%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP051^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:09 AM UTC 2026
% Result : ContradictoryAxioms 2.94s 1.98s
% Output : Refutation 2.94s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP051^2 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.17/0.34 % Computer : n003.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Tue Feb 24 08:24:25 EST 2026
% 0.17/0.34 % CPUTime :
% 0.17/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.31/0.46 ---- Original DTF file ---
% 0.31/0.47 thf(spec,logic,$$dhol).
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 % File : ITP051^2 : TPTP v9.2.1. Released v7.5.0.
% 0.31/0.47 % Domain : Interactive Theorem Proving
% 0.31/0.47 % Problem : Sledgehammer EdmondsKarp_Termination_Abstract problem prob_318__7590952_1
% 0.31/0.47 % Version : Especial.
% 0.31/0.47 % English :
% 0.31/0.47
% 0.31/0.47 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.31/0.47 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.31/0.47 % Source : [Des21]
% 0.31/0.47 % Names : EdmondsKarp_Termination_Abstract/prob_318__7590952_1 [Des21]
% 0.31/0.47
% 0.31/0.47 % Status : ContradictoryAxioms
% 0.31/0.47 % Rating : 0.00 v8.1.0, 0.25 v7.5.0
% 0.31/0.47 % Syntax : Number of formulae : 321 ( 100 unt; 47 typ; 0 def)
% 0.31/0.47 % Number of atoms : 687 ( 144 equ; 0 cnn)
% 0.31/0.47 % Maximal formula atoms : 8 ( 2 avg)
% 0.31/0.47 % Number of connectives : 3279 ( 51 ~; 4 |; 36 &;2868 @)
% 0.31/0.47 % ( 0 <=>; 320 =>; 0 <=; 0 <~>)
% 0.31/0.47 % Maximal formula depth : 19 ( 8 avg)
% 0.31/0.47 % Number of types : 4 ( 3 usr)
% 0.31/0.47 % Number of type conns : 155 ( 155 >; 0 *; 0 +; 0 <<)
% 0.31/0.47 % Number of symbols : 47 ( 44 usr; 7 con; 0-5 aty)
% 0.31/0.47 % Number of variables : 661 ( 33 ^; 563 !; 31 ?; 661 :)
% 0.31/0.47 % ( 34 !>; 0 ?*; 0 @-; 0 @+)
% 0.31/0.47 % SPC : TH1_THM_EQU_NAR
% 0.31/0.47
% 0.31/0.47 % Comments : This file was generated by Sledgehammer 2021-02-23 16:16:42.937
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 % Could-be-implicit typings (6)
% 0.31/0.47 thf(ty_t_Product__Type_Oprod,type,
% 0.31/0.47 product_prod: $tType > $tType > $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_t_List_Olist,type,
% 0.31/0.47 list: $tType > $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_t_Set_Oset,type,
% 0.31/0.47 set: $tType > $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_t_Nat_Onat,type,
% 0.31/0.47 nat: $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_tf_b,type,
% 0.31/0.47 b: $tType ).
% 0.31/0.47
% 0.31/0.47 thf(ty_tf_a,type,
% 0.31/0.47 a: $tType ).
% 0.31/0.47
% 0.31/0.47 % Explicit typings (41)
% 0.31/0.47 thf(sy_cl_HOL_Otype,type,
% 0.31/0.47 type:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Nat_Osize,type,
% 0.31/0.47 size:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Ozero,type,
% 0.31/0.47 zero:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Orderings_Oorder,type,
% 0.31/0.47 order:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.31/0.47 finite_finite:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Orderings_Owellorder,type,
% 0.31/0.47 wellorder:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.31/0.47 linordered_idom:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.31/0.47 canoni770627133id_add:
% 0.31/0.47 !>[A: $tType] : $o ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis,type,
% 0.31/0.47 edmond2129202899alysis:
% 0.31/0.47 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis__defs_OekMeasure,type,
% 0.31/0.47 edmond115586663easure:
% 0.31/0.47 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > nat > nat > nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis__defs_OspEdges,type,
% 0.31/0.47 edmond1803015688pEdges:
% 0.31/0.47 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > nat > nat > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis__defs_OuE,type,
% 0.31/0.47 edmond259086305sis_uE:
% 0.31/0.47 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Finite__Set_Ocard,type,
% 0.31/0.47 finite_card:
% 0.31/0.47 !>[B: $tType] : ( ( set @ B ) > nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Finite__Set_Ofinite,type,
% 0.31/0.47 finite_finite2:
% 0.31/0.47 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Graph_OFinite__Graph,type,
% 0.31/0.47 finite_Graph:
% 0.31/0.47 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Graph_OGraph_OE,type,
% 0.31/0.47 e:
% 0.31/0.47 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Graph_OGraph_OV,type,
% 0.31/0.47 v:
% 0.31/0.47 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Graph_OGraph_Oadjacent__nodes,type,
% 0.31/0.47 adjacent_nodes:
% 0.31/0.47 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ nat ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Graph_OGraph_Oconnected,type,
% 0.31/0.47 connected:
% 0.31/0.47 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Graph_OGraph_Odist,type,
% 0.31/0.47 dist:
% 0.31/0.47 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > nat > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Graph_OGraph_Oincoming_H,type,
% 0.31/0.47 incoming:
% 0.31/0.47 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Graph_OGraph_OisPath,type,
% 0.31/0.47 isPath:
% 0.31/0.47 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Graph_OGraph_OisShortestPath,type,
% 0.31/0.47 isShortestPath:
% 0.31/0.47 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Graph_OGraph_OisSimplePath,type,
% 0.31/0.47 isSimplePath:
% 0.31/0.47 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Graph_OGraph_Omin__dist,type,
% 0.31/0.47 min_dist:
% 0.31/0.47 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Graph_OGraph_Ooutgoing_H,type,
% 0.31/0.47 outgoing:
% 0.31/0.47 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Graph_OGraph_OreachableNodes,type,
% 0.31/0.47 reachableNodes:
% 0.31/0.47 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ nat ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.31/0.47 zero_zero:
% 0.31/0.47 !>[A: $tType] : A ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_List_Olist_Oset,type,
% 0.31/0.47 set2:
% 0.31/0.47 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.31/0.47 size_size:
% 0.31/0.47 !>[A: $tType] : ( A > nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.31/0.47 ord_less:
% 0.31/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.31/0.47 ord_less_eq:
% 0.31/0.47 !>[A: $tType] : ( A > A > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_Set_OCollect,type,
% 0.31/0.47 collect:
% 0.31/0.47 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_c_member,type,
% 0.31/0.47 member:
% 0.31/0.47 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_c,type,
% 0.31/0.47 c: ( product_prod @ nat @ nat ) > a ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_c_H,type,
% 0.31/0.47 c2: ( product_prod @ nat @ nat ) > b ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_edges,type,
% 0.31/0.47 edges: set @ ( product_prod @ nat @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_p,type,
% 0.31/0.47 p: list @ ( product_prod @ nat @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_p_H____,type,
% 0.31/0.47 p2: list @ ( product_prod @ nat @ nat ) ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_s,type,
% 0.31/0.47 s: nat ).
% 0.31/0.47
% 0.31/0.47 thf(sy_v_t,type,
% 0.31/0.47 t: nat ).
% 0.31/0.47
% 0.31/0.47 % Relevant facts (254)
% 0.31/0.47 thf(fact_0_ek__analysis__axioms,axiom,
% 0.31/0.47 edmond2129202899alysis @ a @ c ).
% 0.31/0.47
% 0.31/0.47 % ek_analysis_axioms
% 0.31/0.47 thf(fact_1_LENP,axiom,
% 0.31/0.47 ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p )
% 0.31/0.47 = ( min_dist @ a @ c @ s @ t ) ) ).
% 0.31/0.47
% 0.31/0.47 % LENP
% 0.31/0.47 thf(fact_2_LENP_H,axiom,
% 0.31/0.47 ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 )
% 0.31/0.47 = ( min_dist @ b @ c2 @ s @ t ) ) ).
% 0.31/0.47
% 0.31/0.47 % LENP'
% 0.31/0.47 thf(fact_3_ek__analysis__defs_OekMeasure_Ocong,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ( ( edmond115586663easure @ A )
% 0.31/0.47 = ( edmond115586663easure @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ek_analysis_defs.ekMeasure.cong
% 0.31/0.47 thf(fact_4_g_H_Oek__analysis__axioms,axiom,
% 0.31/0.47 edmond2129202899alysis @ b @ c2 ).
% 0.31/0.47
% 0.31/0.47 % g'.ek_analysis_axioms
% 0.31/0.47 thf(fact_5_SHORTER,axiom,
% 0.31/0.47 ord_less @ nat @ ( min_dist @ b @ c2 @ s @ t ) @ ( min_dist @ a @ c @ s @ t ) ).
% 0.31/0.47
% 0.31/0.47 % SHORTER
% 0.31/0.47 thf(fact_6_ek__analysis__defs_OuE_Ocong,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ( ( edmond259086305sis_uE @ A )
% 0.31/0.47 = ( edmond259086305sis_uE @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ek_analysis_defs.uE.cong
% 0.31/0.47 thf(fact_7_ek__analysis__defs_OspEdges_Ocong,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ( ( edmond1803015688pEdges @ A )
% 0.31/0.47 = ( edmond1803015688pEdges @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ek_analysis_defs.spEdges.cong
% 0.31/0.47 thf(fact_8_g_H_OFinite__Graph__axioms,axiom,
% 0.31/0.47 finite_Graph @ b @ c2 ).
% 0.31/0.47
% 0.31/0.47 % g'.Finite_Graph_axioms
% 0.31/0.47 thf(fact_9_Finite__Graph__axioms,axiom,
% 0.31/0.47 finite_Graph @ a @ c ).
% 0.31/0.47
% 0.31/0.47 % Finite_Graph_axioms
% 0.31/0.47 thf(fact_10__092_060open_062length_Ap_A_060_Alength_Ap_H_092_060close_062,axiom,
% 0.31/0.47 ord_less @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>length p < length p'\<close>
% 0.31/0.47 thf(fact_11_uE__eq,axiom,
% 0.31/0.47 ( ( edmond259086305sis_uE @ b @ c2 )
% 0.31/0.47 = ( edmond259086305sis_uE @ a @ c ) ) ).
% 0.31/0.47
% 0.31/0.47 % uE_eq
% 0.31/0.47 thf(fact_12_SP,axiom,
% 0.31/0.47 isShortestPath @ a @ c @ s @ p @ t ).
% 0.31/0.47
% 0.31/0.47 % SP
% 0.31/0.47 thf(fact_13_CONN2,axiom,
% 0.31/0.47 connected @ b @ c2 @ s @ t ).
% 0.31/0.47
% 0.31/0.47 % CONN2
% 0.31/0.47 thf(fact_14_SV,axiom,
% 0.31/0.47 member @ nat @ s @ ( v @ a @ c ) ).
% 0.31/0.47
% 0.31/0.47 % SV
% 0.31/0.47 thf(fact_15_P_H,axiom,
% 0.31/0.47 isPath @ b @ c2 @ s @ p2 @ t ).
% 0.31/0.47
% 0.31/0.47 % P'
% 0.31/0.47 thf(fact_16_ek__analysis__def,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ( ( edmond2129202899alysis @ A )
% 0.31/0.47 = ( finite_Graph @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ek_analysis_def
% 0.31/0.47 thf(fact_17_ek__analysis_Ointro,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > A] :
% 0.31/0.47 ( ( finite_Graph @ A @ C )
% 0.31/0.47 => ( edmond2129202899alysis @ A @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ek_analysis.intro
% 0.31/0.47 thf(fact_18_ek__analysis_Oaxioms,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > A] :
% 0.31/0.47 ( ( edmond2129202899alysis @ A @ C )
% 0.31/0.47 => ( finite_Graph @ A @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ek_analysis.axioms
% 0.31/0.47 thf(fact_19__092_060open_062_092_060not_062_Ag_H_Oconnected_As_At_A_092_060Longrightarrow_062_Ag_H_OekMeasure_A_060_AekMeasure_092_060close_062,axiom,
% 0.31/0.47 ( ~ ( connected @ b @ c2 @ s @ t )
% 0.31/0.47 => ( ord_less @ nat @ ( edmond115586663easure @ b @ c2 @ s @ t ) @ ( edmond115586663easure @ a @ c @ s @ t ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>\<not> g'.connected s t \<Longrightarrow> g'.ekMeasure < ekMeasure\<close>
% 0.31/0.47 thf(fact_20__092_060open_062g_H_Odist_As_A_Ig_H_Omin__dist_As_At_J_At_092_060close_062,axiom,
% 0.31/0.47 dist @ b @ c2 @ s @ ( min_dist @ b @ c2 @ s @ t ) @ t ).
% 0.31/0.47
% 0.31/0.47 % \<open>g'.dist s (g'.min_dist s t) t\<close>
% 0.31/0.47 thf(fact_21__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062p_H_O_A_092_060lbrakk_062g_H_OisPath_As_Ap_H_At_059_Alength_Ap_H_A_061_Ag_H_Omin__dist_As_At_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.31/0.47 ~ ! [P: list @ ( product_prod @ nat @ nat )] :
% 0.31/0.47 ( ( isPath @ b @ c2 @ s @ P @ t )
% 0.31/0.47 => ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P )
% 0.31/0.47 != ( min_dist @ b @ c2 @ s @ t ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>\<And>thesis. (\<And>p'. \<lbrakk>g'.isPath s p' t; length p' = g'.min_dist s t\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.31/0.47 thf(fact_22_CONN,axiom,
% 0.31/0.47 connected @ a @ c @ s @ t ).
% 0.31/0.47
% 0.31/0.47 % CONN
% 0.31/0.47 thf(fact_23_min__dist__less,axiom,
% 0.31/0.47 ! [Src: nat,V: nat,D: nat,D2: nat] :
% 0.31/0.47 ( ( connected @ a @ c @ Src @ V )
% 0.31/0.47 => ( ( ( min_dist @ a @ c @ Src @ V )
% 0.31/0.47 = D )
% 0.31/0.47 => ( ( ord_less @ nat @ D2 @ D )
% 0.31/0.47 => ? [V2: nat] :
% 0.31/0.47 ( ( connected @ a @ c @ Src @ V2 )
% 0.31/0.47 & ( ( min_dist @ a @ c @ Src @ V2 )
% 0.31/0.47 = D2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % min_dist_less
% 0.31/0.47 thf(fact_24_g_H_Omin__dist__less,axiom,
% 0.31/0.47 ! [Src: nat,V: nat,D: nat,D2: nat] :
% 0.31/0.47 ( ( connected @ b @ c2 @ Src @ V )
% 0.31/0.47 => ( ( ( min_dist @ b @ c2 @ Src @ V )
% 0.31/0.47 = D )
% 0.31/0.47 => ( ( ord_less @ nat @ D2 @ D )
% 0.31/0.47 => ? [V2: nat] :
% 0.31/0.47 ( ( connected @ b @ c2 @ Src @ V2 )
% 0.31/0.47 & ( ( min_dist @ b @ c2 @ Src @ V2 )
% 0.31/0.47 = D2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.min_dist_less
% 0.31/0.47 thf(fact_25_length__induct,axiom,
% 0.31/0.47 ! [A: $tType,P2: ( list @ A ) > $o,Xs: list @ A] :
% 0.31/0.47 ( ! [Xs2: list @ A] :
% 0.31/0.47 ( ! [Ys: list @ A] :
% 0.31/0.47 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 0.31/0.47 => ( P2 @ Ys ) )
% 0.31/0.47 => ( P2 @ Xs2 ) )
% 0.31/0.47 => ( P2 @ Xs ) ) ).
% 0.31/0.47
% 0.31/0.47 % length_induct
% 0.31/0.47 thf(fact_26_finite__spEdges,axiom,
% 0.31/0.47 finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) ).
% 0.31/0.47
% 0.31/0.47 % finite_spEdges
% 0.31/0.47 thf(fact_27_g_H_Ofinite__spEdges,axiom,
% 0.31/0.47 finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ b @ c2 @ s @ t ) ).
% 0.31/0.47
% 0.31/0.47 % g'.finite_spEdges
% 0.31/0.47 thf(fact_28_min__dist__z,axiom,
% 0.31/0.47 ! [V: nat] :
% 0.31/0.47 ( ( min_dist @ a @ c @ V @ V )
% 0.31/0.47 = ( zero_zero @ nat ) ) ).
% 0.31/0.47
% 0.31/0.47 % min_dist_z
% 0.31/0.47 thf(fact_29_g_H_Omin__dist__z,axiom,
% 0.31/0.47 ! [V: nat] :
% 0.31/0.47 ( ( min_dist @ b @ c2 @ V @ V )
% 0.31/0.47 = ( zero_zero @ nat ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.min_dist_z
% 0.31/0.47 thf(fact_30_g_H_Oconnected__def,axiom,
% 0.31/0.47 ! [U: nat,V: nat] :
% 0.31/0.47 ( ( connected @ b @ c2 @ U @ V )
% 0.31/0.47 = ( ? [P3: list @ ( product_prod @ nat @ nat )] : ( isPath @ b @ c2 @ U @ P3 @ V ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.connected_def
% 0.31/0.47 thf(fact_31_connected__inV__iff,axiom,
% 0.31/0.47 ! [U: nat,V: nat] :
% 0.31/0.47 ( ( connected @ a @ c @ U @ V )
% 0.31/0.47 => ( ( member @ nat @ V @ ( v @ a @ c ) )
% 0.31/0.47 = ( member @ nat @ U @ ( v @ a @ c ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % connected_inV_iff
% 0.31/0.47 thf(fact_32_g_H_Oconnected__by__dist,axiom,
% 0.31/0.47 ! [V: nat,V3: nat] :
% 0.31/0.47 ( ( connected @ b @ c2 @ V @ V3 )
% 0.31/0.47 = ( ? [D3: nat] : ( dist @ b @ c2 @ V @ D3 @ V3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.connected_by_dist
% 0.31/0.47 thf(fact_33_obtain__shortest__path,axiom,
% 0.31/0.47 ! [U: nat,V: nat] :
% 0.31/0.47 ( ( connected @ a @ c @ U @ V )
% 0.31/0.47 => ~ ! [P4: list @ ( product_prod @ nat @ nat )] :
% 0.31/0.47 ~ ( isShortestPath @ a @ c @ U @ P4 @ V ) ) ).
% 0.31/0.47
% 0.31/0.47 % obtain_shortest_path
% 0.31/0.47 thf(fact_34_g_H_OisPath__distD,axiom,
% 0.31/0.47 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.31/0.47 ( ( isPath @ b @ c2 @ U @ P5 @ V )
% 0.31/0.47 => ( dist @ b @ c2 @ U @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ V ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.isPath_distD
% 0.31/0.47 thf(fact_35_g_H_Odist__def,axiom,
% 0.31/0.47 ! [V: nat,D: nat,V3: nat] :
% 0.31/0.47 ( ( dist @ b @ c2 @ V @ D @ V3 )
% 0.31/0.47 = ( ? [P3: list @ ( product_prod @ nat @ nat )] :
% 0.31/0.47 ( ( isPath @ b @ c2 @ V @ P3 @ V3 )
% 0.31/0.47 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P3 )
% 0.31/0.47 = D ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.dist_def
% 0.31/0.47 thf(fact_36_g_H_Omin__dist__is__dist,axiom,
% 0.31/0.47 ! [V: nat,V3: nat] :
% 0.31/0.47 ( ( connected @ b @ c2 @ V @ V3 )
% 0.31/0.47 => ( dist @ b @ c2 @ V @ ( min_dist @ b @ c2 @ V @ V3 ) @ V3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.min_dist_is_dist
% 0.31/0.47 thf(fact_37_g_H_Oconnected__refl,axiom,
% 0.31/0.47 ! [V: nat] : ( connected @ b @ c2 @ V @ V ) ).
% 0.31/0.47
% 0.31/0.47 % g'.connected_refl
% 0.31/0.47 thf(fact_38_connected__refl,axiom,
% 0.31/0.47 ! [V: nat] : ( connected @ a @ c @ V @ V ) ).
% 0.31/0.47
% 0.31/0.47 % connected_refl
% 0.31/0.47 thf(fact_39_g_H_Odist__z__iff,axiom,
% 0.31/0.47 ! [V: nat,V3: nat] :
% 0.31/0.47 ( ( dist @ b @ c2 @ V @ ( zero_zero @ nat ) @ V3 )
% 0.31/0.47 = ( V3 = V ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.dist_z_iff
% 0.31/0.47 thf(fact_40_g_H_Odist__z,axiom,
% 0.31/0.47 ! [V: nat] : ( dist @ b @ c2 @ V @ ( zero_zero @ nat ) @ V ) ).
% 0.31/0.47
% 0.31/0.47 % g'.dist_z
% 0.31/0.47 thf(fact_41_g_H_Oconnected__distI,axiom,
% 0.31/0.47 ! [V: nat,D: nat,V3: nat] :
% 0.31/0.47 ( ( dist @ b @ c2 @ V @ D @ V3 )
% 0.31/0.47 => ( connected @ b @ c2 @ V @ V3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.connected_distI
% 0.31/0.47 thf(fact_42_g_H_Ofinite__uE,axiom,
% 0.31/0.47 finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond259086305sis_uE @ b @ c2 ) ).
% 0.31/0.47
% 0.31/0.47 % g'.finite_uE
% 0.31/0.47 thf(fact_43_finite__uE,axiom,
% 0.31/0.47 finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond259086305sis_uE @ a @ c ) ).
% 0.31/0.47
% 0.31/0.47 % finite_uE
% 0.31/0.47 thf(fact_44_g_H_Omin__dist__z__iff,axiom,
% 0.31/0.47 ! [V: nat,V3: nat] :
% 0.31/0.47 ( ( connected @ b @ c2 @ V @ V3 )
% 0.31/0.47 => ( ( ( min_dist @ b @ c2 @ V @ V3 )
% 0.31/0.47 = ( zero_zero @ nat ) )
% 0.31/0.47 = ( V3 = V ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.min_dist_z_iff
% 0.31/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.31/0.47 ! [A: $tType,A2: A,P2: A > $o] :
% 0.31/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 0.31/0.47 = ( P2 @ A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % mem_Collect_eq
% 0.31/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A] :
% 0.31/0.47 ( ( collect @ A
% 0.31/0.47 @ ^ [X: A] : ( member @ A @ X @ A3 ) )
% 0.31/0.47 = A3 ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_mem_eq
% 0.31/0.47 thf(fact_47_Collect__cong,axiom,
% 0.31/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ( P2 @ X2 )
% 0.31/0.47 = ( Q @ X2 ) )
% 0.31/0.47 => ( ( collect @ A @ P2 )
% 0.31/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_cong
% 0.31/0.47 thf(fact_48_ext,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ( F @ X2 )
% 0.31/0.47 = ( G @ X2 ) )
% 0.31/0.47 => ( F = G ) ) ).
% 0.31/0.47
% 0.31/0.47 % ext
% 0.31/0.47 thf(fact_49_min__dist__z__iff,axiom,
% 0.31/0.47 ! [V: nat,V3: nat] :
% 0.31/0.47 ( ( connected @ a @ c @ V @ V3 )
% 0.31/0.47 => ( ( ( min_dist @ a @ c @ V @ V3 )
% 0.31/0.47 = ( zero_zero @ nat ) )
% 0.31/0.47 = ( V3 = V ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % min_dist_z_iff
% 0.31/0.47 thf(fact_50_ek__analysis_Ofinite__uE,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > A] :
% 0.31/0.47 ( ( edmond2129202899alysis @ A @ C )
% 0.31/0.47 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond259086305sis_uE @ A @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ek_analysis.finite_uE
% 0.31/0.47 thf(fact_51_ek__analysis_Ofinite__spEdges,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat] :
% 0.31/0.47 ( ( edmond2129202899alysis @ A @ C )
% 0.31/0.47 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ A @ C @ S @ T ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ek_analysis.finite_spEdges
% 0.31/0.47 thf(fact_52_neq__if__length__neq,axiom,
% 0.31/0.47 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.31/0.47 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.31/0.47 != ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.31/0.47 => ( Xs != Ys2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % neq_if_length_neq
% 0.31/0.47 thf(fact_53_Ex__list__of__length,axiom,
% 0.31/0.47 ! [A: $tType,N: nat] :
% 0.31/0.47 ? [Xs2: list @ A] :
% 0.31/0.47 ( ( size_size @ ( list @ A ) @ Xs2 )
% 0.31/0.47 = N ) ).
% 0.31/0.47
% 0.31/0.47 % Ex_list_of_length
% 0.31/0.47 thf(fact_54_bot__nat__0_Onot__eq__extremum,axiom,
% 0.31/0.47 ! [A2: nat] :
% 0.31/0.47 ( ( A2
% 0.31/0.47 != ( zero_zero @ nat ) )
% 0.31/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % bot_nat_0.not_eq_extremum
% 0.31/0.47 thf(fact_55_less__nat__zero__code,axiom,
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_nat_zero_code
% 0.31/0.47 thf(fact_56_neq0__conv,axiom,
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ( ( N
% 0.31/0.47 != ( zero_zero @ nat ) )
% 0.31/0.47 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % neq0_conv
% 0.31/0.47 thf(fact_57_not__gr__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [N: A] :
% 0.31/0.47 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.31/0.47 = ( N
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_gr_zero
% 0.31/0.47 thf(fact_58_Graph_OisShortestPath__min__dist__def,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( isShortestPath @ Capacity )
% 0.31/0.47 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U2: nat,P3: list @ ( product_prod @ nat @ nat ),V4: nat] :
% 0.31/0.47 ( ( isPath @ Capacity @ C2 @ U2 @ P3 @ V4 )
% 0.31/0.47 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P3 )
% 0.31/0.47 = ( min_dist @ Capacity @ C2 @ U2 @ V4 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.isShortestPath_min_dist_def
% 0.31/0.47 thf(fact_59_g_H_OisShortestPath__min__dist__def,axiom,
% 0.31/0.47 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.31/0.47 ( ( isShortestPath @ b @ c2 @ U @ P5 @ V )
% 0.31/0.47 = ( ( isPath @ b @ c2 @ U @ P5 @ V )
% 0.31/0.47 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 )
% 0.31/0.47 = ( min_dist @ b @ c2 @ U @ V ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.isShortestPath_min_dist_def
% 0.31/0.47 thf(fact_60_g_H_Omin__distI2,axiom,
% 0.31/0.47 ! [V: nat,V3: nat,Q: nat > $o] :
% 0.31/0.47 ( ( connected @ b @ c2 @ V @ V3 )
% 0.31/0.47 => ( ! [D4: nat] :
% 0.31/0.47 ( ( dist @ b @ c2 @ V @ D4 @ V3 )
% 0.31/0.47 => ( ! [D5: nat] :
% 0.31/0.47 ( ( dist @ b @ c2 @ V @ D5 @ V3 )
% 0.31/0.47 => ( ord_less_eq @ nat @ D4 @ D5 ) )
% 0.31/0.47 => ( Q @ D4 ) ) )
% 0.31/0.47 => ( Q @ ( min_dist @ b @ c2 @ V @ V3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.min_distI2
% 0.31/0.47 thf(fact_61_isShortestPath__length__less__V,axiom,
% 0.31/0.47 ! [S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 0.31/0.47 ( ( member @ nat @ S @ ( v @ a @ c ) )
% 0.31/0.47 => ( ( isShortestPath @ a @ c @ S @ P5 @ T )
% 0.31/0.47 => ( ord_less @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ ( finite_card @ nat @ ( v @ a @ c ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % isShortestPath_length_less_V
% 0.31/0.47 thf(fact_62_min__dist__less__V,axiom,
% 0.31/0.47 ! [S: nat,T: nat] :
% 0.31/0.47 ( ( member @ nat @ S @ ( v @ a @ c ) )
% 0.31/0.47 => ( ( connected @ a @ c @ S @ T )
% 0.31/0.47 => ( ord_less @ nat @ ( min_dist @ a @ c @ S @ T ) @ ( finite_card @ nat @ ( v @ a @ c ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % min_dist_less_V
% 0.31/0.47 thf(fact_63_connected__def,axiom,
% 0.31/0.47 ! [U: nat,V: nat] :
% 0.31/0.47 ( ( connected @ a @ c @ U @ V )
% 0.31/0.47 = ( ? [P3: list @ ( product_prod @ nat @ nat )] : ( isPath @ a @ c @ U @ P3 @ V ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % connected_def
% 0.31/0.47 thf(fact_64_shortestPath__is__path,axiom,
% 0.31/0.47 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.31/0.47 ( ( isShortestPath @ a @ c @ U @ P5 @ V )
% 0.31/0.47 => ( isPath @ a @ c @ U @ P5 @ V ) ) ).
% 0.31/0.47
% 0.31/0.47 % shortestPath_is_path
% 0.31/0.47 thf(fact_65_g_H_OshortestPath__is__path,axiom,
% 0.31/0.47 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.31/0.47 ( ( isShortestPath @ b @ c2 @ U @ P5 @ V )
% 0.31/0.47 => ( isPath @ b @ c2 @ U @ P5 @ V ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.shortestPath_is_path
% 0.31/0.47 thf(fact_66_g_H_Oobtain__shortest__path,axiom,
% 0.31/0.47 ! [U: nat,V: nat] :
% 0.31/0.47 ( ( connected @ b @ c2 @ U @ V )
% 0.31/0.47 => ~ ! [P4: list @ ( product_prod @ nat @ nat )] :
% 0.31/0.47 ~ ( isShortestPath @ b @ c2 @ U @ P4 @ V ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.obtain_shortest_path
% 0.31/0.47 thf(fact_67_g_H_Omin__dist__le,axiom,
% 0.31/0.47 ! [Src: nat,V: nat,D2: nat] :
% 0.31/0.47 ( ( connected @ b @ c2 @ Src @ V )
% 0.31/0.47 => ( ( ord_less_eq @ nat @ D2 @ ( min_dist @ b @ c2 @ Src @ V ) )
% 0.31/0.47 => ? [V2: nat] :
% 0.31/0.47 ( ( connected @ b @ c2 @ Src @ V2 )
% 0.31/0.47 & ( ( min_dist @ b @ c2 @ Src @ V2 )
% 0.31/0.47 = D2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.min_dist_le
% 0.31/0.47 thf(fact_68_g_H_Omin__distI__eq,axiom,
% 0.31/0.47 ! [V: nat,D: nat,V3: nat] :
% 0.31/0.47 ( ( dist @ b @ c2 @ V @ D @ V3 )
% 0.31/0.47 => ( ! [D6: nat] :
% 0.31/0.47 ( ( dist @ b @ c2 @ V @ D6 @ V3 )
% 0.31/0.47 => ( ord_less_eq @ nat @ D @ D6 ) )
% 0.31/0.47 => ( ( min_dist @ b @ c2 @ V @ V3 )
% 0.31/0.47 = D ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.min_distI_eq
% 0.31/0.47 thf(fact_69_g_H_Omin__dist__minD,axiom,
% 0.31/0.47 ! [V: nat,D: nat,V3: nat] :
% 0.31/0.47 ( ( dist @ b @ c2 @ V @ D @ V3 )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( min_dist @ b @ c2 @ V @ V3 ) @ D ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.min_dist_minD
% 0.31/0.47 thf(fact_70_min__dist__le,axiom,
% 0.31/0.47 ! [Src: nat,V: nat,D2: nat] :
% 0.31/0.47 ( ( connected @ a @ c @ Src @ V )
% 0.31/0.47 => ( ( ord_less_eq @ nat @ D2 @ ( min_dist @ a @ c @ Src @ V ) )
% 0.31/0.47 => ? [V2: nat] :
% 0.31/0.47 ( ( connected @ a @ c @ Src @ V2 )
% 0.31/0.47 & ( ( min_dist @ a @ c @ Src @ V2 )
% 0.31/0.47 = D2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % min_dist_le
% 0.31/0.47 thf(fact_71_isShortestPath__def,axiom,
% 0.31/0.47 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.31/0.47 ( ( isShortestPath @ a @ c @ U @ P5 @ V )
% 0.31/0.47 = ( ( isPath @ a @ c @ U @ P5 @ V )
% 0.31/0.47 & ! [P6: list @ ( product_prod @ nat @ nat )] :
% 0.31/0.47 ( ( isPath @ a @ c @ U @ P6 @ V )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P6 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % isShortestPath_def
% 0.31/0.47 thf(fact_72_isShortestPath__min__dist__def,axiom,
% 0.31/0.47 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.31/0.47 ( ( isShortestPath @ a @ c @ U @ P5 @ V )
% 0.31/0.47 = ( ( isPath @ a @ c @ U @ P5 @ V )
% 0.31/0.47 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 )
% 0.31/0.47 = ( min_dist @ a @ c @ U @ V ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % isShortestPath_min_dist_def
% 0.31/0.47 thf(fact_73_g_H_OisShortestPath__def,axiom,
% 0.31/0.47 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.31/0.47 ( ( isShortestPath @ b @ c2 @ U @ P5 @ V )
% 0.31/0.47 = ( ( isPath @ b @ c2 @ U @ P5 @ V )
% 0.31/0.47 & ! [P6: list @ ( product_prod @ nat @ nat )] :
% 0.31/0.47 ( ( isPath @ b @ c2 @ U @ P6 @ V )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P6 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.isShortestPath_def
% 0.31/0.47 thf(fact_74_le__zero__eq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [N: A] :
% 0.31/0.47 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.31/0.47 = ( N
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_zero_eq
% 0.31/0.47 thf(fact_75_le0,axiom,
% 0.31/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.31/0.47
% 0.31/0.47 % le0
% 0.31/0.47 thf(fact_76_bot__nat__0_Oextremum,axiom,
% 0.31/0.47 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 0.31/0.47
% 0.31/0.47 % bot_nat_0.extremum
% 0.31/0.47 thf(fact_77_zero__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [X3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_le
% 0.31/0.47 thf(fact_78_le__refl,axiom,
% 0.31/0.47 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.31/0.47
% 0.31/0.47 % le_refl
% 0.31/0.47 thf(fact_79_le__trans,axiom,
% 0.31/0.47 ! [I: nat,J: nat,K: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ I @ J )
% 0.31/0.47 => ( ( ord_less_eq @ nat @ J @ K )
% 0.31/0.47 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_trans
% 0.31/0.47 thf(fact_80_eq__imp__le,axiom,
% 0.31/0.47 ! [M: nat,N: nat] :
% 0.31/0.47 ( ( M = N )
% 0.31/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % eq_imp_le
% 0.31/0.47 thf(fact_81_le__antisym,axiom,
% 0.31/0.47 ! [M: nat,N: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.31/0.47 => ( ( ord_less_eq @ nat @ N @ M )
% 0.31/0.47 => ( M = N ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_antisym
% 0.31/0.47 thf(fact_82_nat__le__linear,axiom,
% 0.31/0.47 ! [M: nat,N: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.31/0.47 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat_le_linear
% 0.31/0.47 thf(fact_83_Nat_Oex__has__greatest__nat,axiom,
% 0.31/0.47 ! [P2: nat > $o,K: nat,B2: nat] :
% 0.31/0.47 ( ( P2 @ K )
% 0.31/0.47 => ( ! [Y: nat] :
% 0.31/0.47 ( ( P2 @ Y )
% 0.31/0.47 => ( ord_less_eq @ nat @ Y @ B2 ) )
% 0.31/0.47 => ? [X2: nat] :
% 0.31/0.47 ( ( P2 @ X2 )
% 0.31/0.47 & ! [Y2: nat] :
% 0.31/0.47 ( ( P2 @ Y2 )
% 0.31/0.47 => ( ord_less_eq @ nat @ Y2 @ X2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Nat.ex_has_greatest_nat
% 0.31/0.47 thf(fact_84_less__eq__nat_Osimps_I1_J,axiom,
% 0.31/0.47 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 0.31/0.47
% 0.31/0.47 % less_eq_nat.simps(1)
% 0.31/0.47 thf(fact_85_le__0__eq,axiom,
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 0.31/0.47 = ( N
% 0.31/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_0_eq
% 0.31/0.47 thf(fact_86_bot__nat__0_Oextremum__unique,axiom,
% 0.31/0.47 ! [A2: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.31/0.47 = ( A2
% 0.31/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % bot_nat_0.extremum_unique
% 0.31/0.47 thf(fact_87_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.31/0.47 ! [A2: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 0.31/0.47 => ( A2
% 0.31/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % bot_nat_0.extremum_uniqueI
% 0.31/0.47 thf(fact_88_nat__less__le,axiom,
% 0.31/0.47 ( ( ord_less @ nat )
% 0.31/0.47 = ( ^ [M2: nat,N2: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.31/0.47 & ( M2 != N2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat_less_le
% 0.31/0.47 thf(fact_89_less__imp__le__nat,axiom,
% 0.31/0.47 ! [M: nat,N: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ M @ N )
% 0.31/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_imp_le_nat
% 0.31/0.47 thf(fact_90_le__eq__less__or__eq,axiom,
% 0.31/0.47 ( ( ord_less_eq @ nat )
% 0.31/0.47 = ( ^ [M2: nat,N2: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ M2 @ N2 )
% 0.31/0.47 | ( M2 = N2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_eq_less_or_eq
% 0.31/0.47 thf(fact_91_less__or__eq__imp__le,axiom,
% 0.31/0.47 ! [M: nat,N: nat] :
% 0.31/0.47 ( ( ( ord_less @ nat @ M @ N )
% 0.31/0.47 | ( M = N ) )
% 0.31/0.47 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_or_eq_imp_le
% 0.31/0.47 thf(fact_92_le__neq__implies__less,axiom,
% 0.31/0.47 ! [M: nat,N: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ M @ N )
% 0.31/0.47 => ( ( M != N )
% 0.31/0.47 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % le_neq_implies_less
% 0.31/0.47 thf(fact_93_less__mono__imp__le__mono,axiom,
% 0.31/0.47 ! [F: nat > nat,I: nat,J: nat] :
% 0.31/0.47 ( ! [I2: nat,J2: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ I2 @ J2 )
% 0.31/0.47 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.31/0.47 => ( ( ord_less_eq @ nat @ I @ J )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_mono_imp_le_mono
% 0.31/0.47 thf(fact_94_ex__least__nat__le,axiom,
% 0.31/0.47 ! [P2: nat > $o,N: nat] :
% 0.31/0.47 ( ( P2 @ N )
% 0.31/0.47 => ( ~ ( P2 @ ( zero_zero @ nat ) )
% 0.31/0.47 => ? [K2: nat] :
% 0.31/0.47 ( ( ord_less_eq @ nat @ K2 @ N )
% 0.31/0.47 & ! [I3: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ I3 @ K2 )
% 0.31/0.47 => ~ ( P2 @ I3 ) )
% 0.31/0.47 & ( P2 @ K2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ex_least_nat_le
% 0.31/0.47 thf(fact_95_Graph_Omin__dist__le,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,Src: nat,V: nat,D2: nat] :
% 0.31/0.47 ( ( connected @ Capacity @ C @ Src @ V )
% 0.31/0.47 => ( ( ord_less_eq @ nat @ D2 @ ( min_dist @ Capacity @ C @ Src @ V ) )
% 0.31/0.47 => ? [V2: nat] :
% 0.31/0.47 ( ( connected @ Capacity @ C @ Src @ V2 )
% 0.31/0.47 & ( ( min_dist @ Capacity @ C @ Src @ V2 )
% 0.31/0.47 = D2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.min_dist_le
% 0.31/0.47 thf(fact_96_Graph_Omin__dist__minD,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,D: nat,V3: nat] :
% 0.31/0.47 ( ( dist @ Capacity @ C @ V @ D @ V3 )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( min_dist @ Capacity @ C @ V @ V3 ) @ D ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.min_dist_minD
% 0.31/0.47 thf(fact_97_Graph_Omin__distI__eq,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,D: nat,V3: nat] :
% 0.31/0.47 ( ( dist @ Capacity @ C @ V @ D @ V3 )
% 0.31/0.47 => ( ! [D6: nat] :
% 0.31/0.47 ( ( dist @ Capacity @ C @ V @ D6 @ V3 )
% 0.31/0.47 => ( ord_less_eq @ nat @ D @ D6 ) )
% 0.31/0.47 => ( ( min_dist @ Capacity @ C @ V @ V3 )
% 0.31/0.47 = D ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.min_distI_eq
% 0.31/0.47 thf(fact_98_Graph_Omin__distI2,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,V3: nat,Q: nat > $o] :
% 0.31/0.47 ( ( connected @ Capacity @ C @ V @ V3 )
% 0.31/0.47 => ( ! [D4: nat] :
% 0.31/0.47 ( ( dist @ Capacity @ C @ V @ D4 @ V3 )
% 0.31/0.47 => ( ! [D5: nat] :
% 0.31/0.47 ( ( dist @ Capacity @ C @ V @ D5 @ V3 )
% 0.31/0.47 => ( ord_less_eq @ nat @ D4 @ D5 ) )
% 0.31/0.47 => ( Q @ D4 ) ) )
% 0.31/0.47 => ( Q @ ( min_dist @ Capacity @ C @ V @ V3 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.min_distI2
% 0.31/0.47 thf(fact_99_Graph_OisShortestPath__def,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( isShortestPath @ Capacity )
% 0.31/0.47 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U2: nat,P3: list @ ( product_prod @ nat @ nat ),V4: nat] :
% 0.31/0.47 ( ( isPath @ Capacity @ C2 @ U2 @ P3 @ V4 )
% 0.31/0.47 & ! [P6: list @ ( product_prod @ nat @ nat )] :
% 0.31/0.47 ( ( isPath @ Capacity @ C2 @ U2 @ P6 @ V4 )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P3 ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P6 ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.isShortestPath_def
% 0.31/0.47 thf(fact_100_zero__reorient,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( zero @ A )
% 0.31/0.47 => ! [X3: A] :
% 0.31/0.47 ( ( ( zero_zero @ A )
% 0.31/0.47 = X3 )
% 0.31/0.47 = ( X3
% 0.31/0.47 = ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_reorient
% 0.31/0.47 thf(fact_101_measure__induct,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( wellorder @ B )
% 0.31/0.47 => ! [F: A > B,P2: A > $o,A2: A] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ! [Y2: A] :
% 0.31/0.47 ( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X2 ) )
% 0.31/0.47 => ( P2 @ Y2 ) )
% 0.31/0.47 => ( P2 @ X2 ) )
% 0.31/0.47 => ( P2 @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % measure_induct
% 0.31/0.47 thf(fact_102_measure__induct__rule,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType] :
% 0.31/0.47 ( ( wellorder @ B )
% 0.31/0.47 => ! [F: A > B,P2: A > $o,A2: A] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ! [Y2: A] :
% 0.31/0.47 ( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X2 ) )
% 0.31/0.47 => ( P2 @ Y2 ) )
% 0.31/0.47 => ( P2 @ X2 ) )
% 0.31/0.47 => ( P2 @ A2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % measure_induct_rule
% 0.31/0.47 thf(fact_103_nat__neq__iff,axiom,
% 0.31/0.47 ! [M: nat,N: nat] :
% 0.31/0.47 ( ( M != N )
% 0.31/0.47 = ( ( ord_less @ nat @ M @ N )
% 0.31/0.47 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat_neq_iff
% 0.31/0.47 thf(fact_104_less__not__refl,axiom,
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ~ ( ord_less @ nat @ N @ N ) ).
