%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP053^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n029.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:10 AM UTC 2026
% Result : Theorem 9.44s 2.76s
% Output : Refutation 9.44s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : ITP053^2 : TPTP v9.2.1. Released v7.5.0.
% 0.11/0.12 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.34 % Computer : n029.cluster.edu
% 0.15/0.34 % Model : x86_64 x86_64
% 0.15/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34 % Memory : 8042.1875MB
% 0.15/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34 % CPULimit : 300
% 0.15/0.34 % WCLimit : 300
% 0.15/0.34 % DateTime : Tue Feb 24 08:24:44 EST 2026
% 0.15/0.34 % CPUTime :
% 0.15/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.29/0.46 ---- Original DTF file ---
% 0.29/0.46 thf(spec,logic,$$dhol).
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 % File : ITP053^2 : TPTP v9.2.1. Released v7.5.0.
% 0.29/0.46 % Domain : Interactive Theorem Proving
% 0.29/0.46 % Problem : Sledgehammer EdmondsKarp_Termination_Abstract problem prob_81__7582066_1
% 0.29/0.46 % Version : Especial.
% 0.29/0.46 % English :
% 0.29/0.46
% 0.29/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.29/0.46 % Source : [Des21]
% 0.29/0.46 % Names : EdmondsKarp_Termination_Abstract/prob_81__7582066_1 [Des21]
% 0.29/0.46
% 0.29/0.46 % Status : Theorem
% 0.29/0.46 % Rating : 0.33 v9.1.0, 0.00 v8.1.0, 0.25 v7.5.0
% 0.29/0.46 % Syntax : Number of formulae : 359 ( 86 unt; 54 typ; 0 def)
% 0.29/0.46 % Number of atoms : 837 ( 194 equ; 0 cnn)
% 0.29/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.29/0.46 % Number of connectives : 4403 ( 84 ~; 6 |; 39 &;3821 @)
% 0.29/0.46 % ( 0 <=>; 453 =>; 0 <=; 0 <~>)
% 0.29/0.46 % Maximal formula depth : 28 ( 9 avg)
% 0.29/0.46 % Number of types : 3 ( 2 usr)
% 0.29/0.46 % Number of type conns : 164 ( 164 >; 0 *; 0 +; 0 <<)
% 0.29/0.46 % Number of symbols : 53 ( 52 usr; 8 con; 0-5 aty)
% 0.29/0.46 % Number of variables : 1167 ( 50 ^;1043 !; 30 ?;1167 :)
% 0.29/0.46 % ( 44 !>; 0 ?*; 0 @-; 0 @+)
% 0.29/0.46 % SPC : TH1_THM_EQU_NAR
% 0.29/0.46
% 0.29/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 16:15:24.178
% 0.29/0.46 %------------------------------------------------------------------------------
% 0.29/0.46 % Could-be-implicit typings (5)
% 0.29/0.46 thf(ty_t_Product__Type_Oprod,type,
% 0.29/0.46 product_prod: $tType > $tType > $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_List_Olist,type,
% 0.29/0.46 list: $tType > $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_tf_capacity,type,
% 0.29/0.46 capacity: $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_Set_Oset,type,
% 0.29/0.46 set: $tType > $tType ).
% 0.29/0.46
% 0.29/0.46 thf(ty_t_Nat_Onat,type,
% 0.29/0.46 nat: $tType ).
% 0.29/0.46
% 0.29/0.46 % Explicit typings (49)
% 0.29/0.46 thf(sy_cl_HOL_Otype,type,
% 0.29/0.46 type:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Nat_Osize,type,
% 0.29/0.46 size:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Oone,type,
% 0.29/0.46 one:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.29/0.46 ord:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.29/0.46 order:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.29/0.46 group_add:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.29/0.46 linorder:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.29/0.46 preorder:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Orderings_Owellorder,type,
% 0.29/0.46 wellorder:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.29/0.46 semigroup_add:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.29/0.46 linordered_idom:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.29/0.46 comm_monoid_add:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.29/0.46 ab_semigroup_add:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Rings_Olinordered__semidom,type,
% 0.29/0.46 linordered_semidom:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 0.29/0.46 cancel_semigroup_add:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 0.29/0.46 ordere779506340up_add:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 0.29/0.46 linord1659791738miring:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 0.29/0.46 canoni770627133id_add:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 0.29/0.46 ordere236663937imp_le:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 0.29/0.46 ordere223160158up_add:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 0.29/0.46 strict2144017051up_add:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
% 0.29/0.46 unique1598680935umeral:
% 0.29/0.46 !>[A: $tType] : $o ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Graph_OGraph_Oconnected,type,
% 0.29/0.46 connected:
% 0.29/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Graph_OGraph_Odist,type,
% 0.29/0.46 dist:
% 0.29/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > nat > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Graph_OGraph_OisPath,type,
% 0.29/0.46 isPath:
% 0.29/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Graph_OGraph_OisShortestPath,type,
% 0.29/0.46 isShortestPath:
% 0.29/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Graph_OGraph_OisSimplePath,type,
% 0.29/0.46 isSimplePath:
% 0.29/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Graph_OGraph_Omin__dist,type,
% 0.29/0.46 min_dist:
% 0.29/0.46 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.29/0.46 one_one:
% 0.29/0.46 !>[A: $tType] : A ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.29/0.46 plus_plus:
% 0.29/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_List_Olist_ONil,type,
% 0.29/0.46 nil:
% 0.29/0.46 !>[A: $tType] : ( list @ A ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_List_Olist_Oset,type,
% 0.29/0.46 set2:
% 0.29/0.46 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.29/0.46 size_size:
% 0.29/0.46 !>[A: $tType] : ( A > nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.29/0.46 ord_less:
% 0.29/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.29/0.46 ord_less_eq:
% 0.29/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_OPair,type,
% 0.29/0.46 product_Pair:
% 0.29/0.46 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.29/0.46 product_rec_prod:
% 0.29/0.46 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.29/0.46 product_swap:
% 0.29/0.46 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_Set_OCollect,type,
% 0.29/0.46 collect:
% 0.29/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_c_member,type,
% 0.29/0.46 member:
% 0.29/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_c,type,
% 0.29/0.46 c: ( product_prod @ nat @ nat ) > capacity ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_p,type,
% 0.29/0.46 p: list @ ( product_prod @ nat @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_p1____,type,
% 0.29/0.46 p1: list @ ( product_prod @ nat @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_p2____,type,
% 0.29/0.46 p2: list @ ( product_prod @ nat @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_p_H,type,
% 0.29/0.46 p3: list @ ( product_prod @ nat @ nat ) ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_s,type,
% 0.29/0.46 s: nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_t,type,
% 0.29/0.46 t: nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_u,type,
% 0.29/0.46 u: nat ).
% 0.29/0.46
% 0.29/0.46 thf(sy_v_v,type,
% 0.29/0.46 v: nat ).
% 0.29/0.46
% 0.29/0.46 % Relevant facts (255)
% 0.29/0.46 thf(fact_0_MIN_H,axiom,
% 0.29/0.46 ( ( min_dist @ capacity @ c @ s @ t )
% 0.29/0.46 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1 ) @ ( one_one @ nat ) ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % MIN'
% 0.29/0.46 thf(fact_1__092_060open_062min__dist_As_At_A_061_Alength_Ap1_A_L_A_I1_A_L_Alength_Ap2_J_A_092_060Longrightarrow_062_Amin__dist_As_Au_A_061_Alength_Ap1_092_060close_062,axiom,
% 0.29/0.46 ( ( ( min_dist @ capacity @ c @ s @ t )
% 0.29/0.46 = ( plus_plus @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 ) ) ) )
% 0.29/0.46 => ( ( min_dist @ capacity @ c @ s @ u )
% 0.29/0.46 = ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % \<open>min_dist s t = length p1 + (1 + length p2) \<Longrightarrow> min_dist s u = length p1\<close>
% 0.29/0.46 thf(fact_2__092_060open_062min__dist_As_At_A_061_Alength_Ap1_A_L_A_I1_A_L_Alength_Ap2_J_A_092_060Longrightarrow_062_Amin__dist_Au_At_A_061_A1_A_L_Alength_Ap2_092_060close_062,axiom,
% 0.29/0.46 ( ( ( min_dist @ capacity @ c @ s @ t )
% 0.29/0.46 = ( plus_plus @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 ) ) ) )
% 0.29/0.46 => ( ( min_dist @ capacity @ c @ u @ t )
% 0.29/0.46 = ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % \<open>min_dist s t = length p1 + (1 + length p2) \<Longrightarrow> min_dist u t = 1 + length p2\<close>
% 0.29/0.46 thf(fact_3__092_060open_062min__dist_Av_At_A_061_Alength_Ap2_092_060close_062,axiom,
% 0.29/0.46 ( ( min_dist @ capacity @ c @ v @ t )
% 0.29/0.46 = ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % \<open>min_dist v t = length p2\<close>
% 0.29/0.46 thf(fact_4_MIN,axiom,
% 0.29/0.46 ( ( min_dist @ capacity @ c @ s @ t )
% 0.29/0.46 = ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p ) ) ).
% 0.29/0.46
% 0.29/0.46 % MIN
% 0.29/0.46 thf(fact_5__092_060open_062u_A_092_060noteq_062_Av_092_060close_062,axiom,
% 0.29/0.46 u != v ).
% 0.29/0.46
% 0.29/0.46 % \<open>u \<noteq> v\<close>
% 0.29/0.46 thf(fact_6_MDSV,axiom,
% 0.29/0.46 ( ( min_dist @ capacity @ c @ s @ v )
% 0.29/0.46 = ( plus_plus @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1 ) @ ( one_one @ nat ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % MDSV
% 0.29/0.46 thf(fact_7_min__dist__split_I2_J,axiom,
% 0.29/0.46 ! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 0.29/0.46 ( ( dist @ capacity @ c @ U @ D1 @ W )
% 0.29/0.46 => ( ( dist @ capacity @ c @ W @ D2 @ V )
% 0.29/0.46 => ( ( ( min_dist @ capacity @ c @ U @ V )
% 0.29/0.46 = ( plus_plus @ nat @ D1 @ D2 ) )
% 0.29/0.46 => ( ( min_dist @ capacity @ c @ W @ V )
% 0.29/0.46 = D2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % min_dist_split(2)
% 0.29/0.46 thf(fact_8_min__dist__split_I1_J,axiom,
% 0.29/0.46 ! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 0.29/0.46 ( ( dist @ capacity @ c @ U @ D1 @ W )
% 0.29/0.46 => ( ( dist @ capacity @ c @ W @ D2 @ V )
% 0.29/0.46 => ( ( ( min_dist @ capacity @ c @ U @ V )
% 0.29/0.46 = ( plus_plus @ nat @ D1 @ D2 ) )
% 0.29/0.46 => ( ( min_dist @ capacity @ c @ U @ W )
% 0.29/0.46 = D1 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % min_dist_split(1)
% 0.29/0.46 thf(fact_9_add__left__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel_semigroup_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.46 = ( B2 = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_left_cancel
% 0.29/0.46 thf(fact_10_add__right__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel_semigroup_add @ A )
% 0.29/0.46 => ! [B2: A,A2: A,C: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.29/0.46 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/0.46 = ( B2 = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_right_cancel
% 0.29/0.46 thf(fact_11_dist__trans,axiom,
% 0.29/0.46 ! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 0.29/0.46 ( ( dist @ capacity @ c @ U @ D1 @ W )
% 0.29/0.46 => ( ( dist @ capacity @ c @ W @ D2 @ V )
% 0.29/0.46 => ( dist @ capacity @ c @ U @ ( plus_plus @ nat @ D1 @ D2 ) @ V ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dist_trans
% 0.29/0.46 thf(fact_12_assms_I3_J,axiom,
% 0.29/0.46 isPath @ capacity @ c @ s @ p3 @ t ).
% 0.29/0.46
% 0.29/0.46 % assms(3)
% 0.29/0.46 thf(fact_13_DISTS_I1_J,axiom,
% 0.29/0.46 dist @ capacity @ c @ s @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1 ) @ u ).
% 0.29/0.46
% 0.29/0.46 % DISTS(1)
% 0.29/0.46 thf(fact_14_isPath__distD,axiom,
% 0.29/0.46 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.29/0.46 ( ( isPath @ capacity @ c @ U @ P @ V )
% 0.29/0.46 => ( dist @ capacity @ c @ U @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P ) @ V ) ) ).
% 0.29/0.46
% 0.29/0.46 % isPath_distD
% 0.29/0.46 thf(fact_15_dist__def,axiom,
% 0.29/0.46 ! [V: nat,D: nat,V2: nat] :
% 0.29/0.46 ( ( dist @ capacity @ c @ V @ D @ V2 )
% 0.29/0.46 = ( ? [P2: list @ ( product_prod @ nat @ nat )] :
% 0.29/0.46 ( ( isPath @ capacity @ c @ V @ P2 @ V2 )
% 0.29/0.46 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 )
% 0.29/0.46 = D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dist_def
% 0.29/0.46 thf(fact_16_DISTS_I2_J,axiom,
% 0.29/0.46 dist @ capacity @ c @ u @ ( one_one @ nat ) @ v ).
% 0.29/0.46
% 0.29/0.46 % DISTS(2)
% 0.29/0.46 thf(fact_17_P,axiom,
% 0.29/0.46 isPath @ capacity @ c @ s @ p @ t ).
% 0.29/0.46
% 0.29/0.46 % P
% 0.29/0.46 thf(fact_18_DISTS_I3_J,axiom,
% 0.29/0.46 dist @ capacity @ c @ v @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 ) @ t ).
% 0.29/0.46
% 0.29/0.46 % DISTS(3)
% 0.29/0.46 thf(fact_19_assms_I1_J,axiom,
% 0.29/0.46 isShortestPath @ capacity @ c @ s @ p @ t ).
% 0.29/0.46
% 0.29/0.46 % assms(1)
% 0.29/0.46 thf(fact_20_add__right__imp__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel_semigroup_add @ A )
% 0.29/0.46 => ! [B2: A,A2: A,C: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.29/0.46 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/0.46 => ( B2 = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_right_imp_eq
% 0.29/0.46 thf(fact_21_add__left__imp__eq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( cancel_semigroup_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.46 => ( B2 = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_left_imp_eq
% 0.29/0.46 thf(fact_22_add_Oleft__commute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ab_semigroup_add @ A )
% 0.29/0.46 => ! [B2: A,A2: A,C: A] :
% 0.29/0.46 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.left_commute
% 0.29/0.46 thf(fact_23_add_Ocommute,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ab_semigroup_add @ A )
% 0.29/0.46 => ( ( plus_plus @ A )
% 0.29/0.46 = ( ^ [A3: A,B3: A] : ( plus_plus @ A @ B3 @ A3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.commute
% 0.29/0.46 thf(fact_24_add_Oright__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [B2: A,A2: A,C: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ B2 @ A2 )
% 0.29/0.46 = ( plus_plus @ A @ C @ A2 ) )
% 0.29/0.46 = ( B2 = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.right_cancel
% 0.29/0.46 thf(fact_25_add_Oleft__cancel,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( group_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ( plus_plus @ A @ A2 @ B2 )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ C ) )
% 0.29/0.46 = ( B2 = C ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.left_cancel
% 0.29/0.46 thf(fact_26_add_Oassoc,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( semigroup_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add.assoc
% 0.29/0.46 thf(fact_27_group__cancel_Oadd2,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comm_monoid_add @ A )
% 0.29/0.46 => ! [B4: A,K: A,B2: A,A2: A] :
% 0.29/0.46 ( ( B4
% 0.29/0.46 = ( plus_plus @ A @ K @ B2 ) )
% 0.29/0.46 => ( ( plus_plus @ A @ A2 @ B4 )
% 0.29/0.46 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % group_cancel.add2
% 0.29/0.46 thf(fact_28_group__cancel_Oadd1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( comm_monoid_add @ A )
% 0.29/0.46 => ! [A4: A,K: A,A2: A,B2: A] :
% 0.29/0.46 ( ( A4
% 0.29/0.46 = ( plus_plus @ A @ K @ A2 ) )
% 0.29/0.46 => ( ( plus_plus @ A @ A4 @ B2 )
% 0.29/0.46 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % group_cancel.add1
% 0.29/0.46 thf(fact_29_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( I = J )
% 0.29/0.46 & ( K = L ) )
% 0.29/0.46 => ( ( plus_plus @ A @ I @ K )
% 0.29/0.46 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(4)
% 0.29/0.46 thf(fact_30_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ab_semigroup_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 0.29/0.46 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.29/0.46 thf(fact_31_one__reorient,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( one @ A )
% 0.29/0.46 => ! [X: A] :
% 0.29/0.46 ( ( ( one_one @ A )
% 0.29/0.46 = X )
% 0.29/0.46 = ( X
% 0.29/0.46 = ( one_one @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % one_reorient
% 0.29/0.46 thf(fact_32_isShortestPath__min__dist__def,axiom,
% 0.29/0.46 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.29/0.46 ( ( isShortestPath @ capacity @ c @ U @ P @ V )
% 0.29/0.46 = ( ( isPath @ capacity @ c @ U @ P @ V )
% 0.29/0.46 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P )
% 0.29/0.46 = ( min_dist @ capacity @ c @ U @ V ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % isShortestPath_min_dist_def
% 0.29/0.46 thf(fact_33_Graph_OisPath__distD,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.29/0.46 ( ( isPath @ Capacity @ C @ U @ P @ V )
% 0.29/0.46 => ( dist @ Capacity @ C @ U @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P ) @ V ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isPath_distD
% 0.29/0.46 thf(fact_34_Graph_Odist__def,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ( ( dist @ Capacity )
% 0.29/0.46 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,V3: nat,D3: nat,V4: nat] :
% 0.29/0.46 ? [P2: list @ ( product_prod @ nat @ nat )] :
% 0.29/0.46 ( ( isPath @ Capacity @ C2 @ V3 @ P2 @ V4 )
% 0.29/0.46 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 )
% 0.29/0.46 = D3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.dist_def
% 0.29/0.46 thf(fact_35_min__dist__minD,axiom,
% 0.29/0.46 ! [V: nat,D: nat,V2: nat] :
% 0.29/0.46 ( ( dist @ capacity @ c @ V @ D @ V2 )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( min_dist @ capacity @ c @ V @ V2 ) @ D ) ) ).
% 0.29/0.46
% 0.29/0.46 % min_dist_minD
% 0.29/0.46 thf(fact_36_min__distI__eq,axiom,
% 0.29/0.46 ! [V: nat,D: nat,V2: nat] :
% 0.29/0.46 ( ( dist @ capacity @ c @ V @ D @ V2 )
% 0.29/0.46 => ( ! [D4: nat] :
% 0.29/0.46 ( ( dist @ capacity @ c @ V @ D4 @ V2 )
% 0.29/0.46 => ( ord_less_eq @ nat @ D @ D4 ) )
% 0.29/0.46 => ( ( min_dist @ capacity @ c @ V @ V2 )
% 0.29/0.46 = D ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % min_distI_eq
% 0.29/0.46 thf(fact_37_Graph_Omin__dist__split_I1_J,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 0.29/0.46 ( ( dist @ Capacity @ C @ U @ D1 @ W )
% 0.29/0.46 => ( ( dist @ Capacity @ C @ W @ D2 @ V )
% 0.29/0.46 => ( ( ( min_dist @ Capacity @ C @ U @ V )
% 0.29/0.46 = ( plus_plus @ nat @ D1 @ D2 ) )
% 0.29/0.46 => ( ( min_dist @ Capacity @ C @ U @ W )
% 0.29/0.46 = D1 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.min_dist_split(1)
% 0.29/0.46 thf(fact_38_Graph_Omin__dist__split_I2_J,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 0.29/0.46 ( ( dist @ Capacity @ C @ U @ D1 @ W )
% 0.29/0.46 => ( ( dist @ Capacity @ C @ W @ D2 @ V )
% 0.29/0.46 => ( ( ( min_dist @ Capacity @ C @ U @ V )
% 0.29/0.46 = ( plus_plus @ nat @ D1 @ D2 ) )
% 0.29/0.46 => ( ( min_dist @ Capacity @ C @ W @ V )
% 0.29/0.46 = D2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.min_dist_split(2)
% 0.29/0.46 thf(fact_39_min__dist__is__dist,axiom,
% 0.29/0.46 ! [V: nat,V2: nat] :
% 0.29/0.46 ( ( connected @ capacity @ c @ V @ V2 )
% 0.29/0.46 => ( dist @ capacity @ c @ V @ ( min_dist @ capacity @ c @ V @ V2 ) @ V2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % min_dist_is_dist
% 0.29/0.46 thf(fact_40_shortestPath__is__path,axiom,
% 0.29/0.46 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.29/0.46 ( ( isShortestPath @ capacity @ c @ U @ P @ V )
% 0.29/0.46 => ( isPath @ capacity @ c @ U @ P @ V ) ) ).
% 0.29/0.46
% 0.29/0.46 % shortestPath_is_path
% 0.29/0.46 thf(fact_41_assms_I4_J,axiom,
% 0.29/0.46 member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ v @ u ) @ ( set2 @ ( product_prod @ nat @ nat ) @ p3 ) ).
% 0.29/0.46
% 0.29/0.46 % assms(4)
% 0.29/0.46 thf(fact_42_connected__def,axiom,
% 0.29/0.46 ! [U: nat,V: nat] :
% 0.29/0.46 ( ( connected @ capacity @ c @ U @ V )
% 0.29/0.46 = ( ? [P2: list @ ( product_prod @ nat @ nat )] : ( isPath @ capacity @ c @ U @ P2 @ V ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % connected_def
% 0.29/0.46 thf(fact_43_connected__by__dist,axiom,
% 0.29/0.46 ! [V: nat,V2: nat] :
% 0.29/0.46 ( ( connected @ capacity @ c @ V @ V2 )
% 0.29/0.46 = ( ? [D3: nat] : ( dist @ capacity @ c @ V @ D3 @ V2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % connected_by_dist
% 0.29/0.46 thf(fact_44_mem__Collect__eq,axiom,
% 0.29/0.46 ! [A: $tType,A2: A,P3: A > $o] :
% 0.29/0.46 ( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
% 0.29/0.46 = ( P3 @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % mem_Collect_eq
% 0.29/0.46 thf(fact_45_Collect__mem__eq,axiom,
% 0.29/0.46 ! [A: $tType,A4: set @ A] :
% 0.29/0.46 ( ( collect @ A
% 0.29/0.46 @ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
% 0.29/0.46 = A4 ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_mem_eq
% 0.29/0.46 thf(fact_46_Collect__cong,axiom,
% 0.29/0.46 ! [A: $tType,P3: A > $o,Q: A > $o] :
% 0.29/0.46 ( ! [X3: A] :
% 0.29/0.46 ( ( P3 @ X3 )
% 0.29/0.46 = ( Q @ X3 ) )
% 0.29/0.46 => ( ( collect @ A @ P3 )
% 0.29/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Collect_cong
% 0.29/0.46 thf(fact_47_ext,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.29/0.46 ( ! [X3: A] :
% 0.29/0.46 ( ( F @ X3 )
% 0.29/0.46 = ( G @ X3 ) )
% 0.29/0.46 => ( F = G ) ) ).
% 0.29/0.46
% 0.29/0.46 % ext
% 0.29/0.46 thf(fact_48_obtain__shortest__path,axiom,
% 0.29/0.46 ! [U: nat,V: nat] :
% 0.29/0.46 ( ( connected @ capacity @ c @ U @ V )
% 0.29/0.46 => ~ ! [P4: list @ ( product_prod @ nat @ nat )] :
% 0.29/0.46 ~ ( isShortestPath @ capacity @ c @ U @ P4 @ V ) ) ).
% 0.29/0.46
% 0.29/0.46 % obtain_shortest_path
% 0.29/0.46 thf(fact_49_assms_I2_J,axiom,
% 0.29/0.46 member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ u @ v ) @ ( set2 @ ( product_prod @ nat @ nat ) @ p ) ).
% 0.29/0.46
% 0.29/0.46 % assms(2)
% 0.29/0.46 thf(fact_50_isPath__ex__edge1,axiom,
% 0.29/0.46 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
% 0.29/0.46 ( ( isPath @ capacity @ c @ U @ P @ V )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( ( U1 != U )
% 0.29/0.46 => ? [U2: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U2 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % isPath_ex_edge1
% 0.29/0.46 thf(fact_51_isPath__ex__edge2,axiom,
% 0.29/0.46 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
% 0.29/0.46 ( ( isPath @ capacity @ c @ U @ P @ V )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( ( V1 != V )
% 0.29/0.46 => ? [V22: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V1 @ V22 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % isPath_ex_edge2
% 0.29/0.46 thf(fact_52_min__dist__le,axiom,
% 0.29/0.46 ! [Src: nat,V: nat,D5: nat] :
% 0.29/0.46 ( ( connected @ capacity @ c @ Src @ V )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ D5 @ ( min_dist @ capacity @ c @ Src @ V ) )
% 0.29/0.46 => ? [V5: nat] :
% 0.29/0.46 ( ( connected @ capacity @ c @ Src @ V5 )
% 0.29/0.46 & ( ( min_dist @ capacity @ c @ Src @ V5 )
% 0.29/0.46 = D5 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % min_dist_le
% 0.29/0.46 thf(fact_53_min__distI2,axiom,
% 0.29/0.46 ! [V: nat,V2: nat,Q: nat > $o] :
% 0.29/0.46 ( ( connected @ capacity @ c @ V @ V2 )
% 0.29/0.46 => ( ! [D6: nat] :
% 0.29/0.46 ( ( dist @ capacity @ c @ V @ D6 @ V2 )
% 0.29/0.46 => ( ! [D7: nat] :
% 0.29/0.46 ( ( dist @ capacity @ c @ V @ D7 @ V2 )
% 0.29/0.46 => ( ord_less_eq @ nat @ D6 @ D7 ) )
% 0.29/0.46 => ( Q @ D6 ) ) )
% 0.29/0.46 => ( Q @ ( min_dist @ capacity @ c @ V @ V2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % min_distI2
% 0.29/0.46 thf(fact_54_isShortestPath__def,axiom,
% 0.29/0.46 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.29/0.46 ( ( isShortestPath @ capacity @ c @ U @ P @ V )
% 0.29/0.46 = ( ( isPath @ capacity @ c @ U @ P @ V )
% 0.29/0.46 & ! [P5: list @ ( product_prod @ nat @ nat )] :
% 0.29/0.46 ( ( isPath @ capacity @ c @ U @ P5 @ V )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % isShortestPath_def
% 0.29/0.46 thf(fact_55_isShortestPath__level__edge_I1_J,axiom,
% 0.29/0.46 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 0.29/0.46 ( ( isShortestPath @ capacity @ c @ S @ P @ T2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( connected @ capacity @ c @ S @ U ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % isShortestPath_level_edge(1)
% 0.29/0.46 thf(fact_56_isShortestPath__level__edge_I2_J,axiom,
% 0.29/0.46 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 0.29/0.46 ( ( isShortestPath @ capacity @ c @ S @ P @ T2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( connected @ capacity @ c @ U @ V ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % isShortestPath_level_edge(2)
% 0.29/0.46 thf(fact_57_isShortestPath__level__edge_I3_J,axiom,
% 0.29/0.46 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 0.29/0.46 ( ( isShortestPath @ capacity @ c @ S @ P @ T2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( connected @ capacity @ c @ V @ T2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % isShortestPath_level_edge(3)
% 0.29/0.46 thf(fact_58_add__le__cancel__left,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [C: A,A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 0.29/0.46 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_cancel_left
% 0.29/0.46 thf(fact_59_add__le__cancel__right,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [A2: A,C: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 0.29/0.46 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_cancel_right
% 0.29/0.46 thf(fact_60_isShortestPath__level__edge_I6_J,axiom,
% 0.29/0.46 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 0.29/0.46 ( ( isShortestPath @ capacity @ c @ S @ P @ T2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( ( min_dist @ capacity @ c @ S @ T2 )
% 0.29/0.46 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( min_dist @ capacity @ c @ S @ U ) @ ( one_one @ nat ) ) @ ( min_dist @ capacity @ c @ V @ T2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % isShortestPath_level_edge(6)
% 0.29/0.46 thf(fact_61_isShortestPath__level__edge_I5_J,axiom,
% 0.29/0.46 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 0.29/0.46 ( ( isShortestPath @ capacity @ c @ S @ P @ T2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( ( min_dist @ capacity @ c @ U @ T2 )
% 0.29/0.46 = ( plus_plus @ nat @ ( one_one @ nat ) @ ( min_dist @ capacity @ c @ V @ T2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % isShortestPath_level_edge(5)
% 0.29/0.46 thf(fact_62_isShortestPath__level__edge_I4_J,axiom,
% 0.29/0.46 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 0.29/0.46 ( ( isShortestPath @ capacity @ c @ S @ P @ T2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( ( min_dist @ capacity @ c @ S @ V )
% 0.29/0.46 = ( plus_plus @ nat @ ( min_dist @ capacity @ c @ S @ U ) @ ( one_one @ nat ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % isShortestPath_level_edge(4)
% 0.29/0.46 thf(fact_63_connected__refl,axiom,
% 0.29/0.46 ! [V: nat] : ( connected @ capacity @ c @ V @ V ) ).
% 0.29/0.46
% 0.29/0.46 % connected_refl
% 0.29/0.46 thf(fact_64_connected__distI,axiom,
% 0.29/0.46 ! [V: nat,D: nat,V2: nat] :
% 0.29/0.46 ( ( dist @ capacity @ c @ V @ D @ V2 )
% 0.29/0.46 => ( connected @ capacity @ c @ V @ V2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % connected_distI
% 0.29/0.46 thf(fact_65_Graph_OisPath__ex__edge1,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
% 0.29/0.46 ( ( isPath @ Capacity @ C @ U @ P @ V )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( ( U1 != U )
% 0.29/0.46 => ? [U2: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U2 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isPath_ex_edge1
% 0.29/0.46 thf(fact_66_Graph_OisPath__ex__edge2,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
% 0.29/0.46 ( ( isPath @ Capacity @ C @ U @ P @ V )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( ( V1 != V )
% 0.29/0.46 => ? [V22: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V1 @ V22 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isPath_ex_edge2
% 0.29/0.46 thf(fact_67_Graph_Oobtain__shortest__path,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat] :
% 0.29/0.46 ( ( connected @ Capacity @ C @ U @ V )
% 0.29/0.46 => ~ ! [P4: list @ ( product_prod @ nat @ nat )] :
% 0.29/0.46 ~ ( isShortestPath @ Capacity @ C @ U @ P4 @ V ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.obtain_shortest_path
% 0.29/0.46 thf(fact_68_Graph_OisShortestPath_Ocong,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ( ( isShortestPath @ Capacity )
% 0.29/0.46 = ( isShortestPath @ Capacity ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isShortestPath.cong
% 0.29/0.46 thf(fact_69_Graph_Oconnected__refl,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat] : ( connected @ Capacity @ C @ V @ V ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.connected_refl
% 0.29/0.46 thf(fact_70_Graph_Oconnected_Ocong,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ( ( connected @ Capacity )
% 0.29/0.46 = ( connected @ Capacity ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.connected.cong
% 0.29/0.46 thf(fact_71_Graph_Omin__dist__le,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,Src: nat,V: nat,D5: nat] :
% 0.29/0.46 ( ( connected @ Capacity @ C @ Src @ V )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ D5 @ ( min_dist @ Capacity @ C @ Src @ V ) )
% 0.29/0.46 => ? [V5: nat] :
% 0.29/0.46 ( ( connected @ Capacity @ C @ Src @ V5 )
% 0.29/0.46 & ( ( min_dist @ Capacity @ C @ Src @ V5 )
% 0.29/0.46 = D5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.min_dist_le
% 0.29/0.46 thf(fact_72_Graph_OisShortestPath__level__edge_I1_J,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 0.29/0.46 ( ( isShortestPath @ Capacity @ C @ S @ P @ T2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( connected @ Capacity @ C @ S @ U ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isShortestPath_level_edge(1)
% 0.29/0.46 thf(fact_73_Graph_OisShortestPath__level__edge_I2_J,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 0.29/0.46 ( ( isShortestPath @ Capacity @ C @ S @ P @ T2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( connected @ Capacity @ C @ U @ V ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isShortestPath_level_edge(2)
% 0.29/0.46 thf(fact_74_Graph_OisShortestPath__level__edge_I3_J,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 0.29/0.46 ( ( isShortestPath @ Capacity @ C @ S @ P @ T2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( connected @ Capacity @ C @ V @ T2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isShortestPath_level_edge(3)
% 0.29/0.46 thf(fact_75_Graph_Oconnected__def,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ( ( connected @ Capacity )
% 0.29/0.46 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U3: nat,V3: nat] :
% 0.29/0.46 ? [P2: list @ ( product_prod @ nat @ nat )] : ( isPath @ Capacity @ C2 @ U3 @ P2 @ V3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.connected_def
% 0.29/0.46 thf(fact_76_Graph_Oconnected__by__dist,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ( ( connected @ Capacity )
% 0.29/0.46 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,V3: nat,V4: nat] :
% 0.29/0.46 ? [D3: nat] : ( dist @ Capacity @ C2 @ V3 @ D3 @ V4 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.connected_by_dist
% 0.29/0.46 thf(fact_77_Graph_Oconnected__distI,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,D: nat,V2: nat] :
% 0.29/0.46 ( ( dist @ Capacity @ C @ V @ D @ V2 )
% 0.29/0.46 => ( connected @ Capacity @ C @ V @ V2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.connected_distI
% 0.29/0.46 thf(fact_78_Graph_OshortestPath__is__path,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.29/0.46 ( ( isShortestPath @ Capacity @ C @ U @ P @ V )
% 0.29/0.46 => ( isPath @ Capacity @ C @ U @ P @ V ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.shortestPath_is_path
% 0.29/0.46 thf(fact_79_Graph_OisShortestPath__level__edge_I6_J,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 0.29/0.46 ( ( isShortestPath @ Capacity @ C @ S @ P @ T2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( ( min_dist @ Capacity @ C @ S @ T2 )
% 0.29/0.46 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( min_dist @ Capacity @ C @ S @ U ) @ ( one_one @ nat ) ) @ ( min_dist @ Capacity @ C @ V @ T2 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isShortestPath_level_edge(6)
% 0.29/0.46 thf(fact_80_Graph_OisShortestPath__level__edge_I5_J,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 0.29/0.46 ( ( isShortestPath @ Capacity @ C @ S @ P @ T2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( ( min_dist @ Capacity @ C @ U @ T2 )
% 0.29/0.46 = ( plus_plus @ nat @ ( one_one @ nat ) @ ( min_dist @ Capacity @ C @ V @ T2 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isShortestPath_level_edge(5)
% 0.29/0.46 thf(fact_81_Graph_OisShortestPath__level__edge_I4_J,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 0.29/0.46 ( ( isShortestPath @ Capacity @ C @ S @ P @ T2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( ( min_dist @ Capacity @ C @ S @ V )
% 0.29/0.46 = ( plus_plus @ nat @ ( min_dist @ Capacity @ C @ S @ U ) @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isShortestPath_level_edge(4)
% 0.29/0.46 thf(fact_82_Graph_Omin__distI2,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,V2: nat,Q: nat > $o] :
% 0.29/0.46 ( ( connected @ Capacity @ C @ V @ V2 )
% 0.29/0.46 => ( ! [D6: nat] :
% 0.29/0.46 ( ( dist @ Capacity @ C @ V @ D6 @ V2 )
% 0.29/0.46 => ( ! [D7: nat] :
% 0.29/0.46 ( ( dist @ Capacity @ C @ V @ D7 @ V2 )
% 0.29/0.46 => ( ord_less_eq @ nat @ D6 @ D7 ) )
% 0.29/0.46 => ( Q @ D6 ) ) )
% 0.29/0.46 => ( Q @ ( min_dist @ Capacity @ C @ V @ V2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.min_distI2
% 0.29/0.46 thf(fact_83_Graph_OisShortestPath__def,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ( ( isShortestPath @ Capacity )
% 0.29/0.46 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U3: nat,P2: list @ ( product_prod @ nat @ nat ),V3: nat] :
% 0.29/0.46 ( ( isPath @ Capacity @ C2 @ U3 @ P2 @ V3 )
% 0.29/0.46 & ! [P5: list @ ( product_prod @ nat @ nat )] :
% 0.29/0.46 ( ( isPath @ Capacity @ C2 @ U3 @ P5 @ V3 )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isShortestPath_def
% 0.29/0.46 thf(fact_84_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ I @ J )
% 0.29/0.46 & ( K = L ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(3)
% 0.29/0.46 thf(fact_85_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( I = J )
% 0.29/0.46 & ( ord_less_eq @ A @ K @ L ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(2)
% 0.29/0.46 thf(fact_86_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ I @ J )
% 0.29/0.46 & ( ord_less_eq @ A @ K @ L ) )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_semiring(1)
% 0.29/0.46 thf(fact_87_add__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A,D: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ C @ D )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono
% 0.29/0.46 thf(fact_88_add__left__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_left_mono
% 0.29/0.46 thf(fact_89_less__eqE,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 => ~ ! [C3: A] :
% 0.29/0.46 ( B2
% 0.29/0.46 != ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_eqE
% 0.29/0.46 thf(fact_90_add__right__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere779506340up_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_right_mono
% 0.29/0.46 thf(fact_91_le__iff__add,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( canoni770627133id_add @ A )
% 0.29/0.46 => ( ( ord_less_eq @ A )
% 0.29/0.46 = ( ^ [A3: A,B3: A] :
% 0.29/0.46 ? [C2: A] :
% 0.29/0.46 ( B3
% 0.29/0.46 = ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_iff_add
% 0.29/0.46 thf(fact_92_add__le__imp__le__left,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [C: A,A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 0.29/0.46 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_imp_le_left
% 0.29/0.46 thf(fact_93_add__le__imp__le__right,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [A2: A,C: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 0.29/0.46 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_imp_le_right
% 0.29/0.46 thf(fact_94_Graph_Omin__dist__minD,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,D: nat,V2: nat] :
% 0.29/0.46 ( ( dist @ Capacity @ C @ V @ D @ V2 )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( min_dist @ Capacity @ C @ V @ V2 ) @ D ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.min_dist_minD
% 0.29/0.46 thf(fact_95_Graph_Omin__distI__eq,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,D: nat,V2: nat] :
% 0.29/0.46 ( ( dist @ Capacity @ C @ V @ D @ V2 )
% 0.29/0.46 => ( ! [D4: nat] :
% 0.29/0.46 ( ( dist @ Capacity @ C @ V @ D4 @ V2 )
% 0.29/0.46 => ( ord_less_eq @ nat @ D @ D4 ) )
% 0.29/0.46 => ( ( min_dist @ Capacity @ C @ V @ V2 )
% 0.29/0.46 = D ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.min_distI_eq
% 0.29/0.46 thf(fact_96_Graph_Omin__dist__is__dist,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,V2: nat] :
% 0.29/0.46 ( ( connected @ Capacity @ C @ V @ V2 )
% 0.29/0.46 => ( dist @ Capacity @ C @ V @ ( min_dist @ Capacity @ C @ V @ V2 ) @ V2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.min_dist_is_dist
% 0.29/0.46 thf(fact_97_Graph_OisShortestPath__min__dist__def,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ( ( isShortestPath @ Capacity )
% 0.29/0.46 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U3: nat,P2: list @ ( product_prod @ nat @ nat ),V3: nat] :
% 0.29/0.46 ( ( isPath @ Capacity @ C2 @ U3 @ P2 @ V3 )
% 0.29/0.46 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 )
% 0.29/0.46 = ( min_dist @ Capacity @ C2 @ U3 @ V3 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isShortestPath_min_dist_def
% 0.29/0.46 thf(fact_98_Graph_OisPath_Ocong,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ( ( isPath @ Capacity )
% 0.29/0.46 = ( isPath @ Capacity ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isPath.cong
% 0.29/0.46 thf(fact_99_Graph_Omin__dist_Ocong,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ( ( min_dist @ Capacity )
% 0.29/0.46 = ( min_dist @ Capacity ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.min_dist.cong
% 0.29/0.46 thf(fact_100_Graph_Odist_Ocong,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ( ( dist @ Capacity )
% 0.29/0.46 = ( dist @ Capacity ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.dist.cong
% 0.29/0.46 thf(fact_101_Graph_Odist__trans,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 0.29/0.46 ( ( dist @ Capacity @ C @ U @ D1 @ W )
% 0.29/0.46 => ( ( dist @ Capacity @ C @ W @ D2 @ V )
% 0.29/0.46 => ( dist @ Capacity @ C @ U @ ( plus_plus @ nat @ D1 @ D2 ) @ V ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.dist_trans
% 0.29/0.46 thf(fact_102_nat__add__left__cancel__le,axiom,
% 0.29/0.46 ! [K: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.29/0.46 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_add_left_cancel_le
% 0.29/0.46 thf(fact_103_isShortestPath__alt,axiom,
% 0.29/0.46 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
% 0.29/0.46 ( ( isShortestPath @ capacity @ c @ U @ P @ V )
% 0.29/0.46 = ( ( isSimplePath @ capacity @ c @ U @ P @ V )
% 0.29/0.46 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P )
% 0.29/0.46 = ( min_dist @ capacity @ c @ U @ V ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % isShortestPath_alt
% 0.29/0.46 thf(fact_104_min__dist__less,axiom,
% 0.29/0.46 ! [Src: nat,V: nat,D: nat,D5: nat] :
% 0.29/0.46 ( ( connected @ capacity @ c @ Src @ V )
% 0.29/0.46 => ( ( ( min_dist @ capacity @ c @ Src @ V )
% 0.29/0.46 = D )
% 0.29/0.46 => ( ( ord_less @ nat @ D5 @ D )
% 0.29/0.46 => ? [V5: nat] :
% 0.29/0.46 ( ( connected @ capacity @ c @ Src @ V5 )
% 0.29/0.46 & ( ( min_dist @ capacity @ c @ Src @ V5 )
% 0.29/0.46 = D5 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % min_dist_less
% 0.29/0.46 thf(fact_105_isSPath__no__selfloop,axiom,
% 0.29/0.46 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat] :
% 0.29/0.46 ( ( isSimplePath @ capacity @ c @ U @ P @ V )
% 0.29/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % isSPath_no_selfloop
% 0.29/0.46 thf(fact_106_isSPath__sg__incoming,axiom,
% 0.29/0.46 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,U22: nat] :
% 0.29/0.46 ( ( isSimplePath @ capacity @ c @ U @ P @ V )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( ( U1 != U22 )
% 0.29/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U22 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % isSPath_sg_incoming
% 0.29/0.46 thf(fact_107_isSPath__sg__outgoing,axiom,
% 0.29/0.46 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,V23: nat] :
% 0.29/0.46 ( ( isSimplePath @ capacity @ c @ U @ P @ V )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( ( V1 != V23 )
% 0.29/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V23 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % isSPath_sg_outgoing
% 0.29/0.46 thf(fact_108_old_Oprod_Oinject,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B5: B] :
% 0.29/0.46 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.29/0.46 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.29/0.46 = ( ( A2 = A5 )
% 0.29/0.46 & ( B2 = B5 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.prod.inject
% 0.29/0.46 thf(fact_109_prod_Oinject,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 0.29/0.46 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.29/0.46 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.29/0.46 = ( ( X1 = Y1 )
% 0.29/0.46 & ( X22 = Y2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod.inject
% 0.29/0.46 thf(fact_110_isSPath__pathLE,axiom,
% 0.29/0.46 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat] :
% 0.29/0.46 ( ( isPath @ capacity @ c @ S @ P @ T2 )
% 0.29/0.46 => ? [P6: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ capacity @ c @ S @ P6 @ T2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % isSPath_pathLE
% 0.29/0.46 thf(fact_111_shortestPath__is__simple,axiom,
% 0.29/0.46 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat] :
% 0.29/0.46 ( ( isShortestPath @ capacity @ c @ S @ P @ T2 )
% 0.29/0.46 => ( isSimplePath @ capacity @ c @ S @ P @ T2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % shortestPath_is_simple
% 0.29/0.46 thf(fact_112_add__less__cancel__right,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [A2: A,C: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 0.29/0.46 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_cancel_right
% 0.29/0.46 thf(fact_113_add__less__cancel__left,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [C: A,A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 0.29/0.46 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_cancel_left
% 0.29/0.46 thf(fact_114_nat__add__left__cancel__less,axiom,
% 0.29/0.46 ! [K: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 0.29/0.46 = ( ord_less @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_add_left_cancel_less
% 0.29/0.46 thf(fact_115_Graph_OisSimplePath_Ocong,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ( ( isSimplePath @ Capacity )
% 0.29/0.46 = ( isSimplePath @ Capacity ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isSimplePath.cong
% 0.29/0.46 thf(fact_116_less__mono__imp__le__mono,axiom,
% 0.29/0.46 ! [F: nat > nat,I: nat,J: nat] :
% 0.29/0.46 ( ! [I2: nat,J2: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I2 @ J2 )
% 0.29/0.46 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ I @ J )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_mono_imp_le_mono
% 0.29/0.46 thf(fact_117_le__neq__implies__less,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 => ( ( M != N )
% 0.29/0.46 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_neq_implies_less
% 0.29/0.46 thf(fact_118_less__or__eq__imp__le,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ( ord_less @ nat @ M @ N )
% 0.29/0.46 | ( M = N ) )
% 0.29/0.46 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_or_eq_imp_le
% 0.29/0.46 thf(fact_119_le__eq__less__or__eq,axiom,
% 0.29/0.46 ( ( ord_less_eq @ nat )
% 0.29/0.46 = ( ^ [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M2 @ N2 )
% 0.29/0.46 | ( M2 = N2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_eq_less_or_eq
% 0.29/0.46 thf(fact_120_less__imp__le__nat,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M @ N )
% 0.29/0.46 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_imp_le_nat
% 0.29/0.46 thf(fact_121_nat__less__le,axiom,
% 0.29/0.46 ( ( ord_less @ nat )
% 0.29/0.46 = ( ^ [M2: nat,N2: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 0.29/0.46 & ( M2 != N2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_less_le
% 0.29/0.46 thf(fact_122_add__lessD1,axiom,
% 0.29/0.46 ! [I: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
% 0.29/0.46 => ( ord_less @ nat @ I @ K ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_lessD1
% 0.29/0.46 thf(fact_123_add__less__mono,axiom,
% 0.29/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I @ J )
% 0.29/0.46 => ( ( ord_less @ nat @ K @ L )
% 0.29/0.46 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_mono
% 0.29/0.46 thf(fact_124_not__add__less1,axiom,
% 0.29/0.46 ! [I: nat,J: nat] :
% 0.29/0.46 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% 0.29/0.46
% 0.29/0.46 % not_add_less1
% 0.29/0.46 thf(fact_125_not__add__less2,axiom,
% 0.29/0.46 ! [J: nat,I: nat] :
% 0.29/0.46 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% 0.29/0.46
% 0.29/0.46 % not_add_less2
% 0.29/0.46 thf(fact_126_add__less__mono1,axiom,
% 0.29/0.46 ! [I: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I @ J )
% 0.29/0.46 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_mono1
% 0.29/0.46 thf(fact_127_trans__less__add1,axiom,
% 0.29/0.46 ! [I: nat,J: nat,M: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I @ J )
% 0.29/0.46 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % trans_less_add1
% 0.29/0.46 thf(fact_128_trans__less__add2,axiom,
% 0.29/0.46 ! [I: nat,J: nat,M: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ I @ J )
% 0.29/0.46 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % trans_less_add2
% 0.29/0.46 thf(fact_129_less__add__eq__less,axiom,
% 0.29/0.46 ! [K: nat,L: nat,M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ K @ L )
% 0.29/0.46 => ( ( ( plus_plus @ nat @ M @ L )
% 0.29/0.46 = ( plus_plus @ nat @ K @ N ) )
% 0.29/0.46 => ( ord_less @ nat @ M @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_add_eq_less
% 0.29/0.46 thf(fact_130_nat__neq__iff,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( M != N )
% 0.29/0.46 = ( ( ord_less @ nat @ M @ N )
% 0.29/0.46 | ( ord_less @ nat @ N @ M ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_neq_iff
% 0.29/0.46 thf(fact_131_less__not__refl,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ~ ( ord_less @ nat @ N @ N ) ).
