TPTP Problem File: ITP048^1.p
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%------------------------------------------------------------------------------
% File : ITP048^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer EdmondsKarp_Termination_Abstract problem prob_146__7582994_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_146__7582994_1 [Des21]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 315 ( 110 unt; 41 typ; 0 def)
% Number of atoms : 715 ( 188 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 2319 ( 60 ~; 10 |; 41 &;1894 @)
% ( 0 <=>; 314 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 160 ( 160 >; 0 *; 0 +; 0 <<)
% Number of symbols : 36 ( 35 usr; 12 con; 0-4 aty)
% Number of variables : 826 ( 87 ^; 705 !; 34 ?; 826 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:30:24.947
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_tf__capacity,type,
capacity: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (35)
thf(sy_c_Graph_OGraph_Oconnected_001tf__a,type,
connected_a: ( product_prod_nat_nat > a ) > nat > nat > $o ).
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__a,type,
dist_a: ( product_prod_nat_nat > a ) > nat > 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__a,type,
isPath_a: ( product_prod_nat_nat > a ) > nat > list_P559422087at_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__a,type,
isShortestPath_a: ( product_prod_nat_nat > a ) > 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__a,type,
isSimplePath_a: ( product_prod_nat_nat > a ) > 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__a,type,
min_dist_a: ( product_prod_nat_nat > a ) > nat > nat > nat ).
thf(sy_c_Graph_OGraph_Omin__dist_001tf__capacity,type,
min_dist_capacity: ( product_prod_nat_nat > capacity ) > nat > nat > nat ).
thf(sy_c_Graph_OGraph_OreachableNodes_001tf__a,type,
reachableNodes_a: ( product_prod_nat_nat > a ) > nat > set_nat ).
thf(sy_c_Graph_OGraph_OreachableNodes_001tf__capacity,type,
reacha1693770334pacity: ( product_prod_nat_nat > capacity ) > nat > set_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > 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_OLeast_001t__Nat__Onat,type,
ord_Least_nat: ( nat > $o ) > 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_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_v_c,type,
c: product_prod_nat_nat > capacity ).
thf(sy_v_c_H,type,
c2: product_prod_nat_nat > a ).
thf(sy_v_p,type,
p: list_P559422087at_nat ).
thf(sy_v_p1____,type,
p1: list_P559422087at_nat ).
thf(sy_v_p1a____,type,
p1a: list_P559422087at_nat ).
thf(sy_v_p2_H____,type,
p2: list_P559422087at_nat ).
thf(sy_v_p2_Ha____,type,
p2_a: 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_ua____,type,
ua: nat ).
thf(sy_v_v____,type,
v: nat ).
thf(sy_v_va____,type,
va: nat ).
% Relevant facts (273)
thf(fact_0__092_060open_062min__dist_As_At_A_061_Amin__dist_As_Au_A_L_Amin__dist_Au_At_092_060close_062,axiom,
( ( min_dist_capacity @ c @ s @ t )
= ( plus_plus_nat @ ( min_dist_capacity @ c @ s @ ua ) @ ( min_dist_capacity @ c @ ua @ t ) ) ) ).
% \<open>min_dist s t = min_dist s u + min_dist u t\<close>
thf(fact_1__092_060open_062min__dist_As_Au_A_060_Alength_Ap1_092_060close_062,axiom,
ord_less_nat @ ( min_dist_capacity @ c @ s @ ua ) @ ( size_s1990949619at_nat @ p1a ) ).
% \<open>min_dist s u < length p1\<close>
thf(fact_2__092_060open_062min__dist_Au_At_A_092_060le_062_Alength_Ap2_H_092_060close_062,axiom,
ord_less_eq_nat @ ( min_dist_capacity @ c @ ua @ t ) @ ( size_s1990949619at_nat @ p2_a ) ).
% \<open>min_dist u t \<le> length p2'\<close>
thf(fact_3_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_4_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_5_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_6__C1_Oprems_C_I2_J,axiom,
isPath_capacity @ c @ s @ p1a @ va ).
% "1.prems"(2)
thf(fact_7_SP,axiom,
isShor1936442771pacity @ c @ s @ p @ t ).
% SP
thf(fact_8_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_9_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_10_min__dist__less,axiom,
! [Src: nat,V: nat,D: nat,D3: nat] :
( ( connected_capacity @ c @ Src @ V )
=> ( ( ( min_dist_capacity @ c @ Src @ V )
= D )
=> ( ( ord_less_nat @ D3 @ D )
=> ? [V2: nat] :
( ( connected_capacity @ c @ Src @ V2 )
& ( ( min_dist_capacity @ c @ Src @ V2 )
= D3 ) ) ) ) ) ).
% min_dist_less
thf(fact_11_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_12_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_13__092_060open_062isPath_Au_Ap2_H_At_092_060close_062,axiom,
isPath_capacity @ c @ ua @ p2_a @ t ).
% \<open>isPath u p2' t\<close>
thf(fact_14_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_15_connected__def,axiom,
! [U: nat,V: nat] :
( ( connected_capacity @ c @ U @ V )
= ( ? [P: list_P559422087at_nat] : ( isPath_capacity @ c @ U @ P @ V ) ) ) ).
% connected_def
thf(fact_16_shortestPath__is__path,axiom,
! [U: nat,P2: list_P559422087at_nat,V: nat] :
( ( isShor1936442771pacity @ c @ U @ P2 @ V )
=> ( isPath_capacity @ c @ U @ P2 @ V ) ) ).
% shortestPath_is_path
thf(fact_17_connected__by__dist,axiom,
! [V: nat,V3: nat] :
( ( connected_capacity @ c @ V @ V3 )
= ( ? [D4: nat] : ( dist_capacity @ c @ V @ D4 @ V3 ) ) ) ).
% connected_by_dist
thf(fact_18_obtain__shortest__path,axiom,
! [U: nat,V: nat] :
( ( connected_capacity @ c @ U @ V )
=> ~ ! [P3: list_P559422087at_nat] :
~ ( isShor1936442771pacity @ c @ U @ P3 @ V ) ) ).
% obtain_shortest_path
thf(fact_19_min__dist__le,axiom,
! [Src: nat,V: nat,D3: nat] :
( ( connected_capacity @ c @ Src @ V )
=> ( ( ord_less_eq_nat @ D3 @ ( min_dist_capacity @ c @ Src @ V ) )
=> ? [V2: nat] :
( ( connected_capacity @ c @ Src @ V2 )
& ( ( min_dist_capacity @ c @ Src @ V2 )
= D3 ) ) ) ) ).
