgoing_H,axiom,
% 2.07/1.13 ! [U2: set @ nat] :
% 2.07/1.13 ( ( finite_finite2 @ nat @ ( v @ a @ c ) )
% 2.07/1.13 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing2 @ a @ c @ U2 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_63_finite__incoming_H,axiom,
% 2.07/1.13 ! [U2: set @ nat] :
% 2.07/1.13 ( ( finite_finite2 @ nat @ ( v @ a @ c ) )
% 2.07/1.13 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming2 @ a @ c @ U2 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_64_adjacent__nodes__ss__V,axiom,
% 2.07/1.13 ! [U: nat] : ( ord_less_eq @ ( set @ nat ) @ ( adjacent_nodes @ a @ c @ U ) @ ( v @ a @ c ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_65_Graph_Oincoming__def,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( incoming @ Capacity )
% 2.07/1.13 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,V2: nat] :
% 2.07/1.13 ( collect @ ( product_prod @ nat @ nat )
% 2.07/1.13 @ ^ [Uu: product_prod @ nat @ nat] :
% 2.07/1.13 ? [U3: nat] :
% 2.07/1.13 ( ( Uu
% 2.07/1.13 = ( product_Pair @ nat @ nat @ U3 @ V2 ) )
% 2.07/1.13 & ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U3 @ V2 ) @ ( e @ Capacity @ C3 ) ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_66_Graph_Oincoming_H__def,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( incoming2 @ Capacity )
% 2.07/1.13 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,K2: set @ nat] :
% 2.07/1.13 ( collect @ ( product_prod @ nat @ nat )
% 2.07/1.13 @ ^ [Uu: product_prod @ nat @ nat] :
% 2.07/1.13 ? [U3: nat,V2: nat] :
% 2.07/1.13 ( ( Uu
% 2.07/1.13 = ( product_Pair @ nat @ nat @ U3 @ V2 ) )
% 2.07/1.13 & ~ ( member @ nat @ U3 @ K2 )
% 2.07/1.13 & ( member @ nat @ V2 @ K2 )
% 2.07/1.13 & ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U3 @ V2 ) @ ( e @ Capacity @ C3 ) ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_67_Graph_Ooutgoing_H__def,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( outgoing2 @ Capacity )
% 2.07/1.13 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,K2: set @ nat] :
% 2.07/1.13 ( collect @ ( product_prod @ nat @ nat )
% 2.07/1.13 @ ^ [Uu: product_prod @ nat @ nat] :
% 2.07/1.13 ? [U3: nat,V2: nat] :
% 2.07/1.13 ( ( Uu
% 2.07/1.13 = ( product_Pair @ nat @ nat @ V2 @ U3 ) )
% 2.07/1.13 & ~ ( member @ nat @ U3 @ K2 )
% 2.07/1.13 & ( member @ nat @ V2 @ K2 )
% 2.07/1.13 & ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V2 @ U3 ) @ ( e @ Capacity @ C3 ) ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_68_Graph_Oreachable__ss__V,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [S: nat,C2: ( product_prod @ nat @ nat ) > Capacity] :
% 2.07/1.13 ( ( member @ nat @ S @ ( v @ Capacity @ C2 ) )
% 2.07/1.13 => ( ord_less_eq @ ( set @ nat ) @ ( reachableNodes @ Capacity @ C2 @ S ) @ ( v @ Capacity @ C2 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_69_Graph_OreachableNodes__append__edge,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [U: nat,C2: ( product_prod @ nat @ nat ) > Capacity,S: nat,V: nat] :
% 2.07/1.13 ( ( member @ nat @ U @ ( reachableNodes @ Capacity @ C2 @ S ) )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( e @ Capacity @ C2 ) )
% 2.07/1.13 => ( member @ nat @ V @ ( reachableNodes @ Capacity @ C2 @ S ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_70_Graph_OreachableNodes_Ocong,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( reachableNodes @ Capacity )
% 2.07/1.13 = ( reachableNodes @ Capacity ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_71_Graph_Ooutgoing_H_Ocong,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( outgoing2 @ Capacity )
% 2.07/1.13 = ( outgoing2 @ Capacity ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_72_Graph_Oincoming_H_Ocong,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( incoming2 @ Capacity )
% 2.07/1.13 = ( incoming2 @ Capacity ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_73_Graph_Oincoming_Ocong,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( incoming @ Capacity )
% 2.07/1.13 = ( incoming @ Capacity ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_74_pred__equals__eq2,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( ( ( ^ [X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R2 ) )
% 2.07/1.13 = ( ^ [X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ S3 ) ) )
% 2.07/1.13 = ( R2 = S3 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_75_pred__subset__eq2,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( ( ord_less_eq @ ( A > B > $o )
% 2.07/1.13 @ ^ [X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R2 )
% 2.07/1.13 @ ^ [X3: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ S3 ) )
% 2.07/1.13 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_76_Graph_OV__def,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( v @ Capacity )
% 2.07/1.13 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity] :
% 2.07/1.13 ( collect @ nat
% 2.07/1.13 @ ^ [U3: nat] :
% 2.07/1.13 ? [V2: nat] :
% 2.07/1.13 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U3 @ V2 ) @ ( e @ Capacity @ C3 ) )
% 2.07/1.13 | ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V2 @ U3 ) @ ( e @ Capacity @ C3 ) ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_77_subrelI,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( ! [X: A,Y2: B] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ R )
% 2.07/1.13 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y2 ) @ S ) )
% 2.07/1.13 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_78_Domain_Oinducts,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,X4: A,R: set @ ( product_prod @ A @ B ),P: A > $o] :
% 2.07/1.13 ( ( member @ A @ X4 @ ( domain @ A @ B @ R ) )
% 2.07/1.13 => ( ! [A6: A,B4: B] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B4 ) @ R )
% 2.07/1.13 => ( P @ A6 ) )
% 2.07/1.13 => ( P @ X4 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_79_Domain_ODomainI,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A5: A,B5: B,R: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R )
% 2.07/1.13 => ( member @ A @ A5 @ ( domain @ A @ B @ R ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_80_Domain_Osimps,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A5: A,R: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( ( member @ A @ A5 @ ( domain @ A @ B @ R ) )
% 2.07/1.13 = ( ? [A4: A,B6: B] :
% 2.07/1.13 ( ( A5 = A4 )
% 2.07/1.13 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B6 ) @ R ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_81_Domain_Ocases,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A5: A,R: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( ( member @ A @ A5 @ ( domain @ A @ B @ R ) )
% 2.07/1.13 => ~ ! [B4: B] :
% 2.07/1.13 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ R ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_82_Domain__iff,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,A5: A,R: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( ( member @ A @ A5 @ ( domain @ A @ B @ R ) )
% 2.07/1.13 = ( ? [Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ Y ) @ R ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_83_DomainE,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A5: A,R: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( ( member @ A @ A5 @ ( domain @ A @ B @ R ) )
% 2.07/1.13 => ~ ! [B4: B] :
% 2.07/1.13 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B4 ) @ R ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_84_Range_Oinducts,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,X4: B,R: set @ ( product_prod @ A @ B ),P: B > $o] :
% 2.07/1.13 ( ( member @ B @ X4 @ ( range @ A @ B @ R ) )
% 2.07/1.13 => ( ! [A6: A,B4: B] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B4 ) @ R )
% 2.07/1.13 => ( P @ B4 ) )
% 2.07/1.13 => ( P @ X4 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_85_Range_Ointros,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A5: A,B5: B,R: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R )
% 2.07/1.13 => ( member @ B @ B5 @ ( range @ A @ B @ R ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_86_Range_Osimps,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A5: B,R: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( ( member @ B @ A5 @ ( range @ A @ B @ R ) )
% 2.07/1.13 = ( ? [A4: A,B6: B] :
% 2.07/1.13 ( ( A5 = B6 )
% 2.07/1.13 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B6 ) @ R ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_87_Range_Ocases,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A5: B,R: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( ( member @ B @ A5 @ ( range @ A @ B @ R ) )
% 2.07/1.13 => ~ ! [A6: A] :
% 2.07/1.13 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ A5 ) @ R ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_88_Range__iff,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,A5: A,R: set @ ( product_prod @ B @ A )] :
% 2.07/1.13 ( ( member @ A @ A5 @ ( range @ B @ A @ R ) )
% 2.07/1.13 = ( ? [Y: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y @ A5 ) @ R ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_89_RangeE,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,B5: A,R: set @ ( product_prod @ B @ A )] :
% 2.07/1.13 ( ( member @ A @ B5 @ ( range @ B @ A @ R ) )
% 2.07/1.13 => ~ ! [A6: B] :
% 2.07/1.13 ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A6 @ B5 ) @ R ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_90_Graph_OreachableNodes__def,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( reachableNodes @ Capacity )
% 2.07/1.13 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,U3: nat] : ( collect @ nat @ ( connected @ Capacity @ C3 @ U3 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_91_pred__subset__eq,axiom,
% 2.07/1.13 ! [A: $tType,R2: set @ A,S3: set @ A] :
% 2.07/1.13 ( ( ord_less_eq @ ( A > $o )
% 2.07/1.13 @ ^ [X3: A] : ( member @ A @ X3 @ R2 )
% 2.07/1.13 @ ^ [X3: A] : ( member @ A @ X3 @ S3 ) )
% 2.07/1.13 = ( ord_less_eq @ ( set @ A ) @ R2 @ S3 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_92_Graph_Oconnected__append__edge,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat,W: nat] :
% 2.07/1.13 ( ( connected @ Capacity @ C2 @ U @ V )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V @ W ) @ ( e @ Capacity @ C2 ) )
% 2.07/1.13 => ( connected @ Capacity @ C2 @ U @ W ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_93_Graph_Oincoming__edges,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming @ Capacity @ C2 @ U ) @ ( e @ Capacity @ C2 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_94_Graph_OisShortestPath__level__edge_I1_J,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat,U: nat,V: nat] :
% 2.07/1.13 ( ( isShortestPath @ Capacity @ C2 @ S @ P4 @ T4 )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ( connected @ Capacity @ C2 @ S @ U ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_95_Graph_OisShortestPath__level__edge_I2_J,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat,U: nat,V: nat] :