% 0.29/0.46
% 0.29/0.46 % less_not_refl
% 0.29/0.46 thf(fact_132_less__not__refl2,axiom,
% 0.29/0.46 ! [N: nat,M: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ N @ M )
% 0.29/0.46 => ( M != N ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_not_refl2
% 0.29/0.46 thf(fact_133_less__not__refl3,axiom,
% 0.29/0.46 ! [S: nat,T2: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ S @ T2 )
% 0.29/0.46 => ( S != T2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_not_refl3
% 0.29/0.46 thf(fact_134_measure__induct,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( wellorder @ B )
% 0.29/0.46 => ! [F: A > B,P3: A > $o,A2: A] :
% 0.29/0.46 ( ! [X3: A] :
% 0.29/0.46 ( ! [Y: A] :
% 0.29/0.46 ( ( ord_less @ B @ ( F @ Y ) @ ( F @ X3 ) )
% 0.29/0.46 => ( P3 @ Y ) )
% 0.29/0.46 => ( P3 @ X3 ) )
% 0.29/0.46 => ( P3 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % measure_induct
% 0.29/0.46 thf(fact_135_less__irrefl__nat,axiom,
% 0.29/0.46 ! [N: nat] :
% 0.29/0.46 ~ ( ord_less @ nat @ N @ N ) ).
% 0.29/0.46
% 0.29/0.46 % less_irrefl_nat
% 0.29/0.46 thf(fact_136_nat__less__induct,axiom,
% 0.29/0.46 ! [P3: nat > $o,N: nat] :
% 0.29/0.46 ( ! [N3: nat] :
% 0.29/0.46 ( ! [M3: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M3 @ N3 )
% 0.29/0.46 => ( P3 @ M3 ) )
% 0.29/0.46 => ( P3 @ N3 ) )
% 0.29/0.46 => ( P3 @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_less_induct
% 0.29/0.46 thf(fact_137_infinite__descent,axiom,
% 0.29/0.46 ! [P3: nat > $o,N: nat] :
% 0.29/0.46 ( ! [N3: nat] :
% 0.29/0.46 ( ~ ( P3 @ N3 )
% 0.29/0.46 => ? [M3: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M3 @ N3 )
% 0.29/0.46 & ~ ( P3 @ M3 ) ) )
% 0.29/0.46 => ( P3 @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_descent
% 0.29/0.46 thf(fact_138_linorder__neqE__nat,axiom,
% 0.29/0.46 ! [X: nat,Y3: nat] :
% 0.29/0.46 ( ( X != Y3 )
% 0.29/0.46 => ( ~ ( ord_less @ nat @ X @ Y3 )
% 0.29/0.46 => ( ord_less @ nat @ Y3 @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_neqE_nat
% 0.29/0.46 thf(fact_139_measure__induct__rule,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType] :
% 0.29/0.46 ( ( wellorder @ B )
% 0.29/0.46 => ! [F: A > B,P3: A > $o,A2: A] :
% 0.29/0.46 ( ! [X3: A] :
% 0.29/0.46 ( ! [Y: A] :
% 0.29/0.46 ( ( ord_less @ B @ ( F @ Y ) @ ( F @ X3 ) )
% 0.29/0.46 => ( P3 @ Y ) )
% 0.29/0.46 => ( P3 @ X3 ) )
% 0.29/0.46 => ( P3 @ A2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % measure_induct_rule
% 0.29/0.46 thf(fact_140_infinite__descent__measure,axiom,
% 0.29/0.46 ! [A: $tType,P3: A > $o,V6: A > nat,X: A] :
% 0.29/0.46 ( ! [X3: A] :
% 0.29/0.46 ( ~ ( P3 @ X3 )
% 0.29/0.46 => ? [Y: A] :
% 0.29/0.46 ( ( ord_less @ nat @ ( V6 @ Y ) @ ( V6 @ X3 ) )
% 0.29/0.46 & ~ ( P3 @ Y ) ) )
% 0.29/0.46 => ( P3 @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % infinite_descent_measure
% 0.29/0.46 thf(fact_141_add__less__imp__less__right,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [A2: A,C: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 0.29/0.46 => ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_imp_less_right
% 0.29/0.46 thf(fact_142_add__less__imp__less__left,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere236663937imp_le @ A )
% 0.29/0.46 => ! [C: A,A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 0.29/0.46 => ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_imp_less_left
% 0.29/0.46 thf(fact_143_add__strict__right__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_strict_right_mono
% 0.29/0.46 thf(fact_144_add__strict__left__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_strict_left_mono
% 0.29/0.46 thf(fact_145_add__strict__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( strict2144017051up_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A,D: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less @ A @ C @ D )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_strict_mono
% 0.29/0.46 thf(fact_146_add__mono__thms__linordered__field_I1_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( ord_less @ A @ I @ J )
% 0.29/0.46 & ( K = L ) )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_field(1)
% 0.29/0.46 thf(fact_147_add__mono__thms__linordered__field_I2_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( I = J )
% 0.29/0.46 & ( ord_less @ A @ K @ L ) )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_field(2)
% 0.29/0.46 thf(fact_148_add__mono__thms__linordered__field_I5_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( ord_less @ A @ I @ J )
% 0.29/0.46 & ( ord_less @ A @ K @ L ) )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_field(5)
% 0.29/0.46 thf(fact_149_mono__nat__linear__lb,axiom,
% 0.29/0.46 ! [F: nat > nat,M: nat,K: nat] :
% 0.29/0.46 ( ! [M4: nat,N3: nat] :
% 0.29/0.46 ( ( ord_less @ nat @ M4 @ N3 )
% 0.29/0.46 => ( ord_less @ nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % mono_nat_linear_lb
% 0.29/0.46 thf(fact_150_Graph_OisSPath__pathLE,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat] :
% 0.29/0.46 ( ( isPath @ Capacity @ C @ S @ P @ T2 )
% 0.29/0.46 => ? [P6: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ Capacity @ C @ S @ P6 @ T2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isSPath_pathLE
% 0.29/0.46 thf(fact_151_Graph_OshortestPath__is__simple,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat] :
% 0.29/0.46 ( ( isShortestPath @ Capacity @ C @ S @ P @ T2 )
% 0.29/0.46 => ( isSimplePath @ Capacity @ C @ S @ P @ T2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.shortestPath_is_simple
% 0.29/0.46 thf(fact_152_add__mono__thms__linordered__field_I4_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ I @ J )
% 0.29/0.46 & ( ord_less @ A @ K @ L ) )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_field(4)
% 0.29/0.46 thf(fact_153_add__mono__thms__linordered__field_I3_J,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [I: A,J: A,K: A,L: A] :
% 0.29/0.46 ( ( ( ord_less @ A @ I @ J )
% 0.29/0.46 & ( ord_less_eq @ A @ K @ L ) )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono_thms_linordered_field(3)
% 0.29/0.46 thf(fact_154_add__le__less__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A,D: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less @ A @ C @ D )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_less_mono
% 0.29/0.46 thf(fact_155_add__less__le__mono,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ordere223160158up_add @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A,D: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ C @ D )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_less_le_mono
% 0.29/0.46 thf(fact_156_Graph_Omin__dist__less,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,Src: nat,V: nat,D: nat,D5: nat] :
% 0.29/0.46 ( ( connected @ Capacity @ C @ Src @ V )
% 0.29/0.46 => ( ( ( min_dist @ Capacity @ C @ Src @ V )
% 0.29/0.46 = D )
% 0.29/0.46 => ( ( ord_less @ nat @ D5 @ D )
% 0.29/0.46 => ? [V5: nat] :
% 0.29/0.46 ( ( connected @ Capacity @ C @ Src @ V5 )
% 0.29/0.46 & ( ( min_dist @ Capacity @ C @ Src @ V5 )
% 0.29/0.46 = D5 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.min_dist_less
% 0.29/0.46 thf(fact_157_Graph_OisSPath__no__selfloop,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat] :
% 0.29/0.46 ( ( isSimplePath @ Capacity @ C @ U @ P @ V )
% 0.29/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isSPath_no_selfloop
% 0.29/0.46 thf(fact_158_Graph_OisSPath__sg__incoming,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,U22: nat] :
% 0.29/0.46 ( ( isSimplePath @ Capacity @ C @ U @ P @ V )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( ( U1 != U22 )
% 0.29/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U22 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isSPath_sg_incoming
% 0.29/0.46 thf(fact_159_Graph_OisSPath__sg__outgoing,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,V23: nat] :
% 0.29/0.46 ( ( isSimplePath @ Capacity @ C @ U @ P @ V )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ( ( V1 != V23 )
% 0.29/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V23 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isSPath_sg_outgoing
% 0.29/0.46 thf(fact_160_surj__pair,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 0.29/0.46 ? [X3: A,Y4: B] :
% 0.29/0.46 ( P
% 0.29/0.46 = ( product_Pair @ A @ B @ X3 @ Y4 ) ) ).
% 0.29/0.46
% 0.29/0.46 % surj_pair
% 0.29/0.46 thf(fact_161_prod__cases,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 0.29/0.46 ( ! [A6: A,B6: B] : ( P3 @ ( product_Pair @ A @ B @ A6 @ B6 ) )
% 0.29/0.46 => ( P3 @ P ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_cases
% 0.29/0.46 thf(fact_162_Pair__inject,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B5: B] :
% 0.29/0.46 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 0.29/0.46 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 0.29/0.46 => ~ ( ( A2 = A5 )
% 0.29/0.46 => ( B2 != B5 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Pair_inject
% 0.29/0.46 thf(fact_163_prod__cases3,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C4: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C4 )] :
% 0.29/0.46 ~ ! [A6: A,B6: B,C3: C4] :
% 0.29/0.46 ( Y3
% 0.29/0.46 != ( product_Pair @ A @ ( product_prod @ B @ C4 ) @ A6 @ ( product_Pair @ B @ C4 @ B6 @ C3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_cases3
% 0.29/0.46 thf(fact_164_prod__cases4,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C4: $tType,D8: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ D8 ) )] :
% 0.29/0.46 ~ ! [A6: A,B6: B,C3: C4,D6: D8] :
% 0.29/0.46 ( Y3
% 0.29/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C4 @ D8 ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C4 @ D8 ) @ B6 @ ( product_Pair @ C4 @ D8 @ C3 @ D6 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_cases4
% 0.29/0.46 thf(fact_165_prod__cases5,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C4: $tType,D8: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ E ) ) )] :
% 0.29/0.46 ~ ! [A6: A,B6: B,C3: C4,D6: D8,E2: E] :
% 0.29/0.46 ( Y3
% 0.29/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ E ) ) @ B6 @ ( product_Pair @ C4 @ ( product_prod @ D8 @ E ) @ C3 @ ( product_Pair @ D8 @ E @ D6 @ E2 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_cases5
% 0.29/0.46 thf(fact_166_prod__cases6,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C4: $tType,D8: $tType,E: $tType,F2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.29/0.46 ~ ! [A6: A,B6: B,C3: C4,D6: D8,E2: E,F3: F2] :
% 0.29/0.46 ( Y3
% 0.29/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) ) @ B6 @ ( product_Pair @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D8 @ ( product_prod @ E @ F2 ) @ D6 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_cases6
% 0.29/0.46 thf(fact_167_prod__cases7,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,C4: $tType,D8: $tType,E: $tType,F2: $tType,G2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 0.29/0.46 ~ ! [A6: A,B6: B,C3: C4,D6: D8,E2: E,F3: F2,G3: G2] :
% 0.29/0.46 ( Y3
% 0.29/0.46 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B6 @ ( product_Pair @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D6 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_cases7
% 0.29/0.46 thf(fact_168_prod__induct3,axiom,
% 0.29/0.46 ! [C4: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ C4 ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C4 )] :
% 0.29/0.46 ( ! [A6: A,B6: B,C3: C4] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ C4 ) @ A6 @ ( product_Pair @ B @ C4 @ B6 @ C3 ) ) )
% 0.29/0.46 => ( P3 @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_induct3
% 0.29/0.46 thf(fact_169_prod__induct4,axiom,
% 0.29/0.46 ! [D8: $tType,C4: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ D8 ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ D8 ) )] :
% 0.29/0.46 ( ! [A6: A,B6: B,C3: C4,D6: D8] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C4 @ D8 ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C4 @ D8 ) @ B6 @ ( product_Pair @ C4 @ D8 @ C3 @ D6 ) ) ) )
% 0.29/0.46 => ( P3 @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_induct4
% 0.29/0.46 thf(fact_170_prod__induct5,axiom,
% 0.29/0.46 ! [E: $tType,D8: $tType,C4: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ E ) ) )] :
% 0.29/0.46 ( ! [A6: A,B6: B,C3: C4,D6: D8,E2: E] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ E ) ) @ B6 @ ( product_Pair @ C4 @ ( product_prod @ D8 @ E ) @ C3 @ ( product_Pair @ D8 @ E @ D6 @ E2 ) ) ) ) )
% 0.29/0.46 => ( P3 @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_induct5
% 0.29/0.46 thf(fact_171_prod__induct6,axiom,
% 0.29/0.46 ! [F2: $tType,E: $tType,D8: $tType,C4: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) ) )] :
% 0.29/0.46 ( ! [A6: A,B6: B,C3: C4,D6: D8,E2: E,F3: F2] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) ) @ B6 @ ( product_Pair @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D8 @ ( product_prod @ E @ F2 ) @ D6 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
% 0.29/0.46 => ( P3 @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_induct6
% 0.29/0.46 thf(fact_172_prod__induct7,axiom,
% 0.29/0.46 ! [G2: $tType,F2: $tType,E: $tType,D8: $tType,C4: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 0.29/0.46 ( ! [A6: A,B6: B,C3: C4,D6: D8,E2: E,F3: F2,G3: G2] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B6 @ ( product_Pair @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D6 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
% 0.29/0.46 => ( P3 @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % prod_induct7
% 0.29/0.46 thf(fact_173_old_Oprod_Oexhaust,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 0.29/0.46 ~ ! [A6: A,B6: B] :
% 0.29/0.46 ( Y3
% 0.29/0.46 != ( product_Pair @ A @ B @ A6 @ B6 ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.prod.exhaust
% 0.29/0.46 thf(fact_174_old_Oprod_Oinducts,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.29/0.46 ( ! [A6: A,B6: B] : ( P3 @ ( product_Pair @ A @ B @ A6 @ B6 ) )
% 0.29/0.46 => ( P3 @ Prod ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.prod.inducts
% 0.29/0.46 thf(fact_175_size__neq__size__imp__neq,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( size @ A )
% 0.29/0.46 => ! [X: A,Y3: A] :
% 0.29/0.46 ( ( ( size_size @ A @ X )
% 0.29/0.46 != ( size_size @ A @ Y3 ) )
% 0.29/0.46 => ( X != Y3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % size_neq_size_imp_neq
% 0.29/0.46 thf(fact_176_le__refl,axiom,
% 0.29/0.46 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 0.29/0.46
% 0.29/0.46 % le_refl
% 0.29/0.46 thf(fact_177_le__trans,axiom,
% 0.29/0.46 ! [I: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I @ J )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ J @ K )
% 0.29/0.46 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_trans
% 0.29/0.46 thf(fact_178_eq__imp__le,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( M = N )
% 0.29/0.46 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % eq_imp_le
% 0.29/0.46 thf(fact_179_le__antisym,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ N @ M )
% 0.29/0.46 => ( M = N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_antisym
% 0.29/0.46 thf(fact_180_nat__le__linear,axiom,
% 0.29/0.46 ! [M: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_le_linear
% 0.29/0.46 thf(fact_181_Nat_Oex__has__greatest__nat,axiom,
% 0.29/0.46 ! [P3: nat > $o,K: nat,B2: nat] :
% 0.29/0.46 ( ( P3 @ K )
% 0.29/0.46 => ( ! [Y4: nat] :
% 0.29/0.46 ( ( P3 @ Y4 )
% 0.29/0.46 => ( ord_less_eq @ nat @ Y4 @ B2 ) )
% 0.29/0.46 => ? [X3: nat] :
% 0.29/0.46 ( ( P3 @ X3 )
% 0.29/0.46 & ! [Y: nat] :
% 0.29/0.46 ( ( P3 @ Y )
% 0.29/0.46 => ( ord_less_eq @ nat @ Y @ X3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Nat.ex_has_greatest_nat
% 0.29/0.46 thf(fact_182_Graph_OisShortestPath__alt,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ( ( isShortestPath @ Capacity )
% 0.29/0.46 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U3: nat,P2: list @ ( product_prod @ nat @ nat ),V3: nat] :
% 0.29/0.46 ( ( isSimplePath @ Capacity @ C2 @ U3 @ P2 @ V3 )
% 0.29/0.46 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 )
% 0.29/0.46 = ( min_dist @ Capacity @ C2 @ U3 @ V3 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isShortestPath_alt
% 0.29/0.46 thf(fact_183_nat__le__iff__add,axiom,
% 0.29/0.46 ( ( ord_less_eq @ nat )
% 0.29/0.46 = ( ^ [M2: nat,N2: nat] :
% 0.29/0.46 ? [K2: nat] :
% 0.29/0.46 ( N2
% 0.29/0.46 = ( plus_plus @ nat @ M2 @ K2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % nat_le_iff_add
% 0.29/0.46 thf(fact_184_trans__le__add2,axiom,
% 0.29/0.46 ! [I: nat,J: nat,M: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I @ J )
% 0.29/0.46 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % trans_le_add2
% 0.29/0.46 thf(fact_185_trans__le__add1,axiom,
% 0.29/0.46 ! [I: nat,J: nat,M: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I @ J )
% 0.29/0.46 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % trans_le_add1
% 0.29/0.46 thf(fact_186_add__le__mono1,axiom,
% 0.29/0.46 ! [I: nat,J: nat,K: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I @ J )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_mono1
% 0.29/0.46 thf(fact_187_add__le__mono,axiom,
% 0.29/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ I @ J )
% 0.29/0.46 => ( ( ord_less_eq @ nat @ K @ L )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_le_mono
% 0.29/0.46 thf(fact_188_le__Suc__ex,axiom,
% 0.29/0.46 ! [K: nat,L: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ K @ L )
% 0.29/0.46 => ? [N3: nat] :
% 0.29/0.46 ( L
% 0.29/0.46 = ( plus_plus @ nat @ K @ N3 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_Suc_ex
% 0.29/0.46 thf(fact_189_add__leD2,axiom,
% 0.29/0.46 ! [M: nat,K: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 0.29/0.46 => ( ord_less_eq @ nat @ K @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_leD2
% 0.29/0.46 thf(fact_190_add__leD1,axiom,
% 0.29/0.46 ! [M: nat,K: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 0.29/0.46 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_leD1
% 0.29/0.46 thf(fact_191_le__add2,axiom,
% 0.29/0.46 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_add2
% 0.29/0.46 thf(fact_192_le__add1,axiom,
% 0.29/0.46 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_add1
% 0.29/0.46 thf(fact_193_add__leE,axiom,
% 0.29/0.46 ! [M: nat,K: nat,N: nat] :
% 0.29/0.46 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 0.29/0.46 => ~ ( ( ord_less_eq @ nat @ M @ N )
% 0.29/0.46 => ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_leE
% 0.29/0.46 thf(fact_194_isSPath__nt__parallel,axiom,
% 0.29/0.46 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,E3: product_prod @ nat @ nat] :
% 0.29/0.46 ( ( isSimplePath @ capacity @ c @ S @ P @ T2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ E3 @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ E3 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % isSPath_nt_parallel
% 0.29/0.46 thf(fact_195_discrete,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( unique1598680935umeral @ A )
% 0.29/0.46 => ( ( ord_less @ A )
% 0.29/0.46 = ( ^ [A3: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % discrete
% 0.29/0.46 thf(fact_196_old_Oprod_Orec,axiom,
% 0.29/0.46 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 0.29/0.46 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 0.29/0.46 = ( F1 @ A2 @ B2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % old.prod.rec
% 0.29/0.46 thf(fact_197_fold__atLeastAtMost__nat_Oinduct,axiom,
% 0.29/0.46 ! [A: $tType,P3: ( nat > A > A ) > nat > nat > A > $o,A0: nat > A > A,A1: nat,A22: nat,A32: A] :
% 0.29/0.46 ( ! [F3: nat > A > A,A6: nat,B6: nat,Acc: A] :
% 0.29/0.46 ( ( ~ ( ord_less @ nat @ B6 @ A6 )
% 0.29/0.46 => ( P3 @ F3 @ ( plus_plus @ nat @ A6 @ ( one_one @ nat ) ) @ B6 @ ( F3 @ A6 @ Acc ) ) )
% 0.29/0.46 => ( P3 @ F3 @ A6 @ B6 @ Acc ) )
% 0.29/0.46 => ( P3 @ A0 @ A1 @ A22 @ A32 ) ) ).
% 0.29/0.46
% 0.29/0.46 % fold_atLeastAtMost_nat.induct
% 0.29/0.46 thf(fact_198_ex__has__greatest__nat__lemma,axiom,
% 0.29/0.46 ! [A: $tType,P3: A > $o,K: A,F: A > nat,N: nat] :
% 0.29/0.46 ( ( P3 @ K )
% 0.29/0.46 => ( ! [X3: A] :
% 0.29/0.46 ( ( P3 @ X3 )
% 0.29/0.46 => ? [Y: A] :
% 0.29/0.46 ( ( P3 @ Y )
% 0.29/0.46 & ~ ( ord_less_eq @ nat @ ( F @ Y ) @ ( F @ X3 ) ) ) )
% 0.29/0.46 => ? [Y4: A] :
% 0.29/0.46 ( ( P3 @ Y4 )
% 0.29/0.46 & ~ ( ord_less @ nat @ ( F @ Y4 ) @ ( plus_plus @ nat @ ( F @ K ) @ N ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ex_has_greatest_nat_lemma
% 0.29/0.46 thf(fact_199_swap__swap,axiom,
% 0.29/0.46 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 0.29/0.46 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
% 0.29/0.46 = P ) ).
% 0.29/0.46
% 0.29/0.46 % swap_swap
% 0.29/0.46 thf(fact_200_swap__simp,axiom,
% 0.29/0.46 ! [A: $tType,B: $tType,X: B,Y3: A] :
% 0.29/0.46 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) )
% 0.29/0.46 = ( product_Pair @ A @ B @ Y3 @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % swap_simp
% 0.29/0.46 thf(fact_201_ex__has__least__nat,axiom,
% 0.29/0.46 ! [A: $tType,P3: A > $o,K: A,M: A > nat] :
% 0.29/0.46 ( ( P3 @ K )
% 0.29/0.46 => ? [X3: A] :
% 0.29/0.46 ( ( P3 @ X3 )
% 0.29/0.46 & ! [Y: A] :
% 0.29/0.46 ( ( P3 @ Y )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ex_has_least_nat
% 0.29/0.46 thf(fact_202_bounded__Max__nat,axiom,
% 0.29/0.46 ! [P3: nat > $o,X: nat,M5: nat] :
% 0.29/0.46 ( ( P3 @ X )
% 0.29/0.46 => ( ! [X3: nat] :
% 0.29/0.46 ( ( P3 @ X3 )
% 0.29/0.46 => ( ord_less_eq @ nat @ X3 @ M5 ) )
% 0.29/0.46 => ~ ! [M4: nat] :
% 0.29/0.46 ( ( P3 @ M4 )
% 0.29/0.46 => ~ ! [X4: nat] :
% 0.29/0.46 ( ( P3 @ X4 )
% 0.29/0.46 => ( ord_less_eq @ nat @ X4 @ M4 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % bounded_Max_nat
% 0.29/0.46 thf(fact_203_fold__atLeastAtMost__nat_Ocases,axiom,
% 0.29/0.46 ! [A: $tType,X: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
% 0.29/0.46 ~ ! [F3: nat > A > A,A6: nat,B6: nat,Acc: A] :
% 0.29/0.46 ( X
% 0.29/0.46 != ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A6 @ ( product_Pair @ nat @ A @ B6 @ Acc ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % fold_atLeastAtMost_nat.cases
% 0.29/0.46 thf(fact_204_Graph_OisSPath__nt__parallel,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,E3: product_prod @ nat @ nat] :
% 0.29/0.46 ( ( isSimplePath @ Capacity @ C @ S @ P @ T2 )
% 0.29/0.46 => ( ( member @ ( product_prod @ nat @ nat ) @ E3 @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 0.29/0.46 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ E3 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isSPath_nt_parallel
% 0.29/0.46 thf(fact_205_Lattices__Big_Oex__has__greatest__nat,axiom,
% 0.29/0.46 ! [A: $tType,P3: A > $o,K: A,F: A > nat,B2: nat] :
% 0.29/0.46 ( ( P3 @ K )
% 0.29/0.46 => ( ! [Y4: A] :
% 0.29/0.46 ( ( P3 @ Y4 )
% 0.29/0.46 => ( ord_less @ nat @ ( F @ Y4 ) @ B2 ) )
% 0.29/0.46 => ? [X3: A] :
% 0.29/0.46 ( ( P3 @ X3 )
% 0.29/0.46 & ! [Y: A] :
% 0.29/0.46 ( ( P3 @ Y )
% 0.29/0.46 => ( ord_less_eq @ nat @ ( F @ Y ) @ ( F @ X3 ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Lattices_Big.ex_has_greatest_nat
% 0.29/0.46 thf(fact_206_less__add__one,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_semidom @ A )
% 0.29/0.46 => ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % less_add_one
% 0.29/0.46 thf(fact_207_add__mono1,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linord1659791738miring @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.46 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % add_mono1
% 0.29/0.46 thf(fact_208_linorder__neqE__linordered__idom,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linordered_idom @ A )
% 0.29/0.46 => ! [X: A,Y3: A] :
% 0.29/0.46 ( ( X != Y3 )
% 0.29/0.46 => ( ~ ( ord_less @ A @ X @ Y3 )
% 0.29/0.46 => ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_neqE_linordered_idom
% 0.29/0.46 thf(fact_209_isPath_Osimps_I1_J,axiom,
% 0.29/0.46 ! [U: nat,V: nat] :
% 0.29/0.46 ( ( isPath @ capacity @ c @ U @ ( nil @ ( product_prod @ nat @ nat ) ) @ V )
% 0.29/0.46 = ( U = V ) ) ).
% 0.29/0.46
% 0.29/0.46 % isPath.simps(1)
% 0.29/0.46 thf(fact_210_order__refl,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A )
% 0.29/0.46 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_refl
% 0.29/0.46 thf(fact_211_simplePath__empty__conv,axiom,
% 0.29/0.46 ! [S: nat,T2: nat] :
% 0.29/0.46 ( ( isSimplePath @ capacity @ c @ S @ ( nil @ ( product_prod @ nat @ nat ) ) @ T2 )
% 0.29/0.46 = ( S = T2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % simplePath_empty_conv
% 0.29/0.46 thf(fact_212_simplePath__same__conv,axiom,
% 0.29/0.46 ! [S: nat,P: list @ ( product_prod @ nat @ nat )] :
% 0.29/0.46 ( ( isSimplePath @ capacity @ c @ S @ P @ S )
% 0.29/0.46 = ( P
% 0.29/0.46 = ( nil @ ( product_prod @ nat @ nat ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % simplePath_same_conv
% 0.29/0.46 thf(fact_213_Graph_OsimplePath__empty__conv,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T2: nat] :
% 0.29/0.46 ( ( isSimplePath @ Capacity @ C @ S @ ( nil @ ( product_prod @ nat @ nat ) ) @ T2 )
% 0.29/0.46 = ( S = T2 ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.simplePath_empty_conv
% 0.29/0.46 thf(fact_214_Graph_OsimplePath__same__conv,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat )] :
% 0.29/0.46 ( ( isSimplePath @ Capacity @ C @ S @ P @ S )
% 0.29/0.46 = ( P
% 0.29/0.46 = ( nil @ ( product_prod @ nat @ nat ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.simplePath_same_conv
% 0.29/0.46 thf(fact_215_Graph_OisPath_Osimps_I1_J,axiom,
% 0.29/0.46 ! [Capacity: $tType] :
% 0.29/0.46 ( ( linordered_idom @ Capacity )
% 0.29/0.46 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat] :
% 0.29/0.46 ( ( isPath @ Capacity @ C @ U @ ( nil @ ( product_prod @ nat @ nat ) ) @ V )
% 0.29/0.46 = ( U = V ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % Graph.isPath.simps(1)
% 0.29/0.46 thf(fact_216_dual__order_Oantisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [B2: A,A2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.antisym
% 0.29/0.46 thf(fact_217_dual__order_Oeq__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 0.29/0.46 = ( ^ [A3: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.29/0.46 & ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.eq_iff
% 0.29/0.46 thf(fact_218_dual__order_Otrans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [B2: A,A2: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ C @ B2 )
% 0.29/0.46 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.trans
% 0.29/0.46 thf(fact_219_linorder__wlog,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [P3: A > A > $o,A2: A,B2: A] :
% 0.29/0.46 ( ! [A6: A,B6: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A6 @ B6 )
% 0.29/0.46 => ( P3 @ A6 @ B6 ) )
% 0.29/0.46 => ( ! [A6: A,B6: A] :
% 0.29/0.46 ( ( P3 @ B6 @ A6 )
% 0.29/0.46 => ( P3 @ A6 @ B6 ) )
% 0.29/0.46 => ( P3 @ A2 @ B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % linorder_wlog
% 0.29/0.46 thf(fact_220_dual__order_Orefl,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.29/0.46
% 0.29/0.46 % dual_order.refl
% 0.29/0.46 thf(fact_221_order__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( preorder @ A )
% 0.29/0.46 => ! [X: A,Y3: A,Z2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.29/0.46 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_trans
% 0.29/0.46 thf(fact_222_order__class_Oorder_Oantisym,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A2: A,B2: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.29/0.46 => ( A2 = B2 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.antisym
% 0.29/0.46 thf(fact_223_ord__le__eq__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 => ( ( B2 = C )
% 0.29/0.46 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_le_eq_trans
% 0.29/0.46 thf(fact_224_ord__eq__le__trans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( ord @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( A2 = B2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.29/0.46 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % ord_eq_le_trans
% 0.29/0.46 thf(fact_225_order__class_Oorder_Oeq__iff,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 0.29/0.46 = ( ^ [A3: A,B3: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.29/0.46 & ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order_class.order.eq_iff
% 0.29/0.46 thf(fact_226_antisym__conv,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [Y3: A,X: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ Y3 @ X )
% 0.29/0.46 => ( ( ord_less_eq @ A @ X @ Y3 )
% 0.29/0.46 = ( X = Y3 ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % antisym_conv
% 0.29/0.46 thf(fact_227_le__cases3,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [X: A,Y3: A,Z2: A] :
% 0.29/0.46 ( ( ( ord_less_eq @ A @ X @ Y3 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Y3 @ X )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Y3 @ X ) )
% 0.29/0.46 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 0.29/0.46 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 0.29/0.46 => ~ ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_cases3
% 0.29/0.46 thf(fact_228_order_Otrans,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( order @ A )
% 0.29/0.46 => ! [A2: A,B2: A,C: A] :
% 0.29/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.46 => ( ( ord_less_eq @ A @ B2 @ C )
% 0.29/0.46 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % order.trans
% 0.29/0.46 thf(fact_229_le__cases,axiom,
% 0.29/0.46 ! [A: $tType] :
% 0.29/0.46 ( ( linorder @ A )
% 0.29/0.46 => ! [X: A,Y3: A] :
% 0.29/0.46 ( ~ ( ord_less_eq @ A @ X @ Y3 )
% 0.29/0.46 => ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 0.29/0.46
% 0.29/0.46 % le_cases
% 0.29/0.46 thf(fact_230_eq__refl,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X: A,Y3: A] :
% 0.29/0.47 ( ( X = Y3 )
% 0.29/0.47 => ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % eq_refl
% 0.29/0.47 thf(fact_231_linear,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X: A,Y3: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.29/0.47 | ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % linear
% 0.29/0.47 thf(fact_232_antisym,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [X: A,Y3: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.29/0.47 => ( ( ord_less_eq @ A @ Y3 @ X )
% 0.29/0.47 => ( X = Y3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % antisym
% 0.29/0.47 thf(fact_233_eq__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 0.29/0.47 = ( ^ [X2: A,Y6: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X2 @ Y6 )
% 0.29/0.47 & ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % eq_iff
% 0.29/0.47 thf(fact_234_ord__le__eq__subst,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType] :
% 0.29/0.47 ( ( ( ord @ B )
% 0.29/0.47 & ( ord @ A ) )
% 0.29/0.47 => ! [A2: A,B2: A,F: A > B,C: B] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.47 => ( ( ( F @ B2 )
% 0.29/0.47 = C )
% 0.29/0.47 => ( ! [X3: A,Y4: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.29/0.47 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_le_eq_subst
% 0.29/0.47 thf(fact_235_ord__eq__le__subst,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType] :
% 0.29/0.47 ( ( ( ord @ B )
% 0.29/0.47 & ( ord @ A ) )
% 0.29/0.47 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.29/0.47 ( ( A2
% 0.29/0.47 = ( F @ B2 ) )
% 0.29/0.47 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.29/0.47 => ( ! [X3: B,Y4: B] :
% 0.29/0.47 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 0.29/0.47 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % ord_eq_le_subst
% 0.29/0.47 thf(fact_236_order__subst2,axiom,
% 0.29/0.47 ! [A: $tType,C4: $tType] :
% 0.29/0.47 ( ( ( order @ C4 )
% 0.29/0.47 & ( order @ A ) )
% 0.29/0.47 => ! [A2: A,B2: A,F: A > C4,C: C4] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.29/0.47 => ( ( ord_less_eq @ C4 @ ( F @ B2 ) @ C )
% 0.29/0.47 => ( ! [X3: A,Y4: A] :
% 0.29/0.47 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 0.29/0.47 => ( ord_less_eq @ C4 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_eq @ C4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_subst2
% 0.29/0.47 thf(fact_237_order__subst1,axiom,
% 0.29/0.47 ! [A: $tType,B: $tType] :
% 0.29/0.47 ( ( ( order @ B )
% 0.29/0.47 & ( order @ A ) )
% 0.29/0.47 => ! [A2: A,F: B > A,B2: B,C: B] :
% 0.29/0.47 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 0.29/0.47 => ( ( ord_less_eq @ B @ B2 @ C )
% 0.29/0.47 => ( ! [X3: B,Y4: B] :
% 0.29/0.47 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 0.29/0.47 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 0.29/0.47 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order_subst1
% 0.29/0.47 thf(fact_238_le__fun__def,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType] :
% 0.29/0.47 ( ( ord @ B )
% 0.29/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.29/0.47 = ( ^ [F4: A > B,G4: A > B] :
% 0.29/0.47 ! [X2: A] : ( ord_less_eq @ B @ ( F4 @ X2 ) @ ( G4 @ X2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_fun_def
% 0.29/0.47 thf(fact_239_le__funI,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType] :
% 0.29/0.47 ( ( ord @ B )
% 0.29/0.47 => ! [F: A > B,G: A > B] :
% 0.29/0.47 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.29/0.47 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_funI
% 0.29/0.47 thf(fact_240_le__funE,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType] :
% 0.29/0.47 ( ( ord @ B )
% 0.29/0.47 => ! [F: A > B,G: A > B,X: A] :
% 0.29/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.29/0.47 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_funE
% 0.29/0.47 thf(fact_241_le__funD,axiom,
% 0.29/0.47 ! [B: $tType,A: $tType] :
% 0.29/0.47 ( ( ord @ B )
% 0.29/0.47 => ! [F: A > B,G: A > B,X: A] :
% 0.29/0.47 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.29/0.47 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % le_funD
% 0.29/0.47 thf(fact_242_dual__order_Ostrict__implies__not__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [B2: A,A2: A] :
% 0.29/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.29/0.47 => ( A2 != B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.strict_implies_not_eq
% 0.29/0.47 thf(fact_243_order_Ostrict__implies__not__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A,B2: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.47 => ( A2 != B2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order.strict_implies_not_eq
% 0.29/0.47 thf(fact_244_not__less__iff__gr__or__eq,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X: A,Y3: A] :
% 0.29/0.47 ( ( ~ ( ord_less @ A @ X @ Y3 ) )
% 0.29/0.47 = ( ( ord_less @ A @ Y3 @ X )
% 0.29/0.47 | ( X = Y3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % not_less_iff_gr_or_eq
% 0.29/0.47 thf(fact_245_dual__order_Ostrict__trans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [B2: A,A2: A,C: A] :
% 0.29/0.47 ( ( ord_less @ A @ B2 @ A2 )
% 0.29/0.47 => ( ( ord_less @ A @ C @ B2 )
% 0.29/0.47 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.strict_trans
% 0.29/0.47 thf(fact_246_linorder__less__wlog,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [P3: A > A > $o,A2: A,B2: A] :
% 0.29/0.47 ( ! [A6: A,B6: A] :
% 0.29/0.47 ( ( ord_less @ A @ A6 @ B6 )
% 0.29/0.47 => ( P3 @ A6 @ B6 ) )
% 0.29/0.47 => ( ! [A6: A] : ( P3 @ A6 @ A6 )
% 0.29/0.47 => ( ! [A6: A,B6: A] :
% 0.29/0.47 ( ( P3 @ B6 @ A6 )
% 0.29/0.47 => ( P3 @ A6 @ B6 ) )
% 0.29/0.47 => ( P3 @ A2 @ B2 ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % linorder_less_wlog
% 0.29/0.47 thf(fact_247_exists__least__iff,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( wellorder @ A )
% 0.29/0.47 => ( ( ^ [P7: A > $o] :
% 0.29/0.47 ? [X5: A] : ( P7 @ X5 ) )
% 0.29/0.47 = ( ^ [P8: A > $o] :
% 0.29/0.47 ? [N2: A] :
% 0.29/0.47 ( ( P8 @ N2 )
% 0.29/0.47 & ! [M2: A] :
% 0.29/0.47 ( ( ord_less @ A @ M2 @ N2 )
% 0.29/0.47 => ~ ( P8 @ M2 ) ) ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % exists_least_iff
% 0.29/0.47 thf(fact_248_less__imp__not__less,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X: A,Y3: A] :
% 0.29/0.47 ( ( ord_less @ A @ X @ Y3 )
% 0.29/0.47 => ~ ( ord_less @ A @ Y3 @ X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_imp_not_less
% 0.29/0.47 thf(fact_249_order_Ostrict__trans,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A,B2: A,C: A] :
% 0.29/0.47 ( ( ord_less @ A @ A2 @ B2 )
% 0.29/0.47 => ( ( ord_less @ A @ B2 @ C )
% 0.29/0.47 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % order.strict_trans
% 0.29/0.47 thf(fact_250_dual__order_Oirrefl,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [A2: A] :
% 0.29/0.47 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 0.29/0.47
% 0.29/0.47 % dual_order.irrefl
% 0.29/0.47 thf(fact_251_linorder__cases,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [X: A,Y3: A] :
% 0.29/0.47 ( ~ ( ord_less @ A @ X @ Y3 )
% 0.29/0.47 => ( ( X != Y3 )
% 0.29/0.47 => ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % linorder_cases
% 0.29/0.47 thf(fact_252_less__imp__triv,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( preorder @ A )
% 0.29/0.47 => ! [X: A,Y3: A,P3: $o] :
% 0.29/0.47 ( ( ord_less @ A @ X @ Y3 )
% 0.29/0.47 => ( ( ord_less @ A @ Y3 @ X )
% 0.29/0.47 => P3 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_imp_triv
% 0.29/0.47 thf(fact_253_less__imp__not__eq2,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( order @ A )
% 0.29/0.47 => ! [X: A,Y3: A] :
% 0.29/0.47 ( ( ord_less @ A @ X @ Y3 )
% 0.29/0.47 => ( Y3 != X ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % less_imp_not_eq2
% 0.29/0.47 thf(fact_254_antisym__conv3,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linorder @ A )
% 0.29/0.47 => ! [Y3: A,X: A] :
% 0.29/0.47 ( ~ ( ord_less @ A @ Y3 @ X )
% 0.29/0.47 => ( ( ~ ( ord_less @ A @ X @ Y3 ) )
% 0.29/0.47 = ( X = Y3 ) ) ) ) ).
