%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP053^1 : 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 : n026.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:10 AM UTC 2026
% Result : Theorem 2.36s 1.62s
% Output : Refutation 2.36s
% Verified :
% SZS Type : Refutation
% Derivation depth : 8
% Number of leaves : 6
% Syntax : Number of formulae : 23 ( 11 unt; 0 typ; 2 def)
% Number of atoms : 35 ( 25 equ; 0 cnn)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 182 ( 10 ~; 7 |; 0 &; 160 @)
% ( 2 <=>; 3 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 3 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 43 ( 41 usr; 13 con; 0-4 aty)
% Number of variables : 19 ( 0 ^; 19 !; 0 ?; 19 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
product_prod_nat_nat: $tType ).
thf(type_def_6,type,
capacity: $tType ).
thf(type_def_8,type,
nat: $tType ).
thf(type_def_9,type,
list_P559422087at_nat: $tType ).
thf(type_def_10,type,
set_Pr1986765409at_nat: $tType ).
thf(type_def_11,type,
produc1695820582at_nat: $tType ).
thf(type_def_12,type,
produc2082277813at_nat: $tType ).
thf(type_def_13,type,
produc1190591575at_nat: $tType ).
thf(type_def_14,type,
produc1864880952at_nat: $tType ).
thf(type_def_15,type,
produc1271302400at_nat: $tType ).
thf(type_def_16,type,
produc787001653at_nat: $tType ).
thf(func_def_0,type,
produc1864880952at_nat: $tType ).
thf(func_def_1,type,
produc787001653at_nat: $tType ).
thf(func_def_2,type,
produc2082277813at_nat: $tType ).
thf(func_def_3,type,
produc1190591575at_nat: $tType ).
thf(func_def_4,type,
produc1271302400at_nat: $tType ).
thf(func_def_5,type,
produc1695820582at_nat: $tType ).
thf(func_def_6,type,
list_P559422087at_nat: $tType ).
thf(func_def_7,type,
set_Pr1986765409at_nat: $tType ).
thf(func_def_8,type,
product_prod_nat_nat: $tType ).
thf(func_def_9,type,
capacity: $tType ).
thf(func_def_10,type,
nat: $tType ).
thf(func_def_11,type,
connected_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > $o ).
thf(func_def_12,type,
dist_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > nat > $o ).
thf(func_def_13,type,
isPath_capacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).
thf(func_def_14,type,
isShor1936442771pacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).
thf(func_def_15,type,
isSimp1359852763pacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).
thf(func_def_16,type,
min_dist_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > nat ).
thf(func_def_17,type,
one_one_nat: nat ).
thf(func_def_18,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_19,type,
nil_Pr1308055047at_nat: list_P559422087at_nat ).
thf(func_def_20,type,
set_Pr2131844118at_nat: list_P559422087at_nat > set_Pr1986765409at_nat ).
thf(func_def_21,type,
size_s1990949619at_nat: list_P559422087at_nat > nat ).
thf(func_def_22,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_23,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_24,type,
produc1682925677at_nat: ( nat > nat > nat ) > produc1695820582at_nat > produc2082277813at_nat ).
thf(func_def_25,type,
produc696256106at_nat: ( nat > product_prod_nat_nat > product_prod_nat_nat ) > produc1190591575at_nat > produc1864880952at_nat ).
thf(func_def_26,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_27,type,
produc1933845336at_nat: nat > product_prod_nat_nat > produc1695820582at_nat ).
thf(func_def_28,type,
produc869658639at_nat: nat > produc1695820582at_nat > produc1190591575at_nat ).
thf(func_def_29,type,
produc947540346at_nat: product_prod_nat_nat > nat > produc1271302400at_nat ).
thf(func_def_30,type,
produc365964781at_nat: produc1695820582at_nat > ( nat > nat > nat ) > produc787001653at_nat ).
thf(func_def_31,type,
produc1728816653at_nat: produc2082277813at_nat > produc787001653at_nat ).
thf(func_def_32,type,
product_swap_nat_nat: product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_33,type,
produc857968056at_nat: produc1695820582at_nat > produc1271302400at_nat ).
thf(func_def_34,type,
produc2019146714at_nat: produc1271302400at_nat > produc1695820582at_nat ).
thf(func_def_35,type,
produc411855757at_nat: produc787001653at_nat > produc2082277813at_nat ).
thf(func_def_36,type,
collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
thf(func_def_37,type,
member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
thf(func_def_38,type,
c: product_prod_nat_nat > capacity ).
thf(func_def_39,type,
p: list_P559422087at_nat ).
thf(func_def_40,type,
p1: list_P559422087at_nat ).
thf(func_def_41,type,
p2: list_P559422087at_nat ).
thf(func_def_42,type,
p3: list_P559422087at_nat ).
thf(func_def_43,type,
s: nat ).
thf(func_def_44,type,
t: nat ).
thf(func_def_45,type,
u: nat ).
thf(func_def_46,type,
v: nat ).
thf(func_def_63,type,
sK0: nat > nat > nat > list_P559422087at_nat ).
thf(func_def_64,type,
sK1: nat > list_P559422087at_nat > nat > list_P559422087at_nat ).
thf(func_def_66,type,
ph3:
!>[X0: $tType] : X0 ).
thf(f842,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f785,f792,f841]) ).
thf(f841,plain,
spl2_1,
inference(avatar_contradiction_clause,[status(thm)],[f840]) ).
thf(f840,plain,
( $false
| spl2_1 ),
inference(subsumption_resolution,[status(thm)],[f829,f780]) ).
thf(f780,plain,
( ( ( min_dist_capacity @ c @ s @ t )
!= ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) )
| spl2_1 ),
inference(avatar_component_clause,[status(thm)],[f778]) ).
thf(f778,definition,
( spl2_1
<=> ( ( min_dist_capacity @ c @ s @ t )
= ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) ) ),
introduced(definition,[new_symbols(naming,[spl2_1])],[avatar_definition]) ).
thf(f829,plain,
( ( min_dist_capacity @ c @ s @ t )
= ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) ),
inference(superposition,[status(thm)],[f715,f772]) ).
thf(f772,plain,
! [X3: nat,X0: nat,X1: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ X1 @ X3 ) @ X0 )
= ( plus_plus_nat @ X1 @ ( plus_plus_nat @ X3 @ X0 ) ) ),
inference(equality_resolution,[status(thm)],[f757]) ).
thf(f757,plain,
! [X2: nat,X3: nat,X0: nat,X1: nat] :
( ( ( plus_plus_nat @ X1 @ X3 )
!= X2 )
| ( ( plus_plus_nat @ X2 @ X0 )
= ( plus_plus_nat @ X1 @ ( plus_plus_nat @ X3 @ X0 ) ) ) ),
inference(cnf_transformation,[status(thm)],[f684]) ).
thf(f684,plain,
! [X0: nat,X1: nat,X2: nat,X3: nat] :
( ( ( plus_plus_nat @ X1 @ X3 )
!= X2 )
| ( ( plus_plus_nat @ X2 @ X0 )
= ( plus_plus_nat @ X1 @ ( plus_plus_nat @ X3 @ X0 ) ) ) ),
inference(ennf_transformation,[status(thm)],[f673]) ).
thf(f673,plain,
! [X0: nat,X1: nat,X2: nat,X3: nat] :
( ( ( plus_plus_nat @ X1 @ X3 )
= X2 )
=> ( ( plus_plus_nat @ X2 @ X0 )
= ( plus_plus_nat @ X1 @ ( plus_plus_nat @ X3 @ X0 ) ) ) ),
inference(rectify,[status(thm)],[f27]) ).
thf(f27,axiom,
! [X6: nat,X15: nat,X16: nat,X5: nat] :
( ( ( plus_plus_nat @ X15 @ X5 )
= X16 )
=> ( ( plus_plus_nat @ X15 @ ( plus_plus_nat @ X5 @ X6 ) )
= ( plus_plus_nat @ X16 @ X6 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.avufQdrk0c/DTF2THF_22776.p',fact_26_group__cancel_Oadd1) ).
thf(f715,plain,
( ( min_dist_capacity @ c @ s @ t )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ one_one_nat ) @ ( size_s1990949619at_nat @ p2 ) ) ),
inference(cnf_transformation,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( min_dist_capacity @ c @ s @ t )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ one_one_nat ) @ ( size_s1990949619at_nat @ p2 ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.avufQdrk0c/DTF2THF_22776.p',fact_0_MIN_H) ).
thf(f792,plain,
~ spl2_2,
inference(avatar_split_clause,[status(thm)],[f722,f782]) ).
thf(f782,definition,
( spl2_2
<=> ( ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) )
= ( min_dist_capacity @ c @ u @ t ) ) ),
introduced(definition,[new_symbols(naming,[spl2_2])],[avatar_definition]) ).
thf(f722,plain,
( ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) )
!= ( min_dist_capacity @ c @ u @ t ) ),
inference(cnf_transformation,[status(thm)],[f666]) ).
thf(f666,plain,
( ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) )
!= ( min_dist_capacity @ c @ u @ t ) ),
inference(flattening,[status(thm)],[f256]) ).
thf(f256,negated_conjecture,
( ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) )
!= ( min_dist_capacity @ c @ u @ t ) ),
inference(negated_conjecture,[status(cth)],[f255]) ).
thf(f255,conjecture,
( ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) )
= ( min_dist_capacity @ c @ u @ t ) ),
file('/export/starexec/sandbox2/tmp/tmp.avufQdrk0c/DTF2THF_22776.p',conj_0) ).
thf(f785,plain,
( ~ spl2_1
| spl2_2 ),
inference(avatar_split_clause,[status(thm)],[f735,f782,f778]) ).
thf(f735,plain,
( ( ( min_dist_capacity @ c @ s @ t )
!= ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) )
| ( ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) )
= ( min_dist_capacity @ c @ u @ t ) ) ),
inference(cnf_transformation,[status(thm)],[f696]) ).
thf(f696,plain,
( ( ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) )
= ( min_dist_capacity @ c @ u @ t ) )
| ( ( min_dist_capacity @ c @ s @ t )
!= ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) ) ),
inference(ennf_transformation,[status(thm)],[f3]) ).
thf(f3,axiom,
( ( ( min_dist_capacity @ c @ s @ t )
= ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) )
=> ( ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) )
= ( min_dist_capacity @ c @ u @ t ) ) ),
file('/export/starexec/sandbox2/tmp/tmp.avufQdrk0c/DTF2THF_22776.p',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) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP053^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.15/0.33 % Computer : n026.cluster.edu
% 0.15/0.33 % Model : x86_64 x86_64
% 0.15/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.33 % Memory : 8042.1875MB
% 0.15/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.15/0.33 % CPULimit : 300
% 0.15/0.33 % WCLimit : 300
% 0.15/0.33 % DateTime : Tue Feb 24 08:24:55 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
% 0.20/0.44 ---- Original DTF file ---
% 0.20/0.45 thf(spec,logic,$$dhol).
% 0.20/0.45 %------------------------------------------------------------------------------
% 0.20/0.45 % File : ITP053^1 : TPTP v9.2.1. Released v7.5.0.
% 0.20/0.45 % Domain : Interactive Theorem Proving
% 0.20/0.45 % Problem : Sledgehammer EdmondsKarp_Termination_Abstract problem prob_81__7582066_1
% 0.20/0.45 % Version : Especial.
% 0.20/0.45 % English :
% 0.20/0.45
% 0.20/0.45 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.45 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.20/0.45 % Source : [Des21]
% 0.20/0.45 % Names : EdmondsKarp_Termination_Abstract/prob_81__7582066_1 [Des21]
% 0.20/0.45
% 0.20/0.45 % Status : Theorem
% 0.20/0.45 % Rating : 0.22 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% 0.20/0.45 % Syntax : Number of formulae : 302 ( 104 unt; 47 typ; 0 def)
% 0.20/0.45 % Number of atoms : 633 ( 212 equ; 0 cnn)
% 0.20/0.45 % Maximal formula atoms : 12 ( 2 avg)
% 0.20/0.45 % Number of connectives : 2205 ( 83 ~; 6 |; 34 &;1799 @)
% 0.20/0.45 % ( 0 <=>; 283 =>; 0 <=; 0 <~>)
% 0.20/0.45 % Maximal formula depth : 16 ( 7 avg)
% 0.20/0.45 % Number of types : 12 ( 11 usr)
% 0.20/0.45 % Number of type conns : 169 ( 169 >; 0 *; 0 +; 0 <<)
% 0.20/0.45 % Number of symbols : 37 ( 36 usr; 10 con; 0-4 aty)
% 0.20/0.45 % Number of variables : 832 ( 48 ^; 755 !; 29 ?; 832 :)
% 0.20/0.45 % SPC : TH0_THM_EQU_NAR
% 0.20/0.45
% 0.20/0.45 % Comments : This file was generated by Sledgehammer 2021-02-23 15:30:03.954
% 0.20/0.45 %------------------------------------------------------------------------------
% 0.20/0.45 % Could-be-implicit typings (11)
% 0.20/0.45 thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.20/0.45 produc1864880952at_nat: $tType ).
% 0.20/0.45
% 0.20/0.45 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.20/0.45 produc787001653at_nat: $tType ).
% 0.20/0.45
% 0.20/0.45 thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.20/0.45 produc2082277813at_nat: $tType ).
% 0.20/0.45
% 0.20/0.45 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.20/0.45 produc1190591575at_nat: $tType ).
% 0.20/0.45
% 0.20/0.45 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
% 0.20/0.45 produc1271302400at_nat: $tType ).
% 0.20/0.45
% 0.20/0.45 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.20/0.45 produc1695820582at_nat: $tType ).
% 0.20/0.45
% 0.20/0.45 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.20/0.45 list_P559422087at_nat: $tType ).
% 0.20/0.45
% 0.20/0.45 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.20/0.45 set_Pr1986765409at_nat: $tType ).
% 0.20/0.45
% 0.20/0.45 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.20/0.45 product_prod_nat_nat: $tType ).
% 0.20/0.45
% 0.20/0.45 thf(ty_n_tf__capacity,type,
% 0.20/0.45 capacity: $tType ).
% 0.20/0.45
% 0.20/0.45 thf(ty_n_t__Nat__Onat,type,
% 0.20/0.45 nat: $tType ).
% 0.20/0.45
% 0.20/0.45 % Explicit typings (36)
% 0.20/0.45 thf(sy_c_Graph_OGraph_Oconnected_001tf__capacity,type,
% 0.20/0.45 connected_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Graph_OGraph_Odist_001tf__capacity,type,
% 0.20/0.45 dist_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > nat > $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Graph_OGraph_OisPath_001tf__capacity,type,
% 0.20/0.45 isPath_capacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Graph_OGraph_OisShortestPath_001tf__capacity,type,
% 0.20/0.45 isShor1936442771pacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Graph_OGraph_OisSimplePath_001tf__capacity,type,
% 0.20/0.45 isSimp1359852763pacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Graph_OGraph_Omin__dist_001tf__capacity,type,
% 0.20/0.45 min_dist_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 0.20/0.45 one_one_nat: nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.20/0.45 plus_plus_nat: nat > nat > nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.20/0.45 nil_Pr1308055047at_nat: list_P559422087at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.20/0.45 set_Pr2131844118at_nat: list_P559422087at_nat > set_Pr1986765409at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.20/0.45 size_s1990949619at_nat: list_P559422087at_nat > nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.20/0.45 ord_less_nat: nat > nat > $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.20/0.45 ord_less_eq_nat: nat > nat > $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.20/0.45 produc1682925677at_nat: ( nat > nat > nat ) > produc1695820582at_nat > produc2082277813at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.20/0.45 produc696256106at_nat: ( nat > product_prod_nat_nat > product_prod_nat_nat ) > produc1190591575at_nat > produc1864880952at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.20/0.45 product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.20/0.45 produc1933845336at_nat: nat > product_prod_nat_nat > produc1695820582at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.20/0.45 produc869658639at_nat: nat > produc1695820582at_nat > produc1190591575at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 0.20/0.45 produc947540346at_nat: product_prod_nat_nat > nat > produc1271302400at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.20/0.45 produc365964781at_nat: produc1695820582at_nat > ( nat > nat > nat ) > produc787001653at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Product__Type_Oprod_Oswap_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.20/0.45 produc1728816653at_nat: produc2082277813at_nat > produc787001653at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.20/0.45 product_swap_nat_nat: product_prod_nat_nat > product_prod_nat_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.20/0.45 produc857968056at_nat: produc1695820582at_nat > produc1271302400at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Product__Type_Oprod_Oswap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 0.20/0.45 produc2019146714at_nat: produc1271302400at_nat > produc1695820582at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Product__Type_Oprod_Oswap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.20/0.45 produc411855757at_nat: produc787001653at_nat > produc2082277813at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.20/0.45 collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.20/0.45 member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
% 0.20/0.45
% 0.20/0.45 thf(sy_v_c,type,
% 0.20/0.45 c: product_prod_nat_nat > capacity ).
% 0.20/0.45
% 0.20/0.45 thf(sy_v_p,type,
% 0.20/0.45 p: list_P559422087at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_v_p1____,type,
% 0.20/0.45 p1: list_P559422087at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_v_p2____,type,
% 0.20/0.45 p2: list_P559422087at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_v_p_H,type,
% 0.20/0.45 p3: list_P559422087at_nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_v_s,type,
% 0.20/0.45 s: nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_v_t,type,
% 0.20/0.45 t: nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_v_u,type,
% 0.20/0.45 u: nat ).
% 0.20/0.45
% 0.20/0.45 thf(sy_v_v,type,
% 0.20/0.45 v: nat ).
% 0.20/0.45
% 0.20/0.45 % Relevant facts (254)
% 0.20/0.45 thf(fact_0_MIN_H,axiom,
% 0.20/0.45 ( ( min_dist_capacity @ c @ s @ t )
% 0.20/0.45 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ one_one_nat ) @ ( size_s1990949619at_nat @ p2 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % MIN'
% 0.20/0.45 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,
% 0.20/0.45 ( ( ( min_dist_capacity @ c @ s @ t )
% 0.20/0.45 = ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) )
% 0.20/0.45 => ( ( min_dist_capacity @ c @ s @ u )
% 0.20/0.45 = ( size_s1990949619at_nat @ p1 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % \<open>min_dist s t = length p1 + (1 + length p2) \<Longrightarrow> min_dist s u = length p1\<close>
% 0.20/0.45 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,
% 0.20/0.45 ( ( ( min_dist_capacity @ c @ s @ t )
% 0.20/0.45 = ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) )
% 0.20/0.45 => ( ( min_dist_capacity @ c @ u @ t )
% 0.20/0.45 = ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % \<open>min_dist s t = length p1 + (1 + length p2) \<Longrightarrow> min_dist u t = 1 + length p2\<close>
% 0.20/0.45 thf(fact_3__092_060open_062min__dist_Av_At_A_061_Alength_Ap2_092_060close_062,axiom,
% 0.20/0.45 ( ( min_dist_capacity @ c @ v @ t )
% 0.20/0.45 = ( size_s1990949619at_nat @ p2 ) ) ).
% 0.20/0.45
% 0.20/0.45 % \<open>min_dist v t = length p2\<close>
% 0.20/0.45 thf(fact_4_MIN,axiom,
% 0.20/0.45 ( ( min_dist_capacity @ c @ s @ t )
% 0.20/0.45 = ( size_s1990949619at_nat @ p ) ) ).
% 0.20/0.45
% 0.20/0.45 % MIN
% 0.20/0.45 thf(fact_5__092_060open_062u_A_092_060noteq_062_Av_092_060close_062,axiom,
% 0.20/0.45 u != v ).
% 0.20/0.45
% 0.20/0.45 % \<open>u \<noteq> v\<close>
% 0.20/0.45 thf(fact_6_MDSV,axiom,
% 0.20/0.45 ( ( min_dist_capacity @ c @ s @ v )
% 0.20/0.45 = ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ one_one_nat ) ) ).
% 0.20/0.45
% 0.20/0.45 % MDSV
% 0.20/0.45 thf(fact_7_min__dist__split_I2_J,axiom,
% 0.20/0.45 ! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 0.20/0.45 ( ( dist_capacity @ c @ U @ D1 @ W )
% 0.20/0.45 => ( ( dist_capacity @ c @ W @ D2 @ V )
% 0.20/0.45 => ( ( ( min_dist_capacity @ c @ U @ V )
% 0.20/0.45 = ( plus_plus_nat @ D1 @ D2 ) )
% 0.20/0.45 => ( ( min_dist_capacity @ c @ W @ V )
% 0.20/0.45 = D2 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % min_dist_split(2)
% 0.20/0.45 thf(fact_8_min__dist__split_I1_J,axiom,
% 0.20/0.45 ! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 0.20/0.45 ( ( dist_capacity @ c @ U @ D1 @ W )
% 0.20/0.45 => ( ( dist_capacity @ c @ W @ D2 @ V )
% 0.20/0.45 => ( ( ( min_dist_capacity @ c @ U @ V )
% 0.20/0.45 = ( plus_plus_nat @ D1 @ D2 ) )
% 0.20/0.45 => ( ( min_dist_capacity @ c @ U @ W )
% 0.20/0.45 = D1 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % min_dist_split(1)
% 0.20/0.45 thf(fact_9_add__left__cancel,axiom,
% 0.20/0.45 ! [A: nat,B: nat,C: nat] :
% 0.20/0.45 ( ( ( plus_plus_nat @ A @ B )
% 0.20/0.45 = ( plus_plus_nat @ A @ C ) )
% 0.20/0.45 = ( B = C ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_left_cancel
% 0.20/0.45 thf(fact_10_add__right__cancel,axiom,
% 0.20/0.45 ! [B: nat,A: nat,C: nat] :
% 0.20/0.45 ( ( ( plus_plus_nat @ B @ A )
% 0.20/0.45 = ( plus_plus_nat @ C @ A ) )
% 0.20/0.45 = ( B = C ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_right_cancel
% 0.20/0.45 thf(fact_11_dist__trans,axiom,
% 0.20/0.45 ! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 0.20/0.45 ( ( dist_capacity @ c @ U @ D1 @ W )
% 0.20/0.45 => ( ( dist_capacity @ c @ W @ D2 @ V )
% 0.20/0.45 => ( dist_capacity @ c @ U @ ( plus_plus_nat @ D1 @ D2 ) @ V ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % dist_trans
% 0.20/0.45 thf(fact_12_assms_I3_J,axiom,
% 0.20/0.45 isPath_capacity @ c @ s @ p3 @ t ).
% 0.20/0.45
% 0.20/0.45 % assms(3)
% 0.20/0.45 thf(fact_13_DISTS_I1_J,axiom,
% 0.20/0.45 dist_capacity @ c @ s @ ( size_s1990949619at_nat @ p1 ) @ u ).
% 0.20/0.45
% 0.20/0.45 % DISTS(1)
% 0.20/0.45 thf(fact_14_isPath__distD,axiom,
% 0.20/0.45 ! [U: nat,P: list_P559422087at_nat,V: nat] :
% 0.20/0.45 ( ( isPath_capacity @ c @ U @ P @ V )
% 0.20/0.45 => ( dist_capacity @ c @ U @ ( size_s1990949619at_nat @ P ) @ V ) ) ).
% 0.20/0.45
% 0.20/0.45 % isPath_distD
% 0.20/0.45 thf(fact_15_dist__def,axiom,
% 0.20/0.45 ! [V: nat,D: nat,V2: nat] :
% 0.20/0.45 ( ( dist_capacity @ c @ V @ D @ V2 )
% 0.20/0.45 = ( ? [P2: list_P559422087at_nat] :
% 0.20/0.45 ( ( isPath_capacity @ c @ V @ P2 @ V2 )
% 0.20/0.45 & ( ( size_s1990949619at_nat @ P2 )
% 0.20/0.45 = D ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % dist_def
% 0.20/0.45 thf(fact_16_DISTS_I2_J,axiom,
% 0.20/0.45 dist_capacity @ c @ u @ one_one_nat @ v ).
% 0.20/0.45
% 0.20/0.45 % DISTS(2)
% 0.20/0.45 thf(fact_17_P,axiom,
% 0.20/0.45 isPath_capacity @ c @ s @ p @ t ).
% 0.20/0.45
% 0.20/0.45 % P
% 0.20/0.45 thf(fact_18_DISTS_I3_J,axiom,
% 0.20/0.45 dist_capacity @ c @ v @ ( size_s1990949619at_nat @ p2 ) @ t ).
% 0.20/0.45
% 0.20/0.45 % DISTS(3)
% 0.20/0.45 thf(fact_19_assms_I1_J,axiom,
% 0.20/0.45 isShor1936442771pacity @ c @ s @ p @ t ).
% 0.20/0.45
% 0.20/0.45 % assms(1)
% 0.20/0.45 thf(fact_20_add__right__imp__eq,axiom,
% 0.20/0.45 ! [B: nat,A: nat,C: nat] :
% 0.20/0.45 ( ( ( plus_plus_nat @ B @ A )
% 0.20/0.45 = ( plus_plus_nat @ C @ A ) )
% 0.20/0.45 => ( B = C ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_right_imp_eq
% 0.20/0.45 thf(fact_21_add__left__imp__eq,axiom,
% 0.20/0.45 ! [A: nat,B: nat,C: nat] :
% 0.20/0.45 ( ( ( plus_plus_nat @ A @ B )
% 0.20/0.45 = ( plus_plus_nat @ A @ C ) )
% 0.20/0.45 => ( B = C ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_left_imp_eq
% 0.20/0.45 thf(fact_22_add_Oleft__commute,axiom,
% 0.20/0.45 ! [B: nat,A: nat,C: nat] :
% 0.20/0.45 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 0.20/0.45 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add.left_commute
% 0.20/0.45 thf(fact_23_add_Ocommute,axiom,
% 0.20/0.45 ( plus_plus_nat
% 0.20/0.45 = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add.commute
% 0.20/0.45 thf(fact_24_add_Oassoc,axiom,
% 0.20/0.45 ! [A: nat,B: nat,C: nat] :
% 0.20/0.45 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.20/0.45 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add.assoc
% 0.20/0.45 thf(fact_25_group__cancel_Oadd2,axiom,
% 0.20/0.45 ! [B3: nat,K: nat,B: nat,A: nat] :
% 0.20/0.45 ( ( B3
% 0.20/0.45 = ( plus_plus_nat @ K @ B ) )
% 0.20/0.45 => ( ( plus_plus_nat @ A @ B3 )
% 0.20/0.45 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % group_cancel.add2
% 0.20/0.45 thf(fact_26_group__cancel_Oadd1,axiom,
% 0.20/0.45 ! [A3: nat,K: nat,A: nat,B: nat] :
% 0.20/0.45 ( ( A3
% 0.20/0.45 = ( plus_plus_nat @ K @ A ) )
% 0.20/0.45 => ( ( plus_plus_nat @ A3 @ B )
% 0.20/0.45 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % group_cancel.add1
% 0.20/0.45 thf(fact_27_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.20/0.45 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.20/0.45 ( ( ( I = J )
% 0.20/0.45 & ( K = L ) )
% 0.20/0.45 => ( ( plus_plus_nat @ I @ K )
% 0.20/0.45 = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_mono_thms_linordered_semiring(4)
% 0.20/0.45 thf(fact_28_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.20/0.45 ! [A: nat,B: nat,C: nat] :
% 0.20/0.45 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.20/0.45 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % ab_semigroup_add_class.add_ac(1)
% 0.20/0.45 thf(fact_29_one__reorient,axiom,
% 0.20/0.45 ! [X: nat] :
% 0.20/0.45 ( ( one_one_nat = X )
% 0.20/0.45 = ( X = one_one_nat ) ) ).
