TPTP Problem File: ITP053^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP053^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer EdmondsKarp_Termination_Abstract problem prob_81__7582066_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : EdmondsKarp_Termination_Abstract/prob_81__7582066_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 302 ( 104 unt; 47 typ; 0 def)
% Number of atoms : 633 ( 212 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 2205 ( 83 ~; 6 |; 34 &;1799 @)
% ( 0 <=>; 283 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 169 ( 169 >; 0 *; 0 +; 0 <<)
% Number of symbols : 37 ( 36 usr; 10 con; 0-4 aty)
% Number of variables : 832 ( 48 ^; 755 !; 29 ?; 832 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:30:03.954
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
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,
produc1864880952at_nat: $tType ).
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,
produc787001653at_nat: $tType ).
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,
produc2082277813at_nat: $tType ).
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,
produc1190591575at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
produc1271302400at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc1695820582at_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P559422087at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1986765409at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_tf__capacity,type,
capacity: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (36)
thf(sy_c_Graph_OGraph_Oconnected_001tf__capacity,type,
connected_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > $o ).
thf(sy_c_Graph_OGraph_Odist_001tf__capacity,type,
dist_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > nat > $o ).
thf(sy_c_Graph_OGraph_OisPath_001tf__capacity,type,
isPath_capacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).
thf(sy_c_Graph_OGraph_OisShortestPath_001tf__capacity,type,
isShor1936442771pacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).
thf(sy_c_Graph_OGraph_OisSimplePath_001tf__capacity,type,
isSimp1359852763pacity: ( product_prod_nat_nat > capacity ) > nat > list_P559422087at_nat > nat > $o ).
thf(sy_c_Graph_OGraph_Omin__dist_001tf__capacity,type,
min_dist_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
nil_Pr1308055047at_nat: list_P559422087at_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
set_Pr2131844118at_nat: list_P559422087at_nat > set_Pr1986765409at_nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s1990949619at_nat: list_P559422087at_nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
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,
produc1682925677at_nat: ( nat > nat > nat ) > produc1695820582at_nat > produc2082277813at_nat ).
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,
produc696256106at_nat: ( nat > product_prod_nat_nat > product_prod_nat_nat ) > produc1190591575at_nat > produc1864880952at_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc1933845336at_nat: nat > product_prod_nat_nat > produc1695820582at_nat ).
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,
produc869658639at_nat: nat > produc1695820582at_nat > produc1190591575at_nat ).
thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc947540346at_nat: product_prod_nat_nat > nat > produc1271302400at_nat ).
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,
produc365964781at_nat: produc1695820582at_nat > ( nat > nat > nat ) > produc787001653at_nat ).
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,
produc1728816653at_nat: produc2082277813at_nat > produc787001653at_nat ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001t__Nat__Onat,type,
product_swap_nat_nat: product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
produc857968056at_nat: produc1695820582at_nat > produc1271302400at_nat ).
thf(sy_c_Product__Type_Oprod_Oswap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc2019146714at_nat: produc1271302400at_nat > produc1695820582at_nat ).
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,
produc411855757at_nat: produc787001653at_nat > produc2082277813at_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
collec7649004at_nat: ( product_prod_nat_nat > $o ) > set_Pr1986765409at_nat ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
thf(sy_v_c,type,
c: product_prod_nat_nat > capacity ).
thf(sy_v_p,type,
p: list_P559422087at_nat ).
thf(sy_v_p1____,type,
p1: list_P559422087at_nat ).
thf(sy_v_p2____,type,
p2: list_P559422087at_nat ).
thf(sy_v_p_H,type,
p3: list_P559422087at_nat ).
thf(sy_v_s,type,
s: nat ).
thf(sy_v_t,type,
t: nat ).
thf(sy_v_u,type,
u: nat ).
thf(sy_v_v,type,
v: nat ).
% Relevant facts (254)
thf(fact_0_MIN_H,axiom,
( ( min_dist_capacity @ c @ s @ t )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ one_one_nat ) @ ( size_s1990949619at_nat @ p2 ) ) ) ).
% MIN'
thf(fact_1__092_060open_062min__dist_As_At_A_061_Alength_Ap1_A_L_A_I1_A_L_Alength_Ap2_J_A_092_060Longrightarrow_062_Amin__dist_As_Au_A_061_Alength_Ap1_092_060close_062,axiom,
( ( ( min_dist_capacity @ c @ s @ t )
= ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) )
=> ( ( min_dist_capacity @ c @ s @ u )
= ( size_s1990949619at_nat @ p1 ) ) ) ).
% \<open>min_dist s t = length p1 + (1 + length p2) \<Longrightarrow> min_dist s u = length p1\<close>
thf(fact_2__092_060open_062min__dist_As_At_A_061_Alength_Ap1_A_L_A_I1_A_L_Alength_Ap2_J_A_092_060Longrightarrow_062_Amin__dist_Au_At_A_061_A1_A_L_Alength_Ap2_092_060close_062,axiom,
( ( ( min_dist_capacity @ c @ s @ t )
= ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) )
=> ( ( min_dist_capacity @ c @ u @ t )
= ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) ) ).
% \<open>min_dist s t = length p1 + (1 + length p2) \<Longrightarrow> min_dist u t = 1 + length p2\<close>
thf(fact_3__092_060open_062min__dist_Av_At_A_061_Alength_Ap2_092_060close_062,axiom,
( ( min_dist_capacity @ c @ v @ t )
= ( size_s1990949619at_nat @ p2 ) ) ).
% \<open>min_dist v t = length p2\<close>
thf(fact_4_MIN,axiom,
( ( min_dist_capacity @ c @ s @ t )
= ( size_s1990949619at_nat @ p ) ) ).
% MIN
thf(fact_5__092_060open_062u_A_092_060noteq_062_Av_092_060close_062,axiom,
u != v ).
% \<open>u \<noteq> v\<close>
thf(fact_6_MDSV,axiom,
( ( min_dist_capacity @ c @ s @ v )
= ( plus_plus_nat @ ( size_s1990949619at_nat @ p1 ) @ one_one_nat ) ) ).
