TSTP Solution File: ITP132^2 by Leo-III---1.7.12
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%------------------------------------------------------------------------------
% File : Leo-III---1.7.12
% Problem : ITP132^2 : TPTP v8.2.0. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : run_Leo-III %s %d
% Computer : n014.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Mon May 20 22:23:14 EDT 2024
% Result : Theorem 58.75s 41.68s
% Output : Refutation 59.77s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 356
% Syntax : Number of formulae : 654 ( 182 unt; 60 typ; 0 def)
% Number of atoms : 1786 ( 555 equ; 8 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 6726 ( 134 ~; 50 |; 92 &;5576 @)
% ( 0 <=>; 874 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 7 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 224 ( 224 >; 0 *; 0 +; 0 <<)
% Number of symbols : 64 ( 61 usr; 6 con; 0-6 aty)
% Number of variables : 1860 ( 111 ^1676 !; 16 ?;1860 :)
% ( 57 !>; 0 ?*; 0 @-; 0 @+)
% Comments :
%------------------------------------------------------------------------------
thf(nat_type,type,
nat: $tType ).
thf(dvd_type,type,
dvd:
!>[TA: $tType] : $o ).
thf(one_type,type,
one:
!>[TA: $tType] : $o ).
thf(ring_type,type,
ring:
!>[TA: $tType] : $o ).
thf(zero_type,type,
zero:
!>[TA: $tType] : $o ).
thf(minus_type,type,
minus:
!>[TA: $tType] : $o ).
thf(ord_type,type,
ord:
!>[TA: $tType] : $o ).
thf(neg_numeral_type,type,
neg_numeral:
!>[TA: $tType] : $o ).
thf(order_type,type,
order:
!>[TA: $tType] : $o ).
thf(comm_ring_type,type,
comm_ring:
!>[TA: $tType] : $o ).
thf(mult_zero_type,type,
mult_zero:
!>[TA: $tType] : $o ).
thf(group_add_type,type,
group_add:
!>[TA: $tType] : $o ).
thf(no_bot_type,type,
no_bot:
!>[TA: $tType] : $o ).
thf(no_top_type,type,
no_top:
!>[TA: $tType] : $o ).
thf(semiring_0_type,type,
semiring_0:
!>[TA: $tType] : $o ).
thf(monoid_mult_type,type,
monoid_mult:
!>[TA: $tType] : $o ).
thf(linorder_type,type,
linorder:
!>[TA: $tType] : $o ).
thf(preorder_type,type,
preorder:
!>[TA: $tType] : $o ).
thf(ordered_ring_type,type,
ordered_ring:
!>[TA: $tType] : $o ).
thf(zero_neq_one_type,type,
zero_neq_one:
!>[TA: $tType] : $o ).
thf(ab_group_add_type,type,
ab_group_add:
!>[TA: $tType] : $o ).
thf(wellorder_type,type,
wellorder:
!>[TA: $tType] : $o ).
thf(semigroup_mult_type,type,
semigroup_mult:
!>[TA: $tType] : $o ).
thf(dense_order_type,type,
dense_order:
!>[TA: $tType] : $o ).
thf(linordered_idom_type,type,
linordered_idom:
!>[TA: $tType] : $o ).
thf(linordered_ring_type,type,
linordered_ring:
!>[TA: $tType] : $o ).
thf(comm_monoid_add_type,type,
comm_monoid_add:
!>[TA: $tType] : $o ).
thf(ordered_semiring_type,type,
ordered_semiring:
!>[TA: $tType] : $o ).
thf(comm_monoid_diff_type,type,
comm_monoid_diff:
!>[TA: $tType] : $o ).
thf(comm_monoid_mult_type,type,
comm_monoid_mult:
!>[TA: $tType] : $o ).
thf(ab_semigroup_mult_type,type,
ab_semigroup_mult:
!>[TA: $tType] : $o ).
thf(ordered_semiring_0_type,type,
ordered_semiring_0:
!>[TA: $tType] : $o ).
thf(ordered_ab_group_add_type,type,
ordered_ab_group_add:
!>[TA: $tType] : $o ).
thf(ordere1490568538miring_type,type,
ordere1490568538miring:
!>[TA: $tType] : $o ).
thf(comm_s1003936772cancel_type,type,
comm_s1003936772cancel:
!>[TA: $tType] : $o ).
thf(linord581940658strict_type,type,
linord581940658strict:
!>[TA: $tType] : $o ).
thf(cancel1352612707id_add_type,type,
cancel1352612707id_add:
!>[TA: $tType] : $o ).
thf(ring_11004092258visors_type,type,
ring_11004092258visors:
!>[TA: $tType] : $o ).
thf(cancel146912293up_add_type,type,
cancel146912293up_add:
!>[TA: $tType] : $o ).
thf(ordere216010020id_add_type,type,
ordere216010020id_add:
!>[TA: $tType] : $o ).
thf(semiri1193490041visors_type,type,
semiri1193490041visors:
!>[TA: $tType] : $o ).
thf(linord1659791738miring_type,type,
linord1659791738miring:
!>[TA: $tType] : $o ).
thf(canoni770627133id_add_type,type,
canoni770627133id_add:
!>[TA: $tType] : $o ).
thf(semiri1923998003cancel_type,type,
semiri1923998003cancel:
!>[TA: $tType] : $o ).
thf(fun_upd_type,type,
fun_upd:
!>[TA: $tType,TB: $tType] : ( ( TB > TA ) > TB > TA > TB > TA ) ).
thf(minus_minus_type,type,
minus_minus:
!>[TA: $tType] : ( TA > TA > TA ) ).
thf(one_one_type,type,
one_one:
!>[TA: $tType] : TA ).
thf(times_times_type,type,
times_times:
!>[TA: $tType] : ( TA > TA > TA ) ).
thf(zero_zero_type,type,
zero_zero:
!>[TA: $tType] : TA ).
thf(groups1340683514dd_sum_type,type,
groups1340683514dd_sum:
!>[TA: $tType,TB: $tType] : ( ( TB > TA ) > ( set @ TB ) > TA ) ).
thf(if_type,type,
if:
!>[TA: $tType] : ( $o > TA > TA > TA ) ).
thf(number2016821345itions_type,type,
number2016821345itions: ( nat > nat ) > nat > $o ).
thf(ord_less_type,type,
ord_less:
!>[TA: $tType] : ( TA > TA > $o ) ).
thf(ord_less_eq_type,type,
ord_less_eq:
!>[TA: $tType] : ( TA > TA > $o ) ).
thf(collect_type,type,
collect:
!>[TA: $tType] : ( ( TA > $o ) > ( set @ TA ) ) ).
thf(set_ord_atMost_type,type,
set_ord_atMost:
!>[TA: $tType] : ( TA > ( set @ TA ) ) ).
thf(member_type,type,
member:
!>[TA: $tType] : ( TA > ( set @ TA ) > $o ) ).
thf(k_type,type,
k: nat ).
thf(n_type,type,
n: nat ).
thf(p_type,type,
p: nat > nat ).
thf(28,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_95_zero__le) ).
thf(365,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[28]) ).
thf(23,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_159_ord__le__eq__subst) ).
thf(347,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)],[23]) ).
thf(17,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_247_less__numeral__extra_I4_J) ).
thf(333,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[17]) ).
thf(70,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_192_subsetI) ).
thf(518,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( member @ TA @ C @ B ) )
=> ( ord_less_eq @ ( set @ TA ) @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[70]) ).
thf(226,axiom,
! [TA: $tType] :
( ( comm_monoid_mult @ TA )
=> ! [A: TA] :
( ( times_times @ TA @ A @ ( one_one @ TA ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_190_mult_Ocomm__neutral) ).
thf(962,plain,
! [TA: $tType] :
( ( comm_monoid_mult @ TA )
=> ! [A: TA] :
( ( times_times @ TA @ A @ ( one_one @ TA ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[226]) ).
thf(219,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ ( one_one @ TA ) )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ( ord_less_eq @ TA @ B @ ( one_one @ TA ) )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( one_one @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_135_mult__le__one) ).
thf(941,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ ( one_one @ TA ) )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ( ord_less_eq @ TA @ B @ ( one_one @ TA ) )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( one_one @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[219]) ).
thf(85,axiom,
! [TA: $tType,TB: $tType] :
( ( preorder @ TA )
=> ( preorder @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Opreorder) ).
thf(555,plain,
! [TA: $tType,TB: $tType] :
( ( preorder @ TA )
=> ( preorder @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[85]) ).
thf(118,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ C @ B ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
=> ( ord_less_eq @ nat @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_209_mult__le__cancel2) ).
thf(666,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ C @ B ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
=> ( ord_less_eq @ nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[118]) ).
thf(279,axiom,
one @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oone) ).
thf(1110,plain,
one @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[279]) ).
thf(291,axiom,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_108_le__0__eq) ).
thf(1142,plain,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[291]) ).
thf(67,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_141_dual__order_Oantisym) ).
thf(511,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)],[67]) ).
thf(72,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_145_dual__order_Orefl) ).
thf(523,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[72]) ).
thf(297,axiom,
canoni770627133id_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add) ).
thf(1158,plain,
canoni770627133id_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[297]) ).
thf(144,axiom,
! [A: nat,B: nat] :
( ( ( minus_minus @ nat @ A @ B )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ B @ A )
= ( zero_zero @ nat ) )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_110_diffs0__imp__equal) ).
thf(741,plain,
! [A: nat,B: nat] :
( ( ( minus_minus @ nat @ A @ B )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ B @ A )
= ( zero_zero @ nat ) )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[144]) ).
thf(264,axiom,
! [TA: $tType] :
( ( mult_zero @ TA )
=> ( ( ^ [A: TA] : ( zero_zero @ TA ) )
= ( times_times @ TA @ ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_103_lambda__zero) ).
thf(1073,plain,
! [TA: $tType] :
( ( mult_zero @ TA )
=> ( ( ^ [A: TA] : ( zero_zero @ TA ) )
= ( times_times @ TA @ ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[264]) ).
thf(249,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ nat @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_19_le__trans) ).
thf(1041,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ord_less_eq @ nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[249]) ).
thf(189,axiom,
! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
( ! [D: TA] :
( ~ ( A @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
& ~ ( A @ E ) ) )
=> ( A @ C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_248_infinite__descent__measure) ).
thf(861,plain,
! [TA: $tType,A: TA > $o,B: TA > nat,C: TA] :
( ! [D: TA] :
( ~ ( A @ D )
=> ? [E: TA] :
( ( ord_less @ nat @ ( B @ E ) @ ( B @ D ) )
& ~ ( A @ E ) ) )
=> ( A @ C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[189]) ).
thf(266,axiom,
number2016821345itions @ p @ n,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_0_partitions) ).
thf(1078,plain,
number2016821345itions @ p @ n,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[266]) ).
thf(35,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( B != A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_222_less__imp__not__eq2) ).
thf(384,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( B != A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[35]) ).
thf(77,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_148_ord__le__eq__trans) ).
thf(533,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)],[77]) ).
thf(167,axiom,
! [TA: $tType,TB: $tType] :
( ( ordere216010020id_add @ TB )
=> ! [A: set @ TA,B: TA > TB,C: TA > TB] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ord_less_eq @ TB @ ( B @ D ) @ ( C @ D ) ) )
=> ( ord_less_eq @ TB @ ( groups1340683514dd_sum @ TA @ TB @ B @ A ) @ ( groups1340683514dd_sum @ TA @ TB @ C @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_40_sum__mono) ).
thf(797,plain,
! [TA: $tType,TB: $tType] :
( ( ordere216010020id_add @ TB )
=> ! [A: set @ TA,B: TA > TB,C: TA > TB] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ord_less_eq @ TB @ ( B @ D ) @ ( C @ D ) ) )
=> ( ord_less_eq @ TB @ ( groups1340683514dd_sum @ TA @ TB @ B @ A ) @ ( groups1340683514dd_sum @ TA @ TB @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[167]) ).
thf(60,axiom,
! [TA: $tType,TB: $tType] :
( ( fun_upd @ TA @ TB )
= ( ^ [A: TA > TB,B: TA,C: TB,D: TA] : ( if @ TB @ ( D = B ) @ C @ ( A @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_180_fun__upd__def) ).
thf(472,plain,
! [TA: $tType,TB: $tType] :
( ( fun_upd @ TA @ TB )
= ( ^ [A: TA > TB,B: TA,C: TB,D: TA] : ( if @ TB @ ( D = B ) @ C @ ( A @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[60]) ).
thf(259,axiom,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_107_bot__nat__0_Oextremum__unique) ).
thf(1063,plain,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[259]) ).
thf(146,axiom,
ab_semigroup_mult @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oab__semigroup__mult) ).
thf(747,plain,
ab_semigroup_mult @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[146]) ).
thf(207,axiom,
ord @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oord_3) ).
thf(913,plain,
ord @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[207]) ).
thf(42,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_227_dual__order_Oasym) ).
thf(404,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[42]) ).
thf(202,axiom,
! [TA: $tType,TB: $tType,TC: $tType] :
( ( comm_monoid_add @ TC )
=> ! [A: TB > TA > TC,B: set @ TA,C: set @ TB] :
( ( groups1340683514dd_sum @ TB @ TC
@ ^ [D: TB] : ( groups1340683514dd_sum @ TA @ TC @ ( A @ D ) @ B )
@ C )
= ( groups1340683514dd_sum @ TA @ TC
@ ^ [D: TA] :
( groups1340683514dd_sum @ TB @ TC
@ ^ [E: TB] : ( A @ E @ D )
@ C )
@ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_54_sum_Oswap) ).
thf(898,plain,
! [TA: $tType,TB: $tType,TC: $tType] :
( ( comm_monoid_add @ TC )
=> ! [A: TB > TA > TC,B: set @ TA,C: set @ TB] :
( ( groups1340683514dd_sum @ TB @ TC
@ ^ [D: TB] : ( groups1340683514dd_sum @ TA @ TC @ ( A @ D ) @ B )
@ C )
= ( groups1340683514dd_sum @ TA @ TC
@ ^ [D: TA] :
( groups1340683514dd_sum @ TB @ TC
@ ^ [E: TB] : ( A @ E @ D )
@ C )
@ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[202]) ).
thf(89,axiom,
! [TA: $tType,TB: $tType,A: TA > TB,B: TA,C: TB] :
( ( ( A @ B )
= C )
=> ( ( fun_upd @ TA @ TB @ A @ B @ C )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_178_fun__upd__idem) ).
thf(565,plain,
! [TA: $tType,TB: $tType,A: TA > TB,B: TA,C: TB] :
( ( ( A @ B )
= C )
=> ( ( fun_upd @ TA @ TB @ A @ B @ C )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[89]) ).
thf(53,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_161_order__subst2) ).
thf(436,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)],[53]) ).
thf(112,axiom,
! [A: nat > nat,B: nat,C: nat] :
( ( number2016821345itions @ A @ B )
=> ( ord_less_eq @ nat @ ( A @ C ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_25_partitions__bounds) ).
thf(651,plain,
! [A: nat > nat,B: nat,C: nat] :
( ( number2016821345itions @ A @ B )
=> ( ord_less_eq @ nat @ ( A @ C ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[112]) ).
thf(137,axiom,
! [TA: $tType] :
( ( semiri1193490041visors @ TA )
=> ! [A: TA,B: TA] :
( ( ( times_times @ TA @ A @ B )
= ( zero_zero @ TA ) )
=> ( ( A
= ( zero_zero @ TA ) )
| ( B
= ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_97_divisors__zero) ).
thf(724,plain,
! [TA: $tType] :
( ( semiri1193490041visors @ TA )
=> ! [A: TA,B: TA] :
( ( ( times_times @ TA @ A @ B )
= ( zero_zero @ TA ) )
=> ( ( A
= ( zero_zero @ TA ) )
| ( B
= ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[137]) ).