% 0.31/0.47
% 0.31/0.47 % less_not_refl
% 0.31/0.47 thf(fact_105_less__not__refl2,axiom,
% 0.31/0.47 ! [N: nat,M: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ N @ M )
% 0.31/0.47 => ( M != N ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_not_refl2
% 0.31/0.47 thf(fact_106_less__not__refl3,axiom,
% 0.31/0.47 ! [S: nat,T: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ S @ T )
% 0.31/0.47 => ( S != T ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_not_refl3
% 0.31/0.47 thf(fact_107_less__irrefl__nat,axiom,
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ~ ( ord_less @ nat @ N @ N ) ).
% 0.31/0.47
% 0.31/0.47 % less_irrefl_nat
% 0.31/0.47 thf(fact_108_nat__less__induct,axiom,
% 0.31/0.47 ! [P2: nat > $o,N: nat] :
% 0.31/0.47 ( ! [N3: nat] :
% 0.31/0.47 ( ! [M3: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ M3 @ N3 )
% 0.31/0.47 => ( P2 @ M3 ) )
% 0.31/0.47 => ( P2 @ N3 ) )
% 0.31/0.47 => ( P2 @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % nat_less_induct
% 0.31/0.47 thf(fact_109_infinite__descent,axiom,
% 0.31/0.47 ! [P2: nat > $o,N: nat] :
% 0.31/0.47 ( ! [N3: nat] :
% 0.31/0.47 ( ~ ( P2 @ N3 )
% 0.31/0.47 => ? [M3: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ M3 @ N3 )
% 0.31/0.47 & ~ ( P2 @ M3 ) ) )
% 0.31/0.47 => ( P2 @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % infinite_descent
% 0.31/0.47 thf(fact_110_linorder__neqE__nat,axiom,
% 0.31/0.47 ! [X3: nat,Y3: nat] :
% 0.31/0.47 ( ( X3 != Y3 )
% 0.31/0.47 => ( ~ ( ord_less @ nat @ X3 @ Y3 )
% 0.31/0.47 => ( ord_less @ nat @ Y3 @ X3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % linorder_neqE_nat
% 0.31/0.47 thf(fact_111_infinite__descent__measure,axiom,
% 0.31/0.47 ! [A: $tType,P2: A > $o,V5: A > nat,X3: A] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ~ ( P2 @ X2 )
% 0.31/0.47 => ? [Y2: A] :
% 0.31/0.47 ( ( ord_less @ nat @ ( V5 @ Y2 ) @ ( V5 @ X2 ) )
% 0.31/0.47 & ~ ( P2 @ Y2 ) ) )
% 0.31/0.47 => ( P2 @ X3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % infinite_descent_measure
% 0.31/0.47 thf(fact_112_size__neq__size__imp__neq,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( size @ A )
% 0.31/0.47 => ! [X3: A,Y3: A] :
% 0.31/0.47 ( ( ( size_size @ A @ X3 )
% 0.31/0.47 != ( size_size @ A @ Y3 ) )
% 0.31/0.47 => ( X3 != Y3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % size_neq_size_imp_neq
% 0.31/0.47 thf(fact_113_Graph_OisPath_Ocong,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( isPath @ Capacity )
% 0.31/0.47 = ( isPath @ Capacity ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.isPath.cong
% 0.31/0.47 thf(fact_114_Graph_OV_Ocong,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( v @ Capacity )
% 0.31/0.47 = ( v @ Capacity ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.V.cong
% 0.31/0.47 thf(fact_115_Graph_Omin__dist_Ocong,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( min_dist @ Capacity )
% 0.31/0.47 = ( min_dist @ Capacity ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.min_dist.cong
% 0.31/0.47 thf(fact_116_Graph_Oconnected_Ocong,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( connected @ Capacity )
% 0.31/0.47 = ( connected @ Capacity ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.connected.cong
% 0.31/0.47 thf(fact_117_Graph_Oconnected__refl,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat] : ( connected @ Capacity @ C @ V @ V ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.connected_refl
% 0.31/0.47 thf(fact_118_Graph_Odist_Ocong,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( dist @ Capacity )
% 0.31/0.47 = ( dist @ Capacity ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.dist.cong
% 0.31/0.47 thf(fact_119_Graph_OisShortestPath_Ocong,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( isShortestPath @ Capacity )
% 0.31/0.47 = ( isShortestPath @ Capacity ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.isShortestPath.cong
% 0.31/0.47 thf(fact_120_Finite__Graph_Omin__dist__less__V,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat] :
% 0.31/0.47 ( ( finite_Graph @ A @ C )
% 0.31/0.47 => ( ( member @ nat @ S @ ( v @ A @ C ) )
% 0.31/0.47 => ( ( connected @ A @ C @ S @ T )
% 0.31/0.47 => ( ord_less @ nat @ ( min_dist @ A @ C @ S @ T ) @ ( finite_card @ nat @ ( v @ A @ C ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Finite_Graph.min_dist_less_V
% 0.31/0.47 thf(fact_121_Finite__Graph_OisShortestPath__length__less__V,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 0.31/0.47 ( ( finite_Graph @ A @ C )
% 0.31/0.47 => ( ( member @ nat @ S @ ( v @ A @ C ) )
% 0.31/0.47 => ( ( isShortestPath @ A @ C @ S @ P5 @ T )
% 0.31/0.47 => ( ord_less @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ ( finite_card @ nat @ ( v @ A @ C ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Finite_Graph.isShortestPath_length_less_V
% 0.31/0.47 thf(fact_122_gr__zeroI,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [N: A] :
% 0.31/0.47 ( ( N
% 0.31/0.47 != ( zero_zero @ A ) )
% 0.31/0.47 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % gr_zeroI
% 0.31/0.47 thf(fact_123_not__less__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [N: A] :
% 0.31/0.47 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_less_zero
% 0.31/0.47 thf(fact_124_gr__implies__not__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [M: A,N: A] :
% 0.31/0.47 ( ( ord_less @ A @ M @ N )
% 0.31/0.47 => ( N
% 0.31/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % gr_implies_not_zero
% 0.31/0.47 thf(fact_125_zero__less__iff__neq__zero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( canoni770627133id_add @ A )
% 0.31/0.47 => ! [N: A] :
% 0.31/0.47 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.31/0.47 = ( N
% 0.31/0.47 != ( zero_zero @ A ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % zero_less_iff_neq_zero
% 0.31/0.47 thf(fact_126_gr0I,axiom,
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ( ( N
% 0.31/0.47 != ( zero_zero @ nat ) )
% 0.31/0.47 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 0.31/0.47
% 0.31/0.47 % gr0I
% 0.31/0.47 thf(fact_127_not__gr0,axiom,
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.31/0.47 = ( N
% 0.31/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_gr0
% 0.31/0.47 thf(fact_128_not__less0,axiom,
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.31/0.47
% 0.31/0.47 % not_less0
% 0.31/0.47 thf(fact_129_less__zeroE,axiom,
% 0.31/0.47 ! [N: nat] :
% 0.31/0.47 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 0.31/0.47
% 0.31/0.47 % less_zeroE
% 0.31/0.47 thf(fact_130_gr__implies__not0,axiom,
% 0.31/0.47 ! [M: nat,N: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ M @ N )
% 0.31/0.47 => ( N
% 0.31/0.47 != ( zero_zero @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % gr_implies_not0
% 0.31/0.47 thf(fact_131_infinite__descent0,axiom,
% 0.31/0.47 ! [P2: nat > $o,N: nat] :
% 0.31/0.47 ( ( P2 @ ( zero_zero @ nat ) )
% 0.31/0.47 => ( ! [N3: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 0.31/0.47 => ( ~ ( P2 @ N3 )
% 0.31/0.47 => ? [M3: nat] :
% 0.31/0.47 ( ( ord_less @ nat @ M3 @ N3 )
% 0.31/0.47 & ~ ( P2 @ M3 ) ) ) )
% 0.31/0.47 => ( P2 @ N ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % infinite_descent0
% 0.31/0.47 thf(fact_132_bot__nat__0_Oextremum__strict,axiom,
% 0.31/0.47 ! [A2: nat] :
% 0.31/0.47 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 0.31/0.47
% 0.31/0.47 % bot_nat_0.extremum_strict
% 0.31/0.47 thf(fact_133_infinite__descent0__measure,axiom,
% 0.31/0.47 ! [A: $tType,V5: A > nat,P2: A > $o,X3: A] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ( ( V5 @ X2 )
% 0.31/0.47 = ( zero_zero @ nat ) )
% 0.31/0.47 => ( P2 @ X2 ) )
% 0.31/0.47 => ( ! [X2: A] :
% 0.31/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V5 @ X2 ) )
% 0.31/0.47 => ( ~ ( P2 @ X2 )
% 0.31/0.47 => ? [Y2: A] :
% 0.31/0.47 ( ( ord_less @ nat @ ( V5 @ Y2 ) @ ( V5 @ X2 ) )
% 0.31/0.47 & ~ ( P2 @ Y2 ) ) ) )
% 0.31/0.47 => ( P2 @ X3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % infinite_descent0_measure
% 0.31/0.47 thf(fact_134_Graph_Omin__dist__z,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat] :
% 0.31/0.47 ( ( min_dist @ Capacity @ C @ V @ V )
% 0.31/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.min_dist_z
% 0.31/0.47 thf(fact_135_Graph_Odist__z,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat] : ( dist @ Capacity @ C @ V @ ( zero_zero @ nat ) @ V ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.dist_z
% 0.31/0.47 thf(fact_136_Graph_Odist__z__iff,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,V3: nat] :
% 0.31/0.47 ( ( dist @ Capacity @ C @ V @ ( zero_zero @ nat ) @ V3 )
% 0.31/0.47 = ( V3 = V ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.dist_z_iff
% 0.31/0.47 thf(fact_137_Graph_Oconnected__def,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( connected @ Capacity )
% 0.31/0.47 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U2: nat,V4: nat] :
% 0.31/0.47 ? [P3: list @ ( product_prod @ nat @ nat )] : ( isPath @ Capacity @ C2 @ U2 @ P3 @ V4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.connected_def
% 0.31/0.47 thf(fact_138_Graph_Oconnected__inV__iff,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat] :
% 0.31/0.47 ( ( connected @ Capacity @ C @ U @ V )
% 0.31/0.47 => ( ( member @ nat @ V @ ( v @ Capacity @ C ) )
% 0.31/0.47 = ( member @ nat @ U @ ( v @ Capacity @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.connected_inV_iff
% 0.31/0.47 thf(fact_139_Graph_OshortestPath__is__path,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.31/0.47 ( ( isShortestPath @ Capacity @ C @ U @ P5 @ V )
% 0.31/0.47 => ( isPath @ Capacity @ C @ U @ P5 @ V ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.shortestPath_is_path
% 0.31/0.47 thf(fact_140_Graph_Oconnected__distI,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,D: nat,V3: nat] :
% 0.31/0.47 ( ( dist @ Capacity @ C @ V @ D @ V3 )
% 0.31/0.47 => ( connected @ Capacity @ C @ V @ V3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.connected_distI
% 0.31/0.47 thf(fact_141_Graph_Oconnected__by__dist,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( connected @ Capacity )
% 0.31/0.47 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,V4: nat,V6: nat] :
% 0.31/0.47 ? [D3: nat] : ( dist @ Capacity @ C2 @ V4 @ D3 @ V6 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.connected_by_dist
% 0.31/0.47 thf(fact_142_Graph_Oobtain__shortest__path,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat] :
% 0.31/0.47 ( ( connected @ Capacity @ C @ U @ V )
% 0.31/0.47 => ~ ! [P4: list @ ( product_prod @ nat @ nat )] :
% 0.31/0.47 ~ ( isShortestPath @ Capacity @ C @ U @ P4 @ V ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.obtain_shortest_path
% 0.31/0.47 thf(fact_143_Graph_Omin__dist__z__iff,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,V3: nat] :
% 0.31/0.47 ( ( connected @ Capacity @ C @ V @ V3 )
% 0.31/0.47 => ( ( ( min_dist @ Capacity @ C @ V @ V3 )
% 0.31/0.47 = ( zero_zero @ nat ) )
% 0.31/0.47 = ( V3 = V ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.min_dist_z_iff
% 0.31/0.47 thf(fact_144_Graph_Omin__dist__less,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,Src: nat,V: nat,D: nat,D2: nat] :
% 0.31/0.47 ( ( connected @ Capacity @ C @ Src @ V )
% 0.31/0.47 => ( ( ( min_dist @ Capacity @ C @ Src @ V )
% 0.31/0.47 = D )
% 0.31/0.47 => ( ( ord_less @ nat @ D2 @ D )
% 0.31/0.47 => ? [V2: nat] :
% 0.31/0.47 ( ( connected @ Capacity @ C @ Src @ V2 )
% 0.31/0.47 & ( ( min_dist @ Capacity @ C @ Src @ V2 )
% 0.31/0.47 = D2 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.min_dist_less
% 0.31/0.47 thf(fact_145_Graph_Omin__dist__is__dist,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,V3: nat] :
% 0.31/0.47 ( ( connected @ Capacity @ C @ V @ V3 )
% 0.31/0.47 => ( dist @ Capacity @ C @ V @ ( min_dist @ Capacity @ C @ V @ V3 ) @ V3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.min_dist_is_dist
% 0.31/0.47 thf(fact_146_Graph_OisPath__distD,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.31/0.47 ( ( isPath @ Capacity @ C @ U @ P5 @ V )
% 0.31/0.47 => ( dist @ Capacity @ C @ U @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ V ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.isPath_distD
% 0.31/0.47 thf(fact_147_Graph_Odist__def,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( dist @ Capacity )
% 0.31/0.47 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,V4: nat,D3: nat,V6: nat] :
% 0.31/0.47 ? [P3: list @ ( product_prod @ nat @ nat )] :
% 0.31/0.47 ( ( isPath @ Capacity @ C2 @ V4 @ P3 @ V6 )
% 0.31/0.47 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P3 )
% 0.31/0.47 = D3 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.dist_def
% 0.31/0.47 thf(fact_148_card_Oinfinite,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A] :
% 0.31/0.47 ( ~ ( finite_finite2 @ A @ A3 )
% 0.31/0.47 => ( ( finite_card @ A @ A3 )
% 0.31/0.47 = ( zero_zero @ nat ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % card.infinite
% 0.31/0.47 thf(fact_149_card__spEdges__le,axiom,
% 0.31/0.47 ord_less_eq @ nat @ ( finite_card @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) ) @ ( finite_card @ ( product_prod @ nat @ nat ) @ ( edmond259086305sis_uE @ a @ c ) ) ).
% 0.31/0.47
% 0.31/0.47 % card_spEdges_le
% 0.31/0.47 thf(fact_150_g_H_Ocard__spEdges__le,axiom,
% 0.31/0.47 ord_less_eq @ nat @ ( finite_card @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ b @ c2 @ s @ t ) ) @ ( finite_card @ ( product_prod @ nat @ nat ) @ ( edmond259086305sis_uE @ b @ c2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.card_spEdges_le
% 0.31/0.47 thf(fact_151_simplePath__length__less__V,axiom,
% 0.31/0.47 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.31/0.47 ( ( member @ nat @ U @ ( v @ a @ c ) )
% 0.31/0.47 => ( ( isSimplePath @ a @ c @ U @ P5 @ V )
% 0.31/0.47 => ( ord_less @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ ( finite_card @ nat @ ( v @ a @ c ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % simplePath_length_less_V
% 0.31/0.47 thf(fact_152_card__ge__0__finite,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A] :
% 0.31/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) )
% 0.31/0.47 => ( finite_finite2 @ A @ A3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % card_ge_0_finite
% 0.31/0.47 thf(fact_153_min__distI2,axiom,
% 0.31/0.47 ! [V: nat,V3: nat,Q: nat > $o] :
% 0.31/0.47 ( ( connected @ a @ c @ V @ V3 )
% 0.31/0.47 => ( ! [D4: nat] :
% 0.31/0.47 ( ( dist @ a @ c @ V @ D4 @ V3 )
% 0.31/0.47 => ( ! [D5: nat] :
% 0.31/0.47 ( ( dist @ a @ c @ V @ D5 @ V3 )
% 0.31/0.47 => ( ord_less_eq @ nat @ D4 @ D5 ) )
% 0.31/0.47 => ( Q @ D4 ) ) )
% 0.31/0.47 => ( Q @ ( min_dist @ a @ c @ V @ V3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % min_distI2
% 0.31/0.47 thf(fact_154_g_H_OisShortestPath__alt,axiom,
% 0.31/0.47 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.31/0.47 ( ( isShortestPath @ b @ c2 @ U @ P5 @ V )
% 0.31/0.47 = ( ( isSimplePath @ b @ c2 @ U @ P5 @ V )
% 0.31/0.47 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 )
% 0.31/0.47 = ( min_dist @ b @ c2 @ U @ V ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.isShortestPath_alt
% 0.31/0.47 thf(fact_155_g_H_Oconnected__inV__iff,axiom,
% 0.31/0.47 ! [U: nat,V: nat] :
% 0.31/0.47 ( ( connected @ b @ c2 @ U @ V )
% 0.31/0.47 => ( ( member @ nat @ V @ ( v @ b @ c2 ) )
% 0.31/0.47 = ( member @ nat @ U @ ( v @ b @ c2 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.connected_inV_iff
% 0.31/0.47 thf(fact_156_g_H_OisSPath__pathLE,axiom,
% 0.31/0.47 ! [S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 0.31/0.47 ( ( isPath @ b @ c2 @ S @ P5 @ T )
% 0.31/0.47 => ? [P: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ b @ c2 @ S @ P @ T ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.isSPath_pathLE
% 0.31/0.47 thf(fact_157_isSPath__pathLE,axiom,
% 0.31/0.47 ! [S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 0.31/0.47 ( ( isPath @ a @ c @ S @ P5 @ T )
% 0.31/0.47 => ? [P: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ a @ c @ S @ P @ T ) ) ).
% 0.31/0.47
% 0.31/0.47 % isSPath_pathLE
% 0.31/0.47 thf(fact_158_connected__by__dist,axiom,
% 0.31/0.47 ! [V: nat,V3: nat] :
% 0.31/0.47 ( ( connected @ a @ c @ V @ V3 )
% 0.31/0.47 = ( ? [D3: nat] : ( dist @ a @ c @ V @ D3 @ V3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % connected_by_dist
% 0.31/0.47 thf(fact_159_shortestPath__is__simple,axiom,
% 0.31/0.47 ! [S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 0.31/0.47 ( ( isShortestPath @ a @ c @ S @ P5 @ T )
% 0.31/0.47 => ( isSimplePath @ a @ c @ S @ P5 @ T ) ) ).
% 0.31/0.47
% 0.31/0.47 % shortestPath_is_simple
% 0.31/0.47 thf(fact_160_g_H_OshortestPath__is__simple,axiom,
% 0.31/0.47 ! [S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 0.31/0.47 ( ( isShortestPath @ b @ c2 @ S @ P5 @ T )
% 0.31/0.47 => ( isSimplePath @ b @ c2 @ S @ P5 @ T ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.shortestPath_is_simple
% 0.31/0.47 thf(fact_161_finite__code,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( finite_finite @ A )
% 0.31/0.47 => ( ( finite_finite2 @ A )
% 0.31/0.47 = ( ^ [A4: set @ A] : $true ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_code
% 0.31/0.47 thf(fact_162_min__dist__minD,axiom,
% 0.31/0.47 ! [V: nat,D: nat,V3: nat] :
% 0.31/0.47 ( ( dist @ a @ c @ V @ D @ V3 )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( min_dist @ a @ c @ V @ V3 ) @ D ) ) ).
% 0.31/0.47
% 0.31/0.47 % min_dist_minD
% 0.31/0.47 thf(fact_163_min__distI__eq,axiom,
% 0.31/0.47 ! [V: nat,D: nat,V3: nat] :
% 0.31/0.47 ( ( dist @ a @ c @ V @ D @ V3 )
% 0.31/0.47 => ( ! [D6: nat] :
% 0.31/0.47 ( ( dist @ a @ c @ V @ D6 @ V3 )
% 0.31/0.47 => ( ord_less_eq @ nat @ D @ D6 ) )
% 0.31/0.47 => ( ( min_dist @ a @ c @ V @ V3 )
% 0.31/0.47 = D ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % min_distI_eq
% 0.31/0.47 thf(fact_164_isPath__distD,axiom,
% 0.31/0.47 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.31/0.47 ( ( isPath @ a @ c @ U @ P5 @ V )
% 0.31/0.47 => ( dist @ a @ c @ U @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ V ) ) ).
% 0.31/0.47
% 0.31/0.47 % isPath_distD
% 0.31/0.47 thf(fact_165_dist__def,axiom,
% 0.31/0.47 ! [V: nat,D: nat,V3: nat] :
% 0.31/0.47 ( ( dist @ a @ c @ V @ D @ V3 )
% 0.31/0.47 = ( ? [P3: list @ ( product_prod @ nat @ nat )] :
% 0.31/0.47 ( ( isPath @ a @ c @ V @ P3 @ V3 )
% 0.31/0.47 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P3 )
% 0.31/0.47 = D ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % dist_def
% 0.31/0.47 thf(fact_166_min__dist__is__dist,axiom,
% 0.31/0.47 ! [V: nat,V3: nat] :
% 0.31/0.47 ( ( connected @ a @ c @ V @ V3 )
% 0.31/0.47 => ( dist @ a @ c @ V @ ( min_dist @ a @ c @ V @ V3 ) @ V3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % min_dist_is_dist
% 0.31/0.47 thf(fact_167_g_H_OsimplePath__length__less__V,axiom,
% 0.31/0.47 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.31/0.47 ( ( member @ nat @ U @ ( v @ b @ c2 ) )
% 0.31/0.47 => ( ( isSimplePath @ b @ c2 @ U @ P5 @ V )
% 0.31/0.47 => ( ord_less @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ ( finite_card @ nat @ ( v @ b @ c2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.simplePath_length_less_V
% 0.31/0.47 thf(fact_168_g_H_Omin__dist__less__V,axiom,
% 0.31/0.47 ! [S: nat,T: nat] :
% 0.31/0.47 ( ( member @ nat @ S @ ( v @ b @ c2 ) )
% 0.31/0.47 => ( ( connected @ b @ c2 @ S @ T )
% 0.31/0.47 => ( ord_less @ nat @ ( min_dist @ b @ c2 @ S @ T ) @ ( finite_card @ nat @ ( v @ b @ c2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.min_dist_less_V
% 0.31/0.47 thf(fact_169_g_H_OisShortestPath__length__less__V,axiom,
% 0.31/0.47 ! [S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 0.31/0.47 ( ( member @ nat @ S @ ( v @ b @ c2 ) )
% 0.31/0.47 => ( ( isShortestPath @ b @ c2 @ S @ P5 @ T )
% 0.31/0.47 => ( ord_less @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ ( finite_card @ nat @ ( v @ b @ c2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.isShortestPath_length_less_V
% 0.31/0.47 thf(fact_170_isShortestPath__alt,axiom,
% 0.31/0.47 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.31/0.47 ( ( isShortestPath @ a @ c @ U @ P5 @ V )
% 0.31/0.47 = ( ( isSimplePath @ a @ c @ U @ P5 @ V )
% 0.31/0.47 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 )
% 0.31/0.47 = ( min_dist @ a @ c @ U @ V ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % isShortestPath_alt
% 0.31/0.47 thf(fact_171_g_H_Ofinite__V,axiom,
% 0.31/0.47 finite_finite2 @ nat @ ( v @ b @ c2 ) ).
% 0.31/0.47
% 0.31/0.47 % g'.finite_V
% 0.31/0.47 thf(fact_172_finite__V,axiom,
% 0.31/0.47 finite_finite2 @ nat @ ( v @ a @ c ) ).
% 0.31/0.47
% 0.31/0.47 % finite_V
% 0.31/0.47 thf(fact_173_dist__z__iff,axiom,
% 0.31/0.47 ! [V: nat,V3: nat] :
% 0.31/0.47 ( ( dist @ a @ c @ V @ ( zero_zero @ nat ) @ V3 )
% 0.31/0.47 = ( V3 = V ) ) ).
% 0.31/0.47
% 0.31/0.47 % dist_z_iff
% 0.31/0.47 thf(fact_174_dist__z,axiom,
% 0.31/0.47 ! [V: nat] : ( dist @ a @ c @ V @ ( zero_zero @ nat ) @ V ) ).
% 0.31/0.47
% 0.31/0.47 % dist_z
% 0.31/0.47 thf(fact_175_connected__distI,axiom,
% 0.31/0.47 ! [V: nat,D: nat,V3: nat] :
% 0.31/0.47 ( ( dist @ a @ c @ V @ D @ V3 )
% 0.31/0.47 => ( connected @ a @ c @ V @ V3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % connected_distI
% 0.31/0.47 thf(fact_176_Veq,axiom,
% 0.31/0.47 ( ( v @ b @ c2 )
% 0.31/0.47 = ( v @ a @ c ) ) ).
% 0.31/0.47
% 0.31/0.47 % Veq
% 0.31/0.47 thf(fact_177_g_H_Oadjacent__nodes__ss__V,axiom,
% 0.31/0.47 ! [U: nat] : ( ord_less_eq @ ( set @ nat ) @ ( adjacent_nodes @ b @ c2 @ U ) @ ( v @ b @ c2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.adjacent_nodes_ss_V
% 0.31/0.47 thf(fact_178_adjacent__nodes__ss__V,axiom,
% 0.31/0.47 ! [U: nat] : ( ord_less_eq @ ( set @ nat ) @ ( adjacent_nodes @ a @ c @ U ) @ ( v @ a @ c ) ) ).
% 0.31/0.47
% 0.31/0.47 % adjacent_nodes_ss_V
% 0.31/0.47 thf(fact_179_Graph_OisSimplePath_Ocong,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( isSimplePath @ Capacity )
% 0.31/0.47 = ( isSimplePath @ Capacity ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.isSimplePath.cong
% 0.31/0.47 thf(fact_180_finite__subset,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.31/0.47 => ( ( finite_finite2 @ A @ B3 )
% 0.31/0.47 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_subset
% 0.31/0.47 thf(fact_181_infinite__super,axiom,
% 0.31/0.47 ! [A: $tType,S2: set @ A,T2: set @ A] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ S2 @ T2 )
% 0.31/0.47 => ( ~ ( finite_finite2 @ A @ S2 )
% 0.31/0.47 => ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % infinite_super
% 0.31/0.47 thf(fact_182_rev__finite__subset,axiom,
% 0.31/0.47 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.31/0.47 ( ( finite_finite2 @ A @ B3 )
% 0.31/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.31/0.47 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % rev_finite_subset
% 0.31/0.47 thf(fact_183_card__subset__eq,axiom,
% 0.31/0.47 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.31/0.47 ( ( finite_finite2 @ A @ B3 )
% 0.31/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.31/0.47 => ( ( ( finite_card @ A @ A3 )
% 0.31/0.47 = ( finite_card @ A @ B3 ) )
% 0.31/0.47 => ( A3 = B3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % card_subset_eq
% 0.31/0.47 thf(fact_184_infinite__arbitrarily__large,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,N: nat] :
% 0.31/0.47 ( ~ ( finite_finite2 @ A @ A3 )
% 0.31/0.47 => ? [B4: set @ A] :
% 0.31/0.47 ( ( finite_finite2 @ A @ B4 )
% 0.31/0.47 & ( ( finite_card @ A @ B4 )
% 0.31/0.47 = N )
% 0.31/0.47 & ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % infinite_arbitrarily_large
% 0.31/0.47 thf(fact_185_Graph_OisSPath__pathLE,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 0.31/0.47 ( ( isPath @ Capacity @ C @ S @ P5 @ T )
% 0.31/0.47 => ? [P: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ Capacity @ C @ S @ P @ T ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.isSPath_pathLE
% 0.31/0.47 thf(fact_186_Graph_OshortestPath__is__simple,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 0.31/0.47 ( ( isShortestPath @ Capacity @ C @ S @ P5 @ T )
% 0.31/0.47 => ( isSimplePath @ Capacity @ C @ S @ P5 @ T ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.shortestPath_is_simple
% 0.31/0.47 thf(fact_187_finite__maxlen,axiom,
% 0.31/0.47 ! [A: $tType,M4: set @ ( list @ A )] :
% 0.31/0.47 ( ( finite_finite2 @ ( list @ A ) @ M4 )
% 0.31/0.47 => ? [N3: nat] :
% 0.31/0.47 ! [X4: list @ A] :
% 0.31/0.47 ( ( member @ ( list @ A ) @ X4 @ M4 )
% 0.31/0.47 => ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X4 ) @ N3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_maxlen
% 0.31/0.47 thf(fact_188_card__psubset,axiom,
% 0.31/0.47 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.31/0.47 ( ( finite_finite2 @ A @ B3 )
% 0.31/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.31/0.47 => ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) )
% 0.31/0.47 => ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % card_psubset
% 0.31/0.47 thf(fact_189_card__mono,axiom,
% 0.31/0.47 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.31/0.47 ( ( finite_finite2 @ A @ B3 )
% 0.31/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % card_mono
% 0.31/0.47 thf(fact_190_card__seteq,axiom,
% 0.31/0.47 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.31/0.47 ( ( finite_finite2 @ A @ B3 )
% 0.31/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.31/0.47 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ B3 ) @ ( finite_card @ A @ A3 ) )
% 0.31/0.47 => ( A3 = B3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % card_seteq
% 0.31/0.47 thf(fact_191_finite__if__finite__subsets__card__bdd,axiom,
% 0.31/0.47 ! [A: $tType,F2: set @ A,C3: nat] :
% 0.31/0.47 ( ! [G2: set @ A] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ G2 @ F2 )
% 0.31/0.47 => ( ( finite_finite2 @ A @ G2 )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( finite_card @ A @ G2 ) @ C3 ) ) )
% 0.31/0.47 => ( ( finite_finite2 @ A @ F2 )
% 0.31/0.47 & ( ord_less_eq @ nat @ ( finite_card @ A @ F2 ) @ C3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_if_finite_subsets_card_bdd
% 0.31/0.47 thf(fact_192_Finite__Graph__def,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ( ( finite_Graph @ A )
% 0.31/0.47 = ( ^ [C2: ( product_prod @ nat @ nat ) > A] : ( finite_finite2 @ nat @ ( v @ A @ C2 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Finite_Graph_def
% 0.31/0.47 thf(fact_193_Finite__Graph_Ointro,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > A] :
% 0.31/0.47 ( ( finite_finite2 @ nat @ ( v @ A @ C ) )
% 0.31/0.47 => ( finite_Graph @ A @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Finite_Graph.intro
% 0.31/0.47 thf(fact_194_Finite__Graph_Ofinite__V,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > A] :
% 0.31/0.47 ( ( finite_Graph @ A @ C )
% 0.31/0.47 => ( finite_finite2 @ nat @ ( v @ A @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Finite_Graph.finite_V
% 0.31/0.47 thf(fact_195_finite,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( finite_finite @ A )
% 0.31/0.47 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite
% 0.31/0.47 thf(fact_196_finite__set__choice,axiom,
% 0.31/0.47 ! [B: $tType,A: $tType,A3: set @ A,P2: A > B > $o] :
% 0.31/0.47 ( ( finite_finite2 @ A @ A3 )
% 0.31/0.47 => ( ! [X2: A] :
% 0.31/0.47 ( ( member @ A @ X2 @ A3 )
% 0.31/0.47 => ? [X_1: B] : ( P2 @ X2 @ X_1 ) )
% 0.31/0.47 => ? [F3: A > B] :
% 0.31/0.47 ! [X4: A] :
% 0.31/0.47 ( ( member @ A @ X4 @ A3 )
% 0.31/0.47 => ( P2 @ X4 @ ( F3 @ X4 ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_set_choice
% 0.31/0.47 thf(fact_197_finite__psubset__induct,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,P2: ( set @ A ) > $o] :
% 0.31/0.47 ( ( finite_finite2 @ A @ A3 )
% 0.31/0.47 => ( ! [A5: set @ A] :
% 0.31/0.47 ( ( finite_finite2 @ A @ A5 )
% 0.31/0.47 => ( ! [B5: set @ A] :
% 0.31/0.47 ( ( ord_less @ ( set @ A ) @ B5 @ A5 )
% 0.31/0.47 => ( P2 @ B5 ) )
% 0.31/0.47 => ( P2 @ A5 ) ) )
% 0.31/0.47 => ( P2 @ A3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_psubset_induct
% 0.31/0.47 thf(fact_198_Graph_OisShortestPath__alt,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( isShortestPath @ Capacity )
% 0.31/0.47 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U2: nat,P3: list @ ( product_prod @ nat @ nat ),V4: nat] :
% 0.31/0.47 ( ( isSimplePath @ Capacity @ C2 @ U2 @ P3 @ V4 )
% 0.31/0.47 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P3 )
% 0.31/0.47 = ( min_dist @ Capacity @ C2 @ U2 @ V4 ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.isShortestPath_alt
% 0.31/0.47 thf(fact_199_ek__analysis_Ocard__spEdges__le,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat] :
% 0.31/0.47 ( ( edmond2129202899alysis @ A @ C )
% 0.31/0.47 => ( ord_less_eq @ nat @ ( finite_card @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ A @ C @ S @ T ) ) @ ( finite_card @ ( product_prod @ nat @ nat ) @ ( edmond259086305sis_uE @ A @ C ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ek_analysis.card_spEdges_le
% 0.31/0.47 thf(fact_200_finite__has__minimal2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [A3: set @ A,A2: A] :
% 0.31/0.47 ( ( finite_finite2 @ A @ A3 )
% 0.31/0.47 => ( ( member @ A @ A2 @ A3 )
% 0.31/0.47 => ? [X2: A] :
% 0.31/0.47 ( ( member @ A @ X2 @ A3 )
% 0.31/0.47 & ( ord_less_eq @ A @ X2 @ A2 )
% 0.31/0.47 & ! [Xa: A] :
% 0.31/0.47 ( ( member @ A @ Xa @ A3 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ Xa @ X2 )
% 0.31/0.47 => ( X2 = Xa ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_has_minimal2
% 0.31/0.47 thf(fact_201_finite__has__maximal2,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( order @ A )
% 0.31/0.47 => ! [A3: set @ A,A2: A] :
% 0.31/0.47 ( ( finite_finite2 @ A @ A3 )
% 0.31/0.47 => ( ( member @ A @ A2 @ A3 )
% 0.31/0.47 => ? [X2: A] :
% 0.31/0.47 ( ( member @ A @ X2 @ A3 )
% 0.31/0.47 & ( ord_less_eq @ A @ A2 @ X2 )
% 0.31/0.47 & ! [Xa: A] :
% 0.31/0.47 ( ( member @ A @ Xa @ A3 )
% 0.31/0.47 => ( ( ord_less_eq @ A @ X2 @ Xa )
% 0.31/0.47 => ( X2 = Xa ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_has_maximal2
% 0.31/0.47 thf(fact_202_Finite__Graph_OsimplePath__length__less__V,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > A,U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.31/0.47 ( ( finite_Graph @ A @ C )
% 0.31/0.47 => ( ( member @ nat @ U @ ( v @ A @ C ) )
% 0.31/0.47 => ( ( isSimplePath @ A @ C @ U @ P5 @ V )
% 0.31/0.47 => ( ord_less @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ ( finite_card @ nat @ ( v @ A @ C ) ) ) ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Finite_Graph.simplePath_length_less_V
% 0.31/0.47 thf(fact_203_psubset__card__mono,axiom,
% 0.31/0.47 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.31/0.47 ( ( finite_finite2 @ A @ B3 )
% 0.31/0.47 => ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 0.31/0.47 => ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % psubset_card_mono
% 0.31/0.47 thf(fact_204_reachable__ss__V,axiom,
% 0.31/0.47 ! [S: nat] :
% 0.31/0.47 ( ( member @ nat @ S @ ( v @ a @ c ) )
% 0.31/0.47 => ( ord_less_eq @ ( set @ nat ) @ ( reachableNodes @ a @ c @ S ) @ ( v @ a @ c ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % reachable_ss_V
% 0.31/0.47 thf(fact_205_g_H_Oreachable__ss__V,axiom,
% 0.31/0.47 ! [S: nat] :
% 0.31/0.47 ( ( member @ nat @ S @ ( v @ b @ c2 ) )
% 0.31/0.47 => ( ord_less_eq @ ( set @ nat ) @ ( reachableNodes @ b @ c2 @ S ) @ ( v @ b @ c2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.reachable_ss_V
% 0.31/0.47 thf(fact_206_spEdges__ss__E,axiom,
% 0.31/0.47 ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) @ ( e @ a @ c ) ).