% 0.20/0.45
% 0.20/0.45 % one_reorient
% 0.20/0.45 thf(fact_30_isShortestPath__min__dist__def,axiom,
% 0.20/0.45 ! [U: nat,P: list_P559422087at_nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ c @ U @ P @ V )
% 0.20/0.45 = ( ( isPath_capacity @ c @ U @ P @ V )
% 0.20/0.45 & ( ( size_s1990949619at_nat @ P )
% 0.20/0.45 = ( min_dist_capacity @ c @ U @ V ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % isShortestPath_min_dist_def
% 0.20/0.45 thf(fact_31_Graph_OisPath__distD,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat] :
% 0.20/0.45 ( ( isPath_capacity @ C @ U @ P @ V )
% 0.20/0.45 => ( dist_capacity @ C @ U @ ( size_s1990949619at_nat @ P ) @ V ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isPath_distD
% 0.20/0.45 thf(fact_32_Graph_Odist__def,axiom,
% 0.20/0.45 ( dist_capacity
% 0.20/0.45 = ( ^ [C2: product_prod_nat_nat > capacity,V3: nat,D3: nat,V4: nat] :
% 0.20/0.45 ? [P2: list_P559422087at_nat] :
% 0.20/0.45 ( ( isPath_capacity @ C2 @ V3 @ P2 @ V4 )
% 0.20/0.45 & ( ( size_s1990949619at_nat @ P2 )
% 0.20/0.45 = D3 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.dist_def
% 0.20/0.45 thf(fact_33_min__dist__minD,axiom,
% 0.20/0.45 ! [V: nat,D: nat,V2: nat] :
% 0.20/0.45 ( ( dist_capacity @ c @ V @ D @ V2 )
% 0.20/0.45 => ( ord_less_eq_nat @ ( min_dist_capacity @ c @ V @ V2 ) @ D ) ) ).
% 0.20/0.45
% 0.20/0.45 % min_dist_minD
% 0.20/0.45 thf(fact_34_min__distI__eq,axiom,
% 0.20/0.45 ! [V: nat,D: nat,V2: nat] :
% 0.20/0.45 ( ( dist_capacity @ c @ V @ D @ V2 )
% 0.20/0.45 => ( ! [D4: nat] :
% 0.20/0.45 ( ( dist_capacity @ c @ V @ D4 @ V2 )
% 0.20/0.45 => ( ord_less_eq_nat @ D @ D4 ) )
% 0.20/0.45 => ( ( min_dist_capacity @ c @ V @ V2 )
% 0.20/0.45 = D ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % min_distI_eq
% 0.20/0.45 thf(fact_35_Graph_Omin__dist__split_I1_J,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 0.20/0.45 ( ( dist_capacity @ C @ U @ D1 @ W )
% 0.20/0.45 => ( ( dist_capacity @ C @ W @ D2 @ V )
% 0.20/0.45 => ( ( ( min_dist_capacity @ C @ U @ V )
% 0.20/0.45 = ( plus_plus_nat @ D1 @ D2 ) )
% 0.20/0.45 => ( ( min_dist_capacity @ C @ U @ W )
% 0.20/0.45 = D1 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.min_dist_split(1)
% 0.20/0.45 thf(fact_36_Graph_Omin__dist__split_I2_J,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 0.20/0.45 ( ( dist_capacity @ C @ U @ D1 @ W )
% 0.20/0.45 => ( ( dist_capacity @ C @ W @ D2 @ V )
% 0.20/0.45 => ( ( ( min_dist_capacity @ C @ U @ V )
% 0.20/0.45 = ( plus_plus_nat @ D1 @ D2 ) )
% 0.20/0.45 => ( ( min_dist_capacity @ C @ W @ V )
% 0.20/0.45 = D2 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.min_dist_split(2)
% 0.20/0.45 thf(fact_37_min__dist__is__dist,axiom,
% 0.20/0.45 ! [V: nat,V2: nat] :
% 0.20/0.45 ( ( connected_capacity @ c @ V @ V2 )
% 0.20/0.45 => ( dist_capacity @ c @ V @ ( min_dist_capacity @ c @ V @ V2 ) @ V2 ) ) ).
% 0.20/0.45
% 0.20/0.45 % min_dist_is_dist
% 0.20/0.45 thf(fact_38_shortestPath__is__path,axiom,
% 0.20/0.45 ! [U: nat,P: list_P559422087at_nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ c @ U @ P @ V )
% 0.20/0.45 => ( isPath_capacity @ c @ U @ P @ V ) ) ).
% 0.20/0.45
% 0.20/0.45 % shortestPath_is_path
% 0.20/0.45 thf(fact_39_assms_I4_J,axiom,
% 0.20/0.45 member701585322at_nat @ ( product_Pair_nat_nat @ v @ u ) @ ( set_Pr2131844118at_nat @ p3 ) ).
% 0.20/0.45
% 0.20/0.45 % assms(4)
% 0.20/0.45 thf(fact_40_connected__def,axiom,
% 0.20/0.45 ! [U: nat,V: nat] :
% 0.20/0.45 ( ( connected_capacity @ c @ U @ V )
% 0.20/0.45 = ( ? [P2: list_P559422087at_nat] : ( isPath_capacity @ c @ U @ P2 @ V ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % connected_def
% 0.20/0.45 thf(fact_41_connected__by__dist,axiom,
% 0.20/0.45 ! [V: nat,V2: nat] :
% 0.20/0.45 ( ( connected_capacity @ c @ V @ V2 )
% 0.20/0.45 = ( ? [D3: nat] : ( dist_capacity @ c @ V @ D3 @ V2 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % connected_by_dist
% 0.20/0.45 thf(fact_42_mem__Collect__eq,axiom,
% 0.20/0.45 ! [A: product_prod_nat_nat,P3: product_prod_nat_nat > $o] :
% 0.20/0.45 ( ( member701585322at_nat @ A @ ( collec7649004at_nat @ P3 ) )
% 0.20/0.45 = ( P3 @ A ) ) ).
% 0.20/0.45
% 0.20/0.45 % mem_Collect_eq
% 0.20/0.45 thf(fact_43_Collect__mem__eq,axiom,
% 0.20/0.45 ! [A3: set_Pr1986765409at_nat] :
% 0.20/0.45 ( ( collec7649004at_nat
% 0.20/0.45 @ ^ [X2: product_prod_nat_nat] : ( member701585322at_nat @ X2 @ A3 ) )
% 0.20/0.45 = A3 ) ).
% 0.20/0.45
% 0.20/0.45 % Collect_mem_eq
% 0.20/0.45 thf(fact_44_Collect__cong,axiom,
% 0.20/0.45 ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.20/0.45 ( ! [X3: product_prod_nat_nat] :
% 0.20/0.45 ( ( P3 @ X3 )
% 0.20/0.45 = ( Q @ X3 ) )
% 0.20/0.45 => ( ( collec7649004at_nat @ P3 )
% 0.20/0.45 = ( collec7649004at_nat @ Q ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Collect_cong
% 0.20/0.45 thf(fact_45_obtain__shortest__path,axiom,
% 0.20/0.45 ! [U: nat,V: nat] :
% 0.20/0.45 ( ( connected_capacity @ c @ U @ V )
% 0.20/0.45 => ~ ! [P4: list_P559422087at_nat] :
% 0.20/0.45 ~ ( isShor1936442771pacity @ c @ U @ P4 @ V ) ) ).
% 0.20/0.45
% 0.20/0.45 % obtain_shortest_path
% 0.20/0.45 thf(fact_46_assms_I2_J,axiom,
% 0.20/0.45 member701585322at_nat @ ( product_Pair_nat_nat @ u @ v ) @ ( set_Pr2131844118at_nat @ p ) ).
% 0.20/0.45
% 0.20/0.45 % assms(2)
% 0.20/0.45 thf(fact_47_isPath__ex__edge1,axiom,
% 0.20/0.45 ! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
% 0.20/0.45 ( ( isPath_capacity @ c @ U @ P @ V )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( ( U1 != U )
% 0.20/0.45 => ? [U2: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ U2 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % isPath_ex_edge1
% 0.20/0.45 thf(fact_48_isPath__ex__edge2,axiom,
% 0.20/0.45 ! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
% 0.20/0.45 ( ( isPath_capacity @ c @ U @ P @ V )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( ( V1 != V )
% 0.20/0.45 => ? [V22: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ V1 @ V22 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % isPath_ex_edge2
% 0.20/0.45 thf(fact_49_min__dist__le,axiom,
% 0.20/0.45 ! [Src: nat,V: nat,D5: nat] :
% 0.20/0.45 ( ( connected_capacity @ c @ Src @ V )
% 0.20/0.45 => ( ( ord_less_eq_nat @ D5 @ ( min_dist_capacity @ c @ Src @ V ) )
% 0.20/0.45 => ? [V5: nat] :
% 0.20/0.45 ( ( connected_capacity @ c @ Src @ V5 )
% 0.20/0.45 & ( ( min_dist_capacity @ c @ Src @ V5 )
% 0.20/0.45 = D5 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % min_dist_le
% 0.20/0.45 thf(fact_50_min__distI2,axiom,
% 0.20/0.45 ! [V: nat,V2: nat,Q: nat > $o] :
% 0.20/0.45 ( ( connected_capacity @ c @ V @ V2 )
% 0.20/0.45 => ( ! [D6: nat] :
% 0.20/0.45 ( ( dist_capacity @ c @ V @ D6 @ V2 )
% 0.20/0.45 => ( ! [D7: nat] :
% 0.20/0.45 ( ( dist_capacity @ c @ V @ D7 @ V2 )
% 0.20/0.45 => ( ord_less_eq_nat @ D6 @ D7 ) )
% 0.20/0.45 => ( Q @ D6 ) ) )
% 0.20/0.45 => ( Q @ ( min_dist_capacity @ c @ V @ V2 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % min_distI2
% 0.20/0.45 thf(fact_51_isShortestPath__def,axiom,
% 0.20/0.45 ! [U: nat,P: list_P559422087at_nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ c @ U @ P @ V )
% 0.20/0.45 = ( ( isPath_capacity @ c @ U @ P @ V )
% 0.20/0.45 & ! [P5: list_P559422087at_nat] :
% 0.20/0.45 ( ( isPath_capacity @ c @ U @ P5 @ V )
% 0.20/0.45 => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P ) @ ( size_s1990949619at_nat @ P5 ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % isShortestPath_def
% 0.20/0.45 thf(fact_52_isShortestPath__level__edge_I1_J,axiom,
% 0.20/0.45 ! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ c @ S @ P @ T )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( connected_capacity @ c @ S @ U ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % isShortestPath_level_edge(1)
% 0.20/0.45 thf(fact_53_isShortestPath__level__edge_I2_J,axiom,
% 0.20/0.45 ! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ c @ S @ P @ T )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( connected_capacity @ c @ U @ V ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % isShortestPath_level_edge(2)
% 0.20/0.45 thf(fact_54_isShortestPath__level__edge_I3_J,axiom,
% 0.20/0.45 ! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ c @ S @ P @ T )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( connected_capacity @ c @ V @ T ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % isShortestPath_level_edge(3)
% 0.20/0.45 thf(fact_55_add__le__cancel__left,axiom,
% 0.20/0.45 ! [C: nat,A: nat,B: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.20/0.45 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_le_cancel_left
% 0.20/0.45 thf(fact_56_add__le__cancel__right,axiom,
% 0.20/0.45 ! [A: nat,C: nat,B: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.20/0.45 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_le_cancel_right
% 0.20/0.45 thf(fact_57_isShortestPath__level__edge_I6_J,axiom,
% 0.20/0.45 ! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ c @ S @ P @ T )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( ( min_dist_capacity @ c @ S @ T )
% 0.20/0.45 = ( plus_plus_nat @ ( plus_plus_nat @ ( min_dist_capacity @ c @ S @ U ) @ one_one_nat ) @ ( min_dist_capacity @ c @ V @ T ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % isShortestPath_level_edge(6)
% 0.20/0.45 thf(fact_58_isShortestPath__level__edge_I5_J,axiom,
% 0.20/0.45 ! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ c @ S @ P @ T )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( ( min_dist_capacity @ c @ U @ T )
% 0.20/0.45 = ( plus_plus_nat @ one_one_nat @ ( min_dist_capacity @ c @ V @ T ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % isShortestPath_level_edge(5)
% 0.20/0.45 thf(fact_59_isShortestPath__level__edge_I4_J,axiom,
% 0.20/0.45 ! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ c @ S @ P @ T )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( ( min_dist_capacity @ c @ S @ V )
% 0.20/0.45 = ( plus_plus_nat @ ( min_dist_capacity @ c @ S @ U ) @ one_one_nat ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % isShortestPath_level_edge(4)
% 0.20/0.45 thf(fact_60_connected__refl,axiom,
% 0.20/0.45 ! [V: nat] : ( connected_capacity @ c @ V @ V ) ).
% 0.20/0.45
% 0.20/0.45 % connected_refl
% 0.20/0.45 thf(fact_61_connected__distI,axiom,
% 0.20/0.45 ! [V: nat,D: nat,V2: nat] :
% 0.20/0.45 ( ( dist_capacity @ c @ V @ D @ V2 )
% 0.20/0.45 => ( connected_capacity @ c @ V @ V2 ) ) ).
% 0.20/0.45
% 0.20/0.45 % connected_distI
% 0.20/0.45 thf(fact_62_Graph_OisPath__ex__edge1,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
% 0.20/0.45 ( ( isPath_capacity @ C @ U @ P @ V )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( ( U1 != U )
% 0.20/0.45 => ? [U2: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ U2 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isPath_ex_edge1
% 0.20/0.45 thf(fact_63_Graph_OisPath__ex__edge2,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
% 0.20/0.45 ( ( isPath_capacity @ C @ U @ P @ V )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( ( V1 != V )
% 0.20/0.45 => ? [V22: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ V1 @ V22 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isPath_ex_edge2
% 0.20/0.45 thf(fact_64_Graph_Oobtain__shortest__path,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,U: nat,V: nat] :
% 0.20/0.45 ( ( connected_capacity @ C @ U @ V )
% 0.20/0.45 => ~ ! [P4: list_P559422087at_nat] :
% 0.20/0.45 ~ ( isShor1936442771pacity @ C @ U @ P4 @ V ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.obtain_shortest_path
% 0.20/0.45 thf(fact_65_Graph_OisShortestPath_Ocong,axiom,
% 0.20/0.45 isShor1936442771pacity = isShor1936442771pacity ).
% 0.20/0.45
% 0.20/0.45 % Graph.isShortestPath.cong
% 0.20/0.45 thf(fact_66_Graph_Oconnected__refl,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,V: nat] : ( connected_capacity @ C @ V @ V ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.connected_refl
% 0.20/0.45 thf(fact_67_Graph_Oconnected_Ocong,axiom,
% 0.20/0.45 connected_capacity = connected_capacity ).
% 0.20/0.45
% 0.20/0.45 % Graph.connected.cong
% 0.20/0.45 thf(fact_68_Graph_Omin__dist__le,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,Src: nat,V: nat,D5: nat] :
% 0.20/0.45 ( ( connected_capacity @ C @ Src @ V )
% 0.20/0.45 => ( ( ord_less_eq_nat @ D5 @ ( min_dist_capacity @ C @ Src @ V ) )
% 0.20/0.45 => ? [V5: nat] :
% 0.20/0.45 ( ( connected_capacity @ C @ Src @ V5 )
% 0.20/0.45 & ( ( min_dist_capacity @ C @ Src @ V5 )
% 0.20/0.45 = D5 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.min_dist_le
% 0.20/0.45 thf(fact_69_Graph_OisShortestPath__level__edge_I1_J,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ C @ S @ P @ T )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( connected_capacity @ C @ S @ U ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isShortestPath_level_edge(1)
% 0.20/0.45 thf(fact_70_Graph_OisShortestPath__level__edge_I2_J,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ C @ S @ P @ T )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( connected_capacity @ C @ U @ V ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isShortestPath_level_edge(2)
% 0.20/0.45 thf(fact_71_Graph_OisShortestPath__level__edge_I3_J,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ C @ S @ P @ T )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( connected_capacity @ C @ V @ T ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isShortestPath_level_edge(3)
% 0.20/0.45 thf(fact_72_Graph_Oconnected__def,axiom,
% 0.20/0.45 ( connected_capacity
% 0.20/0.45 = ( ^ [C2: product_prod_nat_nat > capacity,U3: nat,V3: nat] :
% 0.20/0.45 ? [P2: list_P559422087at_nat] : ( isPath_capacity @ C2 @ U3 @ P2 @ V3 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.connected_def
% 0.20/0.45 thf(fact_73_Graph_Oconnected__by__dist,axiom,
% 0.20/0.45 ( connected_capacity
% 0.20/0.45 = ( ^ [C2: product_prod_nat_nat > capacity,V3: nat,V4: nat] :
% 0.20/0.45 ? [D3: nat] : ( dist_capacity @ C2 @ V3 @ D3 @ V4 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.connected_by_dist
% 0.20/0.45 thf(fact_74_Graph_Oconnected__distI,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,V: nat,D: nat,V2: nat] :
% 0.20/0.45 ( ( dist_capacity @ C @ V @ D @ V2 )
% 0.20/0.45 => ( connected_capacity @ C @ V @ V2 ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.connected_distI
% 0.20/0.45 thf(fact_75_Graph_OshortestPath__is__path,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ C @ U @ P @ V )
% 0.20/0.45 => ( isPath_capacity @ C @ U @ P @ V ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.shortestPath_is_path
% 0.20/0.45 thf(fact_76_Graph_OisShortestPath__level__edge_I6_J,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ C @ S @ P @ T )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( ( min_dist_capacity @ C @ S @ T )
% 0.20/0.45 = ( plus_plus_nat @ ( plus_plus_nat @ ( min_dist_capacity @ C @ S @ U ) @ one_one_nat ) @ ( min_dist_capacity @ C @ V @ T ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isShortestPath_level_edge(6)
% 0.20/0.45 thf(fact_77_Graph_OisShortestPath__level__edge_I5_J,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ C @ S @ P @ T )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( ( min_dist_capacity @ C @ U @ T )
% 0.20/0.45 = ( plus_plus_nat @ one_one_nat @ ( min_dist_capacity @ C @ V @ T ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isShortestPath_level_edge(5)
% 0.20/0.45 thf(fact_78_Graph_OisShortestPath__level__edge_I4_J,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ C @ S @ P @ T )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( ( min_dist_capacity @ C @ S @ V )
% 0.20/0.45 = ( plus_plus_nat @ ( min_dist_capacity @ C @ S @ U ) @ one_one_nat ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isShortestPath_level_edge(4)
% 0.20/0.45 thf(fact_79_Graph_Omin__distI2,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,V: nat,V2: nat,Q: nat > $o] :
% 0.20/0.45 ( ( connected_capacity @ C @ V @ V2 )
% 0.20/0.45 => ( ! [D6: nat] :
% 0.20/0.45 ( ( dist_capacity @ C @ V @ D6 @ V2 )
% 0.20/0.45 => ( ! [D7: nat] :
% 0.20/0.45 ( ( dist_capacity @ C @ V @ D7 @ V2 )
% 0.20/0.45 => ( ord_less_eq_nat @ D6 @ D7 ) )
% 0.20/0.45 => ( Q @ D6 ) ) )
% 0.20/0.45 => ( Q @ ( min_dist_capacity @ C @ V @ V2 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.min_distI2
% 0.20/0.45 thf(fact_80_Graph_OisShortestPath__def,axiom,
% 0.20/0.45 ( isShor1936442771pacity
% 0.20/0.45 = ( ^ [C2: product_prod_nat_nat > capacity,U3: nat,P2: list_P559422087at_nat,V3: nat] :
% 0.20/0.45 ( ( isPath_capacity @ C2 @ U3 @ P2 @ V3 )
% 0.20/0.45 & ! [P5: list_P559422087at_nat] :
% 0.20/0.45 ( ( isPath_capacity @ C2 @ U3 @ P5 @ V3 )
% 0.20/0.45 => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P2 ) @ ( size_s1990949619at_nat @ P5 ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isShortestPath_def
% 0.20/0.45 thf(fact_81_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.20/0.45 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.20/0.45 ( ( ( ord_less_eq_nat @ I @ J )
% 0.20/0.45 & ( K = L ) )
% 0.20/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_mono_thms_linordered_semiring(3)
% 0.20/0.45 thf(fact_82_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.20/0.45 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.20/0.45 ( ( ( I = J )
% 0.20/0.45 & ( ord_less_eq_nat @ K @ L ) )
% 0.20/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_mono_thms_linordered_semiring(2)
% 0.20/0.45 thf(fact_83_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.20/0.45 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.20/0.45 ( ( ( ord_less_eq_nat @ I @ J )
% 0.20/0.45 & ( ord_less_eq_nat @ K @ L ) )
% 0.20/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_mono_thms_linordered_semiring(1)
% 0.20/0.45 thf(fact_84_add__mono,axiom,
% 0.20/0.45 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.45 => ( ( ord_less_eq_nat @ C @ D )
% 0.20/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_mono
% 0.20/0.45 thf(fact_85_add__left__mono,axiom,
% 0.20/0.45 ! [A: nat,B: nat,C: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_left_mono
% 0.20/0.45 thf(fact_86_less__eqE,axiom,
% 0.20/0.45 ! [A: nat,B: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.45 => ~ ! [C3: nat] :
% 0.20/0.45 ( B
% 0.20/0.45 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % less_eqE
% 0.20/0.45 thf(fact_87_add__right__mono,axiom,
% 0.20/0.45 ! [A: nat,B: nat,C: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_right_mono
% 0.20/0.45 thf(fact_88_le__iff__add,axiom,
% 0.20/0.45 ( ord_less_eq_nat
% 0.20/0.45 = ( ^ [A2: nat,B2: nat] :
% 0.20/0.45 ? [C2: nat] :
% 0.20/0.45 ( B2
% 0.20/0.45 = ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % le_iff_add
% 0.20/0.45 thf(fact_89_add__le__imp__le__left,axiom,
% 0.20/0.45 ! [C: nat,A: nat,B: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.20/0.45 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_le_imp_le_left
% 0.20/0.45 thf(fact_90_add__le__imp__le__right,axiom,
% 0.20/0.45 ! [A: nat,C: nat,B: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.20/0.45 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_le_imp_le_right
% 0.20/0.45 thf(fact_91_Graph_Omin__dist__minD,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,V: nat,D: nat,V2: nat] :
% 0.20/0.45 ( ( dist_capacity @ C @ V @ D @ V2 )
% 0.20/0.45 => ( ord_less_eq_nat @ ( min_dist_capacity @ C @ V @ V2 ) @ D ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.min_dist_minD
% 0.20/0.45 thf(fact_92_Graph_Omin__distI__eq,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,V: nat,D: nat,V2: nat] :
% 0.20/0.45 ( ( dist_capacity @ C @ V @ D @ V2 )
% 0.20/0.45 => ( ! [D4: nat] :
% 0.20/0.45 ( ( dist_capacity @ C @ V @ D4 @ V2 )
% 0.20/0.45 => ( ord_less_eq_nat @ D @ D4 ) )
% 0.20/0.45 => ( ( min_dist_capacity @ C @ V @ V2 )
% 0.20/0.45 = D ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.min_distI_eq
% 0.20/0.45 thf(fact_93_Graph_Omin__dist__is__dist,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,V: nat,V2: nat] :
% 0.20/0.45 ( ( connected_capacity @ C @ V @ V2 )
% 0.20/0.45 => ( dist_capacity @ C @ V @ ( min_dist_capacity @ C @ V @ V2 ) @ V2 ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.min_dist_is_dist
% 0.20/0.45 thf(fact_94_Graph_OisShortestPath__min__dist__def,axiom,
% 0.20/0.45 ( isShor1936442771pacity
% 0.20/0.45 = ( ^ [C2: product_prod_nat_nat > capacity,U3: nat,P2: list_P559422087at_nat,V3: nat] :
% 0.20/0.45 ( ( isPath_capacity @ C2 @ U3 @ P2 @ V3 )
% 0.20/0.45 & ( ( size_s1990949619at_nat @ P2 )
% 0.20/0.45 = ( min_dist_capacity @ C2 @ U3 @ V3 ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isShortestPath_min_dist_def
% 0.20/0.45 thf(fact_95_Graph_OisPath_Ocong,axiom,
% 0.20/0.45 isPath_capacity = isPath_capacity ).
% 0.20/0.45
% 0.20/0.45 % Graph.isPath.cong
% 0.20/0.45 thf(fact_96_Graph_Omin__dist_Ocong,axiom,
% 0.20/0.45 min_dist_capacity = min_dist_capacity ).
% 0.20/0.45
% 0.20/0.45 % Graph.min_dist.cong
% 0.20/0.45 thf(fact_97_Graph_Odist_Ocong,axiom,
% 0.20/0.45 dist_capacity = dist_capacity ).
