%------------------------------------------------------------------------------
% File : Leo-III---1.8.0
% Problem : ITP049^2 : TPTP v9.3.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% Computer : n020.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Sep 30 08:06:33 AM UTC 2026
% Result : Theorem 205.21s 173.45s
% Output : Refutation 206.32s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 280
% Syntax : Number of formulae : 562 ( 136 unt; 0 typ; 0 def)
% Number of atoms : 2336 ( 267 equ; 6 cnn)
% Maximal formula atoms : 21 ( 4 avg)
% Number of connectives : 10432 ( 164 ~; 14 |; 100 &;9339 @)
% ( 0 <=>; 815 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 10 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 308 ( 308 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 42 usr; 8 con; 0-5 aty)
% Number of variables : 2265 ( 152 ^;2013 !; 100 ?;2265 :)
% Comments :
%------------------------------------------------------------------------------
thf(nat_type,type,
nat: $tType ).
thf(a_type,type,
a: $tType ).
thf(type_decl,type,
type:
!>[TA: $tType] : $o ).
thf(ord_decl,type,
ord:
!>[TA: $tType] : $o ).
thf(order_decl,type,
order:
!>[TA: $tType] : $o ).
thf(finite_finite_decl,type,
finite_finite:
!>[TA: $tType] : $o ).
thf(linorder_decl,type,
linorder:
!>[TA: $tType] : $o ).
thf(preorder_decl,type,
preorder:
!>[TA: $tType] : $o ).
thf(linordered_idom_decl,type,
linordered_idom:
!>[TA: $tType] : $o ).
thf(bNF_Ca1785829860lChain_decl,type,
bNF_Ca1785829860lChain:
!>[TA: $tType,TB: $tType] : ( ( set @ ( product_prod @ TB @ TB ) ) > ( TB > TA ) > $o ) ).
thf(edmond2129202899alysis_decl,type,
edmond2129202899alysis:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > $o ) ).
thf(edmond1803015688pEdges_decl,type,
edmond1803015688pEdges:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > nat > ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(finite_finite2_decl,type,
finite_finite2:
!>[TA: $tType] : ( ( set @ TA ) > $o ) ).
thf(finite_Graph_decl,type,
finite_Graph:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > $o ) ).
thf(e_decl,type,
e:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(v_decl,type,
v:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > ( set @ nat ) ) ).
thf(adjacent_nodes_decl,type,
adjacent_nodes:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > ( set @ nat ) ) ).
thf(connected_decl,type,
connected:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > nat > $o ) ).
thf(incoming_decl,type,
incoming:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(incoming2_decl,type,
incoming2:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > ( set @ nat ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(isPath_decl,type,
isPath:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
thf(isShortestPath_decl,type,
isShortestPath:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
thf(isSimplePath_decl,type,
isSimplePath:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > ( list @ ( product_prod @ nat @ nat ) ) > nat > $o ) ).
thf(outgoing_decl,type,
outgoing:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(outgoing2_decl,type,
outgoing2:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > ( set @ nat ) > ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(reachableNodes_decl,type,
reachableNodes:
!>[TA: $tType] : ( ( ( product_prod @ nat @ nat ) > TA ) > nat > ( set @ nat ) ) ).
thf(set2_decl,type,
set2:
!>[TA: $tType] : ( ( list @ TA ) > ( set @ TA ) ) ).
thf(ord_less_eq_decl,type,
ord_less_eq:
!>[TA: $tType] : ( TA > TA > $o ) ).
thf(product_Pair_decl,type,
product_Pair:
!>[TA: $tType,TB: $tType] : ( TB > TA > ( product_prod @ TB @ TA ) ) ).
thf(product_Sigma_decl,type,
product_Sigma:
!>[TA: $tType,TB: $tType] : ( ( set @ TB ) > ( TB > ( set @ TA ) ) > ( set @ ( product_prod @ TB @ TA ) ) ) ).
thf(product_rec_prod_decl,type,
product_rec_prod:
!>[TA: $tType,TB: $tType,TC: $tType] : ( ( TC > TB > TA ) > ( product_prod @ TC @ TB ) > TA ) ).
thf(product_swap_decl,type,
product_swap:
!>[TA: $tType,TB: $tType] : ( ( product_prod @ TB @ TA ) > ( product_prod @ TA @ TB ) ) ).
thf(domain_decl,type,
domain:
!>[TA: $tType,TB: $tType] : ( ( set @ ( product_prod @ TB @ TA ) ) > ( set @ TB ) ) ).
thf(image_decl,type,
image:
!>[TA: $tType,TB: $tType] : ( ( set @ ( product_prod @ TB @ TA ) ) > ( set @ TB ) > ( set @ TA ) ) ).
thf(range_decl,type,
range:
!>[TA: $tType,TB: $tType] : ( ( set @ ( product_prod @ TB @ TA ) ) > ( set @ TA ) ) ).
thf(collect_decl,type,
collect:
!>[TA: $tType] : ( ( TA > $o ) > ( set @ TA ) ) ).
thf(transitive_rtrancl_decl,type,
transitive_rtrancl:
!>[TA: $tType] : ( ( set @ ( product_prod @ TA @ TA ) ) > ( set @ ( product_prod @ TA @ TA ) ) ) ).
thf(member_decl,type,
member:
!>[TA: $tType] : ( TA > ( set @ TA ) > $o ) ).
thf(c_decl,type,
c: ( product_prod @ nat @ nat ) > a ).
thf(s_decl,type,
s: nat ).
thf(t_decl,type,
t: nat ).
thf(264,axiom,
! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ ( product_prod @ TA @ TB )] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ C )
=> ( member @ TB @ B @ ( range @ TA @ TB @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_85_Range_Ointros) ).
thf(1036,plain,
! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ ( product_prod @ TA @ TB )] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ C )
=> ( member @ TB @ B @ ( range @ TA @ TB @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).
thf(189,axiom,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TB @ TA ),B: set @ TB,C: set @ TA,D: set @ TB] :
( ( ord_less_eq @ ( set @ ( product_prod @ TB @ TA ) ) @ A
@ ( product_Sigma @ TB @ TA @ B
@ ^ [E: TB] : C ) )
=> ( ord_less_eq @ ( set @ TA ) @ ( image @ TB @ TA @ A @ D ) @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_Image__subset) ).
thf(809,plain,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TB @ TA ),B: set @ TB,C: set @ TA,D: set @ TB] :
( ( ord_less_eq @ ( set @ ( product_prod @ TB @ TA ) ) @ A
@ ( product_Sigma @ TB @ TA @ B
@ ^ [E: TB] : C ) )
=> ( ord_less_eq @ ( set @ TA ) @ ( image @ TB @ TA @ A @ D ) @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(260,axiom,
! [A: set @ nat] :
( ( finite_finite2 @ nat @ ( v @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming2 @ a @ c @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_finite__incoming_H) ).
thf(1024,plain,
! [A: set @ nat] :
( ( finite_finite2 @ nat @ ( v @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming2 @ a @ c @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).
thf(28,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/sandbox/benchmark/theBenchmark.p',fact_239_le__funI) ).
thf(347,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)],[28]) ).
thf(238,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing2 @ TA @ A @ B ) @ ( e @ TA @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_97_Graph_Ooutgoing_H__edges) ).
thf(962,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing2 @ TA @ A @ B ) @ ( e @ TA @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).
thf(137,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TB] :
( ( finite_finite2 @ TA @ A )
=> ( ( finite_finite2 @ TB @ B )
=> ( finite_finite2 @ ( product_prod @ TA @ TB )
@ ( product_Sigma @ TA @ TB @ A
@ ^ [C: TA] : B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_141_finite__cartesian__product) ).
thf(666,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TB] :
( ( finite_finite2 @ TA @ A )
=> ( ( finite_finite2 @ TB @ B )
=> ( finite_finite2 @ ( product_prod @ TA @ TB )
@ ( product_Sigma @ TA @ TB @ A
@ ^ [C: TA] : B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(46,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ( edmond2129202899alysis @ TA @ A )
=> ( finite_Graph @ TA @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_ek__analysis_Oaxioms) ).
thf(408,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ( edmond2129202899alysis @ TA @ A )
=> ( finite_Graph @ TA @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(110,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isPath @ a @ c @ A @ B @ C )
=> ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A @ C ) @ ( transitive_rtrancl @ nat @ ( e @ a @ c ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_249_isPath__rtc) ).
thf(582,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isPath @ a @ c @ A @ B @ C )
=> ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A @ C ) @ ( transitive_rtrancl @ nat @ ( e @ a @ c ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(12,axiom,
! [TA: $tType] : ( ord @ ( set @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oord_7) ).
thf(308,plain,
! [TA: $tType] : ( ord @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[12]) ).
thf(65,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( order @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Orderings_Oorder) ).
thf(449,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( order @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(203,axiom,
! [TA: $tType,A: set @ TA] :
( ( finite_finite2 @ TA @ A )
=> ( finite_finite2 @ ( set @ TA )
@ ( collect @ ( set @ TA )
@ ^ [B: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_finite__Collect__subsets) ).
thf(861,plain,
! [TA: $tType,A: set @ TA] :
( ( finite_finite2 @ TA @ A )
=> ( finite_finite2 @ ( set @ TA )
@ ( collect @ ( set @ TA )
@ ^ [B: set @ TA] : ( ord_less_eq @ ( set @ TA ) @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(60,axiom,
! [TA: $tType] : ( preorder @ ( set @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Opreorder_4) ).
thf(440,plain,
! [TA: $tType] : ( preorder @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(133,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] :
( ( finite_finite2 @ nat @ ( v @ TA @ A ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing @ TA @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_150_Graph_Ofinite__outgoing) ).
thf(655,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] :
( ( finite_finite2 @ nat @ ( v @ TA @ A ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing @ TA @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(39,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/sandbox/benchmark/theBenchmark.p',fact_225_ord__le__eq__trans) ).
thf(385,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)],[39]) ).
thf(95,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: set @ nat] :
( ( finite_finite2 @ nat @ ( v @ TA @ A ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing2 @ TA @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_105_Graph_Ofinite__outgoing_H) ).
thf(533,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: set @ nat] :
( ( finite_finite2 @ nat @ ( v @ TA @ A ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing2 @ TA @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[95]) ).
thf(83,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( connected @ TA )
= ( connected @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_104_Graph_Oconnected_Ocong) ).
thf(500,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(151,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isPath @ a @ c @ A @ B @ C )
=> ? [D: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ a @ c @ A @ D @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_243_isSPath__pathLE) ).
thf(715,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isPath @ a @ c @ A @ B @ C )
=> ? [D: list @ ( product_prod @ nat @ nat )] : ( isSimplePath @ a @ c @ A @ D @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(232,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( finite_Graph @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA] : ( finite_finite2 @ nat @ ( v @ TA @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_114_Finite__Graph__def) ).
thf(945,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( finite_Graph @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA] : ( finite_finite2 @ nat @ ( v @ TA @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(228,axiom,
! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ TA,D: TA > ( set @ TB )] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ ( product_Sigma @ TA @ TB @ C @ D ) )
= ( ( member @ TA @ A @ C )
& ( member @ TB @ B @ ( D @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_153_mem__Sigma__iff) ).
thf(930,plain,
! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ TA,D: TA > ( set @ TB )] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ ( product_Sigma @ TA @ TB @ C @ D ) )
= ( ( member @ TA @ A @ C )
& ( member @ TB @ B @ ( D @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).
thf(251,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: nat,E: nat] :
( ( isShortestPath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ( connected @ a @ c @ D @ E ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50_isShortestPath__level__edge_I2_J) ).
thf(1001,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: nat,E: nat] :
( ( isShortestPath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ( connected @ a @ c @ D @ E ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).
thf(149,axiom,
! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB )] :
( ( member @ TB @ A @ ( range @ TA @ TB @ B ) )
= ( ? [C: TA,D: TB] :
( ( A = D )
& ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_86_Range_Osimps) ).
thf(709,plain,
! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB )] :
( ( member @ TB @ A @ ( range @ TA @ TB @ B ) )
= ( ? [C: TA,D: TB] :
( ( A = D )
& ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(211,axiom,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ( ( finite_finite2 @ TA )
= ( ^ [A: set @ TA] : $true ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_finite__code) ).
thf(885,plain,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ( ( finite_finite2 @ TA )
= ( ^ [A: set @ TA] : $true ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(269,axiom,
! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB )] :
( ( member @ TB @ A @ ( range @ TA @ TB @ B ) )
= ( ? [C: TA] : ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ A ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_Range__iff) ).
thf(1046,plain,
! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB )] :
( ( member @ TB @ A @ ( range @ TA @ TB @ B ) )
= ( ? [C: TA] : ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ A ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).
thf(40,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( edmond2129202899alysis @ TA )
= ( finite_Graph @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29_ek__analysis__def) ).
thf(389,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( edmond2129202899alysis @ TA )
= ( finite_Graph @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[40]) ).
thf(273,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/sandbox/benchmark/theBenchmark.p',fact_90_Graph_OreachableNodes__def) ).
thf(1055,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)],[273]) ).
thf(165,axiom,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB )] :
( ( finite_finite2 @ ( product_prod @ TA @ TB ) @ A )
=> ( finite_finite2 @ TA @ ( domain @ TA @ TB @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_finite__Domain) ).
thf(751,plain,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB )] :
( ( finite_finite2 @ ( product_prod @ TA @ TB ) @ A )
=> ( finite_finite2 @ TA @ ( domain @ TA @ TB @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(30,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( reachableNodes @ TA )
= ( reachableNodes @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_Graph_OreachableNodes_Ocong) ).
thf(357,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(70,axiom,
! [TA: $tType,TB: $tType] :
( ( ( finite_finite @ TB )
& ( finite_finite @ TA ) )
=> ( finite_finite @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Finite__Set_Ofinite) ).
thf(460,plain,
! [TA: $tType,TB: $tType] :
( ( ( finite_finite @ TB )
& ( finite_finite @ TA ) )
=> ( finite_finite @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(185,axiom,
! [A: nat] :
( ( member @ nat @ A @ ( v @ a @ c ) )
=> ( ord_less_eq @ ( set @ nat ) @ ( reachableNodes @ a @ c @ A ) @ ( v @ a @ c ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_reachable__ss__V) ).
thf(802,plain,
! [A: nat] :
( ( member @ nat @ A @ ( v @ a @ c ) )
=> ( ord_less_eq @ ( set @ nat ) @ ( reachableNodes @ a @ c @ A ) @ ( v @ a @ c ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(26,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_le__cases) ).
thf(344,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)],[26]) ).
thf(216,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: set @ TA,B: TA] :
( ( finite_finite2 @ TA @ A )
=> ( ( member @ TA @ B @ A )
=> ? [C: TA] :
( ( member @ TA @ C @ A )
& ( ord_less_eq @ TA @ C @ B )
& ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ( ord_less_eq @ TA @ D @ C )
=> ( C = D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_finite__has__minimal2) ).
thf(902,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: set @ TA,B: TA] :
( ( finite_finite2 @ TA @ A )
=> ( ( member @ TA @ B @ A )
=> ? [C: TA] :
( ( member @ TA @ C @ A )
& ( ord_less_eq @ TA @ C @ B )
& ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ( ord_less_eq @ TA @ D @ C )
=> ( C = D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(104,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: product_prod @ nat @ nat] :
( ( isPath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ E @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ( member @ ( product_prod @ nat @ nat ) @ E @ ( e @ TA @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_252_Graph_OisPath__edgeset) ).
thf(559,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: product_prod @ nat @ nat] :
( ( isPath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ E @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ( member @ ( product_prod @ nat @ nat ) @ E @ ( e @ TA @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(200,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA ),D: set @ TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TA ) ) @ C
@ ( product_Sigma @ TA @ TA @ D
@ ^ [E: TA] : D ) )
=> ( ( A = B )
| ( member @ TA @ A @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_155_trancl__subset__Sigma__aux) ).
