TPTP Problem File: ITP048^2.p
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%------------------------------------------------------------------------------
% File : ITP048^2 : 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 : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 356 ( 82 unt; 55 typ; 0 def)
% Number of atoms : 868 ( 171 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3538 ( 62 ~; 10 |; 48 &;2953 @)
% ( 0 <=>; 465 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 135 ( 135 >; 0 *; 0 +; 0 <<)
% Number of symbols : 53 ( 52 usr; 12 con; 0-5 aty)
% Number of variables : 1002 ( 61 ^; 872 !; 34 ?;1002 :)
% ( 35 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:15:44.477
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_tf_capacity,type,
capacity: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (49)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[A: $tType] : $o ).
thf(sy_c_Graph_OGraph_Oconnected,type,
connected:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > $o ) ).
thf(sy_c_Graph_OGraph_Odist,type,
dist:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > nat > $o ) ).
thf(sy_c_Graph_OGraph_OisPath,type,
isPath:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
thf(sy_c_Graph_OGraph_OisShortestPath,type,
isShortestPath:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
thf(sy_c_Graph_OGraph_OisSimplePath,type,
isSimplePath:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
thf(sy_c_Graph_OGraph_Omin__dist,type,
min_dist:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > nat > nat ) ).
thf(sy_c_Graph_OGraph_OreachableNodes,type,
reachableNodes:
!>[Capacity: $tType] : ( ( ( product_prod @ nat @ nat ) > Capacity ) > nat > ( set @ nat ) ) ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $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 @ ( product_prod @ nat @ nat ) ).
thf(sy_v_p1____,type,
p1: list @ ( product_prod @ nat @ nat ) ).
thf(sy_v_p1a____,type,
p1a: list @ ( product_prod @ nat @ nat ) ).
thf(sy_v_p2_H____,type,
p2: list @ ( product_prod @ nat @ nat ) ).
thf(sy_v_p2_Ha____,type,
p2_a: list @ ( product_prod @ nat @ nat ) ).
thf(sy_v_p_H,type,
p3: list @ ( product_prod @ nat @ 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 (256)
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_size @ ( list @ ( product_prod @ nat @ 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_size @ ( list @ ( product_prod @ nat @ 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,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
= ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_cancel_left
thf(fact_5_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
= ( ord_less @ A @ A2 @ 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,
isShortestPath @ capacity @ 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: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
= ( B = C ) ) ) ).
% add_left_cancel
thf(fact_12_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( 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 @ ( product_prod @ nat @ nat )] : ( isPath @ capacity @ c @ U @ P @ V ) ) ) ).
% connected_def
thf(fact_16_shortestPath__is__path,axiom,
! [U: nat,P2: list @ ( product_prod @ nat @ nat ),V: nat] :
( ( isShortestPath @ capacity @ 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 @ ( product_prod @ nat @ nat )] :
~ ( isShortestPath @ capacity @ 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 @ ( product_prod @ nat @ nat ),V: nat] :
( ( isPath @ capacity @ c @ U @ P2 @ V )
=> ( dist @ capacity @ c @ U @ ( size_size @ ( list @ ( product_prod @ nat @ 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 @ ( product_prod @ nat @ nat )] :
( ( isPath @ capacity @ c @ V @ P @ V3 )
& ( ( size_size @ ( list @ ( product_prod @ nat @ 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 @ ( product_prod @ nat @ nat ),V: nat] :
( ( isShortestPath @ capacity @ c @ U @ P2 @ V )
= ( ( isPath @ capacity @ c @ U @ P2 @ V )
& ! [P4: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ capacity @ c @ U @ P4 @ V )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 ) @ ( size_size @ ( list @ ( product_prod @ nat @ 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 @ ( product_prod @ nat @ nat ),V: nat] :
( ( isShortestPath @ capacity @ c @ U @ P2 @ V )
= ( ( isPath @ capacity @ c @ U @ P2 @ V )
& ( ( size_size @ ( list @ ( product_prod @ nat @ 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: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
= ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_cancel_right
thf(fact_31_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
= ( ord_less_eq @ A @ A2 @ 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: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
=> ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_imp_le_right
thf(fact_43_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
=> ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_imp_le_left
thf(fact_44_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B2: A] :
? [C2: A] :
( B2
= ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% le_iff_add
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P5: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P5 ) )
= ( P5 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P5: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P5 @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P5 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B3: $tType,A: $tType,F: A > B3,G: A > B3] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add_right_mono
thf(fact_50_less__eqE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ~ ! [C3: A] :
( B
!= ( plus_plus @ A @ A2 @ C3 ) ) ) ) ).