% MDSV
thf(fact_7_min__dist__split_I2_J,axiom,
! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
( ( dist_capacity @ c @ U @ D1 @ W )
=> ( ( dist_capacity @ c @ W @ D2 @ V )
=> ( ( ( min_dist_capacity @ c @ U @ V )
= ( plus_plus_nat @ D1 @ D2 ) )
=> ( ( min_dist_capacity @ c @ W @ V )
= D2 ) ) ) ) ).
% min_dist_split(2)
thf(fact_8_min__dist__split_I1_J,axiom,
! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
( ( dist_capacity @ c @ U @ D1 @ W )
=> ( ( dist_capacity @ c @ W @ D2 @ V )
=> ( ( ( min_dist_capacity @ c @ U @ V )
= ( plus_plus_nat @ D1 @ D2 ) )
=> ( ( min_dist_capacity @ c @ U @ W )
= D1 ) ) ) ) ).
% min_dist_split(1)
thf(fact_9_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_10_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_11_dist__trans,axiom,
! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
( ( dist_capacity @ c @ U @ D1 @ W )
=> ( ( dist_capacity @ c @ W @ D2 @ V )
=> ( dist_capacity @ c @ U @ ( plus_plus_nat @ D1 @ D2 ) @ V ) ) ) ).
% dist_trans
thf(fact_12_assms_I3_J,axiom,
isPath_capacity @ c @ s @ p3 @ t ).
% assms(3)
thf(fact_13_DISTS_I1_J,axiom,
dist_capacity @ c @ s @ ( size_s1990949619at_nat @ p1 ) @ u ).
% DISTS(1)
thf(fact_14_isPath__distD,axiom,
! [U: nat,P: list_P559422087at_nat,V: nat] :
( ( isPath_capacity @ c @ U @ P @ V )
=> ( dist_capacity @ c @ U @ ( size_s1990949619at_nat @ P ) @ V ) ) ).
% isPath_distD
thf(fact_15_dist__def,axiom,
! [V: nat,D: nat,V2: nat] :
( ( dist_capacity @ c @ V @ D @ V2 )
= ( ? [P2: list_P559422087at_nat] :
( ( isPath_capacity @ c @ V @ P2 @ V2 )
& ( ( size_s1990949619at_nat @ P2 )
= D ) ) ) ) ).
% dist_def
thf(fact_16_DISTS_I2_J,axiom,
dist_capacity @ c @ u @ one_one_nat @ v ).
% DISTS(2)
thf(fact_17_P,axiom,
isPath_capacity @ c @ s @ p @ t ).
% P
thf(fact_18_DISTS_I3_J,axiom,
dist_capacity @ c @ v @ ( size_s1990949619at_nat @ p2 ) @ t ).
% DISTS(3)
thf(fact_19_assms_I1_J,axiom,
isShor1936442771pacity @ c @ s @ p @ t ).
% assms(1)
thf(fact_20_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_21_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_22_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_23_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_24_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_25_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_26_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_27_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_28_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_29_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_30_isShortestPath__min__dist__def,axiom,
! [U: nat,P: list_P559422087at_nat,V: nat] :
( ( isShor1936442771pacity @ c @ U @ P @ V )
= ( ( isPath_capacity @ c @ U @ P @ V )
& ( ( size_s1990949619at_nat @ P )
= ( min_dist_capacity @ c @ U @ V ) ) ) ) ).
% isShortestPath_min_dist_def
thf(fact_31_Graph_OisPath__distD,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat] :
( ( isPath_capacity @ C @ U @ P @ V )
=> ( dist_capacity @ C @ U @ ( size_s1990949619at_nat @ P ) @ V ) ) ).
% Graph.isPath_distD
thf(fact_32_Graph_Odist__def,axiom,
( dist_capacity
= ( ^ [C2: product_prod_nat_nat > capacity,V3: nat,D3: nat,V4: nat] :
? [P2: list_P559422087at_nat] :
( ( isPath_capacity @ C2 @ V3 @ P2 @ V4 )
& ( ( size_s1990949619at_nat @ P2 )
= D3 ) ) ) ) ).
% Graph.dist_def
thf(fact_33_min__dist__minD,axiom,
! [V: nat,D: nat,V2: nat] :
( ( dist_capacity @ c @ V @ D @ V2 )
=> ( ord_less_eq_nat @ ( min_dist_capacity @ c @ V @ V2 ) @ D ) ) ).
% min_dist_minD
thf(fact_34_min__distI__eq,axiom,
! [V: nat,D: nat,V2: nat] :
( ( dist_capacity @ c @ V @ D @ V2 )
=> ( ! [D4: nat] :
( ( dist_capacity @ c @ V @ D4 @ V2 )
=> ( ord_less_eq_nat @ D @ D4 ) )
=> ( ( min_dist_capacity @ c @ V @ V2 )
= D ) ) ) ).
% min_distI_eq
thf(fact_35_Graph_Omin__dist__split_I1_J,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
( ( dist_capacity @ C @ U @ D1 @ W )
=> ( ( dist_capacity @ C @ W @ D2 @ V )
=> ( ( ( min_dist_capacity @ C @ U @ V )
= ( plus_plus_nat @ D1 @ D2 ) )
=> ( ( min_dist_capacity @ C @ U @ W )
= D1 ) ) ) ) ).
% Graph.min_dist_split(1)
thf(fact_36_Graph_Omin__dist__split_I2_J,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
( ( dist_capacity @ C @ U @ D1 @ W )
=> ( ( dist_capacity @ C @ W @ D2 @ V )
=> ( ( ( min_dist_capacity @ C @ U @ V )
= ( plus_plus_nat @ D1 @ D2 ) )
=> ( ( min_dist_capacity @ C @ W @ V )
= D2 ) ) ) ) ).
% Graph.min_dist_split(2)
thf(fact_37_min__dist__is__dist,axiom,
! [V: nat,V2: nat] :
( ( connected_capacity @ c @ V @ V2 )
=> ( dist_capacity @ c @ V @ ( min_dist_capacity @ c @ V @ V2 ) @ V2 ) ) ).
% min_dist_is_dist
thf(fact_38_shortestPath__is__path,axiom,
! [U: nat,P: list_P559422087at_nat,V: nat] :
( ( isShor1936442771pacity @ c @ U @ P @ V )
=> ( isPath_capacity @ c @ U @ P @ V ) ) ).