% 0.31/0.47
% 0.31/0.47 % spEdges_ss_E
% 0.31/0.47 thf(fact_207_E__ss__uE,axiom,
% 0.31/0.47 ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( e @ a @ c ) @ ( edmond259086305sis_uE @ a @ c ) ).
% 0.31/0.47
% 0.31/0.47 % E_ss_uE
% 0.31/0.47 thf(fact_208_Finite__Graph__EI,axiom,
% 0.31/0.47 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
% 0.31/0.47 => ( finite_Graph @ a @ c ) ) ).
% 0.31/0.47
% 0.31/0.47 % Finite_Graph_EI
% 0.31/0.47 thf(fact_209_Efin__imp__Vfin,axiom,
% 0.31/0.47 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
% 0.31/0.47 => ( finite_finite2 @ nat @ ( v @ a @ c ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Efin_imp_Vfin
% 0.31/0.47 thf(fact_210_finite__E,axiom,
% 0.31/0.47 finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) ).
% 0.31/0.47
% 0.31/0.47 % finite_E
% 0.31/0.47 thf(fact_211_g_H_Oadjacent__nodes__finite,axiom,
% 0.31/0.47 ! [U: nat] : ( finite_finite2 @ nat @ ( adjacent_nodes @ b @ c2 @ U ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.adjacent_nodes_finite
% 0.31/0.47 thf(fact_212_adjacent__nodes__finite,axiom,
% 0.31/0.47 ! [U: nat] : ( finite_finite2 @ nat @ ( adjacent_nodes @ a @ c @ U ) ) ).
% 0.31/0.47
% 0.31/0.47 % adjacent_nodes_finite
% 0.31/0.47 thf(fact_213_Vfin__imp__Efin,axiom,
% 0.31/0.47 ( ( finite_finite2 @ nat @ ( v @ a @ c ) )
% 0.31/0.47 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Vfin_imp_Efin
% 0.31/0.47 thf(fact_214_Graph_OreachableNodes_Ocong,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( reachableNodes @ Capacity )
% 0.31/0.47 = ( reachableNodes @ Capacity ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.reachableNodes.cong
% 0.31/0.47 thf(fact_215_Graph_Oadjacent__nodes_Ocong,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( adjacent_nodes @ Capacity )
% 0.31/0.47 = ( adjacent_nodes @ Capacity ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.adjacent_nodes.cong
% 0.31/0.47 thf(fact_216_Graph_OE_Ocong,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( e @ Capacity )
% 0.31/0.47 = ( e @ Capacity ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.E.cong
% 0.31/0.47 thf(fact_217_Graph_Oreachable__ss__V,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [S: nat,C: ( product_prod @ nat @ nat ) > Capacity] :
% 0.31/0.47 ( ( member @ nat @ S @ ( v @ Capacity @ C ) )
% 0.31/0.47 => ( ord_less_eq @ ( set @ nat ) @ ( reachableNodes @ Capacity @ C @ S ) @ ( v @ Capacity @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.reachable_ss_V
% 0.31/0.47 thf(fact_218_Graph_OEfin__imp__Vfin,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity] :
% 0.31/0.47 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ Capacity @ C ) )
% 0.31/0.47 => ( finite_finite2 @ nat @ ( v @ Capacity @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.Efin_imp_Vfin
% 0.31/0.47 thf(fact_219_Graph_OVfin__imp__Efin,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity] :
% 0.31/0.47 ( ( finite_finite2 @ nat @ ( v @ Capacity @ C ) )
% 0.31/0.47 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ Capacity @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.Vfin_imp_Efin
% 0.31/0.47 thf(fact_220_Graph_OFinite__Graph__EI,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity] :
% 0.31/0.47 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ Capacity @ C ) )
% 0.31/0.47 => ( finite_Graph @ Capacity @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.Finite_Graph_EI
% 0.31/0.47 thf(fact_221_Finite__Graph_Ofinite__E,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > A] :
% 0.31/0.47 ( ( finite_Graph @ A @ C )
% 0.31/0.47 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ A @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Finite_Graph.finite_E
% 0.31/0.47 thf(fact_222_ek__analysis_OE__ss__uE,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > A] :
% 0.31/0.47 ( ( edmond2129202899alysis @ A @ C )
% 0.31/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( e @ A @ C ) @ ( edmond259086305sis_uE @ A @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ek_analysis.E_ss_uE
% 0.31/0.47 thf(fact_223_ek__analysis_OspEdges__ss__E,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat] :
% 0.31/0.47 ( ( edmond2129202899alysis @ A @ C )
% 0.31/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( edmond1803015688pEdges @ A @ C @ S @ T ) @ ( e @ A @ C ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % ek_analysis.spEdges_ss_E
% 0.31/0.47 thf(fact_224_Graph_Oadjacent__nodes__ss__V,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat] : ( ord_less_eq @ ( set @ nat ) @ ( adjacent_nodes @ Capacity @ C @ U ) @ ( v @ Capacity @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.adjacent_nodes_ss_V
% 0.31/0.47 thf(fact_225_Finite__Graph_Oadjacent__nodes__finite,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > A,U: nat] :
% 0.31/0.47 ( ( finite_Graph @ A @ C )
% 0.31/0.47 => ( finite_finite2 @ nat @ ( adjacent_nodes @ A @ C @ U ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Finite_Graph.adjacent_nodes_finite
% 0.31/0.47 thf(fact_226_incoming_H__edges,axiom,
% 0.31/0.47 ! [U3: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming @ a @ c @ U3 ) @ ( e @ a @ c ) ) ).
% 0.31/0.47
% 0.31/0.47 % incoming'_edges
% 0.31/0.47 thf(fact_227_outgoing_H__edges,axiom,
% 0.31/0.47 ! [U3: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing @ a @ c @ U3 ) @ ( e @ a @ c ) ) ).
% 0.31/0.47
% 0.31/0.47 % outgoing'_edges
% 0.31/0.47 thf(fact_228_g_H_OE__ss__uE,axiom,
% 0.31/0.47 ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( e @ b @ c2 ) @ ( edmond259086305sis_uE @ b @ c2 ) ).
% 0.31/0.47
% 0.31/0.47 % g'.E_ss_uE
% 0.31/0.47 thf(fact_229_g_H_OFinite__Graph__EI,axiom,
% 0.31/0.47 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ b @ c2 ) )
% 0.31/0.47 => ( finite_Graph @ b @ c2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.Finite_Graph_EI
% 0.31/0.47 thf(fact_230_g_H_OEfin__imp__Vfin,axiom,
% 0.31/0.47 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ b @ c2 ) )
% 0.31/0.47 => ( finite_finite2 @ nat @ ( v @ b @ c2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.Efin_imp_Vfin
% 0.31/0.47 thf(fact_231_g_H_OspEdges__ss__E,axiom,
% 0.31/0.47 ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( edmond1803015688pEdges @ b @ c2 @ s @ t ) @ ( e @ b @ c2 ) ).
% 0.31/0.47
% 0.31/0.47 % g'.spEdges_ss_E
% 0.31/0.47 thf(fact_232_g_H_Ofinite__E,axiom,
% 0.31/0.47 finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ b @ c2 ) ).
% 0.31/0.47
% 0.31/0.47 % g'.finite_E
% 0.31/0.47 thf(fact_233_g_H_OVfin__imp__Efin,axiom,
% 0.31/0.47 ( ( finite_finite2 @ nat @ ( v @ b @ c2 ) )
% 0.31/0.47 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ b @ c2 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.Vfin_imp_Efin
% 0.31/0.47 thf(fact_234_finite__outgoing_H,axiom,
% 0.31/0.47 ! [U3: set @ nat] :
% 0.31/0.47 ( ( finite_finite2 @ nat @ ( v @ a @ c ) )
% 0.31/0.47 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing @ a @ c @ U3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_outgoing'
% 0.31/0.47 thf(fact_235_finite__incoming_H,axiom,
% 0.31/0.47 ! [U3: set @ nat] :
% 0.31/0.47 ( ( finite_finite2 @ nat @ ( v @ a @ c ) )
% 0.31/0.47 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming @ a @ c @ U3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % finite_incoming'
% 0.31/0.47 thf(fact_236_Graph_Oincoming_H_Ocong,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( incoming @ Capacity )
% 0.31/0.47 = ( incoming @ Capacity ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.incoming'.cong
% 0.31/0.47 thf(fact_237_Graph_Ooutgoing_H_Ocong,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ( ( outgoing @ Capacity )
% 0.31/0.47 = ( outgoing @ Capacity ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.outgoing'.cong
% 0.31/0.47 thf(fact_238_Graph_Ooutgoing_H__edges,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U3: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing @ Capacity @ C @ U3 ) @ ( e @ Capacity @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.outgoing'_edges
% 0.31/0.47 thf(fact_239_Graph_Oincoming_H__edges,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U3: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming @ Capacity @ C @ U3 ) @ ( e @ Capacity @ C ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.incoming'_edges
% 0.31/0.47 thf(fact_240_Graph_Ofinite__outgoing_H,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U3: set @ nat] :
% 0.31/0.47 ( ( finite_finite2 @ nat @ ( v @ Capacity @ C ) )
% 0.31/0.47 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing @ Capacity @ C @ U3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.finite_outgoing'
% 0.31/0.47 thf(fact_241_Graph_Ofinite__incoming_H,axiom,
% 0.31/0.47 ! [Capacity: $tType] :
% 0.31/0.47 ( ( linordered_idom @ Capacity )
% 0.31/0.47 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U3: set @ nat] :
% 0.31/0.47 ( ( finite_finite2 @ nat @ ( v @ Capacity @ C ) )
% 0.31/0.47 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming @ Capacity @ C @ U3 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Graph.finite_incoming'
% 0.31/0.47 thf(fact_242_g_H_Oincoming_H__edges,axiom,
% 0.31/0.47 ! [U3: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming @ b @ c2 @ U3 ) @ ( e @ b @ c2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.incoming'_edges
% 0.31/0.47 thf(fact_243_g_H_Ooutgoing_H__edges,axiom,
% 0.31/0.47 ! [U3: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing @ b @ c2 @ U3 ) @ ( e @ b @ c2 ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.outgoing'_edges
% 0.31/0.47 thf(fact_244_g_H_Ofinite__outgoing_H,axiom,
% 0.31/0.47 ! [U3: set @ nat] :
% 0.31/0.47 ( ( finite_finite2 @ nat @ ( v @ b @ c2 ) )
% 0.31/0.47 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing @ b @ c2 @ U3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.finite_outgoing'
% 0.31/0.47 thf(fact_245_g_H_Ofinite__incoming_H,axiom,
% 0.31/0.47 ! [U3: set @ nat] :
% 0.31/0.47 ( ( finite_finite2 @ nat @ ( v @ b @ c2 ) )
% 0.31/0.47 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming @ b @ c2 @ U3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % g'.finite_incoming'
% 0.31/0.47 thf(fact_246__092_060open_062edges_A_092_060subseteq_062_AE_092_060close_062,axiom,
% 0.31/0.47 ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ edges @ ( e @ a @ c ) ).
% 0.31/0.47
% 0.31/0.47 % \<open>edges \<subseteq> E\<close>
% 0.31/0.47 thf(fact_247_psubsetI,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.31/0.47 => ( ( A3 != B3 )
% 0.31/0.47 => ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % psubsetI
% 0.31/0.47 thf(fact_248_subset__antisym,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.31/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 0.31/0.47 => ( A3 = B3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_antisym
% 0.31/0.47 thf(fact_249_subsetI,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.31/0.47 ( ! [X2: A] :
% 0.31/0.47 ( ( member @ A @ X2 @ A3 )
% 0.31/0.47 => ( member @ A @ X2 @ B3 ) )
% 0.31/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.31/0.47
% 0.31/0.47 % subsetI
% 0.31/0.47 thf(fact_250_SP__EDGES,axiom,
% 0.31/0.47 ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ edges @ ( set2 @ ( product_prod @ nat @ nat ) @ p ) ).
% 0.31/0.47
% 0.31/0.47 % SP_EDGES
% 0.31/0.47 thf(fact_251_Collect__mono__iff,axiom,
% 0.31/0.47 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
% 0.31/0.47 = ( ! [X: A] :
% 0.31/0.47 ( ( P2 @ X )
% 0.31/0.47 => ( Q @ X ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % Collect_mono_iff
% 0.31/0.47 thf(fact_252_set__eq__subset,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( ^ [Y4: set @ A,Z: set @ A] : ( Y4 = Z ) )
% 0.31/0.47 = ( ^ [A4: set @ A,B6: set @ A] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 0.31/0.47 & ( ord_less_eq @ ( set @ A ) @ B6 @ A4 ) ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % set_eq_subset
% 0.31/0.47 thf(fact_253_subset__trans,axiom,
% 0.31/0.47 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 0.31/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.31/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
% 0.31/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 % subset_trans
% 0.31/0.47
% 0.31/0.47 % Subclasses (3)
% 0.31/0.47 thf(subcl_Rings_Olinordered__idom___HOL_Otype,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ( type @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Ozero,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ( zero @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(subcl_Rings_Olinordered__idom___Orderings_Oorder,axiom,
% 0.31/0.47 ! [A: $tType] :
% 0.31/0.47 ( ( linordered_idom @ A )
% 0.31/0.47 => ( order @ A ) ) ).
% 0.31/0.47
% 0.31/0.47 % Type constructors (14)
% 0.31/0.47 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.31/0.47 ! [A6: $tType,A7: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A6 )
% 0.31/0.47 & ( finite_finite @ A7 ) )
% 0.31/0.47 => ( finite_finite @ ( A6 > A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.31/0.47 ! [A6: $tType,A7: $tType] :
% 0.31/0.47 ( ( order @ A7 )
% 0.31/0.47 => ( order @ ( A6 > A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.31/0.47 canoni770627133id_add @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.31/0.47 wellorder @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
% 0.31/0.47 order @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 0.31/0.47 zero @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 0.31/0.47 size @ nat ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Finite__Set_Ofinite_2,axiom,
% 0.31/0.47 ! [A6: $tType] :
% 0.31/0.47 ( ( finite_finite @ A6 )
% 0.31/0.47 => ( finite_finite @ ( set @ A6 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
% 0.31/0.47 ! [A6: $tType] : ( order @ ( set @ A6 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Finite__Set_Ofinite_4,axiom,
% 0.31/0.47 finite_finite @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
% 0.31/0.47 order @ $o ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_List_Olist___Nat_Osize_6,axiom,
% 0.31/0.47 ! [A6: $tType] : ( size @ ( list @ A6 ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_7,axiom,
% 0.31/0.47 ! [A6: $tType,A7: $tType] :
% 0.31/0.47 ( ( ( finite_finite @ A6 )
% 0.31/0.47 & ( finite_finite @ A7 ) )
% 0.31/0.47 => ( finite_finite @ ( product_prod @ A6 @ A7 ) ) ) ).
% 0.31/0.47
% 0.31/0.47 thf(tcon_Product__Type_Oprod___Nat_Osize_8,axiom,
% 0.31/0.47 ! [A6: $tType,A7: $tType] : ( size @ ( product_prod @ A6 @ A7 ) ) ).
% 0.31/0.47
% 0.31/0.47 % Free types (2)
% 0.31/0.47 thf(tfree_0,hypothesis,
% 0.31/0.47 linordered_idom @ a ).
% 0.31/0.47
% 0.31/0.47 thf(tfree_1,hypothesis,
% 0.31/0.47 linordered_idom @ b ).
% 0.31/0.47
% 0.31/0.47 % Conjectures (1)
% 0.31/0.47 thf(conj_0,conjecture,
% 0.31/0.47 $false ).
% 0.31/0.47
% 0.31/0.47 %------------------------------------------------------------------------------
% 0.31/0.47 ------------------------
% 2.88/1.52 ---- Embedded in THF ---
% 2.88/1.52 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.88/1.52 %%% Generated on Tue Feb 24 08:24:26 EST 2026
% 2.88/1.52 %%% using '$$dhol' embedding, version 1.3.0.
% 2.88/1.52 %%% Logic specification used:
% 2.88/1.52 %%% thf(spec, logic, $$dhol).
% 2.88/1.52
% 2.88/1.52 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.AUIVmKJLkR/DTF2DTF13330.p
% 2.88/1.52 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.88/1.52 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.88/1.52 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.88/1.52 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.88/1.52 thf(ty_tf_b, type, b: $tType).
% 2.88/1.52 thf(ty_tf_a, type, a: $tType).
% 2.88/1.52 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.88/1.52 thf(sy_cl_Nat_Osize, type, size: (!> [A:$tType]: ($o))).
% 2.88/1.52 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 2.88/1.52 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.88/1.52 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 2.88/1.52 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 2.88/1.52 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 2.88/1.52 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.88/1.52 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis, type, edmond2129202899alysis: (!> [A:$tType]: (((((product_prod @ nat) @ nat) > A) > $o)))).
% 2.88/1.52 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis__defs_OekMeasure, type, edmond115586663easure: (!> [A:$tType]: (((((product_prod @ nat) @ nat) > A) > (nat > (nat > nat)))))).
% 2.88/1.52 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis__defs_OspEdges, type, edmond1803015688pEdges: (!> [A:$tType]: (((((product_prod @ nat) @ nat) > A) > (nat > (nat > (set @ ((product_prod @ nat) @ nat)))))))).
% 2.88/1.52 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis__defs_OuE, type, edmond259086305sis_uE: (!> [A:$tType]: (((((product_prod @ nat) @ nat) > A) > (set @ ((product_prod @ nat) @ nat)))))).
% 2.88/1.52 thf(sy_c_Finite__Set_Ocard, type, finite_card: (!> [B:$tType]: (((set @ B) > nat)))).
% 2.88/1.52 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.88/1.52 thf(sy_c_Graph_OFinite__Graph, type, finite_Graph: (!> [A:$tType]: (((((product_prod @ nat) @ nat) > A) > $o)))).
% 2.88/1.52 thf(sy_c_Graph_OGraph_OE, type, e: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (set @ ((product_prod @ nat) @ nat)))))).
% 2.88/1.52 thf(sy_c_Graph_OGraph_OV, type, v: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (set @ nat))))).
% 2.88/1.52 thf(sy_c_Graph_OGraph_Oadjacent__nodes, type, adjacent_nodes: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (set @ nat)))))).
% 2.88/1.52 thf(sy_c_Graph_OGraph_Oconnected, type, connected: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > $o)))))).
% 2.88/1.52 thf(sy_c_Graph_OGraph_Odist, type, dist: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > (nat > $o))))))).
% 2.88/1.52 thf(sy_c_Graph_OGraph_Oincoming_H, type, incoming: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > ((set @ nat) > (set @ ((product_prod @ nat) @ nat))))))).
% 2.88/1.52 thf(sy_c_Graph_OGraph_OisPath, type, isPath: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > ((list @ ((product_prod @ nat) @ nat)) > (nat > $o))))))).
% 2.88/1.52 thf(sy_c_Graph_OGraph_OisShortestPath, type, isShortestPath: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > ((list @ ((product_prod @ nat) @ nat)) > (nat > $o))))))).
% 2.88/1.52 thf(sy_c_Graph_OGraph_OisSimplePath, type, isSimplePath: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > ((list @ ((product_prod @ nat) @ nat)) > (nat > $o))))))).
% 2.88/1.52 thf(sy_c_Graph_OGraph_Omin__dist, type, min_dist: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > nat)))))).
% 2.88/1.52 thf(sy_c_Graph_OGraph_Ooutgoing_H, type, outgoing: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > ((set @ nat) > (set @ ((product_prod @ nat) @ nat))))))).
% 2.88/1.52 thf(sy_c_Graph_OGraph_OreachableNodes, type, reachableNodes: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (set @ nat)))))).
% 2.88/1.52 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 2.88/1.52 thf(sy_c_List_Olist_Oset, type, set2: (!> [A:$tType]: (((list @ A) > (set @ A))))).
% 2.88/1.52 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.88/1.52 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.88/1.52 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.88/1.52 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.88/1.52 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.88/1.52 thf(sy_v_c, type, c: (((product_prod @ nat) @ nat) > a)).
% 2.88/1.52 thf(sy_v_c_H, type, c2: (((product_prod @ nat) @ nat) > b)).
% 2.88/1.52 thf(sy_v_edges, type, edges: (set @ ((product_prod @ nat) @ nat))).
% 2.88/1.52 thf(sy_v_p, type, p: (list @ ((product_prod @ nat) @ nat))).
% 2.88/1.52 thf(sy_v_p_H____, type, p2: (list @ ((product_prod @ nat) @ nat))).
% 2.88/1.52 thf(sy_v_s, type, s: nat).
% 2.88/1.52 thf(sy_v_t, type, t: nat).
% 2.88/1.52 thf(fact_0_ek__analysis__axioms, axiom, ((edmond2129202899alysis @ a) @ c)).
% 2.88/1.52 thf(fact_1_LENP, axiom, (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p) = ((((min_dist @ a) @ c) @ s) @ t))).
% 2.88/1.52 thf(fact_2_LENP_H, axiom, (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p2) = ((((min_dist @ b) @ c2) @ s) @ t))).
% 2.88/1.52 thf(fact_3_ek__analysis__defs_OekMeasure_Ocong, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((edmond115586663easure @ A) = (edmond115586663easure @ A)))))).
% 2.88/1.52 thf(fact_4_g_H_Oek__analysis__axioms, axiom, ((edmond2129202899alysis @ b) @ c2)).
% 2.88/1.52 thf(fact_5_SHORTER, axiom, (((ord_less @ nat) @ ((((min_dist @ b) @ c2) @ s) @ t)) @ ((((min_dist @ a) @ c) @ s) @ t))).
% 2.88/1.52 thf(fact_6_ek__analysis__defs_OuE_Ocong, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((edmond259086305sis_uE @ A) = (edmond259086305sis_uE @ A)))))).
% 2.88/1.52 thf(fact_7_ek__analysis__defs_OspEdges_Ocong, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((edmond1803015688pEdges @ A) = (edmond1803015688pEdges @ A)))))).
% 2.88/1.52 thf(fact_8_g_H_OFinite__Graph__axioms, axiom, ((finite_Graph @ b) @ c2)).
% 2.88/1.52 thf(fact_9_Finite__Graph__axioms, axiom, ((finite_Graph @ a) @ c)).
% 2.88/1.52 thf(fact_10__092_060open_062length_Ap_A_060_Alength_Ap_H_092_060close_062, axiom, (((ord_less @ nat) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p)) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p2))).
% 2.88/1.52 thf(fact_11_uE__eq, axiom, (((edmond259086305sis_uE @ b) @ c2) = ((edmond259086305sis_uE @ a) @ c))).
% 2.88/1.52 thf(fact_12_SP, axiom, (((((isShortestPath @ a) @ c) @ s) @ p) @ t)).
% 2.88/1.52 thf(fact_13_CONN2, axiom, ((((connected @ b) @ c2) @ s) @ t)).
% 2.88/1.52 thf(fact_14_SV, axiom, (((member @ nat) @ s) @ ((v @ a) @ c))).
% 2.88/1.52 thf(fact_15_P_H, axiom, (((((isPath @ b) @ c2) @ s) @ p2) @ t)).
% 2.88/1.52 thf(fact_16_ek__analysis__def, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((edmond2129202899alysis @ A) = (finite_Graph @ A)))))).
% 2.88/1.52 thf(fact_17_ek__analysis_Ointro, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A)]: ((((finite_Graph @ A) @ C) => ((edmond2129202899alysis @ A) @ C)))))))).
% 2.88/1.52 thf(fact_18_ek__analysis_Oaxioms, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A)]: ((((edmond2129202899alysis @ A) @ C) => ((finite_Graph @ A) @ C)))))))).
% 2.88/1.52 thf(fact_19__092_060open_062_092_060not_062_Ag_H_Oconnected_As_At_A_092_060Longrightarrow_062_Ag_H_OekMeasure_A_060_AekMeasure_092_060close_062, axiom, ((~ (((((connected @ b) @ c2) @ s) @ t))) => (((ord_less @ nat) @ ((((edmond115586663easure @ b) @ c2) @ s) @ t)) @ ((((edmond115586663easure @ a) @ c) @ s) @ t)))).
% 2.88/1.52 thf(fact_20__092_060open_062g_H_Odist_As_A_Ig_H_Omin__dist_As_At_J_At_092_060close_062, axiom, (((((dist @ b) @ c2) @ s) @ ((((min_dist @ b) @ c2) @ s) @ t)) @ t)).
% 2.88/1.52 thf(fact_21__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062p_H_O_A_092_060lbrakk_062g_H_OisPath_As_Ap_H_At_059_Alength_Ap_H_A_061_Ag_H_Omin__dist_As_At_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom, (~ ((! [P:(list @ ((product_prod @ nat) @ nat))]: (((((((isPath @ b) @ c2) @ s) @ P) @ t) => (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P) != ((((min_dist @ b) @ c2) @ s) @ t)))))))).
% 2.88/1.52 thf(fact_22_CONN, axiom, ((((connected @ a) @ c) @ s) @ t)).
% 2.88/1.52 thf(fact_23_min__dist__less, axiom, (! [Src:nat,V:nat,D:nat,D2:nat]: ((((((connected @ a) @ c) @ Src) @ V) => ((((((min_dist @ a) @ c) @ Src) @ V) = D) => ((((ord_less @ nat) @ D2) @ D) => (? [V2:nat]: ((((((connected @ a) @ c) @ Src) @ V2) & (((((min_dist @ a) @ c) @ Src) @ V2) = D2)))))))))).
% 2.88/1.52 thf(fact_24_g_H_Omin__dist__less, axiom, (! [Src:nat,V:nat,D:nat,D2:nat]: ((((((connected @ b) @ c2) @ Src) @ V) => ((((((min_dist @ b) @ c2) @ Src) @ V) = D) => ((((ord_less @ nat) @ D2) @ D) => (? [V2:nat]: ((((((connected @ b) @ c2) @ Src) @ V2) & (((((min_dist @ b) @ c2) @ Src) @ V2) = D2)))))))))).
% 2.88/1.52 thf(fact_25_length__induct, axiom, (! [A:$tType,P2:((list @ A) > $o),Xs:(list @ A)]: (((! [Xs2:(list @ A)]: (((! [Ys:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ys)) @ ((size_size @ (list @ A)) @ Xs2)) => (P2 @ Ys)))) => (P2 @ Xs2)))) => (P2 @ Xs))))).
% 2.88/1.52 thf(fact_26_finite__spEdges, axiom, ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((((edmond1803015688pEdges @ a) @ c) @ s) @ t))).
% 2.88/1.52 thf(fact_27_g_H_Ofinite__spEdges, axiom, ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((((edmond1803015688pEdges @ b) @ c2) @ s) @ t))).
% 2.88/1.52 thf(fact_28_min__dist__z, axiom, (! [V:nat]: ((((((min_dist @ a) @ c) @ V) @ V) = (zero_zero @ nat))))).
% 2.88/1.52 thf(fact_29_g_H_Omin__dist__z, axiom, (! [V:nat]: ((((((min_dist @ b) @ c2) @ V) @ V) = (zero_zero @ nat))))).
% 2.88/1.52 thf(fact_30_g_H_Oconnected__def, axiom, (! [U:nat,V:nat]: ((((((connected @ b) @ c2) @ U) @ V) = (? [P3:(list @ ((product_prod @ nat) @ nat))]: ((((((isPath @ b) @ c2) @ U) @ P3) @ V))))))).
% 2.88/1.52 thf(fact_31_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))))))).
% 2.88/1.52 thf(fact_32_g_H_Oconnected__by__dist, axiom, (! [V:nat,V3:nat]: ((((((connected @ b) @ c2) @ V) @ V3) = (? [D3:nat]: ((((((dist @ b) @ c2) @ V) @ D3) @ V3))))))).
% 2.88/1.52 thf(fact_33_obtain__shortest__path, axiom, (! [U:nat,V:nat]: ((((((connected @ a) @ c) @ U) @ V) => (~ ((! [P4:(list @ ((product_prod @ nat) @ nat))]: ((~ ((((((isShortestPath @ a) @ c) @ U) @ P4) @ V))))))))))).
% 2.88/1.52 thf(fact_34_g_H_OisPath__distD, axiom, (! [U:nat,P5:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isPath @ b) @ c2) @ U) @ P5) @ V) => (((((dist @ b) @ c2) @ U) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5)) @ V))))).
% 2.88/1.52 thf(fact_35_g_H_Odist__def, axiom, (! [V:nat,D:nat,V3:nat]: (((((((dist @ b) @ c2) @ V) @ D) @ V3) = (? [P3:(list @ ((product_prod @ nat) @ nat))]: (((((((isPath @ b) @ c2) @ V) @ P3) @ V3) & (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P3) = D)))))))).
% 2.88/1.52 thf(fact_36_g_H_Omin__dist__is__dist, axiom, (! [V:nat,V3:nat]: ((((((connected @ b) @ c2) @ V) @ V3) => (((((dist @ b) @ c2) @ V) @ ((((min_dist @ b) @ c2) @ V) @ V3)) @ V3))))).
% 2.88/1.52 thf(fact_37_g_H_Oconnected__refl, axiom, (! [V:nat]: (((((connected @ b) @ c2) @ V) @ V)))).
% 2.88/1.52 thf(fact_38_connected__refl, axiom, (! [V:nat]: (((((connected @ a) @ c) @ V) @ V)))).
% 2.88/1.52 thf(fact_39_g_H_Odist__z__iff, axiom, (! [V:nat,V3:nat]: (((((((dist @ b) @ c2) @ V) @ (zero_zero @ nat)) @ V3) = (V3 = V))))).
% 2.88/1.52 thf(fact_40_g_H_Odist__z, axiom, (! [V:nat]: ((((((dist @ b) @ c2) @ V) @ (zero_zero @ nat)) @ V)))).
% 2.88/1.52 thf(fact_41_g_H_Oconnected__distI, axiom, (! [V:nat,D:nat,V3:nat]: (((((((dist @ b) @ c2) @ V) @ D) @ V3) => ((((connected @ b) @ c2) @ V) @ V3))))).
% 2.88/1.52 thf(fact_42_g_H_Ofinite__uE, axiom, ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((edmond259086305sis_uE @ b) @ c2))).
% 2.88/1.52 thf(fact_43_finite__uE, axiom, ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((edmond259086305sis_uE @ a) @ c))).
% 2.88/1.52 thf(fact_44_g_H_Omin__dist__z__iff, axiom, (! [V:nat,V3:nat]: ((((((connected @ b) @ c2) @ V) @ V3) => ((((((min_dist @ b) @ c2) @ V) @ V3) = (zero_zero @ nat)) = (V3 = V)))))).
% 2.88/1.52 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P2:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P2)) = (P2 @ A2))))).
% 2.88/1.52 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X:A]: ((((member @ A) @ X) @ A3)))) = A3)))).
% 2.88/1.52 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P2 @ X2) = (Q @ X2)))) => (((collect @ A) @ P2) = ((collect @ A) @ Q)))))).
% 2.88/1.52 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 2.88/1.52 thf(fact_49_min__dist__z__iff, axiom, (! [V:nat,V3:nat]: ((((((connected @ a) @ c) @ V) @ V3) => ((((((min_dist @ a) @ c) @ V) @ V3) = (zero_zero @ nat)) = (V3 = V)))))).
% 2.88/1.52 thf(fact_50_ek__analysis_Ofinite__uE, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A)]: ((((edmond2129202899alysis @ A) @ C) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((edmond259086305sis_uE @ A) @ C))))))))).
% 2.88/1.52 thf(fact_51_ek__analysis_Ofinite__spEdges, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,T:nat]: ((((edmond2129202899alysis @ A) @ C) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((((edmond1803015688pEdges @ A) @ C) @ S) @ T))))))))).
% 2.88/1.52 thf(fact_52_neq__if__length__neq, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((((size_size @ (list @ A)) @ Xs) != ((size_size @ (list @ A)) @ Ys2)) => (Xs != Ys2))))).
% 2.88/1.52 thf(fact_53_Ex__list__of__length, axiom, (! [A:$tType,N:nat]: ((? [Xs2:(list @ A)]: ((((size_size @ (list @ A)) @ Xs2) = N)))))).
% 2.88/1.52 thf(fact_54_bot__nat__0_Onot__eq__extremum, axiom, (! [A2:nat]: (((A2 != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ A2))))).
% 2.88/1.52 thf(fact_55_less__nat__zero__code, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.88/1.52 thf(fact_56_neq0__conv, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) = (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.88/1.52 thf(fact_57_not__gr__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 2.88/1.52 thf(fact_58_Graph_OisShortestPath__min__dist__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isShortestPath @ Capacity) = (^ [C2:(((product_prod @ nat) @ nat) > Capacity),U2:nat,P3:(list @ ((product_prod @ nat) @ nat)),V4:nat]: (((((((isPath @ Capacity) @ C2) @ U2) @ P3) @ V4) & (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P3) = ((((min_dist @ Capacity) @ C2) @ U2) @ V4)))))))))).
% 2.88/1.52 thf(fact_59_g_H_OisShortestPath__min__dist__def, axiom, (! [U:nat,P5:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isShortestPath @ b) @ c2) @ U) @ P5) @ V) = ((((((isPath @ b) @ c2) @ U) @ P5) @ V) & (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5) = ((((min_dist @ b) @ c2) @ U) @ V))))))).
% 2.88/1.52 thf(fact_60_g_H_Omin__distI2, axiom, (! [V:nat,V3:nat,Q:(nat > $o)]: ((((((connected @ b) @ c2) @ V) @ V3) => ((! [D4:nat]: (((((((dist @ b) @ c2) @ V) @ D4) @ V3) => ((! [D5:nat]: (((((((dist @ b) @ c2) @ V) @ D5) @ V3) => (((ord_less_eq @ nat) @ D4) @ D5)))) => (Q @ D4))))) => (Q @ ((((min_dist @ b) @ c2) @ V) @ V3))))))).
% 2.88/1.52 thf(fact_61_isShortestPath__length__less__V, axiom, (! [S:nat,P5:(list @ ((product_prod @ nat) @ nat)),T:nat]: (((((member @ nat) @ S) @ ((v @ a) @ c)) => ((((((isShortestPath @ a) @ c) @ S) @ P5) @ T) => (((ord_less @ nat) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5)) @ ((finite_card @ nat) @ ((v @ a) @ c)))))))).
% 2.88/1.52 thf(fact_62_min__dist__less__V, axiom, (! [S:nat,T:nat]: (((((member @ nat) @ S) @ ((v @ a) @ c)) => (((((connected @ a) @ c) @ S) @ T) => (((ord_less @ nat) @ ((((min_dist @ a) @ c) @ S) @ T)) @ ((finite_card @ nat) @ ((v @ a) @ c)))))))).
% 2.88/1.52 thf(fact_63_connected__def, axiom, (! [U:nat,V:nat]: ((((((connected @ a) @ c) @ U) @ V) = (? [P3:(list @ ((product_prod @ nat) @ nat))]: ((((((isPath @ a) @ c) @ U) @ P3) @ V))))))).
% 2.88/1.52 thf(fact_64_shortestPath__is__path, axiom, (! [U:nat,P5:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isShortestPath @ a) @ c) @ U) @ P5) @ V) => (((((isPath @ a) @ c) @ U) @ P5) @ V))))).
% 2.88/1.52 thf(fact_65_g_H_OshortestPath__is__path, axiom, (! [U:nat,P5:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isShortestPath @ b) @ c2) @ U) @ P5) @ V) => (((((isPath @ b) @ c2) @ U) @ P5) @ V))))).
% 2.88/1.52 thf(fact_66_g_H_Oobtain__shortest__path, axiom, (! [U:nat,V:nat]: ((((((connected @ b) @ c2) @ U) @ V) => (~ ((! [P4:(list @ ((product_prod @ nat) @ nat))]: ((~ ((((((isShortestPath @ b) @ c2) @ U) @ P4) @ V))))))))))).
% 2.88/1.52 thf(fact_67_g_H_Omin__dist__le, axiom, (! [Src:nat,V:nat,D2:nat]: ((((((connected @ b) @ c2) @ Src) @ V) => ((((ord_less_eq @ nat) @ D2) @ ((((min_dist @ b) @ c2) @ Src) @ V)) => (? [V2:nat]: ((((((connected @ b) @ c2) @ Src) @ V2) & (((((min_dist @ b) @ c2) @ Src) @ V2) = D2))))))))).
% 2.88/1.52 thf(fact_68_g_H_Omin__distI__eq, axiom, (! [V:nat,D:nat,V3:nat]: (((((((dist @ b) @ c2) @ V) @ D) @ V3) => ((! [D6:nat]: (((((((dist @ b) @ c2) @ V) @ D6) @ V3) => (((ord_less_eq @ nat) @ D) @ D6)))) => (((((min_dist @ b) @ c2) @ V) @ V3) = D)))))).
% 2.88/1.52 thf(fact_69_g_H_Omin__dist__minD, axiom, (! [V:nat,D:nat,V3:nat]: (((((((dist @ b) @ c2) @ V) @ D) @ V3) => (((ord_less_eq @ nat) @ ((((min_dist @ b) @ c2) @ V) @ V3)) @ D))))).
% 2.88/1.52 thf(fact_70_min__dist__le, axiom, (! [Src:nat,V:nat,D2:nat]: ((((((connected @ a) @ c) @ Src) @ V) => ((((ord_less_eq @ nat) @ D2) @ ((((min_dist @ a) @ c) @ Src) @ V)) => (? [V2:nat]: ((((((connected @ a) @ c) @ Src) @ V2) & (((((min_dist @ a) @ c) @ Src) @ V2) = D2))))))))).