% min_dist_le
thf(fact_20_min__dist__minD,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist_capacity @ c @ V @ D @ V3 )
=> ( ord_less_eq_nat @ ( min_dist_capacity @ c @ V @ V3 ) @ D ) ) ).
% min_dist_minD
thf(fact_21_min__distI__eq,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist_capacity @ c @ V @ D @ V3 )
=> ( ! [D5: nat] :
( ( dist_capacity @ c @ V @ D5 @ V3 )
=> ( ord_less_eq_nat @ D @ D5 ) )
=> ( ( min_dist_capacity @ c @ V @ V3 )
= D ) ) ) ).
% min_distI_eq
thf(fact_22_isPath__distD,axiom,
! [U: nat,P2: list_P559422087at_nat,V: nat] :
( ( isPath_capacity @ c @ U @ P2 @ V )
=> ( dist_capacity @ c @ U @ ( size_s1990949619at_nat @ P2 ) @ V ) ) ).
% isPath_distD
thf(fact_23_dist__def,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist_capacity @ c @ V @ D @ V3 )
= ( ? [P: list_P559422087at_nat] :
( ( isPath_capacity @ c @ V @ P @ V3 )
& ( ( size_s1990949619at_nat @ P )
= D ) ) ) ) ).
% dist_def
thf(fact_24__092_060open_062isPath_As_Ap1_____Av_____092_060close_062,axiom,
isPath_capacity @ c @ s @ p1 @ v ).
% \<open>isPath s p1__ v__\<close>
thf(fact_25_min__dist__is__dist,axiom,
! [V: nat,V3: nat] :
( ( connected_capacity @ c @ V @ V3 )
=> ( dist_capacity @ c @ V @ ( min_dist_capacity @ c @ V @ V3 ) @ V3 ) ) ).
% min_dist_is_dist
thf(fact_26_isShortestPath__def,axiom,
! [U: nat,P2: list_P559422087at_nat,V: nat] :
( ( isShor1936442771pacity @ c @ U @ P2 @ V )
= ( ( isPath_capacity @ c @ U @ P2 @ V )
& ! [P4: list_P559422087at_nat] :
( ( isPath_capacity @ c @ U @ P4 @ V )
=> ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P2 ) @ ( size_s1990949619at_nat @ P4 ) ) ) ) ) ).
% isShortestPath_def
thf(fact_27_min__distI2,axiom,
! [V: nat,V3: nat,Q: nat > $o] :
( ( connected_capacity @ c @ V @ V3 )
=> ( ! [D6: nat] :
( ( dist_capacity @ c @ V @ D6 @ V3 )
=> ( ! [D7: nat] :
( ( dist_capacity @ c @ V @ D7 @ V3 )
=> ( ord_less_eq_nat @ D6 @ D7 ) )
=> ( Q @ D6 ) ) )
=> ( Q @ ( min_dist_capacity @ c @ V @ V3 ) ) ) ) ).
% min_distI2
thf(fact_28_isShortestPath__min__dist__def,axiom,
! [U: nat,P2: list_P559422087at_nat,V: nat] :
( ( isShor1936442771pacity @ c @ U @ P2 @ V )
= ( ( isPath_capacity @ c @ U @ P2 @ V )
& ( ( size_s1990949619at_nat @ P2 )
= ( min_dist_capacity @ c @ U @ V ) ) ) ) ).
% isShortestPath_min_dist_def
thf(fact_29__092_060open_062connected_As_Au_092_060close_062,axiom,
connected_capacity @ c @ s @ ua ).
% \<open>connected s u\<close>
thf(fact_30_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_31_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_32_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_33_P1,axiom,
isPath_capacity @ c @ s @ p1a @ va ).
% P1
thf(fact_34_connected__refl,axiom,
! [V: nat] : ( connected_capacity @ c @ V @ V ) ).
% connected_refl
thf(fact_35_connected__distI,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist_capacity @ c @ V @ D @ V3 )
=> ( connected_capacity @ c @ V @ V3 ) ) ).
% connected_distI
thf(fact_36_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_37_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_38_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_39_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_40_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_41_Nat_Oex__has__greatest__nat,axiom,
! [P5: nat > $o,K: nat,B: nat] :
( ( P5 @ K )
=> ( ! [Y: nat] :
( ( P5 @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X: nat] :
( ( P5 @ X )
& ! [Y2: nat] :
( ( P5 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_42_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_43_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_44_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_45_mem__Collect__eq,axiom,
! [A: nat,P5: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P5 ) )
= ( P5 @ A ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A3: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [P5: nat > $o,Q: nat > $o] :
( ! [X: nat] :
( ( P5 @ X )
= ( Q @ X ) )
=> ( ( collect_nat @ P5 )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_48_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_49_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_50_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_51_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_52_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_53_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_54_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_55_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_56_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_57_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_58_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_59_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_60_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_61_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_62_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_63_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_64_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_65_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_66_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_67_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_68_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_69_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_70_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_71_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_72_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_73_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_74_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_75_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_76_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_77_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_78_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_79_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_80_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_81_linorder__neqE__nat,axiom,
! [X3: nat,Y3: nat] :
( ( X3 != Y3 )
=> ( ~ ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_82_infinite__descent,axiom,
! [P5: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P5 @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P5 @ M3 ) ) )
=> ( P5 @ N ) ) ).
% infinite_descent
thf(fact_83_nat__less__induct,axiom,
! [P5: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P5 @ M3 ) )
=> ( P5 @ N3 ) )
=> ( P5 @ N ) ) ).
% nat_less_induct
thf(fact_84_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_85_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_86_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_87_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_88_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_89_size__neq__size__imp__neq,axiom,
! [X3: list_P559422087at_nat,Y3: list_P559422087at_nat] :
( ( ( size_s1990949619at_nat @ X3 )
!= ( size_s1990949619at_nat @ Y3 ) )
=> ( X3 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_90_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_91_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_92_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_93_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_94_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_95_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_96_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_97_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_98_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_99_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_100_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_101_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_102_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_103_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_104_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_105_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_106_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_107_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_108_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_109_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_110_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_111_reachableNodes__def,axiom,
! [U: nat] :
( ( reacha1693770334pacity @ c @ U )
= ( collect_nat @ ( connected_capacity @ c @ U ) ) ) ).