thf(853,plain,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA ),D: set @ TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TA ) ) @ C
@ ( product_Sigma @ TA @ TA @ D
@ ^ [E: TA] : D ) )
=> ( ( A = B )
| ( member @ TA @ A @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(53,axiom,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
=> ~ ( ( A = C )
=> ( B != D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_Pair__inject) ).
thf(421,plain,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
=> ~ ( ( A = C )
=> ( B != D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[53]) ).
thf(119,axiom,
! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TB @ TA )] :
( ( member @ TB @ A @ ( domain @ TB @ TA @ B ) )
= ( ? [C: TA] : ( member @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ A @ C ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_Domain__iff) ).
thf(606,plain,
! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TB @ TA )] :
( ( member @ TB @ A @ ( domain @ TB @ TA @ B ) )
= ( ? [C: TA] : ( member @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ A @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(19,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o,C: TA] :
( ( ord_less_eq @ ( TA > $o ) @ A @ B )
=> ( ( A @ C )
=> ( B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_predicate1D) ).
thf(325,plain,
! [TA: $tType,A: TA > $o,B: TA > $o,C: TA] :
( ( ord_less_eq @ ( TA > $o ) @ A @ B )
=> ( ( A @ C )
=> ( B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(5,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( A = B )
=> ( ord_less_eq @ TA @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_218_eq__refl) ).
thf(289,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)],[5]) ).
thf(64,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( isShortestPath @ TA )
= ( isShortestPath @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_106_Graph_OisShortestPath_Ocong) ).
thf(448,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(225,axiom,
! [TA: $tType,TB: $tType,A: TA,B: set @ ( product_prod @ TA @ TB )] :
( ( member @ TA @ A @ ( domain @ TA @ TB @ B ) )
= ( ? [C: TA,D: TB] :
( ( A = C )
& ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_Domain_Osimps) ).
thf(924,plain,
! [TA: $tType,TB: $tType,A: TA,B: set @ ( product_prod @ TA @ TB )] :
( ( member @ TA @ A @ ( domain @ TA @ TB @ B ) )
= ( ? [C: TA,D: TB] :
( ( A = C )
& ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(167,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] :
( ( finite_Graph @ TA @ A )
=> ( finite_finite2 @ nat @ ( adjacent_nodes @ TA @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_151_Finite__Graph_Oadjacent__nodes__finite) ).
thf(756,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] :
( ( finite_Graph @ TA @ A )
=> ( finite_finite2 @ nat @ ( adjacent_nodes @ TA @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(180,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: nat,E: nat] :
( ( isPath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ( ( D != A )
=> ? [F: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ F @ D ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_isPath__ex__edge1) ).
thf(789,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: nat,E: nat] :
( ( isPath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ( ( D != A )
=> ? [F: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ F @ D ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(62,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) )
=> ( ord_less_eq @ ( TA > $o ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_predicate1I) ).
thf(443,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
=> ( B @ C ) )
=> ( ord_less_eq @ ( TA > $o ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[62]) ).
thf(36,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/sandbox/benchmark/theBenchmark.p',fact_230_dual__order_Otrans) ).
thf(376,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)],[36]) ).
thf(187,axiom,
! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: set @ ( product_prod @ TA @ TA )] :
( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TA ) ) @ A @ ( transitive_rtrancl @ TA @ B ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ TA @ TA ) ) @ ( transitive_rtrancl @ TA @ A ) @ ( transitive_rtrancl @ TA @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_rtrancl__subset__rtrancl) ).
thf(806,plain,
! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: set @ ( product_prod @ TA @ TA )] :
( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TA ) ) @ A @ ( transitive_rtrancl @ TA @ B ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ TA @ TA ) ) @ ( transitive_rtrancl @ TA @ A ) @ ( transitive_rtrancl @ TA @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(240,axiom,
! [TA: $tType,TB: $tType,A: TA,B: set @ TA,C: TB,D: TA > ( set @ TB )] :
( ( member @ TA @ A @ B )
=> ( ( member @ TB @ C @ ( D @ A ) )
=> ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ C ) @ ( product_Sigma @ TA @ TB @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_154_SigmaI) ).
thf(966,plain,
! [TA: $tType,TB: $tType,A: TA,B: set @ TA,C: TB,D: TA > ( set @ TB )] :
( ( member @ TA @ A @ B )
=> ( ( member @ TB @ C @ ( D @ A ) )
=> ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ C ) @ ( product_Sigma @ TA @ TB @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).
thf(271,axiom,
order @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oorder_2) ).
thf(1051,plain,
order @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).
thf(266,axiom,
! [A: nat,B: nat,C: nat] :
( ( connected @ a @ c @ A @ B )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ B @ C ) @ ( e @ a @ c ) )
=> ( connected @ a @ c @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_connected__append__edge) ).
thf(1040,plain,
! [A: nat,B: nat,C: nat] :
( ( connected @ a @ c @ A @ B )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ B @ C ) @ ( e @ a @ c ) )
=> ( connected @ a @ c @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).
thf(17,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/sandbox/benchmark/theBenchmark.p',fact_137_Graph_OshortestPath__is__simple) ).
thf(321,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)],[17]) ).
thf(43,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_order__refl) ).
thf(397,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(135,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: nat,F: nat] :
( ( isPath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ( ( F != D )
=> ? [G: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ F @ G ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_250_Graph_OisPath__ex__edge2) ).
thf(660,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: nat,F: nat] :
( ( isPath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ( ( F != D )
=> ? [G: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ F @ G ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(48,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_228_dual__order_Orefl) ).
thf(411,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(55,axiom,
! [TA: $tType,TB: $tType,A: ( product_prod @ TA @ TB ) > $o,B: product_prod @ TA @ TB] :
( ! [C: TA,D: TB] : ( A @ ( product_Pair @ TA @ TB @ C @ D ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_165_old_Oprod_Oinducts) ).
thf(428,plain,
! [TA: $tType,TB: $tType,A: ( product_prod @ TA @ TB ) > $o,B: product_prod @ TA @ TB] :
( ! [C: TA,D: TB] : ( A @ ( product_Pair @ TA @ TB @ C @ D ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(198,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ( finite_Graph @ TA @ A )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ TA @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_Finite__Graph_Ofinite__E) ).
thf(849,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ( finite_Graph @ TA @ A )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ TA @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(6,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( type @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___HOL_Otype) ).
thf(293,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( type @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(237,axiom,
! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA,B: set @ TB,C: TB > ( set @ TA )] :
( ( member @ ( product_prod @ TB @ TA ) @ A @ ( product_Sigma @ TB @ TA @ B @ C ) )
=> ~ ! [D: TB] :
( ( member @ TB @ D @ B )
=> ! [E: TA] :
( ( member @ TA @ E @ ( C @ D ) )
=> ( A
!= ( product_Pair @ TB @ TA @ D @ E ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_SigmaE) ).
thf(956,plain,
! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA,B: set @ TB,C: TB > ( set @ TA )] :
( ( member @ ( product_prod @ TB @ TA ) @ A @ ( product_Sigma @ TB @ TA @ B @ C ) )
=> ~ ! [D: TB] :
( ( member @ TB @ D @ B )
=> ! [E: TA] :
( ( member @ TA @ E @ ( C @ D ) )
=> ( A
!= ( product_Pair @ TB @ TA @ D @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).
thf(278,axiom,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB )] :
( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ A @ B )
=> ( ord_less_eq @ ( set @ TB ) @ ( range @ TA @ TB @ A ) @ ( range @ TA @ TB @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_Range__mono) ).
thf(1072,plain,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB )] :
( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ A @ B )
=> ( ord_less_eq @ ( set @ TB ) @ ( range @ TA @ TB @ A ) @ ( range @ TA @ TB @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).
thf(67,axiom,
! [TA: $tType,TB: $tType,A: TA,B: TB] :
( ( product_swap @ TA @ TB @ ( product_Pair @ TA @ TB @ A @ B ) )
= ( product_Pair @ TB @ TA @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_161_swap__simp) ).
thf(453,plain,
! [TA: $tType,TB: $tType,A: TA,B: TB] :
( ( product_swap @ TA @ TB @ ( product_Pair @ TA @ TB @ A @ B ) )
= ( product_Pair @ TB @ TA @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[67]) ).
thf(279,axiom,
! [A: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [B: nat] : ( ord_less_eq @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_finite__Collect__le__nat) ).
thf(1074,plain,
! [A: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [B: nat] : ( ord_less_eq @ nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).
thf(223,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: nat,F: nat] :
( ( isShortestPath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ( connected @ TA @ A @ E @ F ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_Graph_OisShortestPath__level__edge_I2_J) ).
thf(920,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: nat,F: nat] :
( ( isShortestPath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ( connected @ TA @ A @ E @ F ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).
thf(86,axiom,
! [TA: $tType,TB: $tType,A: product_prod @ TA @ TB] :
( ( product_swap @ TB @ TA @ ( product_swap @ TA @ TB @ A ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_160_swap__swap) ).
thf(512,plain,
! [TA: $tType,TB: $tType,A: product_prod @ TA @ TB] :
( ( product_swap @ TB @ TA @ ( product_swap @ TA @ TB @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[86]) ).
thf(38,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/sandbox/benchmark/theBenchmark.p',fact_211_order__subst1) ).
thf(381,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)],[38]) ).
thf(32,axiom,
! [TA: $tType,TB: $tType] :
( ( order @ TA )
=> ( order @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oorder) ).
thf(366,plain,
! [TA: $tType,TB: $tType] :
( ( order @ TA )
=> ( order @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(276,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( connected @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ B @ C ) @ ( transitive_rtrancl @ nat @ ( e @ TA @ A ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_148_Graph_Oconnected__edgeRtc) ).
thf(1067,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( connected @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ B @ C ) @ ( transitive_rtrancl @ nat @ ( e @ TA @ A ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).
thf(183,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ a @ c @ A @ B @ C )
=> ( isPath @ a @ c @ A @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_245_shortestPath__is__path) ).
thf(798,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ a @ c @ A @ B @ C )
=> ( isPath @ a @ c @ A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(99,axiom,
! [A: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming @ a @ c @ A ) @ ( e @ a @ c ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_incoming__edges) ).
thf(546,plain,
! [A: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming @ a @ c @ A ) @ ( e @ a @ c ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[99]) ).
thf(181,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( incoming2 @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: set @ nat] :
( collect @ ( product_prod @ nat @ nat )
@ ^ [C: product_prod @ nat @ nat] :
? [D: nat,E: nat] :
( ( C
= ( product_Pair @ nat @ nat @ D @ E ) )
& ~ ( member @ nat @ D @ B )
& ( member @ nat @ E @ B )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ E ) @ ( e @ TA @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_66_Graph_Oincoming_H__def) ).
thf(793,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( incoming2 @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: set @ nat] :
( collect @ ( product_prod @ nat @ nat )
@ ^ [C: product_prod @ nat @ nat] :
? [D: nat,E: nat] :
( ( C
= ( product_Pair @ nat @ nat @ D @ E ) )
& ~ ( member @ nat @ D @ B )
& ( member @ nat @ E @ B )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ E ) @ ( e @ TA @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(22,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/sandbox/benchmark/theBenchmark.p',fact_238_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)],[22]) ).
thf(218,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( edmond1803015688pEdges @ TA )
= ( edmond1803015688pEdges @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_ek__analysis__defs_OspEdges_Ocong) ).
thf(910,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(213,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: nat,F: nat,G: nat] :
( ( isSimplePath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ( ( F != G )
=> ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ G ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_142_Graph_OisSPath__sg__outgoing) ).
thf(894,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: nat,F: nat,G: nat] :
( ( isSimplePath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ( ( F != G )
=> ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ G ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(259,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) )] :
~ ! [B: TD,C: TC,D: TB,E: TA] :
( A
!= ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ B @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ C @ ( product_Pair @ TB @ TA @ D @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_prod__cases4) ).
thf(1021,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) )] :
~ ! [B: TD,C: TC,D: TB,E: TA] :
( A
!= ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ B @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ C @ ( product_Pair @ TB @ TA @ D @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).
thf(234,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: product_prod @ nat @ nat] :
( ( isSimplePath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ E @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_180_Graph_OisSPath__nt__parallel) ).
thf(952,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: product_prod @ nat @ nat] :
( ( isSimplePath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ E @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).
thf(29,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TA,C: TA > ( set @ TB ),D: TA > ( set @ TB )] :
( ( A = B )
=> ( ! [E: TA] :
( ( member @ TA @ E @ B )
=> ( ( C @ E )
= ( D @ E ) ) )
=> ( ( product_Sigma @ TA @ TB @ A @ C )
= ( product_Sigma @ TA @ TB @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_181_Sigma__cong) ).
thf(349,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TA,C: TA > ( set @ TB ),D: TA > ( set @ TB )] :
( ( A = B )
=> ( ! [E: TA] :
( ( member @ TA @ E @ B )
=> ( ( C @ E )
= ( D @ E ) ) )
=> ( ( product_Sigma @ TA @ TB @ A @ C )
= ( product_Sigma @ TA @ TB @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(105,axiom,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA,E: set @ ( product_prod @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) ),F: TB > TA > $o] :
( ( member @ ( product_prod @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) ) @ ( product_Pair @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ A @ B ) @ ( product_Pair @ TB @ TA @ C @ D ) ) @ ( transitive_rtrancl @ ( product_prod @ TB @ TA ) @ E ) )
=> ( ( F @ A @ B )
=> ( ! [G: TB,H: TA,I: TB,J: TA] :
( ( member @ ( product_prod @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) ) @ ( product_Pair @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ A @ B ) @ ( product_Pair @ TB @ TA @ G @ H ) ) @ ( transitive_rtrancl @ ( product_prod @ TB @ TA ) @ E ) )
=> ( ( member @ ( product_prod @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) ) @ ( product_Pair @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ G @ H ) @ ( product_Pair @ TB @ TA @ I @ J ) ) @ E )
=> ( ( F @ G @ H )
=> ( F @ I @ J ) ) ) )
=> ( F @ C @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_164_rtrancl__induct2) ).
thf(561,plain,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA,E: set @ ( product_prod @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) ),F: TB > TA > $o] :
( ( member @ ( product_prod @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) ) @ ( product_Pair @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ A @ B ) @ ( product_Pair @ TB @ TA @ C @ D ) ) @ ( transitive_rtrancl @ ( product_prod @ TB @ TA ) @ E ) )
=> ( ( F @ A @ B )
=> ( ! [G: TB,H: TA,I: TB,J: TA] :
( ( member @ ( product_prod @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) ) @ ( product_Pair @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ A @ B ) @ ( product_Pair @ TB @ TA @ G @ H ) ) @ ( transitive_rtrancl @ ( product_prod @ TB @ TA ) @ E ) )
=> ( ( member @ ( product_prod @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) ) @ ( product_Pair @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ G @ H ) @ ( product_Pair @ TB @ TA @ I @ J ) ) @ E )
=> ( ( F @ G @ H )
=> ( F @ I @ J ) ) ) )
=> ( F @ C @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(41,axiom,
! [TA: $tType,TB: $tType,A: TB > TA > $o,B: TB > TA > $o,C: TB,D: TA] :
( ( ord_less_eq @ ( TB > TA > $o ) @ A @ B )
=> ( ( A @ C @ D )
=> ( B @ C @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_233_predicate2D) ).
thf(392,plain,
! [TA: $tType,TB: $tType,A: TB > TA > $o,B: TB > TA > $o,C: TB,D: TA] :
( ( ord_less_eq @ ( TB > TA > $o ) @ A @ B )
=> ( ( A @ C @ D )
=> ( B @ C @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(142,axiom,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB )] :
( ( finite_finite2 @ ( product_prod @ TA @ TB ) @ A )
=> ( finite_finite2 @ TB @ ( range @ TA @ TB @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_finite__Range) ).
thf(693,plain,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB )] :
( ( finite_finite2 @ ( product_prod @ TA @ TB ) @ A )
=> ( finite_finite2 @ TB @ ( range @ TA @ TB @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(192,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > ( set @ TB )] :
( ( finite_finite2 @ TA @ A )
=> ( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( finite_finite2 @ TB @ ( B @ C ) ) )
=> ( finite_finite2 @ ( product_prod @ TA @ TB ) @ ( product_Sigma @ TA @ TB @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_127_finite__SigmaI) ).