% 2.07/1.13 ( ( isShortestPath @ Capacity @ C2 @ S @ P4 @ T4 )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ( connected @ Capacity @ C2 @ U @ V ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_96_Graph_OisShortestPath__level__edge_I3_J,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat,U: nat,V: nat] :
% 2.07/1.13 ( ( isShortestPath @ Capacity @ C2 @ S @ P4 @ T4 )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ( connected @ Capacity @ C2 @ V @ T4 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_97_Graph_Ooutgoing_H__edges,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U2: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing2 @ Capacity @ C2 @ U2 ) @ ( e @ Capacity @ C2 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_98_Graph_Oincoming_H__edges,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U2: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming2 @ Capacity @ C2 @ U2 ) @ ( e @ Capacity @ C2 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_99_Domain__unfold,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType] :
% 2.07/1.13 ( ( domain @ A @ B )
% 2.07/1.13 = ( ^ [R3: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( collect @ A
% 2.07/1.13 @ ^ [X3: A] :
% 2.07/1.13 ? [Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R3 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_100_Graph_Ofinite__incoming,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat] :
% 2.07/1.13 ( ( finite_finite2 @ nat @ ( v @ Capacity @ C2 ) )
% 2.07/1.13 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming @ Capacity @ C2 @ U ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_101_Graph_OE_Ocong,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( e @ Capacity )
% 2.07/1.13 = ( e @ Capacity ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_102_Graph_OV_Ocong,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( v @ Capacity )
% 2.07/1.13 = ( v @ Capacity ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_103_Graph_Oconnected__refl,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,V: nat] : ( connected @ Capacity @ C2 @ V @ V ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_104_Graph_Oconnected_Ocong,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( connected @ Capacity )
% 2.07/1.13 = ( connected @ Capacity ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_105_Graph_Ofinite__outgoing_H,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U2: set @ nat] :
% 2.07/1.13 ( ( finite_finite2 @ nat @ ( v @ Capacity @ C2 ) )
% 2.07/1.13 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing2 @ Capacity @ C2 @ U2 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_106_Graph_OisShortestPath_Ocong,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( isShortestPath @ Capacity )
% 2.07/1.13 = ( isShortestPath @ Capacity ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_107_Graph_Ofinite__incoming_H,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U2: set @ nat] :
% 2.07/1.13 ( ( finite_finite2 @ nat @ ( v @ Capacity @ C2 ) )
% 2.07/1.13 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming2 @ Capacity @ C2 @ U2 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_108_finite__list,axiom,
% 2.07/1.13 ! [A: $tType,A3: set @ A] :
% 2.07/1.13 ( ( finite_finite2 @ A @ A3 )
% 2.07/1.13 => ? [Xs2: list @ A] :
% 2.07/1.13 ( ( set2 @ A @ Xs2 )
% 2.07/1.13 = A3 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_109_subset__code_I1_J,axiom,
% 2.07/1.13 ! [A: $tType,Xs: list @ A,B3: set @ A] :
% 2.07/1.13 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B3 )
% 2.07/1.13 = ( ! [X3: A] :
% 2.07/1.13 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 2.07/1.13 => ( member @ A @ X3 @ B3 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_110_Graph_Oconnected__inV__iff,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat] :
% 2.07/1.13 ( ( connected @ Capacity @ C2 @ U @ V )
% 2.07/1.13 => ( ( member @ nat @ V @ ( v @ Capacity @ C2 ) )
% 2.07/1.13 = ( member @ nat @ U @ ( v @ Capacity @ C2 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_111_Graph_Oobtain__shortest__path,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat] :
% 2.07/1.13 ( ( connected @ Capacity @ C2 @ U @ V )
% 2.07/1.13 => ~ ! [P3: list @ ( product_prod @ nat @ nat )] :
% 2.07/1.13 ~ ( isShortestPath @ Capacity @ C2 @ U @ P3 @ V ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_112_Finite__Graph_Ofinite__V,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linordered_idom @ A )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > A] :
% 2.07/1.13 ( ( finite_Graph @ A @ C2 )
% 2.07/1.13 => ( finite_finite2 @ nat @ ( v @ A @ C2 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_113_Finite__Graph_Ointro,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linordered_idom @ A )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > A] :
% 2.07/1.13 ( ( finite_finite2 @ nat @ ( v @ A @ C2 ) )
% 2.07/1.13 => ( finite_Graph @ A @ C2 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_114_Finite__Graph__def,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linordered_idom @ A )
% 2.07/1.13 => ( ( finite_Graph @ A )
% 2.07/1.13 = ( ^ [C3: ( product_prod @ nat @ nat ) > A] : ( finite_finite2 @ nat @ ( v @ A @ C3 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_115_Finite__Graph_Ofinite__E,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linordered_idom @ A )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > A] :
% 2.07/1.13 ( ( finite_Graph @ A @ C2 )
% 2.07/1.13 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ A @ C2 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_116_Graph_OFinite__Graph__EI,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity] :
% 2.07/1.13 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ Capacity @ C2 ) )
% 2.07/1.13 => ( finite_Graph @ Capacity @ C2 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_117_outgoing__def,axiom,
% 2.07/1.13 ! [V: nat] :
% 2.07/1.13 ( ( outgoing @ a @ c @ V )
% 2.07/1.13 = ( collect @ ( product_prod @ nat @ nat )
% 2.07/1.13 @ ^ [Uu: product_prod @ nat @ nat] :
% 2.07/1.13 ? [U3: nat] :
% 2.07/1.13 ( ( Uu
% 2.07/1.13 = ( product_Pair @ nat @ nat @ V @ U3 ) )
% 2.07/1.13 & ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V @ U3 ) @ ( e @ a @ c ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_118_outgoing__edges,axiom,
% 2.07/1.13 ! [U: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing @ a @ c @ U ) @ ( e @ a @ c ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_119_E__ss__VxV,axiom,
% 2.07/1.13 ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( e @ a @ c )
% 2.07/1.13 @ ( product_Sigma @ nat @ nat @ ( v @ a @ c )
% 2.07/1.13 @ ^ [Uu: nat] : ( v @ a @ c ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_120_connected__edgeRtc,axiom,
% 2.07/1.13 ! [U: nat,V: nat] :
% 2.07/1.13 ( ( connected @ a @ c @ U @ V )
% 2.07/1.13 = ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( transitive_rtrancl @ nat @ ( e @ a @ c ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_121_isSPath__no__selfloop,axiom,
% 2.07/1.13 ! [U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat] :
% 2.07/1.13 ( ( isSimplePath @ a @ c @ U @ P4 @ V )
% 2.07/1.13 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_122_isSPath__sg__incoming,axiom,
% 2.07/1.13 ! [U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,U22: nat] :
% 2.07/1.13 ( ( isSimplePath @ a @ c @ U @ P4 @ V )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ( ( U1 != U22 )
% 2.07/1.13 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U22 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_123_isSPath__sg__outgoing,axiom,
% 2.07/1.13 ! [U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,V22: nat] :
% 2.07/1.13 ( ( isSimplePath @ a @ c @ U @ P4 @ V )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ( ( V1 != V22 )
% 2.07/1.13 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V22 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_124_reachableNodes__E__closed,axiom,
% 2.07/1.13 ! [S: nat] : ( ord_less_eq @ ( set @ nat ) @ ( image @ nat @ nat @ ( e @ a @ c ) @ ( reachableNodes @ a @ c @ S ) ) @ ( reachableNodes @ a @ c @ S ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_125_shortestPath__is__simple,axiom,
% 2.07/1.13 ! [S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat] :
% 2.07/1.13 ( ( isShortestPath @ a @ c @ S @ P4 @ T4 )
% 2.07/1.13 => ( isSimplePath @ a @ c @ S @ P4 @ T4 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_126_ImageI,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A5: A,B5: B,R: set @ ( product_prod @ A @ B ),A3: set @ A] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R )
% 2.07/1.13 => ( ( member @ A @ A5 @ A3 )
% 2.07/1.13 => ( member @ B @ B5 @ ( image @ A @ B @ R @ A3 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_127_finite__SigmaI,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A3: set @ A,B3: A > ( set @ B )] :
% 2.07/1.13 ( ( finite_finite2 @ A @ A3 )
% 2.07/1.13 => ( ! [A6: A] :
% 2.07/1.13 ( ( member @ A @ A6 @ A3 )
% 2.07/1.13 => ( finite_finite2 @ B @ ( B3 @ A6 ) ) )
% 2.07/1.13 => ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_128_adjacent__nodes__finite,axiom,
% 2.07/1.13 ! [U: nat] : ( finite_finite2 @ nat @ ( adjacent_nodes @ a @ c @ U ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_129_finite__outgoing,axiom,
% 2.07/1.13 ! [U: nat] :
% 2.07/1.13 ( ( finite_finite2 @ nat @ ( v @ a @ c ) )
% 2.07/1.13 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing @ a @ c @ U ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_130_Graph_Oadjacent__nodes_Ocong,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( adjacent_nodes @ Capacity )
% 2.07/1.13 = ( adjacent_nodes @ Capacity ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_131_Graph_OisSimplePath_Ocong,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( isSimplePath @ Capacity )
% 2.07/1.13 = ( isSimplePath @ Capacity ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_132_Graph_Ooutgoing_Ocong,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( outgoing @ Capacity )
% 2.07/1.13 = ( outgoing @ Capacity ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_133_rev__ImageI,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A5: A,A3: set @ A,B5: B,R: set @ ( product_prod @ A @ B )] :
% 2.07/1.13 ( ( member @ A @ A5 @ A3 )
% 2.07/1.13 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ R )
% 2.07/1.13 => ( member @ B @ B5 @ ( image @ A @ B @ R @ A3 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_134_Image__iff,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,B5: A,R: set @ ( product_prod @ B @ A ),A3: set @ B] :
% 2.07/1.13 ( ( member @ A @ B5 @ ( image @ B @ A @ R @ A3 ) )
% 2.07/1.13 = ( ? [X3: B] :
% 2.07/1.13 ( ( member @ B @ X3 @ A3 )
% 2.07/1.13 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ B5 ) @ R ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_135_ImageE,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,B5: A,R: set @ ( product_prod @ B @ A ),A3: set @ B] :