% reachableNodes_def
thf(fact_112_min__dist__def,axiom,
! [V: nat,V3: nat] :
( ( min_dist_capacity @ c @ V @ V3 )
= ( ord_Least_nat
@ ^ [D4: nat] : ( dist_capacity @ c @ V @ D4 @ V3 ) ) ) ).
% min_dist_def
thf(fact_113_Graph_OisShortestPath__min__dist__def,axiom,
( isShor1936442771pacity
= ( ^ [C2: product_prod_nat_nat > capacity,U2: nat,P: list_P559422087at_nat,V4: nat] :
( ( isPath_capacity @ C2 @ U2 @ P @ V4 )
& ( ( size_s1990949619at_nat @ P )
= ( min_dist_capacity @ C2 @ U2 @ V4 ) ) ) ) ) ).
% Graph.isShortestPath_min_dist_def
thf(fact_114_Graph_OisShortestPath__min__dist__def,axiom,
( isShortestPath_a
= ( ^ [C2: product_prod_nat_nat > a,U2: nat,P: list_P559422087at_nat,V4: nat] :
( ( isPath_a @ C2 @ U2 @ P @ V4 )
& ( ( size_s1990949619at_nat @ P )
= ( min_dist_a @ C2 @ U2 @ V4 ) ) ) ) ) ).
% Graph.isShortestPath_min_dist_def
thf(fact_115_Graph_OisShortestPath__def,axiom,
( isShor1936442771pacity
= ( ^ [C2: product_prod_nat_nat > capacity,U2: nat,P: list_P559422087at_nat,V4: nat] :
( ( isPath_capacity @ C2 @ U2 @ P @ V4 )
& ! [P4: list_P559422087at_nat] :
( ( isPath_capacity @ C2 @ U2 @ P4 @ V4 )
=> ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P ) @ ( size_s1990949619at_nat @ P4 ) ) ) ) ) ) ).
% Graph.isShortestPath_def
thf(fact_116_Graph_OisShortestPath__def,axiom,
( isShortestPath_a
= ( ^ [C2: product_prod_nat_nat > a,U2: nat,P: list_P559422087at_nat,V4: nat] :
( ( isPath_a @ C2 @ U2 @ P @ V4 )
& ! [P4: list_P559422087at_nat] :
( ( isPath_a @ C2 @ U2 @ P4 @ V4 )
=> ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P ) @ ( size_s1990949619at_nat @ P4 ) ) ) ) ) ) ).
% Graph.isShortestPath_def
thf(fact_117_Graph_Omin__distI2,axiom,
! [C: product_prod_nat_nat > capacity,V: nat,V3: nat,Q: nat > $o] :
( ( connected_capacity @ C @ V @ V3 )
=> ( ! [D6: nat] :
( ( dist_capacity @ C @ V @ D6 @ V3 )
=> ( ! [D7: nat] :
( ( dist_capacity @ C @ V @ D7 @ V3 )
=> ( ord_less_eq_nat @ D6 @ D7 ) )
=> ( Q @ D6 ) ) )
=> ( Q @ ( min_dist_capacity @ C @ V @ V3 ) ) ) ) ).
% Graph.min_distI2
thf(fact_118_Graph_Omin__distI2,axiom,
! [C: product_prod_nat_nat > a,V: nat,V3: nat,Q: nat > $o] :
( ( connected_a @ C @ V @ V3 )
=> ( ! [D6: nat] :
( ( dist_a @ C @ V @ D6 @ V3 )
=> ( ! [D7: nat] :
( ( dist_a @ C @ V @ D7 @ V3 )
=> ( ord_less_eq_nat @ D6 @ D7 ) )
=> ( Q @ D6 ) ) )
=> ( Q @ ( min_dist_a @ C @ V @ V3 ) ) ) ) ).
% Graph.min_distI2
thf(fact_119_isShortestPath__alt,axiom,
! [U: nat,P2: list_P559422087at_nat,V: nat] :
( ( isShor1936442771pacity @ c @ U @ P2 @ V )
= ( ( isSimp1359852763pacity @ c @ U @ P2 @ V )
& ( ( size_s1990949619at_nat @ P2 )
= ( min_dist_capacity @ c @ U @ V ) ) ) ) ).
% isShortestPath_alt
thf(fact_120_Graph_Odist__def,axiom,
( dist_capacity
= ( ^ [C2: product_prod_nat_nat > capacity,V4: nat,D4: nat,V5: nat] :
? [P: list_P559422087at_nat] :
( ( isPath_capacity @ C2 @ V4 @ P @ V5 )
& ( ( size_s1990949619at_nat @ P )
= D4 ) ) ) ) ).
% Graph.dist_def
thf(fact_121_Graph_Odist__def,axiom,
( dist_a
= ( ^ [C2: product_prod_nat_nat > a,V4: nat,D4: nat,V5: nat] :
? [P: list_P559422087at_nat] :
( ( isPath_a @ C2 @ V4 @ P @ V5 )
& ( ( size_s1990949619at_nat @ P )
= D4 ) ) ) ) ).
% Graph.dist_def
thf(fact_122_Graph_OisPath__distD,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,P2: list_P559422087at_nat,V: nat] :
( ( isPath_capacity @ C @ U @ P2 @ V )
=> ( dist_capacity @ C @ U @ ( size_s1990949619at_nat @ P2 ) @ V ) ) ).
% Graph.isPath_distD
thf(fact_123_Graph_OisPath__distD,axiom,
! [C: product_prod_nat_nat > a,U: nat,P2: list_P559422087at_nat,V: nat] :
( ( isPath_a @ C @ U @ P2 @ V )
=> ( dist_a @ C @ U @ ( size_s1990949619at_nat @ P2 ) @ V ) ) ).
% Graph.isPath_distD
thf(fact_124_Graph_Omin__dist__is__dist,axiom,
! [C: product_prod_nat_nat > capacity,V: nat,V3: nat] :
( ( connected_capacity @ C @ V @ V3 )
=> ( dist_capacity @ C @ V @ ( min_dist_capacity @ C @ V @ V3 ) @ V3 ) ) ).
