TSTP Solution File: ITP048^2 by Leo-III---1.7.12
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%------------------------------------------------------------------------------
% File : Leo-III---1.7.12
% Problem : ITP048^2 : TPTP v8.2.0. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n018.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Mon May 20 22:22:23 EDT 2024
% Result : Theorem 65.45s 42.58s
% Output : Refutation 66.10s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 352
% Syntax : Number of formulae : 654 ( 156 unt; 52 typ; 0 def)
% Number of atoms : 1821 ( 331 equ; 6 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 7022 ( 126 ~; 20 |; 96 &;5859 @)
% ( 0 <=>; 921 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 210 ( 210 >; 0 *; 0 +; 0 <<)
% Number of symbols : 57 ( 53 usr; 19 con; 0-5 aty)
% Number of variables : 1936 ( 108 ^1727 !; 66 ?;1936 :)
% ( 35 !>; 0 ?*; 0 @-; 0 @+)
% Comments :
%------------------------------------------------------------------------------
thf(capacity_type,type,
capacity: $tType ).
thf(nat_type,type,
nat: $tType ).
thf(a_type,type,
a: $tType ).
thf(type_type,type,
type:
!>[TA: $tType] : $o ).
thf(size_type,type,
size:
!>[TA: $tType] : $o ).
thf(ord_type,type,
ord:
!>[TA: $tType] : $o ).
thf(order_type,type,
order:
!>[TA: $tType] : $o ).
thf(group_add_type,type,
group_add:
!>[TA: $tType] : $o ).
thf(no_bot_type,type,
no_bot:
!>[TA: $tType] : $o ).
thf(no_top_type,type,
no_top:
!>[TA: $tType] : $o ).
thf(linorder_type,type,
linorder:
!>[TA: $tType] : $o ).
thf(preorder_type,type,
preorder:
!>[TA: $tType] : $o ).
thf(wellorder_type,type,
wellorder:
!>[TA: $tType] : $o ).
thf(semigroup_add_type,type,
semigroup_add:
!>[TA: $tType] : $o ).
thf(dense_order_type,type,
dense_order:
!>[TA: $tType] : $o ).
thf(linordered_idom_type,type,
linordered_idom:
!>[TA: $tType] : $o ).
thf(comm_monoid_add_type,type,
comm_monoid_add:
!>[TA: $tType] : $o ).
thf(ab_semigroup_add_type,type,
ab_semigroup_add:
!>[TA: $tType] : $o ).
thf(cancel_semigroup_add_type,type,
cancel_semigroup_add:
!>[TA: $tType] : $o ).
thf(ordere779506340up_add_type,type,
ordere779506340up_add:
!>[TA: $tType] : $o ).
thf(canoni770627133id_add_type,type,
canoni770627133id_add:
!>[TA: $tType] : $o ).
thf(ordere236663937imp_le_type,type,
ordere236663937imp_le:
!>[TA: $tType] : $o ).
thf(ordere223160158up_add_type,type,
ordere223160158up_add:
!>[TA: $tType] : $o ).
thf(strict2144017051up_add_type,type,
strict2144017051up_add:
!>[TA: $tType] : $o ).
thf(connected_type,type,
connected:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > nat > $o ) ).
thf(dist_type,type,
dist:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > nat > nat > $o ) ).
thf(isPath_type,type,
isPath:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
thf(isShortestPath_type,type,
isShortestPath:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
thf(isSimplePath_type,type,
isSimplePath:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
thf(min_dist_type,type,
min_dist:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > nat > nat ) ).
thf(reachableNodes_type,type,
reachableNodes:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > ( set @ nat ) ) ).
thf(plus_plus_type,type,
plus_plus:
!>[TA: $tType] : ( TA > TA > TA ) ).
thf(size_size_type,type,
size_size:
!>[TA: $tType] : ( TA > nat ) ).
thf(ord_Least_type,type,
ord_Least:
!>[TA: $tType] : ( ( TA > $o ) > TA ) ).
thf(ord_less_type,type,
ord_less:
!>[TA: $tType] : ( TA > TA > $o ) ).
thf(ord_less_eq_type,type,
ord_less_eq:
!>[TA: $tType] : ( TA > TA > $o ) ).
thf(collect_type,type,
collect:
!>[TA: $tType] : ( ( TA > $o ) > ( set @ TA ) ) ).
thf(member_type,type,
member:
!>[TA: $tType] : ( TA > ( set @ TA ) > $o ) ).
thf(c_type,type,
c: ( product_prod @ nat @ nat ) > capacity ).
thf(c2_type,type,
c2: ( product_prod @ nat @ nat ) > a ).
thf(p_type,type,
p: list @ ( product_prod @ nat @ nat ) ).
thf(p1_type,type,
p1: list @ ( product_prod @ nat @ nat ) ).
thf(p1a_type,type,
p1a: list @ ( product_prod @ nat @ nat ) ).
thf(p2_type,type,
p2: list @ ( product_prod @ nat @ nat ) ).
thf(p2_a_type,type,
p2_a: list @ ( product_prod @ nat @ nat ) ).
thf(p3_type,type,
p3: list @ ( product_prod @ nat @ nat ) ).
thf(s_type,type,
s: nat ).
thf(t_type,type,
t: nat ).
thf(u_type,type,
u: nat ).
thf(ua_type,type,
ua: nat ).
thf(v_type,type,
v: nat ).
thf(va_type,type,
va: nat ).
thf(201,axiom,
! [A: nat,B: nat,C: nat] :
( ( dist @ a @ c2 @ A @ B @ C )
=> ( ! [D: nat] :
( ( dist @ a @ c2 @ A @ D @ C )
=> ( ord_less_eq @ nat @ B @ D ) )
=> ( ( min_dist @ a @ c2 @ A @ C )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_164_g_H_Omin__distI__eq) ).
thf(809,plain,
! [A: nat,B: nat,C: nat] :
( ( dist @ a @ c2 @ A @ B @ C )
=> ( ! [D: nat] :
( ( dist @ a @ c2 @ A @ D @ C )
=> ( ord_less_eq @ nat @ B @ D ) )
=> ( ( min_dist @ a @ c2 @ A @ C )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).
thf(185,axiom,
! [A: nat,B: nat] :
( ( min_dist @ capacity @ c @ A @ B )
= ( ord_Least @ nat
@ ^ [C: nat] : ( dist @ capacity @ c @ A @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_118_min__dist__def) ).
thf(760,plain,
! [A: nat,B: nat] :
( ( min_dist @ capacity @ c @ A @ B )
= ( ord_Least @ nat
@ ^ [C: nat] : ( dist @ capacity @ c @ A @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(296,axiom,
! [TA: $tType] :
( ( dense_order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ? [C: TA] :
( ( ord_less @ TA @ A @ C )
& ( ord_less @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_236_dense) ).
thf(1105,plain,
! [TA: $tType] :
( ( dense_order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ? [C: TA] :
( ( ord_less @ TA @ A @ C )
& ( ord_less @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).
thf(103,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_232_less__trans) ).
thf(555,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(226,axiom,
! [TA: $tType] :
( ( no_top @ TA )
=> ! [A: TA] :
? [B: TA] : ( ord_less @ TA @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_240_gt__ex) ).
thf(909,plain,
! [TA: $tType] :
( ( no_top @ TA )
=> ! [A: TA] :
? [B: TA] : ( ord_less @ TA @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).
thf(18,axiom,
linordered_idom @ a,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tfree_1) ).
thf(333,plain,
linordered_idom @ a,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[18]) ).
thf(249,axiom,
isShortestPath @ capacity @ c @ s @ p @ t,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_7_SP) ).
thf(977,plain,
isShortestPath @ capacity @ c @ s @ p @ t,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(286,axiom,
! [A: nat,B: nat,C: nat] :
( ( dist @ capacity @ c @ A @ B @ C )
=> ( connected @ capacity @ c @ A @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_35_connected__distI) ).
thf(1073,plain,
! [A: nat,B: nat,C: nat] :
( ( dist @ capacity @ c @ A @ B @ C )
=> ( connected @ capacity @ c @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).
thf(220,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA > TA > $o,B: TA,C: TA] :
( ! [D: TA,E: TA] :
( ( ord_less @ TA @ D @ E )
=> ( A @ D @ E ) )
=> ( ! [D: TA] : ( A @ D @ D )
=> ( ! [D: TA,E: TA] :
( ( A @ E @ D )
=> ( A @ D @ E ) )
=> ( A @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_215_linorder__less__wlog) ).
thf(873,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA > TA > $o,B: TA,C: TA] :
( ! [D: TA,E: TA] :
( ( ord_less @ TA @ D @ E )
=> ( A @ D @ E ) )
=> ( ! [D: TA] : ( A @ D @ D )
=> ( ! [D: TA,E: TA] :
( ( A @ E @ D )
=> ( A @ D @ E ) )
=> ( A @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(40,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ A )
= ( B = A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_200_antisym__conv) ).
thf(379,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ A )
= ( B = A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(63,axiom,
! [TA: $tType,TB: $tType] :
( ( ( ord @ TA )
& ( ord @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( A
= ( B @ C ) )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less_eq @ TA @ E @ F )
=> ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_191_ord__eq__le__subst) ).
thf(440,plain,
! [TA: $tType,TB: $tType] :
( ( ( ord @ TA )
& ( ord @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( A
= ( B @ C ) )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less_eq @ TA @ E @ F )
=> ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(129,axiom,
! [A: nat,B: nat] :
( ( connected @ a @ c2 @ A @ B )
=> ( dist @ a @ c2 @ A @ ( min_dist @ a @ c2 @ A @ B ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_158_g_H_Omin__dist__is__dist) ).
thf(614,plain,
! [A: nat,B: nat] :
( ( connected @ a @ c2 @ A @ B )
=> ( dist @ a @ c2 @ A @ ( min_dist @ a @ c2 @ A @ B ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(121,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isPath @ capacity @ c @ A @ B @ C )
=> ? [D: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ capacity @ c @ A @ D @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_127_isSPath__pathLE) ).
thf(597,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isPath @ capacity @ c @ A @ B @ C )
=> ? [D: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ capacity @ c @ A @ D @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(91,axiom,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ( ( plus_plus @ TA )
= ( ^ [A: TA,B: TA] : ( plus_plus @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_78_add_Ocommute) ).
thf(523,plain,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ( ( plus_plus @ TA )
= ( ^ [A: TA,B: TA] : ( plus_plus @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(114,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( min_dist @ TA )
= ( min_dist @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_137_Graph_Omin__dist_Ocong) ).
thf(582,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(273,axiom,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( dist @ a @ c2 @ A @ B @ C )
=> ( ( dist @ a @ c2 @ C @ D @ E )
=> ( dist @ a @ c2 @ A @ ( plus_plus @ nat @ B @ D ) @ E ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_169_g_H_Odist__trans) ).
thf(1041,plain,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( dist @ a @ c2 @ A @ B @ C )
=> ( ( dist @ a @ c2 @ C @ D @ E )
=> ( dist @ a @ c2 @ A @ ( plus_plus @ nat @ B @ D ) @ E ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).
thf(217,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( connected @ a @ c2 @ A @ B )
=> ( ( ( min_dist @ a @ c2 @ A @ B )
= C )
=> ( ( ord_less @ nat @ D @ C )
=> ? [E: nat] :
( ( connected @ a @ c2 @ A @ E )
& ( ( min_dist @ a @ c2 @ A @ E )
= D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_162_g_H_Omin__dist__less) ).
thf(861,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( connected @ a @ c2 @ A @ B )
=> ( ( ( min_dist @ a @ c2 @ A @ B )
= C )
=> ( ( ord_less @ nat @ D @ C )
=> ? [E: nat] :
( ( connected @ a @ c2 @ A @ E )
& ( ( min_dist @ a @ c2 @ A @ E )
= D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).
thf(95,axiom,
! [A: nat,B: nat] :
( ( ( ord_less @ nat @ A @ B )
| ( A = B ) )
=> ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_58_less__or__eq__imp__le) ).
thf(533,plain,
! [A: nat,B: nat] :
( ( ( ord_less @ nat @ A @ B )
| ( A = B ) )
=> ( ord_less_eq @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(76,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ( ( (=) @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ B @ A )
& ( ord_less_eq @ TA @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_209_dual__order_Oeq__iff) ).
thf(472,plain,
! [TA: $tType] :
( ( order @ TA )
=> ( ( (=) @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ B @ A )
& ( ord_less_eq @ TA @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(202,axiom,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A = B )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_229_ord__eq__less__trans) ).
thf(814,plain,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A = B )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).
thf(160,axiom,
semigroup_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Osemigroup__add) ).
thf(698,plain,
semigroup_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(181,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( B != A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_222_less__imp__not__eq2) ).
thf(747,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( B != A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(96,axiom,
! [A: nat,B: nat] :
( ( A != B )
= ( ( ord_less @ nat @ A @ B )
| ( ord_less @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_94_nat__neq__iff) ).
thf(538,plain,
! [A: nat,B: nat] :
( ( A != B )
= ( ( ord_less @ nat @ A @ B )
| ( ord_less @ nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(281,axiom,
! [A: nat,B: nat,C: nat] :
( ( dist @ a @ c2 @ A @ B @ C )
=> ( ord_less_eq @ nat @ ( min_dist @ a @ c2 @ A @ C ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_166_g_H_Omin__dist__minD) ).