% 2.07/1.13 ( ( member @ A @ B5 @ ( image @ B @ A @ R @ A3 ) )
% 2.07/1.13 => ~ ! [X: B] :
% 2.07/1.13 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ B5 ) @ R )
% 2.07/1.13 => ~ ( member @ B @ X @ A3 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_136_Image__subset,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),A3: set @ A,B3: set @ B,C4: set @ A] :
% 2.07/1.13 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R
% 2.07/1.13 @ ( product_Sigma @ A @ B @ A3
% 2.07/1.13 @ ^ [Uu: A] : B3 ) )
% 2.07/1.13 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R @ C4 ) @ B3 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_137_Graph_OshortestPath__is__simple,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat] :
% 2.07/1.13 ( ( isShortestPath @ Capacity @ C2 @ S @ P4 @ T4 )
% 2.07/1.13 => ( isSimplePath @ Capacity @ C2 @ S @ P4 @ T4 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_138_Image__mono,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),R: set @ ( product_prod @ A @ B ),A7: set @ A,A3: set @ A] :
% 2.07/1.13 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ R )
% 2.07/1.13 => ( ( ord_less_eq @ ( set @ A ) @ A7 @ A3 )
% 2.07/1.13 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R4 @ A7 ) @ ( image @ A @ B @ R @ A3 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_139_Relation_Ofinite__Image,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),A3: set @ A] :
% 2.07/1.13 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ R2 )
% 2.07/1.13 => ( finite_finite2 @ B @ ( image @ A @ B @ R2 @ A3 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_140_Graph_OreachableNodes__E__closed,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,S: nat] : ( ord_less_eq @ ( set @ nat ) @ ( image @ nat @ nat @ ( e @ Capacity @ C2 ) @ ( reachableNodes @ Capacity @ C2 @ S ) ) @ ( reachableNodes @ Capacity @ C2 @ S ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_141_finite__cartesian__product,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B] :
% 2.07/1.13 ( ( finite_finite2 @ A @ A3 )
% 2.07/1.13 => ( ( finite_finite2 @ B @ B3 )
% 2.07/1.13 => ( finite_finite2 @ ( product_prod @ A @ B )
% 2.07/1.13 @ ( product_Sigma @ A @ B @ A3
% 2.07/1.13 @ ^ [Uu: A] : B3 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_142_Graph_OisSPath__sg__outgoing,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,V22: nat] :
% 2.07/1.13 ( ( isSimplePath @ Capacity @ C2 @ U @ P4 @ V )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ( ( V1 != V22 )
% 2.07/1.13 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V22 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_143_Graph_OisSPath__sg__incoming,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,U22: nat] :
% 2.07/1.13 ( ( isSimplePath @ Capacity @ C2 @ U @ P4 @ V )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ( ( U1 != U22 )
% 2.07/1.13 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U22 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_144_Graph_OisSPath__no__selfloop,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat] :
% 2.07/1.13 ( ( isSimplePath @ Capacity @ C2 @ U @ P4 @ V )
% 2.07/1.13 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_145_Graph_Oadjacent__nodes__ss__V,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat] : ( ord_less_eq @ ( set @ nat ) @ ( adjacent_nodes @ Capacity @ C2 @ U ) @ ( v @ Capacity @ C2 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_146_Graph_Ooutgoing__edges,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing @ Capacity @ C2 @ U ) @ ( e @ Capacity @ C2 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_147_Graph_Ooutgoing__def,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( outgoing @ Capacity )
% 2.07/1.13 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,V2: nat] :
% 2.07/1.13 ( collect @ ( product_prod @ nat @ nat )
% 2.07/1.13 @ ^ [Uu: product_prod @ nat @ nat] :
% 2.07/1.13 ? [U3: nat] :
% 2.07/1.13 ( ( Uu
% 2.07/1.13 = ( product_Pair @ nat @ nat @ V2 @ U3 ) )
% 2.07/1.13 & ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V2 @ U3 ) @ ( e @ Capacity @ C3 ) ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_148_Graph_Oconnected__edgeRtc,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( connected @ Capacity )
% 2.07/1.13 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,U3: nat,V2: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U3 @ V2 ) @ ( transitive_rtrancl @ nat @ ( e @ Capacity @ C3 ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_149_Graph_OE__ss__VxV,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity] :
% 2.07/1.13 ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( e @ Capacity @ C2 )
% 2.07/1.13 @ ( product_Sigma @ nat @ nat @ ( v @ Capacity @ C2 )
% 2.07/1.13 @ ^ [Uu: nat] : ( v @ Capacity @ C2 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_150_Graph_Ofinite__outgoing,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat] :
% 2.07/1.13 ( ( finite_finite2 @ nat @ ( v @ Capacity @ C2 ) )
% 2.07/1.13 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing @ Capacity @ C2 @ U ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_151_Finite__Graph_Oadjacent__nodes__finite,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linordered_idom @ A )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > A,U: nat] :
% 2.07/1.13 ( ( finite_Graph @ A @ C2 )
% 2.07/1.13 => ( finite_finite2 @ nat @ ( adjacent_nodes @ A @ C2 @ U ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_152_isSPath__nt__parallel,axiom,
% 2.07/1.13 ! [S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat,E2: product_prod @ nat @ nat] :
% 2.07/1.13 ( ( isSimplePath @ a @ c @ S @ P4 @ T4 )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ E2 @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ E2 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_153_mem__Sigma__iff,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A5: A,B5: B,A3: set @ A,B3: A > ( set @ B )] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 2.07/1.13 = ( ( member @ A @ A5 @ A3 )
% 2.07/1.13 & ( member @ B @ B5 @ ( B3 @ A5 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_154_SigmaI,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A5: A,A3: set @ A,B5: B,B3: A > ( set @ B )] :
% 2.07/1.13 ( ( member @ A @ A5 @ A3 )
% 2.07/1.13 => ( ( member @ B @ B5 @ ( B3 @ A5 ) )
% 2.07/1.13 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_155_trancl__subset__Sigma__aux,axiom,
% 2.07/1.13 ! [A: $tType,A5: A,B5: A,R: set @ ( product_prod @ A @ A ),A3: set @ A] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
% 2.07/1.13 @ ( product_Sigma @ A @ A @ A3
% 2.07/1.13 @ ^ [Uu: A] : A3 ) )
% 2.07/1.13 => ( ( A5 = B5 )
% 2.07/1.13 | ( member @ A @ A5 @ A3 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_156_old_Oprod_Oinject,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,A5: A,B5: B,A8: A,B7: B] :
% 2.07/1.13 ( ( ( product_Pair @ A @ B @ A5 @ B5 )
% 2.07/1.13 = ( product_Pair @ A @ B @ A8 @ B7 ) )
% 2.07/1.13 = ( ( A5 = A8 )
% 2.07/1.13 & ( B5 = B7 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_157_prod_Oinject,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
% 2.07/1.13 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 2.07/1.13 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 2.07/1.13 = ( ( X1 = Y1 )
% 2.07/1.13 & ( X22 = Y22 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_158_r__into__rtrancl,axiom,
% 2.07/1.13 ! [A: $tType,P4: product_prod @ A @ A,R: set @ ( product_prod @ A @ A )] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ P4 @ R )
% 2.07/1.13 => ( member @ ( product_prod @ A @ A ) @ P4 @ ( transitive_rtrancl @ A @ R ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_159_rtrancl__idemp,axiom,
% 2.07/1.13 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 2.07/1.13 ( ( transitive_rtrancl @ A @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 = ( transitive_rtrancl @ A @ R ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_160_swap__swap,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,P4: product_prod @ A @ B] :
% 2.07/1.13 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P4 ) )
% 2.07/1.13 = P4 ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_161_swap__simp,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,X4: B,Y3: A] :
% 2.07/1.13 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X4 @ Y3 ) )
% 2.07/1.13 = ( product_Pair @ A @ B @ Y3 @ X4 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_162_converse__rtrancl__induct2,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
% 2.07/1.13 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 2.07/1.13 => ( ( P @ Bx @ By )
% 2.07/1.13 => ( ! [A6: A,B4: B,Aa: A,Ba: B] :
% 2.07/1.13 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B4 ) @ ( product_Pair @ A @ B @ Aa @ Ba ) ) @ R )
% 2.07/1.13 => ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Aa @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 2.07/1.13 => ( ( P @ Aa @ Ba )
% 2.07/1.13 => ( P @ A6 @ B4 ) ) ) )
% 2.07/1.13 => ( P @ Ax @ Ay ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_163_converse__rtranclE2,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,Xa2: A,Xb: B,Za: A,Zb: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )] :
% 2.07/1.13 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa2 @ Xb ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 2.07/1.13 => ( ( ( product_Pair @ A @ B @ Xa2 @ Xb )
% 2.07/1.13 != ( product_Pair @ A @ B @ Za @ Zb ) )
% 2.07/1.13 => ~ ! [A6: A,B4: B] :
% 2.07/1.13 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa2 @ Xb ) @ ( product_Pair @ A @ B @ A6 @ B4 ) ) @ R )
% 2.07/1.13 => ~ ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B4 ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_164_rtrancl__induct2,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
% 2.07/1.13 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 2.07/1.13 => ( ( P @ Ax @ Ay )
% 2.07/1.13 => ( ! [A6: A,B4: B,Aa: A,Ba: B] :
% 2.07/1.13 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A6 @ B4 ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 2.07/1.13 => ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B4 ) @ ( product_Pair @ A @ B @ Aa @ Ba ) ) @ R )
% 2.07/1.13 => ( ( P @ A6 @ B4 )
% 2.07/1.13 => ( P @ Aa @ Ba ) ) ) )
% 2.07/1.13 => ( P @ Bx @ By ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_165_old_Oprod_Oinducts,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.07/1.13 ( ! [A6: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B4 ) )
% 2.07/1.13 => ( P @ Prod ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_166_old_Oprod_Oexhaust,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 2.07/1.13 ~ ! [A6: A,B4: B] :
% 2.07/1.13 ( Y3
% 2.07/1.13 != ( product_Pair @ A @ B @ A6 @ B4 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_167_prod__induct7,axiom,