thf(815,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > ( set @ TB )] :
( ( finite_finite2 @ TA @ A )
=> ( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( finite_finite2 @ TB @ ( B @ C ) ) )
=> ( finite_finite2 @ ( product_prod @ TA @ TB ) @ ( product_Sigma @ TA @ TB @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(156,axiom,
( ( finite_finite2 @ nat @ ( v @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_Vfin__imp__Efin) ).
thf(729,plain,
( ( finite_finite2 @ nat @ ( v @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(58,axiom,
! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ ( image @ TA @ TA @ A @ B ) @ B )
=> ( ( image @ TA @ TA @ ( transitive_rtrancl @ TA @ A ) @ B )
= B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_Image__closed__trancl) ).
thf(435,plain,
! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ ( image @ TA @ TA @ A @ B ) @ B )
=> ( ( image @ TA @ TA @ ( transitive_rtrancl @ TA @ A ) @ B )
= B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(80,axiom,
ord @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oord_12) ).
thf(493,plain,
ord @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[80]) ).
thf(261,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: product_prod @ nat @ nat] :
( ( isPath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ D @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ( member @ ( product_prod @ nat @ nat ) @ D @ ( e @ a @ c ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_248_isPath__edgeset) ).
thf(1027,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: product_prod @ nat @ nat] :
( ( isPath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ D @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ( member @ ( product_prod @ nat @ nat ) @ D @ ( e @ a @ c ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).
thf(101,axiom,
! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: set @ TA] :
( ( finite_finite2 @ ( product_prod @ TA @ TA ) @ A )
=> ( ( finite_finite2 @ TA @ B )
=> ( finite_finite2 @ TA @ ( image @ TA @ TA @ ( transitive_rtrancl @ TA @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_finite__rtrancl__Image) ).
thf(551,plain,
! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: set @ TA] :
( ( finite_finite2 @ ( product_prod @ TA @ TA ) @ A )
=> ( ( finite_finite2 @ TA @ B )
=> ( finite_finite2 @ TA @ ( image @ TA @ TA @ ( transitive_rtrancl @ TA @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(68,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/sandbox/benchmark/theBenchmark.p',fact_216_antisym) ).
thf(456,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)],[68]) ).
thf(243,axiom,
! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ TA,D: TA > ( set @ TB )] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ ( product_Sigma @ TA @ TB @ C @ D ) )
=> ~ ( ( member @ TA @ A @ C )
=> ~ ( member @ TB @ B @ ( D @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_SigmaE2) ).
thf(974,plain,
! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ TA,D: TA > ( set @ TB )] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ ( product_Sigma @ TA @ TB @ C @ D ) )
=> ~ ( ( member @ TA @ A @ C )
=> ~ ( member @ TB @ B @ ( D @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).
thf(255,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: nat,E: nat] :
( ( isShortestPath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ( connected @ a @ c @ E @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_isShortestPath__level__edge_I3_J) ).
thf(1012,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: nat,E: nat] :
( ( isShortestPath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ( connected @ a @ c @ E @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).
thf(221,axiom,
! [A: nat] : ( finite_finite2 @ nat @ ( adjacent_nodes @ a @ c @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_adjacent__nodes__finite) ).
thf(916,plain,
! [A: nat] : ( finite_finite2 @ nat @ ( adjacent_nodes @ a @ c @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).
thf(74,axiom,
! [TA: $tType,TB: $tType] :
( ( ord @ TA )
=> ( ord @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oord) ).
thf(467,plain,
! [TA: $tType,TB: $tType] :
( ( ord @ TA )
=> ( ord @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(144,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ( finite_finite2 @ nat @ ( v @ TA @ A ) )
=> ( finite_Graph @ TA @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_Finite__Graph_Ointro) ).
thf(697,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ( finite_finite2 @ nat @ ( v @ TA @ A ) )
=> ( finite_Graph @ TA @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(208,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA )] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( A != B )
=> ~ ! [D: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ D ) @ ( transitive_rtrancl @ TA @ C ) )
=> ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_rtranclE) ).
thf(874,plain,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA )] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( A != B )
=> ~ ! [D: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ D ) @ ( transitive_rtrancl @ TA @ C ) )
=> ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(16,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/sandbox/benchmark/theBenchmark.p',fact_224_ord__eq__le__trans) ).
thf(317,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)],[16]) ).
thf(257,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ a @ c @ A @ B @ C )
=> ( isSimplePath @ a @ c @ A @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_shortestPath__is__simple) ).
thf(1017,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat] :
( ( isShortestPath @ a @ c @ A @ B @ C )
=> ( isSimplePath @ a @ c @ A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).
thf(123,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( ord_less_eq @ ( set @ nat ) @ ( adjacent_nodes @ TA @ A @ B ) @ ( v @ TA @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_Graph_Oadjacent__nodes__ss__V) ).
thf(615,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( ord_less_eq @ ( set @ nat ) @ ( adjacent_nodes @ TA @ A @ B ) @ ( v @ TA @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(245,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: set @ TA,B: TA] :
( ( finite_finite2 @ TA @ A )
=> ( ( member @ TA @ B @ A )
=> ? [C: TA] :
( ( member @ TA @ C @ A )
& ( ord_less_eq @ TA @ B @ C )
& ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( C = D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_finite__has__maximal2) ).
thf(980,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: set @ TA,B: TA] :
( ( finite_finite2 @ TA @ A )
=> ( ( member @ TA @ B @ A )
=> ? [C: TA] :
( ( member @ TA @ C @ A )
& ( ord_less_eq @ TA @ B @ C )
& ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( C = D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(253,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: nat,B: ( product_prod @ nat @ nat ) > TA] :
( ( member @ nat @ A @ ( v @ TA @ B ) )
=> ( ord_less_eq @ ( set @ nat ) @ ( reachableNodes @ TA @ B @ A ) @ ( v @ TA @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_68_Graph_Oreachable__ss__V) ).
thf(1006,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: nat,B: ( product_prod @ nat @ nat ) > TA] :
( ( member @ nat @ A @ ( v @ TA @ B ) )
=> ( ord_less_eq @ ( set @ nat ) @ ( reachableNodes @ TA @ B @ A ) @ ( v @ TA @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(115,axiom,
! [A: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing2 @ a @ c @ A ) @ ( e @ a @ c ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_outgoing_H__edges) ).
thf(593,plain,
! [A: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing2 @ a @ c @ A ) @ ( e @ a @ c ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(247,axiom,
! [TA: $tType,TB: $tType,A: TB,B: set @ TB,C: set @ TA,D: set @ TA] :
( ( member @ TB @ A @ B )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) )
@ ( product_Sigma @ TA @ TB @ C
@ ^ [E: TA] : B )
@ ( product_Sigma @ TA @ TB @ D
@ ^ [E: TA] : B ) )
= ( ord_less_eq @ ( set @ TA ) @ C @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_Times__subset__cancel2) ).
thf(989,plain,
! [TA: $tType,TB: $tType,A: TB,B: set @ TB,C: set @ TA,D: set @ TA] :
( ( member @ TB @ A @ B )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) )
@ ( product_Sigma @ TA @ TB @ C
@ ^ [E: TA] : B )
@ ( product_Sigma @ TA @ TB @ D
@ ^ [E: TA] : B ) )
= ( ord_less_eq @ ( set @ TA ) @ C @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).
thf(89,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( linorder @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Orderings_Olinorder) ).
thf(519,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( linorder @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(162,axiom,
finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_finite__E) ).
thf(742,plain,
finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(131,axiom,
! [TA: $tType,TB: $tType,A: TA > $o,B: TB > TA > $o] :
( ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
=> ( ( finite_finite2 @ TB
@ ( collect @ TB
@ ^ [C: TB] :
? [D: TA] :
( ( A @ D )
& ( B @ C @ D ) ) ) )
= ( ! [C: TA] :
( ( A @ C )
=> ( finite_finite2 @ TB
@ ( collect @ TB
@ ^ [D: TB] : ( B @ D @ C ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_finite__Collect__bounded__ex) ).
thf(650,plain,
! [TA: $tType,TB: $tType,A: TA > $o,B: TB > TA > $o] :
( ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
=> ( ( finite_finite2 @ TB
@ ( collect @ TB
@ ^ [C: TB] :
? [D: TA] :
( ( A @ D )
& ( B @ C @ D ) ) ) )
= ( ! [C: TA] :
( ( A @ C )
=> ( finite_finite2 @ TB
@ ( collect @ TB
@ ^ [D: TB] : ( B @ D @ C ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(233,axiom,
! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB ),C: set @ TA] :
( ( member @ TB @ A @ ( image @ TA @ TB @ B @ C ) )
=> ~ ! [D: TA] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ D @ A ) @ B )
=> ~ ( member @ TA @ D @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_ImageE) ).
thf(948,plain,
! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB ),C: set @ TA] :
( ( member @ TB @ A @ ( image @ TA @ TB @ B @ C ) )
=> ~ ! [D: TA] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ D @ A ) @ B )
=> ~ ( member @ TA @ D @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).
thf(178,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
( ( connected @ TA @ A @ B @ C )
=> ( ( member @ nat @ C @ ( v @ TA @ A ) )
= ( member @ nat @ B @ ( v @ TA @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_Graph_Oconnected__inV__iff) ).
thf(783,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
( ( connected @ TA @ A @ B @ C )
=> ( ( member @ nat @ C @ ( v @ TA @ A ) )
= ( member @ nat @ B @ ( v @ TA @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(52,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( v @ TA )
= ( v @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_Graph_OV_Ocong) ).
thf(420,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(76,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/sandbox/benchmark/theBenchmark.p',fact_221_le__cases3) ).
thf(471,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)],[76]) ).
thf(78,axiom,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_157_prod_Oinject) ).
thf(488,plain,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(229,axiom,
! [TA: $tType,TB: $tType,A: TA,B: TB,C: TA,D: TB,E: set @ ( product_prod @ ( product_prod @ TA @ TB ) @ ( product_prod @ TA @ TB ) )] :
( ( member @ ( product_prod @ ( product_prod @ TA @ TB ) @ ( product_prod @ TA @ TB ) ) @ ( product_Pair @ ( product_prod @ TA @ TB ) @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ ( product_Pair @ TA @ TB @ C @ D ) ) @ ( transitive_rtrancl @ ( product_prod @ TA @ TB ) @ E ) )
=> ( ( ( product_Pair @ TA @ TB @ A @ B )
!= ( product_Pair @ TA @ TB @ C @ D ) )
=> ~ ! [F: TA,G: TB] :
( ( member @ ( product_prod @ ( product_prod @ TA @ TB ) @ ( product_prod @ TA @ TB ) ) @ ( product_Pair @ ( product_prod @ TA @ TB ) @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ ( product_Pair @ TA @ TB @ F @ G ) ) @ E )
=> ~ ( member @ ( product_prod @ ( product_prod @ TA @ TB ) @ ( product_prod @ TA @ TB ) ) @ ( product_Pair @ ( product_prod @ TA @ TB ) @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ F @ G ) @ ( product_Pair @ TA @ TB @ C @ D ) ) @ ( transitive_rtrancl @ ( product_prod @ TA @ TB ) @ E ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_163_converse__rtranclE2) ).
thf(933,plain,
! [TA: $tType,TB: $tType,A: TA,B: TB,C: TA,D: TB,E: set @ ( product_prod @ ( product_prod @ TA @ TB ) @ ( product_prod @ TA @ TB ) )] :
( ( member @ ( product_prod @ ( product_prod @ TA @ TB ) @ ( product_prod @ TA @ TB ) ) @ ( product_Pair @ ( product_prod @ TA @ TB ) @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ ( product_Pair @ TA @ TB @ C @ D ) ) @ ( transitive_rtrancl @ ( product_prod @ TA @ TB ) @ E ) )
=> ( ( ( product_Pair @ TA @ TB @ A @ B )
!= ( product_Pair @ TA @ TB @ C @ D ) )
=> ~ ! [F: TA,G: TB] :
( ( member @ ( product_prod @ ( product_prod @ TA @ TB ) @ ( product_prod @ TA @ TB ) ) @ ( product_Pair @ ( product_prod @ TA @ TB ) @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ ( product_Pair @ TA @ TB @ F @ G ) ) @ E )
=> ~ ( member @ ( product_prod @ ( product_prod @ TA @ TB ) @ ( product_prod @ TA @ TB ) ) @ ( product_Pair @ ( product_prod @ TA @ TB ) @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ F @ G ) @ ( product_Pair @ TA @ TB @ C @ D ) ) @ ( transitive_rtrancl @ ( product_prod @ TA @ TB ) @ E ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).
thf(166,axiom,
! [A: set @ nat] :
( ( incoming2 @ a @ c @ A )
= ( collect @ ( product_prod @ nat @ nat )
@ ^ [B: product_prod @ nat @ nat] :
? [C: nat,D: nat] :
( ( B
= ( product_Pair @ nat @ nat @ C @ D ) )
& ~ ( member @ nat @ C @ A )
& ( member @ nat @ D @ A )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ C @ D ) @ ( e @ a @ c ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_incoming_H__def) ).
thf(753,plain,
! [A: set @ nat] :
( ( incoming2 @ a @ c @ A )
= ( collect @ ( product_prod @ nat @ nat )
@ ^ [B: product_prod @ nat @ nat] :
? [C: nat,D: nat] :
( ( B
= ( product_Pair @ nat @ nat @ C @ D ) )
& ~ ( member @ nat @ C @ A )
& ( member @ nat @ D @ A )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ C @ D ) @ ( e @ a @ c ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(128,axiom,
! [A: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming2 @ a @ c @ A ) @ ( e @ a @ c ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_incoming_H__edges) ).
thf(642,plain,
! [A: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming2 @ a @ c @ A ) @ ( e @ a @ c ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(92,axiom,
! [TA: $tType,TB: $tType] :
( ( preorder @ TA )
=> ( preorder @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Opreorder) ).
thf(526,plain,
! [TA: $tType,TB: $tType] :
( ( preorder @ TA )
=> ( preorder @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(113,axiom,
! [TA: $tType,TB: $tType,A: TA,B: set @ ( product_prod @ TA @ TB )] :
( ( member @ TA @ A @ ( domain @ TA @ TB @ B ) )
=> ~ ! [C: TB] :
~ ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ C ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_83_DomainE) ).
thf(590,plain,
! [TA: $tType,TB: $tType,A: TA,B: set @ ( product_prod @ TA @ TB )] :
( ( member @ TA @ A @ ( domain @ TA @ TB @ B ) )
=> ~ ! [C: TB] :
~ ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ C ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(130,axiom,
! [A: nat] :
( ( reachableNodes @ a @ c @ A )
= ( collect @ nat @ ( connected @ a @ c @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_reachableNodes__def) ).
thf(647,plain,
! [A: nat] :
( ( reachableNodes @ a @ c @ A )
= ( collect @ nat @ ( connected @ a @ c @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(87,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( incoming @ TA )
= ( incoming @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_Graph_Oincoming_Ocong) ).
thf(515,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(219,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( incoming @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat] :
( collect @ ( product_prod @ nat @ nat )
@ ^ [C: product_prod @ nat @ nat] :
? [D: nat] :
( ( C
= ( product_Pair @ nat @ nat @ D @ B ) )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ B ) @ ( e @ TA @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_Graph_Oincoming__def) ).
thf(911,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( incoming @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat] :
( collect @ ( product_prod @ nat @ nat )
@ ^ [C: product_prod @ nat @ nat] :
? [D: nat] :
( ( C
= ( product_Pair @ nat @ nat @ D @ B ) )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ B ) @ ( e @ TA @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).
thf(176,axiom,
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1__092_060open_062finite_AE_A_092_060Longrightarrow_062_Afinite_AspEdges_092_060close_062) ).