% 0.29/0.47
% 0.29/0.47 % antisym_conv3
% 0.29/0.47
% 0.29/0.47 % Subclasses (17)
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___HOL_Otype,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( type @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Oone,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( one @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Orderings_Oord,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( ord @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Orderings_Oorder,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( order @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Ogroup__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( group_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Orderings_Olinorder,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( linorder @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Orderings_Opreorder,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( preorder @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Osemigroup__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( semigroup_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Ocomm__monoid__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( comm_monoid_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Oab__semigroup__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( ab_semigroup_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Rings_Olinordered__semidom,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( linordered_semidom @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Ocancel__semigroup__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( cancel_semigroup_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( ordere779506340up_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Rings_Olinordered__nonzero__semiring,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( linord1659791738miring @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( ordere236663937imp_le @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( ordere223160158up_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(subcl_Rings_Olinordered__idom___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 0.29/0.47 ! [A: $tType] :
% 0.29/0.47 ( ( linordered_idom @ A )
% 0.29/0.47 => ( strict2144017051up_add @ A ) ) ).
% 0.29/0.47
% 0.29/0.47 % Type constructors (31)
% 0.29/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.29/0.47 ! [A7: $tType,A8: $tType] :
% 0.29/0.47 ( ( preorder @ A8 )
% 0.29/0.47 => ( preorder @ ( A7 > A8 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.29/0.47 ! [A7: $tType,A8: $tType] :
% 0.29/0.47 ( ( order @ A8 )
% 0.29/0.47 => ( order @ ( A7 > A8 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.29/0.47 ! [A7: $tType,A8: $tType] :
% 0.29/0.47 ( ( ord @ A8 )
% 0.29/0.47 => ( ord @ ( A7 > A8 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral,axiom,
% 0.29/0.47 unique1598680935umeral @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 0.29/0.47 strict2144017051up_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 0.29/0.47 ordere223160158up_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.29/0.47 ordere236663937imp_le @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 0.29/0.47 canoni770627133id_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 0.29/0.47 linord1659791738miring @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 0.29/0.47 ordere779506340up_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 0.29/0.47 cancel_semigroup_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
% 0.29/0.47 linordered_semidom @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 0.29/0.47 ab_semigroup_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 0.29/0.47 comm_monoid_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 0.29/0.47 semigroup_add @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 0.29/0.47 wellorder @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 0.29/0.47 preorder @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 0.29/0.47 linorder @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 0.29/0.47 order @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 0.29/0.47 ord @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 0.29/0.47 one @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 0.29/0.47 size @ nat ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.29/0.47 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 0.29/0.47 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 0.29/0.47 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 0.29/0.47 preorder @ $o ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
% 0.29/0.47 linorder @ $o ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 0.29/0.47 order @ $o ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 0.29/0.47 ord @ $o ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_List_Olist___Nat_Osize_11,axiom,
% 0.29/0.47 ! [A7: $tType] : ( size @ ( list @ A7 ) ) ).
% 0.29/0.47
% 0.29/0.47 thf(tcon_Product__Type_Oprod___Nat_Osize_12,axiom,
% 0.29/0.47 ! [A7: $tType,A8: $tType] : ( size @ ( product_prod @ A7 @ A8 ) ) ).
% 0.29/0.47
% 0.29/0.47 % Free types (1)
% 0.29/0.47 thf(tfree_0,hypothesis,
% 0.29/0.47 linordered_idom @ capacity ).
% 0.29/0.47
% 0.29/0.47 % Conjectures (1)
% 0.29/0.47 thf(conj_0,conjecture,
% 0.29/0.47 ( ( min_dist @ capacity @ c @ u @ t )
% 0.29/0.47 = ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 ) ) ) ).
% 0.29/0.47
% 0.29/0.47 %------------------------------------------------------------------------------
% 0.29/0.47 ------------------------
% 2.83/1.51 ---- Embedded in THF ---
% 2.83/1.51 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.83/1.51 %%% Generated on Tue Feb 24 08:24:45 EST 2026
% 2.83/1.51 %%% using '$$dhol' embedding, version 1.3.0.
% 2.83/1.51 %%% Logic specification used:
% 2.83/1.51 %%% thf(spec, logic, $$dhol).
% 2.83/1.51
% 2.83/1.51 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.dNuoTsUl6x/DTF2DTF2221.p
% 2.83/1.51 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.83/1.51 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 2.83/1.51 thf(ty_tf_capacity, type, capacity: $tType).
% 2.83/1.51 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.83/1.51 thf(ty_t_Nat_Onat, type, nat: $tType).
% 2.83/1.51 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Nat_Osize, type, size: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Orderings_Owellorder, type, wellorder: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere779506340up_add: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord1659791738miring: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni770627133id_add: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere236663937imp_le: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add, type, ordere223160158up_add: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add, type, strict2144017051up_add: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral, type, unique1598680935umeral: (!> [A:$tType]: ($o))).
% 2.83/1.51 thf(sy_c_Graph_OGraph_Oconnected, type, connected: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > $o)))))).
% 2.83/1.51 thf(sy_c_Graph_OGraph_Odist, type, dist: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > (nat > $o))))))).
% 2.83/1.51 thf(sy_c_Graph_OGraph_OisPath, type, isPath: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > ((list @ ((product_prod @ nat) @ nat)) > (nat > $o))))))).
% 2.83/1.51 thf(sy_c_Graph_OGraph_OisShortestPath, type, isShortestPath: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > ((list @ ((product_prod @ nat) @ nat)) > (nat > $o))))))).
% 2.83/1.51 thf(sy_c_Graph_OGraph_OisSimplePath, type, isSimplePath: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > ((list @ ((product_prod @ nat) @ nat)) > (nat > $o))))))).
% 2.83/1.51 thf(sy_c_Graph_OGraph_Omin__dist, type, min_dist: (!> [Capacity:$tType]: (((((product_prod @ nat) @ nat) > Capacity) > (nat > (nat > nat)))))).
% 2.83/1.51 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 2.83/1.51 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 2.83/1.51 thf(sy_c_List_Olist_ONil, type, nil: (!> [A:$tType]: ((list @ A)))).
% 2.83/1.51 thf(sy_c_List_Olist_Oset, type, set2: (!> [A:$tType]: (((list @ A) > (set @ A))))).
% 2.83/1.51 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 2.83/1.51 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.83/1.51 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.83/1.51 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.83/1.51 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.83/1.51 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 2.83/1.51 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.83/1.51 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.83/1.51 thf(sy_v_c, type, c: (((product_prod @ nat) @ nat) > capacity)).
% 2.83/1.51 thf(sy_v_p, type, p: (list @ ((product_prod @ nat) @ nat))).
% 2.83/1.51 thf(sy_v_p1____, type, p1: (list @ ((product_prod @ nat) @ nat))).
% 2.83/1.51 thf(sy_v_p2____, type, p2: (list @ ((product_prod @ nat) @ nat))).
% 2.83/1.51 thf(sy_v_p_H, type, p3: (list @ ((product_prod @ nat) @ nat))).
% 2.83/1.51 thf(sy_v_s, type, s: nat).
% 2.83/1.51 thf(sy_v_t, type, t: nat).
% 2.83/1.51 thf(sy_v_u, type, u: nat).
% 2.83/1.51 thf(sy_v_v, type, v: nat).
% 2.83/1.51 thf(fact_0_MIN_H, axiom, (((((min_dist @ capacity) @ c) @ s) @ t) = (((plus_plus @ nat) @ (((plus_plus @ nat) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p1)) @ (one_one @ nat))) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p2)))).
% 2.83/1.51 thf(fact_1__092_060open_062min__dist_As_At_A_061_Alength_Ap1_A_L_A_I1_A_L_Alength_Ap2_J_A_092_060Longrightarrow_062_Amin__dist_As_Au_A_061_Alength_Ap1_092_060close_062, axiom, ((((((min_dist @ capacity) @ c) @ s) @ t) = (((plus_plus @ nat) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p1)) @ (((plus_plus @ nat) @ (one_one @ nat)) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p2)))) => (((((min_dist @ capacity) @ c) @ s) @ u) = ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p1)))).
% 2.83/1.51 thf(fact_2__092_060open_062min__dist_As_At_A_061_Alength_Ap1_A_L_A_I1_A_L_Alength_Ap2_J_A_092_060Longrightarrow_062_Amin__dist_Au_At_A_061_A1_A_L_Alength_Ap2_092_060close_062, axiom, ((((((min_dist @ capacity) @ c) @ s) @ t) = (((plus_plus @ nat) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p1)) @ (((plus_plus @ nat) @ (one_one @ nat)) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p2)))) => (((((min_dist @ capacity) @ c) @ u) @ t) = (((plus_plus @ nat) @ (one_one @ nat)) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p2))))).
% 2.83/1.51 thf(fact_3__092_060open_062min__dist_Av_At_A_061_Alength_Ap2_092_060close_062, axiom, (((((min_dist @ capacity) @ c) @ v) @ t) = ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p2))).
% 2.83/1.51 thf(fact_4_MIN, axiom, (((((min_dist @ capacity) @ c) @ s) @ t) = ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p))).
% 2.83/1.51 thf(fact_5__092_060open_062u_A_092_060noteq_062_Av_092_060close_062, axiom, (u != v)).
% 2.83/1.51 thf(fact_6_MDSV, axiom, (((((min_dist @ capacity) @ c) @ s) @ v) = (((plus_plus @ nat) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p1)) @ (one_one @ nat)))).
% 2.83/1.51 thf(fact_7_min__dist__split_I2_J, axiom, (! [U:nat,D1:nat,W:nat,D2:nat,V:nat]: (((((((dist @ capacity) @ c) @ U) @ D1) @ W) => ((((((dist @ capacity) @ c) @ W) @ D2) @ V) => ((((((min_dist @ capacity) @ c) @ U) @ V) = (((plus_plus @ nat) @ D1) @ D2)) => (((((min_dist @ capacity) @ c) @ W) @ V) = D2))))))).
% 2.83/1.51 thf(fact_8_min__dist__split_I1_J, axiom, (! [U:nat,D1:nat,W:nat,D2:nat,V:nat]: (((((((dist @ capacity) @ c) @ U) @ D1) @ W) => ((((((dist @ capacity) @ c) @ W) @ D2) @ V) => ((((((min_dist @ capacity) @ c) @ U) @ V) = (((plus_plus @ nat) @ D1) @ D2)) => (((((min_dist @ capacity) @ c) @ U) @ W) = D1))))))).
% 2.83/1.51 thf(fact_9_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B2:A,C:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C)) = (B2 = C)))))))).
% 2.83/1.51 thf(fact_10_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A2:A,C:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B2 = C)))))))).
% 2.83/1.51 thf(fact_11_dist__trans, axiom, (! [U:nat,D1:nat,W:nat,D2:nat,V:nat]: (((((((dist @ capacity) @ c) @ U) @ D1) @ W) => ((((((dist @ capacity) @ c) @ W) @ D2) @ V) => (((((dist @ capacity) @ c) @ U) @ (((plus_plus @ nat) @ D1) @ D2)) @ V)))))).
% 2.83/1.51 thf(fact_12_assms_I3_J, axiom, (((((isPath @ capacity) @ c) @ s) @ p3) @ t)).
% 2.83/1.51 thf(fact_13_DISTS_I1_J, axiom, (((((dist @ capacity) @ c) @ s) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p1)) @ u)).
% 2.83/1.51 thf(fact_14_isPath__distD, axiom, (! [U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isPath @ capacity) @ c) @ U) @ P) @ V) => (((((dist @ capacity) @ c) @ U) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P)) @ V))))).
% 2.83/1.51 thf(fact_15_dist__def, axiom, (! [V:nat,D:nat,V2:nat]: (((((((dist @ capacity) @ c) @ V) @ D) @ V2) = (? [P2:(list @ ((product_prod @ nat) @ nat))]: (((((((isPath @ capacity) @ c) @ V) @ P2) @ V2) & (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P2) = D)))))))).
% 2.83/1.51 thf(fact_16_DISTS_I2_J, axiom, (((((dist @ capacity) @ c) @ u) @ (one_one @ nat)) @ v)).
% 2.83/1.51 thf(fact_17_P, axiom, (((((isPath @ capacity) @ c) @ s) @ p) @ t)).
% 2.83/1.51 thf(fact_18_DISTS_I3_J, axiom, (((((dist @ capacity) @ c) @ v) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p2)) @ t)).
% 2.83/1.51 thf(fact_19_assms_I1_J, axiom, (((((isShortestPath @ capacity) @ c) @ s) @ p) @ t)).
% 2.83/1.51 thf(fact_20_add__right__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A2:A,C:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C) @ A2)) => (B2 = C)))))))).
% 2.83/1.51 thf(fact_21_add__left__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A2:A,B2:A,C:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C)) => (B2 = C)))))))).
% 2.83/1.51 thf(fact_22_add_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [B2:A,A2:A,C:A]: (((((plus_plus @ A) @ B2) @ (((plus_plus @ A) @ A2) @ C)) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.83/1.51 thf(fact_23_add_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((plus_plus @ A) = (^ [A3:A,B3:A]: ((((plus_plus @ A) @ B3) @ A3)))))))).
% 2.83/1.51 thf(fact_24_add_Oright__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [B2:A,A2:A,C:A]: ((((((plus_plus @ A) @ B2) @ A2) = (((plus_plus @ A) @ C) @ A2)) = (B2 = C)))))))).
% 2.83/1.51 thf(fact_25_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A2:A,B2:A,C:A]: ((((((plus_plus @ A) @ A2) @ B2) = (((plus_plus @ A) @ A2) @ C)) = (B2 = C)))))))).
% 2.83/1.51 thf(fact_26_add_Oassoc, axiom, (! [A:$tType]: (((semigroup_add @ A) => (! [A2:A,B2:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.83/1.51 thf(fact_27_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B4:A,K:A,B2:A,A2:A]: (((B4 = (((plus_plus @ A) @ K) @ B2)) => ((((plus_plus @ A) @ A2) @ B4) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.83/1.51 thf(fact_28_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A4:A,K:A,A2:A,B2:A]: (((A4 = (((plus_plus @ A) @ K) @ A2)) => ((((plus_plus @ A) @ A4) @ B2) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A2) @ B2)))))))))).
% 2.83/1.51 thf(fact_29_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (K = L)) => ((((plus_plus @ A) @ I) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 2.83/1.51 thf(fact_30_ab__semigroup__add__class_Oadd__ac_I1_J, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A2:A,B2:A,C:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A2) @ B2)) @ C) = (((plus_plus @ A) @ A2) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.83/1.51 thf(fact_31_one__reorient, axiom, (! [A:$tType]: (((one @ A) => (! [X:A]: ((((one_one @ A) = X) = (X = (one_one @ A))))))))).
% 2.83/1.51 thf(fact_32_isShortestPath__min__dist__def, axiom, (! [U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isShortestPath @ capacity) @ c) @ U) @ P) @ V) = ((((((isPath @ capacity) @ c) @ U) @ P) @ V) & (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P) = ((((min_dist @ capacity) @ c) @ U) @ V))))))).
% 2.83/1.51 thf(fact_33_Graph_OisPath__distD, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isPath @ Capacity) @ C) @ U) @ P) @ V) => (((((dist @ Capacity) @ C) @ U) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P)) @ V)))))))).
% 2.83/1.51 thf(fact_34_Graph_Odist__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((dist @ Capacity) = (^ [C2:(((product_prod @ nat) @ nat) > Capacity),V3:nat,D3:nat,V4:nat]: ((? [P2:(list @ ((product_prod @ nat) @ nat))]: (((((((isPath @ Capacity) @ C2) @ V3) @ P2) @ V4) & (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P2) = D3))))))))))).
% 2.83/1.51 thf(fact_35_min__dist__minD, axiom, (! [V:nat,D:nat,V2:nat]: (((((((dist @ capacity) @ c) @ V) @ D) @ V2) => (((ord_less_eq @ nat) @ ((((min_dist @ capacity) @ c) @ V) @ V2)) @ D))))).
% 2.83/1.51 thf(fact_36_min__distI__eq, axiom, (! [V:nat,D:nat,V2:nat]: (((((((dist @ capacity) @ c) @ V) @ D) @ V2) => ((! [D4:nat]: (((((((dist @ capacity) @ c) @ V) @ D4) @ V2) => (((ord_less_eq @ nat) @ D) @ D4)))) => (((((min_dist @ capacity) @ c) @ V) @ V2) = D)))))).
% 2.83/1.51 thf(fact_37_Graph_Omin__dist__split_I1_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,D1:nat,W:nat,D2:nat,V:nat]: (((((((dist @ Capacity) @ C) @ U) @ D1) @ W) => ((((((dist @ Capacity) @ C) @ W) @ D2) @ V) => ((((((min_dist @ Capacity) @ C) @ U) @ V) = (((plus_plus @ nat) @ D1) @ D2)) => (((((min_dist @ Capacity) @ C) @ U) @ W) = D1)))))))))).
% 2.83/1.51 thf(fact_38_Graph_Omin__dist__split_I2_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,D1:nat,W:nat,D2:nat,V:nat]: (((((((dist @ Capacity) @ C) @ U) @ D1) @ W) => ((((((dist @ Capacity) @ C) @ W) @ D2) @ V) => ((((((min_dist @ Capacity) @ C) @ U) @ V) = (((plus_plus @ nat) @ D1) @ D2)) => (((((min_dist @ Capacity) @ C) @ W) @ V) = D2)))))))))).
% 2.83/1.51 thf(fact_39_min__dist__is__dist, axiom, (! [V:nat,V2:nat]: ((((((connected @ capacity) @ c) @ V) @ V2) => (((((dist @ capacity) @ c) @ V) @ ((((min_dist @ capacity) @ c) @ V) @ V2)) @ V2))))).
% 2.83/1.51 thf(fact_40_shortestPath__is__path, axiom, (! [U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isShortestPath @ capacity) @ c) @ U) @ P) @ V) => (((((isPath @ capacity) @ c) @ U) @ P) @ V))))).
% 2.83/1.51 thf(fact_41_assms_I4_J, axiom, (((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ v) @ u)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ p3))).
% 2.83/1.51 thf(fact_42_connected__def, axiom, (! [U:nat,V:nat]: ((((((connected @ capacity) @ c) @ U) @ V) = (? [P2:(list @ ((product_prod @ nat) @ nat))]: ((((((isPath @ capacity) @ c) @ U) @ P2) @ V))))))).
% 2.83/1.51 thf(fact_43_connected__by__dist, axiom, (! [V:nat,V2:nat]: ((((((connected @ capacity) @ c) @ V) @ V2) = (? [D3:nat]: ((((((dist @ capacity) @ c) @ V) @ D3) @ V2))))))).
% 2.83/1.51 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P3:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P3)) = (P3 @ A2))))).
% 2.83/1.51 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A4)))) = A4)))).
% 2.83/1.51 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P3:(A > $o),Q:(A > $o)]: (((! [X3:A]: (((P3 @ X3) = (Q @ X3)))) => (((collect @ A) @ P3) = ((collect @ A) @ Q)))))).
% 2.83/1.51 thf(fact_47_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X3:A]: (((F @ X3) = (G @ X3)))) => (F = G))))).
% 2.83/1.51 thf(fact_48_obtain__shortest__path, axiom, (! [U:nat,V:nat]: ((((((connected @ capacity) @ c) @ U) @ V) => (~ ((! [P4:(list @ ((product_prod @ nat) @ nat))]: ((~ ((((((isShortestPath @ capacity) @ c) @ U) @ P4) @ V))))))))))).
% 2.83/1.51 thf(fact_49_assms_I2_J, axiom, (((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ u) @ v)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ p))).
% 2.83/1.51 thf(fact_50_isPath__ex__edge1, axiom, (! [U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat]: (((((((isPath @ capacity) @ c) @ U) @ P) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((U1 != U) => (? [U2:nat]: ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U2) @ U1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)))))))))).
% 2.83/1.51 thf(fact_51_isPath__ex__edge2, axiom, (! [U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat]: (((((((isPath @ capacity) @ c) @ U) @ P) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((V1 != V) => (? [V22:nat]: ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ V1) @ V22)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)))))))))).
% 2.83/1.51 thf(fact_52_min__dist__le, axiom, (! [Src:nat,V:nat,D5:nat]: ((((((connected @ capacity) @ c) @ Src) @ V) => ((((ord_less_eq @ nat) @ D5) @ ((((min_dist @ capacity) @ c) @ Src) @ V)) => (? [V5:nat]: ((((((connected @ capacity) @ c) @ Src) @ V5) & (((((min_dist @ capacity) @ c) @ Src) @ V5) = D5))))))))).
% 2.83/1.51 thf(fact_53_min__distI2, axiom, (! [V:nat,V2:nat,Q:(nat > $o)]: ((((((connected @ capacity) @ c) @ V) @ V2) => ((! [D6:nat]: (((((((dist @ capacity) @ c) @ V) @ D6) @ V2) => ((! [D7:nat]: (((((((dist @ capacity) @ c) @ V) @ D7) @ V2) => (((ord_less_eq @ nat) @ D6) @ D7)))) => (Q @ D6))))) => (Q @ ((((min_dist @ capacity) @ c) @ V) @ V2))))))).
% 2.83/1.51 thf(fact_54_isShortestPath__def, axiom, (! [U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isShortestPath @ capacity) @ c) @ U) @ P) @ V) = ((((((isPath @ capacity) @ c) @ U) @ P) @ V) & (! [P5:(list @ ((product_prod @ nat) @ nat))]: (((((((isPath @ capacity) @ c) @ U) @ P5) @ V) => (((ord_less_eq @ nat) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P)) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5)))))))))).
% 2.83/1.51 thf(fact_55_isShortestPath__level__edge_I1_J, axiom, (! [S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat,U:nat,V:nat]: (((((((isShortestPath @ capacity) @ c) @ S) @ P) @ T2) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((((connected @ capacity) @ c) @ S) @ U)))))).
% 2.83/1.51 thf(fact_56_isShortestPath__level__edge_I2_J, axiom, (! [S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat,U:nat,V:nat]: (((((((isShortestPath @ capacity) @ c) @ S) @ P) @ T2) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((((connected @ capacity) @ c) @ U) @ V)))))).
% 2.83/1.51 thf(fact_57_isShortestPath__level__edge_I3_J, axiom, (! [S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat,U:nat,V:nat]: (((((((isShortestPath @ capacity) @ c) @ S) @ P) @ T2) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((((connected @ capacity) @ c) @ V) @ T2)))))).
% 2.83/1.51 thf(fact_58_add__le__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.83/1.51 thf(fact_59_add__le__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C)) = (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.83/1.51 thf(fact_60_isShortestPath__level__edge_I6_J, axiom, (! [S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat,U:nat,V:nat]: (((((((isShortestPath @ capacity) @ c) @ S) @ P) @ T2) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => (((((min_dist @ capacity) @ c) @ S) @ T2) = (((plus_plus @ nat) @ (((plus_plus @ nat) @ ((((min_dist @ capacity) @ c) @ S) @ U)) @ (one_one @ nat))) @ ((((min_dist @ capacity) @ c) @ V) @ T2)))))))).
% 2.83/1.51 thf(fact_61_isShortestPath__level__edge_I5_J, axiom, (! [S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat,U:nat,V:nat]: (((((((isShortestPath @ capacity) @ c) @ S) @ P) @ T2) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => (((((min_dist @ capacity) @ c) @ U) @ T2) = (((plus_plus @ nat) @ (one_one @ nat)) @ ((((min_dist @ capacity) @ c) @ V) @ T2)))))))).
% 2.83/1.51 thf(fact_62_isShortestPath__level__edge_I4_J, axiom, (! [S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat,U:nat,V:nat]: (((((((isShortestPath @ capacity) @ c) @ S) @ P) @ T2) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => (((((min_dist @ capacity) @ c) @ S) @ V) = (((plus_plus @ nat) @ ((((min_dist @ capacity) @ c) @ S) @ U)) @ (one_one @ nat)))))))).
% 2.83/1.51 thf(fact_63_connected__refl, axiom, (! [V:nat]: (((((connected @ capacity) @ c) @ V) @ V)))).
% 2.83/1.51 thf(fact_64_connected__distI, axiom, (! [V:nat,D:nat,V2:nat]: (((((((dist @ capacity) @ c) @ V) @ D) @ V2) => ((((connected @ capacity) @ c) @ V) @ V2))))).
% 2.83/1.51 thf(fact_65_Graph_OisPath__ex__edge1, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat]: (((((((isPath @ Capacity) @ C) @ U) @ P) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((U1 != U) => (? [U2:nat]: ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U2) @ U1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P))))))))))))).
% 2.83/1.51 thf(fact_66_Graph_OisPath__ex__edge2, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat]: (((((((isPath @ Capacity) @ C) @ U) @ P) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((V1 != V) => (? [V22:nat]: ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ V1) @ V22)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P))))))))))))).
% 2.83/1.51 thf(fact_67_Graph_Oobtain__shortest__path, axiom, (! [Capacity:$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)))))))))))))).
% 2.83/1.51 thf(fact_68_Graph_OisShortestPath_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isShortestPath @ Capacity) = (isShortestPath @ Capacity)))))).
% 2.83/1.51 thf(fact_69_Graph_Oconnected__refl, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),V:nat]: (((((connected @ Capacity) @ C) @ V) @ V))))))).
% 2.83/1.51 thf(fact_70_Graph_Oconnected_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((connected @ Capacity) = (connected @ Capacity)))))).
% 2.83/1.51 thf(fact_71_Graph_Omin__dist__le, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),Src:nat,V:nat,D5:nat]: ((((((connected @ Capacity) @ C) @ Src) @ V) => ((((ord_less_eq @ nat) @ D5) @ ((((min_dist @ Capacity) @ C) @ Src) @ V)) => (? [V5:nat]: ((((((connected @ Capacity) @ C) @ Src) @ V5) & (((((min_dist @ Capacity) @ C) @ Src) @ V5) = D5)))))))))))).
% 2.83/1.51 thf(fact_72_Graph_OisShortestPath__level__edge_I1_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat,U:nat,V:nat]: (((((((isShortestPath @ Capacity) @ C) @ S) @ P) @ T2) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((((connected @ Capacity) @ C) @ S) @ U))))))))).
% 2.83/1.51 thf(fact_73_Graph_OisShortestPath__level__edge_I2_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat,U:nat,V:nat]: (((((((isShortestPath @ Capacity) @ C) @ S) @ P) @ T2) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((((connected @ Capacity) @ C) @ U) @ V))))))))).
% 2.83/1.51 thf(fact_74_Graph_OisShortestPath__level__edge_I3_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat,U:nat,V:nat]: (((((((isShortestPath @ Capacity) @ C) @ S) @ P) @ T2) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((((connected @ Capacity) @ C) @ V) @ T2))))))))).
% 2.83/1.51 thf(fact_75_Graph_Oconnected__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((connected @ Capacity) = (^ [C2:(((product_prod @ nat) @ nat) > Capacity),U3:nat,V3:nat]: ((? [P2:(list @ ((product_prod @ nat) @ nat))]: ((((((isPath @ Capacity) @ C2) @ U3) @ P2) @ V3)))))))))).
% 2.83/1.51 thf(fact_76_Graph_Oconnected__by__dist, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((connected @ Capacity) = (^ [C2:(((product_prod @ nat) @ nat) > Capacity),V3:nat,V4:nat]: ((? [D3:nat]: ((((((dist @ Capacity) @ C2) @ V3) @ D3) @ V4)))))))))).
% 2.83/1.51 thf(fact_77_Graph_Oconnected__distI, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),V:nat,D:nat,V2:nat]: (((((((dist @ Capacity) @ C) @ V) @ D) @ V2) => ((((connected @ Capacity) @ C) @ V) @ V2)))))))).
% 2.83/1.51 thf(fact_78_Graph_OshortestPath__is__path, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isShortestPath @ Capacity) @ C) @ U) @ P) @ V) => (((((isPath @ Capacity) @ C) @ U) @ P) @ V)))))))).
% 2.83/1.51 thf(fact_79_Graph_OisShortestPath__level__edge_I6_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat,U:nat,V:nat]: (((((((isShortestPath @ Capacity) @ C) @ S) @ P) @ T2) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => (((((min_dist @ Capacity) @ C) @ S) @ T2) = (((plus_plus @ nat) @ (((plus_plus @ nat) @ ((((min_dist @ Capacity) @ C) @ S) @ U)) @ (one_one @ nat))) @ ((((min_dist @ Capacity) @ C) @ V) @ T2))))))))))).
% 2.83/1.51 thf(fact_80_Graph_OisShortestPath__level__edge_I5_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat,U:nat,V:nat]: (((((((isShortestPath @ Capacity) @ C) @ S) @ P) @ T2) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => (((((min_dist @ Capacity) @ C) @ U) @ T2) = (((plus_plus @ nat) @ (one_one @ nat)) @ ((((min_dist @ Capacity) @ C) @ V) @ T2))))))))))).
% 2.83/1.51 thf(fact_81_Graph_OisShortestPath__level__edge_I4_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat,U:nat,V:nat]: (((((((isShortestPath @ Capacity) @ C) @ S) @ P) @ T2) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => (((((min_dist @ Capacity) @ C) @ S) @ V) = (((plus_plus @ nat) @ ((((min_dist @ Capacity) @ C) @ S) @ U)) @ (one_one @ nat))))))))))).
% 2.83/1.51 thf(fact_82_Graph_Omin__distI2, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),V:nat,V2:nat,Q:(nat > $o)]: ((((((connected @ Capacity) @ C) @ V) @ V2) => ((! [D6:nat]: (((((((dist @ Capacity) @ C) @ V) @ D6) @ V2) => ((! [D7:nat]: (((((((dist @ Capacity) @ C) @ V) @ D7) @ V2) => (((ord_less_eq @ nat) @ D6) @ D7)))) => (Q @ D6))))) => (Q @ ((((min_dist @ Capacity) @ C) @ V) @ V2)))))))))).
% 2.83/1.51 thf(fact_83_Graph_OisShortestPath__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isShortestPath @ Capacity) = (^ [C2:(((product_prod @ nat) @ nat) > Capacity),U3:nat,P2:(list @ ((product_prod @ nat) @ nat)),V3:nat]: (((((((isPath @ Capacity) @ C2) @ U3) @ P2) @ V3) & (! [P5:(list @ ((product_prod @ nat) @ nat))]: (((((((isPath @ Capacity) @ C2) @ U3) @ P5) @ V3) => (((ord_less_eq @ nat) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P2)) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P5))))))))))))).
% 2.83/1.51 thf(fact_84_add__mono__thms__linordered__semiring_I3_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (K = L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.83/1.51 thf(fact_85_add__mono__thms__linordered__semiring_I2_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.83/1.51 thf(fact_86_add__mono__thms__linordered__semiring_I1_J, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.83/1.51 thf(fact_87_add__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B2:A,C:A,D:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C) @ D) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ D)))))))))).
% 2.83/1.51 thf(fact_88_add__left__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2))))))))).
% 2.83/1.51 thf(fact_89_less__eqE, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (~ ((! [C3:A]: ((B2 != (((plus_plus @ A) @ A2) @ C3))))))))))))).
% 2.83/1.51 thf(fact_90_add__right__mono, axiom, (! [A:$tType]: (((ordere779506340up_add @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.83/1.51 thf(fact_91_le__iff__add, axiom, (! [A:$tType]: (((canoni770627133id_add @ A) => ((ord_less_eq @ A) = (^ [A3:A,B3:A]: ((? [C2:A]: ((B3 = (((plus_plus @ A) @ A3) @ C2))))))))))).
% 2.83/1.51 thf(fact_92_add__le__imp__le__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2)) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.83/1.51 thf(fact_93_add__le__imp__le__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C)) => (((ord_less_eq @ A) @ A2) @ B2)))))))).
% 2.83/1.51 thf(fact_94_Graph_Omin__dist__minD, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),V:nat,D:nat,V2:nat]: (((((((dist @ Capacity) @ C) @ V) @ D) @ V2) => (((ord_less_eq @ nat) @ ((((min_dist @ Capacity) @ C) @ V) @ V2)) @ D)))))))).
% 2.83/1.51 thf(fact_95_Graph_Omin__distI__eq, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),V:nat,D:nat,V2:nat]: (((((((dist @ Capacity) @ C) @ V) @ D) @ V2) => ((! [D4:nat]: (((((((dist @ Capacity) @ C) @ V) @ D4) @ V2) => (((ord_less_eq @ nat) @ D) @ D4)))) => (((((min_dist @ Capacity) @ C) @ V) @ V2) = D))))))))).
% 2.83/1.51 thf(fact_96_Graph_Omin__dist__is__dist, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),V:nat,V2:nat]: ((((((connected @ Capacity) @ C) @ V) @ V2) => (((((dist @ Capacity) @ C) @ V) @ ((((min_dist @ Capacity) @ C) @ V) @ V2)) @ V2)))))))).
% 2.83/1.51 thf(fact_97_Graph_OisShortestPath__min__dist__def, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isShortestPath @ Capacity) = (^ [C2:(((product_prod @ nat) @ nat) > Capacity),U3:nat,P2:(list @ ((product_prod @ nat) @ nat)),V3:nat]: (((((((isPath @ Capacity) @ C2) @ U3) @ P2) @ V3) & (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P2) = ((((min_dist @ Capacity) @ C2) @ U3) @ V3)))))))))).
% 2.83/1.51 thf(fact_98_Graph_OisPath_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isPath @ Capacity) = (isPath @ Capacity)))))).
% 2.83/1.51 thf(fact_99_Graph_Omin__dist_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((min_dist @ Capacity) = (min_dist @ Capacity)))))).
% 2.83/1.51 thf(fact_100_Graph_Odist_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((dist @ Capacity) = (dist @ Capacity)))))).
% 2.83/1.51 thf(fact_101_Graph_Odist__trans, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,D1:nat,W:nat,D2:nat,V:nat]: (((((((dist @ Capacity) @ C) @ U) @ D1) @ W) => ((((((dist @ Capacity) @ C) @ W) @ D2) @ V) => (((((dist @ Capacity) @ C) @ U) @ (((plus_plus @ nat) @ D1) @ D2)) @ V))))))))).
% 2.83/1.51 thf(fact_102_nat__add__left__cancel__le, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N)) = (((ord_less_eq @ nat) @ M) @ N))))).
% 2.83/1.51 thf(fact_103_isShortestPath__alt, axiom, (! [U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat]: (((((((isShortestPath @ capacity) @ c) @ U) @ P) @ V) = ((((((isSimplePath @ capacity) @ c) @ U) @ P) @ V) & (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P) = ((((min_dist @ capacity) @ c) @ U) @ V))))))).
% 2.83/1.51 thf(fact_104_min__dist__less, axiom, (! [Src:nat,V:nat,D:nat,D5:nat]: ((((((connected @ capacity) @ c) @ Src) @ V) => ((((((min_dist @ capacity) @ c) @ Src) @ V) = D) => ((((ord_less @ nat) @ D5) @ D) => (? [V5:nat]: ((((((connected @ capacity) @ c) @ Src) @ V5) & (((((min_dist @ capacity) @ c) @ Src) @ V5) = D5)))))))))).
% 2.83/1.51 thf(fact_105_isSPath__no__selfloop, axiom, (! [U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat]: (((((((isSimplePath @ capacity) @ c) @ U) @ P) @ V) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ U1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)))))))).
% 2.83/1.51 thf(fact_106_isSPath__sg__incoming, axiom, (! [U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat,U22:nat]: (((((((isSimplePath @ capacity) @ c) @ U) @ P) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((U1 != U22) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U22) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)))))))))).
% 2.83/1.51 thf(fact_107_isSPath__sg__outgoing, axiom, (! [U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat,V23:nat]: (((((((isSimplePath @ capacity) @ c) @ U) @ P) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((V1 != V23) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V23)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)))))))))).
% 2.83/1.51 thf(fact_108_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A5:A,B5:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A5) @ B5)) = ((A2 = A5) & (B2 = B5)))))).
% 2.83/1.51 thf(fact_109_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X22 = Y2)))))).
% 2.83/1.51 thf(fact_110_isSPath__pathLE, axiom, (! [S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat]: (((((((isPath @ capacity) @ c) @ S) @ P) @ T2) => (? [P6:(list @ ((product_prod @ nat) @ nat))]: ((((((isSimplePath @ capacity) @ c) @ S) @ P6) @ T2))))))).
% 2.83/1.51 thf(fact_111_shortestPath__is__simple, axiom, (! [S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat]: (((((((isShortestPath @ capacity) @ c) @ S) @ P) @ T2) => (((((isSimplePath @ capacity) @ c) @ S) @ P) @ T2))))).
% 2.83/1.51 thf(fact_112_add__less__cancel__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C)) = (((ord_less @ A) @ A2) @ B2)))))))).
% 2.83/1.51 thf(fact_113_add__less__cancel__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2)) = (((ord_less @ A) @ A2) @ B2)))))))).
% 2.83/1.51 thf(fact_114_nat__add__left__cancel__less, axiom, (! [K:nat,M:nat,N:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ K) @ M)) @ (((plus_plus @ nat) @ K) @ N)) = (((ord_less @ nat) @ M) @ N))))).
% 2.83/1.51 thf(fact_115_Graph_OisSimplePath_Ocong, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isSimplePath @ Capacity) = (isSimplePath @ Capacity)))))).
% 2.83/1.51 thf(fact_116_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))))))).
% 2.83/1.51 thf(fact_117_le__neq__implies__less, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((M != N) => (((ord_less @ nat) @ M) @ N)))))).
% 2.83/1.51 thf(fact_118_less__or__eq__imp__le, axiom, (! [M:nat,N:nat]: ((((((ord_less @ nat) @ M) @ N) | (M = N)) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.83/1.51 thf(fact_119_le__eq__less__or__eq, axiom, ((ord_less_eq @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less @ nat) @ M2) @ N2) | (M2 = N2)))))).
% 2.83/1.51 thf(fact_120_less__imp__le__nat, axiom, (! [M:nat,N:nat]: (((((ord_less @ nat) @ M) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.83/1.51 thf(fact_121_nat__less__le, axiom, ((ord_less @ nat) = (^ [M2:nat,N2:nat]: (((((ord_less_eq @ nat) @ M2) @ N2) & (M2 != N2)))))).
% 2.83/1.51 thf(fact_122_add__lessD1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ K) => (((ord_less @ nat) @ I) @ K))))).