thf(1063,plain,
! [A: nat,B: nat,C: nat] :
( ( dist @ a @ c2 @ A @ B @ C )
=> ( ord_less_eq @ nat @ ( min_dist @ a @ c2 @ A @ C ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).
thf(22,axiom,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ! [A: TA > TB,B: TA > TB] :
( ! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) )
=> ( ord_less_eq @ ( TA > TB ) @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_187_le__funI) ).
thf(340,plain,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ! [A: TA > TB,B: TA > TB] :
( ! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) )
=> ( ord_less_eq @ ( TA > TB ) @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[22]) ).
thf(292,axiom,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( dist @ capacity @ c @ A @ B @ C )
=> ( ( dist @ capacity @ c @ C @ D @ E )
=> ( dist @ capacity @ c @ A @ ( plus_plus @ nat @ B @ D ) @ E ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_14_dist__trans) ).
thf(1095,plain,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( dist @ capacity @ c @ A @ B @ C )
=> ( ( dist @ capacity @ c @ C @ D @ E )
=> ( dist @ capacity @ c @ A @ ( plus_plus @ nat @ B @ D ) @ E ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).
thf(223,axiom,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
= ( ord_less_eq @ TA @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_31_add__le__cancel__left) ).
thf(898,plain,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
= ( ord_less_eq @ TA @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).
thf(193,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( A != B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_226_less__imp__not__eq) ).
thf(782,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( A != B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).
thf(209,axiom,
ord @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oord_3) ).
thf(841,plain,
ord @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(248,axiom,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( B = C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_228_ord__less__eq__trans) ).
thf(973,plain,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( B = C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).
thf(275,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ capacity @ c @ A @ B @ C )
=> ( isSimplePath @ capacity @ c @ A @ B @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_128_shortestPath__is__simple) ).
thf(1046,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ capacity @ c @ A @ B @ C )
=> ( isSimplePath @ capacity @ c @ A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).
thf(43,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( preorder @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Orderings_Opreorder) ).
thf(389,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( preorder @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(278,axiom,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat,E: nat] :
( ( ord_less @ nat @ D @ E )
=> ( ord_less @ nat @ ( A @ D ) @ ( A @ E ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( A @ B ) @ C ) @ ( A @ ( plus_plus @ nat @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_100_mono__nat__linear__lb) ).
thf(1055,plain,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat,E: nat] :
( ( ord_less @ nat @ D @ E )
=> ( ord_less @ nat @ ( A @ D ) @ ( A @ E ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( A @ B ) @ C ) @ ( A @ ( plus_plus @ nat @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).
thf(110,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_49_add__right__mono) ).
thf(570,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(208,axiom,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
= ( ord_less @ TA @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_5_add__less__cancel__right) ).
thf(838,plain,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
= ( ord_less @ TA @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(24,axiom,
! [TA: $tType] :
( ( wellorder @ TA )
=> ! [A: TA > $o] :
( ? [B: TA] : ( A @ B )
=> ( A @ ( ord_Least @ TA @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_248_LeastI__ex) ).
thf(344,plain,
! [TA: $tType] :
( ( wellorder @ TA )
=> ! [A: TA > $o] :
( ? [B: TA] : ( A @ B )
=> ( A @ ( ord_Least @ TA @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(30,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
( ( dist @ TA @ A @ B @ C @ D )
=> ( connected @ TA @ A @ B @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_146_Graph_Oconnected__distI) ).
thf(357,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
( ( dist @ TA @ A @ B @ C @ D )
=> ( connected @ TA @ A @ B @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(279,axiom,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
=> ( ord_less_eq @ TA @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_42_add__le__imp__le__right) ).
thf(1059,plain,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
=> ( ord_less_eq @ TA @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).
thf(239,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
( ( isPath @ TA @ A @ B @ C @ D )
=> ( dist @ TA @ A @ B @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ C ) @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_124_Graph_OisPath__distD) ).
thf(949,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
( ( isPath @ TA @ A @ B @ C @ D )
=> ( dist @ TA @ A @ B @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ C ) @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).
thf(70,axiom,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( ord_less_eq @ TB @ A @ ( B @ C ) )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less_eq @ TA @ E @ F )
=> ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_189_order__subst1) ).
thf(460,plain,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( ord_less_eq @ TB @ A @ ( B @ C ) )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less_eq @ TA @ E @ F )
=> ( ord_less_eq @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less_eq @ TB @ A @ ( B @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(162,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ C @ A ) @ ( plus_plus @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_51_add__left__mono) ).
thf(702,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ C @ A ) @ ( plus_plus @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(182,axiom,
! [TA: $tType,A: TA > $o,B: TA,C: TA > nat,D: nat] :
( ( A @ B )
=> ( ! [E: TA] :
( ( A @ E )
=> ? [F: TA] :
( ( A @ F )
& ~ ( ord_less_eq @ nat @ ( C @ F ) @ ( C @ E ) ) ) )
=> ? [E: TA] :
( ( A @ E )
& ~ ( ord_less @ nat @ ( C @ E ) @ ( plus_plus @ nat @ ( C @ B ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_154_ex__has__greatest__nat__lemma) ).
thf(751,plain,
! [TA: $tType,A: TA > $o,B: TA,C: TA > nat,D: nat] :
( ( A @ B )
=> ( ! [E: TA] :
( ( A @ E )
=> ? [F: TA] :
( ( A @ F )
& ~ ( ord_less_eq @ nat @ ( C @ F ) @ ( C @ E ) ) ) )
=> ? [E: TA] :
( ( A @ E )
& ~ ( ord_less @ nat @ ( C @ E ) @ ( plus_plus @ nat @ ( C @ B ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(54,axiom,
! [TA: $tType,TB: $tType] :
( ( ( ord @ TA )
& ( ord @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less_eq @ TB @ A @ B )
=> ( ( ( C @ B )
= D )
=> ( ! [E: TB,F: TB] :
( ( ord_less_eq @ TB @ E @ F )
=> ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_192_ord__le__eq__subst) ).
thf(416,plain,
! [TA: $tType,TB: $tType] :
( ( ( ord @ TA )
& ( ord @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less_eq @ TB @ A @ B )
=> ( ( ( C @ B )
= D )
=> ( ! [E: TB,F: TB] :
( ( ord_less_eq @ TB @ E @ F )
=> ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(65,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ a @ c2 @ A @ B @ C )
=> ( isSimplePath @ a @ c2 @ A @ B @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_160_g_H_OshortestPath__is__simple) ).
thf(448,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ a @ c2 @ A @ B @ C )
=> ( isSimplePath @ a @ c2 @ A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(9,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( type @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___HOL_Otype) ).
thf(316,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( type @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(136,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( isShortestPath @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
( ( isPath @ TA @ A @ B @ C @ D )
& ! [E: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ TA @ A @ B @ E @ D )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ C ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ E ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_120_Graph_OisShortestPath__def) ).
thf(638,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( isShortestPath @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
( ( isPath @ TA @ A @ B @ C @ D )
& ! [E: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ TA @ A @ B @ E @ D )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ C ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ E ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(214,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ~ ( ord_less @ TA @ A @ B ) )
= ( ( ord_less @ TA @ B @ A )
| ( A = B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_213_not__less__iff__gr__or__eq) ).
thf(853,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ~ ( ord_less @ TA @ A @ B ) )
= ( ( ord_less @ TA @ B @ A )
| ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(299,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( min_dist @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
( ord_Least @ nat
@ ^ [D: nat] : ( dist @ TA @ A @ B @ D @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_134_Graph_Omin__dist__def) ).
thf(1115,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( min_dist @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
( ord_Least @ nat
@ ^ [D: nat] : ( dist @ TA @ A @ B @ D @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[299]) ).
thf(14,axiom,
! [TA: $tType] :
( ( wellorder @ TA )
=> ! [A: TA > $o,B: TA] :
( ( A @ B )
=> ( A @ ( ord_Least @ TA @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_250_LeastI) ).
thf(325,plain,
! [TA: $tType] :
( ( wellorder @ TA )
=> ! [A: TA > $o,B: TA] :
( ( A @ B )
=> ( A @ ( ord_Least @ TA @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[14]) ).
thf(77,axiom,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less_eq @ TB @ A @ B )
=> ( ( ord_less_eq @ TA @ ( C @ B ) @ D )
=> ( ! [E: TB,F: TB] :
( ( ord_less_eq @ TB @ E @ F )
=> ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_190_order__subst2) ).
thf(475,plain,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less_eq @ TB @ A @ B )
=> ( ( ord_less_eq @ TA @ ( C @ B ) @ D )
=> ( ! [E: TB,F: TB] :
( ( ord_less_eq @ TB @ E @ F )
=> ( ord_less_eq @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less_eq @ TA @ ( C @ A ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[77]) ).
thf(183,axiom,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
= ( ord_less_eq @ TA @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_30_add__le__cancel__right) ).
thf(756,plain,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
= ( ord_less_eq @ TA @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(195,axiom,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less @ TA @ A @ B )
& ( ord_less @ TA @ C @ D ) )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_108_add__mono__thms__linordered__field_I5_J) ).
thf(789,plain,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less @ TA @ A @ B )
& ( ord_less @ TA @ C @ D ) )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(12,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ordere236663937imp_le @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add__imp__le) ).
thf(321,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ordere236663937imp_le @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(140,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( isShortestPath @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
( ( isSimplePath @ TA @ A @ B @ C @ D )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ C )
= ( min_dist @ TA @ A @ B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_135_Graph_OisShortestPath__alt) ).
thf(646,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( isShortestPath @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
( ( isSimplePath @ TA @ A @ B @ C @ D )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ C )
= ( min_dist @ TA @ A @ B @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(235,axiom,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
= ( ord_less @ TA @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_4_add__less__cancel__left) ).
thf(937,plain,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
= ( ord_less @ TA @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).
thf(200,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ( ( ord_less_eq @ TA )
= ( ^ [A: TA,B: TA] :
? [C: TA] :
( B
= ( plus_plus @ TA @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_44_le__iff__add) ).
thf(806,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ( ( ord_less_eq @ TA )
= ( ^ [A: TA,B: TA] :
? [C: TA] :
( B
= ( plus_plus @ TA @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(28,axiom,
! [A: nat,B: nat,C: nat] :
( ( dist @ a @ c2 @ A @ B @ C )
=> ( connected @ a @ c2 @ A @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_182_g_H_Oconnected__distI) ).
thf(353,plain,
! [A: nat,B: nat,C: nat] :
( ( dist @ a @ c2 @ A @ B @ C )
=> ( connected @ a @ c2 @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(79,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ( ( (=) @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_193_eq__iff) ).
thf(480,plain,
! [TA: $tType] :
( ( order @ TA )
=> ( ( (=) @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(266,axiom,
! [TA: $tType,A: TA > $o,B: TA,C: TA > nat] :
( ( A @ B )
=> ? [D: TA] :
( ( A @ D )
& ! [E: TA] :
( ( A @ E )
=> ( ord_less_eq @ nat @ ( C @ D ) @ ( C @ E ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_246_ex__has__least__nat) ).
thf(1020,plain,
! [TA: $tType,A: TA > $o,B: TA,C: TA > nat] :
( ( A @ B )
=> ? [D: TA] :
( ( A @ D )
& ! [E: TA] :
( ( A @ E )
=> ( ord_less_eq @ nat @ ( C @ D ) @ ( C @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).
thf(26,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ A )
=> ( A = B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_204_order__class_Oorder_Oantisym) ).
thf(348,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ A )
=> ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(156,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ C @ A )
=> ( ord_less @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_214_dual__order_Ostrict__trans) ).
thf(687,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ C @ A )
=> ( ord_less @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(158,axiom,
! [TA: $tType] :
( ( wellorder @ TA )
=> ( ( '?' @ TA )
= ( ^ [A: TA > $o] :
? [B: TA] :
( ( A @ B )
& ! [C: TA] :
( ( ord_less @ TA @ C @ B )
=> ~ ( A @ C ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_216_exists__least__iff) ).
thf(692,plain,
! [TA: $tType] :
( ( wellorder @ TA )
=> ( ( '?' @ TA )
= ( ^ [A: TA > $o] :
? [B: TA] :
( ( A @ B )
& ! [C: TA] :
( ( ord_less @ TA @ C @ B )
=> ~ ( A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(176,axiom,
ordere223160158up_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add) ).
thf(734,plain,
ordere223160158up_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(67,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
= ( B @ C ) )
=> ( ( collect @ TA @ A )
= ( collect @ TA @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_47_Collect__cong) ).
thf(453,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
= ( B @ C ) )
=> ( ( collect @ TA @ A )
= ( collect @ TA @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(259,axiom,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ D )
=> ( ord_less_eq @ nat @ D @ C ) )
=> ? [D: nat] :
( ( A @ D )
& ! [E: nat] :
( ( A @ E )
=> ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_41_Nat_Oex__has__greatest__nat) ).
thf(998,plain,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ D )
=> ( ord_less_eq @ nat @ D @ C ) )
=> ? [D: nat] :
( ( A @ D )
& ! [E: nat] :
( ( A @ E )
=> ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).
thf(85,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat,E: nat,F: nat] :
( ( dist @ TA @ A @ B @ C @ D )
=> ( ( dist @ TA @ A @ D @ E @ F )
=> ( ( ( min_dist @ TA @ A @ B @ F )
= ( plus_plus @ nat @ C @ E ) )
=> ( ( min_dist @ TA @ A @ B @ D )
= C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_126_Graph_Omin__dist__split_I1_J) ).