thf(779,plain,
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(14,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/sandbox/benchmark/theBenchmark.p',fact_226_order__class_Oorder_Oantisym) ).
thf(311,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)],[14]) ).
thf(97,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( e @ TA @ A )
@ ( product_Sigma @ nat @ nat @ ( v @ TA @ A )
@ ^ [B: nat] : ( v @ TA @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_149_Graph_OE__ss__VxV) ).
thf(541,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( e @ TA @ A )
@ ( product_Sigma @ nat @ nat @ ( v @ TA @ A )
@ ^ [B: nat] : ( v @ TA @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(134,axiom,
! [A: nat,B: nat] :
( ( connected @ a @ c @ A @ B )
= ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A @ B ) @ ( transitive_rtrancl @ nat @ ( e @ a @ c ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_connected__edgeRtc) ).
thf(657,plain,
! [A: nat,B: nat] :
( ( connected @ a @ c @ A @ B )
= ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A @ B ) @ ( transitive_rtrancl @ nat @ ( e @ a @ c ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(170,axiom,
! [TA: $tType,TB: $tType,A: TB > $o,B: TB > TA] :
( ( finite_finite2 @ TB @ ( collect @ TB @ A ) )
=> ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [C: TA] :
? [D: TB] :
( ( C
= ( B @ D ) )
& ( A @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_finite__image__set) ).
thf(765,plain,
! [TA: $tType,TB: $tType,A: TB > $o,B: TB > TA] :
( ( finite_finite2 @ TB @ ( collect @ TB @ A ) )
=> ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [C: TA] :
? [D: TB] :
( ( C
= ( B @ D ) )
& ( A @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(94,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: product_prod @ TC @ ( product_prod @ TB @ TA )] :
~ ! [B: TC,C: TB,D: TA] :
( A
!= ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ B @ ( product_Pair @ TB @ TA @ C @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_prod__cases3) ).
thf(530,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: product_prod @ TC @ ( product_prod @ TB @ TA )] :
~ ! [B: TC,C: TB,D: TA] :
( A
!= ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ B @ ( product_Pair @ TB @ TA @ C @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(174,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA )] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
= ( ? [D: TA] :
( ( A = D )
& ( B = D ) )
| ? [D: TA,E: TA,F: TA] :
( ( A = D )
& ( B = F )
& ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ E ) @ ( transitive_rtrancl @ TA @ C ) )
& ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ F ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_rtrancl_Osimps) ).
thf(773,plain,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA )] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
= ( ? [D: TA] :
( ( A = D )
& ( B = D ) )
| ? [D: TA,E: TA,F: TA] :
( ( A = D )
& ( B = F )
& ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ E ) @ ( transitive_rtrancl @ TA @ C ) )
& ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ F ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(212,axiom,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA,E: set @ ( product_prod @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) ),F: TB > TA > $o] :
( ( member @ ( product_prod @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) ) @ ( product_Pair @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ A @ B ) @ ( product_Pair @ TB @ TA @ C @ D ) ) @ ( transitive_rtrancl @ ( product_prod @ TB @ TA ) @ E ) )
=> ( ( F @ C @ D )
=> ( ! [G: TB,H: TA,I: TB,J: TA] :
( ( member @ ( product_prod @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) ) @ ( product_Pair @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ G @ H ) @ ( product_Pair @ TB @ TA @ I @ J ) ) @ E )
=> ( ( member @ ( product_prod @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) ) @ ( product_Pair @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ I @ J ) @ ( product_Pair @ TB @ TA @ C @ D ) ) @ ( transitive_rtrancl @ ( product_prod @ TB @ TA ) @ E ) )
=> ( ( F @ I @ J )
=> ( F @ G @ H ) ) ) )
=> ( F @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_162_converse__rtrancl__induct2) ).
thf(888,plain,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA,E: set @ ( product_prod @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) ),F: TB > TA > $o] :
( ( member @ ( product_prod @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) ) @ ( product_Pair @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ A @ B ) @ ( product_Pair @ TB @ TA @ C @ D ) ) @ ( transitive_rtrancl @ ( product_prod @ TB @ TA ) @ E ) )
=> ( ( F @ C @ D )
=> ( ! [G: TB,H: TA,I: TB,J: TA] :
( ( member @ ( product_prod @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) ) @ ( product_Pair @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ G @ H ) @ ( product_Pair @ TB @ TA @ I @ J ) ) @ E )
=> ( ( member @ ( product_prod @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) ) @ ( product_Pair @ ( product_prod @ TB @ TA ) @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ I @ J ) @ ( product_Pair @ TB @ TA @ C @ D ) ) @ ( transitive_rtrancl @ ( product_prod @ TB @ TA ) @ E ) )
=> ( ( F @ I @ J )
=> ( F @ G @ H ) ) ) )
=> ( F @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(195,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) )] :
( ! [C: TA,D: TB,E: TC,F: TD] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ TD ) @ D @ ( product_Pair @ TC @ TD @ E @ F ) ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_prod__induct4) ).
thf(824,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) )] :
( ! [C: TA,D: TB,E: TC,F: TD] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ TD ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ TD ) @ D @ ( product_Pair @ TC @ TD @ E @ F ) ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(201,axiom,
! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: set @ ( product_prod @ TA @ TA )] :
( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TA ) ) @ A @ B )
=> ( ord_less_eq @ ( set @ ( product_prod @ TA @ TA ) ) @ ( transitive_rtrancl @ TA @ A ) @ ( transitive_rtrancl @ TA @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_rtrancl__mono) ).
thf(856,plain,
! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: set @ ( product_prod @ TA @ TA )] :
( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TA ) ) @ A @ B )
=> ( ord_less_eq @ ( set @ ( product_prod @ TA @ TA ) ) @ ( transitive_rtrancl @ TA @ A ) @ ( transitive_rtrancl @ TA @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).
thf(158,axiom,
! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB )] :
( ( member @ TB @ A @ ( range @ TA @ TB @ B ) )
=> ~ ! [C: TA] :
~ ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ A ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_RangeE) ).
thf(733,plain,
! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB )] :
( ( member @ TB @ A @ ( range @ TA @ TB @ B ) )
=> ~ ! [C: TA] :
~ ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(117,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ~ ( finite_finite2 @ TA @ A )
=> ~ ( finite_finite2 @ TA @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_infinite__super) ).
thf(601,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ~ ( finite_finite2 @ TA @ A )
=> ~ ( finite_finite2 @ TA @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(140,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming2 @ TA @ A @ B ) @ ( e @ TA @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_Graph_Oincoming_H__edges) ).
thf(689,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: set @ nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming2 @ TA @ A @ B ) @ ( e @ TA @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(8,axiom,
! [TA: $tType,A: set @ ( product_prod @ TA @ TA )] :
( ( transitive_rtrancl @ TA @ ( transitive_rtrancl @ TA @ A ) )
= ( transitive_rtrancl @ TA @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_159_rtrancl__idemp) ).
thf(296,plain,
! [TA: $tType,A: set @ ( product_prod @ TA @ TA )] :
( ( transitive_rtrancl @ TA @ ( transitive_rtrancl @ TA @ A ) )
= ( transitive_rtrancl @ TA @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[8]) ).
thf(111,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: nat,E: nat] :
( ( isPath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ( ( E != C )
=> ? [F: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_isPath__ex__edge2) ).
thf(584,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: nat,E: nat] :
( ( isPath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ( ( E != C )
=> ? [F: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(193,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TA,C: TA > ( set @ TB ),D: TA > ( set @ TB )] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( ord_less_eq @ ( set @ TB ) @ ( C @ E ) @ ( D @ E ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ ( product_Sigma @ TA @ TB @ A @ C ) @ ( product_Sigma @ TA @ TB @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_Sigma__mono) ).
thf(819,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TA,C: TA > ( set @ TB ),D: TA > ( set @ TB )] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ! [E: TA] :
( ( member @ TA @ E @ A )
=> ( ord_less_eq @ ( set @ TB ) @ ( C @ E ) @ ( D @ E ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ ( product_Sigma @ TA @ TB @ A @ C ) @ ( product_Sigma @ TA @ TB @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).
thf(4,axiom,
! [TA: $tType,A: set @ TA] :
( ( collect @ TA
@ ^ [B: TA] : ( member @ TA @ B @ A ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_46_Collect__mem__eq) ).
thf(286,plain,
! [TA: $tType,A: set @ TA] :
( ( collect @ TA
@ ^ [B: TA] : ( member @ TA @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[4]) ).
thf(163,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TB,C: TA > TB > $o] :
( ~ ( finite_finite2 @ TA @ A )
=> ( ( finite_finite2 @ TB @ B )
=> ( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ? [E: TB] :
( ( member @ TB @ E @ B )
& ( C @ D @ E ) ) )
=> ? [D: TB] :
( ( member @ TB @ D @ B )
& ~ ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [E: TA] :
( ( member @ TA @ E @ A )
& ( C @ E @ D ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_pigeonhole__infinite__rel) ).
thf(743,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TB,C: TA > TB > $o] :
( ~ ( finite_finite2 @ TA @ A )
=> ( ( finite_finite2 @ TB @ B )
=> ( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ? [E: TB] :
( ( member @ TB @ E @ B )
& ( C @ D @ E ) ) )
=> ? [D: TB] :
( ( member @ TB @ D @ B )
& ~ ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [E: TA] :
( ( member @ TA @ E @ A )
& ( C @ E @ D ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(108,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
( ( edmond2129202899alysis @ TA @ A )
=> ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( edmond1803015688pEdges @ TA @ A @ B @ C ) @ ( e @ TA @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_ek__analysis_OspEdges__ss__E) ).
thf(577,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
( ( edmond2129202899alysis @ TA @ A )
=> ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( edmond1803015688pEdges @ TA @ A @ B @ C ) @ ( e @ TA @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(109,axiom,
! [TA: $tType,A: set @ TA] :
( ( finite_finite2 @ TA @ A )
=> ? [B: list @ TA] :
( ( set2 @ TA @ B )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_finite__list) ).
thf(579,plain,
! [TA: $tType,A: set @ TA] :
( ( finite_finite2 @ TA @ A )
=> ? [B: list @ TA] :
( ( set2 @ TA @ B )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(49,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/sandbox/benchmark/theBenchmark.p',fact_111_Graph_Oobtain__shortest__path) ).
thf(413,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)],[49]) ).
thf(121,axiom,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ TA] :
( ( finite_finite2 @ ( product_prod @ TA @ TB ) @ A )
=> ( finite_finite2 @ TB @ ( image @ TA @ TB @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_Relation_Ofinite__Image) ).
thf(611,plain,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ TA] :
( ( finite_finite2 @ ( product_prod @ TA @ TB ) @ A )
=> ( finite_finite2 @ TB @ ( image @ TA @ TB @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(274,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: nat,E: nat,F: nat] :
( ( isSimplePath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ( ( D != F )
=> ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ F @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_isSPath__sg__incoming) ).
thf(1058,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: nat,E: nat,F: nat] :
( ( isSimplePath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ( ( D != F )
=> ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ F @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).
thf(186,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ( finite_finite2 @ nat @ ( v @ TA @ A ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ TA @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_Graph_OVfin__imp__Efin) ).
thf(804,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ( finite_finite2 @ nat @ ( v @ TA @ A ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ TA @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(169,axiom,
linorder @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Olinorder) ).
thf(764,plain,
linorder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(215,axiom,
! [A: nat] : ( ord_less_eq @ ( set @ nat ) @ ( adjacent_nodes @ a @ c @ A ) @ ( v @ a @ c ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_adjacent__nodes__ss__V) ).
thf(900,plain,
! [A: nat] : ( ord_less_eq @ ( set @ nat ) @ ( adjacent_nodes @ a @ c @ A ) @ ( v @ a @ c ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(197,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA ),D: TA > TA > $o] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ! [E: TA] : ( D @ E @ E )
=> ( ! [E: TA,F: TA,G: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ F ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( D @ E @ F )
=> ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ F @ G ) @ C )
=> ( D @ E @ G ) ) ) )
=> ( D @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_rtrancl_Oinducts) ).
thf(831,plain,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA ),D: TA > TA > $o] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ! [E: TA] : ( D @ E @ E )
=> ( ! [E: TA,F: TA,G: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ F ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( D @ E @ F )
=> ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ F @ G ) @ C )
=> ( D @ E @ G ) ) ) )
=> ( D @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).
thf(249,axiom,
! [A: nat,B: nat] :
( ( connected @ a @ c @ A @ B )
=> ( ( member @ nat @ B @ ( v @ a @ c ) )
= ( member @ nat @ A @ ( v @ a @ c ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_connected__inV__iff) ).
thf(996,plain,
! [A: nat,B: nat] :
( ( connected @ a @ c @ A @ B )
=> ( ( member @ nat @ B @ ( v @ a @ c ) )
= ( member @ nat @ A @ ( v @ a @ c ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(160,axiom,
! [A: nat] :
( ( incoming @ a @ c @ A )
= ( collect @ ( product_prod @ nat @ nat )
@ ^ [B: product_prod @ nat @ nat] :
? [C: nat] :
( ( B
= ( product_Pair @ nat @ nat @ C @ A ) )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ C @ A ) @ ( e @ a @ c ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_incoming__def) ).
thf(737,plain,
! [A: nat] :
( ( incoming @ a @ c @ A )
= ( collect @ ( product_prod @ nat @ nat )
@ ^ [B: product_prod @ nat @ nat] :
? [C: nat] :
( ( B
= ( product_Pair @ nat @ nat @ C @ A ) )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ C @ A ) @ ( e @ a @ c ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(10,axiom,
! [TA: $tType,A: TA > $o,B: TA,C: TA > $o] :
( ( A @ B )
=> ( ( ord_less_eq @ ( TA > $o ) @ A @ C )
=> ( C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_236_rev__predicate1D) ).
thf(303,plain,
! [TA: $tType,A: TA > $o,B: TA,C: TA > $o] :
( ( A @ B )
=> ( ( ord_less_eq @ ( TA > $o ) @ A @ C )
=> ( C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(204,axiom,
! [TA: $tType,A: TA > $o] :
( ~ ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
=> ? [B: TA] : ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_not__finite__existsD) ).
thf(863,plain,
! [TA: $tType,A: TA > $o] :
( ~ ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
=> ? [B: TA] : ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(66,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/sandbox/benchmark/theBenchmark.p',fact_254_Graph_OshortestPath__is__path) ).
thf(451,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)],[66]) ).
thf(106,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ TC ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ TC )] :
( ! [C: TA,D: TB,E: TC] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ TC ) @ C @ ( product_Pair @ TB @ TC @ D @ E ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_171_prod__induct3) ).
thf(567,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ TC ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ TC )] :
( ! [C: TA,D: TB,E: TC] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ TC ) @ C @ ( product_Pair @ TB @ TC @ D @ E ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(85,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/sandbox/benchmark/theBenchmark.p',fact_222_antisym__conv) ).
thf(509,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)],[85]) ).
thf(209,axiom,
! [A: nat,B: nat] :
( ( connected @ a @ c @ A @ B )
= ( ? [C: list @ ( product_prod @ nat @ nat )] : ( isPath @ a @ c @ A @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_244_connected__def) ).
thf(880,plain,
! [A: nat,B: nat] :
( ( connected @ a @ c @ A @ B )
= ( ? [C: list @ ( product_prod @ nat @ nat )] : ( isPath @ a @ c @ A @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(267,axiom,
linordered_idom @ a,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tfree_0) ).
thf(1043,plain,
linordered_idom @ a,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).
thf(57,axiom,
finite_finite @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Finite__Set_Ofinite_10) ).
thf(434,plain,
finite_finite @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(190,axiom,
! [A: nat,B: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A @ B ) @ ( transitive_rtrancl @ nat @ ( e @ a @ c ) ) )
=> ? [C: list @ ( product_prod @ nat @ nat )] : ( isPath @ a @ c @ A @ C @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_242_rtc__isPath) ).