thf(135,axiom,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_78_le0) ).
thf(719,plain,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[135]) ).
thf(161,axiom,
mult_zero @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Omult__zero) ).
thf(785,plain,
mult_zero @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[161]) ).
thf(66,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_160_ord__eq__le__subst) ).
thf(504,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)],[66]) ).
thf(22,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_163_le__fun__def) ).
thf(344,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)],[22]) ).
thf(97,axiom,
! [TA: $tType] :
( ( dense_order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ? [C: TA] :
( ( ord_less @ TA @ A @ C )
& ( ord_less @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_236_dense) ).
thf(587,plain,
! [TA: $tType] :
( ( dense_order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ? [C: TA] :
( ( ord_less @ TA @ A @ C )
& ( ord_less @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[97]) ).
thf(108,axiom,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less @ nat @ D @ C )
& ~ ( A @ D ) ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_251_infinite__descent) ).
thf(619,plain,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ~ ( A @ C )
=> ? [D: nat] :
( ( ord_less @ nat @ D @ C )
& ~ ( A @ D ) ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[108]) ).
thf(129,axiom,
! [TA: $tType] :
( ( linord581940658strict @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) )
= ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
& ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) )
| ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
& ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_114_zero__le__mult__iff) ).
thf(702,plain,
! [TA: $tType] :
( ( linord581940658strict @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) )
= ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
& ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) )
| ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
& ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[129]) ).
thf(148,axiom,
! [TA: $tType] :
( ( ordered_semiring @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_121_mult__right__mono) ).
thf(751,plain,
! [TA: $tType] :
( ( ordered_semiring @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[148]) ).
thf(47,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_61_order__refl) ).
thf(421,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[47]) ).
thf(103,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_232_less__trans) ).
thf(606,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[103]) ).
thf(13,axiom,
minus @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Groups_Ominus_13) ).
thf(325,plain,
minus @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[13]) ).
thf(10,axiom,
! [TA: $tType,TB: $tType,A: TA > TB,B: TA] :
( ( fun_upd @ TA @ TB @ A @ B @ ( A @ B ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_59_fun__upd__triv) ).
thf(316,plain,
! [TA: $tType,TB: $tType,A: TA > TB,B: TA] :
( ( fun_upd @ TA @ TB @ A @ B @ ( A @ B ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[10]) ).
thf(275,axiom,
! [TA: $tType,TB: $tType] :
( ( semiring_0 @ TB )
=> ! [A: TA > TB,B: set @ TA,C: TB] :
( ( times_times @ TB @ ( groups1340683514dd_sum @ TA @ TB @ A @ B ) @ C )
= ( groups1340683514dd_sum @ TA @ TB
@ ^ [D: TA] : ( times_times @ TB @ ( A @ D ) @ C )
@ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_9_sum__distrib__right) ).
thf(1100,plain,
! [TA: $tType,TB: $tType] :
( ( semiring_0 @ TB )
=> ! [A: TA > TB,B: set @ TA,C: TB] :
( ( times_times @ TB @ ( groups1340683514dd_sum @ TA @ TB @ A @ B ) @ C )
= ( groups1340683514dd_sum @ TA @ TB
@ ^ [D: TA] : ( times_times @ TB @ ( A @ D ) @ C )
@ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[275]) ).
thf(61,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_152_le__cases3) ).
thf(475,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)],[61]) ).
thf(63,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_158_eq__iff) ).
thf(492,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)],[63]) ).
thf(150,axiom,
! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_18_le__refl) ).
thf(756,plain,
! [A: nat] : ( ord_less_eq @ nat @ A @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[150]) ).
thf(163,axiom,
ordered_semiring_0 @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Oordered__semiring__0) ).
thf(789,plain,
ordered_semiring_0 @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[163]) ).
thf(55,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( B != A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_211_dual__order_Ostrict__implies__not__eq) ).
thf(443,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( B != A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[55]) ).
thf(68,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ( ord_less_eq @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_196_le__numeral__extra_I4_J) ).
thf(514,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ( ord_less_eq @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[68]) ).
thf(170,axiom,
no_top @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Ono__top) ).
thf(806,plain,
no_top @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[170]) ).
thf(176,axiom,
! [A: nat > nat] :
( ( number2016821345itions @ A @ ( zero_zero @ nat ) )
= ( A
= ( ^ [B: nat] : ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_133_partitions__zero) ).
thf(821,plain,
! [A: nat > nat] :
( ( number2016821345itions @ A @ ( zero_zero @ nat ) )
= ( A
= ( ^ [B: nat] : ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[176]) ).
thf(289,axiom,
! [TA: $tType] :
( ( mult_zero @ TA )
=> ! [A: TA] :
( ( times_times @ TA @ A @ ( zero_zero @ TA ) )
= ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_68_mult__zero__right) ).
thf(1136,plain,
! [TA: $tType] :
( ( mult_zero @ TA )
=> ! [A: TA] :
( ( times_times @ TA @ A @ ( zero_zero @ TA ) )
= ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[289]) ).
thf(179,axiom,
! [A: nat] : ( ord_less_eq @ nat @ A @ ( times_times @ nat @ A @ ( times_times @ nat @ A @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_37_le__cube) ).
thf(834,plain,
! [A: nat] : ( ord_less_eq @ nat @ A @ ( times_times @ nat @ A @ ( times_times @ nat @ A @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[179]) ).
thf(102,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( A != B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_226_less__imp__not__eq) ).
thf(602,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( A != B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[102]) ).
thf(111,axiom,
! [TA: $tType,TB: $tType,TC: $tType] :
( ( comm_monoid_add @ TB )
=> ! [A: set @ TC,B: TA > TC,C: TC > TA,D: set @ TA,E: TA > TB,F: TC > TB] :
( ! [G: TC] :
( ( member @ TC @ G @ A )
=> ( ( B @ ( C @ G ) )
= G ) )
=> ( ! [G: TC] :
( ( member @ TC @ G @ A )
=> ( member @ TA @ ( C @ G ) @ D ) )
=> ( ! [G: TA] :
( ( member @ TA @ G @ D )
=> ( ( C @ ( B @ G ) )
= G ) )
=> ( ! [G: TA] :
( ( member @ TA @ G @ D )
=> ( member @ TC @ ( B @ G ) @ A ) )
=> ( ! [G: TC] :
( ( member @ TC @ G @ A )
=> ( ( E @ ( C @ G ) )
= ( F @ G ) ) )
=> ( ( groups1340683514dd_sum @ TC @ TB @ F @ A )
= ( groups1340683514dd_sum @ TA @ TB @ E @ D ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_49_sum_Oreindex__bij__witness) ).
thf(634,plain,
! [TA: $tType,TB: $tType,TC: $tType] :
( ( comm_monoid_add @ TB )
=> ! [A: set @ TC,B: TA > TC,C: TC > TA,D: set @ TA,E: TA > TB,F: TC > TB] :
( ! [G: TC] :
( ( member @ TC @ G @ A )
=> ( ( B @ ( C @ G ) )
= G ) )
=> ( ! [G: TC] :
( ( member @ TC @ G @ A )
=> ( member @ TA @ ( C @ G ) @ D ) )
=> ( ! [G: TA] :
( ( member @ TA @ G @ D )
=> ( ( C @ ( B @ G ) )
= G ) )
=> ( ! [G: TA] :
( ( member @ TA @ G @ D )
=> ( member @ TC @ ( B @ G ) @ A ) )
=> ( ! [G: TC] :
( ( member @ TC @ G @ A )
=> ( ( E @ ( C @ G ) )
= ( F @ G ) ) )
=> ( ( groups1340683514dd_sum @ TC @ TB @ F @ A )
= ( groups1340683514dd_sum @ TA @ TB @ E @ D ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[111]) ).
thf(140,axiom,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
=> ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_106_bot__nat__0_Oextremum__uniqueI) ).
thf(734,plain,
! [A: nat] :
( ( ord_less_eq @ nat @ A @ ( zero_zero @ nat ) )
=> ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[140]) ).
thf(224,axiom,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ A @ A )
= ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_76_cancel__comm__monoid__add__class_Odiff__cancel) ).
thf(956,plain,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ A @ A )
= ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[224]) ).
thf(283,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
= ( minus_minus @ nat @ ( minus_minus @ nat @ A @ C ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_53_diff__commute) ).
thf(1119,plain,
! [A: nat,B: nat,C: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
= ( minus_minus @ nat @ ( minus_minus @ nat @ A @ C ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[283]) ).
thf(116,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times @ nat @ A @ ( minus_minus @ nat @ B @ C ) )
= ( minus_minus @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_11_diff__mult__distrib2) ).
thf(660,plain,
! [A: nat,B: nat,C: nat] :
( ( times_times @ nat @ A @ ( minus_minus @ nat @ B @ C ) )
= ( minus_minus @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[116]) ).
thf(50,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_66_le__zero__eq) ).
thf(428,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[50]) ).
thf(263,axiom,
semiri1193490041visors @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors) ).
thf(1072,plain,
semiri1193490041visors @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[263]) ).
thf(58,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_132_zero__le__one) ).
thf(453,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[58]) ).
thf(6,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_234_less__asym) ).
thf(307,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[6]) ).
thf(211,axiom,
! [TA: $tType] :
( ( ordered_semiring @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_129_mult__mono) ).
thf(920,plain,
! [TA: $tType] :
( ( ordered_semiring @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[211]) ).
thf(54,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( A != B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_212_order_Ostrict__implies__not__eq) ).
thf(439,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( A != B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[54]) ).
thf(81,axiom,
! [TA: $tType,TB: $tType] :
( ( order @ TA )
=> ( order @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oorder) ).
thf(546,plain,
! [TA: $tType,TB: $tType] :
( ( order @ TA )
=> ( order @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[81]) ).
thf(213,axiom,
zero_neq_one @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Ozero__neq__one) ).
thf(925,plain,
zero_neq_one @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[213]) ).
thf(152,axiom,
! [TA: $tType] :
( ( ordered_ring @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ ( zero_zero @ TA ) )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ C @ B ) @ ( times_times @ TA @ C @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_125_mult__left__mono__neg) ).
thf(761,plain,
! [TA: $tType] :
( ( ordered_ring @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ ( zero_zero @ TA ) )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ C @ B ) @ ( times_times @ TA @ C @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[152]) ).
thf(172,axiom,
! [A: nat,B: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ B ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_28_diff__le__self) ).
thf(810,plain,
! [A: nat,B: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ B ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[172]) ).
thf(34,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
=> ( ~ ( ord_less @ TA @ A @ B )
=> ( ord_less @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_239_neqE) ).
thf(381,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
=> ( ~ ( ord_less @ TA @ A @ B )
=> ( ord_less @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[34]) ).
thf(261,axiom,
! [TA: $tType] :
( ( linord581940658strict @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) )
= ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
& ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) )
| ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
& ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_120_mult__le__0__iff) ).
thf(1067,plain,
! [TA: $tType] :
( ( linord581940658strict @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) )
= ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
& ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) )
| ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
& ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[261]) ).
thf(251,axiom,
! [TA: $tType] :
( ( ordered_semiring_0 @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ B @ A ) @ ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_115_mult__nonneg__nonpos2) ).
thf(1047,plain,
! [TA: $tType] :
( ( ordered_semiring_0 @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ B @ A ) @ ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[251]) ).
thf(257,axiom,
! [A: nat] :
( ( minus_minus @ nat @ A @ A )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_80_diff__self__eq__0) ).
thf(1057,plain,
! [A: nat] :
( ( minus_minus @ nat @ A @ A )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[257]) ).
thf(206,axiom,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( minus_minus @ TA @ A @ B )
= ( minus_minus @ TA @ C @ D ) )
=> ( ( ord_less_eq @ TA @ A @ B )
= ( ord_less_eq @ TA @ C @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_184_diff__eq__diff__less__eq) ).
thf(910,plain,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( minus_minus @ TA @ A @ B )
= ( minus_minus @ TA @ C @ D ) )
=> ( ( ord_less_eq @ TA @ A @ B )
= ( ord_less_eq @ TA @ C @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[206]) ).
thf(126,axiom,
! [TA: $tType] :
( ( ring_11004092258visors @ TA )
=> ! [A: TA,B: TA] :
( ( ( times_times @ TA @ A @ B )
= B )
= ( ( B
= ( zero_zero @ TA ) )
| ( A
= ( one_one @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_87_mult__cancel__right2) ).
thf(693,plain,
! [TA: $tType] :
( ( ring_11004092258visors @ TA )
=> ! [A: TA,B: TA] :
( ( ( times_times @ TA @ A @ B )
= B )
= ( ( B
= ( zero_zero @ TA ) )
| ( A
= ( one_one @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[126]) ).
thf(174,axiom,
! [A: nat,B: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ A @ B ) )
= ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_17_nat__1__eq__mult__iff) ).
thf(815,plain,
! [A: nat,B: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ A @ B ) )
= ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[174]) ).
thf(209,axiom,
! [TA: $tType] :
( ( ring_11004092258visors @ TA )
=> ! [A: TA,B: TA] :
( ( A
= ( times_times @ TA @ A @ B ) )
= ( ( A
= ( zero_zero @ TA ) )
| ( B
= ( one_one @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_90_mult__cancel__left1) ).
thf(916,plain,
! [TA: $tType] :
( ( ring_11004092258visors @ TA )
=> ! [A: TA,B: TA] :
( ( A
= ( times_times @ TA @ A @ B ) )
= ( ( A
= ( zero_zero @ TA ) )
| ( B
= ( one_one @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[209]) ).
thf(196,axiom,
! [A: nat] :
( ( times_times @ nat @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_82_mult__0__right) ).
thf(879,plain,
! [A: nat] :
( ( times_times @ nat @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[196]) ).
thf(200,axiom,
! [A: nat > nat,B: nat] :
( ( number2016821345itions @ A @ B )
=> ~ ( ! [C: nat] :
( ( ( A @ C )
!= ( zero_zero @ nat ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ C )
& ( ord_less_eq @ nat @ C @ B ) ) )
=> ( ( groups1340683514dd_sum @ nat @ nat
@ ^ [C: nat] : ( times_times @ nat @ ( A @ C ) @ C )
@ ( set_ord_atMost @ nat @ B ) )
!= B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_65_partitionsE) ).
thf(889,plain,
! [A: nat > nat,B: nat] :
( ( number2016821345itions @ A @ B )
=> ~ ( ! [C: nat] :
( ( ( A @ C )
!= ( zero_zero @ nat ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ C )
& ( ord_less_eq @ nat @ C @ B ) ) )
=> ( ( groups1340683514dd_sum @ nat @ nat
@ ^ [C: nat] : ( times_times @ nat @ ( A @ C ) @ C )
@ ( set_ord_atMost @ nat @ B ) )
!= B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[200]) ).
thf(267,axiom,
! [TA: $tType] :
( ( comm_s1003936772cancel @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( times_times @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
= ( minus_minus @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_187_left__diff__distrib_H) ).
thf(1079,plain,
! [TA: $tType] :
( ( comm_s1003936772cancel @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( times_times @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
= ( minus_minus @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[267]) ).
thf(122,axiom,
! [A: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ A )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_39_nat__mult__1) ).
thf(682,plain,
! [A: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ A )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[122]) ).
thf(83,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_233_less__asym_H) ).
thf(551,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[83]) ).
thf(277,axiom,
semiring_0 @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Osemiring__0) ).
thf(1106,plain,
semiring_0 @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[277]) ).
thf(93,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less_eq @ TA @ ( one_one @ TA ) @ ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_131_not__one__le__zero) ).
thf(578,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less_eq @ TA @ ( one_one @ TA ) @ ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[93]) ).