% 2.07/1.13 ! [G2: $tType,F3: $tType,E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
% 2.07/1.13 ( ! [A6: A,B4: B,C5: C,D2: D,E4: E3,F: F3,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) @ D2 @ ( product_Pair @ E3 @ ( product_prod @ F3 @ G2 ) @ E4 @ ( product_Pair @ F3 @ G2 @ F @ G3 ) ) ) ) ) ) )
% 2.07/1.13 => ( P @ X4 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_168_prod__induct6,axiom,
% 2.07/1.13 ! [F3: $tType,E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) ) )] :
% 2.07/1.13 ( ! [A6: A,B4: B,C5: C,D2: D,E4: E3,F: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ F3 ) @ D2 @ ( product_Pair @ E3 @ F3 @ E4 @ F ) ) ) ) ) )
% 2.07/1.13 => ( P @ X4 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_169_prod__induct5,axiom,
% 2.07/1.13 ! [E3: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) )] :
% 2.07/1.13 ( ! [A6: A,B4: B,C5: C,D2: D,E4: E3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E3 ) @ C5 @ ( product_Pair @ D @ E3 @ D2 @ E4 ) ) ) ) )
% 2.07/1.13 => ( P @ X4 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_170_prod__induct4,axiom,
% 2.07/1.13 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.07/1.13 ( ! [A6: A,B4: B,C5: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C5 @ D2 ) ) ) )
% 2.07/1.13 => ( P @ X4 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_171_prod__induct3,axiom,
% 2.07/1.13 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X4: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.07/1.13 ( ! [A6: A,B4: B,C5: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B4 @ C5 ) ) )
% 2.07/1.13 => ( P @ X4 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_172_prod__cases7,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,F3: $tType,G2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) ) )] :
% 2.07/1.13 ~ ! [A6: A,B4: B,C5: C,D2: D,E4: E3,F: F3,G3: G2] :
% 2.07/1.13 ( Y3
% 2.07/1.13 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ ( product_prod @ F3 @ G2 ) ) @ D2 @ ( product_Pair @ E3 @ ( product_prod @ F3 @ G2 ) @ E4 @ ( product_Pair @ F3 @ G2 @ F @ G3 ) ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_173_prod__cases6,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,F3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) ) )] :
% 2.07/1.13 ~ ! [A6: A,B4: B,C5: C,D2: D,E4: E3,F: F3] :
% 2.07/1.13 ( Y3
% 2.07/1.13 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E3 @ F3 ) ) @ C5 @ ( product_Pair @ D @ ( product_prod @ E3 @ F3 ) @ D2 @ ( product_Pair @ E3 @ F3 @ E4 @ F ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_174_prod__cases5,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,C: $tType,D: $tType,E3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) )] :
% 2.07/1.13 ~ ! [A6: A,B4: B,C5: C,D2: D,E4: E3] :
% 2.07/1.13 ( Y3
% 2.07/1.13 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E3 ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E3 ) @ C5 @ ( product_Pair @ D @ E3 @ D2 @ E4 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_175_prod__cases4,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 2.07/1.13 ~ ! [A6: A,B4: B,C5: C,D2: D] :
% 2.07/1.13 ( Y3
% 2.07/1.13 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C5 @ D2 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_176_prod__cases3,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.07/1.13 ~ ! [A6: A,B4: B,C5: C] :
% 2.07/1.13 ( Y3
% 2.07/1.13 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A6 @ ( product_Pair @ B @ C @ B4 @ C5 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_177_Pair__inject,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,A5: A,B5: B,A8: A,B7: B] :
% 2.07/1.13 ( ( ( product_Pair @ A @ B @ A5 @ B5 )
% 2.07/1.13 = ( product_Pair @ A @ B @ A8 @ B7 ) )
% 2.07/1.13 => ~ ( ( A5 = A8 )
% 2.07/1.13 => ( B5 != B7 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_178_prod__cases,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] :
% 2.07/1.13 ( ! [A6: A,B4: B] : ( P @ ( product_Pair @ A @ B @ A6 @ B4 ) )
% 2.07/1.13 => ( P @ P4 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_179_surj__pair,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,P4: product_prod @ A @ B] :
% 2.07/1.13 ? [X: A,Y2: B] :
% 2.07/1.13 ( P4
% 2.07/1.13 = ( product_Pair @ A @ B @ X @ Y2 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_180_Graph_OisSPath__nt__parallel,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat,E2: product_prod @ nat @ nat] :
% 2.07/1.13 ( ( isSimplePath @ Capacity @ C2 @ S @ P4 @ T4 )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ E2 @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ E2 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_181_Sigma__cong,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ A,C4: A > ( set @ B ),D3: A > ( set @ B )] :
% 2.07/1.13 ( ( A3 = B3 )
% 2.07/1.13 => ( ! [X: A] :
% 2.07/1.13 ( ( member @ A @ X @ B3 )
% 2.07/1.13 => ( ( C4 @ X )
% 2.07/1.13 = ( D3 @ X ) ) )
% 2.07/1.13 => ( ( product_Sigma @ A @ B @ A3 @ C4 )
% 2.07/1.13 = ( product_Sigma @ A @ B @ B3 @ D3 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_182_Times__eq__cancel2,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,X4: A,C4: set @ A,A3: set @ B,B3: set @ B] :
% 2.07/1.13 ( ( member @ A @ X4 @ C4 )
% 2.07/1.13 => ( ( ( product_Sigma @ B @ A @ A3
% 2.07/1.13 @ ^ [Uu: B] : C4 )
% 2.07/1.13 = ( product_Sigma @ B @ A @ B3
% 2.07/1.13 @ ^ [Uu: B] : C4 ) )
% 2.07/1.13 = ( A3 = B3 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_183_SigmaE,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,C2: product_prod @ A @ B,A3: set @ A,B3: A > ( set @ B )] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 2.07/1.13 => ~ ! [X: A] :
% 2.07/1.13 ( ( member @ A @ X @ A3 )
% 2.07/1.13 => ! [Y2: B] :
% 2.07/1.13 ( ( member @ B @ Y2 @ ( B3 @ X ) )
% 2.07/1.13 => ( C2
% 2.07/1.13 != ( product_Pair @ A @ B @ X @ Y2 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_184_SigmaD1,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A5: A,B5: B,A3: set @ A,B3: A > ( set @ B )] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 2.07/1.13 => ( member @ A @ A5 @ A3 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_185_SigmaD2,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A5: A,B5: B,A3: set @ A,B3: A > ( set @ B )] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 2.07/1.13 => ( member @ B @ B5 @ ( B3 @ A5 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_186_SigmaE2,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A5: A,B5: B,A3: set @ A,B3: A > ( set @ B )] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B5 ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) )
% 2.07/1.13 => ~ ( ( member @ A @ A5 @ A3 )
% 2.07/1.13 => ~ ( member @ B @ B5 @ ( B3 @ A5 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_187_rtranclE,axiom,
% 2.07/1.13 ! [A: $tType,A5: A,B5: A,R: set @ ( product_prod @ A @ A )] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 => ( ( A5 != B5 )
% 2.07/1.13 => ~ ! [Y2: A] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ Y2 ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ B5 ) @ R ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_188_rtrancl_Ocases,axiom,
% 2.07/1.13 ! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 => ( ( A22 != A1 )
% 2.07/1.13 => ~ ! [B4: A] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B4 ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A22 ) @ R ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_189_rtrancl_Osimps,axiom,
% 2.07/1.13 ! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 = ( ? [A4: A] :
% 2.07/1.13 ( ( A1 = A4 )
% 2.07/1.13 & ( A22 = A4 ) )
% 2.07/1.13 | ? [A4: A,B6: A,C3: A] :
% 2.07/1.13 ( ( A1 = A4 )
% 2.07/1.13 & ( A22 = C3 )
% 2.07/1.13 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B6 ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B6 @ C3 ) @ R ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_190_rtrancl__trans,axiom,
% 2.07/1.13 ! [A: $tType,X4: A,Y3: A,R: set @ ( product_prod @ A @ A ),Z: A] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_191_rtrancl__induct,axiom,
% 2.07/1.13 ! [A: $tType,A5: A,B5: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 => ( ( P @ A5 )
% 2.07/1.13 => ( ! [Y2: A,Z2: A] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ Y2 ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ R )
% 2.07/1.13 => ( ( P @ Y2 )
% 2.07/1.13 => ( P @ Z2 ) ) ) )
% 2.07/1.13 => ( P @ B5 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_192_rtrancl_Oinducts,axiom,
% 2.07/1.13 ! [A: $tType,X1: A,X22: A,R: set @ ( product_prod @ A @ A ),P: A > A > $o] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X1 @ X22 ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 => ( ! [A6: A] : ( P @ A6 @ A6 )
% 2.07/1.13 => ( ! [A6: A,B4: A,C5: A] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B4 ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 => ( ( P @ A6 @ B4 )
% 2.07/1.13 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C5 ) @ R )
% 2.07/1.13 => ( P @ A6 @ C5 ) ) ) )
% 2.07/1.13 => ( P @ X1 @ X22 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_193_converse__rtranclE,axiom,
% 2.07/1.13 ! [A: $tType,X4: A,Z: A,R: set @ ( product_prod @ A @ A )] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 => ( ( X4 != Z )
% 2.07/1.13 => ~ ! [Y2: A] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y2 ) @ R )
% 2.07/1.13 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_194_rtrancl_Ortrancl__refl,axiom,
% 2.07/1.13 ! [A: $tType,A5: A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ ( transitive_rtrancl @ A @ R ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_195_converse__rtrancl__induct,axiom,
% 2.07/1.13 ! [A: $tType,A5: A,B5: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 => ( ( P @ B5 )
% 2.07/1.13 => ( ! [Y2: A,Z2: A] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ R )
% 2.07/1.13 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z2 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 => ( ( P @ Z2 )
% 2.07/1.13 => ( P @ Y2 ) ) ) )
% 2.07/1.13 => ( P @ A5 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_196_rtrancl_Ortrancl__into__rtrancl,axiom,
% 2.07/1.13 ! [A: $tType,A5: A,B5: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C2 ) @ R )
% 2.07/1.13 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ C2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_197_converse__rtrancl__into__rtrancl,axiom,
% 2.07/1.13 ! [A: $tType,A5: A,B5: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B5 ) @ R )
% 2.07/1.13 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C2 ) @ ( transitive_rtrancl @ A @ R ) )
% 2.07/1.13 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ C2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_198_rtrancl__mono,axiom,
% 2.07/1.13 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
% 2.07/1.13 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
% 2.07/1.13 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R ) @ ( transitive_rtrancl @ A @ S ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_199_rtrancl__subset,axiom,