% Graph.min_dist_is_dist
thf(fact_125_Graph_Omin__dist__is__dist,axiom,
! [C: product_prod_nat_nat > a,V: nat,V3: nat] :
( ( connected_a @ C @ V @ V3 )
=> ( dist_a @ C @ V @ ( min_dist_a @ C @ V @ V3 ) @ V3 ) ) ).
% Graph.min_dist_is_dist
thf(fact_126_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_127_Graph_Omin__dist__split_I1_J,axiom,
! [C: product_prod_nat_nat > a,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
( ( dist_a @ C @ U @ D1 @ W )
=> ( ( dist_a @ C @ W @ D2 @ V )
=> ( ( ( min_dist_a @ C @ U @ V )
= ( plus_plus_nat @ D1 @ D2 ) )
=> ( ( min_dist_a @ C @ U @ W )
= D1 ) ) ) ) ).
% Graph.min_dist_split(1)
thf(fact_128_isSPath__pathLE,axiom,
! [S: nat,P2: list_P559422087at_nat,T: nat] :
( ( isPath_capacity @ c @ S @ P2 @ T )
=> ? [P6: list_P559422087at_nat] : ( isSimp1359852763pacity @ c @ S @ P6 @ T ) ) ).
% isSPath_pathLE
thf(fact_129_shortestPath__is__simple,axiom,
! [S: nat,P2: list_P559422087at_nat,T: nat] :
( ( isShor1936442771pacity @ c @ S @ P2 @ T )
=> ( isSimp1359852763pacity @ c @ S @ P2 @ T ) ) ).
% shortestPath_is_simple
thf(fact_130_Graph_OreachableNodes_Ocong,axiom,
reacha1693770334pacity = reacha1693770334pacity ).
% Graph.reachableNodes.cong
thf(fact_131_Graph_OreachableNodes_Ocong,axiom,
reachableNodes_a = reachableNodes_a ).
% Graph.reachableNodes.cong
thf(fact_132_Graph_OisSimplePath_Ocong,axiom,
isSimp1359852763pacity = isSimp1359852763pacity ).
% Graph.isSimplePath.cong
thf(fact_133_Graph_OisSimplePath_Ocong,axiom,
isSimplePath_a = isSimplePath_a ).
% Graph.isSimplePath.cong
thf(fact_134_Graph_OisSPath__pathLE,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,P2: list_P559422087at_nat,T: nat] :
( ( isPath_capacity @ C @ S @ P2 @ T )
=> ? [P6: list_P559422087at_nat] : ( isSimp1359852763pacity @ C @ S @ P6 @ T ) ) ).
% Graph.isSPath_pathLE
thf(fact_135_Graph_OisSPath__pathLE,axiom,
! [C: product_prod_nat_nat > a,S: nat,P2: list_P559422087at_nat,T: nat] :
( ( isPath_a @ C @ S @ P2 @ T )
=> ? [P6: list_P559422087at_nat] : ( isSimplePath_a @ C @ S @ P6 @ T ) ) ).
% Graph.isSPath_pathLE
thf(fact_136_Graph_OshortestPath__is__simple,axiom,
! [C: product_prod_nat_nat > capacity,S: nat,P2: list_P559422087at_nat,T: nat] :
( ( isShor1936442771pacity @ C @ S @ P2 @ T )
=> ( isSimp1359852763pacity @ C @ S @ P2 @ T ) ) ).
% Graph.shortestPath_is_simple
thf(fact_137_Graph_OshortestPath__is__simple,axiom,
! [C: product_prod_nat_nat > a,S: nat,P2: list_P559422087at_nat,T: nat] :
( ( isShortestPath_a @ C @ S @ P2 @ T )
=> ( isSimplePath_a @ C @ S @ P2 @ T ) ) ).
% Graph.shortestPath_is_simple
thf(fact_138_Graph_OreachableNodes__def,axiom,
( reacha1693770334pacity
= ( ^ [C2: product_prod_nat_nat > capacity,U2: nat] : ( collect_nat @ ( connected_capacity @ C2 @ U2 ) ) ) ) ).
% Graph.reachableNodes_def
thf(fact_139_Graph_OreachableNodes__def,axiom,
( reachableNodes_a
= ( ^ [C2: product_prod_nat_nat > a,U2: nat] : ( collect_nat @ ( connected_a @ C2 @ U2 ) ) ) ) ).
% Graph.reachableNodes_def
thf(fact_140_Graph_Omin__dist__def,axiom,
( min_dist_capacity
= ( ^ [C2: product_prod_nat_nat > capacity,V4: nat,V5: nat] :
( ord_Least_nat
@ ^ [D4: nat] : ( dist_capacity @ C2 @ V4 @ D4 @ V5 ) ) ) ) ).
% Graph.min_dist_def
thf(fact_141_Graph_Omin__dist__def,axiom,
( min_dist_a
= ( ^ [C2: product_prod_nat_nat > a,V4: nat,V5: nat] :
( ord_Least_nat
@ ^ [D4: nat] : ( dist_a @ C2 @ V4 @ D4 @ V5 ) ) ) ) ).
% Graph.min_dist_def
thf(fact_142_Graph_OisShortestPath__alt,axiom,
( isShor1936442771pacity
= ( ^ [C2: product_prod_nat_nat > capacity,U2: nat,P: list_P559422087at_nat,V4: nat] :
( ( isSimp1359852763pacity @ C2 @ U2 @ P @ V4 )
& ( ( size_s1990949619at_nat @ P )
= ( min_dist_capacity @ C2 @ U2 @ V4 ) ) ) ) ) ).
% Graph.isShortestPath_alt
thf(fact_143_Graph_OisShortestPath__alt,axiom,
( isShortestPath_a
= ( ^ [C2: product_prod_nat_nat > a,U2: nat,P: list_P559422087at_nat,V4: nat] :
( ( isSimplePath_a @ C2 @ U2 @ P @ V4 )
& ( ( size_s1990949619at_nat @ P )
= ( min_dist_a @ C2 @ U2 @ V4 ) ) ) ) ) ).
% Graph.isShortestPath_alt
thf(fact_144_Graph_OisPath_Ocong,axiom,
isPath_capacity = isPath_capacity ).
% Graph.isPath.cong
thf(fact_145_Graph_OisPath_Ocong,axiom,
isPath_a = isPath_a ).
% Graph.isPath.cong
thf(fact_146_Graph_Omin__dist_Ocong,axiom,
min_dist_capacity = min_dist_capacity ).