% shortestPath_is_path
thf(fact_39_assms_I4_J,axiom,
member701585322at_nat @ ( product_Pair_nat_nat @ v @ u ) @ ( set_Pr2131844118at_nat @ p3 ) ).
% assms(4)
thf(fact_40_connected__def,axiom,
! [U: nat,V: nat] :
( ( connected_capacity @ c @ U @ V )
= ( ? [P2: list_P559422087at_nat] : ( isPath_capacity @ c @ U @ P2 @ V ) ) ) ).
% connected_def
thf(fact_41_connected__by__dist,axiom,
! [V: nat,V2: nat] :
( ( connected_capacity @ c @ V @ V2 )
= ( ? [D3: nat] : ( dist_capacity @ c @ V @ D3 @ V2 ) ) ) ).
% connected_by_dist
thf(fact_42_mem__Collect__eq,axiom,
! [A: product_prod_nat_nat,P3: product_prod_nat_nat > $o] :
( ( member701585322at_nat @ A @ ( collec7649004at_nat @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A3: set_Pr1986765409at_nat] :
( ( collec7649004at_nat
@ ^ [X2: product_prod_nat_nat] : ( member701585322at_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_44_Collect__cong,axiom,
! [P3: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
( ! [X3: product_prod_nat_nat] :
( ( P3 @ X3 )
= ( Q @ X3 ) )
=> ( ( collec7649004at_nat @ P3 )
= ( collec7649004at_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_45_obtain__shortest__path,axiom,
! [U: nat,V: nat] :
( ( connected_capacity @ c @ U @ V )
=> ~ ! [P4: list_P559422087at_nat] :
~ ( isShor1936442771pacity @ c @ U @ P4 @ V ) ) ).
% obtain_shortest_path
thf(fact_46_assms_I2_J,axiom,
member701585322at_nat @ ( product_Pair_nat_nat @ u @ v ) @ ( set_Pr2131844118at_nat @ p ) ).
% assms(2)
thf(fact_47_isPath__ex__edge1,axiom,
! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
( ( isPath_capacity @ c @ U @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( U1 != U )
=> ? [U2: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ U2 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% isPath_ex_edge1
thf(fact_48_isPath__ex__edge2,axiom,
! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
( ( isPath_capacity @ c @ U @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( V1 != V )
=> ? [V22: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ V1 @ V22 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% isPath_ex_edge2
thf(fact_49_min__dist__le,axiom,
! [Src: nat,V: nat,D5: nat] :
( ( connected_capacity @ c @ Src @ V )
=> ( ( ord_less_eq_nat @ D5 @ ( min_dist_capacity @ c @ Src @ V ) )
=> ? [V5: nat] :
( ( connected_capacity @ c @ Src @ V5 )
& ( ( min_dist_capacity @ c @ Src @ V5 )
= D5 ) ) ) ) ).
% min_dist_le
thf(fact_50_min__distI2,axiom,
! [V: nat,V2: nat,Q: nat > $o] :
( ( connected_capacity @ c @ V @ V2 )
=> ( ! [D6: nat] :
( ( dist_capacity @ c @ V @ D6 @ V2 )
=> ( ! [D7: nat] :
( ( dist_capacity @ c @ V @ D7 @ V2 )
=> ( ord_less_eq_nat @ D6 @ D7 ) )
=> ( Q @ D6 ) ) )
=> ( Q @ ( min_dist_capacity @ c @ V @ V2 ) ) ) ) ).
% min_distI2
thf(fact_51_isShortestPath__def,axiom,
! [U: nat,P: list_P559422087at_nat,V: nat] :
( ( isShor1936442771pacity @ c @ U @ P @ V )
= ( ( isPath_capacity @ c @ U @ P @ V )
& ! [P5: list_P559422087at_nat] :
( ( isPath_capacity @ c @ U @ P5 @ V )
=> ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P ) @ ( size_s1990949619at_nat @ P5 ) ) ) ) ) ).
% isShortestPath_def
thf(fact_52_isShortestPath__level__edge_I1_J,axiom,
! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
( ( isShor1936442771pacity @ c @ S @ P @ T )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( connected_capacity @ c @ S @ U ) ) ) ).
% isShortestPath_level_edge(1)
thf(fact_53_isShortestPath__level__edge_I2_J,axiom,
! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
( ( isShor1936442771pacity @ c @ S @ P @ T )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( connected_capacity @ c @ U @ V ) ) ) ).
% isShortestPath_level_edge(2)
thf(fact_54_isShortestPath__level__edge_I3_J,axiom,
! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
( ( isShor1936442771pacity @ c @ S @ P @ T )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( connected_capacity @ c @ V @ T ) ) ) ).
% isShortestPath_level_edge(3)
thf(fact_55_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_56_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_57_isShortestPath__level__edge_I6_J,axiom,
! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
( ( isShor1936442771pacity @ c @ S @ P @ T )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( min_dist_capacity @ c @ S @ T )
= ( plus_plus_nat @ ( plus_plus_nat @ ( min_dist_capacity @ c @ S @ U ) @ one_one_nat ) @ ( min_dist_capacity @ c @ V @ T ) ) ) ) ) ).
% isShortestPath_level_edge(6)
thf(fact_58_isShortestPath__level__edge_I5_J,axiom,
! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
( ( isShor1936442771pacity @ c @ S @ P @ T )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( min_dist_capacity @ c @ U @ T )
= ( plus_plus_nat @ one_one_nat @ ( min_dist_capacity @ c @ V @ T ) ) ) ) ) ).
% isShortestPath_level_edge(5)
thf(fact_59_isShortestPath__level__edge_I4_J,axiom,
! [S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
( ( isShor1936442771pacity @ c @ S @ P @ T )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( min_dist_capacity @ c @ S @ V )
= ( plus_plus_nat @ ( min_dist_capacity @ c @ S @ U ) @ one_one_nat ) ) ) ) ).
% isShortestPath_level_edge(4)
thf(fact_60_connected__refl,axiom,
! [V: nat] : ( connected_capacity @ c @ V @ V ) ).