thf(507,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat,E: nat,F: nat] :
( ( dist @ TA @ A @ B @ C @ D )
=> ( ( dist @ TA @ A @ D @ E @ F )
=> ( ( ( min_dist @ TA @ A @ B @ F )
= ( plus_plus @ nat @ C @ E ) )
=> ( ( min_dist @ TA @ A @ B @ D )
= C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(7,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ( ( (=) @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_201_order__class_Oorder_Oeq__iff) ).
thf(312,plain,
! [TA: $tType] :
( ( order @ TA )
=> ( ( (=) @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(251,axiom,
! [TA: $tType] : ( size @ ( list @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_List_Olist___Nat_Osize_11) ).
thf(981,plain,
! [TA: $tType] : ( size @ ( list @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).
thf(187,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
| ( A = B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_59_le__eq__less__or__eq) ).
thf(766,plain,
( ( ord_less_eq @ nat )
= ( ^ [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
| ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(5,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_198_order_Otrans) ).
thf(307,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(62,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( dist @ TA )
= ( dist @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_140_Graph_Odist_Ocong) ).
thf(439,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(147,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_225_less__not__sym) ).
thf(661,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(99,axiom,
size @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Nat_Osize) ).
thf(546,plain,
size @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(283,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
=> ( ord_less_eq @ nat @ B @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_68_add__leD2) ).
thf(1067,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
=> ( ord_less_eq @ nat @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).
thf(167,axiom,
! [TA: $tType] :
( ( no_bot @ TA )
=> ! [A: TA] :
? [B: TA] : ( ord_less @ TA @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_241_lt__ex) ).
thf(714,plain,
! [TA: $tType] :
( ( no_bot @ TA )
=> ! [A: TA] :
? [B: TA] : ( ord_less @ TA @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(19,axiom,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ! [A: TA > TB,B: TA > TB,C: TA] :
( ( ord_less_eq @ ( TA > TB ) @ A @ B )
=> ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_186_le__funE) ).
thf(334,plain,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ! [A: TA > TB,B: TA > TB,C: TA] :
( ( ord_less_eq @ ( TA > TB ) @ A @ B )
=> ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(243,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( A
= ( plus_plus @ TA @ B @ C ) )
=> ( ( plus_plus @ TA @ D @ A )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ D @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_82_group__cancel_Oadd2) ).
thf(957,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( A
= ( plus_plus @ TA @ B @ C ) )
=> ( ( plus_plus @ TA @ D @ A )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ D @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).
thf(205,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ a @ c2 @ A @ B @ C )
= ( ( isSimplePath @ a @ c2 @ A @ B @ C )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B )
= ( min_dist @ a @ c2 @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_173_g_H_OisShortestPath__alt) ).
thf(825,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ a @ c2 @ A @ B @ C )
= ( ( isSimplePath @ a @ c2 @ A @ B @ C )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B )
= ( min_dist @ a @ c2 @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(191,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ( ( ord_less @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ( B != A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_253_dual__order_Ostrict__iff__order) ).
thf(778,plain,
! [TA: $tType] :
( ( order @ TA )
=> ( ( ord_less @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
& ( B != A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(1,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 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',conj_0) ).
thf(2,negated_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 ) ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(302,plain,
~ ( 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 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(204,axiom,
! [A: nat,B: nat,C: nat > $o] :
( ( connected @ capacity @ c @ A @ B )
=> ( ! [D: nat] :
( ( dist @ capacity @ c @ A @ D @ B )
=> ( ! [E: nat] :
( ( dist @ capacity @ c @ A @ E @ B )
=> ( ord_less_eq @ nat @ D @ E ) )
=> ( C @ D ) ) )
=> ( C @ ( min_dist @ capacity @ c @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_27_min__distI2) ).
thf(820,plain,
! [A: nat,B: nat,C: nat > $o] :
( ( connected @ capacity @ c @ A @ B )
=> ( ! [D: nat] :
( ( dist @ capacity @ c @ A @ D @ B )
=> ( ! [E: nat] :
( ( dist @ capacity @ c @ A @ E @ B )
=> ( ord_less_eq @ nat @ D @ E ) )
=> ( C @ D ) ) )
=> ( C @ ( min_dist @ capacity @ c @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(51,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( cancel_semigroup_add @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Ocancel__semigroup__add) ).
thf(407,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( cancel_semigroup_add @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(134,axiom,
! [A: nat,B: nat,C: nat] :
( ( dist @ a @ c2 @ A @ B @ C )
= ( ? [D: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ a @ c2 @ A @ D @ C )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ D )
= B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_175_g_H_Odist__def) ).
thf(630,plain,
! [A: nat,B: nat,C: nat] :
( ( dist @ a @ c2 @ A @ B @ C )
= ( ? [D: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ a @ c2 @ A @ D @ C )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ D )
= B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(81,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ordere223160158up_add @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Oordered__cancel__ab__semigroup__add) ).
thf(484,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ordere223160158up_add @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(32,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ A )
=> ( A = B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_194_antisym) ).
thf(361,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ A )
=> ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(133,axiom,
! [A: nat,B: nat,C: nat] :
( ( connected @ capacity @ c @ A @ B )
=> ( ( ord_less_eq @ nat @ C @ ( min_dist @ capacity @ c @ A @ B ) )
=> ? [D: nat] :
( ( connected @ capacity @ c @ A @ D )
& ( ( min_dist @ capacity @ c @ A @ D )
= C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_19_min__dist__le) ).
thf(625,plain,
! [A: nat,B: nat,C: nat] :
( ( connected @ capacity @ c @ A @ B )
=> ( ( ord_less_eq @ nat @ C @ ( min_dist @ capacity @ c @ A @ B ) )
=> ? [D: nat] :
( ( connected @ capacity @ c @ A @ D )
& ( ( min_dist @ capacity @ c @ A @ D )
= C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(60,axiom,
! [TA: $tType,TB: $tType] :
( ( ord @ TA )
=> ( ord @ ( TB > TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_fun___Orderings_Oord) ).
thf(435,plain,
! [TA: $tType,TB: $tType] :
( ( ord @ TA )
=> ( ord @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(16,axiom,
order @ $o,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oorder_9) ).
thf(329,plain,
order @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(115,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 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_37_le__trans) ).
thf(583,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(84,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
=> ~ ( ord_less_eq @ TA @ B @ C ) )
=> ( ( ( ord_less_eq @ TA @ B @ A )
=> ~ ( ord_less_eq @ TA @ A @ C ) )
=> ( ( ( ord_less_eq @ TA @ A @ C )
=> ~ ( ord_less_eq @ TA @ C @ B ) )
=> ( ( ( ord_less_eq @ TA @ C @ B )
=> ~ ( ord_less_eq @ TA @ B @ A ) )
=> ( ( ( ord_less_eq @ TA @ B @ C )
=> ~ ( ord_less_eq @ TA @ C @ A ) )
=> ~ ( ( ord_less_eq @ TA @ C @ A )
=> ~ ( ord_less_eq @ TA @ A @ B ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_199_le__cases3) ).
thf(493,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
=> ~ ( ord_less_eq @ TA @ B @ C ) )
=> ( ( ( ord_less_eq @ TA @ B @ A )
=> ~ ( ord_less_eq @ TA @ A @ C ) )
=> ( ( ( ord_less_eq @ TA @ A @ C )
=> ~ ( ord_less_eq @ TA @ C @ B ) )
=> ( ( ( ord_less_eq @ TA @ C @ B )
=> ~ ( ord_less_eq @ TA @ B @ A ) )
=> ( ( ( ord_less_eq @ TA @ B @ C )
=> ~ ( ord_less_eq @ TA @ C @ A ) )
=> ~ ( ( ord_less_eq @ TA @ C @ A )
=> ~ ( ord_less_eq @ TA @ A @ B ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[84]) ).
thf(154,axiom,
( ( min_dist @ capacity @ c @ s @ t )
= ( plus_plus @ nat @ ( min_dist @ capacity @ c @ s @ ua ) @ ( min_dist @ capacity @ c @ ua @ t ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_0__092_060open_062min__dist_As_At_A_061_Amin__dist_As_Au_A_L_Amin__dist_Au_At_092_060close_062) ).
thf(682,plain,
( ( min_dist @ capacity @ c @ s @ t )
= ( plus_plus @ nat @ ( min_dist @ capacity @ c @ s @ ua ) @ ( min_dist @ capacity @ c @ ua @ t ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(102,axiom,
! [A: nat,B: nat] :
( ( A = B )
=> ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_38_eq__imp__le) ).
thf(551,plain,
! [A: nat,B: nat] :
( ( A = B )
=> ( ord_less_eq @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(165,axiom,
! [A: nat,B: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_114_not__add__less1) ).
thf(710,plain,
! [A: nat,B: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(229,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
| ( ord_less_eq @ nat @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_40_nat__le__linear) ).
thf(916,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
| ( ord_less_eq @ nat @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).
thf(73,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( linorder @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Orderings_Olinorder) ).
thf(467,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( linorder @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(68,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
( ( isShortestPath @ TA @ A @ B @ C @ D )
=> ( isPath @ TA @ A @ B @ C @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_144_Graph_OshortestPath__is__path) ).
thf(456,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
( ( isShortestPath @ TA @ A @ B @ C @ D )
=> ( isPath @ TA @ A @ B @ C @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(210,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ capacity @ c @ A @ B @ C )
= ( ( isPath @ capacity @ c @ A @ B @ C )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B )
= ( min_dist @ capacity @ c @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_28_isShortestPath__min__dist__def) ).
thf(842,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ capacity @ c @ A @ B @ C )
= ( ( isPath @ capacity @ c @ A @ B @ C )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B )
= ( min_dist @ capacity @ c @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).
thf(290,axiom,
ab_semigroup_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oab__semigroup__add) ).
thf(1089,plain,
ab_semigroup_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).
thf(142,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_252_dual__order_Ostrict__implies__order) ).
thf(650,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(298,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( B != A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_92_less__not__refl2) ).
thf(1111,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( B != A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[298]) ).
thf(172,axiom,
connected @ capacity @ c @ s @ ua,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_29__092_060open_062connected_As_Au_092_060close_062) ).
thf(725,plain,
connected @ capacity @ c @ s @ ua,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(49,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( order @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Orderings_Oorder) ).
thf(403,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( order @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[49]) ).
thf(218,axiom,
! [TA: $tType] :
( ( strict2144017051up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ C @ D )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_105_add__strict__mono) ).
thf(869,plain,
! [TA: $tType] :
( ( strict2144017051up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ C @ D )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(127,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA,C: $o] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ A )
=> C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_221_less__imp__triv) ).
thf(610,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA,C: $o] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ A )
=> C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(300,axiom,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( dist @ capacity @ c @ A @ B @ C )
=> ( ( dist @ capacity @ c @ C @ D @ E )
=> ( ( ( min_dist @ capacity @ c @ A @ E )
= ( plus_plus @ nat @ B @ D ) )
=> ( ( min_dist @ capacity @ c @ C @ E )
= D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_8_min__dist__split_I2_J) ).
thf(1118,plain,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( dist @ capacity @ c @ A @ B @ C )
=> ( ( dist @ capacity @ c @ C @ D @ E )
=> ( ( ( min_dist @ capacity @ c @ A @ E )
= ( plus_plus @ nat @ B @ D ) )
=> ( ( min_dist @ capacity @ c @ C @ E )
= D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[300]) ).
thf(89,axiom,
strict2144017051up_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add) ).
thf(519,plain,
strict2144017051up_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(179,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_93_less__not__refl) ).
thf(742,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(233,axiom,
! [A: nat] : ( connected @ capacity @ c @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_34_connected__refl) ).
thf(933,plain,
! [A: nat] : ( connected @ capacity @ c @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).
thf(138,axiom,
isPath @ a @ c2 @ u @ p2 @ t,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_181__092_060open_062g_H_OisPath_Au_____Ap2_H_____At_092_060close_062) ).
thf(642,plain,
isPath @ a @ c2 @ u @ p2 @ t,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(92,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A: nat,B: nat] :
? [C: nat] :
( B
= ( plus_plus @ nat @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_62_nat__le__iff__add) ).
thf(526,plain,
( ( ord_less_eq @ nat )
= ( ^ [A: nat,B: nat] :
? [C: nat] :
( B
= ( plus_plus @ nat @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(265,axiom,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( dist @ capacity @ c @ A @ B @ C )
=> ( ( dist @ capacity @ c @ C @ D @ E )
=> ( ( ( min_dist @ capacity @ c @ A @ E )
= ( plus_plus @ nat @ B @ D ) )
=> ( ( min_dist @ capacity @ c @ A @ C )
= B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_9_min__dist__split_I1_J) ).
thf(1016,plain,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( dist @ capacity @ c @ A @ B @ C )
=> ( ( dist @ capacity @ c @ C @ D @ E )
=> ( ( ( min_dist @ capacity @ c @ A @ E )
= ( plus_plus @ nat @ B @ D ) )
=> ( ( min_dist @ capacity @ c @ A @ C )
= B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).
thf(100,axiom,
ordere236663937imp_le @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le) ).
thf(547,plain,
ordere236663937imp_le @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(224,axiom,
! [A: nat,B: nat] :
( ( min_dist @ a @ c2 @ A @ B )
= ( ord_Least @ nat
@ ^ [C: nat] : ( dist @ a @ c2 @ A @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_174_g_H_Omin__dist__def) ).