% 2.88/1.52 thf(fact_71_isShortestPath__def, axiom, (! [U:nat,P5:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isShortestPath @ a) @ c) @ U) @ P5) @ V) = ((((((isPath @ a) @ c) @ U) @ P5) @ V) & (! [P6:(list @ ((product_prod @ nat) @ nat))]: (((((((isPath @ a) @ c) @ U) @ P6) @ V) => (((ord_less_eq @ nat) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5)) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P6)))))))))).
% 2.88/1.52 thf(fact_72_isShortestPath__min__dist__def, axiom, (! [U:nat,P5:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isShortestPath @ a) @ c) @ U) @ P5) @ V) = ((((((isPath @ a) @ c) @ U) @ P5) @ V) & (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5) = ((((min_dist @ a) @ c) @ U) @ V))))))).
% 2.88/1.52 thf(fact_73_g_H_OisShortestPath__def, axiom, (! [U:nat,P5:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isShortestPath @ b) @ c2) @ U) @ P5) @ V) = ((((((isPath @ b) @ c2) @ U) @ P5) @ V) & (! [P6:(list @ ((product_prod @ nat) @ nat))]: (((((((isPath @ b) @ c2) @ U) @ P6) @ V) => (((ord_less_eq @ nat) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5)) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P6)))))))))).
% 2.88/1.52 thf(fact_74_le__zero__eq, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 2.88/1.52 thf(fact_75_le0, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.88/1.52 thf(fact_76_bot__nat__0_Oextremum, axiom, (! [A2:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ A2)))).
% 2.88/1.52 thf(fact_77_zero__le, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X3))))))).
% 2.88/1.52 thf(fact_78_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.88/1.52 thf(fact_79_le__trans, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ J) @ K) => (((ord_less_eq @ nat) @ I) @ K)))))).
% 2.88/1.52 thf(fact_80_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.88/1.52 thf(fact_81_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.88/1.52 thf(fact_82_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.88/1.52 thf(fact_83_Nat_Oex__has__greatest__nat, axiom, (! [P2:(nat > $o),K:nat,B2:nat]: (((P2 @ K) => ((! [Y:nat]: (((P2 @ Y) => (((ord_less_eq @ nat) @ Y) @ B2)))) => (? [X2:nat]: (((P2 @ X2) & (! [Y2:nat]: (((P2 @ Y2) => (((ord_less_eq @ nat) @ Y2) @ X2)))))))))))).
% 2.88/1.52 thf(fact_84_less__eq__nat_Osimps_I1_J, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ N)))).
% 2.88/1.52 thf(fact_85_le__0__eq, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (zero_zero @ nat)) = (N = (zero_zero @ nat)))))).
% 2.88/1.52 thf(fact_86_bot__nat__0_Oextremum__unique, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) = (A2 = (zero_zero @ nat)))))).
% 2.88/1.52 thf(fact_87_bot__nat__0_Oextremum__uniqueI, axiom, (! [A2:nat]: (((((ord_less_eq @ nat) @ A2) @ (zero_zero @ nat)) => (A2 = (zero_zero @ nat)))))).
% 2.88/1.52 thf(fact_88_nat__less__le, axiom, ((ord_less @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) & (M2 != N2)))))).
% 2.88/1.52 thf(fact_89_less__imp__le__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.88/1.52 thf(fact_90_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) | (M2 = N2)))))).
% 2.88/1.52 thf(fact_91_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.88/1.52 thf(fact_92_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 2.88/1.52 thf(fact_93_less__mono__imp__le__mono, axiom, (! [F:(nat > nat),I:nat,J:nat]: (((! [I2:nat,J2:nat]: (((((ord_less @ nat) @ I2) @ J2) => (((ord_less @ nat) @ (F @ I2)) @ (F @ J2))))) => ((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (F @ I)) @ (F @ J))))))).
% 2.88/1.52 thf(fact_94_ex__least__nat__le, axiom, (! [P2:(nat > $o),N:nat]: (((P2 @ N) => ((~ ((P2 @ (zero_zero @ nat)))) => (? [K2:nat]: (((((ord_less_eq @ nat) @ K2) @ N) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ K2) => (~ ((P2 @ I3)))))) & (P2 @ K2)))))))))).
% 2.88/1.52 thf(fact_95_Graph_Omin__dist__le, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),Src:nat,V:nat,D2:nat]: ((((((connected @ Capacity) @ C) @ Src) @ V) => ((((ord_less_eq @ nat) @ D2) @ ((((min_dist @ Capacity) @ C) @ Src) @ V)) => (? [V2:nat]: ((((((connected @ Capacity) @ C) @ Src) @ V2) & (((((min_dist @ Capacity) @ C) @ Src) @ V2) = D2)))))))))))).
% 2.88/1.52 thf(fact_96_Graph_Omin__dist__minD, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),V:nat,D:nat,V3:nat]: (((((((dist @ Capacity) @ C) @ V) @ D) @ V3) => (((ord_less_eq @ nat) @ ((((min_dist @ Capacity) @ C) @ V) @ V3)) @ D)))))))).
% 2.88/1.52 thf(fact_97_Graph_Omin__distI__eq, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),V:nat,D:nat,V3:nat]: (((((((dist @ Capacity) @ C) @ V) @ D) @ V3) => ((! [D6:nat]: (((((((dist @ Capacity) @ C) @ V) @ D6) @ V3) => (((ord_less_eq @ nat) @ D) @ D6)))) => (((((min_dist @ Capacity) @ C) @ V) @ V3) = D))))))))).
% 2.88/1.52 thf(fact_98_Graph_Omin__distI2, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),V:nat,V3:nat,Q:(nat > $o)]: ((((((connected @ Capacity) @ C) @ V) @ V3) => ((! [D4:nat]: (((((((dist @ Capacity) @ C) @ V) @ D4) @ V3) => ((! [D5:nat]: (((((((dist @ Capacity) @ C) @ V) @ D5) @ V3) => (((ord_less_eq @ nat) @ D4) @ D5)))) => (Q @ D4))))) => (Q @ ((((min_dist @ Capacity) @ C) @ V) @ V3)))))))))).
% 2.88/1.52 thf(fact_99_Graph_OisShortestPath__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isShortestPath @ Capacity) = (^ [C2:(((product_prod @ nat) @ nat) > Capacity),U2:nat,P3:(list @ ((product_prod @ nat) @ nat)),V4:nat]: (((((((isPath @ Capacity) @ C2) @ U2) @ P3) @ V4) & (! [P6:(list @ ((product_prod @ nat) @ nat))]: (((((((isPath @ Capacity) @ C2) @ U2) @ P6) @ V4) => (((ord_less_eq @ nat) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P3)) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P6))))))))))))).
% 2.88/1.52 thf(fact_100_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X3:A]: ((((zero_zero @ A) = X3) = (X3 = (zero_zero @ A))))))))).
% 2.88/1.52 thf(fact_101_measure__induct, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P2:(A > $o),A2:A]: (((! [X2:A]: (((! [Y2:A]: (((((ord_less @ B) @ (F @ Y2)) @ (F @ X2)) => (P2 @ Y2)))) => (P2 @ X2)))) => (P2 @ A2)))))))).
% 2.88/1.52 thf(fact_102_measure__induct__rule, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P2:(A > $o),A2:A]: (((! [X2:A]: (((! [Y2:A]: (((((ord_less @ B) @ (F @ Y2)) @ (F @ X2)) => (P2 @ Y2)))) => (P2 @ X2)))) => (P2 @ A2)))))))).
% 2.88/1.52 thf(fact_103_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 2.88/1.52 thf(fact_104_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.88/1.52 thf(fact_105_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 2.88/1.52 thf(fact_106_less__not__refl3, axiom, (! [S:nat,T:nat]: (((((ord_less @ nat) @ S) @ T) => (S != T))))).
% 2.88/1.52 thf(fact_107_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.88/1.52 thf(fact_108_nat__less__induct, axiom, (! [P2:(nat > $o),N:nat]: (((! [N3:nat]: (((! [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) => (P2 @ M3)))) => (P2 @ N3)))) => (P2 @ N))))).
% 2.88/1.52 thf(fact_109_infinite__descent, axiom, (! [P2:(nat > $o),N:nat]: (((! [N3:nat]: (((~ ((P2 @ N3))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) & (~ ((P2 @ M3))))))))) => (P2 @ N))))).
% 2.88/1.52 thf(fact_110_linorder__neqE__nat, axiom, (! [X3:nat,Y3:nat]: (((X3 != Y3) => ((~ ((((ord_less @ nat) @ X3) @ Y3))) => (((ord_less @ nat) @ Y3) @ X3)))))).
% 2.88/1.52 thf(fact_111_infinite__descent__measure, axiom, (! [A:$tType,P2:(A > $o),V5:(A > nat),X3:A]: (((! [X2:A]: (((~ ((P2 @ X2))) => (? [Y2:A]: (((((ord_less @ nat) @ (V5 @ Y2)) @ (V5 @ X2)) & (~ ((P2 @ Y2))))))))) => (P2 @ X3))))).
% 2.88/1.52 thf(fact_112_size__neq__size__imp__neq, axiom, (! [A:$tType]: (((size @ A) => (! [X3:A,Y3:A]: (((((size_size @ A) @ X3) != ((size_size @ A) @ Y3)) => (X3 != Y3)))))))).
% 2.88/1.52 thf(fact_113_Graph_OisPath_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isPath @ Capacity) = (isPath @ Capacity)))))).
% 2.88/1.52 thf(fact_114_Graph_OV_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((v @ Capacity) = (v @ Capacity)))))).
% 2.88/1.52 thf(fact_115_Graph_Omin__dist_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((min_dist @ Capacity) = (min_dist @ Capacity)))))).
% 2.88/1.52 thf(fact_116_Graph_Oconnected_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((connected @ Capacity) = (connected @ Capacity)))))).
% 2.88/1.52 thf(fact_117_Graph_Oconnected__refl, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),V:nat]: (((((connected @ Capacity) @ C) @ V) @ V))))))).
% 2.88/1.52 thf(fact_118_Graph_Odist_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((dist @ Capacity) = (dist @ Capacity)))))).
% 2.88/1.52 thf(fact_119_Graph_OisShortestPath_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isShortestPath @ Capacity) = (isShortestPath @ Capacity)))))).
% 2.88/1.52 thf(fact_120_Finite__Graph_Omin__dist__less__V, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,T:nat]: ((((finite_Graph @ A) @ C) => ((((member @ nat) @ S) @ ((v @ A) @ C)) => (((((connected @ A) @ C) @ S) @ T) => (((ord_less @ nat) @ ((((min_dist @ A) @ C) @ S) @ T)) @ ((finite_card @ nat) @ ((v @ A) @ C)))))))))))).
% 2.88/1.52 thf(fact_121_Finite__Graph_OisShortestPath__length__less__V, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,P5:(list @ ((product_prod @ nat) @ nat)),T:nat]: ((((finite_Graph @ A) @ C) => ((((member @ nat) @ S) @ ((v @ A) @ C)) => ((((((isShortestPath @ A) @ C) @ S) @ P5) @ T) => (((ord_less @ nat) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5)) @ ((finite_card @ nat) @ ((v @ A) @ C)))))))))))).
% 2.88/1.52 thf(fact_122_gr__zeroI, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 2.88/1.52 thf(fact_123_not__less__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 2.88/1.52 thf(fact_124_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [M:A,N:A]: (((((ord_less @ A) @ M) @ N) => (N != (zero_zero @ A))))))))).
% 2.88/1.52 thf(fact_125_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 2.88/1.52 thf(fact_126_gr0I, axiom, (! [N:nat]: (((N != (zero_zero @ nat)) => (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))).
% 2.88/1.52 thf(fact_127_not__gr0, axiom, (! [N:nat]: (((~ ((((ord_less @ nat) @ (zero_zero @ nat)) @ N))) = (N = (zero_zero @ nat)))))).
% 2.88/1.52 thf(fact_128_not__less0, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.88/1.52 thf(fact_129_less__zeroE, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ (zero_zero @ nat))))))).
% 2.88/1.52 thf(fact_130_gr__implies__not0, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (N != (zero_zero @ nat)))))).
% 2.88/1.52 thf(fact_131_infinite__descent0, axiom, (! [P2:(nat > $o),N:nat]: (((P2 @ (zero_zero @ nat)) => ((! [N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) => ((~ ((P2 @ N3))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) & (~ ((P2 @ M3)))))))))) => (P2 @ N)))))).
% 2.88/1.52 thf(fact_132_bot__nat__0_Oextremum__strict, axiom, (! [A2:nat]: ((~ ((((ord_less @ nat) @ A2) @ (zero_zero @ nat))))))).
% 2.88/1.52 thf(fact_133_infinite__descent0__measure, axiom, (! [A:$tType,V5:(A > nat),P2:(A > $o),X3:A]: (((! [X2:A]: ((((V5 @ X2) = (zero_zero @ nat)) => (P2 @ X2)))) => ((! [X2:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (V5 @ X2)) => ((~ ((P2 @ X2))) => (? [Y2:A]: (((((ord_less @ nat) @ (V5 @ Y2)) @ (V5 @ X2)) & (~ ((P2 @ Y2)))))))))) => (P2 @ X3)))))).
% 2.88/1.52 thf(fact_134_Graph_Omin__dist__z, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),V:nat]: ((((((min_dist @ Capacity) @ C) @ V) @ V) = (zero_zero @ nat)))))))).
% 2.88/1.52 thf(fact_135_Graph_Odist__z, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),V:nat]: ((((((dist @ Capacity) @ C) @ V) @ (zero_zero @ nat)) @ V))))))).
% 2.88/1.52 thf(fact_136_Graph_Odist__z__iff, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),V:nat,V3:nat]: (((((((dist @ Capacity) @ C) @ V) @ (zero_zero @ nat)) @ V3) = (V3 = V)))))))).
% 2.88/1.52 thf(fact_137_Graph_Oconnected__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((connected @ Capacity) = (^ [C2:(((product_prod @ nat) @ nat) > Capacity),U2:nat,V4:nat]: ((? [P3:(list @ ((product_prod @ nat) @ nat))]: ((((((isPath @ Capacity) @ C2) @ U2) @ P3) @ V4)))))))))).
% 2.88/1.52 thf(fact_138_Graph_Oconnected__inV__iff, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,V:nat]: ((((((connected @ Capacity) @ C) @ U) @ V) => ((((member @ nat) @ V) @ ((v @ Capacity) @ C)) = (((member @ nat) @ U) @ ((v @ Capacity) @ C)))))))))).
% 2.88/1.52 thf(fact_139_Graph_OshortestPath__is__path, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,P5:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isShortestPath @ Capacity) @ C) @ U) @ P5) @ V) => (((((isPath @ Capacity) @ C) @ U) @ P5) @ V)))))))).
% 2.88/1.52 thf(fact_140_Graph_Oconnected__distI, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),V:nat,D:nat,V3:nat]: (((((((dist @ Capacity) @ C) @ V) @ D) @ V3) => ((((connected @ Capacity) @ C) @ V) @ V3)))))))).
% 2.88/1.52 thf(fact_141_Graph_Oconnected__by__dist, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((connected @ Capacity) = (^ [C2:(((product_prod @ nat) @ nat) > Capacity),V4:nat,V6:nat]: ((? [D3:nat]: ((((((dist @ Capacity) @ C2) @ V4) @ D3) @ V6)))))))))).
% 2.88/1.52 thf(fact_142_Graph_Oobtain__shortest__path, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,V:nat]: ((((((connected @ Capacity) @ C) @ U) @ V) => (~ ((! [P4:(list @ ((product_prod @ nat) @ nat))]: ((~ ((((((isShortestPath @ Capacity) @ C) @ U) @ P4) @ V)))))))))))))).
% 2.88/1.52 thf(fact_143_Graph_Omin__dist__z__iff, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),V:nat,V3:nat]: ((((((connected @ Capacity) @ C) @ V) @ V3) => ((((((min_dist @ Capacity) @ C) @ V) @ V3) = (zero_zero @ nat)) = (V3 = V))))))))).
% 2.88/1.52 thf(fact_144_Graph_Omin__dist__less, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),Src:nat,V:nat,D:nat,D2:nat]: ((((((connected @ Capacity) @ C) @ Src) @ V) => ((((((min_dist @ Capacity) @ C) @ Src) @ V) = D) => ((((ord_less @ nat) @ D2) @ D) => (? [V2:nat]: ((((((connected @ Capacity) @ C) @ Src) @ V2) & (((((min_dist @ Capacity) @ C) @ Src) @ V2) = D2))))))))))))).
% 2.88/1.52 thf(fact_145_Graph_Omin__dist__is__dist, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),V:nat,V3:nat]: ((((((connected @ Capacity) @ C) @ V) @ V3) => (((((dist @ Capacity) @ C) @ V) @ ((((min_dist @ Capacity) @ C) @ V) @ V3)) @ V3)))))))).
% 2.88/1.52 thf(fact_146_Graph_OisPath__distD, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,P5:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isPath @ Capacity) @ C) @ U) @ P5) @ V) => (((((dist @ Capacity) @ C) @ U) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5)) @ V)))))))).
% 2.88/1.52 thf(fact_147_Graph_Odist__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((dist @ Capacity) = (^ [C2:(((product_prod @ nat) @ nat) > Capacity),V4:nat,D3:nat,V6:nat]: ((? [P3:(list @ ((product_prod @ nat) @ nat))]: (((((((isPath @ Capacity) @ C2) @ V4) @ P3) @ V6) & (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P3) = D3))))))))))).
% 2.88/1.52 thf(fact_148_card_Oinfinite, axiom, (! [A:$tType,A3:(set @ A)]: (((~ (((finite_finite2 @ A) @ A3))) => (((finite_card @ A) @ A3) = (zero_zero @ nat)))))).
% 2.88/1.52 thf(fact_149_card__spEdges__le, axiom, (((ord_less_eq @ nat) @ ((finite_card @ ((product_prod @ nat) @ nat)) @ ((((edmond1803015688pEdges @ a) @ c) @ s) @ t))) @ ((finite_card @ ((product_prod @ nat) @ nat)) @ ((edmond259086305sis_uE @ a) @ c)))).
% 2.88/1.52 thf(fact_150_g_H_Ocard__spEdges__le, axiom, (((ord_less_eq @ nat) @ ((finite_card @ ((product_prod @ nat) @ nat)) @ ((((edmond1803015688pEdges @ b) @ c2) @ s) @ t))) @ ((finite_card @ ((product_prod @ nat) @ nat)) @ ((edmond259086305sis_uE @ b) @ c2)))).
% 2.88/1.52 thf(fact_151_simplePath__length__less__V, axiom, (! [U:nat,P5:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((member @ nat) @ U) @ ((v @ a) @ c)) => ((((((isSimplePath @ a) @ c) @ U) @ P5) @ V) => (((ord_less @ nat) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5)) @ ((finite_card @ nat) @ ((v @ a) @ c)))))))).
% 2.88/1.52 thf(fact_152_card__ge__0__finite, axiom, (! [A:$tType,A3:(set @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ ((finite_card @ A) @ A3)) => ((finite_finite2 @ A) @ A3))))).
% 2.88/1.52 thf(fact_153_min__distI2, axiom, (! [V:nat,V3:nat,Q:(nat > $o)]: ((((((connected @ a) @ c) @ V) @ V3) => ((! [D4:nat]: (((((((dist @ a) @ c) @ V) @ D4) @ V3) => ((! [D5:nat]: (((((((dist @ a) @ c) @ V) @ D5) @ V3) => (((ord_less_eq @ nat) @ D4) @ D5)))) => (Q @ D4))))) => (Q @ ((((min_dist @ a) @ c) @ V) @ V3))))))).
% 2.88/1.52 thf(fact_154_g_H_OisShortestPath__alt, axiom, (! [U:nat,P5:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isShortestPath @ b) @ c2) @ U) @ P5) @ V) = ((((((isSimplePath @ b) @ c2) @ U) @ P5) @ V) & (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5) = ((((min_dist @ b) @ c2) @ U) @ V))))))).
% 2.88/1.52 thf(fact_155_g_H_Oconnected__inV__iff, axiom, (! [U:nat,V:nat]: ((((((connected @ b) @ c2) @ U) @ V) => ((((member @ nat) @ V) @ ((v @ b) @ c2)) = (((member @ nat) @ U) @ ((v @ b) @ c2))))))).
% 2.88/1.52 thf(fact_156_g_H_OisSPath__pathLE, axiom, (! [S:nat,P5:(list @ ((product_prod @ nat) @ nat)),T:nat]: (((((((isPath @ b) @ c2) @ S) @ P5) @ T) => (? [P:(list @ ((product_prod @ nat) @ nat))]: ((((((isSimplePath @ b) @ c2) @ S) @ P) @ T))))))).
% 2.88/1.52 thf(fact_157_isSPath__pathLE, axiom, (! [S:nat,P5:(list @ ((product_prod @ nat) @ nat)),T:nat]: (((((((isPath @ a) @ c) @ S) @ P5) @ T) => (? [P:(list @ ((product_prod @ nat) @ nat))]: ((((((isSimplePath @ a) @ c) @ S) @ P) @ T))))))).
% 2.88/1.52 thf(fact_158_connected__by__dist, axiom, (! [V:nat,V3:nat]: ((((((connected @ a) @ c) @ V) @ V3) = (? [D3:nat]: ((((((dist @ a) @ c) @ V) @ D3) @ V3))))))).
% 2.88/1.52 thf(fact_159_shortestPath__is__simple, axiom, (! [S:nat,P5:(list @ ((product_prod @ nat) @ nat)),T:nat]: (((((((isShortestPath @ a) @ c) @ S) @ P5) @ T) => (((((isSimplePath @ a) @ c) @ S) @ P5) @ T))))).
% 2.88/1.52 thf(fact_160_g_H_OshortestPath__is__simple, axiom, (! [S:nat,P5:(list @ ((product_prod @ nat) @ nat)),T:nat]: (((((((isShortestPath @ b) @ c2) @ S) @ P5) @ T) => (((((isSimplePath @ b) @ c2) @ S) @ P5) @ T))))).
% 2.88/1.52 thf(fact_161_finite__code, axiom, (! [A:$tType]: (((finite_finite @ A) => ((finite_finite2 @ A) = (^ [A4:(set @ A)]: ($true))))))).
% 2.88/1.52 thf(fact_162_min__dist__minD, axiom, (! [V:nat,D:nat,V3:nat]: (((((((dist @ a) @ c) @ V) @ D) @ V3) => (((ord_less_eq @ nat) @ ((((min_dist @ a) @ c) @ V) @ V3)) @ D))))).
% 2.88/1.52 thf(fact_163_min__distI__eq, axiom, (! [V:nat,D:nat,V3:nat]: (((((((dist @ a) @ c) @ V) @ D) @ V3) => ((! [D6:nat]: (((((((dist @ a) @ c) @ V) @ D6) @ V3) => (((ord_less_eq @ nat) @ D) @ D6)))) => (((((min_dist @ a) @ c) @ V) @ V3) = D)))))).
% 2.88/1.52 thf(fact_164_isPath__distD, axiom, (! [U:nat,P5:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isPath @ a) @ c) @ U) @ P5) @ V) => (((((dist @ a) @ c) @ U) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5)) @ V))))).
% 2.88/1.52 thf(fact_165_dist__def, axiom, (! [V:nat,D:nat,V3:nat]: (((((((dist @ a) @ c) @ V) @ D) @ V3) = (? [P3:(list @ ((product_prod @ nat) @ nat))]: (((((((isPath @ a) @ c) @ V) @ P3) @ V3) & (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P3) = D)))))))).
% 2.88/1.52 thf(fact_166_min__dist__is__dist, axiom, (! [V:nat,V3:nat]: ((((((connected @ a) @ c) @ V) @ V3) => (((((dist @ a) @ c) @ V) @ ((((min_dist @ a) @ c) @ V) @ V3)) @ V3))))).
% 2.88/1.52 thf(fact_167_g_H_OsimplePath__length__less__V, axiom, (! [U:nat,P5:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((member @ nat) @ U) @ ((v @ b) @ c2)) => ((((((isSimplePath @ b) @ c2) @ U) @ P5) @ V) => (((ord_less @ nat) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5)) @ ((finite_card @ nat) @ ((v @ b) @ c2)))))))).
% 2.88/1.52 thf(fact_168_g_H_Omin__dist__less__V, axiom, (! [S:nat,T:nat]: (((((member @ nat) @ S) @ ((v @ b) @ c2)) => (((((connected @ b) @ c2) @ S) @ T) => (((ord_less @ nat) @ ((((min_dist @ b) @ c2) @ S) @ T)) @ ((finite_card @ nat) @ ((v @ b) @ c2)))))))).
% 2.88/1.52 thf(fact_169_g_H_OisShortestPath__length__less__V, axiom, (! [S:nat,P5:(list @ ((product_prod @ nat) @ nat)),T:nat]: (((((member @ nat) @ S) @ ((v @ b) @ c2)) => ((((((isShortestPath @ b) @ c2) @ S) @ P5) @ T) => (((ord_less @ nat) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5)) @ ((finite_card @ nat) @ ((v @ b) @ c2)))))))).
% 2.88/1.52 thf(fact_170_isShortestPath__alt, axiom, (! [U:nat,P5:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isShortestPath @ a) @ c) @ U) @ P5) @ V) = ((((((isSimplePath @ a) @ c) @ U) @ P5) @ V) & (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5) = ((((min_dist @ a) @ c) @ U) @ V))))))).
% 2.88/1.52 thf(fact_171_g_H_Ofinite__V, axiom, ((finite_finite2 @ nat) @ ((v @ b) @ c2))).
% 2.88/1.52 thf(fact_172_finite__V, axiom, ((finite_finite2 @ nat) @ ((v @ a) @ c))).
% 2.88/1.52 thf(fact_173_dist__z__iff, axiom, (! [V:nat,V3:nat]: (((((((dist @ a) @ c) @ V) @ (zero_zero @ nat)) @ V3) = (V3 = V))))).
% 2.88/1.52 thf(fact_174_dist__z, axiom, (! [V:nat]: ((((((dist @ a) @ c) @ V) @ (zero_zero @ nat)) @ V)))).
% 2.88/1.52 thf(fact_175_connected__distI, axiom, (! [V:nat,D:nat,V3:nat]: (((((((dist @ a) @ c) @ V) @ D) @ V3) => ((((connected @ a) @ c) @ V) @ V3))))).
% 2.88/1.52 thf(fact_176_Veq, axiom, (((v @ b) @ c2) = ((v @ a) @ c))).
% 2.88/1.52 thf(fact_177_g_H_Oadjacent__nodes__ss__V, axiom, (! [U:nat]: ((((ord_less_eq @ (set @ nat)) @ (((adjacent_nodes @ b) @ c2) @ U)) @ ((v @ b) @ c2))))).
% 2.88/1.52 thf(fact_178_adjacent__nodes__ss__V, axiom, (! [U:nat]: ((((ord_less_eq @ (set @ nat)) @ (((adjacent_nodes @ a) @ c) @ U)) @ ((v @ a) @ c))))).
% 2.88/1.52 thf(fact_179_Graph_OisSimplePath_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isSimplePath @ Capacity) = (isSimplePath @ Capacity)))))).
% 2.88/1.52 thf(fact_180_finite__subset, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((finite_finite2 @ A) @ B3) => ((finite_finite2 @ A) @ A3)))))).
% 2.88/1.52 thf(fact_181_infinite__super, axiom, (! [A:$tType,S2:(set @ A),T2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S2) @ T2) => ((~ (((finite_finite2 @ A) @ S2))) => (~ (((finite_finite2 @ A) @ T2)))))))).
% 2.88/1.52 thf(fact_182_rev__finite__subset, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((finite_finite2 @ A) @ A3)))))).
% 2.88/1.52 thf(fact_183_card__subset__eq, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((finite_card @ A) @ A3) = ((finite_card @ A) @ B3)) => (A3 = B3))))))).
% 2.88/1.52 thf(fact_184_infinite__arbitrarily__large, axiom, (! [A:$tType,A3:(set @ A),N:nat]: (((~ (((finite_finite2 @ A) @ A3))) => (? [B4:(set @ A)]: ((((finite_finite2 @ A) @ B4) & ((((finite_card @ A) @ B4) = N) & (((ord_less_eq @ (set @ A)) @ B4) @ A3))))))))).
% 2.88/1.52 thf(fact_185_Graph_OisSPath__pathLE, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,P5:(list @ ((product_prod @ nat) @ nat)),T:nat]: (((((((isPath @ Capacity) @ C) @ S) @ P5) @ T) => (? [P:(list @ ((product_prod @ nat) @ nat))]: ((((((isSimplePath @ Capacity) @ C) @ S) @ P) @ T)))))))))).
% 2.88/1.52 thf(fact_186_Graph_OshortestPath__is__simple, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,P5:(list @ ((product_prod @ nat) @ nat)),T:nat]: (((((((isShortestPath @ Capacity) @ C) @ S) @ P5) @ T) => (((((isSimplePath @ Capacity) @ C) @ S) @ P5) @ T)))))))).
% 2.88/1.52 thf(fact_187_finite__maxlen, axiom, (! [A:$tType,M4:(set @ (list @ A))]: ((((finite_finite2 @ (list @ A)) @ M4) => (? [N3:nat]: ((! [X4:(list @ A)]: (((((member @ (list @ A)) @ X4) @ M4) => (((ord_less @ nat) @ ((size_size @ (list @ A)) @ X4)) @ N3)))))))))).
% 2.88/1.52 thf(fact_188_card__psubset, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3)) => (((ord_less @ (set @ A)) @ A3) @ B3))))))).
% 2.88/1.52 thf(fact_189_card__mono, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3))))))).
% 2.88/1.52 thf(fact_190_card__seteq, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ B3)) @ ((finite_card @ A) @ A3)) => (A3 = B3))))))).
% 2.88/1.52 thf(fact_191_finite__if__finite__subsets__card__bdd, axiom, (! [A:$tType,F2:(set @ A),C3:nat]: (((! [G2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ G2) @ F2) => (((finite_finite2 @ A) @ G2) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ G2)) @ C3))))) => (((finite_finite2 @ A) @ F2) & (((ord_less_eq @ nat) @ ((finite_card @ A) @ F2)) @ C3)))))).
% 2.88/1.52 thf(fact_192_Finite__Graph__def, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((finite_Graph @ A) = (^ [C2:(((product_prod @ nat) @ nat) > A)]: (((finite_finite2 @ nat) @ ((v @ A) @ C2))))))))).
% 2.88/1.52 thf(fact_193_Finite__Graph_Ointro, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A)]: ((((finite_finite2 @ nat) @ ((v @ A) @ C)) => ((finite_Graph @ A) @ C)))))))).
% 2.88/1.52 thf(fact_194_Finite__Graph_Ofinite__V, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A)]: ((((finite_Graph @ A) @ C) => ((finite_finite2 @ nat) @ ((v @ A) @ C))))))))).
% 2.88/1.52 thf(fact_195_finite, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [A3:(set @ A)]: (((finite_finite2 @ A) @ A3))))))).
% 2.88/1.52 thf(fact_196_finite__set__choice, axiom, (! [B:$tType,A:$tType,A3:(set @ A),P2:(A > (B > $o))]: ((((finite_finite2 @ A) @ A3) => ((! [X2:A]: (((((member @ A) @ X2) @ A3) => (? [X_1:B]: (((P2 @ X2) @ X_1)))))) => (? [F3:(A > B)]: ((! [X4:A]: (((((member @ A) @ X4) @ A3) => ((P2 @ X4) @ (F3 @ X4)))))))))))).
% 2.88/1.52 thf(fact_197_finite__psubset__induct, axiom, (! [A:$tType,A3:(set @ A),P2:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A3) => ((! [A5:(set @ A)]: ((((finite_finite2 @ A) @ A5) => ((! [B5:(set @ A)]: (((((ord_less @ (set @ A)) @ B5) @ A5) => (P2 @ B5)))) => (P2 @ A5))))) => (P2 @ A3)))))).
% 2.88/1.52 thf(fact_198_Graph_OisShortestPath__alt, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isShortestPath @ Capacity) = (^ [C2:(((product_prod @ nat) @ nat) > Capacity),U2:nat,P3:(list @ ((product_prod @ nat) @ nat)),V4:nat]: (((((((isSimplePath @ Capacity) @ C2) @ U2) @ P3) @ V4) & (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P3) = ((((min_dist @ Capacity) @ C2) @ U2) @ V4)))))))))).
% 2.88/1.52 thf(fact_199_ek__analysis_Ocard__spEdges__le, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,T:nat]: ((((edmond2129202899alysis @ A) @ C) => (((ord_less_eq @ nat) @ ((finite_card @ ((product_prod @ nat) @ nat)) @ ((((edmond1803015688pEdges @ A) @ C) @ S) @ T))) @ ((finite_card @ ((product_prod @ nat) @ nat)) @ ((edmond259086305sis_uE @ A) @ C)))))))))).
% 2.88/1.52 thf(fact_200_finite__has__minimal2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A2) @ A3) => (? [X2:A]: (((((member @ A) @ X2) @ A3) & ((((ord_less_eq @ A) @ X2) @ A2) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ Xa) @ X2) => (X2 = Xa))))))))))))))))).
% 2.88/1.52 thf(fact_201_finite__has__maximal2, axiom, (! [A:$tType]: (((order @ A) => (! [A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((((member @ A) @ A2) @ A3) => (? [X2:A]: (((((member @ A) @ X2) @ A3) & ((((ord_less_eq @ A) @ A2) @ X2) & (! [Xa:A]: (((((member @ A) @ Xa) @ A3) => ((((ord_less_eq @ A) @ X2) @ Xa) => (X2 = Xa))))))))))))))))).
% 2.88/1.52 thf(fact_202_Finite__Graph_OsimplePath__length__less__V, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),U:nat,P5:(list @ ((product_prod @ nat) @ nat)),V:nat]: ((((finite_Graph @ A) @ C) => ((((member @ nat) @ U) @ ((v @ A) @ C)) => ((((((isSimplePath @ A) @ C) @ U) @ P5) @ V) => (((ord_less @ nat) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5)) @ ((finite_card @ nat) @ ((v @ A) @ C)))))))))))).
% 2.88/1.52 thf(fact_203_psubset__card__mono, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less @ (set @ A)) @ A3) @ B3) => (((ord_less @ nat) @ ((finite_card @ A) @ A3)) @ ((finite_card @ A) @ B3))))))).
% 2.88/1.52 thf(fact_204_reachable__ss__V, axiom, (! [S:nat]: (((((member @ nat) @ S) @ ((v @ a) @ c)) => (((ord_less_eq @ (set @ nat)) @ (((reachableNodes @ a) @ c) @ S)) @ ((v @ a) @ c)))))).
% 2.88/1.52 thf(fact_205_g_H_Oreachable__ss__V, axiom, (! [S:nat]: (((((member @ nat) @ S) @ ((v @ b) @ c2)) => (((ord_less_eq @ (set @ nat)) @ (((reachableNodes @ b) @ c2) @ S)) @ ((v @ b) @ c2)))))).
% 2.88/1.52 thf(fact_206_spEdges__ss__E, axiom, (((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ ((((edmond1803015688pEdges @ a) @ c) @ s) @ t)) @ ((e @ a) @ c))).
% 2.88/1.52 thf(fact_207_E__ss__uE, axiom, (((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ ((e @ a) @ c)) @ ((edmond259086305sis_uE @ a) @ c))).
% 2.88/1.52 thf(fact_208_Finite__Graph__EI, axiom, (((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ a) @ c)) => ((finite_Graph @ a) @ c))).
% 2.88/1.52 thf(fact_209_Efin__imp__Vfin, axiom, (((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ a) @ c)) => ((finite_finite2 @ nat) @ ((v @ a) @ c)))).
% 2.88/1.52 thf(fact_210_finite__E, axiom, ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ a) @ c))).
% 2.88/1.52 thf(fact_211_g_H_Oadjacent__nodes__finite, axiom, (! [U:nat]: (((finite_finite2 @ nat) @ (((adjacent_nodes @ b) @ c2) @ U))))).
% 2.88/1.52 thf(fact_212_adjacent__nodes__finite, axiom, (! [U:nat]: (((finite_finite2 @ nat) @ (((adjacent_nodes @ a) @ c) @ U))))).
% 2.88/1.52 thf(fact_213_Vfin__imp__Efin, axiom, (((finite_finite2 @ nat) @ ((v @ a) @ c)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ a) @ c)))).
% 2.88/1.52 thf(fact_214_Graph_OreachableNodes_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((reachableNodes @ Capacity) = (reachableNodes @ Capacity)))))).
% 2.88/1.52 thf(fact_215_Graph_Oadjacent__nodes_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((adjacent_nodes @ Capacity) = (adjacent_nodes @ Capacity)))))).
% 2.88/1.52 thf(fact_216_Graph_OE_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((e @ Capacity) = (e @ Capacity)))))).
% 2.88/1.52 thf(fact_217_Graph_Oreachable__ss__V, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [S:nat,C:(((product_prod @ nat) @ nat) > Capacity)]: (((((member @ nat) @ S) @ ((v @ Capacity) @ C)) => (((ord_less_eq @ (set @ nat)) @ (((reachableNodes @ Capacity) @ C) @ S)) @ ((v @ Capacity) @ C))))))))).
% 2.88/1.52 thf(fact_218_Graph_OEfin__imp__Vfin, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity)]: ((((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ Capacity) @ C)) => ((finite_finite2 @ nat) @ ((v @ Capacity) @ C))))))))).
% 2.88/1.52 thf(fact_219_Graph_OVfin__imp__Efin, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity)]: ((((finite_finite2 @ nat) @ ((v @ Capacity) @ C)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ Capacity) @ C))))))))).
% 2.88/1.52 thf(fact_220_Graph_OFinite__Graph__EI, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity)]: ((((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ Capacity) @ C)) => ((finite_Graph @ Capacity) @ C)))))))).
% 2.88/1.52 thf(fact_221_Finite__Graph_Ofinite__E, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A)]: ((((finite_Graph @ A) @ C) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ A) @ C))))))))).
% 2.88/1.52 thf(fact_222_ek__analysis_OE__ss__uE, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A)]: ((((edmond2129202899alysis @ A) @ C) => (((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ ((e @ A) @ C)) @ ((edmond259086305sis_uE @ A) @ C))))))))).
% 2.88/1.52 thf(fact_223_ek__analysis_OspEdges__ss__E, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),S:nat,T:nat]: ((((edmond2129202899alysis @ A) @ C) => (((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ ((((edmond1803015688pEdges @ A) @ C) @ S) @ T)) @ ((e @ A) @ C))))))))).
% 2.88/1.52 thf(fact_224_Graph_Oadjacent__nodes__ss__V, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat]: ((((ord_less_eq @ (set @ nat)) @ (((adjacent_nodes @ Capacity) @ C) @ U)) @ ((v @ Capacity) @ C)))))))).
% 2.88/1.52 thf(fact_225_Finite__Graph_Oadjacent__nodes__finite, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C:(((product_prod @ nat) @ nat) > A),U:nat]: ((((finite_Graph @ A) @ C) => ((finite_finite2 @ nat) @ (((adjacent_nodes @ A) @ C) @ U))))))))).