thf(811,plain,
! [A: nat,B: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A @ B ) @ ( transitive_rtrancl @ nat @ ( e @ a @ c ) ) )
=> ? [C: list @ ( product_prod @ nat @ nat )] : ( isPath @ a @ c @ A @ C @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(168,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA )] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( B != A )
=> ~ ! [D: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ D ) @ ( transitive_rtrancl @ TA @ C ) )
=> ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ B ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_rtrancl_Ocases) ).
thf(758,plain,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA )] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( B != A )
=> ~ ! [D: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ D ) @ ( transitive_rtrancl @ TA @ C ) )
=> ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ B ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(202,axiom,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB )] :
( ( ord_less_eq @ ( TA > TB > $o )
@ ^ [C: TA,D: TB] : ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ A )
@ ^ [C: TA,D: TB] : ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ B ) )
= ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_pred__subset__eq2) ).
thf(858,plain,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB )] :
( ( ord_less_eq @ ( TA > TB > $o )
@ ^ [C: TA,D: TB] : ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ A )
@ ^ [C: TA,D: TB] : ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ B ) )
= ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).
thf(96,axiom,
! [TA: $tType,TB: $tType,A: TA,B: set @ ( product_prod @ TA @ TB ),C: TA > $o] :
( ( member @ TA @ A @ ( domain @ TA @ TB @ B ) )
=> ( ! [D: TA,E: TB] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ D @ E ) @ B )
=> ( C @ D ) )
=> ( C @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_Domain_Oinducts) ).
thf(535,plain,
! [TA: $tType,TB: $tType,A: TA,B: set @ ( product_prod @ TA @ TB ),C: TA > $o] :
( ( member @ TA @ A @ ( domain @ TA @ TB @ B ) )
=> ( ! [D: TA,E: TB] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ D @ E ) @ B )
=> ( C @ D ) )
=> ( C @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(120,axiom,
! [TA: $tType,TB: $tType] :
( ( ( finite_finite @ TB )
& ( finite_finite @ TA ) )
=> ( finite_finite @ ( product_prod @ TB @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Product__Type_Oprod___Finite__Set_Ofinite_13) ).
thf(609,plain,
! [TA: $tType,TB: $tType] :
( ( ( finite_finite @ TB )
& ( finite_finite @ TA ) )
=> ( finite_finite @ ( product_prod @ TB @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(18,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/sandbox/benchmark/theBenchmark.p',fact_237_le__funD) ).
thf(323,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)],[18]) ).
thf(155,axiom,
! [TA: $tType,A: TA,B: set @ ( product_prod @ TA @ TA ),C: TA] :
( ~ ( member @ TA @ A @ ( domain @ TA @ TA @ B ) )
=> ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ C ) @ ( transitive_rtrancl @ TA @ B ) )
= ( A = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_Not__Domain__rtrancl) ).
thf(726,plain,
! [TA: $tType,A: TA,B: set @ ( product_prod @ TA @ TA ),C: TA] :
( ~ ( member @ TA @ A @ ( domain @ TA @ TA @ B ) )
=> ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ C ) @ ( transitive_rtrancl @ TA @ B ) )
= ( A = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(126,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,A: product_prod @ TG @ ( product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) )] :
~ ! [B: TG,C: TF,D: TE,E: TD,F: TC,G: TB,H: TA] :
( A
!= ( product_Pair @ TG @ ( product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) ) @ B @ ( product_Pair @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) @ C @ ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ D @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ E @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ F @ ( product_Pair @ TB @ TA @ G @ H ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_prod__cases7) ).
thf(621,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,A: product_prod @ TG @ ( product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) )] :
~ ! [B: TG,C: TF,D: TE,E: TD,F: TC,G: TB,H: TA] :
( A
!= ( product_Pair @ TG @ ( product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) ) @ B @ ( product_Pair @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) @ C @ ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ D @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ E @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ F @ ( product_Pair @ TB @ TA @ G @ H ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(239,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: TC > $o,B: TB > $o,C: TC > TB > TA] :
( ( finite_finite2 @ TC @ ( collect @ TC @ A ) )
=> ( ( finite_finite2 @ TB @ ( collect @ TB @ B ) )
=> ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [D: TA] :
? [E: TC,F: TB] :
( ( D
= ( C @ E @ F ) )
& ( A @ E )
& ( B @ F ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26_finite__image__set2) ).
thf(964,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: TC > $o,B: TB > $o,C: TC > TB > TA] :
( ( finite_finite2 @ TC @ ( collect @ TC @ A ) )
=> ( ( finite_finite2 @ TB @ ( collect @ TB @ B ) )
=> ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [D: TA] :
? [E: TC,F: TB] :
( ( D
= ( C @ E @ F ) )
& ( A @ E )
& ( B @ F ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).
thf(172,axiom,
! [A: nat] : ( ord_less_eq @ ( set @ nat ) @ ( image @ nat @ nat @ ( e @ a @ c ) @ ( reachableNodes @ a @ c @ A ) ) @ ( reachableNodes @ a @ c @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_reachableNodes__E__closed) ).
thf(769,plain,
! [A: nat] : ( ord_less_eq @ ( set @ nat ) @ ( image @ nat @ nat @ ( e @ a @ c ) @ ( reachableNodes @ a @ c @ A ) ) @ ( reachableNodes @ a @ c @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(231,axiom,
! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ ( product_prod @ TA @ TB )] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ C )
=> ( member @ TA @ A @ ( domain @ TA @ TB @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_Domain_ODomainI) ).
thf(943,plain,
! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ ( product_prod @ TA @ TB )] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ C )
=> ( member @ TA @ A @ ( domain @ TA @ TB @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).
thf(138,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) )] :
~ ! [B: TF,C: TE,D: TD,E: TC,F: TB,G: TA] :
( A
!= ( product_Pair @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) @ B @ ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ C @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ E @ ( product_Pair @ TB @ TA @ F @ G ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_prod__cases6) ).
thf(668,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: product_prod @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) )] :
~ ! [B: TF,C: TE,D: TD,E: TC,F: TB,G: TA] :
( A
!= ( product_Pair @ TF @ ( product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) ) @ B @ ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ C @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ E @ ( product_Pair @ TB @ TA @ F @ G ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(21,axiom,
! [TA: $tType,TB: $tType,A: ( product_prod @ TA @ TB ) > $o,B: product_prod @ TA @ TB] :
( ! [C: TA,D: TB] : ( A @ ( product_Pair @ TA @ TB @ C @ D ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_prod__cases) ).
thf(330,plain,
! [TA: $tType,TB: $tType,A: ( product_prod @ TA @ TB ) > $o,B: product_prod @ TA @ TB] :
( ! [C: TA,D: TB] : ( A @ ( product_Pair @ TA @ TB @ C @ D ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(71,axiom,
! [TA: $tType] : ( order @ ( set @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oorder_6) ).
thf(462,plain,
! [TA: $tType] : ( order @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[71]) ).
thf(81,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/sandbox/benchmark/theBenchmark.p',fact_253_Graph_Oconnected__def) ).
thf(494,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)],[81]) ).
thf(281,axiom,
! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ TA,D: TA > ( set @ TB )] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ ( product_Sigma @ TA @ TB @ C @ D ) )
=> ( member @ TB @ B @ ( D @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_SigmaD2) ).
thf(1080,plain,
! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ TA,D: TA > ( set @ TB )] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ ( product_Sigma @ TA @ TB @ C @ D ) )
=> ( member @ TB @ B @ ( D @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).
thf(90,axiom,
! [TA: $tType,TB: $tType,A: TA > TB > $o,B: TA > TB > $o] :
( ! [C: TA,D: TB] :
( ( A @ C @ D )
=> ( B @ C @ D ) )
=> ( ord_less_eq @ ( TA > TB > $o ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_predicate2I) ).
thf(521,plain,
! [TA: $tType,TB: $tType,A: TA > TB > $o,B: TA > TB > $o] :
( ! [C: TA,D: TB] :
( ( A @ C @ D )
=> ( B @ C @ D ) )
=> ( ord_less_eq @ ( TA > TB > $o ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[90]) ).
thf(44,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/sandbox/benchmark/theBenchmark.p',fact_229_linorder__wlog) ).
thf(399,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)],[44]) ).
thf(206,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: nat,F: nat,G: nat] :
( ( isSimplePath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ( ( E != G )
=> ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ G @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_143_Graph_OisSPath__sg__incoming) ).
thf(868,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: nat,F: nat,G: nat] :
( ( isSimplePath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ( ( E != G )
=> ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ G @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(210,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA ),D: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ D ) @ C )
=> ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ D ) @ ( transitive_rtrancl @ TA @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_rtrancl_Ortrancl__into__rtrancl) ).
thf(883,plain,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA ),D: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ D ) @ C )
=> ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ D ) @ ( transitive_rtrancl @ TA @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).
thf(256,axiom,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB )] :
( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ A @ B )
=> ( ord_less_eq @ ( set @ TA ) @ ( domain @ TA @ TB @ A ) @ ( domain @ TA @ TB @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_Domain__mono) ).
thf(1015,plain,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB )] :
( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ A @ B )
=> ( ord_less_eq @ ( set @ TA ) @ ( domain @ TA @ TB @ A ) @ ( domain @ TA @ TB @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).
thf(9,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/sandbox/benchmark/theBenchmark.p',fact_212_order__subst2) ).
thf(299,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)],[9]) ).
thf(199,axiom,
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
=> ( finite_Graph @ a @ c ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_Finite__Graph__EI) ).
thf(851,plain,
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
=> ( finite_Graph @ a @ c ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(59,axiom,
! [TA: $tType,TB: $tType,A: TB > TA > $o,B: TB,C: TA,D: TB > TA > $o] :
( ( A @ B @ C )
=> ( ( ord_less_eq @ ( TB > TA > $o ) @ A @ D )
=> ( D @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_234_rev__predicate2D) ).
thf(438,plain,
! [TA: $tType,TB: $tType,A: TB > TA > $o,B: TB,C: TA,D: TB > TA > $o] :
( ( A @ B @ C )
=> ( ( ord_less_eq @ ( TB > TA > $o ) @ A @ D )
=> ( D @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(132,axiom,
! [TA: $tType,TB: $tType,A: TA,B: set @ ( product_prod @ TA @ TB )] :
( ( member @ TA @ A @ ( domain @ TA @ TB @ B ) )
=> ~ ! [C: TB] :
~ ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ C ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_Domain_Ocases) ).
thf(653,plain,
! [TA: $tType,TB: $tType,A: TA,B: set @ ( product_prod @ TA @ TB )] :
( ( member @ TA @ A @ ( domain @ TA @ TB @ B ) )
=> ~ ! [C: TB] :
~ ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ C ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(224,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: nat] :
( ( isSimplePath @ TA @ A @ B @ C @ D )
=> ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_144_Graph_OisSPath__no__selfloop) ).
thf(922,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: nat] :
( ( isSimplePath @ TA @ A @ B @ C @ D )
=> ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).
thf(268,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA ),D: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ D ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ D ) @ ( transitive_rtrancl @ TA @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_rtrancl__trans) ).
thf(1044,plain,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA ),D: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ D ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ D ) @ ( transitive_rtrancl @ TA @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).
thf(143,axiom,
! [TA: $tType,A: TA,B: set @ ( product_prod @ TA @ TA )] : ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ A ) @ ( transitive_rtrancl @ TA @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194_rtrancl_Ortrancl__refl) ).
thf(695,plain,
! [TA: $tType,A: TA,B: set @ ( product_prod @ TA @ TA )] : ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ A ) @ ( transitive_rtrancl @ TA @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(42,axiom,
! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA] :
? [B: TB,C: TA] :
( A
= ( product_Pair @ TB @ TA @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_surj__pair) ).
thf(394,plain,
! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA] :
? [B: TB,C: TA] :
( A
= ( product_Pair @ TB @ TA @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(246,axiom,
! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB ),C: set @ TA] :
( ( member @ TB @ A @ ( image @ TA @ TB @ B @ C ) )
= ( ? [D: TA] :
( ( member @ TA @ D @ C )
& ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ D @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_Image__iff) ).
thf(986,plain,
! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB ),C: set @ TA] :
( ( member @ TB @ A @ ( image @ TA @ TB @ B @ C ) )
= ( ? [D: TA] :
( ( member @ TA @ D @ C )
& ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ D @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).
thf(280,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
| ( finite_finite2 @ TA @ ( collect @ TA @ B ) ) )
=> ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [C: TA] :
( ( A @ C )
& ( B @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_finite__Collect__conjI) ).
thf(1076,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
| ( finite_finite2 @ TA @ ( collect @ TA @ B ) ) )
=> ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [C: TA] :
( ( A @ C )
& ( B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).
thf(226,axiom,
finite_finite2 @ nat @ ( v @ a @ c ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_finite__V) ).
thf(927,plain,
finite_finite2 @ nat @ ( v @ a @ c ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).
thf(164,axiom,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ! [A: set @ TA] : ( finite_finite2 @ TA @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_finite) ).
thf(749,plain,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ! [A: set @ TA] : ( finite_finite2 @ TA @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(230,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE,H: TF] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) @ E @ ( product_Pair @ TD @ ( product_prod @ TE @ TF ) @ F @ ( product_Pair @ TE @ TF @ G @ H ) ) ) ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_168_prod__induct6) ).
thf(939,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE,H: TF] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ TF ) ) @ E @ ( product_Pair @ TD @ ( product_prod @ TE @ TF ) @ F @ ( product_Pair @ TE @ TF @ G @ H ) ) ) ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).
thf(102,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: nat,E: nat,F: nat] :
( ( isSimplePath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ( ( E != F )
=> ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_isSPath__sg__outgoing) ).
thf(553,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: nat,E: nat,F: nat] :
( ( isSimplePath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ( ( E != F )
=> ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(270,axiom,
( ( edmond1803015688pEdges @ a @ c @ s @ t )
= ( collect @ ( product_prod @ nat @ nat )
@ ^ [A: product_prod @ nat @ nat] :
? [B: list @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ A @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
& ( isShortestPath @ a @ c @ s @ B @ t ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_spEdges__def) ).
thf(1049,plain,
( ( edmond1803015688pEdges @ a @ c @ s @ t )
= ( collect @ ( product_prod @ nat @ nat )
@ ^ [A: product_prod @ nat @ nat] :
? [B: list @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ A @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
& ( isShortestPath @ a @ c @ s @ B @ t ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).
thf(136,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: nat] :
( ( isSimplePath @ a @ c @ A @ B @ C )
=> ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ D ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_isSPath__no__selfloop) ).
thf(663,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: nat] :
( ( isSimplePath @ a @ c @ A @ B @ C )
=> ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ D ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(100,axiom,
! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: set @ ( product_prod @ TA @ TA )] :
( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TA ) ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TA ) ) @ B @ ( transitive_rtrancl @ TA @ A ) )
=> ( ( transitive_rtrancl @ TA @ B )
= ( transitive_rtrancl @ TA @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_rtrancl__subset) ).
thf(548,plain,
! [TA: $tType,A: set @ ( product_prod @ TA @ TA ),B: set @ ( product_prod @ TA @ TA )] :
( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TA ) ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TA ) ) @ B @ ( transitive_rtrancl @ TA @ A ) )
=> ( ( transitive_rtrancl @ TA @ B )
= ( transitive_rtrancl @ TA @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(124,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( edmond1803015688pEdges @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
( collect @ ( product_prod @ nat @ nat )
@ ^ [D: product_prod @ nat @ nat] :
? [E: list @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ D @ ( set2 @ ( product_prod @ nat @ nat ) @ E ) )
& ( isShortestPath @ TA @ A @ B @ E @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_ek__analysis__defs_OspEdges__def) ).
thf(617,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( edmond1803015688pEdges @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat] :
( collect @ ( product_prod @ nat @ nat )
@ ^ [D: product_prod @ nat @ nat] :
? [E: list @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ D @ ( set2 @ ( product_prod @ nat @ nat ) @ E ) )
& ( isShortestPath @ TA @ A @ B @ E @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(88,axiom,
! [TA: $tType,TB: $tType,TC: $tType,A: TC > TA > TB,B: TC,C: TA] :
( ( product_rec_prod @ TC @ TA @ TB @ A @ ( product_Pair @ TC @ TA @ B @ C ) )
= ( A @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_old_Oprod_Orec) ).