% Graph.min_dist.cong
thf(fact_147_Graph_Omin__dist_Ocong,axiom,
min_dist_a = min_dist_a ).
% Graph.min_dist.cong
thf(fact_148_Graph_Oconnected__refl,axiom,
! [C: product_prod_nat_nat > capacity,V: nat] : ( connected_capacity @ C @ V @ V ) ).
% Graph.connected_refl
thf(fact_149_Graph_Oconnected__refl,axiom,
! [C: product_prod_nat_nat > a,V: nat] : ( connected_a @ C @ V @ V ) ).
% Graph.connected_refl
thf(fact_150_Graph_Oconnected_Ocong,axiom,
connected_capacity = connected_capacity ).
% Graph.connected.cong
thf(fact_151_Graph_Oconnected_Ocong,axiom,
connected_a = connected_a ).
% Graph.connected.cong
thf(fact_152_Graph_Odist_Ocong,axiom,
dist_capacity = dist_capacity ).
% Graph.dist.cong
thf(fact_153_Graph_Odist_Ocong,axiom,
dist_a = dist_a ).
% Graph.dist.cong
thf(fact_154_Graph_OisShortestPath_Ocong,axiom,
isShor1936442771pacity = isShor1936442771pacity ).
% Graph.isShortestPath.cong
thf(fact_155_Graph_OisShortestPath_Ocong,axiom,
isShortestPath_a = isShortestPath_a ).
% Graph.isShortestPath.cong
thf(fact_156_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_157_Graph_Odist__trans,axiom,
! [C: product_prod_nat_nat > a,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
( ( dist_a @ C @ U @ D1 @ W )
=> ( ( dist_a @ C @ W @ D2 @ V )
=> ( dist_a @ C @ U @ ( plus_plus_nat @ D1 @ D2 ) @ V ) ) ) ).
% Graph.dist_trans
thf(fact_158_Graph_Oconnected__def,axiom,
( connected_capacity
= ( ^ [C2: product_prod_nat_nat > capacity,U2: nat,V4: nat] :
? [P: list_P559422087at_nat] : ( isPath_capacity @ C2 @ U2 @ P @ V4 ) ) ) ).
% Graph.connected_def
thf(fact_159_Graph_Oconnected__def,axiom,
( connected_a
= ( ^ [C2: product_prod_nat_nat > a,U2: nat,V4: nat] :
? [P: list_P559422087at_nat] : ( isPath_a @ C2 @ U2 @ P @ V4 ) ) ) ).
% Graph.connected_def
thf(fact_160_Graph_OshortestPath__is__path,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,P2: list_P559422087at_nat,V: nat] :
( ( isShor1936442771pacity @ C @ U @ P2 @ V )
=> ( isPath_capacity @ C @ U @ P2 @ V ) ) ).
% Graph.shortestPath_is_path
thf(fact_161_Graph_OshortestPath__is__path,axiom,
! [C: product_prod_nat_nat > a,U: nat,P2: list_P559422087at_nat,V: nat] :
( ( isShortestPath_a @ C @ U @ P2 @ V )
=> ( isPath_a @ C @ U @ P2 @ V ) ) ).
% Graph.shortestPath_is_path
thf(fact_162_Graph_Oconnected__by__dist,axiom,
( connected_capacity
= ( ^ [C2: product_prod_nat_nat > capacity,V4: nat,V5: nat] :
? [D4: nat] : ( dist_capacity @ C2 @ V4 @ D4 @ V5 ) ) ) ).
% Graph.connected_by_dist
thf(fact_163_Graph_Oconnected__by__dist,axiom,
( connected_a
= ( ^ [C2: product_prod_nat_nat > a,V4: nat,V5: nat] :
? [D4: nat] : ( dist_a @ C2 @ V4 @ D4 @ V5 ) ) ) ).
% Graph.connected_by_dist
thf(fact_164_Graph_Oconnected__distI,axiom,
! [C: product_prod_nat_nat > capacity,V: nat,D: nat,V3: nat] :
( ( dist_capacity @ C @ V @ D @ V3 )
=> ( connected_capacity @ C @ V @ V3 ) ) ).
% Graph.connected_distI
thf(fact_165_Graph_Oconnected__distI,axiom,
! [C: product_prod_nat_nat > a,V: nat,D: nat,V3: nat] :
( ( dist_a @ C @ V @ D @ V3 )
=> ( connected_a @ C @ V @ V3 ) ) ).
% Graph.connected_distI
thf(fact_166_Graph_Oobtain__shortest__path,axiom,
! [C: product_prod_nat_nat > capacity,U: nat,V: nat] :
( ( connected_capacity @ C @ U @ V )
=> ~ ! [P3: list_P559422087at_nat] :
~ ( isShor1936442771pacity @ C @ U @ P3 @ V ) ) ).
% Graph.obtain_shortest_path
thf(fact_167_Graph_Oobtain__shortest__path,axiom,
! [C: product_prod_nat_nat > a,U: nat,V: nat] :
( ( connected_a @ C @ U @ V )
=> ~ ! [P3: list_P559422087at_nat] :
~ ( isShortestPath_a @ C @ U @ P3 @ V ) ) ).
% Graph.obtain_shortest_path
thf(fact_168_Graph_Omin__dist__less,axiom,
! [C: product_prod_nat_nat > capacity,Src: nat,V: nat,D: nat,D3: nat] :
( ( connected_capacity @ C @ Src @ V )
=> ( ( ( min_dist_capacity @ C @ Src @ V )
= D )
=> ( ( ord_less_nat @ D3 @ D )
=> ? [V2: nat] :
( ( connected_capacity @ C @ Src @ V2 )
& ( ( min_dist_capacity @ C @ Src @ V2 )
= D3 ) ) ) ) ) ).
% Graph.min_dist_less
thf(fact_169_Graph_Omin__dist__less,axiom,
! [C: product_prod_nat_nat > a,Src: nat,V: nat,D: nat,D3: nat] :
( ( connected_a @ C @ Src @ V )
=> ( ( ( min_dist_a @ C @ Src @ V )
= D )
=> ( ( ord_less_nat @ D3 @ D )
=> ? [V2: nat] :
( ( connected_a @ C @ Src @ V2 )
& ( ( min_dist_a @ C @ Src @ V2 )
= D3 ) ) ) ) ) ).