% connected_refl
thf(fact_61_connected__distI,axiom,
! [V: nat,D: nat,V2: nat] :
( ( dist_capacity @ c @ V @ D @ V2 )
=> ( connected_capacity @ c @ V @ V2 ) ) ).
% connected_distI
thf(fact_62_Graph_OisPath__ex__edge1,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
( ( isPath_capacity @ C @ U @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( U1 != U )
=> ? [U2: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ U2 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% Graph.isPath_ex_edge1
thf(fact_63_Graph_OisPath__ex__edge2,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat] :
( ( isPath_capacity @ C @ U @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( V1 != V )
=> ? [V22: nat] : ( member701585322at_nat @ ( product_Pair_nat_nat @ V1 @ V22 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% Graph.isPath_ex_edge2
thf(fact_64_Graph_Oobtain__shortest__path,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,V: nat] :
( ( connected_capacity @ C @ U @ V )
=> ~ ! [P4: list_P559422087at_nat] :
~ ( isShor1936442771pacity @ C @ U @ P4 @ V ) ) ).
% Graph.obtain_shortest_path
thf(fact_65_Graph_OisShortestPath_Ocong,axiom,
isShor1936442771pacity = isShor1936442771pacity ).
% Graph.isShortestPath.cong
thf(fact_66_Graph_Oconnected__refl,axiom,
! [C: product_prod_nat_nat > capacity,V: nat] : ( connected_capacity @ C @ V @ V ) ).
% Graph.connected_refl
thf(fact_67_Graph_Oconnected_Ocong,axiom,
connected_capacity = connected_capacity ).
% Graph.connected.cong
thf(fact_68_Graph_Omin__dist__le,axiom,
! [C: product_prod_nat_nat > capacity,Src: nat,V: nat,D5: nat] :
( ( connected_capacity @ C @ Src @ V )
=> ( ( ord_less_eq_nat @ D5 @ ( min_dist_capacity @ C @ Src @ V ) )
=> ? [V5: nat] :
( ( connected_capacity @ C @ Src @ V5 )
& ( ( min_dist_capacity @ C @ Src @ V5 )
= D5 ) ) ) ) ).
% Graph.min_dist_le
thf(fact_69_Graph_OisShortestPath__level__edge_I1_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
( ( isShor1936442771pacity @ C @ S @ P @ T )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( connected_capacity @ C @ S @ U ) ) ) ).
% Graph.isShortestPath_level_edge(1)
thf(fact_70_Graph_OisShortestPath__level__edge_I2_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
( ( isShor1936442771pacity @ C @ S @ P @ T )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( connected_capacity @ C @ U @ V ) ) ) ).
% Graph.isShortestPath_level_edge(2)
thf(fact_71_Graph_OisShortestPath__level__edge_I3_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
( ( isShor1936442771pacity @ C @ S @ P @ T )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( connected_capacity @ C @ V @ T ) ) ) ).
% Graph.isShortestPath_level_edge(3)
thf(fact_72_Graph_Oconnected__def,axiom,
( connected_capacity
= ( ^ [C2: product_prod_nat_nat > capacity,U3: nat,V3: nat] :
? [P2: list_P559422087at_nat] : ( isPath_capacity @ C2 @ U3 @ P2 @ V3 ) ) ) ).
% Graph.connected_def
thf(fact_73_Graph_Oconnected__by__dist,axiom,
( connected_capacity
= ( ^ [C2: product_prod_nat_nat > capacity,V3: nat,V4: nat] :
? [D3: nat] : ( dist_capacity @ C2 @ V3 @ D3 @ V4 ) ) ) ).
% Graph.connected_by_dist
thf(fact_74_Graph_Oconnected__distI,axiom,
! [C: product_prod_nat_nat > capacity,V: nat,D: nat,V2: nat] :
( ( dist_capacity @ C @ V @ D @ V2 )
=> ( connected_capacity @ C @ V @ V2 ) ) ).
% Graph.connected_distI
thf(fact_75_Graph_OshortestPath__is__path,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat] :
( ( isShor1936442771pacity @ C @ U @ P @ V )
=> ( isPath_capacity @ C @ U @ P @ V ) ) ).
% Graph.shortestPath_is_path
thf(fact_76_Graph_OisShortestPath__level__edge_I6_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
( ( isShor1936442771pacity @ C @ S @ P @ T )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( min_dist_capacity @ C @ S @ T )
= ( plus_plus_nat @ ( plus_plus_nat @ ( min_dist_capacity @ C @ S @ U ) @ one_one_nat ) @ ( min_dist_capacity @ C @ V @ T ) ) ) ) ) ).
% Graph.isShortestPath_level_edge(6)
thf(fact_77_Graph_OisShortestPath__level__edge_I5_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
( ( isShor1936442771pacity @ C @ S @ P @ T )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( min_dist_capacity @ C @ U @ T )
= ( plus_plus_nat @ one_one_nat @ ( min_dist_capacity @ C @ V @ T ) ) ) ) ) ).
% Graph.isShortestPath_level_edge(5)
thf(fact_78_Graph_OisShortestPath__level__edge_I4_J,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,U: nat,V: nat] :
( ( isShor1936442771pacity @ C @ S @ P @ T )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U @ V ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( min_dist_capacity @ C @ S @ V )
= ( plus_plus_nat @ ( min_dist_capacity @ C @ S @ U ) @ one_one_nat ) ) ) ) ).
% Graph.isShortestPath_level_edge(4)
thf(fact_79_Graph_Omin__distI2,axiom,
! [C: product_prod_nat_nat > capacity,V: nat,V2: nat,Q: nat > $o] :
( ( connected_capacity @ C @ V @ V2 )
=> ( ! [D6: nat] :
( ( dist_capacity @ C @ V @ D6 @ V2 )
=> ( ! [D7: nat] :
( ( dist_capacity @ C @ V @ D7 @ V2 )
=> ( ord_less_eq_nat @ D6 @ D7 ) )
=> ( Q @ D6 ) ) )
=> ( Q @ ( min_dist_capacity @ C @ V @ V2 ) ) ) ) ).