% 2.88/1.52 thf(fact_226_incoming_H__edges, axiom, (! [U3:(set @ nat)]: ((((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ (((incoming @ a) @ c) @ U3)) @ ((e @ a) @ c))))).
% 2.88/1.52 thf(fact_227_outgoing_H__edges, axiom, (! [U3:(set @ nat)]: ((((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ (((outgoing @ a) @ c) @ U3)) @ ((e @ a) @ c))))).
% 2.88/1.52 thf(fact_228_g_H_OE__ss__uE, axiom, (((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ ((e @ b) @ c2)) @ ((edmond259086305sis_uE @ b) @ c2))).
% 2.88/1.52 thf(fact_229_g_H_OFinite__Graph__EI, axiom, (((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ b) @ c2)) => ((finite_Graph @ b) @ c2))).
% 2.88/1.52 thf(fact_230_g_H_OEfin__imp__Vfin, axiom, (((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ b) @ c2)) => ((finite_finite2 @ nat) @ ((v @ b) @ c2)))).
% 2.88/1.52 thf(fact_231_g_H_OspEdges__ss__E, axiom, (((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ ((((edmond1803015688pEdges @ b) @ c2) @ s) @ t)) @ ((e @ b) @ c2))).
% 2.88/1.52 thf(fact_232_g_H_Ofinite__E, axiom, ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ b) @ c2))).
% 2.88/1.52 thf(fact_233_g_H_OVfin__imp__Efin, axiom, (((finite_finite2 @ nat) @ ((v @ b) @ c2)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ ((e @ b) @ c2)))).
% 2.88/1.52 thf(fact_234_finite__outgoing_H, axiom, (! [U3:(set @ nat)]: ((((finite_finite2 @ nat) @ ((v @ a) @ c)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ (((outgoing @ a) @ c) @ U3)))))).
% 2.88/1.52 thf(fact_235_finite__incoming_H, axiom, (! [U3:(set @ nat)]: ((((finite_finite2 @ nat) @ ((v @ a) @ c)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ (((incoming @ a) @ c) @ U3)))))).
% 2.88/1.52 thf(fact_236_Graph_Oincoming_H_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((incoming @ Capacity) = (incoming @ Capacity)))))).
% 2.88/1.52 thf(fact_237_Graph_Ooutgoing_H_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((outgoing @ Capacity) = (outgoing @ Capacity)))))).
% 2.88/1.52 thf(fact_238_Graph_Ooutgoing_H__edges, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U3:(set @ nat)]: ((((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ (((outgoing @ Capacity) @ C) @ U3)) @ ((e @ Capacity) @ C)))))))).
% 2.88/1.52 thf(fact_239_Graph_Oincoming_H__edges, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U3:(set @ nat)]: ((((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ (((incoming @ Capacity) @ C) @ U3)) @ ((e @ Capacity) @ C)))))))).
% 2.88/1.52 thf(fact_240_Graph_Ofinite__outgoing_H, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U3:(set @ nat)]: ((((finite_finite2 @ nat) @ ((v @ Capacity) @ C)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ (((outgoing @ Capacity) @ C) @ U3))))))))).
% 2.88/1.52 thf(fact_241_Graph_Ofinite__incoming_H, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U3:(set @ nat)]: ((((finite_finite2 @ nat) @ ((v @ Capacity) @ C)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ (((incoming @ Capacity) @ C) @ U3))))))))).
% 2.88/1.52 thf(fact_242_g_H_Oincoming_H__edges, axiom, (! [U3:(set @ nat)]: ((((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ (((incoming @ b) @ c2) @ U3)) @ ((e @ b) @ c2))))).
% 2.88/1.52 thf(fact_243_g_H_Ooutgoing_H__edges, axiom, (! [U3:(set @ nat)]: ((((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ (((outgoing @ b) @ c2) @ U3)) @ ((e @ b) @ c2))))).
% 2.88/1.52 thf(fact_244_g_H_Ofinite__outgoing_H, axiom, (! [U3:(set @ nat)]: ((((finite_finite2 @ nat) @ ((v @ b) @ c2)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ (((outgoing @ b) @ c2) @ U3)))))).
% 2.88/1.52 thf(fact_245_g_H_Ofinite__incoming_H, axiom, (! [U3:(set @ nat)]: ((((finite_finite2 @ nat) @ ((v @ b) @ c2)) => ((finite_finite2 @ ((product_prod @ nat) @ nat)) @ (((incoming @ b) @ c2) @ U3)))))).
% 2.88/1.52 thf(fact_246__092_060open_062edges_A_092_060subseteq_062_AE_092_060close_062, axiom, (((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ edges) @ ((e @ a) @ c))).
% 2.88/1.52 thf(fact_247_psubsetI, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((A3 != B3) => (((ord_less @ (set @ A)) @ A3) @ B3)))))).
% 2.88/1.52 thf(fact_248_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => (A3 = B3)))))).
% 2.88/1.52 thf(fact_249_subsetI, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ A3) => (((member @ A) @ X2) @ B3)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 2.88/1.52 thf(fact_250_SP__EDGES, axiom, (((ord_less_eq @ (set @ ((product_prod @ nat) @ nat))) @ edges) @ ((set2 @ ((product_prod @ nat) @ nat)) @ p))).
% 2.88/1.52 thf(fact_251_Collect__mono__iff, axiom, (! [A:$tType,P2:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P2)) @ ((collect @ A) @ Q)) = (! [X:A]: (((P2 @ X) => (Q @ X)))))))).
% 2.88/1.52 thf(fact_252_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y4:(set @ A),Z:(set @ A)]: ((Y4 = Z))) = (^ [A4:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B6) & (((ord_less_eq @ (set @ A)) @ B6) @ A4)))))))).
% 2.88/1.52 thf(fact_253_subset__trans, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C3) => (((ord_less_eq @ (set @ A)) @ A3) @ C3)))))).
% 2.88/1.52 thf(subcl_Rings_Olinordered__idom___HOL_Otype, axiom, (! [A:$tType]: (((linordered_idom @ A) => (type @ A))))).
% 2.88/1.52 thf(subcl_Rings_Olinordered__idom___Groups_Ozero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (zero @ A))))).
% 2.88/1.52 thf(subcl_Rings_Olinordered__idom___Orderings_Oorder, axiom, (! [A:$tType]: (((linordered_idom @ A) => (order @ A))))).
% 2.88/1.52 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A6:$tType,A7:$tType]: ((((finite_finite @ A6) & (finite_finite @ A7)) => (finite_finite @ (A6 > A7)))))).
% 2.88/1.52 thf(tcon_fun___Orderings_Oorder, axiom, (! [A6:$tType,A7:$tType]: (((order @ A7) => (order @ (A6 > A7)))))).
% 2.88/1.52 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.88/1.52 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 2.88/1.52 thf(tcon_Nat_Onat___Orderings_Oorder_1, axiom, (order @ nat)).
% 2.88/1.52 thf(tcon_Nat_Onat___Groups_Ozero, axiom, (zero @ nat)).
% 2.88/1.52 thf(tcon_Nat_Onat___Nat_Osize, axiom, (size @ nat)).
% 2.88/1.52 thf(tcon_Set_Oset___Finite__Set_Ofinite_2, axiom, (! [A6:$tType]: (((finite_finite @ A6) => (finite_finite @ (set @ A6)))))).
% 2.88/1.52 thf(tcon_Set_Oset___Orderings_Oorder_3, axiom, (! [A6:$tType]: ((order @ (set @ A6))))).
% 2.88/1.52 thf(tcon_HOL_Obool___Finite__Set_Ofinite_4, axiom, (finite_finite @ $o)).
% 2.88/1.52 thf(tcon_HOL_Obool___Orderings_Oorder_5, axiom, (order @ $o)).
% 2.88/1.52 thf(tcon_List_Olist___Nat_Osize_6, axiom, (! [A6:$tType]: ((size @ (list @ A6))))).
% 2.88/1.52 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_7, axiom, (! [A6:$tType,A7:$tType]: ((((finite_finite @ A6) & (finite_finite @ A7)) => (finite_finite @ ((product_prod @ A6) @ A7)))))).
% 2.88/1.52 thf(tcon_Product__Type_Oprod___Nat_Osize_8, axiom, (! [A6:$tType,A7:$tType]: ((size @ ((product_prod @ A6) @ A7))))).
% 2.88/1.52 thf(tfree_0, hypothesis, (linordered_idom @ a)).
% 2.88/1.52 thf(tfree_1, hypothesis, (linordered_idom @ b)).
% 2.88/1.52 thf(conj_0, conjecture, $false).
% 2.88/1.52 % SZS output end ListOfTHF for /export/starexec/sandbox/tmp/tmp.AUIVmKJLkR/DTF2DTF13330.p
% 2.88/1.52 ------------------------
% 2.88/1.55 ---- Cleaned THF ---
% 2.88/1.55 thf(ty_t_Product__Type_Oprod,type,
% 2.88/1.55 product_prod: $tType > $tType > $tType ).
% 2.88/1.55
% 2.88/1.55 thf(ty_t_List_Olist,type,
% 2.88/1.55 list: $tType > $tType ).
% 2.88/1.55
% 2.88/1.55 thf(ty_t_Set_Oset,type,
% 2.88/1.55 set: $tType > $tType ).
% 2.88/1.55
% 2.88/1.55 thf(ty_t_Nat_Onat,type,
% 2.88/1.55 nat: $tType ).
% 2.88/1.55
% 2.88/1.55 thf(ty_tf_b,type,
% 2.88/1.55 b: $tType ).
% 2.88/1.55
% 2.88/1.55 thf(ty_tf_a,type,
% 2.88/1.55 a: $tType ).
% 2.88/1.55
% 2.88/1.55 thf(sy_cl_HOL_Otype,type,
% 2.88/1.55 type:
% 2.88/1.55 !>[A: $tType] : $o ).
% 2.88/1.55
% 2.88/1.55 thf(sy_cl_Nat_Osize,type,
% 2.88/1.55 size:
% 2.88/1.55 !>[A: $tType] : $o ).
% 2.88/1.55
% 2.88/1.55 thf(sy_cl_Groups_Ozero,type,
% 2.88/1.55 zero:
% 2.88/1.55 !>[A: $tType] : $o ).
% 2.88/1.55
% 2.88/1.55 thf(sy_cl_Orderings_Oorder,type,
% 2.88/1.55 order:
% 2.88/1.55 !>[A: $tType] : $o ).
% 2.88/1.55
% 2.88/1.55 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.88/1.55 finite_finite:
% 2.88/1.55 !>[A: $tType] : $o ).
% 2.88/1.55
% 2.88/1.55 thf(sy_cl_Orderings_Owellorder,type,
% 2.88/1.55 wellorder:
% 2.88/1.55 !>[A: $tType] : $o ).
% 2.88/1.55
% 2.88/1.55 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.88/1.55 linordered_idom:
% 2.88/1.55 !>[A: $tType] : $o ).
% 2.88/1.55
% 2.88/1.55 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.88/1.55 canoni770627133id_add:
% 2.88/1.55 !>[A: $tType] : $o ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis,type,
% 2.88/1.55 edmond2129202899alysis:
% 2.88/1.55 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > $o ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis__defs_OekMeasure,type,
% 2.88/1.55 edmond115586663easure:
% 2.88/1.55 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > nat > nat > nat ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis__defs_OspEdges,type,
% 2.88/1.55 edmond1803015688pEdges:
% 2.88/1.55 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > nat > nat > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_EdmondsKarp__Termination__Abstract__Mirabelle__klniualtan_Oek__analysis__defs_OuE,type,
% 2.88/1.55 edmond259086305sis_uE:
% 2.88/1.55 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Finite__Set_Ocard,type,
% 2.88/1.55 finite_card:
% 2.88/1.55 !>[B: $tType] : ( ( set @ B ) > nat ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Finite__Set_Ofinite,type,
% 2.88/1.55 finite_finite2:
% 2.88/1.55 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Graph_OFinite__Graph,type,
% 2.88/1.55 finite_Graph:
% 2.88/1.55 !>[A: $tType] : ( ( ( product_prod @ nat @ nat ) > A ) > $o ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Graph_OGraph_OE,type,
% 2.88/1.55 e:
% 2.88/1.55 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Graph_OGraph_OV,type,
% 2.88/1.55 v:
% 2.88/1.55 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Graph_OGraph_Oadjacent__nodes,type,
% 2.88/1.55 adjacent_nodes:
% 2.88/1.55 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ nat ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Graph_OGraph_Oconnected,type,
% 2.88/1.55 connected:
% 2.88/1.55 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > $o ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Graph_OGraph_Odist,type,
% 2.88/1.55 dist:
% 2.88/1.55 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > nat > $o ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Graph_OGraph_Oincoming_H,type,
% 2.88/1.55 incoming:
% 2.88/1.55 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Graph_OGraph_OisPath,type,
% 2.88/1.55 isPath:
% 2.88/1.55 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Graph_OGraph_OisShortestPath,type,
% 2.88/1.55 isShortestPath:
% 2.88/1.55 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Graph_OGraph_OisSimplePath,type,
% 2.88/1.55 isSimplePath:
% 2.88/1.55 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Graph_OGraph_Omin__dist,type,
% 2.88/1.55 min_dist:
% 2.88/1.55 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > nat ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Graph_OGraph_Ooutgoing_H,type,
% 2.88/1.55 outgoing:
% 2.88/1.55 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > ( set @ nat ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Graph_OGraph_OreachableNodes,type,
% 2.88/1.55 reachableNodes:
% 2.88/1.55 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ nat ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 2.88/1.55 zero_zero:
% 2.88/1.55 !>[A: $tType] : A ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_List_Olist_Oset,type,
% 2.88/1.55 set2:
% 2.88/1.55 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Nat_Osize__class_Osize,type,
% 2.88/1.55 size_size:
% 2.88/1.55 !>[A: $tType] : ( A > nat ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.88/1.55 ord_less:
% 2.88/1.55 !>[A: $tType] : ( A > A > $o ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.88/1.55 ord_less_eq:
% 2.88/1.55 !>[A: $tType] : ( A > A > $o ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_Set_OCollect,type,
% 2.88/1.55 collect:
% 2.88/1.55 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_c_member,type,
% 2.88/1.55 member:
% 2.88/1.55 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_v_c,type,
% 2.88/1.55 c: ( product_prod @ nat @ nat ) > a ).
% 2.88/1.55
% 2.88/1.55 thf(sy_v_c_H,type,
% 2.88/1.55 c2: ( product_prod @ nat @ nat ) > b ).
% 2.88/1.55
% 2.88/1.55 thf(sy_v_edges,type,
% 2.88/1.55 edges: set @ ( product_prod @ nat @ nat ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_v_p,type,
% 2.88/1.55 p: list @ ( product_prod @ nat @ nat ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_v_p_H____,type,
% 2.88/1.55 p2: list @ ( product_prod @ nat @ nat ) ).
% 2.88/1.55
% 2.88/1.55 thf(sy_v_s,type,
% 2.88/1.55 s: nat ).
% 2.88/1.55
% 2.88/1.55 thf(sy_v_t,type,
% 2.88/1.55 t: nat ).
% 2.88/1.55
% 2.88/1.55 thf(fact_0_ek__analysis__axioms,axiom,
% 2.88/1.55 edmond2129202899alysis @ a @ c ).
% 2.88/1.55
% 2.88/1.55 thf(fact_1_LENP,axiom,
% 2.88/1.55 ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p )
% 2.88/1.55 = ( min_dist @ a @ c @ s @ t ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_2_LENP_H,axiom,
% 2.88/1.55 ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 )
% 2.88/1.55 = ( min_dist @ b @ c2 @ s @ t ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_3_ek__analysis__defs_OekMeasure_Ocong,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ( ( edmond115586663easure @ A )
% 2.88/1.55 = ( edmond115586663easure @ A ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_4_g_H_Oek__analysis__axioms,axiom,
% 2.88/1.55 edmond2129202899alysis @ b @ c2 ).
% 2.88/1.55
% 2.88/1.55 thf(fact_5_SHORTER,axiom,
% 2.88/1.55 ord_less @ nat @ ( min_dist @ b @ c2 @ s @ t ) @ ( min_dist @ a @ c @ s @ t ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_6_ek__analysis__defs_OuE_Ocong,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ( ( edmond259086305sis_uE @ A )
% 2.88/1.55 = ( edmond259086305sis_uE @ A ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_7_ek__analysis__defs_OspEdges_Ocong,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ( ( edmond1803015688pEdges @ A )
% 2.88/1.55 = ( edmond1803015688pEdges @ A ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_8_g_H_OFinite__Graph__axioms,axiom,
% 2.88/1.55 finite_Graph @ b @ c2 ).
% 2.88/1.55
% 2.88/1.55 thf(fact_9_Finite__Graph__axioms,axiom,
% 2.88/1.55 finite_Graph @ a @ c ).
% 2.88/1.55
% 2.88/1.55 thf(fact_10__092_060open_062length_Ap_A_060_Alength_Ap_H_092_060close_062,axiom,
% 2.88/1.55 ord_less @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_11_uE__eq,axiom,
% 2.88/1.55 ( ( edmond259086305sis_uE @ b @ c2 )
% 2.88/1.55 = ( edmond259086305sis_uE @ a @ c ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_12_SP,axiom,
% 2.88/1.55 isShortestPath @ a @ c @ s @ p @ t ).
% 2.88/1.55
% 2.88/1.55 thf(fact_13_CONN2,axiom,
% 2.88/1.55 connected @ b @ c2 @ s @ t ).
% 2.88/1.55
% 2.88/1.55 thf(fact_14_SV,axiom,
% 2.88/1.55 member @ nat @ s @ ( v @ a @ c ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_15_P_H,axiom,
% 2.88/1.55 isPath @ b @ c2 @ s @ p2 @ t ).
% 2.88/1.55
% 2.88/1.55 thf(fact_16_ek__analysis__def,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ( ( edmond2129202899alysis @ A )
% 2.88/1.55 = ( finite_Graph @ A ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_17_ek__analysis_Ointro,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > A] :
% 2.88/1.55 ( ( finite_Graph @ A @ C )
% 2.88/1.55 => ( edmond2129202899alysis @ A @ C ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_18_ek__analysis_Oaxioms,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > A] :
% 2.88/1.55 ( ( edmond2129202899alysis @ A @ C )
% 2.88/1.55 => ( finite_Graph @ A @ C ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_19__092_060open_062_092_060not_062_Ag_H_Oconnected_As_At_A_092_060Longrightarrow_062_Ag_H_OekMeasure_A_060_AekMeasure_092_060close_062,axiom,
% 2.88/1.55 ( ~ ( connected @ b @ c2 @ s @ t )
% 2.88/1.55 => ( ord_less @ nat @ ( edmond115586663easure @ b @ c2 @ s @ t ) @ ( edmond115586663easure @ a @ c @ s @ t ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_20__092_060open_062g_H_Odist_As_A_Ig_H_Omin__dist_As_At_J_At_092_060close_062,axiom,
% 2.88/1.55 dist @ b @ c2 @ s @ ( min_dist @ b @ c2 @ s @ t ) @ t ).
% 2.88/1.55
% 2.88/1.55 thf(fact_21__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062p_H_O_A_092_060lbrakk_062g_H_OisPath_As_Ap_H_At_059_Alength_Ap_H_A_061_Ag_H_Omin__dist_As_At_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 2.88/1.55 ~ ! [P: list @ ( product_prod @ nat @ nat )] :
% 2.88/1.55 ( ( isPath @ b @ c2 @ s @ P @ t )
% 2.88/1.55 => ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P )
% 2.88/1.55 != ( min_dist @ b @ c2 @ s @ t ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_22_CONN,axiom,
% 2.88/1.55 connected @ a @ c @ s @ t ).
% 2.88/1.55
% 2.88/1.55 thf(fact_23_min__dist__less,axiom,
% 2.88/1.55 ! [Src: nat,V: nat,D: nat,D2: nat] :
% 2.88/1.55 ( ( connected @ a @ c @ Src @ V )
% 2.88/1.55 => ( ( ( min_dist @ a @ c @ Src @ V )
% 2.88/1.55 = D )
% 2.88/1.55 => ( ( ord_less @ nat @ D2 @ D )
% 2.88/1.55 => ? [V2: nat] :
% 2.88/1.55 ( ( connected @ a @ c @ Src @ V2 )
% 2.88/1.55 & ( ( min_dist @ a @ c @ Src @ V2 )
% 2.88/1.55 = D2 ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_24_g_H_Omin__dist__less,axiom,
% 2.88/1.55 ! [Src: nat,V: nat,D: nat,D2: nat] :
% 2.88/1.55 ( ( connected @ b @ c2 @ Src @ V )
% 2.88/1.55 => ( ( ( min_dist @ b @ c2 @ Src @ V )
% 2.88/1.55 = D )
% 2.88/1.55 => ( ( ord_less @ nat @ D2 @ D )
% 2.88/1.55 => ? [V2: nat] :
% 2.88/1.55 ( ( connected @ b @ c2 @ Src @ V2 )
% 2.88/1.55 & ( ( min_dist @ b @ c2 @ Src @ V2 )
% 2.88/1.55 = D2 ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_25_length__induct,axiom,
% 2.88/1.55 ! [A: $tType,P2: ( list @ A ) > $o,Xs: list @ A] :
% 2.88/1.55 ( ! [Xs2: list @ A] :
% 2.88/1.55 ( ! [Ys: list @ A] :
% 2.88/1.55 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
% 2.88/1.55 => ( P2 @ Ys ) )
% 2.88/1.55 => ( P2 @ Xs2 ) )
% 2.88/1.55 => ( P2 @ Xs ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_26_finite__spEdges,axiom,
% 2.88/1.55 finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_27_g_H_Ofinite__spEdges,axiom,
% 2.88/1.55 finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ b @ c2 @ s @ t ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_28_min__dist__z,axiom,
% 2.88/1.55 ! [V: nat] :
% 2.88/1.55 ( ( min_dist @ a @ c @ V @ V )
% 2.88/1.55 = ( zero_zero @ nat ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_29_g_H_Omin__dist__z,axiom,
% 2.88/1.55 ! [V: nat] :
% 2.88/1.55 ( ( min_dist @ b @ c2 @ V @ V )
% 2.88/1.55 = ( zero_zero @ nat ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_30_g_H_Oconnected__def,axiom,
% 2.88/1.55 ! [U: nat,V: nat] :
% 2.88/1.55 ( ( connected @ b @ c2 @ U @ V )
% 2.88/1.55 = ( ? [P3: list @ ( product_prod @ nat @ nat )] : ( isPath @ b @ c2 @ U @ P3 @ V ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_31_connected__inV__iff,axiom,
% 2.88/1.55 ! [U: nat,V: nat] :
% 2.88/1.55 ( ( connected @ a @ c @ U @ V )
% 2.88/1.55 => ( ( member @ nat @ V @ ( v @ a @ c ) )
% 2.88/1.55 = ( member @ nat @ U @ ( v @ a @ c ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_32_g_H_Oconnected__by__dist,axiom,
% 2.88/1.55 ! [V: nat,V3: nat] :
% 2.88/1.55 ( ( connected @ b @ c2 @ V @ V3 )
% 2.88/1.55 = ( ? [D3: nat] : ( dist @ b @ c2 @ V @ D3 @ V3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_33_obtain__shortest__path,axiom,
% 2.88/1.55 ! [U: nat,V: nat] :
% 2.88/1.55 ( ( connected @ a @ c @ U @ V )
% 2.88/1.55 => ~ ! [P4: list @ ( product_prod @ nat @ nat )] :
% 2.88/1.55 ~ ( isShortestPath @ a @ c @ U @ P4 @ V ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_34_g_H_OisPath__distD,axiom,
% 2.88/1.55 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.88/1.55 ( ( isPath @ b @ c2 @ U @ P5 @ V )
% 2.88/1.55 => ( dist @ b @ c2 @ U @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ V ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_35_g_H_Odist__def,axiom,
% 2.88/1.55 ! [V: nat,D: nat,V3: nat] :
% 2.88/1.55 ( ( dist @ b @ c2 @ V @ D @ V3 )
% 2.88/1.55 = ( ? [P3: list @ ( product_prod @ nat @ nat )] :
% 2.88/1.55 ( ( isPath @ b @ c2 @ V @ P3 @ V3 )
% 2.88/1.55 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P3 )
% 2.88/1.55 = D ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_36_g_H_Omin__dist__is__dist,axiom,
% 2.88/1.55 ! [V: nat,V3: nat] :
% 2.88/1.55 ( ( connected @ b @ c2 @ V @ V3 )
% 2.88/1.55 => ( dist @ b @ c2 @ V @ ( min_dist @ b @ c2 @ V @ V3 ) @ V3 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_37_g_H_Oconnected__refl,axiom,
% 2.88/1.55 ! [V: nat] : ( connected @ b @ c2 @ V @ V ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_38_connected__refl,axiom,
% 2.88/1.55 ! [V: nat] : ( connected @ a @ c @ V @ V ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_39_g_H_Odist__z__iff,axiom,
% 2.88/1.55 ! [V: nat,V3: nat] :
% 2.88/1.55 ( ( dist @ b @ c2 @ V @ ( zero_zero @ nat ) @ V3 )
% 2.88/1.55 = ( V3 = V ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_40_g_H_Odist__z,axiom,
% 2.88/1.55 ! [V: nat] : ( dist @ b @ c2 @ V @ ( zero_zero @ nat ) @ V ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_41_g_H_Oconnected__distI,axiom,
% 2.88/1.55 ! [V: nat,D: nat,V3: nat] :
% 2.88/1.55 ( ( dist @ b @ c2 @ V @ D @ V3 )
% 2.88/1.55 => ( connected @ b @ c2 @ V @ V3 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_42_g_H_Ofinite__uE,axiom,
% 2.88/1.55 finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond259086305sis_uE @ b @ c2 ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_43_finite__uE,axiom,
% 2.88/1.55 finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond259086305sis_uE @ a @ c ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_44_g_H_Omin__dist__z__iff,axiom,
% 2.88/1.55 ! [V: nat,V3: nat] :
% 2.88/1.55 ( ( connected @ b @ c2 @ V @ V3 )
% 2.88/1.55 => ( ( ( min_dist @ b @ c2 @ V @ V3 )
% 2.88/1.55 = ( zero_zero @ nat ) )
% 2.88/1.55 = ( V3 = V ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_45_mem__Collect__eq,axiom,
% 2.88/1.55 ! [A: $tType,A2: A,P2: A > $o] :
% 2.88/1.55 ( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
% 2.88/1.55 = ( P2 @ A2 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_46_Collect__mem__eq,axiom,
% 2.88/1.55 ! [A: $tType,A3: set @ A] :
% 2.88/1.55 ( ( collect @ A
% 2.88/1.55 @ ^ [X: A] : ( member @ A @ X @ A3 ) )
% 2.88/1.55 = A3 ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_47_Collect__cong,axiom,
% 2.88/1.55 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.88/1.55 ( ! [X2: A] :
% 2.88/1.55 ( ( P2 @ X2 )
% 2.88/1.55 = ( Q @ X2 ) )
% 2.88/1.55 => ( ( collect @ A @ P2 )
% 2.88/1.55 = ( collect @ A @ Q ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_48_ext,axiom,
% 2.88/1.55 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.88/1.55 ( ! [X2: A] :
% 2.88/1.55 ( ( F @ X2 )
% 2.88/1.55 = ( G @ X2 ) )
% 2.88/1.55 => ( F = G ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_49_min__dist__z__iff,axiom,
% 2.88/1.55 ! [V: nat,V3: nat] :
% 2.88/1.55 ( ( connected @ a @ c @ V @ V3 )
% 2.88/1.55 => ( ( ( min_dist @ a @ c @ V @ V3 )
% 2.88/1.55 = ( zero_zero @ nat ) )
% 2.88/1.55 = ( V3 = V ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_50_ek__analysis_Ofinite__uE,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > A] :
% 2.88/1.55 ( ( edmond2129202899alysis @ A @ C )
% 2.88/1.55 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond259086305sis_uE @ A @ C ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_51_ek__analysis_Ofinite__spEdges,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat] :
% 2.88/1.55 ( ( edmond2129202899alysis @ A @ C )
% 2.88/1.55 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ A @ C @ S @ T ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_52_neq__if__length__neq,axiom,
% 2.88/1.55 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 2.88/1.55 ( ( ( size_size @ ( list @ A ) @ Xs )
% 2.88/1.55 != ( size_size @ ( list @ A ) @ Ys2 ) )
% 2.88/1.55 => ( Xs != Ys2 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_53_Ex__list__of__length,axiom,
% 2.88/1.55 ! [A: $tType,N: nat] :
% 2.88/1.55 ? [Xs2: list @ A] :
% 2.88/1.55 ( ( size_size @ ( list @ A ) @ Xs2 )
% 2.88/1.55 = N ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_54_bot__nat__0_Onot__eq__extremum,axiom,
% 2.88/1.55 ! [A2: nat] :
% 2.88/1.55 ( ( A2
% 2.88/1.55 != ( zero_zero @ nat ) )
% 2.88/1.55 = ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_55_less__nat__zero__code,axiom,
% 2.88/1.55 ! [N: nat] :
% 2.88/1.55 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_56_neq0__conv,axiom,
% 2.88/1.55 ! [N: nat] :
% 2.88/1.55 ( ( N
% 2.88/1.55 != ( zero_zero @ nat ) )
% 2.88/1.55 = ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_57_not__gr__zero,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( canoni770627133id_add @ A )
% 2.88/1.55 => ! [N: A] :
% 2.88/1.55 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 2.88/1.55 = ( N
% 2.88/1.55 = ( zero_zero @ A ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_58_Graph_OisShortestPath__min__dist__def,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( isShortestPath @ Capacity )
% 2.88/1.55 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U2: nat,P3: list @ ( product_prod @ nat @ nat ),V4: nat] :
% 2.88/1.55 ( ( isPath @ Capacity @ C2 @ U2 @ P3 @ V4 )
% 2.88/1.55 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P3 )
% 2.88/1.55 = ( min_dist @ Capacity @ C2 @ U2 @ V4 ) ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_59_g_H_OisShortestPath__min__dist__def,axiom,
% 2.88/1.55 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.88/1.55 ( ( isShortestPath @ b @ c2 @ U @ P5 @ V )
% 2.88/1.55 = ( ( isPath @ b @ c2 @ U @ P5 @ V )
% 2.88/1.55 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 )
% 2.88/1.55 = ( min_dist @ b @ c2 @ U @ V ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_60_g_H_Omin__distI2,axiom,
% 2.88/1.55 ! [V: nat,V3: nat,Q: nat > $o] :
% 2.88/1.55 ( ( connected @ b @ c2 @ V @ V3 )
% 2.88/1.55 => ( ! [D4: nat] :
% 2.88/1.55 ( ( dist @ b @ c2 @ V @ D4 @ V3 )
% 2.88/1.55 => ( ! [D5: nat] :
% 2.88/1.55 ( ( dist @ b @ c2 @ V @ D5 @ V3 )
% 2.88/1.55 => ( ord_less_eq @ nat @ D4 @ D5 ) )
% 2.88/1.55 => ( Q @ D4 ) ) )
% 2.88/1.55 => ( Q @ ( min_dist @ b @ c2 @ V @ V3 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_61_isShortestPath__length__less__V,axiom,
% 2.88/1.55 ! [S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 2.88/1.55 ( ( member @ nat @ S @ ( v @ a @ c ) )
% 2.88/1.55 => ( ( isShortestPath @ a @ c @ S @ P5 @ T )
% 2.88/1.55 => ( ord_less @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ ( finite_card @ nat @ ( v @ a @ c ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_62_min__dist__less__V,axiom,
% 2.88/1.55 ! [S: nat,T: nat] :
% 2.88/1.55 ( ( member @ nat @ S @ ( v @ a @ c ) )
% 2.88/1.55 => ( ( connected @ a @ c @ S @ T )
% 2.88/1.55 => ( ord_less @ nat @ ( min_dist @ a @ c @ S @ T ) @ ( finite_card @ nat @ ( v @ a @ c ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_63_connected__def,axiom,
% 2.88/1.55 ! [U: nat,V: nat] :
% 2.88/1.55 ( ( connected @ a @ c @ U @ V )
% 2.88/1.55 = ( ? [P3: list @ ( product_prod @ nat @ nat )] : ( isPath @ a @ c @ U @ P3 @ V ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_64_shortestPath__is__path,axiom,
% 2.88/1.55 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.88/1.55 ( ( isShortestPath @ a @ c @ U @ P5 @ V )
% 2.88/1.55 => ( isPath @ a @ c @ U @ P5 @ V ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_65_g_H_OshortestPath__is__path,axiom,
% 2.88/1.55 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.88/1.55 ( ( isShortestPath @ b @ c2 @ U @ P5 @ V )
% 2.88/1.55 => ( isPath @ b @ c2 @ U @ P5 @ V ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_66_g_H_Oobtain__shortest__path,axiom,
% 2.88/1.55 ! [U: nat,V: nat] :
% 2.88/1.55 ( ( connected @ b @ c2 @ U @ V )
% 2.88/1.55 => ~ ! [P4: list @ ( product_prod @ nat @ nat )] :
% 2.88/1.55 ~ ( isShortestPath @ b @ c2 @ U @ P4 @ V ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_67_g_H_Omin__dist__le,axiom,
% 2.88/1.55 ! [Src: nat,V: nat,D2: nat] :
% 2.88/1.55 ( ( connected @ b @ c2 @ Src @ V )
% 2.88/1.55 => ( ( ord_less_eq @ nat @ D2 @ ( min_dist @ b @ c2 @ Src @ V ) )
% 2.88/1.55 => ? [V2: nat] :
% 2.88/1.55 ( ( connected @ b @ c2 @ Src @ V2 )
% 2.88/1.55 & ( ( min_dist @ b @ c2 @ Src @ V2 )
% 2.88/1.55 = D2 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_68_g_H_Omin__distI__eq,axiom,
% 2.88/1.55 ! [V: nat,D: nat,V3: nat] :
% 2.88/1.55 ( ( dist @ b @ c2 @ V @ D @ V3 )
% 2.88/1.55 => ( ! [D6: nat] :
% 2.88/1.55 ( ( dist @ b @ c2 @ V @ D6 @ V3 )
% 2.88/1.55 => ( ord_less_eq @ nat @ D @ D6 ) )
% 2.88/1.55 => ( ( min_dist @ b @ c2 @ V @ V3 )
% 2.88/1.55 = D ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_69_g_H_Omin__dist__minD,axiom,
% 2.88/1.55 ! [V: nat,D: nat,V3: nat] :
% 2.88/1.55 ( ( dist @ b @ c2 @ V @ D @ V3 )
% 2.88/1.55 => ( ord_less_eq @ nat @ ( min_dist @ b @ c2 @ V @ V3 ) @ D ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_70_min__dist__le,axiom,
% 2.88/1.55 ! [Src: nat,V: nat,D2: nat] :
% 2.88/1.55 ( ( connected @ a @ c @ Src @ V )
% 2.88/1.55 => ( ( ord_less_eq @ nat @ D2 @ ( min_dist @ a @ c @ Src @ V ) )
% 2.88/1.55 => ? [V2: nat] :
% 2.88/1.55 ( ( connected @ a @ c @ Src @ V2 )
% 2.88/1.55 & ( ( min_dist @ a @ c @ Src @ V2 )
% 2.88/1.55 = D2 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_71_isShortestPath__def,axiom,
% 2.88/1.55 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.88/1.55 ( ( isShortestPath @ a @ c @ U @ P5 @ V )
% 2.88/1.55 = ( ( isPath @ a @ c @ U @ P5 @ V )
% 2.88/1.55 & ! [P6: list @ ( product_prod @ nat @ nat )] :
% 2.88/1.55 ( ( isPath @ a @ c @ U @ P6 @ V )
% 2.88/1.55 => ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P6 ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_72_isShortestPath__min__dist__def,axiom,
% 2.88/1.55 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.88/1.55 ( ( isShortestPath @ a @ c @ U @ P5 @ V )
% 2.88/1.55 = ( ( isPath @ a @ c @ U @ P5 @ V )
% 2.88/1.55 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 )
% 2.88/1.55 = ( min_dist @ a @ c @ U @ V ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_73_g_H_OisShortestPath__def,axiom,
% 2.88/1.55 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.88/1.55 ( ( isShortestPath @ b @ c2 @ U @ P5 @ V )
% 2.88/1.55 = ( ( isPath @ b @ c2 @ U @ P5 @ V )
% 2.88/1.55 & ! [P6: list @ ( product_prod @ nat @ nat )] :
% 2.88/1.55 ( ( isPath @ b @ c2 @ U @ P6 @ V )
% 2.88/1.55 => ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P6 ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_74_le__zero__eq,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( canoni770627133id_add @ A )
% 2.88/1.55 => ! [N: A] :
% 2.88/1.55 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 2.88/1.55 = ( N
% 2.88/1.55 = ( zero_zero @ A ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_75_le0,axiom,
% 2.88/1.55 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_76_bot__nat__0_Oextremum,axiom,
% 2.88/1.55 ! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_77_zero__le,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( canoni770627133id_add @ A )
% 2.88/1.55 => ! [X3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_78_le__refl,axiom,
% 2.88/1.55 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_79_le__trans,axiom,
% 2.88/1.55 ! [I: nat,J: nat,K: nat] :
% 2.88/1.55 ( ( ord_less_eq @ nat @ I @ J )
% 2.88/1.55 => ( ( ord_less_eq @ nat @ J @ K )
% 2.88/1.55 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_80_eq__imp__le,axiom,
% 2.88/1.55 ! [M: nat,N: nat] :
% 2.88/1.55 ( ( M = N )
% 2.88/1.55 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_81_le__antisym,axiom,
% 2.88/1.55 ! [M: nat,N: nat] :
% 2.88/1.55 ( ( ord_less_eq @ nat @ M @ N )
% 2.88/1.55 => ( ( ord_less_eq @ nat @ N @ M )
% 2.88/1.55 => ( M = N ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_82_nat__le__linear,axiom,
% 2.88/1.55 ! [M: nat,N: nat] :
% 2.88/1.55 ( ( ord_less_eq @ nat @ M @ N )
% 2.88/1.55 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_83_Nat_Oex__has__greatest__nat,axiom,
% 2.88/1.55 ! [P2: nat > $o,K: nat,B2: nat] :
% 2.88/1.55 ( ( P2 @ K )
% 2.88/1.55 => ( ! [Y: nat] :
% 2.88/1.55 ( ( P2 @ Y )
% 2.88/1.55 => ( ord_less_eq @ nat @ Y @ B2 ) )
% 2.88/1.55 => ? [X2: nat] :
% 2.88/1.55 ( ( P2 @ X2 )
% 2.88/1.55 & ! [Y2: nat] :
% 2.88/1.55 ( ( P2 @ Y2 )
% 2.88/1.55 => ( ord_less_eq @ nat @ Y2 @ X2 ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_84_less__eq__nat_Osimps_I1_J,axiom,
% 2.88/1.55 ! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_85_le__0__eq,axiom,
% 2.88/1.55 ! [N: nat] :
% 2.88/1.55 ( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
% 2.88/1.55 = ( N
% 2.88/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_86_bot__nat__0_Oextremum__unique,axiom,
% 2.88/1.55 ! [A2: nat] :
% 2.88/1.55 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.88/1.55 = ( A2
% 2.88/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_87_bot__nat__0_Oextremum__uniqueI,axiom,
% 2.88/1.55 ! [A2: nat] :
% 2.88/1.55 ( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
% 2.88/1.55 => ( A2
% 2.88/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_88_nat__less__le,axiom,
% 2.88/1.55 ( ( ord_less @ nat )
% 2.88/1.55 = ( ^ [M2: nat,N2: nat] :
% 2.88/1.55 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.88/1.55 & ( M2 != N2 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_89_less__imp__le__nat,axiom,
% 2.88/1.55 ! [M: nat,N: nat] :
% 2.88/1.55 ( ( ord_less @ nat @ M @ N )
% 2.88/1.55 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_90_le__eq__less__or__eq,axiom,
% 2.88/1.55 ( ( ord_less_eq @ nat )
% 2.88/1.55 = ( ^ [M2: nat,N2: nat] :
% 2.88/1.55 ( ( ord_less @ nat @ M2 @ N2 )
% 2.88/1.55 | ( M2 = N2 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_91_less__or__eq__imp__le,axiom,
% 2.88/1.55 ! [M: nat,N: nat] :
% 2.88/1.55 ( ( ( ord_less @ nat @ M @ N )
% 2.88/1.55 | ( M = N ) )
% 2.88/1.55 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_92_le__neq__implies__less,axiom,
% 2.88/1.55 ! [M: nat,N: nat] :
% 2.88/1.55 ( ( ord_less_eq @ nat @ M @ N )
% 2.88/1.55 => ( ( M != N )
% 2.88/1.55 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_93_less__mono__imp__le__mono,axiom,
% 2.88/1.55 ! [F: nat > nat,I: nat,J: nat] :
% 2.88/1.55 ( ! [I2: nat,J2: nat] :
% 2.88/1.55 ( ( ord_less @ nat @ I2 @ J2 )
% 2.88/1.55 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 2.88/1.55 => ( ( ord_less_eq @ nat @ I @ J )
% 2.88/1.55 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_94_ex__least__nat__le,axiom,
% 2.88/1.55 ! [P2: nat > $o,N: nat] :
% 2.88/1.55 ( ( P2 @ N )
% 2.88/1.55 => ( ~ ( P2 @ ( zero_zero @ nat ) )
% 2.88/1.55 => ? [K2: nat] :
% 2.88/1.55 ( ( ord_less_eq @ nat @ K2 @ N )
% 2.88/1.55 & ! [I3: nat] :
% 2.88/1.55 ( ( ord_less @ nat @ I3 @ K2 )
% 2.88/1.55 => ~ ( P2 @ I3 ) )
% 2.88/1.55 & ( P2 @ K2 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_95_Graph_Omin__dist__le,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,Src: nat,V: nat,D2: nat] :
% 2.88/1.55 ( ( connected @ Capacity @ C @ Src @ V )
% 2.88/1.55 => ( ( ord_less_eq @ nat @ D2 @ ( min_dist @ Capacity @ C @ Src @ V ) )
% 2.88/1.55 => ? [V2: nat] :
% 2.88/1.55 ( ( connected @ Capacity @ C @ Src @ V2 )
% 2.88/1.55 & ( ( min_dist @ Capacity @ C @ Src @ V2 )
% 2.88/1.55 = D2 ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_96_Graph_Omin__dist__minD,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,D: nat,V3: nat] :
% 2.88/1.55 ( ( dist @ Capacity @ C @ V @ D @ V3 )
% 2.88/1.55 => ( ord_less_eq @ nat @ ( min_dist @ Capacity @ C @ V @ V3 ) @ D ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_97_Graph_Omin__distI__eq,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,D: nat,V3: nat] :
% 2.88/1.55 ( ( dist @ Capacity @ C @ V @ D @ V3 )
% 2.88/1.55 => ( ! [D6: nat] :
% 2.88/1.55 ( ( dist @ Capacity @ C @ V @ D6 @ V3 )
% 2.88/1.55 => ( ord_less_eq @ nat @ D @ D6 ) )
% 2.88/1.55 => ( ( min_dist @ Capacity @ C @ V @ V3 )
% 2.88/1.55 = D ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_98_Graph_Omin__distI2,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,V3: nat,Q: nat > $o] :
% 2.88/1.55 ( ( connected @ Capacity @ C @ V @ V3 )
% 2.88/1.55 => ( ! [D4: nat] :
% 2.88/1.55 ( ( dist @ Capacity @ C @ V @ D4 @ V3 )
% 2.88/1.55 => ( ! [D5: nat] :
% 2.88/1.55 ( ( dist @ Capacity @ C @ V @ D5 @ V3 )
% 2.88/1.55 => ( ord_less_eq @ nat @ D4 @ D5 ) )
% 2.88/1.55 => ( Q @ D4 ) ) )
% 2.88/1.55 => ( Q @ ( min_dist @ Capacity @ C @ V @ V3 ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_99_Graph_OisShortestPath__def,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( isShortestPath @ Capacity )
% 2.88/1.55 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U2: nat,P3: list @ ( product_prod @ nat @ nat ),V4: nat] :
% 2.88/1.55 ( ( isPath @ Capacity @ C2 @ U2 @ P3 @ V4 )
% 2.88/1.55 & ! [P6: list @ ( product_prod @ nat @ nat )] :
% 2.88/1.55 ( ( isPath @ Capacity @ C2 @ U2 @ P6 @ V4 )
% 2.88/1.55 => ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P3 ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P6 ) ) ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_100_zero__reorient,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( zero @ A )
% 2.88/1.55 => ! [X3: A] :
% 2.88/1.55 ( ( ( zero_zero @ A )
% 2.88/1.55 = X3 )
% 2.88/1.55 = ( X3
% 2.88/1.55 = ( zero_zero @ A ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_101_measure__induct,axiom,
% 2.88/1.55 ! [B: $tType,A: $tType] :
% 2.88/1.55 ( ( wellorder @ B )
% 2.88/1.55 => ! [F: A > B,P2: A > $o,A2: A] :
% 2.88/1.55 ( ! [X2: A] :
% 2.88/1.55 ( ! [Y2: A] :
% 2.88/1.55 ( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X2 ) )
% 2.88/1.55 => ( P2 @ Y2 ) )
% 2.88/1.55 => ( P2 @ X2 ) )
% 2.88/1.55 => ( P2 @ A2 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_102_measure__induct__rule,axiom,
% 2.88/1.55 ! [B: $tType,A: $tType] :
% 2.88/1.55 ( ( wellorder @ B )
% 2.88/1.55 => ! [F: A > B,P2: A > $o,A2: A] :
% 2.88/1.55 ( ! [X2: A] :
% 2.88/1.55 ( ! [Y2: A] :
% 2.88/1.55 ( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X2 ) )
% 2.88/1.55 => ( P2 @ Y2 ) )
% 2.88/1.55 => ( P2 @ X2 ) )
% 2.88/1.55 => ( P2 @ A2 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_103_nat__neq__iff,axiom,
% 2.88/1.55 ! [M: nat,N: nat] :
% 2.88/1.55 ( ( M != N )
% 2.88/1.55 = ( ( ord_less @ nat @ M @ N )
% 2.88/1.55 | ( ord_less @ nat @ N @ M ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_104_less__not__refl,axiom,
% 2.88/1.55 ! [N: nat] :
% 2.88/1.55 ~ ( ord_less @ nat @ N @ N ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_105_less__not__refl2,axiom,
% 2.88/1.55 ! [N: nat,M: nat] :
% 2.88/1.55 ( ( ord_less @ nat @ N @ M )
% 2.88/1.55 => ( M != N ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_106_less__not__refl3,axiom,
% 2.88/1.55 ! [S: nat,T: nat] :
% 2.88/1.55 ( ( ord_less @ nat @ S @ T )
% 2.88/1.55 => ( S != T ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_107_less__irrefl__nat,axiom,
% 2.88/1.55 ! [N: nat] :
% 2.88/1.55 ~ ( ord_less @ nat @ N @ N ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_108_nat__less__induct,axiom,
% 2.88/1.55 ! [P2: nat > $o,N: nat] :
% 2.88/1.55 ( ! [N3: nat] :
% 2.88/1.55 ( ! [M3: nat] :
% 2.88/1.55 ( ( ord_less @ nat @ M3 @ N3 )
% 2.88/1.55 => ( P2 @ M3 ) )
% 2.88/1.55 => ( P2 @ N3 ) )