% 0.20/0.45
% 0.20/0.45 % Graph.dist.cong
% 0.20/0.45 thf(fact_98_Graph_Odist__trans,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 0.20/0.45 ( ( dist_capacity @ C @ U @ D1 @ W )
% 0.20/0.45 => ( ( dist_capacity @ C @ W @ D2 @ V )
% 0.20/0.45 => ( dist_capacity @ C @ U @ ( plus_plus_nat @ D1 @ D2 ) @ V ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.dist_trans
% 0.20/0.45 thf(fact_99_nat__add__left__cancel__le,axiom,
% 0.20/0.45 ! [K: nat,M: nat,N: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.20/0.45 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.20/0.45
% 0.20/0.45 % nat_add_left_cancel_le
% 0.20/0.45 thf(fact_100_isShortestPath__alt,axiom,
% 0.20/0.45 ! [U: nat,P: list_P559422087at_nat,V: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ c @ U @ P @ V )
% 0.20/0.45 = ( ( isSimp1359852763pacity @ c @ U @ P @ V )
% 0.20/0.45 & ( ( size_s1990949619at_nat @ P )
% 0.20/0.45 = ( min_dist_capacity @ c @ U @ V ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % isShortestPath_alt
% 0.20/0.45 thf(fact_101_min__dist__less,axiom,
% 0.20/0.45 ! [Src: nat,V: nat,D: nat,D5: nat] :
% 0.20/0.45 ( ( connected_capacity @ c @ Src @ V )
% 0.20/0.45 => ( ( ( min_dist_capacity @ c @ Src @ V )
% 0.20/0.45 = D )
% 0.20/0.45 => ( ( ord_less_nat @ D5 @ D )
% 0.20/0.45 => ? [V5: nat] :
% 0.20/0.45 ( ( connected_capacity @ c @ Src @ V5 )
% 0.20/0.45 & ( ( min_dist_capacity @ c @ Src @ V5 )
% 0.20/0.45 = D5 ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % min_dist_less
% 0.20/0.45 thf(fact_102_isSPath__no__selfloop,axiom,
% 0.20/0.45 ! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat] :
% 0.20/0.45 ( ( isSimp1359852763pacity @ c @ U @ P @ V )
% 0.20/0.45 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % isSPath_no_selfloop
% 0.20/0.45 thf(fact_103_isSPath__sg__incoming,axiom,
% 0.20/0.45 ! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,U22: nat] :
% 0.20/0.45 ( ( isSimp1359852763pacity @ c @ U @ P @ V )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( ( U1 != U22 )
% 0.20/0.45 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U22 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % isSPath_sg_incoming
% 0.20/0.45 thf(fact_104_isSPath__sg__outgoing,axiom,
% 0.20/0.45 ! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,V23: nat] :
% 0.20/0.45 ( ( isSimp1359852763pacity @ c @ U @ P @ V )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( ( V1 != V23 )
% 0.20/0.45 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V23 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % isSPath_sg_outgoing
% 0.20/0.45 thf(fact_105_old_Oprod_Oinject,axiom,
% 0.20/0.45 ! [A: nat > nat > nat,B: produc1695820582at_nat,A4: nat > nat > nat,B4: produc1695820582at_nat] :
% 0.20/0.45 ( ( ( produc1682925677at_nat @ A @ B )
% 0.20/0.45 = ( produc1682925677at_nat @ A4 @ B4 ) )
% 0.20/0.45 = ( ( A = A4 )
% 0.20/0.45 & ( B = B4 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % old.prod.inject
% 0.20/0.45 thf(fact_106_old_Oprod_Oinject,axiom,
% 0.20/0.45 ! [A: nat,B: product_prod_nat_nat,A4: nat,B4: product_prod_nat_nat] :
% 0.20/0.45 ( ( ( produc1933845336at_nat @ A @ B )
% 0.20/0.45 = ( produc1933845336at_nat @ A4 @ B4 ) )
% 0.20/0.45 = ( ( A = A4 )
% 0.20/0.45 & ( B = B4 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % old.prod.inject
% 0.20/0.45 thf(fact_107_old_Oprod_Oinject,axiom,
% 0.20/0.45 ! [A: nat,B: nat,A4: nat,B4: nat] :
% 0.20/0.45 ( ( ( product_Pair_nat_nat @ A @ B )
% 0.20/0.45 = ( product_Pair_nat_nat @ A4 @ B4 ) )
% 0.20/0.45 = ( ( A = A4 )
% 0.20/0.45 & ( B = B4 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % old.prod.inject
% 0.20/0.45 thf(fact_108_prod_Oinject,axiom,
% 0.20/0.45 ! [X1: nat > nat > nat,X22: produc1695820582at_nat,Y1: nat > nat > nat,Y2: produc1695820582at_nat] :
% 0.20/0.45 ( ( ( produc1682925677at_nat @ X1 @ X22 )
% 0.20/0.45 = ( produc1682925677at_nat @ Y1 @ Y2 ) )
% 0.20/0.45 = ( ( X1 = Y1 )
% 0.20/0.45 & ( X22 = Y2 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % prod.inject
% 0.20/0.45 thf(fact_109_prod_Oinject,axiom,
% 0.20/0.45 ! [X1: nat,X22: product_prod_nat_nat,Y1: nat,Y2: product_prod_nat_nat] :
% 0.20/0.45 ( ( ( produc1933845336at_nat @ X1 @ X22 )
% 0.20/0.45 = ( produc1933845336at_nat @ Y1 @ Y2 ) )
% 0.20/0.45 = ( ( X1 = Y1 )
% 0.20/0.45 & ( X22 = Y2 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % prod.inject
% 0.20/0.45 thf(fact_110_prod_Oinject,axiom,
% 0.20/0.45 ! [X1: nat,X22: nat,Y1: nat,Y2: nat] :
% 0.20/0.45 ( ( ( product_Pair_nat_nat @ X1 @ X22 )
% 0.20/0.45 = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
% 0.20/0.45 = ( ( X1 = Y1 )
% 0.20/0.45 & ( X22 = Y2 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % prod.inject
% 0.20/0.45 thf(fact_111_isSPath__pathLE,axiom,
% 0.20/0.45 ! [S: nat,P: list_P559422087at_nat,T: nat] :
% 0.20/0.45 ( ( isPath_capacity @ c @ S @ P @ T )
% 0.20/0.45 => ? [P6: list_P559422087at_nat] : ( isSimp1359852763pacity @ c @ S @ P6 @ T ) ) ).
% 0.20/0.45
% 0.20/0.45 % isSPath_pathLE
% 0.20/0.45 thf(fact_112_shortestPath__is__simple,axiom,
% 0.20/0.45 ! [S: nat,P: list_P559422087at_nat,T: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ c @ S @ P @ T )
% 0.20/0.45 => ( isSimp1359852763pacity @ c @ S @ P @ T ) ) ).
% 0.20/0.45
% 0.20/0.45 % shortestPath_is_simple
% 0.20/0.45 thf(fact_113_add__less__cancel__right,axiom,
% 0.20/0.45 ! [A: nat,C: nat,B: nat] :
% 0.20/0.45 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.20/0.45 = ( ord_less_nat @ A @ B ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_less_cancel_right
% 0.20/0.45 thf(fact_114_add__less__cancel__left,axiom,
% 0.20/0.45 ! [C: nat,A: nat,B: nat] :
% 0.20/0.45 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.20/0.45 = ( ord_less_nat @ A @ B ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_less_cancel_left
% 0.20/0.45 thf(fact_115_nat__add__left__cancel__less,axiom,
% 0.20/0.45 ! [K: nat,M: nat,N: nat] :
% 0.20/0.45 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.20/0.45 = ( ord_less_nat @ M @ N ) ) ).
% 0.20/0.45
% 0.20/0.45 % nat_add_left_cancel_less
% 0.20/0.45 thf(fact_116_Graph_OisSimplePath_Ocong,axiom,
% 0.20/0.45 isSimp1359852763pacity = isSimp1359852763pacity ).
% 0.20/0.45
% 0.20/0.45 % Graph.isSimplePath.cong
% 0.20/0.45 thf(fact_117_less__mono__imp__le__mono,axiom,
% 0.20/0.45 ! [F: nat > nat,I: nat,J: nat] :
% 0.20/0.45 ( ! [I2: nat,J2: nat] :
% 0.20/0.45 ( ( ord_less_nat @ I2 @ J2 )
% 0.20/0.45 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.20/0.45 => ( ( ord_less_eq_nat @ I @ J )
% 0.20/0.45 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % less_mono_imp_le_mono
% 0.20/0.45 thf(fact_118_le__neq__implies__less,axiom,
% 0.20/0.45 ! [M: nat,N: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.20/0.45 => ( ( M != N )
% 0.20/0.45 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % le_neq_implies_less
% 0.20/0.45 thf(fact_119_less__or__eq__imp__le,axiom,
% 0.20/0.45 ! [M: nat,N: nat] :
% 0.20/0.45 ( ( ( ord_less_nat @ M @ N )
% 0.20/0.45 | ( M = N ) )
% 0.20/0.45 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.20/0.45
% 0.20/0.45 % less_or_eq_imp_le
% 0.20/0.45 thf(fact_120_le__eq__less__or__eq,axiom,
% 0.20/0.45 ( ord_less_eq_nat
% 0.20/0.45 = ( ^ [M2: nat,N2: nat] :
% 0.20/0.45 ( ( ord_less_nat @ M2 @ N2 )
% 0.20/0.45 | ( M2 = N2 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % le_eq_less_or_eq
% 0.20/0.45 thf(fact_121_less__imp__le__nat,axiom,
% 0.20/0.45 ! [M: nat,N: nat] :
% 0.20/0.45 ( ( ord_less_nat @ M @ N )
% 0.20/0.45 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.20/0.45
% 0.20/0.45 % less_imp_le_nat
% 0.20/0.45 thf(fact_122_nat__less__le,axiom,
% 0.20/0.45 ( ord_less_nat
% 0.20/0.45 = ( ^ [M2: nat,N2: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.20/0.45 & ( M2 != N2 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % nat_less_le
% 0.20/0.45 thf(fact_123_add__lessD1,axiom,
% 0.20/0.45 ! [I: nat,J: nat,K: nat] :
% 0.20/0.45 ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.20/0.45 => ( ord_less_nat @ I @ K ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_lessD1
% 0.20/0.45 thf(fact_124_add__less__mono,axiom,
% 0.20/0.45 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.20/0.45 ( ( ord_less_nat @ I @ J )
% 0.20/0.45 => ( ( ord_less_nat @ K @ L )
% 0.20/0.45 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_less_mono
% 0.20/0.45 thf(fact_125_not__add__less1,axiom,
% 0.20/0.45 ! [I: nat,J: nat] :
% 0.20/0.45 ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 0.20/0.45
% 0.20/0.45 % not_add_less1
% 0.20/0.45 thf(fact_126_not__add__less2,axiom,
% 0.20/0.45 ! [J: nat,I: nat] :
% 0.20/0.45 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 0.20/0.45
% 0.20/0.45 % not_add_less2
% 0.20/0.45 thf(fact_127_add__less__mono1,axiom,
% 0.20/0.45 ! [I: nat,J: nat,K: nat] :
% 0.20/0.45 ( ( ord_less_nat @ I @ J )
% 0.20/0.45 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_less_mono1
% 0.20/0.45 thf(fact_128_trans__less__add1,axiom,
% 0.20/0.45 ! [I: nat,J: nat,M: nat] :
% 0.20/0.45 ( ( ord_less_nat @ I @ J )
% 0.20/0.45 => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % trans_less_add1
% 0.20/0.45 thf(fact_129_trans__less__add2,axiom,
% 0.20/0.45 ! [I: nat,J: nat,M: nat] :
% 0.20/0.45 ( ( ord_less_nat @ I @ J )
% 0.20/0.45 => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % trans_less_add2
% 0.20/0.45 thf(fact_130_less__add__eq__less,axiom,
% 0.20/0.45 ! [K: nat,L: nat,M: nat,N: nat] :
% 0.20/0.45 ( ( ord_less_nat @ K @ L )
% 0.20/0.45 => ( ( ( plus_plus_nat @ M @ L )
% 0.20/0.45 = ( plus_plus_nat @ K @ N ) )
% 0.20/0.45 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % less_add_eq_less
% 0.20/0.45 thf(fact_131_nat__neq__iff,axiom,
% 0.20/0.45 ! [M: nat,N: nat] :
% 0.20/0.45 ( ( M != N )
% 0.20/0.45 = ( ( ord_less_nat @ M @ N )
% 0.20/0.45 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % nat_neq_iff
% 0.20/0.45 thf(fact_132_less__not__refl,axiom,
% 0.20/0.45 ! [N: nat] :
% 0.20/0.45 ~ ( ord_less_nat @ N @ N ) ).
% 0.20/0.45
% 0.20/0.45 % less_not_refl
% 0.20/0.45 thf(fact_133_less__not__refl2,axiom,
% 0.20/0.45 ! [N: nat,M: nat] :
% 0.20/0.45 ( ( ord_less_nat @ N @ M )
% 0.20/0.45 => ( M != N ) ) ).
% 0.20/0.45
% 0.20/0.45 % less_not_refl2
% 0.20/0.45 thf(fact_134_less__not__refl3,axiom,
% 0.20/0.45 ! [S: nat,T: nat] :
% 0.20/0.45 ( ( ord_less_nat @ S @ T )
% 0.20/0.45 => ( S != T ) ) ).
% 0.20/0.45
% 0.20/0.45 % less_not_refl3
% 0.20/0.45 thf(fact_135_less__irrefl__nat,axiom,
% 0.20/0.45 ! [N: nat] :
% 0.20/0.45 ~ ( ord_less_nat @ N @ N ) ).
% 0.20/0.45
% 0.20/0.45 % less_irrefl_nat
% 0.20/0.45 thf(fact_136_nat__less__induct,axiom,
% 0.20/0.45 ! [P3: nat > $o,N: nat] :
% 0.20/0.45 ( ! [N3: nat] :
% 0.20/0.45 ( ! [M3: nat] :
% 0.20/0.45 ( ( ord_less_nat @ M3 @ N3 )
% 0.20/0.45 => ( P3 @ M3 ) )
% 0.20/0.45 => ( P3 @ N3 ) )
% 0.20/0.45 => ( P3 @ N ) ) ).
% 0.20/0.45
% 0.20/0.45 % nat_less_induct
% 0.20/0.45 thf(fact_137_infinite__descent,axiom,
% 0.20/0.45 ! [P3: nat > $o,N: nat] :
% 0.20/0.45 ( ! [N3: nat] :
% 0.20/0.45 ( ~ ( P3 @ N3 )
% 0.20/0.45 => ? [M3: nat] :
% 0.20/0.45 ( ( ord_less_nat @ M3 @ N3 )
% 0.20/0.45 & ~ ( P3 @ M3 ) ) )
% 0.20/0.45 => ( P3 @ N ) ) ).
% 0.20/0.45
% 0.20/0.45 % infinite_descent
% 0.20/0.45 thf(fact_138_linorder__neqE__nat,axiom,
% 0.20/0.45 ! [X: nat,Y: nat] :
% 0.20/0.45 ( ( X != Y )
% 0.20/0.45 => ( ~ ( ord_less_nat @ X @ Y )
% 0.20/0.45 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % linorder_neqE_nat
% 0.20/0.45 thf(fact_139_add__less__imp__less__right,axiom,
% 0.20/0.45 ! [A: nat,C: nat,B: nat] :
% 0.20/0.45 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.20/0.45 => ( ord_less_nat @ A @ B ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_less_imp_less_right
% 0.20/0.45 thf(fact_140_add__less__imp__less__left,axiom,
% 0.20/0.45 ! [C: nat,A: nat,B: nat] :
% 0.20/0.45 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.20/0.45 => ( ord_less_nat @ A @ B ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_less_imp_less_left
% 0.20/0.45 thf(fact_141_add__strict__right__mono,axiom,
% 0.20/0.45 ! [A: nat,B: nat,C: nat] :
% 0.20/0.45 ( ( ord_less_nat @ A @ B )
% 0.20/0.45 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_strict_right_mono
% 0.20/0.45 thf(fact_142_add__strict__left__mono,axiom,
% 0.20/0.45 ! [A: nat,B: nat,C: nat] :
% 0.20/0.45 ( ( ord_less_nat @ A @ B )
% 0.20/0.45 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_strict_left_mono
% 0.20/0.45 thf(fact_143_add__strict__mono,axiom,
% 0.20/0.45 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.20/0.45 ( ( ord_less_nat @ A @ B )
% 0.20/0.45 => ( ( ord_less_nat @ C @ D )
% 0.20/0.45 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_strict_mono
% 0.20/0.45 thf(fact_144_add__mono__thms__linordered__field_I1_J,axiom,
% 0.20/0.45 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.20/0.45 ( ( ( ord_less_nat @ I @ J )
% 0.20/0.45 & ( K = L ) )
% 0.20/0.45 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_mono_thms_linordered_field(1)
% 0.20/0.45 thf(fact_145_add__mono__thms__linordered__field_I2_J,axiom,
% 0.20/0.45 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.20/0.45 ( ( ( I = J )
% 0.20/0.45 & ( ord_less_nat @ K @ L ) )
% 0.20/0.45 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_mono_thms_linordered_field(2)
% 0.20/0.45 thf(fact_146_add__mono__thms__linordered__field_I5_J,axiom,
% 0.20/0.45 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.20/0.45 ( ( ( ord_less_nat @ I @ J )
% 0.20/0.45 & ( ord_less_nat @ K @ L ) )
% 0.20/0.45 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_mono_thms_linordered_field(5)
% 0.20/0.45 thf(fact_147_mono__nat__linear__lb,axiom,
% 0.20/0.45 ! [F: nat > nat,M: nat,K: nat] :
% 0.20/0.45 ( ! [M4: nat,N3: nat] :
% 0.20/0.45 ( ( ord_less_nat @ M4 @ N3 )
% 0.20/0.45 => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
% 0.20/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % mono_nat_linear_lb
% 0.20/0.45 thf(fact_148_Graph_OisSPath__pathLE,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat] :
% 0.20/0.45 ( ( isPath_capacity @ C @ S @ P @ T )
% 0.20/0.45 => ? [P6: list_P559422087at_nat] : ( isSimp1359852763pacity @ C @ S @ P6 @ T ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isSPath_pathLE
% 0.20/0.45 thf(fact_149_Graph_OshortestPath__is__simple,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat] :
% 0.20/0.45 ( ( isShor1936442771pacity @ C @ S @ P @ T )
% 0.20/0.45 => ( isSimp1359852763pacity @ C @ S @ P @ T ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.shortestPath_is_simple
% 0.20/0.45 thf(fact_150_add__mono__thms__linordered__field_I4_J,axiom,
% 0.20/0.45 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.20/0.45 ( ( ( ord_less_eq_nat @ I @ J )
% 0.20/0.45 & ( ord_less_nat @ K @ L ) )
% 0.20/0.45 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_mono_thms_linordered_field(4)
% 0.20/0.45 thf(fact_151_add__mono__thms__linordered__field_I3_J,axiom,
% 0.20/0.45 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.20/0.45 ( ( ( ord_less_nat @ I @ J )
% 0.20/0.45 & ( ord_less_eq_nat @ K @ L ) )
% 0.20/0.45 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_mono_thms_linordered_field(3)
% 0.20/0.45 thf(fact_152_add__le__less__mono,axiom,
% 0.20/0.45 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.45 => ( ( ord_less_nat @ C @ D )
% 0.20/0.45 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_le_less_mono
% 0.20/0.45 thf(fact_153_add__less__le__mono,axiom,
% 0.20/0.45 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.20/0.45 ( ( ord_less_nat @ A @ B )
% 0.20/0.45 => ( ( ord_less_eq_nat @ C @ D )
% 0.20/0.45 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_less_le_mono
% 0.20/0.45 thf(fact_154_Graph_Omin__dist__less,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,Src: nat,V: nat,D: nat,D5: nat] :
% 0.20/0.45 ( ( connected_capacity @ C @ Src @ V )
% 0.20/0.45 => ( ( ( min_dist_capacity @ C @ Src @ V )
% 0.20/0.45 = D )
% 0.20/0.45 => ( ( ord_less_nat @ D5 @ D )
% 0.20/0.45 => ? [V5: nat] :
% 0.20/0.45 ( ( connected_capacity @ C @ Src @ V5 )
% 0.20/0.45 & ( ( min_dist_capacity @ C @ Src @ V5 )
% 0.20/0.45 = D5 ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.min_dist_less
% 0.20/0.45 thf(fact_155_Graph_OisSPath__no__selfloop,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat] :
% 0.20/0.45 ( ( isSimp1359852763pacity @ C @ U @ P @ V )
% 0.20/0.45 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isSPath_no_selfloop
% 0.20/0.45 thf(fact_156_Graph_OisSPath__sg__incoming,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,U22: nat] :
% 0.20/0.45 ( ( isSimp1359852763pacity @ C @ U @ P @ V )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( ( U1 != U22 )
% 0.20/0.45 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U22 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isSPath_sg_incoming
% 0.20/0.45 thf(fact_157_Graph_OisSPath__sg__outgoing,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,V23: nat] :
% 0.20/0.45 ( ( isSimp1359852763pacity @ C @ U @ P @ V )
% 0.20/0.45 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ( ( V1 != V23 )
% 0.20/0.45 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V23 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isSPath_sg_outgoing
% 0.20/0.45 thf(fact_158_surj__pair,axiom,
% 0.20/0.45 ! [P: produc2082277813at_nat] :
% 0.20/0.45 ? [X3: nat > nat > nat,Y3: produc1695820582at_nat] :
% 0.20/0.45 ( P
% 0.20/0.45 = ( produc1682925677at_nat @ X3 @ Y3 ) ) ).
% 0.20/0.45
% 0.20/0.45 % surj_pair
% 0.20/0.45 thf(fact_159_surj__pair,axiom,
% 0.20/0.45 ! [P: produc1695820582at_nat] :
% 0.20/0.45 ? [X3: nat,Y3: product_prod_nat_nat] :
% 0.20/0.45 ( P
% 0.20/0.45 = ( produc1933845336at_nat @ X3 @ Y3 ) ) ).
% 0.20/0.45
% 0.20/0.45 % surj_pair
% 0.20/0.45 thf(fact_160_surj__pair,axiom,
% 0.20/0.45 ! [P: product_prod_nat_nat] :
% 0.20/0.45 ? [X3: nat,Y3: nat] :
% 0.20/0.45 ( P
% 0.20/0.45 = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).
% 0.20/0.45
% 0.20/0.45 % surj_pair
% 0.20/0.45 thf(fact_161_prod__cases,axiom,
% 0.20/0.45 ! [P3: produc2082277813at_nat > $o,P: produc2082277813at_nat] :
% 0.20/0.45 ( ! [A5: nat > nat > nat,B5: produc1695820582at_nat] : ( P3 @ ( produc1682925677at_nat @ A5 @ B5 ) )
% 0.20/0.45 => ( P3 @ P ) ) ).
% 0.20/0.45
% 0.20/0.45 % prod_cases
% 0.20/0.45 thf(fact_162_prod__cases,axiom,
% 0.20/0.45 ! [P3: produc1695820582at_nat > $o,P: produc1695820582at_nat] :
% 0.20/0.45 ( ! [A5: nat,B5: product_prod_nat_nat] : ( P3 @ ( produc1933845336at_nat @ A5 @ B5 ) )
% 0.20/0.45 => ( P3 @ P ) ) ).
% 0.20/0.45
% 0.20/0.45 % prod_cases
% 0.20/0.45 thf(fact_163_prod__cases,axiom,
% 0.20/0.45 ! [P3: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
% 0.20/0.45 ( ! [A5: nat,B5: nat] : ( P3 @ ( product_Pair_nat_nat @ A5 @ B5 ) )
% 0.20/0.45 => ( P3 @ P ) ) ).
% 0.20/0.45
% 0.20/0.45 % prod_cases
% 0.20/0.45 thf(fact_164_Pair__inject,axiom,
% 0.20/0.45 ! [A: nat > nat > nat,B: produc1695820582at_nat,A4: nat > nat > nat,B4: produc1695820582at_nat] :
% 0.20/0.45 ( ( ( produc1682925677at_nat @ A @ B )
% 0.20/0.45 = ( produc1682925677at_nat @ A4 @ B4 ) )
% 0.20/0.45 => ~ ( ( A = A4 )
% 0.20/0.45 => ( B != B4 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Pair_inject
% 0.20/0.45 thf(fact_165_Pair__inject,axiom,
% 0.20/0.45 ! [A: nat,B: product_prod_nat_nat,A4: nat,B4: product_prod_nat_nat] :
% 0.20/0.45 ( ( ( produc1933845336at_nat @ A @ B )
% 0.20/0.45 = ( produc1933845336at_nat @ A4 @ B4 ) )
% 0.20/0.45 => ~ ( ( A = A4 )
% 0.20/0.45 => ( B != B4 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Pair_inject
% 0.20/0.45 thf(fact_166_Pair__inject,axiom,
% 0.20/0.45 ! [A: nat,B: nat,A4: nat,B4: nat] :
% 0.20/0.45 ( ( ( product_Pair_nat_nat @ A @ B )
% 0.20/0.45 = ( product_Pair_nat_nat @ A4 @ B4 ) )
% 0.20/0.45 => ~ ( ( A = A4 )
% 0.20/0.45 => ( B != B4 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Pair_inject
% 0.20/0.45 thf(fact_167_prod__cases3,axiom,
% 0.20/0.45 ! [Y: produc2082277813at_nat] :
% 0.20/0.45 ~ ! [A5: nat > nat > nat,B5: nat,C3: product_prod_nat_nat] :
% 0.20/0.45 ( Y
% 0.20/0.45 != ( produc1682925677at_nat @ A5 @ ( produc1933845336at_nat @ B5 @ C3 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % prod_cases3
% 0.20/0.45 thf(fact_168_prod__cases3,axiom,
% 0.20/0.45 ! [Y: produc1695820582at_nat] :
% 0.20/0.45 ~ ! [A5: nat,B5: nat,C3: nat] :
% 0.20/0.45 ( Y
% 0.20/0.45 != ( produc1933845336at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ C3 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % prod_cases3
% 0.20/0.45 thf(fact_169_prod__cases4,axiom,
% 0.20/0.45 ! [Y: produc2082277813at_nat] :
% 0.20/0.45 ~ ! [A5: nat > nat > nat,B5: nat,C3: nat,D6: nat] :
% 0.20/0.45 ( Y
% 0.20/0.45 != ( produc1682925677at_nat @ A5 @ ( produc1933845336at_nat @ B5 @ ( product_Pair_nat_nat @ C3 @ D6 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % prod_cases4
% 0.20/0.45 thf(fact_170_prod__induct3,axiom,
% 0.20/0.45 ! [P3: produc2082277813at_nat > $o,X: produc2082277813at_nat] :
% 0.20/0.45 ( ! [A5: nat > nat > nat,B5: nat,C3: product_prod_nat_nat] : ( P3 @ ( produc1682925677at_nat @ A5 @ ( produc1933845336at_nat @ B5 @ C3 ) ) )
% 0.20/0.45 => ( P3 @ X ) ) ).
% 0.20/0.45
% 0.20/0.45 % prod_induct3
% 0.20/0.45 thf(fact_171_prod__induct3,axiom,
% 0.20/0.45 ! [P3: produc1695820582at_nat > $o,X: produc1695820582at_nat] :
% 0.20/0.45 ( ! [A5: nat,B5: nat,C3: nat] : ( P3 @ ( produc1933845336at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ C3 ) ) )
% 0.20/0.45 => ( P3 @ X ) ) ).
% 0.20/0.45
% 0.20/0.45 % prod_induct3
% 0.20/0.45 thf(fact_172_prod__induct4,axiom,
% 0.20/0.45 ! [P3: produc2082277813at_nat > $o,X: produc2082277813at_nat] :
% 0.20/0.45 ( ! [A5: nat > nat > nat,B5: nat,C3: nat,D6: nat] : ( P3 @ ( produc1682925677at_nat @ A5 @ ( produc1933845336at_nat @ B5 @ ( product_Pair_nat_nat @ C3 @ D6 ) ) ) )
% 0.20/0.45 => ( P3 @ X ) ) ).
% 0.20/0.45
% 0.20/0.45 % prod_induct4
% 0.20/0.45 thf(fact_173_old_Oprod_Oexhaust,axiom,
% 0.20/0.45 ! [Y: produc2082277813at_nat] :
% 0.20/0.45 ~ ! [A5: nat > nat > nat,B5: produc1695820582at_nat] :
% 0.20/0.45 ( Y
% 0.20/0.45 != ( produc1682925677at_nat @ A5 @ B5 ) ) ).
% 0.20/0.45
% 0.20/0.45 % old.prod.exhaust
% 0.20/0.45 thf(fact_174_old_Oprod_Oexhaust,axiom,
% 0.20/0.45 ! [Y: produc1695820582at_nat] :
% 0.20/0.45 ~ ! [A5: nat,B5: product_prod_nat_nat] :
% 0.20/0.45 ( Y
% 0.20/0.45 != ( produc1933845336at_nat @ A5 @ B5 ) ) ).
% 0.20/0.45
% 0.20/0.45 % old.prod.exhaust
% 0.20/0.45 thf(fact_175_old_Oprod_Oexhaust,axiom,
% 0.20/0.45 ! [Y: product_prod_nat_nat] :
% 0.20/0.45 ~ ! [A5: nat,B5: nat] :
% 0.20/0.45 ( Y
% 0.20/0.45 != ( product_Pair_nat_nat @ A5 @ B5 ) ) ).
% 0.20/0.45
% 0.20/0.45 % old.prod.exhaust
% 0.20/0.45 thf(fact_176_old_Oprod_Oinducts,axiom,
% 0.20/0.45 ! [P3: produc2082277813at_nat > $o,Prod: produc2082277813at_nat] :
% 0.20/0.45 ( ! [A5: nat > nat > nat,B5: produc1695820582at_nat] : ( P3 @ ( produc1682925677at_nat @ A5 @ B5 ) )
% 0.20/0.45 => ( P3 @ Prod ) ) ).
% 0.20/0.45
% 0.20/0.45 % old.prod.inducts
% 0.20/0.45 thf(fact_177_old_Oprod_Oinducts,axiom,
% 0.20/0.45 ! [P3: produc1695820582at_nat > $o,Prod: produc1695820582at_nat] :
% 0.20/0.45 ( ! [A5: nat,B5: product_prod_nat_nat] : ( P3 @ ( produc1933845336at_nat @ A5 @ B5 ) )
% 0.20/0.45 => ( P3 @ Prod ) ) ).
% 0.20/0.45
% 0.20/0.45 % old.prod.inducts
% 0.20/0.45 thf(fact_178_old_Oprod_Oinducts,axiom,
% 0.20/0.45 ! [P3: product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
% 0.20/0.45 ( ! [A5: nat,B5: nat] : ( P3 @ ( product_Pair_nat_nat @ A5 @ B5 ) )
% 0.20/0.45 => ( P3 @ Prod ) ) ).