% less_eqE
thf(fact_51_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% add_left_mono
thf(fact_52_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_mono
thf(fact_53_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_54_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_55_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_56_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_57_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_58_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_59_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_60_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_61_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_62_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_63_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_64_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_65_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_66_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_67_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_68_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_69_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_70_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_71_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_72_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_73_measure__induct__rule,axiom,
! [B3: $tType,A: $tType] :
( ( wellorder @ B3 )
=> ! [F: A > B3,P5: A > $o,A2: A] :
( ! [X: A] :
( ! [Y2: A] :
( ( ord_less @ B3 @ ( F @ Y2 ) @ ( F @ X ) )
=> ( P5 @ Y2 ) )
=> ( P5 @ X ) )
=> ( P5 @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_74_measure__induct,axiom,
! [B3: $tType,A: $tType] :
( ( wellorder @ B3 )
=> ! [F: A > B3,P5: A > $o,A2: A] :
( ! [X: A] :
( ! [Y2: A] :
( ( ord_less @ B3 @ ( F @ Y2 ) @ ( F @ X ) )
=> ( P5 @ Y2 ) )
=> ( P5 @ X ) )
=> ( P5 @ A2 ) ) ) ).
% measure_induct
thf(fact_75_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
=> ( B = C ) ) ) ).
% add_right_imp_eq
thf(fact_76_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
=> ( B = C ) ) ) ).
% add_left_imp_eq
thf(fact_77_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B: A,A2: A,C: A] :
( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add.left_commute
thf(fact_78_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,B2: A] : ( plus_plus @ A @ B2 @ A3 ) ) ) ) ).
% add.commute
thf(fact_79_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B = C ) ) ) ).
% add.right_cancel
thf(fact_80_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
= ( B = C ) ) ) ).
% add.left_cancel
thf(fact_81_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add.assoc
thf(fact_82_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B4: A,K: A,B: A,A2: A] :
( ( B4
= ( plus_plus @ A @ K @ B ) )
=> ( ( plus_plus @ A @ A2 @ B4 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% group_cancel.add2
thf(fact_83_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A,K: A,A2: A,B: A] :
( ( A4
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A4 @ B )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B ) ) ) ) ) ).
% group_cancel.add1
thf(fact_84_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_85_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_86_infinite__descent__measure,axiom,
! [A: $tType,P5: A > $o,V4: A > nat,X3: A] :
( ! [X: A] :
( ~ ( P5 @ X )
=> ? [Y2: A] :
( ( ord_less @ nat @ ( V4 @ Y2 ) @ ( V4 @ X ) )
& ~ ( P5 @ Y2 ) ) )
=> ( P5 @ X3 ) ) ).
% infinite_descent_measure
thf(fact_87_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_88_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_89_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_90_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_91_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less @ nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_92_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_93_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_94_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_95_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X3: A,Y3: A] :
( ( ( size_size @ A @ X3 )
!= ( size_size @ A @ Y3 ) )
=> ( X3 != Y3 ) ) ) ).
% size_neq_size_imp_neq
thf(fact_96_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_less_le_mono
thf(fact_97_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_le_less_mono
thf(fact_98_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_99_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_100_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_101_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
=> ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_imp_less_right
thf(fact_102_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A2: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
=> ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_imp_less_left
thf(fact_103_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) ) ) ) ).
% add_strict_right_mono
thf(fact_104_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) ) ) ) ).
% add_strict_left_mono
thf(fact_105_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A )
=> ! [A2: A,B: A,C: A,D: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ D ) ) ) ) ) ).
% add_strict_mono
thf(fact_106_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_107_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_108_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_109_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_110_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_111_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_112_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_113_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_114_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_115_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_116_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_117_reachableNodes__def,axiom,
! [U: nat] :
( ( reachableNodes @ capacity @ c @ U )
= ( collect @ nat @ ( connected @ capacity @ c @ U ) ) ) ).