% 2.07/1.13 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
% 2.07/1.13 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 )
% 2.07/1.13 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S3 @ ( transitive_rtrancl @ A @ R2 ) )
% 2.07/1.13 => ( ( transitive_rtrancl @ A @ S3 )
% 2.07/1.13 = ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_200_rtrancl__subset__rtrancl,axiom,
% 2.07/1.13 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
% 2.07/1.13 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( transitive_rtrancl @ A @ S ) )
% 2.07/1.13 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R ) @ ( transitive_rtrancl @ A @ S ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_201_Sigma__mono,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A3: set @ A,C4: set @ A,B3: A > ( set @ B ),D3: A > ( set @ B )] :
% 2.07/1.13 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C4 )
% 2.07/1.13 => ( ! [X: A] :
% 2.07/1.13 ( ( member @ A @ X @ A3 )
% 2.07/1.13 => ( ord_less_eq @ ( set @ B ) @ ( B3 @ X ) @ ( D3 @ X ) ) )
% 2.07/1.13 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A3 @ B3 ) @ ( product_Sigma @ A @ B @ C4 @ D3 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_202_Image__closed__trancl,axiom,
% 2.07/1.13 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X5: set @ A] :
% 2.07/1.13 ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ X5 ) @ X5 )
% 2.07/1.13 => ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ X5 )
% 2.07/1.13 = X5 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_203_Not__Domain__rtrancl,axiom,
% 2.07/1.13 ! [A: $tType,X4: A,R2: set @ ( product_prod @ A @ A ),Y3: A] :
% 2.07/1.13 ( ~ ( member @ A @ X4 @ ( domain @ A @ A @ R2 ) )
% 2.07/1.13 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ ( transitive_rtrancl @ A @ R2 ) )
% 2.07/1.13 = ( X4 = Y3 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_204_Times__subset__cancel2,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,X4: A,C4: set @ A,A3: set @ B,B3: set @ B] :
% 2.07/1.13 ( ( member @ A @ X4 @ C4 )
% 2.07/1.13 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
% 2.07/1.13 @ ( product_Sigma @ B @ A @ A3
% 2.07/1.13 @ ^ [Uu: B] : C4 )
% 2.07/1.13 @ ( product_Sigma @ B @ A @ B3
% 2.07/1.13 @ ^ [Uu: B] : C4 ) )
% 2.07/1.13 = ( ord_less_eq @ ( set @ B ) @ A3 @ B3 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_205_finite__rtrancl__Image,axiom,
% 2.07/1.13 ! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: set @ A] :
% 2.07/1.13 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
% 2.07/1.13 => ( ( finite_finite2 @ A @ A3 )
% 2.07/1.13 => ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R2 ) @ A3 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_206_infinite__cartesian__product,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,A3: set @ A,B3: set @ B] :
% 2.07/1.13 ( ~ ( finite_finite2 @ A @ A3 )
% 2.07/1.13 => ( ~ ( finite_finite2 @ B @ B3 )
% 2.07/1.13 => ~ ( finite_finite2 @ ( product_prod @ A @ B )
% 2.07/1.13 @ ( product_Sigma @ A @ B @ A3
% 2.07/1.13 @ ^ [Uu: A] : B3 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_207_old_Oprod_Orec,axiom,
% 2.07/1.13 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A5: A,B5: B] :
% 2.07/1.13 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.07/1.13 = ( F1 @ A5 @ B5 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_208_predicate1I,axiom,
% 2.07/1.13 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.07/1.13 ( ! [X: A] :
% 2.07/1.13 ( ( P @ X )
% 2.07/1.13 => ( Q @ X ) )
% 2.07/1.13 => ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_209_order__refl,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( preorder @ A )
% 2.07/1.13 => ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_210_predicate2I,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
% 2.07/1.13 ( ! [X: A,Y2: B] :
% 2.07/1.13 ( ( P @ X @ Y2 )
% 2.07/1.13 => ( Q @ X @ Y2 ) )
% 2.07/1.13 => ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_211_order__subst1,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType] :
% 2.07/1.13 ( ( ( order @ B )
% 2.07/1.13 & ( order @ A ) )
% 2.07/1.13 => ! [A5: A,F2: B > A,B5: B,C2: B] :
% 2.07/1.13 ( ( ord_less_eq @ A @ A5 @ ( F2 @ B5 ) )
% 2.07/1.13 => ( ( ord_less_eq @ B @ B5 @ C2 )
% 2.07/1.13 => ( ! [X: B,Y2: B] :
% 2.07/1.13 ( ( ord_less_eq @ B @ X @ Y2 )
% 2.07/1.13 => ( ord_less_eq @ A @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
% 2.07/1.13 => ( ord_less_eq @ A @ A5 @ ( F2 @ C2 ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_212_order__subst2,axiom,
% 2.07/1.13 ! [A: $tType,C: $tType] :
% 2.07/1.13 ( ( ( order @ C )
% 2.07/1.13 & ( order @ A ) )
% 2.07/1.13 => ! [A5: A,B5: A,F2: A > C,C2: C] :
% 2.07/1.13 ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.07/1.13 => ( ( ord_less_eq @ C @ ( F2 @ B5 ) @ C2 )
% 2.07/1.13 => ( ! [X: A,Y2: A] :
% 2.07/1.13 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.07/1.13 => ( ord_less_eq @ C @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
% 2.07/1.13 => ( ord_less_eq @ C @ ( F2 @ A5 ) @ C2 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_213_ord__eq__le__subst,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType] :
% 2.07/1.13 ( ( ( ord @ B )
% 2.07/1.13 & ( ord @ A ) )
% 2.07/1.13 => ! [A5: A,F2: B > A,B5: B,C2: B] :
% 2.07/1.13 ( ( A5
% 2.07/1.13 = ( F2 @ B5 ) )
% 2.07/1.13 => ( ( ord_less_eq @ B @ B5 @ C2 )
% 2.07/1.13 => ( ! [X: B,Y2: B] :
% 2.07/1.13 ( ( ord_less_eq @ B @ X @ Y2 )
% 2.07/1.13 => ( ord_less_eq @ A @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
% 2.07/1.13 => ( ord_less_eq @ A @ A5 @ ( F2 @ C2 ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_214_ord__le__eq__subst,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType] :
% 2.07/1.13 ( ( ( ord @ B )
% 2.07/1.13 & ( ord @ A ) )
% 2.07/1.13 => ! [A5: A,B5: A,F2: A > B,C2: B] :
% 2.07/1.13 ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.07/1.13 => ( ( ( F2 @ B5 )
% 2.07/1.13 = C2 )
% 2.07/1.13 => ( ! [X: A,Y2: A] :
% 2.07/1.13 ( ( ord_less_eq @ A @ X @ Y2 )
% 2.07/1.13 => ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y2 ) ) )
% 2.07/1.13 => ( ord_less_eq @ B @ ( F2 @ A5 ) @ C2 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_215_eq__iff,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( order @ A )
% 2.07/1.13 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 2.07/1.13 = ( ^ [X3: A,Y: A] :
% 2.07/1.13 ( ( ord_less_eq @ A @ X3 @ Y )
% 2.07/1.13 & ( ord_less_eq @ A @ Y @ X3 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_216_antisym,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( order @ A )
% 2.07/1.13 => ! [X4: A,Y3: A] :
% 2.07/1.13 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.07/1.13 => ( ( ord_less_eq @ A @ Y3 @ X4 )
% 2.07/1.13 => ( X4 = Y3 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_217_linear,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linorder @ A )
% 2.07/1.13 => ! [X4: A,Y3: A] :
% 2.07/1.13 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.07/1.13 | ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_218_eq__refl,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( preorder @ A )
% 2.07/1.13 => ! [X4: A,Y3: A] :
% 2.07/1.13 ( ( X4 = Y3 )
% 2.07/1.13 => ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_219_le__cases,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linorder @ A )
% 2.07/1.13 => ! [X4: A,Y3: A] :
% 2.07/1.13 ( ~ ( ord_less_eq @ A @ X4 @ Y3 )
% 2.07/1.13 => ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_220_order_Otrans,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( order @ A )
% 2.07/1.13 => ! [A5: A,B5: A,C2: A] :
% 2.07/1.13 ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.07/1.13 => ( ( ord_less_eq @ A @ B5 @ C2 )
% 2.07/1.13 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_221_le__cases3,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linorder @ A )
% 2.07/1.13 => ! [X4: A,Y3: A,Z: A] :
% 2.07/1.13 ( ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.07/1.13 => ~ ( ord_less_eq @ A @ Y3 @ Z ) )
% 2.07/1.13 => ( ( ( ord_less_eq @ A @ Y3 @ X4 )
% 2.07/1.13 => ~ ( ord_less_eq @ A @ X4 @ Z ) )
% 2.07/1.13 => ( ( ( ord_less_eq @ A @ X4 @ Z )
% 2.07/1.13 => ~ ( ord_less_eq @ A @ Z @ Y3 ) )
% 2.07/1.13 => ( ( ( ord_less_eq @ A @ Z @ Y3 )
% 2.07/1.13 => ~ ( ord_less_eq @ A @ Y3 @ X4 ) )
% 2.07/1.13 => ( ( ( ord_less_eq @ A @ Y3 @ Z )
% 2.07/1.13 => ~ ( ord_less_eq @ A @ Z @ X4 ) )
% 2.07/1.13 => ~ ( ( ord_less_eq @ A @ Z @ X4 )
% 2.07/1.13 => ~ ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_222_antisym__conv,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( order @ A )
% 2.07/1.13 => ! [Y3: A,X4: A] :
% 2.07/1.13 ( ( ord_less_eq @ A @ Y3 @ X4 )
% 2.07/1.13 => ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.07/1.13 = ( X4 = Y3 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_223_order__class_Oorder_Oeq__iff,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( order @ A )
% 2.07/1.13 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 2.07/1.13 = ( ^ [A4: A,B6: A] :
% 2.07/1.13 ( ( ord_less_eq @ A @ A4 @ B6 )
% 2.07/1.13 & ( ord_less_eq @ A @ B6 @ A4 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_224_ord__eq__le__trans,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( ord @ A )
% 2.07/1.13 => ! [A5: A,B5: A,C2: A] :
% 2.07/1.13 ( ( A5 = B5 )
% 2.07/1.13 => ( ( ord_less_eq @ A @ B5 @ C2 )
% 2.07/1.13 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_225_ord__le__eq__trans,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( ord @ A )
% 2.07/1.13 => ! [A5: A,B5: A,C2: A] :
% 2.07/1.13 ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.07/1.13 => ( ( B5 = C2 )
% 2.07/1.13 => ( ord_less_eq @ A @ A5 @ C2 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_226_order__class_Oorder_Oantisym,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( order @ A )
% 2.07/1.13 => ! [A5: A,B5: A] :
% 2.07/1.13 ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.07/1.13 => ( ( ord_less_eq @ A @ B5 @ A5 )
% 2.07/1.13 => ( A5 = B5 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_227_order__trans,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( preorder @ A )
% 2.07/1.13 => ! [X4: A,Y3: A,Z: A] :
% 2.07/1.13 ( ( ord_less_eq @ A @ X4 @ Y3 )
% 2.07/1.13 => ( ( ord_less_eq @ A @ Y3 @ Z )
% 2.07/1.13 => ( ord_less_eq @ A @ X4 @ Z ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_228_dual__order_Orefl,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( order @ A )
% 2.07/1.13 => ! [A5: A] : ( ord_less_eq @ A @ A5 @ A5 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_229_linorder__wlog,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linorder @ A )
% 2.07/1.13 => ! [P: A > A > $o,A5: A,B5: A] :
% 2.07/1.13 ( ! [A6: A,B4: A] :
% 2.07/1.13 ( ( ord_less_eq @ A @ A6 @ B4 )
% 2.07/1.13 => ( P @ A6 @ B4 ) )
% 2.07/1.13 => ( ! [A6: A,B4: A] :