thf(901,plain,
! [A: nat,B: nat] :
( ( min_dist @ a @ c2 @ A @ B )
= ( ord_Least @ nat
@ ^ [C: nat] : ( dist @ a @ c2 @ A @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).
thf(287,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
( ( dist @ TA @ A @ B @ C @ D )
=> ( ord_less_eq @ nat @ ( min_dist @ TA @ A @ B @ D ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_151_Graph_Omin__dist__minD) ).
thf(1075,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
( ( dist @ TA @ A @ B @ C @ D )
=> ( ord_less_eq @ nat @ ( min_dist @ TA @ A @ B @ D ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).
thf(153,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
= ( ord_less_eq @ nat @ B @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_32_nat__add__left__cancel__le) ).
thf(679,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
= ( ord_less_eq @ nat @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(247,axiom,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
= ( A = C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_79_add_Oright__cancel) ).
thf(970,plain,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
= ( A = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).
thf(132,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ? [C: nat] :
( B
= ( plus_plus @ nat @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_67_le__Suc__ex) ).
thf(622,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ? [C: nat] :
( B
= ( plus_plus @ nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(45,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ A )
=> ( ord_less_eq @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_208_dual__order_Otrans) ).
thf(393,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ A )
=> ( ord_less_eq @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(258,axiom,
isPath @ a @ c2 @ ua @ p2_a @ t,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_184_P2_H) ).
thf(997,plain,
isPath @ a @ c2 @ ua @ p2_a @ t,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).
thf(227,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_110_trans__less__add2) ).
thf(911,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).
thf(109,axiom,
ordere779506340up_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add) ).
thf(569,plain,
ordere779506340up_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(36,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( group_add @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Ogroup__add) ).
thf(371,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( group_add @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(135,axiom,
! [A: nat,B: nat,C: nat] :
( ( dist @ capacity @ c @ A @ B @ C )
=> ( ! [D: nat] :
( ( dist @ capacity @ c @ A @ D @ C )
=> ( ord_less_eq @ nat @ B @ D ) )
=> ( ( min_dist @ capacity @ c @ A @ C )
= B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_21_min__distI__eq) ).
thf(633,plain,
! [A: nat,B: nat,C: nat] :
( ( dist @ capacity @ c @ A @ B @ C )
=> ( ! [D: nat] :
( ( dist @ capacity @ c @ A @ D @ C )
=> ( ord_less_eq @ nat @ B @ D ) )
=> ( ( min_dist @ capacity @ c @ A @ C )
= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(97,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ capacity @ c @ A @ B @ C )
= ( ( isPath @ capacity @ c @ A @ B @ C )
& ! [D: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ capacity @ c @ A @ D @ C )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_26_isShortestPath__def) ).
thf(541,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ capacity @ c @ A @ B @ C )
= ( ( isPath @ capacity @ c @ A @ B @ C )
& ! [D: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ capacity @ c @ A @ D @ C )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(145,axiom,
comm_monoid_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__add) ).
thf(657,plain,
comm_monoid_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(268,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat > $o] :
( ( connected @ TA @ A @ B @ C )
=> ( ! [E: nat] :
( ( dist @ TA @ A @ B @ E @ C )
=> ( ! [F: nat] :
( ( dist @ TA @ A @ B @ F @ C )
=> ( ord_less_eq @ nat @ E @ F ) )
=> ( D @ E ) ) )
=> ( D @ ( min_dist @ TA @ A @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_121_Graph_Omin__distI2) ).
thf(1026,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat > $o] :
( ( connected @ TA @ A @ B @ C )
=> ( ! [E: nat] :
( ( dist @ TA @ A @ B @ E @ C )
=> ( ! [F: nat] :
( ( dist @ TA @ A @ B @ F @ C )
=> ( ord_less_eq @ nat @ E @ F ) )
=> ( D @ E ) ) )
=> ( D @ ( min_dist @ TA @ A @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).
thf(241,axiom,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
=> ( ord_less @ TA @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_102_add__less__imp__less__left) ).
thf(953,plain,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
=> ( ord_less @ TA @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).
thf(169,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
& ( C = D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_55_add__mono__thms__linordered__semiring_I3_J) ).
thf(718,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
& ( C = D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(232,axiom,
! [TA: $tType,TB: $tType] :
( ( ( ord @ TA )
& ( ord @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less @ TB @ A @ B )
=> ( ( ( C @ B )
= D )
=> ( ! [E: TB,F: TB] :
( ( ord_less @ TB @ E @ F )
=> ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_244_ord__less__eq__subst) ).
thf(926,plain,
! [TA: $tType,TB: $tType] :
( ( ( ord @ TA )
& ( ord @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less @ TB @ A @ B )
=> ( ( ( C @ B )
= D )
=> ( ! [E: TB,F: TB] :
( ( ord_less @ TB @ E @ F )
=> ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(47,axiom,
preorder @ $o,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Opreorder_7) ).
thf(398,plain,
preorder @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(271,axiom,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
=> ( B = C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_76_add__left__imp__eq) ).
thf(1035,plain,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
=> ( B = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).
thf(105,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_218_order_Ostrict__trans) ).
thf(558,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(285,axiom,
! [A: nat,B: nat] : ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_70_le__add2) ).
thf(1071,plain,
! [A: nat,B: nat] : ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).
thf(197,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
=> ( ord_less @ nat @ A @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_116_add__lessD1) ).
thf(794,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
=> ( ord_less @ nat @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).
thf(116,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat,E: nat,F: nat] :
( ( dist @ TA @ A @ B @ C @ D )
=> ( ( dist @ TA @ A @ D @ E @ F )
=> ( dist @ TA @ A @ B @ ( plus_plus @ nat @ C @ E ) @ F ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_142_Graph_Odist__trans) ).
thf(586,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat,E: nat,F: nat] :
( ( dist @ TA @ A @ B @ C @ D )
=> ( ( dist @ TA @ A @ D @ E @ F )
=> ( dist @ TA @ A @ B @ ( plus_plus @ nat @ C @ E ) @ F ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(215,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_237_order_Oasym) ).
thf(856,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(180,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isPath @ capacity @ c @ A @ B @ C )
=> ( dist @ capacity @ c @ A @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B ) @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_22_isPath__distD) ).
thf(745,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isPath @ capacity @ c @ A @ B @ C )
=> ( dist @ capacity @ c @ A @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(157,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ a @ c2 @ A @ B @ C )
= ( ( isPath @ a @ c2 @ A @ B @ C )
& ! [D: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ a @ c2 @ A @ D @ C )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_179_g_H_OisShortestPath__def) ).
thf(689,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ a @ c2 @ A @ B @ C )
= ( ( isPath @ a @ c2 @ A @ B @ C )
& ! [D: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ a @ c2 @ A @ D @ C )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(38,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ordere779506340up_add @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add) ).
thf(375,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ordere779506340up_add @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(123,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
= ( ( ord_less @ TA @ A @ B )
| ( ord_less @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_238_neq__iff) ).
thf(601,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
= ( ( ord_less @ TA @ A @ B )
| ( ord_less @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(87,axiom,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( A
= ( plus_plus @ TA @ B @ C ) )
=> ( ( plus_plus @ TA @ A @ D )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ C @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_83_group__cancel_Oadd1) ).
thf(513,plain,
! [TA: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( A
= ( plus_plus @ TA @ B @ C ) )
=> ( ( plus_plus @ TA @ A @ D )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ C @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(58,axiom,
! [TA: $tType] : ( ord @ ( set @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oord_6) ).
thf(430,plain,
! [TA: $tType] : ( ord @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(225,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
( ( dist @ TA @ A @ B @ C @ D )
=> ( ! [E: nat] :
( ( dist @ TA @ A @ B @ E @ D )
=> ( ord_less_eq @ nat @ C @ E ) )
=> ( ( min_dist @ TA @ A @ B @ D )
= C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_150_Graph_Omin__distI__eq) ).
thf(904,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
( ( dist @ TA @ A @ B @ C @ D )
=> ( ! [E: nat] :
( ( dist @ TA @ A @ B @ E @ D )
=> ( ord_less_eq @ nat @ C @ E ) )
=> ( ( min_dist @ TA @ A @ B @ D )
= C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(55,axiom,
! [A: nat,B: nat] :
( ( connected @ a @ c2 @ A @ B )
=> ~ ! [C: list @ ( product_prod @ nat @ nat )] :
~ ( isShortestPath @ a @ c2 @ A @ C @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_159_g_H_Oobtain__shortest__path) ).
thf(423,plain,
! [A: nat,B: nat] :
( ( connected @ a @ c2 @ A @ B )
=> ~ ! [C: list @ ( product_prod @ nat @ nat )] :
~ ( isShortestPath @ a @ c2 @ A @ C @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(113,axiom,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less @ nat @ D @ C )
& ~ ( A @ D ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_88_infinite__descent) ).
thf(577,plain,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less @ nat @ D @ C )
& ~ ( A @ D ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(137,axiom,
ord_less_eq @ nat @ ( min_dist @ capacity @ c @ ua @ t ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2_a ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_2__092_060open_062min__dist_Au_At_A_092_060le_062_Alength_Ap2_H_092_060close_062) ).
thf(641,plain,
ord_less_eq @ nat @ ( min_dist @ capacity @ c @ ua @ t ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p2_a ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(44,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
( ( isShortestPath @ TA @ A @ B @ C @ D )
=> ( isSimplePath @ TA @ A @ B @ C @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_132_Graph_OshortestPath__is__simple) ).
thf(391,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
( ( isShortestPath @ TA @ A @ B @ C @ D )
=> ( isSimplePath @ TA @ A @ B @ C @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(41,axiom,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_203_ord__le__eq__trans) ).
thf(382,plain,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(111,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_90_less__irrefl__nat) ).
thf(572,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(166,axiom,
linorder @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Olinorder) ).
thf(713,plain,
linorder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(291,axiom,
! [A: nat,B: nat,C: nat > $o] :
( ( connected @ a @ c2 @ A @ B )
=> ( ! [D: nat] :
( ( dist @ a @ c2 @ A @ D @ B )
=> ( ! [E: nat] :
( ( dist @ a @ c2 @ A @ E @ B )
=> ( ord_less_eq @ nat @ D @ E ) )
=> ( C @ D ) ) )
=> ( C @ ( min_dist @ a @ c2 @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_163_g_H_Omin__distI2) ).
thf(1090,plain,
! [A: nat,B: nat,C: nat > $o] :
( ( connected @ a @ c2 @ A @ B )
=> ( ! [D: nat] :
( ( dist @ a @ c2 @ A @ D @ B )
=> ( ! [E: nat] :
( ( dist @ a @ c2 @ A @ E @ B )
=> ( ord_less_eq @ nat @ D @ E ) )
=> ( C @ D ) ) )
=> ( C @ ( min_dist @ a @ c2 @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).
thf(163,axiom,
! [A: nat,B: nat,C: nat] :
( ( dist @ capacity @ c @ A @ B @ C )
= ( ? [D: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ capacity @ c @ A @ D @ C )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ D )
= B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_23_dist__def) ).
thf(704,plain,
! [A: nat,B: nat,C: nat] :
( ( dist @ capacity @ c @ A @ B @ C )
= ( ? [D: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ capacity @ c @ A @ D @ C )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ D )
= B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(238,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isPath @ a @ c2 @ A @ B @ C )
=> ( dist @ a @ c2 @ A @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B ) @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_176_g_H_OisPath__distD) ).
thf(947,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isPath @ a @ c2 @ A @ B @ C )
=> ( dist @ a @ c2 @ A @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).
thf(124,axiom,
! [A: nat,B: nat] :
( ( connected @ capacity @ c @ A @ B )
= ( ? [C: nat] : ( dist @ capacity @ c @ A @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_17_connected__by__dist) ).
thf(604,plain,
! [A: nat,B: nat] :
( ( connected @ capacity @ c @ A @ B )
= ( ? [C: nat] : ( dist @ capacity @ c @ A @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(207,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat,E: nat] :
( ( connected @ TA @ A @ B @ C )
=> ( ( ( min_dist @ TA @ A @ B @ C )
= D )
=> ( ( ord_less @ nat @ E @ D )
=> ? [F: nat] :
( ( connected @ TA @ A @ B @ F )
& ( ( min_dist @ TA @ A @ B @ F )
= E ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_148_Graph_Omin__dist__less) ).
thf(831,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat,E: nat] :
( ( connected @ TA @ A @ B @ C )
=> ( ( ( min_dist @ TA @ A @ B @ C )
= D )
=> ( ( ord_less @ nat @ E @ D )
=> ? [F: nat] :
( ( connected @ TA @ A @ B @ F )
& ( ( min_dist @ TA @ A @ B @ F )
= E ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(262,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ~ ! [C: TA] :
( B
!= ( plus_plus @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_50_less__eqE) ).
thf(1011,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ~ ! [C: TA] :
( B
!= ( plus_plus @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).
thf(276,axiom,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( dist @ a @ c2 @ A @ B @ C )
=> ( ( dist @ a @ c2 @ C @ D @ E )
=> ( ( ( min_dist @ a @ c2 @ A @ E )
= ( plus_plus @ nat @ B @ D ) )
=> ( ( min_dist @ a @ c2 @ C @ E )
= D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_167_g_H_Omin__dist__split_I2_J) ).