% 2.83/1.51 thf(fact_123_add__less__mono, axiom, (! [I:nat,J:nat,K:nat,L:nat]: (((((ord_less @ nat) @ I) @ J) => ((((ord_less @ nat) @ K) @ L) => (((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ L))))))).
% 2.83/1.51 thf(fact_124_not__add__less1, axiom, (! [I:nat,J:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ I)))))).
% 2.83/1.51 thf(fact_125_not__add__less2, axiom, (! [J:nat,I:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ J) @ I)) @ I)))))).
% 2.83/1.51 thf(fact_126_add__less__mono1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ K)))))).
% 2.83/1.51 thf(fact_127_trans__less__add1, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ I) @ (((plus_plus @ nat) @ J) @ M)))))).
% 2.83/1.51 thf(fact_128_trans__less__add2, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ I) @ (((plus_plus @ nat) @ M) @ J)))))).
% 2.83/1.51 thf(fact_129_less__add__eq__less, axiom, (! [K:nat,L:nat,M:nat,N:nat]: (((((ord_less @ nat) @ K) @ L) => (((((plus_plus @ nat) @ M) @ L) = (((plus_plus @ nat) @ K) @ N)) => (((ord_less @ nat) @ M) @ N)))))).
% 2.83/1.51 thf(fact_130_nat__neq__iff, axiom, (! [M:nat,N:nat]: (((M != N) = ((((ord_less @ nat) @ M) @ N) | (((ord_less @ nat) @ N) @ M)))))).
% 2.83/1.51 thf(fact_131_less__not__refl, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.83/1.51 thf(fact_132_less__not__refl2, axiom, (! [N:nat,M:nat]: (((((ord_less @ nat) @ N) @ M) => (M != N))))).
% 2.83/1.51 thf(fact_133_less__not__refl3, axiom, (! [S:nat,T2:nat]: (((((ord_less @ nat) @ S) @ T2) => (S != T2))))).
% 2.83/1.51 thf(fact_134_measure__induct, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P3:(A > $o),A2:A]: (((! [X3:A]: (((! [Y:A]: (((((ord_less @ B) @ (F @ Y)) @ (F @ X3)) => (P3 @ Y)))) => (P3 @ X3)))) => (P3 @ A2)))))))).
% 2.83/1.51 thf(fact_135_less__irrefl__nat, axiom, (! [N:nat]: ((~ ((((ord_less @ nat) @ N) @ N)))))).
% 2.83/1.51 thf(fact_136_nat__less__induct, axiom, (! [P3:(nat > $o),N:nat]: (((! [N3:nat]: (((! [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) => (P3 @ M3)))) => (P3 @ N3)))) => (P3 @ N))))).
% 2.83/1.51 thf(fact_137_infinite__descent, axiom, (! [P3:(nat > $o),N:nat]: (((! [N3:nat]: (((~ ((P3 @ N3))) => (? [M3:nat]: (((((ord_less @ nat) @ M3) @ N3) & (~ ((P3 @ M3))))))))) => (P3 @ N))))).
% 2.83/1.51 thf(fact_138_linorder__neqE__nat, axiom, (! [X:nat,Y3:nat]: (((X != Y3) => ((~ ((((ord_less @ nat) @ X) @ Y3))) => (((ord_less @ nat) @ Y3) @ X)))))).
% 2.83/1.51 thf(fact_139_measure__induct__rule, axiom, (! [B:$tType,A:$tType]: (((wellorder @ B) => (! [F:(A > B),P3:(A > $o),A2:A]: (((! [X3:A]: (((! [Y:A]: (((((ord_less @ B) @ (F @ Y)) @ (F @ X3)) => (P3 @ Y)))) => (P3 @ X3)))) => (P3 @ A2)))))))).
% 2.83/1.51 thf(fact_140_infinite__descent__measure, axiom, (! [A:$tType,P3:(A > $o),V6:(A > nat),X:A]: (((! [X3:A]: (((~ ((P3 @ X3))) => (? [Y:A]: (((((ord_less @ nat) @ (V6 @ Y)) @ (V6 @ X3)) & (~ ((P3 @ Y))))))))) => (P3 @ X))))).
% 2.83/1.51 thf(fact_141_add__less__imp__less__right, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [A2:A,C:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C)) => (((ord_less @ A) @ A2) @ B2)))))))).
% 2.83/1.51 thf(fact_142_add__less__imp__less__left, axiom, (! [A:$tType]: (((ordere236663937imp_le @ A) => (! [C:A,A2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2)) => (((ord_less @ A) @ A2) @ B2)))))))).
% 2.83/1.51 thf(fact_143_add__strict__right__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ C))))))))).
% 2.83/1.51 thf(fact_144_add__strict__left__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ (((plus_plus @ A) @ C) @ A2)) @ (((plus_plus @ A) @ C) @ B2))))))))).
% 2.83/1.51 thf(fact_145_add__strict__mono, axiom, (! [A:$tType]: (((strict2144017051up_add @ A) => (! [A2:A,B2:A,C:A,D:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ C) @ D) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ D)))))))))).
% 2.83/1.51 thf(fact_146_add__mono__thms__linordered__field_I1_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (K = L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.83/1.51 thf(fact_147_add__mono__thms__linordered__field_I2_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.83/1.51 thf(fact_148_add__mono__thms__linordered__field_I5_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.83/1.51 thf(fact_149_mono__nat__linear__lb, axiom, (! [F:(nat > nat),M:nat,K:nat]: (((! [M4:nat,N3:nat]: (((((ord_less @ nat) @ M4) @ N3) => (((ord_less @ nat) @ (F @ M4)) @ (F @ N3))))) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (F @ M)) @ K)) @ (F @ (((plus_plus @ nat) @ M) @ K))))))).
% 2.83/1.51 thf(fact_150_Graph_OisSPath__pathLE, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat]: (((((((isPath @ Capacity) @ C) @ S) @ P) @ T2) => (? [P6:(list @ ((product_prod @ nat) @ nat))]: ((((((isSimplePath @ Capacity) @ C) @ S) @ P6) @ T2)))))))))).
% 2.83/1.51 thf(fact_151_Graph_OshortestPath__is__simple, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat]: (((((((isShortestPath @ Capacity) @ C) @ S) @ P) @ T2) => (((((isSimplePath @ Capacity) @ C) @ S) @ P) @ T2)))))))).
% 2.83/1.51 thf(fact_152_add__mono__thms__linordered__field_I4_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.83/1.51 thf(fact_153_add__mono__thms__linordered__field_I3_J, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 2.83/1.51 thf(fact_154_add__le__less__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A2:A,B2:A,C:A,D:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less @ A) @ C) @ D) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ D)))))))))).
% 2.83/1.51 thf(fact_155_add__less__le__mono, axiom, (! [A:$tType]: (((ordere223160158up_add @ A) => (! [A2:A,B2:A,C:A,D:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ C) @ D) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ C)) @ (((plus_plus @ A) @ B2) @ D)))))))))).
% 2.83/1.51 thf(fact_156_Graph_Omin__dist__less, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),Src:nat,V:nat,D:nat,D5:nat]: ((((((connected @ Capacity) @ C) @ Src) @ V) => ((((((min_dist @ Capacity) @ C) @ Src) @ V) = D) => ((((ord_less @ nat) @ D5) @ D) => (? [V5:nat]: ((((((connected @ Capacity) @ C) @ Src) @ V5) & (((((min_dist @ Capacity) @ C) @ Src) @ V5) = D5))))))))))))).
% 2.83/1.51 thf(fact_157_Graph_OisSPath__no__selfloop, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat]: (((((((isSimplePath @ Capacity) @ C) @ U) @ P) @ V) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ U1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P))))))))))).
% 2.83/1.51 thf(fact_158_Graph_OisSPath__sg__incoming, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat,U22:nat]: (((((((isSimplePath @ Capacity) @ C) @ U) @ P) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((U1 != U22) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U22) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P))))))))))))).
% 2.83/1.51 thf(fact_159_Graph_OisSPath__sg__outgoing, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,P:(list @ ((product_prod @ nat) @ nat)),V:nat,U1:nat,V1:nat,V23:nat]: (((((((isSimplePath @ Capacity) @ C) @ U) @ P) @ V) => ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V1)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => ((V1 != V23) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ U1) @ V23)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P))))))))))))).
% 2.83/1.51 thf(fact_160_surj__pair, axiom, (! [A:$tType,B:$tType,P:((product_prod @ A) @ B)]: ((? [X3:A,Y4:B]: ((P = ((((product_Pair @ A) @ B) @ X3) @ Y4))))))).
% 2.83/1.51 thf(fact_161_prod__cases, axiom, (! [B:$tType,A:$tType,P3:(((product_prod @ A) @ B) > $o),P:((product_prod @ A) @ B)]: (((! [A6:A,B6:B]: ((P3 @ ((((product_Pair @ A) @ B) @ A6) @ B6)))) => (P3 @ P))))).
% 2.83/1.51 thf(fact_162_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B2:B,A5:A,B5:B]: (((((((product_Pair @ A) @ B) @ A2) @ B2) = ((((product_Pair @ A) @ B) @ A5) @ B5)) => (~ (((A2 = A5) => (B2 != B5)))))))).
% 2.83/1.51 thf(fact_163_prod__cases3, axiom, (! [A:$tType,B:$tType,C4:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ C4))]: ((~ ((! [A6:A,B6:B,C3:C4]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ C4)) @ A6) @ ((((product_Pair @ B) @ C4) @ B6) @ C3)))))))))).
% 2.83/1.51 thf(fact_164_prod__cases4, axiom, (! [A:$tType,B:$tType,C4:$tType,D8:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ D8)))]: ((~ ((! [A6:A,B6:B,C3:C4,D6:D8]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ D8))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C4) @ D8)) @ B6) @ ((((product_Pair @ C4) @ D8) @ C3) @ D6))))))))))).
% 2.83/1.51 thf(fact_165_prod__cases5, axiom, (! [A:$tType,B:$tType,C4:$tType,D8:$tType,E:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ E))))]: ((~ ((! [A6:A,B6:B,C3:C4,D6:D8,E2:E]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ E)))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ E))) @ B6) @ ((((product_Pair @ C4) @ ((product_prod @ D8) @ E)) @ C3) @ ((((product_Pair @ D8) @ E) @ D6) @ E2)))))))))))).
% 2.83/1.51 thf(fact_166_prod__cases6, axiom, (! [A:$tType,B:$tType,C4:$tType,D8:$tType,E:$tType,F2:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ F2)))))]: ((~ ((! [A6:A,B6:B,C3:C4,D6:D8,E2:E,F3:F2]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ F2))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ F2)))) @ B6) @ ((((product_Pair @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ F2))) @ C3) @ ((((product_Pair @ D8) @ ((product_prod @ E) @ F2)) @ D6) @ ((((product_Pair @ E) @ F2) @ E2) @ F3))))))))))))).
% 2.83/1.51 thf(fact_167_prod__cases7, axiom, (! [A:$tType,B:$tType,C4:$tType,D8:$tType,E:$tType,F2:$tType,G2:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))))]: ((~ ((! [A6:A,B6:B,C3:C4,D6:D8,E2:E,F3:F2,G3:G2]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))) @ B6) @ ((((product_Pair @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))) @ C3) @ ((((product_Pair @ D8) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))) @ D6) @ ((((product_Pair @ E) @ ((product_prod @ F2) @ G2)) @ E2) @ ((((product_Pair @ F2) @ G2) @ F3) @ G3)))))))))))))).
% 2.83/1.51 thf(fact_168_prod__induct3, axiom, (! [C4:$tType,B:$tType,A:$tType,P3:(((product_prod @ A) @ ((product_prod @ B) @ C4)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C4))]: (((! [A6:A,B6:B,C3:C4]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ C4)) @ A6) @ ((((product_Pair @ B) @ C4) @ B6) @ C3))))) => (P3 @ X))))).
% 2.83/1.51 thf(fact_169_prod__induct4, axiom, (! [D8:$tType,C4:$tType,B:$tType,A:$tType,P3:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ D8))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ D8)))]: (((! [A6:A,B6:B,C3:C4,D6:D8]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ D8))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C4) @ D8)) @ B6) @ ((((product_Pair @ C4) @ D8) @ C3) @ D6)))))) => (P3 @ X))))).
% 2.83/1.51 thf(fact_170_prod__induct5, axiom, (! [E:$tType,D8:$tType,C4:$tType,B:$tType,A:$tType,P3:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ E))))]: (((! [A6:A,B6:B,C3:C4,D6:D8,E2:E]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ E)))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ E))) @ B6) @ ((((product_Pair @ C4) @ ((product_prod @ D8) @ E)) @ C3) @ ((((product_Pair @ D8) @ E) @ D6) @ E2))))))) => (P3 @ X))))).
% 2.83/1.51 thf(fact_171_prod__induct6, axiom, (! [F2:$tType,E:$tType,D8:$tType,C4:$tType,B:$tType,A:$tType,P3:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ F2))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ F2)))))]: (((! [A6:A,B6:B,C3:C4,D6:D8,E2:E,F3:F2]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ F2))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ F2)))) @ B6) @ ((((product_Pair @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ F2))) @ C3) @ ((((product_Pair @ D8) @ ((product_prod @ E) @ F2)) @ D6) @ ((((product_Pair @ E) @ F2) @ E2) @ F3)))))))) => (P3 @ X))))).
% 2.83/1.51 thf(fact_172_prod__induct7, axiom, (! [G2:$tType,F2:$tType,E:$tType,D8:$tType,C4:$tType,B:$tType,A:$tType,P3:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))))]: (((! [A6:A,B6:B,C3:C4,D6:D8,E2:E,F3:F2,G3:G2]: ((P3 @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))))) @ A6) @ ((((product_Pair @ B) @ ((product_prod @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))))) @ B6) @ ((((product_Pair @ C4) @ ((product_prod @ D8) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2)))) @ C3) @ ((((product_Pair @ D8) @ ((product_prod @ E) @ ((product_prod @ F2) @ G2))) @ D6) @ ((((product_Pair @ E) @ ((product_prod @ F2) @ G2)) @ E2) @ ((((product_Pair @ F2) @ G2) @ F3) @ G3))))))))) => (P3 @ X))))).
% 2.83/1.51 thf(fact_173_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y3:((product_prod @ A) @ B)]: ((~ ((! [A6:A,B6:B]: ((Y3 != ((((product_Pair @ A) @ B) @ A6) @ B6))))))))).
% 2.83/1.51 thf(fact_174_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P3:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A6:A,B6:B]: ((P3 @ ((((product_Pair @ A) @ B) @ A6) @ B6)))) => (P3 @ Prod))))).
% 2.83/1.51 thf(fact_175_size__neq__size__imp__neq, axiom, (! [A:$tType]: (((size @ A) => (! [X:A,Y3:A]: (((((size_size @ A) @ X) != ((size_size @ A) @ Y3)) => (X != Y3)))))))).
% 2.83/1.51 thf(fact_176_le__refl, axiom, (! [N:nat]: ((((ord_less_eq @ nat) @ N) @ N)))).
% 2.83/1.51 thf(fact_177_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)))))).
% 2.83/1.51 thf(fact_178_eq__imp__le, axiom, (! [M:nat,N:nat]: (((M = N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.83/1.51 thf(fact_179_le__antisym, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) => ((((ord_less_eq @ nat) @ N) @ M) => (M = N)))))).
% 2.83/1.51 thf(fact_180_nat__le__linear, axiom, (! [M:nat,N:nat]: (((((ord_less_eq @ nat) @ M) @ N) | (((ord_less_eq @ nat) @ N) @ M))))).
% 2.83/1.51 thf(fact_181_Nat_Oex__has__greatest__nat, axiom, (! [P3:(nat > $o),K:nat,B2:nat]: (((P3 @ K) => ((! [Y4:nat]: (((P3 @ Y4) => (((ord_less_eq @ nat) @ Y4) @ B2)))) => (? [X3:nat]: (((P3 @ X3) & (! [Y:nat]: (((P3 @ Y) => (((ord_less_eq @ nat) @ Y) @ X3)))))))))))).
% 2.83/1.51 thf(fact_182_Graph_OisShortestPath__alt, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => ((isShortestPath @ Capacity) = (^ [C2:(((product_prod @ nat) @ nat) > Capacity),U3:nat,P2:(list @ ((product_prod @ nat) @ nat)),V3:nat]: (((((((isSimplePath @ Capacity) @ C2) @ U3) @ P2) @ V3) & (((size_size @ (list @ ((product_prod @ nat) @ nat))) @ P2) = ((((min_dist @ Capacity) @ C2) @ U3) @ V3)))))))))).
% 2.83/1.51 thf(fact_183_nat__le__iff__add, axiom, ((ord_less_eq @ nat) = (^ [M2:nat,N2:nat]: ((? [K2:nat]: ((N2 = (((plus_plus @ nat) @ M2) @ K2)))))))).
% 2.83/1.51 thf(fact_184_trans__le__add2, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ I) @ (((plus_plus @ nat) @ M) @ J)))))).
% 2.83/1.51 thf(fact_185_trans__le__add1, axiom, (! [I:nat,J:nat,M:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ I) @ (((plus_plus @ nat) @ J) @ M)))))).
% 2.83/1.51 thf(fact_186_add__le__mono1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ K)))))).
% 2.83/1.51 thf(fact_187_add__le__mono, axiom, (! [I:nat,J:nat,K:nat,L:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ K) @ L) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ L))))))).
% 2.83/1.51 thf(fact_188_le__Suc__ex, axiom, (! [K:nat,L:nat]: (((((ord_less_eq @ nat) @ K) @ L) => (? [N3:nat]: ((L = (((plus_plus @ nat) @ K) @ N3)))))))).
% 2.83/1.51 thf(fact_189_add__leD2, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ N) => (((ord_less_eq @ nat) @ K) @ N))))).
% 2.83/1.51 thf(fact_190_add__leD1, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ N) => (((ord_less_eq @ nat) @ M) @ N))))).
% 2.83/1.51 thf(fact_191_le__add2, axiom, (! [N:nat,M:nat]: ((((ord_less_eq @ nat) @ N) @ (((plus_plus @ nat) @ M) @ N))))).
% 2.83/1.51 thf(fact_192_le__add1, axiom, (! [N:nat,M:nat]: ((((ord_less_eq @ nat) @ N) @ (((plus_plus @ nat) @ N) @ M))))).
% 2.83/1.51 thf(fact_193_add__leE, axiom, (! [M:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M) @ K)) @ N) => (~ (((((ord_less_eq @ nat) @ M) @ N) => (~ ((((ord_less_eq @ nat) @ K) @ N)))))))))).
% 2.83/1.51 thf(fact_194_isSPath__nt__parallel, axiom, (! [S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat,E3:((product_prod @ nat) @ nat)]: (((((((isSimplePath @ capacity) @ c) @ S) @ P) @ T2) => ((((member @ ((product_prod @ nat) @ nat)) @ E3) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ (((product_swap @ nat) @ nat) @ E3)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P))))))))).
% 2.83/1.51 thf(fact_195_discrete, axiom, (! [A:$tType]: (((unique1598680935umeral @ A) => ((ord_less @ A) = (^ [A3:A]: (((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A)))))))))).
% 2.83/1.51 thf(fact_196_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B2)) = ((F1 @ A2) @ B2))))).
% 2.83/1.51 thf(fact_197_fold__atLeastAtMost__nat_Oinduct, axiom, (! [A:$tType,P3:((nat > (A > A)) > (nat > (nat > (A > $o)))),A0:(nat > (A > A)),A1:nat,A22:nat,A32:A]: (((! [F3:(nat > (A > A)),A6:nat,B6:nat,Acc:A]: ((((~ ((((ord_less @ nat) @ B6) @ A6))) => ((((P3 @ F3) @ (((plus_plus @ nat) @ A6) @ (one_one @ nat))) @ B6) @ ((F3 @ A6) @ Acc))) => ((((P3 @ F3) @ A6) @ B6) @ Acc)))) => ((((P3 @ A0) @ A1) @ A22) @ A32))))).
% 2.83/1.51 thf(fact_198_ex__has__greatest__nat__lemma, axiom, (! [A:$tType,P3:(A > $o),K:A,F:(A > nat),N:nat]: (((P3 @ K) => ((! [X3:A]: (((P3 @ X3) => (? [Y:A]: (((P3 @ Y) & (~ ((((ord_less_eq @ nat) @ (F @ Y)) @ (F @ X3)))))))))) => (? [Y4:A]: (((P3 @ Y4) & (~ ((((ord_less @ nat) @ (F @ Y4)) @ (((plus_plus @ nat) @ (F @ K)) @ N)))))))))))).
% 2.83/1.51 thf(fact_199_swap__swap, axiom, (! [B:$tType,A:$tType,P:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P)) = P)))).
% 2.83/1.51 thf(fact_200_swap__simp, axiom, (! [A:$tType,B:$tType,X:B,Y3:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y3)) = ((((product_Pair @ A) @ B) @ Y3) @ X))))).
% 2.83/1.51 thf(fact_201_ex__has__least__nat, axiom, (! [A:$tType,P3:(A > $o),K:A,M:(A > nat)]: (((P3 @ K) => (? [X3:A]: (((P3 @ X3) & (! [Y:A]: (((P3 @ Y) => (((ord_less_eq @ nat) @ (M @ X3)) @ (M @ Y)))))))))))).
% 2.83/1.51 thf(fact_202_bounded__Max__nat, axiom, (! [P3:(nat > $o),X:nat,M5:nat]: (((P3 @ X) => ((! [X3:nat]: (((P3 @ X3) => (((ord_less_eq @ nat) @ X3) @ M5)))) => (~ ((! [M4:nat]: (((P3 @ M4) => (~ ((! [X4:nat]: (((P3 @ X4) => (((ord_less_eq @ nat) @ X4) @ M4)))))))))))))))).
% 2.83/1.51 thf(fact_203_fold__atLeastAtMost__nat_Ocases, axiom, (! [A:$tType,X:((product_prod @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A)))]: ((~ ((! [F3:(nat > (A > A)),A6:nat,B6:nat,Acc:A]: ((X != ((((product_Pair @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A))) @ F3) @ ((((product_Pair @ nat) @ ((product_prod @ nat) @ A)) @ A6) @ ((((product_Pair @ nat) @ A) @ B6) @ Acc))))))))))).
% 2.83/1.51 thf(fact_204_Graph_OisSPath__nt__parallel, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,P:(list @ ((product_prod @ nat) @ nat)),T2:nat,E3:((product_prod @ nat) @ nat)]: (((((((isSimplePath @ Capacity) @ C) @ S) @ P) @ T2) => ((((member @ ((product_prod @ nat) @ nat)) @ E3) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)) => (~ ((((member @ ((product_prod @ nat) @ nat)) @ (((product_swap @ nat) @ nat) @ E3)) @ ((set2 @ ((product_prod @ nat) @ nat)) @ P)))))))))))).
% 2.83/1.51 thf(fact_205_Lattices__Big_Oex__has__greatest__nat, axiom, (! [A:$tType,P3:(A > $o),K:A,F:(A > nat),B2:nat]: (((P3 @ K) => ((! [Y4:A]: (((P3 @ Y4) => (((ord_less @ nat) @ (F @ Y4)) @ B2)))) => (? [X3:A]: (((P3 @ X3) & (! [Y:A]: (((P3 @ Y) => (((ord_less_eq @ nat) @ (F @ Y)) @ (F @ X3))))))))))))).
% 2.83/1.51 thf(fact_206_less__add__one, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A2:A]: ((((ord_less @ A) @ A2) @ (((plus_plus @ A) @ A2) @ (one_one @ A))))))))).
% 2.83/1.51 thf(fact_207_add__mono1, axiom, (! [A:$tType]: (((linord1659791738miring @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ (((plus_plus @ A) @ A2) @ (one_one @ A))) @ (((plus_plus @ A) @ B2) @ (one_one @ A)))))))))).
% 2.83/1.51 thf(fact_208_linorder__neqE__linordered__idom, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y3:A]: (((X != Y3) => ((~ ((((ord_less @ A) @ X) @ Y3))) => (((ord_less @ A) @ Y3) @ X))))))))).
% 2.83/1.51 thf(fact_209_isPath_Osimps_I1_J, axiom, (! [U:nat,V:nat]: (((((((isPath @ capacity) @ c) @ U) @ (nil @ ((product_prod @ nat) @ nat))) @ V) = (U = V))))).
% 2.83/1.51 thf(fact_210_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less_eq @ A) @ X) @ X))))))).
% 2.83/1.51 thf(fact_211_simplePath__empty__conv, axiom, (! [S:nat,T2:nat]: (((((((isSimplePath @ capacity) @ c) @ S) @ (nil @ ((product_prod @ nat) @ nat))) @ T2) = (S = T2))))).
% 2.83/1.51 thf(fact_212_simplePath__same__conv, axiom, (! [S:nat,P:(list @ ((product_prod @ nat) @ nat))]: (((((((isSimplePath @ capacity) @ c) @ S) @ P) @ S) = (P = (nil @ ((product_prod @ nat) @ nat))))))).
% 2.83/1.51 thf(fact_213_Graph_OsimplePath__empty__conv, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,T2:nat]: (((((((isSimplePath @ Capacity) @ C) @ S) @ (nil @ ((product_prod @ nat) @ nat))) @ T2) = (S = T2)))))))).
% 2.83/1.51 thf(fact_214_Graph_OsimplePath__same__conv, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),S:nat,P:(list @ ((product_prod @ nat) @ nat))]: (((((((isSimplePath @ Capacity) @ C) @ S) @ P) @ S) = (P = (nil @ ((product_prod @ nat) @ nat)))))))))).
% 2.83/1.51 thf(fact_215_Graph_OisPath_Osimps_I1_J, axiom, (! [Capacity:$tType]: (((linordered_idom @ Capacity) => (! [C:(((product_prod @ nat) @ nat) > Capacity),U:nat,V:nat]: (((((((isPath @ Capacity) @ C) @ U) @ (nil @ ((product_prod @ nat) @ nat))) @ V) = (U = V)))))))).
% 2.83/1.51 thf(fact_216_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ A2) @ B2) => (A2 = B2))))))))).
% 2.83/1.51 thf(fact_217_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z:A]: ((Y5 = Z))) = (^ [A3:A,B3:A]: (((((ord_less_eq @ A) @ B3) @ A3) & (((ord_less_eq @ A) @ A3) @ B3))))))))).
% 2.83/1.51 thf(fact_218_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C:A]: (((((ord_less_eq @ A) @ B2) @ A2) => ((((ord_less_eq @ A) @ C) @ B2) => (((ord_less_eq @ A) @ C) @ A2))))))))).
% 2.83/1.51 thf(fact_219_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P3:(A > (A > $o)),A2:A,B2:A]: (((! [A6:A,B6:A]: (((((ord_less_eq @ A) @ A6) @ B6) => ((P3 @ A6) @ B6)))) => ((! [A6:A,B6:A]: ((((P3 @ B6) @ A6) => ((P3 @ A6) @ B6)))) => ((P3 @ A2) @ B2))))))))).
% 2.83/1.51 thf(fact_220_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.83/1.51 thf(fact_221_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y3:A,Z2:A]: (((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z2) => (((ord_less_eq @ A) @ X) @ Z2))))))))).
% 2.83/1.51 thf(fact_222_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ A2) => (A2 = B2))))))))).
% 2.83/1.51 thf(fact_223_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((B2 = C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.83/1.51 thf(fact_224_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A2:A,B2:A,C:A]: (((A2 = B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.83/1.51 thf(fact_225_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z:A]: ((Y5 = Z))) = (^ [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) & (((ord_less_eq @ A) @ B3) @ A3))))))))).
% 2.83/1.51 thf(fact_226_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y3:A,X:A]: (((((ord_less_eq @ A) @ Y3) @ X) => ((((ord_less_eq @ A) @ X) @ Y3) = (X = Y3))))))))).
% 2.83/1.51 thf(fact_227_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A,Z2:A]: ((((((ord_less_eq @ A) @ X) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ Z2)))) => (((((ord_less_eq @ A) @ Y3) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Z2)))) => (((((ord_less_eq @ A) @ X) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y3)))) => (((((ord_less_eq @ A) @ Z2) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ X)))) => (((((ord_less_eq @ A) @ Y3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X) => (~ ((((ord_less_eq @ A) @ X) @ Y3))))))))))))))))).
% 2.83/1.51 thf(fact_228_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ B2) @ C) => (((ord_less_eq @ A) @ A2) @ C))))))))).
% 2.83/1.51 thf(fact_229_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X)))))))).
% 2.83/1.51 thf(fact_230_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y3:A]: (((X = Y3) => (((ord_less_eq @ A) @ X) @ Y3)))))))).
% 2.83/1.51 thf(fact_231_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) | (((ord_less_eq @ A) @ Y3) @ X)))))))).
% 2.83/1.51 thf(fact_232_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ X) => (X = Y3))))))))).
% 2.83/1.51 thf(fact_233_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y5:A,Z:A]: ((Y5 = Z))) = (^ [X2:A,Y6:A]: (((((ord_less_eq @ A) @ X2) @ Y6) & (((ord_less_eq @ A) @ Y6) @ X2))))))))).
% 2.83/1.51 thf(fact_234_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,B2:A,F:(A > B),C:B]: (((((ord_less_eq @ A) @ A2) @ B2) => (((F @ B2) = C) => ((! [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) => (((ord_less_eq @ B) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less_eq @ B) @ (F @ A2)) @ C)))))))))).
% 2.83/1.51 thf(fact_235_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A2:A,F:(B > A),B2:B,C:B]: (((A2 = (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X3:B,Y4:B]: (((((ord_less_eq @ B) @ X3) @ Y4) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.83/1.51 thf(fact_236_order__subst2, axiom, (! [A:$tType,C4:$tType]: ((((order @ C4) & (order @ A)) => (! [A2:A,B2:A,F:(A > C4),C:C4]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ C4) @ (F @ B2)) @ C) => ((! [X3:A,Y4:A]: (((((ord_less_eq @ A) @ X3) @ Y4) => (((ord_less_eq @ C4) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less_eq @ C4) @ (F @ A2)) @ C)))))))))).
% 2.83/1.51 thf(fact_237_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A2:A,F:(B > A),B2:B,C:B]: (((((ord_less_eq @ A) @ A2) @ (F @ B2)) => ((((ord_less_eq @ B) @ B2) @ C) => ((! [X3:B,Y4:B]: (((((ord_less_eq @ B) @ X3) @ Y4) => (((ord_less_eq @ A) @ (F @ X3)) @ (F @ Y4))))) => (((ord_less_eq @ A) @ A2) @ (F @ C))))))))))).
% 2.83/1.51 thf(fact_238_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F4:(A > B),G4:(A > B)]: ((! [X2:A]: ((((ord_less_eq @ B) @ (F4 @ X2)) @ (G4 @ X2))))))))))).
% 2.83/1.51 thf(fact_239_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X3:A]: ((((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.83/1.51 thf(fact_240_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.83/1.51 thf(fact_241_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X)) @ (G @ X))))))))).
% 2.83/1.51 thf(fact_242_dual__order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A]: (((((ord_less @ A) @ B2) @ A2) => (A2 != B2)))))))).
% 2.83/1.51 thf(fact_243_order_Ostrict__implies__not__eq, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A]: (((((ord_less @ A) @ A2) @ B2) => (A2 != B2)))))))).
% 2.83/1.51 thf(fact_244_not__less__iff__gr__or__eq, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((~ ((((ord_less @ A) @ X) @ Y3))) = ((((ord_less @ A) @ Y3) @ X) | (X = Y3))))))))).
% 2.83/1.51 thf(fact_245_dual__order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A2:A,C:A]: (((((ord_less @ A) @ B2) @ A2) => ((((ord_less @ A) @ C) @ B2) => (((ord_less @ A) @ C) @ A2))))))))).
% 2.83/1.51 thf(fact_246_linorder__less__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P3:(A > (A > $o)),A2:A,B2:A]: (((! [A6:A,B6:A]: (((((ord_less @ A) @ A6) @ B6) => ((P3 @ A6) @ B6)))) => ((! [A6:A]: (((P3 @ A6) @ A6))) => ((! [A6:A,B6:A]: ((((P3 @ B6) @ A6) => ((P3 @ A6) @ B6)))) => ((P3 @ A2) @ B2)))))))))).
% 2.83/1.51 thf(fact_247_exists__least__iff, axiom, (! [A:$tType]: (((wellorder @ A) => ((^ [P7:(A > $o)]: ((? [X5:A]: ((P7 @ X5))))) = (^ [P8:(A > $o)]: ((? [N2:A]: (((P8 @ N2) & (! [M2:A]: (((((ord_less @ A) @ M2) @ N2) => (~ ((P8 @ M2)))))))))))))))).
% 2.83/1.51 thf(fact_248_less__imp__not__less, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) => (~ ((((ord_less @ A) @ Y3) @ X)))))))))).
% 2.83/1.51 thf(fact_249_order_Ostrict__trans, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A,B2:A,C:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ B2) @ C) => (((ord_less @ A) @ A2) @ C))))))))).
% 2.83/1.51 thf(fact_250_dual__order_Oirrefl, axiom, (! [A:$tType]: (((order @ A) => (! [A2:A]: ((~ ((((ord_less @ A) @ A2) @ A2))))))))).
% 2.83/1.51 thf(fact_251_linorder__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y3:A]: (((~ ((((ord_less @ A) @ X) @ Y3))) => ((X != Y3) => (((ord_less @ A) @ Y3) @ X))))))))).
% 2.83/1.51 thf(fact_252_less__imp__triv, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y3:A,P3:$o]: (((((ord_less @ A) @ X) @ Y3) => ((((ord_less @ A) @ Y3) @ X) => P3)))))))).
% 2.83/1.51 thf(fact_253_less__imp__not__eq2, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y3:A]: (((((ord_less @ A) @ X) @ Y3) => (Y3 != X)))))))).
% 2.83/1.51 thf(fact_254_antisym__conv3, axiom, (! [A:$tType]: (((linorder @ A) => (! [Y3:A,X:A]: (((~ ((((ord_less @ A) @ Y3) @ X))) => ((~ ((((ord_less @ A) @ X) @ Y3))) = (X = Y3))))))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___HOL_Otype, axiom, (! [A:$tType]: (((linordered_idom @ A) => (type @ A))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___Groups_Oone, axiom, (! [A:$tType]: (((linordered_idom @ A) => (one @ A))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___Orderings_Oord, axiom, (! [A:$tType]: (((linordered_idom @ A) => (ord @ A))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___Orderings_Oorder, axiom, (! [A:$tType]: (((linordered_idom @ A) => (order @ A))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___Groups_Ogroup__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (group_add @ A))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___Orderings_Olinorder, axiom, (! [A:$tType]: (((linordered_idom @ A) => (linorder @ A))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___Orderings_Opreorder, axiom, (! [A:$tType]: (((linordered_idom @ A) => (preorder @ A))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___Groups_Osemigroup__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (semigroup_add @ A))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___Groups_Ocomm__monoid__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (comm_monoid_add @ A))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___Groups_Oab__semigroup__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (ab_semigroup_add @ A))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___Rings_Olinordered__semidom, axiom, (! [A:$tType]: (((linordered_idom @ A) => (linordered_semidom @ A))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___Groups_Ocancel__semigroup__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (cancel_semigroup_add @ A))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (ordere779506340up_add @ A))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___Rings_Olinordered__nonzero__semiring, axiom, (! [A:$tType]: (((linordered_idom @ A) => (linord1659791738miring @ A))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (! [A:$tType]: (((linordered_idom @ A) => (ordere236663937imp_le @ A))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__cancel__ab__semigroup__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (ordere223160158up_add @ A))))).
% 2.83/1.51 thf(subcl_Rings_Olinordered__idom___Groups_Ostrict__ordered__ab__semigroup__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (strict2144017051up_add @ A))))).
% 2.83/1.51 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A7:$tType,A8:$tType]: (((preorder @ A8) => (preorder @ (A7 > A8)))))).
% 2.83/1.51 thf(tcon_fun___Orderings_Oorder, axiom, (! [A7:$tType,A8:$tType]: (((order @ A8) => (order @ (A7 > A8)))))).
% 2.83/1.51 thf(tcon_fun___Orderings_Oord, axiom, (! [A7:$tType,A8:$tType]: (((ord @ A8) => (ord @ (A7 > A8)))))).
% 2.83/1.51 thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral, axiom, (unique1598680935umeral @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add, axiom, (strict2144017051up_add @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add, axiom, (ordere223160158up_add @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (ordere236663937imp_le @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni770627133id_add @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, axiom, (linord1659791738miring @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, axiom, (ordere779506340up_add @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Rings_Olinordered__semidom, axiom, (linordered_semidom @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add, axiom, (ab_semigroup_add @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add, axiom, (comm_monoid_add @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Groups_Osemigroup__add, axiom, (semigroup_add @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Orderings_Owellorder, axiom, (wellorder @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Orderings_Opreorder_1, axiom, (preorder @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Orderings_Olinorder, axiom, (linorder @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Orderings_Oorder_2, axiom, (order @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Orderings_Oord_3, axiom, (ord @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Groups_Oone, axiom, (one @ nat)).
% 2.83/1.51 thf(tcon_Nat_Onat___Nat_Osize, axiom, (size @ nat)).
% 2.83/1.51 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A7:$tType]: ((preorder @ (set @ A7))))).
% 2.83/1.51 thf(tcon_Set_Oset___Orderings_Oorder_5, axiom, (! [A7:$tType]: ((order @ (set @ A7))))).
% 2.83/1.51 thf(tcon_Set_Oset___Orderings_Oord_6, axiom, (! [A7:$tType]: ((ord @ (set @ A7))))).
% 2.83/1.51 thf(tcon_HOL_Obool___Orderings_Opreorder_7, axiom, (preorder @ $o)).
% 2.83/1.51 thf(tcon_HOL_Obool___Orderings_Olinorder_8, axiom, (linorder @ $o)).
% 2.83/1.51 thf(tcon_HOL_Obool___Orderings_Oorder_9, axiom, (order @ $o)).
% 2.83/1.51 thf(tcon_HOL_Obool___Orderings_Oord_10, axiom, (ord @ $o)).
% 2.83/1.51 thf(tcon_List_Olist___Nat_Osize_11, axiom, (! [A7:$tType]: ((size @ (list @ A7))))).
% 2.83/1.51 thf(tcon_Product__Type_Oprod___Nat_Osize_12, axiom, (! [A7:$tType,A8:$tType]: ((size @ ((product_prod @ A7) @ A8))))).
% 2.83/1.51 thf(tfree_0, hypothesis, (linordered_idom @ capacity)).
% 2.83/1.51 thf(conj_0, conjecture, (((((min_dist @ capacity) @ c) @ u) @ t) = (((plus_plus @ nat) @ (one_one @ nat)) @ ((size_size @ (list @ ((product_prod @ nat) @ nat))) @ p2)))).
% 2.83/1.51 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.dNuoTsUl6x/DTF2DTF2221.p
% 2.83/1.51 ------------------------
% 2.83/1.56 ---- Cleaned THF ---
% 2.83/1.56 thf(ty_t_Product__Type_Oprod,type,
% 2.83/1.56 product_prod: $tType > $tType > $tType ).
% 2.83/1.56
% 2.83/1.56 thf(ty_t_List_Olist,type,
% 2.83/1.56 list: $tType > $tType ).
% 2.83/1.56
% 2.83/1.56 thf(ty_tf_capacity,type,
% 2.83/1.56 capacity: $tType ).
% 2.83/1.56
% 2.83/1.56 thf(ty_t_Set_Oset,type,
% 2.83/1.56 set: $tType > $tType ).
% 2.83/1.56
% 2.83/1.56 thf(ty_t_Nat_Onat,type,
% 2.83/1.56 nat: $tType ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_HOL_Otype,type,
% 2.83/1.56 type:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Nat_Osize,type,
% 2.83/1.56 size:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Groups_Oone,type,
% 2.83/1.56 one:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Orderings_Oord,type,
% 2.83/1.56 ord:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Orderings_Oorder,type,
% 2.83/1.56 order:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Groups_Ogroup__add,type,
% 2.83/1.56 group_add:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Orderings_Olinorder,type,
% 2.83/1.56 linorder:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Orderings_Opreorder,type,
% 2.83/1.56 preorder:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Orderings_Owellorder,type,
% 2.83/1.56 wellorder:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Groups_Osemigroup__add,type,
% 2.83/1.56 semigroup_add:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Rings_Olinordered__idom,type,
% 2.83/1.56 linordered_idom:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 2.83/1.56 comm_monoid_add:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 2.83/1.56 ab_semigroup_add:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Rings_Olinordered__semidom,type,
% 2.83/1.56 linordered_semidom:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 2.83/1.56 cancel_semigroup_add:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 2.83/1.56 ordere779506340up_add:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 2.83/1.56 linord1659791738miring:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 2.83/1.56 canoni770627133id_add:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 2.83/1.56 ordere236663937imp_le:
% 2.83/1.56 !>[A: $tType] : $o ).
% 2.83/1.56
% 2.83/1.56 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 2.83/1.56 ordere223160158up_add:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 2.83/1.57 strict2144017051up_add:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
% 2.83/1.57 unique1598680935umeral:
% 2.83/1.57 !>[A: $tType] : $o ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Graph_OGraph_Oconnected,type,
% 2.83/1.57 connected:
% 2.83/1.57 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > $o ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Graph_OGraph_Odist,type,
% 2.83/1.57 dist:
% 2.83/1.57 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > nat > $o ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Graph_OGraph_OisPath,type,
% 2.83/1.57 isPath:
% 2.83/1.57 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Graph_OGraph_OisShortestPath,type,
% 2.83/1.57 isShortestPath:
% 2.83/1.57 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Graph_OGraph_OisSimplePath,type,
% 2.83/1.57 isSimplePath:
% 2.83/1.57 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Graph_OGraph_Omin__dist,type,
% 2.83/1.57 min_dist:
% 2.83/1.57 !>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > nat ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Groups_Oone__class_Oone,type,
% 2.83/1.57 one_one:
% 2.83/1.57 !>[A: $tType] : A ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 2.83/1.57 plus_plus:
% 2.83/1.57 !>[A: $tType] : ( A > A > A ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_List_Olist_ONil,type,
% 2.83/1.57 nil:
% 2.83/1.57 !>[A: $tType] : ( list @ A ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_List_Olist_Oset,type,
% 2.83/1.57 set2:
% 2.83/1.57 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Nat_Osize__class_Osize,type,
% 2.83/1.57 size_size:
% 2.83/1.57 !>[A: $tType] : ( A > nat ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.83/1.57 ord_less:
% 2.83/1.57 !>[A: $tType] : ( A > A > $o ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.83/1.57 ord_less_eq:
% 2.83/1.57 !>[A: $tType] : ( A > A > $o ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Product__Type_OPair,type,
% 2.83/1.57 product_Pair:
% 2.83/1.57 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.83/1.57 product_rec_prod:
% 2.83/1.57 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 2.83/1.57 product_swap:
% 2.83/1.57 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_Set_OCollect,type,
% 2.83/1.57 collect:
% 2.83/1.57 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_c_member,type,
% 2.83/1.57 member:
% 2.83/1.57 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_v_c,type,
% 2.83/1.57 c: ( product_prod @ nat @ nat ) > capacity ).