% 2.88/1.55 => ( P2 @ N ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_109_infinite__descent,axiom,
% 2.88/1.55 ! [P2: nat > $o,N: nat] :
% 2.88/1.55 ( ! [N3: nat] :
% 2.88/1.55 ( ~ ( P2 @ N3 )
% 2.88/1.55 => ? [M3: nat] :
% 2.88/1.55 ( ( ord_less @ nat @ M3 @ N3 )
% 2.88/1.55 & ~ ( P2 @ M3 ) ) )
% 2.88/1.55 => ( P2 @ N ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_110_linorder__neqE__nat,axiom,
% 2.88/1.55 ! [X3: nat,Y3: nat] :
% 2.88/1.55 ( ( X3 != Y3 )
% 2.88/1.55 => ( ~ ( ord_less @ nat @ X3 @ Y3 )
% 2.88/1.55 => ( ord_less @ nat @ Y3 @ X3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_111_infinite__descent__measure,axiom,
% 2.88/1.55 ! [A: $tType,P2: A > $o,V5: A > nat,X3: A] :
% 2.88/1.55 ( ! [X2: A] :
% 2.88/1.55 ( ~ ( P2 @ X2 )
% 2.88/1.55 => ? [Y2: A] :
% 2.88/1.55 ( ( ord_less @ nat @ ( V5 @ Y2 ) @ ( V5 @ X2 ) )
% 2.88/1.55 & ~ ( P2 @ Y2 ) ) )
% 2.88/1.55 => ( P2 @ X3 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_112_size__neq__size__imp__neq,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( size @ A )
% 2.88/1.55 => ! [X3: A,Y3: A] :
% 2.88/1.55 ( ( ( size_size @ A @ X3 )
% 2.88/1.55 != ( size_size @ A @ Y3 ) )
% 2.88/1.55 => ( X3 != Y3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_113_Graph_OisPath_Ocong,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( isPath @ Capacity )
% 2.88/1.55 = ( isPath @ Capacity ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_114_Graph_OV_Ocong,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( v @ Capacity )
% 2.88/1.55 = ( v @ Capacity ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_115_Graph_Omin__dist_Ocong,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( min_dist @ Capacity )
% 2.88/1.55 = ( min_dist @ Capacity ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_116_Graph_Oconnected_Ocong,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( connected @ Capacity )
% 2.88/1.55 = ( connected @ Capacity ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_117_Graph_Oconnected__refl,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat] : ( connected @ Capacity @ C @ V @ V ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_118_Graph_Odist_Ocong,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( dist @ Capacity )
% 2.88/1.55 = ( dist @ Capacity ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_119_Graph_OisShortestPath_Ocong,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( isShortestPath @ Capacity )
% 2.88/1.55 = ( isShortestPath @ Capacity ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_120_Finite__Graph_Omin__dist__less__V,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat] :
% 2.88/1.55 ( ( finite_Graph @ A @ C )
% 2.88/1.55 => ( ( member @ nat @ S @ ( v @ A @ C ) )
% 2.88/1.55 => ( ( connected @ A @ C @ S @ T )
% 2.88/1.55 => ( ord_less @ nat @ ( min_dist @ A @ C @ S @ T ) @ ( finite_card @ nat @ ( v @ A @ C ) ) ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_121_Finite__Graph_OisShortestPath__length__less__V,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 2.88/1.55 ( ( finite_Graph @ A @ C )
% 2.88/1.55 => ( ( member @ nat @ S @ ( v @ A @ C ) )
% 2.88/1.55 => ( ( isShortestPath @ A @ C @ S @ P5 @ T )
% 2.88/1.55 => ( ord_less @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ ( finite_card @ nat @ ( v @ A @ C ) ) ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_122_gr__zeroI,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( canoni770627133id_add @ A )
% 2.88/1.55 => ! [N: A] :
% 2.88/1.55 ( ( N
% 2.88/1.55 != ( zero_zero @ A ) )
% 2.88/1.55 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_123_not__less__zero,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( canoni770627133id_add @ A )
% 2.88/1.55 => ! [N: A] :
% 2.88/1.55 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_124_gr__implies__not__zero,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( canoni770627133id_add @ A )
% 2.88/1.55 => ! [M: A,N: A] :
% 2.88/1.55 ( ( ord_less @ A @ M @ N )
% 2.88/1.55 => ( N
% 2.88/1.55 != ( zero_zero @ A ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_125_zero__less__iff__neq__zero,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( canoni770627133id_add @ A )
% 2.88/1.55 => ! [N: A] :
% 2.88/1.55 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 2.88/1.55 = ( N
% 2.88/1.55 != ( zero_zero @ A ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_126_gr0I,axiom,
% 2.88/1.55 ! [N: nat] :
% 2.88/1.55 ( ( N
% 2.88/1.55 != ( zero_zero @ nat ) )
% 2.88/1.55 => ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_127_not__gr0,axiom,
% 2.88/1.55 ! [N: nat] :
% 2.88/1.55 ( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
% 2.88/1.55 = ( N
% 2.88/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_128_not__less0,axiom,
% 2.88/1.55 ! [N: nat] :
% 2.88/1.55 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_129_less__zeroE,axiom,
% 2.88/1.55 ! [N: nat] :
% 2.88/1.55 ~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_130_gr__implies__not0,axiom,
% 2.88/1.55 ! [M: nat,N: nat] :
% 2.88/1.55 ( ( ord_less @ nat @ M @ N )
% 2.88/1.55 => ( N
% 2.88/1.55 != ( zero_zero @ nat ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_131_infinite__descent0,axiom,
% 2.88/1.55 ! [P2: nat > $o,N: nat] :
% 2.88/1.55 ( ( P2 @ ( zero_zero @ nat ) )
% 2.88/1.55 => ( ! [N3: nat] :
% 2.88/1.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 2.88/1.55 => ( ~ ( P2 @ N3 )
% 2.88/1.55 => ? [M3: nat] :
% 2.88/1.55 ( ( ord_less @ nat @ M3 @ N3 )
% 2.88/1.55 & ~ ( P2 @ M3 ) ) ) )
% 2.88/1.55 => ( P2 @ N ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_132_bot__nat__0_Oextremum__strict,axiom,
% 2.88/1.55 ! [A2: nat] :
% 2.88/1.55 ~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_133_infinite__descent0__measure,axiom,
% 2.88/1.55 ! [A: $tType,V5: A > nat,P2: A > $o,X3: A] :
% 2.88/1.55 ( ! [X2: A] :
% 2.88/1.55 ( ( ( V5 @ X2 )
% 2.88/1.55 = ( zero_zero @ nat ) )
% 2.88/1.55 => ( P2 @ X2 ) )
% 2.88/1.55 => ( ! [X2: A] :
% 2.88/1.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V5 @ X2 ) )
% 2.88/1.55 => ( ~ ( P2 @ X2 )
% 2.88/1.55 => ? [Y2: A] :
% 2.88/1.55 ( ( ord_less @ nat @ ( V5 @ Y2 ) @ ( V5 @ X2 ) )
% 2.88/1.55 & ~ ( P2 @ Y2 ) ) ) )
% 2.88/1.55 => ( P2 @ X3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_134_Graph_Omin__dist__z,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat] :
% 2.88/1.55 ( ( min_dist @ Capacity @ C @ V @ V )
% 2.88/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_135_Graph_Odist__z,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat] : ( dist @ Capacity @ C @ V @ ( zero_zero @ nat ) @ V ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_136_Graph_Odist__z__iff,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,V3: nat] :
% 2.88/1.55 ( ( dist @ Capacity @ C @ V @ ( zero_zero @ nat ) @ V3 )
% 2.88/1.55 = ( V3 = V ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_137_Graph_Oconnected__def,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( connected @ Capacity )
% 2.88/1.55 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U2: nat,V4: nat] :
% 2.88/1.55 ? [P3: list @ ( product_prod @ nat @ nat )] : ( isPath @ Capacity @ C2 @ U2 @ P3 @ V4 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_138_Graph_Oconnected__inV__iff,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat] :
% 2.88/1.55 ( ( connected @ Capacity @ C @ U @ V )
% 2.88/1.55 => ( ( member @ nat @ V @ ( v @ Capacity @ C ) )
% 2.88/1.55 = ( member @ nat @ U @ ( v @ Capacity @ C ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_139_Graph_OshortestPath__is__path,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.88/1.55 ( ( isShortestPath @ Capacity @ C @ U @ P5 @ V )
% 2.88/1.55 => ( isPath @ Capacity @ C @ U @ P5 @ V ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_140_Graph_Oconnected__distI,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,D: nat,V3: nat] :
% 2.88/1.55 ( ( dist @ Capacity @ C @ V @ D @ V3 )
% 2.88/1.55 => ( connected @ Capacity @ C @ V @ V3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_141_Graph_Oconnected__by__dist,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( connected @ Capacity )
% 2.88/1.55 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,V4: nat,V6: nat] :
% 2.88/1.55 ? [D3: nat] : ( dist @ Capacity @ C2 @ V4 @ D3 @ V6 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_142_Graph_Oobtain__shortest__path,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat] :
% 2.88/1.55 ( ( connected @ Capacity @ C @ U @ V )
% 2.88/1.55 => ~ ! [P4: list @ ( product_prod @ nat @ nat )] :
% 2.88/1.55 ~ ( isShortestPath @ Capacity @ C @ U @ P4 @ V ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_143_Graph_Omin__dist__z__iff,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,V3: nat] :
% 2.88/1.55 ( ( connected @ Capacity @ C @ V @ V3 )
% 2.88/1.55 => ( ( ( min_dist @ Capacity @ C @ V @ V3 )
% 2.88/1.55 = ( zero_zero @ nat ) )
% 2.88/1.55 = ( V3 = V ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_144_Graph_Omin__dist__less,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,Src: nat,V: nat,D: nat,D2: nat] :
% 2.88/1.55 ( ( connected @ Capacity @ C @ Src @ V )
% 2.88/1.55 => ( ( ( min_dist @ Capacity @ C @ Src @ V )
% 2.88/1.55 = D )
% 2.88/1.55 => ( ( ord_less @ nat @ D2 @ D )
% 2.88/1.55 => ? [V2: nat] :
% 2.88/1.55 ( ( connected @ Capacity @ C @ Src @ V2 )
% 2.88/1.55 & ( ( min_dist @ Capacity @ C @ Src @ V2 )
% 2.88/1.55 = D2 ) ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_145_Graph_Omin__dist__is__dist,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,V3: nat] :
% 2.88/1.55 ( ( connected @ Capacity @ C @ V @ V3 )
% 2.88/1.55 => ( dist @ Capacity @ C @ V @ ( min_dist @ Capacity @ C @ V @ V3 ) @ V3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_146_Graph_OisPath__distD,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.88/1.55 ( ( isPath @ Capacity @ C @ U @ P5 @ V )
% 2.88/1.55 => ( dist @ Capacity @ C @ U @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ V ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_147_Graph_Odist__def,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( dist @ Capacity )
% 2.88/1.55 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,V4: nat,D3: nat,V6: nat] :
% 2.88/1.55 ? [P3: list @ ( product_prod @ nat @ nat )] :
% 2.88/1.55 ( ( isPath @ Capacity @ C2 @ V4 @ P3 @ V6 )
% 2.88/1.55 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P3 )
% 2.88/1.55 = D3 ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_148_card_Oinfinite,axiom,
% 2.88/1.55 ! [A: $tType,A3: set @ A] :
% 2.88/1.55 ( ~ ( finite_finite2 @ A @ A3 )
% 2.88/1.55 => ( ( finite_card @ A @ A3 )
% 2.88/1.55 = ( zero_zero @ nat ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_149_card__spEdges__le,axiom,
% 2.88/1.55 ord_less_eq @ nat @ ( finite_card @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) ) @ ( finite_card @ ( product_prod @ nat @ nat ) @ ( edmond259086305sis_uE @ a @ c ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_150_g_H_Ocard__spEdges__le,axiom,
% 2.88/1.55 ord_less_eq @ nat @ ( finite_card @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ b @ c2 @ s @ t ) ) @ ( finite_card @ ( product_prod @ nat @ nat ) @ ( edmond259086305sis_uE @ b @ c2 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_151_simplePath__length__less__V,axiom,
% 2.88/1.55 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.88/1.55 ( ( member @ nat @ U @ ( v @ a @ c ) )
% 2.88/1.55 => ( ( isSimplePath @ a @ c @ U @ P5 @ V )
% 2.88/1.55 => ( ord_less @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ ( finite_card @ nat @ ( v @ a @ c ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_152_card__ge__0__finite,axiom,
% 2.88/1.55 ! [A: $tType,A3: set @ A] :
% 2.88/1.55 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) )
% 2.88/1.55 => ( finite_finite2 @ A @ A3 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_153_min__distI2,axiom,
% 2.88/1.55 ! [V: nat,V3: nat,Q: nat > $o] :
% 2.88/1.55 ( ( connected @ a @ c @ V @ V3 )
% 2.88/1.55 => ( ! [D4: nat] :
% 2.88/1.55 ( ( dist @ a @ c @ V @ D4 @ V3 )
% 2.88/1.55 => ( ! [D5: nat] :
% 2.88/1.55 ( ( dist @ a @ c @ V @ D5 @ V3 )
% 2.88/1.55 => ( ord_less_eq @ nat @ D4 @ D5 ) )
% 2.88/1.55 => ( Q @ D4 ) ) )
% 2.88/1.55 => ( Q @ ( min_dist @ a @ c @ V @ V3 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_154_g_H_OisShortestPath__alt,axiom,
% 2.88/1.55 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.88/1.55 ( ( isShortestPath @ b @ c2 @ U @ P5 @ V )
% 2.88/1.55 = ( ( isSimplePath @ b @ c2 @ U @ P5 @ V )
% 2.88/1.55 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 )
% 2.88/1.55 = ( min_dist @ b @ c2 @ U @ V ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_155_g_H_Oconnected__inV__iff,axiom,
% 2.88/1.55 ! [U: nat,V: nat] :
% 2.88/1.55 ( ( connected @ b @ c2 @ U @ V )
% 2.88/1.55 => ( ( member @ nat @ V @ ( v @ b @ c2 ) )
% 2.88/1.55 = ( member @ nat @ U @ ( v @ b @ c2 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_156_g_H_OisSPath__pathLE,axiom,
% 2.88/1.55 ! [S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 2.88/1.55 ( ( isPath @ b @ c2 @ S @ P5 @ T )
% 2.88/1.55 => ? [P: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ b @ c2 @ S @ P @ T ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_157_isSPath__pathLE,axiom,
% 2.88/1.55 ! [S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 2.88/1.55 ( ( isPath @ a @ c @ S @ P5 @ T )
% 2.88/1.55 => ? [P: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ a @ c @ S @ P @ T ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_158_connected__by__dist,axiom,
% 2.88/1.55 ! [V: nat,V3: nat] :
% 2.88/1.55 ( ( connected @ a @ c @ V @ V3 )
% 2.88/1.55 = ( ? [D3: nat] : ( dist @ a @ c @ V @ D3 @ V3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_159_shortestPath__is__simple,axiom,
% 2.88/1.55 ! [S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 2.88/1.55 ( ( isShortestPath @ a @ c @ S @ P5 @ T )
% 2.88/1.55 => ( isSimplePath @ a @ c @ S @ P5 @ T ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_160_g_H_OshortestPath__is__simple,axiom,
% 2.88/1.55 ! [S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 2.88/1.55 ( ( isShortestPath @ b @ c2 @ S @ P5 @ T )
% 2.88/1.55 => ( isSimplePath @ b @ c2 @ S @ P5 @ T ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_161_finite__code,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( finite_finite @ A )
% 2.88/1.55 => ( ( finite_finite2 @ A )
% 2.88/1.55 = ( ^ [A4: set @ A] : $true ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_162_min__dist__minD,axiom,
% 2.88/1.55 ! [V: nat,D: nat,V3: nat] :
% 2.88/1.55 ( ( dist @ a @ c @ V @ D @ V3 )
% 2.88/1.55 => ( ord_less_eq @ nat @ ( min_dist @ a @ c @ V @ V3 ) @ D ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_163_min__distI__eq,axiom,
% 2.88/1.55 ! [V: nat,D: nat,V3: nat] :
% 2.88/1.55 ( ( dist @ a @ c @ V @ D @ V3 )
% 2.88/1.55 => ( ! [D6: nat] :
% 2.88/1.55 ( ( dist @ a @ c @ V @ D6 @ V3 )
% 2.88/1.55 => ( ord_less_eq @ nat @ D @ D6 ) )
% 2.88/1.55 => ( ( min_dist @ a @ c @ V @ V3 )
% 2.88/1.55 = D ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_164_isPath__distD,axiom,
% 2.88/1.55 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.88/1.55 ( ( isPath @ a @ c @ U @ P5 @ V )
% 2.88/1.55 => ( dist @ a @ c @ U @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ V ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_165_dist__def,axiom,
% 2.88/1.55 ! [V: nat,D: nat,V3: nat] :
% 2.88/1.55 ( ( dist @ a @ c @ V @ D @ V3 )
% 2.88/1.55 = ( ? [P3: list @ ( product_prod @ nat @ nat )] :
% 2.88/1.55 ( ( isPath @ a @ c @ V @ P3 @ V3 )
% 2.88/1.55 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P3 )
% 2.88/1.55 = D ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_166_min__dist__is__dist,axiom,
% 2.88/1.55 ! [V: nat,V3: nat] :
% 2.88/1.55 ( ( connected @ a @ c @ V @ V3 )
% 2.88/1.55 => ( dist @ a @ c @ V @ ( min_dist @ a @ c @ V @ V3 ) @ V3 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_167_g_H_OsimplePath__length__less__V,axiom,
% 2.88/1.55 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.88/1.55 ( ( member @ nat @ U @ ( v @ b @ c2 ) )
% 2.88/1.55 => ( ( isSimplePath @ b @ c2 @ U @ P5 @ V )
% 2.88/1.55 => ( ord_less @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ ( finite_card @ nat @ ( v @ b @ c2 ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_168_g_H_Omin__dist__less__V,axiom,
% 2.88/1.55 ! [S: nat,T: nat] :
% 2.88/1.55 ( ( member @ nat @ S @ ( v @ b @ c2 ) )
% 2.88/1.55 => ( ( connected @ b @ c2 @ S @ T )
% 2.88/1.55 => ( ord_less @ nat @ ( min_dist @ b @ c2 @ S @ T ) @ ( finite_card @ nat @ ( v @ b @ c2 ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_169_g_H_OisShortestPath__length__less__V,axiom,
% 2.88/1.55 ! [S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 2.88/1.55 ( ( member @ nat @ S @ ( v @ b @ c2 ) )
% 2.88/1.55 => ( ( isShortestPath @ b @ c2 @ S @ P5 @ T )
% 2.88/1.55 => ( ord_less @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ ( finite_card @ nat @ ( v @ b @ c2 ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_170_isShortestPath__alt,axiom,
% 2.88/1.55 ! [U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.88/1.55 ( ( isShortestPath @ a @ c @ U @ P5 @ V )
% 2.88/1.55 = ( ( isSimplePath @ a @ c @ U @ P5 @ V )
% 2.88/1.55 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 )
% 2.88/1.55 = ( min_dist @ a @ c @ U @ V ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_171_g_H_Ofinite__V,axiom,
% 2.88/1.55 finite_finite2 @ nat @ ( v @ b @ c2 ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_172_finite__V,axiom,
% 2.88/1.55 finite_finite2 @ nat @ ( v @ a @ c ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_173_dist__z__iff,axiom,
% 2.88/1.55 ! [V: nat,V3: nat] :
% 2.88/1.55 ( ( dist @ a @ c @ V @ ( zero_zero @ nat ) @ V3 )
% 2.88/1.55 = ( V3 = V ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_174_dist__z,axiom,
% 2.88/1.55 ! [V: nat] : ( dist @ a @ c @ V @ ( zero_zero @ nat ) @ V ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_175_connected__distI,axiom,
% 2.88/1.55 ! [V: nat,D: nat,V3: nat] :
% 2.88/1.55 ( ( dist @ a @ c @ V @ D @ V3 )
% 2.88/1.55 => ( connected @ a @ c @ V @ V3 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_176_Veq,axiom,
% 2.88/1.55 ( ( v @ b @ c2 )
% 2.88/1.55 = ( v @ a @ c ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_177_g_H_Oadjacent__nodes__ss__V,axiom,
% 2.88/1.55 ! [U: nat] : ( ord_less_eq @ ( set @ nat ) @ ( adjacent_nodes @ b @ c2 @ U ) @ ( v @ b @ c2 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_178_adjacent__nodes__ss__V,axiom,
% 2.88/1.55 ! [U: nat] : ( ord_less_eq @ ( set @ nat ) @ ( adjacent_nodes @ a @ c @ U ) @ ( v @ a @ c ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_179_Graph_OisSimplePath_Ocong,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( isSimplePath @ Capacity )
% 2.88/1.55 = ( isSimplePath @ Capacity ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_180_finite__subset,axiom,
% 2.88/1.55 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.88/1.55 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.88/1.55 => ( ( finite_finite2 @ A @ B3 )
% 2.88/1.55 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_181_infinite__super,axiom,
% 2.88/1.55 ! [A: $tType,S2: set @ A,T2: set @ A] :
% 2.88/1.55 ( ( ord_less_eq @ ( set @ A ) @ S2 @ T2 )
% 2.88/1.55 => ( ~ ( finite_finite2 @ A @ S2 )
% 2.88/1.55 => ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_182_rev__finite__subset,axiom,
% 2.88/1.55 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 2.88/1.55 ( ( finite_finite2 @ A @ B3 )
% 2.88/1.55 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.88/1.55 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_183_card__subset__eq,axiom,
% 2.88/1.55 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 2.88/1.55 ( ( finite_finite2 @ A @ B3 )
% 2.88/1.55 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.88/1.55 => ( ( ( finite_card @ A @ A3 )
% 2.88/1.55 = ( finite_card @ A @ B3 ) )
% 2.88/1.55 => ( A3 = B3 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_184_infinite__arbitrarily__large,axiom,
% 2.88/1.55 ! [A: $tType,A3: set @ A,N: nat] :
% 2.88/1.55 ( ~ ( finite_finite2 @ A @ A3 )
% 2.88/1.55 => ? [B4: set @ A] :
% 2.88/1.55 ( ( finite_finite2 @ A @ B4 )
% 2.88/1.55 & ( ( finite_card @ A @ B4 )
% 2.88/1.55 = N )
% 2.88/1.55 & ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_185_Graph_OisSPath__pathLE,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 2.88/1.55 ( ( isPath @ Capacity @ C @ S @ P5 @ T )
% 2.88/1.55 => ? [P: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ Capacity @ C @ S @ P @ T ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_186_Graph_OshortestPath__is__simple,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P5: list @ ( product_prod @ nat @ nat ),T: nat] :
% 2.88/1.55 ( ( isShortestPath @ Capacity @ C @ S @ P5 @ T )
% 2.88/1.55 => ( isSimplePath @ Capacity @ C @ S @ P5 @ T ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_187_finite__maxlen,axiom,
% 2.88/1.55 ! [A: $tType,M4: set @ ( list @ A )] :
% 2.88/1.55 ( ( finite_finite2 @ ( list @ A ) @ M4 )
% 2.88/1.55 => ? [N3: nat] :
% 2.88/1.55 ! [X4: list @ A] :
% 2.88/1.55 ( ( member @ ( list @ A ) @ X4 @ M4 )
% 2.88/1.55 => ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X4 ) @ N3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_188_card__psubset,axiom,
% 2.88/1.55 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 2.88/1.55 ( ( finite_finite2 @ A @ B3 )
% 2.88/1.55 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.88/1.55 => ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) )
% 2.88/1.55 => ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_189_card__mono,axiom,
% 2.88/1.55 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 2.88/1.55 ( ( finite_finite2 @ A @ B3 )
% 2.88/1.55 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.88/1.55 => ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_190_card__seteq,axiom,
% 2.88/1.55 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 2.88/1.55 ( ( finite_finite2 @ A @ B3 )
% 2.88/1.55 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.88/1.55 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ B3 ) @ ( finite_card @ A @ A3 ) )
% 2.88/1.55 => ( A3 = B3 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_191_finite__if__finite__subsets__card__bdd,axiom,
% 2.88/1.55 ! [A: $tType,F2: set @ A,C3: nat] :
% 2.88/1.55 ( ! [G2: set @ A] :
% 2.88/1.55 ( ( ord_less_eq @ ( set @ A ) @ G2 @ F2 )
% 2.88/1.55 => ( ( finite_finite2 @ A @ G2 )
% 2.88/1.55 => ( ord_less_eq @ nat @ ( finite_card @ A @ G2 ) @ C3 ) ) )
% 2.88/1.55 => ( ( finite_finite2 @ A @ F2 )
% 2.88/1.55 & ( ord_less_eq @ nat @ ( finite_card @ A @ F2 ) @ C3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_192_Finite__Graph__def,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ( ( finite_Graph @ A )
% 2.88/1.55 = ( ^ [C2: ( product_prod @ nat @ nat ) > A] : ( finite_finite2 @ nat @ ( v @ A @ C2 ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_193_Finite__Graph_Ointro,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > A] :
% 2.88/1.55 ( ( finite_finite2 @ nat @ ( v @ A @ C ) )
% 2.88/1.55 => ( finite_Graph @ A @ C ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_194_Finite__Graph_Ofinite__V,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > A] :
% 2.88/1.55 ( ( finite_Graph @ A @ C )
% 2.88/1.55 => ( finite_finite2 @ nat @ ( v @ A @ C ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_195_finite,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( finite_finite @ A )
% 2.88/1.55 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_196_finite__set__choice,axiom,
% 2.88/1.55 ! [B: $tType,A: $tType,A3: set @ A,P2: A > B > $o] :
% 2.88/1.55 ( ( finite_finite2 @ A @ A3 )
% 2.88/1.55 => ( ! [X2: A] :
% 2.88/1.55 ( ( member @ A @ X2 @ A3 )
% 2.88/1.55 => ? [X_1: B] : ( P2 @ X2 @ X_1 ) )
% 2.88/1.55 => ? [F3: A > B] :
% 2.88/1.55 ! [X4: A] :
% 2.88/1.55 ( ( member @ A @ X4 @ A3 )
% 2.88/1.55 => ( P2 @ X4 @ ( F3 @ X4 ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_197_finite__psubset__induct,axiom,
% 2.88/1.55 ! [A: $tType,A3: set @ A,P2: ( set @ A ) > $o] :
% 2.88/1.55 ( ( finite_finite2 @ A @ A3 )
% 2.88/1.55 => ( ! [A5: set @ A] :
% 2.88/1.55 ( ( finite_finite2 @ A @ A5 )
% 2.88/1.55 => ( ! [B5: set @ A] :
% 2.88/1.55 ( ( ord_less @ ( set @ A ) @ B5 @ A5 )
% 2.88/1.55 => ( P2 @ B5 ) )
% 2.88/1.55 => ( P2 @ A5 ) ) )
% 2.88/1.55 => ( P2 @ A3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_198_Graph_OisShortestPath__alt,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( isShortestPath @ Capacity )
% 2.88/1.55 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U2: nat,P3: list @ ( product_prod @ nat @ nat ),V4: nat] :
% 2.88/1.55 ( ( isSimplePath @ Capacity @ C2 @ U2 @ P3 @ V4 )
% 2.88/1.55 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P3 )
% 2.88/1.55 = ( min_dist @ Capacity @ C2 @ U2 @ V4 ) ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_199_ek__analysis_Ocard__spEdges__le,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat] :
% 2.88/1.55 ( ( edmond2129202899alysis @ A @ C )
% 2.88/1.55 => ( ord_less_eq @ nat @ ( finite_card @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ A @ C @ S @ T ) ) @ ( finite_card @ ( product_prod @ nat @ nat ) @ ( edmond259086305sis_uE @ A @ C ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_200_finite__has__minimal2,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( order @ A )
% 2.88/1.55 => ! [A3: set @ A,A2: A] :
% 2.88/1.55 ( ( finite_finite2 @ A @ A3 )
% 2.88/1.55 => ( ( member @ A @ A2 @ A3 )
% 2.88/1.55 => ? [X2: A] :
% 2.88/1.55 ( ( member @ A @ X2 @ A3 )
% 2.88/1.55 & ( ord_less_eq @ A @ X2 @ A2 )
% 2.88/1.55 & ! [Xa: A] :
% 2.88/1.55 ( ( member @ A @ Xa @ A3 )
% 2.88/1.55 => ( ( ord_less_eq @ A @ Xa @ X2 )
% 2.88/1.55 => ( X2 = Xa ) ) ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_201_finite__has__maximal2,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( order @ A )
% 2.88/1.55 => ! [A3: set @ A,A2: A] :
% 2.88/1.55 ( ( finite_finite2 @ A @ A3 )
% 2.88/1.55 => ( ( member @ A @ A2 @ A3 )
% 2.88/1.55 => ? [X2: A] :
% 2.88/1.55 ( ( member @ A @ X2 @ A3 )
% 2.88/1.55 & ( ord_less_eq @ A @ A2 @ X2 )
% 2.88/1.55 & ! [Xa: A] :
% 2.88/1.55 ( ( member @ A @ Xa @ A3 )
% 2.88/1.55 => ( ( ord_less_eq @ A @ X2 @ Xa )
% 2.88/1.55 => ( X2 = Xa ) ) ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_202_Finite__Graph_OsimplePath__length__less__V,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > A,U: nat,P5: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.88/1.55 ( ( finite_Graph @ A @ C )
% 2.88/1.55 => ( ( member @ nat @ U @ ( v @ A @ C ) )
% 2.88/1.55 => ( ( isSimplePath @ A @ C @ U @ P5 @ V )
% 2.88/1.55 => ( ord_less @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) @ ( finite_card @ nat @ ( v @ A @ C ) ) ) ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_203_psubset__card__mono,axiom,
% 2.88/1.55 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 2.88/1.55 ( ( finite_finite2 @ A @ B3 )
% 2.88/1.55 => ( ( ord_less @ ( set @ A ) @ A3 @ B3 )
% 2.88/1.55 => ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B3 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_204_reachable__ss__V,axiom,
% 2.88/1.55 ! [S: nat] :
% 2.88/1.55 ( ( member @ nat @ S @ ( v @ a @ c ) )
% 2.88/1.55 => ( ord_less_eq @ ( set @ nat ) @ ( reachableNodes @ a @ c @ S ) @ ( v @ a @ c ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_205_g_H_Oreachable__ss__V,axiom,
% 2.88/1.55 ! [S: nat] :
% 2.88/1.55 ( ( member @ nat @ S @ ( v @ b @ c2 ) )
% 2.88/1.55 => ( ord_less_eq @ ( set @ nat ) @ ( reachableNodes @ b @ c2 @ S ) @ ( v @ b @ c2 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_206_spEdges__ss__E,axiom,
% 2.88/1.55 ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) @ ( e @ a @ c ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_207_E__ss__uE,axiom,
% 2.88/1.55 ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( e @ a @ c ) @ ( edmond259086305sis_uE @ a @ c ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_208_Finite__Graph__EI,axiom,
% 2.88/1.55 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
% 2.88/1.55 => ( finite_Graph @ a @ c ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_209_Efin__imp__Vfin,axiom,
% 2.88/1.55 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
% 2.88/1.55 => ( finite_finite2 @ nat @ ( v @ a @ c ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_210_finite__E,axiom,
% 2.88/1.55 finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_211_g_H_Oadjacent__nodes__finite,axiom,
% 2.88/1.55 ! [U: nat] : ( finite_finite2 @ nat @ ( adjacent_nodes @ b @ c2 @ U ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_212_adjacent__nodes__finite,axiom,
% 2.88/1.55 ! [U: nat] : ( finite_finite2 @ nat @ ( adjacent_nodes @ a @ c @ U ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_213_Vfin__imp__Efin,axiom,
% 2.88/1.55 ( ( finite_finite2 @ nat @ ( v @ a @ c ) )
% 2.88/1.55 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_214_Graph_OreachableNodes_Ocong,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( reachableNodes @ Capacity )
% 2.88/1.55 = ( reachableNodes @ Capacity ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_215_Graph_Oadjacent__nodes_Ocong,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( adjacent_nodes @ Capacity )
% 2.88/1.55 = ( adjacent_nodes @ Capacity ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_216_Graph_OE_Ocong,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( e @ Capacity )
% 2.88/1.55 = ( e @ Capacity ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_217_Graph_Oreachable__ss__V,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [S: nat,C: ( product_prod @ nat @ nat ) > Capacity] :
% 2.88/1.55 ( ( member @ nat @ S @ ( v @ Capacity @ C ) )
% 2.88/1.55 => ( ord_less_eq @ ( set @ nat ) @ ( reachableNodes @ Capacity @ C @ S ) @ ( v @ Capacity @ C ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_218_Graph_OEfin__imp__Vfin,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity] :
% 2.88/1.55 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ Capacity @ C ) )
% 2.88/1.55 => ( finite_finite2 @ nat @ ( v @ Capacity @ C ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_219_Graph_OVfin__imp__Efin,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity] :
% 2.88/1.55 ( ( finite_finite2 @ nat @ ( v @ Capacity @ C ) )
% 2.88/1.55 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ Capacity @ C ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_220_Graph_OFinite__Graph__EI,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity] :
% 2.88/1.55 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ Capacity @ C ) )
% 2.88/1.55 => ( finite_Graph @ Capacity @ C ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_221_Finite__Graph_Ofinite__E,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > A] :
% 2.88/1.55 ( ( finite_Graph @ A @ C )
% 2.88/1.55 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ A @ C ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_222_ek__analysis_OE__ss__uE,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > A] :
% 2.88/1.55 ( ( edmond2129202899alysis @ A @ C )
% 2.88/1.55 => ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( e @ A @ C ) @ ( edmond259086305sis_uE @ A @ C ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_223_ek__analysis_OspEdges__ss__E,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > A,S: nat,T: nat] :
% 2.88/1.55 ( ( edmond2129202899alysis @ A @ C )
% 2.88/1.55 => ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( edmond1803015688pEdges @ A @ C @ S @ T ) @ ( e @ A @ C ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_224_Graph_Oadjacent__nodes__ss__V,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat] : ( ord_less_eq @ ( set @ nat ) @ ( adjacent_nodes @ Capacity @ C @ U ) @ ( v @ Capacity @ C ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_225_Finite__Graph_Oadjacent__nodes__finite,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > A,U: nat] :
% 2.88/1.55 ( ( finite_Graph @ A @ C )
% 2.88/1.55 => ( finite_finite2 @ nat @ ( adjacent_nodes @ A @ C @ U ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_226_incoming_H__edges,axiom,
% 2.88/1.55 ! [U3: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming @ a @ c @ U3 ) @ ( e @ a @ c ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_227_outgoing_H__edges,axiom,
% 2.88/1.55 ! [U3: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing @ a @ c @ U3 ) @ ( e @ a @ c ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_228_g_H_OE__ss__uE,axiom,
% 2.88/1.55 ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( e @ b @ c2 ) @ ( edmond259086305sis_uE @ b @ c2 ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_229_g_H_OFinite__Graph__EI,axiom,
% 2.88/1.55 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ b @ c2 ) )
% 2.88/1.55 => ( finite_Graph @ b @ c2 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_230_g_H_OEfin__imp__Vfin,axiom,
% 2.88/1.55 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ b @ c2 ) )
% 2.88/1.55 => ( finite_finite2 @ nat @ ( v @ b @ c2 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_231_g_H_OspEdges__ss__E,axiom,
% 2.88/1.55 ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( edmond1803015688pEdges @ b @ c2 @ s @ t ) @ ( e @ b @ c2 ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_232_g_H_Ofinite__E,axiom,
% 2.88/1.55 finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ b @ c2 ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_233_g_H_OVfin__imp__Efin,axiom,
% 2.88/1.55 ( ( finite_finite2 @ nat @ ( v @ b @ c2 ) )
% 2.88/1.55 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ b @ c2 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_234_finite__outgoing_H,axiom,
% 2.88/1.55 ! [U3: set @ nat] :
% 2.88/1.55 ( ( finite_finite2 @ nat @ ( v @ a @ c ) )
% 2.88/1.55 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing @ a @ c @ U3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_235_finite__incoming_H,axiom,
% 2.88/1.55 ! [U3: set @ nat] :
% 2.88/1.55 ( ( finite_finite2 @ nat @ ( v @ a @ c ) )
% 2.88/1.55 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming @ a @ c @ U3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_236_Graph_Oincoming_H_Ocong,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( incoming @ Capacity )
% 2.88/1.55 = ( incoming @ Capacity ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_237_Graph_Ooutgoing_H_Ocong,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ( ( outgoing @ Capacity )
% 2.88/1.55 = ( outgoing @ Capacity ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_238_Graph_Ooutgoing_H__edges,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U3: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing @ Capacity @ C @ U3 ) @ ( e @ Capacity @ C ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_239_Graph_Oincoming_H__edges,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U3: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming @ Capacity @ C @ U3 ) @ ( e @ Capacity @ C ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_240_Graph_Ofinite__outgoing_H,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U3: set @ nat] :
% 2.88/1.55 ( ( finite_finite2 @ nat @ ( v @ Capacity @ C ) )
% 2.88/1.55 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing @ Capacity @ C @ U3 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_241_Graph_Ofinite__incoming_H,axiom,
% 2.88/1.55 ! [Capacity: $tType] :
% 2.88/1.55 ( ( linordered_idom @ Capacity )
% 2.88/1.55 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U3: set @ nat] :
% 2.88/1.55 ( ( finite_finite2 @ nat @ ( v @ Capacity @ C ) )
% 2.88/1.55 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming @ Capacity @ C @ U3 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_242_g_H_Oincoming_H__edges,axiom,
% 2.88/1.55 ! [U3: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming @ b @ c2 @ U3 ) @ ( e @ b @ c2 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_243_g_H_Ooutgoing_H__edges,axiom,
% 2.88/1.55 ! [U3: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing @ b @ c2 @ U3 ) @ ( e @ b @ c2 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_244_g_H_Ofinite__outgoing_H,axiom,
% 2.88/1.55 ! [U3: set @ nat] :
% 2.88/1.55 ( ( finite_finite2 @ nat @ ( v @ b @ c2 ) )
% 2.88/1.55 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing @ b @ c2 @ U3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_245_g_H_Ofinite__incoming_H,axiom,
% 2.88/1.55 ! [U3: set @ nat] :
% 2.88/1.55 ( ( finite_finite2 @ nat @ ( v @ b @ c2 ) )
% 2.88/1.55 => ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming @ b @ c2 @ U3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_246__092_060open_062edges_A_092_060subseteq_062_AE_092_060close_062,axiom,
% 2.88/1.55 ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ edges @ ( e @ a @ c ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_247_psubsetI,axiom,
% 2.88/1.55 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.88/1.55 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.88/1.55 => ( ( A3 != B3 )
% 2.88/1.55 => ( ord_less @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_248_subset__antisym,axiom,
% 2.88/1.55 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.88/1.55 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.88/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 2.88/1.55 => ( A3 = B3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_249_subsetI,axiom,
% 2.88/1.55 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.88/1.55 ( ! [X2: A] :
% 2.88/1.55 ( ( member @ A @ X2 @ A3 )
% 2.88/1.55 => ( member @ A @ X2 @ B3 ) )
% 2.88/1.55 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_250_SP__EDGES,axiom,
% 2.88/1.55 ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ edges @ ( set2 @ ( product_prod @ nat @ nat ) @ p ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_251_Collect__mono__iff,axiom,
% 2.88/1.55 ! [A: $tType,P2: A > $o,Q: A > $o] :
% 2.88/1.55 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
% 2.88/1.55 = ( ! [X: A] :
% 2.88/1.55 ( ( P2 @ X )
% 2.88/1.55 => ( Q @ X ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_252_set__eq__subset,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( ^ [Y4: set @ A,Z: set @ A] : ( Y4 = Z ) )
% 2.88/1.55 = ( ^ [A4: set @ A,B6: set @ A] :
% 2.88/1.55 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 2.88/1.55 & ( ord_less_eq @ ( set @ A ) @ B6 @ A4 ) ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(fact_253_subset__trans,axiom,
% 2.88/1.55 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 2.88/1.55 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.88/1.55 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
% 2.88/1.55 => ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(subcl_Rings_Olinordered__idom___HOL_Otype,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ( type @ A ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(subcl_Rings_Olinordered__idom___Groups_Ozero,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ( zero @ A ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(subcl_Rings_Olinordered__idom___Orderings_Oorder,axiom,
% 2.88/1.55 ! [A: $tType] :
% 2.88/1.55 ( ( linordered_idom @ A )
% 2.88/1.55 => ( order @ A ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.88/1.55 ! [A6: $tType,A7: $tType] :
% 2.88/1.55 ( ( ( finite_finite @ A6 )
% 2.88/1.55 & ( finite_finite @ A7 ) )
% 2.88/1.55 => ( finite_finite @ ( A6 > A7 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.88/1.55 ! [A6: $tType,A7: $tType] :
% 2.88/1.55 ( ( order @ A7 )
% 2.88/1.55 => ( order @ ( A6 > A7 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.88/1.55 canoni770627133id_add @ nat ).