thf(516,plain,
! [TA: $tType,TB: $tType,TC: $tType,A: TC > TA > TB,B: TC,C: TA] :
( ( product_rec_prod @ TC @ TA @ TB @ A @ ( product_Pair @ TC @ TA @ B @ C ) )
= ( A @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(241,axiom,
! [A: nat,B: nat] :
( ( connected @ a @ c @ A @ B )
=> ~ ! [C: list @ ( product_prod @ nat @ nat )] :
~ ( isShortestPath @ a @ c @ A @ C @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_obtain__shortest__path) ).
thf(968,plain,
! [A: nat,B: nat] :
( ( connected @ a @ c @ A @ B )
=> ~ ! [C: list @ ( product_prod @ nat @ nat )] :
~ ( isShortestPath @ a @ c @ A @ C @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).
thf(194,axiom,
ord @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oord_3) ).
thf(823,plain,
ord @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(175,axiom,
! [A: set @ nat] :
( ( finite_finite2 @ nat @ ( v @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing2 @ a @ c @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_finite__outgoing_H) ).
thf(776,plain,
! [A: set @ nat] :
( ( finite_finite2 @ nat @ ( v @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing2 @ a @ c @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(91,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( outgoing @ TA )
= ( outgoing @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_Graph_Ooutgoing_Ocong) ).
thf(525,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(7,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( isSimplePath @ TA )
= ( isSimplePath @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_Graph_OisSimplePath_Ocong) ).
thf(295,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(277,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( finite_finite2 @ TA @ A )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
=> ( finite_finite2 @ TA @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_rev__finite__subset) ).
thf(1070,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( finite_finite2 @ TA @ A )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
=> ( finite_finite2 @ TA @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).
thf(148,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ TA @ A ) )
=> ( finite_finite2 @ nat @ ( v @ TA @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_Graph_OEfin__imp__Vfin) ).
thf(707,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ TA @ A ) )
=> ( finite_finite2 @ nat @ ( v @ TA @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(242,axiom,
! [A: nat] :
( ( outgoing @ a @ c @ A )
= ( collect @ ( product_prod @ nat @ nat )
@ ^ [B: product_prod @ nat @ nat] :
? [C: nat] :
( ( B
= ( product_Pair @ nat @ nat @ A @ C ) )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A @ C ) @ ( e @ a @ c ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_outgoing__def) ).
thf(971,plain,
! [A: nat] :
( ( outgoing @ a @ c @ A )
= ( collect @ ( product_prod @ nat @ nat )
@ ^ [B: product_prod @ nat @ nat] :
? [C: nat] :
( ( B
= ( product_Pair @ nat @ nat @ A @ C ) )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A @ C ) @ ( e @ a @ c ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).
thf(20,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ! [C: TA] :
( ( A @ C )
= ( B @ C ) )
=> ( ( collect @ TA @ A )
= ( collect @ TA @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_47_Collect__cong) ).
thf(327,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)],[20]) ).
thf(235,axiom,
preorder @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Opreorder_1) ).
thf(954,plain,
preorder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).
thf(13,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( adjacent_nodes @ TA )
= ( adjacent_nodes @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_Graph_Oadjacent__nodes_Ocong) ).
thf(310,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(150,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( v @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA] :
( collect @ nat
@ ^ [B: nat] :
? [C: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ B @ C ) @ ( e @ TA @ A ) )
| ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ C @ B ) @ ( e @ TA @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_Graph_OV__def) ).
thf(712,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( v @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA] :
( collect @ nat
@ ^ [B: nat] :
? [C: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ B @ C ) @ ( e @ TA @ A ) )
| ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ C @ B ) @ ( e @ TA @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(103,axiom,
! [TA: $tType,A: product_prod @ TA @ TA,B: set @ ( product_prod @ TA @ TA )] :
( ( member @ ( product_prod @ TA @ TA ) @ A @ B )
=> ( member @ ( product_prod @ TA @ TA ) @ A @ ( transitive_rtrancl @ TA @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_158_r__into__rtrancl) ).
thf(557,plain,
! [TA: $tType,A: product_prod @ TA @ TA,B: set @ ( product_prod @ TA @ TA )] :
( ( member @ ( product_prod @ TA @ TA ) @ A @ B )
=> ( member @ ( product_prod @ TA @ TA ) @ A @ ( transitive_rtrancl @ TA @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(147,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: set @ nat] :
( ( finite_finite2 @ nat @ ( v @ TA @ A ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming2 @ TA @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_Graph_Ofinite__incoming_H) ).
thf(705,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: set @ nat] :
( ( finite_finite2 @ nat @ ( v @ TA @ A ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming2 @ TA @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(1,conjecture,
finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).
thf(2,negated_conjecture,
~ ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(282,plain,
~ ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(152,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) )] :
~ ! [B: TE,C: TD,D: TC,E: TB,F: TA] :
( A
!= ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ B @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ C @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ D @ ( product_Pair @ TB @ TA @ E @ F ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_prod__cases5) ).
thf(717,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: product_prod @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) )] :
~ ! [B: TE,C: TD,D: TC,E: TB,F: TA] :
( A
!= ( product_Pair @ TE @ ( product_prod @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) ) @ B @ ( product_Pair @ TD @ ( product_prod @ TC @ ( product_prod @ TB @ TA ) ) @ C @ ( product_Pair @ TC @ ( product_prod @ TB @ TA ) @ D @ ( product_Pair @ TB @ TA @ E @ F ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(24,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( TA > $o )
@ ^ [C: TA] : ( member @ TA @ C @ A )
@ ^ [C: TA] : ( member @ TA @ C @ B ) )
= ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_pred__subset__eq) ).
thf(338,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( TA > $o )
@ ^ [C: TA] : ( member @ TA @ C @ A )
@ ^ [C: TA] : ( member @ TA @ C @ B ) )
= ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(188,axiom,
edmond2129202899alysis @ a @ c,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0_ek__analysis__axioms) ).
thf(808,plain,
edmond2129202899alysis @ a @ c,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(265,axiom,
! [TA: $tType,TB: $tType,A: TA,B: set @ TA,C: TB,D: set @ ( product_prod @ TA @ TB )] :
( ( member @ TA @ A @ B )
=> ( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ C ) @ D )
=> ( member @ TB @ C @ ( image @ TA @ TB @ D @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_rev__ImageI) ).
thf(1038,plain,
! [TA: $tType,TB: $tType,A: TA,B: set @ TA,C: TB,D: set @ ( product_prod @ TA @ TB )] :
( ( member @ TA @ A @ B )
=> ( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ C ) @ D )
=> ( member @ TB @ C @ ( image @ TA @ TB @ D @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).
thf(34,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/sandbox/benchmark/theBenchmark.p',fact_231_dual__order_Oeq__iff) ).
thf(370,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)],[34]) ).
thf(23,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/sandbox/benchmark/theBenchmark.p',fact_227_order__trans) ).
thf(336,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)],[23]) ).
thf(146,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ( finite_Graph @ TA @ A )
=> ( finite_finite2 @ nat @ ( v @ TA @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_112_Finite__Graph_Ofinite__V) ).
thf(703,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ( finite_Graph @ TA @ A )
=> ( finite_finite2 @ nat @ ( v @ TA @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(122,axiom,
! [A: nat] : ( connected @ a @ c @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_connected__refl) ).
thf(613,plain,
! [A: nat] : ( connected @ a @ c @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(272,axiom,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [C: TA] :
( ( A @ C )
| ( B @ C ) ) ) )
= ( ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
& ( finite_finite2 @ TA @ ( collect @ TA @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_finite__Collect__disjI) ).
thf(1052,plain,
! [TA: $tType,A: TA > $o,B: TA > $o] :
( ( finite_finite2 @ TA
@ ( collect @ TA
@ ^ [C: TA] :
( ( A @ C )
| ( B @ C ) ) ) )
= ( ( finite_finite2 @ TA @ ( collect @ TA @ A ) )
& ( finite_finite2 @ TA @ ( collect @ TA @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).
thf(244,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] :
( ( finite_finite2 @ nat @ ( v @ TA @ A ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming @ TA @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_Graph_Ofinite__incoming) ).
thf(978,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] :
( ( finite_finite2 @ nat @ ( v @ TA @ A ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming @ TA @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).
thf(254,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ TE ) @ E @ ( product_Pair @ TD @ TE @ F @ G ) ) ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_prod__induct5) ).
thf(1008,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ TE ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ TE ) @ E @ ( product_Pair @ TD @ TE @ F @ G ) ) ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).
thf(15,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/sandbox/benchmark/theBenchmark.p',fact_223_order__class_Oorder_Oeq__iff) ).
thf(314,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)],[15]) ).
thf(107,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA )] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( A != B )
=> ~ ! [D: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ D ) @ C )
=> ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ B ) @ ( transitive_rtrancl @ TA @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_converse__rtranclE) ).
thf(571,plain,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA )] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( A != B )
=> ~ ! [D: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ D ) @ C )
=> ~ ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ D @ B ) @ ( transitive_rtrancl @ TA @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(116,axiom,
! [TA: $tType,TB: $tType,A: TA,B: set @ ( product_prod @ TB @ TA ),C: TA > $o] :
( ( member @ TA @ A @ ( range @ TB @ TA @ B ) )
=> ( ! [D: TB,E: TA] :
( ( member @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ D @ E ) @ B )
=> ( C @ E ) )
=> ( C @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_84_Range_Oinducts) ).
thf(595,plain,
! [TA: $tType,TB: $tType,A: TA,B: set @ ( product_prod @ TB @ TA ),C: TA > $o] :
( ( member @ TA @ A @ ( range @ TB @ TA @ B ) )
=> ( ! [D: TB,E: TA] :
( ( member @ ( product_prod @ TB @ TA ) @ ( product_Pair @ TB @ TA @ D @ E ) @ B )
=> ( C @ E ) )
=> ( C @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(82,axiom,
! [TA: $tType,TB: $tType,A: TB,B: set @ TB,C: set @ TA,D: set @ TA] :
( ( member @ TB @ A @ B )
=> ( ( ( product_Sigma @ TA @ TB @ C
@ ^ [E: TA] : B )
= ( product_Sigma @ TA @ TB @ D
@ ^ [E: TA] : B ) )
= ( C = D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_182_Times__eq__cancel2) ).
thf(497,plain,
! [TA: $tType,TB: $tType,A: TB,B: set @ TB,C: set @ TA,D: set @ TA] :
( ( member @ TB @ A @ B )
=> ( ( ( product_Sigma @ TA @ TB @ C
@ ^ [E: TA] : B )
= ( product_Sigma @ TA @ TB @ D
@ ^ [E: TA] : B ) )
= ( C = D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(73,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ord @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Orderings_Oord) ).
thf(465,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ord @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(139,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA ),D: TA > $o] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( D @ A )
=> ( ! [E: TA,F: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ E ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ F ) @ C )
=> ( ( D @ E )
=> ( D @ F ) ) ) )
=> ( D @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_rtrancl__induct) ).
thf(671,plain,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA ),D: TA > $o] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( D @ A )
=> ( ! [E: TA,F: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ E ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ F ) @ C )
=> ( ( D @ E )
=> ( D @ F ) ) ) )
=> ( D @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(33,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( connected @ TA @ A @ B @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_103_Graph_Oconnected__refl) ).
thf(368,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)],[33]) ).
thf(173,axiom,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB ),C: set @ TA,D: set @ TA] :
( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ D )
=> ( ord_less_eq @ ( set @ TB ) @ ( image @ TA @ TB @ A @ C ) @ ( image @ TA @ TB @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_Image__mono) ).
thf(771,plain,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB ),C: set @ TA,D: set @ TA] :
( ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ C @ D )
=> ( ord_less_eq @ ( set @ TB ) @ ( image @ TA @ TB @ A @ C ) @ ( image @ TA @ TB @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(191,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TB] :
( ~ ( finite_finite2 @ TA @ A )
=> ( ~ ( finite_finite2 @ TB @ B )
=> ~ ( finite_finite2 @ ( product_prod @ TA @ TB )
@ ( product_Sigma @ TA @ TB @ A
@ ^ [C: TA] : B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_infinite__cartesian__product) ).
thf(813,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: set @ TB] :
( ~ ( finite_finite2 @ TA @ A )
=> ( ~ ( finite_finite2 @ TB @ B )
=> ~ ( finite_finite2 @ ( product_prod @ TA @ TB )
@ ( product_Sigma @ TA @ TB @ A
@ ^ [C: TA] : B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(275,axiom,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB > $o] :
( ( finite_finite2 @ TA @ A )
=> ( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ? [D: TB] : ( B @ C @ D ) )
=> ? [C: TA > TB] :
! [D: TA] :
( ( member @ TA @ D @ A )
=> ( B @ D @ ( C @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8_finite__set__choice) ).
thf(1062,plain,
! [TA: $tType,TB: $tType,A: set @ TA,B: TA > TB > $o] :
( ( finite_finite2 @ TA @ A )
=> ( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ? [D: TB] : ( B @ C @ D ) )
=> ? [C: TA > TB] :
! [D: TA] :
( ( member @ TA @ D @ A )
=> ( B @ D @ ( C @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).
thf(177,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming @ TA @ A @ B ) @ ( e @ TA @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_Graph_Oincoming__edges) ).
thf(781,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( incoming @ TA @ A @ B ) @ ( e @ TA @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(112,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
( ( connected @ TA @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ C @ D ) @ ( e @ TA @ A ) )
=> ( connected @ TA @ A @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_Graph_Oconnected__append__edge) ).
thf(588,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: nat,D: nat] :
( ( connected @ TA @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ C @ D ) @ ( e @ TA @ A ) )
=> ( connected @ TA @ A @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(145,axiom,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE,H: TF,I: TG] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) @ E @ ( product_Pair @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) @ F @ ( product_Pair @ TE @ ( product_prod @ TF @ TG ) @ G @ ( product_Pair @ TF @ TG @ H @ I ) ) ) ) ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_167_prod__induct7) ).
thf(699,plain,
! [TA: $tType,TB: $tType,TC: $tType,TD: $tType,TE: $tType,TF: $tType,TG: $tType,A: ( product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) ) ) > $o,B: product_prod @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) )] :
( ! [C: TA,D: TB,E: TC,F: TD,G: TE,H: TF,I: TG] : ( A @ ( product_Pair @ TA @ ( product_prod @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) ) @ C @ ( product_Pair @ TB @ ( product_prod @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) ) @ D @ ( product_Pair @ TC @ ( product_prod @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) ) @ E @ ( product_Pair @ TD @ ( product_prod @ TE @ ( product_prod @ TF @ TG ) ) @ F @ ( product_Pair @ TE @ ( product_prod @ TF @ TG ) @ G @ ( product_Pair @ TF @ TG @ H @ I ) ) ) ) ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(157,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ TA @ A ) )
=> ( finite_Graph @ TA @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_Graph_OFinite__Graph__EI) ).
thf(731,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ TA @ A ) )
=> ( finite_Graph @ TA @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(51,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( preorder @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',subcl_Rings_Olinordered__idom___Orderings_Opreorder) ).
thf(418,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( preorder @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(179,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( outgoing @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat] :
( collect @ ( product_prod @ nat @ nat )
@ ^ [C: product_prod @ nat @ nat] :
? [D: nat] :
( ( C
= ( product_Pair @ nat @ nat @ B @ D ) )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ B @ D ) @ ( e @ TA @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_147_Graph_Ooutgoing__def) ).