% reachableNodes_def
thf(fact_118_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_119_Graph_OisShortestPath__min__dist__def,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( isShortestPath @ Capacity )
= ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U2: nat,P: list @ ( product_prod @ nat @ nat ),V5: nat] :
( ( isPath @ Capacity @ C2 @ U2 @ P @ V5 )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P )
= ( min_dist @ Capacity @ C2 @ U2 @ V5 ) ) ) ) ) ) ).
% Graph.isShortestPath_min_dist_def
thf(fact_120_Graph_OisShortestPath__def,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( isShortestPath @ Capacity )
= ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U2: nat,P: list @ ( product_prod @ nat @ nat ),V5: nat] :
( ( isPath @ Capacity @ C2 @ U2 @ P @ V5 )
& ! [P4: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ Capacity @ C2 @ U2 @ P4 @ V5 )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P4 ) ) ) ) ) ) ) ).
% Graph.isShortestPath_def
thf(fact_121_Graph_Omin__distI2,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,V3: nat,Q: nat > $o] :
( ( connected @ Capacity @ C @ V @ V3 )
=> ( ! [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_122_isShortestPath__alt,axiom,
! [U: nat,P2: list @ ( product_prod @ nat @ nat ),V: nat] :
( ( isShortestPath @ capacity @ c @ U @ P2 @ V )
= ( ( isSimplePath @ capacity @ c @ U @ P2 @ V )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 )
= ( min_dist @ capacity @ c @ U @ V ) ) ) ) ).
% isShortestPath_alt
thf(fact_123_Graph_Odist__def,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( dist @ Capacity )
= ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,V5: nat,D4: nat,V6: nat] :
? [P: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ Capacity @ C2 @ V5 @ P @ V6 )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P )
= D4 ) ) ) ) ) ).
% Graph.dist_def
thf(fact_124_Graph_OisPath__distD,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P2: list @ ( product_prod @ nat @ nat ),V: nat] :
( ( isPath @ Capacity @ C @ U @ P2 @ V )
=> ( dist @ Capacity @ C @ U @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 ) @ V ) ) ) ).
% Graph.isPath_distD
thf(fact_125_Graph_Omin__dist__is__dist,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,V3: nat] :
( ( connected @ Capacity @ C @ V @ V3 )
=> ( dist @ Capacity @ C @ V @ ( min_dist @ Capacity @ C @ V @ V3 ) @ V3 ) ) ) ).
% Graph.min_dist_is_dist
thf(fact_126_Graph_Omin__dist__split_I1_J,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
( ( dist @ Capacity @ C @ U @ D1 @ W )
=> ( ( dist @ Capacity @ C @ W @ D2 @ V )
=> ( ( ( min_dist @ Capacity @ C @ U @ V )
= ( plus_plus @ nat @ D1 @ D2 ) )
=> ( ( min_dist @ Capacity @ C @ U @ W )
= D1 ) ) ) ) ) ).
% Graph.min_dist_split(1)
thf(fact_127_isSPath__pathLE,axiom,
! [S: nat,P2: list @ ( product_prod @ nat @ nat ),T: nat] :
( ( isPath @ capacity @ c @ S @ P2 @ T )
=> ? [P6: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ capacity @ c @ S @ P6 @ T ) ) ).
% isSPath_pathLE
thf(fact_128_shortestPath__is__simple,axiom,
! [S: nat,P2: list @ ( product_prod @ nat @ nat ),T: nat] :
( ( isShortestPath @ capacity @ c @ S @ P2 @ T )
=> ( isSimplePath @ capacity @ c @ S @ P2 @ T ) ) ).
% shortestPath_is_simple
thf(fact_129_Graph_OreachableNodes_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( reachableNodes @ Capacity )
= ( reachableNodes @ Capacity ) ) ) ).
% Graph.reachableNodes.cong
thf(fact_130_Graph_OisSimplePath_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( isSimplePath @ Capacity )
= ( isSimplePath @ Capacity ) ) ) ).
% Graph.isSimplePath.cong
thf(fact_131_Graph_OisSPath__pathLE,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P2: list @ ( product_prod @ nat @ nat ),T: nat] :
( ( isPath @ Capacity @ C @ S @ P2 @ T )
=> ? [P6: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ Capacity @ C @ S @ P6 @ T ) ) ) ).