thf(194,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ( ord_less_eq @ TA @ B @ ( one_one @ TA ) )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ B @ A ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_137_mult__left__le__one__le) ).
thf(874,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ( ord_less_eq @ TA @ B @ ( one_one @ TA ) )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ B @ A ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[194]) ).
thf(242,axiom,
! [A: nat > nat,B: nat,C: nat,D: nat] :
( ( number2016821345itions @ A @ B )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( A @ C ) )
=> ( ( ( fun_upd @ nat @ nat @ A @ C @ ( minus_minus @ nat @ ( A @ C ) @ ( one_one @ nat ) ) @ D )
!= ( zero_zero @ nat ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ D )
& ( ord_less_eq @ nat @ D @ ( minus_minus @ nat @ B @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_194_partitions__remove1__bounds) ).
thf(1017,plain,
! [A: nat > nat,B: nat,C: nat,D: nat] :
( ( number2016821345itions @ A @ B )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( A @ C ) )
=> ( ( ( fun_upd @ nat @ nat @ A @ C @ ( minus_minus @ nat @ ( A @ C ) @ ( one_one @ nat ) ) @ D )
!= ( zero_zero @ nat ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ D )
& ( ord_less_eq @ nat @ D @ ( minus_minus @ nat @ B @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[242]) ).
thf(106,axiom,
! [TA: $tType] :
( ( no_bot @ TA )
=> ! [A: TA] :
? [B: TA] : ( ord_less @ TA @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_241_lt__ex) ).
thf(615,plain,
! [TA: $tType] :
( ( no_bot @ TA )
=> ! [A: TA] :
? [B: TA] : ( ord_less @ TA @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[106]) ).
thf(280,axiom,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_109_less__eq__nat_Osimps_I1_J) ).
thf(1111,plain,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[280]) ).
thf(229,axiom,
( ( groups1340683514dd_sum @ nat @ nat
@ ^ [A: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ A ) @ A )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ n @ k ) ) )
= ( minus_minus @ nat
@ ( groups1340683514dd_sum @ nat @ nat
@ ^ [A: nat] : ( times_times @ nat @ ( p @ A ) @ A )
@ ( set_ord_atMost @ nat @ n ) )
@ k ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_5_calculation) ).
thf(980,plain,
( ( groups1340683514dd_sum @ nat @ nat
@ ^ [A: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ A ) @ A )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ n @ k ) ) )
= ( minus_minus @ nat
@ ( groups1340683514dd_sum @ nat @ nat
@ ^ [A: nat] : ( times_times @ nat @ ( p @ A ) @ A )
@ ( set_ord_atMost @ nat @ n ) )
@ k ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[229]) ).
thf(239,axiom,
! [TA: $tType] :
( ( monoid_mult @ TA )
=> ( ( ^ [A: TA] : A )
= ( times_times @ TA @ ( one_one @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_62_lambda__one) ).
thf(1006,plain,
! [TA: $tType] :
( ( monoid_mult @ TA )
=> ( ( ^ [A: TA] : A )
= ( times_times @ TA @ ( one_one @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[239]) ).
thf(36,axiom,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA] :
( ( ( zero_zero @ TA )
= A )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_94_zero__reorient) ).
thf(388,plain,
! [TA: $tType] :
( ( zero @ TA )
=> ! [A: TA] :
( ( ( zero_zero @ TA )
= A )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[36]) ).
thf(25,axiom,
! [TA: $tType] :
( ( wellorder @ TA )
=> ! [A: TA > $o,B: TA] :
( ! [C: TA] :
( ! [D: TA] :
( ( ord_less @ TA @ D @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_224_less__induct) ).
thf(356,plain,
! [TA: $tType] :
( ( wellorder @ TA )
=> ! [A: TA > $o,B: TA] :
( ! [C: TA] :
( ! [D: TA] :
( ( ord_less @ TA @ D @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[25]) ).
thf(15,axiom,
! [TA: $tType,TB: $tType,A: TA,B: TA,C: TA > TB,D: TB] :
( ( A != B )
=> ( ( fun_upd @ TA @ TB @ C @ B @ D @ A )
= ( C @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_176_fun__upd__other) ).
thf(328,plain,
! [TA: $tType,TB: $tType,A: TA,B: TA,C: TA > TB,D: TB] :
( ( A != B )
=> ( ( fun_upd @ TA @ TB @ C @ B @ D @ A )
= ( C @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[15]) ).
thf(268,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
& ( ord_less @ nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_204_nat__mult__less__cancel__disj) ).
thf(1082,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
& ( ord_less @ nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[268]) ).
thf(20,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_151_antisym__conv) ).
thf(339,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)],[20]) ).
thf(149,axiom,
! [A: nat] :
( ( minus_minus @ nat @ A @ ( zero_zero @ nat ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_111_minus__nat_Odiff__0) ).
thf(753,plain,
! [A: nat] :
( ( minus_minus @ nat @ A @ ( zero_zero @ nat ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[149]) ).
thf(78,axiom,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A = B )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_229_ord__eq__less__trans) ).
thf(537,plain,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A = B )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[78]) ).
thf(119,axiom,
! [A: nat] : ( ord_less_eq @ nat @ A @ ( times_times @ nat @ A @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_36_le__square) ).
thf(669,plain,
! [A: nat] : ( ord_less_eq @ nat @ A @ ( times_times @ nat @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[119]) ).
thf(203,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( times_times @ nat @ A @ B )
= ( times_times @ nat @ A @ C ) )
= ( ( A
= ( zero_zero @ nat ) )
| ( B = C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_195_nat__mult__eq__cancel__disj) ).
thf(901,plain,
! [A: nat,B: nat,C: nat] :
( ( ( times_times @ nat @ A @ B )
= ( times_times @ nat @ A @ C ) )
= ( ( A
= ( zero_zero @ nat ) )
| ( B = C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[203]) ).
thf(62,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_162_order__subst1) ).
thf(489,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)],[62]) ).
thf(98,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_218_order_Ostrict__trans) ).
thf(590,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[98]) ).
thf(296,axiom,
! [TA: $tType,TB: $tType] :
( ( semiring_0 @ TB )
=> ! [A: TB,B: TA > TB,C: set @ TA] :
( ( times_times @ TB @ A @ ( groups1340683514dd_sum @ TA @ TB @ B @ C ) )
= ( groups1340683514dd_sum @ TA @ TB
@ ^ [D: TA] : ( times_times @ TB @ A @ ( B @ D ) )
@ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_8_sum__distrib__left) ).
thf(1155,plain,
! [TA: $tType,TB: $tType] :
( ( semiring_0 @ TB )
=> ! [A: TB,B: TA > TB,C: set @ TA] :
( ( times_times @ TB @ A @ ( groups1340683514dd_sum @ TA @ TB @ B @ C ) )
= ( groups1340683514dd_sum @ TA @ TB
@ ^ [D: TA] : ( times_times @ TB @ A @ ( B @ D ) )
@ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[296]) ).
thf(38,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ C @ A )
=> ( ord_less @ TA @ C @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_214_dual__order_Ostrict__trans) ).
thf(393,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ C @ A )
=> ( ord_less @ TA @ C @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[38]) ).
thf(236,axiom,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( minus_minus @ TA @ A @ B )
= ( minus_minus @ TA @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_171_diff__eq__diff__eq) ).
thf(997,plain,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ( minus_minus @ TA @ A @ B )
= ( minus_minus @ TA @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[236]) ).
thf(231,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( p @ k ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_93_gr0) ).
thf(985,plain,
ord_less @ nat @ ( zero_zero @ nat ) @ ( p @ k ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[231]) ).
thf(142,axiom,
ordered_semiring @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Oordered__semiring) ).
thf(738,plain,
ordered_semiring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[142]) ).
thf(136,axiom,
! [A: nat,B: nat] :
( ( ( times_times @ nat @ A @ B )
= ( one_one @ nat ) )
= ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_16_nat__mult__eq__1__iff) ).
thf(721,plain,
! [A: nat,B: nat] :
( ( ( times_times @ nat @ A @ B )
= ( one_one @ nat ) )
= ( ( A
= ( one_one @ nat ) )
& ( B
= ( one_one @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[136]) ).
thf(90,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(569,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)],[90]) ).
thf(159,axiom,
ordere1490568538miring @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Oordered__comm__semiring) ).
thf(779,plain,
ordere1490568538miring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[159]) ).
thf(287,axiom,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_77_bot__nat__0_Oextremum) ).
thf(1131,plain,
! [A: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[287]) ).
thf(113,axiom,
! [TA: $tType] :
( ( ( comm_ring @ TA )
& ( dvd @ TA ) )
=> ! [A: TA > $o,B: TA,C: TA > $o] :
( ! [D: TA,E: TA] :
( ( A @ D )
= ( A @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) )
=> ( ! [D: TA,E: TA] :
( ( C @ D )
= ( C @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) )
=> ! [D: TA,E: TA] :
( ( ( A @ D )
& ( C @ D ) )
= ( ( A @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) )
& ( C @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_55_inf__period_I1_J) ).
thf(654,plain,
! [TA: $tType] :
( ( ( comm_ring @ TA )
& ( dvd @ TA ) )
=> ! [A: TA > $o,B: TA,C: TA > $o] :
( ! [D: TA,E: TA] :
( ( A @ D )
= ( A @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) )
=> ( ! [D: TA,E: TA] :
( ( C @ D )
= ( C @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) )
=> ! [D: TA,E: TA] :
( ( ( A @ D )
& ( C @ D ) )
= ( ( A @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) )
& ( C @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[113]) ).
thf(258,axiom,
! [TA: $tType] :
( ( ab_semigroup_mult @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( times_times @ TA @ ( times_times @ TA @ A @ B ) @ C )
= ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_167_ab__semigroup__mult__class_Omult__ac_I1_J) ).
thf(1060,plain,
! [TA: $tType] :
( ( ab_semigroup_mult @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( times_times @ TA @ ( times_times @ TA @ A @ B ) @ C )
= ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[258]) ).
thf(253,axiom,
! [TA: $tType] :
( ( ordered_semiring @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ C @ A ) @ ( times_times @ TA @ C @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_123_mult__left__mono) ).
thf(1050,plain,
! [TA: $tType] :
( ( ordered_semiring @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ C @ A ) @ ( times_times @ TA @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[253]) ).
thf(218,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( minus_minus @ nat @ B @ ( minus_minus @ nat @ B @ A ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_15_diff__diff__cancel) ).
thf(938,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( minus_minus @ nat @ B @ ( minus_minus @ nat @ B @ A ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[218]) ).
thf(27,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_144_linorder__wlog) ).
thf(360,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)],[27]) ).
thf(8,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_146_order__trans) ).
thf(311,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)],[8]) ).
thf(79,axiom,
! [TA: $tType] : ( order @ ( set @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oorder_6) ).
thf(541,plain,
! [TA: $tType] : ( order @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[79]) ).
thf(114,axiom,
ordere216010020id_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Oordered__comm__monoid__add) ).
thf(657,plain,
ordere216010020id_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[114]) ).
thf(256,axiom,
! [TA: $tType,TB: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: set @ TB,B: TB > TA] :
( ! [C: TB] :
( ( member @ TB @ C @ A )
=> ( ord_less_eq @ TA @ ( B @ C ) @ ( zero_zero @ TA ) ) )
=> ( ord_less_eq @ TA @ ( groups1340683514dd_sum @ TB @ TA @ B @ A ) @ ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_138_sum__nonpos) ).
thf(1054,plain,
! [TA: $tType,TB: $tType] :
( ( ordere216010020id_add @ TA )
=> ! [A: set @ TB,B: TB > TA] :
( ! [C: TB] :
( ( member @ TB @ C @ A )
=> ( ord_less_eq @ TA @ ( B @ C ) @ ( zero_zero @ TA ) ) )
=> ( ord_less_eq @ TA @ ( groups1340683514dd_sum @ TB @ TA @ B @ A ) @ ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[256]) ).
thf(121,axiom,
! [TA: $tType,TB: $tType,TC: $tType] :
( ( comm_monoid_add @ TB )
=> ! [A: set @ TA,B: TA > TC,C: set @ TC,D: TC > TA,E: TA > TB,F: TC > TB] :
( ! [G: TA] :
( ( member @ TA @ G @ A )
=> ( ( member @ TC @ ( B @ G ) @ C )
& ( ( D @ ( B @ G ) )
= G ) ) )
=> ( ! [G: TC] :
( ( member @ TC @ G @ C )
=> ( ( member @ TA @ ( D @ G ) @ A )
& ( ( B @ ( D @ G ) )
= G )
& ( ( E @ ( D @ G ) )
= ( F @ G ) ) ) )
=> ( ( groups1340683514dd_sum @ TC @ TB @ F @ C )
= ( groups1340683514dd_sum @ TA @ TB @ E @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_50_sum_Oeq__general__inverses) ).
thf(674,plain,
! [TA: $tType,TB: $tType,TC: $tType] :
( ( comm_monoid_add @ TB )
=> ! [A: set @ TA,B: TA > TC,C: set @ TC,D: TC > TA,E: TA > TB,F: TC > TB] :
( ! [G: TA] :
( ( member @ TA @ G @ A )
=> ( ( member @ TC @ ( B @ G ) @ C )
& ( ( D @ ( B @ G ) )
= G ) ) )
=> ( ! [G: TC] :
( ( member @ TC @ G @ C )
=> ( ( member @ TA @ ( D @ G ) @ A )
& ( ( B @ ( D @ G ) )
= G )
& ( ( E @ ( D @ G ) )
= ( F @ G ) ) ) )
=> ( ( groups1340683514dd_sum @ TC @ TB @ F @ C )
= ( groups1340683514dd_sum @ TA @ TB @ E @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[121]) ).
thf(244,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ A @ C )
=> ( ( ( minus_minus @ nat @ B @ A )
= ( minus_minus @ nat @ C @ A ) )
= ( B = C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_32_eq__diff__iff) ).
thf(1026,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ A @ C )
=> ( ( ( minus_minus @ nat @ B @ A )
= ( minus_minus @ nat @ C @ A ) )
= ( B = C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[244]) ).
thf(223,axiom,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
= ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_202_diff__gt__0__iff__gt) ).
thf(953,plain,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
= ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[223]) ).
thf(214,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ C @ D )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ A @ C ) @ ( times_times @ nat @ B @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_35_mult__le__mono) ).
thf(926,plain,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ C @ D )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ A @ C ) @ ( times_times @ nat @ B @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[214]) ).
thf(40,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_149_ord__eq__le__trans) ).
thf(398,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)],[40]) ).
thf(247,axiom,
! [TA: $tType,TB: $tType] :
( ( ab_group_add @ TB )
=> ! [A: TA > TB,B: TA > TB,C: set @ TA] :
( ( groups1340683514dd_sum @ TA @ TB
@ ^ [D: TA] : ( minus_minus @ TB @ ( A @ D ) @ ( B @ D ) )
@ C )
= ( minus_minus @ TB @ ( groups1340683514dd_sum @ TA @ TB @ A @ C ) @ ( groups1340683514dd_sum @ TA @ TB @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_6_sum__subtractf) ).
thf(1035,plain,
! [TA: $tType,TB: $tType] :
( ( ab_group_add @ TB )
=> ! [A: TA > TB,B: TA > TB,C: set @ TA] :
( ( groups1340683514dd_sum @ TA @ TB
@ ^ [D: TA] : ( minus_minus @ TB @ ( A @ D ) @ ( B @ D ) )
@ C )
= ( minus_minus @ TB @ ( groups1340683514dd_sum @ TA @ TB @ A @ C ) @ ( groups1340683514dd_sum @ TA @ TB @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[247]) ).
thf(156,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ ( one_one @ TA ) )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ B @ A ) @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_134_mult__left__le) ).