thf(1048,plain,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( dist @ a @ c2 @ A @ B @ C )
=> ( ( dist @ a @ c2 @ C @ D @ E )
=> ( ( ( min_dist @ a @ c2 @ A @ E )
= ( plus_plus @ nat @ B @ D ) )
=> ( ( min_dist @ a @ c2 @ C @ E )
= D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).
thf(108,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
=> ( ord_less_eq @ nat @ A @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_69_add__leD1) ).
thf(567,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
=> ( ord_less_eq @ nat @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(10,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( connected @ TA )
= ( connected @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_139_Graph_Oconnected_Ocong) ).
thf(318,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(141,axiom,
isPath @ a @ c2 @ s @ p3 @ t,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_180_assms_I5_J) ).
thf(649,plain,
isPath @ a @ c2 @ s @ p3 @ t,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(122,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_60_less__imp__le__nat) ).
thf(599,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(39,axiom,
! [A: nat] : ( connected @ a @ c2 @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_183_g_H_Oconnected__refl) ).
thf(377,plain,
! [A: nat] : ( connected @ a @ c2 @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(254,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA] :
~ ( ord_less @ TA @ A @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_230_less__irrefl) ).
thf(990,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA] :
~ ( ord_less @ TA @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).
thf(257,axiom,
isPath @ capacity @ c @ s @ p1 @ v,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_24__092_060open_062isPath_As_Ap1_____Av_____092_060close_062) ).
thf(996,plain,
isPath @ capacity @ c @ s @ p1 @ v,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).
thf(130,axiom,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less @ TA @ A @ B )
& ( C = D ) )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_106_add__mono__thms__linordered__field_I1_J) ).
thf(616,plain,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less @ TA @ A @ B )
& ( C = D ) )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(3,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( isSimplePath @ TA )
= ( isSimplePath @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_130_Graph_OisSimplePath_Ocong) ).
thf(303,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(301,axiom,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less @ TA @ A @ B )
& ( ord_less_eq @ TA @ C @ D ) )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_98_add__mono__thms__linordered__field_I3_J) ).
thf(1122,plain,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less @ TA @ A @ B )
& ( ord_less_eq @ TA @ C @ D ) )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[301]) ).
thf(221,axiom,
! [A: nat,B: nat,C: nat] :
( ( connected @ a @ c2 @ A @ B )
=> ( ( ord_less_eq @ nat @ C @ ( min_dist @ a @ c2 @ A @ B ) )
=> ? [D: nat] :
( ( connected @ a @ c2 @ A @ D )
& ( ( min_dist @ a @ c2 @ A @ D )
= C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_165_g_H_Omin__dist__le) ).
thf(890,plain,
! [A: nat,B: nat,C: nat] :
( ( connected @ a @ c2 @ A @ B )
=> ( ( ord_less_eq @ nat @ C @ ( min_dist @ a @ c2 @ A @ B ) )
=> ? [D: nat] :
( ( connected @ a @ c2 @ A @ D )
& ( ( min_dist @ a @ c2 @ A @ D )
= C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).
thf(126,axiom,
ord_less @ nat @ ( min_dist @ capacity @ c @ s @ ua ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1a ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_1__092_060open_062min__dist_As_Au_A_060_Alength_Ap1_092_060close_062) ).
thf(609,plain,
ord_less @ nat @ ( min_dist @ capacity @ c @ s @ ua ) @ ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ p1a ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(13,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_197_le__cases) ).
thf(323,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(80,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( isShortestPath @ TA )
= ( isShortestPath @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_141_Graph_OisShortestPath_Ocong) ).
thf(483,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(222,axiom,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
=> ( A = C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_75_add__right__imp__eq) ).
thf(895,plain,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
=> ( A = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).
thf(17,axiom,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ( ( ord_less_eq @ ( TA > TB ) )
= ( ^ [A: TA > TB,B: TA > TB] :
! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_188_le__fun__def) ).
thf(330,plain,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ( ( ord_less_eq @ ( TA > TB ) )
= ( ^ [A: TA > TB,B: TA > TB] :
! [C: TA] : ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(280,axiom,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
=> ( ord_less_eq @ TA @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_43_add__le__imp__le__left) ).
thf(1061,plain,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ A @ C ) )
=> ( ord_less_eq @ TA @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).
thf(216,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_111_trans__less__add1) ).
thf(858,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(34,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_206_dual__order_Orefl) ).
thf(367,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(199,axiom,
! [TA: $tType,TB: $tType] :
( ( ( ord @ TA )
& ( ord @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( A
= ( B @ C ) )
=> ( ( ord_less @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less @ TA @ E @ F )
=> ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_245_ord__eq__less__subst) ).
thf(799,plain,
! [TA: $tType,TB: $tType] :
( ( ( ord @ TA )
& ( ord @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( A
= ( B @ C ) )
=> ( ( ord_less @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less @ TA @ E @ F )
=> ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(173,axiom,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less @ TA @ C @ D )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_97_add__le__less__mono) ).
thf(726,plain,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less @ TA @ C @ D )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(212,axiom,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
= ( B = C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_80_add_Oleft__cancel) ).
thf(848,plain,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
= ( B = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(90,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( A != B )
=> ( ord_less @ nat @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_57_le__neq__implies__less) ).
thf(520,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( A != B )
=> ( ord_less @ nat @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(252,axiom,
! [A: nat] :
( ( reachableNodes @ a @ c2 @ A )
= ( collect @ nat @ ( connected @ a @ c2 @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_161_g_H_OreachableNodes__def) ).
thf(983,plain,
! [A: nat] :
( ( reachableNodes @ a @ c2 @ A )
= ( collect @ nat @ ( connected @ a @ c2 @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).
thf(48,axiom,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A = B )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_202_ord__eq__le__trans) ).
thf(399,plain,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A = B )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(188,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ( ( ord_less_eq @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
| ( B = A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_254_dual__order_Oorder__iff__strict) ).
thf(768,plain,
! [TA: $tType] :
( ( order @ TA )
=> ( ( ord_less_eq @ TA )
= ( ^ [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
| ( B = A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(264,axiom,
isPath @ capacity @ c @ ua @ p2_a @ t,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_13__092_060open_062isPath_Au_Ap2_H_At_092_060close_062) ).
thf(1015,plain,
isPath @ capacity @ c @ ua @ p2_a @ t,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).
thf(253,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
( ( connected @ TA @ A @ B @ C )
=> ( ( ord_less_eq @ nat @ D @ ( min_dist @ TA @ A @ B @ C ) )
=> ? [E: nat] :
( ( connected @ TA @ A @ B @ E )
& ( ( min_dist @ TA @ A @ B @ E )
= D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_149_Graph_Omin__dist__le) ).
thf(986,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
( ( connected @ TA @ A @ B @ C )
=> ( ( ord_less_eq @ nat @ D @ ( min_dist @ TA @ A @ B @ C ) )
=> ? [E: nat] :
( ( connected @ TA @ A @ B @ E )
& ( ( min_dist @ TA @ A @ B @ E )
= D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(144,axiom,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat,E: nat] :
( ( ord_less @ nat @ D @ E )
=> ( ord_less @ nat @ ( A @ D ) @ ( A @ E ) ) )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ nat @ ( A @ B ) @ ( A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_56_less__mono__imp__le__mono) ).
thf(653,plain,
! [A: nat > nat,B: nat,C: nat] :
( ! [D: nat,E: nat] :
( ( ord_less @ nat @ D @ E )
=> ( ord_less @ nat @ ( A @ D ) @ ( A @ E ) ) )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ nat @ ( A @ B ) @ ( A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(231,axiom,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( A = B )
& ( ord_less @ TA @ C @ D ) )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_107_add__mono__thms__linordered__field_I2_J) ).
thf(922,plain,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( A = B )
& ( ord_less @ TA @ C @ D ) )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).
thf(146,axiom,
! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
( ! [D: TA] :
( ~ ( A @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
& ~ ( A @ E ) ) )
=> ( A @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_86_infinite__descent__measure) ).
thf(658,plain,
! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
( ! [D: TA] :
( ~ ( A @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
& ~ ( A @ E ) ) )
=> ( A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(274,axiom,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_103_add__strict__right__mono) ).
thf(1044,plain,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).
thf(159,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ capacity @ c @ A @ B @ C )
= ( ( isSimplePath @ capacity @ c @ A @ B @ C )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B )
= ( min_dist @ capacity @ c @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_122_isShortestPath__alt) ).
thf(695,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ capacity @ c @ A @ B @ C )
= ( ( isSimplePath @ capacity @ c @ A @ B @ C )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B )
= ( min_dist @ capacity @ c @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(213,axiom,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
& ( ord_less @ TA @ C @ D ) )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_99_add__mono__thms__linordered__field_I4_J) ).
thf(851,plain,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
& ( ord_less @ TA @ C @ D ) )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(295,axiom,
! [A: nat,B: nat] :
( ( connected @ capacity @ c @ A @ B )
=> ~ ! [C: list @ ( product_prod @ nat @ nat )] :
~ ( isShortestPath @ capacity @ c @ A @ C @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_18_obtain__shortest__path) ).
thf(1102,plain,
! [A: nat,B: nat] :
( ( connected @ capacity @ c @ A @ B )
=> ~ ! [C: list @ ( product_prod @ nat @ nat )] :
~ ( isShortestPath @ capacity @ c @ A @ C @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).
thf(170,axiom,
no_top @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Ono__top) ).
thf(722,plain,
no_top @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(83,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( connected @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
? [D: list @ ( product_prod @ nat @ nat )] : ( isPath @ TA @ A @ B @ D @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_143_Graph_Oconnected__def) ).
thf(490,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( connected @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
? [D: list @ ( product_prod @ nat @ nat )] : ( isPath @ TA @ A @ B @ D @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(56,axiom,
ord @ $o,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oord_10) ).
thf(426,plain,
ord @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(118,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_63_trans__le__add2) ).
thf(589,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(289,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( connected @ capacity @ c @ A @ B )
=> ( ( ( min_dist @ capacity @ c @ A @ B )
= C )
=> ( ( ord_less @ nat @ D @ C )
=> ? [E: nat] :
( ( connected @ capacity @ c @ A @ E )
& ( ( min_dist @ capacity @ c @ A @ E )
= D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_10_min__dist__less) ).
thf(1081,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( connected @ capacity @ c @ A @ B )
=> ( ( ( min_dist @ capacity @ c @ A @ B )
= C )
=> ( ( ord_less @ nat @ D @ C )
=> ? [E: nat] :
( ( connected @ capacity @ c @ A @ E )
& ( ( min_dist @ capacity @ c @ A @ E )
= D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).
thf(23,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
( ( isPath @ TA @ A @ B @ C @ D )
=> ? [E: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ TA @ A @ B @ E @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_131_Graph_OisSPath__pathLE) ).
thf(342,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
( ( isPath @ TA @ A @ B @ C @ D )
=> ? [E: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ TA @ A @ B @ E @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[23]) ).
thf(29,axiom,
! [TA: $tType] : ( preorder @ ( set @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Opreorder_4) ).
thf(355,plain,
! [TA: $tType] : ( preorder @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(69,axiom,
! [TA: $tType] :
( ( wellorder @ TA )
=> ! [A: TA > $o,B: TA] :
( ( A @ B )
=> ( ord_less_eq @ TA @ ( ord_Least @ TA @ A ) @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_156_Least__le) ).
thf(458,plain,
! [TA: $tType] :
( ( wellorder @ TA )
=> ! [A: TA > $o,B: TA] :
( ( A @ B )
=> ( ord_less_eq @ TA @ ( ord_Least @ TA @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(94,axiom,
! [TA: $tType,TB: $tType] : ( size @ ( product_prod @ TB @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Product__Type_Oprod___Nat_Osize_12) ).
thf(531,plain,
! [TA: $tType,TB: $tType] : ( size @ ( product_prod @ TB @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(237,axiom,
! [TA: $tType,TB: $tType] :
( ( wellorder @ TB )
=> ! [A: TA > TB,B: TA > $o,C: TA] :
( ! [D: TA] :
( ! [E: TA] :
( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
=> ( B @ E ) )
=> ( B @ D ) )
=> ( B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_73_measure__induct__rule) ).
thf(944,plain,
! [TA: $tType,TB: $tType] :
( ( wellorder @ TB )
=> ! [A: TA > TB,B: TA > $o,C: TA] :
( ! [D: TA] :
( ! [E: TA] :
( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
=> ( B @ E ) )
=> ( B @ D ) )
=> ( B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).
thf(148,axiom,
isPath @ capacity @ c @ s @ p1a @ va,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_6__C1_Oprems_C_I2_J) ).
thf(663,plain,
isPath @ capacity @ c @ s @ p1a @ va,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(101,axiom,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
= ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_85_ab__semigroup__add__class_Oadd__ac_I1_J) ).
thf(548,plain,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
= ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(104,axiom,
cancel_semigroup_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__semigroup__add) ).
thf(557,plain,
cancel_semigroup_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(164,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_64_trans__le__add1) ).
thf(707,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(53,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ab_semigroup_add @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Oab__semigroup__add) ).
thf(414,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ab_semigroup_add @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(15,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
| ( ord_less_eq @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_195_linear) ).