% Graph.isSPath_pathLE
thf(fact_132_Graph_OshortestPath__is__simple,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,S: nat,P2: list @ ( product_prod @ nat @ nat ),T: nat] :
( ( isShortestPath @ Capacity @ C @ S @ P2 @ T )
=> ( isSimplePath @ Capacity @ C @ S @ P2 @ T ) ) ) ).
% Graph.shortestPath_is_simple
thf(fact_133_Graph_OreachableNodes__def,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( reachableNodes @ Capacity )
= ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U2: nat] : ( collect @ nat @ ( connected @ Capacity @ C2 @ U2 ) ) ) ) ) ).
% Graph.reachableNodes_def
thf(fact_134_Graph_Omin__dist__def,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( min_dist @ Capacity )
= ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,V5: nat,V6: nat] :
( ord_Least @ nat
@ ^ [D4: nat] : ( dist @ Capacity @ C2 @ V5 @ D4 @ V6 ) ) ) ) ) ).
% Graph.min_dist_def
thf(fact_135_Graph_OisShortestPath__alt,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( isShortestPath @ Capacity )
= ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U2: nat,P: list @ ( product_prod @ nat @ nat ),V5: nat] :
( ( isSimplePath @ Capacity @ C2 @ U2 @ P @ V5 )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P )
= ( min_dist @ Capacity @ C2 @ U2 @ V5 ) ) ) ) ) ) ).
% Graph.isShortestPath_alt
thf(fact_136_Graph_OisPath_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( isPath @ Capacity )
= ( isPath @ Capacity ) ) ) ).
% Graph.isPath.cong
thf(fact_137_Graph_Omin__dist_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( min_dist @ Capacity )
= ( min_dist @ Capacity ) ) ) ).
% Graph.min_dist.cong
thf(fact_138_Graph_Oconnected__refl,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat] : ( connected @ Capacity @ C @ V @ V ) ) ).
% Graph.connected_refl
thf(fact_139_Graph_Oconnected_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( connected @ Capacity )
= ( connected @ Capacity ) ) ) ).
% Graph.connected.cong
thf(fact_140_Graph_Odist_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( dist @ Capacity )
= ( dist @ Capacity ) ) ) ).
% Graph.dist.cong
thf(fact_141_Graph_OisShortestPath_Ocong,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( isShortestPath @ Capacity )
= ( isShortestPath @ Capacity ) ) ) ).
% Graph.isShortestPath.cong
thf(fact_142_Graph_Odist__trans,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
( ( dist @ Capacity @ C @ U @ D1 @ W )
=> ( ( dist @ Capacity @ C @ W @ D2 @ V )
=> ( dist @ Capacity @ C @ U @ ( plus_plus @ nat @ D1 @ D2 ) @ V ) ) ) ) ).
% Graph.dist_trans
thf(fact_143_Graph_Oconnected__def,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( connected @ Capacity )
= ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,U2: nat,V5: nat] :
? [P: list @ ( product_prod @ nat @ nat )] : ( isPath @ Capacity @ C2 @ U2 @ P @ V5 ) ) ) ) ).
% Graph.connected_def
thf(fact_144_Graph_OshortestPath__is__path,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,P2: list @ ( product_prod @ nat @ nat ),V: nat] :
( ( isShortestPath @ Capacity @ C @ U @ P2 @ V )
=> ( isPath @ Capacity @ C @ U @ P2 @ V ) ) ) ).
% Graph.shortestPath_is_path
thf(fact_145_Graph_Oconnected__by__dist,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ( ( connected @ Capacity )
= ( ^ [C2: ( product_prod @ nat @ nat ) > Capacity,V5: nat,V6: nat] :
? [D4: nat] : ( dist @ Capacity @ C2 @ V5 @ D4 @ V6 ) ) ) ) ).
% Graph.connected_by_dist
thf(fact_146_Graph_Oconnected__distI,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,D: nat,V3: nat] :
( ( dist @ Capacity @ C @ V @ D @ V3 )
=> ( connected @ Capacity @ C @ V @ V3 ) ) ) ).
% Graph.connected_distI
thf(fact_147_Graph_Oobtain__shortest__path,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,V: nat] :
( ( connected @ Capacity @ C @ U @ V )
=> ~ ! [P3: list @ ( product_prod @ nat @ nat )] :
~ ( isShortestPath @ Capacity @ C @ U @ P3 @ V ) ) ) ).