% 0.20/0.45
% 0.20/0.45 % old.prod.inducts
% 0.20/0.45 thf(fact_179_size__neq__size__imp__neq,axiom,
% 0.20/0.45 ! [X: list_P559422087at_nat,Y: list_P559422087at_nat] :
% 0.20/0.45 ( ( ( size_s1990949619at_nat @ X )
% 0.20/0.45 != ( size_s1990949619at_nat @ Y ) )
% 0.20/0.45 => ( X != Y ) ) ).
% 0.20/0.45
% 0.20/0.45 % size_neq_size_imp_neq
% 0.20/0.45 thf(fact_180_le__refl,axiom,
% 0.20/0.45 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.20/0.45
% 0.20/0.45 % le_refl
% 0.20/0.45 thf(fact_181_le__trans,axiom,
% 0.20/0.45 ! [I: nat,J: nat,K: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ I @ J )
% 0.20/0.45 => ( ( ord_less_eq_nat @ J @ K )
% 0.20/0.45 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % le_trans
% 0.20/0.45 thf(fact_182_eq__imp__le,axiom,
% 0.20/0.45 ! [M: nat,N: nat] :
% 0.20/0.45 ( ( M = N )
% 0.20/0.45 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.20/0.45
% 0.20/0.45 % eq_imp_le
% 0.20/0.45 thf(fact_183_le__antisym,axiom,
% 0.20/0.45 ! [M: nat,N: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.20/0.45 => ( ( ord_less_eq_nat @ N @ M )
% 0.20/0.45 => ( M = N ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % le_antisym
% 0.20/0.45 thf(fact_184_nat__le__linear,axiom,
% 0.20/0.45 ! [M: nat,N: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ M @ N )
% 0.20/0.45 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.20/0.45
% 0.20/0.45 % nat_le_linear
% 0.20/0.45 thf(fact_185_Nat_Oex__has__greatest__nat,axiom,
% 0.20/0.45 ! [P3: nat > $o,K: nat,B: nat] :
% 0.20/0.45 ( ( P3 @ K )
% 0.20/0.45 => ( ! [Y3: nat] :
% 0.20/0.45 ( ( P3 @ Y3 )
% 0.20/0.45 => ( ord_less_eq_nat @ Y3 @ B ) )
% 0.20/0.45 => ? [X3: nat] :
% 0.20/0.45 ( ( P3 @ X3 )
% 0.20/0.45 & ! [Y4: nat] :
% 0.20/0.45 ( ( P3 @ Y4 )
% 0.20/0.45 => ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Nat.ex_has_greatest_nat
% 0.20/0.45 thf(fact_186_Graph_OisShortestPath__alt,axiom,
% 0.20/0.45 ( isShor1936442771pacity
% 0.20/0.45 = ( ^ [C2: product_prod_nat_nat > capacity,U3: nat,P2: list_P559422087at_nat,V3: nat] :
% 0.20/0.45 ( ( isSimp1359852763pacity @ C2 @ U3 @ P2 @ V3 )
% 0.20/0.45 & ( ( size_s1990949619at_nat @ P2 )
% 0.20/0.45 = ( min_dist_capacity @ C2 @ U3 @ V3 ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isShortestPath_alt
% 0.20/0.45 thf(fact_187_nat__le__iff__add,axiom,
% 0.20/0.45 ( ord_less_eq_nat
% 0.20/0.45 = ( ^ [M2: nat,N2: nat] :
% 0.20/0.45 ? [K2: nat] :
% 0.20/0.45 ( N2
% 0.20/0.45 = ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % nat_le_iff_add
% 0.20/0.45 thf(fact_188_trans__le__add2,axiom,
% 0.20/0.45 ! [I: nat,J: nat,M: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ I @ J )
% 0.20/0.45 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % trans_le_add2
% 0.20/0.45 thf(fact_189_trans__le__add1,axiom,
% 0.20/0.45 ! [I: nat,J: nat,M: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ I @ J )
% 0.20/0.45 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % trans_le_add1
% 0.20/0.45 thf(fact_190_add__le__mono1,axiom,
% 0.20/0.45 ! [I: nat,J: nat,K: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ I @ J )
% 0.20/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_le_mono1
% 0.20/0.45 thf(fact_191_add__le__mono,axiom,
% 0.20/0.45 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ I @ J )
% 0.20/0.45 => ( ( ord_less_eq_nat @ K @ L )
% 0.20/0.45 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_le_mono
% 0.20/0.45 thf(fact_192_le__Suc__ex,axiom,
% 0.20/0.45 ! [K: nat,L: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ K @ L )
% 0.20/0.45 => ? [N3: nat] :
% 0.20/0.45 ( L
% 0.20/0.45 = ( plus_plus_nat @ K @ N3 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % le_Suc_ex
% 0.20/0.45 thf(fact_193_add__leD2,axiom,
% 0.20/0.45 ! [M: nat,K: nat,N: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.20/0.45 => ( ord_less_eq_nat @ K @ N ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_leD2
% 0.20/0.45 thf(fact_194_add__leD1,axiom,
% 0.20/0.45 ! [M: nat,K: nat,N: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.20/0.45 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_leD1
% 0.20/0.45 thf(fact_195_le__add2,axiom,
% 0.20/0.45 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 0.20/0.45
% 0.20/0.45 % le_add2
% 0.20/0.45 thf(fact_196_le__add1,axiom,
% 0.20/0.45 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 0.20/0.45
% 0.20/0.45 % le_add1
% 0.20/0.45 thf(fact_197_add__leE,axiom,
% 0.20/0.45 ! [M: nat,K: nat,N: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.20/0.45 => ~ ( ( ord_less_eq_nat @ M @ N )
% 0.20/0.45 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_leE
% 0.20/0.45 thf(fact_198_isSPath__nt__parallel,axiom,
% 0.20/0.45 ! [S: nat,P: list_P559422087at_nat,T: nat,E: product_prod_nat_nat] :
% 0.20/0.45 ( ( isSimp1359852763pacity @ c @ S @ P @ T )
% 0.20/0.45 => ( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ~ ( member701585322at_nat @ ( product_swap_nat_nat @ E ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % isSPath_nt_parallel
% 0.20/0.45 thf(fact_199_discrete,axiom,
% 0.20/0.45 ( ord_less_nat
% 0.20/0.45 = ( ^ [A2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % discrete
% 0.20/0.45 thf(fact_200_swap__swap,axiom,
% 0.20/0.45 ! [P: product_prod_nat_nat] :
% 0.20/0.45 ( ( product_swap_nat_nat @ ( product_swap_nat_nat @ P ) )
% 0.20/0.45 = P ) ).
% 0.20/0.45
% 0.20/0.45 % swap_swap
% 0.20/0.45 thf(fact_201_swap__simp,axiom,
% 0.20/0.45 ! [X: produc1695820582at_nat,Y: nat > nat > nat] :
% 0.20/0.45 ( ( produc411855757at_nat @ ( produc365964781at_nat @ X @ Y ) )
% 0.20/0.45 = ( produc1682925677at_nat @ Y @ X ) ) ).
% 0.20/0.45
% 0.20/0.45 % swap_simp
% 0.20/0.45 thf(fact_202_swap__simp,axiom,
% 0.20/0.45 ! [X: product_prod_nat_nat,Y: nat] :
% 0.20/0.45 ( ( produc2019146714at_nat @ ( produc947540346at_nat @ X @ Y ) )
% 0.20/0.45 = ( produc1933845336at_nat @ Y @ X ) ) ).
% 0.20/0.45
% 0.20/0.45 % swap_simp
% 0.20/0.45 thf(fact_203_swap__simp,axiom,
% 0.20/0.45 ! [X: nat > nat > nat,Y: produc1695820582at_nat] :
% 0.20/0.45 ( ( produc1728816653at_nat @ ( produc1682925677at_nat @ X @ Y ) )
% 0.20/0.45 = ( produc365964781at_nat @ Y @ X ) ) ).
% 0.20/0.45
% 0.20/0.45 % swap_simp
% 0.20/0.45 thf(fact_204_swap__simp,axiom,
% 0.20/0.45 ! [X: nat,Y: product_prod_nat_nat] :
% 0.20/0.45 ( ( produc857968056at_nat @ ( produc1933845336at_nat @ X @ Y ) )
% 0.20/0.45 = ( produc947540346at_nat @ Y @ X ) ) ).
% 0.20/0.45
% 0.20/0.45 % swap_simp
% 0.20/0.45 thf(fact_205_swap__simp,axiom,
% 0.20/0.45 ! [X: nat,Y: nat] :
% 0.20/0.45 ( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
% 0.20/0.45 = ( product_Pair_nat_nat @ Y @ X ) ) ).
% 0.20/0.45
% 0.20/0.45 % swap_simp
% 0.20/0.45 thf(fact_206_bounded__Max__nat,axiom,
% 0.20/0.45 ! [P3: nat > $o,X: nat,M5: nat] :
% 0.20/0.45 ( ( P3 @ X )
% 0.20/0.45 => ( ! [X3: nat] :
% 0.20/0.45 ( ( P3 @ X3 )
% 0.20/0.45 => ( ord_less_eq_nat @ X3 @ M5 ) )
% 0.20/0.45 => ~ ! [M4: nat] :
% 0.20/0.45 ( ( P3 @ M4 )
% 0.20/0.45 => ~ ! [X4: nat] :
% 0.20/0.45 ( ( P3 @ X4 )
% 0.20/0.45 => ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % bounded_Max_nat
% 0.20/0.45 thf(fact_207_fold__atLeastAtMost__nat_Ocases,axiom,
% 0.20/0.45 ! [X: produc1864880952at_nat] :
% 0.20/0.45 ~ ! [F2: nat > product_prod_nat_nat > product_prod_nat_nat,A5: nat,B5: nat,Acc: product_prod_nat_nat] :
% 0.20/0.45 ( X
% 0.20/0.45 != ( produc696256106at_nat @ F2 @ ( produc869658639at_nat @ A5 @ ( produc1933845336at_nat @ B5 @ Acc ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % fold_atLeastAtMost_nat.cases
% 0.20/0.45 thf(fact_208_fold__atLeastAtMost__nat_Ocases,axiom,
% 0.20/0.45 ! [X: produc2082277813at_nat] :
% 0.20/0.45 ~ ! [F2: nat > nat > nat,A5: nat,B5: nat,Acc: nat] :
% 0.20/0.45 ( X
% 0.20/0.45 != ( produc1682925677at_nat @ F2 @ ( produc1933845336at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ Acc ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % fold_atLeastAtMost_nat.cases
% 0.20/0.45 thf(fact_209_Graph_OisSPath__nt__parallel,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,E: product_prod_nat_nat] :
% 0.20/0.45 ( ( isSimp1359852763pacity @ C @ S @ P @ T )
% 0.20/0.45 => ( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
% 0.20/0.45 => ~ ( member701585322at_nat @ ( product_swap_nat_nat @ E ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isSPath_nt_parallel
% 0.20/0.45 thf(fact_210_less__add__one,axiom,
% 0.20/0.45 ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 0.20/0.45
% 0.20/0.45 % less_add_one
% 0.20/0.45 thf(fact_211_add__mono1,axiom,
% 0.20/0.45 ! [A: nat,B: nat] :
% 0.20/0.45 ( ( ord_less_nat @ A @ B )
% 0.20/0.45 => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % add_mono1
% 0.20/0.45 thf(fact_212_isPath_Osimps_I1_J,axiom,
% 0.20/0.45 ! [U: nat,V: nat] :
% 0.20/0.45 ( ( isPath_capacity @ c @ U @ nil_Pr1308055047at_nat @ V )
% 0.20/0.45 = ( U = V ) ) ).
% 0.20/0.45
% 0.20/0.45 % isPath.simps(1)
% 0.20/0.45 thf(fact_213_order__refl,axiom,
% 0.20/0.45 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.20/0.45
% 0.20/0.45 % order_refl
% 0.20/0.45 thf(fact_214_simplePath__empty__conv,axiom,
% 0.20/0.45 ! [S: nat,T: nat] :
% 0.20/0.45 ( ( isSimp1359852763pacity @ c @ S @ nil_Pr1308055047at_nat @ T )
% 0.20/0.45 = ( S = T ) ) ).
% 0.20/0.45
% 0.20/0.45 % simplePath_empty_conv
% 0.20/0.45 thf(fact_215_simplePath__same__conv,axiom,
% 0.20/0.45 ! [S: nat,P: list_P559422087at_nat] :
% 0.20/0.45 ( ( isSimp1359852763pacity @ c @ S @ P @ S )
% 0.20/0.45 = ( P = nil_Pr1308055047at_nat ) ) ).
% 0.20/0.45
% 0.20/0.45 % simplePath_same_conv
% 0.20/0.45 thf(fact_216_Graph_OsimplePath__empty__conv,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,S: nat,T: nat] :
% 0.20/0.45 ( ( isSimp1359852763pacity @ C @ S @ nil_Pr1308055047at_nat @ T )
% 0.20/0.45 = ( S = T ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.simplePath_empty_conv
% 0.20/0.45 thf(fact_217_Graph_OsimplePath__same__conv,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat] :
% 0.20/0.45 ( ( isSimp1359852763pacity @ C @ S @ P @ S )
% 0.20/0.45 = ( P = nil_Pr1308055047at_nat ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.simplePath_same_conv
% 0.20/0.45 thf(fact_218_Graph_OisPath_Osimps_I1_J,axiom,
% 0.20/0.45 ! [C: product_prod_nat_nat > capacity,U: nat,V: nat] :
% 0.20/0.45 ( ( isPath_capacity @ C @ U @ nil_Pr1308055047at_nat @ V )
% 0.20/0.45 = ( U = V ) ) ).
% 0.20/0.45
% 0.20/0.45 % Graph.isPath.simps(1)
% 0.20/0.45 thf(fact_219_dual__order_Oantisym,axiom,
% 0.20/0.45 ! [B: nat,A: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ B @ A )
% 0.20/0.45 => ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.45 => ( A = B ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % dual_order.antisym
% 0.20/0.45 thf(fact_220_dual__order_Oeq__iff,axiom,
% 0.20/0.45 ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
% 0.20/0.45 = ( ^ [A2: nat,B2: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ B2 @ A2 )
% 0.20/0.45 & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % dual_order.eq_iff
% 0.20/0.45 thf(fact_221_dual__order_Otrans,axiom,
% 0.20/0.45 ! [B: nat,A: nat,C: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ B @ A )
% 0.20/0.45 => ( ( ord_less_eq_nat @ C @ B )
% 0.20/0.45 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % dual_order.trans
% 0.20/0.45 thf(fact_222_linorder__wlog,axiom,
% 0.20/0.45 ! [P3: nat > nat > $o,A: nat,B: nat] :
% 0.20/0.45 ( ! [A5: nat,B5: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ A5 @ B5 )
% 0.20/0.45 => ( P3 @ A5 @ B5 ) )
% 0.20/0.45 => ( ! [A5: nat,B5: nat] :
% 0.20/0.45 ( ( P3 @ B5 @ A5 )
% 0.20/0.45 => ( P3 @ A5 @ B5 ) )
% 0.20/0.45 => ( P3 @ A @ B ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % linorder_wlog
% 0.20/0.45 thf(fact_223_dual__order_Orefl,axiom,
% 0.20/0.45 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.20/0.45
% 0.20/0.45 % dual_order.refl
% 0.20/0.45 thf(fact_224_order__trans,axiom,
% 0.20/0.45 ! [X: nat,Y: nat,Z2: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ X @ Y )
% 0.20/0.45 => ( ( ord_less_eq_nat @ Y @ Z2 )
% 0.20/0.45 => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % order_trans
% 0.20/0.45 thf(fact_225_order__class_Oorder_Oantisym,axiom,
% 0.20/0.45 ! [A: nat,B: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.45 => ( ( ord_less_eq_nat @ B @ A )
% 0.20/0.45 => ( A = B ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % order_class.order.antisym
% 0.20/0.45 thf(fact_226_ord__le__eq__trans,axiom,
% 0.20/0.45 ! [A: nat,B: nat,C: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.45 => ( ( B = C )
% 0.20/0.45 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % ord_le_eq_trans
% 0.20/0.45 thf(fact_227_ord__eq__le__trans,axiom,
% 0.20/0.45 ! [A: nat,B: nat,C: nat] :
% 0.20/0.45 ( ( A = B )
% 0.20/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.20/0.45 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % ord_eq_le_trans
% 0.20/0.45 thf(fact_228_order__class_Oorder_Oeq__iff,axiom,
% 0.20/0.45 ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
% 0.20/0.45 = ( ^ [A2: nat,B2: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ A2 @ B2 )
% 0.20/0.45 & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % order_class.order.eq_iff
% 0.20/0.45 thf(fact_229_antisym__conv,axiom,
% 0.20/0.45 ! [Y: nat,X: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ Y @ X )
% 0.20/0.45 => ( ( ord_less_eq_nat @ X @ Y )
% 0.20/0.45 = ( X = Y ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % antisym_conv
% 0.20/0.45 thf(fact_230_le__cases3,axiom,
% 0.20/0.45 ! [X: nat,Y: nat,Z2: nat] :
% 0.20/0.45 ( ( ( ord_less_eq_nat @ X @ Y )
% 0.20/0.45 => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
% 0.20/0.45 => ( ( ( ord_less_eq_nat @ Y @ X )
% 0.20/0.45 => ~ ( ord_less_eq_nat @ X @ Z2 ) )
% 0.20/0.45 => ( ( ( ord_less_eq_nat @ X @ Z2 )
% 0.20/0.45 => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
% 0.20/0.45 => ( ( ( ord_less_eq_nat @ Z2 @ Y )
% 0.20/0.45 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 0.20/0.45 => ( ( ( ord_less_eq_nat @ Y @ Z2 )
% 0.20/0.45 => ~ ( ord_less_eq_nat @ Z2 @ X ) )
% 0.20/0.45 => ~ ( ( ord_less_eq_nat @ Z2 @ X )
% 0.20/0.45 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % le_cases3
% 0.20/0.45 thf(fact_231_order_Otrans,axiom,
% 0.20/0.45 ! [A: nat,B: nat,C: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.20/0.45 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % order.trans
% 0.20/0.45 thf(fact_232_le__cases,axiom,
% 0.20/0.45 ! [X: nat,Y: nat] :
% 0.20/0.45 ( ~ ( ord_less_eq_nat @ X @ Y )
% 0.20/0.45 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.20/0.45
% 0.20/0.45 % le_cases
% 0.20/0.45 thf(fact_233_eq__refl,axiom,
% 0.20/0.45 ! [X: nat,Y: nat] :
% 0.20/0.45 ( ( X = Y )
% 0.20/0.45 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.20/0.45
% 0.20/0.45 % eq_refl
% 0.20/0.45 thf(fact_234_linear,axiom,
% 0.20/0.45 ! [X: nat,Y: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ X @ Y )
% 0.20/0.45 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.20/0.45
% 0.20/0.45 % linear
% 0.20/0.45 thf(fact_235_antisym,axiom,
% 0.20/0.45 ! [X: nat,Y: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ X @ Y )
% 0.20/0.45 => ( ( ord_less_eq_nat @ Y @ X )
% 0.20/0.45 => ( X = Y ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % antisym
% 0.20/0.45 thf(fact_236_eq__iff,axiom,
% 0.20/0.45 ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
% 0.20/0.45 = ( ^ [X2: nat,Y6: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ X2 @ Y6 )
% 0.20/0.45 & ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % eq_iff
% 0.20/0.45 thf(fact_237_ord__le__eq__subst,axiom,
% 0.20/0.45 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.45 => ( ( ( F @ B )
% 0.20/0.45 = C )
% 0.20/0.45 => ( ! [X3: nat,Y3: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.20/0.45 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.20/0.45 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % ord_le_eq_subst
% 0.20/0.45 thf(fact_238_ord__eq__le__subst,axiom,
% 0.20/0.45 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.20/0.45 ( ( A
% 0.20/0.45 = ( F @ B ) )
% 0.20/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.20/0.45 => ( ! [X3: nat,Y3: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.20/0.45 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.20/0.45 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % ord_eq_le_subst
% 0.20/0.45 thf(fact_239_order__subst2,axiom,
% 0.20/0.45 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ A @ B )
% 0.20/0.45 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.20/0.45 => ( ! [X3: nat,Y3: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.20/0.45 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.20/0.45 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % order_subst2
% 0.20/0.45 thf(fact_240_order__subst1,axiom,
% 0.20/0.45 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.20/0.45 => ( ( ord_less_eq_nat @ B @ C )
% 0.20/0.45 => ( ! [X3: nat,Y3: nat] :
% 0.20/0.45 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.20/0.45 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 0.20/0.45 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % order_subst1
% 0.20/0.45 thf(fact_241_dual__order_Ostrict__implies__not__eq,axiom,
% 0.20/0.45 ! [B: nat,A: nat] :
% 0.20/0.45 ( ( ord_less_nat @ B @ A )
% 0.20/0.45 => ( A != B ) ) ).
% 0.20/0.45
% 0.20/0.45 % dual_order.strict_implies_not_eq
% 0.20/0.45 thf(fact_242_order_Ostrict__implies__not__eq,axiom,
% 0.20/0.45 ! [A: nat,B: nat] :
% 0.20/0.45 ( ( ord_less_nat @ A @ B )
% 0.20/0.45 => ( A != B ) ) ).
% 0.20/0.45
% 0.20/0.45 % order.strict_implies_not_eq
% 0.20/0.45 thf(fact_243_not__less__iff__gr__or__eq,axiom,
% 0.20/0.45 ! [X: nat,Y: nat] :
% 0.20/0.45 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.20/0.45 = ( ( ord_less_nat @ Y @ X )
% 0.20/0.45 | ( X = Y ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % not_less_iff_gr_or_eq
% 0.20/0.45 thf(fact_244_dual__order_Ostrict__trans,axiom,
% 0.20/0.45 ! [B: nat,A: nat,C: nat] :
% 0.20/0.45 ( ( ord_less_nat @ B @ A )
% 0.20/0.45 => ( ( ord_less_nat @ C @ B )
% 0.20/0.45 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % dual_order.strict_trans
% 0.20/0.45 thf(fact_245_linorder__less__wlog,axiom,
% 0.20/0.45 ! [P3: nat > nat > $o,A: nat,B: nat] :
% 0.20/0.45 ( ! [A5: nat,B5: nat] :
% 0.20/0.45 ( ( ord_less_nat @ A5 @ B5 )
% 0.20/0.45 => ( P3 @ A5 @ B5 ) )
% 0.20/0.45 => ( ! [A5: nat] : ( P3 @ A5 @ A5 )
% 0.20/0.45 => ( ! [A5: nat,B5: nat] :
% 0.20/0.45 ( ( P3 @ B5 @ A5 )
% 0.20/0.45 => ( P3 @ A5 @ B5 ) )
% 0.20/0.45 => ( P3 @ A @ B ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % linorder_less_wlog
% 0.20/0.45 thf(fact_246_exists__least__iff,axiom,
% 0.20/0.45 ( ( ^ [P7: nat > $o] :
% 0.20/0.45 ? [X5: nat] : ( P7 @ X5 ) )
% 0.20/0.45 = ( ^ [P8: nat > $o] :
% 0.20/0.45 ? [N2: nat] :
% 0.20/0.45 ( ( P8 @ N2 )
% 0.20/0.45 & ! [M2: nat] :
% 0.20/0.45 ( ( ord_less_nat @ M2 @ N2 )
% 0.20/0.45 => ~ ( P8 @ M2 ) ) ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % exists_least_iff
% 0.20/0.45 thf(fact_247_less__imp__not__less,axiom,
% 0.20/0.45 ! [X: nat,Y: nat] :
% 0.20/0.45 ( ( ord_less_nat @ X @ Y )
% 0.20/0.45 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.20/0.45
% 0.20/0.45 % less_imp_not_less
% 0.20/0.45 thf(fact_248_order_Ostrict__trans,axiom,
% 0.20/0.45 ! [A: nat,B: nat,C: nat] :
% 0.20/0.45 ( ( ord_less_nat @ A @ B )
% 0.20/0.45 => ( ( ord_less_nat @ B @ C )
% 0.20/0.45 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % order.strict_trans
% 0.20/0.45 thf(fact_249_dual__order_Oirrefl,axiom,
% 0.20/0.45 ! [A: nat] :
% 0.20/0.45 ~ ( ord_less_nat @ A @ A ) ).
% 0.20/0.45
% 0.20/0.45 % dual_order.irrefl
% 0.20/0.45 thf(fact_250_linorder__cases,axiom,
% 0.20/0.45 ! [X: nat,Y: nat] :
% 0.20/0.45 ( ~ ( ord_less_nat @ X @ Y )
% 0.20/0.45 => ( ( X != Y )
% 0.20/0.45 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % linorder_cases
% 0.20/0.45 thf(fact_251_less__imp__triv,axiom,
% 0.20/0.45 ! [X: nat,Y: nat,P3: $o] :
% 0.20/0.45 ( ( ord_less_nat @ X @ Y )
% 0.20/0.45 => ( ( ord_less_nat @ Y @ X )
% 0.20/0.45 => P3 ) ) ).
% 0.20/0.45
% 0.20/0.45 % less_imp_triv
% 0.20/0.45 thf(fact_252_less__imp__not__eq2,axiom,
% 0.20/0.45 ! [X: nat,Y: nat] :
% 0.20/0.45 ( ( ord_less_nat @ X @ Y )
% 0.20/0.45 => ( Y != X ) ) ).
% 0.20/0.45
% 0.20/0.45 % less_imp_not_eq2
% 0.20/0.45 thf(fact_253_antisym__conv3,axiom,
% 0.20/0.45 ! [Y: nat,X: nat] :
% 0.20/0.45 ( ~ ( ord_less_nat @ Y @ X )
% 0.20/0.45 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.20/0.45 = ( X = Y ) ) ) ).
% 0.20/0.45
% 0.20/0.45 % antisym_conv3
% 0.20/0.45
% 0.20/0.45 % Conjectures (1)
% 0.20/0.45 thf(conj_0,conjecture,
% 0.20/0.45 ( ( min_dist_capacity @ c @ u @ t )
% 0.20/0.45 = ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) ).
% 0.20/0.45
% 0.20/0.45 %------------------------------------------------------------------------------
% 0.20/0.45 ------------------------
% 2.13/1.46 ---- Embedded in THF ---
% 2.13/1.46 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.13/1.46 %%% Generated on Tue Feb 24 08:24:56 EST 2026
% 2.13/1.46 %%% using '$$dhol' embedding, version 1.3.0.
% 2.13/1.46 %%% Logic specification used:
% 2.13/1.46 %%% thf(spec, logic, $$dhol).
% 2.13/1.46
% 2.13/1.46 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.avufQdrk0c/DTF2DTF22776.p
% See solution above
% 2.13/1.46 ------------------------
% 2.33/1.50 ---- Cleaned THF ---
% 2.33/1.50 thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 2.33/1.50 produc1864880952at_nat: $tType ).
% 2.33/1.50
% 2.33/1.50 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 2.33/1.50 produc787001653at_nat: $tType ).
% 2.33/1.50
% 2.33/1.50 thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 2.33/1.50 produc2082277813at_nat: $tType ).
% 2.33/1.50
% 2.33/1.50 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 2.33/1.50 produc1190591575at_nat: $tType ).
% 2.33/1.50
% 2.33/1.50 thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
% 2.33/1.50 produc1271302400at_nat: $tType ).
% 2.33/1.50
% 2.33/1.50 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.33/1.50 produc1695820582at_nat: $tType ).
% 2.33/1.50
% 2.33/1.50 thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.33/1.50 list_P559422087at_nat: $tType ).