thf(327,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
| ( ord_less_eq @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(178,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat,E: nat,F: nat] :
( ( dist @ TA @ A @ B @ C @ D )
=> ( ( dist @ TA @ A @ D @ E @ F )
=> ( ( ( min_dist @ TA @ A @ B @ F )
= ( plus_plus @ nat @ C @ E ) )
=> ( ( min_dist @ TA @ A @ D @ F )
= E ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_152_Graph_Omin__dist__split_I2_J) ).
thf(739,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat,E: nat,F: nat] :
( ( dist @ TA @ A @ B @ C @ D )
=> ( ( dist @ TA @ A @ D @ E @ F )
=> ( ( ( min_dist @ TA @ A @ B @ F )
= ( plus_plus @ nat @ C @ E ) )
=> ( ( min_dist @ TA @ A @ D @ F )
= E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(52,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA > TA > $o,B: TA,C: TA] :
( ! [D: TA,E: TA] :
( ( ord_less_eq @ TA @ D @ E )
=> ( A @ D @ E ) )
=> ( ! [D: TA,E: TA] :
( ( A @ E @ D )
=> ( A @ D @ E ) )
=> ( A @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_207_linorder__wlog) ).
thf(409,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA > TA > $o,B: TA,C: TA] :
( ! [D: TA,E: TA] :
( ( ord_less_eq @ TA @ D @ E )
=> ( A @ D @ E ) )
=> ( ! [D: TA,E: TA] :
( ( A @ E @ D )
=> ( A @ D @ E ) )
=> ( A @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(293,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
| ( A = B )
| ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_231_less__linear) ).
thf(1098,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
| ( A = B )
| ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).
thf(196,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less @ TA @ A @ B )
=> ( ( A != B )
=> ( ord_less @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_220_linorder__cases) ).
thf(791,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less @ TA @ A @ B )
=> ( ( A != B )
=> ( ord_less @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(272,axiom,
! [TA: $tType,TB: $tType] :
( ( wellorder @ TB )
=> ! [A: TA > TB,B: TA > $o,C: TA] :
( ! [D: TA] :
( ! [E: TA] :
( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
=> ( B @ E ) )
=> ( B @ D ) )
=> ( B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_74_measure__induct) ).
thf(1038,plain,
! [TA: $tType,TB: $tType] :
( ( wellorder @ TB )
=> ! [A: TA > TB,B: TA > $o,C: TA] :
( ! [D: TA] :
( ! [E: TA] :
( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
=> ( B @ E ) )
=> ( B @ D ) )
=> ( B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).
thf(71,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( strict2144017051up_add @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Ostrict__ordered__ab__semigroup__add) ).
thf(463,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( strict2144017051up_add @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(21,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( comm_monoid_add @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Ocomm__monoid__add) ).
thf(338,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( comm_monoid_add @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(75,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_178_order__refl) ).
thf(470,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(240,axiom,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_96_add__less__le__mono) ).
thf(951,plain,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ord_less @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).
thf(31,axiom,
! [TA: $tType] : ( order @ ( set @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oorder_5) ).
thf(359,plain,
! [TA: $tType] : ( order @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(139,axiom,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( ord_less @ TB @ A @ ( B @ C ) )
=> ( ( ord_less @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less @ TA @ E @ F )
=> ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_243_order__less__subst1) ).
thf(643,plain,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( ord_less @ TB @ A @ ( B @ C ) )
=> ( ( ord_less @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less @ TA @ E @ F )
=> ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(98,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
( ( connected @ TA @ A @ B @ C )
=> ( dist @ TA @ A @ B @ ( min_dist @ TA @ A @ B @ C ) @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_125_Graph_Omin__dist__is__dist) ).
thf(544,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
( ( connected @ TA @ A @ B @ C )
=> ( dist @ TA @ A @ B @ ( min_dist @ TA @ A @ B @ C ) @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(20,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( semigroup_add @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Groups_Osemigroup__add) ).
thf(336,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( semigroup_add @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[20]) ).
thf(267,axiom,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
= ( B = C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_11_add__left__cancel) ).
thf(1023,plain,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ A @ C ) )
= ( B = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).
thf(277,axiom,
! [TA: $tType] :
( ( wellorder @ TA )
=> ! [A: TA > $o,B: TA] :
( ! [C: TA] :
( ! [D: TA] :
( ( ord_less @ TA @ D @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_224_less__induct) ).
thf(1052,plain,
! [TA: $tType] :
( ( wellorder @ TA )
=> ! [A: TA > $o,B: TA] :
( ! [C: TA] :
( ! [D: TA] :
( ( ord_less @ TA @ D @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).
thf(261,axiom,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less @ TB @ A @ B )
=> ( ( ord_less @ TA @ ( C @ B ) @ D )
=> ( ! [E: TB,F: TB] :
( ( ord_less @ TB @ E @ F )
=> ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_242_order__less__subst2) ).
thf(1008,plain,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less @ TB @ A @ B )
=> ( ( ord_less @ TA @ ( C @ B ) @ D )
=> ( ! [E: TB,F: TB] :
( ( ord_less @ TB @ E @ F )
=> ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).
thf(120,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA] :
~ ( ord_less @ TA @ A @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_219_dual__order_Oirrefl) ).
thf(595,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA] :
~ ( ord_less @ TA @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(128,axiom,
( ( ord_less @ nat )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
& ( A != B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_61_nat__less__le) ).
thf(612,plain,
( ( ord_less @ nat )
= ( ^ [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
& ( A != B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(250,axiom,
! [A: nat] :
( ( reachableNodes @ capacity @ c @ A )
= ( collect @ nat @ ( connected @ capacity @ c @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_117_reachableNodes__def) ).
thf(978,plain,
! [A: nat] :
( ( reachableNodes @ capacity @ c @ A )
= ( collect @ nat @ ( connected @ capacity @ c @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).
thf(42,axiom,
! [A: nat,B: nat] :
( ( connected @ a @ c2 @ A @ B )
= ( ? [C: list @ ( product_prod @ nat @ nat )] : ( isPath @ a @ c2 @ A @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_170_g_H_Oconnected__def) ).
thf(386,plain,
! [A: nat,B: nat] :
( ( connected @ a @ c2 @ A @ B )
= ( ? [C: list @ ( product_prod @ nat @ nat )] : ( isPath @ a @ c2 @ A @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(211,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
= ( ord_less @ nat @ B @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_3_nat__add__left__cancel__less) ).
thf(845,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ ( plus_plus @ nat @ A @ C ) )
= ( ord_less @ nat @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(72,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isPath @ a @ c2 @ A @ B @ C )
=> ? [D: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ a @ c2 @ A @ D @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_171_g_H_OisSPath__pathLE) ).
thf(465,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isPath @ a @ c2 @ A @ B @ C )
=> ? [D: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ a @ c2 @ A @ D @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(66,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ A )
=> ( B = A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_210_dual__order_Oantisym) ).
thf(450,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ A )
=> ( B = A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[66]) ).
thf(151,axiom,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ! [D: nat] :
( ( ord_less @ nat @ D @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_89_nat__less__induct) ).
thf(671,plain,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ! [D: nat] :
( ( ord_less @ nat @ D @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(184,axiom,
isPath @ a @ c2 @ ua @ p2_a @ t,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_153__C1_Oprems_C_I3_J) ).
thf(759,plain,
isPath @ a @ c2 @ ua @ p2_a @ t,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(189,axiom,
! [TA: $tType] :
( ( size @ TA )
=> ! [A: TA,B: TA] :
( ( ( size_size @ TA @ A )
!= ( size_size @ TA @ B ) )
=> ( A != B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_95_size__neq__size__imp__neq) ).
thf(771,plain,
! [TA: $tType] :
( ( size @ TA )
=> ! [A: TA,B: TA] :
( ( ( size_size @ TA @ A )
!= ( size_size @ TA @ B ) )
=> ( A != B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(297,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( reachableNodes @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( collect @ nat @ ( connected @ TA @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_133_Graph_OreachableNodes__def) ).
thf(1108,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( reachableNodes @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( collect @ nat @ ( connected @ TA @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).
thf(244,axiom,
! [A: nat,B: nat] :
( ( connected @ capacity @ c @ A @ B )
=> ( dist @ capacity @ c @ A @ ( min_dist @ capacity @ c @ A @ B ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_25_min__dist__is__dist) ).
thf(961,plain,
! [A: nat,B: nat] :
( ( connected @ capacity @ c @ A @ B )
=> ( dist @ capacity @ c @ A @ ( min_dist @ capacity @ c @ A @ B ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).
thf(11,axiom,
! [TA: $tType,TB: $tType] :
( ( order @ TA )
=> ( order @ ( TB > TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_fun___Orderings_Oorder) ).
thf(319,plain,
! [TA: $tType,TB: $tType] :
( ( order @ TA )
=> ( order @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(4,axiom,
! [TA: $tType] :
( ( wellorder @ TA )
=> ! [A: TA > $o,B: TA > $o] :
( ? [C: TA] : ( A @ C )
=> ( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) )
=> ( B @ ( ord_Least @ TA @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_249_LeastI2__ex) ).
thf(304,plain,
! [TA: $tType] :
( ( wellorder @ TA )
=> ! [A: TA > $o,B: TA > $o] :
( ? [C: TA] : ( A @ C )
=> ( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) )
=> ( B @ ( ord_Least @ TA @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(294,axiom,
order @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oorder_2) ).
thf(1101,plain,
order @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).
thf(192,axiom,
isPath @ capacity @ c @ s @ p1a @ va,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_33_P1) ).
thf(781,plain,
isPath @ capacity @ c @ s @ p1a @ va,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(256,axiom,
wellorder @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Owellorder) ).
thf(995,plain,
wellorder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).
thf(117,axiom,
canoni770627133id_add @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add) ).
thf(588,plain,
canoni770627133id_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(161,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 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_112_add__less__mono1) ).
thf(699,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ ( plus_plus @ nat @ A @ C ) @ ( plus_plus @ nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(245,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( A = B )
& ( ord_less_eq @ TA @ C @ D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_54_add__mono__thms__linordered__semiring_I2_J) ).
thf(963,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( A = B )
& ( ord_less_eq @ TA @ C @ D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(88,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_234_less__asym) ).
thf(517,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(143,axiom,
linordered_idom @ capacity,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tfree_0) ).
thf(652,plain,
linordered_idom @ capacity,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(27,axiom,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ! [A: TA > TB,B: TA > TB,C: TA] :
( ( ord_less_eq @ ( TA > TB ) @ A @ B )
=> ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_185_le__funD) ).
thf(351,plain,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ! [A: TA > TB,B: TA > TB,C: TA] :
( ( ord_less_eq @ ( TA > TB ) @ A @ B )
=> ( ord_less_eq @ TB @ ( A @ C ) @ ( B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(175,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 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_65_add__le__mono1) ).
thf(731,plain,
! [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 ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(194,axiom,
! [A: nat,B: nat] :
( ( connected @ capacity @ c @ A @ B )
= ( ? [C: list @ ( product_prod @ nat @ nat )] : ( isPath @ capacity @ c @ A @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_15_connected__def) ).
thf(786,plain,
! [A: nat,B: nat] :
( ( connected @ capacity @ c @ A @ B )
= ( ? [C: list @ ( product_prod @ nat @ nat )] : ( isPath @ capacity @ c @ A @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(269,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ a @ c2 @ A @ B @ C )
= ( ( isPath @ a @ c2 @ A @ B @ C )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B )
= ( min_dist @ a @ c2 @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_177_g_H_OisShortestPath__min__dist__def) ).
thf(1030,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ a @ c2 @ A @ B @ C )
= ( ( isPath @ a @ c2 @ A @ B @ C )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ B )
= ( min_dist @ a @ c2 @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).
thf(228,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_255_order_Ostrict__implies__order) ).
thf(914,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).
thf(86,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
=> ( ~ ( ord_less @ TA @ A @ B )
=> ( ord_less @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_239_neqE) ).
thf(510,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
=> ( ~ ( ord_less @ TA @ A @ B )
=> ( ord_less @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(150,axiom,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( dist @ a @ c2 @ A @ B @ C )
=> ( ( dist @ a @ c2 @ C @ D @ E )
=> ( ( ( min_dist @ a @ c2 @ A @ E )
= ( plus_plus @ nat @ B @ D ) )
=> ( ( min_dist @ a @ c2 @ A @ C )
= B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_168_g_H_Omin__dist__split_I1_J) ).
thf(667,plain,
! [A: nat,B: nat,C: nat,D: nat,E: nat] :
( ( dist @ a @ c2 @ A @ B @ C )
=> ( ( dist @ a @ c2 @ C @ D @ E )
=> ( ( ( min_dist @ a @ c2 @ A @ E )
= ( plus_plus @ nat @ B @ D ) )
=> ( ( min_dist @ a @ c2 @ A @ C )
= B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(186,axiom,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
= ( A = C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_12_add__right__cancel) ).
thf(763,plain,
! [TA: $tType] :
( ( cancel_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( plus_plus @ TA @ A @ B )
= ( plus_plus @ TA @ C @ B ) )
= ( A = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(155,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 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_115_add__less__mono) ).
thf(684,plain,
! [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 ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(25,axiom,
! [TA: $tType,TB: $tType] :
( ( preorder @ TA )
=> ( preorder @ ( TB > TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_fun___Orderings_Opreorder) ).
thf(346,plain,
! [TA: $tType,TB: $tType] :
( ( preorder @ TA )
=> ( preorder @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(168,axiom,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ord_less @ TA @ ( plus_plus @ TA @ C @ A ) @ ( plus_plus @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_104_add__strict__left__mono) ).