% Graph.obtain_shortest_path
thf(fact_148_Graph_Omin__dist__less,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [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_149_Graph_Omin__dist__le,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [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_150_Graph_Omin__distI__eq,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,D: nat,V3: nat] :
( ( dist @ Capacity @ C @ V @ D @ V3 )
=> ( ! [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_151_Graph_Omin__dist__minD,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,V: nat,D: nat,V3: nat] :
( ( dist @ Capacity @ C @ V @ D @ V3 )
=> ( ord_less_eq @ nat @ ( min_dist @ Capacity @ C @ V @ V3 ) @ D ) ) ) ).
% Graph.min_dist_minD
thf(fact_152_Graph_Omin__dist__split_I2_J,axiom,
! [Capacity: $tType] :
( ( linordered_idom @ Capacity )
=> ! [C: ( product_prod @ nat @ nat ) > Capacity,U: nat,D1: nat,W: nat,D2: nat,V: nat] :
( ( dist @ Capacity @ C @ U @ D1 @ W )
=> ( ( dist @ Capacity @ C @ W @ D2 @ V )
=> ( ( ( min_dist @ Capacity @ C @ U @ V )
= ( plus_plus @ nat @ D1 @ D2 ) )
=> ( ( min_dist @ Capacity @ C @ W @ V )
= D2 ) ) ) ) ) ).
% Graph.min_dist_split(2)
thf(fact_153__C1_Oprems_C_I3_J,axiom,
isPath @ a @ c2 @ ua @ p2_a @ t ).
% "1.prems"(3)
thf(fact_154_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P5: A > $o,K: A,F: A > nat,N: nat] :
( ( P5 @ K )
=> ( ! [X: A] :
( ( P5 @ X )
=> ? [Y2: A] :
( ( P5 @ Y2 )
& ~ ( ord_less_eq @ nat @ ( F @ Y2 ) @ ( F @ X ) ) ) )
=> ? [Y: A] :
( ( P5 @ Y )
& ~ ( ord_less @ nat @ ( F @ Y ) @ ( plus_plus @ nat @ ( F @ K ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_155_not__less__Least,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [K: A,P5: A > $o] :
( ( ord_less @ A @ K @ ( ord_Least @ A @ P5 ) )
=> ~ ( P5 @ K ) ) ) ).
% not_less_Least
thf(fact_156_Least__le,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P5: A > $o,K: A] :
( ( P5 @ K )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P5 ) @ K ) ) ) ).
% Least_le
thf(fact_157_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_158_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_159_g_H_Oobtain__shortest__path,axiom,
! [U: nat,V: nat] :
( ( connected @ a @ c2 @ U @ V )
=> ~ ! [P3: list @ ( product_prod @ nat @ nat )] :
~ ( isShortestPath @ a @ c2 @ U @ P3 @ V ) ) ).
% g'.obtain_shortest_path
thf(fact_160_g_H_OshortestPath__is__simple,axiom,
! [S: nat,P2: list @ ( product_prod @ nat @ nat ),T: nat] :
( ( isShortestPath @ a @ c2 @ S @ P2 @ T )
=> ( isSimplePath @ a @ c2 @ S @ P2 @ T ) ) ).
% g'.shortestPath_is_simple
thf(fact_161_g_H_OreachableNodes__def,axiom,
! [U: nat] :
( ( reachableNodes @ a @ c2 @ U )
= ( collect @ nat @ ( connected @ a @ c2 @ U ) ) ) ).
% g'.reachableNodes_def
thf(fact_162_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_163_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_164_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_165_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_166_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_167_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_168_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_169_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_170_g_H_Oconnected__def,axiom,
! [U: nat,V: nat] :
( ( connected @ a @ c2 @ U @ V )
= ( ? [P: list @ ( product_prod @ nat @ nat )] : ( isPath @ a @ c2 @ U @ P @ V ) ) ) ).
% g'.connected_def
thf(fact_171_g_H_OisSPath__pathLE,axiom,
! [S: nat,P2: list @ ( product_prod @ nat @ nat ),T: nat] :
( ( isPath @ a @ c2 @ S @ P2 @ T )
=> ? [P6: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ a @ c2 @ S @ P6 @ T ) ) ).