% 2.07/1.13 ( ( P @ B4 @ A6 )
% 2.07/1.13 => ( P @ A6 @ B4 ) )
% 2.07/1.13 => ( P @ A5 @ B5 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_230_dual__order_Otrans,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( order @ A )
% 2.07/1.13 => ! [B5: A,A5: A,C2: A] :
% 2.07/1.13 ( ( ord_less_eq @ A @ B5 @ A5 )
% 2.07/1.13 => ( ( ord_less_eq @ A @ C2 @ B5 )
% 2.07/1.13 => ( ord_less_eq @ A @ C2 @ A5 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_231_dual__order_Oeq__iff,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( order @ A )
% 2.07/1.13 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 2.07/1.13 = ( ^ [A4: A,B6: A] :
% 2.07/1.13 ( ( ord_less_eq @ A @ B6 @ A4 )
% 2.07/1.13 & ( ord_less_eq @ A @ A4 @ B6 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_232_dual__order_Oantisym,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( order @ A )
% 2.07/1.13 => ! [B5: A,A5: A] :
% 2.07/1.13 ( ( ord_less_eq @ A @ B5 @ A5 )
% 2.07/1.13 => ( ( ord_less_eq @ A @ A5 @ B5 )
% 2.07/1.13 => ( A5 = B5 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_233_predicate2D,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X4: A,Y3: B] :
% 2.07/1.13 ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 2.07/1.13 => ( ( P @ X4 @ Y3 )
% 2.07/1.13 => ( Q @ X4 @ Y3 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_234_rev__predicate2D,axiom,
% 2.07/1.13 ! [A: $tType,B: $tType,P: A > B > $o,X4: A,Y3: B,Q: A > B > $o] :
% 2.07/1.13 ( ( P @ X4 @ Y3 )
% 2.07/1.13 => ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
% 2.07/1.13 => ( Q @ X4 @ Y3 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_235_predicate1D,axiom,
% 2.07/1.13 ! [A: $tType,P: A > $o,Q: A > $o,X4: A] :
% 2.07/1.13 ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 2.07/1.13 => ( ( P @ X4 )
% 2.07/1.13 => ( Q @ X4 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_236_rev__predicate1D,axiom,
% 2.07/1.13 ! [A: $tType,P: A > $o,X4: A,Q: A > $o] :
% 2.07/1.13 ( ( P @ X4 )
% 2.07/1.13 => ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
% 2.07/1.13 => ( Q @ X4 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_237_le__funD,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType] :
% 2.07/1.13 ( ( ord @ B )
% 2.07/1.13 => ! [F2: A > B,G: A > B,X4: A] :
% 2.07/1.13 ( ( ord_less_eq @ ( A > B ) @ F2 @ G )
% 2.07/1.13 => ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_238_le__funE,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType] :
% 2.07/1.13 ( ( ord @ B )
% 2.07/1.13 => ! [F2: A > B,G: A > B,X4: A] :
% 2.07/1.13 ( ( ord_less_eq @ ( A > B ) @ F2 @ G )
% 2.07/1.13 => ( ord_less_eq @ B @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_239_le__funI,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType] :
% 2.07/1.13 ( ( ord @ B )
% 2.07/1.13 => ! [F2: A > B,G: A > B] :
% 2.07/1.13 ( ! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) )
% 2.07/1.13 => ( ord_less_eq @ ( A > B ) @ F2 @ G ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_240_le__fun__def,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType] :
% 2.07/1.13 ( ( ord @ B )
% 2.07/1.13 => ( ( ord_less_eq @ ( A > B ) )
% 2.07/1.13 = ( ^ [F4: A > B,G4: A > B] :
% 2.07/1.13 ! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_241_relChain__def,axiom,
% 2.07/1.13 ! [B: $tType,A: $tType] :
% 2.07/1.13 ( ( ord @ B )
% 2.07/1.13 => ( ( bNF_Ca1785829860lChain @ A @ B )
% 2.07/1.13 = ( ^ [R3: set @ ( product_prod @ A @ A ),As: A > B] :
% 2.07/1.13 ! [I: A,J: A] :
% 2.07/1.13 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R3 )
% 2.07/1.13 => ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_242_rtc__isPath,axiom,
% 2.07/1.13 ! [U: nat,V: nat] :
% 2.07/1.13 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( transitive_rtrancl @ nat @ ( e @ a @ c ) ) )
% 2.07/1.13 => ? [P3: list @ ( product_prod @ nat @ nat )] : ( isPath @ a @ c @ U @ P3 @ V ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_243_isSPath__pathLE,axiom,
% 2.07/1.13 ! [S: nat,P4: list @ ( product_prod @ nat @ nat ),T4: nat] :
% 2.07/1.13 ( ( isPath @ a @ c @ S @ P4 @ T4 )
% 2.07/1.13 => ? [P5: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ a @ c @ S @ P5 @ T4 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_244_connected__def,axiom,
% 2.07/1.13 ! [U: nat,V: nat] :
% 2.07/1.13 ( ( connected @ a @ c @ U @ V )
% 2.07/1.13 = ( ? [P2: list @ ( product_prod @ nat @ nat )] : ( isPath @ a @ c @ U @ P2 @ V ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_245_shortestPath__is__path,axiom,
% 2.07/1.13 ! [U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.07/1.13 ( ( isShortestPath @ a @ c @ U @ P4 @ V )
% 2.07/1.13 => ( isPath @ a @ c @ U @ P4 @ V ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_246_isPath__ex__edge1,axiom,
% 2.07/1.13 ! [U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
% 2.07/1.13 ( ( isPath @ a @ c @ U @ P4 @ V )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ( ( U1 != U )
% 2.07/1.13 => ? [U23: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U23 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_247_isPath__ex__edge2,axiom,
% 2.07/1.13 ! [U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
% 2.07/1.13 ( ( isPath @ a @ c @ U @ P4 @ V )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ( ( V1 != V )
% 2.07/1.13 => ? [V23: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V1 @ V23 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_248_isPath__edgeset,axiom,
% 2.07/1.13 ! [U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,E2: product_prod @ nat @ nat] :
% 2.07/1.13 ( ( isPath @ a @ c @ U @ P4 @ V )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ E2 @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ( member @ ( product_prod @ nat @ nat ) @ E2 @ ( e @ a @ c ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_249_isPath__rtc,axiom,
% 2.07/1.13 ! [U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.07/1.13 ( ( isPath @ a @ c @ U @ P4 @ V )
% 2.07/1.13 => ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( transitive_rtrancl @ nat @ ( e @ a @ c ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_250_Graph_OisPath__ex__edge2,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
% 2.07/1.13 ( ( isPath @ Capacity @ C2 @ U @ P4 @ V )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ( ( V1 != V )
% 2.07/1.13 => ? [V23: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V1 @ V23 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_251_Graph_OisPath__ex__edge1,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
% 2.07/1.13 ( ( isPath @ Capacity @ C2 @ U @ P4 @ V )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ( ( U1 != U )
% 2.07/1.13 => ? [U23: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U23 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_252_Graph_OisPath__edgeset,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat,E2: product_prod @ nat @ nat] :
% 2.07/1.13 ( ( isPath @ Capacity @ C2 @ U @ P4 @ V )
% 2.07/1.13 => ( ( member @ ( product_prod @ nat @ nat ) @ E2 @ ( set2 @ ( product_prod @ nat @ nat ) @ P4 ) )
% 2.07/1.13 => ( member @ ( product_prod @ nat @ nat ) @ E2 @ ( e @ Capacity @ C2 ) ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_253_Graph_Oconnected__def,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( connected @ Capacity )
% 2.07/1.13 = ( ^ [C3: ( product_prod @ nat @ nat ) > Capacity,U3: nat,V2: nat] :
% 2.07/1.13 ? [P2: list @ ( product_prod @ nat @ nat )] : ( isPath @ Capacity @ C3 @ U3 @ P2 @ V2 ) ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_254_Graph_OshortestPath__is__path,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ! [C2: ( product_prod @ nat @ nat ) > Capacity,U: nat,P4: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.07/1.13 ( ( isShortestPath @ Capacity @ C2 @ U @ P4 @ V )
% 2.07/1.13 => ( isPath @ Capacity @ C2 @ U @ P4 @ V ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(fact_255_Graph_OisPath_Ocong,axiom,
% 2.07/1.13 ! [Capacity: $tType] :
% 2.07/1.13 ( ( linordered_idom @ Capacity )
% 2.07/1.13 => ( ( isPath @ Capacity )
% 2.07/1.13 = ( isPath @ Capacity ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(subcl_Rings_Olinordered__idom___HOL_Otype,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linordered_idom @ A )
% 2.07/1.13 => ( type @ A ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(subcl_Rings_Olinordered__idom___Orderings_Oord,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linordered_idom @ A )
% 2.07/1.13 => ( ord @ A ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(subcl_Rings_Olinordered__idom___Orderings_Oorder,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linordered_idom @ A )
% 2.07/1.13 => ( order @ A ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(subcl_Rings_Olinordered__idom___Orderings_Olinorder,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linordered_idom @ A )
% 2.07/1.13 => ( linorder @ A ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(subcl_Rings_Olinordered__idom___Orderings_Opreorder,axiom,
% 2.07/1.13 ! [A: $tType] :
% 2.07/1.13 ( ( linordered_idom @ A )
% 2.07/1.13 => ( preorder @ A ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.07/1.13 ! [A9: $tType,A10: $tType] :
% 2.07/1.13 ( ( preorder @ A10 )
% 2.07/1.13 => ( preorder @ ( A9 > A10 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.07/1.13 ! [A9: $tType,A10: $tType] :
% 2.07/1.13 ( ( ( finite_finite @ A9 )
% 2.07/1.13 & ( finite_finite @ A10 ) )
% 2.07/1.13 => ( finite_finite @ ( A9 > A10 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.07/1.13 ! [A9: $tType,A10: $tType] :
% 2.07/1.13 ( ( order @ A10 )
% 2.07/1.13 => ( order @ ( A9 > A10 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_fun___Orderings_Oord,axiom,
% 2.07/1.13 ! [A9: $tType,A10: $tType] :
% 2.07/1.13 ( ( ord @ A10 )
% 2.07/1.13 => ( ord @ ( A9 > A10 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 2.07/1.13 preorder @ nat ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.07/1.13 linorder @ nat ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 2.07/1.13 order @ nat ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 2.07/1.13 ord @ nat ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 2.07/1.13 ! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
% 2.07/1.13 ! [A9: $tType] :
% 2.07/1.13 ( ( finite_finite @ A9 )
% 2.07/1.13 => ( finite_finite @ ( set @ A9 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 2.07/1.13 ! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
% 2.07/1.13 ! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
% 2.07/1.13 preorder @ $o ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_HOL_Obool___Orderings_Olinorder_9,axiom,
% 2.07/1.13 linorder @ $o ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_HOL_Obool___Finite__Set_Ofinite_10,axiom,
% 2.07/1.13 finite_finite @ $o ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
% 2.07/1.13 order @ $o ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
% 2.07/1.13 ord @ $o ).