% 2.33/1.50
% 2.33/1.50 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.33/1.50 set_Pr1986765409at_nat: $tType ).
% 2.33/1.50
% 2.33/1.50 thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 2.33/1.50 product_prod_nat_nat: $tType ).
% 2.33/1.50
% 2.33/1.50 thf(ty_n_tf__capacity,type,
% 2.33/1.50 capacity: $tType ).
% 2.33/1.50
% 2.33/1.50 thf(ty_n_t__Nat__Onat,type,
% 2.33/1.50 nat: $tType ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Graph_OGraph_Oconnected_001tf__capacity,type,
% 2.33/1.50 connected_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > $o ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Graph_OGraph_Odist_001tf__capacity,type,
% 2.33/1.50 dist_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > nat > $o ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Graph_OGraph_OisPath_001tf__capacity,type,
% 2.33/1.50 isPath_capacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Graph_OGraph_OisShortestPath_001tf__capacity,type,
% 2.33/1.50 isShor1936442771pacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Graph_OGraph_OisSimplePath_001tf__capacity,type,
% 2.33/1.50 isSimp1359852763pacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Graph_OGraph_Omin__dist_001tf__capacity,type,
% 2.33/1.50 min_dist_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 2.33/1.50 one_one_nat: nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 2.33/1.50 plus_plus_nat: nat > nat > nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 2.33/1.50 nil_Pr1308055047at_nat: list_P559422087at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 2.33/1.50 set_Pr2131844118at_nat: list_P559422087at_nat > set_Pr1986765409at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.33/1.50 size_s1990949619at_nat: list_P559422087at_nat > nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 2.33/1.50 ord_less_nat: nat > nat > $o ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 2.33/1.50 ord_less_eq_nat: nat > nat > $o ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.33/1.50 produc1682925677at_nat: ( nat > nat > nat ) > produc1695820582at_nat > produc2082277813at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 2.33/1.50 produc696256106at_nat: ( nat > product_prod_nat_nat > product_prod_nat_nat ) > produc1190591575at_nat > produc1864880952at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.33/1.50 product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 2.33/1.50 produc1933845336at_nat: nat > product_prod_nat_nat > produc1695820582at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.33/1.50 produc869658639at_nat: nat > produc1695820582at_nat > produc1190591575at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 2.33/1.50 produc947540346at_nat: product_prod_nat_nat > nat > produc1271302400at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.33/1.50 produc365964781at_nat: produc1695820582at_nat > ( nat > nat > nat ) > produc787001653at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Product__Type_Oprod_Oswap_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.33/1.50 produc1728816653at_nat: produc2082277813at_nat > produc787001653at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001t__Nat__Onat,type,
% 2.33/1.50 product_swap_nat_nat: product_prod_nat_nat > product_prod_nat_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 2.33/1.50 produc857968056at_nat: produc1695820582at_nat > produc1271302400at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Product__Type_Oprod_Oswap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
% 2.33/1.50 produc2019146714at_nat: produc1271302400at_nat > produc1695820582at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Product__Type_Oprod_Oswap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 2.33/1.50 produc411855757at_nat: produc787001653at_nat > produc2082277813at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 2.33/1.50 collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 2.33/1.50 member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
% 2.33/1.50
% 2.33/1.50 thf(sy_v_c,type,
% 2.33/1.50 c: product_prod_nat_nat > capacity ).
% 2.33/1.50
% 2.33/1.50 thf(sy_v_p,type,
% 2.33/1.50 p: list_P559422087at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_v_p1____,type,
% 2.33/1.50 p1: list_P559422087at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_v_p2____,type,
% 2.33/1.50 p2: list_P559422087at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_v_p_H,type,
% 2.33/1.50 p3: list_P559422087at_nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_v_s,type,
% 2.33/1.50 s: nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_v_t,type,
% 2.33/1.50 t: nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_v_u,type,
% 2.33/1.50 u: nat ).
% 2.33/1.50
% 2.33/1.50 thf(sy_v_v,type,
% 2.33/1.50 v: nat ).
% 2.33/1.50
% 2.33/1.50 thf(fact_0_MIN_H,axiom,
% 2.33/1.50 ( ( min_dist_capacity @ c @ s @ t )
% 2.33/1.50 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ one_one_nat ) @ ( size_s1990949619at_nat @ p2 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 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,
% 2.33/1.50 ( ( ( min_dist_capacity @ c @ s @ t )
% 2.33/1.50 = ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) )
% 2.33/1.50 => ( ( min_dist_capacity @ c @ s @ u )
% 2.33/1.50 = ( size_s1990949619at_nat @ p1 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 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,
% 2.33/1.50 ( ( ( min_dist_capacity @ c @ s @ t )
% 2.33/1.50 = ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) )
% 2.33/1.50 => ( ( min_dist_capacity @ c @ u @ t )
% 2.33/1.50 = ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_3__092_060open_062min__dist_Av_At_A_061_Alength_Ap2_092_060close_062,axiom,
% 2.33/1.50 ( ( min_dist_capacity @ c @ v @ t )
% 2.33/1.50 = ( size_s1990949619at_nat @ p2 ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_4_MIN,axiom,
% 2.33/1.50 ( ( min_dist_capacity @ c @ s @ t )
% 2.33/1.50 = ( size_s1990949619at_nat @ p ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_5__092_060open_062u_A_092_060noteq_062_Av_092_060close_062,axiom,
% 2.33/1.50 u != v ).
% 2.33/1.50
% 2.33/1.50 thf(fact_6_MDSV,axiom,
% 2.33/1.50 ( ( min_dist_capacity @ c @ s @ v )
% 2.33/1.50 = ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ one_one_nat ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_7_min__dist__split_I2_J,axiom,
% 2.33/1.50 ! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 2.33/1.50 ( ( dist_capacity @ c @ U @ D1 @ W )
% 2.33/1.50 => ( ( dist_capacity @ c @ W @ D2 @ V )
% 2.33/1.50 => ( ( ( min_dist_capacity @ c @ U @ V )
% 2.33/1.50 = ( plus_plus_nat @ D1 @ D2 ) )
% 2.33/1.50 => ( ( min_dist_capacity @ c @ W @ V )
% 2.33/1.50 = D2 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_8_min__dist__split_I1_J,axiom,
% 2.33/1.50 ! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 2.33/1.50 ( ( dist_capacity @ c @ U @ D1 @ W )
% 2.33/1.50 => ( ( dist_capacity @ c @ W @ D2 @ V )
% 2.33/1.50 => ( ( ( min_dist_capacity @ c @ U @ V )
% 2.33/1.50 = ( plus_plus_nat @ D1 @ D2 ) )
% 2.33/1.50 => ( ( min_dist_capacity @ c @ U @ W )
% 2.33/1.50 = D1 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_9_add__left__cancel,axiom,
% 2.33/1.50 ! [A: nat,B: nat,C: nat] :
% 2.33/1.50 ( ( ( plus_plus_nat @ A @ B )
% 2.33/1.50 = ( plus_plus_nat @ A @ C ) )
% 2.33/1.50 = ( B = C ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_10_add__right__cancel,axiom,
% 2.33/1.50 ! [B: nat,A: nat,C: nat] :
% 2.33/1.50 ( ( ( plus_plus_nat @ B @ A )
% 2.33/1.50 = ( plus_plus_nat @ C @ A ) )
% 2.33/1.50 = ( B = C ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_11_dist__trans,axiom,
% 2.33/1.50 ! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 2.33/1.50 ( ( dist_capacity @ c @ U @ D1 @ W )
% 2.33/1.50 => ( ( dist_capacity @ c @ W @ D2 @ V )
% 2.33/1.50 => ( dist_capacity @ c @ U @ ( plus_plus_nat @ D1 @ D2 ) @ V ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_12_assms_I3_J,axiom,
% 2.33/1.50 isPath_capacity @ c @ s @ p3 @ t ).
% 2.33/1.50
% 2.33/1.50 thf(fact_13_DISTS_I1_J,axiom,
% 2.33/1.50 dist_capacity @ c @ s @ ( size_s1990949619at_nat @ p1 ) @ u ).
% 2.33/1.50
% 2.33/1.50 thf(fact_14_isPath__distD,axiom,
% 2.33/1.50 ! [U: nat,P: list_P559422087at_nat,V: nat] :
% 2.33/1.50 ( ( isPath_capacity @ c @ U @ P @ V )
% 2.33/1.50 => ( dist_capacity @ c @ U @ ( size_s1990949619at_nat @ P ) @ V ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_15_dist__def,axiom,
% 2.33/1.50 ! [V: nat,D: nat,V2: nat] :
% 2.33/1.50 ( ( dist_capacity @ c @ V @ D @ V2 )
% 2.33/1.50 = ( ? [P2: list_P559422087at_nat] :
% 2.33/1.50 ( ( isPath_capacity @ c @ V @ P2 @ V2 )
% 2.33/1.50 & ( ( size_s1990949619at_nat @ P2 )
% 2.33/1.50 = D ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_16_DISTS_I2_J,axiom,
% 2.33/1.50 dist_capacity @ c @ u @ one_one_nat @ v ).
% 2.33/1.50
% 2.33/1.50 thf(fact_17_P,axiom,
% 2.33/1.50 isPath_capacity @ c @ s @ p @ t ).
% 2.33/1.50
% 2.33/1.50 thf(fact_18_DISTS_I3_J,axiom,
% 2.33/1.50 dist_capacity @ c @ v @ ( size_s1990949619at_nat @ p2 ) @ t ).
% 2.33/1.50
% 2.33/1.50 thf(fact_19_assms_I1_J,axiom,
% 2.33/1.50 isShor1936442771pacity @ c @ s @ p @ t ).
% 2.33/1.50
% 2.33/1.50 thf(fact_20_add__right__imp__eq,axiom,
% 2.33/1.50 ! [B: nat,A: nat,C: nat] :
% 2.33/1.50 ( ( ( plus_plus_nat @ B @ A )
% 2.33/1.50 = ( plus_plus_nat @ C @ A ) )
% 2.33/1.50 => ( B = C ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_21_add__left__imp__eq,axiom,
% 2.33/1.50 ! [A: nat,B: nat,C: nat] :
% 2.33/1.50 ( ( ( plus_plus_nat @ A @ B )
% 2.33/1.50 = ( plus_plus_nat @ A @ C ) )
% 2.33/1.50 => ( B = C ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_22_add_Oleft__commute,axiom,
% 2.33/1.50 ! [B: nat,A: nat,C: nat] :
% 2.33/1.50 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 2.33/1.50 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_23_add_Ocommute,axiom,
% 2.33/1.50 ( plus_plus_nat
% 2.33/1.50 = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_24_add_Oassoc,axiom,
% 2.33/1.50 ! [A: nat,B: nat,C: nat] :
% 2.33/1.50 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 2.33/1.50 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_25_group__cancel_Oadd2,axiom,
% 2.33/1.50 ! [B3: nat,K: nat,B: nat,A: nat] :
% 2.33/1.50 ( ( B3
% 2.33/1.50 = ( plus_plus_nat @ K @ B ) )
% 2.33/1.50 => ( ( plus_plus_nat @ A @ B3 )
% 2.33/1.50 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_26_group__cancel_Oadd1,axiom,
% 2.33/1.50 ! [A3: nat,K: nat,A: nat,B: nat] :
% 2.33/1.50 ( ( A3
% 2.33/1.50 = ( plus_plus_nat @ K @ A ) )
% 2.33/1.50 => ( ( plus_plus_nat @ A3 @ B )
% 2.33/1.50 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_27_add__mono__thms__linordered__semiring_I4_J,axiom,
% 2.33/1.50 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.33/1.50 ( ( ( I = J )
% 2.33/1.50 & ( K = L ) )
% 2.33/1.50 => ( ( plus_plus_nat @ I @ K )
% 2.33/1.50 = ( plus_plus_nat @ J @ L ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_28_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 2.33/1.50 ! [A: nat,B: nat,C: nat] :
% 2.33/1.50 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 2.33/1.50 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_29_one__reorient,axiom,
% 2.33/1.50 ! [X: nat] :
% 2.33/1.50 ( ( one_one_nat = X )
% 2.33/1.50 = ( X = one_one_nat ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_30_isShortestPath__min__dist__def,axiom,
% 2.33/1.50 ! [U: nat,P: list_P559422087at_nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ c @ U @ P @ V )
% 2.33/1.50 = ( ( isPath_capacity @ c @ U @ P @ V )
% 2.33/1.50 & ( ( size_s1990949619at_nat @ P )
% 2.33/1.50 = ( min_dist_capacity @ c @ U @ V ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_31_Graph_OisPath__distD,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat] :
% 2.33/1.50 ( ( isPath_capacity @ C @ U @ P @ V )
% 2.33/1.50 => ( dist_capacity @ C @ U @ ( size_s1990949619at_nat @ P ) @ V ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_32_Graph_Odist__def,axiom,
% 2.33/1.50 ( dist_capacity
% 2.33/1.50 = ( ^ [C2: product_prod_nat_nat > capacity,V3: nat,D3: nat,V4: nat] :
% 2.33/1.50 ? [P2: list_P559422087at_nat] :
% 2.33/1.50 ( ( isPath_capacity @ C2 @ V3 @ P2 @ V4 )
% 2.33/1.50 & ( ( size_s1990949619at_nat @ P2 )
% 2.33/1.50 = D3 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_33_min__dist__minD,axiom,
% 2.33/1.50 ! [V: nat,D: nat,V2: nat] :
% 2.33/1.50 ( ( dist_capacity @ c @ V @ D @ V2 )
% 2.33/1.50 => ( ord_less_eq_nat @ ( min_dist_capacity @ c @ V @ V2 ) @ D ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_34_min__distI__eq,axiom,
% 2.33/1.50 ! [V: nat,D: nat,V2: nat] :
% 2.33/1.50 ( ( dist_capacity @ c @ V @ D @ V2 )
% 2.33/1.50 => ( ! [D4: nat] :
% 2.33/1.50 ( ( dist_capacity @ c @ V @ D4 @ V2 )
% 2.33/1.50 => ( ord_less_eq_nat @ D @ D4 ) )
% 2.33/1.50 => ( ( min_dist_capacity @ c @ V @ V2 )
% 2.33/1.50 = D ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_35_Graph_Omin__dist__split_I1_J,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 2.33/1.50 ( ( dist_capacity @ C @ U @ D1 @ W )
% 2.33/1.50 => ( ( dist_capacity @ C @ W @ D2 @ V )
% 2.33/1.50 => ( ( ( min_dist_capacity @ C @ U @ V )
% 2.33/1.50 = ( plus_plus_nat @ D1 @ D2 ) )
% 2.33/1.50 => ( ( min_dist_capacity @ C @ U @ W )
% 2.33/1.50 = D1 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_36_Graph_Omin__dist__split_I2_J,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 2.33/1.50 ( ( dist_capacity @ C @ U @ D1 @ W )
% 2.33/1.50 => ( ( dist_capacity @ C @ W @ D2 @ V )
% 2.33/1.50 => ( ( ( min_dist_capacity @ C @ U @ V )
% 2.33/1.50 = ( plus_plus_nat @ D1 @ D2 ) )
% 2.33/1.50 => ( ( min_dist_capacity @ C @ W @ V )
% 2.33/1.50 = D2 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_37_min__dist__is__dist,axiom,
% 2.33/1.50 ! [V: nat,V2: nat] :
% 2.33/1.50 ( ( connected_capacity @ c @ V @ V2 )
% 2.33/1.50 => ( dist_capacity @ c @ V @ ( min_dist_capacity @ c @ V @ V2 ) @ V2 ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_38_shortestPath__is__path,axiom,
% 2.33/1.50 ! [U: nat,P: list_P559422087at_nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ c @ U @ P @ V )
% 2.33/1.50 => ( isPath_capacity @ c @ U @ P @ V ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_39_assms_I4_J,axiom,
% 2.33/1.50 member701585322at_nat @ ( product_Pair_nat_nat @ v @ u ) @ ( set_Pr2131844118at_nat @ p3 ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_40_connected__def,axiom,
% 2.33/1.50 ! [U: nat,V: nat] :
% 2.33/1.50 ( ( connected_capacity @ c @ U @ V )
% 2.33/1.50 = ( ? [P2: list_P559422087at_nat] : ( isPath_capacity @ c @ U @ P2 @ V ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_41_connected__by__dist,axiom,
% 2.33/1.50 ! [V: nat,V2: nat] :
% 2.33/1.50 ( ( connected_capacity @ c @ V @ V2 )
% 2.33/1.50 = ( ? [D3: nat] : ( dist_capacity @ c @ V @ D3 @ V2 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_42_mem__Collect__eq,axiom,
% 2.33/1.50 ! [A: product_prod_nat_nat,P3: product_prod_nat_nat > $o] :
% 2.33/1.50 ( ( member701585322at_nat @ A @ ( collec7649004at_nat @ P3 ) )
% 2.33/1.50 = ( P3 @ A ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_43_Collect__mem__eq,axiom,
% 2.33/1.50 ! [A3: set_Pr1986765409at_nat] :
% 2.33/1.50 ( ( collec7649004at_nat
% 2.33/1.50 @ ^ [X2: product_prod_nat_nat] : ( member701585322at_nat @ X2 @ A3 ) )
% 2.33/1.50 = A3 ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_44_Collect__cong,axiom,
% 2.33/1.50 ! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 2.33/1.50 ( ! [X3: product_prod_nat_nat] :
% 2.33/1.50 ( ( P3 @ X3 )
% 2.33/1.50 = ( Q @ X3 ) )
% 2.33/1.50 => ( ( collec7649004at_nat @ P3 )
% 2.33/1.50 = ( collec7649004at_nat @ Q ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_45_obtain__shortest__path,axiom,
% 2.33/1.50 ! [U: nat,V: nat] :
% 2.33/1.50 ( ( connected_capacity @ c @ U @ V )
% 2.33/1.50 => ~ ! [P4: list_P559422087at_nat] :
% 2.33/1.50 ~ ( isShor1936442771pacity @ c @ U @ P4 @ V ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_46_assms_I2_J,axiom,
% 2.33/1.50 member701585322at_nat @ ( product_Pair_nat_nat @ u @ v ) @ ( set_Pr2131844118at_nat @ p ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_47_isPath__ex__edge1,axiom,
% 2.33/1.50 ! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
% 2.33/1.50 ( ( isPath_capacity @ c @ U @ P @ V )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( ( U1 != U )
% 2.33/1.50 => ? [U2: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ U2 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_48_isPath__ex__edge2,axiom,
% 2.33/1.50 ! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
% 2.33/1.50 ( ( isPath_capacity @ c @ U @ P @ V )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( ( V1 != V )
% 2.33/1.50 => ? [V22: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ V1 @ V22 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_49_min__dist__le,axiom,
% 2.33/1.50 ! [Src: nat,V: nat,D5: nat] :
% 2.33/1.50 ( ( connected_capacity @ c @ Src @ V )
% 2.33/1.50 => ( ( ord_less_eq_nat @ D5 @ ( min_dist_capacity @ c @ Src @ V ) )
% 2.33/1.50 => ? [V5: nat] :
% 2.33/1.50 ( ( connected_capacity @ c @ Src @ V5 )
% 2.33/1.50 & ( ( min_dist_capacity @ c @ Src @ V5 )
% 2.33/1.50 = D5 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_50_min__distI2,axiom,
% 2.33/1.50 ! [V: nat,V2: nat,Q: nat > $o] :
% 2.33/1.50 ( ( connected_capacity @ c @ V @ V2 )
% 2.33/1.50 => ( ! [D6: nat] :
% 2.33/1.50 ( ( dist_capacity @ c @ V @ D6 @ V2 )
% 2.33/1.50 => ( ! [D7: nat] :
% 2.33/1.50 ( ( dist_capacity @ c @ V @ D7 @ V2 )
% 2.33/1.50 => ( ord_less_eq_nat @ D6 @ D7 ) )
% 2.33/1.50 => ( Q @ D6 ) ) )
% 2.33/1.50 => ( Q @ ( min_dist_capacity @ c @ V @ V2 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_51_isShortestPath__def,axiom,
% 2.33/1.50 ! [U: nat,P: list_P559422087at_nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ c @ U @ P @ V )
% 2.33/1.50 = ( ( isPath_capacity @ c @ U @ P @ V )
% 2.33/1.50 & ! [P5: list_P559422087at_nat] :
% 2.33/1.50 ( ( isPath_capacity @ c @ U @ P5 @ V )
% 2.33/1.50 => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P ) @ ( size_s1990949619at_nat @ P5 ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_52_isShortestPath__level__edge_I1_J,axiom,
% 2.33/1.50 ! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ c @ S @ P @ T )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( connected_capacity @ c @ S @ U ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_53_isShortestPath__level__edge_I2_J,axiom,
% 2.33/1.50 ! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ c @ S @ P @ T )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( connected_capacity @ c @ U @ V ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_54_isShortestPath__level__edge_I3_J,axiom,
% 2.33/1.50 ! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ c @ S @ P @ T )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( connected_capacity @ c @ V @ T ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_55_add__le__cancel__left,axiom,
% 2.33/1.50 ! [C: nat,A: nat,B: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.33/1.50 = ( ord_less_eq_nat @ A @ B ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_56_add__le__cancel__right,axiom,
% 2.33/1.50 ! [A: nat,C: nat,B: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.33/1.50 = ( ord_less_eq_nat @ A @ B ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_57_isShortestPath__level__edge_I6_J,axiom,
% 2.33/1.50 ! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ c @ S @ P @ T )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( ( min_dist_capacity @ c @ S @ T )
% 2.33/1.50 = ( plus_plus_nat @ ( plus_plus_nat @ ( min_dist_capacity @ c @ S @ U ) @ one_one_nat ) @ ( min_dist_capacity @ c @ V @ T ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_58_isShortestPath__level__edge_I5_J,axiom,
% 2.33/1.50 ! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ c @ S @ P @ T )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( ( min_dist_capacity @ c @ U @ T )
% 2.33/1.50 = ( plus_plus_nat @ one_one_nat @ ( min_dist_capacity @ c @ V @ T ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_59_isShortestPath__level__edge_I4_J,axiom,
% 2.33/1.50 ! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ c @ S @ P @ T )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( ( min_dist_capacity @ c @ S @ V )
% 2.33/1.50 = ( plus_plus_nat @ ( min_dist_capacity @ c @ S @ U ) @ one_one_nat ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_60_connected__refl,axiom,
% 2.33/1.50 ! [V: nat] : ( connected_capacity @ c @ V @ V ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_61_connected__distI,axiom,
% 2.33/1.50 ! [V: nat,D: nat,V2: nat] :
% 2.33/1.50 ( ( dist_capacity @ c @ V @ D @ V2 )
% 2.33/1.50 => ( connected_capacity @ c @ V @ V2 ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_62_Graph_OisPath__ex__edge1,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
% 2.33/1.50 ( ( isPath_capacity @ C @ U @ P @ V )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( ( U1 != U )
% 2.33/1.50 => ? [U2: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ U2 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_63_Graph_OisPath__ex__edge2,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
% 2.33/1.50 ( ( isPath_capacity @ C @ U @ P @ V )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( ( V1 != V )
% 2.33/1.50 => ? [V22: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ V1 @ V22 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_64_Graph_Oobtain__shortest__path,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,U: nat,V: nat] :
% 2.33/1.50 ( ( connected_capacity @ C @ U @ V )
% 2.33/1.50 => ~ ! [P4: list_P559422087at_nat] :
% 2.33/1.50 ~ ( isShor1936442771pacity @ C @ U @ P4 @ V ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_65_Graph_OisShortestPath_Ocong,axiom,
% 2.33/1.50 isShor1936442771pacity = isShor1936442771pacity ).
% 2.33/1.50
% 2.33/1.50 thf(fact_66_Graph_Oconnected__refl,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,V: nat] : ( connected_capacity @ C @ V @ V ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_67_Graph_Oconnected_Ocong,axiom,
% 2.33/1.50 connected_capacity = connected_capacity ).