thf(716,plain,
! [TA: $tType] :
( ( ordere223160158up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ord_less @ TA @ ( plus_plus @ TA @ C @ A ) @ ( plus_plus @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(288,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( A != B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_91_less__not__refl3) ).
thf(1077,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ A @ B )
=> ( A != B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).
thf(78,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ a @ c2 @ A @ B @ C )
=> ( isPath @ a @ c2 @ A @ B @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_172_g_H_OshortestPath__is__path) ).
thf(478,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ a @ c2 @ A @ B @ C )
=> ( isPath @ a @ c2 @ A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(131,axiom,
! [TA: $tType] :
( ( wellorder @ TA )
=> ! [A: TA,B: TA > $o] :
( ( ord_less @ TA @ A @ ( ord_Least @ TA @ B ) )
=> ~ ( B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_155_not__less__Least) ).
thf(620,plain,
! [TA: $tType] :
( ( wellorder @ TA )
=> ! [A: TA,B: TA > $o] :
( ( ord_less @ TA @ A @ ( ord_Least @ TA @ B ) )
=> ~ ( B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(64,axiom,
linorder @ $o,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Olinorder_8) ).
thf(447,plain,
linorder @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(8,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( reachableNodes @ TA )
= ( reachableNodes @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_129_Graph_OreachableNodes_Ocong) ).
thf(315,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(59,axiom,
! [TA: $tType] :
( ( wellorder @ TA )
=> ! [A: TA > $o,B: TA,C: TA > $o] :
( ( A @ B )
=> ( ! [D: TA] :
( ( A @ D )
=> ( C @ D ) )
=> ( C @ ( ord_Least @ TA @ A ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_247_LeastI2) ).
thf(432,plain,
! [TA: $tType] :
( ( wellorder @ TA )
=> ! [A: TA > $o,B: TA,C: TA > $o] :
( ( A @ B )
=> ( ! [D: TA] :
( ( A @ D )
=> ( C @ D ) )
=> ( C @ ( ord_Least @ TA @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(246,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( dist @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
? [E: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ TA @ A @ B @ E @ D )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ E )
= C ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_123_Graph_Odist__def) ).
thf(967,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( dist @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
? [E: list @ ( product_prod @ nat @ nat )] :
( ( isPath @ TA @ A @ B @ E @ D )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ E )
= C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).
thf(203,axiom,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
=> ( ord_less @ TA @ A @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_101_add__less__imp__less__right) ).
thf(818,plain,
! [TA: $tType] :
( ( ordere236663937imp_le @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ ( plus_plus @ TA @ A @ B ) @ ( plus_plus @ TA @ C @ B ) )
=> ( ord_less @ TA @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(50,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
( ( connected @ TA @ A @ B @ C )
=> ~ ! [D: list @ ( product_prod @ nat @ nat )] :
~ ( isShortestPath @ TA @ A @ B @ D @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_147_Graph_Oobtain__shortest__path) ).
thf(405,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
( ( connected @ TA @ A @ B @ C )
=> ~ ! [D: list @ ( product_prod @ nat @ nat )] :
~ ( isShortestPath @ TA @ A @ B @ D @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(37,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_205_order__trans) ).
thf(373,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ B @ C )
=> ( ord_less_eq @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(177,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( A != B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_212_order_Ostrict__implies__not__eq) ).
thf(735,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( A != B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(107,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less @ TA @ A @ B )
=> ( ( ~ ( ord_less @ TA @ B @ A ) )
= ( B = A ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_223_antisym__conv3) ).
thf(564,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less @ TA @ A @ B )
=> ( ( ~ ( ord_less @ TA @ B @ A ) )
= ( B = A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(230,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ( ( plus_plus @ nat @ C @ B )
= ( plus_plus @ nat @ A @ D ) )
=> ( ord_less @ nat @ C @ D ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_109_less__add__eq__less) ).
thf(918,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less @ nat @ A @ B )
=> ( ( ( plus_plus @ nat @ C @ B )
= ( plus_plus @ nat @ A @ D ) )
=> ( ord_less @ nat @ C @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).
thf(61,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ord @ TA ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Orderings_Oord) ).
thf(437,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ord @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(149,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ B @ A )
=> ( A = B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_39_le__antisym) ).
thf(664,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ B @ A )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(270,axiom,
! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_36_le__refl) ).
thf(1033,plain,
! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).
thf(206,axiom,
! [A: nat,B: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ B ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_113_not__add__less2) ).
thf(828,plain,
! [A: nat,B: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ A @ B ) @ B ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(282,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_52_add__mono) ).
thf(1065,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).
thf(242,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_217_less__imp__not__less) ).
thf(955,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).
thf(198,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( isShortestPath @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
( ( isPath @ TA @ A @ B @ C @ D )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ C )
= ( min_dist @ TA @ A @ B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_119_Graph_OisShortestPath__min__dist__def) ).
thf(796,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( isShortestPath @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat] :
( ( isPath @ TA @ A @ B @ C @ D )
& ( ( size_size @ ( list @ ( product_prod @ nat @ nat ) ) @ C )
= ( min_dist @ TA @ A @ B @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(46,axiom,
! [TA: $tType,A: set @ TA] :
( ( collect @ TA
@ ^ [B: TA] : ( member @ TA @ B @ A ) )
= A ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_46_Collect__mem__eq) ).
thf(395,plain,
! [TA: $tType,A: set @ TA] :
( ( collect @ TA
@ ^ [B: TA] : ( member @ TA @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(174,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
=> ~ ( ( ord_less_eq @ nat @ A @ C )
=> ~ ( ord_less_eq @ nat @ B @ C ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_72_add__leE) ).
thf(728,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ A @ B ) @ C )
=> ~ ( ( ord_less_eq @ nat @ A @ C )
=> ~ ( ord_less_eq @ nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(112,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_227_dual__order_Oasym) ).
thf(575,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(57,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( connected @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
? [D: nat] : ( dist @ TA @ A @ B @ D @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_145_Graph_Oconnected__by__dist) ).
thf(427,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( connected @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
? [D: nat] : ( dist @ TA @ A @ B @ D @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(93,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ~ ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_87_linorder__neqE__nat) ).
thf(528,plain,
! [A: nat,B: nat] :
( ( A != B )
=> ( ~ ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(255,axiom,
! [TA: $tType] :
( ( semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
= ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_81_add_Oassoc) ).
thf(992,plain,
! [TA: $tType] :
( ( semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ ( plus_plus @ TA @ A @ B ) @ C )
= ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).
thf(171,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_233_less__asym_H) ).
thf(723,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(260,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( B != A ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_211_dual__order_Ostrict__implies__not__eq) ).
thf(1004,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( B != A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).
thf(219,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
& ( ord_less_eq @ TA @ C @ D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_53_add__mono__thms__linordered__semiring_I1_J) ).
thf(871,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( ord_less_eq @ TA @ A @ B )
& ( ord_less_eq @ TA @ C @ D ) )
=> ( ord_less_eq @ TA @ ( plus_plus @ TA @ A @ C ) @ ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).
thf(35,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( connected @ TA @ A @ B @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_138_Graph_Oconnected__refl) ).
thf(369,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( connected @ TA @ A @ B @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(125,axiom,
! [A: nat,B: nat,C: nat] :
( ( dist @ capacity @ c @ A @ B @ C )
=> ( ord_less_eq @ nat @ ( min_dist @ capacity @ c @ A @ C ) @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_20_min__dist__minD) ).
thf(607,plain,
! [A: nat,B: nat,C: nat] :
( ( dist @ capacity @ c @ A @ B @ C )
=> ( ord_less_eq @ nat @ ( min_dist @ capacity @ c @ A @ C ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(82,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( A = B )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_196_eq__refl) ).
thf(486,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( A = B )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(234,axiom,
! [A: nat,B: nat] : ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ A @ B ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_71_le__add1) ).
thf(935,plain,
! [A: nat,B: nat] : ( ord_less_eq @ nat @ A @ ( plus_plus @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).
thf(6,axiom,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_45_mem__Collect__eq) ).
thf(309,plain,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(152,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
=> ( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less @ TA @ A @ B ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_251_order_Onot__eq__order__implies__strict) ).
thf(676,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
=> ( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less @ TA @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(119,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 ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_66_add__le__mono) ).
thf(592,plain,
! [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 ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(284,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ capacity @ c @ A @ B @ C )
=> ( isPath @ capacity @ c @ A @ B @ C ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_16_shortestPath__is__path) ).
thf(1069,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ capacity @ c @ A @ B @ C )
=> ( isPath @ capacity @ c @ A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).
thf(33,axiom,
! [A: nat,B: nat] :
( ( connected @ a @ c2 @ A @ B )
= ( ? [C: nat] : ( dist @ a @ c2 @ A @ C @ B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_157_g_H_Oconnected__by__dist) ).
thf(364,plain,
! [A: nat,B: nat] :
( ( connected @ a @ c2 @ A @ B )
= ( ? [C: nat] : ( dist @ a @ c2 @ A @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(263,axiom,
preorder @ nat,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Opreorder_1) ).
thf(1014,plain,
preorder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).
thf(74,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( isPath @ TA )
= ( isPath @ TA ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_136_Graph_OisPath_Ocong) ).
thf(469,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(236,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( A != B ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_235_less__imp__neq) ).
thf(940,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( A != B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).
thf(190,axiom,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_77_add_Oleft__commute) ).
thf(775,plain,
! [TA: $tType] :
( ( ab_semigroup_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( plus_plus @ TA @ A @ ( plus_plus @ TA @ B @ C ) )
= ( plus_plus @ TA @ B @ ( plus_plus @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(106,axiom,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( A = B )
& ( C = D ) )
=> ( ( plus_plus @ TA @ A @ C )
= ( plus_plus @ TA @ B @ D ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',fact_84_add__mono__thms__linordered__semiring_I4_J) ).
thf(560,plain,
! [TA: $tType] :
( ( ordere779506340up_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( A = B )
& ( C = D ) )
=> ( ( plus_plus @ TA @ A @ C )
= ( plus_plus @ TA @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(1124,plain,
$false,
inference(cvc4,[status(thm)],[809,760,1105,555,909,333,977,1073,873,379,440,614,597,523,582,1041,861,533,472,814,698,747,538,1063,340,1095,898,782,841,973,1046,389,1055,570,838,344,357,1059,949,460,702,751,416,448,316,638,853,1115,325,475,756,789,321,646,937,806,353,480,1020,348,687,692,734,453,998,507,312,981,766,307,439,661,546,1067,714,334,957,825,778,302,820,407,630,484,361,625,435,329,583,493,682,551,710,916,467,456,842,1089,650,1111,725,403,869,610,1118,519,742,933,642,526,1016,547,901,1075,679,970,622,393,997,911,569,371,633,541,657,1026,953,718,926,398,1035,558,1071,794,586,856,745,689,375,601,513,430,904,423,577,641,391,382,572,713,1090,704,947,604,831,1011,1048,567,318,649,599,377,990,996,616,303,1122,890,609,323,483,895,330,1061,858,367,799,726,848,520,983,399,768,1015,986,653,922,658,1044,695,851,1102,722,490,426,589,1081,342,355,458,531,944,663,548,557,707,414,327,739,409,1098,791,1038,463,338,470,951,359,643,544,336,1023,1052,1008,595,612,978,386,845,465,450,671,759,771,1108,961,319,304,1101,781,995,588,699,963,517,652,351,731,786,1030,914,510,667,763,684,346,716,1077,478,620,447,315,432,967,818,405,373,735,564,918,437,664,1033,828,1065,955,796,395,728,575,427,528,992,723,1004,871,369,607,486,935,309,676,592,1069,364,1014,469,940,775,560]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.14 % Problem : ITP048^2 : TPTP v8.2.0. Released v7.5.0.
% 0.13/0.17 % Command : run_Leo-III %s %d
% 0.18/0.38 % Computer : n018.cluster.edu
% 0.18/0.38 % Model : x86_64 x86_64
% 0.18/0.38 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.38 % Memory : 8042.1875MB
% 0.18/0.38 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.38 % CPULimit : 300
% 0.18/0.38 % WCLimit : 300
% 0.18/0.38 % DateTime : Sat May 18 17:05:09 EDT 2024
% 0.18/0.39 % CPUTime :
% 1.00/0.90 % [INFO] Parsing problem /export/starexec/sandbox2/benchmark/theBenchmark.p ...
% 1.63/1.15 % [INFO] Parsing done (246ms).
% 1.99/1.16 % [INFO] Running in sequential loop mode.
% 2.67/1.38 % [INFO] eprover registered as external prover.
% 2.67/1.38 % [INFO] cvc4 registered as external prover.
% 2.67/1.39 % [INFO] Scanning for conjecture ...
% 2.89/1.48 % [INFO] Found a conjecture (or negated_conjecture) and 300 axioms. Running axiom selection ...
% 3.17/1.58 % [INFO] Axiom selection finished. Selected 299 axioms (removed 1 axioms).
% 3.75/1.77 % [INFO] Problem is higher-order (TPTP THF).
% 4.07/1.78 % [INFO] Type checking passed.
% 4.07/1.79 % [CONFIG] Using configuration: timeout(300) with strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>. Searching for refutation ...
% 65.45/42.57 % External prover 'cvc4' found a proof!
% 65.45/42.57 % [INFO] Killing All external provers ...