% 2.83/1.57
% 2.83/1.57 thf(sy_v_p,type,
% 2.83/1.57 p: list @ ( product_prod @ nat @ nat ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_v_p1____,type,
% 2.83/1.57 p1: list @ ( product_prod @ nat @ nat ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_v_p2____,type,
% 2.83/1.57 p2: list @ ( product_prod @ nat @ nat ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_v_p_H,type,
% 2.83/1.57 p3: list @ ( product_prod @ nat @ nat ) ).
% 2.83/1.57
% 2.83/1.57 thf(sy_v_s,type,
% 2.83/1.57 s: nat ).
% 2.83/1.57
% 2.83/1.57 thf(sy_v_t,type,
% 2.83/1.57 t: nat ).
% 2.83/1.57
% 2.83/1.57 thf(sy_v_u,type,
% 2.83/1.57 u: nat ).
% 2.83/1.57
% 2.83/1.57 thf(sy_v_v,type,
% 2.83/1.57 v: nat ).
% 2.83/1.57
% 2.83/1.57 thf(fact_0_MIN_H,axiom,
% 2.83/1.57 ( ( min_dist @ capacity @ c @ s @ t )
% 2.83/1.57 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1 ) @ ( one_one @ nat ) ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_1__092_060open_062min__dist_As_At_A_061_Alength_Ap1_A_L_A_I1_A_L_Alength_Ap2_J_A_092_060Longrightarrow_062_Amin__dist_As_Au_A_061_Alength_Ap1_092_060close_062,axiom,
% 2.83/1.57 ( ( ( min_dist @ capacity @ c @ s @ t )
% 2.83/1.57 = ( plus_plus @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 ) ) ) )
% 2.83/1.57 => ( ( min_dist @ capacity @ c @ s @ u )
% 2.83/1.57 = ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_2__092_060open_062min__dist_As_At_A_061_Alength_Ap1_A_L_A_I1_A_L_Alength_Ap2_J_A_092_060Longrightarrow_062_Amin__dist_Au_At_A_061_A1_A_L_Alength_Ap2_092_060close_062,axiom,
% 2.83/1.57 ( ( ( min_dist @ capacity @ c @ s @ t )
% 2.83/1.57 = ( plus_plus @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1 ) @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 ) ) ) )
% 2.83/1.57 => ( ( min_dist @ capacity @ c @ u @ t )
% 2.83/1.57 = ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_3__092_060open_062min__dist_Av_At_A_061_Alength_Ap2_092_060close_062,axiom,
% 2.83/1.57 ( ( min_dist @ capacity @ c @ v @ t )
% 2.83/1.57 = ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_4_MIN,axiom,
% 2.83/1.57 ( ( min_dist @ capacity @ c @ s @ t )
% 2.83/1.57 = ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_5__092_060open_062u_A_092_060noteq_062_Av_092_060close_062,axiom,
% 2.83/1.57 u != v ).
% 2.83/1.57
% 2.83/1.57 thf(fact_6_MDSV,axiom,
% 2.83/1.57 ( ( min_dist @ capacity @ c @ s @ v )
% 2.83/1.57 = ( plus_plus @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1 ) @ ( one_one @ nat ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_7_min__dist__split_I2_J,axiom,
% 2.83/1.57 ! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 2.83/1.57 ( ( dist @ capacity @ c @ U @ D1 @ W )
% 2.83/1.57 => ( ( dist @ capacity @ c @ W @ D2 @ V )
% 2.83/1.57 => ( ( ( min_dist @ capacity @ c @ U @ V )
% 2.83/1.57 = ( plus_plus @ nat @ D1 @ D2 ) )
% 2.83/1.57 => ( ( min_dist @ capacity @ c @ W @ V )
% 2.83/1.57 = D2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_8_min__dist__split_I1_J,axiom,
% 2.83/1.57 ! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 2.83/1.57 ( ( dist @ capacity @ c @ U @ D1 @ W )
% 2.83/1.57 => ( ( dist @ capacity @ c @ W @ D2 @ V )
% 2.83/1.57 => ( ( ( min_dist @ capacity @ c @ U @ V )
% 2.83/1.57 = ( plus_plus @ nat @ D1 @ D2 ) )
% 2.83/1.57 => ( ( min_dist @ capacity @ c @ U @ W )
% 2.83/1.57 = D1 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_9_add__left__cancel,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( cancel_semigroup_add @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A] :
% 2.83/1.57 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.83/1.57 = ( plus_plus @ A @ A2 @ C ) )
% 2.83/1.57 = ( B2 = C ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_10_add__right__cancel,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( cancel_semigroup_add @ A )
% 2.83/1.57 => ! [B2: A,A2: A,C: A] :
% 2.83/1.57 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.83/1.57 = ( plus_plus @ A @ C @ A2 ) )
% 2.83/1.57 = ( B2 = C ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_11_dist__trans,axiom,
% 2.83/1.57 ! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 2.83/1.57 ( ( dist @ capacity @ c @ U @ D1 @ W )
% 2.83/1.57 => ( ( dist @ capacity @ c @ W @ D2 @ V )
% 2.83/1.57 => ( dist @ capacity @ c @ U @ ( plus_plus @ nat @ D1 @ D2 ) @ V ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_12_assms_I3_J,axiom,
% 2.83/1.57 isPath @ capacity @ c @ s @ p3 @ t ).
% 2.83/1.57
% 2.83/1.57 thf(fact_13_DISTS_I1_J,axiom,
% 2.83/1.57 dist @ capacity @ c @ s @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1 ) @ u ).
% 2.83/1.57
% 2.83/1.57 thf(fact_14_isPath__distD,axiom,
% 2.83/1.57 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.83/1.57 ( ( isPath @ capacity @ c @ U @ P @ V )
% 2.83/1.57 => ( dist @ capacity @ c @ U @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P ) @ V ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_15_dist__def,axiom,
% 2.83/1.57 ! [V: nat,D: nat,V2: nat] :
% 2.83/1.57 ( ( dist @ capacity @ c @ V @ D @ V2 )
% 2.83/1.57 = ( ? [P2: list @ ( product_prod @ nat @ nat )] :
% 2.83/1.57 ( ( isPath @ capacity @ c @ V @ P2 @ V2 )
% 2.83/1.57 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 )
% 2.83/1.57 = D ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_16_DISTS_I2_J,axiom,
% 2.83/1.57 dist @ capacity @ c @ u @ ( one_one @ nat ) @ v ).
% 2.83/1.57
% 2.83/1.57 thf(fact_17_P,axiom,
% 2.83/1.57 isPath @ capacity @ c @ s @ p @ t ).
% 2.83/1.57
% 2.83/1.57 thf(fact_18_DISTS_I3_J,axiom,
% 2.83/1.57 dist @ capacity @ c @ v @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 ) @ t ).
% 2.83/1.57
% 2.83/1.57 thf(fact_19_assms_I1_J,axiom,
% 2.83/1.57 isShortestPath @ capacity @ c @ s @ p @ t ).
% 2.83/1.57
% 2.83/1.57 thf(fact_20_add__right__imp__eq,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( cancel_semigroup_add @ A )
% 2.83/1.57 => ! [B2: A,A2: A,C: A] :
% 2.83/1.57 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.83/1.57 = ( plus_plus @ A @ C @ A2 ) )
% 2.83/1.57 => ( B2 = C ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_21_add__left__imp__eq,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( cancel_semigroup_add @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A] :
% 2.83/1.57 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.83/1.57 = ( plus_plus @ A @ A2 @ C ) )
% 2.83/1.57 => ( B2 = C ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_22_add_Oleft__commute,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ab_semigroup_add @ A )
% 2.83/1.57 => ! [B2: A,A2: A,C: A] :
% 2.83/1.57 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C ) )
% 2.83/1.57 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_23_add_Ocommute,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ab_semigroup_add @ A )
% 2.83/1.57 => ( ( plus_plus @ A )
% 2.83/1.57 = ( ^ [A3: A,B3: A] : ( plus_plus @ A @ B3 @ A3 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_24_add_Oright__cancel,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( group_add @ A )
% 2.83/1.57 => ! [B2: A,A2: A,C: A] :
% 2.83/1.57 ( ( ( plus_plus @ A @ B2 @ A2 )
% 2.83/1.57 = ( plus_plus @ A @ C @ A2 ) )
% 2.83/1.57 = ( B2 = C ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_25_add_Oleft__cancel,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( group_add @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A] :
% 2.83/1.57 ( ( ( plus_plus @ A @ A2 @ B2 )
% 2.83/1.57 = ( plus_plus @ A @ A2 @ C ) )
% 2.83/1.57 = ( B2 = C ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_26_add_Oassoc,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( semigroup_add @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A] :
% 2.83/1.57 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 2.83/1.57 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_27_group__cancel_Oadd2,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( comm_monoid_add @ A )
% 2.83/1.57 => ! [B4: A,K: A,B2: A,A2: A] :
% 2.83/1.57 ( ( B4
% 2.83/1.57 = ( plus_plus @ A @ K @ B2 ) )
% 2.83/1.57 => ( ( plus_plus @ A @ A2 @ B4 )
% 2.83/1.57 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_28_group__cancel_Oadd1,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( comm_monoid_add @ A )
% 2.83/1.57 => ! [A4: A,K: A,A2: A,B2: A] :
% 2.83/1.57 ( ( A4
% 2.83/1.57 = ( plus_plus @ A @ K @ A2 ) )
% 2.83/1.57 => ( ( plus_plus @ A @ A4 @ B2 )
% 2.83/1.57 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_29_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere779506340up_add @ A )
% 2.83/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.83/1.57 ( ( ( I = J )
% 2.83/1.57 & ( K = L ) )
% 2.83/1.57 => ( ( plus_plus @ A @ I @ K )
% 2.83/1.57 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_30_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ab_semigroup_add @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A] :
% 2.83/1.57 ( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C )
% 2.83/1.57 = ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_31_one__reorient,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( one @ A )
% 2.83/1.57 => ! [X: A] :
% 2.83/1.57 ( ( ( one_one @ A )
% 2.83/1.57 = X )
% 2.83/1.57 = ( X
% 2.83/1.57 = ( one_one @ A ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_32_isShortestPath__min__dist__def,axiom,
% 2.83/1.57 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.83/1.57 ( ( isShortestPath @ capacity @ c @ U @ P @ V )
% 2.83/1.57 = ( ( isPath @ capacity @ c @ U @ P @ V )
% 2.83/1.57 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P )
% 2.83/1.57 = ( min_dist @ capacity @ c @ U @ V ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_33_Graph_OisPath__distD,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.83/1.57 ( ( isPath @ Capacity @ C @ U @ P @ V )
% 2.83/1.57 => ( dist @ Capacity @ C @ U @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P ) @ V ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_34_Graph_Odist__def,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ( ( dist @ Capacity )
% 2.83/1.57 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,V3: nat,D3: nat,V4: nat] :
% 2.83/1.57 ? [P2: list @ ( product_prod @ nat @ nat )] :
% 2.83/1.57 ( ( isPath @ Capacity @ C2 @ V3 @ P2 @ V4 )
% 2.83/1.57 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 )
% 2.83/1.57 = D3 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_35_min__dist__minD,axiom,
% 2.83/1.57 ! [V: nat,D: nat,V2: nat] :
% 2.83/1.57 ( ( dist @ capacity @ c @ V @ D @ V2 )
% 2.83/1.57 => ( ord_less_eq @ nat @ ( min_dist @ capacity @ c @ V @ V2 ) @ D ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_36_min__distI__eq,axiom,
% 2.83/1.57 ! [V: nat,D: nat,V2: nat] :
% 2.83/1.57 ( ( dist @ capacity @ c @ V @ D @ V2 )
% 2.83/1.57 => ( ! [D4: nat] :
% 2.83/1.57 ( ( dist @ capacity @ c @ V @ D4 @ V2 )
% 2.83/1.57 => ( ord_less_eq @ nat @ D @ D4 ) )
% 2.83/1.57 => ( ( min_dist @ capacity @ c @ V @ V2 )
% 2.83/1.57 = D ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_37_Graph_Omin__dist__split_I1_J,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 2.83/1.57 ( ( dist @ Capacity @ C @ U @ D1 @ W )
% 2.83/1.57 => ( ( dist @ Capacity @ C @ W @ D2 @ V )
% 2.83/1.57 => ( ( ( min_dist @ Capacity @ C @ U @ V )
% 2.83/1.57 = ( plus_plus @ nat @ D1 @ D2 ) )
% 2.83/1.57 => ( ( min_dist @ Capacity @ C @ U @ W )
% 2.83/1.57 = D1 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_38_Graph_Omin__dist__split_I2_J,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 2.83/1.57 ( ( dist @ Capacity @ C @ U @ D1 @ W )
% 2.83/1.57 => ( ( dist @ Capacity @ C @ W @ D2 @ V )
% 2.83/1.57 => ( ( ( min_dist @ Capacity @ C @ U @ V )
% 2.83/1.57 = ( plus_plus @ nat @ D1 @ D2 ) )
% 2.83/1.57 => ( ( min_dist @ Capacity @ C @ W @ V )
% 2.83/1.57 = D2 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_39_min__dist__is__dist,axiom,
% 2.83/1.57 ! [V: nat,V2: nat] :
% 2.83/1.57 ( ( connected @ capacity @ c @ V @ V2 )
% 2.83/1.57 => ( dist @ capacity @ c @ V @ ( min_dist @ capacity @ c @ V @ V2 ) @ V2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_40_shortestPath__is__path,axiom,
% 2.83/1.57 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.83/1.57 ( ( isShortestPath @ capacity @ c @ U @ P @ V )
% 2.83/1.57 => ( isPath @ capacity @ c @ U @ P @ V ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_41_assms_I4_J,axiom,
% 2.83/1.57 member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ v @ u ) @ ( set2 @ ( product_prod @ nat @ nat ) @ p3 ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_42_connected__def,axiom,
% 2.83/1.57 ! [U: nat,V: nat] :
% 2.83/1.57 ( ( connected @ capacity @ c @ U @ V )
% 2.83/1.57 = ( ? [P2: list @ ( product_prod @ nat @ nat )] : ( isPath @ capacity @ c @ U @ P2 @ V ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_43_connected__by__dist,axiom,
% 2.83/1.57 ! [V: nat,V2: nat] :
% 2.83/1.57 ( ( connected @ capacity @ c @ V @ V2 )
% 2.83/1.57 = ( ? [D3: nat] : ( dist @ capacity @ c @ V @ D3 @ V2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_44_mem__Collect__eq,axiom,
% 2.83/1.57 ! [A: $tType,A2: A,P3: A > $o] :
% 2.83/1.57 ( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
% 2.83/1.57 = ( P3 @ A2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_45_Collect__mem__eq,axiom,
% 2.83/1.57 ! [A: $tType,A4: set @ A] :
% 2.83/1.57 ( ( collect @ A
% 2.83/1.57 @ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
% 2.83/1.57 = A4 ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_46_Collect__cong,axiom,
% 2.83/1.57 ! [A: $tType,P3: A > $o,Q: A > $o] :
% 2.83/1.57 ( ! [X3: A] :
% 2.83/1.57 ( ( P3 @ X3 )
% 2.83/1.57 = ( Q @ X3 ) )
% 2.83/1.57 => ( ( collect @ A @ P3 )
% 2.83/1.57 = ( collect @ A @ Q ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_47_ext,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.83/1.57 ( ! [X3: A] :
% 2.83/1.57 ( ( F @ X3 )
% 2.83/1.57 = ( G @ X3 ) )
% 2.83/1.57 => ( F = G ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_48_obtain__shortest__path,axiom,
% 2.83/1.57 ! [U: nat,V: nat] :
% 2.83/1.57 ( ( connected @ capacity @ c @ U @ V )
% 2.83/1.57 => ~ ! [P4: list @ ( product_prod @ nat @ nat )] :
% 2.83/1.57 ~ ( isShortestPath @ capacity @ c @ U @ P4 @ V ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_49_assms_I2_J,axiom,
% 2.83/1.57 member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ u @ v ) @ ( set2 @ ( product_prod @ nat @ nat ) @ p ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_50_isPath__ex__edge1,axiom,
% 2.83/1.57 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
% 2.83/1.57 ( ( isPath @ capacity @ c @ U @ P @ V )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( ( U1 != U )
% 2.83/1.57 => ? [U2: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U2 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_51_isPath__ex__edge2,axiom,
% 2.83/1.57 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
% 2.83/1.57 ( ( isPath @ capacity @ c @ U @ P @ V )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( ( V1 != V )
% 2.83/1.57 => ? [V22: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V1 @ V22 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_52_min__dist__le,axiom,
% 2.83/1.57 ! [Src: nat,V: nat,D5: nat] :
% 2.83/1.57 ( ( connected @ capacity @ c @ Src @ V )
% 2.83/1.57 => ( ( ord_less_eq @ nat @ D5 @ ( min_dist @ capacity @ c @ Src @ V ) )
% 2.83/1.57 => ? [V5: nat] :
% 2.83/1.57 ( ( connected @ capacity @ c @ Src @ V5 )
% 2.83/1.57 & ( ( min_dist @ capacity @ c @ Src @ V5 )
% 2.83/1.57 = D5 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_53_min__distI2,axiom,
% 2.83/1.57 ! [V: nat,V2: nat,Q: nat > $o] :
% 2.83/1.57 ( ( connected @ capacity @ c @ V @ V2 )
% 2.83/1.57 => ( ! [D6: nat] :
% 2.83/1.57 ( ( dist @ capacity @ c @ V @ D6 @ V2 )
% 2.83/1.57 => ( ! [D7: nat] :
% 2.83/1.57 ( ( dist @ capacity @ c @ V @ D7 @ V2 )
% 2.83/1.57 => ( ord_less_eq @ nat @ D6 @ D7 ) )
% 2.83/1.57 => ( Q @ D6 ) ) )
% 2.83/1.57 => ( Q @ ( min_dist @ capacity @ c @ V @ V2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_54_isShortestPath__def,axiom,
% 2.83/1.57 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.83/1.57 ( ( isShortestPath @ capacity @ c @ U @ P @ V )
% 2.83/1.57 = ( ( isPath @ capacity @ c @ U @ P @ V )
% 2.83/1.57 & ! [P5: list @ ( product_prod @ nat @ nat )] :
% 2.83/1.57 ( ( isPath @ capacity @ c @ U @ P5 @ V )
% 2.83/1.57 => ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_55_isShortestPath__level__edge_I1_J,axiom,
% 2.83/1.57 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 2.83/1.57 ( ( isShortestPath @ capacity @ c @ S @ P @ T2 )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( connected @ capacity @ c @ S @ U ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_56_isShortestPath__level__edge_I2_J,axiom,
% 2.83/1.57 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 2.83/1.57 ( ( isShortestPath @ capacity @ c @ S @ P @ T2 )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( connected @ capacity @ c @ U @ V ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_57_isShortestPath__level__edge_I3_J,axiom,
% 2.83/1.57 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 2.83/1.57 ( ( isShortestPath @ capacity @ c @ S @ P @ T2 )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( connected @ capacity @ c @ V @ T2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_58_add__le__cancel__left,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere236663937imp_le @ A )
% 2.83/1.57 => ! [C: A,A2: A,B2: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 2.83/1.57 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_59_add__le__cancel__right,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere236663937imp_le @ A )
% 2.83/1.57 => ! [A2: A,C: A,B2: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 2.83/1.57 = ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_60_isShortestPath__level__edge_I6_J,axiom,
% 2.83/1.57 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 2.83/1.57 ( ( isShortestPath @ capacity @ c @ S @ P @ T2 )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( ( min_dist @ capacity @ c @ S @ T2 )
% 2.83/1.57 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( min_dist @ capacity @ c @ S @ U ) @ ( one_one @ nat ) ) @ ( min_dist @ capacity @ c @ V @ T2 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_61_isShortestPath__level__edge_I5_J,axiom,
% 2.83/1.57 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 2.83/1.57 ( ( isShortestPath @ capacity @ c @ S @ P @ T2 )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( ( min_dist @ capacity @ c @ U @ T2 )
% 2.83/1.57 = ( plus_plus @ nat @ ( one_one @ nat ) @ ( min_dist @ capacity @ c @ V @ T2 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_62_isShortestPath__level__edge_I4_J,axiom,
% 2.83/1.57 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 2.83/1.57 ( ( isShortestPath @ capacity @ c @ S @ P @ T2 )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( ( min_dist @ capacity @ c @ S @ V )
% 2.83/1.57 = ( plus_plus @ nat @ ( min_dist @ capacity @ c @ S @ U ) @ ( one_one @ nat ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_63_connected__refl,axiom,
% 2.83/1.57 ! [V: nat] : ( connected @ capacity @ c @ V @ V ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_64_connected__distI,axiom,
% 2.83/1.57 ! [V: nat,D: nat,V2: nat] :
% 2.83/1.57 ( ( dist @ capacity @ c @ V @ D @ V2 )
% 2.83/1.57 => ( connected @ capacity @ c @ V @ V2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_65_Graph_OisPath__ex__edge1,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
% 2.83/1.57 ( ( isPath @ Capacity @ C @ U @ P @ V )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( ( U1 != U )
% 2.83/1.57 => ? [U2: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U2 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_66_Graph_OisPath__ex__edge2,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat] :
% 2.83/1.57 ( ( isPath @ Capacity @ C @ U @ P @ V )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( ( V1 != V )
% 2.83/1.57 => ? [V22: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ V1 @ V22 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_67_Graph_Oobtain__shortest__path,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat] :
% 2.83/1.57 ( ( connected @ Capacity @ C @ U @ V )
% 2.83/1.57 => ~ ! [P4: list @ ( product_prod @ nat @ nat )] :
% 2.83/1.57 ~ ( isShortestPath @ Capacity @ C @ U @ P4 @ V ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_68_Graph_OisShortestPath_Ocong,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ( ( isShortestPath @ Capacity )
% 2.83/1.57 = ( isShortestPath @ Capacity ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_69_Graph_Oconnected__refl,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat] : ( connected @ Capacity @ C @ V @ V ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_70_Graph_Oconnected_Ocong,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ( ( connected @ Capacity )
% 2.83/1.57 = ( connected @ Capacity ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_71_Graph_Omin__dist__le,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,Src: nat,V: nat,D5: nat] :
% 2.83/1.57 ( ( connected @ Capacity @ C @ Src @ V )
% 2.83/1.57 => ( ( ord_less_eq @ nat @ D5 @ ( min_dist @ Capacity @ C @ Src @ V ) )
% 2.83/1.57 => ? [V5: nat] :
% 2.83/1.57 ( ( connected @ Capacity @ C @ Src @ V5 )
% 2.83/1.57 & ( ( min_dist @ Capacity @ C @ Src @ V5 )
% 2.83/1.57 = D5 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_72_Graph_OisShortestPath__level__edge_I1_J,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 2.83/1.57 ( ( isShortestPath @ Capacity @ C @ S @ P @ T2 )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( connected @ Capacity @ C @ S @ U ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_73_Graph_OisShortestPath__level__edge_I2_J,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 2.83/1.57 ( ( isShortestPath @ Capacity @ C @ S @ P @ T2 )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( connected @ Capacity @ C @ U @ V ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_74_Graph_OisShortestPath__level__edge_I3_J,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 2.83/1.57 ( ( isShortestPath @ Capacity @ C @ S @ P @ T2 )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( connected @ Capacity @ C @ V @ T2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_75_Graph_Oconnected__def,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ( ( connected @ Capacity )
% 2.83/1.57 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U3: nat,V3: nat] :
% 2.83/1.57 ? [P2: list @ ( product_prod @ nat @ nat )] : ( isPath @ Capacity @ C2 @ U3 @ P2 @ V3 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_76_Graph_Oconnected__by__dist,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ( ( connected @ Capacity )
% 2.83/1.57 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,V3: nat,V4: nat] :
% 2.83/1.57 ? [D3: nat] : ( dist @ Capacity @ C2 @ V3 @ D3 @ V4 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_77_Graph_Oconnected__distI,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,D: nat,V2: nat] :
% 2.83/1.57 ( ( dist @ Capacity @ C @ V @ D @ V2 )
% 2.83/1.57 => ( connected @ Capacity @ C @ V @ V2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_78_Graph_OshortestPath__is__path,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.83/1.57 ( ( isShortestPath @ Capacity @ C @ U @ P @ V )
% 2.83/1.57 => ( isPath @ Capacity @ C @ U @ P @ V ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_79_Graph_OisShortestPath__level__edge_I6_J,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 2.83/1.57 ( ( isShortestPath @ Capacity @ C @ S @ P @ T2 )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( ( min_dist @ Capacity @ C @ S @ T2 )
% 2.83/1.57 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( min_dist @ Capacity @ C @ S @ U ) @ ( one_one @ nat ) ) @ ( min_dist @ Capacity @ C @ V @ T2 ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_80_Graph_OisShortestPath__level__edge_I5_J,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 2.83/1.57 ( ( isShortestPath @ Capacity @ C @ S @ P @ T2 )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( ( min_dist @ Capacity @ C @ U @ T2 )
% 2.83/1.57 = ( plus_plus @ nat @ ( one_one @ nat ) @ ( min_dist @ Capacity @ C @ V @ T2 ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_81_Graph_OisShortestPath__level__edge_I4_J,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,U: nat,V: nat] :
% 2.83/1.57 ( ( isShortestPath @ Capacity @ C @ S @ P @ T2 )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U @ V ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( ( min_dist @ Capacity @ C @ S @ V )
% 2.83/1.57 = ( plus_plus @ nat @ ( min_dist @ Capacity @ C @ S @ U ) @ ( one_one @ nat ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_82_Graph_Omin__distI2,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,V2: nat,Q: nat > $o] :
% 2.83/1.57 ( ( connected @ Capacity @ C @ V @ V2 )
% 2.83/1.57 => ( ! [D6: nat] :
% 2.83/1.57 ( ( dist @ Capacity @ C @ V @ D6 @ V2 )
% 2.83/1.57 => ( ! [D7: nat] :
% 2.83/1.57 ( ( dist @ Capacity @ C @ V @ D7 @ V2 )
% 2.83/1.57 => ( ord_less_eq @ nat @ D6 @ D7 ) )
% 2.83/1.57 => ( Q @ D6 ) ) )
% 2.83/1.57 => ( Q @ ( min_dist @ Capacity @ C @ V @ V2 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_83_Graph_OisShortestPath__def,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ( ( isShortestPath @ Capacity )
% 2.83/1.57 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U3: nat,P2: list @ ( product_prod @ nat @ nat ),V3: nat] :
% 2.83/1.57 ( ( isPath @ Capacity @ C2 @ U3 @ P2 @ V3 )
% 2.83/1.57 & ! [P5: list @ ( product_prod @ nat @ nat )] :
% 2.83/1.57 ( ( isPath @ Capacity @ C2 @ U3 @ P5 @ V3 )
% 2.83/1.57 => ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P5 ) ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_84_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere779506340up_add @ A )
% 2.83/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.83/1.57 ( ( ( ord_less_eq @ A @ I @ J )
% 2.83/1.57 & ( K = L ) )
% 2.83/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_85_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere779506340up_add @ A )
% 2.83/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.83/1.57 ( ( ( I = J )
% 2.83/1.57 & ( ord_less_eq @ A @ K @ L ) )
% 2.83/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_86_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere779506340up_add @ A )
% 2.83/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.83/1.57 ( ( ( ord_less_eq @ A @ I @ J )
% 2.83/1.57 & ( ord_less_eq @ A @ K @ L ) )
% 2.83/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_87_add__mono,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere779506340up_add @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A,D: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.57 => ( ( ord_less_eq @ A @ C @ D )
% 2.83/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_88_add__left__mono,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere779506340up_add @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_89_less__eqE,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( canoni770627133id_add @ A )
% 2.83/1.57 => ! [A2: A,B2: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.57 => ~ ! [C3: A] :
% 2.83/1.57 ( B2
% 2.83/1.57 != ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_90_add__right__mono,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere779506340up_add @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.57 => ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_91_le__iff__add,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( canoni770627133id_add @ A )
% 2.83/1.57 => ( ( ord_less_eq @ A )
% 2.83/1.57 = ( ^ [A3: A,B3: A] :
% 2.83/1.57 ? [C2: A] :
% 2.83/1.57 ( B3
% 2.83/1.57 = ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_92_add__le__imp__le__left,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere236663937imp_le @ A )
% 2.83/1.57 => ! [C: A,A2: A,B2: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 2.83/1.57 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_93_add__le__imp__le__right,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere236663937imp_le @ A )
% 2.83/1.57 => ! [A2: A,C: A,B2: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 2.83/1.57 => ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_94_Graph_Omin__dist__minD,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,D: nat,V2: nat] :
% 2.83/1.57 ( ( dist @ Capacity @ C @ V @ D @ V2 )
% 2.83/1.57 => ( ord_less_eq @ nat @ ( min_dist @ Capacity @ C @ V @ V2 ) @ D ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_95_Graph_Omin__distI__eq,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,D: nat,V2: nat] :
% 2.83/1.57 ( ( dist @ Capacity @ C @ V @ D @ V2 )
% 2.83/1.57 => ( ! [D4: nat] :
% 2.83/1.57 ( ( dist @ Capacity @ C @ V @ D4 @ V2 )
% 2.83/1.57 => ( ord_less_eq @ nat @ D @ D4 ) )
% 2.83/1.57 => ( ( min_dist @ Capacity @ C @ V @ V2 )
% 2.83/1.57 = D ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_96_Graph_Omin__dist__is__dist,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,V2: nat] :
% 2.83/1.57 ( ( connected @ Capacity @ C @ V @ V2 )
% 2.83/1.57 => ( dist @ Capacity @ C @ V @ ( min_dist @ Capacity @ C @ V @ V2 ) @ V2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_97_Graph_OisShortestPath__min__dist__def,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ( ( isShortestPath @ Capacity )
% 2.83/1.57 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U3: nat,P2: list @ ( product_prod @ nat @ nat ),V3: nat] :
% 2.83/1.57 ( ( isPath @ Capacity @ C2 @ U3 @ P2 @ V3 )
% 2.83/1.57 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 )
% 2.83/1.57 = ( min_dist @ Capacity @ C2 @ U3 @ V3 ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_98_Graph_OisPath_Ocong,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ( ( isPath @ Capacity )
% 2.83/1.57 = ( isPath @ Capacity ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_99_Graph_Omin__dist_Ocong,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ( ( min_dist @ Capacity )
% 2.83/1.57 = ( min_dist @ Capacity ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_100_Graph_Odist_Ocong,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ( ( dist @ Capacity )
% 2.83/1.57 = ( dist @ Capacity ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_101_Graph_Odist__trans,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 2.83/1.57 ( ( dist @ Capacity @ C @ U @ D1 @ W )
% 2.83/1.57 => ( ( dist @ Capacity @ C @ W @ D2 @ V )
% 2.83/1.57 => ( dist @ Capacity @ C @ U @ ( plus_plus @ nat @ D1 @ D2 ) @ V ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_102_nat__add__left__cancel__le,axiom,
% 2.83/1.57 ! [K: nat,M: nat,N: nat] :
% 2.83/1.57 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 2.83/1.57 = ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_103_isShortestPath__alt,axiom,
% 2.83/1.57 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat] :
% 2.83/1.57 ( ( isShortestPath @ capacity @ c @ U @ P @ V )
% 2.83/1.57 = ( ( isSimplePath @ capacity @ c @ U @ P @ V )
% 2.83/1.57 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P )
% 2.83/1.57 = ( min_dist @ capacity @ c @ U @ V ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_104_min__dist__less,axiom,
% 2.83/1.57 ! [Src: nat,V: nat,D: nat,D5: nat] :
% 2.83/1.57 ( ( connected @ capacity @ c @ Src @ V )
% 2.83/1.57 => ( ( ( min_dist @ capacity @ c @ Src @ V )
% 2.83/1.57 = D )
% 2.83/1.57 => ( ( ord_less @ nat @ D5 @ D )
% 2.83/1.57 => ? [V5: nat] :
% 2.83/1.57 ( ( connected @ capacity @ c @ Src @ V5 )
% 2.83/1.57 & ( ( min_dist @ capacity @ c @ Src @ V5 )
% 2.83/1.57 = D5 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_105_isSPath__no__selfloop,axiom,
% 2.83/1.57 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat] :
% 2.83/1.57 ( ( isSimplePath @ capacity @ c @ U @ P @ V )
% 2.83/1.57 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_106_isSPath__sg__incoming,axiom,
% 2.83/1.57 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,U22: nat] :
% 2.83/1.57 ( ( isSimplePath @ capacity @ c @ U @ P @ V )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( ( U1 != U22 )
% 2.83/1.57 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U22 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_107_isSPath__sg__outgoing,axiom,
% 2.83/1.57 ! [U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,V23: nat] :
% 2.83/1.57 ( ( isSimplePath @ capacity @ c @ U @ P @ V )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( ( V1 != V23 )
% 2.83/1.57 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V23 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_108_old_Oprod_Oinject,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B5: B] :
% 2.83/1.57 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.83/1.57 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.83/1.57 = ( ( A2 = A5 )
% 2.83/1.57 & ( B2 = B5 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_109_prod_Oinject,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 2.83/1.57 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 2.83/1.57 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 2.83/1.57 = ( ( X1 = Y1 )
% 2.83/1.57 & ( X22 = Y2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_110_isSPath__pathLE,axiom,
% 2.83/1.57 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat] :
% 2.83/1.57 ( ( isPath @ capacity @ c @ S @ P @ T2 )
% 2.83/1.57 => ? [P6: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ capacity @ c @ S @ P6 @ T2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_111_shortestPath__is__simple,axiom,
% 2.83/1.57 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat] :
% 2.83/1.57 ( ( isShortestPath @ capacity @ c @ S @ P @ T2 )
% 2.83/1.57 => ( isSimplePath @ capacity @ c @ S @ P @ T2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_112_add__less__cancel__right,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere236663937imp_le @ A )
% 2.83/1.57 => ! [A2: A,C: A,B2: A] :
% 2.83/1.57 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 2.83/1.57 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_113_add__less__cancel__left,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere236663937imp_le @ A )
% 2.83/1.57 => ! [C: A,A2: A,B2: A] :
% 2.83/1.57 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 2.83/1.57 = ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_114_nat__add__left__cancel__less,axiom,
% 2.83/1.57 ! [K: nat,M: nat,N: nat] :
% 2.83/1.57 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
% 2.83/1.57 = ( ord_less @ nat @ M @ N ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_115_Graph_OisSimplePath_Ocong,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ( ( isSimplePath @ Capacity )
% 2.83/1.57 = ( isSimplePath @ Capacity ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_116_less__mono__imp__le__mono,axiom,
% 2.83/1.57 ! [F: nat > nat,I: nat,J: nat] :
% 2.83/1.57 ( ! [I2: nat,J2: nat] :
% 2.83/1.57 ( ( ord_less @ nat @ I2 @ J2 )
% 2.83/1.57 => ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 2.83/1.57 => ( ( ord_less_eq @ nat @ I @ J )
% 2.83/1.57 => ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_117_le__neq__implies__less,axiom,
% 2.83/1.57 ! [M: nat,N: nat] :
% 2.83/1.57 ( ( ord_less_eq @ nat @ M @ N )
% 2.83/1.57 => ( ( M != N )
% 2.83/1.57 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_118_less__or__eq__imp__le,axiom,
% 2.83/1.57 ! [M: nat,N: nat] :
% 2.83/1.57 ( ( ( ord_less @ nat @ M @ N )
% 2.83/1.57 | ( M = N ) )
% 2.83/1.57 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_119_le__eq__less__or__eq,axiom,
% 2.83/1.57 ( ( ord_less_eq @ nat )
% 2.83/1.57 = ( ^ [M2: nat,N2: nat] :
% 2.83/1.57 ( ( ord_less @ nat @ M2 @ N2 )
% 2.83/1.57 | ( M2 = N2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_120_less__imp__le__nat,axiom,
% 2.83/1.57 ! [M: nat,N: nat] :
% 2.83/1.57 ( ( ord_less @ nat @ M @ N )
% 2.83/1.57 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_121_nat__less__le,axiom,
% 2.83/1.57 ( ( ord_less @ nat )
% 2.83/1.57 = ( ^ [M2: nat,N2: nat] :
% 2.83/1.57 ( ( ord_less_eq @ nat @ M2 @ N2 )
% 2.83/1.57 & ( M2 != N2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_122_add__lessD1,axiom,
% 2.83/1.57 ! [I: nat,J: nat,K: nat] :
% 2.83/1.57 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
% 2.83/1.57 => ( ord_less @ nat @ I @ K ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_123_add__less__mono,axiom,
% 2.83/1.57 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.83/1.57 ( ( ord_less @ nat @ I @ J )
% 2.83/1.57 => ( ( ord_less @ nat @ K @ L )
% 2.83/1.57 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_124_not__add__less1,axiom,
% 2.83/1.57 ! [I: nat,J: nat] :
% 2.83/1.57 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_125_not__add__less2,axiom,
% 2.83/1.57 ! [J: nat,I: nat] :
% 2.83/1.57 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_126_add__less__mono1,axiom,
% 2.83/1.57 ! [I: nat,J: nat,K: nat] :
% 2.83/1.57 ( ( ord_less @ nat @ I @ J )
% 2.83/1.57 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_127_trans__less__add1,axiom,
% 2.83/1.57 ! [I: nat,J: nat,M: nat] :
% 2.83/1.57 ( ( ord_less @ nat @ I @ J )
% 2.83/1.57 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_128_trans__less__add2,axiom,
% 2.83/1.57 ! [I: nat,J: nat,M: nat] :
% 2.83/1.57 ( ( ord_less @ nat @ I @ J )
% 2.83/1.57 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_129_less__add__eq__less,axiom,
% 2.83/1.57 ! [K: nat,L: nat,M: nat,N: nat] :
% 2.83/1.57 ( ( ord_less @ nat @ K @ L )
% 2.83/1.57 => ( ( ( plus_plus @ nat @ M @ L )
% 2.83/1.57 = ( plus_plus @ nat @ K @ N ) )
% 2.83/1.57 => ( ord_less @ nat @ M @ N ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_130_nat__neq__iff,axiom,
% 2.83/1.57 ! [M: nat,N: nat] :
% 2.83/1.57 ( ( M != N )
% 2.83/1.57 = ( ( ord_less @ nat @ M @ N )
% 2.83/1.57 | ( ord_less @ nat @ N @ M ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_131_less__not__refl,axiom,
% 2.83/1.57 ! [N: nat] :
% 2.83/1.57 ~ ( ord_less @ nat @ N @ N ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_132_less__not__refl2,axiom,
% 2.83/1.57 ! [N: nat,M: nat] :
% 2.83/1.57 ( ( ord_less @ nat @ N @ M )
% 2.83/1.57 => ( M != N ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_133_less__not__refl3,axiom,
% 2.83/1.57 ! [S: nat,T2: nat] :
% 2.83/1.57 ( ( ord_less @ nat @ S @ T2 )
% 2.83/1.57 => ( S != T2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_134_measure__induct,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType] :
% 2.83/1.57 ( ( wellorder @ B )
% 2.83/1.57 => ! [F: A > B,P3: A > $o,A2: A] :
% 2.83/1.57 ( ! [X3: A] :
% 2.83/1.57 ( ! [Y: A] :
% 2.83/1.57 ( ( ord_less @ B @ ( F @ Y ) @ ( F @ X3 ) )
% 2.83/1.57 => ( P3 @ Y ) )
% 2.83/1.57 => ( P3 @ X3 ) )
% 2.83/1.57 => ( P3 @ A2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_135_less__irrefl__nat,axiom,
% 2.83/1.57 ! [N: nat] :
% 2.83/1.57 ~ ( ord_less @ nat @ N @ N ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_136_nat__less__induct,axiom,
% 2.83/1.57 ! [P3: nat > $o,N: nat] :
% 2.83/1.57 ( ! [N3: nat] :
% 2.83/1.57 ( ! [M3: nat] :
% 2.83/1.57 ( ( ord_less @ nat @ M3 @ N3 )
% 2.83/1.57 => ( P3 @ M3 ) )
% 2.83/1.57 => ( P3 @ N3 ) )
% 2.83/1.57 => ( P3 @ N ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_137_infinite__descent,axiom,
% 2.83/1.57 ! [P3: nat > $o,N: nat] :
% 2.83/1.57 ( ! [N3: nat] :
% 2.83/1.57 ( ~ ( P3 @ N3 )
% 2.83/1.57 => ? [M3: nat] :
% 2.83/1.57 ( ( ord_less @ nat @ M3 @ N3 )
% 2.83/1.57 & ~ ( P3 @ M3 ) ) )
% 2.83/1.57 => ( P3 @ N ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_138_linorder__neqE__nat,axiom,
% 2.83/1.57 ! [X: nat,Y3: nat] :
% 2.83/1.57 ( ( X != Y3 )
% 2.83/1.57 => ( ~ ( ord_less @ nat @ X @ Y3 )
% 2.83/1.57 => ( ord_less @ nat @ Y3 @ X ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_139_measure__induct__rule,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType] :
% 2.83/1.57 ( ( wellorder @ B )
% 2.83/1.57 => ! [F: A > B,P3: A > $o,A2: A] :
% 2.83/1.57 ( ! [X3: A] :
% 2.83/1.57 ( ! [Y: A] :
% 2.83/1.57 ( ( ord_less @ B @ ( F @ Y ) @ ( F @ X3 ) )
% 2.83/1.57 => ( P3 @ Y ) )
% 2.83/1.57 => ( P3 @ X3 ) )
% 2.83/1.57 => ( P3 @ A2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_140_infinite__descent__measure,axiom,
% 2.83/1.57 ! [A: $tType,P3: A > $o,V6: A > nat,X: A] :
% 2.83/1.57 ( ! [X3: A] :
% 2.83/1.57 ( ~ ( P3 @ X3 )
% 2.83/1.57 => ? [Y: A] :
% 2.83/1.57 ( ( ord_less @ nat @ ( V6 @ Y ) @ ( V6 @ X3 ) )
% 2.83/1.57 & ~ ( P3 @ Y ) ) )
% 2.83/1.57 => ( P3 @ X ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_141_add__less__imp__less__right,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere236663937imp_le @ A )
% 2.83/1.57 => ! [A2: A,C: A,B2: A] :
% 2.83/1.57 ( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
% 2.83/1.57 => ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_142_add__less__imp__less__left,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere236663937imp_le @ A )
% 2.83/1.57 => ! [C: A,A2: A,B2: A] :
% 2.83/1.57 ( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
% 2.83/1.57 => ( ord_less @ A @ A2 @ B2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_143_add__strict__right__mono,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere223160158up_add @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A] :
% 2.83/1.57 ( ( ord_less @ A @ A2 @ B2 )
% 2.83/1.57 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_144_add__strict__left__mono,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere223160158up_add @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A] :
% 2.83/1.57 ( ( ord_less @ A @ A2 @ B2 )