thf(786,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( outgoing @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: nat] :
( collect @ ( product_prod @ nat @ nat )
@ ^ [C: product_prod @ nat @ nat] :
? [D: nat] :
( ( C
= ( product_Pair @ nat @ nat @ B @ D ) )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ B @ D ) @ ( e @ TA @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(262,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( outgoing2 @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: set @ nat] :
( collect @ ( product_prod @ nat @ nat )
@ ^ [C: product_prod @ nat @ nat] :
? [D: nat,E: nat] :
( ( C
= ( product_Pair @ nat @ nat @ E @ D ) )
& ~ ( member @ nat @ D @ B )
& ( member @ nat @ E @ B )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ D ) @ ( e @ TA @ A ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_67_Graph_Ooutgoing_H__def) ).
thf(1030,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( outgoing2 @ TA )
= ( ^ [A: ( product_prod @ nat @ nat ) > TA,B: set @ nat] :
( collect @ ( product_prod @ nat @ nat )
@ ^ [C: product_prod @ nat @ nat] :
? [D: nat,E: nat] :
( ( C
= ( product_Pair @ nat @ nat @ E @ D ) )
& ~ ( member @ nat @ D @ B )
& ( member @ nat @ E @ B )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ D ) @ ( e @ TA @ A ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).
thf(27,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( outgoing2 @ TA )
= ( outgoing2 @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_Graph_Ooutgoing_H_Ocong) ).
thf(346,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[27]) ).
thf(184,axiom,
! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB )] :
( ( member @ TB @ A @ ( range @ TA @ TB @ B ) )
=> ~ ! [C: TA] :
~ ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ A ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_87_Range_Ocases) ).
thf(800,plain,
! [TA: $tType,TB: $tType,A: TB,B: set @ ( product_prod @ TA @ TB )] :
( ( member @ TB @ A @ ( range @ TA @ TB @ B ) )
=> ~ ! [C: TA] :
~ ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ A ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(220,axiom,
! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ ( product_prod @ TA @ TB ),D: set @ TA] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ C )
=> ( ( member @ TA @ A @ D )
=> ( member @ TB @ B @ ( image @ TA @ TB @ C @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_126_ImageI) ).
thf(914,plain,
! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ ( product_prod @ TA @ TB ),D: set @ TA] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ C )
=> ( ( member @ TA @ A @ D )
=> ( member @ TB @ B @ ( image @ TA @ TB @ C @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(37,axiom,
! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA] :
~ ! [B: TB,C: TA] :
( A
!= ( product_Pair @ TB @ TA @ B @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_166_old_Oprod_Oexhaust) ).
thf(378,plain,
! [TA: $tType,TB: $tType,A: product_prod @ TB @ TA] :
~ ! [B: TB,C: TA] :
( A
!= ( product_Pair @ TB @ TA @ B @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[37]) ).
thf(63,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( isPath @ TA )
= ( isPath @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_255_Graph_OisPath_Ocong) ).
thf(447,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[63]) ).
thf(56,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ( finite_Graph @ TA @ A )
=> ( edmond2129202899alysis @ TA @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_ek__analysis_Ointro) ).
thf(432,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA] :
( ( finite_Graph @ TA @ A )
=> ( edmond2129202899alysis @ TA @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(47,axiom,
preorder @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Opreorder_8) ).
thf(410,plain,
preorder @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(45,axiom,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_45_mem__Collect__eq) ).
thf(405,plain,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(205,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: nat,F: nat] :
( ( isPath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ( ( E != B )
=> ? [G: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ G @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_251_Graph_OisPath__ex__edge1) ).
thf(865,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: nat,F: nat] :
( ( isPath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ( ( E != B )
=> ? [G: nat] : ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ G @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(159,axiom,
( ( v @ a @ c )
= ( collect @ nat
@ ^ [A: nat] :
? [B: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A @ B ) @ ( e @ a @ c ) )
| ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ B @ A ) @ ( e @ a @ c ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_V__def) ).
thf(735,plain,
( ( v @ a @ c )
= ( collect @ nat
@ ^ [A: nat] :
? [B: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A @ B ) @ ( e @ a @ c ) )
| ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ B @ A ) @ ( e @ a @ c ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(250,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: nat,B: ( product_prod @ nat @ nat ) > TA,C: nat,D: nat] :
( ( member @ nat @ A @ ( reachableNodes @ TA @ B @ C ) )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A @ D ) @ ( e @ TA @ B ) )
=> ( member @ nat @ D @ ( reachableNodes @ TA @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_Graph_OreachableNodes__append__edge) ).
thf(999,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: nat,B: ( product_prod @ nat @ nat ) > TA,C: nat,D: nat] :
( ( member @ nat @ A @ ( reachableNodes @ TA @ B @ C ) )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A @ D ) @ ( e @ TA @ B ) )
=> ( member @ nat @ D @ ( reachableNodes @ TA @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).
thf(125,axiom,
( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( e @ a @ c )
@ ( product_Sigma @ nat @ nat @ ( v @ a @ c )
@ ^ [A: nat] : ( v @ a @ c ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_E__ss__VxV) ).
thf(620,plain,
( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( e @ a @ c )
@ ( product_Sigma @ nat @ nat @ ( v @ a @ c )
@ ^ [A: nat] : ( v @ a @ c ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(35,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/sandbox/benchmark/theBenchmark.p',fact_240_le__fun__def) ).
thf(373,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)],[35]) ).
thf(118,axiom,
! [TA: $tType,TB: $tType] :
( ( domain @ TA @ TB )
= ( ^ [A: set @ ( product_prod @ TA @ TB )] :
( collect @ TA
@ ^ [B: TA] :
? [C: TB] : ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ B @ C ) @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_Domain__unfold) ).
thf(603,plain,
! [TA: $tType,TB: $tType] :
( ( domain @ TA @ TB )
= ( ^ [A: set @ ( product_prod @ TA @ TB )] :
( collect @ TA
@ ^ [B: TA] :
? [C: TB] : ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ B @ C ) @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(11,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/sandbox/benchmark/theBenchmark.p',fact_232_dual__order_Oantisym) ).
thf(305,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)],[11]) ).
thf(222,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA ),D: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ C )
=> ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ D ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ D ) @ ( transitive_rtrancl @ TA @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_converse__rtrancl__into__rtrancl) ).
thf(918,plain,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA ),D: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ C )
=> ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ B @ D ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ D ) @ ( transitive_rtrancl @ TA @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).
thf(25,axiom,
! [TA: $tType,A: list @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ ( set2 @ TA @ A ) @ B )
= ( ! [C: TA] :
( ( member @ TA @ C @ ( set2 @ TA @ A ) )
=> ( member @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_subset__code_I1_J) ).
thf(341,plain,
! [TA: $tType,A: list @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ ( set2 @ TA @ A ) @ B )
= ( ! [C: TA] :
( ( member @ TA @ C @ ( set2 @ TA @ A ) )
=> ( member @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(31,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/sandbox/benchmark/theBenchmark.p',fact_214_ord__le__eq__subst) ).
thf(358,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)],[31]) ).
thf(263,axiom,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB )] :
( ( ( ^ [C: TA,D: TB] : ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ A ) )
= ( ^ [C: TA,D: TB] : ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ B ) ) )
= ( A = B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_74_pred__equals__eq2) ).
thf(1033,plain,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB )] :
( ( ( ^ [C: TA,D: TB] : ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ A ) )
= ( ^ [C: TA,D: TB] : ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ B ) ) )
= ( A = B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).
thf(196,axiom,
! [A: nat] :
( ( finite_finite2 @ nat @ ( v @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming @ a @ c @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_finite__incoming) ).
thf(828,plain,
! [A: nat] :
( ( finite_finite2 @ nat @ ( v @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( incoming @ a @ c @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(153,axiom,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ( ( bNF_Ca1785829860lChain @ TA @ TB )
= ( ^ [A: set @ ( product_prod @ TA @ TA ),B: TA > TB] :
! [C: TA,D: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ A )
=> ( ord_less_eq @ TB @ ( B @ C ) @ ( B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_relChain__def) ).
thf(720,plain,
! [TA: $tType,TB: $tType] :
( ( ord @ TB )
=> ( ( bNF_Ca1785829860lChain @ TA @ TB )
= ( ^ [A: set @ ( product_prod @ TA @ TA ),B: TA > TB] :
! [C: TA,D: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ C @ D ) @ A )
=> ( ord_less_eq @ TB @ ( B @ C ) @ ( B @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(236,axiom,
finite_Graph @ a @ c,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5_Finite__Graph__axioms) ).
thf(955,plain,
finite_Graph @ a @ c,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).
thf(214,axiom,
! [A: nat] :
( ( finite_finite2 @ nat @ ( v @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing @ a @ c @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_finite__outgoing) ).
thf(897,plain,
! [A: nat] :
( ( finite_finite2 @ nat @ ( v @ a @ c ) )
=> ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( outgoing @ a @ c @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(171,axiom,
! [A: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing @ a @ c @ A ) @ ( e @ a @ c ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_outgoing__edges) ).
thf(767,plain,
! [A: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing @ a @ c @ A ) @ ( e @ a @ c ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(182,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: nat,F: nat] :
( ( isShortestPath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ( connected @ TA @ A @ F @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_Graph_OisShortestPath__level__edge_I3_J) ).
thf(796,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: nat,F: nat] :
( ( isShortestPath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ( connected @ TA @ A @ F @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(3,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/sandbox/benchmark/theBenchmark.p',fact_215_eq__iff) ).
thf(283,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)],[3]) ).
thf(227,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( ord_less_eq @ ( set @ nat ) @ ( image @ nat @ nat @ ( e @ TA @ A ) @ ( reachableNodes @ TA @ A @ B ) ) @ ( reachableNodes @ TA @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_Graph_OreachableNodes__E__closed) ).
thf(928,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( ord_less_eq @ ( set @ nat ) @ ( image @ nat @ nat @ ( e @ TA @ A ) @ ( reachableNodes @ TA @ A @ B ) ) @ ( reachableNodes @ TA @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).
thf(50,axiom,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_156_old_Oprod_Oinject) ).
thf(415,plain,
! [TA: $tType,TB: $tType,A: TB,B: TA,C: TB,D: TA] :
( ( ( product_Pair @ TB @ TA @ A @ B )
= ( product_Pair @ TB @ TA @ C @ D ) )
= ( ( A = C )
& ( B = D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(127,axiom,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA ),D: TA > $o] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( D @ B )
=> ( ! [E: TA,F: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ F ) @ C )
=> ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ F @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( D @ F )
=> ( D @ E ) ) ) )
=> ( D @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_195_converse__rtrancl__induct) ).
thf(624,plain,
! [TA: $tType,A: TA,B: TA,C: set @ ( product_prod @ TA @ TA ),D: TA > $o] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ A @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( D @ B )
=> ( ! [E: TA,F: TA] :
( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ E @ F ) @ C )
=> ( ( member @ ( product_prod @ TA @ TA ) @ ( product_Pair @ TA @ TA @ F @ B ) @ ( transitive_rtrancl @ TA @ C ) )
=> ( ( D @ F )
=> ( D @ E ) ) ) )
=> ( D @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(161,axiom,
! [TA: $tType,A: list @ TA] : ( finite_finite2 @ TA @ ( set2 @ TA @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_List_Ofinite__set) ).
thf(740,plain,
! [TA: $tType,A: list @ TA] : ( finite_finite2 @ TA @ ( set2 @ TA @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(72,axiom,
linorder @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Olinorder_9) ).
thf(464,plain,
linorder @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(54,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( e @ TA )
= ( e @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_101_Graph_OE_Ocong) ).
thf(427,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(258,axiom,
! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ TA,D: TA > ( set @ TB )] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ ( product_Sigma @ TA @ TB @ C @ D ) )
=> ( member @ TA @ A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_184_SigmaD1) ).
thf(1019,plain,
! [TA: $tType,TB: $tType,A: TA,B: TB,C: set @ TA,D: TA > ( set @ TB )] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ A @ B ) @ ( product_Sigma @ TA @ TB @ C @ D ) )
=> ( member @ TA @ A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).
thf(93,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing @ TA @ A @ B ) @ ( e @ TA @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_146_Graph_Ooutgoing__edges) ).
thf(528,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat] : ( ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( outgoing @ TA @ A @ B ) @ ( e @ TA @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(129,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: nat,E: nat] :
( ( isShortestPath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ( connected @ a @ c @ A @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_isShortestPath__level__edge_I1_J) ).
thf(644,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: nat,E: nat] :
( ( isShortestPath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ E ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ( connected @ a @ c @ A @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(69,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ( ( incoming2 @ TA )
= ( incoming2 @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_Graph_Oincoming_H_Ocong) ).
thf(459,plain,
$true,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(154,axiom,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: product_prod @ nat @ nat] :
( ( isSimplePath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ D @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ D ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_152_isSPath__nt__parallel) ).
thf(723,plain,
! [A: nat,B: list @ ( product_prod @ nat @ nat ),C: nat,D: product_prod @ nat @ nat] :
( ( isSimplePath @ a @ c @ A @ B @ C )
=> ( ( member @ ( product_prod @ nat @ nat ) @ D @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) )
=> ~ ( member @ ( product_prod @ nat @ nat ) @ ( product_swap @ nat @ nat @ D ) @ ( set2 @ ( product_prod @ nat @ nat ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(252,axiom,
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
=> ( finite_finite2 @ nat @ ( v @ a @ c ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_Efin__imp__Vfin) ).
thf(1004,plain,
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ ( e @ a @ c ) )
=> ( finite_finite2 @ nat @ ( v @ a @ c ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).
thf(141,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: nat,F: nat] :
( ( isShortestPath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ( connected @ TA @ A @ B @ E ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_Graph_OisShortestPath__level__edge_I1_J) ).
thf(691,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: ( product_prod @ nat @ nat ) > TA,B: nat,C: list @ ( product_prod @ nat @ nat ),D: nat,E: nat,F: nat] :
( ( isShortestPath @ TA @ A @ B @ C @ D )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ E @ F ) @ ( set2 @ ( product_prod @ nat @ nat ) @ C ) )
=> ( connected @ TA @ A @ B @ E ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(61,axiom,
order @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oorder_11) ).
thf(442,plain,
order @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[61]) ).
thf(207,axiom,
! [A: nat,B: nat,C: nat] :
( ( member @ nat @ A @ ( reachableNodes @ a @ c @ B ) )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A @ C ) @ ( e @ a @ c ) )
=> ( member @ nat @ C @ ( reachableNodes @ a @ c @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_reachableNodes__append__edge) ).
thf(871,plain,
! [A: nat,B: nat,C: nat] :
( ( member @ nat @ A @ ( reachableNodes @ a @ c @ B ) )
=> ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A @ C ) @ ( e @ a @ c ) )
=> ( member @ nat @ C @ ( reachableNodes @ a @ c @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(84,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/sandbox/benchmark/theBenchmark.p',fact_213_ord__eq__le__subst) ).
thf(501,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)],[84]) ).
thf(217,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( finite_finite2 @ TA @ B )
=> ( finite_finite2 @ TA @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_finite__subset) ).
thf(908,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( finite_finite2 @ TA @ B )
=> ( finite_finite2 @ TA @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).
thf(248,axiom,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB )] :
( ! [C: TA,D: TB] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ A )
=> ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ B ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_subrelI) ).
thf(992,plain,
! [TA: $tType,TB: $tType,A: set @ ( product_prod @ TA @ TB ),B: set @ ( product_prod @ TA @ TB )] :
( ! [C: TA,D: TB] :
( ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ A )
=> ( member @ ( product_prod @ TA @ TB ) @ ( product_Pair @ TA @ TB @ C @ D ) @ B ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ TA @ TB ) ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).
thf(77,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/sandbox/benchmark/theBenchmark.p',fact_220_order_Otrans) ).
thf(486,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)],[77]) ).
thf(75,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ B )
| ( ord_less_eq @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_217_linear) ).
thf(469,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)],[75]) ).
thf(98,axiom,
! [A: set @ nat] :
( ( outgoing2 @ a @ c @ A )
= ( collect @ ( product_prod @ nat @ nat )
@ ^ [B: product_prod @ nat @ nat] :
? [C: nat,D: nat] :
( ( B
= ( product_Pair @ nat @ nat @ D @ C ) )
& ~ ( member @ nat @ C @ A )
& ( member @ nat @ D @ A )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ C ) @ ( e @ a @ c ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_outgoing_H__def) ).