thf(773,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ ( one_one @ TA ) )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ B @ A ) @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[156]) ).
thf(166,axiom,
! [TA: $tType] :
( ( semiri1923998003cancel @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A
!= ( zero_zero @ TA ) )
=> ( ( ( times_times @ TA @ A @ B )
= ( times_times @ TA @ A @ C ) )
= ( B = C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_99_mult__left__cancel) ).
thf(794,plain,
! [TA: $tType] :
( ( semiri1923998003cancel @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A
!= ( zero_zero @ TA ) )
=> ( ( ( times_times @ TA @ A @ B )
= ( times_times @ TA @ A @ C ) )
= ( B = C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[166]) ).
thf(1,conjecture,
( ( minus_minus @ nat
@ ( groups1340683514dd_sum @ nat @ nat
@ ^ [A: nat] : ( times_times @ nat @ ( p @ A ) @ A )
@ ( set_ord_atMost @ nat @ n ) )
@ k )
= ( minus_minus @ nat @ n @ k ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',conj_0) ).
thf(2,negated_conjecture,
( ( minus_minus @ nat
@ ( groups1340683514dd_sum @ nat @ nat
@ ^ [A: nat] : ( times_times @ nat @ ( p @ A ) @ A )
@ ( set_ord_atMost @ nat @ n ) )
@ k )
!= ( minus_minus @ nat @ n @ k ) ),
inference(neg_conjecture,[status(cth)],[1]) ).
thf(298,plain,
( ( minus_minus @ nat
@ ( groups1340683514dd_sum @ nat @ nat
@ ^ [A: nat] : ( times_times @ nat @ ( p @ A ) @ A )
@ ( set_ord_atMost @ nat @ n ) )
@ k )
!= ( minus_minus @ nat @ n @ k ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[2]) ).
thf(45,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_217_less__imp__not__less) ).
thf(412,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[45]) ).
thf(227,axiom,
! [TA: $tType] :
( ( ring @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( times_times @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
= ( minus_minus @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_185_left__diff__distrib) ).
thf(965,plain,
! [TA: $tType] :
( ( ring @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( times_times @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
= ( minus_minus @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[227]) ).
thf(293,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ A @ B ) )
= ( ord_less @ nat @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_203_zero__less__diff) ).
thf(1146,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ A @ B ) )
= ( ord_less @ nat @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[293]) ).
thf(276,axiom,
! [TA: $tType] :
( ( neg_numeral @ TA )
=> ( ( minus_minus @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) )
= ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_191_diff__numeral__special_I9_J) ).
thf(1103,plain,
! [TA: $tType] :
( ( neg_numeral @ TA )
=> ( ( minus_minus @ TA @ ( one_one @ TA ) @ ( one_one @ TA ) )
= ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[276]) ).
thf(177,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ C @ A ) @ ( times_times @ nat @ C @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_33_mult__le__mono2) ).
thf(824,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ C @ A ) @ ( times_times @ nat @ C @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[177]) ).
thf(187,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ B ) @ ( minus_minus @ nat @ C @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_26_diff__le__mono2) ).
thf(856,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ B ) @ ( minus_minus @ nat @ C @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[187]) ).
thf(32,axiom,
! [TA: $tType] :
( ( zero_neq_one @ TA )
=> ( ( zero_zero @ TA )
!= ( one_one @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_102_zero__neq__one) ).
thf(375,plain,
! [TA: $tType] :
( ( zero_neq_one @ TA )
=> ( ( zero_zero @ TA )
!= ( one_one @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[32]) ).
thf(96,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less @ TA @ A @ B )
=> ( ( A != B )
=> ( ord_less @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_220_linorder__cases) ).
thf(584,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less @ TA @ A @ B )
=> ( ( A != B )
=> ( ord_less @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[96]) ).
thf(204,axiom,
! [TA: $tType] :
( ( monoid_mult @ TA )
=> ! [A: TA] :
( ( times_times @ TA @ ( one_one @ TA ) @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_57_mult_Oleft__neutral) ).
thf(904,plain,
! [TA: $tType] :
( ( monoid_mult @ TA )
=> ! [A: TA] :
( ( times_times @ TA @ ( one_one @ TA ) @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[204]) ).
thf(48,axiom,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( ord_less @ TB @ A @ ( B @ C ) )
=> ( ( ord_less @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less @ TA @ E @ F )
=> ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_243_order__less__subst1) ).
thf(423,plain,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( ord_less @ TB @ A @ ( B @ C ) )
=> ( ( ord_less @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less @ TA @ E @ F )
=> ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[48]) ).
thf(41,axiom,
! [TA: $tType] : ( preorder @ ( set @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Opreorder_5) ).
thf(402,plain,
! [TA: $tType] : ( preorder @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[41]) ).
thf(155,axiom,
comm_monoid_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__add) ).
thf(772,plain,
comm_monoid_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[155]) ).
thf(132,axiom,
minus @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ominus_4) ).
thf(711,plain,
minus @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[132]) ).
thf(285,axiom,
! [A: nat > nat,B: nat] :
( ! [C: nat] :
( ( ( A @ C )
!= ( zero_zero @ nat ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ C )
& ( ord_less_eq @ nat @ C @ B ) ) )
=> ( ( ( groups1340683514dd_sum @ nat @ nat
@ ^ [C: nat] : ( times_times @ nat @ ( A @ C ) @ C )
@ ( set_ord_atMost @ nat @ B ) )
= B )
=> ( number2016821345itions @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_64_partitionsI) ).
thf(1124,plain,
! [A: nat > nat,B: nat] :
( ! [C: nat] :
( ( ( A @ C )
!= ( zero_zero @ nat ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ C )
& ( ord_less_eq @ nat @ C @ B ) ) )
=> ( ( ( groups1340683514dd_sum @ nat @ nat
@ ^ [C: nat] : ( times_times @ nat @ ( A @ C ) @ C )
@ ( set_ord_atMost @ nat @ B ) )
= B )
=> ( number2016821345itions @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[285]) ).
thf(91,axiom,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( B = C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_228_ord__less__eq__trans) ).
thf(572,plain,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less @ TA @ A @ B )
=> ( ( B = C )
=> ( ord_less @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[91]) ).
thf(9,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
| ( A = B )
| ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_231_less__linear) ).
thf(313,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
| ( A = B )
| ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[9]) ).
thf(105,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( A != B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_235_less__imp__neq) ).
thf(611,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ( A != B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[105]) ).
thf(278,axiom,
! [TA: $tType] :
( ( ordered_ring @ TA )
=> ! [A: TA,B: TA] :
( ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
& ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) )
| ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
& ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_126_split__mult__pos__le) ).
thf(1107,plain,
! [TA: $tType] :
( ( ordered_ring @ TA )
=> ! [A: TA,B: TA] :
( ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
& ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) )
| ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
& ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) ) )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[278]) ).
thf(18,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_153_order_Otrans) ).
thf(335,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)],[18]) ).
thf(271,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ B @ A )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_21_le__antisym) ).
thf(1090,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ B @ A )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[271]) ).
thf(210,axiom,
dvd @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Odvd) ).
thf(919,plain,
dvd @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[210]) ).
thf(37,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_165_le__funE) ).
thf(391,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)],[37]) ).
thf(290,axiom,
! [A: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ A )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_112_mult__0) ).
thf(1139,plain,
! [A: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ A )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[290]) ).
thf(43,axiom,
! [TA: $tType,TB: $tType,A: TB,B: TB,C: TB > TA,D: TA,E: TA] :
( ( A != B )
=> ( ( fun_upd @ TB @ TA @ ( fun_upd @ TB @ TA @ C @ A @ D ) @ B @ E )
= ( fun_upd @ TB @ TA @ ( fun_upd @ TB @ TA @ C @ B @ E ) @ A @ D ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_175_fun__upd__twist) ).
thf(406,plain,
! [TA: $tType,TB: $tType,A: TB,B: TB,C: TB > TA,D: TA,E: TA] :
( ( A != B )
=> ( ( fun_upd @ TB @ TA @ ( fun_upd @ TB @ TA @ C @ A @ D ) @ B @ E )
= ( fun_upd @ TB @ TA @ ( fun_upd @ TB @ TA @ C @ B @ E ) @ A @ D ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[43]) ).
thf(141,axiom,
cancel146912293up_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add) ).
thf(737,plain,
cancel146912293up_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[141]) ).
thf(101,axiom,
! [TA: $tType] :
( ( wellorder @ TA )
=> ( ( '?' @ TA )
= ( ^ [A: TA > $o] :
? [B: TA] :
( ( A @ B )
& ! [C: TA] :
( ( ord_less @ TA @ C @ B )
=> ~ ( A @ C ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_216_exists__least__iff) ).
thf(599,plain,
! [TA: $tType] :
( ( wellorder @ TA )
=> ( ( '?' @ TA )
= ( ^ [A: TA > $o] :
? [B: TA] :
( ( A @ B )
& ! [C: TA] :
( ( ord_less @ TA @ C @ B )
=> ~ ( A @ C ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[101]) ).
thf(233,axiom,
! [TA: $tType] :
( ( comm_monoid_mult @ TA )
=> ! [A: TA] :
( ( times_times @ TA @ ( one_one @ TA ) @ A )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_189_comm__monoid__mult__class_Omult__1) ).
thf(990,plain,
! [TA: $tType] :
( ( comm_monoid_mult @ TA )
=> ! [A: TA] :
( ( times_times @ TA @ ( one_one @ TA ) @ A )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[233]) ).
thf(235,axiom,
comm_monoid_mult @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__mult) ).
thf(996,plain,
comm_monoid_mult @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[235]) ).
thf(4,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_164_le__funI) ).
thf(303,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)],[4]) ).
thf(284,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
| ( ord_less_eq @ nat @ B @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_22_nat__le__linear) ).
thf(1122,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
| ( ord_less_eq @ nat @ B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[284]) ).
thf(110,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
= ( minus_minus @ nat @ ( times_times @ nat @ A @ C ) @ ( times_times @ nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_10_diff__mult__distrib) ).
thf(631,plain,
! [A: nat,B: nat,C: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ A @ B ) @ C )
= ( minus_minus @ nat @ ( times_times @ nat @ A @ C ) @ ( times_times @ nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[110]) ).
thf(151,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ C @ B ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
& ( ord_less @ nat @ A @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_205_mult__less__cancel2) ).
thf(758,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ C @ B ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ B )
& ( ord_less @ nat @ A @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[151]) ).
thf(29,axiom,
! [TA: $tType] :
( ( one @ TA )
=> ! [A: TA] :
( ( ( one_one @ TA )
= A )
= ( A
= ( one_one @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_173_one__reorient) ).
thf(367,plain,
! [TA: $tType] :
( ( one @ TA )
=> ! [A: TA] :
( ( ( one_one @ TA )
= A )
= ( A
= ( one_one @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[29]) ).
thf(294,axiom,
! [TA: $tType] :
( ( semiri1923998003cancel @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A
!= ( zero_zero @ TA ) )
=> ( ( ( times_times @ TA @ B @ A )
= ( times_times @ TA @ C @ A ) )
= ( B = C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_100_mult__right__cancel) ).
thf(1149,plain,
! [TA: $tType] :
( ( semiri1923998003cancel @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( A
!= ( zero_zero @ TA ) )
=> ( ( ( times_times @ TA @ B @ A )
= ( times_times @ TA @ C @ A ) )
= ( B = C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[294]) ).
thf(220,axiom,
! [TA: $tType] :
( ( ordered_semiring_0 @ TA )
=> ! [A: TA,B: TA] :
( ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
& ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) )
| ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
& ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_119_split__mult__neg__le) ).
thf(943,plain,
! [TA: $tType] :
( ( ordered_semiring_0 @ TA )
=> ! [A: TA,B: TA] :
( ( ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
& ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) ) )
| ( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
& ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B ) ) )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[220]) ).
thf(216,axiom,
! [TA: $tType,TB: $tType,TC: $tType] :
( ( semiring_0 @ TC )
=> ! [A: TA > TC,B: set @ TA,C: TB > TC,D: set @ TB] :
( ( times_times @ TC @ ( groups1340683514dd_sum @ TA @ TC @ A @ B ) @ ( groups1340683514dd_sum @ TB @ TC @ C @ D ) )
= ( groups1340683514dd_sum @ TA @ TC
@ ^ [E: TA] :
( groups1340683514dd_sum @ TB @ TC
@ ^ [F: TB] : ( times_times @ TC @ ( A @ E ) @ ( C @ F ) )
@ D )
@ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_7_sum__product) ).
thf(932,plain,
! [TA: $tType,TB: $tType,TC: $tType] :
( ( semiring_0 @ TC )
=> ! [A: TA > TC,B: set @ TA,C: TB > TC,D: set @ TB] :
( ( times_times @ TC @ ( groups1340683514dd_sum @ TA @ TC @ A @ B ) @ ( groups1340683514dd_sum @ TB @ TC @ C @ D ) )
= ( groups1340683514dd_sum @ TA @ TC
@ ^ [E: TA] :
( groups1340683514dd_sum @ TB @ TC
@ ^ [F: TB] : ( times_times @ TC @ ( A @ E ) @ ( C @ F ) )
@ D )
@ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[216]) ).
thf(184,axiom,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ( ( ord_less_eq @ TA )
= ( ^ [A: TA,B: TA] : ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_130_le__iff__diff__le__0) ).
thf(848,plain,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ( ( ord_less_eq @ TA )
= ( ^ [A: TA,B: TA] : ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[184]) ).
thf(128,axiom,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ D ) @ ( minus_minus @ TA @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_181_diff__mono) ).
thf(700,plain,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ D ) @ ( minus_minus @ TA @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[128]) ).
thf(225,axiom,
! [A: nat,B: nat] :
( ( ( minus_minus @ nat @ A @ B )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_92_diff__is__0__eq) ).
thf(959,plain,
! [A: nat,B: nat] :
( ( ( minus_minus @ nat @ A @ B )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[225]) ).
thf(250,axiom,
! [TA: $tType,TB: $tType] :
( ( minus @ TB )
=> ( ( minus_minus @ ( TA > TB ) )
= ( ^ [A: TA > TB,B: TA > TB,C: TA] : ( minus_minus @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_3_minus__apply) ).
thf(1044,plain,
! [TA: $tType,TB: $tType] :
( ( minus @ TB )
=> ( ( minus_minus @ ( TA > TB ) )
= ( ^ [A: TA > TB,B: TA > TB,C: TA] : ( minus_minus @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[250]) ).
thf(160,axiom,
! [TA: $tType,TB: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TB > TA,B: set @ TB] :
( ( ( groups1340683514dd_sum @ TB @ TA @ A @ B )
!= ( zero_zero @ TA ) )
=> ~ ! [C: TB] :
( ( member @ TB @ C @ B )
=> ( ( A @ C )
= ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_105_sum_Onot__neutral__contains__not__neutral) ).
thf(780,plain,
! [TA: $tType,TB: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: TB > TA,B: set @ TB] :
( ( ( groups1340683514dd_sum @ TB @ TA @ A @ B )
!= ( zero_zero @ TA ) )
=> ~ ! [C: TB] :
( ( member @ TB @ C @ B )
=> ( ( A @ C )
= ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[160]) ).
thf(104,axiom,
! [TA: $tType,TB: $tType,A: TB > TA,B: TB,C: TA] :
( ( fun_upd @ TB @ TA @ A @ B @ C @ B )
= C ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_177_fun__upd__same) ).
thf(608,plain,
! [TA: $tType,TB: $tType,A: TB > TA,B: TB,C: TA] :
( ( fun_upd @ TB @ TA @ A @ B @ C @ B )
= C ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[104]) ).
thf(232,axiom,
! [A: nat,B: nat] :
( ( A = B )
=> ( ord_less_eq @ nat @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_20_eq__imp__le) ).