% 2.83/1.57 => ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_145_add__strict__mono,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( strict2144017051up_add @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A,D: A] :
% 2.83/1.57 ( ( ord_less @ A @ A2 @ B2 )
% 2.83/1.57 => ( ( ord_less @ A @ C @ D )
% 2.83/1.57 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_146_add__mono__thms__linordered__field_I1_J,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere223160158up_add @ A )
% 2.83/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.83/1.57 ( ( ( ord_less @ A @ I @ J )
% 2.83/1.57 & ( K = L ) )
% 2.83/1.57 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_147_add__mono__thms__linordered__field_I2_J,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere223160158up_add @ A )
% 2.83/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.83/1.57 ( ( ( I = J )
% 2.83/1.57 & ( ord_less @ A @ K @ L ) )
% 2.83/1.57 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_148_add__mono__thms__linordered__field_I5_J,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere223160158up_add @ A )
% 2.83/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.83/1.57 ( ( ( ord_less @ A @ I @ J )
% 2.83/1.57 & ( ord_less @ A @ K @ L ) )
% 2.83/1.57 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_149_mono__nat__linear__lb,axiom,
% 2.83/1.57 ! [F: nat > nat,M: nat,K: nat] :
% 2.83/1.57 ( ! [M4: nat,N3: nat] :
% 2.83/1.57 ( ( ord_less @ nat @ M4 @ N3 )
% 2.83/1.57 => ( ord_less @ nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
% 2.83/1.57 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_150_Graph_OisSPath__pathLE,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat] :
% 2.83/1.57 ( ( isPath @ Capacity @ C @ S @ P @ T2 )
% 2.83/1.57 => ? [P6: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ Capacity @ C @ S @ P6 @ T2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_151_Graph_OshortestPath__is__simple,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat] :
% 2.83/1.57 ( ( isShortestPath @ Capacity @ C @ S @ P @ T2 )
% 2.83/1.57 => ( isSimplePath @ Capacity @ C @ S @ P @ T2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_152_add__mono__thms__linordered__field_I4_J,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere223160158up_add @ A )
% 2.83/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.83/1.57 ( ( ( ord_less_eq @ A @ I @ J )
% 2.83/1.57 & ( ord_less @ A @ K @ L ) )
% 2.83/1.57 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_153_add__mono__thms__linordered__field_I3_J,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere223160158up_add @ A )
% 2.83/1.57 => ! [I: A,J: A,K: A,L: A] :
% 2.83/1.57 ( ( ( ord_less @ A @ I @ J )
% 2.83/1.57 & ( ord_less_eq @ A @ K @ L ) )
% 2.83/1.57 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_154_add__le__less__mono,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere223160158up_add @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A,D: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.57 => ( ( ord_less @ A @ C @ D )
% 2.83/1.57 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_155_add__less__le__mono,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ordere223160158up_add @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A,D: A] :
% 2.83/1.57 ( ( ord_less @ A @ A2 @ B2 )
% 2.83/1.57 => ( ( ord_less_eq @ A @ C @ D )
% 2.83/1.57 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_156_Graph_Omin__dist__less,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,Src: nat,V: nat,D: nat,D5: nat] :
% 2.83/1.57 ( ( connected @ Capacity @ C @ Src @ V )
% 2.83/1.57 => ( ( ( min_dist @ Capacity @ C @ Src @ V )
% 2.83/1.57 = D )
% 2.83/1.57 => ( ( ord_less @ nat @ D5 @ D )
% 2.83/1.57 => ? [V5: nat] :
% 2.83/1.57 ( ( connected @ Capacity @ C @ Src @ V5 )
% 2.83/1.57 & ( ( min_dist @ Capacity @ C @ Src @ V5 )
% 2.83/1.57 = D5 ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_157_Graph_OisSPath__no__selfloop,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat] :
% 2.83/1.57 ( ( isSimplePath @ Capacity @ C @ U @ P @ V )
% 2.83/1.57 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ U1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_158_Graph_OisSPath__sg__incoming,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,U22: nat] :
% 2.83/1.57 ( ( isSimplePath @ Capacity @ C @ U @ P @ V )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( ( U1 != U22 )
% 2.83/1.57 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U22 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_159_Graph_OisSPath__sg__outgoing,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P: list @ ( product_prod @ nat @ nat ),V: nat,U1: nat,V1: nat,V23: nat] :
% 2.83/1.57 ( ( isSimplePath @ Capacity @ C @ U @ P @ V )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V1 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ( ( V1 != V23 )
% 2.83/1.57 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ U1 @ V23 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_160_surj__pair,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,P: product_prod @ A @ B] :
% 2.83/1.57 ? [X3: A,Y4: B] :
% 2.83/1.57 ( P
% 2.83/1.57 = ( product_Pair @ A @ B @ X3 @ Y4 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_161_prod__cases,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
% 2.83/1.57 ( ! [A6: A,B6: B] : ( P3 @ ( product_Pair @ A @ B @ A6 @ B6 ) )
% 2.83/1.57 => ( P3 @ P ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_162_Pair__inject,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B5: B] :
% 2.83/1.57 ( ( ( product_Pair @ A @ B @ A2 @ B2 )
% 2.83/1.57 = ( product_Pair @ A @ B @ A5 @ B5 ) )
% 2.83/1.57 => ~ ( ( A2 = A5 )
% 2.83/1.57 => ( B2 != B5 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_163_prod__cases3,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,C4: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C4 )] :
% 2.83/1.57 ~ ! [A6: A,B6: B,C3: C4] :
% 2.83/1.57 ( Y3
% 2.83/1.57 != ( product_Pair @ A @ ( product_prod @ B @ C4 ) @ A6 @ ( product_Pair @ B @ C4 @ B6 @ C3 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_164_prod__cases4,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,C4: $tType,D8: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ D8 ) )] :
% 2.83/1.57 ~ ! [A6: A,B6: B,C3: C4,D6: D8] :
% 2.83/1.57 ( Y3
% 2.83/1.57 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C4 @ D8 ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C4 @ D8 ) @ B6 @ ( product_Pair @ C4 @ D8 @ C3 @ D6 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_165_prod__cases5,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,C4: $tType,D8: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ E ) ) )] :
% 2.83/1.57 ~ ! [A6: A,B6: B,C3: C4,D6: D8,E2: E] :
% 2.83/1.57 ( Y3
% 2.83/1.57 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ E ) ) @ B6 @ ( product_Pair @ C4 @ ( product_prod @ D8 @ E ) @ C3 @ ( product_Pair @ D8 @ E @ D6 @ E2 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_166_prod__cases6,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,C4: $tType,D8: $tType,E: $tType,F2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) ) )] :
% 2.83/1.57 ~ ! [A6: A,B6: B,C3: C4,D6: D8,E2: E,F3: F2] :
% 2.83/1.57 ( Y3
% 2.83/1.57 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) ) @ B6 @ ( product_Pair @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D8 @ ( product_prod @ E @ F2 ) @ D6 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_167_prod__cases7,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,C4: $tType,D8: $tType,E: $tType,F2: $tType,G2: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 2.83/1.57 ~ ! [A6: A,B6: B,C3: C4,D6: D8,E2: E,F3: F2,G3: G2] :
% 2.83/1.57 ( Y3
% 2.83/1.57 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B6 @ ( product_Pair @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D6 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_168_prod__induct3,axiom,
% 2.83/1.57 ! [C4: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ C4 ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C4 )] :
% 2.83/1.57 ( ! [A6: A,B6: B,C3: C4] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ C4 ) @ A6 @ ( product_Pair @ B @ C4 @ B6 @ C3 ) ) )
% 2.83/1.57 => ( P3 @ X ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_169_prod__induct4,axiom,
% 2.83/1.57 ! [D8: $tType,C4: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ D8 ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ D8 ) )] :
% 2.83/1.57 ( ! [A6: A,B6: B,C3: C4,D6: D8] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C4 @ D8 ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C4 @ D8 ) @ B6 @ ( product_Pair @ C4 @ D8 @ C3 @ D6 ) ) ) )
% 2.83/1.57 => ( P3 @ X ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_170_prod__induct5,axiom,
% 2.83/1.57 ! [E: $tType,D8: $tType,C4: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ E ) ) )] :
% 2.83/1.57 ( ! [A6: A,B6: B,C3: C4,D6: D8,E2: E] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ E ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ E ) ) @ B6 @ ( product_Pair @ C4 @ ( product_prod @ D8 @ E ) @ C3 @ ( product_Pair @ D8 @ E @ D6 @ E2 ) ) ) ) )
% 2.83/1.57 => ( P3 @ X ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_171_prod__induct6,axiom,
% 2.83/1.57 ! [F2: $tType,E: $tType,D8: $tType,C4: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) ) )] :
% 2.83/1.57 ( ! [A6: A,B6: B,C3: C4,D6: D8,E2: E,F3: F2] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) ) @ B6 @ ( product_Pair @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D8 @ ( product_prod @ E @ F2 ) @ D6 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
% 2.83/1.57 => ( P3 @ X ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_172_prod__induct7,axiom,
% 2.83/1.57 ! [G2: $tType,F2: $tType,E: $tType,D8: $tType,C4: $tType,B: $tType,A: $tType,P3: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
% 2.83/1.57 ( ! [A6: A,B6: B,C3: C4,D6: D8,E2: E,F3: F2,G3: G2] : ( P3 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A6 @ ( product_Pair @ B @ ( product_prod @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B6 @ ( product_Pair @ C4 @ ( product_prod @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D8 @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D6 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
% 2.83/1.57 => ( P3 @ X ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_173_old_Oprod_Oexhaust,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 2.83/1.57 ~ ! [A6: A,B6: B] :
% 2.83/1.57 ( Y3
% 2.83/1.57 != ( product_Pair @ A @ B @ A6 @ B6 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_174_old_Oprod_Oinducts,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,P3: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.83/1.57 ( ! [A6: A,B6: B] : ( P3 @ ( product_Pair @ A @ B @ A6 @ B6 ) )
% 2.83/1.57 => ( P3 @ Prod ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_175_size__neq__size__imp__neq,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( size @ A )
% 2.83/1.57 => ! [X: A,Y3: A] :
% 2.83/1.57 ( ( ( size_size @ A @ X )
% 2.83/1.57 != ( size_size @ A @ Y3 ) )
% 2.83/1.57 => ( X != Y3 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_176_le__refl,axiom,
% 2.83/1.57 ! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_177_le__trans,axiom,
% 2.83/1.57 ! [I: nat,J: nat,K: nat] :
% 2.83/1.57 ( ( ord_less_eq @ nat @ I @ J )
% 2.83/1.57 => ( ( ord_less_eq @ nat @ J @ K )
% 2.83/1.57 => ( ord_less_eq @ nat @ I @ K ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_178_eq__imp__le,axiom,
% 2.83/1.57 ! [M: nat,N: nat] :
% 2.83/1.57 ( ( M = N )
% 2.83/1.57 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_179_le__antisym,axiom,
% 2.83/1.57 ! [M: nat,N: nat] :
% 2.83/1.57 ( ( ord_less_eq @ nat @ M @ N )
% 2.83/1.57 => ( ( ord_less_eq @ nat @ N @ M )
% 2.83/1.57 => ( M = N ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_180_nat__le__linear,axiom,
% 2.83/1.57 ! [M: nat,N: nat] :
% 2.83/1.57 ( ( ord_less_eq @ nat @ M @ N )
% 2.83/1.57 | ( ord_less_eq @ nat @ N @ M ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_181_Nat_Oex__has__greatest__nat,axiom,
% 2.83/1.57 ! [P3: nat > $o,K: nat,B2: nat] :
% 2.83/1.57 ( ( P3 @ K )
% 2.83/1.57 => ( ! [Y4: nat] :
% 2.83/1.57 ( ( P3 @ Y4 )
% 2.83/1.57 => ( ord_less_eq @ nat @ Y4 @ B2 ) )
% 2.83/1.57 => ? [X3: nat] :
% 2.83/1.57 ( ( P3 @ X3 )
% 2.83/1.57 & ! [Y: nat] :
% 2.83/1.57 ( ( P3 @ Y )
% 2.83/1.57 => ( ord_less_eq @ nat @ Y @ X3 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_182_Graph_OisShortestPath__alt,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ( ( isShortestPath @ Capacity )
% 2.83/1.57 = ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U3: nat,P2: list @ ( product_prod @ nat @ nat ),V3: nat] :
% 2.83/1.57 ( ( isSimplePath @ Capacity @ C2 @ U3 @ P2 @ V3 )
% 2.83/1.57 & ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 )
% 2.83/1.57 = ( min_dist @ Capacity @ C2 @ U3 @ V3 ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_183_nat__le__iff__add,axiom,
% 2.83/1.57 ( ( ord_less_eq @ nat )
% 2.83/1.57 = ( ^ [M2: nat,N2: nat] :
% 2.83/1.57 ? [K2: nat] :
% 2.83/1.57 ( N2
% 2.83/1.57 = ( plus_plus @ nat @ M2 @ K2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_184_trans__le__add2,axiom,
% 2.83/1.57 ! [I: nat,J: nat,M: nat] :
% 2.83/1.57 ( ( ord_less_eq @ nat @ I @ J )
% 2.83/1.57 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_185_trans__le__add1,axiom,
% 2.83/1.57 ! [I: nat,J: nat,M: nat] :
% 2.83/1.57 ( ( ord_less_eq @ nat @ I @ J )
% 2.83/1.57 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_186_add__le__mono1,axiom,
% 2.83/1.57 ! [I: nat,J: nat,K: nat] :
% 2.83/1.57 ( ( ord_less_eq @ nat @ I @ J )
% 2.83/1.57 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_187_add__le__mono,axiom,
% 2.83/1.57 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.83/1.57 ( ( ord_less_eq @ nat @ I @ J )
% 2.83/1.57 => ( ( ord_less_eq @ nat @ K @ L )
% 2.83/1.57 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_188_le__Suc__ex,axiom,
% 2.83/1.57 ! [K: nat,L: nat] :
% 2.83/1.57 ( ( ord_less_eq @ nat @ K @ L )
% 2.83/1.57 => ? [N3: nat] :
% 2.83/1.57 ( L
% 2.83/1.57 = ( plus_plus @ nat @ K @ N3 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_189_add__leD2,axiom,
% 2.83/1.57 ! [M: nat,K: nat,N: nat] :
% 2.83/1.57 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 2.83/1.57 => ( ord_less_eq @ nat @ K @ N ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_190_add__leD1,axiom,
% 2.83/1.57 ! [M: nat,K: nat,N: nat] :
% 2.83/1.57 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 2.83/1.57 => ( ord_less_eq @ nat @ M @ N ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_191_le__add2,axiom,
% 2.83/1.57 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_192_le__add1,axiom,
% 2.83/1.57 ! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_193_add__leE,axiom,
% 2.83/1.57 ! [M: nat,K: nat,N: nat] :
% 2.83/1.57 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
% 2.83/1.57 => ~ ( ( ord_less_eq @ nat @ M @ N )
% 2.83/1.57 => ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_194_isSPath__nt__parallel,axiom,
% 2.83/1.57 ! [S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,E3: product_prod @ nat @ nat] :
% 2.83/1.57 ( ( isSimplePath @ capacity @ c @ S @ P @ T2 )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ E3 @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ E3 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_195_discrete,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( unique1598680935umeral @ A )
% 2.83/1.57 => ( ( ord_less @ A )
% 2.83/1.57 = ( ^ [A3: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_196_old_Oprod_Orec,axiom,
% 2.83/1.57 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
% 2.83/1.57 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
% 2.83/1.57 = ( F1 @ A2 @ B2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_197_fold__atLeastAtMost__nat_Oinduct,axiom,
% 2.83/1.57 ! [A: $tType,P3: ( nat > A > A ) > nat > nat > A > $o,A0: nat > A > A,A1: nat,A22: nat,A32: A] :
% 2.83/1.57 ( ! [F3: nat > A > A,A6: nat,B6: nat,Acc: A] :
% 2.83/1.57 ( ( ~ ( ord_less @ nat @ B6 @ A6 )
% 2.83/1.57 => ( P3 @ F3 @ ( plus_plus @ nat @ A6 @ ( one_one @ nat ) ) @ B6 @ ( F3 @ A6 @ Acc ) ) )
% 2.83/1.57 => ( P3 @ F3 @ A6 @ B6 @ Acc ) )
% 2.83/1.57 => ( P3 @ A0 @ A1 @ A22 @ A32 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_198_ex__has__greatest__nat__lemma,axiom,
% 2.83/1.57 ! [A: $tType,P3: A > $o,K: A,F: A > nat,N: nat] :
% 2.83/1.57 ( ( P3 @ K )
% 2.83/1.57 => ( ! [X3: A] :
% 2.83/1.57 ( ( P3 @ X3 )
% 2.83/1.57 => ? [Y: A] :
% 2.83/1.57 ( ( P3 @ Y )
% 2.83/1.57 & ~ ( ord_less_eq @ nat @ ( F @ Y ) @ ( F @ X3 ) ) ) )
% 2.83/1.57 => ? [Y4: A] :
% 2.83/1.57 ( ( P3 @ Y4 )
% 2.83/1.57 & ~ ( ord_less @ nat @ ( F @ Y4 ) @ ( plus_plus @ nat @ ( F @ K ) @ N ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_199_swap__swap,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType,P: product_prod @ A @ B] :
% 2.83/1.57 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
% 2.83/1.57 = P ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_200_swap__simp,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType,X: B,Y3: A] :
% 2.83/1.57 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) )
% 2.83/1.57 = ( product_Pair @ A @ B @ Y3 @ X ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_201_ex__has__least__nat,axiom,
% 2.83/1.57 ! [A: $tType,P3: A > $o,K: A,M: A > nat] :
% 2.83/1.57 ( ( P3 @ K )
% 2.83/1.57 => ? [X3: A] :
% 2.83/1.57 ( ( P3 @ X3 )
% 2.83/1.57 & ! [Y: A] :
% 2.83/1.57 ( ( P3 @ Y )
% 2.83/1.57 => ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_202_bounded__Max__nat,axiom,
% 2.83/1.57 ! [P3: nat > $o,X: nat,M5: nat] :
% 2.83/1.57 ( ( P3 @ X )
% 2.83/1.57 => ( ! [X3: nat] :
% 2.83/1.57 ( ( P3 @ X3 )
% 2.83/1.57 => ( ord_less_eq @ nat @ X3 @ M5 ) )
% 2.83/1.57 => ~ ! [M4: nat] :
% 2.83/1.57 ( ( P3 @ M4 )
% 2.83/1.57 => ~ ! [X4: nat] :
% 2.83/1.57 ( ( P3 @ X4 )
% 2.83/1.57 => ( ord_less_eq @ nat @ X4 @ M4 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_203_fold__atLeastAtMost__nat_Ocases,axiom,
% 2.83/1.57 ! [A: $tType,X: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
% 2.83/1.57 ~ ! [F3: nat > A > A,A6: nat,B6: nat,Acc: A] :
% 2.83/1.57 ( X
% 2.83/1.57 != ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A6 @ ( product_Pair @ nat @ A @ B6 @ Acc ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_204_Graph_OisSPath__nt__parallel,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat ),T2: nat,E3: product_prod @ nat @ nat] :
% 2.83/1.57 ( ( isSimplePath @ Capacity @ C @ S @ P @ T2 )
% 2.83/1.57 => ( ( member @ ( product_prod @ nat @ nat ) @ E3 @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) )
% 2.83/1.57 => ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ E3 ) @ ( set2 @ ( product_prod @ nat @ nat ) @ P ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_205_Lattices__Big_Oex__has__greatest__nat,axiom,
% 2.83/1.57 ! [A: $tType,P3: A > $o,K: A,F: A > nat,B2: nat] :
% 2.83/1.57 ( ( P3 @ K )
% 2.83/1.57 => ( ! [Y4: A] :
% 2.83/1.57 ( ( P3 @ Y4 )
% 2.83/1.57 => ( ord_less @ nat @ ( F @ Y4 ) @ B2 ) )
% 2.83/1.57 => ? [X3: A] :
% 2.83/1.57 ( ( P3 @ X3 )
% 2.83/1.57 & ! [Y: A] :
% 2.83/1.57 ( ( P3 @ Y )
% 2.83/1.57 => ( ord_less_eq @ nat @ ( F @ Y ) @ ( F @ X3 ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_206_less__add__one,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_semidom @ A )
% 2.83/1.57 => ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_207_add__mono1,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linord1659791738miring @ A )
% 2.83/1.57 => ! [A2: A,B2: A] :
% 2.83/1.57 ( ( ord_less @ A @ A2 @ B2 )
% 2.83/1.57 => ( ord_less @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_208_linorder__neqE__linordered__idom,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ! [X: A,Y3: A] :
% 2.83/1.57 ( ( X != Y3 )
% 2.83/1.57 => ( ~ ( ord_less @ A @ X @ Y3 )
% 2.83/1.57 => ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_209_isPath_Osimps_I1_J,axiom,
% 2.83/1.57 ! [U: nat,V: nat] :
% 2.83/1.57 ( ( isPath @ capacity @ c @ U @ ( nil @ ( product_prod @ nat @ nat ) ) @ V )
% 2.83/1.57 = ( U = V ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_210_order__refl,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( preorder @ A )
% 2.83/1.57 => ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_211_simplePath__empty__conv,axiom,
% 2.83/1.57 ! [S: nat,T2: nat] :
% 2.83/1.57 ( ( isSimplePath @ capacity @ c @ S @ ( nil @ ( product_prod @ nat @ nat ) ) @ T2 )
% 2.83/1.57 = ( S = T2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_212_simplePath__same__conv,axiom,
% 2.83/1.57 ! [S: nat,P: list @ ( product_prod @ nat @ nat )] :
% 2.83/1.57 ( ( isSimplePath @ capacity @ c @ S @ P @ S )
% 2.83/1.57 = ( P
% 2.83/1.57 = ( nil @ ( product_prod @ nat @ nat ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_213_Graph_OsimplePath__empty__conv,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,T2: nat] :
% 2.83/1.57 ( ( isSimplePath @ Capacity @ C @ S @ ( nil @ ( product_prod @ nat @ nat ) ) @ T2 )
% 2.83/1.57 = ( S = T2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_214_Graph_OsimplePath__same__conv,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P: list @ ( product_prod @ nat @ nat )] :
% 2.83/1.57 ( ( isSimplePath @ Capacity @ C @ S @ P @ S )
% 2.83/1.57 = ( P
% 2.83/1.57 = ( nil @ ( product_prod @ nat @ nat ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_215_Graph_OisPath_Osimps_I1_J,axiom,
% 2.83/1.57 ! [Capacity: $tType] :
% 2.83/1.57 ( ( linordered_idom @ Capacity )
% 2.83/1.57 => ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat] :
% 2.83/1.57 ( ( isPath @ Capacity @ C @ U @ ( nil @ ( product_prod @ nat @ nat ) ) @ V )
% 2.83/1.57 = ( U = V ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_216_dual__order_Oantisym,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [B2: A,A2: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.83/1.57 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.57 => ( A2 = B2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_217_dual__order_Oeq__iff,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 2.83/1.57 = ( ^ [A3: A,B3: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.83/1.57 & ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_218_dual__order_Otrans,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [B2: A,A2: A,C: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.83/1.57 => ( ( ord_less_eq @ A @ C @ B2 )
% 2.83/1.57 => ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_219_linorder__wlog,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linorder @ A )
% 2.83/1.57 => ! [P3: A > A > $o,A2: A,B2: A] :
% 2.83/1.57 ( ! [A6: A,B6: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ A6 @ B6 )
% 2.83/1.57 => ( P3 @ A6 @ B6 ) )
% 2.83/1.57 => ( ! [A6: A,B6: A] :
% 2.83/1.57 ( ( P3 @ B6 @ A6 )
% 2.83/1.57 => ( P3 @ A6 @ B6 ) )
% 2.83/1.57 => ( P3 @ A2 @ B2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_220_dual__order_Orefl,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_221_order__trans,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( preorder @ A )
% 2.83/1.57 => ! [X: A,Y3: A,Z2: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ X @ Y3 )
% 2.83/1.57 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 2.83/1.57 => ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_222_order__class_Oorder_Oantisym,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [A2: A,B2: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.57 => ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.83/1.57 => ( A2 = B2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_223_ord__le__eq__trans,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ord @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.57 => ( ( B2 = C )
% 2.83/1.57 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_224_ord__eq__le__trans,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( ord @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A] :
% 2.83/1.57 ( ( A2 = B2 )
% 2.83/1.57 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.83/1.57 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_225_order__class_Oorder_Oeq__iff,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 2.83/1.57 = ( ^ [A3: A,B3: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.83/1.57 & ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_226_antisym__conv,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [Y3: A,X: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ Y3 @ X )
% 2.83/1.57 => ( ( ord_less_eq @ A @ X @ Y3 )
% 2.83/1.57 = ( X = Y3 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_227_le__cases3,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linorder @ A )
% 2.83/1.57 => ! [X: A,Y3: A,Z2: A] :
% 2.83/1.57 ( ( ( ord_less_eq @ A @ X @ Y3 )
% 2.83/1.57 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 2.83/1.57 => ( ( ( ord_less_eq @ A @ Y3 @ X )
% 2.83/1.57 => ~ ( ord_less_eq @ A @ X @ Z2 ) )
% 2.83/1.57 => ( ( ( ord_less_eq @ A @ X @ Z2 )
% 2.83/1.57 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 2.83/1.57 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 2.83/1.57 => ~ ( ord_less_eq @ A @ Y3 @ X ) )
% 2.83/1.57 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 2.83/1.57 => ~ ( ord_less_eq @ A @ Z2 @ X ) )
% 2.83/1.57 => ~ ( ( ord_less_eq @ A @ Z2 @ X )
% 2.83/1.57 => ~ ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_228_order_Otrans,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.57 => ( ( ord_less_eq @ A @ B2 @ C )
% 2.83/1.57 => ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_229_le__cases,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linorder @ A )
% 2.83/1.57 => ! [X: A,Y3: A] :
% 2.83/1.57 ( ~ ( ord_less_eq @ A @ X @ Y3 )
% 2.83/1.57 => ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_230_eq__refl,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( preorder @ A )
% 2.83/1.57 => ! [X: A,Y3: A] :
% 2.83/1.57 ( ( X = Y3 )
% 2.83/1.57 => ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_231_linear,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linorder @ A )
% 2.83/1.57 => ! [X: A,Y3: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ X @ Y3 )
% 2.83/1.57 | ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_232_antisym,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [X: A,Y3: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ X @ Y3 )
% 2.83/1.57 => ( ( ord_less_eq @ A @ Y3 @ X )
% 2.83/1.57 => ( X = Y3 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_233_eq__iff,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ( ( ^ [Y5: A,Z: A] : ( Y5 = Z ) )
% 2.83/1.57 = ( ^ [X2: A,Y6: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ X2 @ Y6 )
% 2.83/1.57 & ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_234_ord__le__eq__subst,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType] :
% 2.83/1.57 ( ( ( ord @ B )
% 2.83/1.57 & ( ord @ A ) )
% 2.83/1.57 => ! [A2: A,B2: A,F: A > B,C: B] :
% 2.83/1.57 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.57 => ( ( ( F @ B2 )
% 2.83/1.57 = C )
% 2.83/1.57 => ( ! [X3: A,Y4: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 2.83/1.57 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.83/1.57 => ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_235_ord__eq__le__subst,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType] :
% 2.83/1.57 ( ( ( ord @ B )
% 2.83/1.57 & ( ord @ A ) )
% 2.83/1.57 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.83/1.57 ( ( A2
% 2.83/1.57 = ( F @ B2 ) )
% 2.83/1.57 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.83/1.57 => ( ! [X3: B,Y4: B] :
% 2.83/1.57 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 2.83/1.57 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.83/1.57 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_236_order__subst2,axiom,
% 2.83/1.57 ! [A: $tType,C4: $tType] :
% 2.83/1.57 ( ( ( order @ C4 )
% 2.83/1.57 & ( order @ A ) )
% 2.83/1.57 => ! [A2: A,B2: A,F: A > C4,C: C4] :
% 2.83/1.57 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.83/1.57 => ( ( ord_less_eq @ C4 @ ( F @ B2 ) @ C )
% 2.83/1.57 => ( ! [X3: A,Y4: A] :
% 2.83/1.57 ( ( ord_less_eq @ A @ X3 @ Y4 )
% 2.83/1.57 => ( ord_less_eq @ C4 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.83/1.57 => ( ord_less_eq @ C4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_237_order__subst1,axiom,
% 2.83/1.57 ! [A: $tType,B: $tType] :
% 2.83/1.57 ( ( ( order @ B )
% 2.83/1.57 & ( order @ A ) )
% 2.83/1.57 => ! [A2: A,F: B > A,B2: B,C: B] :
% 2.83/1.57 ( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
% 2.83/1.57 => ( ( ord_less_eq @ B @ B2 @ C )
% 2.83/1.57 => ( ! [X3: B,Y4: B] :
% 2.83/1.57 ( ( ord_less_eq @ B @ X3 @ Y4 )
% 2.83/1.57 => ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
% 2.83/1.57 => ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_238_le__fun__def,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType] :
% 2.83/1.57 ( ( ord @ B )
% 2.83/1.57 => ( ( ord_less_eq @ ( A > B ) )
% 2.83/1.57 = ( ^ [F4: A > B,G4: A > B] :
% 2.83/1.57 ! [X2: A] : ( ord_less_eq @ B @ ( F4 @ X2 ) @ ( G4 @ X2 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_239_le__funI,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType] :
% 2.83/1.57 ( ( ord @ B )
% 2.83/1.57 => ! [F: A > B,G: A > B] :
% 2.83/1.57 ( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 2.83/1.57 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_240_le__funE,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType] :
% 2.83/1.57 ( ( ord @ B )
% 2.83/1.57 => ! [F: A > B,G: A > B,X: A] :
% 2.83/1.57 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.83/1.57 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_241_le__funD,axiom,
% 2.83/1.57 ! [B: $tType,A: $tType] :
% 2.83/1.57 ( ( ord @ B )
% 2.83/1.57 => ! [F: A > B,G: A > B,X: A] :
% 2.83/1.57 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.83/1.57 => ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_242_dual__order_Ostrict__implies__not__eq,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [B2: A,A2: A] :
% 2.83/1.57 ( ( ord_less @ A @ B2 @ A2 )
% 2.83/1.57 => ( A2 != B2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_243_order_Ostrict__implies__not__eq,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [A2: A,B2: A] :
% 2.83/1.57 ( ( ord_less @ A @ A2 @ B2 )
% 2.83/1.57 => ( A2 != B2 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_244_not__less__iff__gr__or__eq,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linorder @ A )
% 2.83/1.57 => ! [X: A,Y3: A] :
% 2.83/1.57 ( ( ~ ( ord_less @ A @ X @ Y3 ) )
% 2.83/1.57 = ( ( ord_less @ A @ Y3 @ X )
% 2.83/1.57 | ( X = Y3 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_245_dual__order_Ostrict__trans,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [B2: A,A2: A,C: A] :
% 2.83/1.57 ( ( ord_less @ A @ B2 @ A2 )
% 2.83/1.57 => ( ( ord_less @ A @ C @ B2 )
% 2.83/1.57 => ( ord_less @ A @ C @ A2 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_246_linorder__less__wlog,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linorder @ A )
% 2.83/1.57 => ! [P3: A > A > $o,A2: A,B2: A] :
% 2.83/1.57 ( ! [A6: A,B6: A] :
% 2.83/1.57 ( ( ord_less @ A @ A6 @ B6 )
% 2.83/1.57 => ( P3 @ A6 @ B6 ) )
% 2.83/1.57 => ( ! [A6: A] : ( P3 @ A6 @ A6 )
% 2.83/1.57 => ( ! [A6: A,B6: A] :
% 2.83/1.57 ( ( P3 @ B6 @ A6 )
% 2.83/1.57 => ( P3 @ A6 @ B6 ) )
% 2.83/1.57 => ( P3 @ A2 @ B2 ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_247_exists__least__iff,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( wellorder @ A )
% 2.83/1.57 => ( ( ^ [P7: A > $o] :
% 2.83/1.57 ? [X5: A] : ( P7 @ X5 ) )
% 2.83/1.57 = ( ^ [P8: A > $o] :
% 2.83/1.57 ? [N2: A] :
% 2.83/1.57 ( ( P8 @ N2 )
% 2.83/1.57 & ! [M2: A] :
% 2.83/1.57 ( ( ord_less @ A @ M2 @ N2 )
% 2.83/1.57 => ~ ( P8 @ M2 ) ) ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_248_less__imp__not__less,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( preorder @ A )
% 2.83/1.57 => ! [X: A,Y3: A] :
% 2.83/1.57 ( ( ord_less @ A @ X @ Y3 )
% 2.83/1.57 => ~ ( ord_less @ A @ Y3 @ X ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_249_order_Ostrict__trans,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [A2: A,B2: A,C: A] :
% 2.83/1.57 ( ( ord_less @ A @ A2 @ B2 )
% 2.83/1.57 => ( ( ord_less @ A @ B2 @ C )
% 2.83/1.57 => ( ord_less @ A @ A2 @ C ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_250_dual__order_Oirrefl,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [A2: A] :
% 2.83/1.57 ~ ( ord_less @ A @ A2 @ A2 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_251_linorder__cases,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linorder @ A )
% 2.83/1.57 => ! [X: A,Y3: A] :
% 2.83/1.57 ( ~ ( ord_less @ A @ X @ Y3 )
% 2.83/1.57 => ( ( X != Y3 )
% 2.83/1.57 => ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_252_less__imp__triv,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( preorder @ A )
% 2.83/1.57 => ! [X: A,Y3: A,P3: $o] :
% 2.83/1.57 ( ( ord_less @ A @ X @ Y3 )
% 2.83/1.57 => ( ( ord_less @ A @ Y3 @ X )
% 2.83/1.57 => P3 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_253_less__imp__not__eq2,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( order @ A )
% 2.83/1.57 => ! [X: A,Y3: A] :
% 2.83/1.57 ( ( ord_less @ A @ X @ Y3 )
% 2.83/1.57 => ( Y3 != X ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(fact_254_antisym__conv3,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linorder @ A )
% 2.83/1.57 => ! [Y3: A,X: A] :
% 2.83/1.57 ( ~ ( ord_less @ A @ Y3 @ X )
% 2.83/1.57 => ( ( ~ ( ord_less @ A @ X @ Y3 ) )
% 2.83/1.57 = ( X = Y3 ) ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___HOL_Otype,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( type @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___Groups_Oone,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( one @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___Orderings_Oord,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( ord @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___Orderings_Oorder,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( order @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___Groups_Ogroup__add,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( group_add @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___Orderings_Olinorder,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( linorder @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___Orderings_Opreorder,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( preorder @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___Groups_Osemigroup__add,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( semigroup_add @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___Groups_Ocomm__monoid__add,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( comm_monoid_add @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___Groups_Oab__semigroup__add,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( ab_semigroup_add @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___Rings_Olinordered__semidom,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( linordered_semidom @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___Groups_Ocancel__semigroup__add,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( cancel_semigroup_add @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( ordere779506340up_add @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___Rings_Olinordered__nonzero__semiring,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( linord1659791738miring @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( ordere236663937imp_le @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( ordere223160158up_add @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(subcl_Rings_Olinordered__idom___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 2.83/1.57 ! [A: $tType] :
% 2.83/1.57 ( ( linordered_idom @ A )
% 2.83/1.57 => ( strict2144017051up_add @ A ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.83/1.57 ! [A7: $tType,A8: $tType] :
% 2.83/1.57 ( ( preorder @ A8 )
% 2.83/1.57 => ( preorder @ ( A7 > A8 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.83/1.57 ! [A7: $tType,A8: $tType] :
% 2.83/1.57 ( ( order @ A8 )
% 2.83/1.57 => ( order @ ( A7 > A8 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_fun___Orderings_Oord,axiom,
% 2.83/1.57 ! [A7: $tType,A8: $tType] :
% 2.83/1.57 ( ( ord @ A8 )
% 2.83/1.57 => ( ord @ ( A7 > A8 ) ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral,axiom,
% 2.83/1.57 unique1598680935umeral @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 2.83/1.57 strict2144017051up_add @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 2.83/1.57 ordere223160158up_add @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 2.83/1.57 ordere236663937imp_le @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 2.83/1.57 canoni770627133id_add @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
% 2.83/1.57 linord1659791738miring @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
% 2.83/1.57 ordere779506340up_add @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
% 2.83/1.57 cancel_semigroup_add @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
% 2.83/1.57 linordered_semidom @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
% 2.83/1.57 ab_semigroup_add @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
% 2.83/1.57 comm_monoid_add @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
% 2.83/1.57 semigroup_add @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
% 2.83/1.57 wellorder @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
% 2.83/1.57 preorder @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
% 2.83/1.57 linorder @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
% 2.83/1.57 order @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
% 2.83/1.57 ord @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Groups_Oone,axiom,
% 2.83/1.57 one @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Nat_Onat___Nat_Osize,axiom,
% 2.83/1.57 size @ nat ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 2.83/1.57 ! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
% 2.83/1.57 ! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
% 2.83/1.57 ! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
% 2.83/1.57 preorder @ $o ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
% 2.83/1.57 linorder @ $o ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
% 2.83/1.57 order @ $o ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
% 2.83/1.57 ord @ $o ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_List_Olist___Nat_Osize_11,axiom,
% 2.83/1.57 ! [A7: $tType] : ( size @ ( list @ A7 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(tcon_Product__Type_Oprod___Nat_Osize_12,axiom,
% 2.83/1.57 ! [A7: $tType,A8: $tType] : ( size @ ( product_prod @ A7 @ A8 ) ) ).
% 2.83/1.57
% 2.83/1.57 thf(tfree_0,hypothesis,
% 2.83/1.57 linordered_idom @ capacity ).
% 2.83/1.57
% 2.83/1.57 thf(conj_0,conjecture,
% 2.83/1.57 ( ( min_dist @ capacity @ c @ u @ t )
% 2.83/1.57 = ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2 ) ) ) ).
% 2.83/1.57 ------------------------
% 2.90/1.60 % (2444)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_2221 for (2999ds/2Mi)
% 2.90/1.60 % (2444)Instruction limit reached!
% 2.90/1.60 % (2444)------------------------------
% 2.90/1.60 % (2444)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.60 % (2444)Termination reason: Unknown
% 2.90/1.60 % (2444)Termination phase: shuffling
% 2.90/1.60
% 2.90/1.60 % (2444)Memory used [KB]: 1407
% 2.90/1.60 % (2444)Time elapsed: 0.003 s
% 2.90/1.60 % (2444)Instructions burned: 3 (million)
% 2.90/1.60 % (2444)------------------------------
% 2.90/1.60 % (2444)------------------------------
% 2.90/1.61 % (2441)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_2221 for (2999ds/183Mi)
% 2.90/1.61 % (2442)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_2221 for (2999ds/4Mi)
% 2.90/1.61 % (2443)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_2221 for (2999ds/27Mi)
% 2.90/1.61 % (2445)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_2221 for (2999ds/2Mi)
% 2.90/1.61 % (2446)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_2221 for (2999ds/275Mi)
% 2.90/1.61 % (2447)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_2221 for (2999ds/18Mi)
% 2.90/1.61 % (2445)Instruction limit reached!
% 2.90/1.61 % (2445)------------------------------
% 2.90/1.61 % (2445)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.61 % (2445)Termination reason: Unknown
% 2.90/1.61 % (2445)Termination phase: shuffling
% 2.90/1.61
% 2.90/1.61 % (2445)Memory used [KB]: 1407
% 2.90/1.61 % (2445)Time elapsed: 0.004 s
% 2.90/1.61 % (2445)Instructions burned: 3 (million)
% 2.90/1.61 % (2445)------------------------------
% 2.90/1.61 % (2445)------------------------------
% 2.90/1.61 % (2442)Instruction limit reached!
% 2.90/1.61 % (2442)------------------------------
% 2.90/1.61 % (2442)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.61 % (2442)Termination reason: Unknown
% 2.90/1.61 % (2442)Termination phase: shuffling
% 2.90/1.61
% 2.90/1.61 % (2442)Memory used [KB]: 1407
% 2.90/1.61 % (2442)Time elapsed: 0.004 s
% 2.90/1.61 % (2442)Instructions burned: 4 (million)
% 2.90/1.61 % (2442)------------------------------
% 2.90/1.61 % (2442)------------------------------
% 2.90/1.61 % (2448)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_2221 for (2999ds/3Mi)
% 2.90/1.61 % (2448)Instruction limit reached!