% Graph.min_distI2
thf(fact_80_Graph_OisShortestPath__def,axiom,
( isShor1936442771pacity
= ( ^ [C2: product_prod_nat_nat > capacity,U3: nat,P2: list_P559422087at_nat,V3: nat] :
( ( isPath_capacity @ C2 @ U3 @ P2 @ V3 )
& ! [P5: list_P559422087at_nat] :
( ( isPath_capacity @ C2 @ U3 @ P5 @ V3 )
=> ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P2 ) @ ( size_s1990949619at_nat @ P5 ) ) ) ) ) ) ).
% Graph.isShortestPath_def
thf(fact_81_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_82_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_83_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_84_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_85_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_86_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_87_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_88_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C2: nat] :
( B2
= ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_89_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_90_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_91_Graph_Omin__dist__minD,axiom,
! [C: product_prod_nat_nat > capacity,V: nat,D: nat,V2: nat] :
( ( dist_capacity @ C @ V @ D @ V2 )
=> ( ord_less_eq_nat @ ( min_dist_capacity @ C @ V @ V2 ) @ D ) ) ).
% Graph.min_dist_minD
thf(fact_92_Graph_Omin__distI__eq,axiom,
! [C: product_prod_nat_nat > capacity,V: nat,D: nat,V2: nat] :
( ( dist_capacity @ C @ V @ D @ V2 )
=> ( ! [D4: nat] :
( ( dist_capacity @ C @ V @ D4 @ V2 )
=> ( ord_less_eq_nat @ D @ D4 ) )
=> ( ( min_dist_capacity @ C @ V @ V2 )
= D ) ) ) ).
% Graph.min_distI_eq
thf(fact_93_Graph_Omin__dist__is__dist,axiom,
! [C: product_prod_nat_nat > capacity,V: nat,V2: nat] :
( ( connected_capacity @ C @ V @ V2 )
=> ( dist_capacity @ C @ V @ ( min_dist_capacity @ C @ V @ V2 ) @ V2 ) ) ).
% Graph.min_dist_is_dist
thf(fact_94_Graph_OisShortestPath__min__dist__def,axiom,
( isShor1936442771pacity
= ( ^ [C2: product_prod_nat_nat > capacity,U3: nat,P2: list_P559422087at_nat,V3: nat] :
( ( isPath_capacity @ C2 @ U3 @ P2 @ V3 )
& ( ( size_s1990949619at_nat @ P2 )
= ( min_dist_capacity @ C2 @ U3 @ V3 ) ) ) ) ) ).
% Graph.isShortestPath_min_dist_def
thf(fact_95_Graph_OisPath_Ocong,axiom,
isPath_capacity = isPath_capacity ).
% Graph.isPath.cong
thf(fact_96_Graph_Omin__dist_Ocong,axiom,
min_dist_capacity = min_dist_capacity ).
% Graph.min_dist.cong
thf(fact_97_Graph_Odist_Ocong,axiom,
dist_capacity = dist_capacity ).
% Graph.dist.cong
thf(fact_98_Graph_Odist__trans,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
( ( dist_capacity @ C @ U @ D1 @ W )
=> ( ( dist_capacity @ C @ W @ D2 @ V )
=> ( dist_capacity @ C @ U @ ( plus_plus_nat @ D1 @ D2 ) @ V ) ) ) ).
% Graph.dist_trans
thf(fact_99_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_100_isShortestPath__alt,axiom,
! [U: nat,P: list_P559422087at_nat,V: nat] :
( ( isShor1936442771pacity @ c @ U @ P @ V )
= ( ( isSimp1359852763pacity @ c @ U @ P @ V )
& ( ( size_s1990949619at_nat @ P )
= ( min_dist_capacity @ c @ U @ V ) ) ) ) ).
% isShortestPath_alt
thf(fact_101_min__dist__less,axiom,
! [Src: nat,V: nat,D: nat,D5: nat] :
( ( connected_capacity @ c @ Src @ V )
=> ( ( ( min_dist_capacity @ c @ Src @ V )
= D )
=> ( ( ord_less_nat @ D5 @ D )
=> ? [V5: nat] :
( ( connected_capacity @ c @ Src @ V5 )
& ( ( min_dist_capacity @ c @ Src @ V5 )
= D5 ) ) ) ) ) ).
% min_dist_less
thf(fact_102_isSPath__no__selfloop,axiom,
! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat] :
( ( isSimp1359852763pacity @ c @ U @ P @ V )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ).
% isSPath_no_selfloop
thf(fact_103_isSPath__sg__incoming,axiom,
! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,U22: nat] :
( ( isSimp1359852763pacity @ c @ U @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( U1 != U22 )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U22 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% isSPath_sg_incoming
thf(fact_104_isSPath__sg__outgoing,axiom,
! [U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,V23: nat] :
( ( isSimp1359852763pacity @ c @ U @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( V1 != V23 )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V23 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% isSPath_sg_outgoing
thf(fact_105_old_Oprod_Oinject,axiom,
! [A: nat > nat > nat,B: produc1695820582at_nat,A4: nat > nat > nat,B4: produc1695820582at_nat] :
( ( ( produc1682925677at_nat @ A @ B )
= ( produc1682925677at_nat @ A4 @ B4 ) )
= ( ( A = A4 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_106_old_Oprod_Oinject,axiom,
! [A: nat,B: product_prod_nat_nat,A4: nat,B4: product_prod_nat_nat] :
( ( ( produc1933845336at_nat @ A @ B )
= ( produc1933845336at_nat @ A4 @ B4 ) )
= ( ( A = A4 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_107_old_Oprod_Oinject,axiom,
! [A: nat,B: nat,A4: nat,B4: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A4 @ B4 ) )
= ( ( A = A4 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_108_prod_Oinject,axiom,
! [X1: nat > nat > nat,X22: produc1695820582at_nat,Y1: nat > nat > nat,Y2: produc1695820582at_nat] :
( ( ( produc1682925677at_nat @ X1 @ X22 )
= ( produc1682925677at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_109_prod_Oinject,axiom,
! [X1: nat,X22: product_prod_nat_nat,Y1: nat,Y2: product_prod_nat_nat] :
( ( ( produc1933845336at_nat @ X1 @ X22 )
= ( produc1933845336at_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_110_prod_Oinject,axiom,
! [X1: nat,X22: nat,Y1: nat,Y2: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X22 )
= ( product_Pair_nat_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_111_isSPath__pathLE,axiom,
! [S: nat,P: list_P559422087at_nat,T: nat] :
( ( isPath_capacity @ c @ S @ P @ T )
=> ? [P6: list_P559422087at_nat] : ( isSimp1359852763pacity @ c @ S @ P6 @ T ) ) ).