% 2.33/1.50
% 2.33/1.50 thf(fact_68_Graph_Omin__dist__le,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,Src: nat,V: nat,D5: nat] :
% 2.33/1.50 ( ( connected_capacity @ C @ Src @ V )
% 2.33/1.50 => ( ( ord_less_eq_nat @ D5 @ ( min_dist_capacity @ C @ Src @ V ) )
% 2.33/1.50 => ? [V5: nat] :
% 2.33/1.50 ( ( connected_capacity @ C @ Src @ V5 )
% 2.33/1.50 & ( ( min_dist_capacity @ C @ Src @ V5 )
% 2.33/1.50 = D5 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_69_Graph_OisShortestPath__level__edge_I1_J,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ C @ S @ P @ T )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( connected_capacity @ C @ S @ U ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_70_Graph_OisShortestPath__level__edge_I2_J,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ C @ S @ P @ T )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( connected_capacity @ C @ U @ V ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_71_Graph_OisShortestPath__level__edge_I3_J,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ C @ S @ P @ T )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( connected_capacity @ C @ V @ T ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_72_Graph_Oconnected__def,axiom,
% 2.33/1.50 ( connected_capacity
% 2.33/1.50 = ( ^ [C2: product_prod_nat_nat > capacity,U3: nat,V3: nat] :
% 2.33/1.50 ? [P2: list_P559422087at_nat] : ( isPath_capacity @ C2 @ U3 @ P2 @ V3 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_73_Graph_Oconnected__by__dist,axiom,
% 2.33/1.50 ( connected_capacity
% 2.33/1.50 = ( ^ [C2: product_prod_nat_nat > capacity,V3: nat,V4: nat] :
% 2.33/1.50 ? [D3: nat] : ( dist_capacity @ C2 @ V3 @ D3 @ V4 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_74_Graph_Oconnected__distI,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,V: nat,D: nat,V2: nat] :
% 2.33/1.50 ( ( dist_capacity @ C @ V @ D @ V2 )
% 2.33/1.50 => ( connected_capacity @ C @ V @ V2 ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_75_Graph_OshortestPath__is__path,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ C @ U @ P @ V )
% 2.33/1.50 => ( isPath_capacity @ C @ U @ P @ V ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_76_Graph_OisShortestPath__level__edge_I6_J,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ C @ S @ P @ T )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( ( min_dist_capacity @ C @ S @ T )
% 2.33/1.50 = ( plus_plus_nat @ ( plus_plus_nat @ ( min_dist_capacity @ C @ S @ U ) @ one_one_nat ) @ ( min_dist_capacity @ C @ V @ T ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_77_Graph_OisShortestPath__level__edge_I5_J,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ C @ S @ P @ T )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( ( min_dist_capacity @ C @ U @ T )
% 2.33/1.50 = ( plus_plus_nat @ one_one_nat @ ( min_dist_capacity @ C @ V @ T ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_78_Graph_OisShortestPath__level__edge_I4_J,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ C @ S @ P @ T )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( ( min_dist_capacity @ C @ S @ V )
% 2.33/1.50 = ( plus_plus_nat @ ( min_dist_capacity @ C @ S @ U ) @ one_one_nat ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_79_Graph_Omin__distI2,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,V: nat,V2: nat,Q: nat > $o] :
% 2.33/1.50 ( ( connected_capacity @ C @ V @ V2 )
% 2.33/1.50 => ( ! [D6: nat] :
% 2.33/1.50 ( ( dist_capacity @ C @ V @ D6 @ V2 )
% 2.33/1.50 => ( ! [D7: nat] :
% 2.33/1.50 ( ( dist_capacity @ C @ V @ D7 @ V2 )
% 2.33/1.50 => ( ord_less_eq_nat @ D6 @ D7 ) )
% 2.33/1.50 => ( Q @ D6 ) ) )
% 2.33/1.50 => ( Q @ ( min_dist_capacity @ C @ V @ V2 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_80_Graph_OisShortestPath__def,axiom,
% 2.33/1.50 ( isShor1936442771pacity
% 2.33/1.50 = ( ^ [C2: product_prod_nat_nat > capacity,U3: nat,P2: list_P559422087at_nat,V3: nat] :
% 2.33/1.50 ( ( isPath_capacity @ C2 @ U3 @ P2 @ V3 )
% 2.33/1.50 & ! [P5: list_P559422087at_nat] :
% 2.33/1.50 ( ( isPath_capacity @ C2 @ U3 @ P5 @ V3 )
% 2.33/1.50 => ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P2 ) @ ( size_s1990949619at_nat @ P5 ) ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_81_add__mono__thms__linordered__semiring_I3_J,axiom,
% 2.33/1.50 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.33/1.50 ( ( ( ord_less_eq_nat @ I @ J )
% 2.33/1.50 & ( K = L ) )
% 2.33/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_82_add__mono__thms__linordered__semiring_I2_J,axiom,
% 2.33/1.50 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.33/1.50 ( ( ( I = J )
% 2.33/1.50 & ( ord_less_eq_nat @ K @ L ) )
% 2.33/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_83_add__mono__thms__linordered__semiring_I1_J,axiom,
% 2.33/1.50 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.33/1.50 ( ( ( ord_less_eq_nat @ I @ J )
% 2.33/1.50 & ( ord_less_eq_nat @ K @ L ) )
% 2.33/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_84_add__mono,axiom,
% 2.33/1.50 ! [A: nat,B: nat,C: nat,D: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.33/1.50 => ( ( ord_less_eq_nat @ C @ D )
% 2.33/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_85_add__left__mono,axiom,
% 2.33/1.50 ! [A: nat,B: nat,C: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.33/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_86_less__eqE,axiom,
% 2.33/1.50 ! [A: nat,B: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.33/1.50 => ~ ! [C3: nat] :
% 2.33/1.50 ( B
% 2.33/1.50 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_87_add__right__mono,axiom,
% 2.33/1.50 ! [A: nat,B: nat,C: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.33/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_88_le__iff__add,axiom,
% 2.33/1.50 ( ord_less_eq_nat
% 2.33/1.50 = ( ^ [A2: nat,B2: nat] :
% 2.33/1.50 ? [C2: nat] :
% 2.33/1.50 ( B2
% 2.33/1.50 = ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_89_add__le__imp__le__left,axiom,
% 2.33/1.50 ! [C: nat,A: nat,B: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.33/1.50 => ( ord_less_eq_nat @ A @ B ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_90_add__le__imp__le__right,axiom,
% 2.33/1.50 ! [A: nat,C: nat,B: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.33/1.50 => ( ord_less_eq_nat @ A @ B ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_91_Graph_Omin__dist__minD,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,V: nat,D: nat,V2: nat] :
% 2.33/1.50 ( ( dist_capacity @ C @ V @ D @ V2 )
% 2.33/1.50 => ( ord_less_eq_nat @ ( min_dist_capacity @ C @ V @ V2 ) @ D ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_92_Graph_Omin__distI__eq,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,V: nat,D: nat,V2: nat] :
% 2.33/1.50 ( ( dist_capacity @ C @ V @ D @ V2 )
% 2.33/1.50 => ( ! [D4: nat] :
% 2.33/1.50 ( ( dist_capacity @ C @ V @ D4 @ V2 )
% 2.33/1.50 => ( ord_less_eq_nat @ D @ D4 ) )
% 2.33/1.50 => ( ( min_dist_capacity @ C @ V @ V2 )
% 2.33/1.50 = D ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_93_Graph_Omin__dist__is__dist,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,V: nat,V2: nat] :
% 2.33/1.50 ( ( connected_capacity @ C @ V @ V2 )
% 2.33/1.50 => ( dist_capacity @ C @ V @ ( min_dist_capacity @ C @ V @ V2 ) @ V2 ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_94_Graph_OisShortestPath__min__dist__def,axiom,
% 2.33/1.50 ( isShor1936442771pacity
% 2.33/1.50 = ( ^ [C2: product_prod_nat_nat > capacity,U3: nat,P2: list_P559422087at_nat,V3: nat] :
% 2.33/1.50 ( ( isPath_capacity @ C2 @ U3 @ P2 @ V3 )
% 2.33/1.50 & ( ( size_s1990949619at_nat @ P2 )
% 2.33/1.50 = ( min_dist_capacity @ C2 @ U3 @ V3 ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_95_Graph_OisPath_Ocong,axiom,
% 2.33/1.50 isPath_capacity = isPath_capacity ).
% 2.33/1.50
% 2.33/1.50 thf(fact_96_Graph_Omin__dist_Ocong,axiom,
% 2.33/1.50 min_dist_capacity = min_dist_capacity ).
% 2.33/1.50
% 2.33/1.50 thf(fact_97_Graph_Odist_Ocong,axiom,
% 2.33/1.50 dist_capacity = dist_capacity ).
% 2.33/1.50
% 2.33/1.50 thf(fact_98_Graph_Odist__trans,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
% 2.33/1.50 ( ( dist_capacity @ C @ U @ D1 @ W )
% 2.33/1.50 => ( ( dist_capacity @ C @ W @ D2 @ V )
% 2.33/1.50 => ( dist_capacity @ C @ U @ ( plus_plus_nat @ D1 @ D2 ) @ V ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_99_nat__add__left__cancel__le,axiom,
% 2.33/1.50 ! [K: nat,M: nat,N: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 2.33/1.50 = ( ord_less_eq_nat @ M @ N ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_100_isShortestPath__alt,axiom,
% 2.33/1.50 ! [U: nat,P: list_P559422087at_nat,V: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ c @ U @ P @ V )
% 2.33/1.50 = ( ( isSimp1359852763pacity @ c @ U @ P @ V )
% 2.33/1.50 & ( ( size_s1990949619at_nat @ P )
% 2.33/1.50 = ( min_dist_capacity @ c @ U @ V ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_101_min__dist__less,axiom,
% 2.33/1.50 ! [Src: nat,V: nat,D: nat,D5: nat] :
% 2.33/1.50 ( ( connected_capacity @ c @ Src @ V )
% 2.33/1.50 => ( ( ( min_dist_capacity @ c @ Src @ V )
% 2.33/1.50 = D )
% 2.33/1.50 => ( ( ord_less_nat @ D5 @ D )
% 2.33/1.50 => ? [V5: nat] :
% 2.33/1.50 ( ( connected_capacity @ c @ Src @ V5 )
% 2.33/1.50 & ( ( min_dist_capacity @ c @ Src @ V5 )
% 2.33/1.50 = D5 ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_102_isSPath__no__selfloop,axiom,
% 2.33/1.50 ! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat] :
% 2.33/1.50 ( ( isSimp1359852763pacity @ c @ U @ P @ V )
% 2.33/1.50 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_103_isSPath__sg__incoming,axiom,
% 2.33/1.50 ! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,U22: nat] :
% 2.33/1.50 ( ( isSimp1359852763pacity @ c @ U @ P @ V )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( ( U1 != U22 )
% 2.33/1.50 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U22 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_104_isSPath__sg__outgoing,axiom,
% 2.33/1.50 ! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,V23: nat] :
% 2.33/1.50 ( ( isSimp1359852763pacity @ c @ U @ P @ V )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( ( V1 != V23 )
% 2.33/1.50 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V23 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_105_old_Oprod_Oinject,axiom,
% 2.33/1.50 ! [A: nat > nat > nat,B: produc1695820582at_nat,A4: nat > nat > nat,B4: produc1695820582at_nat] :
% 2.33/1.50 ( ( ( produc1682925677at_nat @ A @ B )
% 2.33/1.50 = ( produc1682925677at_nat @ A4 @ B4 ) )
% 2.33/1.50 = ( ( A = A4 )
% 2.33/1.50 & ( B = B4 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_106_old_Oprod_Oinject,axiom,
% 2.33/1.50 ! [A: nat,B: product_prod_nat_nat,A4: nat,B4: product_prod_nat_nat] :
% 2.33/1.50 ( ( ( produc1933845336at_nat @ A @ B )
% 2.33/1.50 = ( produc1933845336at_nat @ A4 @ B4 ) )
% 2.33/1.50 = ( ( A = A4 )
% 2.33/1.50 & ( B = B4 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_107_old_Oprod_Oinject,axiom,
% 2.33/1.50 ! [A: nat,B: nat,A4: nat,B4: nat] :
% 2.33/1.50 ( ( ( product_Pair_nat_nat @ A @ B )
% 2.33/1.50 = ( product_Pair_nat_nat @ A4 @ B4 ) )
% 2.33/1.50 = ( ( A = A4 )
% 2.33/1.50 & ( B = B4 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_108_prod_Oinject,axiom,
% 2.33/1.50 ! [X1: nat > nat > nat,X22: produc1695820582at_nat,Y1: nat > nat > nat,Y2: produc1695820582at_nat] :
% 2.33/1.50 ( ( ( produc1682925677at_nat @ X1 @ X22 )
% 2.33/1.50 = ( produc1682925677at_nat @ Y1 @ Y2 ) )
% 2.33/1.50 = ( ( X1 = Y1 )
% 2.33/1.50 & ( X22 = Y2 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_109_prod_Oinject,axiom,
% 2.33/1.50 ! [X1: nat,X22: product_prod_nat_nat,Y1: nat,Y2: product_prod_nat_nat] :
% 2.33/1.50 ( ( ( produc1933845336at_nat @ X1 @ X22 )
% 2.33/1.50 = ( produc1933845336at_nat @ Y1 @ Y2 ) )
% 2.33/1.50 = ( ( X1 = Y1 )
% 2.33/1.50 & ( X22 = Y2 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_110_prod_Oinject,axiom,
% 2.33/1.50 ! [X1: nat,X22: nat,Y1: nat,Y2: nat] :
% 2.33/1.50 ( ( ( product_Pair_nat_nat @ X1 @ X22 )
% 2.33/1.50 = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
% 2.33/1.50 = ( ( X1 = Y1 )
% 2.33/1.50 & ( X22 = Y2 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_111_isSPath__pathLE,axiom,
% 2.33/1.50 ! [S: nat,P: list_P559422087at_nat,T: nat] :
% 2.33/1.50 ( ( isPath_capacity @ c @ S @ P @ T )
% 2.33/1.50 => ? [P6: list_P559422087at_nat] : ( isSimp1359852763pacity @ c @ S @ P6 @ T ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_112_shortestPath__is__simple,axiom,
% 2.33/1.50 ! [S: nat,P: list_P559422087at_nat,T: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ c @ S @ P @ T )
% 2.33/1.50 => ( isSimp1359852763pacity @ c @ S @ P @ T ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_113_add__less__cancel__right,axiom,
% 2.33/1.50 ! [A: nat,C: nat,B: nat] :
% 2.33/1.50 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.33/1.50 = ( ord_less_nat @ A @ B ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_114_add__less__cancel__left,axiom,
% 2.33/1.50 ! [C: nat,A: nat,B: nat] :
% 2.33/1.50 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.33/1.50 = ( ord_less_nat @ A @ B ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_115_nat__add__left__cancel__less,axiom,
% 2.33/1.50 ! [K: nat,M: nat,N: nat] :
% 2.33/1.50 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 2.33/1.50 = ( ord_less_nat @ M @ N ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_116_Graph_OisSimplePath_Ocong,axiom,
% 2.33/1.50 isSimp1359852763pacity = isSimp1359852763pacity ).
% 2.33/1.50
% 2.33/1.50 thf(fact_117_less__mono__imp__le__mono,axiom,
% 2.33/1.50 ! [F: nat > nat,I: nat,J: nat] :
% 2.33/1.50 ( ! [I2: nat,J2: nat] :
% 2.33/1.50 ( ( ord_less_nat @ I2 @ J2 )
% 2.33/1.50 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 2.33/1.50 => ( ( ord_less_eq_nat @ I @ J )
% 2.33/1.50 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_118_le__neq__implies__less,axiom,
% 2.33/1.50 ! [M: nat,N: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ M @ N )
% 2.33/1.50 => ( ( M != N )
% 2.33/1.50 => ( ord_less_nat @ M @ N ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_119_less__or__eq__imp__le,axiom,
% 2.33/1.50 ! [M: nat,N: nat] :
% 2.33/1.50 ( ( ( ord_less_nat @ M @ N )
% 2.33/1.50 | ( M = N ) )
% 2.33/1.50 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_120_le__eq__less__or__eq,axiom,
% 2.33/1.50 ( ord_less_eq_nat
% 2.33/1.50 = ( ^ [M2: nat,N2: nat] :
% 2.33/1.50 ( ( ord_less_nat @ M2 @ N2 )
% 2.33/1.50 | ( M2 = N2 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_121_less__imp__le__nat,axiom,
% 2.33/1.50 ! [M: nat,N: nat] :
% 2.33/1.50 ( ( ord_less_nat @ M @ N )
% 2.33/1.50 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_122_nat__less__le,axiom,
% 2.33/1.50 ( ord_less_nat
% 2.33/1.50 = ( ^ [M2: nat,N2: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ M2 @ N2 )
% 2.33/1.50 & ( M2 != N2 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_123_add__lessD1,axiom,
% 2.33/1.50 ! [I: nat,J: nat,K: nat] :
% 2.33/1.50 ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 2.33/1.50 => ( ord_less_nat @ I @ K ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_124_add__less__mono,axiom,
% 2.33/1.50 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.33/1.50 ( ( ord_less_nat @ I @ J )
% 2.33/1.50 => ( ( ord_less_nat @ K @ L )
% 2.33/1.50 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_125_not__add__less1,axiom,
% 2.33/1.50 ! [I: nat,J: nat] :
% 2.33/1.50 ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_126_not__add__less2,axiom,
% 2.33/1.50 ! [J: nat,I: nat] :
% 2.33/1.50 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_127_add__less__mono1,axiom,
% 2.33/1.50 ! [I: nat,J: nat,K: nat] :
% 2.33/1.50 ( ( ord_less_nat @ I @ J )
% 2.33/1.50 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_128_trans__less__add1,axiom,
% 2.33/1.50 ! [I: nat,J: nat,M: nat] :
% 2.33/1.50 ( ( ord_less_nat @ I @ J )
% 2.33/1.50 => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_129_trans__less__add2,axiom,
% 2.33/1.50 ! [I: nat,J: nat,M: nat] :
% 2.33/1.50 ( ( ord_less_nat @ I @ J )
% 2.33/1.50 => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_130_less__add__eq__less,axiom,
% 2.33/1.50 ! [K: nat,L: nat,M: nat,N: nat] :
% 2.33/1.50 ( ( ord_less_nat @ K @ L )
% 2.33/1.50 => ( ( ( plus_plus_nat @ M @ L )
% 2.33/1.50 = ( plus_plus_nat @ K @ N ) )
% 2.33/1.50 => ( ord_less_nat @ M @ N ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_131_nat__neq__iff,axiom,
% 2.33/1.50 ! [M: nat,N: nat] :
% 2.33/1.50 ( ( M != N )
% 2.33/1.50 = ( ( ord_less_nat @ M @ N )
% 2.33/1.50 | ( ord_less_nat @ N @ M ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_132_less__not__refl,axiom,
% 2.33/1.50 ! [N: nat] :
% 2.33/1.50 ~ ( ord_less_nat @ N @ N ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_133_less__not__refl2,axiom,
% 2.33/1.50 ! [N: nat,M: nat] :
% 2.33/1.50 ( ( ord_less_nat @ N @ M )
% 2.33/1.50 => ( M != N ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_134_less__not__refl3,axiom,
% 2.33/1.50 ! [S: nat,T: nat] :
% 2.33/1.50 ( ( ord_less_nat @ S @ T )
% 2.33/1.50 => ( S != T ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_135_less__irrefl__nat,axiom,
% 2.33/1.50 ! [N: nat] :
% 2.33/1.50 ~ ( ord_less_nat @ N @ N ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_136_nat__less__induct,axiom,
% 2.33/1.50 ! [P3: nat > $o,N: nat] :
% 2.33/1.50 ( ! [N3: nat] :
% 2.33/1.50 ( ! [M3: nat] :
% 2.33/1.50 ( ( ord_less_nat @ M3 @ N3 )
% 2.33/1.50 => ( P3 @ M3 ) )
% 2.33/1.50 => ( P3 @ N3 ) )
% 2.33/1.50 => ( P3 @ N ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_137_infinite__descent,axiom,
% 2.33/1.50 ! [P3: nat > $o,N: nat] :
% 2.33/1.50 ( ! [N3: nat] :
% 2.33/1.50 ( ~ ( P3 @ N3 )
% 2.33/1.50 => ? [M3: nat] :
% 2.33/1.50 ( ( ord_less_nat @ M3 @ N3 )
% 2.33/1.50 & ~ ( P3 @ M3 ) ) )
% 2.33/1.50 => ( P3 @ N ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_138_linorder__neqE__nat,axiom,
% 2.33/1.50 ! [X: nat,Y: nat] :
% 2.33/1.50 ( ( X != Y )
% 2.33/1.50 => ( ~ ( ord_less_nat @ X @ Y )
% 2.33/1.50 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_139_add__less__imp__less__right,axiom,
% 2.33/1.50 ! [A: nat,C: nat,B: nat] :
% 2.33/1.50 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 2.33/1.50 => ( ord_less_nat @ A @ B ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_140_add__less__imp__less__left,axiom,
% 2.33/1.50 ! [C: nat,A: nat,B: nat] :
% 2.33/1.50 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 2.33/1.50 => ( ord_less_nat @ A @ B ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_141_add__strict__right__mono,axiom,
% 2.33/1.50 ! [A: nat,B: nat,C: nat] :
% 2.33/1.50 ( ( ord_less_nat @ A @ B )
% 2.33/1.50 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_142_add__strict__left__mono,axiom,
% 2.33/1.50 ! [A: nat,B: nat,C: nat] :
% 2.33/1.50 ( ( ord_less_nat @ A @ B )
% 2.33/1.50 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_143_add__strict__mono,axiom,
% 2.33/1.50 ! [A: nat,B: nat,C: nat,D: nat] :
% 2.33/1.50 ( ( ord_less_nat @ A @ B )
% 2.33/1.50 => ( ( ord_less_nat @ C @ D )
% 2.33/1.50 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_144_add__mono__thms__linordered__field_I1_J,axiom,
% 2.33/1.50 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.33/1.50 ( ( ( ord_less_nat @ I @ J )
% 2.33/1.50 & ( K = L ) )
% 2.33/1.50 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_145_add__mono__thms__linordered__field_I2_J,axiom,
% 2.33/1.50 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.33/1.50 ( ( ( I = J )
% 2.33/1.50 & ( ord_less_nat @ K @ L ) )
% 2.33/1.50 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_146_add__mono__thms__linordered__field_I5_J,axiom,
% 2.33/1.50 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.33/1.50 ( ( ( ord_less_nat @ I @ J )
% 2.33/1.50 & ( ord_less_nat @ K @ L ) )
% 2.33/1.50 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_147_mono__nat__linear__lb,axiom,
% 2.33/1.50 ! [F: nat > nat,M: nat,K: nat] :
% 2.33/1.50 ( ! [M4: nat,N3: nat] :
% 2.33/1.50 ( ( ord_less_nat @ M4 @ N3 )
% 2.33/1.50 => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
% 2.33/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_148_Graph_OisSPath__pathLE,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat] :
% 2.33/1.50 ( ( isPath_capacity @ C @ S @ P @ T )
% 2.33/1.50 => ? [P6: list_P559422087at_nat] : ( isSimp1359852763pacity @ C @ S @ P6 @ T ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_149_Graph_OshortestPath__is__simple,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat] :
% 2.33/1.50 ( ( isShor1936442771pacity @ C @ S @ P @ T )
% 2.33/1.50 => ( isSimp1359852763pacity @ C @ S @ P @ T ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_150_add__mono__thms__linordered__field_I4_J,axiom,
% 2.33/1.50 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.33/1.50 ( ( ( ord_less_eq_nat @ I @ J )
% 2.33/1.50 & ( ord_less_nat @ K @ L ) )
% 2.33/1.50 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_151_add__mono__thms__linordered__field_I3_J,axiom,
% 2.33/1.50 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.33/1.50 ( ( ( ord_less_nat @ I @ J )
% 2.33/1.50 & ( ord_less_eq_nat @ K @ L ) )
% 2.33/1.50 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_152_add__le__less__mono,axiom,
% 2.33/1.50 ! [A: nat,B: nat,C: nat,D: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.33/1.50 => ( ( ord_less_nat @ C @ D )
% 2.33/1.50 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_153_add__less__le__mono,axiom,
% 2.33/1.50 ! [A: nat,B: nat,C: nat,D: nat] :
% 2.33/1.50 ( ( ord_less_nat @ A @ B )
% 2.33/1.50 => ( ( ord_less_eq_nat @ C @ D )
% 2.33/1.50 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_154_Graph_Omin__dist__less,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,Src: nat,V: nat,D: nat,D5: nat] :
% 2.33/1.50 ( ( connected_capacity @ C @ Src @ V )
% 2.33/1.50 => ( ( ( min_dist_capacity @ C @ Src @ V )
% 2.33/1.50 = D )
% 2.33/1.50 => ( ( ord_less_nat @ D5 @ D )
% 2.33/1.50 => ? [V5: nat] :
% 2.33/1.50 ( ( connected_capacity @ C @ Src @ V5 )
% 2.33/1.50 & ( ( min_dist_capacity @ C @ Src @ V5 )
% 2.33/1.50 = D5 ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_155_Graph_OisSPath__no__selfloop,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat] :
% 2.33/1.50 ( ( isSimp1359852763pacity @ C @ U @ P @ V )
% 2.33/1.50 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_156_Graph_OisSPath__sg__incoming,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,U22: nat] :
% 2.33/1.50 ( ( isSimp1359852763pacity @ C @ U @ P @ V )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( ( U1 != U22 )
% 2.33/1.50 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U22 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_157_Graph_OisSPath__sg__outgoing,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,V23: nat] :
% 2.33/1.50 ( ( isSimp1359852763pacity @ C @ U @ P @ V )
% 2.33/1.50 => ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ( ( V1 != V23 )
% 2.33/1.50 => ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V23 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_158_surj__pair,axiom,
% 2.33/1.50 ! [P: produc2082277813at_nat] :
% 2.33/1.50 ? [X3: nat > nat > nat,Y3: produc1695820582at_nat] :
% 2.33/1.50 ( P
% 2.33/1.50 = ( produc1682925677at_nat @ X3 @ Y3 ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_159_surj__pair,axiom,
% 2.33/1.50 ! [P: produc1695820582at_nat] :
% 2.33/1.50 ? [X3: nat,Y3: product_prod_nat_nat] :
% 2.33/1.50 ( P
% 2.33/1.50 = ( produc1933845336at_nat @ X3 @ Y3 ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_160_surj__pair,axiom,
% 2.33/1.50 ! [P: product_prod_nat_nat] :
% 2.33/1.50 ? [X3: nat,Y3: nat] :
% 2.33/1.50 ( P
% 2.33/1.50 = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_161_prod__cases,axiom,
% 2.33/1.50 ! [P3: produc2082277813at_nat > $o,P: produc2082277813at_nat] :
% 2.33/1.50 ( ! [A5: nat > nat > nat,B5: produc1695820582at_nat] : ( P3 @ ( produc1682925677at_nat @ A5 @ B5 ) )
% 2.33/1.50 => ( P3 @ P ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_162_prod__cases,axiom,
% 2.33/1.50 ! [P3: produc1695820582at_nat > $o,P: produc1695820582at_nat] :
% 2.33/1.50 ( ! [A5: nat,B5: product_prod_nat_nat] : ( P3 @ ( produc1933845336at_nat @ A5 @ B5 ) )
% 2.33/1.50 => ( P3 @ P ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_163_prod__cases,axiom,
% 2.33/1.50 ! [P3: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
% 2.33/1.50 ( ! [A5: nat,B5: nat] : ( P3 @ ( product_Pair_nat_nat @ A5 @ B5 ) )
% 2.33/1.50 => ( P3 @ P ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_164_Pair__inject,axiom,
% 2.33/1.50 ! [A: nat > nat > nat,B: produc1695820582at_nat,A4: nat > nat > nat,B4: produc1695820582at_nat] :
% 2.33/1.50 ( ( ( produc1682925677at_nat @ A @ B )
% 2.33/1.50 = ( produc1682925677at_nat @ A4 @ B4 ) )
% 2.33/1.50 => ~ ( ( A = A4 )
% 2.33/1.50 => ( B != B4 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_165_Pair__inject,axiom,
% 2.33/1.50 ! [A: nat,B: product_prod_nat_nat,A4: nat,B4: product_prod_nat_nat] :
% 2.33/1.50 ( ( ( produc1933845336at_nat @ A @ B )
% 2.33/1.50 = ( produc1933845336at_nat @ A4 @ B4 ) )
% 2.33/1.50 => ~ ( ( A = A4 )
% 2.33/1.50 => ( B != B4 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_166_Pair__inject,axiom,
% 2.33/1.50 ! [A: nat,B: nat,A4: nat,B4: nat] :
% 2.33/1.50 ( ( ( product_Pair_nat_nat @ A @ B )
% 2.33/1.50 = ( product_Pair_nat_nat @ A4 @ B4 ) )
% 2.33/1.50 => ~ ( ( A = A4 )
% 2.33/1.50 => ( B != B4 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_167_prod__cases3,axiom,
% 2.33/1.50 ! [Y: produc2082277813at_nat] :
% 2.33/1.50 ~ ! [A5: nat > nat > nat,B5: nat,C3: product_prod_nat_nat] :
% 2.33/1.50 ( Y
% 2.33/1.50 != ( produc1682925677at_nat @ A5 @ ( produc1933845336at_nat @ B5 @ C3 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_168_prod__cases3,axiom,
% 2.33/1.50 ! [Y: produc1695820582at_nat] :
% 2.33/1.50 ~ ! [A5: nat,B5: nat,C3: nat] :
% 2.33/1.50 ( Y