% 2.90/1.61 % (2448)------------------------------
% 2.90/1.61 % (2448)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.61 % (2448)Termination reason: Unknown
% 2.90/1.61 % (2448)Termination phase: shuffling
% 2.90/1.61
% 2.90/1.61 % (2448)Memory used [KB]: 1407
% 2.90/1.61 % (2448)Time elapsed: 0.003 s
% 2.90/1.61 % (2448)Instructions burned: 5 (million)
% 2.90/1.61 % (2448)------------------------------
% 2.90/1.61 % (2448)------------------------------
% 2.90/1.62 % (2447)Instruction limit reached!
% 2.90/1.62 % (2447)------------------------------
% 2.90/1.62 % (2447)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.62 % (2447)Termination reason: Unknown
% 2.90/1.62 % (2447)Termination phase: shuffling
% 2.90/1.62
% 2.90/1.62 % (2447)Memory used [KB]: 1663
% 2.90/1.62 % (2447)Time elapsed: 0.013 s
% 2.90/1.62 % (2447)Instructions burned: 18 (million)
% 2.90/1.62 % (2447)------------------------------
% 2.90/1.62 % (2447)------------------------------
% 2.90/1.62 % (2451)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_2221 for (2999ds/3Mi)
% 2.90/1.63 % (2451)Instruction limit reached!
% 2.90/1.63 % (2451)------------------------------
% 2.90/1.63 % (2451)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.63 % (2451)Termination reason: Unknown
% 2.90/1.63 % (2451)Termination phase: shuffling
% 2.90/1.63
% 2.90/1.63 % (2451)Memory used [KB]: 1407
% 2.90/1.63 % (2451)Time elapsed: 0.002 s
% 2.90/1.63 % (2451)Instructions burned: 3 (million)
% 2.90/1.63 % (2451)------------------------------
% 2.90/1.63 % (2451)------------------------------
% 2.90/1.63 % (2443)Instruction limit reached!
% 2.90/1.63 % (2443)------------------------------
% 2.90/1.63 % (2443)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.63 % (2443)Termination reason: Unknown
% 2.90/1.63 % (2443)Termination phase: Property scanning
% 2.90/1.63
% 2.90/1.63 % (2443)Memory used [KB]: 1918
% 2.90/1.63 % (2443)Time elapsed: 0.018 s
% 2.90/1.63 % (2443)Instructions burned: 27 (million)
% 2.90/1.63 % (2443)------------------------------
% 2.90/1.63 % (2443)------------------------------
% 2.90/1.63 % (2449)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_2221 for (2999ds/37Mi)
% 2.90/1.63 % (2450)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_2221 for (2999ds/15Mi)
% 2.90/1.63 % (2453)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_2221 for (2999ds/7Mi)
% 2.90/1.64 % (2452)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_2221 for (2999ds/1041Mi)
% 2.90/1.64 % (2453)Instruction limit reached!
% 2.90/1.64 % (2453)------------------------------
% 2.90/1.64 % (2453)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.64 % (2453)Termination reason: Unknown
% 2.90/1.64 % (2453)Termination phase: shuffling
% 2.90/1.64
% 2.90/1.64 % (2453)Memory used [KB]: 1535
% 2.90/1.64 % (2453)Time elapsed: 0.004 s
% 2.90/1.64 % (2453)Instructions burned: 8 (million)
% 2.90/1.64 % (2453)------------------------------
% 2.90/1.64 % (2453)------------------------------
% 2.90/1.64 % (2454)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_2221 for (2999ds/16Mi)
% 2.90/1.64 % (2450)Instruction limit reached!
% 2.90/1.64 % (2450)------------------------------
% 2.90/1.64 % (2450)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.64 % (2450)Termination reason: Unknown
% 2.90/1.64 % (2450)Termination phase: shuffling
% 2.90/1.64
% 2.90/1.64 % (2450)Memory used [KB]: 1663
% 2.90/1.64 % (2450)Time elapsed: 0.012 s
% 2.90/1.64 % (2450)Instructions burned: 16 (million)
% 2.90/1.64 % (2450)------------------------------
% 2.90/1.64 % (2450)------------------------------
% 2.90/1.64 % (2454)Instruction limit reached!
% 2.90/1.64 % (2454)------------------------------
% 2.90/1.64 % (2454)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.64 % (2454)Termination reason: Unknown
% 2.90/1.64 % (2454)Termination phase: shuffling
% 2.90/1.64
% 2.90/1.64 % (2454)Memory used [KB]: 1663
% 2.90/1.64 % (2454)Time elapsed: 0.008 s
% 2.90/1.64 % (2454)Instructions burned: 17 (million)
% 2.90/1.64 % (2454)------------------------------
% 2.90/1.64 % (2454)------------------------------
% 2.90/1.65 % (2455)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_2221 for (2999ds/3Mi)
% 2.90/1.65 % (2455)Instruction limit reached!
% 2.90/1.65 % (2455)------------------------------
% 2.90/1.65 % (2455)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.65 % (2455)Termination reason: Unknown
% 2.90/1.65 % (2455)Termination phase: shuffling
% 2.90/1.65
% 2.90/1.65 % (2455)Memory used [KB]: 1407
% 2.90/1.65 % (2455)Time elapsed: 0.002 s
% 2.90/1.65 % (2455)Instructions burned: 3 (million)
% 2.90/1.65 % (2455)------------------------------
% 2.90/1.65 % (2455)------------------------------
% 2.90/1.65 % (2449)Instruction limit reached!
% 2.90/1.65 % (2449)------------------------------
% 2.90/1.65 % (2449)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.65 % (2449)Termination reason: Unknown
% 2.90/1.65 % (2449)Termination phase: shuffling
% 2.90/1.65
% 2.90/1.65 % (2449)Memory used [KB]: 2174
% 2.90/1.65 % (2449)Time elapsed: 0.022 s
% 2.90/1.65 % (2449)Instructions burned: 38 (million)
% 2.90/1.65 % (2449)------------------------------
% 2.90/1.65 % (2449)------------------------------
% 2.90/1.66 % (2457)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_2221 for (2999ds/7Mi)
% 2.90/1.66 % (2446)Refutation not found, incomplete strategy
% 2.90/1.66 % (2446)------------------------------
% 2.90/1.66 % (2446)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.66 % (2446)Termination reason: Refutation not found, incomplete strategy
% 2.90/1.66
% 2.90/1.66
% 2.90/1.66 % (2446)Memory used [KB]: 6908
% 2.90/1.66 % (2446)Time elapsed: 0.046 s
% 2.90/1.66 % (2446)Instructions burned: 86 (million)
% 2.90/1.66 % (2446)------------------------------
% 2.90/1.66 % (2446)------------------------------
% 2.90/1.66 % (2458)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_2221 for (2999ds/3Mi)
% 2.90/1.66 % (2456)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_2221 for (2999ds/3Mi)
% 2.90/1.66 % (2457)Instruction limit reached!
% 2.90/1.66 % (2457)------------------------------
% 2.90/1.66 % (2457)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.66 % (2457)Termination reason: Unknown
% 2.90/1.66 % (2457)Termination phase: shuffling
% 2.90/1.66
% 2.90/1.66 % (2457)Memory used [KB]: 1535
% 2.90/1.66 % (2457)Time elapsed: 0.005 s
% 2.90/1.66 % (2457)Instructions burned: 9 (million)
% 2.90/1.66 % (2457)------------------------------
% 2.90/1.66 % (2457)------------------------------
% 2.90/1.66 % (2458)Instruction limit reached!
% 2.90/1.66 % (2458)------------------------------
% 2.90/1.66 % (2458)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.66 % (2458)Termination reason: Unknown
% 2.90/1.66 % (2458)Termination phase: shuffling
% 2.90/1.66
% 2.90/1.66 % (2458)Memory used [KB]: 1407
% 2.90/1.66 % (2458)Time elapsed: 0.002 s
% 2.90/1.66 % (2458)Instructions burned: 3 (million)
% 2.90/1.66 % (2458)------------------------------
% 2.90/1.66 % (2458)------------------------------
% 2.90/1.66 % (2456)Instruction limit reached!
% 2.90/1.66 % (2456)------------------------------
% 2.90/1.66 % (2456)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.66 % (2456)Termination reason: Unknown
% 2.90/1.66 % (2456)Termination phase: shuffling
% 2.90/1.66
% 2.90/1.66 % (2456)Memory used [KB]: 1407
% 2.90/1.66 % (2456)Time elapsed: 0.004 s
% 2.90/1.66 % (2456)Instructions burned: 3 (million)
% 2.90/1.66 % (2456)------------------------------
% 2.90/1.66 % (2456)------------------------------
% 2.90/1.66 % (2459)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_2221 for (2999ds/4Mi)
% 2.90/1.67 % (2460)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_2221 for (2999ds/18Mi)
% 2.90/1.67 % (2459)Instruction limit reached!
% 2.90/1.67 % (2459)------------------------------
% 2.90/1.67 % (2459)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.67 % (2459)Termination reason: Unknown
% 2.90/1.67 % (2459)Termination phase: shuffling
% 2.90/1.67
% 2.90/1.67 % (2459)Memory used [KB]: 1407
% 2.90/1.67 % (2459)Time elapsed: 0.004 s
% 2.90/1.67 % (2459)Instructions burned: 4 (million)
% 2.90/1.67 % (2459)------------------------------
% 2.90/1.67 % (2459)------------------------------
% 2.90/1.67 % (2462)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_2221 for (2999ds/6Mi)
% 2.90/1.67 % (2461)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_2221 for (2999ds/710Mi)
% 2.90/1.67 % (2462)Instruction limit reached!
% 2.90/1.67 % (2462)------------------------------
% 2.90/1.67 % (2462)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.67 % (2462)Termination reason: Unknown
% 2.90/1.67 % (2462)Termination phase: shuffling
% 2.90/1.67
% 2.90/1.67 % (2462)Memory used [KB]: 1535
% 2.90/1.67 % (2462)Time elapsed: 0.004 s
% 2.90/1.67 % (2462)Instructions burned: 8 (million)
% 2.90/1.67 % (2462)------------------------------
% 2.90/1.67 % (2462)------------------------------
% 2.90/1.67 % (2460)Instruction limit reached!
% 2.90/1.67 % (2460)------------------------------
% 2.90/1.67 % (2460)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.67 % (2460)Termination reason: Unknown
% 2.90/1.67 % (2460)Termination phase: shuffling
% 2.90/1.67
% 2.90/1.67 % (2460)Memory used [KB]: 1663
% 2.90/1.67 % (2460)Time elapsed: 0.007 s
% 2.90/1.67 % (2460)Instructions burned: 18 (million)
% 2.90/1.67 % (2460)------------------------------
% 2.90/1.67 % (2460)------------------------------
% 2.90/1.68 % (2463)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_2221 for (2999ds/902Mi)
% 2.90/1.68 % (2464)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_2221 for (2999ds/21Mi)
% 2.90/1.68 % (2465)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_2221 for (2999ds/5Mi)
% 2.90/1.68 % (2466)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_2221 for (2999ds/6Mi)
% 2.90/1.68 % (2465)Instruction limit reached!
% 2.90/1.68 % (2465)------------------------------
% 2.90/1.68 % (2465)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.68 % (2465)Termination reason: Unknown
% 2.90/1.68 % (2465)Termination phase: shuffling
% 2.90/1.68
% 2.90/1.68 % (2465)Memory used [KB]: 1407
% 2.90/1.68 % (2465)Time elapsed: 0.003 s
% 2.90/1.68 % (2465)Instructions burned: 6 (million)
% 2.90/1.68 % (2465)------------------------------
% 2.90/1.68 % (2465)------------------------------
% 2.90/1.68 % (2466)Instruction limit reached!
% 2.90/1.68 % (2466)------------------------------
% 2.90/1.68 % (2466)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.68 % (2466)Termination reason: Unknown
% 2.90/1.68 % (2466)Termination phase: shuffling
% 2.90/1.68
% 2.90/1.68 % (2466)Memory used [KB]: 1407
% 2.90/1.68 % (2466)Time elapsed: 0.003 s
% 2.90/1.68 % (2466)Instructions burned: 6 (million)
% 2.90/1.68 % (2466)------------------------------
% 2.90/1.68 % (2466)------------------------------
% 2.90/1.69 % (2464)Instruction limit reached!
% 2.90/1.69 % (2464)------------------------------
% 2.90/1.69 % (2464)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.69 % (2464)Termination reason: Unknown
% 2.90/1.69 % (2464)Termination phase: shuffling
% 2.90/1.69
% 2.90/1.69 % (2464)Memory used [KB]: 1791
% 2.90/1.69 % (2464)Time elapsed: 0.009 s
% 2.90/1.69 % (2464)Instructions burned: 21 (million)
% 2.90/1.69 % (2464)------------------------------
% 2.90/1.69 % (2464)------------------------------
% 2.90/1.69 % (2441)Instruction limit reached!
% 2.90/1.69 % (2441)------------------------------
% 2.90/1.69 % (2441)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.69 % (2441)Termination reason: Unknown
% 2.90/1.69 % (2441)Termination phase: Saturation
% 2.90/1.69
% 2.90/1.69 % (2441)Memory used [KB]: 7164
% 2.90/1.69 % (2441)Time elapsed: 0.079 s
% 2.90/1.69 % (2441)Instructions burned: 184 (million)
% 2.90/1.69 % (2441)------------------------------
% 2.90/1.69 % (2441)------------------------------
% 2.90/1.69 % (2467)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_2221 for (2998ds/377Mi)
% 2.90/1.69 % (2468)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_2221 for (2998ds/779Mi)
% 2.90/1.70 % (2469)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_2221 for (2998ds/19Mi)
% 2.90/1.70 % (2470)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_2221 for (2998ds/879Mi)
% 2.90/1.70 % (2469)Instruction limit reached!
% 2.90/1.70 % (2469)------------------------------
% 2.90/1.70 % (2469)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.70 % (2469)Termination reason: Unknown
% 2.90/1.70 % (2469)Termination phase: shuffling
% 2.90/1.70
% 2.90/1.70 % (2469)Memory used [KB]: 1791
% 2.90/1.70 % (2469)Time elapsed: 0.008 s
% 2.90/1.70 % (2469)Instructions burned: 21 (million)
% 2.90/1.70 % (2469)------------------------------
% 2.90/1.70 % (2469)------------------------------
% 2.90/1.71 % (2471)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_2221 for (2998ds/17Mi)
% 2.90/1.72 % (2471)Instruction limit reached!
% 2.90/1.72 % (2471)------------------------------
% 2.90/1.72 % (2471)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.72 % (2471)Termination reason: Unknown
% 2.90/1.72 % (2471)Termination phase: shuffling
% 2.90/1.72
% 2.90/1.72 % (2471)Memory used [KB]: 1663
% 2.90/1.72 % (2471)Time elapsed: 0.008 s
% 2.90/1.72 % (2471)Instructions burned: 17 (million)
% 2.90/1.72 % (2471)------------------------------
% 2.90/1.72 % (2471)------------------------------
% 2.90/1.73 % (2472)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_2221 for (2998ds/3Mi)
% 2.90/1.73 % (2472)Instruction limit reached!
% 2.90/1.73 % (2472)------------------------------
% 2.90/1.73 % (2472)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.73 % (2472)Termination reason: Unknown
% 2.90/1.73 % (2472)Termination phase: shuffling
% 2.90/1.73
% 2.90/1.73 % (2472)Memory used [KB]: 1407
% 2.90/1.73 % (2472)Time elapsed: 0.003 s
% 2.90/1.73 % (2472)Instructions burned: 4 (million)
% 2.90/1.73 % (2472)------------------------------
% 2.90/1.73 % (2472)------------------------------
% 2.90/1.73 % (2463)Refutation not found, incomplete strategy
% 2.90/1.73 % (2463)------------------------------
% 2.90/1.73 % (2463)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.73 % (2463)Termination reason: Refutation not found, incomplete strategy
% 2.90/1.73
% 2.90/1.73
% 2.90/1.73 % (2463)Memory used [KB]: 7291
% 2.90/1.73 % (2463)Time elapsed: 0.057 s
% 2.90/1.73 % (2463)Instructions burned: 122 (million)
% 2.90/1.73 % (2463)------------------------------
% 2.90/1.73 % (2463)------------------------------
% 2.90/1.74 % (2473)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_2221 for (2998ds/30Mi)
% 2.90/1.74 % (2474)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_2221 for (2998ds/127Mi)
% 2.90/1.75 % (2473)Instruction limit reached!
% 2.90/1.75 % (2473)------------------------------
% 2.90/1.75 % (2473)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.75 % (2473)Termination reason: Unknown
% 2.90/1.75 % (2473)Termination phase: Property scanning
% 2.90/1.75
% 2.90/1.75 % (2473)Memory used [KB]: 1918
% 2.90/1.75 % (2473)Time elapsed: 0.013 s
% 2.90/1.75 % (2473)Instructions burned: 32 (million)
% 2.90/1.75 % (2473)------------------------------
% 2.90/1.75 % (2473)------------------------------
% 2.90/1.76 % (2475)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_2221 for (2998ds/100Mi)
% 2.90/1.79 % (2474)Instruction limit reached!
% 2.90/1.79 % (2474)------------------------------
% 2.90/1.79 % (2474)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.79 % (2474)Termination reason: Unknown
% 2.90/1.79 % (2474)Termination phase: Saturation
% 2.90/1.79
% 2.90/1.79 % (2474)Memory used [KB]: 6908
% 2.90/1.79 % (2474)Time elapsed: 0.044 s
% 2.90/1.79 % (2474)Instructions burned: 128 (million)
% 2.90/1.79 % (2474)------------------------------
% 2.90/1.79 % (2474)------------------------------
% 2.90/1.79 % (2475)Instruction limit reached!
% 2.90/1.79 % (2475)------------------------------
% 2.90/1.79 % (2475)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.79 % (2475)Termination reason: Unknown
% 2.90/1.79 % (2475)Termination phase: Property scanning
% 2.90/1.79
% 2.90/1.79 % (2475)Memory used [KB]: 2430
% 2.90/1.79 % (2475)Time elapsed: 0.031 s
% 2.90/1.79 % (2475)Instructions burned: 101 (million)
% 2.90/1.79 % (2475)------------------------------
% 2.90/1.79 % (2475)------------------------------
% 2.90/1.80 % (2476)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_2221 for (2998ds/3Mi)
% 2.90/1.80 % (2476)Instruction limit reached!
% 2.90/1.80 % (2476)------------------------------
% 2.90/1.80 % (2476)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.80 % (2476)Termination reason: Unknown
% 2.90/1.80 % (2476)Termination phase: shuffling
% 2.90/1.80
% 2.90/1.80 % (2476)Memory used [KB]: 1407
% 2.90/1.80 % (2476)Time elapsed: 0.003 s
% 2.90/1.80 % (2476)Instructions burned: 4 (million)
% 2.90/1.80 % (2476)------------------------------
% 2.90/1.80 % (2476)------------------------------
% 2.90/1.80 % (2477)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_2221 for (2997ds/20Mi)
% 2.90/1.81 % (2478)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_2221 for (2997ds/86Mi)
% 2.90/1.81 % (2477)Instruction limit reached!
% 2.90/1.81 % (2477)------------------------------
% 2.90/1.81 % (2477)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.81 % (2477)Termination reason: Unknown
% 2.90/1.81 % (2477)Termination phase: shuffling
% 2.90/1.81
% 2.90/1.81 % (2477)Memory used [KB]: 1663
% 2.90/1.81 % (2477)Time elapsed: 0.009 s
% 2.90/1.81 % (2477)Instructions burned: 21 (million)
% 2.90/1.81 % (2477)------------------------------
% 2.90/1.81 % (2477)------------------------------
% 2.90/1.81 % (2467)Instruction limit reached!
% 2.90/1.81 % (2467)------------------------------
% 2.90/1.81 % (2467)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.81 % (2467)Termination reason: Unknown
% 2.90/1.81 % (2467)Termination phase: Saturation
% 2.90/1.81
% 2.90/1.81 % (2467)Memory used [KB]: 8059
% 2.90/1.81 % (2467)Time elapsed: 0.122 s
% 2.90/1.81 % (2467)Instructions burned: 378 (million)
% 2.90/1.81 % (2467)------------------------------
% 2.90/1.81 % (2467)------------------------------
% 2.90/1.82 % (2479)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_2221 for (2997ds/107Mi)
% 2.90/1.82 % (2480)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_2221 for (2997ds/39Mi)
% 2.90/1.83 % (2478)Instruction limit reached!
% 2.90/1.83 % (2478)------------------------------
% 2.90/1.83 % (2478)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.83 % (2478)Termination reason: Unknown
% 2.90/1.83 % (2478)Termination phase: Property scanning
% 2.90/1.83
% 2.90/1.83 % (2478)Memory used [KB]: 2302
% 2.90/1.83 % (2478)Time elapsed: 0.029 s
% 2.90/1.83 % (2478)Instructions burned: 87 (million)
% 2.90/1.83 % (2478)------------------------------
% 2.90/1.83 % (2478)------------------------------
% 2.90/1.83 % (2480)Instruction limit reached!
% 2.90/1.83 % (2480)------------------------------
% 2.90/1.83 % (2480)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.83 % (2480)Termination reason: Unknown
% 2.90/1.83 % (2480)Termination phase: shuffling
% 2.90/1.83
% 2.90/1.83 % (2480)Memory used [KB]: 2174
% 2.90/1.83 % (2480)Time elapsed: 0.013 s
% 2.90/1.83 % (2480)Instructions burned: 42 (million)
% 2.90/1.83 % (2480)------------------------------
% 2.90/1.83 % (2480)------------------------------
% 2.90/1.84 % (2481)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_2221 for (2997ds/448Mi)
% 2.90/1.84 % (2482)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_2221 for (2997ds/46Mi)
% 2.90/1.85 % (2479)Instruction limit reached!
% 2.90/1.85 % (2479)------------------------------
% 2.90/1.85 % (2479)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.85 % (2479)Termination reason: Unknown
% 2.90/1.85 % (2479)Termination phase: Saturation
% 2.90/1.85
% 2.90/1.85 % (2479)Memory used [KB]: 6908
% 2.90/1.85 % (2479)Time elapsed: 0.036 s
% 2.90/1.85 % (2479)Instructions burned: 107 (million)
% 2.90/1.85 % (2479)------------------------------
% 2.90/1.85 % (2479)------------------------------
% 2.90/1.86 % (2482)Instruction limit reached!
% 2.90/1.86 % (2482)------------------------------
% 2.90/1.86 % (2482)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.86 % (2482)Termination reason: Unknown
% 2.90/1.86 % (2482)Termination phase: SInE selection
% 2.90/1.86
% 2.90/1.86 % (2482)Memory used [KB]: 1918
% 2.90/1.86 % (2482)Time elapsed: 0.017 s
% 2.90/1.86 % (2482)Instructions burned: 46 (million)
% 2.90/1.86 % (2482)------------------------------
% 2.90/1.86 % (2482)------------------------------
% 2.90/1.86 % (2483)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_2221 for (2997ds/98Mi)
% 2.90/1.87 % (2484)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_2221 for (2997ds/507Mi)
% 2.90/1.88 % (2461)Instruction limit reached!
% 2.90/1.88 % (2461)------------------------------
% 2.90/1.88 % (2461)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.88 % (2461)Termination reason: Unknown
% 2.90/1.88 % (2461)Termination phase: Saturation
% 2.90/1.88
% 2.90/1.88 % (2461)Memory used [KB]: 7931
% 2.90/1.88 % (2461)Time elapsed: 0.211 s
% 2.90/1.88 % (2461)Instructions burned: 711 (million)
% 2.90/1.88 % (2461)------------------------------
% 2.90/1.88 % (2461)------------------------------
% 2.90/1.89 % (2485)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_2221 for (2996ds/149Mi)
% 2.90/1.89 % (2483)Instruction limit reached!
% 2.90/1.89 % (2483)------------------------------
% 2.90/1.89 % (2483)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.89 % (2483)Termination reason: Unknown
% 2.90/1.89 % (2483)Termination phase: Property scanning
% 2.90/1.89
% 2.90/1.89 % (2483)Memory used [KB]: 2430
% 2.90/1.89 % (2483)Time elapsed: 0.030 s
% 2.90/1.89 % (2483)Instructions burned: 98 (million)
% 2.90/1.89 % (2483)------------------------------
% 2.90/1.89 % (2483)------------------------------
% 2.90/1.90 % (2486)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_2221 for (2996ds/39Mi)
% 2.90/1.91 % (2484)Refutation not found, incomplete strategy
% 2.90/1.91 % (2484)------------------------------
% 2.90/1.91 % (2484)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.91 % (2484)Termination reason: Refutation not found, incomplete strategy
% 2.90/1.91
% 2.90/1.91
% 2.90/1.91 % (2484)Memory used [KB]: 7547
% 2.90/1.91 % (2484)Time elapsed: 0.044 s
% 2.90/1.91 % (2484)Instructions burned: 129 (million)
% 2.90/1.91 % (2484)------------------------------
% 2.90/1.91 % (2484)------------------------------
% 2.90/1.91 % (2486)Instruction limit reached!
% 2.90/1.91 % (2486)------------------------------
% 2.90/1.91 % (2486)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.91 % (2486)Termination reason: Unknown
% 2.90/1.91 % (2486)Termination phase: Preprocessing 1
% 2.90/1.91
% 2.90/1.91 % (2486)Memory used [KB]: 1918
% 2.90/1.91 % (2486)Time elapsed: 0.015 s
% 2.90/1.91 % (2486)Instructions burned: 40 (million)
% 2.90/1.91 % (2486)------------------------------
% 2.90/1.91 % (2486)------------------------------
% 2.90/1.92 % (2487)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_2221 for (2996ds/6Mi)
% 2.90/1.92 % (2488)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_2221 for (2996ds/1087Mi)
% 2.90/1.93 % (2487)Instruction limit reached!
% 2.90/1.93 % (2487)------------------------------
% 2.90/1.93 % (2487)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.93 % (2487)Termination reason: Unknown
% 2.90/1.93 % (2487)Termination phase: shuffling
% 2.90/1.93
% 2.90/1.93 % (2487)Memory used [KB]: 1535
% 2.90/1.93 % (2487)Time elapsed: 0.004 s
% 2.90/1.93 % (2487)Instructions burned: 7 (million)
% 2.90/1.93 % (2487)------------------------------
% 2.90/1.93 % (2487)------------------------------
% 2.90/1.93 % (2468)Instruction limit reached!
% 2.90/1.93 % (2468)------------------------------
% 2.90/1.93 % (2468)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.93 % (2468)Termination reason: Unknown
% 2.90/1.93 % (2468)Termination phase: Saturation
% 2.90/1.93
% 2.90/1.93 % (2468)Memory used [KB]: 9210
% 2.90/1.93 % (2468)Time elapsed: 0.237 s
% 2.90/1.93 % (2468)Instructions burned: 780 (million)
% 2.90/1.93 % (2468)------------------------------
% 2.90/1.93 % (2468)------------------------------
% 2.90/1.93 % (2485)Instruction limit reached!
% 2.90/1.93 % (2485)------------------------------
% 2.90/1.93 % (2485)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.93 % (2485)Termination reason: Unknown
% 2.90/1.93 % (2485)Termination phase: Saturation
% 2.90/1.93
% 2.90/1.93 % (2485)Memory used [KB]: 7036
% 2.90/1.93 % (2485)Time elapsed: 0.047 s
% 2.90/1.93 % (2485)Instructions burned: 150 (million)
% 2.90/1.93 % (2485)------------------------------
% 2.90/1.93 % (2485)------------------------------
% 2.90/1.94 % (2489)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_2221 for (2996ds/47Mi)
% 2.90/1.94 % (2490)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_2221 for (2996ds/44Mi)
% 2.90/1.94 % (2491)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_2221 for (2996ds/432Mi)
% 2.90/1.95 % (2490)Instruction limit reached!
% 2.90/1.95 % (2490)------------------------------
% 2.90/1.95 % (2490)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.95 % (2490)Termination reason: Unknown
% 2.90/1.95 % (2490)Termination phase: Property scanning
% 2.90/1.95
% 2.90/1.95 % (2490)Memory used [KB]: 2302
% 2.90/1.95 % (2490)Time elapsed: 0.014 s
% 2.90/1.95 % (2490)Instructions burned: 46 (million)
% 2.90/1.95 % (2490)------------------------------
% 2.90/1.95 % (2490)------------------------------
% 2.90/1.95 % (2489)Instruction limit reached!
% 2.90/1.95 % (2489)------------------------------
% 2.90/1.95 % (2489)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.95 % (2489)Termination reason: Unknown
% 2.90/1.95 % (2489)Termination phase: Preprocessing 3
% 2.90/1.95
% 2.90/1.95 % (2489)Memory used [KB]: 2174
% 2.90/1.95 % (2489)Time elapsed: 0.018 s
% 2.90/1.95 % (2489)Instructions burned: 48 (million)
% 2.90/1.95 % (2489)------------------------------
% 2.90/1.95 % (2489)------------------------------
% 2.90/1.96 % (2492)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_2221 for (2996ds/62Mi)
% 2.90/1.96 % (2493)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_2221 for (2996ds/31Mi)
% 2.90/1.97 % (2452)Instruction limit reached!
% 2.90/1.97 % (2452)------------------------------
% 2.90/1.97 % (2452)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.97 % (2452)Termination reason: Unknown
% 2.90/1.97 % (2452)Termination phase: Saturation
% 2.90/1.97
% 2.90/1.97 % (2452)Memory used [KB]: 10234
% 2.90/1.97 % (2452)Time elapsed: 0.333 s
% 2.90/1.97 % (2452)Instructions burned: 1042 (million)
% 2.90/1.97 % (2452)------------------------------
% 2.90/1.97 % (2452)------------------------------
% 2.90/1.97 % (2493)Instruction limit reached!
% 2.90/1.97 % (2493)------------------------------
% 2.90/1.97 % (2493)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.97 % (2493)Termination reason: Unknown
% 2.90/1.97 % (2493)Termination phase: Property scanning
% 2.90/1.97
% 2.90/1.97 % (2493)Memory used [KB]: 1918
% 2.90/1.97 % (2493)Time elapsed: 0.013 s
% 2.90/1.97 % (2493)Instructions burned: 33 (million)
% 2.90/1.97 % (2493)------------------------------
% 2.90/1.97 % (2493)------------------------------
% 2.90/1.97 % (2481)Instruction limit reached!
% 2.90/1.97 % (2481)------------------------------
% 2.90/1.97 % (2481)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.97 % (2481)Termination reason: Unknown
% 2.90/1.97 % (2481)Termination phase: Saturation
% 2.90/1.97
% 2.90/1.97 % (2481)Memory used [KB]: 8699
% 2.90/1.97 % (2481)Time elapsed: 0.156 s
% 2.90/1.97 % (2481)Instructions burned: 450 (million)
% 2.90/1.97 % (2481)------------------------------
% 2.90/1.97 % (2481)------------------------------
% 2.90/1.98 % (2492)Instruction limit reached!
% 2.90/1.98 % (2492)------------------------------
% 2.90/1.98 % (2492)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.98 % (2492)Termination reason: Unknown
% 2.90/1.98 % (2492)Termination phase: Property scanning
% 2.90/1.98
% 2.90/1.98 % (2492)Memory used [KB]: 2302
% 2.90/1.98 % (2492)Time elapsed: 0.018 s
% 2.90/1.98 % (2492)Instructions burned: 64 (million)
% 2.90/1.98 % (2492)------------------------------
% 2.90/1.98 % (2492)------------------------------
% 2.90/1.98 % (2494)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_2221 for (2996ds/96Mi)
% 2.90/1.98 % (2495)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_2221 for (2995ds/453Mi)
% 2.90/1.98 % (2496)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_2221 for (2995ds/71Mi)
% 2.90/1.98 % (2470)Instruction limit reached!
% 2.90/1.98 % (2470)------------------------------
% 2.90/1.98 % (2470)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/1.98 % (2470)Termination reason: Unknown
% 2.90/1.98 % (2470)Termination phase: Saturation
% 2.90/1.98
% 2.90/1.98 % (2470)Memory used [KB]: 7803
% 2.90/1.98 % (2470)Time elapsed: 0.289 s
% 2.90/1.98 % (2470)Instructions burned: 881 (million)
% 2.90/1.98 % (2470)------------------------------
% 2.90/1.98 % (2470)------------------------------
% 2.90/1.99 % (2497)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_2221 for (2995ds/83Mi)
% 2.90/1.99 % (2498)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_2221 for (2995ds/11Mi)
% 2.90/2.00 % (2498)Instruction limit reached!
% 2.90/2.00 % (2498)------------------------------
% 2.90/2.00 % (2498)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/2.00 % (2498)Termination reason: Unknown
% 2.90/2.00 % (2498)Termination phase: shuffling
% 2.90/2.00
% 2.90/2.00 % (2498)Memory used [KB]: 1663
% 2.90/2.00 % (2498)Time elapsed: 0.004 s
% 2.90/2.00 % (2498)Instructions burned: 11 (million)
% 2.90/2.00 % (2498)------------------------------
% 2.90/2.00 % (2498)------------------------------
% 2.90/2.01 % (2496)Instruction limit reached!
% 2.90/2.01 % (2496)------------------------------
% 2.90/2.01 % (2496)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.90/2.01 % (2496)Termination reason: Unknown
% 2.90/2.01 % (2496)Termination phase: Property scanning
% 2.90/2.01
% 2.90/2.01 % (2496)Memory used [KB]: 2302
% 2.90/2.01 % (2496)Time elapsed: 0.023 s
% 2.90/2.01 % (2496)Instructions burned: 74 (million)
% 2.90/2.01 % (2496)------------------------------
% 2.90/2.01 % (2496)------------------------------
% 2.90/2.01 % (2499)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_2221 for (2995ds/41Mi)
% 5.79/2.01 % (2497)Instruction limit reached!
% 5.79/2.01 % (2497)------------------------------
% 5.79/2.01 % (2497)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.01 % (2497)Termination reason: Unknown
% 5.79/2.01 % (2497)Termination phase: Saturation
% 5.79/2.01
% 5.79/2.01 % (2497)Memory used [KB]: 6652
% 5.79/2.01 % (2497)Time elapsed: 0.026 s
% 5.79/2.01 % (2497)Instructions burned: 84 (million)
% 5.79/2.01 % (2497)------------------------------
% 5.79/2.01 % (2497)------------------------------
% 5.79/2.02 % (2500)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_2221 for (2995ds/6Mi)
% 5.79/2.02 % (2494)Instruction limit reached!
% 5.79/2.02 % (2494)------------------------------
% 5.79/2.02 % (2494)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.02 % (2494)Termination reason: Unknown
% 5.79/2.02 % (2494)Termination phase: Saturation
% 5.79/2.02
% 5.79/2.02 % (2494)Memory used [KB]: 6780
% 5.79/2.02 % (2494)Time elapsed: 0.040 s
% 5.79/2.02 % (2494)Instructions burned: 97 (million)
% 5.79/2.02 % (2494)------------------------------
% 5.79/2.02 % (2494)------------------------------
% 5.79/2.02 % (2500)Instruction limit reached!
% 5.79/2.02 % (2500)------------------------------
% 5.79/2.02 % (2500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.02 % (2500)Termination reason: Unknown
% 5.79/2.02 % (2500)Termination phase: shuffling
% 5.79/2.02
% 5.79/2.02 % (2500)Memory used [KB]: 1407
% 5.79/2.02 % (2500)Time elapsed: 0.003 s
% 5.79/2.02 % (2500)Instructions burned: 6 (million)
% 5.79/2.02 % (2500)------------------------------
% 5.79/2.02 % (2500)------------------------------
% 5.79/2.02 % (2499)Instruction limit reached!
% 5.79/2.02 % (2499)------------------------------
% 5.79/2.02 % (2499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.02 % (2499)Termination reason: Unknown
% 5.79/2.02 % (2499)Termination phase: shuffling
% 5.79/2.02
% 5.79/2.02 % (2499)Memory used [KB]: 2174
% 5.79/2.02 % (2499)Time elapsed: 0.012 s
% 5.79/2.02 % (2499)Instructions burned: 41 (million)
% 5.79/2.02 % (2499)------------------------------
% 5.79/2.02 % (2499)------------------------------
% 5.79/2.02 % (2501)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_2221 for (2995ds/150Mi)
% 5.79/2.03 % (2503)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_2221 for (2995ds/109Mi)
% 5.79/2.03 % (2504)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_2221 for (2995ds/253Mi)
% 5.79/2.03 % (2502)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_2221 for (2995ds/343Mi)
% 5.79/2.06 % (2503)Instruction limit reached!
% 5.79/2.06 % (2503)------------------------------
% 5.79/2.06 % (2503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.06 % (2503)Termination reason: Unknown
% 5.79/2.06 % (2503)Termination phase: Saturation
% 5.79/2.06
% 5.79/2.06 % (2503)Memory used [KB]: 6780
% 5.79/2.06 % (2503)Time elapsed: 0.033 s
% 5.79/2.06 % (2503)Instructions burned: 109 (million)
% 5.79/2.06 % (2503)------------------------------
% 5.79/2.06 % (2503)------------------------------
% 5.79/2.07 % (2501)Instruction limit reached!
% 5.79/2.07 % (2501)------------------------------
% 5.79/2.07 % (2501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.07 % (2501)Termination reason: Unknown
% 5.79/2.07 % (2501)Termination phase: Saturation
% 5.79/2.07
% 5.79/2.07 % (2501)Memory used [KB]: 7164
% 5.79/2.07 % (2501)Time elapsed: 0.047 s
% 5.79/2.07 % (2501)Instructions burned: 150 (million)
% 5.79/2.07 % (2501)------------------------------
% 5.79/2.07 % (2501)------------------------------
% 5.79/2.07 % (2505)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_2221 for (2995ds/140Mi)
% 5.79/2.07 % (2491)Instruction limit reached!
% 5.79/2.07 % (2491)------------------------------
% 5.79/2.07 % (2491)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.07 % (2491)Termination reason: Unknown
% 5.79/2.07 % (2491)Termination phase: Saturation
% 5.79/2.07
% 5.79/2.07 % (2491)Memory used [KB]: 4733
% 5.79/2.07 % (2491)Time elapsed: 0.131 s
% 5.79/2.07 % (2491)Instructions burned: 435 (million)
% 5.79/2.07 % (2491)------------------------------
% 5.79/2.07 % (2491)------------------------------
% 5.79/2.08 % (2506)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_2221 for (2995ds/614Mi)
% 5.79/2.08 % (2507)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_2221 for (2995ds/202Mi)
% 5.79/2.10 % (2504)Instruction limit reached!
% 5.79/2.10 % (2504)------------------------------
% 5.79/2.10 % (2504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.10 % (2504)Termination reason: Unknown
% 5.79/2.10 % (2504)Termination phase: Saturation
% 5.79/2.10
% 5.79/2.10 % (2504)Memory used [KB]: 7803
% 5.79/2.10 % (2504)Time elapsed: 0.078 s
% 5.79/2.10 % (2504)Instructions burned: 255 (million)
% 5.79/2.10 % (2504)------------------------------
% 5.79/2.10 % (2504)------------------------------
% 5.79/2.11 % (2507)Refutation not found, incomplete strategy
% 5.79/2.11 % (2507)------------------------------
% 5.79/2.11 % (2507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 5.79/2.11 % (2507)Termination reason: Refutation not found, incomplete strategy
% 5.79/2.11
% 5.79/2.11
% 5.79/2.11 % (2507)Memory used [KB]: 6780
% 5.79/2.11 % (2507)Time elapsed: 0.026 s
% 5.79/2.11 % (2507)Instructions burned: 84 (million)
% 5.79/2.11 % (2507)------------------------------
% 5.79/2.11 % (2507)------------------------------
% 6.33/2.11 % (2505)Instruction limit reached!
% 6.33/2.11 % (2505)------------------------------
% 6.33/2.11 % (2505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.33/2.11 % (2505)Termination reason: Unknown
% 6.33/2.11 % (2505)Termination phase: Saturation
% 6.33/2.11
% 6.33/2.11 % (2505)Memory used [KB]: 6908
% 6.33/2.11 % (2505)Time elapsed: 0.046 s
% 6.33/2.11 % (2505)Instructions burned: 142 (million)
% 6.33/2.11 % (2505)------------------------------
% 6.33/2.11 % (2505)------------------------------
% 6.33/2.11 % (2508)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_2221 for (2994ds/339Mi)
% 6.33/2.12 % (2509)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=283:si=on:rtra=on_0 on DTF2THF_2221 for (2994ds/283Mi)
% 6.33/2.12 % (2495)Instruction limit reached!
% 6.33/2.12 % (2495)------------------------------
% 6.33/2.12 % (2495)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.33/2.12 % (2495)Termination reason: Unknown
% 6.33/2.12 % (2495)Termination phase: Saturation
% 6.33/2.12
% 6.33/2.12 % (2495)Memory used [KB]: 7803
% 6.33/2.12 % (2495)Time elapsed: 0.139 s
% 6.33/2.12 % (2495)Instructions burned: 453 (million)
% 6.33/2.12 % (2495)------------------------------
% 6.33/2.12 % (2495)------------------------------
% 6.33/2.12 % (2510)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_2221 for (2994ds/134Mi)
% 6.33/2.13 % (2511)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_2221 for (2994ds/298Mi)
% 6.33/2.16 % (2510)Instruction limit reached!
% 6.33/2.16 % (2510)------------------------------
% 6.33/2.16 % (2510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.33/2.16 % (2510)Termination reason: Unknown
% 6.33/2.16 % (2510)Termination phase: Saturation
% 6.33/2.16
% 6.33/2.16 % (2510)Memory used [KB]: 7036
% 6.33/2.16 % (2510)Time elapsed: 0.042 s
% 6.33/2.16 % (2510)Instructions burned: 137 (million)
% 6.33/2.16 % (2510)------------------------------
% 6.33/2.16 % (2510)------------------------------
% 6.33/2.17 % (2512)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_2221 for (2994ds/192Mi)
% 6.33/2.18 % (2502)Instruction limit reached!
% 6.33/2.18 % (2502)------------------------------
% 6.33/2.18 % (2502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.33/2.18 % (2502)Termination reason: Unknown
% 6.33/2.18 % (2502)Termination phase: Saturation
% 6.33/2.18
% 6.33/2.18 % (2502)Memory used [KB]: 8059
% 6.33/2.18 % (2502)Time elapsed: 0.151 s
% 6.33/2.18 % (2502)Instructions burned: 343 (million)
% 6.33/2.18 % (2502)------------------------------
% 6.33/2.18 % (2502)------------------------------
% 6.33/2.19 % (2513)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_2221 for (2993ds/258Mi)
% 6.70/2.20 % (2509)Instruction limit reached!