% 65.45/42.58 % Time passed: 42024ms (effective reasoning time: 41413ms)
% 65.45/42.58 % Solved by strategy<name(default),share(1.0),primSubst(3),sos(false),unifierCount(4),uniDepth(8),boolExt(true),choice(true),renaming(true),funcspec(false), domConstr(0),specialInstances(39),restrictUniAttempts(true),termOrdering(CPO)>
% 65.45/42.58 % Axioms used in derivation (299): fact_53_add__mono__thms__linordered__semiring_I1_J, fact_37_le__trans, fact_249_LeastI2__ex, fact_22_isPath__distD, fact_2__092_060open_062min__dist_Au_At_A_092_060le_062_Alength_Ap2_H_092_060close_062, fact_156_Least__le, subcl_Rings_Olinordered__idom___Groups_Ogroup__add, fact_42_add__le__imp__le__right, fact_36_le__refl, fact_79_add_Oright__cancel, fact_128_shortestPath__is__simple, subcl_Rings_Olinordered__idom___Orderings_Olinorder, tcon_HOL_Obool___Orderings_Opreorder_7, fact_30_add__le__cancel__right, tfree_0, tcon_Nat_Onat___Orderings_Oorder_2, fact_197_le__cases, fact_18_obtain__shortest__path, fact_204_order__class_Oorder_Oantisym, fact_242_order__less__subst2, fact_149_Graph_Omin__dist__le, fact_51_add__left__mono, fact_200_antisym__conv, fact_80_add_Oleft__cancel, fact_6__C1_Oprems_C_I2_J, fact_92_less__not__refl2, fact_248_LeastI__ex, fact_118_min__dist__def, tcon_Nat_Onat___Orderings_Ono__top, fact_190_order__subst2, subcl_Rings_Olinordered__idom___Groups_Oordered__cancel__ab__semigroup__add, fact_133_Graph_OreachableNodes__def, fact_185_le__funD, tcon_Nat_Onat___Orderings_Olinorder, fact_251_order_Onot__eq__order__implies__strict, fact_75_add__right__imp__eq, tcon_Nat_Onat___Nat_Osize, fact_217_less__imp__not__less, fact_129_Graph_OreachableNodes_Ocong, fact_213_not__less__iff__gr__or__eq, fact_255_order_Ostrict__implies__order, fact_29__092_060open_062connected_As_Au_092_060close_062, fact_3_nat__add__left__cancel__less, fact_66_add__le__mono, fact_96_add__less__le__mono, fact_176_g_H_OisPath__distD, fact_11_add__left__cancel, fact_175_g_H_Odist__def, fact_221_less__imp__triv, fact_178_order__refl, fact_134_Graph_Omin__dist__def, fact_64_trans__le__add1, fact_209_dual__order_Oeq__iff, fact_247_LeastI2, fact_150_Graph_Omin__distI__eq, fact_90_less__irrefl__nat, fact_76_add__left__imp__eq, fact_73_measure__induct__rule, fact_74_measure__induct, fact_218_order_Ostrict__trans, fact_244_ord__less__eq__subst, fact_194_antisym, fact_104_add__strict__left__mono, fact_208_dual__order_Otrans, fact_50_less__eqE, fact_122_isShortestPath__alt, fact_121_Graph_Omin__distI2, tcon_fun___Orderings_Oord, fact_224_less__induct, fact_227_dual__order_Oasym, fact_233_less__asym_H, fact_8_min__dist__split_I2_J, fact_220_linorder__cases, fact_151_Graph_Omin__dist__minD, fact_216_exists__least__iff, fact_14_dist__trans, fact_85_ab__semigroup__add__class_Oadd__ac_I1_J, fact_157_g_H_Oconnected__by__dist, fact_139_Graph_Oconnected_Ocong, subcl_Rings_Olinordered__idom___Orderings_Opreorder, fact_105_add__strict__mono, fact_33_P1, fact_155_not__less__Least, fact_52_add__mono, fact_97_add__le__less__mono, fact_237_order_Oasym, fact_161_g_H_OreachableNodes__def, fact_254_dual__order_Oorder__iff__strict, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add, subcl_Rings_Olinordered__idom___Groups_Ocomm__monoid__add, fact_108_add__mono__thms__linordered__field_I5_J, fact_187_le__funI, fact_241_lt__ex, fact_136_Graph_OisPath_Ocong, fact_192_ord__le__eq__subst, fact_60_less__imp__le__nat, tcon_HOL_Obool___Orderings_Oorder_9, fact_253_dual__order_Ostrict__iff__order, fact_56_less__mono__imp__le__mono, tcon_Product__Type_Oprod___Nat_Osize_12, fact_57_le__neq__implies__less, tcon_List_Olist___Nat_Osize_11, fact_169_g_H_Odist__trans, tcon_Nat_Onat___Groups_Ocomm__monoid__add, fact_23_dist__def, fact_49_add__right__mono, fact_189_order__subst1, fact_145_Graph_Oconnected__by__dist, fact_87_linorder__neqE__nat, subcl_Rings_Olinordered__idom___HOL_Otype, tcon_Nat_Onat___Groups_Ocancel__semigroup__add, fact_83_group__cancel_Oadd1, fact_1__092_060open_062min__dist_As_Au_A_060_Alength_Ap1_092_060close_062, fact_88_infinite__descent, fact_99_add__mono__thms__linordered__field_I4_J, fact_125_Graph_Omin__dist__is__dist, fact_231_less__linear, fact_182_g_H_Oconnected__distI, fact_127_isSPath__pathLE, subcl_Rings_Olinordered__idom___Orderings_Oorder, fact_153__C1_Oprems_C_I3_J, fact_142_Graph_Odist__trans, fact_184_P2_H, fact_41_Nat_Oex__has__greatest__nat, fact_183_g_H_Oconnected__refl, fact_46_Collect__mem__eq, fact_31_add__le__cancel__left, fact_132_Graph_OshortestPath__is__simple, fact_13__092_060open_062isPath_Au_Ap2_H_At_092_060close_062, fact_67_le__Suc__ex, fact_165_g_H_Omin__dist__le, fact_225_less__not__sym, fact_226_less__imp__not__eq, fact_205_order__trans, subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add, fact_34_connected__refl, fact_69_add__leD1, fact_93_less__not__refl, fact_54_add__mono__thms__linordered__semiring_I2_J, tcon_Set_Oset___Orderings_Oord_6, fact_103_add__strict__right__mono, fact_94_nat__neq__iff, fact_68_add__leD2, fact_186_le__funE, fact_65_add__le__mono1, fact_81_add_Oassoc, fact_32_nat__add__left__cancel__le, fact_130_Graph_OisSimplePath_Ocong, fact_235_less__imp__neq, fact_91_less__not__refl3, fact_163_g_H_Omin__distI2, tcon_Nat_Onat___Orderings_Owellorder, tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le, fact_206_dual__order_Orefl, fact_246_ex__has__least__nat, fact_230_less__irrefl, fact_131_Graph_OisSPath__pathLE, fact_196_eq__refl, fact_102_add__less__imp__less__left, fact_111_trans__less__add1, subcl_Rings_Olinordered__idom___Groups_Ocancel__semigroup__add, fact_215_linorder__less__wlog, fact_55_add__mono__thms__linordered__semiring_I3_J, tcon_HOL_Obool___Orderings_Oord_10, fact_45_mem__Collect__eq, fact_84_add__mono__thms__linordered__semiring_I4_J, fact_78_add_Ocommute, fact_164_g_H_Omin__distI__eq, fact_82_group__cancel_Oadd2, fact_77_add_Oleft__commute, fact_20_min__dist__minD, fact_101_add__less__imp__less__right, fact_4_add__less__cancel__left, fact_229_ord__eq__less__trans, fact_243_order__less__subst1, fact_71_le__add1, fact_199_le__cases3, fact_239_neqE, fact_119_Graph_OisShortestPath__min__dist__def, subcl_Rings_Olinordered__idom___Groups_Oordered__ab__semigroup__add__imp__le, fact_106_add__mono__thms__linordered__field_I1_J, fact_70_le__add2, fact_143_Graph_Oconnected__def, fact_202_ord__eq__le__trans, fact_232_less__trans, fact_148_Graph_Omin__dist__less, tcon_Nat_Onat___Orderings_Oord_3, subcl_Rings_Olinordered__idom___Groups_Osemigroup__add, fact_100_mono__nat__linear__lb, fact_89_nat__less__induct, fact_47_Collect__cong, fact_171_g_H_OisSPath__pathLE, fact_5_add__less__cancel__right, fact_252_dual__order_Ostrict__implies__order, fact_110_trans__less__add2, fact_203_ord__le__eq__trans, fact_172_g_H_OshortestPath__is__path, fact_112_add__less__mono1, fact_16_shortestPath__is__path, tcon_Set_Oset___Orderings_Opreorder_4, fact_98_add__mono__thms__linordered__field_I3_J, fact_135_Graph_OisShortestPath__alt, fact_198_order_Otrans, fact_27_min__distI2, fact_158_g_H_Omin__dist__is__dist, fact_228_ord__less__eq__trans, subcl_Rings_Olinordered__idom___Groups_Oab__semigroup__add, fact_214_dual__order_Ostrict__trans, fact_62_nat__le__iff__add, fact_159_g_H_Oobtain__shortest__path, fact_236_dense, fact_191_ord__eq__le__subst, fact_181__092_060open_062g_H_OisPath_Au_____Ap2_H_____At_092_060close_062, fact_38_eq__imp__le, fact_7_SP, fact_146_Graph_Oconnected__distI, fact_39_le__antisym, fact_207_linorder__wlog, tcon_fun___Orderings_Opreorder, tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add, fact_124_Graph_OisPath__distD, fact_250_LeastI, tcon_Set_Oset___Orderings_Oorder_5, fact_116_add__lessD1, fact_120_Graph_OisShortestPath__def, fact_245_ord__eq__less__subst, fact_173_g_H_OisShortestPath__alt, fact_154_ex__has__greatest__nat__lemma, fact_19_min__dist__le, fact_109_less__add__eq__less, fact_137_Graph_Omin__dist_Ocong, tfree_1, tcon_Nat_Onat___Orderings_Opreorder_1, fact_144_Graph_OshortestPath__is__path, fact_210_dual__order_Oantisym, fact_123_Graph_Odist__def, fact_168_g_H_Omin__dist__split_I1_J, fact_15_connected__def, tcon_HOL_Obool___Orderings_Olinorder_8, fact_195_linear, fact_107_add__mono__thms__linordered__field_I2_J, fact_238_neq__iff, fact_25_min__dist__is__dist, fact_44_le__iff__add, fact_59_le__eq__less__or__eq, fact_17_connected__by__dist, fact_219_dual__order_Oirrefl, fact_179_g_H_OisShortestPath__def, fact_113_not__add__less2, tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add, tcon_fun___Orderings_Oorder, fact_170_g_H_Oconnected__def, fact_24__092_060open_062isPath_As_Ap1_____Av_____092_060close_062, tcon_Nat_Onat___Groups_Osemigroup__add, subcl_Rings_Olinordered__idom___Orderings_Oord, fact_21_min__distI__eq, fact_115_add__less__mono, fact_61_nat__less__le, fact_26_isShortestPath__def, fact_162_g_H_Omin__dist__less, tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, fact_86_infinite__descent__measure, fact_147_Graph_Oobtain__shortest__path, fact_166_g_H_Omin__dist__minD, fact_201_order__class_Oorder_Oeq__iff, fact_43_add__le__imp__le__left, fact_152_Graph_Omin__dist__split_I2_J, fact_140_Graph_Odist_Ocong, fact_167_g_H_Omin__dist__split_I2_J, fact_10_min__dist__less, tcon_Nat_Onat___Groups_Oab__semigroup__add, subcl_Rings_Olinordered__idom___Groups_Ostrict__ordered__ab__semigroup__add, fact_138_Graph_Oconnected__refl, fact_63_trans__le__add2, fact_212_order_Ostrict__implies__not__eq, fact_35_connected__distI, fact_141_Graph_OisShortestPath_Ocong, fact_28_isShortestPath__min__dist__def, fact_177_g_H_OisShortestPath__min__dist__def, fact_9_min__dist__split_I1_J, fact_117_reachableNodes__def, fact_0__092_060open_062min__dist_As_At_A_061_Amin__dist_As_Au_A_L_Amin__dist_Au_At_092_060close_062, fact_180_assms_I5_J, fact_223_antisym__conv3, fact_40_nat__le__linear, fact_222_less__imp__not__eq2, fact_95_size__neq__size__imp__neq, fact_114_not__add__less1, fact_12_add__right__cancel, fact_126_Graph_Omin__dist__split_I1_J, fact_160_g_H_OshortestPath__is__simple, fact_193_eq__iff, fact_58_less__or__eq__imp__le, fact_174_g_H_Omin__dist__def, fact_72_add__leE, fact_240_gt__ex, fact_234_less__asym, fact_188_le__fun__def, fact_211_dual__order_Ostrict__implies__not__eq
% 65.45/42.58 % No. of inferences in proof: 602
% 65.45/42.58 % SZS status Theorem for /export/starexec/sandbox2/benchmark/theBenchmark.p : 42024 ms resp. 41413 ms w/o parsing
% 66.10/42.86 % SZS output start Refutation for /export/starexec/sandbox2/benchmark/theBenchmark.p
% See solution above
% 66.10/42.86 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------