thf(986,plain,
! [A: nat,B: nat] :
( ( A = B )
=> ( ord_less_eq @ nat @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[232]) ).
thf(237,axiom,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ A @ A )
= ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_72_diff__self) ).
thf(1000,plain,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ A @ A )
= ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[237]) ).
thf(185,axiom,
! [TA: $tType] :
( ( ring @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( times_times @ TA @ A @ ( minus_minus @ TA @ B @ C ) )
= ( minus_minus @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_186_right__diff__distrib) ).
thf(851,plain,
! [TA: $tType] :
( ( ring @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( times_times @ TA @ A @ ( minus_minus @ TA @ B @ C ) )
= ( minus_minus @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[185]) ).
thf(51,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA] :
~ ( ord_less @ TA @ A @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_230_less__irrefl) ).
thf(431,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA] :
~ ( ord_less @ TA @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[51]) ).
thf(212,axiom,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ A @ B ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_206_nat__0__less__mult__iff) ).
thf(922,plain,
! [A: nat,B: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ A @ B ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[212]) ).
thf(115,axiom,
! [TA: $tType] :
( ( ordered_semiring_0 @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_117_mult__nonneg__nonpos) ).
thf(658,plain,
! [TA: $tType] :
( ( ordered_semiring_0 @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[115]) ).
thf(123,axiom,
! [TA: $tType,TB: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: set @ TB] :
( ( groups1340683514dd_sum @ TB @ TA
@ ^ [B: TB] : ( zero_zero @ TA )
@ A )
= ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_85_sum_Oneutral__const) ).
thf(685,plain,
! [TA: $tType,TB: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: set @ TB] :
( ( groups1340683514dd_sum @ TB @ TA
@ ^ [B: TB] : ( zero_zero @ TA )
@ A )
= ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[123]) ).
thf(71,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_166_le__funD) ).
thf(521,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)],[71]) ).
thf(154,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ A @ C )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
= ( ord_less_eq @ nat @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_31_le__diff__iff) ).
thf(768,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ A @ C )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
= ( ord_less_eq @ nat @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[154]) ).
thf(178,axiom,
! [TA: $tType] :
( ( mult_zero @ TA )
=> ! [A: TA,B: TA] :
( ( ( times_times @ TA @ A @ B )
!= ( zero_zero @ TA ) )
=> ( ( A
!= ( zero_zero @ TA ) )
& ( B
!= ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_96_mult__not__zero) ).
thf(827,plain,
! [TA: $tType] :
( ( mult_zero @ TA )
=> ! [A: TA,B: TA] :
( ( ( times_times @ TA @ A @ B )
!= ( zero_zero @ TA ) )
=> ( ( A
!= ( zero_zero @ TA ) )
& ( B
!= ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[178]) ).
thf(260,axiom,
comm_s1003936772cancel @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel) ).
thf(1066,plain,
comm_s1003936772cancel @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[260]) ).
thf(192,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_199_bot__nat__0_Onot__eq__extremum) ).
thf(868,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[192]) ).
thf(59,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA > TA > $o,B: TA,C: TA] :
( ! [D: TA,E: TA] :
( ( ord_less @ TA @ D @ E )
=> ( A @ D @ E ) )
=> ( ! [D: TA] : ( A @ D @ D )
=> ( ! [D: TA,E: TA] :
( ( A @ E @ D )
=> ( A @ D @ E ) )
=> ( A @ B @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_215_linorder__less__wlog) ).
thf(455,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA > TA > $o,B: TA,C: TA] :
( ! [D: TA,E: TA] :
( ( ord_less @ TA @ D @ E )
=> ( A @ D @ E ) )
=> ( ! [D: TA] : ( A @ D @ D )
=> ( ! [D: TA,E: TA] :
( ( A @ E @ D )
=> ( A @ D @ E ) )
=> ( A @ B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[59]) ).
thf(198,axiom,
! [TA: $tType] :
( ( comm_s1003936772cancel @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( times_times @ TA @ A @ ( minus_minus @ TA @ B @ C ) )
= ( minus_minus @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_188_right__diff__distrib_H) ).
thf(883,plain,
! [TA: $tType] :
( ( comm_s1003936772cancel @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( times_times @ TA @ A @ ( minus_minus @ TA @ B @ C ) )
= ( minus_minus @ TA @ ( times_times @ TA @ A @ B ) @ ( times_times @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[198]) ).
thf(281,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ C ) @ ( minus_minus @ nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_29_diff__le__mono) ).
thf(1113,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ A @ C ) @ ( minus_minus @ nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[281]) ).
thf(88,axiom,
! [TA: $tType] : ( minus @ ( set @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Groups_Ominus_8) ).
thf(563,plain,
! [TA: $tType] : ( minus @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[88]) ).
thf(265,axiom,
! [TA: $tType] :
( ( ordered_semiring_0 @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_118_mult__nonneg__nonneg) ).
thf(1076,plain,
! [TA: $tType] :
( ( ordered_semiring_0 @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[265]) ).
thf(49,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_154_le__cases) ).
thf(426,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)],[49]) ).
thf(75,axiom,
linorder @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Olinorder_10) ).
thf(531,plain,
linorder @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[75]) ).
thf(173,axiom,
! [TA: $tType] :
( ( ring_11004092258visors @ TA )
=> ! [A: TA,B: TA] :
( ( A
= ( times_times @ TA @ B @ A ) )
= ( ( A
= ( zero_zero @ TA ) )
| ( B
= ( one_one @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_88_mult__cancel__right1) ).
thf(812,plain,
! [TA: $tType] :
( ( ring_11004092258visors @ TA )
=> ! [A: TA,B: TA] :
( ( A
= ( times_times @ TA @ B @ A ) )
= ( ( A
= ( zero_zero @ TA ) )
| ( B
= ( one_one @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[173]) ).
thf(21,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_246_less__numeral__extra_I3_J) ).
thf(342,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[21]) ).
thf(292,axiom,
order @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Oorder_2) ).
thf(1145,plain,
order @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[292]) ).
thf(84,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_156_linear) ).
thf(553,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)],[84]) ).
thf(228,axiom,
! [TA: $tType,TB: $tType,TC: $tType] :
( ( comm_monoid_add @ TB )
=> ! [A: set @ TA,B: set @ TC,C: TC > TA,D: TA > TB,E: TC > TB] :
( ! [F: TA] :
( ( member @ TA @ F @ A )
=> ? [G: TC] :
( ( member @ TC @ G @ B )
& ( ( C @ G )
= F )
& ! [H: TC] :
( ( ( member @ TC @ H @ B )
& ( ( C @ H )
= F ) )
=> ( H = G ) ) ) )
=> ( ! [F: TC] :
( ( member @ TC @ F @ B )
=> ( ( member @ TA @ ( C @ F ) @ A )
& ( ( D @ ( C @ F ) )
= ( E @ F ) ) ) )
=> ( ( groups1340683514dd_sum @ TC @ TB @ E @ B )
= ( groups1340683514dd_sum @ TA @ TB @ D @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_51_sum_Oeq__general) ).
thf(968,plain,
! [TA: $tType,TB: $tType,TC: $tType] :
( ( comm_monoid_add @ TB )
=> ! [A: set @ TA,B: set @ TC,C: TC > TA,D: TA > TB,E: TC > TB] :
( ! [F: TA] :
( ( member @ TA @ F @ A )
=> ? [G: TC] :
( ( member @ TC @ G @ B )
& ( ( C @ G )
= F )
& ! [H: TC] :
( ( ( member @ TC @ H @ B )
& ( ( C @ H )
= F ) )
=> ( H = G ) ) ) )
=> ( ! [F: TC] :
( ( member @ TC @ F @ B )
=> ( ( member @ TA @ ( C @ F ) @ A )
& ( ( D @ ( C @ F ) )
= ( E @ F ) ) ) )
=> ( ( groups1340683514dd_sum @ TC @ TB @ E @ B )
= ( groups1340683514dd_sum @ TA @ TB @ D @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[228]) ).
thf(180,axiom,
! [TA: $tType] :
( ( semiri1923998003cancel @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( times_times @ TA @ A @ B )
= ( times_times @ TA @ A @ C ) )
= ( ( A
= ( zero_zero @ TA ) )
| ( B = C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_70_mult__cancel__left) ).
thf(836,plain,
! [TA: $tType] :
( ( semiri1923998003cancel @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( times_times @ TA @ A @ B )
= ( times_times @ TA @ A @ C ) )
= ( ( A
= ( zero_zero @ TA ) )
| ( B = C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[180]) ).
thf(252,axiom,
comm_monoid_diff @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocomm__monoid__diff) ).
thf(1049,plain,
comm_monoid_diff @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[252]) ).
thf(117,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
=> ( ord_less_eq @ nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_208_nat__mult__le__cancel__disj) ).
thf(663,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ A @ B ) @ ( times_times @ nat @ A @ C ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ A )
=> ( ord_less_eq @ nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[117]) ).
thf(82,axiom,
! [TA: $tType,TB: $tType,A: TB > TA,B: TB,C: TA,D: TA] :
( ( fun_upd @ TB @ TA @ ( fun_upd @ TB @ TA @ A @ B @ C ) @ B @ D )
= ( fun_upd @ TB @ TA @ A @ B @ D ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_58_fun__upd__upd) ).
thf(548,plain,
! [TA: $tType,TB: $tType,A: TB > TA,B: TB,C: TA,D: TA] :
( ( fun_upd @ TB @ TA @ ( fun_upd @ TB @ TA @ A @ B @ C ) @ B @ D )
= ( fun_upd @ TB @ TA @ A @ B @ D ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[82]) ).
thf(107,axiom,
( number2016821345itions
= ( ^ [A: nat > nat,B: nat] :
( ! [C: nat] :
( ( ( A @ C )
!= ( zero_zero @ nat ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ C )
& ( ord_less_eq @ nat @ C @ B ) ) )
& ( ( groups1340683514dd_sum @ nat @ nat
@ ^ [C: nat] : ( times_times @ nat @ ( A @ C ) @ C )
@ ( set_ord_atMost @ nat @ B ) )
= B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_63_partitions__def) ).
thf(617,plain,
( number2016821345itions
= ( ^ [A: nat > nat,B: nat] :
( ! [C: nat] :
( ( ( A @ C )
!= ( zero_zero @ nat ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ C )
& ( ord_less_eq @ nat @ C @ B ) ) )
& ( ( groups1340683514dd_sum @ nat @ nat
@ ^ [C: nat] : ( times_times @ nat @ ( A @ C ) @ C )
@ ( set_ord_atMost @ nat @ B ) )
= B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[107]) ).
thf(69,axiom,
! [TA: $tType,TB: $tType] :
( ( ord @ TA )
=> ( ord @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Orderings_Oord) ).
thf(516,plain,
! [TA: $tType,TB: $tType] :
( ( ord @ TA )
=> ( ord @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[69]) ).
thf(158,axiom,
! [A: nat,B: nat] :
( ( A
= ( times_times @ nat @ A @ B ) )
=> ( ( B
= ( one_one @ nat ) )
| ( A
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_140_mult__eq__self__implies__10) ).
thf(776,plain,
! [A: nat,B: nat] :
( ( A
= ( times_times @ nat @ A @ B ) )
=> ( ( B
= ( one_one @ nat ) )
| ( A
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[158]) ).
thf(73,axiom,
! [TA: $tType,TB: $tType] :
( ( wellorder @ TB )
=> ! [A: TA > TB,B: TA > $o,C: TA] :
( ! [D: TA] :
( ! [E: TA] :
( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
=> ( B @ E ) )
=> ( B @ D ) )
=> ( B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_249_measure__induct__rule) ).
thf(525,plain,
! [TA: $tType,TB: $tType] :
( ( wellorder @ TB )
=> ! [A: TA > TB,B: TA > $o,C: TA] :
( ! [D: TA] :
( ! [E: TA] :
( ( ord_less @ TB @ ( A @ E ) @ ( A @ D ) )
=> ( B @ E ) )
=> ( B @ D ) )
=> ( B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[73]) ).
thf(94,axiom,
ord @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oord_12) ).
thf(580,plain,
ord @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[94]) ).
thf(125,axiom,
! [TA: $tType,TB: $tType] :
( ( ordere216010020id_add @ TB )
=> ! [A: set @ TA,B: TA > TB] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( ord_less_eq @ TB @ ( zero_zero @ TB ) @ ( B @ C ) ) )
=> ( ord_less_eq @ TB @ ( zero_zero @ TB ) @ ( groups1340683514dd_sum @ TA @ TB @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_139_sum__nonneg) ).
thf(690,plain,
! [TA: $tType,TB: $tType] :
( ( ordere216010020id_add @ TB )
=> ! [A: set @ TA,B: TA > TB] :
( ! [C: TA] :
( ( member @ TA @ C @ A )
=> ( ord_less_eq @ TB @ ( zero_zero @ TB ) @ ( B @ C ) ) )
=> ( ord_less_eq @ TB @ ( zero_zero @ TB ) @ ( groups1340683514dd_sum @ TA @ TB @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[125]) ).
thf(262,axiom,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ C ) @ ( minus_minus @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_183_diff__right__mono) ).
thf(1070,plain,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ ( minus_minus @ TA @ A @ C ) @ ( minus_minus @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[262]) ).
thf(130,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( times_times @ nat @ A @ B )
= ( times_times @ nat @ C @ B ) )
= ( ( A = C )
| ( B
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_84_mult__cancel2) ).
thf(705,plain,
! [A: nat,B: nat,C: nat] :
( ( ( times_times @ nat @ A @ B )
= ( times_times @ nat @ C @ B ) )
= ( ( A = C )
| ( B
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[130]) ).
thf(46,axiom,
! [TA: $tType,TB: $tType] :
( ( ( ord @ TA )
& ( ord @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less @ TB @ A @ B )
=> ( ( ( C @ B )
= D )
=> ( ! [E: TB,F: TB] :
( ( ord_less @ TB @ E @ F )
=> ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_244_ord__less__eq__subst) ).
thf(414,plain,
! [TA: $tType,TB: $tType] :
( ( ( ord @ TA )
& ( ord @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less @ TB @ A @ B )
=> ( ( ( C @ B )
= D )
=> ( ! [E: TB,F: TB] :
( ( ord_less @ TB @ E @ F )
=> ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[46]) ).
thf(143,axiom,
! [TA: $tType] :
( ( ordered_ring @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ ( zero_zero @ TA ) )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ B @ C ) @ ( times_times @ TA @ A @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_122_mult__right__mono__neg) ).
thf(739,plain,
! [TA: $tType] :
( ( ordered_ring @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ ( zero_zero @ TA ) )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ B @ C ) @ ( times_times @ TA @ A @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[143]) ).
thf(44,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
= ( ( ord_less @ TA @ A @ B )
| ( ord_less @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_238_neq__iff) ).
thf(409,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
= ( ( ord_less @ TA @ A @ B )
| ( ord_less @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[44]) ).
thf(234,axiom,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_200_less__nat__zero__code) ).
thf(993,plain,
! [A: nat] :
~ ( ord_less @ nat @ A @ ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[234]) ).
thf(133,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ A @ C )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
= ( minus_minus @ nat @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_30_Nat_Odiff__diff__eq) ).
thf(712,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ A @ C )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ C @ A ) )
= ( minus_minus @ nat @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[133]) ).
thf(86,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_157_antisym) ).
thf(557,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)],[86]) ).
thf(238,axiom,
! [TA: $tType] :
( ( semiri1193490041visors @ TA )
=> ! [A: TA,B: TA] :
( ( A
!= ( zero_zero @ TA ) )
=> ( ( B
!= ( zero_zero @ TA ) )
=> ( ( times_times @ TA @ A @ B )
!= ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_98_no__zero__divisors) ).