% isSPath_pathLE
thf(fact_112_shortestPath__is__simple,axiom,
! [S: nat,P: list_P559422087at_nat,T: nat] :
( ( isShor1936442771pacity @ c @ S @ P @ T )
=> ( isSimp1359852763pacity @ c @ S @ P @ T ) ) ).
% shortestPath_is_simple
thf(fact_113_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_114_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_115_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_116_Graph_OisSimplePath_Ocong,axiom,
isSimp1359852763pacity = isSimp1359852763pacity ).
% Graph.isSimplePath.cong
thf(fact_117_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_118_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_119_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_120_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_121_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_122_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_123_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_124_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_125_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_126_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_127_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_128_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_129_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_130_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_131_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_132_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_133_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_134_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_135_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_136_nat__less__induct,axiom,
! [P3: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P3 @ M3 ) )
=> ( P3 @ N3 ) )
=> ( P3 @ N ) ) ).
% nat_less_induct
thf(fact_137_infinite__descent,axiom,
! [P3: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P3 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P3 @ M3 ) ) )
=> ( P3 @ N ) ) ).
% infinite_descent
thf(fact_138_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_139_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_140_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_141_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_142_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_143_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_144_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_145_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_146_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_147_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_148_Graph_OisSPath__pathLE,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat] :
( ( isPath_capacity @ C @ S @ P @ T )
=> ? [P6: list_P559422087at_nat] : ( isSimp1359852763pacity @ C @ S @ P6 @ T ) ) ).
% Graph.isSPath_pathLE
thf(fact_149_Graph_OshortestPath__is__simple,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat] :
( ( isShor1936442771pacity @ C @ S @ P @ T )
=> ( isSimp1359852763pacity @ C @ S @ P @ T ) ) ).
% Graph.shortestPath_is_simple
thf(fact_150_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_151_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_152_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_153_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_154_Graph_Omin__dist__less,axiom,
! [C: product_prod_nat_nat > capacity,Src: nat,V: nat,D: nat,D5: nat] :
( ( connected_capacity @ C @ Src @ V )
=> ( ( ( min_dist_capacity @ C @ Src @ V )
= D )
=> ( ( ord_less_nat @ D5 @ D )
=> ? [V5: nat] :
( ( connected_capacity @ C @ Src @ V5 )
& ( ( min_dist_capacity @ C @ Src @ V5 )
= D5 ) ) ) ) ) ).
% Graph.min_dist_less
thf(fact_155_Graph_OisSPath__no__selfloop,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat] :
( ( isSimp1359852763pacity @ C @ U @ P @ V )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ U1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ).
% Graph.isSPath_no_selfloop
thf(fact_156_Graph_OisSPath__sg__incoming,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,U22: nat] :
( ( isSimp1359852763pacity @ C @ U @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( U1 != U22 )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U22 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% Graph.isSPath_sg_incoming
thf(fact_157_Graph_OisSPath__sg__outgoing,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,P: list_P559422087at_nat,V: nat,U1: nat,V1: nat,V23: nat] :
( ( isSimp1359852763pacity @ C @ U @ P @ V )
=> ( ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V1 ) @ ( set_Pr2131844118at_nat @ P ) )
=> ( ( V1 != V23 )
=> ~ ( member701585322at_nat @ ( product_Pair_nat_nat @ U1 @ V23 ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ) ).
% Graph.isSPath_sg_outgoing
thf(fact_158_surj__pair,axiom,
! [P: produc2082277813at_nat] :
? [X3: nat > nat > nat,Y3: produc1695820582at_nat] :
( P
= ( produc1682925677at_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_159_surj__pair,axiom,
! [P: produc1695820582at_nat] :
? [X3: nat,Y3: product_prod_nat_nat] :
( P
= ( produc1933845336at_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_160_surj__pair,axiom,
! [P: product_prod_nat_nat] :
? [X3: nat,Y3: nat] :
( P
= ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_161_prod__cases,axiom,
! [P3: produc2082277813at_nat > $o,P: produc2082277813at_nat] :
( ! [A5: nat > nat > nat,B5: produc1695820582at_nat] : ( P3 @ ( produc1682925677at_nat @ A5 @ B5 ) )
=> ( P3 @ P ) ) ).
% prod_cases
thf(fact_162_prod__cases,axiom,
! [P3: produc1695820582at_nat > $o,P: produc1695820582at_nat] :
( ! [A5: nat,B5: product_prod_nat_nat] : ( P3 @ ( produc1933845336at_nat @ A5 @ B5 ) )
=> ( P3 @ P ) ) ).
% prod_cases
thf(fact_163_prod__cases,axiom,
! [P3: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
( ! [A5: nat,B5: nat] : ( P3 @ ( product_Pair_nat_nat @ A5 @ B5 ) )
=> ( P3 @ P ) ) ).
% prod_cases
thf(fact_164_Pair__inject,axiom,
! [A: nat > nat > nat,B: produc1695820582at_nat,A4: nat > nat > nat,B4: produc1695820582at_nat] :
( ( ( produc1682925677at_nat @ A @ B )
= ( produc1682925677at_nat @ A4 @ B4 ) )
=> ~ ( ( A = A4 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_165_Pair__inject,axiom,
! [A: nat,B: product_prod_nat_nat,A4: nat,B4: product_prod_nat_nat] :
( ( ( produc1933845336at_nat @ A @ B )
= ( produc1933845336at_nat @ A4 @ B4 ) )
=> ~ ( ( A = A4 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_166_Pair__inject,axiom,
! [A: nat,B: nat,A4: nat,B4: nat] :
( ( ( product_Pair_nat_nat @ A @ B )
= ( product_Pair_nat_nat @ A4 @ B4 ) )
=> ~ ( ( A = A4 )
=> ( B != B4 ) ) ) ).