% 2.07/1.13
% 2.07/1.13 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_13,axiom,
% 2.07/1.13 ! [A9: $tType,A10: $tType] :
% 2.07/1.13 ( ( ( finite_finite @ A9 )
% 2.07/1.13 & ( finite_finite @ A10 ) )
% 2.07/1.13 => ( finite_finite @ ( product_prod @ A9 @ A10 ) ) ) ).
% 2.07/1.13
% 2.07/1.13 thf(tfree_0,hypothesis,
% 2.07/1.13 linordered_idom @ a ).
% 2.07/1.13
% 2.07/1.13 thf(conj_0,conjecture,
% 2.07/1.13 finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) ).
% 2.07/1.13 ------------------------
% 2.07/1.15 % (23532)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_23290 for (2999ds/2Mi)
% 2.07/1.15 % (23531)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_23290 for (2999ds/2Mi)
% 2.07/1.15 % (23530)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_23290 for (2999ds/27Mi)
% 2.07/1.15 % (23528)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_23290 for (2999ds/183Mi)
% 2.07/1.15 % (23529)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_23290 for (2999ds/4Mi)
% 2.07/1.15 % (23533)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_23290 for (2999ds/275Mi)
% 2.07/1.15 % (23534)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_23290 for (2999ds/18Mi)
% 2.07/1.15 % (23531)Instruction limit reached!
% 2.07/1.15 % (23531)------------------------------
% 2.07/1.15 % (23531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.07/1.15 % (23531)Termination reason: Unknown
% 2.07/1.15 % (23531)Termination phase: shuffling
% 2.07/1.15
% 2.07/1.15 % (23531)Memory used [KB]: 1407
% 2.07/1.15 % (23531)Time elapsed: 0.002 s
% 2.07/1.15 % (23531)Instructions burned: 2 (million)
% 2.07/1.15 % (23531)------------------------------
% 2.07/1.15 % (23531)------------------------------
% 2.07/1.15 % (23532)Instruction limit reached!
% 2.07/1.15 % (23532)------------------------------
% 2.07/1.15 % (23532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.07/1.15 % (23532)Termination reason: Unknown
% 2.07/1.15 % (23532)Termination phase: shuffling
% 2.07/1.15
% 2.07/1.15 % (23532)Memory used [KB]: 1407
% 2.07/1.15 % (23532)Time elapsed: 0.002 s
% 2.07/1.15 % (23532)Instructions burned: 2 (million)
% 2.07/1.15 % (23532)------------------------------
% 2.07/1.15 % (23532)------------------------------
% 2.07/1.16 % (23529)Instruction limit reached!
% 2.07/1.16 % (23529)------------------------------
% 2.07/1.16 % (23529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.07/1.16 % (23529)Termination reason: Unknown
% 2.07/1.16 % (23529)Termination phase: shuffling
% 2.07/1.16
% 2.07/1.16 % (23529)Memory used [KB]: 1407
% 2.07/1.16 % (23529)Time elapsed: 0.003 s
% 2.07/1.16 % (23529)Instructions burned: 5 (million)
% 2.07/1.16 % (23529)------------------------------
% 2.07/1.16 % (23529)------------------------------
% 2.07/1.16 % (23534)Instruction limit reached!
% 2.07/1.16 % (23534)------------------------------
% 2.07/1.16 % (23534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.07/1.16 % (23534)Termination reason: Unknown
% 2.07/1.16 % (23534)Termination phase: shuffling
% 2.07/1.16
% 2.07/1.16 % (23534)Memory used [KB]: 1791
% 2.07/1.16 % (23534)Time elapsed: 0.008 s
% 2.07/1.16 % (23534)Instructions burned: 20 (million)
% 2.07/1.16 % (23534)------------------------------
% 2.07/1.16 % (23534)------------------------------
% 2.07/1.16 % (23530)Instruction limit reached!
% 2.07/1.16 % (23530)------------------------------
% 2.07/1.16 % (23530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.07/1.16 % (23530)Termination reason: Unknown
% 2.07/1.16 % (23530)Termination phase: shuffling
% 2.07/1.16
% 2.07/1.16 % (23530)Memory used [KB]: 1918
% 2.07/1.16 % (23530)Time elapsed: 0.011 s
% 2.07/1.16 % (23530)Instructions burned: 28 (million)
% 2.07/1.16 % (23530)------------------------------
% 2.07/1.16 % (23530)------------------------------
% 2.07/1.16 % (23535)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_23290 for (2999ds/3Mi)
% 2.07/1.16 % (23536)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_23290 for (2999ds/37Mi)
% 2.07/1.17 % (23537)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_23290 for (2999ds/15Mi)
% 2.07/1.17 % (23535)Instruction limit reached!
% 2.07/1.17 % (23535)------------------------------
% 2.07/1.17 % (23535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.07/1.17 % (23535)Termination reason: Unknown
% 2.07/1.17 % (23535)Termination phase: shuffling
% 2.07/1.17
% 2.07/1.17 % (23535)Memory used [KB]: 1407
% 2.07/1.17 % (23535)Time elapsed: 0.002 s
% 2.07/1.17 % (23535)Instructions burned: 3 (million)
% 2.07/1.17 % (23535)------------------------------
% 2.07/1.17 % (23535)------------------------------
% 2.07/1.17 % (23537)Instruction limit reached!
% 2.07/1.17 % (23537)------------------------------
% 2.07/1.17 % (23537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.07/1.17 % (23537)Termination reason: Unknown
% 2.07/1.17 % (23537)Termination phase: shuffling
% 2.07/1.17
% 2.07/1.17 % (23537)Memory used [KB]: 1663
% 2.07/1.17 % (23537)Time elapsed: 0.006 s
% 2.07/1.17 % (23537)Instructions burned: 15 (million)
% 2.07/1.17 % (23537)------------------------------
% 2.07/1.17 % (23537)------------------------------
% 2.07/1.17 % (23538)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_23290 for (2999ds/3Mi)
% 2.07/1.17 % (23538)Instruction limit reached!
% 2.07/1.17 % (23538)------------------------------
% 2.07/1.17 % (23538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.07/1.17 % (23538)Termination reason: Unknown
% 2.07/1.17 % (23538)Termination phase: shuffling
% 2.07/1.17
% 2.07/1.17 % (23538)Memory used [KB]: 1407
% 2.07/1.17 % (23538)Time elapsed: 0.002 s
% 2.07/1.17 % (23538)Instructions burned: 3 (million)
% 2.07/1.17 % (23538)------------------------------
% 2.07/1.17 % (23538)------------------------------
% 2.07/1.17 % (23539)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_23290 for (2999ds/1041Mi)
% 2.07/1.17 % (23540)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_23290 for (2999ds/7Mi)
% 2.07/1.18 % (23536)Instruction limit reached!
% 2.07/1.18 % (23536)------------------------------
% 2.07/1.18 % (23536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.07/1.18 % (23536)Termination reason: Unknown
% 2.07/1.18 % (23536)Termination phase: shuffling
% 2.07/1.18
% 2.07/1.18 % (23536)Memory used [KB]: 2174
% 2.07/1.18 % (23536)Time elapsed: 0.012 s
% 2.07/1.18 % (23536)Instructions burned: 39 (million)
% 2.07/1.18 % (23536)------------------------------
% 2.07/1.18 % (23536)------------------------------
% 2.07/1.18 % (23533)First to succeed.
% 2.07/1.18 % (23540)Instruction limit reached!
% 2.07/1.18 % (23540)------------------------------
% 2.07/1.18 % (23540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.07/1.18 % (23540)Termination reason: Unknown
% 2.07/1.18 % (23540)Termination phase: shuffling
% 2.07/1.18
% 2.07/1.18 % (23540)Memory used [KB]: 1407
% 2.07/1.18 % (23540)Time elapsed: 0.003 s
% 2.07/1.18 % (23540)Instructions burned: 7 (million)
% 2.07/1.18 % (23540)------------------------------
% 2.07/1.18 % (23540)------------------------------
% 2.07/1.18 % (23533)Refutation found. Thanks to Tanya!
% 2.07/1.18 % SZS status Theorem for DTF2THF_23290
% 2.07/1.18 % SZS output start Proof for DTF2THF_23290
% 2.07/1.18 thf(type_def_5, type, product_prod: ($tType * $tType) > $tType).
% 2.07/1.18 thf(type_def_6, type, set: ($tType) > $tType).
% 2.07/1.18 thf(type_def_8, type, nat: $tType).
% 2.07/1.18 thf(type_def_9, type, list: ($tType) > $tType).
% 2.07/1.18 thf(type_def_10, type, a: $tType).
% 2.07/1.18 thf(func_def_0, type, product_prod: ($tType * $tType) > $tType).
% 2.07/1.18 thf(func_def_1, type, list: ($tType) > $tType).
% 2.07/1.18 thf(func_def_2, type, set: ($tType) > $tType).
% 2.07/1.18 thf(func_def_3, type, nat: $tType).
% 2.07/1.18 thf(func_def_4, type, a: $tType).
% 2.07/1.18 thf(func_def_5, type, type: !>[X0: $tType]:($o)).
% 2.07/1.18 thf(func_def_6, type, ord: !>[X0: $tType]:($o)).
% 2.07/1.18 thf(func_def_7, type, order: !>[X0: $tType]:($o)).
% 2.07/1.18 thf(func_def_8, type, finite_finite: !>[X0: $tType]:($o)).
% 2.07/1.18 thf(func_def_9, type, linorder: !>[X0: $tType]:($o)).
% 2.07/1.18 thf(func_def_10, type, preorder: !>[X0: $tType]:($o)).
% 2.07/1.18 thf(func_def_11, type, linordered_idom: !>[X0: $tType]:($o)).
% 2.07/1.18 thf(func_def_12, type, bNF_Ca1785829860lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 2.07/1.18 thf(func_def_13, type, edmond2129202899alysis: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > $o)).
% 2.07/1.18 thf(func_def_14, type, edmond1803015688pEdges: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > set @ product_prod @ nat @ nat)).
% 2.07/1.18 thf(func_def_15, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 2.07/1.18 thf(func_def_16, type, finite_Graph: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > $o)).
% 2.07/1.18 thf(func_def_17, type, e: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > set @ product_prod @ nat @ nat)).