% Graph.min_dist_less
thf(fact_170_Graph_Omin__dist__le,axiom,
! [C: product_prod_nat_nat > capacity,Src: nat,V: nat,D3: nat] :
( ( connected_capacity @ C @ Src @ V )
=> ( ( ord_less_eq_nat @ D3 @ ( min_dist_capacity @ C @ Src @ V ) )
=> ? [V2: nat] :
( ( connected_capacity @ C @ Src @ V2 )
& ( ( min_dist_capacity @ C @ Src @ V2 )
= D3 ) ) ) ) ).
% Graph.min_dist_le
thf(fact_171_Graph_Omin__dist__le,axiom,
! [C: product_prod_nat_nat > a,Src: nat,V: nat,D3: nat] :
( ( connected_a @ C @ Src @ V )
=> ( ( ord_less_eq_nat @ D3 @ ( min_dist_a @ C @ Src @ V ) )
=> ? [V2: nat] :
( ( connected_a @ C @ Src @ V2 )
& ( ( min_dist_a @ C @ Src @ V2 )
= D3 ) ) ) ) ).
% Graph.min_dist_le
thf(fact_172_Graph_Omin__distI__eq,axiom,
! [C: product_prod_nat_nat > capacity,V: nat,D: nat,V3: nat] :
( ( dist_capacity @ C @ V @ D @ V3 )
=> ( ! [D5: nat] :
( ( dist_capacity @ C @ V @ D5 @ V3 )
=> ( ord_less_eq_nat @ D @ D5 ) )
=> ( ( min_dist_capacity @ C @ V @ V3 )
= D ) ) ) ).
% Graph.min_distI_eq
thf(fact_173_Graph_Omin__distI__eq,axiom,
! [C: product_prod_nat_nat > a,V: nat,D: nat,V3: nat] :
( ( dist_a @ C @ V @ D @ V3 )
=> ( ! [D5: nat] :
( ( dist_a @ C @ V @ D5 @ V3 )
=> ( ord_less_eq_nat @ D @ D5 ) )
=> ( ( min_dist_a @ C @ V @ V3 )
= D ) ) ) ).
% Graph.min_distI_eq
thf(fact_174_Graph_Omin__dist__minD,axiom,
! [C: product_prod_nat_nat > capacity,V: nat,D: nat,V3: nat] :
( ( dist_capacity @ C @ V @ D @ V3 )
=> ( ord_less_eq_nat @ ( min_dist_capacity @ C @ V @ V3 ) @ D ) ) ).
% Graph.min_dist_minD
thf(fact_175_Graph_Omin__dist__minD,axiom,
! [C: product_prod_nat_nat > a,V: nat,D: nat,V3: nat] :
( ( dist_a @ C @ V @ D @ V3 )
=> ( ord_less_eq_nat @ ( min_dist_a @ C @ V @ V3 ) @ D ) ) ).
% Graph.min_dist_minD
thf(fact_176_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_177_Graph_Omin__dist__split_I2_J,axiom,
! [C: product_prod_nat_nat > a,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
( ( dist_a @ C @ U @ D1 @ W )
=> ( ( dist_a @ C @ W @ D2 @ V )
=> ( ( ( min_dist_a @ C @ U @ V )
= ( plus_plus_nat @ D1 @ D2 ) )
=> ( ( min_dist_a @ C @ W @ V )
= D2 ) ) ) ) ).
% Graph.min_dist_split(2)
thf(fact_178__C1_Oprems_C_I3_J,axiom,
isPath_a @ c2 @ ua @ p2_a @ t ).
% "1.prems"(3)
thf(fact_179_not__less__Least,axiom,
! [K: nat,P5: nat > $o] :
( ( ord_less_nat @ K @ ( ord_Least_nat @ P5 ) )
=> ~ ( P5 @ K ) ) ).
% not_less_Least
thf(fact_180_Least__le,axiom,
! [P5: nat > $o,K: nat] :
( ( P5 @ K )
=> ( ord_less_eq_nat @ ( ord_Least_nat @ P5 ) @ K ) ) ).
% Least_le
thf(fact_181_g_H_Oconnected__by__dist,axiom,
! [V: nat,V3: nat] :
( ( connected_a @ c2 @ V @ V3 )
= ( ? [D4: nat] : ( dist_a @ c2 @ V @ D4 @ V3 ) ) ) ).
% g'.connected_by_dist
thf(fact_182_g_H_Omin__dist__is__dist,axiom,
! [V: nat,V3: nat] :
( ( connected_a @ c2 @ V @ V3 )
=> ( dist_a @ c2 @ V @ ( min_dist_a @ c2 @ V @ V3 ) @ V3 ) ) ).
% g'.min_dist_is_dist
thf(fact_183_g_H_Oobtain__shortest__path,axiom,
! [U: nat,V: nat] :
( ( connected_a @ c2 @ U @ V )
=> ~ ! [P3: list_P559422087at_nat] :
~ ( isShortestPath_a @ c2 @ U @ P3 @ V ) ) ).
% g'.obtain_shortest_path
thf(fact_184_g_H_OshortestPath__is__simple,axiom,
! [S: nat,P2: list_P559422087at_nat,T: nat] :
( ( isShortestPath_a @ c2 @ S @ P2 @ T )
=> ( isSimplePath_a @ c2 @ S @ P2 @ T ) ) ).
% g'.shortestPath_is_simple
thf(fact_185_g_H_OreachableNodes__def,axiom,
! [U: nat] :
( ( reachableNodes_a @ c2 @ U )
= ( collect_nat @ ( connected_a @ c2 @ U ) ) ) ).
% g'.reachableNodes_def
thf(fact_186_g_H_Omin__dist__less,axiom,
! [Src: nat,V: nat,D: nat,D3: nat] :
( ( connected_a @ c2 @ Src @ V )
=> ( ( ( min_dist_a @ c2 @ Src @ V )
= D )
=> ( ( ord_less_nat @ D3 @ D )
=> ? [V2: nat] :
( ( connected_a @ c2 @ Src @ V2 )
& ( ( min_dist_a @ c2 @ Src @ V2 )
= D3 ) ) ) ) ) ).