% Pair_inject
thf(fact_167_prod__cases3,axiom,
! [Y: produc2082277813at_nat] :
~ ! [A5: nat > nat > nat,B5: nat,C3: product_prod_nat_nat] :
( Y
!= ( produc1682925677at_nat @ A5 @ ( produc1933845336at_nat @ B5 @ C3 ) ) ) ).
% prod_cases3
thf(fact_168_prod__cases3,axiom,
! [Y: produc1695820582at_nat] :
~ ! [A5: nat,B5: nat,C3: nat] :
( Y
!= ( produc1933845336at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ C3 ) ) ) ).
% prod_cases3
thf(fact_169_prod__cases4,axiom,
! [Y: produc2082277813at_nat] :
~ ! [A5: nat > nat > nat,B5: nat,C3: nat,D6: nat] :
( Y
!= ( produc1682925677at_nat @ A5 @ ( produc1933845336at_nat @ B5 @ ( product_Pair_nat_nat @ C3 @ D6 ) ) ) ) ).
% prod_cases4
thf(fact_170_prod__induct3,axiom,
! [P3: produc2082277813at_nat > $o,X: produc2082277813at_nat] :
( ! [A5: nat > nat > nat,B5: nat,C3: product_prod_nat_nat] : ( P3 @ ( produc1682925677at_nat @ A5 @ ( produc1933845336at_nat @ B5 @ C3 ) ) )
=> ( P3 @ X ) ) ).
% prod_induct3
thf(fact_171_prod__induct3,axiom,
! [P3: produc1695820582at_nat > $o,X: produc1695820582at_nat] :
( ! [A5: nat,B5: nat,C3: nat] : ( P3 @ ( produc1933845336at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ C3 ) ) )
=> ( P3 @ X ) ) ).
% prod_induct3
thf(fact_172_prod__induct4,axiom,
! [P3: produc2082277813at_nat > $o,X: produc2082277813at_nat] :
( ! [A5: nat > nat > nat,B5: nat,C3: nat,D6: nat] : ( P3 @ ( produc1682925677at_nat @ A5 @ ( produc1933845336at_nat @ B5 @ ( product_Pair_nat_nat @ C3 @ D6 ) ) ) )
=> ( P3 @ X ) ) ).
% prod_induct4
thf(fact_173_old_Oprod_Oexhaust,axiom,
! [Y: produc2082277813at_nat] :
~ ! [A5: nat > nat > nat,B5: produc1695820582at_nat] :
( Y
!= ( produc1682925677at_nat @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_174_old_Oprod_Oexhaust,axiom,
! [Y: produc1695820582at_nat] :
~ ! [A5: nat,B5: product_prod_nat_nat] :
( Y
!= ( produc1933845336at_nat @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_175_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_nat_nat] :
~ ! [A5: nat,B5: nat] :
( Y
!= ( product_Pair_nat_nat @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_176_old_Oprod_Oinducts,axiom,
! [P3: produc2082277813at_nat > $o,Prod: produc2082277813at_nat] :
( ! [A5: nat > nat > nat,B5: produc1695820582at_nat] : ( P3 @ ( produc1682925677at_nat @ A5 @ B5 ) )
=> ( P3 @ Prod ) ) ).
% old.prod.inducts
thf(fact_177_old_Oprod_Oinducts,axiom,
! [P3: produc1695820582at_nat > $o,Prod: produc1695820582at_nat] :
( ! [A5: nat,B5: product_prod_nat_nat] : ( P3 @ ( produc1933845336at_nat @ A5 @ B5 ) )
=> ( P3 @ Prod ) ) ).
% old.prod.inducts
thf(fact_178_old_Oprod_Oinducts,axiom,
! [P3: product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ! [A5: nat,B5: nat] : ( P3 @ ( product_Pair_nat_nat @ A5 @ B5 ) )
=> ( P3 @ Prod ) ) ).
% old.prod.inducts
thf(fact_179_size__neq__size__imp__neq,axiom,
! [X: list_P559422087at_nat,Y: list_P559422087at_nat] :
( ( ( size_s1990949619at_nat @ X )
!= ( size_s1990949619at_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_180_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_181_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_182_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_183_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_184_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_185_Nat_Oex__has__greatest__nat,axiom,
! [P3: nat > $o,K: nat,B: nat] :
( ( P3 @ K )
=> ( ! [Y3: nat] :
( ( P3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P3 @ X3 )
& ! [Y4: nat] :
( ( P3 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_186_Graph_OisShortestPath__alt,axiom,
( isShor1936442771pacity
= ( ^ [C2: product_prod_nat_nat > capacity,U3: nat,P2: list_P559422087at_nat,V3: nat] :
( ( isSimp1359852763pacity @ C2 @ U3 @ P2 @ V3 )
& ( ( size_s1990949619at_nat @ P2 )
= ( min_dist_capacity @ C2 @ U3 @ V3 ) ) ) ) ) ).
% Graph.isShortestPath_alt
thf(fact_187_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K2: nat] :
( N2
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_188_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_189_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_190_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_191_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_192_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_193_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_194_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_195_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_196_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_197_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_198_isSPath__nt__parallel,axiom,
! [S: nat,P: list_P559422087at_nat,T: nat,E: product_prod_nat_nat] :
( ( isSimp1359852763pacity @ c @ S @ P @ T )
=> ( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
=> ~ ( member701585322at_nat @ ( product_swap_nat_nat @ E ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ).
% isSPath_nt_parallel
thf(fact_199_discrete,axiom,
( ord_less_nat
= ( ^ [A2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ) ) ).
% discrete
thf(fact_200_swap__swap,axiom,
! [P: product_prod_nat_nat] :
( ( product_swap_nat_nat @ ( product_swap_nat_nat @ P ) )
= P ) ).
% swap_swap
thf(fact_201_swap__simp,axiom,
! [X: produc1695820582at_nat,Y: nat > nat > nat] :
( ( produc411855757at_nat @ ( produc365964781at_nat @ X @ Y ) )
= ( produc1682925677at_nat @ Y @ X ) ) ).