% 2.88/1.55
% 2.88/1.55 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.88/1.55 wellorder @ nat ).
% 2.88/1.55
% 2.88/1.55 thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
% 2.88/1.55 order @ nat ).
% 2.88/1.55
% 2.88/1.55 thf(tcon_Nat_Onat___Groups_Ozero,axiom,
% 2.88/1.55 zero @ nat ).
% 2.88/1.55
% 2.88/1.55 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 2.88/1.55 size @ nat ).
% 2.88/1.55
% 2.88/1.55 thf(tcon_Set_Oset___Finite__Set_Ofinite_2,axiom,
% 2.88/1.55 ! [A6: $tType] :
% 2.88/1.55 ( ( finite_finite @ A6 )
% 2.88/1.55 => ( finite_finite @ ( set @ A6 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
% 2.88/1.55 ! [A6: $tType] : ( order @ ( set @ A6 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(tcon_HOL_Obool___Finite__Set_Ofinite_4,axiom,
% 2.88/1.55 finite_finite @ $o ).
% 2.88/1.55
% 2.88/1.55 thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
% 2.88/1.55 order @ $o ).
% 2.88/1.55
% 2.88/1.55 thf(tcon_List_Olist___Nat_Osize_6,axiom,
% 2.88/1.55 ! [A6: $tType] : ( size @ ( list @ A6 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_7,axiom,
% 2.88/1.55 ! [A6: $tType,A7: $tType] :
% 2.88/1.55 ( ( ( finite_finite @ A6 )
% 2.88/1.55 & ( finite_finite @ A7 ) )
% 2.88/1.55 => ( finite_finite @ ( product_prod @ A6 @ A7 ) ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(tcon_Product__Type_Oprod___Nat_Osize_8,axiom,
% 2.88/1.55 ! [A6: $tType,A7: $tType] : ( size @ ( product_prod @ A6 @ A7 ) ) ).
% 2.88/1.55
% 2.88/1.55 thf(tfree_0,hypothesis,
% 2.88/1.55 linordered_idom @ a ).
% 2.88/1.55
% 2.88/1.55 thf(tfree_1,hypothesis,
% 2.88/1.55 linordered_idom @ b ).
% 2.88/1.55
% 2.88/1.55 thf(conj_0,conjecture,
% 2.88/1.55 $false ).
% 2.88/1.56 ------------------------
% 2.88/1.58 % (13449)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_13330 for (2999ds/27Mi)
% 2.88/1.58 % (13450)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/2Mi)
% 2.88/1.58 % (13451)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_13330 for (2999ds/2Mi)
% 2.88/1.58 % (13448)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_13330 for (2999ds/4Mi)
% 2.88/1.58 % (13450)Instruction limit reached!
% 2.88/1.58 % (13450)------------------------------
% 2.88/1.58 % (13450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.58 % (13450)Termination reason: Unknown
% 2.88/1.58 % (13450)Termination phase: shuffling
% 2.88/1.58
% 2.88/1.58 % (13450)Memory used [KB]: 1279
% 2.88/1.58 % (13450)Time elapsed: 0.003 s
% 2.88/1.58 % (13450)Instructions burned: 4 (million)
% 2.88/1.58 % (13450)------------------------------
% 2.88/1.58 % (13450)------------------------------
% 2.88/1.58 % (13451)Instruction limit reached!
% 2.88/1.58 % (13451)------------------------------
% 2.88/1.58 % (13451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.58 % (13451)Termination reason: Unknown
% 2.88/1.58 % (13451)Termination phase: shuffling
% 2.88/1.58
% 2.88/1.58 % (13451)Memory used [KB]: 1279
% 2.88/1.58 % (13451)Time elapsed: 0.002 s
% 2.88/1.58 % (13451)Instructions burned: 3 (million)
% 2.88/1.58 % (13451)------------------------------
% 2.88/1.58 % (13451)------------------------------
% 2.88/1.58 % (13448)Instruction limit reached!
% 2.88/1.58 % (13448)------------------------------
% 2.88/1.58 % (13448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.58 % (13448)Termination reason: Unknown
% 2.88/1.58 % (13448)Termination phase: shuffling
% 2.88/1.58
% 2.88/1.58 % (13448)Memory used [KB]: 1407
% 2.88/1.58 % (13448)Time elapsed: 0.003 s
% 2.88/1.58 % (13448)Instructions burned: 5 (million)
% 2.88/1.58 % (13448)------------------------------
% 2.88/1.58 % (13448)------------------------------
% 2.88/1.59 % (13453)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/18Mi)
% 2.88/1.59 % (13447)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/183Mi)
% 2.88/1.59 % (13452)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_13330 for (2999ds/275Mi)
% 2.88/1.59 % (13449)Instruction limit reached!
% 2.88/1.59 % (13449)------------------------------
% 2.88/1.59 % (13449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.59 % (13449)Termination reason: Unknown
% 2.88/1.59 % (13449)Termination phase: Property scanning
% 2.88/1.59
% 2.88/1.59 % (13449)Memory used [KB]: 1663
% 2.88/1.59 % (13449)Time elapsed: 0.011 s
% 2.88/1.59 % (13449)Instructions burned: 29 (million)
% 2.88/1.59 % (13449)------------------------------
% 2.88/1.59 % (13449)------------------------------
% 2.88/1.59 % (13454)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/3Mi)
% 2.88/1.59 % (13454)Instruction limit reached!
% 2.88/1.59 % (13454)------------------------------
% 2.88/1.59 % (13454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.59 % (13454)Termination reason: Unknown
% 2.88/1.59 % (13454)Termination phase: shuffling
% 2.88/1.59
% 2.88/1.59 % (13454)Memory used [KB]: 1279
% 2.88/1.59 % (13454)Time elapsed: 0.002 s
% 2.88/1.59 % (13454)Instructions burned: 3 (million)
% 2.88/1.59 % (13454)------------------------------
% 2.88/1.59 % (13454)------------------------------
% 2.88/1.59 % (13456)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_13330 for (2999ds/15Mi)
% 2.88/1.60 % (13455)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_13330 for (2999ds/37Mi)
% 2.88/1.60 % (13453)Instruction limit reached!
% 2.88/1.60 % (13453)------------------------------
% 2.88/1.60 % (13453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.88/1.60 % (13453)Termination reason: Unknown
% 2.88/1.60 % (13453)Termination phase: shuffling
% 2.88/1.60
% 2.94/1.60 % (13453)Memory used [KB]: 1535
% 2.94/1.60 % (13453)Time elapsed: 0.010 s
% 2.94/1.60 % (13453)Instructions burned: 19 (million)
% 2.94/1.60 % (13453)------------------------------
% 2.94/1.60 % (13453)------------------------------
% 2.94/1.60 % (13456)Instruction limit reached!
% 2.94/1.60 % (13456)------------------------------
% 2.94/1.60 % (13456)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.60 % (13456)Termination reason: Unknown
% 2.94/1.60 % (13456)Termination phase: shuffling
% 2.94/1.60
% 2.94/1.60 % (13456)Memory used [KB]: 1535
% 2.94/1.60 % (13456)Time elapsed: 0.007 s
% 2.94/1.60 % (13456)Instructions burned: 15 (million)
% 2.94/1.60 % (13456)------------------------------
% 2.94/1.60 % (13456)------------------------------
% 2.94/1.60 % (13457)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_13330 for (2999ds/3Mi)
% 2.94/1.60 % (13457)Instruction limit reached!
% 2.94/1.60 % (13457)------------------------------
% 2.94/1.60 % (13457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.60 % (13457)Termination reason: Unknown
% 2.94/1.60 % (13457)Termination phase: shuffling
% 2.94/1.60
% 2.94/1.60 % (13457)Memory used [KB]: 1279
% 2.94/1.60 % (13457)Time elapsed: 0.002 s
% 2.94/1.60 % (13457)Instructions burned: 3 (million)
% 2.94/1.60 % (13457)------------------------------
% 2.94/1.60 % (13457)------------------------------
% 2.94/1.60 % (13458)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_13330 for (2999ds/1041Mi)
% 2.94/1.61 % (13459)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/7Mi)
% 2.94/1.61 % (13455)Instruction limit reached!
% 2.94/1.61 % (13455)------------------------------
% 2.94/1.61 % (13455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.61 % (13455)Termination reason: Unknown
% 2.94/1.61 % (13455)Termination phase: Property scanning
% 2.94/1.61
% 2.94/1.61 % (13455)Memory used [KB]: 1918
% 2.94/1.61 % (13455)Time elapsed: 0.013 s
% 2.94/1.61 % (13455)Instructions burned: 38 (million)
% 2.94/1.61 % (13455)------------------------------
% 2.94/1.61 % (13455)------------------------------
% 2.94/1.61 % (13460)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/16Mi)
% 2.94/1.61 % (13459)Instruction limit reached!
% 2.94/1.61 % (13459)------------------------------
% 2.94/1.61 % (13459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.61 % (13459)Termination reason: Unknown
% 2.94/1.61 % (13459)Termination phase: shuffling
% 2.94/1.61
% 2.94/1.61 % (13459)Memory used [KB]: 1407
% 2.94/1.61 % (13459)Time elapsed: 0.004 s
% 2.94/1.61 % (13459)Instructions burned: 7 (million)
% 2.94/1.61 % (13459)------------------------------
% 2.94/1.61 % (13459)------------------------------
% 2.94/1.61 % (13461)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/3Mi)
% 2.94/1.61 % (13452)Refutation not found, incomplete strategy
% 2.94/1.61 % (13452)------------------------------
% 2.94/1.61 % (13452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.61 % (13452)Termination reason: Refutation not found, incomplete strategy
% 2.94/1.61
% 2.94/1.61
% 2.94/1.61 % (13452)Memory used [KB]: 6140
% 2.94/1.61 % (13452)Time elapsed: 0.025 s
% 2.94/1.61 % (13452)Instructions burned: 36 (million)
% 2.94/1.61 % (13452)------------------------------
% 2.94/1.61 % (13452)------------------------------
% 2.94/1.61 % (13461)Instruction limit reached!
% 2.94/1.61 % (13461)------------------------------
% 2.94/1.61 % (13461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.61 % (13461)Termination reason: Unknown
% 2.94/1.61 % (13461)Termination phase: shuffling
% 2.94/1.61
% 2.94/1.61 % (13461)Memory used [KB]: 1407
% 2.94/1.61 % (13461)Time elapsed: 0.003 s
% 2.94/1.61 % (13461)Instructions burned: 6 (million)
% 2.94/1.61 % (13461)------------------------------
% 2.94/1.61 % (13461)------------------------------
% 2.94/1.61 % (13460)Instruction limit reached!
% 2.94/1.61 % (13460)------------------------------
% 2.94/1.61 % (13460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.61 % (13460)Termination reason: Unknown
% 2.94/1.61 % (13460)Termination phase: shuffling
% 2.94/1.61
% 2.94/1.61 % (13460)Memory used [KB]: 1535
% 2.94/1.61 % (13460)Time elapsed: 0.008 s
% 2.94/1.61 % (13460)Instructions burned: 17 (million)
% 2.94/1.61 % (13460)------------------------------
% 2.94/1.61 % (13460)------------------------------
% 2.94/1.62 % (13462)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/3Mi)
% 2.94/1.62 % (13463)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/7Mi)
% 2.94/1.62 % (13462)Instruction limit reached!
% 2.94/1.62 % (13462)------------------------------
% 2.94/1.62 % (13462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.62 % (13462)Termination reason: Unknown
% 2.94/1.62 % (13462)Termination phase: shuffling
% 2.94/1.62
% 2.94/1.62 % (13462)Memory used [KB]: 1279
% 2.94/1.62 % (13462)Time elapsed: 0.003 s
% 2.94/1.62 % (13462)Instructions burned: 3 (million)
% 2.94/1.62 % (13462)------------------------------
% 2.94/1.62 % (13462)------------------------------
% 2.94/1.62 % (13464)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/3Mi)
% 2.94/1.62 % (13463)Instruction limit reached!
% 2.94/1.62 % (13463)------------------------------
% 2.94/1.62 % (13463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.62 % (13463)Termination reason: Unknown
% 2.94/1.62 % (13463)Termination phase: shuffling
% 2.94/1.62
% 2.94/1.62 % (13463)Memory used [KB]: 1407
% 2.94/1.62 % (13463)Time elapsed: 0.004 s
% 2.94/1.62 % (13463)Instructions burned: 9 (million)
% 2.94/1.62 % (13463)------------------------------
% 2.94/1.62 % (13463)------------------------------
% 2.94/1.62 % (13464)Instruction limit reached!
% 2.94/1.62 % (13464)------------------------------
% 2.94/1.62 % (13464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.62 % (13464)Termination reason: Unknown
% 2.94/1.62 % (13464)Termination phase: shuffling
% 2.94/1.62
% 2.94/1.62 % (13464)Memory used [KB]: 1279
% 2.94/1.62 % (13464)Time elapsed: 0.002 s
% 2.94/1.62 % (13464)Instructions burned: 3 (million)
% 2.94/1.62 % (13464)------------------------------
% 2.94/1.62 % (13464)------------------------------
% 2.94/1.62 % (13466)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/18Mi)
% 2.94/1.63 % (13467)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/710Mi)
% 2.94/1.63 % (13465)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/4Mi)
% 2.94/1.63 % (13468)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/6Mi)
% 2.94/1.63 % (13469)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/902Mi)
% 2.94/1.63 % (13465)Instruction limit reached!
% 2.94/1.63 % (13465)------------------------------
% 2.94/1.63 % (13465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (13465)Termination reason: Unknown
% 2.94/1.63 % (13465)Termination phase: shuffling
% 2.94/1.63
% 2.94/1.63 % (13465)Memory used [KB]: 1279
% 2.94/1.63 % (13465)Time elapsed: 0.005 s
% 2.94/1.63 % (13465)Instructions burned: 5 (million)
% 2.94/1.63 % (13465)------------------------------
% 2.94/1.63 % (13465)------------------------------
% 2.94/1.63 % (13466)Instruction limit reached!
% 2.94/1.63 % (13466)------------------------------
% 2.94/1.63 % (13466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (13466)Termination reason: Unknown
% 2.94/1.63 % (13466)Termination phase: shuffling
% 2.94/1.63
% 2.94/1.63 % (13466)Memory used [KB]: 1535
% 2.94/1.63 % (13466)Time elapsed: 0.009 s
% 2.94/1.63 % (13466)Instructions burned: 20 (million)
% 2.94/1.63 % (13466)------------------------------
% 2.94/1.63 % (13466)------------------------------
% 2.94/1.63 % (13468)Instruction limit reached!
% 2.94/1.63 % (13468)------------------------------
% 2.94/1.63 % (13468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.63 % (13468)Termination reason: Unknown
% 2.94/1.63 % (13468)Termination phase: shuffling
% 2.94/1.63
% 2.94/1.63 % (13468)Memory used [KB]: 1407
% 2.94/1.63 % (13468)Time elapsed: 0.003 s
% 2.94/1.63 % (13468)Instructions burned: 6 (million)
% 2.94/1.63 % (13468)------------------------------
% 2.94/1.63 % (13468)------------------------------
% 2.94/1.64 % (13467)Refutation not found, incomplete strategy
% 2.94/1.64 % (13467)------------------------------
% 2.94/1.64 % (13467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.64 % (13467)Termination reason: Refutation not found, incomplete strategy
% 2.94/1.64
% 2.94/1.64
% 2.94/1.64 % (13467)Memory used [KB]: 6140
% 2.94/1.64 % (13467)Time elapsed: 0.014 s
% 2.94/1.64 % (13467)Instructions burned: 36 (million)
% 2.94/1.64 % (13467)------------------------------
% 2.94/1.64 % (13467)------------------------------
% 2.94/1.64 % (13472)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/6Mi)
% 2.94/1.64 % (13471)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/5Mi)
% 2.94/1.64 % (13472)Instruction limit reached!
% 2.94/1.64 % (13472)------------------------------
% 2.94/1.64 % (13472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.64 % (13472)Termination reason: Unknown
% 2.94/1.64 % (13472)Termination phase: shuffling
% 2.94/1.64
% 2.94/1.64 % (13472)Memory used [KB]: 1407
% 2.94/1.64 % (13472)Time elapsed: 0.003 s
% 2.94/1.64 % (13472)Instructions burned: 7 (million)
% 2.94/1.64 % (13472)------------------------------
% 2.94/1.64 % (13472)------------------------------
% 2.94/1.64 % (13471)Instruction limit reached!
% 2.94/1.64 % (13471)------------------------------
% 2.94/1.64 % (13471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.64 % (13471)Termination reason: Unknown
% 2.94/1.64 % (13471)Termination phase: shuffling
% 2.94/1.64
% 2.94/1.64 % (13471)Memory used [KB]: 1407
% 2.94/1.64 % (13471)Time elapsed: 0.004 s
% 2.94/1.64 % (13471)Instructions burned: 7 (million)
% 2.94/1.64 % (13471)------------------------------
% 2.94/1.64 % (13471)------------------------------
% 2.94/1.65 % (13470)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/21Mi)
% 2.94/1.65 % (13473)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/377Mi)
% 2.94/1.65 % (13474)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/779Mi)
% 2.94/1.65 % (13475)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_13330 for (2999ds/19Mi)
% 2.94/1.66 % (13470)Instruction limit reached!
% 2.94/1.66 % (13470)------------------------------
% 2.94/1.66 % (13470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.66 % (13470)Termination reason: Unknown
% 2.94/1.66 % (13470)Termination phase: shuffling
% 2.94/1.66
% 2.94/1.66 % (13470)Memory used [KB]: 1535
% 2.94/1.66 % (13470)Time elapsed: 0.011 s
% 2.94/1.66 % (13470)Instructions burned: 21 (million)
% 2.94/1.66 % (13470)------------------------------
% 2.94/1.66 % (13470)------------------------------
% 2.94/1.66 % (13475)Instruction limit reached!
% 2.94/1.66 % (13475)------------------------------
% 2.94/1.66 % (13475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.66 % (13475)Termination reason: Unknown
% 2.94/1.66 % (13475)Termination phase: shuffling
% 2.94/1.66
% 2.94/1.66 % (13475)Memory used [KB]: 1663
% 2.94/1.66 % (13475)Time elapsed: 0.008 s
% 2.94/1.66 % (13475)Instructions burned: 20 (million)
% 2.94/1.66 % (13475)------------------------------
% 2.94/1.66 % (13475)------------------------------
% 2.94/1.66 % (13469)Refutation not found, incomplete strategy
% 2.94/1.66 % (13469)------------------------------
% 2.94/1.66 % (13469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.66 % (13469)Termination reason: Refutation not found, incomplete strategy
% 2.94/1.66
% 2.94/1.66
% 2.94/1.66 % (13469)Memory used [KB]: 6908
% 2.94/1.66 % (13469)Time elapsed: 0.030 s
% 2.94/1.66 % (13469)Instructions burned: 97 (million)
% 2.94/1.66 % (13469)------------------------------
% 2.94/1.66 % (13469)------------------------------
% 2.94/1.67 % (13476)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_13330 for (2998ds/879Mi)
% 2.94/1.67 % (13478)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_13330 for (2998ds/3Mi)
% 2.94/1.67 % (13477)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_13330 for (2998ds/17Mi)
% 2.94/1.67 % (13478)Instruction limit reached!
% 2.94/1.67 % (13478)------------------------------
% 2.94/1.67 % (13478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.67 % (13478)Termination reason: Unknown
% 2.94/1.67 % (13478)Termination phase: shuffling
% 2.94/1.67
% 2.94/1.67 % (13478)Memory used [KB]: 1279
% 2.94/1.67 % (13478)Time elapsed: 0.002 s
% 2.94/1.67 % (13478)Instructions burned: 3 (million)
% 2.94/1.67 % (13478)------------------------------
% 2.94/1.67 % (13478)------------------------------
% 2.94/1.68 % (13477)Instruction limit reached!
% 2.94/1.68 % (13477)------------------------------
% 2.94/1.68 % (13477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.68 % (13477)Termination reason: Unknown
% 2.94/1.68 % (13477)Termination phase: shuffling
% 2.94/1.68
% 2.94/1.68 % (13477)Memory used [KB]: 1535
% 2.94/1.68 % (13477)Time elapsed: 0.008 s
% 2.94/1.68 % (13477)Instructions burned: 17 (million)
% 2.94/1.68 % (13477)------------------------------
% 2.94/1.68 % (13477)------------------------------
% 2.94/1.68 % (13479)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_13330 for (2998ds/30Mi)
% 2.94/1.68 % (13476)Refutation not found, incomplete strategy
% 2.94/1.68 % (13476)------------------------------
% 2.94/1.68 % (13476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.68 % (13476)Termination reason: Refutation not found, incomplete strategy
% 2.94/1.68
% 2.94/1.68
% 2.94/1.68 % (13476)Memory used [KB]: 6140
% 2.94/1.68 % (13476)Time elapsed: 0.015 s
% 2.94/1.68 % (13476)Instructions burned: 41 (million)
% 2.94/1.68 % (13476)------------------------------
% 2.94/1.68 % (13476)------------------------------
% 2.94/1.69 % (13480)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_13330 for (2998ds/127Mi)
% 2.94/1.69 % (13479)Instruction limit reached!
% 2.94/1.69 % (13479)------------------------------
% 2.94/1.69 % (13479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.69 % (13479)Termination reason: Unknown
% 2.94/1.69 % (13479)Termination phase: Preprocessing 1
% 2.94/1.69
% 2.94/1.69 % (13479)Memory used [KB]: 1663
% 2.94/1.69 % (13479)Time elapsed: 0.011 s
% 2.94/1.69 % (13479)Instructions burned: 33 (million)
% 2.94/1.69 % (13479)------------------------------
% 2.94/1.69 % (13479)------------------------------
% 2.94/1.69 % (13447)Instruction limit reached!
% 2.94/1.69 % (13447)------------------------------
% 2.94/1.69 % (13447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.69 % (13447)Termination reason: Unknown
% 2.94/1.69 % (13447)Termination phase: Saturation
% 2.94/1.69
% 2.94/1.69 % (13447)Memory used [KB]: 6780
% 2.94/1.69 % (13447)Time elapsed: 0.106 s
% 2.94/1.69 % (13447)Instructions burned: 184 (million)
% 2.94/1.69 % (13447)------------------------------
% 2.94/1.69 % (13447)------------------------------
% 2.94/1.69 % (13481)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_13330 for (2998ds/100Mi)
% 2.94/1.70 % (13482)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_13330 for (2998ds/3Mi)
% 2.94/1.70 % (13483)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_13330 for (2998ds/20Mi)
% 2.94/1.70 % (13482)Instruction limit reached!
% 2.94/1.70 % (13482)------------------------------
% 2.94/1.70 % (13482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.70 % (13482)Termination reason: Unknown
% 2.94/1.70 % (13482)Termination phase: shuffling
% 2.94/1.70
% 2.94/1.70 % (13482)Memory used [KB]: 1407
% 2.94/1.70 % (13482)Time elapsed: 0.003 s
% 2.94/1.70 % (13482)Instructions burned: 5 (million)
% 2.94/1.70 % (13482)------------------------------
% 2.94/1.70 % (13482)------------------------------
% 2.94/1.71 % (13483)Instruction limit reached!
% 2.94/1.71 % (13483)------------------------------
% 2.94/1.71 % (13483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.71 % (13483)Termination reason: Unknown
% 2.94/1.71 % (13483)Termination phase: shuffling
% 2.94/1.71
% 2.94/1.71 % (13483)Memory used [KB]: 1535
% 2.94/1.71 % (13483)Time elapsed: 0.008 s
% 2.94/1.71 % (13483)Instructions burned: 20 (million)
% 2.94/1.71 % (13483)------------------------------
% 2.94/1.71 % (13483)------------------------------
% 2.94/1.71 % (13484)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_13330 for (2998ds/86Mi)
% 2.94/1.72 % (13485)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_13330 for (2998ds/107Mi)
% 2.94/1.72 % (13481)Instruction limit reached!
% 2.94/1.72 % (13481)------------------------------
% 2.94/1.72 % (13481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.72 % (13481)Termination reason: Unknown
% 2.94/1.72 % (13481)Termination phase: Saturation
% 2.94/1.72
% 2.94/1.72 % (13481)Memory used [KB]: 6652
% 2.94/1.73 % (13481)Time elapsed: 0.032 s
% 2.94/1.73 % (13481)Instructions burned: 102 (million)
% 2.94/1.73 % (13481)------------------------------
% 2.94/1.73 % (13481)------------------------------
% 2.94/1.73 % (13480)Instruction limit reached!
% 2.94/1.73 % (13480)------------------------------
% 2.94/1.73 % (13480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.73 % (13480)Termination reason: Unknown
% 2.94/1.73 % (13480)Termination phase: Saturation
% 2.94/1.73
% 2.94/1.73 % (13480)Memory used [KB]: 6780
% 2.94/1.73 % (13480)Time elapsed: 0.046 s
% 2.94/1.73 % (13480)Instructions burned: 129 (million)
% 2.94/1.73 % (13480)------------------------------
% 2.94/1.73 % (13480)------------------------------
% 2.94/1.74 % (13486)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_13330 for (2998ds/39Mi)
% 2.94/1.74 % (13484)Instruction limit reached!
% 2.94/1.74 % (13484)------------------------------
% 2.94/1.74 % (13484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.74 % (13484)Termination reason: Unknown
% 2.94/1.74 % (13484)Termination phase: Saturation
% 2.94/1.74
% 2.94/1.74 % (13484)Memory used [KB]: 6524
% 2.94/1.74 % (13484)Time elapsed: 0.027 s
% 2.94/1.74 % (13484)Instructions burned: 87 (million)
% 2.94/1.74 % (13484)------------------------------
% 2.94/1.74 % (13484)------------------------------
% 2.94/1.74 % (13487)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_13330 for (2998ds/448Mi)
% 2.94/1.75 % (13486)Instruction limit reached!
% 2.94/1.75 % (13486)------------------------------
% 2.94/1.75 % (13486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.75 % (13486)Termination reason: Unknown
% 2.94/1.75 % (13486)Termination phase: Property scanning
% 2.94/1.75
% 2.94/1.75 % (13486)Memory used [KB]: 1918
% 2.94/1.75 % (13486)Time elapsed: 0.013 s
% 2.94/1.75 % (13486)Instructions burned: 40 (million)
% 2.94/1.75 % (13486)------------------------------
% 2.94/1.75 % (13486)------------------------------
% 2.94/1.75 % (13488)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_13330 for (2998ds/46Mi)
% 2.94/1.75 % (13485)Instruction limit reached!
% 2.94/1.75 % (13485)------------------------------
% 2.94/1.75 % (13485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.75 % (13485)Termination reason: Unknown
% 2.94/1.75 % (13485)Termination phase: Saturation
% 2.94/1.75
% 2.94/1.75 % (13485)Memory used [KB]: 6652
% 2.94/1.75 % (13485)Time elapsed: 0.036 s
% 2.94/1.75 % (13485)Instructions burned: 107 (million)
% 2.94/1.75 % (13485)------------------------------
% 2.94/1.75 % (13485)------------------------------
% 2.94/1.76 % (13489)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_13330 for (2998ds/98Mi)
% 2.94/1.76 % (13488)Instruction limit reached!
% 2.94/1.76 % (13488)------------------------------
% 2.94/1.76 % (13488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.76 % (13488)Termination reason: Unknown
% 2.94/1.76 % (13488)Termination phase: Preprocessing 3
% 2.94/1.76
% 2.94/1.76 % (13488)Memory used [KB]: 1791
% 2.94/1.76 % (13488)Time elapsed: 0.016 s
% 2.94/1.76 % (13488)Instructions burned: 47 (million)
% 2.94/1.76 % (13488)------------------------------
% 2.94/1.76 % (13488)------------------------------
% 2.94/1.77 % (13490)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_13330 for (2998ds/507Mi)
% 2.94/1.77 % (13473)Instruction limit reached!
% 2.94/1.77 % (13473)------------------------------
% 2.94/1.77 % (13473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.77 % (13473)Termination reason: Unknown
% 2.94/1.77 % (13473)Termination phase: Saturation
% 2.94/1.77
% 2.94/1.77 % (13473)Memory used [KB]: 7419
% 2.94/1.77 % (13473)Time elapsed: 0.121 s
% 2.94/1.77 % (13473)Instructions burned: 377 (million)
% 2.94/1.77 % (13473)------------------------------
% 2.94/1.77 % (13473)------------------------------
% 2.94/1.77 % (13491)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_13330 for (2997ds/149Mi)
% 2.94/1.78 % (13492)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_13330 for (2997ds/39Mi)
% 2.94/1.79 % (13489)Instruction limit reached!