% 2.33/1.50 != ( produc1933845336at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ C3 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_169_prod__cases4,axiom,
% 2.33/1.50 ! [Y: produc2082277813at_nat] :
% 2.33/1.50 ~ ! [A5: nat > nat > nat,B5: nat,C3: nat,D6: nat] :
% 2.33/1.50 ( Y
% 2.33/1.50 != ( produc1682925677at_nat @ A5 @ ( produc1933845336at_nat @ B5 @ ( product_Pair_nat_nat @ C3 @ D6 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_170_prod__induct3,axiom,
% 2.33/1.50 ! [P3: produc2082277813at_nat > $o,X: produc2082277813at_nat] :
% 2.33/1.50 ( ! [A5: nat > nat > nat,B5: nat,C3: product_prod_nat_nat] : ( P3 @ ( produc1682925677at_nat @ A5 @ ( produc1933845336at_nat @ B5 @ C3 ) ) )
% 2.33/1.50 => ( P3 @ X ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_171_prod__induct3,axiom,
% 2.33/1.50 ! [P3: produc1695820582at_nat > $o,X: produc1695820582at_nat] :
% 2.33/1.50 ( ! [A5: nat,B5: nat,C3: nat] : ( P3 @ ( produc1933845336at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ C3 ) ) )
% 2.33/1.50 => ( P3 @ X ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_172_prod__induct4,axiom,
% 2.33/1.50 ! [P3: produc2082277813at_nat > $o,X: produc2082277813at_nat] :
% 2.33/1.50 ( ! [A5: nat > nat > nat,B5: nat,C3: nat,D6: nat] : ( P3 @ ( produc1682925677at_nat @ A5 @ ( produc1933845336at_nat @ B5 @ ( product_Pair_nat_nat @ C3 @ D6 ) ) ) )
% 2.33/1.50 => ( P3 @ X ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_173_old_Oprod_Oexhaust,axiom,
% 2.33/1.50 ! [Y: produc2082277813at_nat] :
% 2.33/1.50 ~ ! [A5: nat > nat > nat,B5: produc1695820582at_nat] :
% 2.33/1.50 ( Y
% 2.33/1.50 != ( produc1682925677at_nat @ A5 @ B5 ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_174_old_Oprod_Oexhaust,axiom,
% 2.33/1.50 ! [Y: produc1695820582at_nat] :
% 2.33/1.50 ~ ! [A5: nat,B5: product_prod_nat_nat] :
% 2.33/1.50 ( Y
% 2.33/1.50 != ( produc1933845336at_nat @ A5 @ B5 ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_175_old_Oprod_Oexhaust,axiom,
% 2.33/1.50 ! [Y: product_prod_nat_nat] :
% 2.33/1.50 ~ ! [A5: nat,B5: nat] :
% 2.33/1.50 ( Y
% 2.33/1.50 != ( product_Pair_nat_nat @ A5 @ B5 ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_176_old_Oprod_Oinducts,axiom,
% 2.33/1.50 ! [P3: produc2082277813at_nat > $o,Prod: produc2082277813at_nat] :
% 2.33/1.50 ( ! [A5: nat > nat > nat,B5: produc1695820582at_nat] : ( P3 @ ( produc1682925677at_nat @ A5 @ B5 ) )
% 2.33/1.50 => ( P3 @ Prod ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_177_old_Oprod_Oinducts,axiom,
% 2.33/1.50 ! [P3: produc1695820582at_nat > $o,Prod: produc1695820582at_nat] :
% 2.33/1.50 ( ! [A5: nat,B5: product_prod_nat_nat] : ( P3 @ ( produc1933845336at_nat @ A5 @ B5 ) )
% 2.33/1.50 => ( P3 @ Prod ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_178_old_Oprod_Oinducts,axiom,
% 2.33/1.50 ! [P3: product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
% 2.33/1.50 ( ! [A5: nat,B5: nat] : ( P3 @ ( product_Pair_nat_nat @ A5 @ B5 ) )
% 2.33/1.50 => ( P3 @ Prod ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_179_size__neq__size__imp__neq,axiom,
% 2.33/1.50 ! [X: list_P559422087at_nat,Y: list_P559422087at_nat] :
% 2.33/1.50 ( ( ( size_s1990949619at_nat @ X )
% 2.33/1.50 != ( size_s1990949619at_nat @ Y ) )
% 2.33/1.50 => ( X != Y ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_180_le__refl,axiom,
% 2.33/1.50 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_181_le__trans,axiom,
% 2.33/1.50 ! [I: nat,J: nat,K: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ I @ J )
% 2.33/1.50 => ( ( ord_less_eq_nat @ J @ K )
% 2.33/1.50 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_182_eq__imp__le,axiom,
% 2.33/1.50 ! [M: nat,N: nat] :
% 2.33/1.50 ( ( M = N )
% 2.33/1.50 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_183_le__antisym,axiom,
% 2.33/1.50 ! [M: nat,N: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ M @ N )
% 2.33/1.50 => ( ( ord_less_eq_nat @ N @ M )
% 2.33/1.50 => ( M = N ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_184_nat__le__linear,axiom,
% 2.33/1.50 ! [M: nat,N: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ M @ N )
% 2.33/1.50 | ( ord_less_eq_nat @ N @ M ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_185_Nat_Oex__has__greatest__nat,axiom,
% 2.33/1.50 ! [P3: nat > $o,K: nat,B: nat] :
% 2.33/1.50 ( ( P3 @ K )
% 2.33/1.50 => ( ! [Y3: nat] :
% 2.33/1.50 ( ( P3 @ Y3 )
% 2.33/1.50 => ( ord_less_eq_nat @ Y3 @ B ) )
% 2.33/1.50 => ? [X3: nat] :
% 2.33/1.50 ( ( P3 @ X3 )
% 2.33/1.50 & ! [Y4: nat] :
% 2.33/1.50 ( ( P3 @ Y4 )
% 2.33/1.50 => ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_186_Graph_OisShortestPath__alt,axiom,
% 2.33/1.50 ( isShor1936442771pacity
% 2.33/1.50 = ( ^ [C2: product_prod_nat_nat > capacity,U3: nat,P2: list_P559422087at_nat,V3: nat] :
% 2.33/1.50 ( ( isSimp1359852763pacity @ C2 @ U3 @ P2 @ V3 )
% 2.33/1.50 & ( ( size_s1990949619at_nat @ P2 )
% 2.33/1.50 = ( min_dist_capacity @ C2 @ U3 @ V3 ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_187_nat__le__iff__add,axiom,
% 2.33/1.50 ( ord_less_eq_nat
% 2.33/1.50 = ( ^ [M2: nat,N2: nat] :
% 2.33/1.50 ? [K2: nat] :
% 2.33/1.50 ( N2
% 2.33/1.50 = ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_188_trans__le__add2,axiom,
% 2.33/1.50 ! [I: nat,J: nat,M: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ I @ J )
% 2.33/1.50 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_189_trans__le__add1,axiom,
% 2.33/1.50 ! [I: nat,J: nat,M: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ I @ J )
% 2.33/1.50 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_190_add__le__mono1,axiom,
% 2.33/1.50 ! [I: nat,J: nat,K: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ I @ J )
% 2.33/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_191_add__le__mono,axiom,
% 2.33/1.50 ! [I: nat,J: nat,K: nat,L: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ I @ J )
% 2.33/1.50 => ( ( ord_less_eq_nat @ K @ L )
% 2.33/1.50 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_192_le__Suc__ex,axiom,
% 2.33/1.50 ! [K: nat,L: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ K @ L )
% 2.33/1.50 => ? [N3: nat] :
% 2.33/1.50 ( L
% 2.33/1.50 = ( plus_plus_nat @ K @ N3 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_193_add__leD2,axiom,
% 2.33/1.50 ! [M: nat,K: nat,N: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 2.33/1.50 => ( ord_less_eq_nat @ K @ N ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_194_add__leD1,axiom,
% 2.33/1.50 ! [M: nat,K: nat,N: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 2.33/1.50 => ( ord_less_eq_nat @ M @ N ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_195_le__add2,axiom,
% 2.33/1.50 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_196_le__add1,axiom,
% 2.33/1.50 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_197_add__leE,axiom,
% 2.33/1.50 ! [M: nat,K: nat,N: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 2.33/1.50 => ~ ( ( ord_less_eq_nat @ M @ N )
% 2.33/1.50 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_198_isSPath__nt__parallel,axiom,
% 2.33/1.50 ! [S: nat,P: list_P559422087at_nat,T: nat,E: product_prod_nat_nat] :
% 2.33/1.50 ( ( isSimp1359852763pacity @ c @ S @ P @ T )
% 2.33/1.50 => ( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ~ ( member701585322at_nat @ ( product_swap_nat_nat @ E ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_199_discrete,axiom,
% 2.33/1.50 ( ord_less_nat
% 2.33/1.50 = ( ^ [A2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_200_swap__swap,axiom,
% 2.33/1.50 ! [P: product_prod_nat_nat] :
% 2.33/1.50 ( ( product_swap_nat_nat @ ( product_swap_nat_nat @ P ) )
% 2.33/1.50 = P ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_201_swap__simp,axiom,
% 2.33/1.50 ! [X: produc1695820582at_nat,Y: nat > nat > nat] :
% 2.33/1.50 ( ( produc411855757at_nat @ ( produc365964781at_nat @ X @ Y ) )
% 2.33/1.50 = ( produc1682925677at_nat @ Y @ X ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_202_swap__simp,axiom,
% 2.33/1.50 ! [X: product_prod_nat_nat,Y: nat] :
% 2.33/1.50 ( ( produc2019146714at_nat @ ( produc947540346at_nat @ X @ Y ) )
% 2.33/1.50 = ( produc1933845336at_nat @ Y @ X ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_203_swap__simp,axiom,
% 2.33/1.50 ! [X: nat > nat > nat,Y: produc1695820582at_nat] :
% 2.33/1.50 ( ( produc1728816653at_nat @ ( produc1682925677at_nat @ X @ Y ) )
% 2.33/1.50 = ( produc365964781at_nat @ Y @ X ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_204_swap__simp,axiom,
% 2.33/1.50 ! [X: nat,Y: product_prod_nat_nat] :
% 2.33/1.50 ( ( produc857968056at_nat @ ( produc1933845336at_nat @ X @ Y ) )
% 2.33/1.50 = ( produc947540346at_nat @ Y @ X ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_205_swap__simp,axiom,
% 2.33/1.50 ! [X: nat,Y: nat] :
% 2.33/1.50 ( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
% 2.33/1.50 = ( product_Pair_nat_nat @ Y @ X ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_206_bounded__Max__nat,axiom,
% 2.33/1.50 ! [P3: nat > $o,X: nat,M5: nat] :
% 2.33/1.50 ( ( P3 @ X )
% 2.33/1.50 => ( ! [X3: nat] :
% 2.33/1.50 ( ( P3 @ X3 )
% 2.33/1.50 => ( ord_less_eq_nat @ X3 @ M5 ) )
% 2.33/1.50 => ~ ! [M4: nat] :
% 2.33/1.50 ( ( P3 @ M4 )
% 2.33/1.50 => ~ ! [X4: nat] :
% 2.33/1.50 ( ( P3 @ X4 )
% 2.33/1.50 => ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_207_fold__atLeastAtMost__nat_Ocases,axiom,
% 2.33/1.50 ! [X: produc1864880952at_nat] :
% 2.33/1.50 ~ ! [F2: nat > product_prod_nat_nat > product_prod_nat_nat,A5: nat,B5: nat,Acc: product_prod_nat_nat] :
% 2.33/1.50 ( X
% 2.33/1.50 != ( produc696256106at_nat @ F2 @ ( produc869658639at_nat @ A5 @ ( produc1933845336at_nat @ B5 @ Acc ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_208_fold__atLeastAtMost__nat_Ocases,axiom,
% 2.33/1.50 ! [X: produc2082277813at_nat] :
% 2.33/1.50 ~ ! [F2: nat > nat > nat,A5: nat,B5: nat,Acc: nat] :
% 2.33/1.50 ( X
% 2.33/1.50 != ( produc1682925677at_nat @ F2 @ ( produc1933845336at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ Acc ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_209_Graph_OisSPath__nt__parallel,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,E: product_prod_nat_nat] :
% 2.33/1.50 ( ( isSimp1359852763pacity @ C @ S @ P @ T )
% 2.33/1.50 => ( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
% 2.33/1.50 => ~ ( member701585322at_nat @ ( product_swap_nat_nat @ E ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_210_less__add__one,axiom,
% 2.33/1.50 ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_211_add__mono1,axiom,
% 2.33/1.50 ! [A: nat,B: nat] :
% 2.33/1.50 ( ( ord_less_nat @ A @ B )
% 2.33/1.50 => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_212_isPath_Osimps_I1_J,axiom,
% 2.33/1.50 ! [U: nat,V: nat] :
% 2.33/1.50 ( ( isPath_capacity @ c @ U @ nil_Pr1308055047at_nat @ V )
% 2.33/1.50 = ( U = V ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_213_order__refl,axiom,
% 2.33/1.50 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_214_simplePath__empty__conv,axiom,
% 2.33/1.50 ! [S: nat,T: nat] :
% 2.33/1.50 ( ( isSimp1359852763pacity @ c @ S @ nil_Pr1308055047at_nat @ T )
% 2.33/1.50 = ( S = T ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_215_simplePath__same__conv,axiom,
% 2.33/1.50 ! [S: nat,P: list_P559422087at_nat] :
% 2.33/1.50 ( ( isSimp1359852763pacity @ c @ S @ P @ S )
% 2.33/1.50 = ( P = nil_Pr1308055047at_nat ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_216_Graph_OsimplePath__empty__conv,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,S: nat,T: nat] :
% 2.33/1.50 ( ( isSimp1359852763pacity @ C @ S @ nil_Pr1308055047at_nat @ T )
% 2.33/1.50 = ( S = T ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_217_Graph_OsimplePath__same__conv,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat] :
% 2.33/1.50 ( ( isSimp1359852763pacity @ C @ S @ P @ S )
% 2.33/1.50 = ( P = nil_Pr1308055047at_nat ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_218_Graph_OisPath_Osimps_I1_J,axiom,
% 2.33/1.50 ! [C: product_prod_nat_nat > capacity,U: nat,V: nat] :
% 2.33/1.50 ( ( isPath_capacity @ C @ U @ nil_Pr1308055047at_nat @ V )
% 2.33/1.50 = ( U = V ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_219_dual__order_Oantisym,axiom,
% 2.33/1.50 ! [B: nat,A: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ B @ A )
% 2.33/1.50 => ( ( ord_less_eq_nat @ A @ B )
% 2.33/1.50 => ( A = B ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_220_dual__order_Oeq__iff,axiom,
% 2.33/1.50 ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
% 2.33/1.50 = ( ^ [A2: nat,B2: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ B2 @ A2 )
% 2.33/1.50 & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_221_dual__order_Otrans,axiom,
% 2.33/1.50 ! [B: nat,A: nat,C: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ B @ A )
% 2.33/1.50 => ( ( ord_less_eq_nat @ C @ B )
% 2.33/1.50 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_222_linorder__wlog,axiom,
% 2.33/1.50 ! [P3: nat > nat > $o,A: nat,B: nat] :
% 2.33/1.50 ( ! [A5: nat,B5: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ A5 @ B5 )
% 2.33/1.50 => ( P3 @ A5 @ B5 ) )
% 2.33/1.50 => ( ! [A5: nat,B5: nat] :
% 2.33/1.50 ( ( P3 @ B5 @ A5 )
% 2.33/1.50 => ( P3 @ A5 @ B5 ) )
% 2.33/1.50 => ( P3 @ A @ B ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_223_dual__order_Orefl,axiom,
% 2.33/1.50 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_224_order__trans,axiom,
% 2.33/1.50 ! [X: nat,Y: nat,Z2: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ X @ Y )
% 2.33/1.50 => ( ( ord_less_eq_nat @ Y @ Z2 )
% 2.33/1.50 => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_225_order__class_Oorder_Oantisym,axiom,
% 2.33/1.50 ! [A: nat,B: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.33/1.50 => ( ( ord_less_eq_nat @ B @ A )
% 2.33/1.50 => ( A = B ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_226_ord__le__eq__trans,axiom,
% 2.33/1.50 ! [A: nat,B: nat,C: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.33/1.50 => ( ( B = C )
% 2.33/1.50 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_227_ord__eq__le__trans,axiom,
% 2.33/1.50 ! [A: nat,B: nat,C: nat] :
% 2.33/1.50 ( ( A = B )
% 2.33/1.50 => ( ( ord_less_eq_nat @ B @ C )
% 2.33/1.50 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_228_order__class_Oorder_Oeq__iff,axiom,
% 2.33/1.50 ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
% 2.33/1.50 = ( ^ [A2: nat,B2: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ A2 @ B2 )
% 2.33/1.50 & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_229_antisym__conv,axiom,
% 2.33/1.50 ! [Y: nat,X: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ Y @ X )
% 2.33/1.50 => ( ( ord_less_eq_nat @ X @ Y )
% 2.33/1.50 = ( X = Y ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_230_le__cases3,axiom,
% 2.33/1.50 ! [X: nat,Y: nat,Z2: nat] :
% 2.33/1.50 ( ( ( ord_less_eq_nat @ X @ Y )
% 2.33/1.50 => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
% 2.33/1.50 => ( ( ( ord_less_eq_nat @ Y @ X )
% 2.33/1.50 => ~ ( ord_less_eq_nat @ X @ Z2 ) )
% 2.33/1.50 => ( ( ( ord_less_eq_nat @ X @ Z2 )
% 2.33/1.50 => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
% 2.33/1.50 => ( ( ( ord_less_eq_nat @ Z2 @ Y )
% 2.33/1.50 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 2.33/1.50 => ( ( ( ord_less_eq_nat @ Y @ Z2 )
% 2.33/1.50 => ~ ( ord_less_eq_nat @ Z2 @ X ) )
% 2.33/1.50 => ~ ( ( ord_less_eq_nat @ Z2 @ X )
% 2.33/1.50 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_231_order_Otrans,axiom,
% 2.33/1.50 ! [A: nat,B: nat,C: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.33/1.50 => ( ( ord_less_eq_nat @ B @ C )
% 2.33/1.50 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_232_le__cases,axiom,
% 2.33/1.50 ! [X: nat,Y: nat] :
% 2.33/1.50 ( ~ ( ord_less_eq_nat @ X @ Y )
% 2.33/1.50 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_233_eq__refl,axiom,
% 2.33/1.50 ! [X: nat,Y: nat] :
% 2.33/1.50 ( ( X = Y )
% 2.33/1.50 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_234_linear,axiom,
% 2.33/1.50 ! [X: nat,Y: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ X @ Y )
% 2.33/1.50 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_235_antisym,axiom,
% 2.33/1.50 ! [X: nat,Y: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ X @ Y )
% 2.33/1.50 => ( ( ord_less_eq_nat @ Y @ X )
% 2.33/1.50 => ( X = Y ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_236_eq__iff,axiom,
% 2.33/1.50 ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
% 2.33/1.50 = ( ^ [X2: nat,Y6: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ X2 @ Y6 )
% 2.33/1.50 & ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_237_ord__le__eq__subst,axiom,
% 2.33/1.50 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.33/1.50 => ( ( ( F @ B )
% 2.33/1.50 = C )
% 2.33/1.50 => ( ! [X3: nat,Y3: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.33/1.50 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.33/1.50 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_238_ord__eq__le__subst,axiom,
% 2.33/1.50 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.33/1.50 ( ( A
% 2.33/1.50 = ( F @ B ) )
% 2.33/1.50 => ( ( ord_less_eq_nat @ B @ C )
% 2.33/1.50 => ( ! [X3: nat,Y3: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.33/1.50 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.33/1.50 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_239_order__subst2,axiom,
% 2.33/1.50 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ A @ B )
% 2.33/1.50 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 2.33/1.50 => ( ! [X3: nat,Y3: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.33/1.50 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.33/1.50 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_240_order__subst1,axiom,
% 2.33/1.50 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 2.33/1.50 => ( ( ord_less_eq_nat @ B @ C )
% 2.33/1.50 => ( ! [X3: nat,Y3: nat] :
% 2.33/1.50 ( ( ord_less_eq_nat @ X3 @ Y3 )
% 2.33/1.50 => ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
% 2.33/1.50 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_241_dual__order_Ostrict__implies__not__eq,axiom,
% 2.33/1.50 ! [B: nat,A: nat] :
% 2.33/1.50 ( ( ord_less_nat @ B @ A )
% 2.33/1.50 => ( A != B ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_242_order_Ostrict__implies__not__eq,axiom,
% 2.33/1.50 ! [A: nat,B: nat] :
% 2.33/1.50 ( ( ord_less_nat @ A @ B )
% 2.33/1.50 => ( A != B ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_243_not__less__iff__gr__or__eq,axiom,
% 2.33/1.50 ! [X: nat,Y: nat] :
% 2.33/1.50 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 2.33/1.50 = ( ( ord_less_nat @ Y @ X )
% 2.33/1.50 | ( X = Y ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_244_dual__order_Ostrict__trans,axiom,
% 2.33/1.50 ! [B: nat,A: nat,C: nat] :
% 2.33/1.50 ( ( ord_less_nat @ B @ A )
% 2.33/1.50 => ( ( ord_less_nat @ C @ B )
% 2.33/1.50 => ( ord_less_nat @ C @ A ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_245_linorder__less__wlog,axiom,
% 2.33/1.50 ! [P3: nat > nat > $o,A: nat,B: nat] :
% 2.33/1.50 ( ! [A5: nat,B5: nat] :
% 2.33/1.50 ( ( ord_less_nat @ A5 @ B5 )
% 2.33/1.50 => ( P3 @ A5 @ B5 ) )
% 2.33/1.50 => ( ! [A5: nat] : ( P3 @ A5 @ A5 )
% 2.33/1.50 => ( ! [A5: nat,B5: nat] :
% 2.33/1.50 ( ( P3 @ B5 @ A5 )
% 2.33/1.50 => ( P3 @ A5 @ B5 ) )
% 2.33/1.50 => ( P3 @ A @ B ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_246_exists__least__iff,axiom,
% 2.33/1.50 ( ( ^ [P7: nat > $o] :
% 2.33/1.50 ? [X5: nat] : ( P7 @ X5 ) )
% 2.33/1.50 = ( ^ [P8: nat > $o] :
% 2.33/1.50 ? [N2: nat] :
% 2.33/1.50 ( ( P8 @ N2 )
% 2.33/1.50 & ! [M2: nat] :
% 2.33/1.50 ( ( ord_less_nat @ M2 @ N2 )
% 2.33/1.50 => ~ ( P8 @ M2 ) ) ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_247_less__imp__not__less,axiom,
% 2.33/1.50 ! [X: nat,Y: nat] :
% 2.33/1.50 ( ( ord_less_nat @ X @ Y )
% 2.33/1.50 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_248_order_Ostrict__trans,axiom,
% 2.33/1.50 ! [A: nat,B: nat,C: nat] :
% 2.33/1.50 ( ( ord_less_nat @ A @ B )
% 2.33/1.50 => ( ( ord_less_nat @ B @ C )
% 2.33/1.50 => ( ord_less_nat @ A @ C ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_249_dual__order_Oirrefl,axiom,
% 2.33/1.50 ! [A: nat] :
% 2.33/1.50 ~ ( ord_less_nat @ A @ A ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_250_linorder__cases,axiom,
% 2.33/1.50 ! [X: nat,Y: nat] :
% 2.33/1.50 ( ~ ( ord_less_nat @ X @ Y )
% 2.33/1.50 => ( ( X != Y )
% 2.33/1.50 => ( ord_less_nat @ Y @ X ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_251_less__imp__triv,axiom,
% 2.33/1.50 ! [X: nat,Y: nat,P3: $o] :
% 2.33/1.50 ( ( ord_less_nat @ X @ Y )
% 2.33/1.50 => ( ( ord_less_nat @ Y @ X )
% 2.33/1.50 => P3 ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_252_less__imp__not__eq2,axiom,
% 2.33/1.50 ! [X: nat,Y: nat] :
% 2.33/1.50 ( ( ord_less_nat @ X @ Y )
% 2.33/1.50 => ( Y != X ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(fact_253_antisym__conv3,axiom,
% 2.33/1.50 ! [Y: nat,X: nat] :
% 2.33/1.50 ( ~ ( ord_less_nat @ Y @ X )
% 2.33/1.50 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 2.33/1.50 = ( X = Y ) ) ) ).
% 2.33/1.50
% 2.33/1.50 thf(conj_0,conjecture,
% 2.33/1.50 ( ( min_dist_capacity @ c @ u @ t )
% 2.33/1.50 = ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) ).
% 2.33/1.50 ------------------------
% 2.33/1.53 % (23016)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_22776 for (2999ds/4Mi)
% 2.33/1.53 % (23018)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22776 for (2999ds/2Mi)
% 2.33/1.53 % (23020)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_22776 for (2999ds/275Mi)
% 2.33/1.53 % (23019)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_22776 for (2999ds/2Mi)
% 2.33/1.53 % (23018)Instruction limit reached!
% 2.33/1.53 % (23018)------------------------------
% 2.33/1.53 % (23018)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.53 % (23018)Termination reason: Unknown
% 2.33/1.53 % (23018)Termination phase: shuffling
% 2.33/1.53
% 2.33/1.53 % (23018)Memory used [KB]: 1279
% 2.33/1.53 % (23018)Time elapsed: 0.002 s
% 2.33/1.53 % (23018)Instructions burned: 2 (million)
% 2.33/1.53 % (23018)------------------------------
% 2.33/1.53 % (23018)------------------------------
% 2.33/1.53 % (23019)Instruction limit reached!
% 2.33/1.53 % (23019)------------------------------
% 2.33/1.53 % (23019)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.53 % (23019)Termination reason: Unknown
% 2.33/1.53 % (23019)Termination phase: shuffling
% 2.33/1.53
% 2.33/1.53 % (23019)Memory used [KB]: 1279
% 2.33/1.53 % (23019)Time elapsed: 0.002 s
% 2.33/1.53 % (23019)Instructions burned: 2 (million)
% 2.33/1.53 % (23019)------------------------------
% 2.33/1.53 % (23019)------------------------------
% 2.33/1.53 % (23016)Instruction limit reached!
% 2.33/1.53 % (23016)------------------------------
% 2.33/1.53 % (23016)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.53 % (23016)Termination reason: Unknown
% 2.33/1.53 % (23016)Termination phase: shuffling
% 2.33/1.53
% 2.33/1.53 % (23016)Memory used [KB]: 1407
% 2.33/1.53 % (23016)Time elapsed: 0.003 s
% 2.33/1.53 % (23016)Instructions burned: 5 (million)
% 2.33/1.53 % (23016)------------------------------
% 2.33/1.53 % (23016)------------------------------
% 2.33/1.54 % (23015)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22776 for (2999ds/183Mi)
% 2.33/1.54 % (23017)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_22776 for (2999ds/27Mi)
% 2.33/1.54 % (23021)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22776 for (2999ds/18Mi)
% 2.33/1.54 % (23025)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22776 for (2999ds/3Mi)
% 2.33/1.54 % (23026)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_22776 for (2999ds/37Mi)
% 2.33/1.54 % (23027)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_22776 for (2999ds/15Mi)
% 2.33/1.54 % (23025)Instruction limit reached!
% 2.33/1.54 % (23025)------------------------------
% 2.33/1.54 % (23025)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.54 % (23025)Termination reason: Unknown
% 2.33/1.54 % (23025)Termination phase: shuffling
% 2.33/1.54
% 2.33/1.54 % (23025)Memory used [KB]: 1407
% 2.33/1.54 % (23025)Time elapsed: 0.003 s
% 2.33/1.54 % (23025)Instructions burned: 5 (million)
% 2.33/1.54 % (23025)------------------------------
% 2.33/1.54 % (23025)------------------------------
% 2.33/1.54 % (23027)Instruction limit reached!
% 2.33/1.54 % (23027)------------------------------
% 2.33/1.54 % (23027)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.54 % (23027)Termination reason: Unknown
% 2.33/1.54 % (23027)Termination phase: shuffling
% 2.33/1.54
% 2.33/1.54 % (23027)Memory used [KB]: 1535
% 2.33/1.54 % (23027)Time elapsed: 0.006 s
% 2.33/1.54 % (23027)Instructions burned: 15 (million)
% 2.33/1.54 % (23027)------------------------------
% 2.33/1.54 % (23027)------------------------------
% 2.33/1.55 % (23021)Instruction limit reached!
% 2.33/1.55 % (23021)------------------------------
% 2.33/1.55 % (23021)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.55 % (23021)Termination reason: Unknown
% 2.33/1.55 % (23021)Termination phase: shuffling
% 2.33/1.55
% 2.33/1.55 % (23021)Memory used [KB]: 1535
% 2.33/1.55 % (23021)Time elapsed: 0.013 s
% 2.33/1.55 % (23021)Instructions burned: 18 (million)
% 2.33/1.55 % (23021)------------------------------
% 2.33/1.55 % (23021)------------------------------
% 2.33/1.55 % (23026)Instruction limit reached!
% 2.33/1.55 % (23026)------------------------------
% 2.33/1.55 % (23026)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.55 % (23026)Termination reason: Unknown
% 2.33/1.55 % (23026)Termination phase: shuffling
% 2.33/1.55
% 2.33/1.55 % (23026)Memory used [KB]: 2174
% 2.33/1.55 % (23026)Time elapsed: 0.012 s
% 2.33/1.55 % (23026)Instructions burned: 40 (million)
% 2.33/1.55 % (23026)------------------------------
% 2.33/1.55 % (23026)------------------------------
% 2.33/1.55 % (23032)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_22776 for (2999ds/3Mi)
% 2.33/1.55 % (23032)Instruction limit reached!