% g'.isSPath_pathLE
thf(fact_172_g_H_OshortestPath__is__path,axiom,
! [U: nat,P2: list @ ( product_prod @ nat @ nat ),V: nat] :
( ( isShortestPath @ a @ c2 @ U @ P2 @ V )
=> ( isPath @ a @ c2 @ U @ P2 @ V ) ) ).
% g'.shortestPath_is_path
thf(fact_173_g_H_OisShortestPath__alt,axiom,
! [U: nat,P2: list @ ( product_prod @ nat @ nat ),V: nat] :
( ( isShortestPath @ a @ c2 @ U @ P2 @ V )
= ( ( isSimplePath @ a @ c2 @ U @ P2 @ V )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 )
= ( min_dist @ a @ c2 @ U @ V ) ) ) ) ).
% g'.isShortestPath_alt
thf(fact_174_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_175_g_H_Odist__def,axiom,
! [V: nat,D: nat,V3: nat] :
( ( dist @ a @ c2 @ V @ D @ V3 )
= ( ? [P: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ a @ c2 @ V @ P @ V3 )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P )
= D ) ) ) ) ).
% g'.dist_def
thf(fact_176_g_H_OisPath__distD,axiom,
! [U: nat,P2: list @ ( product_prod @ nat @ nat ),V: nat] :
( ( isPath @ a @ c2 @ U @ P2 @ V )
=> ( dist @ a @ c2 @ U @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 ) @ V ) ) ).
% g'.isPath_distD
thf(fact_177_g_H_OisShortestPath__min__dist__def,axiom,
! [U: nat,P2: list @ ( product_prod @ nat @ nat ),V: nat] :
( ( isShortestPath @ a @ c2 @ U @ P2 @ V )
= ( ( isPath @ a @ c2 @ U @ P2 @ V )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 )
= ( min_dist @ a @ c2 @ U @ V ) ) ) ) ).
% g'.isShortestPath_min_dist_def
thf(fact_178_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_179_g_H_OisShortestPath__def,axiom,
! [U: nat,P2: list @ ( product_prod @ nat @ nat ),V: nat] :
( ( isShortestPath @ a @ c2 @ U @ P2 @ V )
= ( ( isPath @ a @ c2 @ U @ P2 @ V )
& ! [P4: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ a @ c2 @ U @ P4 @ V )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P2 ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ P4 ) ) ) ) ) ).
% g'.isShortestPath_def
thf(fact_180_assms_I5_J,axiom,
isPath @ a @ c2 @ s @ p3 @ t ).
% assms(5)
thf(fact_181__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_182_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_183_g_H_Oconnected__refl,axiom,
! [V: nat] : ( connected @ a @ c2 @ V @ V ) ).
% g'.connected_refl
thf(fact_184_P2_H,axiom,
isPath @ a @ c2 @ ua @ p2_a @ t ).
% P2'
thf(fact_185_le__funD,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 )
=> ! [F: A > B3,G: A > B3,X3: A] :
( ( ord_less_eq @ ( A > B3 ) @ F @ G )
=> ( ord_less_eq @ B3 @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funD
thf(fact_186_le__funE,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 )
=> ! [F: A > B3,G: A > B3,X3: A] :
( ( ord_less_eq @ ( A > B3 ) @ F @ G )
=> ( ord_less_eq @ B3 @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funE
thf(fact_187_le__funI,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 )
=> ! [F: A > B3,G: A > B3] :
( ! [X: A] : ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) )
=> ( ord_less_eq @ ( A > B3 ) @ F @ G ) ) ) ).
% le_funI
thf(fact_188_le__fun__def,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 )
=> ( ( ord_less_eq @ ( A > B3 ) )
= ( ^ [F2: A > B3,G2: A > B3] :
! [X2: A] : ( ord_less_eq @ B3 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_189_order__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 )
& ( order @ A ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq @ B3 @ B @ C )
=> ( ! [X: B3,Y: B3] :
( ( ord_less_eq @ B3 @ X @ Y )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_190_order__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 )
& ( order @ A ) )
=> ! [A2: A,B: A,F: A > C4,C: C4] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ C4 @ ( F @ B ) @ C )
=> ( ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ C4 @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ C4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_191_ord__eq__le__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 )
& ( ord @ A ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq @ B3 @ B @ C )
=> ( ! [X: B3,Y: B3] :
( ( ord_less_eq @ B3 @ X @ Y )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_192_ord__le__eq__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 )
& ( ord @ A ) )
=> ! [A2: A,B: A,F: A > B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B3 @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_193_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [X2: A,Y5: A] :
( ( ord_less_eq @ A @ X2 @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_194_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ) ).