% 2.07/1.18 thf(func_def_18, type, v: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > set @ nat)).
% 2.07/1.18 thf(func_def_19, type, adjacent_nodes: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > set @ nat)).
% 2.07/1.18 thf(func_def_20, type, connected: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > $o)).
% 2.07/1.18 thf(func_def_21, type, incoming: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > set @ product_prod @ nat @ nat)).
% 2.07/1.18 thf(func_def_22, type, incoming2: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > set @ nat > set @ product_prod @ nat @ nat)).
% 2.07/1.18 thf(func_def_23, type, isPath: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > list @ product_prod @ nat @ nat > nat > $o)).
% 2.07/1.18 thf(func_def_24, type, isShortestPath: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > list @ product_prod @ nat @ nat > nat > $o)).
% 2.07/1.18 thf(func_def_25, type, isSimplePath: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > list @ product_prod @ nat @ nat > nat > $o)).
% 2.07/1.18 thf(func_def_26, type, outgoing: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > set @ product_prod @ nat @ nat)).
% 2.07/1.18 thf(func_def_27, type, outgoing2: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > set @ nat > set @ product_prod @ nat @ nat)).
% 2.07/1.18 thf(func_def_28, type, reachableNodes: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > set @ nat)).
% 2.07/1.18 thf(func_def_29, type, set2: !>[X0: $tType]:(list @ X0 > set @ X0)).
% 2.07/1.18 thf(func_def_30, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.07/1.18 thf(func_def_31, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.07/1.18 thf(func_def_32, type, product_Sigma: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > set @ X1) > set @ product_prod @ X0 @ X1)).
% 2.07/1.18 thf(func_def_33, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.07/1.18 thf(func_def_34, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 2.07/1.18 thf(func_def_35, type, domain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X0)).
% 2.07/1.18 thf(func_def_36, type, image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X0 > set @ X1)).
% 2.07/1.18 thf(func_def_37, type, range: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X1)).
% 2.07/1.18 thf(func_def_38, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.07/1.18 thf(func_def_39, type, transitive_rtrancl: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X0 @ X0)).
% 2.07/1.18 thf(func_def_40, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.07/1.18 thf(func_def_41, type, c: product_prod @ nat @ nat > a).
% 2.07/1.18 thf(func_def_42, type, s: nat).
% 2.07/1.18 thf(func_def_43, type, t: nat).
% 2.07/1.18 thf(func_def_60, type, sK0: !>[X0: $tType]:((X0 > $o) > X0)).
% 2.07/1.18 thf(func_def_61, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1 > X0)).
% 2.07/1.18 thf(func_def_62, type, sK2: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1)).
% 2.07/1.18 thf(func_def_63, type, sK3: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X1 > $o) > X1)).
% 2.07/1.18 thf(func_def_64, type, sK4: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0 > $o) > X1)).
% 2.07/1.18 thf(func_def_65, type, sK5: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > (X1 > X0 > $o) > X0)).
% 2.07/1.18 thf(func_def_66, type, sK6: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X1)).
% 2.07/1.18 thf(func_def_67, type, sK7: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X0)).
% 2.07/1.18 thf(func_def_68, type, sK8: !>[X0: $tType]:(nat > nat > (product_prod @ nat @ nat > X0) > list @ product_prod @ nat @ nat)).
% 2.07/1.18 thf(func_def_69, type, sK9: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X0 > set @ X1 > X1)).
% 2.07/1.18 thf(func_def_70, type, sK10: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X0 > set @ X1 > X0)).
% 2.07/1.18 thf(func_def_71, type, sK11: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.07/1.18 thf(func_def_72, type, sK12: nat > nat > list @ product_prod @ nat @ nat).
% 2.07/1.18 thf(func_def_73, type, sK13: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.07/1.18 thf(func_def_74, type, sK14: !>[X0: $tType]:(set @ X0 > list @ X0 > X0)).
% 2.07/1.18 thf(func_def_75, type, sK15: !>[X0: $tType]:((X0 > $o) > (X0 > $o) > X0)).
% 2.07/1.18 thf(func_def_76, type, sK16: !>[X0: $tType]:(set @ X0 > list @ X0)).
% 2.07/1.18 thf(func_def_78, type, ph18: !>[X0: $tType]:(X0)).
% 2.07/1.18 thf(f963,plain,(
% 2.07/1.18 $false),
% 2.07/1.18 inference(subsumption_resolution,[status(thm)],[f962,f909])).
% 2.07/1.18 thf(f909,plain,(
% 2.07/1.18 ((finite_finite2 @ product_prod @ nat @ nat @ (e @ a @ c)) = $true)),
% 2.07/1.18 inference(cnf_transformation,[status(thm)],[f435])).
% 2.07/1.18 thf(f435,plain,(
% 2.07/1.18 ((finite_finite2 @ product_prod @ nat @ nat @ (e @ a @ c)) = $true)),
% 2.07/1.18 inference(fool_elimination,[status(thm)],[f434])).
% 2.07/1.18 thf(f434,plain,(
% 2.07/1.18 (finite_finite2 @ product_prod @ nat @ nat @ (e @ a @ c))),
% 2.07/1.18 inference(rectify,[status(thm)],[f4])).
% 2.07/1.18 thf(f4,axiom,(
% 2.07/1.18 (finite_finite2 @ product_prod @ nat @ nat @ (e @ a @ c))),
% 2.07/1.18 file('/export/starexec/sandbox2/tmp/tmp.vhIah8fDUX/DTF2THF_23290.p',fact_3_finite__E)).
% 2.07/1.18 thf(f962,plain,(
% 2.07/1.18 ((finite_finite2 @ product_prod @ nat @ nat @ (e @ a @ c)) != $true)),
% 2.07/1.18 inference(trivial_inequality_removal,[status(thm)],[f961])).
% 2.07/1.18 thf(f961,plain,(
% 2.07/1.18 ($true != $true) | ((finite_finite2 @ product_prod @ nat @ nat @ (e @ a @ c)) != $true)),
% 2.07/1.18 inference(superposition,[status(thm)],[f922,f904])).
% 2.07/1.18 thf(f904,plain,(
% 2.07/1.18 ((finite_finite2 @ product_prod @ nat @ nat @ (edmond1803015688pEdges @ a @ c @ s @ t)) = $true) | ((finite_finite2 @ product_prod @ nat @ nat @ (e @ a @ c)) != $true)),
% 2.07/1.18 inference(cnf_transformation,[status(thm)],[f852])).
% 2.07/1.18 thf(f852,plain,(
% 2.07/1.18 ((finite_finite2 @ product_prod @ nat @ nat @ (e @ a @ c)) != $true) | ((finite_finite2 @ product_prod @ nat @ nat @ (edmond1803015688pEdges @ a @ c @ s @ t)) = $true)),
% 2.07/1.18 inference(ennf_transformation,[status(thm)],[f583])).
% 2.07/1.18 thf(f583,plain,(
% 2.07/1.18 ((finite_finite2 @ product_prod @ nat @ nat @ (e @ a @ c)) = $true) => ((finite_finite2 @ product_prod @ nat @ nat @ (edmond1803015688pEdges @ a @ c @ s @ t)) = $true)),
% 2.07/1.18 inference(fool_elimination,[status(thm)],[f582])).
% 2.07/1.18 thf(f582,plain,(
% 2.07/1.18 (finite_finite2 @ product_prod @ nat @ nat @ (e @ a @ c)) => (finite_finite2 @ product_prod @ nat @ nat @ (edmond1803015688pEdges @ a @ c @ s @ t))),
% 2.07/1.18 inference(rectify,[status(thm)],[f2])).
% 2.07/1.18 thf(f2,axiom,(
% 2.07/1.18 (finite_finite2 @ product_prod @ nat @ nat @ (e @ a @ c)) => (finite_finite2 @ product_prod @ nat @ nat @ (edmond1803015688pEdges @ a @ c @ s @ t))),
% 2.07/1.18 file('/export/starexec/sandbox2/tmp/tmp.vhIah8fDUX/DTF2THF_23290.p',fact_1__092_060open_062finite_AE_A_092_060Longrightarrow_062_Afinite_AspEdges_092_060close_062)).
% 2.07/1.18 thf(f922,plain,(
% 2.07/1.18 ((finite_finite2 @ product_prod @ nat @ nat @ (edmond1803015688pEdges @ a @ c @ s @ t)) != $true)),
% 2.07/1.18 inference(cnf_transformation,[status(thm)],[f812])).
% 2.07/1.18 thf(f812,plain,(
% 2.07/1.18 ((finite_finite2 @ product_prod @ nat @ nat @ (edmond1803015688pEdges @ a @ c @ s @ t)) != $true)),
% 2.07/1.18 inference(flattening,[status(thm)],[f387])).
% 2.07/1.18 thf(f387,plain,(
% 2.07/1.18 ~((finite_finite2 @ product_prod @ nat @ nat @ (edmond1803015688pEdges @ a @ c @ s @ t)) = $true)),
% 2.07/1.18 inference(fool_elimination,[status(thm)],[f386])).
% 2.07/1.18 thf(f386,plain,(
% 2.07/1.18 ~(finite_finite2 @ product_prod @ nat @ nat @ (edmond1803015688pEdges @ a @ c @ s @ t))),
% 2.07/1.18 inference(rectify,[status(thm)],[f282])).
% 2.07/1.18 thf(f282,negated_conjecture,(
% 2.07/1.18 ~(finite_finite2 @ product_prod @ nat @ nat @ (edmond1803015688pEdges @ a @ c @ s @ t))),
% 2.07/1.18 inference(negated_conjecture,[status(cth)],[f281])).
% 2.07/1.18 thf(f281,conjecture,(
% 2.07/1.18 (finite_finite2 @ product_prod @ nat @ nat @ (edmond1803015688pEdges @ a @ c @ s @ t))),
% 2.07/1.18 file('/export/starexec/sandbox2/tmp/tmp.vhIah8fDUX/DTF2THF_23290.p',conj_0)).
% 2.07/1.18 % SZS output end Proof for DTF2THF_23290
% 2.07/1.18 % (23533)------------------------------
% 2.07/1.18 % (23533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.07/1.18 % (23533)Termination reason: Refutation
% 2.07/1.18
% 2.07/1.18 % (23533)Memory used [KB]: 6652
% 2.07/1.18 % (23533)Time elapsed: 0.023 s
% 2.07/1.18 % (23533)Instructions burned: 65 (million)
% 2.07/1.18 % (23533)------------------------------
% 2.07/1.18 % (23533)------------------------------
% 2.07/1.18 % (23527)Success in time 0.039 s
%------------------------------------------------------------------------------