% g'.min_dist_less
thf(fact_187_g_H_Omin__distI2,axiom,
! [V: nat,V3: nat,Q: nat > $o] :
( ( connected_a @ c2 @ V @ V3 )
=> ( ! [D6: nat] :
( ( dist_a @ c2 @ V @ D6 @ V3 )
=> ( ! [D7: nat] :
( ( dist_a @ c2 @ V @ D7 @ V3 )
=> ( ord_less_eq_nat @ D6 @ D7 ) )
=> ( Q @ D6 ) ) )
=> ( Q @ ( min_dist_a @ c2 @ V @ V3 ) ) ) ) ).
% g'.min_distI2
thf(fact_188_g_H_Omin__distI__eq,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist_a @ c2 @ V @ D @ V3 )
=> ( ! [D5: nat] :
( ( dist_a @ c2 @ V @ D5 @ V3 )
=> ( ord_less_eq_nat @ D @ D5 ) )
=> ( ( min_dist_a @ c2 @ V @ V3 )
= D ) ) ) ).
% g'.min_distI_eq
thf(fact_189_g_H_Omin__dist__le,axiom,
! [Src: nat,V: nat,D3: nat] :
( ( connected_a @ c2 @ Src @ V )
=> ( ( ord_less_eq_nat @ D3 @ ( min_dist_a @ c2 @ Src @ V ) )
=> ? [V2: nat] :
( ( connected_a @ c2 @ Src @ V2 )
& ( ( min_dist_a @ c2 @ Src @ V2 )
= D3 ) ) ) ) ).
% g'.min_dist_le
thf(fact_190_g_H_Omin__dist__minD,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist_a @ c2 @ V @ D @ V3 )
=> ( ord_less_eq_nat @ ( min_dist_a @ c2 @ V @ V3 ) @ D ) ) ).
% g'.min_dist_minD
thf(fact_191_g_H_Omin__dist__split_I2_J,axiom,
! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
( ( dist_a @ c2 @ U @ D1 @ W )
=> ( ( dist_a @ c2 @ W @ D2 @ V )
=> ( ( ( min_dist_a @ c2 @ U @ V )
= ( plus_plus_nat @ D1 @ D2 ) )
=> ( ( min_dist_a @ c2 @ W @ V )
= D2 ) ) ) ) ).
% g'.min_dist_split(2)
thf(fact_192_g_H_Omin__dist__split_I1_J,axiom,
! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
( ( dist_a @ c2 @ U @ D1 @ W )
=> ( ( dist_a @ c2 @ W @ D2 @ V )
=> ( ( ( min_dist_a @ c2 @ U @ V )
= ( plus_plus_nat @ D1 @ D2 ) )
=> ( ( min_dist_a @ c2 @ U @ W )
= D1 ) ) ) ) ).
% g'.min_dist_split(1)
thf(fact_193_g_H_Odist__trans,axiom,
! [U: nat,D1: nat,W: nat,D2: nat,V: nat] :
( ( dist_a @ c2 @ U @ D1 @ W )
=> ( ( dist_a @ c2 @ W @ D2 @ V )
=> ( dist_a @ c2 @ U @ ( plus_plus_nat @ D1 @ D2 ) @ V ) ) ) ).
% g'.dist_trans
thf(fact_194_g_H_Oconnected__def,axiom,
! [U: nat,V: nat] :
( ( connected_a @ c2 @ U @ V )
= ( ? [P: list_P559422087at_nat] : ( isPath_a @ c2 @ U @ P @ V ) ) ) ).
% g'.connected_def
thf(fact_195_g_H_OisSPath__pathLE,axiom,
! [S: nat,P2: list_P559422087at_nat,T: nat] :
( ( isPath_a @ c2 @ S @ P2 @ T )
=> ? [P6: list_P559422087at_nat] : ( isSimplePath_a @ c2 @ S @ P6 @ T ) ) ).
% g'.isSPath_pathLE
thf(fact_196_g_H_OshortestPath__is__path,axiom,
! [U: nat,P2: list_P559422087at_nat,V: nat] :
( ( isShortestPath_a @ c2 @ U @ P2 @ V )
=> ( isPath_a @ c2 @ U @ P2 @ V ) ) ).
% g'.shortestPath_is_path
thf(fact_197_g_H_OisShortestPath__alt,axiom,
! [U: nat,P2: list_P559422087at_nat,V: nat] :
( ( isShortestPath_a @ c2 @ U @ P2 @ V )
= ( ( isSimplePath_a @ c2 @ U @ P2 @ V )
& ( ( size_s1990949619at_nat @ P2 )
= ( min_dist_a @ c2 @ U @ V ) ) ) ) ).
% g'.isShortestPath_alt
thf(fact_198_g_H_Omin__dist__def,axiom,
! [V: nat,V3: nat] :
( ( min_dist_a @ c2 @ V @ V3 )
= ( ord_Least_nat
@ ^ [D4: nat] : ( dist_a @ c2 @ V @ D4 @ V3 ) ) ) ).
% g'.min_dist_def
thf(fact_199_g_H_Odist__def,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist_a @ c2 @ V @ D @ V3 )
= ( ? [P: list_P559422087at_nat] :
( ( isPath_a @ c2 @ V @ P @ V3 )
& ( ( size_s1990949619at_nat @ P )
= D ) ) ) ) ).
% g'.dist_def
thf(fact_200_g_H_OisPath__distD,axiom,
! [U: nat,P2: list_P559422087at_nat,V: nat] :
( ( isPath_a @ c2 @ U @ P2 @ V )
=> ( dist_a @ c2 @ U @ ( size_s1990949619at_nat @ P2 ) @ V ) ) ).
% g'.isPath_distD
thf(fact_201_g_H_OisShortestPath__min__dist__def,axiom,
! [U: nat,P2: list_P559422087at_nat,V: nat] :
( ( isShortestPath_a @ c2 @ U @ P2 @ V )
= ( ( isPath_a @ c2 @ U @ P2 @ V )
& ( ( size_s1990949619at_nat @ P2 )
= ( min_dist_a @ c2 @ U @ V ) ) ) ) ).
% g'.isShortestPath_min_dist_def
thf(fact_202_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_203_g_H_OisShortestPath__def,axiom,
! [U: nat,P2: list_P559422087at_nat,V: nat] :
( ( isShortestPath_a @ c2 @ U @ P2 @ V )
= ( ( isPath_a @ c2 @ U @ P2 @ V )
& ! [P4: list_P559422087at_nat] :
( ( isPath_a @ c2 @ U @ P4 @ V )
=> ( ord_less_eq_nat @ ( size_s1990949619at_nat @ P2 ) @ ( size_s1990949619at_nat @ P4 ) ) ) ) ) ).