% 6.70/2.20 % (2509)------------------------------
% 6.70/2.20 % (2509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.70/2.20 % (2509)Termination reason: Unknown
% 6.70/2.20 % (2509)Termination phase: Saturation
% 6.70/2.20
% 6.70/2.20 % (2509)Memory used [KB]: 7675
% 6.70/2.20 % (2509)Time elapsed: 0.087 s
% 6.70/2.20 % (2509)Instructions burned: 284 (million)
% 6.70/2.20 % (2509)------------------------------
% 6.70/2.20 % (2509)------------------------------
% 6.70/2.21 % (2514)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_2221 for (2993ds/90Mi)
% 6.70/2.21 % (2511)Instruction limit reached!
% 6.70/2.21 % (2511)------------------------------
% 6.70/2.21 % (2511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.70/2.21 % (2511)Termination reason: Unknown
% 6.70/2.21 % (2511)Termination phase: Saturation
% 6.70/2.21
% 6.70/2.21 % (2511)Memory used [KB]: 8571
% 6.70/2.21 % (2511)Time elapsed: 0.084 s
% 6.70/2.21 % (2511)Instructions burned: 298 (million)
% 6.70/2.21 % (2511)------------------------------
% 6.70/2.21 % (2511)------------------------------
% 6.70/2.22 % (2508)Instruction limit reached!
% 6.70/2.22 % (2508)------------------------------
% 6.70/2.22 % (2508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.70/2.22 % (2508)Termination reason: Unknown
% 6.70/2.22 % (2508)Termination phase: Saturation
% 6.70/2.22
% 6.70/2.22 % (2508)Memory used [KB]: 7419
% 6.70/2.22 % (2508)Time elapsed: 0.108 s
% 6.70/2.22 % (2508)Instructions burned: 340 (million)
% 6.70/2.22 % (2508)------------------------------
% 6.70/2.22 % (2508)------------------------------
% 6.70/2.22 % (2515)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_2221 for (2993ds/231Mi)
% 6.70/2.23 % (2516)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_2221 for (2993ds/458Mi)
% 6.70/2.23 % (2512)Instruction limit reached!
% 6.70/2.23 % (2512)------------------------------
% 6.70/2.23 % (2512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.70/2.23 % (2512)Termination reason: Unknown
% 6.70/2.23 % (2512)Termination phase: Saturation
% 6.70/2.23
% 6.70/2.23 % (2512)Memory used [KB]: 8187
% 6.70/2.23 % (2512)Time elapsed: 0.061 s
% 6.70/2.23 % (2512)Instructions burned: 192 (million)
% 6.70/2.23 % (2512)------------------------------
% 6.70/2.23 % (2512)------------------------------
% 6.70/2.24 % (2514)Instruction limit reached!
% 6.70/2.24 % (2514)------------------------------
% 6.70/2.24 % (2514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.70/2.24 % (2514)Termination reason: Unknown
% 6.70/2.24 % (2514)Termination phase: Saturation
% 6.70/2.24
% 6.70/2.24 % (2514)Memory used [KB]: 6908
% 6.70/2.24 % (2514)Time elapsed: 0.027 s
% 6.70/2.24 % (2514)Instructions burned: 90 (million)
% 6.70/2.24 % (2514)------------------------------
% 6.70/2.24 % (2514)------------------------------
% 6.70/2.24 % (2517)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_2221 for (2993ds/31Mi)
% 6.70/2.25 % (2518)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_2221 for (2993ds/297Mi)
% 6.70/2.25 % (2517)Instruction limit reached!
% 6.70/2.25 % (2517)------------------------------
% 6.70/2.25 % (2517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.70/2.25 % (2517)Termination reason: Unknown
% 6.70/2.25 % (2517)Termination phase: Property scanning
% 6.70/2.25
% 6.70/2.25 % (2517)Memory used [KB]: 1918
% 6.70/2.25 % (2517)Time elapsed: 0.012 s
% 6.70/2.25 % (2517)Instructions burned: 34 (million)
% 6.70/2.25 % (2517)------------------------------
% 6.70/2.25 % (2517)------------------------------
% 6.70/2.26 % (2506)Instruction limit reached!
% 6.70/2.26 % (2506)------------------------------
% 6.70/2.26 % (2506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.70/2.26 % (2506)Termination reason: Unknown
% 6.70/2.26 % (2506)Termination phase: Saturation
% 6.70/2.26
% 6.70/2.26 % (2506)Memory used [KB]: 8187
% 6.70/2.26 % (2506)Time elapsed: 0.186 s
% 6.70/2.26 % (2506)Instructions burned: 617 (million)
% 6.70/2.26 % (2506)------------------------------
% 6.70/2.26 % (2506)------------------------------
% 6.70/2.26 % (2519)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_2221 for (2993ds/987Mi)
% 6.70/2.27 % (2520)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_2221 for (2993ds/182Mi)
% 6.70/2.27 % (2488)Instruction limit reached!
% 6.70/2.27 % (2488)------------------------------
% 6.70/2.27 % (2488)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.70/2.27 % (2488)Termination reason: Unknown
% 6.70/2.27 % (2488)Termination phase: Saturation
% 6.70/2.27
% 6.70/2.27 % (2488)Memory used [KB]: 8315
% 6.70/2.27 % (2488)Time elapsed: 0.350 s
% 6.70/2.27 % (2488)Instructions burned: 1087 (million)
% 6.70/2.27 % (2488)------------------------------
% 6.70/2.27 % (2488)------------------------------
% 6.70/2.28 % (2513)Instruction limit reached!
% 6.70/2.28 % (2513)------------------------------
% 6.70/2.28 % (2513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.70/2.28 % (2513)Termination reason: Unknown
% 6.70/2.28 % (2513)Termination phase: Saturation
% 6.70/2.28
% 6.70/2.28 % (2513)Memory used [KB]: 7675
% 6.70/2.28 % (2513)Time elapsed: 0.088 s
% 6.70/2.28 % (2513)Instructions burned: 260 (million)
% 6.70/2.28 % (2513)------------------------------
% 6.70/2.28 % (2513)------------------------------
% 6.70/2.28 % (2521)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_2221 for (2993ds/182Mi)
% 6.70/2.28 % (2515)Instruction limit reached!
% 6.70/2.28 % (2515)------------------------------
% 6.70/2.28 % (2515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 6.70/2.28 % (2515)Termination reason: Unknown
% 6.70/2.28 % (2515)Termination phase: Saturation
% 6.70/2.28
% 6.70/2.28 % (2515)Memory used [KB]: 7036
% 6.70/2.28 % (2515)Time elapsed: 0.062 s
% 6.70/2.28 % (2515)Instructions burned: 231 (million)
% 6.70/2.28 % (2515)------------------------------
% 6.70/2.28 % (2515)------------------------------
% 6.70/2.29 % (2522)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_2221 for (2993ds/154Mi)
% 6.70/2.29 % (2523)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_2221 for (2993ds/79Mi)
% 7.58/2.32 % (2523)Instruction limit reached!
% 7.58/2.32 % (2523)------------------------------
% 7.58/2.32 % (2523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.58/2.32 % (2523)Termination reason: Unknown
% 7.58/2.32 % (2523)Termination phase: Property scanning
% 7.58/2.32
% 7.58/2.32 % (2523)Memory used [KB]: 2302
% 7.58/2.32 % (2523)Time elapsed: 0.024 s
% 7.58/2.32 % (2523)Instructions burned: 79 (million)
% 7.58/2.32 % (2523)------------------------------
% 7.58/2.32 % (2523)------------------------------
% 7.58/2.33 % (2524)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_2221 for (2992ds/216Mi)
% 7.58/2.33 % (2520)Instruction limit reached!
% 7.58/2.33 % (2520)------------------------------
% 7.58/2.33 % (2520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.58/2.33 % (2520)Termination reason: Unknown
% 7.58/2.33 % (2520)Termination phase: Saturation
% 7.58/2.33
% 7.58/2.33 % (2520)Memory used [KB]: 7291
% 7.58/2.33 % (2520)Time elapsed: 0.057 s
% 7.58/2.33 % (2520)Instructions burned: 184 (million)
% 7.58/2.33 % (2520)------------------------------
% 7.58/2.33 % (2520)------------------------------
% 7.58/2.33 % (2518)Instruction limit reached!
% 7.58/2.33 % (2518)------------------------------
% 7.58/2.33 % (2518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.58/2.33 % (2518)Termination reason: Unknown
% 7.58/2.33 % (2518)Termination phase: Saturation
% 7.58/2.33
% 7.58/2.33 % (2518)Memory used [KB]: 7419
% 7.58/2.33 % (2518)Time elapsed: 0.086 s
% 7.58/2.33 % (2518)Instructions burned: 298 (million)
% 7.58/2.33 % (2518)------------------------------
% 7.58/2.33 % (2518)------------------------------
% 7.58/2.34 % (2525)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_2221 for (2992ds/114Mi)
% 7.58/2.34 % (2522)Instruction limit reached!
% 7.58/2.34 % (2522)------------------------------
% 7.58/2.34 % (2522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.58/2.34 % (2522)Termination reason: Unknown
% 7.58/2.34 % (2522)Termination phase: Saturation
% 7.58/2.34
% 7.58/2.34 % (2522)Memory used [KB]: 2942
% 7.58/2.34 % (2522)Time elapsed: 0.048 s
% 7.58/2.34 % (2522)Instructions burned: 155 (million)
% 7.58/2.34 % (2522)------------------------------
% 7.58/2.34 % (2522)------------------------------
% 7.58/2.34 % (2521)Instruction limit reached!
% 7.58/2.34 % (2521)------------------------------
% 7.58/2.34 % (2521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.58/2.34 % (2521)Termination reason: Unknown
% 7.58/2.34 % (2521)Termination phase: Saturation
% 7.58/2.34
% 7.58/2.34 % (2521)Memory used [KB]: 7291
% 7.58/2.34 % (2521)Time elapsed: 0.058 s
% 7.58/2.34 % (2521)Instructions burned: 184 (million)
% 7.58/2.34 % (2521)------------------------------
% 7.58/2.34 % (2521)------------------------------
% 7.58/2.34 % (2526)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_2221 for (2992ds/922Mi)
% 7.58/2.35 % (2527)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_2221 for (2992ds/182Mi)
% 7.58/2.35 % (2528)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_2221 for (2992ds/164Mi)
% 7.58/2.37 % (2525)Instruction limit reached!
% 7.58/2.37 % (2525)------------------------------
% 7.58/2.37 % (2525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.58/2.37 % (2525)Termination reason: Unknown
% 7.58/2.37 % (2525)Termination phase: Saturation
% 7.58/2.37
% 7.58/2.37 % (2525)Memory used [KB]: 7036
% 7.58/2.37 % (2525)Time elapsed: 0.035 s
% 7.58/2.37 % (2525)Instructions burned: 115 (million)
% 7.58/2.37 % (2525)------------------------------
% 7.58/2.37 % (2525)------------------------------
% 7.58/2.38 % (2516)Instruction limit reached!
% 7.58/2.38 % (2516)------------------------------
% 7.58/2.38 % (2516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.58/2.38 % (2516)Termination reason: Unknown
% 7.58/2.38 % (2516)Termination phase: Saturation
% 7.58/2.38
% 7.58/2.38 % (2516)Memory used [KB]: 8443
% 7.58/2.38 % (2516)Time elapsed: 0.147 s
% 7.58/2.38 % (2516)Instructions burned: 459 (million)
% 7.58/2.38 % (2516)------------------------------
% 7.58/2.38 % (2516)------------------------------
% 7.58/2.38 % (2529)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_2221 for (2992ds/50Mi)
% 7.58/2.39 % (2530)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_2221 for (2991ds/368Mi)
% 7.58/2.39 % (2524)Instruction limit reached!
% 7.58/2.39 % (2524)------------------------------
% 7.58/2.39 % (2524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.58/2.39 % (2524)Termination reason: Unknown
% 7.58/2.39 % (2524)Termination phase: Saturation
% 7.58/2.39
% 7.58/2.39 % (2524)Memory used [KB]: 7419
% 7.58/2.39 % (2524)Time elapsed: 0.062 s
% 7.58/2.39 % (2524)Instructions burned: 219 (million)
% 7.58/2.39 % (2524)------------------------------
% 7.58/2.39 % (2524)------------------------------
% 7.58/2.39 % (2529)Instruction limit reached!
% 7.58/2.39 % (2529)------------------------------
% 7.58/2.39 % (2529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.58/2.39 % (2529)Termination reason: Unknown
% 7.58/2.39 % (2529)Termination phase: Property scanning
% 7.58/2.39
% 7.58/2.39 % (2529)Memory used [KB]: 2302
% 7.58/2.39 % (2529)Time elapsed: 0.015 s
% 7.58/2.39 % (2529)Instructions burned: 51 (million)
% 7.58/2.39 % (2529)------------------------------
% 7.58/2.39 % (2529)------------------------------
% 7.58/2.40 % (2528)Instruction limit reached!
% 7.58/2.40 % (2528)------------------------------
% 7.58/2.40 % (2528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 7.58/2.40 % (2528)Termination reason: Unknown
% 7.58/2.40 % (2528)Termination phase: Saturation
% 7.58/2.40
% 7.58/2.40 % (2528)Memory used [KB]: 2814
% 7.58/2.40 % (2528)Time elapsed: 0.047 s
% 7.58/2.40 % (2528)Instructions burned: 166 (million)
% 7.58/2.40 % (2528)------------------------------
% 7.58/2.40 % (2528)------------------------------
% 8.25/2.40 % (2531)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_2221 for (2991ds/1021Mi)
% 8.25/2.40 % (2527)Instruction limit reached!
% 8.25/2.40 % (2527)------------------------------
% 8.25/2.40 % (2527)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.25/2.40 % (2527)Termination reason: Unknown
% 8.25/2.40 % (2527)Termination phase: Saturation
% 8.25/2.40
% 8.25/2.40 % (2527)Memory used [KB]: 7291
% 8.25/2.40 % (2527)Time elapsed: 0.056 s
% 8.25/2.40 % (2527)Instructions burned: 183 (million)
% 8.25/2.40 % (2527)------------------------------
% 8.25/2.40 % (2527)------------------------------
% 8.25/2.40 % (2532)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_2221 for (2991ds/267Mi)
% 8.25/2.40 % (2533)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_2221 for (2991ds/204Mi)
% 8.25/2.41 % (2534)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_2221 for (2991ds/26Mi)
% 8.25/2.42 % (2534)Instruction limit reached!
% 8.25/2.42 % (2534)------------------------------
% 8.25/2.42 % (2534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.25/2.42 % (2534)Termination reason: Unknown
% 8.25/2.42 % (2534)Termination phase: shuffling
% 8.25/2.42
% 8.25/2.42 % (2534)Memory used [KB]: 2046
% 8.25/2.42 % (2534)Time elapsed: 0.010 s
% 8.25/2.42 % (2534)Instructions burned: 28 (million)
% 8.25/2.42 % (2534)------------------------------
% 8.25/2.42 % (2534)------------------------------
% 8.25/2.43 % (2535)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_2221 for (2991ds/53Mi)
% 8.25/2.45 % (2535)Instruction limit reached!
% 8.25/2.45 % (2535)------------------------------
% 8.25/2.45 % (2535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.25/2.45 % (2535)Termination reason: Unknown
% 8.25/2.45 % (2535)Termination phase: Property scanning
% 8.25/2.45
% 8.25/2.45 % (2535)Memory used [KB]: 2302
% 8.25/2.45 % (2535)Time elapsed: 0.018 s
% 8.25/2.45 % (2535)Instructions burned: 56 (million)
% 8.25/2.45 % (2535)------------------------------
% 8.25/2.45 % (2535)------------------------------
% 8.25/2.46 % (2536)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_2221 for (2991ds/4436Mi)
% 8.25/2.47 % (2533)Instruction limit reached!
% 8.25/2.47 % (2533)------------------------------
% 8.25/2.47 % (2533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.25/2.47 % (2533)Termination reason: Unknown
% 8.25/2.47 % (2533)Termination phase: Saturation
% 8.25/2.47
% 8.25/2.47 % (2533)Memory used [KB]: 7164
% 8.25/2.47 % (2533)Time elapsed: 0.065 s
% 8.25/2.47 % (2533)Instructions burned: 204 (million)
% 8.25/2.47 % (2533)------------------------------
% 8.25/2.47 % (2533)------------------------------
% 8.25/2.48 % (2537)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_2221 for (2991ds/1539Mi)
% 8.25/2.49 % (2532)Instruction limit reached!
% 8.25/2.49 % (2532)------------------------------
% 8.25/2.49 % (2532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.25/2.49 % (2532)Termination reason: Unknown
% 8.25/2.49 % (2532)Termination phase: Saturation
% 8.25/2.49
% 8.25/2.49 % (2532)Memory used [KB]: 7547
% 8.25/2.49 % (2532)Time elapsed: 0.083 s
% 8.25/2.49 % (2532)Instructions burned: 267 (million)
% 8.25/2.49 % (2532)------------------------------
% 8.25/2.49 % (2532)------------------------------
% 8.25/2.50 % (2538)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_2221 for (2990ds/408Mi)
% 8.53/2.50 % (2530)Instruction limit reached!
% 8.53/2.50 % (2530)------------------------------
% 8.53/2.50 % (2530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.53/2.50 % (2530)Termination reason: Unknown
% 8.53/2.50 % (2530)Termination phase: Saturation
% 8.53/2.50
% 8.53/2.50 % (2530)Memory used [KB]: 8699
% 8.53/2.50 % (2530)Time elapsed: 0.120 s
% 8.53/2.50 % (2530)Instructions burned: 370 (million)
% 8.53/2.50 % (2530)------------------------------
% 8.53/2.50 % (2530)------------------------------
% 8.53/2.51 % (2539)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_2221 for (2990ds/5205Mi)
% 8.53/2.56 % (2519)Instruction limit reached!
% 8.53/2.56 % (2519)------------------------------
% 8.53/2.56 % (2519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.53/2.56 % (2519)Termination reason: Unknown
% 8.53/2.56 % (2519)Termination phase: Saturation
% 8.53/2.56
% 8.53/2.56 % (2519)Memory used [KB]: 9850
% 8.53/2.56 % (2519)Time elapsed: 0.300 s
% 8.53/2.56 % (2519)Instructions burned: 990 (million)
% 8.53/2.56 % (2519)------------------------------
% 8.53/2.56 % (2519)------------------------------
% 8.53/2.57 % (2540)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_2221 for (2990ds/1913Mi)
% 8.96/2.62 % (2538)Instruction limit reached!
% 8.96/2.62 % (2538)------------------------------
% 8.96/2.62 % (2538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.96/2.62 % (2538)Termination reason: Unknown
% 8.96/2.62 % (2538)Termination phase: Saturation
% 8.96/2.62
% 8.96/2.62 % (2538)Memory used [KB]: 8443
% 8.96/2.62 % (2538)Time elapsed: 0.127 s
% 8.96/2.62 % (2538)Instructions burned: 410 (million)
% 8.96/2.62 % (2538)------------------------------
% 8.96/2.62 % (2538)------------------------------
% 8.96/2.63 % (2541)lrs+2_1:1_bd=off:cbe=off:piset=equals:sd=1:sgt=20:sos=all:ss=axioms:i=4350:si=on:rtra=on_0 on DTF2THF_2221 for (2989ds/4350Mi)
% 8.96/2.64 % (2526)Instruction limit reached!
% 8.96/2.64 % (2526)------------------------------
% 8.96/2.64 % (2526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.96/2.64 % (2526)Termination reason: Unknown
% 8.96/2.64 % (2526)Termination phase: Saturation
% 8.96/2.64
% 8.96/2.64 % (2526)Memory used [KB]: 7931
% 8.96/2.64 % (2526)Time elapsed: 0.320 s
% 8.96/2.64 % (2526)Instructions burned: 925 (million)
% 8.96/2.64 % (2526)------------------------------
% 8.96/2.64 % (2526)------------------------------
% 8.96/2.65 % (2542)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=68:si=on:rtra=on_0 on DTF2THF_2221 for (2989ds/68Mi)
% 8.96/2.67 % (2542)Instruction limit reached!
% 8.96/2.67 % (2542)------------------------------
% 8.96/2.67 % (2542)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 8.96/2.67 % (2542)Termination reason: Unknown
% 8.96/2.67 % (2542)Termination phase: Property scanning
% 8.96/2.67
% 8.96/2.67 % (2542)Memory used [KB]: 2174
% 8.96/2.67 % (2542)Time elapsed: 0.023 s
% 8.96/2.67 % (2542)Instructions burned: 70 (million)
% 8.96/2.67 % (2542)------------------------------
% 8.96/2.67 % (2542)------------------------------
% 8.96/2.68 % (2543)dis+10_1:1_au=on:nwc=5.0:s2a=on:s2at=2.0:sac=on:sd=7:ss=axioms:st=2.0:i=358:si=on:rtra=on_0 on DTF2THF_2221 for (2989ds/358Mi)
% 9.44/2.70 % (2531)Instruction limit reached!
% 9.44/2.70 % (2531)------------------------------
% 9.44/2.70 % (2531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.44/2.70 % (2531)Termination reason: Unknown
% 9.44/2.70 % (2531)Termination phase: Saturation
% 9.44/2.70
% 9.44/2.70 % (2531)Memory used [KB]: 9338
% 9.44/2.70 % (2531)Time elapsed: 0.306 s
% 9.44/2.70 % (2531)Instructions burned: 1023 (million)
% 9.44/2.70 % (2531)------------------------------
% 9.44/2.70 % (2531)------------------------------
% 9.44/2.71 % (2544)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_2221 for (2988ds/31Mi)
% 9.44/2.72 % (2544)Instruction limit reached!
% 9.44/2.72 % (2544)------------------------------
% 9.44/2.72 % (2544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.44/2.72 % (2544)Termination reason: Unknown
% 9.44/2.72 % (2544)Termination phase: Property scanning
% 9.44/2.72
% 9.44/2.72 % (2544)Memory used [KB]: 1918
% 9.44/2.72 % (2544)Time elapsed: 0.011 s
% 9.44/2.72 % (2544)Instructions burned: 31 (million)
% 9.44/2.72 % (2544)------------------------------
% 9.44/2.72 % (2544)------------------------------
% 9.44/2.73 % (2545)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_2221 for (2988ds/279Mi)
% 9.44/2.76 % (2541)First to succeed.
% 9.44/2.76 % (2541)Refutation found. Thanks to Tanya!
% 9.44/2.76 % SZS status Theorem for DTF2THF_2221
% 9.44/2.76 % SZS output start Proof for DTF2THF_2221
% 9.44/2.76 thf(type_def_5, type, nat: $tType).
% 9.44/2.76 thf(type_def_6, type, product_prod: ($tType * $tType) > $tType).
% 9.44/2.76 thf(type_def_8, type, list: ($tType) > $tType).
% 9.44/2.76 thf(type_def_9, type, set: ($tType) > $tType).
% 9.44/2.76 thf(type_def_10, type, capacity: $tType).
% 9.44/2.76 thf(func_def_0, type, product_prod: ($tType * $tType) > $tType).
% 9.44/2.76 thf(func_def_1, type, list: ($tType) > $tType).
% 9.44/2.76 thf(func_def_2, type, capacity: $tType).
% 9.44/2.76 thf(func_def_3, type, set: ($tType) > $tType).
% 9.44/2.76 thf(func_def_4, type, nat: $tType).
% 9.44/2.76 thf(func_def_5, type, type: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_6, type, size: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_7, type, one: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_8, type, ord: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_9, type, order: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_10, type, group_add: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_11, type, linorder: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_12, type, preorder: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_13, type, wellorder: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_14, type, semigroup_add: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_15, type, linordered_idom: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_16, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_17, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_18, type, linordered_semidom: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_19, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_20, type, ordere779506340up_add: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_21, type, linord1659791738miring: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_22, type, canoni770627133id_add: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_23, type, ordere236663937imp_le: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_24, type, ordere223160158up_add: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_25, type, strict2144017051up_add: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_26, type, unique1598680935umeral: !>[X0: $tType]:($o)).
% 9.44/2.76 thf(func_def_27, type, connected: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > $o)).
% 9.44/2.76 thf(func_def_28, type, dist: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > nat > $o)).
% 9.44/2.76 thf(func_def_29, type, isPath: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > list @ product_prod @ nat @ nat > nat > $o)).
% 9.44/2.76 thf(func_def_30, type, isShortestPath: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > list @ product_prod @ nat @ nat > nat > $o)).
% 9.44/2.76 thf(func_def_31, type, isSimplePath: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > list @ product_prod @ nat @ nat > nat > $o)).
% 9.44/2.76 thf(func_def_32, type, min_dist: !>[X0: $tType]:((product_prod @ nat @ nat > X0) > nat > nat > nat)).
% 9.44/2.76 thf(func_def_33, type, one_one: !>[X0: $tType]:(X0)).
% 9.44/2.76 thf(func_def_34, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 9.44/2.76 thf(func_def_35, type, nil: !>[X0: $tType]:(list @ X0)).
% 9.44/2.76 thf(func_def_36, type, set2: !>[X0: $tType]:(list @ X0 > set @ X0)).
% 9.44/2.76 thf(func_def_37, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 9.44/2.76 thf(func_def_38, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 9.44/2.76 thf(func_def_39, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 9.44/2.76 thf(func_def_40, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 9.44/2.76 thf(func_def_41, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 9.44/2.76 thf(func_def_42, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 9.44/2.76 thf(func_def_43, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 9.44/2.76 thf(func_def_44, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 9.44/2.76 thf(func_def_45, type, c: product_prod @ nat @ nat > capacity).
% 9.44/2.76 thf(func_def_46, type, p: list @ product_prod @ nat @ nat).
% 9.44/2.76 thf(func_def_47, type, p1: list @ product_prod @ nat @ nat).
% 9.44/2.76 thf(func_def_48, type, p2: list @ product_prod @ nat @ nat).
% 9.44/2.76 thf(func_def_49, type, p3: list @ product_prod @ nat @ nat).
% 9.44/2.76 thf(func_def_50, type, s: nat).
% 9.44/2.76 thf(func_def_51, type, t: nat).
% 9.44/2.76 thf(func_def_52, type, u: nat).
% 9.44/2.76 thf(func_def_53, type, v: nat).
% 9.44/2.76 thf(func_def_70, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 9.44/2.76 thf(func_def_71, type, sK1: !>[X0: $tType]:(((nat > X0 > X0) > nat > nat > X0 > $o) > nat > X0 > X0)).
% 9.44/2.76 thf(func_def_72, type, sK2: !>[X0: $tType]:(((nat > X0 > X0) > nat > nat > X0 > $o) > nat)).
% 9.44/2.76 thf(func_def_73, type, sK3: !>[X0: $tType]:(((nat > X0 > X0) > nat > nat > X0 > $o) > nat)).
% 9.44/2.76 thf(func_def_74, type, sK4: !>[X0: $tType]:(((nat > X0 > X0) > nat > nat > X0 > $o) > X0)).
% 9.44/2.76 thf(func_def_75, type, sK5: nat > nat > nat).
% 9.44/2.76 thf(func_def_77, type, ph7: !>[X0: $tType]:(X0)).
% 9.44/2.76 thf(f1282,plain,(
% 9.44/2.76 $false),
% 9.44/2.76 inference(subsumption_resolution,[status(thm)],[f1281,f962])).
% 9.44/2.76 thf(f962,plain,(
% 9.44/2.76 ((plus_plus @ nat @ one_one @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p2)) != (min_dist @ capacity @ c @ u @ t))),
% 9.44/2.76 inference(cnf_transformation,[status(thm)],[f868])).
% 9.44/2.76 thf(f868,plain,(
% 9.44/2.76 ((plus_plus @ nat @ one_one @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p2)) != (min_dist @ capacity @ c @ u @ t))),
% 9.44/2.76 inference(flattening,[status(thm)],[f306])).
% 9.44/2.76 thf(f306,negated_conjecture,(
% 9.44/2.76 ~((plus_plus @ nat @ one_one @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p2)) = (min_dist @ capacity @ c @ u @ t))),
% 9.44/2.76 inference(negated_conjecture,[status(cth)],[f305])).
% 9.44/2.76 thf(f305,conjecture,(
% 9.44/2.76 ((plus_plus @ nat @ one_one @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p2)) = (min_dist @ capacity @ c @ u @ t))),
% 9.44/2.76 file('/export/starexec/sandbox2/tmp/tmp.dNuoTsUl6x/DTF2THF_2221.p',conj_0)).
% 9.44/2.76 thf(f1281,plain,(
% 9.44/2.76 ((plus_plus @ nat @ one_one @ nat @ (size_size @ list @ product_prod @ nat @ nat @ p2)) = (min_dist @ capacity @ c @ u @ t))),
% 9.44/2.76 inference(forward_demodulation,[status(thm)],[f1278,f961])).
% 9.44/2.76 thf(f961,plain,(
% 9.44/2.76 ((size_size @ list @ product_prod @ nat @ nat @ p2) = (min_dist @ capacity @ c @ v @ t))),
% 9.44/2.76 inference(cnf_transformation,[status(thm)],[f4])).
% 9.44/2.76 thf(f4,axiom,(
% 9.44/2.76 ((size_size @ list @ product_prod @ nat @ nat @ p2) = (min_dist @ capacity @ c @ v @ t))),
% 9.44/2.76 file('/export/starexec/sandbox2/tmp/tmp.dNuoTsUl6x/DTF2THF_2221.p',fact_3__092_060open_062min__dist_Av_At_A_061_Alength_Ap2_092_060close_062)).
% 9.44/2.76 thf(f1278,plain,(
% 9.44/2.76 ((min_dist @ capacity @ c @ u @ t) = (plus_plus @ nat @ one_one @ nat @ (min_dist @ capacity @ c @ v @ t)))),
% 9.44/2.76 inference(trivial_inequality_removal,[status(thm)],[f1270])).
% 9.44/2.76 thf(f1270,plain,(
% 9.44/2.76 ($true != $true) | ((min_dist @ capacity @ c @ u @ t) = (plus_plus @ nat @ one_one @ nat @ (min_dist @ capacity @ c @ v @ t)))),
% 9.44/2.76 inference(superposition,[status(thm)],[f1054,f966])).
% 9.44/2.76 thf(f966,plain,(
% 9.44/2.76 ((isShortestPath @ capacity @ c @ s @ p @ t) = $true)),
% 9.44/2.76 inference(cnf_transformation,[status(thm)],[f425])).
% 9.44/2.76 thf(f425,plain,(
% 9.44/2.76 ((isShortestPath @ capacity @ c @ s @ p @ t) = $true)),
% 9.44/2.76 inference(fool_elimination,[status(thm)],[f424])).
% 9.44/2.76 thf(f424,plain,(
% 9.44/2.76 (isShortestPath @ capacity @ c @ s @ p @ t)),
% 9.44/2.76 inference(rectify,[status(thm)],[f20])).
% 9.44/2.76 thf(f20,axiom,(
% 9.44/2.76 (isShortestPath @ capacity @ c @ s @ p @ t)),
% 9.44/2.76 file('/export/starexec/sandbox2/tmp/tmp.dNuoTsUl6x/DTF2THF_2221.p',fact_19_assms_I1_J)).
% 9.44/2.76 thf(f1054,plain,(
% 9.44/2.76 ( ! [X2 : nat,X0 : product_prod @ nat @ nat > capacity,X1 : nat] : (($true != (isShortestPath @ capacity @ X0 @ X2 @ p @ X1)) | ((plus_plus @ nat @ one_one @ nat @ (min_dist @ capacity @ X0 @ v @ X1)) = (min_dist @ capacity @ X0 @ u @ X1))) )),
% 9.44/2.76 inference(trivial_inequality_removal,[status(thm)],[f1046])).
% 9.44/2.76 thf(f1046,plain,(
% 9.44/2.76 ( ! [X2 : nat,X0 : product_prod @ nat @ nat > capacity,X1 : nat] : (($true != (isShortestPath @ capacity @ X0 @ X2 @ p @ X1)) | ((plus_plus @ nat @ one_one @ nat @ (min_dist @ capacity @ X0 @ v @ X1)) = (min_dist @ capacity @ X0 @ u @ X1)) | ($true != $true)) )),
% 9.44/2.76 inference(superposition,[status(thm)],[f986,f939])).
% 9.44/2.76 thf(f939,plain,(
% 9.44/2.76 ((member @ product_prod @ nat @ nat @ (product_Pair @ nat @ nat @ u @ v) @ (set2 @ product_prod @ nat @ nat @ p)) = $true)),
% 9.44/2.76 inference(cnf_transformation,[status(thm)],[f607])).
% 9.44/2.76 thf(f607,plain,(
% 9.44/2.76 ((member @ product_prod @ nat @ nat @ (product_Pair @ nat @ nat @ u @ v) @ (set2 @ product_prod @ nat @ nat @ p)) = $true)),
% 9.44/2.76 inference(fool_elimination,[status(thm)],[f606])).
% 9.44/2.76 thf(f606,plain,(
% 9.44/2.76 (member @ product_prod @ nat @ nat @ (product_Pair @ nat @ nat @ u @ v) @ (set2 @ product_prod @ nat @ nat @ p))),
% 9.44/2.76 inference(rectify,[status(thm)],[f50])).
% 9.44/2.76 thf(f50,axiom,(
% 9.44/2.76 (member @ product_prod @ nat @ nat @ (product_Pair @ nat @ nat @ u @ v) @ (set2 @ product_prod @ nat @ nat @ p))),
% 9.44/2.76 file('/export/starexec/sandbox2/tmp/tmp.dNuoTsUl6x/DTF2THF_2221.p',fact_49_assms_I2_J)).
% 9.44/2.76 thf(f986,plain,(
% 9.44/2.76 ( ! [X2 : list @ product_prod @ nat @ nat,X3 : nat,X0 : product_prod @ nat @ nat > capacity,X1 : nat,X4 : nat,X5 : nat] : (($true != (member @ product_prod @ nat @ nat @ (product_Pair @ nat @ nat @ X4 @ X5) @ (set2 @ product_prod @ nat @ nat @ X2))) | ((min_dist @ capacity @ X0 @ X4 @ X3) = (plus_plus @ nat @ one_one @ nat @ (min_dist @ capacity @ X0 @ X5 @ X3))) | ($true != (isShortestPath @ capacity @ X0 @ X1 @ X2 @ X3))) )),
% 9.44/2.76 inference(trivial_inequality_removal,[status(thm)],[f977])).
% 9.44/2.76 thf(f977,plain,(
% 9.44/2.76 ( ! [X2 : list @ product_prod @ nat @ nat,X3 : nat,X0 : product_prod @ nat @ nat > capacity,X1 : nat,X4 : nat,X5 : nat] : (($true != (member @ product_prod @ nat @ nat @ (product_Pair @ nat @ nat @ X4 @ X5) @ (set2 @ product_prod @ nat @ nat @ X2))) | ((min_dist @ capacity @ X0 @ X4 @ X3) = (plus_plus @ nat @ one_one @ nat @ (min_dist @ capacity @ X0 @ X5 @ X3))) | ($true != $true) | ($true != (isShortestPath @ capacity @ X0 @ X1 @ X2 @ X3))) )),
% 9.44/2.76 inference(superposition,[status(thm)],[f931,f938])).
% 9.44/2.76 thf(f938,plain,(
% 9.44/2.76 (linordered_idom @ capacity = $true)),
% 9.44/2.76 inference(cnf_transformation,[status(thm)],[f461])).
% 9.44/2.76 thf(f461,plain,(
% 9.44/2.76 (linordered_idom @ capacity = $true)),
% 9.44/2.76 inference(fool_elimination,[status(thm)],[f460])).
% 9.44/2.76 thf(f460,plain,(
% 9.44/2.76 linordered_idom @ capacity),
% 9.44/2.76 inference(rectify,[status(thm)],[f304])).
% 9.44/2.76 thf(f304,axiom,(
% 9.44/2.76 linordered_idom @ capacity),
% 9.44/2.76 file('/export/starexec/sandbox2/tmp/tmp.dNuoTsUl6x/DTF2THF_2221.p',tfree_0)).
% 9.44/2.76 thf(f931,plain,(
% 9.44/2.76 ( ! [X0 : $tType,X2 : nat,X3 : list @ product_prod @ nat @ nat,X1 : product_prod @ nat @ nat > X0,X6 : nat,X4 : nat,X5 : nat] : (((isShortestPath @ X0 @ X1 @ X2 @ X3 @ X4) != $true) | ((member @ product_prod @ nat @ nat @ (product_Pair @ nat @ nat @ X5 @ X6) @ (set2 @ product_prod @ nat @ nat @ X3)) != $true) | ((plus_plus @ nat @ one_one @ nat @ (min_dist @ X0 @ X1 @ X6 @ X4)) = (min_dist @ X0 @ X1 @ X5 @ X4)) | (linordered_idom @ X0 != $true)) )),
% 9.44/2.76 inference(cnf_transformation,[status(thm)],[f912])).
% 9.44/2.76 thf(f912,plain,(
% 9.44/2.76 ! [X0 : $tType] : (! [X1 : product_prod @ nat @ nat > X0,X2 : nat,X3 : list @ product_prod @ nat @ nat,X4 : nat,X5 : nat,X6 : nat] : (((plus_plus @ nat @ one_one @ nat @ (min_dist @ X0 @ X1 @ X6 @ X4)) = (min_dist @ X0 @ X1 @ X5 @ X4)) | ((isShortestPath @ X0 @ X1 @ X2 @ X3 @ X4) != $true) | ((member @ product_prod @ nat @ nat @ (product_Pair @ nat @ nat @ X5 @ X6) @ (set2 @ product_prod @ nat @ nat @ X3)) != $true)) | (linordered_idom @ X0 != $true))),
% 9.44/2.76 inference(flattening,[status(thm)],[f911])).
% 9.44/2.76 thf(f911,plain,(
% 9.44/2.76 ! [X0 : $tType] : (! [X1 : product_prod @ nat @ nat > X0,X2 : nat,X3 : list @ product_prod @ nat @ nat,X4 : nat,X5 : nat,X6 : nat] : ((((plus_plus @ nat @ one_one @ nat @ (min_dist @ X0 @ X1 @ X6 @ X4)) = (min_dist @ X0 @ X1 @ X5 @ X4)) | ((member @ product_prod @ nat @ nat @ (product_Pair @ nat @ nat @ X5 @ X6) @ (set2 @ product_prod @ nat @ nat @ X3)) != $true)) | ((isShortestPath @ X0 @ X1 @ X2 @ X3 @ X4) != $true)) | (linordered_idom @ X0 != $true))),
% 9.44/2.76 inference(ennf_transformation,[status(thm)],[f821])).
% 9.44/2.76 thf(f821,plain,(
% 9.44/2.76 ! [X0 : $tType] : ((linordered_idom @ X0 = $true) => ! [X1 : product_prod @ nat @ nat > X0,X2 : nat,X3 : list @ product_prod @ nat @ nat,X4 : nat,X5 : nat,X6 : nat] : (((isShortestPath @ X0 @ X1 @ X2 @ X3 @ X4) = $true) => (((member @ product_prod @ nat @ nat @ (product_Pair @ nat @ nat @ X5 @ X6) @ (set2 @ product_prod @ nat @ nat @ X3)) = $true) => ((plus_plus @ nat @ one_one @ nat @ (min_dist @ X0 @ X1 @ X6 @ X4)) = (min_dist @ X0 @ X1 @ X5 @ X4)))))),
% 9.44/2.76 inference(fool_elimination,[status(thm)],[f820])).
% 9.44/2.76 thf(f820,plain,(
% 9.44/2.76 ! [X0 : $tType] : (linordered_idom @ X0 => ! [X1 : product_prod @ nat @ nat > X0,X2 : nat,X3 : list @ product_prod @ nat @ nat,X4 : nat,X5 : nat,X6 : nat] : ((isShortestPath @ X0 @ X1 @ X2 @ X3 @ X4) => ((member @ product_prod @ nat @ nat @ (product_Pair @ nat @ nat @ X5 @ X6) @ (set2 @ product_prod @ nat @ nat @ X3)) => ((plus_plus @ nat @ one_one @ nat @ (min_dist @ X0 @ X1 @ X6 @ X4)) = (min_dist @ X0 @ X1 @ X5 @ X4)))))),
% 9.44/2.76 inference(rectify,[status(thm)],[f81])).
% 9.44/2.76 thf(f81,axiom,(
% 9.44/2.76 ! [X1 : $tType] : (linordered_idom @ X1 => ! [X11 : product_prod @ nat @ nat > X1,X47 : nat,X12 : list @ product_prod @ nat @ nat,X48 : nat,X4 : nat,X8 : nat] : ((isShortestPath @ X1 @ X11 @ X47 @ X12 @ X48) => ((member @ product_prod @ nat @ nat @ (product_Pair @ nat @ nat @ X4 @ X8) @ (set2 @ product_prod @ nat @ nat @ X12)) => ((min_dist @ X1 @ X11 @ X4 @ X48) = (plus_plus @ nat @ one_one @ nat @ (min_dist @ X1 @ X11 @ X8 @ X48))))))),
% 9.44/2.76 file('/export/starexec/sandbox2/tmp/tmp.dNuoTsUl6x/DTF2THF_2221.p',fact_80_Graph_OisShortestPath__level__edge_I5_J)).
% 9.44/2.76 % SZS output end Proof for DTF2THF_2221
% 9.44/2.76 % (2541)------------------------------
% 9.44/2.76 % (2541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.44/2.76 % (2541)Termination reason: Refutation
% 9.44/2.76
% 9.44/2.76 % (2541)Memory used [KB]: 7547
% 9.44/2.76 % (2541)Time elapsed: 0.128 s
% 9.44/2.76 % (2541)Instructions burned: 444 (million)
% 9.44/2.76 % (2541)------------------------------
% 9.44/2.76 % (2541)------------------------------
% 9.44/2.76 % (2440)Success in time 1.185 s
%------------------------------------------------------------------------------