% 2.33/1.55 % (23032)------------------------------
% 2.33/1.55 % (23032)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.55 % (23032)Termination reason: Unknown
% 2.33/1.55 % (23032)Termination phase: shuffling
% 2.33/1.55
% 2.33/1.55 % (23032)Memory used [KB]: 1279
% 2.33/1.55 % (23032)Time elapsed: 0.002 s
% 2.33/1.55 % (23032)Instructions burned: 3 (million)
% 2.33/1.55 % (23032)------------------------------
% 2.33/1.55 % (23032)------------------------------
% 2.33/1.55 % (23017)Instruction limit reached!
% 2.33/1.55 % (23017)------------------------------
% 2.33/1.55 % (23017)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.55 % (23017)Termination reason: Unknown
% 2.33/1.55 % (23017)Termination phase: Property scanning
% 2.33/1.55
% 2.33/1.55 % (23017)Memory used [KB]: 1663
% 2.33/1.55 % (23017)Time elapsed: 0.018 s
% 2.33/1.55 % (23017)Instructions burned: 27 (million)
% 2.33/1.55 % (23017)------------------------------
% 2.33/1.55 % (23017)------------------------------
% 2.33/1.55 % (23020)Refutation not found, incomplete strategy
% 2.33/1.55 % (23020)------------------------------
% 2.33/1.55 % (23020)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.55 % (23020)Termination reason: Refutation not found, incomplete strategy
% 2.33/1.55
% 2.33/1.55
% 2.33/1.55 % (23020)Memory used [KB]: 6652
% 2.33/1.55 % (23020)Time elapsed: 0.026 s
% 2.33/1.55 % (23020)Instructions burned: 84 (million)
% 2.33/1.55 % (23020)------------------------------
% 2.33/1.55 % (23020)------------------------------
% 2.33/1.55 % (23034)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_22776 for (2999ds/1041Mi)
% 2.33/1.56 % (23037)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_22776 for (2999ds/16Mi)
% 2.33/1.56 % (23039)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_22776 for (2999ds/3Mi)
% 2.33/1.56 % (23039)Instruction limit reached!
% 2.33/1.56 % (23039)------------------------------
% 2.33/1.56 % (23039)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.56 % (23039)Termination reason: Unknown
% 2.33/1.56 % (23039)Termination phase: shuffling
% 2.33/1.56
% 2.33/1.56 % (23039)Memory used [KB]: 1407
% 2.33/1.56 % (23039)Time elapsed: 0.003 s
% 2.33/1.56 % (23039)Instructions burned: 6 (million)
% 2.33/1.56 % (23039)------------------------------
% 2.33/1.56 % (23039)------------------------------
% 2.33/1.56 % (23042)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_22776 for (2999ds/7Mi)
% 2.33/1.56 % (23037)Instruction limit reached!
% 2.33/1.56 % (23037)------------------------------
% 2.33/1.56 % (23037)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.56 % (23037)Termination reason: Unknown
% 2.33/1.56 % (23037)Termination phase: shuffling
% 2.33/1.56
% 2.33/1.56 % (23037)Memory used [KB]: 1535
% 2.33/1.56 % (23037)Time elapsed: 0.006 s
% 2.33/1.56 % (23037)Instructions burned: 16 (million)
% 2.33/1.56 % (23037)------------------------------
% 2.33/1.56 % (23037)------------------------------
% 2.33/1.57 % (23035)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_22776 for (2999ds/7Mi)
% 2.33/1.57 % (23042)Instruction limit reached!
% 2.33/1.57 % (23042)------------------------------
% 2.33/1.57 % (23042)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.57 % (23042)Termination reason: Unknown
% 2.33/1.57 % (23042)Termination phase: shuffling
% 2.33/1.57
% 2.33/1.57 % (23042)Memory used [KB]: 1407
% 2.33/1.57 % (23042)Time elapsed: 0.004 s
% 2.33/1.57 % (23042)Instructions burned: 10 (million)
% 2.33/1.57 % (23042)------------------------------
% 2.33/1.57 % (23042)------------------------------
% 2.33/1.57 % (23035)Instruction limit reached!
% 2.33/1.57 % (23035)------------------------------
% 2.33/1.57 % (23035)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.57 % (23035)Termination reason: Unknown
% 2.33/1.57 % (23035)Termination phase: shuffling
% 2.33/1.57
% 2.33/1.57 % (23035)Memory used [KB]: 1407
% 2.33/1.57 % (23035)Time elapsed: 0.006 s
% 2.33/1.57 % (23035)Instructions burned: 7 (million)
% 2.33/1.57 % (23035)------------------------------
% 2.33/1.57 % (23035)------------------------------
% 2.33/1.57 % (23040)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_22776 for (2999ds/3Mi)
% 2.33/1.57 % (23040)Instruction limit reached!
% 2.33/1.57 % (23040)------------------------------
% 2.33/1.57 % (23040)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.57 % (23040)Termination reason: Unknown
% 2.33/1.57 % (23040)Termination phase: shuffling
% 2.33/1.57
% 2.33/1.57 % (23040)Memory used [KB]: 1279
% 2.33/1.57 % (23040)Time elapsed: 0.004 s
% 2.33/1.57 % (23040)Instructions burned: 3 (million)
% 2.33/1.57 % (23040)------------------------------
% 2.33/1.57 % (23040)------------------------------
% 2.33/1.57 % (23046)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_22776 for (2999ds/3Mi)
% 2.33/1.57 % (23046)Instruction limit reached!
% 2.33/1.57 % (23046)------------------------------
% 2.33/1.57 % (23046)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.57 % (23046)Termination reason: Unknown
% 2.33/1.57 % (23046)Termination phase: shuffling
% 2.33/1.57
% 2.33/1.57 % (23046)Memory used [KB]: 1279
% 2.33/1.57 % (23046)Time elapsed: 0.002 s
% 2.33/1.57 % (23046)Instructions burned: 3 (million)
% 2.33/1.57 % (23046)------------------------------
% 2.33/1.57 % (23046)------------------------------
% 2.33/1.57 % (23047)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_22776 for (2999ds/4Mi)
% 2.33/1.58 % (23047)Instruction limit reached!
% 2.33/1.58 % (23047)------------------------------
% 2.33/1.58 % (23047)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.58 % (23047)Termination reason: Unknown
% 2.33/1.58 % (23047)Termination phase: shuffling
% 2.33/1.58
% 2.33/1.58 % (23047)Memory used [KB]: 1407
% 2.33/1.58 % (23047)Time elapsed: 0.003 s
% 2.33/1.58 % (23047)Instructions burned: 6 (million)
% 2.33/1.58 % (23047)------------------------------
% 2.33/1.58 % (23047)------------------------------
% 2.33/1.58 % (23049)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_22776 for (2999ds/18Mi)
% 2.33/1.58 % (23049)Instruction limit reached!
% 2.33/1.58 % (23049)------------------------------
% 2.33/1.58 % (23049)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.58 % (23049)Termination reason: Unknown
% 2.33/1.58 % (23049)Termination phase: shuffling
% 2.33/1.58
% 2.33/1.58 % (23049)Memory used [KB]: 1535
% 2.33/1.58 % (23049)Time elapsed: 0.007 s
% 2.33/1.58 % (23049)Instructions burned: 19 (million)
% 2.33/1.58 % (23049)------------------------------
% 2.33/1.58 % (23049)------------------------------
% 2.33/1.58 % (23055)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_22776 for (2999ds/902Mi)
% 2.33/1.58 % (23058)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_22776 for (2999ds/21Mi)
% 2.33/1.59 % (23054)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_22776 for (2999ds/6Mi)
% 2.33/1.59 % (23052)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_22776 for (2999ds/710Mi)
% 2.33/1.59 % (23054)Instruction limit reached!
% 2.33/1.59 % (23054)------------------------------
% 2.33/1.59 % (23054)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.59 % (23054)Termination reason: Unknown
% 2.33/1.59 % (23054)Termination phase: shuffling
% 2.33/1.59
% 2.33/1.59 % (23054)Memory used [KB]: 1407
% 2.33/1.59 % (23054)Time elapsed: 0.005 s
% 2.33/1.59 % (23054)Instructions burned: 7 (million)
% 2.33/1.59 % (23054)------------------------------
% 2.33/1.59 % (23054)------------------------------
% 2.33/1.59 % (23061)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_22776 for (2999ds/5Mi)
% 2.33/1.59 % (23058)Instruction limit reached!
% 2.33/1.59 % (23058)------------------------------
% 2.33/1.59 % (23058)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.59 % (23058)Termination reason: Unknown
% 2.33/1.59 % (23058)Termination phase: shuffling
% 2.33/1.59
% 2.33/1.59 % (23058)Memory used [KB]: 1663
% 2.33/1.59 % (23058)Time elapsed: 0.008 s
% 2.33/1.59 % (23058)Instructions burned: 22 (million)
% 2.33/1.59 % (23058)------------------------------
% 2.33/1.59 % (23058)------------------------------
% 2.33/1.59 % (23061)Instruction limit reached!
% 2.33/1.59 % (23061)------------------------------
% 2.33/1.59 % (23061)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.33/1.59 % (23061)Termination reason: Unknown
% 2.33/1.59 % (23061)Termination phase: shuffling
% 2.33/1.59
% 2.33/1.59 % (23061)Memory used [KB]: 1407
% 2.33/1.59 % (23061)Time elapsed: 0.003 s
% 2.33/1.59 % (23061)Instructions burned: 7 (million)
% 2.33/1.59 % (23061)------------------------------
% 2.33/1.59 % (23061)------------------------------
% 2.36/1.60 % (23065)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_22776 for (2999ds/6Mi)
% 2.36/1.60 % (23067)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_22776 for (2999ds/377Mi)
% 2.36/1.60 % (23065)Instruction limit reached!
% 2.36/1.60 % (23065)------------------------------
% 2.36/1.60 % (23065)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.36/1.60 % (23065)Termination reason: Unknown
% 2.36/1.60 % (23065)Termination phase: shuffling
% 2.36/1.60
% 2.36/1.60 % (23065)Memory used [KB]: 1407
% 2.36/1.60 % (23065)Time elapsed: 0.004 s
% 2.36/1.60 % (23065)Instructions burned: 8 (million)
% 2.36/1.60 % (23065)------------------------------
% 2.36/1.60 % (23065)------------------------------
% 2.36/1.60 % (23069)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_22776 for (2999ds/779Mi)
% 2.36/1.61 % (23055)Refutation not found, incomplete strategy
% 2.36/1.61 % (23055)------------------------------
% 2.36/1.61 % (23055)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.36/1.61 % (23055)Termination reason: Refutation not found, incomplete strategy
% 2.36/1.61
% 2.36/1.61
% 2.36/1.61 % (23055)Memory used [KB]: 6780
% 2.36/1.61 % (23055)Time elapsed: 0.029 s
% 2.36/1.61 % (23055)Instructions burned: 92 (million)
% 2.36/1.61 % (23055)------------------------------
% 2.36/1.61 % (23055)------------------------------
% 2.36/1.61 % (23071)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_22776 for (2999ds/19Mi)
% 2.36/1.62 % (23071)Instruction limit reached!
% 2.36/1.62 % (23071)------------------------------
% 2.36/1.62 % (23071)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.36/1.62 % (23071)Termination reason: Unknown
% 2.36/1.62 % (23071)Termination phase: shuffling
% 2.36/1.62
% 2.36/1.62 % (23071)Memory used [KB]: 1663
% 2.36/1.62 % (23071)Time elapsed: 0.007 s
% 2.36/1.62 % (23071)Instructions burned: 21 (million)
% 2.36/1.62 % (23071)------------------------------
% 2.36/1.62 % (23071)------------------------------
% 2.36/1.62 % (23052)First to succeed.
% 2.36/1.62 % (23075)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_22776 for (2999ds/879Mi)
% 2.36/1.62 % (23052)Refutation found. Thanks to Tanya!
% 2.36/1.62 % SZS status Theorem for DTF2THF_22776
% 2.36/1.62 % SZS output start Proof for DTF2THF_22776
% 2.36/1.62 thf(type_def_5, type, product_prod_nat_nat: $tType).
% 2.36/1.62 thf(type_def_6, type, capacity: $tType).
% 2.36/1.62 thf(type_def_8, type, nat: $tType).
% 2.36/1.62 thf(type_def_9, type, list_P559422087at_nat: $tType).
% 2.36/1.62 thf(type_def_10, type, set_Pr1986765409at_nat: $tType).
% 2.36/1.62 thf(type_def_11, type, produc1695820582at_nat: $tType).
% 2.36/1.62 thf(type_def_12, type, produc2082277813at_nat: $tType).
% 2.36/1.62 thf(type_def_13, type, produc1190591575at_nat: $tType).
% 2.36/1.62 thf(type_def_14, type, produc1864880952at_nat: $tType).
% 2.36/1.62 thf(type_def_15, type, produc1271302400at_nat: $tType).
% 2.36/1.62 thf(type_def_16, type, produc787001653at_nat: $tType).
% 2.36/1.62 thf(func_def_0, type, produc1864880952at_nat: $tType).
% 2.36/1.62 thf(func_def_1, type, produc787001653at_nat: $tType).
% 2.36/1.62 thf(func_def_2, type, produc2082277813at_nat: $tType).
% 2.36/1.62 thf(func_def_3, type, produc1190591575at_nat: $tType).
% 2.36/1.62 thf(func_def_4, type, produc1271302400at_nat: $tType).
% 2.36/1.62 thf(func_def_5, type, produc1695820582at_nat: $tType).
% 2.36/1.62 thf(func_def_6, type, list_P559422087at_nat: $tType).
% 2.36/1.62 thf(func_def_7, type, set_Pr1986765409at_nat: $tType).
% 2.36/1.62 thf(func_def_8, type, product_prod_nat_nat: $tType).
% 2.36/1.62 thf(func_def_9, type, capacity: $tType).
% 2.36/1.62 thf(func_def_10, type, nat: $tType).
% 2.36/1.62 thf(func_def_11, type, connected_capacity: (product_prod_nat_nat > capacity) > nat > nat > $o).
% 2.36/1.62 thf(func_def_12, type, dist_capacity: (product_prod_nat_nat > capacity) > nat > nat > nat > $o).
% 2.36/1.62 thf(func_def_13, type, isPath_capacity: (product_prod_nat_nat > capacity) > nat > list_P559422087at_nat > nat > $o).
% 2.36/1.62 thf(func_def_14, type, isShor1936442771pacity: (product_prod_nat_nat > capacity) > nat > list_P559422087at_nat > nat > $o).
% 2.36/1.62 thf(func_def_15, type, isSimp1359852763pacity: (product_prod_nat_nat > capacity) > nat > list_P559422087at_nat > nat > $o).
% 2.36/1.62 thf(func_def_16, type, min_dist_capacity: (product_prod_nat_nat > capacity) > nat > nat > nat).
% 2.36/1.62 thf(func_def_17, type, one_one_nat: nat).
% 2.36/1.62 thf(func_def_18, type, plus_plus_nat: nat > nat > nat).
% 2.36/1.62 thf(func_def_19, type, nil_Pr1308055047at_nat: list_P559422087at_nat).
% 2.36/1.62 thf(func_def_20, type, set_Pr2131844118at_nat: list_P559422087at_nat > set_Pr1986765409at_nat).
% 2.36/1.62 thf(func_def_21, type, size_s1990949619at_nat: list_P559422087at_nat > nat).
% 2.36/1.62 thf(func_def_22, type, ord_less_nat: nat > nat > $o).
% 2.36/1.62 thf(func_def_23, type, ord_less_eq_nat: nat > nat > $o).
% 2.36/1.62 thf(func_def_24, type, produc1682925677at_nat: (nat > nat > nat) > produc1695820582at_nat > produc2082277813at_nat).
% 2.36/1.62 thf(func_def_25, type, produc696256106at_nat: (nat > product_prod_nat_nat > product_prod_nat_nat) > produc1190591575at_nat > produc1864880952at_nat).
% 2.36/1.62 thf(func_def_26, type, product_Pair_nat_nat: nat > nat > product_prod_nat_nat).
% 2.36/1.62 thf(func_def_27, type, produc1933845336at_nat: nat > product_prod_nat_nat > produc1695820582at_nat).
% 2.36/1.62 thf(func_def_28, type, produc869658639at_nat: nat > produc1695820582at_nat > produc1190591575at_nat).
% 2.36/1.62 thf(func_def_29, type, produc947540346at_nat: product_prod_nat_nat > nat > produc1271302400at_nat).
% 2.36/1.62 thf(func_def_30, type, produc365964781at_nat: produc1695820582at_nat > (nat > nat > nat) > produc787001653at_nat).
% 2.36/1.62 thf(func_def_31, type, produc1728816653at_nat: produc2082277813at_nat > produc787001653at_nat).
% 2.36/1.62 thf(func_def_32, type, product_swap_nat_nat: product_prod_nat_nat > product_prod_nat_nat).
% 2.36/1.62 thf(func_def_33, type, produc857968056at_nat: produc1695820582at_nat > produc1271302400at_nat).
% 2.36/1.62 thf(func_def_34, type, produc2019146714at_nat: produc1271302400at_nat > produc1695820582at_nat).
% 2.36/1.62 thf(func_def_35, type, produc411855757at_nat: produc787001653at_nat > produc2082277813at_nat).
% 2.36/1.62 thf(func_def_36, type, collec7649004at_nat: (product_prod_nat_nat > $o) > set_Pr1986765409at_nat).
% 2.36/1.62 thf(func_def_37, type, member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o).
% 2.36/1.62 thf(func_def_38, type, c: product_prod_nat_nat > capacity).
% 2.36/1.62 thf(func_def_39, type, p: list_P559422087at_nat).
% 2.36/1.62 thf(func_def_40, type, p1: list_P559422087at_nat).
% 2.36/1.62 thf(func_def_41, type, p2: list_P559422087at_nat).
% 2.36/1.62 thf(func_def_42, type, p3: list_P559422087at_nat).
% 2.36/1.62 thf(func_def_43, type, s: nat).
% 2.36/1.62 thf(func_def_44, type, t: nat).
% 2.36/1.62 thf(func_def_45, type, u: nat).
% 2.36/1.62 thf(func_def_46, type, v: nat).
% 2.36/1.62 thf(func_def_63, type, sK0: nat > nat > nat > list_P559422087at_nat).
% 2.36/1.62 thf(func_def_64, type, sK1: nat > list_P559422087at_nat > nat > list_P559422087at_nat).
% 2.36/1.62 thf(func_def_66, type, ph3: !>[X0: $tType]:(X0)).
% 2.36/1.62 thf(f842,plain,(
% 2.36/1.62 $false),
% 2.36/1.62 inference(avatar_sat_refutation,[status(thm)],[f785,f792,f841])).
% 2.36/1.62 thf(f841,plain,(
% 2.36/1.62 spl2_1),
% 2.36/1.62 inference(avatar_contradiction_clause,[status(thm)],[f840])).
% 2.36/1.62 thf(f840,plain,(
% 2.36/1.62 $false | spl2_1),
% 2.36/1.62 inference(subsumption_resolution,[status(thm)],[f829,f780])).
% 2.36/1.62 thf(f780,plain,(
% 2.36/1.62 ((min_dist_capacity @ c @ s @ t) != (plus_plus_nat @ (size_s1990949619at_nat @ p1) @ (plus_plus_nat @ one_one_nat @ (size_s1990949619at_nat @ p2)))) | spl2_1),
% 2.36/1.62 inference(avatar_component_clause,[status(thm)],[f778])).
% 2.36/1.62 thf(f778,definition,(
% 2.36/1.62 spl2_1 <=> ((min_dist_capacity @ c @ s @ t) = (plus_plus_nat @ (size_s1990949619at_nat @ p1) @ (plus_plus_nat @ one_one_nat @ (size_s1990949619at_nat @ p2))))),
% 2.36/1.62 introduced(definition,[new_symbols(naming,[spl2_1])],[avatar_definition])).
% 2.36/1.62 thf(f829,plain,(
% 2.36/1.62 ((min_dist_capacity @ c @ s @ t) = (plus_plus_nat @ (size_s1990949619at_nat @ p1) @ (plus_plus_nat @ one_one_nat @ (size_s1990949619at_nat @ p2))))),
% 2.36/1.62 inference(superposition,[status(thm)],[f715,f772])).
% 2.36/1.62 thf(f772,plain,(
% 2.36/1.62 ( ! [X3 : nat,X0 : nat,X1 : nat] : (((plus_plus_nat @ (plus_plus_nat @ X1 @ X3) @ X0) = (plus_plus_nat @ X1 @ (plus_plus_nat @ X3 @ X0)))) )),
% 2.36/1.62 inference(equality_resolution,[status(thm)],[f757])).
% 2.36/1.62 thf(f757,plain,(
% 2.36/1.62 ( ! [X2 : nat,X3 : nat,X0 : nat,X1 : nat] : (((plus_plus_nat @ X1 @ X3) != X2) | ((plus_plus_nat @ X2 @ X0) = (plus_plus_nat @ X1 @ (plus_plus_nat @ X3 @ X0)))) )),
% 2.36/1.62 inference(cnf_transformation,[status(thm)],[f684])).
% 2.36/1.62 thf(f684,plain,(
% 2.36/1.62 ! [X0 : nat,X1 : nat,X2 : nat,X3 : nat] : (((plus_plus_nat @ X1 @ X3) != X2) | ((plus_plus_nat @ X2 @ X0) = (plus_plus_nat @ X1 @ (plus_plus_nat @ X3 @ X0))))),
% 2.36/1.62 inference(ennf_transformation,[status(thm)],[f673])).
% 2.36/1.62 thf(f673,plain,(
% 2.36/1.62 ! [X0 : nat,X1 : nat,X2 : nat,X3 : nat] : (((plus_plus_nat @ X1 @ X3) = X2) => ((plus_plus_nat @ X2 @ X0) = (plus_plus_nat @ X1 @ (plus_plus_nat @ X3 @ X0))))),
% 2.36/1.62 inference(rectify,[status(thm)],[f27])).
% 2.36/1.62 thf(f27,axiom,(
% 2.36/1.62 ! [X6 : nat,X15 : nat,X16 : nat,X5 : nat] : (((plus_plus_nat @ X15 @ X5) = X16) => ((plus_plus_nat @ X15 @ (plus_plus_nat @ X5 @ X6)) = (plus_plus_nat @ X16 @ X6)))),
% 2.36/1.62 file('/export/starexec/sandbox2/tmp/tmp.avufQdrk0c/DTF2THF_22776.p',fact_26_group__cancel_Oadd1)).
% 2.36/1.62 thf(f715,plain,(
% 2.36/1.62 ((min_dist_capacity @ c @ s @ t) = (plus_plus_nat @ (plus_plus_nat @ (size_s1990949619at_nat @ p1) @ one_one_nat) @ (size_s1990949619at_nat @ p2)))),
% 2.36/1.62 inference(cnf_transformation,[status(thm)],[f1])).
% 2.36/1.62 thf(f1,axiom,(
% 2.36/1.62 ((min_dist_capacity @ c @ s @ t) = (plus_plus_nat @ (plus_plus_nat @ (size_s1990949619at_nat @ p1) @ one_one_nat) @ (size_s1990949619at_nat @ p2)))),
% 2.36/1.62 file('/export/starexec/sandbox2/tmp/tmp.avufQdrk0c/DTF2THF_22776.p',fact_0_MIN_H)).
% 2.36/1.62 thf(f792,plain,(
% 2.36/1.62 ~spl2_2),
% 2.36/1.62 inference(avatar_split_clause,[status(thm)],[f722,f782])).
% 2.36/1.62 thf(f782,definition,(
% 2.36/1.62 spl2_2 <=> ((plus_plus_nat @ one_one_nat @ (size_s1990949619at_nat @ p2)) = (min_dist_capacity @ c @ u @ t))),
% 2.36/1.62 introduced(definition,[new_symbols(naming,[spl2_2])],[avatar_definition])).
% 2.36/1.62 thf(f722,plain,(
% 2.36/1.62 ((plus_plus_nat @ one_one_nat @ (size_s1990949619at_nat @ p2)) != (min_dist_capacity @ c @ u @ t))),
% 2.36/1.62 inference(cnf_transformation,[status(thm)],[f666])).
% 2.36/1.62 thf(f666,plain,(
% 2.36/1.62 ((plus_plus_nat @ one_one_nat @ (size_s1990949619at_nat @ p2)) != (min_dist_capacity @ c @ u @ t))),
% 2.36/1.62 inference(flattening,[status(thm)],[f256])).
% 2.36/1.62 thf(f256,negated_conjecture,(
% 2.36/1.62 ~((plus_plus_nat @ one_one_nat @ (size_s1990949619at_nat @ p2)) = (min_dist_capacity @ c @ u @ t))),
% 2.36/1.62 inference(negated_conjecture,[status(cth)],[f255])).
% 2.36/1.62 thf(f255,conjecture,(
% 2.36/1.62 ((plus_plus_nat @ one_one_nat @ (size_s1990949619at_nat @ p2)) = (min_dist_capacity @ c @ u @ t))),
% 2.36/1.62 file('/export/starexec/sandbox2/tmp/tmp.avufQdrk0c/DTF2THF_22776.p',conj_0)).
% 2.36/1.62 thf(f785,plain,(
% 2.36/1.62 ~spl2_1 | spl2_2),
% 2.36/1.62 inference(avatar_split_clause,[status(thm)],[f735,f782,f778])).
% 2.36/1.62 thf(f735,plain,(
% 2.36/1.62 ((min_dist_capacity @ c @ s @ t) != (plus_plus_nat @ (size_s1990949619at_nat @ p1) @ (plus_plus_nat @ one_one_nat @ (size_s1990949619at_nat @ p2)))) | ((plus_plus_nat @ one_one_nat @ (size_s1990949619at_nat @ p2)) = (min_dist_capacity @ c @ u @ t))),
% 2.36/1.62 inference(cnf_transformation,[status(thm)],[f696])).
% 2.36/1.62 thf(f696,plain,(
% 2.36/1.62 ((plus_plus_nat @ one_one_nat @ (size_s1990949619at_nat @ p2)) = (min_dist_capacity @ c @ u @ t)) | ((min_dist_capacity @ c @ s @ t) != (plus_plus_nat @ (size_s1990949619at_nat @ p1) @ (plus_plus_nat @ one_one_nat @ (size_s1990949619at_nat @ p2))))),
% 2.36/1.62 inference(ennf_transformation,[status(thm)],[f3])).
% 2.36/1.62 thf(f3,axiom,(
% 2.36/1.62 ((min_dist_capacity @ c @ s @ t) = (plus_plus_nat @ (size_s1990949619at_nat @ p1) @ (plus_plus_nat @ one_one_nat @ (size_s1990949619at_nat @ p2)))) => ((plus_plus_nat @ one_one_nat @ (size_s1990949619at_nat @ p2)) = (min_dist_capacity @ c @ u @ t))),
% 2.36/1.62 file('/export/starexec/sandbox2/tmp/tmp.avufQdrk0c/DTF2THF_22776.p',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)).
% 2.36/1.62 % SZS output end Proof for DTF2THF_22776
% 2.36/1.62 % (23052)------------------------------
% 2.36/1.62 % (23052)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.36/1.62 % (23052)Termination reason: Refutation
% 2.36/1.62
% 2.36/1.62 % (23052)Memory used [KB]: 6396
% 2.36/1.62 % (23052)Time elapsed: 0.035 s
% 2.36/1.62 % (23052)Instructions burned: 75 (million)
% 2.36/1.62 % (23052)------------------------------
% 2.36/1.62 % (23052)------------------------------
% 2.36/1.62 % (23005)Success in time 0.107 s
%------------------------------------------------------------------------------