thf(1003,plain,
! [TA: $tType] :
( ( semiri1193490041visors @ TA )
=> ! [A: TA,B: TA] :
( ( A
!= ( zero_zero @ TA ) )
=> ( ( B
!= ( zero_zero @ TA ) )
=> ( ( times_times @ TA @ A @ B )
!= ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[238]) ).
thf(255,axiom,
ord_less_eq @ nat @ k @ n,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_1__092_060open_062k_A_092_060le_062_An_092_060close_062) ).
thf(1053,plain,
ord_less_eq @ nat @ k @ n,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[255]) ).
thf(165,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ A @ C ) @ ( times_times @ nat @ B @ C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_34_mult__le__mono1) ).
thf(791,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ A @ C ) @ ( times_times @ nat @ B @ C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[165]) ).
thf(16,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_237_order_Oasym) ).
thf(331,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[16]) ).
thf(222,axiom,
! [A: nat > nat,B: nat] :
( ( number2016821345itions @ A @ B )
=> ( ord_less_eq @ nat @ ( groups1340683514dd_sum @ nat @ nat @ A @ ( set_ord_atMost @ nat @ B ) ) @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_43_partitions__parts__bounded) ).
thf(951,plain,
! [A: nat > nat,B: nat] :
( ( number2016821345itions @ A @ B )
=> ( ord_less_eq @ nat @ ( groups1340683514dd_sum @ nat @ nat @ A @ ( set_ord_atMost @ nat @ B ) ) @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[222]) ).
thf(26,axiom,
preorder @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Opreorder_9) ).
thf(359,plain,
preorder @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[26]) ).
thf(205,axiom,
! [TA: $tType] :
( ( ring_11004092258visors @ TA )
=> ! [A: TA,B: TA] :
( ( ( times_times @ TA @ A @ B )
= A )
= ( ( A
= ( zero_zero @ TA ) )
| ( B
= ( one_one @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_89_mult__cancel__left2) ).
thf(907,plain,
! [TA: $tType] :
( ( ring_11004092258visors @ TA )
=> ! [A: TA,B: TA] :
( ( ( times_times @ TA @ A @ B )
= A )
= ( ( A
= ( zero_zero @ TA ) )
| ( B
= ( one_one @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[205]) ).
thf(248,axiom,
! [TA: $tType] :
( ( semiri1193490041visors @ TA )
=> ! [A: TA,B: TA] :
( ( ( times_times @ TA @ A @ B )
= ( zero_zero @ TA ) )
= ( ( A
= ( zero_zero @ TA ) )
| ( B
= ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_69_mult__eq__0__iff) ).
thf(1038,plain,
! [TA: $tType] :
( ( semiri1193490041visors @ TA )
=> ! [A: TA,B: TA] :
( ( ( times_times @ TA @ A @ B )
= ( zero_zero @ TA ) )
= ( ( A
= ( zero_zero @ TA ) )
| ( B
= ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[248]) ).
thf(30,axiom,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
=> ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_193_subset__antisym) ).
thf(370,plain,
! [TA: $tType,A: set @ TA,B: set @ TA] :
( ( ord_less_eq @ ( set @ TA ) @ A @ B )
=> ( ( ord_less_eq @ ( set @ TA ) @ B @ A )
=> ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[30]) ).
thf(269,axiom,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
= ( ord_less_eq @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_86_diff__ge__0__iff__ge) ).
thf(1085,plain,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( minus_minus @ TA @ A @ B ) )
= ( ord_less_eq @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[269]) ).
thf(92,axiom,
! [TA: $tType] :
( ( no_top @ TA )
=> ! [A: TA] :
? [B: TA] : ( ord_less @ TA @ A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_240_gt__ex) ).
thf(576,plain,
! [TA: $tType] :
( ( no_top @ TA )
=> ! [A: TA] :
? [B: TA] : ( ord_less @ TA @ A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[92]) ).
thf(145,axiom,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_75_diff__zero) ).
thf(744,plain,
! [TA: $tType] :
( ( cancel1352612707id_add @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[145]) ).
thf(243,axiom,
! [A: nat,B: nat] :
( ( ( times_times @ nat @ A @ B )
= ( zero_zero @ nat ) )
= ( ( A
= ( zero_zero @ nat ) )
| ( B
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_81_mult__is__0) ).
thf(1023,plain,
! [A: nat,B: nat] :
( ( ( times_times @ nat @ A @ B )
= ( zero_zero @ nat ) )
= ( ( A
= ( zero_zero @ nat ) )
| ( B
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[243]) ).
thf(254,axiom,
preorder @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Opreorder_1) ).
thf(1052,plain,
preorder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[254]) ).
thf(190,axiom,
cancel1352612707id_add @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add) ).
thf(864,plain,
cancel1352612707id_add @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[190]) ).
thf(138,axiom,
zero @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Ozero) ).
thf(727,plain,
zero @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[138]) ).
thf(273,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( times_times @ nat @ A @ B )
= ( times_times @ nat @ A @ C ) )
= ( ( B = C )
| ( A
= ( zero_zero @ nat ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_83_mult__cancel1) ).
thf(1094,plain,
! [A: nat,B: nat,C: nat] :
( ( ( times_times @ nat @ A @ B )
= ( times_times @ nat @ A @ C ) )
= ( ( B = C )
| ( A
= ( zero_zero @ nat ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[273]) ).
thf(188,axiom,
! [TA: $tType] :
( ( ordere1490568538miring @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ C @ A ) @ ( times_times @ TA @ C @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_113_ordered__comm__semiring__class_Ocomm__mult__left__mono) ).
thf(859,plain,
! [TA: $tType] :
( ( ordere1490568538miring @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ C @ A ) @ ( times_times @ TA @ C @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[188]) ).
thf(65,axiom,
! [TA: $tType,TB: $tType] :
( ( ( ord @ TA )
& ( ord @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( A
= ( B @ C ) )
=> ( ( ord_less @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less @ TA @ E @ F )
=> ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_245_ord__eq__less__subst) ).
thf(497,plain,
! [TA: $tType,TB: $tType] :
( ( ( ord @ TA )
& ( ord @ TB ) )
=> ! [A: TB,B: TA > TB,C: TA,D: TA] :
( ( A
= ( B @ C ) )
=> ( ( ord_less @ TA @ C @ D )
=> ( ! [E: TA,F: TA] :
( ( ord_less @ TA @ E @ F )
=> ( ord_less @ TB @ ( B @ E ) @ ( B @ F ) ) )
=> ( ord_less @ TB @ A @ ( B @ D ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[65]) ).
thf(195,axiom,
! [TA: $tType] :
( ( cancel146912293up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( minus_minus @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
= ( minus_minus @ TA @ ( minus_minus @ TA @ A @ C ) @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_172_cancel__ab__semigroup__add__class_Odiff__right__commute) ).
thf(876,plain,
! [TA: $tType] :
( ( cancel146912293up_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( minus_minus @ TA @ ( minus_minus @ TA @ A @ B ) @ C )
= ( minus_minus @ TA @ ( minus_minus @ TA @ A @ C ) @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[195]) ).
thf(183,axiom,
! [TA: $tType] :
( ( ord @ TA )
=> ( ( set_ord_atMost @ TA )
= ( ^ [A: TA] :
( collect @ TA
@ ^ [B: TA] : ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_41_atMost__def) ).
thf(845,plain,
! [TA: $tType] :
( ( ord @ TA )
=> ( ( set_ord_atMost @ TA )
= ( ^ [A: TA] :
( collect @ TA
@ ^ [B: TA] : ( ord_less_eq @ TA @ B @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[183]) ).
thf(57,axiom,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less @ TB @ A @ B )
=> ( ( ord_less @ TA @ ( C @ B ) @ D )
=> ( ! [E: TB,F: TB] :
( ( ord_less @ TB @ E @ F )
=> ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_242_order__less__subst2) ).
thf(450,plain,
! [TA: $tType,TB: $tType] :
( ( ( order @ TA )
& ( order @ TB ) )
=> ! [A: TB,B: TB,C: TB > TA,D: TA] :
( ( ord_less @ TB @ A @ B )
=> ( ( ord_less @ TA @ ( C @ B ) @ D )
=> ( ! [E: TB,F: TB] :
( ( ord_less @ TB @ E @ F )
=> ( ord_less @ TA @ ( C @ E ) @ ( C @ F ) ) )
=> ( ord_less @ TA @ ( C @ A ) @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[57]) ).
thf(120,axiom,
! [A: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ A )
= ( zero_zero @ nat ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_79_diff__0__eq__0) ).
thf(671,plain,
! [A: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ A )
= ( zero_zero @ nat ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[120]) ).
thf(124,axiom,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ ( minus_minus @ TA @ C @ B ) @ ( minus_minus @ TA @ C @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_182_diff__left__mono) ).
thf(688,plain,
! [TA: $tType] :
( ( ordered_ab_group_add @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ord_less_eq @ TA @ ( minus_minus @ TA @ C @ B ) @ ( minus_minus @ TA @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[124]) ).
thf(95,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_147_order__class_Oorder_Oantisym) ).
thf(581,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)],[95]) ).
thf(64,axiom,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_197_le__numeral__extra_I3_J) ).
thf(495,plain,
! [TA: $tType] :
( ( linord1659791738miring @ TA )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[64]) ).
thf(241,axiom,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ D )
=> ( ord_less_eq @ nat @ D @ C ) )
=> ~ ! [D: nat] :
( ( A @ D )
=> ~ ! [E: nat] :
( ( A @ E )
=> ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_24_bounded__Max__nat) ).
thf(1010,plain,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ D )
=> ( ord_less_eq @ nat @ D @ C ) )
=> ~ ! [D: nat] :
( ( A @ D )
=> ~ ! [E: nat] :
( ( A @ E )
=> ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[241]) ).
thf(182,axiom,
monoid_mult @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Omonoid__mult) ).
thf(844,plain,
monoid_mult @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[182]) ).
thf(11,axiom,
! [TA: $tType,TB: $tType] :
( ( fun_upd @ TA @ TB )
= ( ^ [A: TA > TB,B: TA,C: TB,D: TA] : ( if @ TB @ ( D = B ) @ C @ ( A @ D ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_60_fun__upd__apply) ).
thf(319,plain,
! [TA: $tType,TB: $tType] :
( ( fun_upd @ TA @ TB )
= ( ^ [A: TA > TB,B: TA,C: TB,D: TA] : ( if @ TB @ ( D = B ) @ C @ ( A @ D ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[11]) ).
thf(221,axiom,
! [TA: $tType,A: set @ TA,B: TA > nat,C: TA > nat] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ord_less_eq @ nat @ ( B @ D ) @ ( C @ D ) ) )
=> ( ( groups1340683514dd_sum @ TA @ nat
@ ^ [D: TA] : ( minus_minus @ nat @ ( C @ D ) @ ( B @ D ) )
@ A )
= ( minus_minus @ nat @ ( groups1340683514dd_sum @ TA @ nat @ C @ A ) @ ( groups1340683514dd_sum @ TA @ nat @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_42_sum__subtractf__nat) ).
thf(946,plain,
! [TA: $tType,A: set @ TA,B: TA > nat,C: TA > nat] :
( ! [D: TA] :
( ( member @ TA @ D @ A )
=> ( ord_less_eq @ nat @ ( B @ D ) @ ( C @ D ) ) )
=> ( ( groups1340683514dd_sum @ TA @ nat
@ ^ [D: TA] : ( minus_minus @ nat @ ( C @ D ) @ ( B @ D ) )
@ A )
= ( minus_minus @ nat @ ( groups1340683514dd_sum @ TA @ nat @ C @ A ) @ ( groups1340683514dd_sum @ TA @ nat @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[221]) ).
thf(162,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ~ ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_250_linorder__neqE__nat) ).
thf(786,plain,
! [A: nat,B: nat] :
( ( A != B )
=> ( ~ ( ord_less @ nat @ A @ B )
=> ( ord_less @ nat @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[162]) ).
thf(201,axiom,
! [TA: $tType] :
( ( ordered_semiring_0 @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_116_mult__nonpos__nonneg) ).
thf(896,plain,
! [TA: $tType] :
( ( ordered_semiring_0 @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[201]) ).
thf(147,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( minus_minus @ nat @ A @ B )
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_91_diff__is__0__eq_H) ).
thf(748,plain,
! [A: nat,B: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( minus_minus @ nat @ A @ B )
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[147]) ).
thf(245,axiom,
! [TA: $tType,TB: $tType] :
( ( minus @ TB )
=> ( ( minus_minus @ ( TA > TB ) )
= ( ^ [A: TA > TB,B: TA > TB,C: TA] : ( minus_minus @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_44_fun__diff__def) ).
thf(1030,plain,
! [TA: $tType,TB: $tType] :
( ( minus @ TB )
=> ( ( minus_minus @ ( TA > TB ) )
= ( ^ [A: TA > TB,B: TA > TB,C: TA] : ( minus_minus @ TB @ ( A @ C ) @ ( B @ C ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[245]) ).
thf(52,axiom,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_198_not__gr__zero) ).
thf(433,plain,
! [TA: $tType] :
( ( canoni770627133id_add @ TA )
=> ! [A: TA] :
( ( ~ ( ord_less @ TA @ ( zero_zero @ TA ) @ A ) )
= ( A
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[52]) ).
thf(215,axiom,
! [TA: $tType] :
( ( semiri1923998003cancel @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( times_times @ TA @ A @ B )
= ( times_times @ TA @ C @ B ) )
= ( ( B
= ( zero_zero @ TA ) )
| ( A = C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_71_mult__cancel__right) ).
thf(929,plain,
! [TA: $tType] :
( ( semiri1923998003cancel @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( ( times_times @ TA @ A @ B )
= ( times_times @ TA @ C @ B ) )
= ( ( B
= ( zero_zero @ TA ) )
| ( A = C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[215]) ).
thf(153,axiom,
! [TA: $tType,TB: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: set @ TB,B: TB > TA] :
( ! [C: TB] :
( ( member @ TB @ C @ A )
=> ( ( B @ C )
= ( zero_zero @ TA ) ) )
=> ( ( groups1340683514dd_sum @ TB @ TA @ B @ A )
= ( zero_zero @ TA ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_104_sum_Oneutral) ).
thf(763,plain,
! [TA: $tType,TB: $tType] :
( ( comm_monoid_add @ TA )
=> ! [A: set @ TB,B: TB > TA] :
( ! [C: TB] :
( ( member @ TB @ C @ A )
=> ( ( B @ C )
= ( zero_zero @ TA ) ) )
=> ( ( groups1340683514dd_sum @ TB @ TA @ B @ A )
= ( zero_zero @ TA ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[153]) ).
thf(208,axiom,
! [TA: $tType] :
( ( linordered_ring @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_127_zero__le__square) ).
thf(914,plain,
! [TA: $tType] :
( ( linordered_ring @ TA )
=> ! [A: TA] : ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[208]) ).
thf(33,axiom,
! [TA: $tType,TB: $tType,A: TB > TA,B: TB,C: TA] :
( ( ( fun_upd @ TB @ TA @ A @ B @ C )
= A )
= ( ( A @ B )
= C ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_174_fun__upd__idem__iff) ).
thf(378,plain,
! [TA: $tType,TB: $tType,A: TB > TA,B: TB,C: TA] :
( ( ( fun_upd @ TB @ TA @ A @ B @ C )
= A )
= ( ( A @ B )
= C ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[33]) ).
thf(134,axiom,
! [A: nat] :
( ( ord_less @ nat @ A @ ( one_one @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_207_less__one) ).
thf(716,plain,
! [A: nat] :
( ( ord_less @ nat @ A @ ( one_one @ nat ) )
= ( A
= ( zero_zero @ nat ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[134]) ).