% 2.94/1.79 % (13489)------------------------------
% 2.94/1.79 % (13489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.79 % (13489)Termination reason: Unknown
% 2.94/1.79 % (13489)Termination phase: Saturation
% 2.94/1.79
% 2.94/1.79 % (13489)Memory used [KB]: 2174
% 2.94/1.79 % (13489)Time elapsed: 0.031 s
% 2.94/1.79 % (13489)Instructions burned: 99 (million)
% 2.94/1.79 % (13489)------------------------------
% 2.94/1.79 % (13489)------------------------------
% 2.94/1.79 % (13492)Instruction limit reached!
% 2.94/1.79 % (13492)------------------------------
% 2.94/1.79 % (13492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.79 % (13492)Termination reason: Unknown
% 2.94/1.79 % (13492)Termination phase: Preprocessing 3
% 2.94/1.79
% 2.94/1.79 % (13492)Memory used [KB]: 1791
% 2.94/1.79 % (13492)Time elapsed: 0.014 s
% 2.94/1.79 % (13492)Instructions burned: 41 (million)
% 2.94/1.79 % (13492)------------------------------
% 2.94/1.79 % (13492)------------------------------
% 2.94/1.80 % (13490)Refutation not found, incomplete strategy
% 2.94/1.80 % (13490)------------------------------
% 2.94/1.80 % (13490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.80 % (13490)Termination reason: Refutation not found, incomplete strategy
% 2.94/1.80
% 2.94/1.80
% 2.94/1.80 % (13490)Memory used [KB]: 6908
% 2.94/1.80 % (13490)Time elapsed: 0.033 s
% 2.94/1.80 % (13490)Instructions burned: 94 (million)
% 2.94/1.80 % (13490)------------------------------
% 2.94/1.80 % (13490)------------------------------
% 2.94/1.80 % (13493)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_13330 for (2997ds/6Mi)
% 2.94/1.80 % (13493)Instruction limit reached!
% 2.94/1.80 % (13493)------------------------------
% 2.94/1.80 % (13493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.80 % (13493)Termination reason: Unknown
% 2.94/1.80 % (13493)Termination phase: shuffling
% 2.94/1.80
% 2.94/1.80 % (13493)Memory used [KB]: 1407
% 2.94/1.80 % (13493)Time elapsed: 0.004 s
% 2.94/1.80 % (13493)Instructions burned: 8 (million)
% 2.94/1.80 % (13493)------------------------------
% 2.94/1.80 % (13493)------------------------------
% 2.94/1.80 % (13494)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_13330 for (2997ds/1087Mi)
% 2.94/1.81 % (13495)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_13330 for (2997ds/47Mi)
% 2.94/1.81 % (13496)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_13330 for (2997ds/44Mi)
% 2.94/1.82 % (13491)Instruction limit reached!
% 2.94/1.82 % (13491)------------------------------
% 2.94/1.82 % (13491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.82 % (13491)Termination reason: Unknown
% 2.94/1.82 % (13491)Termination phase: Saturation
% 2.94/1.82
% 2.94/1.82 % (13491)Memory used [KB]: 6780
% 2.94/1.82 % (13491)Time elapsed: 0.048 s
% 2.94/1.82 % (13491)Instructions burned: 149 (million)
% 2.94/1.82 % (13491)------------------------------
% 2.94/1.82 % (13491)------------------------------
% 2.94/1.82 % (13495)Instruction limit reached!
% 2.94/1.82 % (13495)------------------------------
% 2.94/1.82 % (13495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.82 % (13495)Termination reason: Unknown
% 2.94/1.82 % (13495)Termination phase: Property scanning
% 2.94/1.82
% 2.94/1.82 % (13495)Memory used [KB]: 1918
% 2.94/1.82 % (13495)Time elapsed: 0.018 s
% 2.94/1.82 % (13495)Instructions burned: 49 (million)
% 2.94/1.82 % (13495)------------------------------
% 2.94/1.82 % (13495)------------------------------
% 2.94/1.83 % (13496)Instruction limit reached!
% 2.94/1.83 % (13496)------------------------------
% 2.94/1.83 % (13496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.83 % (13496)Termination reason: Unknown
% 2.94/1.83 % (13496)Termination phase: Property scanning
% 2.94/1.83
% 2.94/1.83 % (13496)Memory used [KB]: 1918
% 2.94/1.83 % (13496)Time elapsed: 0.015 s
% 2.94/1.83 % (13496)Instructions burned: 47 (million)
% 2.94/1.83 % (13496)------------------------------
% 2.94/1.83 % (13496)------------------------------
% 2.94/1.83 % (13497)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_13330 for (2997ds/432Mi)
% 2.94/1.83 % (13498)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_13330 for (2997ds/62Mi)
% 2.94/1.84 % (13499)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_13330 for (2997ds/31Mi)
% 2.94/1.85 % (13499)Instruction limit reached!
% 2.94/1.85 % (13499)------------------------------
% 2.94/1.85 % (13499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.85 % (13499)Termination reason: Unknown
% 2.94/1.85 % (13499)Termination phase: Property scanning
% 2.94/1.85
% 2.94/1.85 % (13499)Memory used [KB]: 1535
% 2.94/1.85 % (13499)Time elapsed: 0.012 s
% 2.94/1.85 % (13499)Instructions burned: 32 (million)
% 2.94/1.85 % (13499)------------------------------
% 2.94/1.85 % (13499)------------------------------
% 2.94/1.85 % (13498)Instruction limit reached!
% 2.94/1.85 % (13498)------------------------------
% 2.94/1.85 % (13498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.85 % (13498)Termination reason: Unknown
% 2.94/1.85 % (13498)Termination phase: Property scanning
% 2.94/1.85
% 2.94/1.85 % (13498)Memory used [KB]: 2046
% 2.94/1.85 % (13498)Time elapsed: 0.020 s
% 2.94/1.85 % (13498)Instructions burned: 62 (million)
% 2.94/1.85 % (13498)------------------------------
% 2.94/1.85 % (13498)------------------------------
% 2.94/1.86 % (13500)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_13330 for (2997ds/96Mi)
% 2.94/1.86 % (13501)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_13330 for (2997ds/453Mi)
% 2.94/1.89 % (13474)Instruction limit reached!
% 2.94/1.89 % (13474)------------------------------
% 2.94/1.89 % (13474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.89 % (13474)Termination reason: Unknown
% 2.94/1.89 % (13474)Termination phase: Saturation
% 2.94/1.89
% 2.94/1.89 % (13474)Memory used [KB]: 8955
% 2.94/1.89 % (13474)Time elapsed: 0.234 s
% 2.94/1.89 % (13474)Instructions burned: 781 (million)
% 2.94/1.89 % (13474)------------------------------
% 2.94/1.89 % (13474)------------------------------
% 2.94/1.89 % (13487)Instruction limit reached!
% 2.94/1.89 % (13487)------------------------------
% 2.94/1.89 % (13487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.89 % (13487)Termination reason: Unknown
% 2.94/1.89 % (13487)Termination phase: Saturation
% 2.94/1.89
% 2.94/1.89 % (13487)Memory used [KB]: 7675
% 2.94/1.89 % (13487)Time elapsed: 0.147 s
% 2.94/1.89 % (13487)Instructions burned: 448 (million)
% 2.94/1.89 % (13487)------------------------------
% 2.94/1.89 % (13487)------------------------------
% 2.94/1.89 % (13500)Instruction limit reached!
% 2.94/1.89 % (13500)------------------------------
% 2.94/1.89 % (13500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.89 % (13500)Termination reason: Unknown
% 2.94/1.89 % (13500)Termination phase: Saturation
% 2.94/1.89
% 2.94/1.89 % (13500)Memory used [KB]: 6652
% 2.94/1.89 % (13500)Time elapsed: 0.034 s
% 2.94/1.89 % (13500)Instructions burned: 98 (million)
% 2.94/1.89 % (13500)------------------------------
% 2.94/1.89 % (13500)------------------------------
% 2.94/1.90 % (13502)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_13330 for (2996ds/71Mi)
% 2.94/1.90 % (13503)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_13330 for (2996ds/83Mi)
% 2.94/1.90 % (13504)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_13330 for (2996ds/11Mi)
% 2.94/1.91 % (13504)Instruction limit reached!
% 2.94/1.91 % (13504)------------------------------
% 2.94/1.91 % (13504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.91 % (13504)Termination reason: Unknown
% 2.94/1.91 % (13504)Termination phase: shuffling
% 2.94/1.91
% 2.94/1.91 % (13504)Memory used [KB]: 1535
% 2.94/1.91 % (13504)Time elapsed: 0.005 s
% 2.94/1.91 % (13504)Instructions burned: 12 (million)
% 2.94/1.91 % (13504)------------------------------
% 2.94/1.91 % (13504)------------------------------
% 2.94/1.91 % (13503)Refutation not found, incomplete strategy
% 2.94/1.91 % (13503)------------------------------
% 2.94/1.91 % (13503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.91 % (13503)Termination reason: Refutation not found, incomplete strategy
% 2.94/1.91
% 2.94/1.91
% 2.94/1.91 % (13503)Memory used [KB]: 6140
% 2.94/1.91 % (13503)Time elapsed: 0.015 s
% 2.94/1.91 % (13503)Instructions burned: 40 (million)
% 2.94/1.91 % (13503)------------------------------
% 2.94/1.91 % (13503)------------------------------
% 2.94/1.92 % (13505)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_13330 for (2996ds/41Mi)
% 2.94/1.92 % (13502)Instruction limit reached!
% 2.94/1.92 % (13502)------------------------------
% 2.94/1.92 % (13502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.92 % (13502)Termination reason: Unknown
% 2.94/1.92 % (13502)Termination phase: Saturation
% 2.94/1.92
% 2.94/1.92 % (13502)Memory used [KB]: 6396
% 2.94/1.92 % (13502)Time elapsed: 0.022 s
% 2.94/1.92 % (13502)Instructions burned: 71 (million)
% 2.94/1.92 % (13502)------------------------------
% 2.94/1.92 % (13502)------------------------------
% 2.94/1.92 % (13458)Instruction limit reached!
% 2.94/1.92 % (13458)------------------------------
% 2.94/1.92 % (13458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.92 % (13458)Termination reason: Unknown
% 2.94/1.92 % (13458)Termination phase: Saturation
% 2.94/1.92
% 2.94/1.92 % (13458)Memory used [KB]: 10362
% 2.94/1.92 % (13458)Time elapsed: 0.341 s
% 2.94/1.92 % (13458)Instructions burned: 1043 (million)
% 2.94/1.92 % (13458)------------------------------
% 2.94/1.92 % (13458)------------------------------
% 2.94/1.92 % (13506)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_13330 for (2996ds/6Mi)
% 2.94/1.92 % (13506)Instruction limit reached!
% 2.94/1.92 % (13506)------------------------------
% 2.94/1.92 % (13506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.92 % (13506)Termination reason: Unknown
% 2.94/1.92 % (13506)Termination phase: shuffling
% 2.94/1.92
% 2.94/1.92 % (13506)Memory used [KB]: 1407
% 2.94/1.92 % (13506)Time elapsed: 0.003 s
% 2.94/1.92 % (13506)Instructions burned: 6 (million)
% 2.94/1.92 % (13506)------------------------------
% 2.94/1.92 % (13506)------------------------------
% 2.94/1.93 % (13507)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_13330 for (2996ds/150Mi)
% 2.94/1.93 % (13505)Instruction limit reached!
% 2.94/1.93 % (13505)------------------------------
% 2.94/1.93 % (13505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.93 % (13505)Termination reason: Unknown
% 2.94/1.93 % (13505)Termination phase: Property scanning
% 2.94/1.93
% 2.94/1.93 % (13505)Memory used [KB]: 1918
% 2.94/1.93 % (13505)Time elapsed: 0.014 s
% 2.94/1.93 % (13505)Instructions burned: 43 (million)
% 2.94/1.93 % (13505)------------------------------
% 2.94/1.93 % (13505)------------------------------
% 2.94/1.93 % (13508)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_13330 for (2996ds/343Mi)
% 2.94/1.93 % (13509)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_13330 for (2996ds/109Mi)
% 2.94/1.94 % (13510)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_13330 for (2996ds/253Mi)
% 2.94/1.95 % (13510)Refutation not found, incomplete strategy
% 2.94/1.95 % (13510)------------------------------
% 2.94/1.95 % (13510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.95 % (13510)Termination reason: Refutation not found, incomplete strategy
% 2.94/1.95
% 2.94/1.95
% 2.94/1.95 % (13510)Memory used [KB]: 6140
% 2.94/1.95 % (13510)Time elapsed: 0.014 s
% 2.94/1.95 % (13510)Instructions burned: 36 (million)
% 2.94/1.95 % (13510)------------------------------
% 2.94/1.95 % (13510)------------------------------
% 2.94/1.96 % (13497)Instruction limit reached!
% 2.94/1.96 % (13497)------------------------------
% 2.94/1.96 % (13497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.97 % (13497)Termination reason: Unknown
% 2.94/1.97 % (13497)Termination phase: Saturation
% 2.94/1.97
% 2.94/1.97 % (13497)Memory used [KB]: 4093
% 2.94/1.97 % (13497)Time elapsed: 0.137 s
% 2.94/1.97 % (13497)Instructions burned: 432 (million)
% 2.94/1.97 % (13497)------------------------------
% 2.94/1.97 % (13497)------------------------------
% 2.94/1.97 % (13514)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_13330 for (2995ds/140Mi)
% 2.94/1.97 % (13509)Instruction limit reached!
% 2.94/1.97 % (13509)------------------------------
% 2.94/1.97 % (13509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.97 % (13509)Termination reason: Unknown
% 2.94/1.97 % (13509)Termination phase: Saturation
% 2.94/1.97
% 2.94/1.97 % (13509)Memory used [KB]: 6652
% 2.94/1.97 % (13509)Time elapsed: 0.035 s
% 2.94/1.97 % (13509)Instructions burned: 111 (million)
% 2.94/1.97 % (13509)------------------------------
% 2.94/1.97 % (13509)------------------------------
% 2.94/1.97 % (13501)First to succeed.
% 2.94/1.97 % (13507)Instruction limit reached!
% 2.94/1.97 % (13507)------------------------------
% 2.94/1.97 % (13507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.97 % (13507)Termination reason: Unknown
% 2.94/1.97 % (13507)Termination phase: Saturation
% 2.94/1.97
% 2.94/1.97 % (13507)Memory used [KB]: 6908
% 2.94/1.97 % (13507)Time elapsed: 0.049 s
% 2.94/1.97 % (13507)Instructions burned: 150 (million)
% 2.94/1.97 % (13507)------------------------------
% 2.94/1.97 % (13507)------------------------------
% 2.94/1.98 % (13501)Refutation found. Thanks to Tanya!
% 2.94/1.98 % SZS status ContradictoryAxioms for DTF2THF_13330
% 2.94/1.98 % SZS output start Proof for DTF2THF_13330
% 2.94/1.98 thf(type_def_5, type, nat: $tType).
% 2.94/1.98 thf(type_def_6, type, product_prod: ($tType * $tType) > $tType).
% 2.94/1.98 thf(type_def_8, type, set: ($tType) > $tType).
% 2.94/1.98 thf(type_def_9, type, list: ($tType) > $tType).
% 2.94/1.98 thf(type_def_10, type, a: $tType).
% 2.94/1.98 thf(type_def_11, type, b: $tType).
% 2.94/1.98 thf(func_def_0, type, product_prod: ($tType * $tType) > $tType).
% 2.94/1.98 thf(func_def_1, type, list: ($tType) > $tType).
% 2.94/1.98 thf(func_def_2, type, set: ($tType) > $tType).
% 2.94/1.98 thf(func_def_3, type, nat: $tType).
% 2.94/1.98 thf(func_def_4, type, b: $tType).
% 2.94/1.98 thf(func_def_5, type, a: $tType).
% 2.94/1.98 thf(func_def_6, type, type: !>[X0: $tType]:($o)).
% 2.94/1.98 thf(func_def_7, type, size: !>[X0: $tType]:($o)).
% 2.94/1.98 thf(func_def_8, type, zero: !>[X0: $tType]:($o)).
% 2.94/1.98 thf(func_def_9, type, order: !>[X0: $tType]:($o)).
% 2.94/1.98 thf(func_def_10, type, finite_finite: !>[X0: $tType]:($o)).
% 2.94/1.98 thf(func_def_11, type, wellorder: !>[X0: $tType]:($o)).
% 2.94/1.98 thf(func_def_12, type, linordered_idom: !>[X0: $tType]:($o)).
% 2.94/1.98 thf(func_def_13, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 2.94/1.98 thf(func_def_14, type, edmond2129202899alysis: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > $o)).
% 2.94/1.98 thf(func_def_15, type, edmond115586663easure: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > nat)).
% 2.94/1.98 thf(func_def_16, type, edmond1803015688pEdges: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > set @ product_prod @ nat @ nat)).
% 2.94/1.98 thf(func_def_17, type, edmond259086305sis_uE: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > set @ product_prod @ nat @ nat)).
% 2.94/1.98 thf(func_def_18, type, finite_card: !>[X0: $tType]:(set @ X0 > nat)).
% 2.94/1.98 thf(func_def_19, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 2.94/1.98 thf(func_def_20, type, finite_Graph: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > $o)).
% 2.94/1.98 thf(func_def_21, type, e: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > set @ product_prod @ nat @ nat)).
% 2.94/1.98 thf(func_def_22, type, v: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > set @ nat)).
% 2.94/1.98 thf(func_def_23, type, adjacent_nodes: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > set @ nat)).
% 2.94/1.98 thf(func_def_24, type, connected: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > $o)).
% 2.94/1.98 thf(func_def_25, type, dist: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > nat > $o)).
% 2.94/1.98 thf(func_def_26, type, incoming: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > set @ nat > set @ product_prod @ nat @ nat)).
% 2.94/1.98 thf(func_def_27, type, isPath: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > list @ product_prod @ nat @ nat > nat > $o)).
% 2.94/1.98 thf(func_def_28, type, isShortestPath: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > list @ product_prod @ nat @ nat > nat > $o)).
% 2.94/1.98 thf(func_def_29, type, isSimplePath: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > list @ product_prod @ nat @ nat > nat > $o)).
% 2.94/1.98 thf(func_def_30, type, min_dist: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > nat)).
% 2.94/1.98 thf(func_def_31, type, outgoing: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > set @ nat > set @ product_prod @ nat @ nat)).
% 2.94/1.98 thf(func_def_32, type, reachableNodes: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > set @ nat)).
% 2.94/1.98 thf(func_def_33, type, zero_zero: !>[X0: $tType]:(X0)).
% 2.94/1.98 thf(func_def_34, type, set2: !>[X0: $tType]:(list @ X0 > set @ X0)).
% 2.94/1.98 thf(func_def_35, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 2.94/1.98 thf(func_def_36, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.94/1.98 thf(func_def_37, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.94/1.98 thf(func_def_38, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.94/1.98 thf(func_def_39, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.94/1.98 thf(func_def_40, type, c: product_prod @ nat @ nat > a).
% 2.94/1.98 thf(func_def_41, type, c2: product_prod @ nat @ nat > b).
% 2.94/1.98 thf(func_def_42, type, edges: set @ product_prod @ nat @ nat).
% 2.94/1.98 thf(func_def_43, type, p: list @ product_prod @ nat @ nat).
% 2.94/1.98 thf(func_def_44, type, p2: list @ product_prod @ nat @ nat).
% 2.94/1.98 thf(func_def_45, type, s: nat).
% 2.94/1.98 thf(func_def_46, type, t: nat).
% 2.94/1.98 thf(func_def_65, type, ph1: !>[X0: $tType]:(X0)).
% 2.94/1.98 thf(func_def_66, type, sK2: !>[X0: $tType]:(nat > list @ X0)).
% 2.94/1.98 thf(f1843,plain,(
% 2.94/1.98 $false),
% 2.94/1.98 inference(trivial_inequality_removal,[status(thm)],[f1842])).
% 2.94/1.98 thf(f1842,plain,(
% 2.94/1.98 ($false = $true)),
% 2.94/1.98 inference(backward_demodulation,[status(thm)],[f1689,f1841])).
% 2.94/1.98 thf(f1841,plain,(
% 2.94/1.98 ($false = (ord_less_eq @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p2) @ (size_size @ list @ product_prod @ nat @ nat @ p)))),
% 2.94/1.98 inference(subsumption_resolution,[status(thm)],[f1830,f1537])).
% 2.94/1.98 thf(f1537,plain,(
% 2.94/1.98 ((size_size @ list @ product_prod @ nat @ nat @ p) != (size_size @ list @ product_prod @ nat @ nat @ p2))),
% 2.94/1.98 inference(trivial_inequality_removal,[status(thm)],[f1530])).
% 2.94/1.98 thf(f1530,plain,(
% 2.94/1.98 ($false = $true) | ((size_size @ list @ product_prod @ nat @ nat @ p) != (size_size @ list @ product_prod @ nat @ nat @ p2))),
% 2.94/1.98 inference(superposition,[status(thm)],[f1118,f1467])).
% 2.94/1.98 thf(f1467,plain,(
% 2.94/1.98 ( ! [X2 : nat,X1 : nat] : (($false = (ord_less @ nat @ X2 @ X1)) | (X1 != X2)) )),
% 2.94/1.98 inference(equality_proxy_clausification,[status(thm)],[f1466])).
% 2.94/1.98 thf(f1466,plain,(
% 2.94/1.98 ( ! [X2 : nat,X1 : nat] : (($false = (ord_less @ nat @ X2 @ X1)) | ((X2 = X1) = $false)) )),
% 2.94/1.98 inference(not_proxy_clausification,[status(thm)],[f1465])).
% 2.94/1.98 thf(f1465,plain,(
% 2.94/1.98 ( ! [X2 : nat,X1 : nat] : (((~ (X2 = X1)) = $true) | ($false = (ord_less @ nat @ X2 @ X1))) )),
% 2.94/1.98 inference(binary_proxy_clausification,[status(thm)],[f1464])).
% 2.94/1.98 thf(f1464,plain,(
% 2.94/1.98 ( ! [X2 : nat,X1 : nat] : (($true = ((ord_less @ nat @ X2 @ X1) => (~ (X2 = X1))))) )),
% 2.94/1.98 inference(beta_eta_normalization,[status(thm)],[f1463])).
% 2.94/1.98 thf(f1463,plain,(
% 2.94/1.98 ( ! [X2 : nat,X1 : nat] : ((((^[Y0 : nat]: ((ord_less @ nat @ Y0 @ X1) => (~ (Y0 = X1)))) @ X2) = $true)) )),
% 2.94/1.98 inference(pi_clausification,[status(thm)],[f1460])).
% 2.94/1.98 thf(f1460,plain,(
% 2.94/1.98 ( ! [X1 : nat] : (($true = (!! @ nat @ (^[Y0 : nat]: ((ord_less @ nat @ Y0 @ X1) => (~ (Y0 = X1))))))) )),
% 2.94/1.98 inference(beta_eta_normalization,[status(thm)],[f1459])).
% 2.94/1.98 thf(f1459,plain,(
% 2.94/1.98 ( ! [X1 : nat] : (($true = ((^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less @ nat @ Y1 @ Y0) => (~ (Y1 = Y0)))))) @ X1))) )),
% 2.94/1.98 inference(pi_clausification,[status(thm)],[f1136])).
% 2.94/1.98 thf(f1136,plain,(
% 2.94/1.98 ($true = (!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less @ nat @ Y1 @ Y0) => (~ (Y1 = Y0))))))))),
% 2.94/1.98 inference(cnf_transformation,[status(thm)],[f675])).
% 2.94/1.98 thf(f675,plain,(
% 2.94/1.98 ($true = (!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less @ nat @ Y1 @ Y0) => (~ (Y1 = Y0))))))))),
% 2.94/1.98 inference(fool_elimination,[status(thm)],[f674])).
% 2.94/1.98 thf(f674,plain,(
% 2.94/1.98 ! [X0 : nat,X1 : nat] : ((ord_less @ nat @ X0 @ X1) => (X0 != X1))),
% 2.94/1.98 inference(rectify,[status(thm)],[f106])).
% 2.94/1.98 thf(f106,axiom,(
% 2.94/1.98 ! [X30 : nat,X42 : nat] : ((ord_less @ nat @ X30 @ X42) => (X30 != X42))),
% 2.94/1.98 file('/export/starexec/sandbox/tmp/tmp.AUIVmKJLkR/DTF2THF_13330.p',fact_105_less__not__refl2)).
% 2.94/1.98 thf(f1118,plain,(
% 2.94/1.98 ((ord_less @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p) @ (size_size @ list @ product_prod @ nat @ nat @ p2)) = $true)),
% 2.94/1.98 inference(cnf_transformation,[status(thm)],[f725])).
% 2.94/1.98 thf(f725,plain,(
% 2.94/1.98 ((ord_less @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p) @ (size_size @ list @ product_prod @ nat @ nat @ p2)) = $true)),
% 2.94/1.98 inference(fool_elimination,[status(thm)],[f724])).
% 2.94/1.98 thf(f724,plain,(
% 2.94/1.98 (ord_less @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p) @ (size_size @ list @ product_prod @ nat @ nat @ p2))),
% 2.94/1.98 inference(rectify,[status(thm)],[f11])).
% 2.94/1.98 thf(f11,axiom,(
% 2.94/1.98 (ord_less @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p) @ (size_size @ list @ product_prod @ nat @ nat @ p2))),
% 2.94/1.98 file('/export/starexec/sandbox/tmp/tmp.AUIVmKJLkR/DTF2THF_13330.p',fact_10__092_060open_062length_Ap_A_060_Alength_Ap_H_092_060close_062)).
% 2.94/1.98 thf(f1830,plain,(
% 2.94/1.98 ($false = (ord_less_eq @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p2) @ (size_size @ list @ product_prod @ nat @ nat @ p))) | ((size_size @ list @ product_prod @ nat @ nat @ p) = (size_size @ list @ product_prod @ nat @ nat @ p2))),
% 2.94/1.98 inference(trivial_inequality_removal,[status(thm)],[f1819])).
% 2.94/1.98 thf(f1819,plain,(
% 2.94/1.98 ($false = (ord_less_eq @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p2) @ (size_size @ list @ product_prod @ nat @ nat @ p))) | ((size_size @ list @ product_prod @ nat @ nat @ p) = (size_size @ list @ product_prod @ nat @ nat @ p2)) | ($false = $true)),
% 2.94/1.98 inference(superposition,[status(thm)],[f1682,f1796])).
% 2.94/1.98 thf(f1796,plain,(
% 2.94/1.98 ( ! [X2 : nat,X1 : nat] : (((ord_less_eq @ nat @ X2 @ X1) = $false) | ((ord_less_eq @ nat @ X1 @ X2) = $false) | (X1 = X2)) )),
% 2.94/1.98 inference(equality_proxy_clausification,[status(thm)],[f1795])).
% 2.94/1.98 thf(f1795,plain,(
% 2.94/1.98 ( ! [X2 : nat,X1 : nat] : (((ord_less_eq @ nat @ X2 @ X1) = $false) | ((ord_less_eq @ nat @ X1 @ X2) = $false) | ((X2 = X1) = $true)) )),
% 2.94/1.98 inference(binary_proxy_clausification,[status(thm)],[f1794])).
% 2.94/1.98 thf(f1794,plain,(
% 2.94/1.98 ( ! [X2 : nat,X1 : nat] : (((ord_less_eq @ nat @ X2 @ X1) = $false) | ($true = ((ord_less_eq @ nat @ X1 @ X2) => (X2 = X1)))) )),
% 2.94/1.98 inference(binary_proxy_clausification,[status(thm)],[f1793])).
% 2.94/1.98 thf(f1793,plain,(
% 2.94/1.98 ( ! [X2 : nat,X1 : nat] : ((((ord_less_eq @ nat @ X2 @ X1) => ((ord_less_eq @ nat @ X1 @ X2) => (X2 = X1))) = $true)) )),
% 2.94/1.98 inference(beta_eta_normalization,[status(thm)],[f1792])).
% 2.94/1.98 thf(f1792,plain,(
% 2.94/1.98 ( ! [X2 : nat,X1 : nat] : (($true = ((^[Y0 : nat]: ((ord_less_eq @ nat @ Y0 @ X1) => ((ord_less_eq @ nat @ X1 @ Y0) => (Y0 = X1)))) @ X2))) )),
% 2.94/1.98 inference(pi_clausification,[status(thm)],[f1767])).
% 2.94/1.98 thf(f1767,plain,(
% 2.94/1.98 ( ! [X1 : nat] : (($true = (!! @ nat @ (^[Y0 : nat]: ((ord_less_eq @ nat @ Y0 @ X1) => ((ord_less_eq @ nat @ X1 @ Y0) => (Y0 = X1))))))) )),
% 2.94/1.98 inference(beta_eta_normalization,[status(thm)],[f1766])).
% 2.94/1.98 thf(f1766,plain,(
% 2.94/1.98 ( ! [X1 : nat] : ((((^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less_eq @ nat @ Y1 @ Y0) => ((ord_less_eq @ nat @ Y0 @ Y1) => (Y1 = Y0)))))) @ X1) = $true)) )),
% 2.94/1.98 inference(pi_clausification,[status(thm)],[f1175])).
% 2.94/1.98 thf(f1175,plain,(
% 2.94/1.98 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less_eq @ nat @ Y1 @ Y0) => ((ord_less_eq @ nat @ Y0 @ Y1) => (Y1 = Y0))))))) = $true)),
% 2.94/1.98 inference(cnf_transformation,[status(thm)],[f438])).
% 2.94/1.98 thf(f438,plain,(
% 2.94/1.98 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less_eq @ nat @ Y1 @ Y0) => ((ord_less_eq @ nat @ Y0 @ Y1) => (Y1 = Y0))))))) = $true)),
% 2.94/1.98 inference(fool_elimination,[status(thm)],[f437])).
% 2.94/1.98 thf(f437,plain,(
% 2.94/1.98 ! [X0 : nat,X1 : nat] : ((ord_less_eq @ nat @ X0 @ X1) => ((ord_less_eq @ nat @ X1 @ X0) => (X0 = X1)))),
% 2.94/1.98 inference(rectify,[status(thm)],[f82])).
% 2.94/1.98 thf(f82,axiom,(
% 2.94/1.98 ! [X42 : nat,X30 : nat] : ((ord_less_eq @ nat @ X42 @ X30) => ((ord_less_eq @ nat @ X30 @ X42) => (X30 = X42)))),
% 2.94/1.98 file('/export/starexec/sandbox/tmp/tmp.AUIVmKJLkR/DTF2THF_13330.p',fact_81_le__antisym)).
% 2.94/1.98 thf(f1682,plain,(
% 2.94/1.98 ((ord_less_eq @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p) @ (size_size @ list @ product_prod @ nat @ nat @ p2)) = $true)),
% 2.94/1.98 inference(trivial_inequality_removal,[status(thm)],[f1676])).
% 2.94/1.98 thf(f1676,plain,(
% 2.94/1.98 ((ord_less_eq @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p) @ (size_size @ list @ product_prod @ nat @ nat @ p2)) = $true) | ($false = $true)),
% 2.94/1.98 inference(superposition,[status(thm)],[f1118,f1665])).
% 2.94/1.98 thf(f1665,plain,(
% 2.94/1.98 ( ! [X2 : nat,X1 : nat] : (($false = (ord_less @ nat @ X2 @ X1)) | ((ord_less_eq @ nat @ X2 @ X1) = $true)) )),
% 2.94/1.98 inference(binary_proxy_clausification,[status(thm)],[f1664])).
% 2.94/1.98 thf(f1664,plain,(
% 2.94/1.98 ( ! [X2 : nat,X1 : nat] : (($true = ((ord_less @ nat @ X2 @ X1) => (ord_less_eq @ nat @ X2 @ X1)))) )),
% 2.94/1.98 inference(beta_eta_normalization,[status(thm)],[f1663])).
% 2.94/1.98 thf(f1663,plain,(
% 2.94/1.98 ( ! [X2 : nat,X1 : nat] : (($true = ((^[Y0 : nat]: ((ord_less @ nat @ Y0 @ X1) => (ord_less_eq @ nat @ Y0 @ X1))) @ X2))) )),
% 2.94/1.98 inference(pi_clausification,[status(thm)],[f1662])).
% 2.94/1.98 thf(f1662,plain,(
% 2.94/1.98 ( ! [X1 : nat] : (($true = (!! @ nat @ (^[Y0 : nat]: ((ord_less @ nat @ Y0 @ X1) => (ord_less_eq @ nat @ Y0 @ X1)))))) )),
% 2.94/1.98 inference(beta_eta_normalization,[status(thm)],[f1661])).
% 2.94/1.98 thf(f1661,plain,(
% 2.94/1.98 ( ! [X1 : nat] : (($true = ((^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less @ nat @ Y1 @ Y0) => (ord_less_eq @ nat @ Y1 @ Y0))))) @ X1))) )),
% 2.94/1.98 inference(pi_clausification,[status(thm)],[f998])).
% 2.94/1.98 thf(f998,plain,(
% 2.94/1.98 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less @ nat @ Y1 @ Y0) => (ord_less_eq @ nat @ Y1 @ Y0)))))) = $true)),
% 2.94/1.98 inference(cnf_transformation,[status(thm)],[f363])).
% 2.94/1.98 thf(f363,plain,(
% 2.94/1.98 ((!! @ nat @ (^[Y0 : nat]: (!! @ nat @ (^[Y1 : nat]: ((ord_less @ nat @ Y1 @ Y0) => (ord_less_eq @ nat @ Y1 @ Y0)))))) = $true)),
% 2.94/1.98 inference(fool_elimination,[status(thm)],[f362])).
% 2.94/1.98 thf(f362,plain,(
% 2.94/1.98 ! [X0 : nat,X1 : nat] : ((ord_less @ nat @ X0 @ X1) => (ord_less_eq @ nat @ X0 @ X1))),
% 2.94/1.98 inference(rectify,[status(thm)],[f90])).
% 2.94/1.98 thf(f90,axiom,(
% 2.94/1.98 ! [X42 : nat,X30 : nat] : ((ord_less @ nat @ X42 @ X30) => (ord_less_eq @ nat @ X42 @ X30))),
% 2.94/1.98 file('/export/starexec/sandbox/tmp/tmp.AUIVmKJLkR/DTF2THF_13330.p',fact_89_less__imp__le__nat)).
% 2.94/1.98 thf(f1689,plain,(
% 2.94/1.98 ($true = (ord_less_eq @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p2) @ (size_size @ list @ product_prod @ nat @ nat @ p)))),
% 2.94/1.98 inference(trivial_inequality_removal,[status(thm)],[f1677])).
% 2.94/1.98 thf(f1677,plain,(
% 2.94/1.98 ($true = (ord_less_eq @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p2) @ (size_size @ list @ product_prod @ nat @ nat @ p))) | ($false = $true)),
% 2.94/1.98 inference(superposition,[status(thm)],[f1378,f1665])).
% 2.94/1.98 thf(f1378,plain,(
% 2.94/1.98 ((ord_less @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p2) @ (size_size @ list @ product_prod @ nat @ nat @ p)) = $true)),
% 2.94/1.98 inference(forward_demodulation,[status(thm)],[f1377,f1202])).
% 2.94/1.98 thf(f1202,plain,(
% 2.94/1.98 ((size_size @ list @ product_prod @ nat @ nat @ p) = (min_dist @ a @ c @ s @ t))),
% 2.94/1.98 inference(cnf_transformation,[status(thm)],[f645])).
% 2.94/1.98 thf(f645,plain,(
% 2.94/1.98 ((size_size @ list @ product_prod @ nat @ nat @ p) = (min_dist @ a @ c @ s @ t))),
% 2.94/1.98 inference(fool_elimination,[status(thm)],[f2])).
% 2.94/1.98 thf(f2,axiom,(
% 2.94/1.98 ((size_size @ list @ product_prod @ nat @ nat @ p) = (min_dist @ a @ c @ s @ t))),
% 2.94/1.98 file('/export/starexec/sandbox/tmp/tmp.AUIVmKJLkR/DTF2THF_13330.p',fact_1_LENP)).
% 2.94/1.98 thf(f1377,plain,(
% 2.94/1.98 ($true = (ord_less @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p2) @ (min_dist @ a @ c @ s @ t)))),
% 2.94/1.98 inference(forward_demodulation,[status(thm)],[f1052,f1114])).
% 2.94/1.98 thf(f1114,plain,(
% 2.94/1.98 ((size_size @ list @ product_prod @ nat @ nat @ p2) = (min_dist @ b @ c2 @ s @ t))),
% 2.94/1.98 inference(cnf_transformation,[status(thm)],[f393])).
% 2.94/1.98 thf(f393,plain,(
% 2.94/1.98 ((size_size @ list @ product_prod @ nat @ nat @ p2) = (min_dist @ b @ c2 @ s @ t))),
% 2.94/1.98 inference(fool_elimination,[status(thm)],[f3])).
% 2.94/1.98 thf(f3,axiom,(
% 2.94/1.98 ((size_size @ list @ product_prod @ nat @ nat @ p2) = (min_dist @ b @ c2 @ s @ t))),
% 2.94/1.98 file('/export/starexec/sandbox/tmp/tmp.AUIVmKJLkR/DTF2THF_13330.p',fact_2_LENP_H)).
% 2.94/1.98 thf(f1052,plain,(
% 2.94/1.98 ((ord_less @ nat @ (min_dist @ b @ c2 @ s @ t) @ (min_dist @ a @ c @ s @ t)) = $true)),
% 2.94/1.98 inference(cnf_transformation,[status(thm)],[f484])).
% 2.94/1.98 thf(f484,plain,(
% 2.94/1.98 ((ord_less @ nat @ (min_dist @ b @ c2 @ s @ t) @ (min_dist @ a @ c @ s @ t)) = $true)),
% 2.94/1.98 inference(fool_elimination,[status(thm)],[f483])).
% 2.94/1.98 thf(f483,plain,(
% 2.94/1.98 (ord_less @ nat @ (min_dist @ b @ c2 @ s @ t) @ (min_dist @ a @ c @ s @ t))),
% 2.94/1.98 inference(rectify,[status(thm)],[f6])).
% 2.94/1.98 thf(f6,axiom,(
% 2.94/1.98 (ord_less @ nat @ (min_dist @ b @ c2 @ s @ t) @ (min_dist @ a @ c @ s @ t))),
% 2.94/1.98 file('/export/starexec/sandbox/tmp/tmp.AUIVmKJLkR/DTF2THF_13330.p',fact_5_SHORTER)).
% 2.94/1.98 % SZS output end Proof for DTF2THF_13330
% 2.94/1.98 % (13501)------------------------------
% 2.94/1.98 % (13501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.94/1.98 % (13501)Termination reason: Refutation
% 2.94/1.98
% 2.94/1.98 % (13501)Memory used [KB]: 7419
% 2.94/1.98 % (13501)Time elapsed: 0.133 s
% 2.94/1.98 % (13501)Instructions burned: 307 (million)
% 2.94/1.98 % (13501)------------------------------
% 2.94/1.98 % (13501)------------------------------
% 2.94/1.98 % (13446)Success in time 0.425 s
%------------------------------------------------------------------------------