% g'.isShortestPath_def
thf(fact_204_assms_I5_J,axiom,
isPath_a @ c2 @ s @ p3 @ t ).
% assms(5)
thf(fact_205__092_060open_062g_H_OisPath_Au_____Ap2_H_____At_092_060close_062,axiom,
isPath_a @ c2 @ u @ p2 @ t ).
% \<open>g'.isPath u__ p2'__ t\<close>
thf(fact_206_g_H_Oconnected__distI,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist_a @ c2 @ V @ D @ V3 )
=> ( connected_a @ c2 @ V @ V3 ) ) ).
% g'.connected_distI
thf(fact_207_g_H_Oconnected__refl,axiom,
! [V: nat] : ( connected_a @ c2 @ V @ V ) ).
% g'.connected_refl
thf(fact_208_P2_H,axiom,
isPath_a @ c2 @ ua @ p2_a @ t ).
% P2'
thf(fact_209_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 )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_210_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 )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_211_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_212_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_213_eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [X2: nat,Y5: nat] :
( ( ord_less_eq_nat @ X2 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).
% eq_iff
thf(fact_214_antisym,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% antisym
thf(fact_215_linear,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% linear
thf(fact_216_eq__refl,axiom,
! [X3: nat,Y3: nat] :
( ( X3 = Y3 )
=> ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% eq_refl
thf(fact_217_le__cases,axiom,
! [X3: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% le_cases
thf(fact_218_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_219_le__cases3,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_220_antisym__conv,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% antisym_conv
thf(fact_221_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_222_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_223_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_224_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_225_order__trans,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_226_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_227_linorder__wlog,axiom,
! [P5: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P5 @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P5 @ B4 @ A4 )
=> ( P5 @ A4 @ B4 ) )
=> ( P5 @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_228_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_229_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_230_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_231_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_232_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_233_not__less__iff__gr__or__eq,axiom,
! [X3: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X3 )
| ( X3 = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_234_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_235_linorder__less__wlog,axiom,
! [P5: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P5 @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P5 @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P5 @ B4 @ A4 )
=> ( P5 @ A4 @ B4 ) )
=> ( P5 @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_236_exists__least__iff,axiom,
( ( ^ [P7: nat > $o] :
? [X4: nat] : ( P7 @ X4 ) )
= ( ^ [P8: nat > $o] :
? [N2: nat] :
( ( P8 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P8 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_237_less__imp__not__less,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% less_imp_not_less
thf(fact_238_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_239_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_240_linorder__cases,axiom,
! [X3: nat,Y3: nat] :
( ~ ( ord_less_nat @ X3 @ Y3 )
=> ( ( X3 != Y3 )
=> ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% linorder_cases
thf(fact_241_less__imp__triv,axiom,
! [X3: nat,Y3: nat,P5: $o] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X3 )
=> P5 ) ) ).
% less_imp_triv
thf(fact_242_less__imp__not__eq2,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( Y3 != X3 ) ) ).
% less_imp_not_eq2
thf(fact_243_antisym__conv3,axiom,
! [Y3: nat,X3: nat] :
( ~ ( ord_less_nat @ Y3 @ X3 )
=> ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ).
% antisym_conv3
thf(fact_244_less__induct,axiom,
! [P5: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y2: nat] :
( ( ord_less_nat @ Y2 @ X )
=> ( P5 @ Y2 ) )
=> ( P5 @ X ) )
=> ( P5 @ A ) ) ).
% less_induct
thf(fact_245_less__not__sym,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% less_not_sym
thf(fact_246_less__imp__not__eq,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ).
% less_imp_not_eq
thf(fact_247_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_248_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_249_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_250_less__irrefl,axiom,
! [X3: nat] :
~ ( ord_less_nat @ X3 @ X3 ) ).
% less_irrefl
thf(fact_251_less__linear,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
| ( X3 = Y3 )
| ( ord_less_nat @ Y3 @ X3 ) ) ).
% less_linear
thf(fact_252_less__trans,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z2 )
=> ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% less_trans
thf(fact_253_less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% less_asym'
thf(fact_254_less__asym,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% less_asym
thf(fact_255_less__imp__neq,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ).
% less_imp_neq
thf(fact_256_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_257_neq__iff,axiom,
! [X3: nat,Y3: nat] :
( ( X3 != Y3 )
= ( ( ord_less_nat @ X3 @ Y3 )
| ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% neq_iff
thf(fact_258_neqE,axiom,
! [X3: nat,Y3: nat] :
( ( X3 != Y3 )
=> ( ~ ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% neqE
thf(fact_259_gt__ex,axiom,
! [X3: nat] :
? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% gt_ex
thf(fact_260_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_261_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_262_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_263_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_264_LeastI2,axiom,
! [P5: nat > $o,A: nat,Q: nat > $o] :
( ( P5 @ A )
=> ( ! [X: nat] :
( ( P5 @ X )
=> ( Q @ X ) )
=> ( Q @ ( ord_Least_nat @ P5 ) ) ) ) ).
% LeastI2
thf(fact_265_LeastI__ex,axiom,
! [P5: nat > $o] :
( ? [X_12: nat] : ( P5 @ X_12 )
=> ( P5 @ ( ord_Least_nat @ P5 ) ) ) ).
% LeastI_ex
thf(fact_266_LeastI2__ex,axiom,
! [P5: nat > $o,Q: nat > $o] :
( ? [X_12: nat] : ( P5 @ X_12 )
=> ( ! [X: nat] :
( ( P5 @ X )
=> ( Q @ X ) )
=> ( Q @ ( ord_Least_nat @ P5 ) ) ) ) ).
% LeastI2_ex
thf(fact_267_LeastI,axiom,
! [P5: nat > $o,K: nat] :
( ( P5 @ K )
=> ( P5 @ ( ord_Least_nat @ P5 ) ) ) ).
% LeastI
thf(fact_268_order_Onot__eq__order__implies__strict,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_269_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_270_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_271_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_272_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_nat @ ( min_dist_capacity @ c @ s @ t ) @ ( plus_plus_nat @ ( size_s1990949619at_nat @ p1a ) @ ( size_s1990949619at_nat @ p2_a ) ) ).
%------------------------------------------------------------------------------