thf(168,axiom,
! [TA: $tType] :
( ( comm_monoid_diff @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ ( zero_zero @ TA ) @ A )
= ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_74_zero__diff) ).
thf(800,plain,
! [TA: $tType] :
( ( comm_monoid_diff @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ ( zero_zero @ TA ) @ A )
= ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[168]) ).
thf(282,axiom,
! [A: nat] :
( ( times_times @ nat @ A @ ( one_one @ nat ) )
= A ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_38_nat__mult__1__right) ).
thf(1116,plain,
! [A: nat] :
( ( times_times @ nat @ A @ ( one_one @ nat ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[282]) ).
thf(288,axiom,
! [TA: $tType] :
( ( group_add @ TA )
=> ( ( (=) @ TA )
= ( ^ [A: TA,B: TA] :
( ( minus_minus @ TA @ A @ B )
= ( zero_zero @ TA ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_101_eq__iff__diff__eq__0) ).
thf(1133,plain,
! [TA: $tType] :
( ( group_add @ TA )
=> ( ( (=) @ TA )
= ( ^ [A: TA,B: TA] :
( ( minus_minus @ TA @ A @ B )
= ( zero_zero @ TA ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[288]) ).
thf(3,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ~ ( ord_less @ TA @ A @ B ) )
= ( ( ord_less @ TA @ B @ A )
| ( A = B ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_213_not__less__iff__gr__or__eq) ).
thf(300,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ( ~ ( ord_less @ TA @ A @ B ) )
= ( ( ord_less @ TA @ B @ A )
| ( A = B ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[3]) ).
thf(56,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(447,plain,
! [TA: $tType,A: set @ TA] :
( ( collect @ TA
@ ^ [B: TA] : ( member @ TA @ B @ A ) )
= A ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[56]) ).
thf(274,axiom,
! [TA: $tType] :
( ( ab_semigroup_mult @ TA )
=> ( ( times_times @ TA )
= ( ^ [A: TA,B: TA] : ( times_times @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_169_mult_Ocommute) ).
thf(1097,plain,
! [TA: $tType] :
( ( ab_semigroup_mult @ TA )
=> ( ( times_times @ TA )
= ( ^ [A: TA,B: TA] : ( times_times @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[274]) ).
thf(175,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ ( set @ TA ) @ ( set_ord_atMost @ TA @ A ) @ ( set_ord_atMost @ TA @ B ) )
= ( ord_less_eq @ TA @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_13_atMost__subset__iff) ).
thf(818,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ ( set @ TA ) @ ( set_ord_atMost @ TA @ A ) @ ( set_ord_atMost @ TA @ B ) )
= ( ord_less_eq @ TA @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[175]) ).
thf(186,axiom,
( ( groups1340683514dd_sum @ nat @ nat
@ ^ [A: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ A ) @ A )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ n @ k ) ) )
= ( groups1340683514dd_sum @ nat @ nat
@ ^ [A: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ A ) @ A )
@ ( set_ord_atMost @ nat @ n ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_4__092_060open_062_I_092_060Sum_062i_092_060le_062n_A_N_Ak_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_A_061_A_I_092_060Sum_062i_092_060le_062n_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_092_060close_062) ).
thf(854,plain,
( ( groups1340683514dd_sum @ nat @ nat
@ ^ [A: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ A ) @ A )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ n @ k ) ) )
= ( groups1340683514dd_sum @ nat @ nat
@ ^ [A: nat] : ( times_times @ nat @ ( fun_upd @ nat @ nat @ p @ k @ ( minus_minus @ nat @ ( p @ k ) @ ( one_one @ nat ) ) @ A ) @ A )
@ ( set_ord_atMost @ nat @ n ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[186]) ).
thf(164,axiom,
linorder @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Olinorder) ).
thf(790,plain,
linorder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[164]) ).
thf(191,axiom,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA] :
( ( member @ TA @ A @ ( set_ord_atMost @ TA @ B ) )
= ( ord_less_eq @ TA @ A @ B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_14_atMost__iff) ).
thf(865,plain,
! [TA: $tType] :
( ( ord @ TA )
=> ! [A: TA,B: TA] :
( ( member @ TA @ A @ ( set_ord_atMost @ TA @ B ) )
= ( ord_less_eq @ TA @ A @ B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[191]) ).
thf(31,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA] :
~ ( ord_less @ TA @ A @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_219_dual__order_Oirrefl) ).
thf(373,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA] :
~ ( ord_less @ TA @ A @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[31]) ).
thf(5,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA,C: $o] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ A )
=> C ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_221_less__imp__triv) ).
thf(305,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA,C: $o] :
( ( ord_less @ TA @ A @ B )
=> ( ( ord_less @ TA @ B @ A )
=> C ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[5]) ).
thf(14,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_143_dual__order_Otrans) ).
thf(326,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)],[14]) ).
thf(171,axiom,
! [TA: $tType] :
( ( monoid_mult @ TA )
=> ! [A: TA] :
( ( times_times @ TA @ A @ ( one_one @ TA ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_56_mult_Oright__neutral) ).
thf(807,plain,
! [TA: $tType] :
( ( monoid_mult @ TA )
=> ! [A: TA] :
( ( times_times @ TA @ A @ ( one_one @ TA ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[171]) ).
thf(246,axiom,
! [TA: $tType] :
( ( ordered_semiring @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ D ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_128_mult__mono_H) ).
thf(1033,plain,
! [TA: $tType] :
( ( ordered_semiring @ TA )
=> ! [A: TA,B: TA,C: TA,D: TA] :
( ( ord_less_eq @ TA @ A @ B )
=> ( ( ord_less_eq @ TA @ C @ D )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ C )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ A @ C ) @ ( times_times @ TA @ B @ D ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[246]) ).
thf(76,axiom,
order @ $o,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_HOL_Obool___Orderings_Oorder_11) ).
thf(532,plain,
order @ $o,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[76]) ).
thf(127,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ C @ B )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ B @ C ) )
= ( ord_less_eq @ nat @ C @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_27_le__diff__iff_H) ).
thf(696,plain,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq @ nat @ A @ B )
=> ( ( ord_less_eq @ nat @ C @ B )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ B @ A ) @ ( minus_minus @ nat @ B @ C ) )
= ( ord_less_eq @ nat @ C @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[127]) ).
thf(199,axiom,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_201_neq0__conv) ).
thf(886,plain,
! [A: nat] :
( ( A
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[199]) ).
thf(139,axiom,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ D )
=> ( ord_less_eq @ nat @ D @ C ) )
=> ? [D: nat] :
( ( A @ D )
& ! [E: nat] :
( ( A @ E )
=> ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_23_Nat_Oex__has__greatest__nat) ).
thf(728,plain,
! [A: nat > $o,B: nat,C: nat] :
( ( A @ B )
=> ( ! [D: nat] :
( ( A @ D )
=> ( ord_less_eq @ nat @ D @ C ) )
=> ? [D: nat] :
( ( A @ D )
& ! [E: nat] :
( ( A @ E )
=> ( ord_less_eq @ nat @ E @ D ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[139]) ).
thf(286,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ( ord_less_eq @ TA @ B @ ( one_one @ TA ) )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ A ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_136_mult__right__le__one__le) ).
thf(1129,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ A )
=> ( ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ B )
=> ( ( ord_less_eq @ TA @ B @ ( one_one @ TA ) )
=> ( ord_less_eq @ TA @ ( times_times @ TA @ A @ B ) @ A ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[286]) ).
thf(295,axiom,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ( set_ord_atMost @ TA @ A )
= ( set_ord_atMost @ TA @ B ) )
= ( A = B ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_2_atMost__eq__iff) ).
thf(1152,plain,
! [TA: $tType] :
( ( order @ TA )
=> ! [A: TA,B: TA] :
( ( ( set_ord_atMost @ TA @ A )
= ( set_ord_atMost @ TA @ B ) )
= ( A = B ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[295]) ).
thf(100,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(596,plain,
! [TA: $tType,A: TA,B: TA > $o] :
( ( member @ TA @ A @ ( collect @ TA @ B ) )
= ( B @ A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[100]) ).
thf(74,axiom,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
=> ( ~ ( ord_less @ TA @ A @ B )
=> ( ord_less @ TA @ B @ A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_210_linorder__neqE__linordered__idom) ).
thf(528,plain,
! [TA: $tType] :
( ( linordered_idom @ TA )
=> ! [A: TA,B: TA] :
( ( A != B )
=> ( ~ ( ord_less @ TA @ A @ B )
=> ( ord_less @ TA @ B @ A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[74]) ).
thf(230,axiom,
! [TA: $tType] :
( ( ab_semigroup_mult @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) )
= ( times_times @ TA @ B @ ( times_times @ TA @ A @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_170_mult_Oleft__commute) ).
thf(982,plain,
! [TA: $tType] :
( ( ab_semigroup_mult @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) )
= ( times_times @ TA @ B @ ( times_times @ TA @ A @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[230]) ).
thf(193,axiom,
! [TA: $tType] :
( ( semigroup_mult @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( times_times @ TA @ ( times_times @ TA @ A @ B ) @ C )
= ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_168_mult_Oassoc) ).
thf(871,plain,
! [TA: $tType] :
( ( semigroup_mult @ TA )
=> ! [A: TA,B: TA,C: TA] :
( ( times_times @ TA @ ( times_times @ TA @ A @ B ) @ C )
= ( times_times @ TA @ A @ ( times_times @ TA @ B @ C ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[193]) ).
thf(272,axiom,
semigroup_mult @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Groups_Osemigroup__mult) ).
thf(1093,plain,
semigroup_mult @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[272]) ).
thf(39,axiom,
! [TA: $tType,TB: $tType,A: TB > TA,B: TB,C: TA,D: TB > TA,E: TA] :
( ( ( fun_upd @ TB @ TA @ A @ B @ C )
= ( fun_upd @ TB @ TA @ D @ B @ E ) )
=> ( C = E ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_179_fun__upd__eqD) ).
thf(395,plain,
! [TA: $tType,TB: $tType,A: TB > TA,B: TB,C: TA,D: TB > TA,E: TA] :
( ( ( fun_upd @ TB @ TA @ A @ B @ C )
= ( fun_upd @ TB @ TA @ D @ B @ E ) )
=> ( C = E ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[39]) ).
thf(109,axiom,
! [TA: $tType,TB: $tType] :
( ( comm_monoid_add @ TB )
=> ! [A: set @ TA,B: set @ TA,C: TA > TB,D: TA > TB] :
( ( A = B )
=> ( ! [E: TA] :
( ( member @ TA @ E @ B )
=> ( ( C @ E )
= ( D @ E ) ) )
=> ( ( groups1340683514dd_sum @ TA @ TB @ C @ A )
= ( groups1340683514dd_sum @ TA @ TB @ D @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_52_sum_Ocong) ).
thf(624,plain,
! [TA: $tType,TB: $tType] :
( ( comm_monoid_add @ TB )
=> ! [A: set @ TA,B: set @ TA,C: TA > TB,D: TA > TB] :
( ( A = B )
=> ( ! [E: TA] :
( ( member @ TA @ E @ B )
=> ( ( C @ E )
= ( D @ E ) ) )
=> ( ( groups1340683514dd_sum @ TA @ TB @ C @ A )
= ( groups1340683514dd_sum @ TA @ TB @ D @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[109]) ).
thf(181,axiom,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ! [D: nat] :
( ( ord_less @ nat @ D @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_252_nat__less__induct) ).
thf(839,plain,
! [A: nat > $o,B: nat] :
( ! [C: nat] :
( ! [D: nat] :
( ( ord_less @ nat @ D @ C )
=> ( A @ D ) )
=> ( A @ C ) )
=> ( A @ B ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[181]) ).
thf(12,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_150_order__class_Oorder_Oeq__iff) ).
thf(322,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)],[12]) ).
thf(169,axiom,
! [TA: $tType] :
( ( ( comm_ring @ TA )
& ( dvd @ TA ) )
=> ! [A: TA > $o,B: TA,C: TA > $o] :
( ! [D: TA,E: TA] :
( ( A @ D )
= ( A @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) )
=> ( ! [D: TA,E: TA] :
( ( C @ D )
= ( C @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) )
=> ! [D: TA,E: TA] :
( ( ( A @ D )
| ( C @ D ) )
= ( ( A @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) )
| ( C @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_12_inf__period_I2_J) ).
thf(803,plain,
! [TA: $tType] :
( ( ( comm_ring @ TA )
& ( dvd @ TA ) )
=> ! [A: TA > $o,B: TA,C: TA > $o] :
( ! [D: TA,E: TA] :
( ( A @ D )
= ( A @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) )
=> ( ! [D: TA,E: TA] :
( ( C @ D )
= ( C @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) )
=> ! [D: TA,E: TA] :
( ( ( A @ D )
| ( C @ D ) )
= ( ( A @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) )
| ( C @ ( minus_minus @ TA @ D @ ( times_times @ TA @ E @ B ) ) ) ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[169]) ).
thf(19,axiom,
! [TA: $tType] : ( ord @ ( set @ TA ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Set_Oset___Orderings_Oord_7) ).
thf(337,plain,
! [TA: $tType] : ( ord @ ( set @ TA ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[19]) ).
thf(24,axiom,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_225_less__not__sym) ).
thf(354,plain,
! [TA: $tType] :
( ( preorder @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less @ TA @ A @ B )
=> ~ ( ord_less @ TA @ B @ A ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[24]) ).
thf(7,axiom,
! [TA: $tType,TB: $tType] :
( ( minus @ TA )
=> ( minus @ ( TB > TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_fun___Groups_Ominus) ).
thf(309,plain,
! [TA: $tType,TB: $tType] :
( ( minus @ TA )
=> ( minus @ ( TB > TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[7]) ).
thf(80,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_142_dual__order_Oeq__iff) ).
thf(543,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)],[80]) ).
thf(270,axiom,
! [TA: $tType] :
( ( ordered_ring @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
=> ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_124_mult__nonpos__nonpos) ).
thf(1088,plain,
! [TA: $tType] :
( ( ordered_ring @ TA )
=> ! [A: TA,B: TA] :
( ( ord_less_eq @ TA @ A @ ( zero_zero @ TA ) )
=> ( ( ord_less_eq @ TA @ B @ ( zero_zero @ TA ) )
=> ( ord_less_eq @ TA @ ( zero_zero @ TA ) @ ( times_times @ TA @ A @ B ) ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[270]) ).
thf(99,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_155_eq__refl) ).
thf(592,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)],[99]) ).
thf(131,axiom,
! [TA: $tType] :
( ( mult_zero @ TA )
=> ! [A: TA] :
( ( times_times @ TA @ ( zero_zero @ TA ) @ A )
= ( zero_zero @ TA ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_67_mult__zero__left) ).
thf(708,plain,
! [TA: $tType] :
( ( mult_zero @ TA )
=> ! [A: TA] :
( ( times_times @ TA @ ( zero_zero @ TA ) @ A )
= ( zero_zero @ TA ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[131]) ).
thf(197,axiom,
linord1659791738miring @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring) ).
thf(882,plain,
linord1659791738miring @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[197]) ).
thf(217,axiom,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_73_diff__0__right) ).
thf(935,plain,
! [TA: $tType] :
( ( group_add @ TA )
=> ! [A: TA] :
( ( minus_minus @ TA @ A @ ( zero_zero @ TA ) )
= A ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[217]) ).
thf(240,axiom,
wellorder @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Orderings_Owellorder) ).
thf(1009,plain,
wellorder @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[240]) ).
thf(157,axiom,
semiri1923998003cancel @ nat,
file('/export/starexec/sandbox/benchmark/theBenchmark.p',tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel) ).