% antisym
thf(fact_195_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% linear
thf(fact_196_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( X3 = Y3 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% eq_refl
thf(fact_197_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% le_cases
thf(fact_198_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_199_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_200_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv
thf(fact_201_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_202_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B: A,C: A] :
( ( A2 = B )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_203_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_204_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ A2 )
=> ( A2 = B ) ) ) ) ).
% order_class.order.antisym
thf(fact_205_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_206_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_207_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P5: A > A > $o,A2: A,B: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P5 @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P5 @ B5 @ A5 )
=> ( P5 @ A5 @ B5 ) )
=> ( P5 @ A2 @ B ) ) ) ) ).
% linorder_wlog
thf(fact_208_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ C @ B )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_209_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A3: A,B2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_210_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B )
=> ( A2 = B ) ) ) ) ).
% dual_order.antisym
thf(fact_211_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ B @ A2 )
=> ( A2 != B ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_212_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ( A2 != B ) ) ) ).
% order.strict_implies_not_eq
thf(fact_213_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
= ( ( ord_less @ A @ Y3 @ X3 )
| ( X3 = Y3 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_214_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( ord_less @ A @ C @ B )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_215_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P5: A > A > $o,A2: A,B: A] :
( ! [A5: A,B5: A] :
( ( ord_less @ A @ A5 @ B5 )
=> ( P5 @ A5 @ B5 ) )
=> ( ! [A5: A] : ( P5 @ A5 @ A5 )
=> ( ! [A5: A,B5: A] :
( ( P5 @ B5 @ A5 )
=> ( P5 @ A5 @ B5 ) )
=> ( P5 @ A2 @ B ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_216_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P7: A > $o] :
? [X4: A] : ( P7 @ X4 ) )
= ( ^ [P8: A > $o] :
? [N2: A] :
( ( P8 @ N2 )
& ! [M2: A] :
( ( ord_less @ A @ M2 @ N2 )
=> ~ ( P8 @ M2 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_217_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% less_imp_not_less
thf(fact_218_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_219_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_220_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ( X3 != Y3 )
=> ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% linorder_cases
thf(fact_221_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,P5: $o] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ X3 )
=> P5 ) ) ) ).
% less_imp_triv
thf(fact_222_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( Y3 != X3 ) ) ) ).
% less_imp_not_eq2
thf(fact_223_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X3: A] :
( ~ ( ord_less @ A @ Y3 @ X3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv3
thf(fact_224_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P5: A > $o,A2: A] :
( ! [X: A] :
( ! [Y2: A] :
( ( ord_less @ A @ Y2 @ X )
=> ( P5 @ Y2 ) )
=> ( P5 @ X ) )
=> ( P5 @ A2 ) ) ) ).
% less_induct
thf(fact_225_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% less_not_sym
thf(fact_226_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ) ).
% less_imp_not_eq
thf(fact_227_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ B @ A2 )
=> ~ ( ord_less @ A @ A2 @ B ) ) ) ).
% dual_order.asym
thf(fact_228_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( B = C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_229_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B: A,C: A] :
( ( A2 = B )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_230_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 ) ) ).
% less_irrefl
thf(fact_231_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
| ( X3 = Y3 )
| ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% less_linear
thf(fact_232_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% less_trans
thf(fact_233_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% less_asym'
thf(fact_234_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% less_asym
thf(fact_235_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ) ).
% less_imp_neq
thf(fact_236_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ? [Z3: A] :
( ( ord_less @ A @ X3 @ Z3 )
& ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% dense
thf(fact_237_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% order.asym
thf(fact_238_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( X3 != Y3 )
= ( ( ord_less @ A @ X3 @ Y3 )
| ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% neq_iff
thf(fact_239_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y3: A] :
( ( X3 != Y3 )
=> ( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% neqE
thf(fact_240_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X3: A] :
? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% gt_ex
thf(fact_241_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X3: A] :
? [Y: A] : ( ord_less @ A @ Y @ X3 ) ) ).