% swap_simp
thf(fact_202_swap__simp,axiom,
! [X: product_prod_nat_nat,Y: nat] :
( ( produc2019146714at_nat @ ( produc947540346at_nat @ X @ Y ) )
= ( produc1933845336at_nat @ Y @ X ) ) ).
% swap_simp
thf(fact_203_swap__simp,axiom,
! [X: nat > nat > nat,Y: produc1695820582at_nat] :
( ( produc1728816653at_nat @ ( produc1682925677at_nat @ X @ Y ) )
= ( produc365964781at_nat @ Y @ X ) ) ).
% swap_simp
thf(fact_204_swap__simp,axiom,
! [X: nat,Y: product_prod_nat_nat] :
( ( produc857968056at_nat @ ( produc1933845336at_nat @ X @ Y ) )
= ( produc947540346at_nat @ Y @ X ) ) ).
% swap_simp
thf(fact_205_swap__simp,axiom,
! [X: nat,Y: nat] :
( ( product_swap_nat_nat @ ( product_Pair_nat_nat @ X @ Y ) )
= ( product_Pair_nat_nat @ Y @ X ) ) ).
% swap_simp
thf(fact_206_bounded__Max__nat,axiom,
! [P3: nat > $o,X: nat,M5: nat] :
( ( P3 @ X )
=> ( ! [X3: nat] :
( ( P3 @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) )
=> ~ ! [M4: nat] :
( ( P3 @ M4 )
=> ~ ! [X4: nat] :
( ( P3 @ X4 )
=> ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_207_fold__atLeastAtMost__nat_Ocases,axiom,
! [X: produc1864880952at_nat] :
~ ! [F2: nat > product_prod_nat_nat > product_prod_nat_nat,A5: nat,B5: nat,Acc: product_prod_nat_nat] :
( X
!= ( produc696256106at_nat @ F2 @ ( produc869658639at_nat @ A5 @ ( produc1933845336at_nat @ B5 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_208_fold__atLeastAtMost__nat_Ocases,axiom,
! [X: produc2082277813at_nat] :
~ ! [F2: nat > nat > nat,A5: nat,B5: nat,Acc: nat] :
( X
!= ( produc1682925677at_nat @ F2 @ ( produc1933845336at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_209_Graph_OisSPath__nt__parallel,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat,T: nat,E: product_prod_nat_nat] :
( ( isSimp1359852763pacity @ C @ S @ P @ T )
=> ( ( member701585322at_nat @ E @ ( set_Pr2131844118at_nat @ P ) )
=> ~ ( member701585322at_nat @ ( product_swap_nat_nat @ E ) @ ( set_Pr2131844118at_nat @ P ) ) ) ) ).
% Graph.isSPath_nt_parallel
thf(fact_210_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_211_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_212_isPath_Osimps_I1_J,axiom,
! [U: nat,V: nat] :
( ( isPath_capacity @ c @ U @ nil_Pr1308055047at_nat @ V )
= ( U = V ) ) ).
% isPath.simps(1)
thf(fact_213_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_214_simplePath__empty__conv,axiom,
! [S: nat,T: nat] :
( ( isSimp1359852763pacity @ c @ S @ nil_Pr1308055047at_nat @ T )
= ( S = T ) ) ).
% simplePath_empty_conv
thf(fact_215_simplePath__same__conv,axiom,
! [S: nat,P: list_P559422087at_nat] :
( ( isSimp1359852763pacity @ c @ S @ P @ S )
= ( P = nil_Pr1308055047at_nat ) ) ).
% simplePath_same_conv
thf(fact_216_Graph_OsimplePath__empty__conv,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,T: nat] :
( ( isSimp1359852763pacity @ C @ S @ nil_Pr1308055047at_nat @ T )
= ( S = T ) ) ).
% Graph.simplePath_empty_conv
thf(fact_217_Graph_OsimplePath__same__conv,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,P: list_P559422087at_nat] :
( ( isSimp1359852763pacity @ C @ S @ P @ S )
= ( P = nil_Pr1308055047at_nat ) ) ).
% Graph.simplePath_same_conv
thf(fact_218_Graph_OisPath_Osimps_I1_J,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,V: nat] :
( ( isPath_capacity @ C @ U @ nil_Pr1308055047at_nat @ V )
= ( U = V ) ) ).
% Graph.isPath.simps(1)
thf(fact_219_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_220_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : Y5 = Z )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_221_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_222_linorder__wlog,axiom,
! [P3: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P3 @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P3 @ B5 @ A5 )
=> ( P3 @ A5 @ B5 ) )
=> ( P3 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_223_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_224_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_225_order__class_Oorder_Oantisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_226_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_227_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_228_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : Y5 = Z )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_229_antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_230_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_231_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_232_le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% le_cases
thf(fact_233_eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% eq_refl
thf(fact_234_linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linear
thf(fact_235_antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_236_eq__iff,axiom,
( ( ^ [Y5: nat,Z: nat] : Y5 = Z )
= ( ^ [X2: nat,Y6: nat] :
( ( ord_less_eq_nat @ X2 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ).
% eq_iff
thf(fact_237_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_238_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_239_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_240_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_241_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_242_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_243_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_244_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_245_linorder__less__wlog,axiom,
! [P3: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P3 @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P3 @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P3 @ B5 @ A5 )
=> ( P3 @ A5 @ B5 ) )
=> ( P3 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_246_exists__least__iff,axiom,
( ( ^ [P7: nat > $o] :
? [X5: nat] : ( P7 @ X5 ) )
= ( ^ [P8: nat > $o] :
? [N2: nat] :
( ( P8 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P8 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_247_less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% less_imp_not_less
thf(fact_248_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_249_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_250_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_251_less__imp__triv,axiom,
! [X: nat,Y: nat,P3: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P3 ) ) ).
% less_imp_triv
thf(fact_252_less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% less_imp_not_eq2
thf(fact_253_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
% Conjectures (1)
thf(conj_0,conjecture,
( ( min_dist_capacity @ c @ u @ t )
= ( plus_plus_nat @ one_one_nat @ ( size_s1990949619at_nat @ p2 ) ) ) ).
%------------------------------------------------------------------------------