thf(543,plain,
! [A: set @ nat] :
( ( outgoing2 @ a @ c @ A )
= ( collect @ ( product_prod @ nat @ nat )
@ ^ [B: product_prod @ nat @ nat] :
? [C: nat,D: nat] :
( ( B
= ( product_Pair @ nat @ nat @ D @ C ) )
& ~ ( member @ nat @ C @ A )
& ( member @ nat @ D @ A )
& ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ D @ C ) @ ( e @ a @ c ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(79,axiom,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ( finite_finite @ ( set @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Finite__Set_Ofinite_5) ).
thf(491,plain,
! [TA: $tType] :
( ( finite_finite @ TA )
=> ( finite_finite @ ( set @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(114,axiom,
ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) @ ( e @ a @ c ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4_spEdges__ss__E) ).
thf(592,plain,
ord_less_eq @ ( set @ ( product_prod @ nat @ nat ) ) @ ( edmond1803015688pEdges @ a @ c @ s @ t ) @ ( e @ a @ c ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(1082,plain,
$false,
inference(e,[status(thm)],[1036,809,1024,347,962,666,408,582,308,449,861,440,655,385,533,500,715,945,930,1001,709,885,1046,389,1055,751,357,460,802,344,902,559,853,421,606,325,289,448,924,756,789,443,376,806,966,1051,1040,321,397,660,411,428,849,293,956,1072,453,1074,920,512,381,366,1067,798,546,793,334,910,894,1021,952,349,561,392,693,815,729,435,493,1027,551,456,974,1012,916,467,697,874,317,1017,615,980,1006,593,989,519,742,650,948,783,420,471,488,933,753,642,526,590,647,515,911,779,311,541,657,765,530,773,888,824,856,733,601,689,296,584,819,286,743,577,579,413,611,1058,804,764,900,831,996,737,303,863,451,567,509,880,1043,434,811,758,858,535,609,323,726,621,964,769,943,668,330,462,494,1080,521,399,868,883,1015,299,851,438,653,922,1044,695,394,986,1076,927,749,939,553,1049,663,548,617,516,968,823,776,525,295,1070,707,971,327,954,310,712,557,705,282,717,338,808,1038,370,336,703,613,1052,978,1008,314,571,595,497,465,671,368,771,813,1062,781,588,699,731,418,786,1030,346,800,914,378,447,432,410,405,865,735,999,620,373,603,305,918,341,358,1033,828,720,955,897,767,796,283,928,415,624,740,464,427,1019,528,644,459,723,1004,691,442,871,501,908,992,486,469,543,491,592]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : ITP049^2 : TPTP v9.3.1. Released v7.5.0.
% 0.00/0.08 % Command : java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.11/0.22 % Computer : n020.cluster.edu
% 0.11/0.22 % Model : x86_64 x86_64
% 0.11/0.22 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.22 % Memory : 8046.5625MB
% 0.11/0.22 % OS : Linux 6.8.0-71-generic
% 0.11/0.22 % CPULimit : 300
% 0.11/0.22 % WCLimit : 300
% 0.11/0.22 % DateTime : Tue Sep 29 19:44:18 UTC 2026
% 0.11/0.23 % CPUTime :
% 0.11/0.23 Running java -Xss128m -Xmx2g -Xms1g -jar /export/starexec/sandbox/solver/bin/leo3.jar /export/starexec/sandbox/benchmark/theBenchmark.p -t 300 -p --atp eprover=/export/starexec/sandbox/solver/bin/externals/eprover --instantiate 39
% 0.82/0.65 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.81/0.94 % [INFO] Parsing done (289ms).
% 1.81/0.95 % [INFO] Running in sequential loop mode.
% 2.85/1.15 % [INFO] eprover registered as external prover.
% 2.85/1.15 % [INFO] Scanning for conjecture ...
% 3.16/1.25 % [INFO] Found a conjecture (or negated_conjecture) and 280 axioms. Running axiom selection ...
% 3.57/1.34 % [INFO] Axiom selection finished. Selected 279 axioms (removed 1 axioms).
% 4.99/1.60 % [INFO] Problem is higher-order (TPTP THF).
% 4.99/1.63 % [INFO] Type checking passed.
% 4.99/1.64 % [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 ...
% 205.21/173.44 % External prover 'e' found a proof!
% 205.21/173.44 % [INFO] Killing All external provers ...
% 205.21/173.44 % Time passed: 173076ms (effective reasoning time: 172490ms)
% 205.21/173.44 % 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)>
% 205.21/173.45 % Axioms used in derivation (279): fact_232_dual__order_Oantisym, fact_224_ord__eq__le__trans, fact_79_Domain_ODomainI, fact_9_finite, fact_136_Image__subset, subcl_Rings_Olinordered__idom___Orderings_Olinorder, fact_206_infinite__cartesian__product, fact_128_adjacent__nodes__finite, tfree_0, fact_17_finite__Collect__conjI, tcon_Nat_Onat___Orderings_Oorder_2, fact_158_r__into__rtrancl, fact_142_Graph_OisSPath__sg__outgoing, fact_118_outgoing__edges, fact_31_ek__analysis_Oaxioms, fact_110_Graph_Oconnected__inV__iff, fact_231_dual__order_Oeq__iff, fact_87_Range_Ocases, fact_242_rtc__isPath, fact_120_connected__edgeRtc, fact_33_finite__has__minimal2, fact_151_Finite__Graph_Oadjacent__nodes__finite, fact_93_Graph_Oincoming__edges, fact_90_Graph_OreachableNodes__def, fact_135_ImageE, fact_18_finite__Collect__disjI, fact_192_rtrancl_Oinducts, tcon_fun___Finite__Set_Ofinite, fact_34_finite__has__maximal2, fact_96_Graph_OisShortestPath__level__edge_I3_J, tcon_Nat_Onat___Orderings_Olinorder, fact_132_Graph_Ooutgoing_Ocong, fact_239_le__funI, fact_202_Image__closed__trancl, fact_213_ord__eq__le__subst, fact_0_ek__analysis__axioms, fact_72_Graph_Oincoming_H_Ocong, fact_91_pred__subset__eq, fact_102_Graph_OV_Ocong, fact_83_DomainE, fact_216_antisym, fact_247_isPath__ex__edge2, fact_246_isPath__ex__edge1, fact_205_finite__rtrancl__Image, fact_41_reachableNodes__def, fact_124_reachableNodes__E__closed, fact_115_Finite__Graph_Ofinite__E, fact_117_outgoing__def, tcon_fun___Orderings_Oord, tcon_HOL_Obool___Orderings_Olinorder_9, fact_113_Finite__Graph_Ointro, subcl_Rings_Olinordered__idom___Orderings_Opreorder, fact_190_rtrancl__trans, fact_89_RangeE, fact_161_swap__simp, fact_177_Pair__inject, fact_11_spEdges__def, fact_159_rtrancl__idemp, fact_163_converse__rtranclE2, fact_240_le__fun__def, fact_59_incoming_H__def, fact_199_rtrancl__subset, fact_14_Efin__imp__Vfin, fact_218_eq__refl, fact_103_Graph_Oconnected__refl, fact_15_connected__inV__iff, fact_32_ek__analysis__defs_OspEdges__def, fact_4_spEdges__ss__E, subcl_Rings_Olinordered__idom___HOL_Otype, fact_3_finite__E, fact_145_Graph_Oadjacent__nodes__ss__V, fact_129_finite__outgoing, fact_94_Graph_OisShortestPath__level__edge_I1_J, fact_116_Graph_OFinite__Graph__EI, fact_46_Collect__mem__eq, fact_207_old_Oprod_Orec, fact_147_Graph_Ooutgoing__def, fact_24_Domain__mono, fact_67_Graph_Ooutgoing_H__def, subcl_Rings_Olinordered__idom___Orderings_Oorder, fact_209_order__refl, fact_21_finite__V, fact_27_not__finite__existsD, fact_149_Graph_OE__ss__VxV, fact_222_antisym__conv, fact_26_finite__image__set2, fact_88_Range__iff, tcon_HOL_Obool___Orderings_Oorder_11, fact_29_ek__analysis__def, fact_138_Image__mono, fact_226_order__class_Oorder_Oantisym, fact_235_predicate1D, fact_208_predicate1I, fact_248_isPath__edgeset, fact_85_Range_Ointros, fact_43_Graph_OEfin__imp__Vfin, fact_217_linear, fact_60_finite__Collect__le__nat, fact_220_order_Otrans, fact_62_finite__outgoing_H, fact_210_predicate2I, fact_156_old_Oprod_Oinject, fact_169_prod__induct5, fact_77_subrelI, fact_68_Graph_Oreachable__ss__V, fact_22_Vfin__imp__Efin, fact_221_le__cases3, fact_108_finite__list, fact_189_rtrancl_Osimps, fact_146_Graph_Ooutgoing__edges, fact_160_swap__swap, fact_230_dual__order_Otrans, fact_140_Graph_OreachableNodes__E__closed, fact_245_shortestPath__is__path, fact_58_outgoing_H__def, fact_16_obtain__shortest__path, fact_127_finite__SigmaI, fact_152_isSPath__nt__parallel, tcon_Set_Oset___Orderings_Oorder_6, fact_241_relChain__def, fact_233_predicate2D, fact_243_isSPath__pathLE, fact_51_isShortestPath__level__edge_I3_J, fact_105_Graph_Ofinite__outgoing_H, fact_249_isPath__rtc, fact_114_Finite__Graph__def, fact_157_prod_Oinject, tcon_Set_Oset___Orderings_Oord_7, fact_55_reachableNodes__append__edge, fact_74_pred__equals__eq2, fact_214_ord__le__eq__subst, fact_125_shortestPath__is__simple, fact_203_Not__Domain__rtrancl, fact_45_mem__Collect__eq, fact_100_Graph_Ofinite__incoming, fact_84_Range_Oinducts, fact_139_Relation_Ofinite__Image, fact_36_infinite__super, fact_182_Times__eq__cancel2, fact_92_Graph_Oconnected__append__edge, fact_181_Sigma__cong, fact_37_finite__subset, fact_42_List_Ofinite__set, tcon_Nat_Onat___Orderings_Oord_3, fact_121_isSPath__no__selfloop, fact_107_Graph_Ofinite__incoming_H, fact_143_Graph_OisSPath__sg__incoming, fact_25_finite__image__set, fact_98_Graph_Oincoming_H__edges, fact_111_Graph_Oobtain__shortest__path, fact_101_Graph_OE_Ocong, fact_191_rtrancl__induct, fact_69_Graph_OreachableNodes__append__edge, fact_211_order__subst1, fact_234_rev__predicate2D, fact_7_Finite__Graph__EI, fact_66_Graph_Oincoming_H__def, fact_47_Collect__cong, fact_188_rtrancl_Ocases, tcon_Product__Type_Oprod___Finite__Set_Ofinite_13, fact_137_Graph_OshortestPath__is__simple, fact_104_Graph_Oconnected_Ocong, fact_130_Graph_Oadjacent__nodes_Ocong, fact_63_finite__incoming_H, tcon_Set_Oset___Orderings_Opreorder_4, fact_197_converse__rtrancl__into__rtrancl, fact_76_Graph_OV__def, fact_97_Graph_Ooutgoing_H__edges, fact_78_Domain_Oinducts, fact_155_trancl__subset__Sigma__aux, fact_238_le__funE, fact_150_Graph_Ofinite__outgoing, fact_86_Range_Osimps, fact_141_finite__cartesian__product, fact_13_finite__Domain, fact_162_converse__rtrancl__induct2, fact_244_connected__def, fact_250_Graph_OisPath__ex__edge2, tcon_Set_Oset___Finite__Set_Ofinite_5, fact_80_Domain_Osimps, fact_167_prod__induct7, fact_30_ek__analysis_Ointro, fact_172_prod__cases7, fact_179_surj__pair, fact_166_old_Oprod_Oexhaust, fact_194_rtrancl_Ortrancl__refl, fact_251_Graph_OisPath__ex__edge1, fact_64_adjacent__nodes__ss__V, fact_200_rtrancl__subset__rtrancl, fact_196_rtrancl_Ortrancl__into__rtrancl, fact_123_isSPath__sg__outgoing, fact_215_eq__iff, fact_6_finite__code, fact_81_Domain_Ocases, tcon_fun___Orderings_Opreorder, fact_82_Domain__iff, tcon_Nat_Onat___Orderings_Opreorder_1, fact_122_isSPath__sg__incoming, fact_53_reachable__ss__V, fact_176_prod__cases3, fact_237_le__funD, fact_95_Graph_OisShortestPath__level__edge_I2_J, fact_8_finite__set__choice, fact_171_prod__induct3, fact_106_Graph_OisShortestPath_Ocong, fact_28_pigeonhole__infinite__rel, fact_178_prod__cases, fact_168_prod__induct6, fact_44_Graph_OVfin__imp__Efin, fact_254_Graph_OshortestPath__is__path, fact_204_Times__subset__cancel2, fact_175_prod__cases4, fact_255_Graph_OisPath_Ocong, fact_201_Sigma__mono, fact_228_dual__order_Orefl, fact_40_outgoing_H__edges, fact_186_SigmaE2, fact_183_SigmaE, fact_174_prod__cases5, fact_180_Graph_OisSPath__nt__parallel, fact_112_Finite__Graph_Ofinite__V, tcon_fun___Orderings_Oorder, fact_56_V__def, fact_23_Range__mono, fact_229_linorder__wlog, subcl_Rings_Olinordered__idom___Orderings_Oord, fact_35_rev__finite__subset, fact_20_finite__Collect__subsets, fact_193_converse__rtranclE, fact_173_prod__cases6, fact_52_incoming__edges, fact_54_connected__append__edge, fact_185_SigmaD2, tcon_HOL_Obool___Orderings_Opreorder_8, fact_19_finite__Collect__bounded__ex, fact_109_subset__code_I1_J, fact_12_finite__Range, fact_1__092_060open_062finite_AE_A_092_060Longrightarrow_062_Afinite_AspEdges_092_060close_062, fact_2_ek__analysis__defs_OspEdges_Ocong, fact_38_ek__analysis_OspEdges__ss__E, fact_198_rtrancl__mono, fact_236_rev__predicate1D, fact_184_SigmaD1, fact_71_Graph_Ooutgoing_H_Ocong, fact_153_mem__Sigma__iff, fact_70_Graph_OreachableNodes_Ocong, fact_253_Graph_Oconnected__def, tcon_HOL_Obool___Orderings_Oord_12, fact_134_Image__iff, fact_154_SigmaI, fact_225_ord__le__eq__trans, fact_170_prod__induct4, fact_131_Graph_OisSimplePath_Ocong, fact_61_finite__incoming, fact_39_incoming_H__edges, fact_49_isShortestPath__level__edge_I1_J, fact_73_Graph_Oincoming_Ocong, fact_252_Graph_OisPath__edgeset, fact_144_Graph_OisSPath__no__selfloop, fact_219_le__cases, fact_212_order__subst2, fact_57_incoming__def, fact_195_converse__rtrancl__induct, fact_75_pred__subset__eq2, fact_65_Graph_Oincoming__def, fact_223_order__class_Oorder_Oeq__iff, fact_227_order__trans, fact_148_Graph_Oconnected__edgeRtc, fact_164_rtrancl__induct2, fact_133_rev__ImageI, fact_165_old_Oprod_Oinducts, tcon_HOL_Obool___Finite__Set_Ofinite_10, fact_10_connected__refl, fact_119_E__ss__VxV, fact_5_Finite__Graph__axioms, fact_126_ImageI, fact_99_Domain__unfold, fact_50_isShortestPath__level__edge_I2_J, fact_187_rtranclE
% 205.21/173.45 % No. of inferences in proof: 562
% 205.21/173.45 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 173076 ms resp. 172490 ms w/o parsing
% 206.32/173.71 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 206.32/173.72 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------