% lt_ex
thf(fact_242_order__less__subst2,axiom,
! [A: $tType,C4: $tType] :
( ( ( order @ C4 )
& ( order @ A ) )
=> ! [A2: A,B: A,F: A > C4,C: C4] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ C4 @ ( F @ B ) @ C )
=> ( ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less @ C4 @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less @ C4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_243_order__less__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 )
& ( order @ A ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( ord_less @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less @ B3 @ B @ C )
=> ( ! [X: B3,Y: B3] :
( ( ord_less @ B3 @ X @ Y )
=> ( ord_less @ A @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_244_ord__less__eq__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 )
& ( ord @ A ) )
=> ! [A2: A,B: A,F: A > B3,C: B3] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less @ B3 @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_245_ord__eq__less__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 )
& ( ord @ A ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less @ B3 @ B @ C )
=> ( ! [X: B3,Y: B3] :
( ( ord_less @ B3 @ X @ Y )
=> ( ord_less @ A @ ( F @ X ) @ ( F @ Y ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_246_ex__has__least__nat,axiom,
! [A: $tType,P5: A > $o,K: A,M: A > nat] :
( ( P5 @ K )
=> ? [X: A] :
( ( P5 @ X )
& ! [Y2: A] :
( ( P5 @ Y2 )
=> ( ord_less_eq @ nat @ ( M @ X ) @ ( M @ Y2 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_247_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P5: A > $o,A2: A,Q: A > $o] :
( ( P5 @ A2 )
=> ( ! [X: A] :
( ( P5 @ X )
=> ( Q @ X ) )
=> ( Q @ ( ord_Least @ A @ P5 ) ) ) ) ) ).
% LeastI2
thf(fact_248_LeastI__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P5: A > $o] :
( ? [X_12: A] : ( P5 @ X_12 )
=> ( P5 @ ( ord_Least @ A @ P5 ) ) ) ) ).
% LeastI_ex
thf(fact_249_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P5: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P5 @ X_12 )
=> ( ! [X: A] :
( ( P5 @ X )
=> ( Q @ X ) )
=> ( Q @ ( ord_Least @ A @ P5 ) ) ) ) ) ).
% LeastI2_ex
thf(fact_250_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P5: A > $o,K: A] :
( ( P5 @ K )
=> ( P5 @ ( ord_Least @ A @ P5 ) ) ) ) ).
% LeastI
thf(fact_251_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( A2 != B )
=> ( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less @ A @ A2 @ B ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_252_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_253_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_254_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_255_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% order.strict_implies_order
% Subclasses (14)
thf(subcl_Rings_Olinordered__idom___HOL_Otype,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( type @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Orderings_Oord,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Orderings_Oorder,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( order @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Ogroup__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( group_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Orderings_Olinorder,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( linorder @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Orderings_Opreorder,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( preorder @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Osemigroup__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( semigroup_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Ocomm__monoid__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( comm_monoid_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oab__semigroup__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ab_semigroup_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Ocancel__semigroup__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( cancel_semigroup_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ordere779506340up_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ordere236663937imp_le @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oordered__cancel__ab__semigroup__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ordere223160158up_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( strict2144017051up_add @ A ) ) ).
% Type constructors (28)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A6: $tType,A7: $tType] :
( ( preorder @ A7 )
=> ( preorder @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 )
=> ( order @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 )
=> ( ord @ ( A6 > A7 ) ) ) ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A6: $tType] : ( preorder @ ( set @ A6 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
ord @ $o ).
thf(tcon_List_Olist___Nat_Osize_11,axiom,
! [A6: $tType] : ( size @ ( list @ A6 ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_12,axiom,
! [A6: $tType,A7: $tType] : ( size @ ( product_prod @ A6 @ A7 ) ) ).
% Free types (2)
thf(tfree_0,hypothesis,
linordered_idom @ capacity ).
thf(tfree_1,hypothesis,
linordered_idom @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less @ nat @ ( min_dist @ capacity @ c @ s @ t ) @ ( plus_plus @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1a ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2_a ) ) ).
%------------------------------------------------------------------------------