thf(775,plain,
semiri1923998003cancel @ nat,
inference(defexp_and_simp_and_etaexpand,[status(thm)],[157]) ).
thf(87,axiom,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less @ TA @ A @ B )
=> ( ( ~ ( ord_less @ TA @ B @ A ) )
= ( B = A ) ) ) ),
file('/export/starexec/sandbox/benchmark/theBenchmark.p',fact_223_antisym__conv3) ).
thf(560,plain,
! [TA: $tType] :
( ( linorder @ TA )
=> ! [A: TA,B: TA] :
( ~ ( ord_less @ TA @ A @ B )
=> ( ( ~ ( ord_less @ TA @ B @ A ) )
= ( B = A ) ) ) ),
inference(defexp_and_simp_and_etaexpand,[status(thm)],[87]) ).
thf(1159,plain,
$false,
inference(e,[status(thm)],[365,347,333,518,962,941,555,666,1110,1142,511,523,1158,741,1073,1041,861,1078,384,533,797,472,1063,747,913,404,898,565,436,651,724,719,785,504,344,587,619,702,751,421,606,325,316,1100,475,492,756,789,443,514,806,821,1136,834,602,634,734,956,1119,660,428,1072,453,307,920,439,546,925,761,810,381,1067,1047,1057,910,693,815,916,879,889,1079,682,551,1106,578,874,1017,615,1111,980,1006,388,356,328,1082,339,753,537,669,901,489,590,1155,393,997,985,738,721,569,779,1131,654,1060,1050,938,360,311,541,657,1054,674,1026,953,926,398,1035,773,794,298,412,965,1146,1103,824,856,375,584,904,423,402,772,711,1124,572,313,611,1107,335,1090,919,391,1139,406,737,599,990,996,303,1122,631,758,367,1149,943,932,848,700,959,1044,780,608,986,1000,851,431,922,658,685,521,768,827,1066,868,455,883,1113,563,1076,426,531,812,342,1145,553,968,836,1049,663,548,617,516,776,525,580,690,1070,705,414,739,409,993,712,557,1003,1053,791,331,951,359,907,1038,370,1085,576,744,1023,1052,864,727,1094,859,497,876,845,450,671,688,581,495,1010,844,319,946,786,896,748,1030,433,929,763,914,378,716,800,1116,1133,300,447,1097,818,854,790,865,373,305,326,807,1033,532,696,886,728,1129,1152,596,528,982,871,1093,395,624,839,322,803,337,354,309,543,1088,592,708,882,935,1009,775,560]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.11 % Problem : ITP132^2 : TPTP v8.2.0. Released v7.5.0.
% 0.03/0.14 % Command : run_Leo-III %s %d
% 0.14/0.35 % Computer : n014.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Sat May 18 17:03:09 EDT 2024
% 0.14/0.35 % CPUTime :
% 0.92/0.85 % [INFO] Parsing problem /export/starexec/sandbox/benchmark/theBenchmark.p ...
% 1.69/1.08 % [INFO] Parsing done (225ms).
% 1.89/1.09 % [INFO] Running in sequential loop mode.
% 2.54/1.30 % [INFO] eprover registered as external prover.
% 2.54/1.30 % [INFO] cvc4 registered as external prover.
% 2.54/1.30 % [INFO] Scanning for conjecture ...
% 2.78/1.39 % [INFO] Found a conjecture (or negated_conjecture) and 299 axioms. Running axiom selection ...
% 3.14/1.48 % [INFO] Axiom selection finished. Selected 295 axioms (removed 4 axioms).
% 3.70/1.65 % [INFO] Problem is higher-order (TPTP THF).
% 3.70/1.66 % [INFO] Type checking passed.
% 3.70/1.67 % [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 ...
% 58.75/41.65 % External prover 'e' found a proof!
% 58.75/41.65 % [INFO] Killing All external provers ...
% 58.75/41.66 % Time passed: 41138ms (effective reasoning time: 40565ms)
% 58.75/41.66 % 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)>
% 58.75/41.67 % Axioms used in derivation (295): fact_116_mult__nonpos__nonneg, fact_55_inf__period_I1_J, fact_180_fun__upd__def, fact_62_lambda__one, fact_40_sum__mono, tcon_fun___Groups_Ominus, fact_186_right__diff__distrib, fact_179_fun__upd__eqD, fact_61_order__refl, fact_242_order__less__subst2, fact_70_mult__cancel__left, fact_206_nat__0__less__mult__iff, tcon_Nat_Onat___Orderings_Oorder_2, fact_204_nat__mult__less__cancel__disj, fact_16_nat__mult__eq__1__iff, fact_189_comm__monoid__mult__class_Omult__1, tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel, fact_127_zero__le__square, tcon_Nat_Onat___Orderings_Ono__top, fact_132_zero__le__one, tcon_Nat_Onat___Groups_Ocomm__monoid__diff, fact_14_atMost__iff, tcon_Nat_Onat___Orderings_Olinorder, fact_217_less__imp__not__less, fact_58_fun__upd__upd, fact_213_not__less__iff__gr__or__eq, fact_125_mult__left__mono__neg, fact_210_linorder__neqE__linordered__idom, tcon_Nat_Onat___Rings_Odvd, tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add, fact_64_partitionsI, fact_246_less__numeral__extra_I3_J, fact_25_partitions__bounds, fact_28_diff__le__self, fact_221_less__imp__triv, fact_57_mult_Oleft__neutral, fact_233_less__asym_H, fact_149_ord__eq__le__trans, fact_244_ord__less__eq__subst, tcon_Nat_Onat___Rings_Ozero__neq__one, fact_100_mult__right__cancel, fact_220_linorder__cases, fact_112_mult__0, tcon_fun___Orderings_Oord, fact_23_Nat_Oex__has__greatest__nat, fact_224_less__induct, fact_227_dual__order_Oasym, fact_216_exists__least__iff, fact_218_order_Ostrict__trans, fact_102_zero__neq__one, fact_103_lambda__zero, tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors, fact_10_diff__mult__distrib, fact_75_diff__zero, fact_176_fun__upd__other, fact_208_nat__mult__le__cancel__disj, fact_197_le__numeral__extra_I3_J, fact_89_mult__cancel__left2, fact_191_diff__numeral__special_I9_J, fact_188_right__diff__distrib_H, fact_5_calculation, fact_158_eq__iff, tcon_Nat_Onat___Groups_Ocomm__monoid__mult, fact_3_minus__apply, fact_237_order_Oasym, fact_142_dual__order_Oeq__iff, fact_41_atMost__def, tcon_HOL_Obool___Groups_Ominus_13, tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring, fact_137_mult__left__le__one__le, fact_241_lt__ex, fact_113_ordered__comm__semiring__class_Ocomm__mult__left__mono, tcon_Nat_Onat___Rings_Oordered__semiring, fact_130_le__iff__diff__le__0, fact_129_mult__mono, tcon_Nat_Onat___Groups_Ocomm__monoid__add, fact_161_order__subst2, fact_195_nat__mult__eq__cancel__disj, fact_81_mult__is__0, fact_231_less__linear, tcon_Nat_Onat___Groups_Oone, fact_173_one__reorient, fact_167_ab__semigroup__mult__class_Omult__ac_I1_J, fact_56_mult_Oright__neutral, fact_247_less__numeral__extra_I4_J, fact_131_not__one__le__zero, fact_11_diff__mult__distrib2, fact_13_atMost__subset__iff, fact_17_nat__1__eq__mult__iff, fact_12_inf__period_I2_J, tcon_Nat_Onat___Groups_Oab__semigroup__mult, fact_46_Collect__mem__eq, fact_166_le__funD, fact_141_dual__order_Oantisym, fact_177_fun__upd__same, fact_209_mult__le__cancel2, fact_63_partitions__def, fact_24_bounded__Max__nat, fact_120_mult__le__0__iff, tcon_HOL_Obool___Orderings_Oorder_11, fact_147_order__class_Oorder_Oantisym, tcon_HOL_Obool___Orderings_Olinorder_10, tcon_Nat_Onat___Groups_Ominus_4, fact_140_mult__eq__self__implies__10, fact_108_le__0__eq, fact_199_bot__nat__0_Onot__eq__extremum, fact_37_le__cube, fact_19_le__trans, fact_22_nat__le__linear, fact_182_diff__left__mono, fact_136_mult__right__le__one__le, fact_225_less__not__sym, fact_150_order__class_Oorder_Oeq__iff, fact_226_less__imp__not__eq, tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel, fact_76_cancel__comm__monoid__add__class_Odiff__cancel, fact_164_le__funI, fact_15_diff__diff__cancel, fact_27_le__diff__iff_H, fact_155_eq__refl, fact_36_le__square, fact_172_cancel__ab__semigroup__add__class_Odiff__right__commute, fact_198_not__gr__zero, fact_53_diff__commute, fact_169_mult_Ocommute, fact_98_no__zero__divisors, fact_187_left__diff__distrib_H, fact_92_diff__is__0__eq, fact_79_diff__0__eq__0, fact_235_less__imp__neq, tcon_Set_Oset___Orderings_Oorder_6, fact_67_mult__zero__left, fact_110_diffs0__imp__equal, fact_2_atMost__eq__iff, tcon_Nat_Onat___Groups_Ozero, fact_230_less__irrefl, fact_107_bot__nat__0_Oextremum__unique, tcon_Nat_Onat___Orderings_Owellorder, fact_86_diff__ge__0__iff__ge, fact_65_partitionsE, fact_160_ord__eq__le__subst, fact_145_dual__order_Orefl, fact_215_linorder__less__wlog, fact_59_fun__upd__triv, fact_51_sum_Oeq__general, tcon_Nat_Onat___Groups_Omonoid__mult, fact_45_mem__Collect__eq, fact_123_mult__left__mono, fact_146_order__trans, tcon_Set_Oset___Orderings_Oord_7, fact_202_diff__gt__0__iff__gt, fact_60_fun__upd__apply, fact_72_diff__self, fact_135_mult__le__one, fact_29_diff__le__mono, fact_229_ord__eq__less__trans, fact_99_mult__left__cancel, fact_243_order__less__subst1, fact_30_Nat_Odiff__diff__eq, fact_0_partitions, fact_71_mult__cancel__right, fact_239_neqE, fact_168_mult_Oassoc, fact_96_mult__not__zero, fact_184_diff__eq__diff__less__eq, fact_91_diff__is__0__eq_H, fact_26_diff__le__mono2, fact_32_eq__diff__iff, tcon_Nat_Onat___Orderings_Oord_3, fact_49_sum_Oreindex__bij__witness, fact_104_sum_Oneutral, fact_232_less__trans, tcon_HOL_Obool___Orderings_Opreorder_9, fact_185_left__diff__distrib, fact_97_divisors__zero, fact_66_le__zero__eq, fact_174_fun__upd__idem__iff, fact_69_mult__eq__0__iff, fact_47_Collect__cong, fact_33_mult__le__mono2, fact_193_subset__antisym, fact_207_less__one, tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add, tcon_Nat_Onat___Rings_Omult__zero, tcon_Nat_Onat___Rings_Oordered__semiring__0, fact_133_partitions__zero, fact_156_linear, fact_152_le__cases3, fact_196_le__numeral__extra_I4_J, fact_119_split__mult__neg__le, fact_205_mult__less__cancel2, fact_95_zero__le, fact_38_nat__mult__1__right, fact_163_le__fun__def, fact_159_ord__le__eq__subst, fact_122_mult__right__mono__neg, fact_170_mult_Oleft__commute, fact_228_ord__less__eq__trans, fact_214_dual__order_Ostrict__trans, fact_73_diff__0__right, fact_138_sum__nonpos, fact_236_dense, fact_148_ord__le__eq__trans, fact_80_diff__self__eq__0, fact_128_mult__mono_H, fact_109_less__eq__nat_Osimps_I1_J, fact_54_sum_Oswap, fact_175_fun__upd__twist, fact_124_mult__nonpos__nonpos, fact_21_le__antisym, tcon_fun___Orderings_Opreorder, fact_200_less__nat__zero__code, fact_245_ord__eq__less__subst, fact_134_mult__left__le, fact_35_mult__le__mono, fact_83_mult__cancel1, fact_203_zero__less__diff, fact_157_antisym, tcon_Nat_Onat___Orderings_Opreorder_1, fact_126_split__mult__pos__le, fact_181_diff__mono, fact_201_neq0__conv, fact_252_nat__less__induct, fact_18_le__refl, fact_42_sum__subtractf__nat, fact_101_eq__iff__diff__eq__0, fact_248_infinite__descent__measure, fact_117_mult__nonneg__nonpos, fact_139_sum__nonneg, fact_219_dual__order_Oirrefl, fact_238_neq__iff, fact_106_bot__nat__0_Oextremum__uniqueI, fact_78_le0, fact_8_sum__distrib__left, tcon_Nat_Onat___Rings_Osemiring__0, fact_153_order_Otrans, fact_118_mult__nonneg__nonneg, fact_90_mult__cancel__left1, fact_77_bot__nat__0_Oextremum, fact_9_sum__distrib__right, fact_105_sum_Onot__neutral__contains__not__neutral, fact_20_eq__imp__le, tcon_Nat_Onat___Rings_Oordered__comm__semiring, tcon_fun___Orderings_Oorder, fact_74_zero__diff, fact_115_mult__nonneg__nonpos2, fact_4__092_060open_062_I_092_060Sum_062i_092_060le_062n_A_N_Ak_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_A_061_A_I_092_060Sum_062i_092_060le_062n_O_A_Ip_Ik_A_058_061_Ap_Ak_A_N_A1_J_J_Ai_A_K_Ai_J_092_060close_062, fact_7_sum__product, fact_143_dual__order_Otrans, fact_111_minus__nat_Odiff__0, tcon_Nat_Onat___Groups_Osemigroup__mult, fact_165_le__funE, fact_43_partitions__parts__bounded, fact_194_partitions__remove1__bounds, fact_68_mult__zero__right, tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, fact_6_sum__subtractf, fact_1__092_060open_062k_A_092_060le_062_An_092_060close_062, fact_82_mult__0__right, fact_94_zero__reorient, fact_84_mult__cancel2, tcon_HOL_Obool___Orderings_Oord_12, fact_251_infinite__descent, tcon_Set_Oset___Orderings_Opreorder_5, fact_151_antisym__conv, fact_249_measure__induct__rule, fact_52_sum_Ocong, fact_212_order_Ostrict__implies__not__eq, fact_178_fun__upd__idem, fact_34_mult__le__mono1, fact_183_diff__right__mono, fact_223_antisym__conv3, fact_31_le__diff__iff, fact_44_fun__diff__def, fact_222_less__imp__not__eq2, fact_93_gr0, fact_88_mult__cancel__right1, fact_162_order__subst1, fact_87_mult__cancel__right2, fact_50_sum_Oeq__general__inverses, fact_250_linorder__neqE__nat, fact_171_diff__eq__diff__eq, fact_190_mult_Ocomm__neutral, fact_144_linorder__wlog, fact_240_gt__ex, fact_234_less__asym, fact_211_dual__order_Ostrict__implies__not__eq, tcon_Nat_Onat___Groups_Oordered__comm__monoid__add, fact_121_mult__right__mono, fact_114_zero__le__mult__iff, fact_154_le__cases, fact_39_nat__mult__1, fact_192_subsetI, fact_85_sum_Oneutral__const, tcon_Set_Oset___Groups_Ominus_8
% 58.75/41.67 % No. of inferences in proof: 594
% 58.75/41.68 % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p : 41138 ms resp. 40565 ms w/o parsing
% 59.77/41.91 % SZS output start Refutation for /export/starexec/sandbox/benchmark/theBenchmark.p
% See solution above
% 59.77/41.91 % [INFO] Killing All external provers ...
%------------------------------------------------------------------------------